Ssp-notes.pdf

SIGNALS & STOCHASTIC PROCESS

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SIGNALS AND STOCHASTIC PROCESS II Year B. Tech. ECE – I Sem Prepared by

B. Vamsi Krishna Assistant Professor

Mrs. V. Poornima Assistant Professor

M. Sruthi Assistant Professor

Department of Electronics & Communication Engineering

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EC304ES: SIGNALS AND STOCHASTIC PROCESS B.Tech. II Year I Sem.

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LTPC 3103

UNIT – I Signal Analysis: Analogy between Vectors and Signals, Orthogonal Signal Space, Signal approximation using Orthogonal functions, Mean Square Error, Closed or complete set of orthogonal functions, Orthogonality in Complex functions, Exponential and Sinusoidal signals, Concepts of Impulse function, Unit Step function, Signum function. Signal Transmission through Linear Systems: Linear System, Impulse response, Response of a Linear System, Linear Time Invariant (LTI) System, Linear Time Variant (LTV) System, Transfer function of a LTI system, Filter characteristics of Linear Systems, Distortion less transmission through a system, Signal bandwidth, System bandwidth, Ideal LPF, HPF and BPF characteristics, Causality and Paley-Wiener criterion for physical realization, relationship between Bandwidth and Rise time. Concept of convolution in Time domain and Frequency domain, Graphical representation of Convolution, Convolution property of Fourier Transforms UNIT – II Fourier series, Transforms, and Sampling: Fourier series: Representation of Fourier series, Continuous time periodic signals, Properties of Fourier Series, Dirichlet’s conditions, Trigonometric Fourier Series and Exponential Fourier Series, Complex Fourier spectrum. Fourier Transforms: Deriving Fourier Transform from Fourier series, Fourier Transform of arbitrary signal, Fourier Transform of standard signals, Fourier Transform of Periodic Signals, Properties of Fourier Transform, Fourier Transforms involving Impulse function and Signum function. Sampling: Sampling theorem – Graphical and analytical proof for Band Limited Signals, Reconstruction of signal from its samples, Effect of under sampling – Aliasing. UNIT – III Laplace Transforms and Z–Transforms: Laplace Transforms: Review of Laplace Transforms (L.T), Partial fraction expansion, Inverse Laplace Transform, Concept of Region of Convergence (ROC) for Laplace Transforms, Constraints on ROC for various classes of signals, Properties of L.T, Relation between L.T and F.T of a signal, Laplace Transform of certain signals using waveform synthesis. Z–Transforms: Fundamental difference between Continuous and Discrete time signals, Discrete time signal representation using Complex exponential and Sinusoidal components, Periodicity of Discrete time signal using complex exponential signal, Concept of ZTransform of a Discrete Sequence, Distinction between Laplace, Fourier and Z Transforms, Region of Convergence in Z-Transform, Constraints on ROC for various classes of signals, Inverse Z-transform, Properties of Z-transforms. UNIT – IV Random Processes – Temporal Characteristics: The Random Process Concept, classification of Processes, Deterministic and Nondeterministic Processes, Distribution and Density Functions, concept of Stationarity and Statistical Independence. First-Order Stationary Processes, SecondOrder and Wide-Sense Stationarity, (N-Order) and Strict- Sense Stationarity, Time Averages and Ergodicity, Autocorrelation Function and Its Properties, Cross-Correlation Function and Its Properties, Covariance Functions, Gaussian Random Processes, Poisson Random Process.

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Random Signal, Mean and Mean-squared Value of System Response, autocorrelation Function of Response, Cross-Correlation Functions of Input and Output. UNIT- V: Random Processes – Spectral Characteristics: The Power Spectrum: Properties, Relationship between Power Spectrum and Autocorrelation Function, The Cross-Power Density Spectrum, Properties, Relationship between Cross-Power Spectrum and Cross- Correlation Function. Spectral Characteristics of System Response: Power Density Spectrum of Response, Cross-Power Density Spectrums of Input and Output. TEXT BOOKS: 1. Signals, Systems & Communications - B.P. Lathi, 2013, BSP. 2. Signal and systems principles and applications, shaila dinakar Apten, Cambridez university press, 2016. 3. Probability, Random Variables & Random Signal Principles - Peyton Z. Peebles, MC GRAW HILL EDUCATION, 4th Edition, 2001 REFERENCE BOOKS: 1. Signals and Systems - A.V. Oppenheim, A.S. Willsky and S.H. Nawab, 2 Ed. 2. Signals and Signals – Iyer and K. Satya Prasad, Cengage Learning

COURSE OBJECTIVES: 1. This gives the basics of Signals and Systems required for all Electrical Engineering related courses. 2. This gives concepts of Signals and Systems and its analysis using different transform techniques. 3. This gives basic understanding of random process which is essential for random signals and systems encountered in Communications and Signal Processing areas. COURSE OUTCOMES: Upon completing his course, the student will be able to 1. Represent any arbitrary analog or Digital time domain signal in frequency domain. 2. Understand the importance of sampling, sampling theorem and its effects. 3. Understand the characteristics of linear time invariant systems. 4. Determine the conditions for distortion less transmission through a system. 5. Understand the concepts of Random Process and its Characteristics. 6. Understand the response of linear time Invariant system for a Random Processes.

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INDEX Unit-I Part I: Signal Analysis 1.1 Introduction 1.2 Classification of the Signals 1.3 Analogy between Vectors and signal 1.4 Orthogonal signal space 1.5 Signal approximation using Orthogonal functions 1.6 Mean square Error 1.7 Closed and complete set of orthogonal functions 1.8 Orthogonality in complex functions 1.9 Standard Signals

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Part II: Signal Transmission through Linear Systems 1.10 System 1.11 Classification of Systems 1.12 Transfer Function of an LTI System 1.13 Filter characteristics of Linear Systems 1.14 Distortion less transmission through a system 1.15 Signal bandwidth & System bandwidth 1.16 Ideal LPF, HPF and BPF characteristics 1.17 Causality and Paley-Wiener Criterion 1.18 Relationship between bandwidth and rise time 1.19 Convolution in time and frequency domain 1.20 Graphical representation of convolution 1.21 Convolution properties of Fourier transform

16 16 18 19 19 20 21 22 23 24 25 27

Unit- II Fourier series, Fourier Transforms, and Sampling 2.1 Fourier series representation of Periodic signals 2.2 Dirichlet’s conditions 2.3 Trigonometric Fourier series 2.4 Exponential Fourier series 2.5 Properties of Continuous time Fourier series 2.6 Complex Fourier Spectrum 2.7 Fourier Transform 2.8 Derivation from FS- FT 2.9 Solved Problems 2.10 Fourier Transform of Standard Signals 2.11 Fourier Transform of Periodic Signal 2.12 Properties of the Fourier Transform 2.13 Sampling Theorem 2.14 Aliasing 2.15 Nyquist Rate & Nyquist Interval

30 30 30 31 35 37 38 38 39 43 50 53 55 60 60

Unit- III Part I: Laplace Transforms 3.1 Introduction 3.2 Relation between Laplace and Fourier transform 3.3 Region of Convergence (ROC) of LT 3.4 Properties of ROC 3.5 Properties of Laplace Transform

62 63 67 69 70

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3.6 3.7 3.8

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Inverse Laplace Transform Partial Fraction Expansion Laplace transform using Waveform Synthesis

75 75 77

PART - II Z–Transforms 3.9 Introduction 3.10 Relation between ZT and DTFT 3.11 Relation between Laplace, Fourier and z- transforms 3.12 Problems 3.13 Region of Convergence (ROC) of Z-Transforms 3.14 Properties of ROC 3.15 Properties of Z-Transform 3.16 Inverse Z-Transform

79 79 80 80 83 84 87 89

UNIT – IV Random Processes – Temporal Characteristics 4.1 Introduction 4.2 Classification of Random Processes 4.3 Deterministic and Non-deterministic processes 4.4 Distribution Function and Density function 4.5 Independence and Stationary Random Process 4.6 Time Averages and Ergodicity 4.7 Autocorrelation Function and its Properties 4.8 Properties of Cross Correlation Function 4.9 Covariance functions for random processes 4.10 Gaussian Random Process 4.11 System Response

98 98 100 101 102 105 105 107 108 109 109

UNIT – V Random Process - Spectral Characteristics 5.1 Introduction 5.2 Power Spectral Density of a random Process 5.3 Properties of power density spectrum 5.4 Cross Power density spectrum 5.5 Properties of cross power density spectrum 5.6 Problems 5.7 Spectral characteristics of system response 5.8 Problems

111 111 114 115 117 118 120 122

Question Bank

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UNIT - I Signal Analysis & Transmission through Linear Systems 1.1.

Introduction:

In typical applications of science and engineering, we have to process signals, using systems. While the applications can be varied large communication systems to control systems but the basic analysis and design tools are the same. In a signals and systems course, we study these tools: convolution, Fourier analysis, z-transform, and Laplace transform. The use of these tools in the analysis of linear time-invariant (LTI) systems with deterministic signals. For most practical systems, input and output signals are continuous and these signals can be processed using continuous systems. However, due to advances in digital systems technology and numerical algorithms, it is advantageous to process continuous signals using digital systems by converting the input signal into a digital signal. Therefore, the study of both continuous and digital systems is required. As most practical systems are digital and the concepts are relatively easier to understand, we describe discrete signals and systems, immediately followed by the corresponding description of continuous signals and systems. 1.2.

Classification of the Signals:

The Signals can be classified into several categories depending upon the criteria and for its classification. Broadly the signals are classified into the following categories 1. Continuous, Discrete and Digital Signals 2. Periodic and Aperiodic Signals 3. Even and Odd Signals 4. Power and Energy Signals 5. Deterministic and Random signals 1.2.1 Continuous-time and Discrete-time Signals: Continuous-Time (CT) Signals: They may be defined as continuous in time and continuous in amplitude as shown in Figure 1.1. Ex: Speech, audio signals etc.. Discrete Time (DT) Signals: Discretized in time and Continuous in amplitude. They may also be defined as sampled version of continuous time signals. Ex: Rail track signals. Digital Signals: Discretized in time and quantized in amplitude. They may also be defined as quantized version of discrete signals. 1.2.2 Periodic Signals A CT signal x(t) is said to be periodic if it satisfies the following condition x (t) = x (t + T0) (1.1) The smallest positive value of T0 that satisfies the periodicity condition Eq.(1.1), is referred as the fundamental period of x(t). The reciprocal of fundamental period of a signal is fundamental frequency f0. Likewise, a DT signal x[n] is said to be periodic if it satisfies x(n) = x (n + N0) (1.2)

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Figure 1.1: Description of Continuous, Discrete and Digital Signals The smallest positive value of N0 that satis es the periodicity condition Eq.(1.2) is referred to as the fundamental period of x [n]. Note: All periodic signals are ever lasting signals i.e. they start at -1 and end at +1 as shown in below Figure 1.2.

Figure 1.2: A typical periodic signal Ex.1.1 Consider a periodic signal is a sinusoidal function represented as x (t) = A sin (10t + 20) The time period of the signal T0 is 10 Ex.1.2 CT tangent wave: x (t) = tan (10t) is a periodic signal with period T =10 Note: Amplitude and phase difference will not affect the time period. i.e. 2 sin (3t), 4 sin (3t), 4 sin (3t + 32) will have the same time period 1.2.3 Even and Odd Signals Any signal can be called even signal if it satisfies x(t) = x(-t) or x(n) = x(-n). Similarly any signal can be called odd signal if it not satisfies x(t) = x(-t) or x(n) = x(-n). Figure 1.2, shows an example of an even and odd signal whereas Figure 1.3 shows neither even nor odd signal.

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Any signal X(t) can be expressed in terms of even component Xe(t) and odd component X0(t). X(t) =Xe(t)+Xo(t), Xe(t)+=(X(t) + X(-t)) / 2, Xo(t)+=(X(t) - X(-t)) / 2 1.2.4 Energy and Power signals A signal x(t) (or) x(n) is called an energy signal if total energy has a non - zero finite value i.e. 0 < Ex < 1 and Pavg = 0 A signal is called a power signal if it has non - zero nite power i.e. 0 < Px < 1 and E = 1. A signal can't be both an energy and power signal simultaneously. The term instantaneous power is reserved for the true rate of change of energy in a system. All periodic signals are power signals and all finite durations signals are energy signals.

1.2.5 Deterministic and Random signal A deterministic signal is a signal in which each value of the signal is fixed and can be determined by a mathematical expression, rule, or table. Because of this the future values of the signal can be calculated from past values with complete confidence. On the other hand, a random signal has lot of uncertainty about its behaviour. The future values of a random signal cannot be accurately predicted and can usually only be guessed based on the averages of sets of signals.

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1.3. Analogy between Vectors and Signals Vector A vector contains magnitude and direction. The name of the vector is denoted by bold face type and their magnitude is denoted by light face type. Example: V is a vector with magnitude V. Consider two vectors V1 and V2 as shown in the following diagram. Let the component of V1 along with V2 is given by C12V2. The component of a vector V1 along with the vector V2 can obtained by taking a perpendicular from the end of V1 to the vector V2 as shown in diagram:

The vector V1 can be expressed in terms of vector V2 V1= C12V2 + Ve Where Ve is the error vector. But this is not the only way of expressing vector V1 in terms of V2. The alternate possibilities are: V1=C1V2+Ve1 & V2=C2V2+Ve2

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1.4

1.5 Signal approximation using Orthogonal functions

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1.6.

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1.7

1.8

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1.9. Standard Signals

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PART II - Signal Transmission through Linear Systems 1.10. System:

1.11. Classification of Systems: Systems are classified into the following categories:      

Linear and Non-linear Systems Time Variant and Time Invariant Systems Liner Time variant and Liner Time invariant systems Static and Dynamic Systems Causal and Non-causal Systems Stable and Unstable System

Linear and Non-linear Systems A system is said to be linear when it satisfies superposition and homogeneity principles. Consider two systems with inputs as x1(t), x2(t), and outputs as y1(t), y2(t) respectively. Then, according to the superposition and homogeneity principles, T [a1 x1(t) + a2 x2(t)] = a1 T[x1(t)] + a2 T[x2(t)]

∴ T [a1 x1(t) + a2 x2(t)] = a1 y1(t) + a2 y2(t)

From the above expression, is clear that response of overall system is equal to response of individual system. Example: x (t) = x2(t) Solution: y1 (t) = T[x1(t)] = x12(t) y2 (t) = T[x2(t)] = x22(t) T [a1 x1(t) + a2 x2(t)] = [ a1 x1(t) + a2 x2(t)]2 Which is not equal to a1 y1(t) + a2 y2(t). Hence the system is said to be non linear.

Time Variant and Time Invariant Systems A system is said to be time variant if its input and output characteristics vary with time. Otherwise, the system is considered as time invariant. The condition for time invariant system is: y (n , t) = y(n-t) The condition for time variant system is: y (n , t) ≠ y(n-t) Where y (n , t) = T[x(n-t)] = input change y (n-t) = output change MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Example: y(n) = x(-n)

y(n, t) = T[x(n-t)] = x(-n-t) y(n-t) = x(-(n-t)) = x(-n + t)

∴ y(n, t) ≠ y(n-t). Hence, the system is time variant.

Liner Time variant (LTV) and Liner Time Invariant (LTI) Systems If a system is both liner and time variant, then it is called liner time variant (LTV) system. If a system is both liner and time Invariant then it is called liner time invariant (LTI) system.

Static and Dynamic Systems Static system is memory-less whereas dynamic system is a memory system. Example 1: y(t) = 2 x(t) For present value t=0, the system output is y(0) = 2x(0). Here, the output is only dependent upon present input. Hence it is memory less or static. Example 2: y(t) = 2 x(t) + 3 x(t-3) For present value t=0, the system output is y(0) = 2x(0) + 3x(-3). Here x(-3) is past value for the present input for which the system requires memory to get this output. Hence, the system is a dynamic system.

Causal and Non-Causal Systems A system is said to be causal if its output depends upon present and past inputs, and does not depend upon future input. For non-causal system, the output depends upon future inputs also. Example 1: y(n) = 2 x(t) + 3 x(t-3) For present value t=1, the system output is y(1) = 2x(1) + 3x(-2). Here, the system output only depends upon present and past inputs. Hence, the system is causal. Example 2: y(n) = 2 x(t) + 3 x(t-3) + 6x(t + 3) For present value t=1, the system output is y(1) = 2x(1) + 3x(-2) + 6x(4) Here, the system output depends upon future input. Hence the system is non-causal system.

Stable and Unstable Systems The system is said to be stable only when the output is bounded for bounded input. For a bounded input, if the output is unbounded in the system then it is said to be unstable. Note: For a bounded signal, amplitude is finite. Example 1: y (t) = x2(t) Let the input is u(t) (unit step bounded input) then the output y(t) = u2(t) = u(t) = bounded output. Hence, the system is stable. Example 2: y (t) = ∫x(t)dt

Let the input is u (t) (unit step bounded input) then the output y(t) = ∫u(t)dt = ramp signal (unbounded because amplitude of ramp is not finite, it goes to infinite when t → infinite). Hence, the system is unstable.

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1.12. Transfer Function of an LTI System: The transfer function of a continuous-time LTI system may be defined using Fourier transform or Laplace transform. The transfer function is defined only under zero initial conditions. A continuous time system is shown in fig: i/p signal

The o/p signal h(t)or H(s)or H(ω)

x(t) The i/p spectrum

y(t) The o/p spectrum

X(s) or X(ω)

Y(s) or Y(ω) Fig: A system

The transfer function of a LTI system H(ω) is defined as the ratio of the Fourier transform of the output signal to the Fourier Transform of the input signal when the initial conditions are zero. 𝐻(𝜔) = (𝜔)/ (𝜔)

H(ω) is a complex quantity having magnitude and phase.

