Convolution Theorem - Functions of A Discrete Variable... Sequences

Functions of A Discrete Variable... Sequences

By similar arguments, it can be shown that the discrete convolution of sequences and is given by:



where DTFT represents the discrete-time Fourier transform.

An important special case is the circular convolution of and defined by where is a periodic summation:

It can then be shown that:

\begin{align} x_N * y\ &=\ \scriptstyle{DTFT}^{-1} \displaystyle \big\\ &=\ \scriptstyle{DFT}^{-1} \displaystyle \big, \end{align}

where DFT represents the discrete Fourier transform.

The proof follows from DTFT#Periodic_data, which indicates that can be written as:

The product with is thereby reduced to a discrete-frequency function:

(also using Sampling the DTFT).

The inverse DTFT is:

\begin{align} (x_N * y)\ &=\ \int_{0}^{1} \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\ &=\ \frac{1}{N} \sum_{k=-\infty}^{\infty} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot \int_{0}^{1} \delta\left(f-k/N\right)\cdot e^{i 2 \pi f n} df\\ &=\ \frac{1}{N} \sum_{k=0}^{N-1} \scriptstyle{DFT}\displaystyle\{x_N\}\cdot \scriptstyle{DFT}\displaystyle\{y_N\}\cdot e^{i 2 \pi \frac{n}{N} k}\\ &=\ \scriptstyle{DFT}^{-1} \displaystyle \big, \end{align}

QED.

Read more about this topic:  Convolution Theorem

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