Discussion
The multiplication of two discrete Fourier transforms (DFTs) is equal to the ............... of two sequences in the time domain.
Answer: A
Multiplying two DFTs corresponds to circular convolution in the time domain.
The N-point DFT treats a sequence as one period of a periodic signal. As a result, if \(X(k)\) and \(H(k)\) are N-point DFTs of \(x(n)\) and \(h(n)\), then \(X(k)H(k)\) is the DFT of their N-point circular convolution, in which indices wrap around modulo N.
Linear convolution is obtained this way only after zero-padding both sequences to at least \(L + M - 1\) points, which is how fast convolution uses the FFT. Correlation corresponds to multiplying one DFT by the complex conjugate of the other, not by the DFT itself.
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