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.