| Line 1: | Line 1: | ||
| − | [[Category: | + | [[Link title]][[Category:slecture]] |
| + | [[Category:ECE438Fall2014Boutin]] | ||
| + | [[Category:ECE]] | ||
| + | [[Category:ECE438]] | ||
| + | [[Category:signal processing]] | ||
| − | = | + | <center><font size= 5> |
| + | DTFT of a Cosine Sampled Above and Below the Nyquist Rate | ||
| + | </font size> | ||
| + | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Sahil Sanghani | ||
| + | Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]]. | ||
| + | </center> | ||
| + | ---- | ||
| − | + | == 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: <span class="math"> <em>f</em><sub><em>s</em></sub> = 2 * <em>f</em><sub><em>m</em><em>a</em><em>x</em></sub></span><br />DTFT of a Cosine: <span class="math"> <em>x</em><sub><em>d</em></sub>[<em>n</em>] = <em>c</em><em>o</em><em>s</em>(2<em>π</em><em>n</em><em>T</em>) → <em>X</em>(<em>ω</em>) = <em>π</em>(<em>δ</em>(<em>ω</em> − <em>ω</em><sub><em>o</em></sub>) + <em>δ</em>(<em>ω</em> + <em>ω</em><sub><em>o</em></sub>))</span>, for <span class="math"><em>ω</em> ∈ [ − <em>π</em>, <em>π</em>]</span><br />The DTFT of a sampled signal is periodic with <span class="math">2<em>π</em></span>.</p> | ||
| + | == DTFT of a Cosine Sampled Above the Nyquist Rate == | ||
| + | <p>For our original pure frequency, let’s choose the E below middle C. The E occurs at 330<span class="math">H</em><em>z</em></span>.<br /><br /></p> | ||
| + | <p><br /><span class="math"><em>x</em>(<em>t</em>) = <em>c</em><em>o</em><em>s</em>(2<em>π</em> * 330<em>t</em>)</span><br /></p> | ||
| + | <p>Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is <span class="math"> <em>f</em><sub><em>s</em></sub> = 2 * <em>f</em><sub><em>m</em><em>a</em><em>x</em></sub> = 2 * (330<em>H</em><em>z</em>) = 660<em>H</em><em>z</em></span>. Let’s sample at 990<span class="math"><em>H</em><em>z</em></span>. | ||
| − | [[ | + | [[2014_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014]] |
Revision as of 05:17, 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 * fmax
DTFT of a Cosine: xd[n] = cos(2πnT) → X(ω) = π(δ(ω − ωo) + δ(ω + ωo)), for ω ∈ [ − π, π]
The DTFT of a sampled signal is periodic with 2π.</p>
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 330H</em>z.
x(t) = cos(2π * 330t)
Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is fs = 2 * fmax = 2 * (330Hz) = 660Hz. Let’s sample at 990Hz. Back to ECE438, Fall 2014