𝐻(𝜔) = |𝐻(𝜔)| 𝜃(𝜔) The transfer function in frequency domain H(ω) is also called frequency response of the system. The frequency response is amplitude response plus phase response. |H(𝜔)|= Amplitude response of the system. θ(ω)=⌊𝐻(𝜔)= Phase response of the system. We can say that H(ω) is a frequency domain representation of a system. Since

Y(ω)=H(ω)X(ω) | (𝜔)| = |𝐻(𝜔)|| (𝜔)|

⌊ (𝜔) = ⌊𝐻(𝜔) + ⌊ (𝜔) H(ω) has conjugate symmetry property.

H(-ω)=H*(ω)

i.e. ⌊𝐻(ω)=-⌊𝐻(𝜔)

H(−𝜔)| = |𝐻(𝜔)| and

The impulse response h(t) of a system is the inverse Fourier transform of its transfer function H(ω). H(ω)=F[h(t)]

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&

h(t)=𝐹−1 [H(ω)]

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1.13. Filter characteristics of Linear Systems: A filter is a frequency selective network. It allows transmission of signals of certain frequencies with no attenuation or with very little attenuation, and it rejects or heavily attenuates signals of all other frequencies. Filters are usually classified according to their frequency response characteristics as low pass filter (LPF), high- pass filter (HPF), band-pass filter (BPF) and bandelimination or band stop or band reject filter (BEF, BSF, BRF). The system modifies the spectral density function of the input. The system acts as a kind of filter for various frequency components. Some frequency components are boosted in strength, i.e. they are amplified. Some frequency components are weakened in strength, i.e. they are attenuated and some may remain unaffected. Similarly, each frequency component suffers a different amount of phase shift in the process of transmission. The system, therefore, modifies the spectral density function of the input according to its filter characteristics. The modification is carried out according to the transfer function H(s) or H(ω), which represents the response of the system to various frequency components. H(ω) acts as a weighting function or spectral shaping function to the different frequency components in the input signal. An LTI system, acts as a filter. A filter is a basically a frequency selective network. (i) (ii) (iii) (iv)

Some LTI systems allow the transmission of only low frequency components and stop all high frequency components. They are called low – pass filters (LPFs). Some LTI systems allow the transmission of only high frequency components and stop all low frequency components. They are called high – pass filters (HPFs). Some LTI systems allow the transmission of only a particular band of frequencies and stop all other frequency components. They are called band pass filters (BPFs). Some LTI systems reject the transmission of only a particular band of frequencies and allow all other frequency components. They are called band-rejection filters (BRFs).

The band of frequency that is allowed by the filter is called pass-band. The band of frequency that is severely attenuated and not allowed to pass through the filter is called stop-band or rejection-band. An LTI system may be characterized by its pass-band, stopband and half power band width. 1.14. Distortion less transmission through a system The change of shape of the signal when it is transmitted through a system is called distortion. Transmission of a signal through a system is said to be distortion less if the output is an exact replica of the input signal. This replica may have different magnitude and also it may have different time delay. Mathematically, We can say that a signal x(t) is transmitted without distortion if the output y(t) = kx(t-td) Where k is a constant representing the change in amplitude and td is delay time. Taking Fourier transform on both sides of the equation for y(t) and using the shifting property, we have Y ω = k e-jωtd MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Therefore inverse Fourier transform, the corresponding impulse response must be h(t)=kδ(t-td) The magnitude of the transfer function is given by |H(𝜔)|=k The phase shift is given by

for all w

θ (ω) = ⌊(𝜔)= -ωtd

and it varies linearly with frequency given by

θ (ω) = n -ωtd ( n integral )

So for distortion less transmission of a signal through a system, the magnitude |H(𝜔)|should be a constant, i.e. all the frequency components of the input signal must undergo the same amount of amplification or attenuation, i.e. the system bandwidth is infinite and the phase spectrum should be proportional to frequency as shown in above figure. But, in practice, no system can have infinite bandwidth and hence distortion less conditions are never met exactly. 1.15. Signal bandwidth & System bandwidth: Signal Bandwidth: The spectral components of a signal extend from -∞ ∞. Any practical signal has finite energy. As a result, the spectral components approach zero as ω tends to ∞. Therefore, we neglect the spectral components which have negligible energy and select only a band of frequency components which have most of the signal energy is known as the bandwidth of the signal. Normally, the band is selected such that it contains 95% of total energy depending on the precision.

Fig. Frequency response

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System Bandwidth: For distortion less transmission, we need a system with infinite bandwidth. Due to physical limitations, it is impossible to construct a system with infinite bandwidth. Actually a satisfactory distortion less transmission can be achieved by a system with finite, but fairly large band widths, if the magnitude H ω is constant over this band. The bandwidth of a system is defined as the range of frequencies over which the magnitude H ω remain within 1/√2 times (within 3 dB) of its value at mid band . The bandwidth of a system H ω plot is shown in above figu e is ω2-ω1 whe e ω2 is called the upper cut off frequency or upper 3 dB frequency or upper half powe f e uency and ω1 is called lower cut off frequency or lower 3dB frequency or lower half frequency. The band limited signals can be transmitted without distortion, if the system bandwidth is atleast equal to the signal bandwidth.

1.16. Ideal LPF, HPF and BPF characteristics An ideal filter has very sharp cutoff characteristics, and it passes signals of certain specified band of frequencies exactly and totally rejects signals of frequencies outside this band. Its phase spectrum is linear.

Fig. Ideal filter characteristics Ideal LPF An ideal low-pass filter transmits, without any distortion, all of the signals of frequencies below a certain frequency ⍵c radians per second. The signals of frequencies above ⍵c radians/second are completely attenuated. ⍵c is called the cutoff frequency. The corresponding phase function for distortion less transmission is -⍵td. the transfer function of an ideal LPF is given by H(⍵) = 1, ⍵ ˂⍵c

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= 0, ⍵ >⍵c The frequency response characteristics of an ideal LPF are shown in figure (a). It is a gate function. Ideal HPF An ideal high-pass filter transmits, without any distortion, all of the signals of frequencies above a certain frequency ⍵c radians per second and attenuates completely the signals of frequencies below ⍵c radians per second, where ⍵c is called the cutoff frequency. The corresponding phase function for distortion less transmission is -⍵td. the transfer function of an ideal LPF is given by H ⍵ = , ⍵ ˂⍵c = , ⍵ >⍵c The frequency response characteristics of an ideal HPF are shown in figure (b). Ideal BPF An ideal band-pass filter transmits, without any distortion, all of the signals of frequencies within a certain frequency band ⍵2-⍵1 radians per second and attenuates completely the signals of frequencies outside this band. (⍵2-⍵1) is the bandwidth of the band-pass filter. The corresponding phase function for distortion less transmission is -⍵td. An ideal BPF is given by H(⍵) = 1, ⍵1 ˂⍵˂ ⍵2 = 0, ⍵˂ ⍵1 and ⍵> ⍵2 The frequency response characteristics of an ideal BPF are shown in figure (c). Ideal BRF An ideal band-rejection filter rejects totally all of the signals of frequencies within a certain frequency band ⍵2-⍵1 radians per second and transmits without any distortion all signals of frequencies outside this band. (⍵2-⍵1) is the rejection band. The corresponding phase function for distortion less transmission is -⍵td. An ideal BRF is given by H(⍵) = 0, ⍵1 ˂⍵˂ ⍵2 = 1, ⍵˂ ⍵1 and ⍵> ⍵2 The frequency response characteristics of an ideal BRF are shown in figure (d). All ideal filters are non-causal systems. Hence none of them is physically realizable.

1.17. Causality and Paley-Wiener Criterion For Physical Realization: A system is said to be causal if it does not produce an output before the input is applied. For an LTI system to be causal, the condition to be satisfied is its impulse response must be zero for t < 0, i.e. h(t)=0 for t<0. Physical realizability implies that it is physically possible to construct that system in real time. A physically realizable system cannot have a response before the input is applied. This is known as causality condition. It means the unit impulse response h(t) of a physically realizable system must be causal. This is the time domain criterion of physical realizability. In the frequency domain, this criterion implies that a necessary and sufficient condition for a magnitude function H(ω) to be physically realizable is:

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The magnitude function H(ω) must, however, be square integrable before the Paley-wiener criterion valid, that is,

1.18. Relationship between bandwidth and rise time: Risetime is an easily measured parameter that provides considerable insight into the potential pitfalls in performing a measurement or designing a circuit. Risetime is defined as the time it takes for a signal to rise (or fall for falltime) from 10% to 90% of its final value. We know that the transfer function of an ideal LPF is given by

⍵c is called the cutoff frequency. The impulse response h(t) of the LPF is obtained by taking the inverse Fourier transform of the transfer function H(⍵).

The impulse response of the ideal LPF is shown in below figure. The impulse response has a peak value at t=td. This value ⍵c/𝜋 is proportional to cutoff frequency ⍵c. The width of the main lobe is 2𝜋/⍵c. As ⍵c⟶∞, the LPF becomes an all pass filter. As td⟶0, the output response peak ⟶∞, that is, the output response approaches input. If the impulse response is known, the step response can be obtained by convolution. The step response is given by

We have

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1.19. Convolution in time and frequency domain Convolution of signals may be done either in time domain or frequency domain. So there are following two theorems of convolution associated with Fourier transforms: 1.Time convolution theorem 2.Frequency convolution theorem Time convolution theorem The time convolution theorem states that convolution in time domain is equivalent to multiplication of their spectra in frequency domain. Mathematically, if x1(t)↔X1(⍵), x2(t)↔X2(⍵) then, x1(t) * x2(t)↔ X1(⍵)X2(⍵).

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Frequency convolution theorem The frequency convolution theorem states that the multiplication of two functions in time domain is equivalent to convolution of their spectra in frequency domain. Mathematically, if x1(t)↔X1(⍵), x2(t)↔X2(⍵), then, x (t) x (t) ↔ 1/2𝜋 [ X (⍵) * X (⍵)]

Interchanging the order of integration, we get

1.20 Graphical representation of convolution The convolution of two signals can be performed using graphical method. The procedure is: 1. For the given signals x(t) and h(t), replace the independent variable t by a dummy variable 𝜏 and plot the graph for x(𝜏) and h(𝜏). 2. Keep the function x(𝜏) fixed. visualize the function h(𝜏) as a rigid wire frame and rotate (or invert) this frame about the vertical axis (𝜏 = 0) to obtain h(-𝜏). 3. Shift the frame along the 𝜏-axis by t sec. the shifted frame now represents h(t-𝜏). 4. Plot the graph for x(𝜏) and h(t-𝜏) on the same axis beginning with very large negative time shift t. 5. For a particular value of t=a , integration of the product x(𝜏)h(t-𝜏) represents the area under the product curve (common area). this common area represents the convolution of x(t) and h(t) for a shift of t=a.

6. Increase the time shift t and take the new interval whenever the function either x(𝜏) or h(t-𝜏) changes. the value of t at which the change occurs defines the end of the current interval and the beginning of a new interval. calculate y(t) using step5. 7. The value of convolution obtained at different values of t ( both positive and negative values) may be plotted on a graph to get the combined convolution.

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The above figure shows the plots of x( ) and h(t- ) together on the same time axis. Here the signal h(t- ) is sketched for t<-3. x( ) and h(t- ) do not overlap. Therefore, the product x( ) h(t- ) is equal to zero. y(t)=0 (for t<-3)

Plots of (a) x(𝜏), and (b)h(t-𝜏)when there is no overlap

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Now, increase the time shift t until the signal h(t- ) intersects x( ). Figure below shows the situation for t>-3. Here x( ) and h(t- ) overlapped.[ This overlapping continuous for all values for t>-3 up to t=∞ because x( ) exists for all values of >0]. But x( )=0 to =t+3.

Plot of x(𝜏), and h(t-𝜏) with overlap

1.21 Convolution properties of Fourier transform: With two functions h(t) and g(t), and their corresponding Fourier transforms H(f) and G(f), we can form two special combinations – The convolution, denoted f = g * h, defined by Convolution: g*h is a function of time, and g*h = h*g. The convolution is one member of a transform pair g*h ↔ G(f) H(f) The Fourier transform of the convolution is the product of the two Fourier transforms. This is the Convolution Theorem. Problems: Find the convolution of the signals using Fourier transform. 1 2 3 4 1 MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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2)

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3)

4)

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UNIT – II Fourier series, Fourier Transforms, and Sampling 2.1. Fourier series representation of Periodic signals Introduction: A signal is said to be a continuous time signal if it is available at all instants of time. A real time naturally available signal is in the form of time domain. However, the analysis of a signal is far more convenient in the frequency domain. These are three important classes of transformation methods available for continuous-time systems. They are: 1. Fourier series 2. Fourier Transform 3. Laplace Transform Out of these three methods, the Fourier series is applicable only to periodic signals, i.e. signals which repeat periodically over -∞ < t < ∞. Not all periodic signals can be represented by Fourier series. Fourier series is to project periodic signals onto a set of basic functions. The basis functions are orthogonal and any periodic signal can be written as a weighted sum of these basis functions. Representation of Fourier Series The representation of signals over a certain interval of time in terms of the linear combination of orthogonal functions is called Fourier Series. The Fourier analysis is also sometimes called the harmonic analysis. Fourier series is applicable only for periodic signals. It cannot be applied to non-periodic signals. A periodic signal is one which repeats itself at regular intervals of time, i.e. periodically over -∞ to ∞. Three important classes of Fourier series methods are available. They are: 1. Trigonometric form 2. Cosine form 3. Exponential form 1) The function x(t) is absolutely integrable over one period, that is

2) The function x(t) has only a finite number of maxima and minima. 3) The function x(t) has a finite number of discontinuities. 2.3. Trigonometric Fourier series: A sinusoidal signal, x(t)=A sin ⍵0t is a periodic signal with period T=2𝜋/⍵0. Also, the sum of two sinusoids is periodic provided that their frequencies are integral multiples of a fundamental frequency ⍵0. We can show that a signal x(t), a sum of sine and cosine functions whose frequencies are integral multiples of ⍵0, is a periodic signal.

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The infinite series of sine and cosine terms of frequencies 0, ⍵0, 2⍵0, …… k⍵0 is known as trigonometric form of Fourier series and can be written as:

where an and bn are constants; the coefficient a0 is called the dc component; a1cos⍵0t+b1sin⍵0t the first harmonic, a2cos2⍵0t+b2sin2⍵0t the second harmonic and ⍵0 + ⍵0 the nth harmonic. The constant b0=0 because sin⍵0 = 0 for n=0. 2.4. Exponential Fourier series: The exponential Fourier series is the most widely used form of Fourier series. In this, the function x(t) is expressed as a weighted sum of the complex exponential functions. The complex exponential form is more general and usually more convenient and more compact. So, it is used almost exclusively, and it finds extensive application in communication theory.

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Problems:

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3) Obtain the exponential Fourier Series for the wave form shown in below figure

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Solution: The periodic waveform shown in fig with a period T= 2π can be expressed as:

4) Find the exponential Fourier series for the full wave rectified sine wave given in below figure.

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2.5. Properties of continuous- time Fourier series The Fourier series representation possesses a number of important properties that are useful for various purposes during the transformation of signals from one form to other . Some of the properties are listed below. [x1(t) and x2(t)] are two periodic signals with period T and with Fourier series coefficients Cn and Dn respectively. 1) Linearity property The linearity property states that, if x then Proof:

Ax1(t)+Bx

Cn and x

Dn

+BDn

From the definition of Fourier series, we have FS[Ax1(t)+Bx

x(t - t0)

𝐹

−

𝜔0 0 C n

=ACn+BDn Ax1(t)+Bx 2) Time shifting property Proof: x(t)=

+BDn

The time shifting property states that, if x(t) 𝐹 Cn then

x(tCn From the definition of Fourier series, we have

x(t=FS-1[Cn

− 𝜔0

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3) Time reversal property The time reversal property states that, if x(t) Cn then

x(-

-n

From the definition of Fourier series, we have

Proof: x(t)= x(-

substituting n = -p in the right hand side, we get

x(substituting p = n , we get x(-

]

x(x(-

-n

4) Time scaling property The time scaling property states that, if x(t) Cn then ω0→𝛼ω0 Proof: series, we have

x(𝛼t) Cn with From the definition of Fourier x(t)= x(

]

where ω0→ ω0. x(𝛼t)𝐹 Cn with fundamental frequency of 𝛼ω0

then

5) Time differential property: The time differential property states that, if x(t) 𝐹

Cn

Cn

From the definition of Fourier series, we x(t)= Differentiating both sides with respect to t, we get Proof: have

Cn 6) Time integration property: The time integration property states that, if x(t)

n

then

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Proof: x(t)=

DEPT.OF ECE

From the definition of Fourier series, we have

Interchanging the order of integration and summation, we get

= 2.6. Complex Fourier Spectrum The Fourier spectrum of a periodic signal x(t) is a plot of its Fourier coefficients versus frequency ω. It is in two parts: (a) Amplitude spectrum and (b) phase spectrum. The plot of the amplitude of Fourier coefficients verses frequency is known as the amplitude spectra, and the plot of the phase of Fourier coefficients verses frequency is known as phase spectra. The two plots together are known as Fourier frequency spectra of x(t).This type of representation is also called frequency domain representation. The Fourier spectrum exists only at discrete frequencies nωo, where n=0,1,2,….. Hence it is known as discrete spectrum or line spectrum. The envelope of the spectrum depends only upon the pulse shape, but not upon the period of repetition. The below figure (a) represents the spectrum of a trigonometric Fourier series extending from 0 to ∞, producing a one-sided spectrum as no negative frequencies exist here. The figure (b) represents the spectrum of a complex exponential Fourier series extending from - ∞ ∞, producing a two-sided spectrum. The amplitude spectrum of the exponential Fourier series is symmetrical Fourier series is symmetrical about the vertical axis. This is true for all periodic functions.

Fig: Complex frequency spectrum for (a) Trigonometric Fourier series and (b) complex exponential Fourier series. If Cn is a general complex number, then Cn = Cn

& C-n = Cn

−𝜃

& Cn = C-n

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The magnitude spectrum is symmetrical about the vertical axis passing through the origin, and the phase spectrum is antisymmetrical about the vertical axis passing through the origin. So the magnitude spectrum exhibits even symmetry and phase spectrum exhibits odd symmetry. When x(t) is real , then C, the complex conjugate of Cn.

