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The DFT of a vector [a b c d] is the vector [p q r s]. Consider the product [p q r s] = [a b c d] [ a b c d; d a b c; c d a b; b c d a ]. The DFT of the vector [p q r s] is a scaled version of

1. [a² b² c² d²]
2. [√a √b √c √d]
3. [a b c d a b c d]
4. [a b c d]

Correct answer: [a b c d a b c d]

Solution

The correct option is [a b c d a b c d] because the product of the vector [a b c d] with the given matrix results in a vector that effectively replicates the original input, leading to a DFT that retains the periodicity and structure of the original vector, thus forming a scaled version of the extended vector.

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