## 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)=0.5^{n} u(n)+8^{n}u(-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) x_{1}(n)= cos (0.5 π n + 0.3)

(ii) x_{2}(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 x_{1}(n) = {1, 2, 3, 4} and x_{2}(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)