2.7. Fourier Transform The Fourier transform is used to analyse aperiodic signals and can be use analyse periodic signals also. So it over comes the limitations of Fourier series. Fourier transform is a transformation technique which transforms signals from the continuous-time domain to the corresponding frequency domain and vice versa, and which applies for both periodic as well as aperiodic signals. Fourier transform can be developed by finding the Fourier series of a periodic function and then tending T to infinity. The Fourier transform is an extremely useful mathematical tool and is extensively used in the analysis of linear time-invariant systems, cryptography, signal analysis, signal processing, astronomy, etc. Several applications ranging from RADAR to spread spectrum communication employ Fourier transform. 2.8. Derivation of the Fourier Transform of a non-periodic signal from the Fourier series of a periodic signal Let x(t) be a non-periodic function and, xT (t) be periodic with period T, and let their relation is given by x(t) =

(t) The

Fourier series of a periodic signal x (t) is jn

t and ω

Where

T Let nω0 = ω at T → . As T → , we have ω

→ 0 and the discrete Fourier spectrum

becomes continuos. Further, the summation becomes integral and xT(t) → x(t). Thus, as T → , T X(ω) = Hence, X(ω) is called Fourier transform or the Fourier integral of x(t).

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[n⍵0 = ω, T x(t) = T→∞ As T →

becomes infinitesimally small and may be represented by dω.

Also the summation becomes integration. Hence, x(t) is called the inverse Fourier transform of X(ω) . The equations X(ω) = And For X(ω) and x(t) are known as Fourier transform pair and can be denoted as: X(ω) = F and

x(t) =

The other notation that can be used to represent the Fourier transform pair is x(t) Magnitude and phase representation of Fourier transform The magnitude and phase representation of the Fourier transform is the tool used to analyse the transformed signal. In general, X( ) is a complex valued function of . Therefore, X( ) can be written as: X( ) = XR( ) + jXI(⍵) where XR( ) is real part of X( ) and XI(

2.9. Solved Problems: Problem 1: Find the Fourier transform of x(t) = f(t-2)+f(t+2) Solution: Given

x(t) = f(t-2)+f(t+2)

Using linearity property MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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F[x(t)] = F[f(t-2)+f(t+2)] Using time shifting property [ i.e. x(t

)] , we have

F[x(t)] = F[f(t)] − 2𝜔 + F[f(t)] 2𝜔 = − 2𝜔 F(⍵)+ 2𝜔 F(⍵) F(⍵)[

2𝜔

− 2𝜔 ]

+

Problem 2:Find the Fourier transform of the signal Solution: Given x(t) = − ( ) We know that F[ The signal x(t) =

−

−

()

( )is the time reversal of the signal

reversal property [ i.e. x(-

−

(− ). Therefore, using time

, we have

F[ ∴

Problem 3: Find the Fourier transform of the signals cos ωot u(t) Solution: Given

x(t) = cos ωot u(t)

i.e.

u(t) X( ) = F[cos ωot u(t)] =

With impulses of strength

dt

at ω = ωo and ω = −ωo

X(

Problem 4:Find the Fourier transform of the signals sin ωot u(t) Solution: Given i.e.

x(t) = sin ωot u(t) u(t)

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X( ) = F[sin ωot u(t)] =

With impulses of strength

dt

at ω = ωo and ω = −ωo

X(

Problem 5: Find the Fourier transform of the signals e−tsin5t u(t) Solution: Given

x(t) = e−tsin 5t u(t) x(t) =

u(t)

X( ) = F[e−t sin 5t u(t)] )u(t)] e−jωt dt

[neglecting impulses] Problem 6: Find the Fourier transform of the signals e−2tcos 5t u(t) Solution: Given

x(t) = e−2tcos 5t u(t) x(t) =

u(t)

X( ) = F[e−2t cos 5t u(t)] )u(t)] e−jωt dt

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[neglecting impulses]

Problem 7: Find the Fourier transform of the signals e− t sin 5 t

for all t

Solution: x(t) = e− t sin 5 t

Given i.e.

x(t) =

i.e. ∴

x(t) = − X(

5

for all t

(− ) +

5

()

)u(t)] e−jωt dt

[neglecting impulses] Problem 8: Find the Fourier transform of the signals eat u(-t) Solution: Given

x(t) = eat u(-t) X(

dt dt =

dt =

Problem 9: Find the Fourier transform of the signals teat u(t) Solution: Given

x(t) = teat u(t) X(

dt dt =

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dt =

=

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2.10. Fourier Transform of Standard Signals Introduction: • The Fourier transform of a finite duration signal can be found using the formula

This is called as analysis equation •

The inverse Fourier transform is given by

This is called as synthesis equation Both these equations form the Fourier transform pair. Existence of Fourier Transform: The Fourier Transform does not exist for all aperiodic functions. The condition for a function x(t) to have Fourier Transform, called Dirichlet conditions are: 1.

is absolutely integrable over the interval -∞ to +∞,that is

2.

has a finite number of discontinuities in every finite time interval. Further, each of these discontinuities must be finite. 3. has a finite number of maxima and minima in every finite time interval. 1. Impulse Function Given δ ,

Then =

e−jωt t=0

=

1

Hence , the Fourier Transform of a unit impulse function is unity. w w

w

The impulse function with its magnitude and phase spectra are shown in below figure:

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Similarly,

2. Single Sided Real exponential function 𝐞−𝐚 ( ) Given Then

or Now,

Figure shows the single-sided exponential function with its magnitude and phase spectra.

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3.Double sided real exponential function 𝐞−𝐚 Given

And A Two sided exponential function and its amplitude and phase spectra are shown in figures below:

4. Constant Amplitude (1) Let ∞ ≤t≤ ∞ The waveform of a constant function is shown in below figure .Let us consider a small section of constant function, say, of duration .If we extend the small duration to infinity, we will get back the original function.Therefore

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Where By definition, the Fourier transform of x(t) is: X( ) = F[x(t)] = F

dt =

Using the sampling property of the delta function

, we get

X(

5. Signum function (sgn(t)) The signum function is denoted by sgn(t) and is defined by sgn(t) = This function is not absolutely integrable. So we cannot directly find its Fourier transform. Therefore, let us consider the function e−a t sgn(t) and substitute the limit a 0 to obtain the above sgn(t) Given x(t) = sgn(t) =

sgn(t) =

X( ) = F[sgn(t)] =

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and Figure below shows the signum function and its magnitude and phase spectra

6. Unit step function u(t) The unit step function is defined by u(t) since the unit step function is not absolutely integrable, we cannot directly find its Fourier transform. So express the unit step function in terms of signum function as:

u(t) =

x(t)= u(t) = X( ) = F[u(t)] = F

We know that F[1] = 2𝜋𝛿(𝜔) and F[sgn(t)] = F[u(t)]= u(t) ∴ X(⍵) =∞ at ⍵=0 and is equal to 0 at ⍵=−∞ and ⍵=∞

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7. Rectangular pulse ( Gate pulse)

DEPT.OF ECE

or rect

Consider a rectangular pulse as shown in below figure. This is called a unit gate function and is defined as

x(t) = rect Then X( ) = F[ x(t)] = F

= ∴

F

, that is rect

Figure shows the spectra of the gate function

8. Triangular Pulse Consider the triangular pulse as shown in below figure. It is defined as:

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x

i.e. as

x(t) =

Then X( ) = F[ x(t)] = F

F Or Figure shows the amplitude spectrum of a triangular pulse.

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2.11 Fourier Transform of Periodic Signal The periodic functions can be analysed using Fourier series and that non-periodic function can be analysed using Fourier transform. But we can find the Fourier transform of a periodic function also. This means that the Fourier transform can be used as a universal mathematical tool in the analysis of both non-periodic and periodic waveforms over the entire interval. Fourier transform of periodic functions may be found using the concept of impulse function. We know that using Fourier series , any periodic signal can be represented as a sum of complex exponentials. Therefore, we can represent a periodic signal using the Fourier integral. Let us consider a periodic signal x(t) with period T. Then, we can express x(t) in terms of exponential Fourier series as: x(t) = The Fourier transform of x(t) is: X( ) = F[x(t)] = F

Using the frequency shifting theorem, we have =

=s

X( Where 𝐶

are the Fourier coefficients associated with x(t) and are given by

Thus, the Fourier transform of a periodic function consists of a train of equally spaced impulses. These impulses are located at the harmonic frequencies of the signal and the strength of each impulse is given as 2 .

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Solved Problems: Problem 1:Find the Fourier transform of the signals e3tu(t) Solution: x(t) = e3tu(t)

Given

The given signal is not absolutely integrable. That is

.

Therefore, Fourier transform of x(t) = e3tu(t) does not exist. Problem 2: Find the Fourier transform of the signals cosωotu(t) Solution: Given

x(t) = cosωot u(t)

i.e.

u(t) X( ) = F[cosωot u(t)] =

With impulses of strength

dt

at ω=ωo and ω=−ωo

X(

Problem 3: Find the Fourier transform of the signals sinωot u(t) Solution: Given i.e.

x(t) = sinωot u(t) u(t)

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X( ) = F[sinωot u(t)] =

dt

at ω=ωo and ω=−ωo

With impulses of strength X(

Problem 4: Find the Fourier transform of the signals e−tsin5t u(t) Solution: Given

x(t) = e−tsin5t u(t) x(t) =

u(t)

X( ) = F[e−t sin5t u(t)] )u(t)] e−jωt dt

[neglecting impulses]

Problem 5: Find the Fourier transform of the signals e−2tcos5t u(t) Solution: Given

x(t) = e−2tcos5t u(t) x(t) =

u(t)

X( ) = F[e−2t cos5t u(t)] MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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)u(t)] e−jωt dt

[neglecting impulses]

2.12. Properties of the Fourier Transform These properties provides significant amount of insight into the transform and into the relationship between the time-domain and frequency domain descriptions of a signal. Many of these properties are useful in reducing the complexity Fourier transforms or inverse transforms. Linearity If

f X (jw)

x ( t)

y (t)

f Y (jw)

Then

f (aX (jw) + Y b (jw))

a x(t) + b y(t)

Time Shifting X (jw)

If x (t) I ( Then x (t - t0)

(

- jwt 0

fX (jw)e

To establish this property, consider x(t)=1/2π∫-∞ X(jw)e-jwtO dw Replacing t by t-to in this equation, we obtain x(t-to)=1/2π∫-∞ X(jw)e-jw(t-to)dw x(t)=1/2π∫-∞e-jwto X(jw)ejwt dw Recognizing this as the synthesis equation for x(t-to) ,we conclude that F{x(t-to)}= e-jwto X(jw)

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Conjugation and Conjugate symmetry The conjugation property states that if x (t)

fX(jw)

x* (t)

Then

f X*(-jw)

……………………………………………………(i)

This property follows from the evaluation of the complex conjugate X*(-jw) = [ x(t)

-jwt

dt ]*

∞ = ∫-∞ x*(t)ejwt dt. Replacing w by –w, we see that ∞

X*(-jw)

= ∫-∞ x*(t)ejwt dt. ………………………………………….(ii)

The conjugate property allows us to show that if x(t) is real ,then X(jw) has conjugate symmetry: that is X(-jw)=X*(jw)

x(t) real] …………………………………..(iii)

If x(t) is real so that x*(t) = x(t), we have ,from eq.(ii) X*(-jw)= ∫-∞ x*(t)ejwt dt = X(jw).

Follows by replacing w by –w

Differentiation and Integration If x (t) f X(jw) then differentiating both sides of the Fourier transform synthesis equation we have ∞ dx(t)/dt=1/2π∫-∞jwX(jw)e-jwto dw Therefore, dx(t)/dt

f jwX (jw)

This important property replaces the operation of the differentiation in time domain with that of multiplication by jw in the frequency domain similarly integration should involve division by jwin frequency domain.

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t

DEPT.OF ECE

f ( 1/ jw(X( jw) )+ π X(0 ) δ w ) )

∫-∞ x(t)dt

The impulse term on the right-hand side above equation reflects the dc or average value that can result from integration. Time and Frequency Scaling If x (t) fX(jw) Then x(at)

f1/|a|X(jw/a)

………………………………(v)

Where a is real constant. This property follows directly from the definition of the Fourier transform If a = -1 we have, x (t)

f X ( -jw) -

2.13. Sampling Theorem Statement of the sampling theorem 



A band limited signal of finite energy , which has no frequency components higher than W hertz , is completely described by specifying the values of the signal at instants of time separated by 1/2W seconds and A band limited signal of finite energy, which has no frequency components higher than W hertz, may be completely recovered from the knowledge of its samples taken at the rate of 2W samples per second.

The first part of above statement tells about sampling of the signal and second part tells about reconstruction of the signal. Above statement can be combined and stated alternately as follows: A continuous time signal can be completely represented into samples and recovered back if the sampling frequency is twice of the highest frequency content of the signal i.e., fs≥2W Here fs is the sampling frequency and W is the higher frequency content.

Proof of sampling theorem There are two parts: I) II)

Representation of x(t) in terms of its samples Reconstruction of x(t) from its samples

PART I: Representation of x(t) in its samples x(nTs) Step 1: Define xδ(t) Step 2 : Fourier transform of xδ(t) i.e. Xδ(f) MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Step 3: Relation between X(f) and Xδ(f) Step 4 : Relation between x(t) and x(nTs) Step 1: Define xδ(t) The sampled signal xδ(t) is given as ,

----------- (1) Here, observe that xδ(t) is the product of x(t) and impulse train δ(t) as shown in figure. In the above equation δ(t-nTs) indicates the samples placed at ±Ts,±2Ts,±3Ts… and so on Step 2: Fourier transform of xδ(t)i.e. Xδ(f) Taking FT of equation (1)

---------- (2) We know that FT of product in time domain becomes convolution in frequency domain i.e.,

------------ (3)

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Conclusions:

Figure 1.10 Spectrum of Original Signal & Sampled Signal

Step 3: Relation between X(f) and Xδ(f) Important assumption Let us assume that fs=2W , then as per above diagram

---------- (4) Step 3: Relation between x(t) and x(nTs): From DTFT,

------------ (5)

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Putting above expression in equation (4)

------------ (6) Conclusions:   

Here x(t) is represented completely in terms of x(nTs) Above equation holds for fs=2W.This means if the samples are taken at the rate of 2W or higher, x(t) is completely represented by its samples. First part of the sampling theorem is proved by above two conclusions.

II) Reconstruction of x(t)from its samples Step 1 : Take inverse Fourier transform of X(f) which is in terms of Xδ(f) Step 2 : Show that x(t) is obtained back with the help of interpolation function. Step 1 : Take inverse Fourier transform of equation (6) becomes ,

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Step 2: Let us interpret the above equation and exapanding we get,

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Figure 1.11 Sampled Version of Signal & Reconstruction of x(t) from its samples

Conclusions: The samples x(nTs(nTs)are weighted by sinc functions. The sinc function is the interpolating function above figure shows, how x(t) is interpolated. Step 3: Reconstruction of x(t) by low pass filter When the interpolated signal of equation (6) is passed through the low pass filter of bandwidth W≤f≤W , then the reconstructed waveform shown in figure is obtained. The individual sinc functions are interpolated to get smooth x(t). 2.14. Aliasing When high frequency interferes with low frequency and appears as low frequency, then the phenomenon is called aliasing.

Figure 1.12 Effects of under sampling or aliasing

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Effects of aliasing: 

Since high and low frequencies interfere with each other, distortion is generated.



The data is lost and it cannot be recovered.

Different ways to avoid aliasing: Aliasing can be avoided by two methods  

Sampling rate fs≥2W Strictly band limit the signal to ‘W’

Figure 1.13 Methods to avoid aliasing

2.15. Nyquist Rate & Nyquist Interval

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UNIT – III Laplace Transforms and Z–Transforms 3.1 Introduction: Laplace Transform is a powerful tool for analysis and design of Continuous Time signals and systems. The Laplace Transform differs from Fourier Transform because it covers a broader class of CT signals and systems which may or may not be stable. Most of the LTI Systems act in time domain but they are more clearly described in the frequency domain instead. It is important to understand Fourier analysis in solving many problems involving signals and LTI systems. Now, we shall deal with signals and systems which do not have a Fourier transform. We found that continuous-time Fourier transform is a tool to represent signals as linear combinations of complex exponentials. The exponentials are of the form est with = 𝜔and 𝜔 is an eigen function of the LTI system. Also, we note that the Fourier Transform only exists for signals which can absolutely integrated and have a finite energy. This observation leads to generalization of continuous-time Fourier transform by considering a broader class of signals using the powerful tool of "Laplace transform". Bilateral Laplace Transform The Laplace transform of a general signal x(t) is defined as

It is a function of complex variable„s‟ and is written as =𝜎+ 𝜔, with imaginary parts, respectively. The transform relationship between x(t) and X(s) is indicated as

and , the real and

Existence of Laplace Transform In general,

The ROC consists of those values of „s‟ (i.e., those points in the s-plane) for which X(s) converges i.e., value of s for which

Since =𝜎+ 𝜔 the condition for existence is

Thus, ROC of the Laplace transform of an x(t) consists of all values of s for which x(t) e- t is absolutely integrable.

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3.2. Relation between Laplace and Fourier transform When

corresponds to the Fourier transform of

x(t), i.e., . The Laplace transform also bears a straight forward relationship to the Fourier transform when the complex variable „s‟ is not purely imaginary. To see this relationship, consider X(s) with

or

The real exponential negative.

−𝜎

may be decaying or growing in time, depending on

being positive or

Unilateral Laplace Transform This Transform have considerable value in analyzing causal systems and particularly, systems specified by linear constant coefficient differential equations with nonzero initial conditions( i.e., systems that are not initially at rest) The Unilateral Laplace transform of a continuous time signal x(t) is defined as

Problem 1: Find the Laplace transform of ( )= − ( ) Solution: The Fourier transform X(jω) converges for a>0 and is given by

Now, the Laplace transform is

with

or equivalently, since

= 𝜎+ 𝜔 and 𝜎 =

{ },

, Re{s} > -a

That is

,Re{s} > -a.

