Frequency domain view of the relationship between a signal and a sampling of that signal
A slecture by ECE student Botao Chen
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Outline
- Introduction
- Derivation
- Example
- Conclusion
Introduction
In this slecture I will discuss about the relations between the original signal $ X(f) $ (the CTFT of $ x(t) $ ), sampling continuous time signal $ X_s(f) $ (the CTFT of $ x_s(t) $ ) and sampling discrete time signal $ X_d(\omega) $ (the DTFT of $ x_d[n] $ ) in frequency domain and give a specific example showing the relations.
Derivation
The first thing which need to be clarified is that there two different types of sampling signal: $ x_s(t) $ and $ x_d[n] $. $ x_s(t) $ is created by multiplying a impulse train $ P_T(t) $ with the original signal $ x(t) $ and actually $ x_s(t) $ is $ comb_T(x(t)) $ where T is the sampling period. However the $ x_d[n] $ is $ x(nT) $ where T is the sampling period.
Now we first concentrate on the relationship between $ X(f) $ and $ X_s(f) $.
As I say just now, $ x_s(t) = x(t)*P_T(t) $ Given that $ \mathcal{F}(x(t)) = X(f) $, we can find $ X_s(f) $ using the convolution property.
$ \begin{align} X_s(f) &= X(f)*\mathcal{F}(p_{\frac{1}{f_s}})\\ &= X(f)*\mathcal{F}(\sum_{k = -\infty}^\infty \delta(t-\frac{k}{f_s}))\\ &= X(f)*f_s\sum_{k = -\infty}^\infty \delta(f-kf_s)\\ &= f_s\sum_{k = -\infty}^\infty X(f)*\delta(t-\frac{k}{f_s})\\ &= f_s\sum_{k = -\infty}^\infty X(f-kf_s)\\ \end{align} $
