Downsampling

A slecture by ECE student Yerkebulan Yeshmukhanbetov

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

## Outline

1. Introduction
2. Definition of Downsampling
3. Derivation of DTFT of downsampled signal
4. Example
5. Decimator
6. Conclusion

## Introduction

This slecture provides definition of downsampling, derives DTFT of  downsampled signal and demonstrates it in a frequency domain. Also, it explains process of decimation and why it needs a low-pass filter.

## Definition of Downsampling

Downsampling is an operation which involves throwing away samples from discrete-time signal. Let  x[n] be a digital-time signal shown below:

then y[n] will be produced by downsampling x [n]  by factor D = 3. So, y [n] = x[Dn].

As seen in above graph, y [n] is obtained by throwing away some samples from x [n]. So, y [n] is a downsampled signal from

x [n].

## Derivation of DTFT of downsampled signal

Let x (t) be a continuous time signal. Then x1 [n] = x (T1n) and  x2 [n] = x (T2n). And ratio of sampling periods would be

D = T2/T1,   which is an integer greater than 1. From these equations we obtain realtionship between x1 [n] and x2 [n].

\begin{align} x_2 [n] = x(T_2 n) = x(DT_1 n) = x_1 [nD] \end{align}

Below we derive Discrete-Time Fourier Transform of x2 [n] in terms of DTFT of x1 [n].

\begin{align} &\mathcal{X}_2(\omega)= \mathcal{F}(x_2 [n]) = \mathcal{F}(x_1 [Dn])\\ &= \sum_{n = -\infty}^\infty x_1[Dn] e^{-j \omega n} = \sum_{m = -\infty}^\infty x_1[m] e^{-j \omega {\frac{m}{D}}}\\ &= \sum_{n = -\infty}^\infty s_D[m]* x_1 [m] e^{-j \omega {\frac{m}{D}}}\\ \end{align}

where

$s_D [m]=\left\{ \begin{array}{ll} 1,& \text{ if } n \text{ is a multiple of } D,\\ 0, & \text{ else}. \end{array}\right. = {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}}$

\begin{align} &\mathcal{X}_2(\omega)= \sum_{m = -\infty}^\infty {\frac{1}{D}} \sum_{k = -\infty}^{D-1} e^{jk {\frac{2 \pi}{D} m}} x_1[m] e^{-j \omega {\frac{m}{D}}}\\ &= {\frac{1}{D}} \sum_{k = -\infty}^{D-1} \sum_{m = -\infty}^\infty x_1[m] e^{-jm ({\frac{\omega - 2 \pi k}{D}})} = \\ &= {\frac{1}{D}} \sum_{k = -\infty}^{D-1} \mathcal{X}_1 ({\frac{\omega - 2 \pi k}{D}}) \\ \end{align}

## Example

Let's take a look  at  an original signal X1 (w) and  X2 (w) which is obtained after downsampling X1(w) by factor D = 2 in a frequency domain.

From two graphs it is seen that signal is stretched by D  in frequency domain and  decreased by D in a magnitude after downsampling. Both signals have the frequency of \begin{align} 2\pi \end{align} .

## Decimator

As seen in second graph, if \begin{align} D2\pi T_1f_{max} \end{align} is greater than \begin{align} \pi \end{align} aliasing occurs. Downsampler is a part of a decimator which also has a low-pass filter to  prevent aliasing.  LPF eliminates signal components which has  frequencies higher than cutoff frequency, which can be found from graphs shown above.

\begin{align} & D\omega_c = D 2 \pi T_1 f_{max} < \pi\\ & {\frac{T_2}{T_1}} 2\pi T_1 f_{max} < \pi \\ & 2\pi T_2f_{max} < \pi \\ &f_{max} < {\frac{1}{2T_2}} \end{align}

Thereby, signal needs to be filtered before downsampling if fmax > 1/(2T2) . Complete block diagram of a decimator is shown below:

## Conclusion

To summarize, downsampling is a process of removing samples from signal. After downsampling,  signal decreases by factor D in the magnitude and stretches by D in frequency domain.  In order to downsample a signal, it first should be filtered by LPF to prevent aliasing.  Both LPF and downsampler are parts of a decimator.