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For example a=0, x(t) is the unit step with Laplace transform

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, Re{s} > 0

Problem 2: Find the Laplace transform of Solution: The Laplace transform is

This result converges only when Re{s+a} <0, or Re{s}<-a That is

,Re{s} < -a

Comparing the two results in the above two problems, we see that Laplace transform is identical for both the signals. But the range of values of „s‟ for which the transform converges is different i.e., Region of Convergence (ROC) is different for the above said signals. Note: same X(s) may correspond to different x(t) depending on ROC Problem 3: Find the Laplace transform of ( )=𝛿( ). What is the region of convergence? Solution: The Laplace transform of a general signal x(t) is defined as

From the property of impulse function

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As we know that area under impulse function is unity. Since , which is a constant, Laplace transform converges for all values of s i.e., ROC is entire s-plane Problem 4: What is the Laplace transform of What is its Region of Convergence? Solution: The Laplace transform of a general signal x(t) is defined as

From Euler‟s relation

Problem 5: Find the unilateral Laplace transform of ( )=

− ( +1)

( +1)

Solution: The Unilateral Laplace transform is

ROC: Re{s}>-a Problem 6: Determine the Laplace transform of the ramp function. Solution: The unit ramp function is given as or ( )=

()

The Laplace transform of a general signal x(t) is defined as

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With ROC: Re{s}>0

Problem 7: A damped sine wave is given by ( )= Solution: With the help of Euler‟s Identity,

−

(𝜔 ). Find the LT of this signal

Applying the Laplace transform

ROC : Re{s} > -a Problem 8: Find the Transfer function of a system with impulse response

Solution: Transfer function is obtained by applying Laplace Transform to the impulse response h(t)

Problem 9: Find the Laplace transform of Solution: With the help of Euler‟s Identity,

Therefore,

Problem 10: Find the Unilateral Laplace transform of Solution: The Unilateral Laplace transform of a general signal x(t) is defined as

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3.3 Region of Convergence (ROC) of LT The Laplace transform of a continuous signal x(t) is given by

The Laplace transform has two parts which are, the expression and Region of Convergence respectively. Whether the Laplace transform X(s) of a signal x(t) exists or not depends on the complex variable „s‟ as well as the signal itself. All complex values of „s‟ for which the integral in the definition converges form a region of convergence (ROC) in the s-plane The concept of ROC can be understood easily by finding Laplace transform of two functions given below: a)

with =𝜎+ 𝜔 the integral converges only when therefore

, i.e.,

. The ROC is shown in figure below.

b)

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with =𝜎+ 𝜔 the integral converges only when therefore

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, i.e.,

. The ROC is shown in figure below.

The plane in which ROC is shown is known as s-plane.As =𝜎+ 𝜔, s-plane consists of real and imaginary axes. The region towards left side of the imaginary axis is called Left Half Plane and towards right is called Right Half Plane. Zeros and Poles of the Laplace Transform Laplace transforms in the above examples are rational, i.e., they can be written as a ratio of polynomials of variable „s‟in the general form . N(s) is the numerator polynomial of order M withszk,(k=1,2,…,M) roots D(s) is the denominator polynomial of order N with spk(k=1,2,…,N) roots Roots of numerator polynomial are called zeros and the roots of denominator polynomial are called poles. Poles in s-plane are indicated with „x‟ and zeros with‟o‟. The representation of X(s) through its poles and zeros in the s-plane is referred to as the pole-zero plot of X(s). • •

In general, we assume the order of the numerator polynomial is always lower than that of the denominator polynomial, i.e., M
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3.4. Properties of ROC: Property 1: The ROC of X(s) consists of strips parallel to the jω-axis in the s-plane.

Property 2: For rational Laplace transforms, the ROC does not contain any poles but it is bounded by poles or extends to infinity. Property 3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire splane. Property 4: If x(t) is right sided, and if the line Re{s}= which Re{s}> o will also be in the ROC. Property 5: If x(t) is left sided, and if the line Re{s}= which Re{s}< o will also be in the ROC. Property 6: If x(t) is two sided, and if the line Re{s}= a strip in the s-plane that includes the line Re{s}= o.

o

is in the ROC, then all values of s for

o

is in the ROC, then all values of s for

o

is in the ROC, then the ROC consist of

Property 7: Ifx(t) is right sided and its Laplace transform X(s) is rational, then the ROC in plane is right of the rightmost pole.

s-

Property 8: If x(t) is left sided and its Laplace transform X(s) is rational, then the ROC in s-plane is left of the leftmost pole. Solved Problems: Problem 1: Find the Laplace transform of x(t) = [2e-2t+3e-3t]u(t).Also indicate locations of poles and zeros and Plot Region of Convergence. Solution:The signal given x(t) = [2e-2t+3e-3t]u(t) is a right-sided signal and its Laplace transform is with ROC: Re{s}>-2 ∩ Re{s}>-3 = Re{s}>-2 ROC is right of the right most pole (Property 7) and the plot is shown below

Problem 2: Find the Laplace transform of x(t) = [2e2t+3e3t]u(-t).Also indicate locations of poles and zeros and Plot Region of Convergence.

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Solution:The signal given x(t) = [2e2t+3e3t]u(-t) is a left-sided signal and its Laplace transform is with ROC: Re{s}<2 ∩ Re{s}<3 = Re{s}<2 ROC is left of the left most pole (Property 8) and the plot is shown below

Problem 3: Find the Laplace transform of x(t) = e-tu(t)+e2tu(-t).Also indicate locations of poles and zeros and Plot Region of Convergence. Solution:The signal given x(t) = e-tu(t)+e2tu(-t) is a two-sided signal and its Laplace transform is with ROC: Re{s}>-1 ∩ Re{s}<2 = -1
3.5. Properties of Laplace Transform 1. Linearity of the Laplace Transform Statement: If

with a region of convergence denoted as R1 and with a region of convergence denoted as R2

then

, with ROC containing

1∩ 2

Proof: MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Consider the linear combination of two signals x1(t) and x2(t) as z(t)=ax1(t)+bx2(t). Now, take the Laplace transform of z(t) as

2. Time Shifting: If then Proof:

with ROC= R with ROC= R

Let t- =p

3. Shifting in s-Domain: If R+Re{so} Proof:

4. Time Scaling: If

with ROC= R then

with ROC= R then

with ROC=

with ROC= R1=aR

Proof: To prove this we have to consider two cases: a (real) is positive and a is negative. Case 1: For a>0:

Using the substitution of λ=at; dt=adλ

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Case 2: For a< 0 :

Using the substitution of λ=at; dt=adλ

Combining the two cases, we get

with ROC= R1=aR

5. Convolution Property: If then

with ROC = R1 and , with ROC containing

with ROC = R2

1∩ 2

Proof:

Interchanging the order of integrations

(Since

from

Time

shifting property)

6. Differentiation in the Time Domain: If ROC containing R Proof: Inverse Laplace transform is given by

with ROC= then

with

Differentiating above on both sides with respect to „t‟

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Comparing both equations

is the Laplace transform of

7. Integration in the Time Domain: If then

DEPT.OF ECE

.

with ROC= R

with ROC containing

Proof: This can be derived using convolution property as

8. The Initial and Final Value Theorem: If x(t) and

are Laplace transformable, and under the specific constraints that x(t)=0 for t<0

containing no impulses at the origin,one can directly calculate, from the Laplace transform, the initial value x(0+), i.e., x(t) as t approaches zero from positive values of t. Specifically the initial -value theorem states that Also, if x(t)=0 for t<0 and, in addition, x(t) has a finite limit as t→∞, then the final value theorem says that Proof: To prove these theorems, we need to consider the Unilateral Laplace transform of

Unilateral Laplace transform of

Initial value theorem From the above discussion, we know that

Applying the lim

→∞

ℎ

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Final value theorem we know that

Applying the lim

→0

ℎ

Solved Problems: Problem 1: Find the Laplace transform and ROC of Solution: We know that

Signal

is now delayed by 2 units to get

Therefore, applying time shifting property

Problem 2: Given

, find f(0)

Solution: Consider From the initial value theorem, we know that

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3.6. Inverse Laplace Transform Inverse Laplace transform maps a function in s-domain back to the time domain. One application is to convert a system response to an input signal from s-domain back to the time domain. The Laplace transform converts the differential equations that describe system behaviour to a polynomial. Also the convolution operation which describes the system action on the input signals is converted to a multiplication operation. These two properties make it much easier to do systems analysis in the s-domain. Inverse Laplace transform is performed using Partial Fraction Expansion that split up a complicated fraction into forms that are in the Laplace Transform table. The Laplace transform of a continuous signal x(t) is given by

Since s= +jω

We can recover x(t) from its Laplace transform evaluated for a set of values of s= +jω in the ROC, with fixed and ω varying from -∞ to +∞. Recovering s(t) from X(s) is done by changing the variable of integration in the above equation from ω to s and using the fact that is constant, so that ds=jdω.

The contour of integration in above equation is a straight line in the s-plane corresponding to all points s satisfying Re{s}= . This line is parallel to the jω-axis. Therefore, we can choose any value of such that converges. 3.7. Partial Fraction Expansion As we know that the rational form of X(s) can be expanded into partial fractions, Inverse Laplace transform can be taken according to location of poles and ROC of X(s). The roots of denominator polynomial, i.e., poles can be simple and real, complex or multiple. We know that X(s) is expanded in partial fractions as

Here the roots s0,s1,s2,...sn can be real, complex or multiple. Then the values of k0,k1,k2,...kn constants are calculated accordingly. In order to find the appropriate time domain function, ROC should be indicated for the sdomain function. Otherwise we may have multiple time-domain functions based on different possible ROCs. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Example for Real roots: Problem 1:Find out the partial fraction expansion and hence Inverse Laplace transform of the function

, ROC: Re{s} > 3

Solution: The function

can be written as,

The constants calculated are A=1/3, B=-1/5, C=13/15

From the given ROC: Re{s}>3, the resultant signal x(t) should be right sided. Therefore, Example for Complex roots: Problem 2: Obtain right sided time domain signal for the function Solution: We can write the given function as The constants can be calculated as A=1/8, B=0.874, C=0.5 Therefore,

Finally i.e., Example for Multiple roots: Problem 3: Find out the inverse Laplace transform of

, ROC:Re{s}<-1

Solution: We can write the given function as The constants can be calculated as A=3, B=2, C=2, D=-2 Therefore, From the given ROC: Re{s}<-1, the resultant signal x(t) should be left sided. Finally, MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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From the result of Laplace transform

i.e.

3.8. Laplace transform using Waveform Synthesis In waveform synthesis, the unit step function u(t) and other functions serve as building blocks in constructing other waveforms. Once the waveforms are synthesized in the form of other functions, Laplace transform is found and simplified. For example, we may describe a pulse waveform in terms of unit step functions. A pulse of unit amplitude from t=a to t=b can be formed by taking the difference between the two step functions

i.e., x(t) = u(t -a) - u(t -b)

Hence Examples: Solved Problems: Problem 1: Find the Inverse Laplace transform of

ROC: Re{s}>3

Solution: We know that Writing X(s) in the form of partial fraction expansion

The constants can be calculated as A=4, B=-1 Therefore, From the given ROC: Re{s}>3, the resultant signal x(t) should be right sided. i.e.,

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Problem 2: Find the Inverse Laplace transform of

ROC: Re{s}<-2

Solution: We know that Writing X(s) in the form of partial fraction expansion

The constants can be calculated as A=1, B=-1 Therefore, From the given ROC: Re{s}<-2, the resultant signal x(t) should be left sided. i.e., Problem 3: Find the Inverse Laplace transform of

ROC: -2
Solution: We know that

Writing X(s) in the form of partial fraction expansion

The constants can be calculated as A=1, B=-1

Therefore, From the given ROC: -2-2 which suits for

. Therefore, the resultant signal x(t) should be two-

sided. i.e.,

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PART - II

Z–Transforms 3.9 Introduction: Digital signals are discrete in both time (the independent variable) and amplitude (the dependent variable). Signals that are discrete in time but continuous in amplitude are referred to as discrete-time signals. Z Transform is a powerful tool for analysis and design of Discrete Time signals and systems. The Z Transform differs from Fourier Transform because it covers a broader class of DT signals and systems which may or may not be stable. Fourier Transform only exists for signals which can absolutely integrated and have a finite energy. Z-transforms is a generalization of Discrete-time Fourier transform by considering a broader class of signals. Existence of z Transform In general,

The ROC consists of those values of „z‟ (i.e., those points in the z-plane) for which X(z) converges i.e., value of z for which

Since =

Since

−𝜔

𝜔

the condition for existence is

=1

Therefore, the condition for which z-transform exists and converges is Thus, ROC of the z transform of an x(n) consists of all values of z for which absolutely summable. 3.10. Relation between ZT and Discrete Time Fourier transform

is

When corresponds to the Discrete Time Fourier transform (DTFT) of x(n), i.e., . Thez transform also bears a straight forward relationship to the DTFT when the complex variable = . To see this relationship, consider X(z) with

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Unilateral Z Transform have considerable value in analyzing causal systems and particularly, systems specified by linear constant coefficient difference equations with non-zero initial conditions( i.e., systems that are not initially at rest). The Unilateral z- transform of a discrete time signal x(n) is defined as

3.11. Relation between Laplace, Fourier and z- transforms Let x(t) be a continuous signal sampled with a sampling time of T units. Call this sampled signal as xs(t). We represent this sampled signal by

Applying the Laplace transform to xs(t) results

Interchanging the order of integration and summation

For uniform sampling x(kT)≡x(k) Then Comparing this with the z-transform formula

We get a relation that

z=esT

3.12. Problems Problem 1. Finding the z-transform of a)

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For convergence of X(z), we require that convergence is that range of values of z for which shown in figure below

−1

. Consequently, the region of <1, or equivalently, > and is

Then

b)

This result converges only when

or equivalently, |z| < |a|. The ROC is shown below

If we consider the signals anu(n) and -anu(-n-1), we note that although the signals are differing, their z Transforms are identical which is .Thus, we conclude that to distinguish zTransforms uniquely their ROC's must be specified. Problem 2: Find the z-transform of ( )=𝛿( ). What is the region of convergence? Solution: The z transform of a general signal x( ) is defined as

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Since

Since , which is a constant, the result of z transform converges for all values of „z‟, i.e., ROC is entire z-plane Problem 3: Determine the z-transform of Solution:

Therefore,

Problem 4: Find the z-transform of Solution: We know that AlsoWe know that Therefore, ROC: 2<|z|<3 Problem 5: Find the two sided z-transform of the signal

Solution: Here x(n) can be written as

Therefore, the z-transform of x(n) is

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ROC: 1/3 < |z| < 2 3.13. Region of Convergence (ROC) of Z-Transforms The Z- transform of a discrete signal x(n) is given by

The Z-transform has two parts which are, the expression and Region of Convergence respectively. Whether the Z-transform X(z) of a signal x(n) exists or not depends on the complex variable „z‟ as well as the signal itself. All complex values of „z=rejω‟ for which the summation in the definition converges form a region of convergence (ROC) in the z-plane. A circle with r=1 is called unit circle and the complex variable in z-plane is represented as shown below.

Description : The concept of ROC can be understood easily by finding z transform of two functions given below: a)

For convergence of X(z), we require that convergence is that range of values of z for which shown in figure below

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−1

. Consequently, the region of <1, or equivalently, > and is

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Then

b)

This result converges only when

or equivalently, |z| < |a|. The ROC is shown below

3.14. Properties of ROC Property 1: The ROC of X(z) consists of a ring in the z-plane centered about the origin.

Property 2: If the z-transform X(z) of x(n) is rational, then the ROC does not contain any poles but is bounded by poles or extend to infinity. Property 3: If x(n) is of finite duration, then the ROC is the entire z-plane, except possibly z=0 and / or z=∞ Property 4: If x(nt) is a right sided sequence, and if the circle |z|=ro is in the ROC, then all finite values of z for which |z|>ro will also be in the ROC.

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Property 5: If x(n) is a left sided sequence, and if the circle|z|=ro is in the ROC, then all values of z for which 0<|z|
is right sided

We know that Therefore,

{shown in figure a} and {Shown in figure b}

For convergence of X(z), both sums must converge, which requires that the ROC should be an intersection of and . i.e., {shown in figure c} The pole zero plot and ROC are shown in the figure below

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Problem 2:Draw the pole-zero plot and graph the frequency response for the system whose transfer function is

. Is the system both causal and stable?

Solution: The transfer function can be factored into

The Pole-zero diagram is shown below

The magnitude and phase frequency responses of the system are obtained by substituting z=ejωand varying the frequency variable ω over a range of 2 . The frequency response plots are illustrated in Figure below

The system is both causal and stable because all the poles are inside the unit circle Problem 3: For the following algebraic expression for the z-transform of a signal, determine the number of zeros in the finite z-plane and the number of zeros at infinity

Solution: The given z-transform may be written as

Clearly, X(z) has a zero at z=1/2. Since the order of the denominator polynomial exceeds the order of the numerator polynomial by 1, X(z) has a zero at infinity. Therefore, X(z) has one zero in the finite z-plane and one zero at infinity.

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3.15. Properties of Z-Transform The z transform of a discrete signal x(n) is given by

And inverse z transform is given by

1. Linearity: If

with ROC = R1 and

with ROC = R2 then , with ROC containing

1∩ 2

Proof: Taking the z-transform

2. Time Shifting: If

with ROC= R

then with ROC= R, except for the possible addition or deletion of the origin or infinity Proof:

Let n-m=p

3. Scaling in the z-Domain: If with ROC= R then

with ROC= |zo|R where, |zo|R is the scaled version of R.

Proof:

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4.

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Time Reversal: If

with ROC= R then with ROC=

Proof:

Let -n=p

5. The Initial Value Theorem: If x(n)=0, for n < 0 then initial value of x(n) i.e., We know that Z{x(n)}=

Proof:

as x(n) is causal.

