Opening the paper…
Digital Signal Processing, End Term
(a) Determine the poles and zeros of the system. Is it a minimum phase system? [2 Marks]
(b) The system function can be represented as a cascade of a minimum – phase system and a unity gain all pass system . Determine a choice for and . [3 Marks]
(c) Is the minimum phase system obtained in part (b) an FIR system? Justify. [2 Marks]
(d) Is the minimum – phase system obtained in part (b) a linear phase system? If not can you represent as a cascade of a linear phase system, and a unity gain all – pass system ? [3 Marks]
Suppose the following input is given to the system
Using DFT, find the output of the system.
The following information is given about the filter
a.
b. The frequency response can be written as
Here is real and even.
c. and
i. Justify that it is a Type – I filter. [2 Marks]
ii. Completely specify the impulse response and plot it. [5 Marks]
iii. Find the frequencies which this filter will completely block or stop. [3 Marks]
| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 3 | 2 | 0 | 3 | 0 | 6 | 3 | 4 | 5 |
Let the 14-point DFT be denoted as .
Given that and .
Evaluate the following without computing the DFT
(i) The missing samples, and [3 Marks]
(ii) [2 Marks]
(iii) [3 Marks]
(iv) [2 Marks]
Instructions and questions 1 and 2 including a plot of the triangular Fourier transform X(jΩ) with peak 4 between −200π and 200π 3. 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] 4. 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. 5. 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] 6. Consider a length 14 discrete time sequence defined for $0 \leq n \leq 13$, | $n$ | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | |---|---|---|---|---|---|---|---|---|---|---|---|---|---|---| | $x[n]$ | 3 | $-1$ | 2 | 0 | 3 | $-2$ | 0 | $a$ | $-4$ | 6 | $b$ | 3 | 4 | 5 | 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]