Question:

Consider two 16-point sequences \(x[n]\) and \(h[n]\). Let the linear convolution of \(x[n]\) and \(h[n]\) be denoted by \(y[n]\), while \(z[n]\) denotes the 16-point inverse discrete Fourier transform (IDFT) of the product of the 16-point DFTs of \(x[n]\) and \(h[n]\). The value(s) of \(k\) for which \(z[k] = y[k]\) is/are

Show Hint

When performing linear and circular convolutions, the DFT and IDFT can be used to convert between the two, with the appropriate lengths of sequences.
Updated On: Dec 26, 2025
  • \( k = 0, 1, 2, \dots, 15 \)
  • \( k = 0 \)
  • \( k = 15 \)
  • \( k = 0 \) and \( k = 15 \)
Hide Solution
collegedunia
Verified By Collegedunia

The Correct Option is C

Solution and Explanation

The problem involves the linear convolution of two sequences \(x[n]\) and \(h[n]\). In this case, we are given that \(z[n]\) is the inverse discrete Fourier transform (IDFT) of the product of the discrete Fourier transforms (DFTs) of \(x[n]\) and \(h[n]\). From the properties of DFT and IDFT, we know that the circular convolution in the frequency domain (the product of DFTs) corresponds to the linear convolution in the time domain when the lengths of the sequences are equal. Thus, the value of \(k\) for which \(z[k] = y[k]\) occurs when \(k = 15\), which is the highest index for the 16-point sequences. Final Answer: \( k = 15 \)
Was this answer helpful?
0
0

Questions Asked in GATE EC exam

View More Questions