Digital Signal Processing End Term: 13 April 2025 (January 2025 term)

1. Consider a stable LTI system whose system function is given below

$H(z) = \frac{(1 - 4z^{-2})\left(1 + \frac{1}{2}z^{-1}\right)}{1 - \frac{1}{2}z^{-1}}$

(a) Determine the poles and zeros of the system. Is it a minimum phase system? [2 Marks]

(b) The system function $H(z)$ can be represented as a cascade of a minimum – phase system $H_{min}(z)$ and a unity gain all pass system $H_{ap}(z)$. Determine a choice for $H_{min}(z)$ and $H_{ap}(z)$. [3 Marks]

(c) Is the minimum phase system $H_{min}(z)$ obtained in part (b) an FIR system? Justify. [2 Marks]

(d) Is the minimum – phase system $H_{min}(z)$ obtained in part (b) a linear phase system? If not can you represent $H(z)$ as a cascade of a linear phase system, $H_{lin}(z)$ and a unity gain all – pass system $H_{ap1}(z)$? [3 Marks]

1. Consider a Linear Time Invariant system having an impulse response

$h[n] = \delta[n] - 3\delta[n-1]$

Suppose the following input is given to the system

$x[n] = 2\delta[n] + 3\delta[n-1] + 4\delta[n-2]$

Using DFT, find the output of the system.

1. Consider a discrete time causal LTI FIR filter whose impulse response $h[n]$ is non zero only over five consecutive time samples. The frequency response of the filter is $H\left(e^{j\omega}\right)$.

The following information is given about the filter

a. $\int_{-\pi}^{\pi} H\left(e^{j\omega}\right) d\omega = \pi$

b. The frequency response can be written as

$H\left(e^{j\omega}\right) = A(\omega)e^{-j2\omega}$

Here $A(\omega)$ is real and even.

c. $A(0) = 1$ and $A(\pi) = 3$

i. Justify that it is a Type – I filter. [2 Marks]
ii. Completely specify the impulse response $h[n]$ and plot it. [5 Marks]
iii. Find the frequencies which this filter will completely block or stop. [3 Marks]

1. Consider a length 14 discrete time sequence defined for $0 \leq n \leq 13$,

Let the 14-point DFT be denoted as $X[k], 0 \leq k \leq 13$.
Given that $X[0] = 22$ and $X[7] = -2$.

Evaluate the following without computing the DFT

(i) The missing samples, $a$ and $b$ [3 Marks]
(ii) $\sum_{k=0}^{13} X[k]$ [2 Marks]
(iii) $\sum_{k=0}^{13} e^{-j\frac{5\pi}{7}k}\, X[k]$ [3 Marks]
(iv) $\sum_{k=0}^{13} |X[k]|^2$ [2 Marks]