Expanding the summation

Applying the lim

→∞

฀

i.e.,

6. The Final Value Theorem: If x(n) is causal and X(z) is the z-transform of x(n) and if all the poles of X(z) lie strictly inside the unit circle except possibly for a first order pole at z=1 then

Proof: Consider the z-transform of x(n)-x(n-1)

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Also, the above can be written as

Applying the limit z→1 on both sides

LHS after applying the limit z→1 becomes

All terms cancel except x(N). Therefore,

7.Differentiation in the z-Domain: If with ROC= R then with ROC = R Proof: z transform is given by

Differentiating above on both sides with respect to „z‟

Comparing both equations

is the z transform of ( ).

3.16. Inverse Z-Transform Inverse z-transform maps a function in z-domain back to the time domain. One application is to convert a discrete system response to an input sequence from z-domain back to the time domain. The z-transform converts the Linear Constant Coefficient Difference Equations(LCCDE) that describe system behaviour to a polynomial. Also the convolution operation which describes the system action on the input signals is converted to a multiplication operation. These two properties make it much easier to do systems analysis in the z-domain. Inverse z-transform is performed using Long Division Method(Power Series Expansion method), Partial Fraction Expansion and Residue method (Contour Integral Method). MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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The z- transform of a discrete signal x(n) is given by

Since z=rejω

Long Division Method (Power Series Expansion) Z-transform of the sequence x(n) is given as,

From above expansion of z-transform, the sequence x(n) can be obtained as,

The Power series expansion can be obtained directly or by long division method. Example: Determine inverse z-transform of the following: i)

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ii)

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Partial Fraction Expansion

As we know that the rational form of X(z) can be expanded into partial fractions, Inverse z-transform can be taken according to location of poles and ROC of X(z). Following steps are to be performed for partial fraction expansions: Step 1: Arrange the given X(z) as,

Step 2:

Where Ak for k=1, 2,…N are the constants to be found in partial fractions. Poles may be of multiple order. The coefficients will be calculated accordingly. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Step 3: Above equation can be written as

Step 4: All the terms in above step are of the form

. Depending upon ROC, following

standard pairs must be used. with ROC: |z| >| |,i.e., causal response with ROC: |z| <| |,i.e., non-causal response Example: Determine the inverse z-transform of For ROC i) |z| > 1

ii)|z| < 0.5

iii)0.5 < |z| < 1

Solution: Given Which can be written as After finding the constants as A=2 and B=-1

i)

For ROC |z| > 1 i.e., causal or right sided

ii) For ROC |z| < 0.5 i.e., non-causal or left sided

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iii) For ROC 0.5 < |z| < 1 i.e., two sided The ROC is a circular strip between z>0.5 and z<1 Residue Method (Contour Integral Method) Cauchy integral theorem is used to calculate inverse z-transform. Following steps are to be followed: Step 1: Define the function ( )= ( ) product of poles.

−1

, which is rational and its denominator is expanded into

Here „m‟ is order of the pole Step 2: For poles of order „m‟, the residue of

can be calculated as,

Step 3: i) Using residue theorem, calculate x(n) for poles inside the unit circle, i.e.,

ii) For

poles

outside

the

contour

of

integration,

with n < 0 Example: Determine the inverse z-transform of

,

ROC : |z| > |a| using contour integration Solution: Step 1:

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Step 2: Here the pole is at z=a and it has order m=2. Hence finding the residues at z=a

Step 3: The sequence x(n) is given as

Since ROC:

>| |

Examples: Solved Problems: Problem 1: Find the inverse z-transform of

, |z|>|a|

Solution: log(1-p) is expanded with the help of power series. It is given as

Therefore, for |z| > |a| Hence

i.e.,

Problem 2: Find the inverse z-transform of

using partial fraction method when

ROC is (i) |z| <- 2 (ii) |z| > 3 (iii)

-2 < |z| < 3

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Solution: Given Applying partial fractions

i)

For ROC |z| < -2

ii)

For ROC |z| > 3

iii)

For ROC -2 < |z| <3

Problem 3: Determine the sequence whose z-transform is, ROC : |z|> 1 Solution: To arrange X(z) in proper form suitable for partial fraction expansion The highest power of denominator polynomial should be atleast one less than that of numerator polynomial Let us arrange X(z) as follows:

Now perform one step division so that order of numerator polynomial is reduced by one unit.

Now MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Writing in partial fractions

Finding the constants A =8 and B=-9

As ROC is |z| > 1, representing the signal as causal signal which is right sided

Problem 4: An LTI system is characterized by the system function

Specify the ROC of H(z) and determine h(n) for the following conditions a) b) c)

The system is causal and unstable The system is non-causal and stable The system is non-causal and unstable

Solution: Given that

Using partial fraction expansion, we obtain

Finding the constants A =1 and B=2

The system has poles at z=0.5 and z=3 a) For the system to be causal and unstable, the ROC of H(z) is the region in the z-plane outside the outermost pole and it must not include the unit circle. Therefore, the ROC is the region |z| > 3

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Hence b) For the system to be non-causal and stable, the ROC of H(z) is the ring in the z-plane and it must include the unit circle. Therefore, the ROC is the region, 0.5 < |z| < 3

Hence c) For the system to be non-causal and unstable, the ROC of H(z) is the ring in the zplane inside the inner most pole and it must not include the unit circle. Therefore, the ROC is the region, |z| < 0.5

Hence

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UNIT – IV Random Processes – Temporal Characteristics 4.1. Introduction A Random Variable ‘X’ is defined as a function of the possible outcome ‘s’ of an experiment or whose value is unknown and possibly depends on a set of random events. It is denoted by X(s). The Concept of Random Process is based on enlarging the random variable concept to include time ‘t’ and is denoted by X(t,s) i.e., we assign a time function to every outcome according to some rule. In short, it is represented as X(t). A random process clearly represents a family or ensemble of time functions when t and s are variables. Each member time function is called a sample function or ensemble member. Depending on time ‘t’ and outcome ’s’ fixed or variable. A random process represents a single time function when t is a variable and s is fixed at a specific value. Note: A random process represents a random variable when t is fixed and s is a variable. A random process represents a number when t and s are both fixed. 4.2. Classification of Random Processes A Random Processes X(t) has been classified in to four types as listed below depending on whether random variable X and time t is continuous or discrete. 1. Continuous Random Processes If a random variable X is continuous and time t can have any of a continuum of values, then X(t) is called as a continuous random process. Example: Thermal Noise

Fig 1: Continuous Random Processes

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2. Discrete Random Processes If a random variable X is discrete and time t is continuous, then X(t) is called as a discrete random process. The sample functions will have only two discrete values.

Fig 2: Discrete Random Processes 3. Continuous random Sequence If a random variable X is continuous and time t is discrete, then X(t) is called as a continuous random sequence. Since a continuous random sequence is defined at only discrete times, it is also called as discrete time random process. It can be generated by periodically sampling the ensemble members of continuous random processes. These types of processes are important in the analysis of digital signal processing systems.

Fig 3

: Continuous

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Random

Sequence

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4. Discrete Random Sequence If a random variable X and time t areboth discrete, then X(t) is called as a discrete random sequence. It can be generated by sampling the sample functions of discrete random process or rounding off the samples of continuous random sequence.

Fig 4: Discrete Random Sequence 4.3. Deterministic and Non-deterministic processes In addition to the processes, discussed above a random process can be described by the form of its sample functions. A Process is said to be deterministic process, if future values of any sample function can be predicted from past values. These are also called as regular signals,which have a particular shape. Example: X(t) = A Sin (ωt + ϴ), A, ω and ϴ may be random variables

Fig 5: Example of Deterministic Process

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A Process is said to be non-deterministic process, if future values of any sample function cannot be predicted from past values.

Fig 6: Example of Non-Deterministic Processes 4.4. Distribution Function and Density function Distribution Function: Probability distribution function (PDF) which is also be called as Cumulative Distribution Function (CDF) of a real valued random variable ‘X ‘ is the probability that X will take value less than or equal to X. It is given by In case of random process X(t), for a particular time t, the distribution function associated with the random variable X is denoted as In case of two random variables, X1 = X(t1) and X2 = X (t2), the second order joint distribution function is two dimensional and given by and can be similarly extended to N random variables, called as Nth order joint distribution function Density Function: The probability density function (pdf) in case of random variable is defined as the derivative of the distribution function and is given by

In case of random process, density function is given by

In case of two random functions, two dimensional density function is given by

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4.5 Independence and Stationary Random Process In probability theory, two events are independent, statistically independent, or stochastically independent if the occurrence of one does not affect the probability of occurrence of other i.e., having the probability of their joint occurrence equal to the product of their individual probabilities. A random process becomes a random variable when time is fixed at some particular value. The random variable will possess statistical properties such as a mean value, moments, variance etc. that are related to its density function. If two random variables are obtained from the process for two time instants they will have statistical properties related to their joint density function. More generally, N random variables will possess statistical properties related to their N dimensional joint density function. A random process is said to be stationary if all its statistical properties do not change with time other processes are called non-stationary. It can also be defined as a stationary time series is one whose statistical properties such as mean, variance, autocorrelation, etc. are all constant over time.

Fig 1: Stationary and Non – Stationary Time Series As an example, white noise is stationary. The sound of a cymbal clashing, if hit only once, is not stationary because the acoustic power of the clash (and hence its variance) diminishes with time. Statistical Independence In case of two random variables X and Y, defined by the events A = {X ≤ x}, B = {Y ≤ y}, to be statistically independent, Consider two processes X(t) and Y(t). The two processes, X(t) and Y(t) are said to be statistically independent, if the random variable group X(t1), X(t2), X(t3) …… X(tN) is independent of the group Y(t1’), Y(t2’), Y(t3’)……….Y(tM’). Independence requires that the joint density function be factorable by groups. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Independence requires that the joint density be factorable by groups fx,Y(x1 ,.... xN, y

1 ,....

xN; t1,......tN ) fy(y1 ,.... yN;

Statistical independence means one event conveys no information about the other; statistical dependence means there is some information. Stationary To define Stationary, Distribution and density functions of random process X(t) must be defined. For a particular time t1, the distribution function associated with the random variable X1=X(t1)will be denoted Fx(x1; t1 ). It is defined as Fx(x1; t1)=P{X (t1) ≤x1} For any real number x1. For two random variables X1=X(t1) and X2=X(t2) , the second order joint distribution function is the two dimensional extension of above equation Fx(x1 , x2; t1,t2 )=P{X(t1)≤x1, X(t2)≤x2} In a similar manner, for N random variables Xi=X(ti) , i=1,2,3,....N, the Nth order joint distribution function is Fx(x1 ,.... xN; t1,......tN )=P{X(t1)≤x1,........ X(tN)≤xN} Joint density functions are found from derivatives of the above joint distribution functions fx(x1; t1 )=d Fx(x1; t1 )/dx1 fx(x1 , x2; t1,t2 )=d2Fx(x1 , x2; t1,t2 )/(dx1dx2 ) fx(x1 ,.... xN; t1,......tN )=dNFx(x1 ,.... xN; t1,......tN )/(dx1.......dxN) First order stationary processes A Random process is called stationary to order one, if its first order density function does not change with a shift in time origin. In other words fx(x1; t1 ) =fx(x1; t1+Δ) must be true for any t1 and any real number Δ if X(t) is to be a first order stationary process. If fx(x1; t1) is independent of t1, the process mean value E[X(t)]= =constant. Second Order and wide sense Stationary A process is called stationary to order two if its second order density function satisfies fx(x1 , x2; t1,t2 )= fx(x1 , x2; t1+Δ,t2+Δ ) for all t1 , t2 and Δ. The correlation E[X(t1) X(t2)]of a random process will be a function of t1 and t2. Let us denote the function by RXX(t1,t2) and call it the autocorrelation function of the random process X(t): RXX(t1,t2)= E[X(t1) X(t2)] MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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The autocorrelation function of second order stationary process is a function only of time differences and not absolute time, that is , if =t2-t1, then RXX(t1,t1+ )= E*X(t1) X(t1+ )= RXX( ) A Second order stationary process is wide sense stationary for which the following two conditions are true: E[X(t)]= =constant and E[X(t1) X(t1+ )= RXX( ) N – Order and Strict Sense Stationary A Random process is stationary to order N if its Nth order density function is invariant to a time origin shift ; that is, if fx(x1 ,....... xN; t1,.......tN )= fx(x1 ,....... xN; t1+Δ,......tN+Δ ) for all t1,.......tN and Δ. Stationary of order N implies stationarity to all orders k≤N. A process stationary to all orders N=1,2,3,.... is called Strict sense Stationary.

Example 1.

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4.6 Time Averages and Ergodicity: In probability theory, an ergodic system is one that has the same behaviour averaged over time as averaged over the samples. A random process is ergodic if its time average is the same as its average over the probability space The time average of a quantity is defined as The time autocorrelation function is denoted Rxx=A[ ( ) ( +𝜏)]. These functions are defined by Rxx By taking the expected value on both sides of the above two equations and assuming the expectation can be brought inside the integrals, then E[ ]= E[Rxx( )]= RXX( ) Ergodic Theorem: The Ergodic Theorem states that for any random process X(t), all the time averages of the sample functions of the X(t) are equal to the corresponding statistical or ensemble averages of X(t). i.e., = Rxx( ) = RXX( ) Ergodic Process: Random processes that satisfy the ergodic theorem are called ergodic processes. The analysis of Ergodicity is extremely complex. In most physical applications, it is assumed that all stationary processes are ergodic processes. Mean Ergodic process: A process X(t) with a constant mean value is called mean ergodic or ergodic in the mean, if its statistical average equals the time average of any sample function ( ) with probability 1 for all sample functions; that is, if E[ ]= =A[ ( )] = with probability 1 for all ( ). Correlation Ergodic Processes: Analogous to a mean ergodic process, a stationary continuous process X(t) with autocorrelation function RXX( ) is called autocorrelation ergodic or ergodic in the autocorrelation if, and only if, for all . 4.7. Autocorrelation Function and its Properties: Consider that a random process X(t) is at least WSS and is a function of time difference = t2-t1. Then the following are the properties of the autocorrelation function of X(t). 1. Mean square value of X(t) is E[X2(t)] = RXX(0). It is equal to the power (average) of the process, X(t). Proof: We know that for X(t), RXX( ) = E[X(t) X(t+ )] . If = 0, then RXX(0) = E[X(t) X(t)] = E[X2(t)] hence proved. 2. Autocorrelation function is maximum at the origin i.e. RXX(0). Proof: Consider two random variables X(t1) and X(t2) of X(t) defined at time intervals t1 and t2 respectively. Consider a positive quantity [X(t1) X(t2)]2 0 Taking Expectation on both sides, we get E[X(t1) X(t2)]2 E[X2(t1)+ X2(t2)

2X(t1) X(t2)]

0

0

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E[X2(t1)]+ E[X2(t2)

2E[X(t1) X(t2)]

RXX(0)+ RXX(0) 2 RXX(t1,t2)

DEPT.OF ECE

0

0 [Since E[X2(t)] = RXX(0)]

Given X(t) is WSS and = t2-t1. Therefore 2 RXX(0 2 RXX( ) RXX(0 RXX( )

0

0 or RXX(0) hence proved.

3. RXX( ) is an even function of i.e. RXX(- ) = RXX( ). Proof: We know that RXX( ) = E[X(t) X(t+ )] Let = - then RXX(- ) = E[X(t) X(t- )] Let u=t- or t= u+ Therefore RXX(- ) = E[X(u+ ) X(u)] = E[X(u) X(u+ )] RXX(- ) = RXX( ) hence proved.

4. If a random process X(t) has a non zero mean value, E[X(t)] 0 and Ergodic with no periodic components, then = ̅. Proof : Consider a random variable X(t)with random variables X(t1) and X(t2). Given the mean value is E[X(t)] = ̅ 0 . We know that RXX( ) = E[X(t)X(t+ )] = E[X(t1) X(t2)]. Since the process has no periodic components, as

, the random variable becomes independent, i.e.

= E[X(t1)

X(t2)] = E[X(t1)] E[ X(t2)] Since X(t) is Ergodic E[X(t1)] = E[ X(t2)] = ̅ Therefore

=

hence proved.

5. If X(t) is periodic then its autocorrelation function is also periodic. Proof: Consider a Random process X(t) which is periodic with period T0 Then X(t) = X(t T0) or X(t+ ) = X(t RXX(

T0). Now we have RXX( ) = E[X(t)X(t+ )] then

T0) = E[X(t)X(t+

Given X(t) is WSS, RXX( RXX(

T0)] T0) = E[X(t)X(t+ )]

T0) = RXX( ) Therefore RXX( ) is periodic

hence proved. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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6. If X(t) is Ergodic has zero mean, and no periodic components then = . Proof: It is already proved that

=

. Where ̅ is the mean value of

X(t) which is given as zero. Therefore

= hence proved.

4.8. Properties of Cross Correlation Function: Consider two random processes X(t) and Y(t) are at least jointly WSS. And the cross correlation function is a function of the time difference = t2-t1. Then the following are the properties of cross correlation function. 1. RXY( ) = RYX(- ) is a Symmetrical property. Proof: We know that RXY( ) = E[X(t) Y(t+ )] also RYX( ) = E[Y(t) X(t+ )] Let = - then RYX(- ) = E[Y(t) X(t- )] Let u=t- or t= u+ . then RYX(- ) = E[Y(u+ ) X(u)] = E[X(u) Y(u+ )] Therefore RYX(- ) = RXY( ) hence proved. 2. If RXX( ) and RYY( ) are the autocorrelation functions of X(t) and Y(t) respectively then the cross correlation satisfies the inequality: . Proof: Consider two random processes X(t) and Y(t) with auto correlation functions and . Also consider the inequality E[ E[ E[ We know that E[X2(t)] = RXX(0) and E[Y2(t)] = RYY(0) and E[X(t) X(t+ )] = RXY( ) Therefore

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Or hence proved. 3. If RXX( ) and RYY( ) are the autocorrelation functions of X(t) and Y(t) respectively then the cross correlation satisfies the inequality:

RYY(0)].

