## Digital Signal Processing - Jun 2015

### Electronics & Communication (Semester 5)

TOTAL MARKS: 100

TOTAL TIME: 3 HOURS
(1) Question 1 is compulsory.

(2) Attempt any **four** from the remaining questions.

(3) Assume data wherever required.

(4) Figures to the right indicate full marks.
**1 (a)** Compute the DFT of the sequence $$x(n)=\cos\left ( \dfrac{n\pi}{4} \right )$$ for N=4, plot |x(k)| and ∠x(k).(9 marks)
**1 (b)** Find the DFT of the sequence x(n)=0.5^{n} u(n) for 0 < n <3 by evaluating x(n)=a^{n} for 0< n < N - 1.(7 marks)
**1 (c)** Find the relation between DFT and Z transform.(4 marks)
**2 (a)** State and prove the linearity property of DFT and symmetrical property.(5 marks)
**2 (b)** The five samples of the 8 point DFT x(k) are given as

x(0)=0.25, x(1)=1.25 - j0.3018, x(6)=x(4)=0, x(5)=0.125 - j0.0518.(5 marks)
**2 (c)** For x(n)={1,-2,3,-4,5-6}, without computing its DFT, find the following.

$$i)\ x(o)\ ii)\ \sum_{k=0}^{5}\ iii)\ X(3)\ iv)\ \sum_{k=0}^{5}1 \times(k)|^{2}\ v)\ \sum_{k=0}^{5}(-1)^{k}\times (k)$$(10 marks)
**3 (a)** Consider a FIR filter with impulse response

h(n)={1,1,1}, if the input is

X(n)={1,2,0,-3,4,2,-1,1,-2,3,2,1,-3}. Find the output y(a) using overlap add method.(12 marks)
**3 (b)** What is an plane computation? What is total number of complex additions and multiplication required for N=256 point, if DFT is computed directly and if FFT is used?(3 marks)
**3 (c)** For sequence x(n)={2,0,2,0} determine x(2) using Goertzel Filter. Assume the zero initial conditions.(5 marks)
**4 (a)** Find the circular convolution of x(n)={1,1,1,1} and h(n)={1,0,1,0} using DIF-FFT algorithm.(12 marks)
**4 (b)** Derive DIT-FFT algorithm for N=4. Draw the complete signal How graph?(8 marks)
**5 (a)** Design a Chebyshev analog filter (low pass) that has a-3dB cut-off frequency of 100 rad/sce and a stopband attenuation 25dB or greater for all radian frequencies past 250 rad/sec.(14 marks)
**5 (b)** Compare Butterworth and Chebyshev filters.(3 marks)
**5 (c)** Let $$H(s)=\dfrac{1}{s^{2}+s+1}$$ represent the transfer function of LPF with a passband of 1 rad/sec. Use frequency transformation (Analog to Analog) to find the transfer function of a band pass filter with passband 10 rad/sec and a centre frequency of 100 rad/sec.(3 marks)
**6 (a)** Obtain block diagram of the direct form I and direct from II realization for a digital IIR filter described by the system function.

$$H(z)=\dfrac{8z^{3}-4z^{2}+11z-2}{\left ( z-\dfrac{1}{4} \right )\left ( z^{2}-z+\dfrac{1}{2} \right )}$$(10 marks)
**6 (b)** Find the transfer function and difference equation realization shown in fig Q6(b)

(6 marks)
**6 (c)** Obtain the direct form realization of liner phase FIR system given by

$$H(z)=1+\dfrac{2}{3}z^{-1}+\dfrac{15}{8}z^{-2}$$(4 marks)
**7 (a)** "The desired frequency response of low pass filter is given by

$$H_{d}(e^{jw})=H_{d}(\infty )=\left\{\begin{matrix}
e^{-j3w} &|\infty|\dfrac{3 \pi}{4} \\0
&\dfrac{3\pi}{4}<|\infty|<\pi
\end{matrix}\right$$

Determine the frequency response of the FIR if Hamming window is used with N=7."(10 marks)
**7 (b)** Compare IIR filter and FIR filters.(6 marks)
**7 (c)** Consider the pole-zero plot as shown in FigQ7(c). i) Does it represent an FIR filter? ii) Is it linear phase system?

(4 marks)
**8 (a)** Design a digital filter H(z) that when used in an A/D-H(z)-D/A structure gives an equivalent analog filter with the following specification:

Passband ripple:≤3.01dB

Passband edge : 500Hz

Stopband attenuation : ≥ 15dB

Stopband edge : 750 Hz

Sample Rate : 2 KHz

Use Bilinear transformation to design the filter on an analog system function. Use Butterworth filter prototype. Also obtain the difference equation.(14 marks)
**8 (b)** Transform the analog filter

$$H_{a}(s)=\dfrac{s+1}{s^{2}+5s+6}$$

Into H(z) using impulse invariant transformation Take T=0.1 Sec.(6 marks)