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| − | + | <center><font size="5"></font> | |
| − | + | <font size="5">DTFT of a Cosine Sampled Above and Below the Nyquist Rate </font> | |
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| − | + | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Sahil Sanghani | |
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| − | + | Partly based on the [[2014 Fall ECE 438 Boutin|ECE438 Fall 2014 lecture]] material of [[User:Mboutin|Prof. Mireille Boutin]]. | |
| − | + | </center> | |
| − | Partly based on the [[ | + | |
| − | </center> | + | |
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| − | == Outline == | + | == Outline == |
| − | * Introduction | + | |
| − | * Useful Background | + | *Introduction |
| − | * DTFT Example of a Cosine Sampled Above the Nyquist Rate | + | *Useful Background |
| − | * DTFT Example of a Cosine Sampled Below the Nyquist Rate | + | *DTFT Example of a Cosine Sampled Above the Nyquist Rate |
| − | * Conclusion | + | *DTFT Example of a Cosine Sampled Below the Nyquist Rate |
| − | * References | + | *Conclusion |
| + | *References | ||
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| − | == Introduction == | + | == Introduction == |
| − | In this Slecture, I will walk you through taking the DTFT of a pure frequency sampled above and below the Nyquist Rate. Then I will compare the differences between them. | + | |
| + | In this Slecture, I will walk you through taking the DTFT of a pure frequency sampled above and below the Nyquist Rate. Then I will compare the differences between them. | ||
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| − | == | + | == Useful Background == |
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| − | + | Nyquist Condition: <span class="texhtml">''f''<sub>''s''</sub> = 2 * ''f''<sub>''m''''a''''x''</sub></span> DTFT of a Cosine: <font size="2"><math>x_d[n] = cos(2\pi nT){\leftrightarrow}X(\omega) = \pi(\delta(\omega-\omega_o) + \delta(\omega+\omega_o)){ ,for\ } \omega \in [-\pi,\pi]</math></font> | |
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| + | <font size="2">The DTFT of a sampled signal is periodic with <span class="texhtml">2π</span>.</font> | ||
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| + | <font size="2"> | ||
| + | == DTFT of a Cosine Sampled Above the Nyquist Rate == | ||
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| + | For our original pure frequency, let’s choose the E below middle C. The E occurs at 330''Hz'' <span class="texhtml">''x''(''t'') = cos(2π * 330''t'')</span> <p>Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is: <br><span class="math"> ''f''<sub>''s''</sub> = 2 * ''f''<sub>''m''''a''''x''</sub> = 2 * (330''H''''z'') = 660''H''''z''</span>. <br>Let’s sample at 990<span class="math">''H''''z''</span>. | ||
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| + | <math> \begin{align} \\ | ||
| + | x_d[n] & = x(n*\frac{1}{990Hz})\\ | ||
| + | & = cos(2\pi n *\frac{330}{990}) = \frac{e^{j2\pi n \frac{330}{550}} + e^{-j2\pi n \frac{330}{550}}}{2}\\ | ||
| + | & = cos(\frac{2\pi n}{3}) | ||
| + | \end{align} | ||
| + | </math> | ||
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| + | Because <font style="vertical-align:-150%;"><math>\left | \frac{2\pi}{3}\right | < \pi</math></font>, there is no aliasing occurring in the DTFT, and it can be written as follows: | ||
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| + | <math> \begin{align} \\ | ||
| + | X(\omega) & = \frac{1}{2}(2\pi\delta(\omega - 2\pi \frac{330}{990}) + 2\pi\delta(\omega + 2\pi \frac{330}{990})) , \ \omega \in\ [-\pi,\pi]\\ | ||
| + | & = \frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330)) , \ \omega \in\ [-\pi,\pi]\\ | ||
| + | & = rep_{2\pi}(\frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330))), \forall \omega | ||
| + | \end{align} | ||
| + | </math> | ||
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| + | [[2014 Fall ECE 438 Boutin|Back to ECE438, Fall 2014]] | ||
| − | [[ | + | [[Category:Slecture]] [[Category:ECE438Fall2014Boutin]] [[Category:ECE]] [[Category:ECE438]] [[Category:Signal_processing]] |
Revision as of 07:24, 2 October 2014
DTFT of a Cosine Sampled Above and Below the Nyquist Rate
A slecture by ECE student Sahil Sanghani
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Contents
Outline
- Introduction
- Useful Background
- DTFT Example of a Cosine Sampled Above the Nyquist Rate
- DTFT Example of a Cosine Sampled Below the Nyquist Rate
- Conclusion
- References
Introduction
In this Slecture, I will walk you through taking the DTFT of a pure frequency sampled above and below the Nyquist Rate. Then I will compare the differences between them.
Useful Background
Nyquist Condition: fs = 2 * fm'a'x DTFT of a Cosine: $ x_d[n] = cos(2\pi nT){\leftrightarrow}X(\omega) = \pi(\delta(\omega-\omega_o) + \delta(\omega+\omega_o)){ ,for\ } \omega \in [-\pi,\pi] $
The DTFT of a sampled signal is periodic with 2π.
DTFT of a Cosine Sampled Above the Nyquist Rate
For our original pure frequency, let’s choose the E below middle C. The E occurs at 330Hz x(t) = cos(2π * 330t) <p>Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is:
fs = 2 * fm'a'x = 2 * (330H'z) = 660H'z.
Let’s sample at 990H'z.
$ \begin{align} \\ x_d[n] & = x(n*\frac{1}{990Hz})\\ & = cos(2\pi n *\frac{330}{990}) = \frac{e^{j2\pi n \frac{330}{550}} + e^{-j2\pi n \frac{330}{550}}}{2}\\ & = cos(\frac{2\pi n}{3}) \end{align} $
Because $ \left | \frac{2\pi}{3}\right | < \pi $, there is no aliasing occurring in the DTFT, and it can be written as follows:
$ \begin{align} \\ X(\omega) & = \frac{1}{2}(2\pi\delta(\omega - 2\pi \frac{330}{990}) + 2\pi\delta(\omega + 2\pi \frac{330}{990})) , \ \omega \in\ [-\pi,\pi]\\ & = \frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330)) , \ \omega \in\ [-\pi,\pi]\\ & = rep_{2\pi}(\frac{990}{2}(\delta(\frac{990}{2\pi}\omega - 330) + \delta(\frac{990}{2\pi}\omega + 330))), \forall \omega \end{align} $
