1 (a)
Transfer functions of casual and stable digital filters are given below. State whether these filtes are

Minimum/ Maximum/ Mixed Phase filters

\[\left(i\right)\ H_1\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(1-\frac{1}{4}z\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

\[\left(ii\right)\ H_2\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

\[\left(iii\right)\ \ H_3(Z)=\dfrac{\left(z-\frac{1}{2}\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

Minimum/ Maximum/ Mixed Phase filters

\[\left(i\right)\ H_1\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(1-\frac{1}{4}z\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

\[\left(ii\right)\ H_2\left(Z\right)=\dfrac{\left(1-\frac{1}{2}z\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

\[\left(iii\right)\ \ H_3(Z)=\dfrac{\left(z-\frac{1}{2}\right)\left(z-\frac{1}{4}\right)}{\left(z-\frac{1}{3}\right)\left(z-\frac{1}{5}\right)}\]

5 M

1 (b)
Compute DFT of the sequence X

_{1},(n) = {1,2,4,2} using property and not otherwise compute DFT of x_{2}(n) = {1+j, 2+2j, 4+4j, 2+2j}
5 M

1 (c)
The impulse response of a system is h(n)=a

(i) Stable low pass filter. (ii) Stable High pass filter.

^{n}u(n), a≠0. Determine a and sketch pole zero plot for this system to act as :-(i) Stable low pass filter. (ii) Stable High pass filter.

5 M

1 (d)
Draw direct form structure for a filter with transfer function, H(z) = 1+3z

^{-1}+ 2z+^{-3}- 4z^{-4}
5 M

2 (a)
Consider a filter with impulse response, h(n)={0.5, 1, 0.5}. Sketch its amplitude spectrum. Find its response to the inputs

\[\left(i\right)\ \ x_1\left(n\right)=\cos{\left(\frac{n\pi{}}{2}\right)\ }\]

\[\left(ii\right)\ \ x_2\left(n\right)=3+2\delta{}\left(n\right)-4\cos?(\frac{n\pi{}}{2})\]

\[\left(i\right)\ \ x_1\left(n\right)=\cos{\left(\frac{n\pi{}}{2}\right)\ }\]

\[\left(ii\right)\ \ x_2\left(n\right)=3+2\delta{}\left(n\right)-4\cos?(\frac{n\pi{}}{2})\]

10 M

2 (b)
Determine circular convolution of x(n)= {1,2,1,4} and h(n)= {1,2,3,2} using time domain convolution and radix -2FFT. Also find circular correlation using time domain correlation.

10 M

3 (a)
Explain overlap and add method for long data filtering. Using this method find output of a system with impulse response, h(n)= {1,1,1} and input x(n)= {1,2,3,3,4,5}.

10 M

3 (b)
Compute Dft of a sequence, x(n)= {1,2,2,2,1,0,0,0} using DIF-FFT algorithm. Sketch its magnitude spectrum.

10 M

4 (a)
Draw lattice filter realization for a filter with the following transfer function.

\[ H(Z)=\frac{1}{1+\frac{13}{24}z^{-1}+\frac{5}{8}z^{-2}+\frac{1}{3}z^{-3}} \]

\[ H(Z)=\frac{1}{1+\frac{13}{24}z^{-1}+\frac{5}{8}z^{-2}+\frac{1}{3}z^{-3}} \]

10 M

4 (b)
Design a low pass Buttre worth filter with order 4 and passband cut off frequency of 0.4π. Sketch pole zero plot. Use Bilinear transformation. Draw direct form II structure for the designed filter.

10 M

5 (a)
Design a FIR Bandpass filter with the following specifications :-

Length : 9

stop band cut off frequency : 0.7π

Use Hanning window.

Length : 9

stop band cut off frequency : 0.7π

Use Hanning window.

10 M

5 (b)
The transfer function of a filter has two poles at z=0, two zeroes at z= -1 and a dc gain of 8. Final transfer function and impulse response.

Is this a causal or noncausal filter?

Is this a linear phase filter?

If another zero is added at z=1 find transfer function and check whether it is a linear phase filter or not.

Is this a causal or noncausal filter?

Is this a linear phase filter?

If another zero is added at z=1 find transfer function and check whether it is a linear phase filter or not.

10 M

6 (a)
Transfer function of an FIR filter is given by H(z)=1-z

^{N}. Sketch pole zero plots for N=4 and N=5 prove that it is comb filter.
10 M

6 (b)
Write about frequency sampling realization of FIR filters.

10 M

7 (a)
Explain the process of decimation for reducing sampling rate of signal.

10 M

7 (b)
Compare various windows used for designing FIR filters.

10 M

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