GATE EC 2021 Set 1 — Question 15
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Question
Consider two -point sequences and . Let the linear convolution of and be denoted by , while denotes the -point inverse discrete Fourier transform (IDFT) of the product of the -point DFTs of and . The value(s) of for which is/are
Correct answer
(C) k = 15
Solution
Let and be -point sequences (). Their linear convolution has length . The sequence is the -point IDFT of the product of -point DFTs, which is equivalent to the -point circular convolution of and .The relationship between -point circular convolution and linear convolution is:For , this simplifies to since is only defined for .We want , which implies . Since is non-zero for , only if . Given , the only possible value is .
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