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.
Contents
Outline
- Introduction
- Derivation
- Example
- 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
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] $
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} $
3. Example
In order to prevent Aliasing 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.
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.
Questions and comments
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