GATE EC 2025 Set 1 — Question 38
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Signals & Systems → Continuous-Time Signals → Aliasing & Reconstruction
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Question
Consider a continuous-time finite-energy signal whose Fourier transform vanishes outside the frequency interval , where is in rad/sec.The signal is uniformly sampled to obtain . Here,with being the Dirac impulse, , and . The sampled signal is passed through an ideal lowpass filter with cutoff frequency and passband gain .The output of the filter is given by ________________.
Correct answer
(A) f(t) if Tₛ < π/ω_c
Solution
1.Identify the Sampling Process:
The signal is bandlimited to . It is sampled by a shifted impulse train . The sampled signal is:2.Determine the Filter Output:
The output is the convolution of with the impulse response :3.Apply the Given Filter:
The filter is .At the Nyquist limit , the filter becomes .
Substituting this into the output expression:
4.Reconstruction Principle:
According to the Whittaker-Shannon interpolation formula, a bandlimited signal can be perfectly reconstructed from its samples taken at using the formula:Since our filter output matches this interpolation structure, .
5.Nyquist Criterion:
For perfect reconstruction without aliasing, the sampling frequency must satisfy . Thus, the output is if .
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