## Discrete Time Signal Processing - May 2013

### Electronics & Telecomm. (Semester 7)

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)** Assume that a complex multiplier takes 1 micro sec to perform one multiplication and that the amount of time to compute a DFT is determined by the amount of time to perform all the multiplications.

(i) How much time does it take to compute a 1024 point DFT directly?

(ii) How much time is required is FFT is used? (5 marks)
**1 (b)** Let h[n] be the unit impulse response of a Low Pass filter with a cutoff frequency ω_{c}, what type of filter has a unit sample response g[n]= (-1)^{n} h[n]. (5 marks)
**1 (c) ** A two pole low pass filter has the system function $$H\left(z\right)=\frac{b_0}{{\left(1-pz^{-1}\right)}^z}$$ Determine the values of b_{0} and P(5 marks)
**1 (d)** Consider filter with transfer function. Identify the type of filter and justify it.

$$ H\left(z\right)=\frac{z^{-1}-a}{1-az^{-1}}$$(5 marks)
**2 (a)** The unit sample response of a system is h(n)={3,2,1} use overlap-add method of linear filtering to determine output sequence for the repeating input sequence x[n]= {2,0,-2,0,2,1,0,-2,-1,0}(10 marks)
**2 (b)** For a given sequence x(n)= {2,0,0,1}, perform following operation :

(i) Find out the 4 point DFT of x(n)

(ii) Plot x(n), its periodic extension x_{p}(n) and x_{p}(n-3)

(iii) Find out 4 point DFT of x_{p}(n-3)

(iv) Add phase angel in (i) with factor - $$-\ \ [\frac{2\pi{}rk}{N}]$$ where N=4, r=3, k=0,1,2,3

(v) Comment on the result you had in point (i) and (ii)

(10 marks)
**3 (a) ** The transfer function of discrete time causal system is given below :

$$H\left(z\right)=\frac{1-z^{-1}}{1-0.2z^{-1}-0.15z^{-2}}$$

(i) Find the difference equation

(ii) Draw cascade and parallel realization

(iii) Show pole-zero diagram and then find magnitude at ω = 0 and ω= π

(iv) Calculate the impulse response of the system. (10 marks)
**3 (b)** Obtain the lattice realization for the system :

$$H\left(z\right)=\frac{1+3z^{-1}+3z^{-2}+z^{-3}}{1+\frac{3}{4}z^{-1}+\frac{1}{2}z^{-2}+\frac{1}{4}z^{-3}}$$(10 marks)
**4 (a)** What is a linear phase filter? What conditions are to be satisfied by the impulse response of an FIR system in order to have a linear phase? Plot and justify compulsory zero locations for symmetric even antisymmetric even and antisymmetric odd FIR filters. (10 marks)
**4 (b) ** Determine the zero of be following FIR system and indicate whether the system is minimum phase, maximum phase, or mixed phase

$$H_1\left(z\right)=6+z^{-1}-z^{-2}$$

$$H_2\left(z\right)=1-z^{-1}-6z^{-2}$$

$$H_1\left(z\right)=1-\frac{5}{2}z^{-1}-\frac{3}{2}z^{-2}$$

$$H_1\left(z\right)=1+\frac{5}{3}z^{-1}-\frac{2}{3}z^{-2}$$

Comment on the stability of the minimum and maximum phase system (10 marks)
**5 (a)** A digital low pass filter is required to meet the following specification:

Pass band ripple : ≤1dB

Pass band edge : 4KHz

Stop band attenuation : ≥ 40dB

Stop band edge : 8KHz

Sampling rate : 24KHz

Find order, cutoff frequency and pole locations for Butterworth filter using bilinear transformation.(10 marks)
**5 (b)** Design an FIR digital filter to approximate an ideal low pass filter with passband gain of unity, cut-off frequency of 950Hz and working at a sampling frequency of Fs=5000 Hz. The length of the impulse response should be 5. Use a rectangular window. (10 marks)
**6 (a) ** Explain the need of a low pass filter with a decimator and mathematically prove that ω_{y} = ω_{x}D.(10 marks)
**6 (b)** Why is the direct form FIR structure for a multirate system inefficient? Explain with neat diagrams how this inefficiency is overdone in implementing a decimator and an interpolator.(10 marks)

### Write short notes (any four ) :

**7 (a)** DTMF detection using Geortzel algorithm (5 marks)
**7 (b)** Filter bank (5 marks)
**7 (c) ** Comparison of FIR and IIR filters (5 marks)
**7 (d) ** Split radix FFT (5 marks)
**7 (e) ** Optimum Equiripple Linear phase FIR filter design (5 marks)