Question

Which of the following is/are CORRECT when using the Discrete Fourier Transform to compute convolution without zero padding?

• A. Aliasing occurs
• B. Wraparound error occurs
• C. Convolution is linear
• D. Aliasing can be avoided

Hint:

Without zero padding, DFT multiplication computes circular, not linear, convolution.

Answer 1

The Correct Option is A, B

Step 1: Recall what the DFT actually computes.
The Discrete Fourier Transform (DFT) treats any finite-length sequence as one period of an infinite periodic sequence. So when two sequences \(x[n]\) of length \(N_1\) and \(h[n]\) of length \(N_2\) are multiplied in the DFT domain and inverse transformed, the result is not the ordinary (linear) convolution \(x[n]*h[n]\); it is the circular convolution of the two sequences, computed over a length equal to the DFT size used, which here (with no zero padding) is just the length of the sequences themselves.
Step 2: Explain why this produces wraparound and aliasing.
Circular convolution wraps the tail of the true linear convolution result around to the front of the output block, because the DFT implicitly assumes the sequences repeat periodically outside their given length. This wrap-around contamination of samples is exactly what is called wraparound error, and because it corrupts the correct non-periodic result with overlapping periodic replicas, it is also described as a form of aliasing in the discrete convolution result.
Step 3: Check option (C), Convolution is linear.
Here this option is testing whether the DFT-based product equals ordinary linear convolution. Without zero padding, the DFT computes circular convolution, which is generally not equal to the linear convolution of the two sequences. So the operation performed here does not give the linear convolution result, and option (C) is not a correct description of what happens without zero padding.
Step 4: Check option (D), Aliasing can be avoided.
Aliasing and wraparound in DFT-based convolution are avoided only by zero-padding both sequences to a length of at least \(N_1+N_2-1\) before transforming. The question explicitly states that no zero padding is used, so under this stated condition aliasing cannot be avoided; option (D) is false for the scenario described.
Step 5: Conclusion.
Aliasing and wraparound error are the direct, unavoidable consequence of computing convolution via the DFT without zero padding, so (A) and (B) are correct, while (C) and (D) are not.\[ \boxed{\text{(A) and (B) are correct}} \]