DISCRETE TIME SIGNAL PROCESSING (DTSP) Semester 7 (3 Hours) December 2011

DISCRETE TIME SIGNAL PROCESSING (DTSP) Semester 7 (3 Hours) December  2011
MP-5689     [Total Marks : 100]
N.B. :	(1)	Question No. 1 is compulsory.
	(2)	Attempt any four questions out of remaining six questions.
	(3)	Assume any data wherever required but justify the same.
	(4)	Figures to the right indicate full marks.
1.	(a)	Derive the Parsevel's Energy relation. State the significance of Parsevel's theorem.	05
	(b)	One of zeros of a Causal Linear phase FIR filter is at 0.5 ej	10
	(c)	A two pole pass filter has the system function H(z)= b0/(1-Pz-1)2 Determine the values of b0 and P. Such that the frequency response H(w) satisfies the condition H(0) = 1 and | H(Π/4) |2 = 1/2.	05
	(d)	Consider the signal x(n) = anu (n), |a| < 1 :-      (i) Determine the spectrum.      (ii) The signal x(n) is applied to a decimator that reduces the rate by a factor 2.            Determine the output spectrum.	05
2.	(a)	An analog signal Xa(t) is band limited to the range 900 < F < 1100 Hz. it is used as an input to the system shown in figure. In this system, H(w) is an ideal lowpass filter with cut-off frequency Fc = 125 Hz.   FS= 1/TS =2500                    FY.= 1/Ty=250   (i) Determine and sketch the spectra x(n), w(n), v(n) and y(n).   (ii) Show that it is possible to obtain y(n) by sampling Xa(t) with period T=4         millisecond.	10
	(b)	Derive and draw the FET for N = 6 = 2.3 use DITFFT method.      x(n) = { 1  2  3  1  2  3 }                   ↑ find x(k) using DITFFT for N = 6 = 2.3.	10
3.	(a)	Design a digital Butterworth low pass filter satisfying the following specifications using bilinear transformation. (Assume T = 15)   0.9 <  | H(ejw) | < 1 ; 0 < W < Π/2           | H(ejw) | < 0.2 ; 3 Π/4 < W <  Π	10
	(b)	(i) If x(n) = { 1 +5j, 2 +6j, 3 + 7j, 4 +8j}. Find DFT X(K) using DIFFT.   (ii) Using the results obtained in (i) not otherwise, find DFT of following sequences:            X1(n) = { 1, 2, 3, 4} and  X2(n) = { 5, 6, 7, 8}.	08
4.	(a)	Consider the realization of system shown in figure.   (i) Obtain System Function. 04 (ii) Obtain Difference Equation. 02 (iii) Find the impulse response of system. 03 (iv) Draw pole-zero  plot and comment on system Stability. 03
	(b)	Derive the Expression for impulse invariance technique for obtaining transfer function of digital filter from analog filter. Derive the necessary equation for relationship between frequency of analog and digital filter.	08
5.	(a)	What do you mean by inplace computations in FFT algorithms?	04
	(b)	Find number of Real additions and multiplications required to find DFT for 82 point. Compare them with number of computations required if FFT algorithms is used.	04
	(c)	Design a digital Chebyshev filter to satisfy the following constraints :-   0.707 <  | H(ejw) | < 1 ; 0 < W <  0.2Π           | H(ejw) | < 0.1 ; 0.5 < W <  Π Use bilinear transformation and assume T = 1 second.	12
6.	(a)	Given x(n) = n + 1 and N = 8, find DFT X(K) using DIFFT algorithms.	08
	(b)	Obtain direct form l, Direct form ll realization to second order filter given by --      y(n) = 2b cos w0y(n-1) - b2y (n-2) +x(n) - b cos w0x(n-1).	08
	(c)	Explain the concept of decimation by integer (M) and interpolation by integer factor (L).	04
7.	(a)	Write short note on set top box for digital TV receiver.	04
	(b)	Application of Signal Processing in Radar.	04
	(c)	What is linear phase filter? What condition are to be satisfied by the impulse response of an FIR system in order to have a linear phase? Define phase delay and group delay.	08
	(d)	Short note on Frequency Sampling realization of FIR filters.	04