Proof: We know that the geometric mean of any two positive numbers cannot exceed their arithmetic mean that is if RXX( ) and RYY( ) are two positive quantities then at =0, RYY(0)].We know that Therefore

RYY(0)]. Hence proved.

4.9. Covariance functions for random processes: Auto Covariance function: Consider two random processes X(t) and X(t+ ) at two time intervals t and t+ . The auto covariance function can be expressed as CXX(t, t+ ) = E[(X(t)E[X(t)]) ((X(t+ ) – E[X(t+ )])] or CXX(t, t+ ) = RXX(t, t+ ) - E[(X(t) E[X(t+ )] If X(t) is WSS, then CXX( ) = RXX( ) -

. At = 0, CXX(0) = RXX(0) -

=E[X2]-

=

Therefore at = 0, the auto covariance function becomes the Variance of the random process. The autocorrelation coefficient of the random process, X(t) is defined as (t, t+ ) =

if =0, =1. Also

1.

Cross Covariance Function: If two random processes X(t) and Y(t) have random variables X(t) and Y(t+ ), then the cross covariance function can be defined as CXY(t, t+ ) = E[(X(t)-E[X(t)]) ((Y(t+ ) – E[Y(t+ )])] or CXY(t, t+ ) = RXY(t, t+ ) - E[(X(t) E[Y(t+ )]. If X(t) and Y(t) are jointly WSS, then CXY( ) = RXY( ) - ̅ ̅. If X(t) and Y(t) are Uncorrelated then CXY(t, t+ ) =0. The cross correlation coefficient of random processes X(t) and Y(t) is defined as (t, t+ ) =

if =0, .

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4.10. Gaussian Random Process: Consider a continuous random process X(t). Let N random variables X1=X(t1),X2=X(t2), . . . ,XN =X(tN) be defined at time intervals t1, t2, . . . tN respectively. If random variables are jointly Gaussian for any N=1,2,…. And at any time instants t 1,t2,. . . tN. Then the random process X(t) is called Gaussian random process. The Gaussian density function is given as fX(x1, x2…… xN ; t1, t2,….. tN) = where CXX is a covariance matrix. Poisson’s random process: The Poisson process X(t) is a discrete random process which represents the number of times that some event has occurred as a function of time. If the number of occurrences of an event in any finite time interval is described by a Poisson distribution with the average rate of occurrence is λ, then the probability of exactly occurrences over a time interval (0,t) P[X(t)=K] =

, K=0,1,2, . . .

And the probability density function is

-k).

4.11. System Response Consider a continuous LTI system with impulse response h(t). Assume that the system is always causal and stable. When a continuous time random process X(t) with ensemble members is applied on this system, the output response is also a continuous time random process Y(t) with ensemble members . Let a random process X(t) be applied to a continuous linear time invariant system whose impulse response h(t) . Then the output response is also a random process. It can be expressed by the convolution integral, Y(t)=h(t)*X(t) i.e. the output response is Y(t)=

Mean value of output Response: Consider that the random process X(t) is wide sense stationary process. Mean value of output response = E[Y(t)] Then, E[Y(t)] =E [h(t)*X(t)]

But

=

= constant , since X(t) is wide sense stationary

Then E[Y(t)]=

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Mean Square value of Output Response:

Where 𝜏1and 𝜏2 are the shift in time variables

If the input X(t) is a wide sense stationary random process, then Therefore,

The above expression is independent of time t. It represents the output power. Autocorrelation Function of Output Response: The autocorrelation of Y(t) is

We know that if the input X(t) is a wide sense stationary random process, let the time difference 𝜏 = t2 – t1 and t = t1, Then

Cross Correlation Function of Response: If the input X(t) is a wide sense stationary random process, then the cross correlation function of input X(t) and output Y(t) is

Therefore, This expression show the relationship between the autocorrelation functions and cross correlation functions. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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UNIT – V Random Process - Spectral Characteristics 5.1.

Introduction: To design any LTI filter which is intended to extract or suppress the signal, it is necessary to understand how the strength of a signal is distributed in the frequency domain. There are two immediate challenges in trying to find an appropriate frequency-domain description for a WSS random process. The present module focuses on the expected power in the signal which is a measure of signal strength and will be shown that it meshes nicely with the second moment characterizations of a WSS process. It is possible to define a meaningful PSD for a stationary (at least in the wide sense) random process. For non-stationary processes, PSD does not exist. On these lines, the PSD of a random process is defined as a weighted mean of the PSDs of all sample functions, as it is not known exactly which of the sample functions may occur in a given trial. 5.2.

Power Spectral Density of a random Process:

Consider

As will represent the random process X (t). Applying Fourier Transform, We get Using Parseval’s theorem to find the energy of the signal

Therefore, the power associated with XT (t) is The average power is given by

Where

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is the contribution to the average power at frequency w and represents the power spectral density the power spectral density for As T tends to infinite, the left-hand side in the above expression represents the average power of X (t). Therefore, the PSD (𝜔) of the process X(t) is defined by

Thus, The PSD (𝜔) of a random process X(t) is defined as the ensemble average of the PSDs of all sample functions, Thus,

Relation Between Power-spectral Density and Autocorrelation function of the Random Process (Wiener-Khinchin Relation)

By Taking Inverse FT of PSD, we have

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=

Since, we know that

Similarly, we have

We know that

Let,

Therefore,

Hence, The RHS of the above eq. is the time average of Auto correlation function. Thus, Time average of Autocorrelation function and the PSD form a Fourier Transform Pair. Thus, for a WSS process, Autocorrelation and Power Spectral Density form a Fourier Transform Pair and this is referred to as Wiener-Khintchine relation.

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5.3.Properties of power density spectrum: The properties of the power density spectrum SXX(ω) for a WSS random process X(t) are given as (1) SXX(ω) Proof: From the definition, the expected value of a non negative function E[

] is always

non-negative. Therefore SXX(ω)

hence proved.

(2) The power spectral density at zero frequency is equal to the area under the curve of the autocorrelation RXX ( ) i.e. SXX Proof: From the definition we know that SXX SXX

at ω=0,

hence proved

(3) The power density spectrum of a real process X(t) is an even function i.e. SXX(-ω)= SXX(ω) Proof: Consider a WSS real process X(t), then SXX

also

SXX Substitute = - then SXX Since X(t) is real, from the properties of autocorrelation we know that, RXX (- ) = RXX ( ) Therefore SXX SXX(-ω)= SXX(ω) hence proved. (4) SXX(ω) is always a real function Proof: We know that SXX Since the function

is a real function, SXX(ω) is always a real function hence proved.

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(5) If SXX(ω) is a psd of the WSS random process X(t), then = A {E[X 2(t)]} = RXX (0) or The time average of the mean square value of a WSS random process equals the area under the curve of the power spectral density. Proof: We know that RXX ( ) = A {E[X (t + ) X(t)]} at =0,

RXX (0) = A {E[X2(t)]} =

= Area under the curve of the power spectral density.

Hence proved. 1. The autocorrelation function of a WSS process X(t) is given by

Find the power spectral density of the process. Solution:

5.4 Cross Power density spectrum: Consider two real random processes X(t) and Y(t). which are jointly WSS random processes, then the cross power density spectrum is defined as the Fourier transform of the cross correlation function of X(t) and Y(t).and is expressed as

SXY

and SYX

By inverse Fourier transformation, we can obtain the cross correlation functions as

RXY

and RYX

Therefore the cross psd and cross correlation functions are forms a Fourier transform pair. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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If XT(ω) and YT(ω) are Fourier transforms of X(t) and Y(t) respectively in interval [-T,T], Then the cross power density spectrum is defined as

SXY

and SYX

Relation Between cross Power-spectral Density and Cross Correlation function

Consider the inverse Fourier Transform of cross PSD i.e.



Since,

Similarly,

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Hence

Let 1 = 𝜏 →

Hence,

1=

𝜏

Thus, time average of cross-correlation function and the cross spectral density form a Fourier Transform Pair. Thus, for a two jointly WSS processes, cross-correlation and cross Spectral Density form a Fourier Transform Pair.

5.5. Properties of cross power density spectrum: The properties of the cross power for real random processes X(t) and Y(t) are given by (1) SXY(-ω)= SXY(ω) and SYX(-ω)= SYX(ω) Proof: Consider the cross correlation function

The cross power density spectrum is

SXY Let

= - Then Since RXY(- ) = RXY( )

SXY SXY

Therefore SXY(-ω)= SXY(ω) Similarly SYX(-ω)= SYX(ω) hence proved. (2) The real part of SXY(ω) and real part SYX(ω) are even functions of ω i.e. Re [SXY(ω)] and Re [SYX(ω)] are even functions. Proof: We know that SXY

and also we know that

=cosωt-jsinωt, Re [SXY Since cos ωt is an even function i.e. cos ωt = cos (-ωt) Re [SXY Therefore SXY(ω)= SXY(-ω) Similarly SYX(ω)= SYX(-ω) hence proved.

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(3) The imaginary part of SXY(ω) and imaginary part SYX(ω) are odd functions of ω i.e. Im [SXY(ω)] and Im [SYX(ω)] are odd functions. Proof: We know that SXY

and also we know that

=cosωt-jsinωt, - Im [SXY(ω)]

Im [SXY

Therefore Im [SXY(ω)] = - Im [SXY(ω)] Similarly Im [SYX(ω)] = - Im [SYX(ω)] hence proved. (4) SXY(ω)=0 and SYX(ω)=0 if X(t) and Y(t) are Orthogonal. Proof: From the properties of cross correlation function, We know that the random processes X(t) and Y(t) are said to be orthogonal if their cross correlation function is zero. i.e. RXY( ) = RYX( ) =0. We know that SXY Therefore SXY(ω)=0. Similarly SYX(ω)=0 Hence proved. (5) If X(t) and Y(t) are uncorrelated and have mean values SXY(ω)=2

and , then

.

Proof: We know that SXY

Since X(t) and Y(t) are uncorrelated, we know that = Therefore SXY SXY SXY Therefore SXY(ω)=2

. Hence proved.

5.6 Problems: 1.A random process is defined as , where is a WSS process, 𝜔0 is a real constant and is a uniform random variable over(0,2𝜋) an is independent of X(t). Find the PSD of Y(t). Soln.:

.

]

Since, and X(t) are independent of each other, . 𝐶(𝜔0( +𝜏)+𝜃)] MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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PSD of Y(t) is Fourier Transform of

. i.e

2.A random process is given by X(t) and

, where A and B are real constants and

Y(t) are jointly WSS processes. (i) Find the Power spectrum of Z(t) (ii) Find the cross power spectrum

(𝜔)

Soln.: ]] = Power spectrum of Z(t) is = (ii)

= =

}]

]=

3.A stationary random process X(t) has a spectral density stationary Y(t) has a spectral density

and an independent

. Assuming X(t) and Y(t) are of zero

mean, find the (i) PSD of U(t)=X(t)+Y(t) (ii) Soln.: (i) PSD of U(t)= PSD of X(t) +PSD of Y(t) =1 (ii)

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5.7. Spectral characteristics of system response Let the input for an LTI systemof impulse response h(t), be a WSS process x(t), resulting an output y(t).

LTI System

x(t)

y (t )

Impulse Response h(t)

The response of the system y(t)= x(t)*h(t)=

Mean value of the Output Process Expected Value of the output is

Since x(t) is WSS,

.

Then, For an LTI system, Unit impulse response h(t) and its Transfer function H(𝜔) form a Fourier Transform Pair, i.e. Then, Hence,

.

Thus, the mean of the output process is independent of time and is a constant. Autocorrelation Function of the Output Process

Where 𝜏= 2 − 1 Thus, the Autocorrelation function of the output process is independent of time and is dependent on “𝜏". Hence, y(t) is a stationary process. MALLA REDDY ENGINEERING COLLEGE FOR WOMEN

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Mean Squared Value of the Output Process

Since, y(t) is stationary, the above equation can be written as

The MS value of y(t) is

Cross correlation Function between the input process and output process

. In the above eq.1, replace 𝜏

−𝜏 i.e

.

Since, Autocorrelation function is an even function of .

In the above eq.2, replace second ‘x’ in the prefix by ‘y’ i.e. In the above eq.3, replace 𝜏 It same as

−𝜏, i.e.

.

, since Autocorrelation is having even symmetry.

Since,

----4.

Relation between the PSDs of the input process and output process of an LTI Systems Take Fourier Transform on both sides for eq.2 ---5 Take Fourier Transform on both sides for eq.4 ---6 Substituting

from eq.5 in eq.6 =

The power at the output of the LTI system is the area enclosed by the output PSD i.e. Output Power

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5.8 Problems: 1. The input to an LTI system with Impulse response h(t)=δ(t)+t.exp(-at).U(t) is a WSS process with mean of 3.Find the mean of the output of the system. Soln.:

Hence, 2. A WSS process X(t) is applied to the following System: + Add X(t)

Y(t)

Delay 2T

If the input PSD is K Watts/Hz, find the out

put PSD

Soln.: The system is represented as The Transfer function of the above system is

and the corresponding

(𝜔) 2 =2+2.𝐶 2𝜔 The output PSD is

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INDEX

S.NO

OBJECTIVE

PAGE NO

1.

TUTORIAL SHEETS

2

2.

ASSIGNMENT QUESTIONS

5

3.

UNITWISE IMPORTANT QUESTIONS

7

4.

OBJECTIVE TYPE QUESTIONS

13

5.

INTERNAL QUESTION PAPERS

6.

EXTERNAL QUESTION PAPERS

1

26

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TUTORIAL SHEETS UNIT – I 1. Compute the energy E∞ and the power P∞ of the following continuous-time signal x(t)=e−2πjt. 2. Make a sketch of each of the following signals (a) f(n) = X∞ k=0 (−0.9)k δ(n − 3 k) (b) g(n) = X∞ k=−∞ (−0.9)|k| δ(n − 3 k) (c) x(n) = cos(0.25 π n) u(n) (d) x(n) = cos(0.5 π n) u(n) 3. Classify each of the following discrete-times systems. (a) y(n) = cos(x(n)). (b) y(n) = 2 n 2 x(n) + n x(n + 1). (c) y(n) = max {x(n), x(n + 1)} Note: the notation max{a, b} means for example; max{4, 6} = 6. (d) y(n) = ฀ x(n) when n is even x(n − 1) when n is odd (e) y(n) = x(n) + 2 x(n − 1) − 3 x(n − 2). (f) y(n) = X∞ k=0 (1/2)k x(n − k). That is, y(n) = x(n) + (1/2) x(n − 1) + (1/4) x(n − 2) + · · · y(n) = x(2 n) 4. A discrete-time system is described by the following rule y(n) = ( x(n), when n is an even integer −x(n), when n is an odd integer where x is the input signal, and y the output signal. (a) Sketch the output signal, y(n), produced by the 5-point input signal, x(n) illustrated below. 1 2 3 2 1 -2 -1 0 1 2 3 4 5 6 n x(n) (b) Classify the system as: i. linear/nonlinear ii. time-invariant/time-varying iii. stable/unstable. UNIT – II 1. State whether y =tan x can be expressed as a Fourier series. If so how?. If not why? 2. State the convergence condition on Fourier series. 3. To what value does the sum of Fourier series of f (x) converge at the point of continuity x =a ? 4. To what value does the sum of Fourier series of f ( x) converge at the point of discontinuity x =a ? 5. Write the formulae for Fourier constants for f ( x) in the interval (-p, p).

2

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UNIT – III 1. The Z-transform of the discrete-time signal x(n) is X(z) = −3 z 2 + 2 z −3 Accurately sketch the signal x(n). 2. Define the discrete-time signal x(n) as x(n) = −0.3 δ(n + 2) + 2.0 δ(n) + 1.5 δ(n − 3) − δ(n − 5) (a) Sketch x(n). (b) Write the Z-transform X(z). (c) Define G(z) = z −2 X(z). Sketch g(n). Let x(n) be the length-5 signal x(n) = {1, 2, 3, 2, 1} where x(0) is underlined. Sketch the signal corresponding to each of the following Z-transforms. (a) X(2z) (b) X(z 2 ) (c) X(z) + X(−z) (d) X(1/z) 3. Sketch the discrete-time signal x(n) with the Z-transform X(z) = (1 + 2 z) (1 + 3 z −1 ) (1 − z −1 ). 4. Define three discrete-time signals: a(n) = u(n) − u(n − 4) b(n) = δ(n) + 2 δ(n − 3) c(n) = δ(n) − δ(n − 1) Define three new Z-transforms: D(z) = A(−z), E(z) = A(1/z), F(z) = A(−1/z) (a) Sketch a(n), b(n), c(n) (b) Write the Z-transforms A(z), B(z), C(z) (c) Write the Ztransforms D(z), E(z), F(z) (d) Sketch d(n), e(n), f(n) 5.

Find the Z-transform X(z) of the signal x(n) = 4 ฀ 1 3 ฀n u(n) − ฀ 2 3 ฀n u(n).

