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Digital Signal Processing : Question Paper Dec 2015 - Computer Engineering (Semester 7) | Mumbai University (MU)

## Digital Signal Processing - Dec 2015

### Computer Engineering (Semester 7)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks.
1 (a) State the condition for stability of LTI system, determine the range of values of a and b for which the impulse time - invariant system with following given impulse response is stable. $$h(n)= \left\{\begin{matrix} a^n &m\le 0 \\ b^n & n<0 \end{matrix}\right.$$(5 marks) 1 (b) Find the Energy of the signal x(n)=0.5n u(n)+8nu(-n-1).(5 marks) 1 (c) Find the values of x(n)= cos (0.25 π n) for n=0, 1, 2, 3. Compute the DFT of x(n) using FFT flow graph.(5 marks) 1 (d) Find the cross correlation of the sequence $x(n)= \big \{ \underset{\uparrow}{1} , 2, 3, 4 \big \} \text{ and }h(n)= \big \{ \underset{\uparrow}{2}, 4, 6 \big \}.$(5 marks) 2 (a) Determine whether or not the following signals are periodic If periodic specify its fundamental period.
(i) x1(n)= cos (0.5 π n + 0.3)
(ii) x2(n) = cos (0.3 πn) + 10 sin (0.25 π n).
(10 marks)
2 (b) Compute Linear convolution of causal x(n) and h(n) using overlap and method in time domain.
x(n)={1, 2, 3, 4, 5, 6, 7, 8}, h(n)={1, 1, 1}
(10 marks)
3 (a) Check whether the given system y(n) = x(2n) - x(n-1) is:
i) Static or Dynamic
ii) Linear or non-linear
iii) Shift invariant or variant
iv) Causal or non causal.
v) Stable or unstable.
(10 marks)
3 (b) State the following DFT properties:
i) Linearity property
ii) Periodicity
iii) Time shift
iv) Convolution
v) Time Reversal
(10 marks)
4 (a) For the causal LTI digital filter with impulse response given by h(n)=0.3 δ(n) - δ(n+1) + 0.38 δ(n-3) sketch the magnitude spectrum of the filter. Using DFT.(10 marks) 4 (b) Let X (K) = {20, 0, -4+4j, 0, -4} is the 8 point DFT of a real valued sequence x(n)
i) Find X(K) for K=5, 6, 7.
ii) Find the 8 point DFT P(K) such that p(n) = (-1)n x(n) Using DFT property.
(10 marks)
5 (a) Find circular convolution and linear using circular convolution for the following sequence x1(n) = {1, 2, 3, 4} and x2(n) = {1, 2, 1, 2}. Using Time Domain formula method.(10 marks) 5 (b) Derive radix 2 DITFET flow graph and find the DFT of the sequence x(n) = {0, 1, 2, 3}(10 marks) 6 (a) Write a detailed note on DSP Processor.(10 marks) 6 (b) Write a detailed note on Carl's Correlation Coefficient Algorithm. Justify the necessary of Algorithm by given suitable example.(10 marks)

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