Higher powers in the discrete Plancherel theorem

Question

For a function $f\colon \mathbb{Z}/n\mathbb{Z} \to \mathbb{C}$, the Plancherel Theorem for the discrete Fourier transform states that $$ \sum_{a = 1}^n \lvert f(a)\rvert^2 = \frac{1}{n} \sum_{\chi \text{ mod } n} \lvert \hat{f}(\chi)\rvert^2. $$ If we set $g(a) = \lvert f(a)\rvert^2$, then applying Plancherel to $g$ gives $$ \sum_{a = 1}^n \lvert f(a)\rvert^4 = \frac{1}{n} \sum_{\chi \text{ mod } n} \lvert \hat{g}(\chi)\rvert^2. $$ However, is there a way to relate the sum of fourth powers directly back to the Fourier transform of $f$? Is there an analogous formula that generalizes the Plancherel Theorem for the sum of the $2m$th powers?

Answer 1

Just apply the DFT to the squares of the original function values- they will become (up to constant factors) the circular autocorrelation of the DFT of the original function.