UNIT-IV 1. Differentiate between Random Processes and Random variables with example 2. Prove that the Auto correlation function has maximum value at the origin i.e │RXX( )│= RXX(0) 3. A stationary ergodic random processes has the Auto correlation function with the periodic components as RXX( ) = 25 + 4 1+6𝜏 2 4. Define mean ergodic random processes and correlation ergodic Random processes. 5. Find the mean value of Response of a linear system. 5. a) Define Wide Sense Stationary Process and write it‟s conditions. b) A random process is given as X(t) = At, where A is a uniformly distributed random variable on (0,2). Find whether X(t) is wide sense stationary or not. 6. X(t) is a stationary random process with a mean of 3 and an auto correlation function of 6+5 exp (-0.2 │ │). Find the second central Moment of the random variable Y=ZW, where „Z‟ and „W‟ are the samples of the random process at t=4 sec and t=8 sec respectively. 7. Explain the following i) Stationarity ii) Ergodicity iii) Statistical independence with respect to random processes

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UNIT – V 1. Define wiener khinchine relations 2. State any two properties of cross-power density spectrum. 3. Define cross power spectral density and its examples. 4. Derive the relationship between cross-power spectral density and cross correlation function. a) Check the following power spectral density functions are valid or not ) 𝑜 8(𝜔) 2+𝜔4 ) −(𝜔−1) 2 b) Derive the relation between input PSD and

output PSD of an LTI system

5. X(t) is a stationary random process with a mean of 3 and an auto correlation function of 6+5 exp (-0.2 │ │). Find the second central Moment of the random variable Y=ZW, where ‘Z’ and ‘W’ are the samples of the random process at t=4 sec and t=8 sec respectively. 6. a) Define Wide Sense Stationary Process and write it’s conditions. b) A random process is given as X(t) = At, where A is a uniformly distributed random variable on (0,2). Find whether X(t) is wide sense stationary or not. 1. A stationary process has an autocorrelation function

Find Mean, Mean square and Variance of the process X(t).

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ASSIGNMENT QUESTIONS UNIT-I 1. Define Root Mean Square value of a function. 2. Find the R.M.S value of y =x 2 in (-p, p). 3. (a) For the systems represented by the following functions. Determine whether every system is (1) stable (2) Causal (3) linear (4) Shift invariant (i) T[x(n)]= ex(n) (ii) T[x(n)]=ax(n)+6 4. 2. Determine whether the following systems are static or Dynamic, Linear or Nonlinear,Shift variant or Invarient, Causal or Non-causal, Stable or unstable. (i) y(t) = x(t+10) + 2 x (t) (ii) dy(t)/dt + 10 y(t) = x(t) 5. Explain about the properties of continuous time fourier series. 6. Find the fourier coefficients of the given signal. (4) x(t) = 1+ sin 2wt + 2 cos 2wt + cos (3wt) 7. Determine the Fourier series coefficient of exponential representation of x(t) x(t) = 1, |t| (8) 0, T1< |t|< T/ 2 UNIT-II 1. Find the constant term a0 in the Fourier series corresponding to f (x )= x -x3 in (π, π). 2. If f (x)=x 2 -x4 is expanded as a Fourier series in (-l,l ), find the value of bn . 3. Write the Fourier sine series of k in (0,p). 4. Find the R.M.S value if f ( x ) = x2 in -π £x £.π 5. State the Parseval’s Identity (or) theorem

UNIT-III 1. The signal x is defined as x(n) = a |n| Find X(z) and the ROC. Consider separately the cases: |a| < 1 and |a| ≥ 1 2. Find the Unilateral Z-transform and R.O.C of x(n) = sin ω0 n u(n) 3. A finite sequence x[n]=[5,3,-2,0,4,-3]. Find X[z]and its ROC 4. Find Z- transform of x (n) =cos (now) u (n) 5. Find the z-transform of the following sequences i) x[n] = a-n u[-n-1] ii) x[n] = u[-n] iii)x[-n] = -an u[-n-1] UNIT-IV 1.

Explain about Poisson Random process and also find its mean and variance.

2.

The function of time Z(t) = X1cosω0t- X2sinω0t is a random process. If X1 and X2are independent Gaussian random variables, each with zero mean and variance 2 , find E[Z]. E[Z2 ] and var(z).

5

DEPARTMENT OF ECE

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3.

Briefly explain the distribution and density functions in the context of stationary and independent random processes.

4.

Explain about the following random process (i) Mean ergodic process (ii) Correlation ergodic process (iii) Gaussian random process

5.

State and prove the auto correlation and cross correlation function properties.

6.

a) Define Wide Sense Stationary Process and write it’s conditions. b) A random process is given as X(t) = At, where A is a uniformly distributed random variable on (0,2). Find whether X(t) is wide sense stationary or not. UNIT-V 1. a) Check the following power spectral density functions are valid or not ) 𝑜 8(𝜔) 2+𝜔4 ) −(𝜔−1) 2 b) Derive the relation between input PSD and

output PSD of an LTI system

2. Derive the relationship between cross-power spectral density and cross correlation function. 3. Give the statement of ergodic theorem. 4.

Differentiate between Random Processes and Random variables with example (3 marks) i. Define Power Spectrum Density.

5.

Show that SXX(-ω) = SXX(ω). i.e., Power spectrum density is even function of ω

6. A stationery random process X(t) has spectral density SXX(ω)=25/ (𝜔 2+25) and an independent stationary process Y(t) has the spectral density SYY(ω)= 𝜔 2 / (𝜔 2+25). If X(t) and Y(t) are of zero mean, find the: a) PSD of Z(t)=X(t) + Y(t) b) Cross spectral density of X(t) and Z(t) 7. a) The input to an LTI system with impulse response h(t)= 𝛿 + 2 −𝑎 . U(t) is a WSS process with mean of 3. Find the mean of the output of the system.

8. State and prove Power Spectral density with three properties.

6

DEPARTMENT OF ECE

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UNIT WISE IMPORTANT QUESTION QUESTIONS UNIT-I PART-A-UNIT-I

(2 or 3 marks)

1. State the properties of LTI system. 2. Draw the function π(2t+3) when π(t) = 1 ; for t ≤ ½ 0 ; otherwise 3. Determine whether the following signal is energy or power signal. And calculate its Energy or Power. x(t)=e-2tu(t). 4. Check whether the following system is static or dynamic & also causal or noncausal. y(n)=x(2n). 5. Define Fourier transform pair. 6. Explain how aperiodic signals can be represented by fourier transform. 7. Give the mathematical & graphical representation of CT & DT unit impulse function 8. What are the conditions for a system to be LTI system? 9. State any 2 properties of unit impulse function. 10. What is Aliasing? 11. What are the Conditions for a System to be LTI System? 12. Define time invariant and time varying systems. 13. Is the system describe by the equation y(t) = x(2t) Time invariant or not? Why?

PART-B-UNIT-I (5-10 marks) 1. Derive the expression for component vector of approximating the function f1(t) over f2(t) and also prove that the component vector becomes zero if the f1(t) and f2(t) are orthogonal. 2. Define Linearity and Time-Invariant properties of a system and write with an example. 7

DEPARTMENT OF ECE

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3. Find the convolution of two signals x(n) = { 1, 1,0, 1, 1} and h(n) = { 1, -2, -3, 4} and represent them graphically 4. Find the convolution of x(t) = u(t+1) and h(t) = u(t-2)

UNIT-II PART-A-UNIT-II

(2 or 3 marks)

1. Find the convolution of the two signals x(t)= e -2t u(t) h(t)= u(t+2) 2. Find the Fourier transform of x(t) = t cos ωt 3. Discuss the block diagram representation of an LTI-DT system

4. Consider a causal LTI system as in the fig

a. Determine the differential equation relating x(n) and y(n)

5. State Fourier integral theorem. 6. Show that f(x) = 1, 0 < x < ¥ cannot be represented by a Fourier integral. PART-B-UNIT-II (5-10 marks)

1.

Obtain the cosine fourier series representation of x(t)

2. Find the trigonometric fourier series of the figure shown below

8

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

3. 4. 5. 6.

State and prove Parseval’s Theorem Find the fourier transform of a rectangular pulse of duration T and amplitude A Show that e x2/2 is reciprocal with respect to Fourier transforms State sampling theorem for band limited signals. Prove theorem graphically. What is aliasing effect?

UNIT-III PART-A-UNIT-III (2 or 3 marks) 1. 2. 3. 4.

Prove the following property of the z-transform: zn0x[n]→X(zz0) Compute the inverse z-transform of X(z)=log(1+z),|z|<1. Find the z transform of 3n + 2 × 3 n Find the Unilateral Z-transform and R.O.C of x(n) = sin ω0 n u(n)

5. Find the z-transform of each of the following sequences: (a) x(n)= 2 n u(n)+3(½)n u(n) 0)u(n).(b)x(n)=cos(n) 6. Find the inverse of each of the following z-transforms: 7. Find the inverse z-transform of X(z) = sin z

PART-B-UNIT-III (5-10 marks) 1. Determine the Z=Transform of x1(n)=an and x2(n) =nu(n) 2. Find the inverse laplace transform of X(S) = S / S2+5S+6 3. State and prove initial value and final value theorems. 4. Using convolution theorem find inverse Laplace transform of s/(s2+a2)2 5. State and prove differentiation and integration property of z-transform.

UNIT-IV PART-A-UNIT-IV (2 or 3 marks) 1. Define random process? 2. Define ergodicity? 3. Define mean ergodic process? 4.

Define wide sense stationary random processes. 9

DEPARTMENT OF ECE

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5.

Give the statement of ergodic theorem.

6.

Define the auto covariance & cross covariance functions of Random processes X(t).

7.

When two random processes X(t)& Y(t) are said to be independent.

8.

Define the cross correlation function between two random processes X(t) & Y(t)..

PART-B-UNIT-IV (5-10 marks) 1. Define Wide Sense Stationary Process and write it‟s conditions. 2.

A random process is given as X(t) = At, where A is a uniformly distributed random variable on (0,2). Find whether X(t) is wide sense stationary or not. 2. X(t) is a stationary random process with a mean of 3 and an auto correlation function of 6+5 exp (-0.2 │ │). Find the second central Moment of the random variable Y=Z-W, where „Z‟ and „W‟ are the samples of the random process at t=4 sec and t=8 sec respectively.

3.

Explain the following i) Stationarity ii) Ergodicity iii) Statistical independence with respect to random processes

4.

Given the RP X(t) = A cos(w0t) + B sin (w0t) where ω0 is a constant, and A and B are uncorrelated Zero mean random variables having different density functions but the same variance

5.

Show that X(t) is wide sense stationary. b) Define Covariance of the Random processes with any two properties.

6.

A Gaussian RP has an auto correlation function RXX( )=6 sin (𝜋𝜏 ) 𝜋𝜏 .

Determine a covariance matrix for the Random variable X(t) b) Derive the expression for cross correlation function between the input and output of a LTI system. UNIT-V PART-A-UNIT-V (2 or 3 marks) 1. Define Power Spectrum Density. 2.

Give the statement of Wiener-Khinchin relation.

3.

Define spectrum Band width and RMS bandwidth.

4.

Write any two properties of Power Spectrum Density. 5. Define linear system.

10

DEPARTMENT OF ECE

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5.

Show that SXX(-ω) = SXX(ω). i.e., Power spectrum density is even function of ω.

6.

If the Power spectrum density of x(t) is SXX(ω), find the PSD of

7.

If the Auto correlation function of wide sense stationary X(t) is RXX( )=4+2e -2

𝑥( ).

𝜏 . Find the area enclosed by the power spectrum density curve of X(t).

8. Define linear system and derive the expression for output response. 5. If X(t) & Y(t)are uncorrelated and have constant mean values SXX(ω)= 2Π

& then show that

𝛿(𝜔)

PART-B-UNIT-V (5-10 marks) 1. Check the following power spectral density functions are valid or not ) 𝑜 8(𝜔) 2 + 𝜔4 ) −(𝜔−1)

2. Derive the relation between input PSD and output PSD of an LTI system 3. Derive the relationship between cross-power spectral density and cross correlation function. 3. A stationery random process X(t) has spectral density SXX(ω)=25/ (𝜔 2+25) and an independent stationary process Y(t) has the spectral density SYY(ω)= 𝜔

2 / (𝜔 2+25). If X(t) and Y(t) are of zero mean, find the: a) PSD of Z(t)=X(t) + Y(t) b) Cross spectral density of X(t) and Z(t) 4. The input to an LTI system with impulse response h(t)= 𝛿 + 2 −𝑎 . U(t) is a WSS process with mean of 3. Find the mean of the output of the system.

5. Define Power Spectral density with three properties. 6. A random process Y(t) has the power spectral density SYY(ω)= 9 𝜔2+64 Find i) The average power of the process ii) The Auto correlation function

7. State the properties of power spectral density 8. A random process has the power density spectrum SYY(ω)= 6𝜔 2 1+𝜔4 . Find the average power in the process. 9. Find the auto correlation function of the random process whose psd is 16 𝜔2+4 7. a) Find the cross correlation function corresponding to the cross power spectrum

SXY(ω)= 6 (9+𝜔2)(3+ 𝜔) 2 b) Write short notes on cross power density spectrum.

10. Consider a random process X(t)=cos(𝜔 + 𝜃)where 𝜔 is a real constant and 𝜃is a uniform random variable in (0, π/2). Find the average power in the process.

11. Define and derive the expression for average power of Random process. 11

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

12. The power spectrum density function of a stationary random process is given by SXX(𝜔)= A, -K< 𝜔< K 0, other wise Find the auto correlation function.

13. Derive the expression for power spectrum density. 10. a) Define and derive the expression for average cross power between two random process X(t) and Y(t). 14. Find the cross power spectral density for RXX( )=𝐴 2 2 sin⁡(𝜔0𝜏)

12

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

OBJECTIVE QUESTIONS UNIT-I: 1.Which mathematical notation specifies the condition of periodicity for a continuous time signal? a. x(t) = x(t +T0) b. x(n) = x(n+ N) c. x(t) = e-αt d. None of the above ANSWER: (a) x(t) = x(t +T0) 2. Which among the below specified conditions/cases of discrete time in terms of real constant ‘a’, represents the double-sided decaying exponential signal? a. a > 1 b. 0 < a < 1 c. a < -1 d. -1 < a < 0 ANSWER: (d) -1 < a < 0 3) Damped sinusoids are _____ a. sinusoid signals multiplied by growing exponentials b. sinusoid signals divided by growing exponentials c. sinusoid signals multiplied by decaying exponentials d. sinusoid signals divided by decaying exponentials ANSWER: (c) sinusoid signals 4. An amplitude of sinc function that passes through zero at multiple values of an independent variable ‘x’ ______ a. Decreases with an increase in the magnitude of an independent variable (x) b. Increases with an increase in the magnitude of an independent variable (x) c. Always remains constant irrespective of variation in magnitude of ‘x’ d. Cannot be defined ANSWER: (a) Decreases with an increase in the 5) A system is said to be shift invariant only if______ a. a shift in the input signal also results in the corresponding shift in the output b. a shift in the input signal does not exhibit the corresponding shift in the output c. a shifting level does not vary in an input as well as output d. a shifting at input does not affect the output ANSWER: (a) a shift in the input signal also results in the corresponding shift in the output 6) Which condition determines the causality of the LTI system in terms of its impulse response? a. Only if the value of an impulse response is zero for all negative values of time b. Only if the value of an impulse response is unity for all negative values of time c. Only if the value of an impulse response is infinity for all negative values of time d. Only if the value of an impulse response is negative for all negative values of time 13

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

ANSWER: (a) Only if the value of an impulse response is zero for all negative values of time 7) Under which conditions does an initially relaxed system become unstable? a. only if bounded input generates unbounded output b. only if bounded input generates bounded output c. only if unbounded input generates unbounded output d. only if unbounded input generates bounded output ANSWER: (a) only if bounded input generates unbounded output 8) Which among the following are the stable discrete time systems? 1. y(n) = x(4n) 2. y(n) = x(-n) 3. y(n) = ax(n) + 8 4. y(n) = cos x(n) 9) An equalizer used to compensate the distortion in the communication system by faithful recovery of an original signal is nothing but an illustration of _________ a. Static system b. Dynamic system c. Invertible system d. None of the above ANSWER: (c) Invertible system 10) Which block of the discrete time systems requires memory in order to store the previous input? a. Adder b. Signal Multiplier c. Unit Delay d. Unit Advance ANSWER: (c)Unit Delay 11) Which type/s of discrete-time system do/does not exhibit the necessity of any feedback? a. Recursive Systems b. Non-recursive Systems c. Both a & b d. None of the above ANSWER: (b) Non-recursive Systems 12) Recursive Systems are basically characterized by the dependency of its output on _______ a. Present input b. Past input c. Previous outputs d. All of the above ANSWER: (d) All of the above 13) What does the term y(-1) indicate especially in an equation that represents the behaviour of the causal system? 14

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

a. initial condition of the system b. negative initial condition of the system c. negative feedback condition of the system d. response of the system to its initial input ANSWER: (a) initial condition of the system 14) Which type of system response to its input represents the zero value of its initial condition? a. Zero state response b. Zero input response c. Total response d. Natural response ANSWER: (a) Zero state response 15) Which is/are the essential condition/s to get satisfied for a recursive system to be linear? a. Zero state response should be linear b. Principle of Superposition should be applicable to zero input response c. Total Response of the system should be addition of zero state & zero input responses d. All of the above ANSWER: (d) All of the above 16) Which among the following operations is/are not involved /associated with the computation process of linear convolution? a. Folding Operation b. Shifting Operation c. Multiplication Operation d. Integration Operation ANSWER: (d) Integration Operation 17) A LTI system is said to be initially relaxed system only if ____ a. Zero input produces zero output b. Zero input produces non-zero output c. Zero input produces an output equal to unity d. None of the above ANSWER:(a) Zero input produces zero output 18) What are the number of samples present in an impulse response called as? a. string b. array c. length d. element ANSWER: (c) length 19) Double-sided phase & amplitude spectra _____ a. Possess an odd & even symmetry respectively b. Possess an even & odd symmetry respectively c. Both possess an odd symmetry d. Both possess an even symmetry ANSWER: (a) Possess an odd & even symmetry respectively 15

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

20) Which is/are the mandatory condition/s to get satisfied by the transfer function for the purpose of distortionless transmission? a. Amplitude Response should be constant for all frequencies b. Phase should be linear with frequency passing through zero c. Both a & b d. None of the above ANSWER: (c)Both a & b

