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Exercise 4.1.1

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$w = e^{2 πi ∕ N}$ and $ω = e^{- 2 πi ∕ N} = \bar{w}$ , so we have the complex dot product between the $p$ th row of $F$ and $q$ th column of $Ω$ , we have

\begin{array}{l} {\bar{p}}^{T} q & = 1 + {\bar{w}}^{p} ω^{q} + \dots + {\bar{w}}^{(N - 1) p} ω^{(N - 1) q} \\ = 1 + ω^{p} ω^{q} + \dots + ω^{(N - 1) p} ω^{(N - 1) q} \\ = 1 + ω^{p + q} + \dots + ω^{(N - 1) (p + q)} \\ = \frac{1 - ω^{N (p + q)}}{1 - ω} \\ = \frac{1 - e^{- 2 πi (p + q)}}{1 - ω} \\ = 0 \end{array}

niuers

2020-03-20 00:00

Exercise 4.1.1

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