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}_2(\omega) &= \mathcal{F }\left \{ x_2[n] \right \} = \mathcal{F }\left \{ x_1[Dn] \right \}\\ &= \sum_{n=-\infty}^\infty x_1[Dn]e^{-j\omega n} \end{align} $
make the substitution of $ n=\frac{m}{D} $

$ \begin{align} \mathcal{X}_2(\omega) &= \sum_{m=-\infty}^\infty x_1[m]e^{-j\omega \frac{m}{D}} \end{align} $

Downsampled signal will only be nonzero for m equal to multiples of D so:
$ \begin{align} \mathcal{X}_2(\omega) &= \sum_{m=-\infty}^\infty s_d[m]x_1[m]e^{-j\omega \frac{m}{D}} \text{ where}\\ &s_d[m] = \frac{1}{D}\sum_{k=0}^{D-1} e^{jk2\pi \frac{m}{D}} \end{align} $

Example

Conclusion