R05221902-signals-and-systems

Set No. 1

Code No: R05221902

II B.Tech Supplimentary Examinations, Aug/Sep 2008 SIGNALS AND SYSTEMS ( Common to Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define and sketch the unit step function and signum function bring out the relation between these two functions. (b) Explain the Graphical Evaluation of a component of one function in other function. [6+10] 2. (a) Derive polar Fourier series from the exponential Fourier series representation and hence prove that Dn = 2 |Cn | (b) Show that the magnitude spectrum of every periodic function is Symmetrical about the vertical axis passing through the origin. [8+8] 3. (a) Obtain the Fourier transform of the following functions: i. Impulse function δ(t) ii. DC Signal iii. Unit step function. (b) State and prove time differentiation property of Fourier Transform.

[12+4]

4. (a) Explain how Impulse Response and Transfer Function of a LTI system are related. (b) Consider a stable LTI system that is characterized by the differential equation d2 y(t) +3y (t) = dx(t) +2x (t). Find its response for input x (t) = e−t u (t). +4 dy(t) dt2 dt dt [4+12] 5. (a) If V(t) = Sin ωo t. i. find R(Γ) ii. Find energy spectral density GE (f) = Fourier transform of R(τ )   (b) Applying the convolution theorem find Fourier Transform of A e−|at| sin c 2W t . (c) Use the convolution theorem to find the spectrum of x(t) = A Cos2 ωc t [6+6+4] 6. (a) Determine the Nyquist rate corresponding to each of the following signals. i. x(t) = 1 + cos 2000 pt + sin 4000 πt ii. x(t) = sin 4000πt πt

1 of 2

Set No. 1

Code No: R05221902

(b) The signal. Y(t) is generated by convolving a band limited signal x1 (t) with another band limited signal x2 (t) that is y(t) = x1 (t) * x2 (t) where x1 (jω) = 0 f or |ω| > 1000Π x2 (jω) = 0 f or |ω| > 2000Π Impulse train sampling is performed on y(t) to obtain ∞ P y(nT )δ(t − nT ) yp (t) = n=−∞

Specify the range of values for sampling period T which ensures that y(t) is recoverable from yp (t). [8+8] 7. (a) State the properties of the ROC of L.T. (b) Determine the function of time x(t) for each of the following laplace transforms and their associated regions of convergence. [8+8] i. ii.

(s+1)2 s2 −s+1 s2 −s+1 (s+1)2

Re {S} > 1/2 Re {S} > −1

8. (a) Determine inverse Z transforms of x(z) = when i. ROC : ii. ROC :

1 2−4z −1 +2z 2

by long division method

|z| > 1 |z| < 12

(b) Find Z transform of the following: i. (1/4)4 u (n) − cos (nπ/4) u (n) ii. 2n u(n-2) ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 2

Code No: R05221902

II B.Tech Supplimentary Examinations, Aug/Sep 2008 SIGNALS AND SYSTEMS ( Common to Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Find the even and odd components of the signal x(t) = e−2t Cos t (b) Rectangular function is approximated by a finite set of Sinusoidal function as given by
f (t) = Π4 sin t + 13 sin 3t + 51 sin 5t + 17 sin 7t + ..... Show that the Mean Square error of above function leads to zero for infinite terms. [6+10] 2. (a) State the properties of Fourier series. (b) Obtain the trigonometric fourier series representation for a half wave rectified Sine wave shown in figure 2 [6+10]

Figure 2 3. (a) Explain the concept of Fourier Transform for periodic signals. (b) Find Fourier Transform of a sequence of equidistant impulses of unit strength and separated by T seconds. [8+8] 4. (a) Explain how input and output signals are related to impulse response of a LTI system. (b) Let the system function of a LTI system be system for an input (0.8)t u (t).

1 . jw+2

What is the output of the [8+8]

5. (a) Explain briefly detection of periodic signals in the presence of noise by correlation. (b) Explain briefly extraction of a signal from noise by filtering.

1 of 2

[8+8]

Set No. 2

Code No: R05221902


50Πt 2 which to be sampled with a sampling 6. (a) Consider the signal x(t) = sinΠt frequency of ωs = 150Π to obtain a signal g(t) with Fourier transform G(jω ). Determine the maximum value of ω0 for which it is guaranteed that G(jω) = 75 × (jω) f or |ω| ≤ ω0 where X(jω) is the Fourier transform of x(t). (b) The signal x(t) = u(t + T0 ) − u(t − T0 ) can undergo impulse train sampling without aliasing, provided that the sampling period T< 2T0 . Justify. (c) The signal x(t) with Fourier transform X(jω) = u(ω + ω0 ) − u(ω − ω0 ) can undergo impulse train sampling without aliasing, provided that the sampling period T < π/ω0 . Justify. [6+5+5] 7. (a) Find the step response of series RLC circuits. (b) Find the pulse response of series RL circuit figure 7b

