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| − | <center | + | <center> |
<font size="4">DTFT of a Cosine Signal Sampled Above and Below the Nyquist Frequency </font> | <font size="4">DTFT of a Cosine Signal Sampled Above and Below the Nyquist Frequency </font> | ||
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We will represent this tone as a cosine signal, <math>cos*(2\pi349t)</math> | We will represent this tone as a cosine signal, <math>cos*(2\pi349t)</math> | ||
| − | |||
| + | ---- | ||
| + | <font size="4">Sampling <u>Above</u> Nyquist Frequency</font> | ||
| + | |||
| + | For this signal <math>f_{s} > 2f_{m} = 2(349)Hz = 698Hz</math> or else aliasing will occur. We will choose a sampling frequency of <math>1000Hz = 1/T_{1}</math> | ||
| + | |||
| + | <math> | ||
| + | \begin{align} | ||
| + | x_{1}(n) &= x(nT_{1}) \\ | ||
| + | |||
| + | &= cos(2\pi349nT_{1}) \\ | ||
| + | |||
| + | &= cos(\frac{2\pi349n}{1000}) \\ | ||
| + | |||
| + | &= \frac{1}{2}(e^{\frac{-j2\pi349n}{1000}} + e^{\frac{j2\pi349n}{1000}}) \\ | ||
| + | |||
| + | \\ | ||
| + | \\ | ||
| + | \end{align} | ||
| + | </math> | ||
| + | |||
| + | <math>Note\ that:\ 0 < |\pm2\pi\frac{349}{1000}| < \pi </math> | ||
| + | |||
| + | This means the original signal can be properly represented when sampled at <math>f_{s} = 1000Hz.</math> | ||
| + | |||
| + | Using the discrete-time Fourier transform pair for cosine: | ||
| + | |||
| + | <math>cos(\omega_{0}n) \ \ \ \ -----> \ \ \ \ \pi\sum_{n=-\infty}^\infty (\delta(\omega - \omega_{0}) + \delta(\omega + \omega_{0})) </math> | ||
| + | <math>X_{1}(\omega) = rep_{2\pi}\pi (\delta(\omega - 2\pi\frac{349}{1000}) + \delta(\omega + 2\pi\frac{349}{1000}))</math> | ||
| + | |||
| + | Plot of <math>X_{1}(\omega)</math>: | ||
| + | |||
| + | <font size="4">Sampling <u>Below</u> Nyquist Frequency</font> | ||
| + | For this signal <math>f_{s} > 2f_{m} = 2(349)Hz = 698Hz</math> or else aliasing will occur. We will choose a sampling frequency of <math>1000Hz = 1/T_{1}</math> | ||
| + | |||
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| Line 41: | Line 75: | ||
<math>Note\ that:\ 0 < |\pm2\pi\frac{349}{1000}| < \pi </math> | <math>Note\ that:\ 0 < |\pm2\pi\frac{349}{1000}| < \pi </math> | ||
| − | This means the original signal can be properly represented when sampled at | + | This means the original signal can be properly represented when sampled at <math>f_{s} = 1000Hz.</math> |
<math>X_{1}(\omega) = rep_{2\pi}\pi (\delta(\omega - 2\pi\frac{349}{1000}) + \delta(\omega + 2\pi\frac{349}{1000}))</math> | <math>X_{1}(\omega) = rep_{2\pi}\pi (\delta(\omega - 2\pi\frac{349}{1000}) + \delta(\omega + 2\pi\frac{349}{1000}))</math> | ||
| + | |||
| + | Plot of <math>X_{1}(\omega)</math>: | ||
Revision as of 16:14, 2 October 2014
DTFT of a Cosine Signal Sampled Above and Below the Nyquist Frequency
A slecture by ECE student Andrew Pawling
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
In this slecture we will look at an example that illustrates the Nyquist condition. When a signal is sampled, frequencies above half the sampling rate cannot be properly represented and result in aliasing.
Lets look at a pure tone frequency F4 = 349Hz
We will represent this tone as a cosine signal, $ cos*(2\pi349t) $
Sampling Above Nyquist Frequency
For this signal $ f_{s} > 2f_{m} = 2(349)Hz = 698Hz $ or else aliasing will occur. We will choose a sampling frequency of $ 1000Hz = 1/T_{1} $
$ \begin{align} x_{1}(n) &= x(nT_{1}) \\ &= cos(2\pi349nT_{1}) \\ &= cos(\frac{2\pi349n}{1000}) \\ &= \frac{1}{2}(e^{\frac{-j2\pi349n}{1000}} + e^{\frac{j2\pi349n}{1000}}) \\ \\ \\ \end{align} $
$ Note\ that:\ 0 < |\pm2\pi\frac{349}{1000}| < \pi $
This means the original signal can be properly represented when sampled at $ f_{s} = 1000Hz. $
Using the discrete-time Fourier transform pair for cosine:
$ cos(\omega_{0}n) \ \ \ \ -----> \ \ \ \ \pi\sum_{n=-\infty}^\infty (\delta(\omega - \omega_{0}) + \delta(\omega + \omega_{0})) $ $ X_{1}(\omega) = rep_{2\pi}\pi (\delta(\omega - 2\pi\frac{349}{1000}) + \delta(\omega + 2\pi\frac{349}{1000})) $
Plot of $ X_{1}(\omega) $:
Sampling Below Nyquist Frequency
For this signal $ f_{s} > 2f_{m} = 2(349)Hz = 698Hz $ or else aliasing will occur. We will choose a sampling frequency of $ 1000Hz = 1/T_{1} $
$ \begin{align} x_{1}(n) &= x(nT_{1}) \\ &= cos(2\pi349nT_{1}) \\ &= cos(\frac{2\pi349n}{1000}) \\ &= \frac{1}{2}(e^{\frac{-j2\pi349n}{1000}} + e^{\frac{j2\pi349n}{1000}}) \\ \\ \\ \end{align} $
$ Note\ that:\ 0 < |\pm2\pi\frac{349}{1000}| < \pi $
This means the original signal can be properly represented when sampled at $ f_{s} = 1000Hz. $
$ X_{1}(\omega) = rep_{2\pi}\pi (\delta(\omega - 2\pi\frac{349}{1000}) + \delta(\omega + 2\pi\frac{349}{1000})) $
Plot of $ X_{1}(\omega) $:
