Frequency Downsampling - Rhea

Introduction

Remember for time domain, Downsampling is defined as:

Image1

Now let's describe this process in the frequency domain.

Derivation

First we'll take the Discrete Time Fourier Transform of the original signal and the downsampled version of it.
$ \begin{align} \mathcal{X}(\omega) &= \mathcal{F }\left \{ x_2[n] \right \} = \mathcal{F }\left \{ x_1[Dn] \right \}\\ &= \sum_{n=-\infty}^\infty x_1[Dn]e^{-j2\omega f} \end{align} $
make the substitution of $ n=\frac{m}{\D} $

Example

Conclusion