Figure 7b (c) Find the pulse response of series RC circuit figure 7c

[6+5+5]

Figure 7c 8. (a) Show that if x[n] is a right sided sequence and X[z] converges for some valve of z, then the ROC of X[z] is of the form. |Z| > rmax or ∞ > |Z| > rmax . Where rmax is the maximum magnitude of any of the poles of X(z). (b) Find the Z-transform of x(n) = sin(nω)u(n). ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 3

Code No: R05221902

II B.Tech Supplimentary Examinations, Aug/Sep 2008 SIGNALS AND SYSTEMS ( Common to Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Define mean square error and derive the expression for evaluating mean square errors. (b) A rectangular function f(t) is defined by  1 0
Figure 3a (b) Findthe Fourier Transform of the signal given below cos 10t, −2 ≤ t ≤ 2 y (t) 0, otherwise

[8+8]

4. (a) Explain how input and output signals are related to impulse response of a LTI system. (b) Let the system function of a LTI system be system for an input (0.8)t u (t).

1 . jw+2

What is the output of the [8+8]

5. (a) For the signal g(t) = 2a/(t2 +a2 ),determine the essential Band width B Hz of g(t) such that the energy contained in the spectral components of g(t) of frequencies below B Hz is 99% of signal energy Eg . 1 of 2

Set No. 3

Code No: R05221902

(b) Show that the auto correlation function of g(t)=C cos (ω0 t+ θ0 ) is given by Rg (τ )=(c2 /2) cos ω0 τ ,and the corresponding PSD is Sg (ω) = (c2 π/2) [δ (ω − ω0 ) + δ (ω + ωo ) . [8+8] 6. (a) A low pass signal x(t) has a spectrum x(f) given by 1 − |f | /200 |f | < 200 x(f ) = 0 elsewhere Assume that x(t) is ideally sampled at fs=300 Hz. Sketch the spectrum of xδ (t)f or |f | < 200. (b) The uniform sampling theorem says that a band limited signal x(t) can be completely specified by its sampled values in the time domain. Now consider a time limited signal x(t) that is zero for |t| ≥ T . Show that the spectrum x(f) of x(t) can be completely specified by the sampled values x(kfo ) where f0 ≤ 1/2T . [8+8] 7. (a) Determine the laplace transform of signal shown in figure 7a.

Figure 7a (b) Find the step response of series RL circuit. (c) Find the step response of series RC circuit.

[6+5+5]

8. (a) State the properties of the ROC of Z.T. (b) Find the Z-transform of the sequences. i. δ [n] ii. u[n]. ⋆⋆⋆⋆⋆

2 of 2

[8+8]

Set No. 4

Code No: R05221902

II B.Tech Supplimentary Examinations, Aug/Sep 2008 SIGNALS AND SYSTEMS ( Common to Electronics & Computer Engineering and Instrumentation & Control Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Sketch the following signals
+ Π(t − 1) i. Π t−1 2 ii. f (t) = 3u(t) + tu(t) − (t − 1)u(t − 1) − 5u(t − 2) (b) Evaluate the following Integrals R5 i. δ(t)Sin2Πtdt ii.

0 Rα

[8+8]

−αt2

e δ(t − 10)dt

−α

2. (a) Write short notes on “Exponential Fourier Spectrum”. (b) Find the Fourier series expansion of the periodic triangular wave shown figure 2. [6+10]

Figure 2 3. (a) Determine the Fourier transform of a two sided exponential pulse x (t) = e−|t| (b) Find the Fourier transforms of an even function xe (t) and odd function xo (t) of x(t). [8+8] 4. (a) Explain the difference between the following system. i. Linear and Non-linear systems. ii. Causal and Non-Causal systems. (b) Consider a stable LTI system that is characterized by the differential equation d2 y(t) + 4 dy(t) + 3y (t) = dx(t) + 2x (t) dt2 dt dt Find its impulse response and transfer function. [8+8] 5. (a) State and Prove Properties of auto correlation function?

1 of 2

Set No. 4

Code No: R05221902

(b) A filter has an impulse response h(t) as shown in figure 5b The input to the network is a pulse of unit amplitude extending from t=0 to t=2. By graphical means determine the output of the filter. [8+8]

Figure 5b 6. (a) Explain briefly impulse sampling. (b) Define sampling theorem for time limited signal and find the nyquist rate for the following signals. [8+8] i. rect300t ii. -10 sin 40πt cos 300πt. 7. (a) State the properties of the ROC of L.T. (b) Determine the function of time x(t) for each of the following laplace transforms and their associated regions of convergence. [8+8] i. ii.

(s+1)2 s2 −s+1 s2 −s+1 (s+1)2

Re {S} > 1/2 Re {S} > −1

8. (a) 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] (b) Derive relationship between z and Laplace Transform. ⋆⋆⋆⋆⋆

2 of 2

[8+8]