Downsampling

A slecture by ECE student Yerkebulan Yeshmukhanbetov

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

## Contents

## Outline

- Introduction
- Definition of Downsampling

- Derivation of DTFT of downsampled signal

- Example
- Decimator
- 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 tim*e *signal. Then *x _{1} [n] = x (T_{1}n) *and

*x*. And ratio of sampling periods would be

_{2}[n] = x (T_{2}n)D = T_{2}/T_{1}, which is an integer greater than 1. From these equations we obtain realtionship between *x _{1} [n]* and

*x*.

_{2}[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 *x _{2} [n]* in terms of DTFT of

*x*.

_{1}[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 *X _{1} (w)* and

*X*which is obtained after downsampling X

_{2}(w)_{1}(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 f_{max} > 1/(2T_{2}) . 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.

## Questions and comments

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