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Outline | Outline | ||
| − | + | <br> | |
| − | Derivation | + | Introduction<br>Derivation<br>Example<br>Conclusion |
| − | + | <br> | |
| − | + | Introduction | |
| − | + | Hello! My name is Ryan Johnson! You might be wondering what a slecture is! A slecture is a student lecture that gives a brief overview about a particular topic! In this slecture, I will discuss the relationship between an original signal and a sampling of that original signal. We will also take a look at how this relationship translates to the frequency domain. | |
| + | |||
| + | Derivation | ||
| − | + | F = fourier transform | |
<div style="margin-left: 3em;"> | <div style="margin-left: 3em;"> | ||
<math> | <math> | ||
| Line 17: | Line 19: | ||
comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ | comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ | ||
\end{align} | \end{align} | ||
| − | </math> | + | </math> The comb of a signal is equal to the signal multiplied by an impulse train. <math> |
| − | + | ||
| − | <math> | + | |
\begin{align} | \begin{align} | ||
X_s(f) &= F(x_s(t)) = F(comb_T)\\ | X_s(f) &= F(x_s(t)) = F(comb_T)\\ | ||
&= F(x(t)p_T(t))\\ | &= F(x(t)p_T(t))\\ | ||
| − | &= X(f) * F(p_T( | + | &= X(f) * F(p_T(f)\\ |
| − | &= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ | + | \end{align} |
| + | </math> Multiplication in time is equal to convolution in frequency. <br> <math> | ||
| + | \begin{align} | ||
| + | X_s(f)&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ | ||
&= \frac{1}{T}X(f)*p_\frac{1}{T}(f)\\ | &= \frac{1}{T}X(f)*p_\frac{1}{T}(f)\\ | ||
&= \frac{1}{T}rep_\frac{1}{T}X(f)\\ | &= \frac{1}{T}rep_\frac{1}{T}X(f)\\ | ||
\end{align} | \end{align} | ||
| − | </math> | + | </math> The result is a repetition of the fourier transformed signal. |
</div> | </div> | ||
<font size="size"></font> | <font size="size"></font> | ||
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Example | Example | ||
| − | + | <br> | |
| + | |||
| + | Conclusion | ||
| − | + | Xs(f) is a rep of X(f) in the frequency domain with amplitude of 1/T and period of 1/T. | |
Revision as of 19:31, 6 October 2014
Outline
Introduction
Derivation
Example
Conclusion
Introduction
Hello! My name is Ryan Johnson! You might be wondering what a slecture is! A slecture is a student lecture that gives a brief overview about a particular topic! In this slecture, I will discuss the relationship between an original signal and a sampling of that original signal. We will also take a look at how this relationship translates to the frequency domain.
Derivation
F = fourier transform
$ \begin{align} comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ \end{align} $ The comb of a signal is equal to the signal multiplied by an impulse train. $ \begin{align} X_s(f) &= F(x_s(t)) = F(comb_T)\\ &= F(x(t)p_T(t))\\ &= X(f) * F(p_T(f)\\ \end{align} $ Multiplication in time is equal to convolution in frequency.
$ \begin{align} X_s(f)&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ &= \frac{1}{T}X(f)*p_\frac{1}{T}(f)\\ &= \frac{1}{T}rep_\frac{1}{T}X(f)\\ \end{align} $ The result is a repetition of the fourier transformed signal.
Example
Conclusion
Xs(f) is a rep of X(f) in the frequency domain with amplitude of 1/T and period of 1/T.
