Introduction Signal Processing

Digital Signal Processing By hafsa Ahmed

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Books: • J.G. Proakis and D.G. Manolakis, “Digital Signal Processing. Third Edition or Latest. • M. Ifeachor and B.W. Jarvis, “Digital Signal Processing – A Practical Approach”, Second Edition or Latest. • A.V. Oppenhiem, Digital Signal Processing.

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Lecture 1-2 Basic Concepts

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• Signal: A signal is defined as a function of one or more variables which conveys information on the nature of a physical phenomenon. The value of the function can be a real valued scalar quantity, a complex valued quantity, or perhaps a vector.

• System: A system is defined as an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.

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• Continuos-Time Signal: A signal x(t) is said to be a continuous time signal if it is defined for all time t.

• Discrete-Time Signal: A discrete time signal x[nT] has values specified only at discrete points in time.

• Signal Processing: A system characterized by the type of operation that it performs on the signal. For example, if the operation is linear, the system is called linear. If the operation is nonlinear, the system is said to be non-linear, and so forth. Such operations are usually referred to as “Signal Processing”. 5

Basic Elements of a Signal Processing System Analog input signal

Analog output signal

Analog Signal Processor Analog Signal Processing

Analog input signal

A/D converter

Digital Signal Processor Digital Signal Processing

D/A converter

Analog output signal

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• Advantages of Digital over Analogue Signal Processing: A digital programmable system allows flexibility in reconfiguring the DSP operations simply by changing the program. Reconfiguration of an analogue system usually implies a redesign of hardware, testing and verification that it operates properly. DSP provides better control of accuracy requirements. Digital signals are easily stored on magnetic media (tape or disk). The DSP allows for the implementation of more sophisticated signal processing algorithms. In some cases a digital implementation of the signal processing system is cheaper than its analogue 7 counterpart.

DSP Applications Space

Medical

Space photograph enhancement Data compression Intelligent sensory analysis Diagnostic imaging (MRI, CT, ultrasound, etc.) Electrocardiogram analysis Medical image storage and retrieval

Image and sound compression for Commercial multimedia presentation. Movie special effects Video conference calling

Telephone

Video and data compression echo reduction signal multiplexing filtering 8

DSP Applications (cont.) Military

Radar Sonar Ordnance Guidance Secure communication

Oil and mineral prospecting Industrial Process monitoring and control Non-destructive testing CAD and design tools

Scientific

Earth quick recording and analysis Data acquisition Spectral Analysis Simulation and Modeling

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Classification of Signals •Deterministic Signals A deterministic signal behaves in a fixed known way with respect to time. Thus, it can be modeled by a known function of time t for continuous time signals, or a known function of a sampler number n, and sampling spacing T for discrete time signals.

• Random or Stochastic Signals: In many practical situations, there are signals that either cannot be described to any reasonable degree of accuracy by explicit mathematical formulas, or such a description is too complicated to be of any practical use. The lack of such a relationship implies that such signals evolve in time in an unpredictable manner. We refer to these signals as random. 10

Even and Odd Signals A continuous time signal x(t) is said to an even signal if it satisfies the condition x(-t) = x(t) for all t The signal x(t) is said to be an odd signal if it satisfies the condition x(-t) = -x(t) In other words, even signals are symmetric about the vertical axis or time origin, whereas odd signals are antisymmetric about the time origin. Similar remarks apply to discrete-time signals. Example:

even odd

odd

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Tutorial 1: Q1 Define xe(t) to be even and xo(t) to be odd part of a signal x(t). Show that 1 x e (t ) =

[x(t ) +x( −t )]

2 1 x 0 (t ) = [x( t ) −x( − t )] 2

Q2: Decompose the following signal into its even and odd parts: x(t)

-1

1

Note: The above definitions of even and odd signals assume that the signals are real valued. A complex valued signal x(t) is said to to be conjugate symmetric if it satisfies the condition 12 x(-t) = x*(t)

Periodic Signals A continuous signal x(t) is periodic if and only if there exists a T > 0 such that x(t + T) = x(t) where T is the period of the signal in units of time. f = 1/T is the frequency of the signal in Hz. W = 2π/T is the angular frequency in radians per second. The discrete time signal x[nT] is periodic if and only if there exists an N > 0 such that x[nT + N] = x[nT] where N is the period of the signal in number of sample spacings. Example: Frequency = 5 Hz or 10π rad/s 0