UNIT-II: 1) What does the first term ‘a0‘ in the below stated expression of a line spectrum indicate? x(t) = a0 + a1 cos w0 t + a2 cos2 w0t +……+ b1 sin w0 t + b2 sin w0 t + ….. a. DC component b. Fundamental component c. Second harmonic component d. All of the above ANSWER: (a) DC component 2) Which kind of frequency spectrum/spectra is/are obtained from the line spectrum of a continuous signal on the basis of Polar Fourier Series Method? a. Continuous in nature b. Discrete in nature c. Sampled in nature d. All of the above ANSWER: (b) Discrete in nature 3) Which type/s of Fourier Series allow/s to represent the negative frequencies by plotting the double-sided spectrum for the analysis of periodic signals? a. Trigonometric Fourier Series b. Polar Fourier Series c. Exponential Fourier Series d. All of the above ANSWER: (c) Exponential Fourier Series 4) What does the signalling rate in the digital communication system imply? a. Number of digital pulses transmitted per second b. Number of digital pulses transmitted per minute c. Number of digital pulses received per second d. Number of digital pulses received per minute ANSWER: (a)Number of digital pulses transmitted per second 5) Duality Theorem / Property of Fourier Transform states that _________ a. Shape of signal in time domain & shape of spectrum can be interchangeable b. Shape of signal in frequency domain & shape of spectrum can be interchangeable c. Shape of signal in time domain & shape of spectrum can never be interchangeable d. Shape of signal in time domain & shape of spectrum can never be interchangeable 16

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

ANSWER: (a) Shape of signal in time domain & shape of spectrum can be interchangeable 6) Which property of fourier transform gives rise to an additional phase shift of -2π ftd for the generated time delay in the communication system without affecting an amplitude spectrum? a. Time Scaling b. Linearity c. Time Shifting d. Duality ANSWER: (c) Time Shifting 7) Which among the below assertions is precise in accordance to the effect of time scaling? A : Inverse relationship exists between the time and frequency domain representation of signal B : A signal must be necessarily limited in time as well as frequency domainsa. A is true & B is false b. A is false & B is true c. Both A & B are true d. Both A & B are false ANSWER: (a)A is true & B is false 8) What is/are the crucial purposes of using the Fourier Transform while analyzing any elementary signals at different frequencies? a. Transformation from time domain to frequency domain b. Plotting of amplitude & phase spectrum c. Both a & b d. None of the above ANSWER: (c)Both a & b 9) What is the possible range of frequency spectrum for discrete time fourier series (DTFS)? a. 0 to 2π b. -π to +π c. Both a & b d. None of the above ANSWER: (c)Both a & b 10) Which among the following assertions represents a necessary condition for the existence of Fourier Transform of discrete time signal (DTFT)? a. Discrete Time Signal should be absolutely summable b. Discrete Time Signal should be absolutely multipliable c. Discrete Time Signal should be absolutely integrable d. Discrete Time Signal should be absolutely differentiable ANSWER: (a)Discrete Time Signal should be absolutely summable 11)What is the nature of Fourier representation of a discrete & aperiodic signal? a. Continuous & periodic b. Discrete & aperiodic c. Continuous & aperiodic d. Discrete & periodic ANSWER: (a) Continuous & periodic 17

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

12) Which property of periodic signal in DTFS gets completely clarified / identified by the equation x (n – n0)? a. Conjugation b. Time Shifting c. Frequency Shifting d. Time Reversal ANSWER: (b) Time Shifting 13) Which theorem states that the total average power of a periodic signal is equal to the sum of average powers of the individual fourier coefficients? a. Parseval’s Theorem b. Rayleigh’s Theorem c. Both a & b d. None of the above ANSWER: (a) Parseval’s Theorem 14) Which among the below mentioned transform pairs is/are formed between the auto-correlation function and the energy spectral density, in accordance to the property of Energy Spectral Density (ESD)? a. Laplace Transform b. Z-Transform c. Fourier Transform d. All of the above ANSWER: (c) Fourier Transform

UNIT-III: 1) A Laplace Transform exists when ______ A. The function is piece-wise continuous B. The function is of exponential order C. The function is piecewise discrete D. The function is of differential order a. A & B b. C & D c. A & D d. B & C ANSWER: (a) A & B 2) Where is the ROC defined or specified for the signals containing causal as well as anti-causal terms? a. Greater than the largest pole b. Less than the smallest pole c. Between two poles d. Cannot be defined ANSWER: (c) Between two poles 3) What should be the value of laplace transform for the time-domain signal equation e-at cos ωt.u(t)? a. 1 / s + a with ROC > – a b. ω / (s + a) 2 + ω2 with ROC > – a c. s + a / (s + a)2 + ω2 with ROC > – a d. Aω / s2 + ω2 with ROC > 0 18

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

ANSWER: (c) s + a / (s + a)2 + ω2 with ROC σ > – a 4) According to the time-shifting property of Laplace Transform, shifting the signal in time domain corresponds to the ______ a. Multiplication by e-st0 in the time domain b. Multiplication by e-st0 in the frequency domain c. Multiplication by est0 in the time domain d. Multiplication by est0 in the frequency domain ANSWER: (b) Multiplication by e-st0 in the frequency domain 5) Which result is generated/ obtained by the addition of a step to a ramp function? a. Step Function shifted by an amount equal to ramp b. Ramp Function shifted by an amount equal to step c. Ramp function of zero slope d. Step function of zero slope ANSWER: (b) Ramp Function shifted by an amount equal to step 6) Unilateral Laplace Transform is applicable for the determination of linear constant coefficient differential equations with ________ a. Zero initial condition b. Non-zero initial condition c. Zero final condition d. Non-zero final condition ANSWER: (b) Non-zero initial condition 7) What should be location of poles corresponding to ROC for bilateral Inverse Laplace Transform especially for determining the nature of time domain signal? a. On L.H.S of ROC b. On R.H.S of ROC c. On both sides of ROC d. None of the above ANSWER: (c) On both sides of ROC 8) Generally, the convolution process associated with the Laplace Transform in time domain results into________ a. Simple multiplication in complex frequency domain b. Simple division in complex frequency domain c. Simple multiplication in complex time domain d. Simple division in complex time domain ANSWER: (a) Simple multiplication in complex frequency domain 9) An impulse response of the system at initially rest condition is basically a response to its input & hence also regarded as, a. Black’s function b. Red’s function c. Green’s function d. None of the above ANSWER: (c) Green’s function 10) When is the system said to be causal as well as stable in accordance to pole/zero of ROC specified by system transfer function? a. Only if all the poles of system transfer function lie in left-half of S-plane b. Only if all the poles of system transfer function lie in right-half of S-plane c. Only if all the poles of system transfer function lie at the centre of S-plane d. None of the above ANSWER: (a) Only if all the poles of system transfer function lie in left-half of S-plane 11) Correlogram is a graph of ______ 19

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

a. Amplitude of one signal plotted against the amplitude of another signal b. Frequency of one signal plotted against the frequency of another signal c. Amplitude of one signal plotted against the frequency of another signal d. Frequency of one signal plotted against the time period of another signal ANSWER: (a) Amplitude of one signal plotted against the amplitude of another signal 12) According to Rayleigh’s theorem, it becomes possible to determine the energy of a signal by________ a. Estimating the area under the square root of its amplitude spectrum b. Estimating the area under the square of its amplitude spectrum c. Estimating the area under the one-fourth power of its amplitude spectrum d. Estimating the area exactly half as that of its amplitude spectrum ANSWER: (b) Estimating the area under the square of its amplitude spectrum 13) The ESD of a real valued energy signal is always _________ a. An even (symmetric) function of frequency b. An odd (non-symmetric) function of frequency c. A function that is odd and half-wave symmetric d. None of the above ANSWER: (a) An even (symmetric) function of frequency 14) Which among the below mentioned assertions is /are correct? a. Greater the value of correlation function, higher is the similarity level between two signals b. Greater the value of correlation function, lower is the similarity level between two signals c. Lesser the value of correlation function, higher is the similarity level between two signals d. Lesser the value of correlation function, lower is the similarity level between two signals a. Only C b. Only B c. A & D d. B & C ANSWER: (c) A & D 15) Which property is exhibited by the auto-correlation function of a complex valued signal? a. Commutative property b. Distributive property c. Conjugate property d. Associative property ANSWER: (c) Conjugate property 16) Where does the maximum value of auto-correlation function of a power signal occur? a. At origin b. At extremities c. At unity d. At infinity ANSWER: (a) At origin UNIT-IV: 1. What does the set comprising all possible outcomes of an experiment known as ? a. Null event b. Sure event 20

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

c. Elementary event d. None of the above ANSWER: b. Sure event 2. What does an each outcome in the sample space regarded as ? a. Sample point b. Element c. Both a & b d. None of the above ANSWER: c. Both a & b 3. Mutually Exclusive events ________ a. Contain all sample points b. Contain all common sample points c. Does not contain any common sample point d. Does not contain any sample point ANSWER: c. Does not contain any common sample point 4. What would be the probability of an event 'G' if G denotes its complement, according to the axioms of probability? a. P (G) = 1 / P (G) b. P (G) = 1 - P (G) c. P (G) = 1 + P (G) d. P (G) = 1 * P (G) ANSWER: b. P (G) = 1- P (G) 5. What would happen if the two events are statistically independent ? a. Conditional probability becomes less than the elementary probability b. Conditional probability becomes more than the elementary probability c. Conditional probability becomes equal to the elementary probability d. Conditional as well as elementary probabilities will exhibit no change ANSWER: c. Conditional probability becomes equal to the elementary probability

6. What would be the joint probability of statistically independent events that occur simultaneously ? a. Zero b. Not equal to zero c. Infinite d. None of the above ANSWER: b. Not equal to zero 7. Consider the assertions given below : A : CDF is a monotonously increasing function B : PDF is a derivative of CDF & is always positive 21

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

Which among them is correct according to the properties of PDF? a. A is true & B is false b. A is false & B is true c. Both A & B are true but B is a reason for A d. Both A & B are false since B is not a reason for A ANSWER: c. Both A & B are true but B is a reason for A 8. The Joint Cumulative Density Function (CDF) _____ a. Is a non-negative function b. Is a non-decreasing function of x & y planes c. Is always a continuous function in xy plane d. All of the above ANSWER: d. All of the above 9. What is the value of an area under the conditional PDF ? a. Greater than '0' but less than '1' b. Greater than '1' c. Equal to '1' d. Infinite ANSWER: c. Equal to '1' 10. When do the conditional density functions get converted into the marginally density functions ? a. Only if random variables exhibit statistical dependency b. Only if random variables exhibit statistical independency c. Only if random variables exhibit deviation from its mean value d. None of the above ANSWER: b. Only if random variables exhibit statistical independency 11. Which among the below mentioned standard PDFs is/are applicable to discrete random variables ? a. Gaussian distribution b. Rayleigh distribution c. Poisson distribution d. All of the above ANSWER: c. Poisson distribution 12. A random variable belongs to the category of a uniform PDF only when __________ a. It occurs in a finite range b. It is likely to possess zero value outside the finite range c. Both a & b d. None of the above ANSWER: c. Both a & b 13. What would happen if the value of term [(m-x) / (σ √2)] increases in the expression of Guassian CDF? a. Complementary error function also goes on increasing 22

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

b. Complementary error function goes on decreasing c. Complementary error function remains constant or unchanged d. Cannot predict ANSWER: b. Complementary error function goes on decreasing 14. Which type of standard PDFs has/ have an ability to describe an integer valued random variable concerning to the repeated trials carried /conducted in an experiment? a. Binomial b. Uniform c. Both a & b d. None of the above ANSWER: a. Binomial UNIT-V: 1) What does the set comprising all possible outcomes of an experiment known as? a. Null event b. Sure event c. Elementary event d. None of the above ANSWER: (b) Sure event 2) What does an each outcome in the sample space regarded as? a. Sample point b. Element c. Both a & b d. None of the above ANSWER: (c) Both a & b 3) Mutually Exclusive events ________ a. Contain all sample points b. Contain all common sample points c. Does not contain any common sample point d. Does not contain any sample point ANSWER: (c) Does not contain any common sample point 4) What would be the probability of an event ‘G’ if G denotes its complement, according to the axioms of probability? a. P (G) = 1 / P (G) b. P (G) = 1 – P (G) c. P (G) = 1 + P (G) d. P (G) = 1 * P (G) ANSWER:(b) P (G) = 1 – P (G) 5) What would happen if the two events are statistically independent? a. Conditional probability becomes less than the elementary probability b. Conditional probability becomes more than the elementary probability c. Conditional probability becomes equal to the elementary probability d. Conditional as well as elementary probabilities will exhibit no change 23

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

ANSWER: (c) Conditional probability becomes equal to the elementary probability 6) What would be the joint probability of statistically independent events that occur simultaneously? a. Zero b. Not equal to zero c. Infinite d. None of the above ANSWER: (b) Not equal to zero 7) Consider the assertions given below A : CDF is a monotonously increasing function B : PDF is a derivative of CDF & is always positive Which among them is correct according to the properties of PDF? a. A is true & B is false b. A is false & B is true c. Both A & B are true but B is a reason for A d. Both A & B are false since B is not a reason for A ANSWER: (c) Both A & B are true but B is a reason for A 8) The Joint Cumulative Density Function (CDF) _____ a. Is a non-negative function b. Is a non-decreasing function of x & y planes c. Is always a continuous function in xy plane d. All of the above ANSWER: (d) All of the above 9) What is the value of an area under the conditional PDF? a. Greater than ‘0’ but less than ‘1’ b. Greater than ‘1’ c. Equal to ‘1’ d. Infinite ANSWER: (c) Equal to ‘1’ 10) When do the conditional density functions get converted into the marginally density functions? a. Only if random variables exhibit statistical dependency b. Only if random variables exhibit statistical independency c. Only if random variables exhibit deviation from its mean value d. None of the above ANSWER: (b) Only if random variables exhibit statistical independency 11) Which among the below mentioned standard PDFs is/are applicable to discrete random variables? a. Gaussian distribution b. Rayleigh distribution c. Poisson distribution d. All of the above ANSWER: (c) Poisson distribution 12) A random variable belongs to the category of a uniform PDF only when ______ a. It occurs in a finite range b. It is likely to possess zero value outside the finite range c. Both a & b d. None of the above ANSWER: (c) Both a & b 24

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

13) What would happen if the value of term [(m-x) / (σ √2)] increases in the expression of Guassian CDF? a. Complementary error function also goes on increasing b. Complementary error function goes on decreasing c. Complementary error function remains constant or unchanged d. Cannot predict ANSWER: (b) Complementary error function goes on decreasing 14) Which type of standard PDFs has/ have an ability to describe an integer valued random variable concerning to the repeated trials carried /conducted in an experiment? a. Binomial b. Uniform c. Both a & b d. None of the above ANSWER: (a) Binomial

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DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

INTERNAL QUESTION PAPER PART-A Answer All the following Questions (5*2=10 M) 1. a) A rectangular function f(t) is defined by f(t) =1for 0
3. Determine the Fourier transform of a two sided exponential pulse x (t) = e

.

4. State and prove time convolution and time differentiation properties of Fourier Transform. 5.Find the Fourier transform of a gate pulse of unit height, unit width and centered at t=0. PART-B Answer the following Questions (3*10=30 M) 1. (a) Define autocorrelation function and explain with properties. (b) Statistically Independent zero mean random processes X(t) and Y(t) have auto correlation functions respectively. Find

the auto correlation of W1(t)=X(t)+Y(t). (OR) 2. (a) Define Random process and explain classification of random processes with neat sketch. (b) A random process is described by X(t)=A2.cos2 ωt+θ here A a d ω are o sta ts a d θ is a random variable uniformly distributed between (-π, π . Is X(t) wide sense stationary or not. 3. (a) Check the given function is valid or not. [2M]

(b) Derive the equation for power density spectrum and state any four properties of it. [8M] (OR) 4. State and prove wiener-khintchine relation. 5. (a) Find the inverse z-Transform of

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DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

Using Residue Method. (b) Sate and prove initial and final value theorems of z-Transform (OR) 6. (a) Show that (b) Auto correlation of a WSS random process X(t) constants. Find PSD.

.

27

is where are

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

EXTERNAL QUESTION PAPER

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DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

29

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

SIGNALS & STOCHASTIC PROCESS – PREFINAL PAPER PART-A 1. (a) Define mean square error and give equation for it [3M] (b) Define system bandwidth and signal bandwidth (c) State Dirichlet Conditions[2M] (d) Find the Fourier Transform of a Symmetrical Gate Pulse (e) Define ROC of Laplace Transform and state any 4 Properties (f) Find the z-Transform of

[3M] [3M] [2M] [3M]

(g) Assume that ergodic process has an autocorrelation function Find its mean, mean square and variance.

(h) Define Nyquist rate and Nyquist Interval (i) What is WSS and State its conditions

[3M]

[3M] [2M]

PART-B 1. Obtain the condition under which two signals orthogonal to each other. Hence prove that

are said to are

orthogonal to each other for all integer values of (m,n) [10M] (OR) 2. (a) Derive the relation between rise time and Bandwidth [5M] (b) Explain the method of Graphical approach for Convolution with an example [5M] 4. (a) Define even symmetry and derive the trigonometric fourier series coefficients for odd function [5M] (b) For the Periodic function given below, derive the exponential form of fourier series [5M]

(OR) 5. (a) State and prove the given properties of fourier transform 30

DEPARTMENT OF ECE

SIGNALS AND STOCHASTIC PROCESS

(i) Convolution (ii) Time Shifting (iii) Frequency Shifting (b) What is Aliasing Effect and explain methods to avoid it

[6M] [4M]

6. (a) State Sampling theorem and explain the construction of sampled Signal [5M] (b) Find the Inverse z-Transform of using power

series expansion

[5M] (OR)

7. (a) Find the inverse laplace transform of (b) Prove that the sequences

have same X(z)

and differ only in ROC. Plot their ROC 8. (a) What is Cross Correlaton and explain with Properties [5M] Pro e that the ra do pro ess X t =A. os ωt+θ is WSS of it is assu ed that ω a d A are o sta ts a d θ is a U ifor ly Distri uted Ra do Varia le o er 0,2π [5M] (OR) 9. Prove the relation between Cross PSD and Cross Correlation [10M] 10. State and prove wiener-khintchine relation [10M] (OR) 11. (a) Show that

[5M] (b) Define Cross PSD and explain with Properties

********

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[5M]