## Discrete Time Signal Processing - May 17

### Electronics And Telecomm (Semester 6)

Total marks: 80

Total time: 3 Hours
INSTRUCTIONS

(1) Question 1 is compulsory.

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

(3) Draw neat diagrams wherever necessary.

**1(a)**Derive relationship between DFT and DTFT.

**1(b)**Compare: Impulsive invariant technique and bilinear transformation technique.

**1(c)**Define phase delay and group Delay.

**1(d)**Explain interpolation process with frequency spectrum.

**2(a)**Develop Composite radix DITFFT flow graph for N=6=2X3

**2(b)**Analog filter transform function is $H(S) = \frac{4}{(s+1)(s^2+4s+5)}$ obtain equivalent digital filter transfer function H(Z) using impulse invariant technique by taking T=0.5 sec.

**3(a)**State two important properties of DFT which are used to derive FFT. How computational complexity of DITFFT algorithm is determined from flow graph. Derive necessary formulas.

**3(b)**$$ y(n) = 2x(n) + \frac{4}{5} x(n-1) + \frac{3}{2}x(n-2) + \frac{2}{3} x(n-3) $$

Determine lattice realization.

**4(a)**Using frequency sampling method, design FIR band pass filter for following specifications

Sampling frequency = 800 Hz

Cut Off frequency = $f_{c2}$ = 3000 Hz

Cut Off frequency = $f_{c1}$ = 1000 Hz

Determine filter coefficients for N=7

**4(b)**Write short note on: Dual tone multi frequency detection

**4(c)**What is multi rate DSP? State its applications.

**5(a)**Design a Butterworth digital IIR filter using BLT by taking T=0.1 sec to satisfy following specifications

$$ 0.6 \le |H (e^{jwr}) | \le 1.0 \space \space \space 0 \le w \le 0.35 \pi$$

$$ |H (e^{jwr}) | \le 1.0 \space \space \space \space \space \space 0.7 \pi \le w \le \pi$$

**5(b)**x(n) = {2,3,4,5} and y(n) = {5,2,3,4}

(i) Find circular convolution using time domain method

(ii) Find circular convolution using frequency domain method

(iii) Compute linear convolution. Comment on your results.

**6(a)**The transfer function for discrete time causal system is given by $H(z) = \frac{1-z^{-1}}{1-0.2Z^{-1} - 0.15Z^{-1}}$

(i) Find difference equation

(ii) Draw Direct Form-I and Direct Form-II realization structure.

(iii) Draw cascade and parallel realization.

**6(b)**Explain the effects of coefficients quantization in FIR filters.

**6(c)**State Parserval's theorem. Verify it for x(n) = {1,2,3,4}