0.2

0.4

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Continuous Time Sinusoidal Signals A simple harmonic oscillation is mathematically described as x(t) = Acos(wt + θ) This signal is completely characterized by three parameters: A = amplitude, w = 2πf = frequency in rad/s, and θ = phase in radians. A 

T=1/f 

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Discrete Time Sinusoidal Signals A discrete time sinusoidal signal may be expressed as x[n] = Acos(wn + θ) -∞ < n < ∞ Properties: • A discrete time sinusoid is periodic only if its frequency is a rational number. • Discrete time sinusoids whose frequencies are separated by an integer multiple of 2π are identical. • The highest rate of oscillation in a discrete time sinusoid is attained when w = π ( or w = -π ), or equivalently f = 1/2 (or f = -1/2). 1 0 ­1

0

2

4

6

8

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Energy and Power Signals The total energy of a continuous time signal x(t) is defined as T

∞

E x = lim ∫ x ( t )dt = ∫ x 2 ( t )dt T →∞ 2

−T

−∞

And its average power is Px

1 = lim T→ ∞T

T/ 2

2 ∫x ( t )dt

− T/ 2

In the case of a discrete time signal x[nT], the total energy of the ∞ E dx signal is = T ∑ x 2 [n ] n =−∞

And its average power is defined by Pdx

 1  N = lim   ∑ x[nT] N →∞ 2N + 1 n =−N

2

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Energy and Power Signals •A signal is referred to as an energy signal, if and only if the total energy of the signal satisfies the condition 0<E<∞ •On the other hand, it is referred to as a power signal, if and only if the average power of the signal satisfies the condition 0
Tutorial 1: Q3 Determine the signal energy and signal power for each of the given signals and indicate whether it is an energy signal or a power signal? (a)

y[nT] = 3( −0.2)n u[n − 3],

(b) z[nT] = 4(1.1) n u[n + 1] (c)

π   (3π w[nT ] = 6 sin  n / 8 )−  3 

T = 2 ms T = 0.02 s T = 1.5 s

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Basic Operations on Signals (a) Operations performed on dependent variables 1. Amplitude Scaling: let x(t) denote a continuous time signal. The signal y(t) resulting from amplitude scaling applied to x(t) is defined by y(t) = cx(t) where c is the scale factor. In a similar manner to the above equation, for discrete time signals we write 2x(t) y[nT] = cx[nT] x(t) 19

2. Addition: Let x1 [n] and x2[[n] denote a pair of discrete time signals. The signal y[n] obtained by the addition of x1[n] + x2[n] is defined as y[n] = x1[n] + x2[n]

Example: audio mixer 3. Multiplication: Let x1[n] and x2[n] denote a pair of discrete-time signals. The signal y[n] resulting from the multiplication of the x1[n] and x2[n] is defined by y[n] = x1[n].x2[n]

Example: AM Radio Signal 20

(b) Operations performed on independent variable • Time Scaling: Let y(t) is a compressed version of x(t). The signal y(t) obtained by scaling the independent variable, time t, by a factor k is defined by y(t) = x(kt) – if k > 1, the signal y(t) is a compressed version of x(t). – If, on the other hand, 0 < k < 1, the signal y(t) is an expanded (stretched) version of x(t).

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Example of time scaling 1 0.9 0.8 0.7

Expansion and compression of the signal e-t. exp(­t) 

0.6 0.5 0.4

exp(­2t)  exp(­0.5t) 

0.3 0.2 0.1 0 0

5

10

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Time scaling of discrete time systems

x[n]

10 5

0 ­3 10

­1

0

1

2

3

0 ­1.5 5

­1

­0.5

0

0.5

1

1.5

0 ­6

­4

­2

0 n

2

4

6

5

x[2n]

x[0.5n]

­2

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Time Reversal

x[­n]

x[n]

• This operation reflects the signal about t = 0 and thus reverses the signal on the time scale. 5 4 3 2 1 00 0.5 1 1.5 2 5 4 3 2 1 0­5 ­4.5 ­4 ­3.5 ­3

2.5

3

3.5

4

4.5

5

­2.5 ­2 ­1.5 ­1 n

­0.5

0

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Time Shift A signal may be shifted in time by replacing the independent variable n by n-k, where k is an integer. If k is a positive integer, the time shift results in a delay of the signal by k units of time. If k is a negative integer, the time shift results in an advance of the signal by |k| units in time.

x[n­3]

x[n+3]

x[n]

1 0.5 0 ­2 1 0.5 0 ­2 1 0.5 0 ­2

0

2

4

6

8

10

0

2

4

6

8

10

0

2

n4

6

8

10

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