Digital Signal Processing 2025

(a) Non-parametric spectral estimation uses:

1. Autocorrelation function
2. ARMA model
3. Data modeling
4. Linear prediction

(b) Compute Z-transform of δ[n − 2]

1. z⁻²
2. 1
3. z²
4. None

(c) Find the Nyquist rate for a continuous-time signal

x(t) = sin(200πt) + cos(600πt)

1. 100 Hz
2. 200 Hz
3. 600 Hz
4. 300 Hz

(d) For a signal x(n) = {1,2,3}, what is the energy of the signal?

1. 6
2. 14
3. 12
4. 9

(e) For a system described by y(n) = x(n) + 2x(n − 1), what is the output for x(n) = {1,2,3}?

1. {1,4,7}
2. {1,2,3}
3. {3,4,1}
4. {2,4,6}

(f) For a system with transfer function H(z) = 1/(1 − 0.5z⁻¹), what is the impulse response?

1. 0.5ⁿu(n)
2. 0.5⁽ⁿ⁻¹⁾u(n − 1)
3. 2ⁿu(n)
4. 2⁽ⁿ⁻¹⁾u(n − 1)

(g) The system H(z) = 1/(1 − 0.5z⁻¹) is causal if:

1. ROC is |z| > 0.5
2. ROC is |z| < 0.5
3. ROC is |z| > 1
4. ROC is |z| < 1

(h) What is the impulse response duration of an IIR filter?

1. Finite
2. Infinite
3. Zero
4. One sample

(i) The DFT of x(n) = {1,2,3,4} is:

1. {10, -2+2j, -2, -2-2j}
2. {10, 2, 2, 2}
3. {10, -2-2j, -2, -2+2j}
4. {10, -2, -2, -2}

(j) The IDFT of X(k) = {3,0,0} is:

1. {1,1,1}
2. {3,0,0}
3. {1,0,0}
4. {0,0,1}

(a) Find IDFT of X[k] = {4, -2j, 0, 2j}.

(b) Solve the following difference equation using the unilateral Z-transform:

with initial conditions

and

(a) Compare the magnitude responses of FIR and IIR filters designed for the same frequencies.

(b) Compute the cross-correlation between x[n] = {1,2,3} and y[n] = {3,2,1}.

(a) Explain the effects of finite register length in FIR filter implementations. Discuss quantization of filter coefficients, round-off noise, accumulator overflow on filter stability and frequency response.

(b) Given two sequences x₁[n] = u[n] and x₂[n] = (-1)^n u[n], find their orthogonal representation and calculate their inner product for n = 0 to 4.

(a) A linear shift-invariant system is described by the difference equation

with

Find:

1. The natural response of the system.
2. The forced response of the system for a unit step input.

(b) Find the linear convolution of the sequences x(n) = {1,2} and h(n) = {2,1} using DFT.

(a) Given an input x[n] = cos(0.2πn) to a system with y[n] = x[n] - 0.9x[n-1], find the output response.

(b) Design a Butterworth digital filter using the bilinear transformation. The specifications of the desired low-pass filter are:

with T = 1 sec.

(a) Apply the FFT radix-2 algorithm to compute the DFT of

x[n] = {1,2,3,4,4,3,2,1}

(b) Design a digital IIR low-pass filter with a passband edge at 1000 Hz and a stopband edge at 1500 Hz for a sampling frequency of 5000 Hz. The filter is to have a passband ripple of 0.5 dB and a stopband ripple below 30 dB. Design a Butterworth filter using bilinear transformation.

(a) Design a Chebyshev IIR digital low-pass filter to satisfy the constraints

(b) Given signal power 5 W and noise power 0.5 W, find the output SNR improvement after Wiener filtering.

Write short notes on any two of the following:

(a) Properties of IDFT

(b) Sampling theorem

(c) Properties of Z-transform

(d) Auto-correlation and cross-correlation