BE Computer Engineering (IOE, TU) Digital Signal Analysis and Processing (IOE, EX 701 / ENEX 416) Question Paper 2078

(a) Define a discrete-time LTI system. State and prove the conditions on the impulse response $h[n]$ for the system to be (i) causal and (ii) BIBO stable. [6]

(b) A discrete-time LTI system has impulse response $h[n] = \left(\tfrac{1}{2}\right)^n u[n]$. Determine the output $y[n]$ when the input is $x[n] = u[n] - u[n-4]$ using linear convolution, and comment on whether the system is stable. [6]

Consider an LTI system described by the difference equation

(a) Determine the system function $H(z)$ and its region of convergence assuming the system is causal. [5]

(b) Plot the pole-zero diagram and comment on the stability of the system. [4]

(c) Find the unit-impulse response $h[n]$ using the inverse Z-transform (partial fraction method). [5]

(d) Sketch the general shape of the magnitude response $|H(e^{j\omega})|$ and identify whether the system behaves as a low-pass or high-pass filter. [2]

(a) Explain the impulse invariance method and the bilinear transformation method of designing IIR digital filters. Clearly state the mapping equations and discuss the problem of frequency warping. [6]

(b) Design a first-order digital low-pass IIR filter using the bilinear transformation from the analog prototype $H_a(s) = \dfrac{\Omega_c}{s + \Omega_c}$ with a 3 dB cut-off frequency of $\omega_c = 0.2\pi$ rad/sample. Assume $T = 1$ s and give the resulting transfer function $H(z)$. [6]

(a) Derive the decimation-in-time (DIT) radix-2 FFT algorithm for an $N$-point DFT and draw the complete signal-flow graph (butterfly diagram) for $N = 8$. [8]

(b) Compare the number of complex multiplications and additions required for direct DFT computation versus the radix-2 FFT for $N = 1024$, and comment on the computational savings. [4]

Compute the 4-point DFT of the sequence $x[n] = \{1,\ 2,\ 3,\ 4\}$ using the DFT definition. Show the magnitude and phase of each DFT coefficient.

State the Nyquist sampling theorem. An analog signal $x_a(t) = 3\cos(2\pi \cdot 1000\,t) + 2\cos(2\pi \cdot 3000\,t)$ is sampled at $f_s = 4000$ Hz. Determine which frequency components are aliased and write the reconstructed signal's frequency content.

Explain the window method of FIR filter design. Compare the rectangular, Hamming and Blackman windows in terms of main-lobe width and peak side-lobe level, and discuss their effect on the transition band and stop-band attenuation of the designed filter.

Distinguish between linear convolution and correlation of two discrete-time sequences. Compute the cross-correlation $r_{xy}[l]$ of $x[n] = \{1,\ 2,\ 1\}$ and $y[n] = \{1,\ -1,\ 2\}$ for all lags $l$.

(a) Determine the Z-transform and the region of convergence of $x[n] = a^n u[n] + b^n u[-n-1]$, stating the condition on $a$ and $b$ for the ROC to exist. [4]

(b) List two important properties of the region of convergence of the Z-transform. [2]

For the FIR system $y[n] = x[n] - x[n-1]$, determine the frequency response $H(e^{j\omega})$, and obtain expressions for its magnitude and phase response. Sketch $|H(e^{j\omega})|$ over $0 \le \omega \le \pi$ and state whether the system is linear phase.

Classify the following signals/systems with justification: (a) Is $x[n] = \cos(0.3\pi n)$ periodic? If so, find its fundamental period. (b) Is the system $y[n] = n\,x[n]$ linear, time-invariant, and stable?

Explain how the Discrete Fourier Transform (DFT) is related to the Discrete-Time Fourier Transform (DTFT). State two important properties of the DFT (e.g., circular shift, Parseval's theorem) with their mathematical expressions.