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<font size="4">Frequency domain view of the relationship between a signal and a sampling of that signal </font> | <font size="4">Frequency domain view of the relationship between a signal and a sampling of that signal </font> | ||
| − | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Evan Stockrahm | + | A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Evan Stockrahm |
Partly based on the [[2014 Fall ECE 438 Boutin|ECE438 Fall 2014 lecture]] material of [[User:Mboutin|Prof. Mireille Boutin]]. | Partly based on the [[2014 Fall ECE 438 Boutin|ECE438 Fall 2014 lecture]] material of [[User:Mboutin|Prof. Mireille Boutin]]. | ||
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== Introduction == | == Introduction == | ||
| − | + | This slecture will discuss the frequency domain view of the relationship between a signal, and a sampling of that signal. Essentially, given a signal x(t), we are going to take a look at the similarities and differences in X(f) and X<sub>s</sub>(f). X<sub>s</sub>(f) is the Fourier Transform of the sampling, x<sub>s</sub>(t), of x(t). | |
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== Derivation == | == Derivation == | ||
| − | The | + | Given an arbitrary signal x(t), its Fourier Transform is X(f) |
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| + | The sampling of signal x(t), is the comb of x(t), which is equivalent to multiplying x(t) by the impulse train p<sub>T</sub>(t). | ||
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| + | So, x<sub>s</sub>(t) = x(t) x pT(t) | ||
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== example == | == example == | ||
| − | jfidosa | + | jfidosa |
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== Derivation == | == Derivation == | ||
| − | Then we are going to find<span style="line-height: 1.5em;">etween </span><span class="texhtml" style="line-height: 1.5em;">''X''<sub>''s''</sub>(''f'')</span><span style="line-height: 1.5em;"> and </span><span class="texhtml" style="line-height: 1.5em;">''X''<sub>''d''</sub>(ω)</span><span style="line-height: 1.5em;"> and the relationship is showed in graph as below:</span> | + | Then we are going to find<span style="line-height: 1.5em;">etween </span><span class="texhtml" style="line-height: 1.5em;">''X''<sub>''s''</sub>(''f'')</span><span style="line-height: 1.5em;"> and </span><span class="texhtml" style="line-height: 1.5em;">''X''<sub>''d''</sub>(ω)</span><span style="line-height: 1.5em;"> and the relationship is showed in graph as below:</span> <font size="size"><font size="size"><font size="size"><font size="size"></font></font></font></font> |
| − | <font size="size"><font size="size"><font size="size"><font size="size"> | + | <font size="size"><font size="size"><font size="size"> |
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== example == | == example == | ||
| − | <br> | + | <br> |
| − | fd | + | fd |
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| − | </font></font></font></font></font></font> | + | </font></font></font></font></font> |
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Revision as of 18:15, 6 October 2014
Frequency domain view of the relationship between a signal and a sampling of that signal
A slecture by ECE student Evan Stockrahm
Partly based on the ECE438 Fall 2014 lecture material of Prof. Mireille Boutin.
Outline
- Introduction
- Derivation
- Example
- Conclusion
Introduction
This slecture will discuss the frequency domain view of the relationship between a signal, and a sampling of that signal. Essentially, given a signal x(t), we are going to take a look at the similarities and differences in X(f) and Xs(f). Xs(f) is the Fourier Transform of the sampling, xs(t), of x(t).
Derivation
Given an arbitrary signal x(t), its Fourier Transform is X(f)
The sampling of signal x(t), is the comb of x(t), which is equivalent to multiplying x(t) by the impulse train pT(t).
So, xs(t) = x(t) x pT(t)
<span style="line-height: 1.5em;" />
<span style="line-height: 1.5em;" />
example
jfidosa
Derivation
Then we are going to findetween Xs(f) and Xd(ω) and the relationship is showed in graph as below:
example
fd
conclusion
So t2πT
