BE Computer Engineering (IOE, TU) Digital Signal Analysis and Processing (IOE, EX 701 / ENEX 416) Question Paper 2079

(a) Define a linear time-invariant (LTI) discrete-time system. State and prove the conditions for stability and causality of an LTI system in terms of its impulse response $h[n]$.

(b) The impulse response of a discrete-time LTI system is given by $h[n] = (0.5)^n u[n]$. Determine the response $y[n]$ of the system to the input $x[n] = u[n]$ using the convolution sum, and comment on whether the system is stable.

(a) State the definition of the Z-transform and explain the significance of the region of convergence (ROC). List the properties of the ROC.

(b) Consider a causal LTI system described by the difference equation

Find the system function $H(z)$, sketch its pole-zero plot, and determine the impulse response $h[n]$ using the inverse Z-transform (partial fraction expansion).

(a) Explain the steps involved in designing a digital IIR filter from an analog prototype using the bilinear transformation method. Discuss the problem of frequency warping and how prewarping addresses it.

(b) Design a first-order digital low-pass Butterworth filter with a 3-dB cutoff frequency of $\omega_c = 0.2\pi$ rad/sample using the bilinear transformation. Assume a sampling period $T = 1$ s and obtain the transfer function $H(z)$.

(a) Explain why FIR filters can be designed to have exactly linear phase. State the symmetry conditions on the impulse response $h[n]$ that guarantee linear phase.

(b) Design a linear-phase FIR low-pass filter of length $N = 7$ with cutoff frequency $\omega_c = \pi/2$ using the Hamming window method. Determine the filter coefficients $h[n]$ and comment on the trade-off between main-lobe width and side-lobe attenuation when choosing a window.

(a) Compute the 4-point DFT of the sequence $x[n] = \{1,\, 2,\, 3,\, 4\}$.

(b) Distinguish between linear convolution and circular convolution, and explain how the DFT can be used to perform linear convolution of two finite-length sequences.

Draw the signal-flow graph (butterfly diagram) of an 8-point decimation-in-time (DIT) radix-2 FFT algorithm. Show the input bit-reversal ordering and the twiddle factors $W_8^k$. Compare the number of complex multiplications and additions required by direct DFT computation versus the radix-2 FFT for $N = 8$.

State and explain the sampling theorem. A continuous-time signal $x_a(t) = \cos(2\pi \cdot 2000 t) + \cos(2\pi \cdot 5000 t)$ is sampled at $f_s = 6\,\text{kHz}$. Determine which frequency components are aliased and state the apparent (alias) frequencies that appear in the sampled signal.

(a) Define the cross-correlation and autocorrelation of discrete-time sequences and explain their physical significance in signal processing.

(b) Compute the linear convolution of $x[n] = \{1,\, -1,\, 2\}$ and $h[n] = \{2,\, 1,\, 1\}$.

A discrete-time system has the system function $H(z) = \dfrac{1 - z^{-1}}{1 - 0.5 z^{-1}}$. Obtain expressions for the magnitude response $|H(e^{j\omega})|$ and the phase response $\angle H(e^{j\omega})$, and sketch the magnitude response over $0 \le \omega \le \pi$. State whether the system behaves as a low-pass or high-pass filter.

(a) Classify discrete-time signals as energy signals and power signals, giving the defining equations for energy and power.

(b) Determine whether the signal $x[n] = \cos\!\left(\dfrac{\pi n}{4}\right)$ is periodic; if so, find its fundamental period $N$. Also test the system $y[n] = n\,x[n]$ for linearity and time-invariance.

The system function of a discrete-time LTI system is $H(z) = \dfrac{z}{z - 1.5}$. Using the pole locations and the ROC, determine whether the system can be both causal and stable. Justify your answer.