Discrete-time Fourier transform (DTFT) of a sampled cosine
A slecture by ECE student Yijun Han
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Contents
outline
- Introduction
- Sampling rate above Nyquist rate
- Sampling rate below Nyquist rate
- Conclusion
- References
Introduction
\qquad Consider a CT cosine signal (a pure frequency), and sample that signal with a rate above or below Nyquist rate. In this slecture, I will talk about how does the discrete-time Fourier transform of the sampling of this signal look like. Suppose the cosine signal is $ x(t)=cos(2pi*440t) $.
Sampling rate above Nyquist rate
The Nyquist sampling rate $ fs=2fM=880 $,so we pick a sample frequency 1000 which is above the Nyquist rate.
$ x1[n]=x(n/1000) $ $ x1[n]=cos(2pi*440*n/1000) $ $ x1[n]=1/2(e^(j2pi440*n/1000)+e^(-j2pi440*n/1000)) $
Sampling rate below Nyquist rate
Conclusion
References
[1].Mireille Boutin, "ECE438 Digital Signal Processing with Applications," Purdue University August 26,2009
Questions and comments
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