Frequency Domain View of Downsampling

A Text slecture by ECE David Klouda

Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.

## Outline

1. Introduction
2. Derivation
3. Example
4. Conclusion

## 1. Introduction

In this slecture, the Frequency Domain view of Downsampling will be discussed. It will begin with the derivation of the formulas and explaining the terms involved. It will then show an example using the DTFT and finish with an explanation as to why filtering is necessary when decimating.

## 2. Derivation

Let there be two discrete signals $x_1[n]$ and $x_2[n]$ with $x_2$ being a scalar multiple D of $x_1$

Taking the Discrete Time Fourier Transform of each we get

\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}

substitute using $m= D*n$ resulting with:

$\sum_{m = -\infty}^\infty x_1[m] e^{-j \omega {\frac{m}{D}}}$

Introduce $S_D[m]$ which is a piece wise function that equals 1 at multiples of D:

$\mathcal{F }\left \{ x_2[m] \right \} = \sum_{m = -\infty}^\infty S_D[m]*x_1[m]e^{-j\omega n}$

The Fourier representation $S_D = {\frac{1}{D}} \sum_{k = 0}^{D-1} e^{j\omega \frac{2\pi}{D}mk}$

Therefore: $\mathcal{F }\left \{ x_2[n] \right \} = {\frac{1}{D}} \sum_{k = 0}^{D-1} \sum_{m=-\infty}^\infty X_1[m]e^{-jm{\frac{w-2\pi k}{D}}}$

The segment within the m sum reduces to $X_1 ({\frac{w-2\pi k}{D}})$

Which leads to ${\frac{1}{D}} \sum_{k = 0}^{D-1} X_1 ({\frac{w-2\pi k}{D}})$

## 4. Conclusion

This slecture demonstrated the use of downsampling as seen from the Fourier domain. It showed that if a signal is below a certain threshold, then it must be filtered before downsampling to eliminate the possibility of aliasing and distorting the reconstructed signal.

NOTE:

Downsampling is always equivalent to resampling with the larger period.

Begin with x(t) as a continuous time signal with $x_1[n]= x(T_1*n)$ being its discrete time sampling.

Let $x_2[n]=x(T_2*n)=x_1[T_2/T_1*n]$

with Downsampling factor $D=T_2/T_1$

This lets us define $x_2$ as $x_1[D*n]$

In order to prevent Aliasing within this context you need to have $D*2\pi*T_1*f_{MAX} < \pi$

${T_2/T_1}*2\pi*T_1*f_{MAX} < \pi$

$2\pi*T_2*f_{MAX} < \pi$

$f_{MAX} < 1/{2*T_2}$

If $f_{MAX} > 1/{2*T_2}$ is true, then you must use a low-pass filter before downsampling.