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== Useful Background == | == 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>. | + | 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>. |
== DTFT of a Cosine Sampled Above the Nyquist Rate == | == 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>. | + | <p>For our original pure frequency, let’s choose the E below middle C. The E occurs at 330<span class="math"><em>H</em><em>z</em></span>.</p> |
| − | <p | + | <p><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>. | + | <p>Now let’s sample this pure cosine at a frequency above the Nyquist Rate. The Nyquist Rate is: <br /><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>. <br />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]] | [[2014_Fall_ECE_438_Boutin|Back to ECE438, Fall 2014]] | ||
Revision as of 05:36, 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.
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)
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
