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Let the discrete Fourier transform be $$ \mathcal{F}_N\mathbf{a}=\hat{\mathbf{a}},\quad \hat{a}_m=\sum_{n=0}^{N-1}e^{-2\pi i m n/N}a_n $$

and let the discrete convolution be $$ (\mathbf{a}*\mathbf{b})_n=\sum_{k=0}^{N-1} a_k b_{n-k} $$

where $n$ and $k$ are taken to be integers modulo $N$. Prove that $$ \mathcal{F}_N(\mathbf{a}*\mathbf{b})=(\hat{a}_0\hat{b}_0,...,\hat{a}_{N-1}\hat{b}_{N-1}). $$

My book leaves it to the reader to do this proof since it is supposedly simple, alas I can't figure it out. I tried to substitute the expression of the convolution into the expression of the discrete Fourier transform and writing out a few terms of that, but it didn't leave me any wiser.

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1 Answer 1

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First, let me say that you can show this pretty easily be writing the convolution as a matrix vector product, with the matrix being a circulant one defined by one of the vectors. The result falls out due to the DFT diagnolizing circulant matrices.

Anyway, you can also show this directly substituting the discrete convolution formula, and playing with indices. We need one result before we do this, however. I'm assuming from here out that we're in $\mathbb{R}^N$, and that addition is mod $N$.

Proposition

Let $x^m$ be the vector $x$ shifted by $m$, i.e., $x^m_j = x_{m+j}$. Then $\langle x, y \rangle = \langle x^m, y^m \rangle$.

Note, this says that

$$ \langle x, y \rangle = \sum_{j=0}^{N-1} x_j y_j = \sum_{j=-m}^{N-1-m} x_{j+m} y_{j+m} = \sum_{j=0}^{N-1} x_{j+m} y_{j+m} = \langle x^m, y^m \rangle $$

proof

Without loss of generality, assume $m \in \{0, 1, \ldots N-1\}$. Let $P^m$ be the identity matrix whose rows are circularly permuted by $m$.

To be precise, rows $m, m+1,\ldots, N-1$ of $P^m$ are the rows $0, 1, \ldots, N-1-m$ rows of the identity, while rows $0,1, \ldots, m-1$ are rows $N-m, \ldots, N-1$ of the identity.

For example, $$ P^2 = \begin{bmatrix} e_{N-2} \\ e_{N-1} \\ e_{0} \\ e_{1} \\ \vdots \\ e_{N-3} \end{bmatrix} $$ where $e_j$ is the $j$th row of the identity.

It should be clear that $x^m = P^m x$. Furthermore, $P^m$ is unitary so it preserves inner products, i.e. $\langle x, y \rangle = \langle P^mx, P^my \rangle$. $$\tag*{$\blacksquare$}$$

This should be intuitively clear. Since we're working with mod $N$, the shift just rearranges the terms in the sum circularly, not changing the overall value of the sum.

Now, back to the main result. Let's call $c_n = (\mathbf{a}*\mathbf{b})_n$.

\begin{align*} \hat{c}_m &= \sum_{n=0}^{N-1} \exp(-2\pi imn / N) c_n \\ &= \sum_{n=0}^{N-1} \exp(-2\pi imn / N) \sum_{k=0}^{N-1} a_kb_{n-k} \\ &= \sum_{k=0}^{N-1} a_k \sum_{n=0}^{N-1} \exp(-2\pi imn / N) b_{n-k} \end{align*}

Now, let $n'=n-k$. So,

\begin{align*} \hat{c}_m &= \sum_{k=0}^{N-1} a_k \sum_{n'=-k}^{N-1-k} \exp(-2\pi im(n'+k) / N) b_{n'} \\ &= \sum_{k=0}^{N-1} a_k \exp(-2\pi imk / N) \sum_{n'=-k}^{N-1-k} \exp(-2\pi imn' / N) b_{n'} \\ &= \left( \sum_{k=0}^{N-1} a_k \exp(-2\pi imk / N) \right) \left(\sum_{n=0}^{N-1} \exp(-2\pi imn / N) b_{n} \right) \\ &= \hat{a}_m \hat{b}_m \end{align*} where the second to last line is due to the above proposition.

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