## Discrete Time Signal Processing - May 2012

### Electronics & Telecomm. (Semester 6)

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)** Show that FIR filters are linear phase filters. Define group delay and phase delay. (8 marks)
**1 (b)** find the response of the system given by difference equation y(n)-5y(n-1)+6y(n-2) = x(n) for (i) x(n) = ?(n) and (ii) x(n) = U(n)(6 marks)
**1 (c) ** Draw direct form - I and Direct form-II realization of the system given by

y(n)-3/4?y(n?1) + 1/8?y(n-2) = x(n) + 1/3?x(n-1) (6 marks)
**2 (a)** Derive Radix-2 Decimation in Time Fast Fourier Transform and draw its signal flow graph (10 marks)
**2 (b)** Design FIR digital filter by using window method for the following specification : -

$$H\left(e^{j\omega{}}\right)=e^{-j3\omega{}}\ \ \ \ -\frac{3\pi{}}{4}\leq{}\omega{}\leq{}\frac{3\pi{}}{4}$$

$$H\left(e^{j\omega{}}\right)=0,\dfrac{3\pi{}}{4}\leq{}\left\vert{}\omega{}\right\vert{}\leq{}\pi{}$$

Use Hamming window of lenght = 7(10 marks)
**3 (a) ** If H_{d}(?)=1 for 0? f ? 800 Hz and

H_{d} (?)=0 for f ? 800 Hz

Given that sampling frequency Fs= 5000Hz Design FIR filter for length M=5 using Bartiett window (10 marks)
**3 (b)** Find 8-point FFT of x(n)= { 1,2,2,2,1 } using signal flow graph of Radix- 2 Decimation in frequency FFT (10 marks)
**4 (a)** Design a digital Butterworth filter satisfying the following constraints :

$$0.9=\left\{\begin{array}{l}\leq{}\left\vert{}H\left(e^{j\omega{}}\right)\right\vert{}\leq{}1\\ \ \ \ \ \ \ \ \&0\leq{}\omega{}\leq{}\frac{\pi{}}{2} \\\leq{}\left\vert{}H\left(e^{j\omega{}}\right)\right\vert{}\leq{}0.2\ \ \ \ \ \\&\frac{3\pi{}}{4}\leq{}\omega{}\leq{}\pi{}\end{array}\right.$$

with sampling period T=1 sec, use Bilinear transformation method.(12 marks)
**4 (b) ** State and derive Geortzel algorithm, also state its application (8 marks)
**5 (a)** For the analog tansfer function $$H\left(s\right)=\frac{3}{\left(s+2\right)\left(s+3\right)}\$$ Determine H(z) with sampling period T=0.1 sec using

(i) Impulse Invariant method

(ii) Bilinear Transformation method (10 marks)
**5 (b)** Describe the application Geortzel algorithm in dual tone multi- frequency detection. (10 marks)
**6 (a) ** With a suitable block diagram describe sub-band coding of speech signals. (10 marks)
**6 (b)** With a neat diagram describe frequency sampling realization of FIR filters. (10 marks)
**7 (a)** Explain up sampling by non-integer factor, with a neat diagram and waveforms (10 marks)
**7 (b)** Explain the application of DTSP in adaptive echo cancellation. (5 marks)
**7 (c) ** Describe the concept of digital resonator (5 marks)