1(a) Determine the zeros of the following systems and indicate whether the system is minimum, maximum or mixed phase.

1) $H_1(z) = 6+Z^{-1} + 6Z^{-2}$

1) $H_2(z) = 1-Z^{-1} - 6Z^{-2}$

1(b) What is multirate DSP? State its applications.

1(c) Compare BLT and impulse invariant method.

1(d) Explain concept of decimation by integer D.

1(e) If X(K) = {16, -4, 0, -4}, determine x[n] using IFFT.

2(a) If x(n) = {1,2,3} and h(n) = {1,0}

1. Find linear convolution using circular convolution.
2. Find circular convolution using DFT-IDFT.

2(b) Show the mapping from S plane to Z plane using impluse invariant method. Explain its limitations. Using this method determine H(z) if

$$H(s) = \frac{2}{(s+1) (s+2)} \space\space\space\space if \space T_s=1s$$

3(a) Compute DFT of sequence x(n) = {1,2,3,4,5,6,7,8} using DIT-FFT algorithm.

3(b) Design low pass IIR Butterworth filter for following specifications

Passband attenuation = 1 dB

Stopband attenuation = 40dB

Passband edge frequency = 200 Hz

Stopband edge frequency = 540 Hz

Sampling frequency = 8 KHz

Use Bilinear transformation method.

4(a) A low pass filter is to be designed with following desired frequency response.

$$ Hd(e^{j\omega}) \space\space\space\space \frac{-\pi}{4} \le w \le \frac{\pi}{4}$$

$$ = 0\space\space\space\space \frac{\pi}{4} \le w \le \pi$$

Detemine the filter coefficients $h_d(n)$ if the window function is defined as

$$ w(n) = 1 \space \space \space \space 0 \le n \le 4 $$

$$ = 0 \space \space \space \space otherwise$$

Also determine the frequency response $H(e^{j\omega})$ of the designed filter.

4(b) Find DFT of x(n) = {1,2,3,4} Using these results not otherwise find DFT.

i) $x_1(n)$ = {4,1,2,3}

ii) $x_2(n)$ = {2,3,4,1}

iii) $x_3(n)$ = {6,4,6,4}

5(a) Explain subband coding of speech signal as a application of multirate signal processing.

5(b) Determine the Direct form-I and Direct form-II realization for the system

$y(n) = -0.1y(n-1) + 0.2y(n-2) + 3x(n) + 3.6x(n-1) + 0.6x9n-2)$

6(a) Dual Tone Multifrequency Detection using Goertzel's algorithm

6(b) The effects of coefficients quantization in FIR filters.

6(c) Concept of interpolation by integer factor I