| Line 9: | Line 9: | ||
Conclusion | Conclusion | ||
| − | Introduction Hello! My name is Ryan Johnson! You might be wondering was 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 | + | Introduction Hello! My name is Ryan Johnson! You might be wondering was 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 | Derivation F = fourier transform | ||
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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> |
Explanation 1 | Explanation 1 | ||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| − | X_s(f) &= F(x_s(t)) | + | X_s(f) &= F(x_s(t)) = F(comb_T)\\ |
| + | &= F(x(t)p_T(t))\\ | ||
| + | &= X(f) * F(p_T(t)\\ | ||
&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ | &= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\ | ||
| − | &= \frac{1}{T}X(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} | ||
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Example | Example | ||
| − | + | [[Image:Example.jpg]] | |
| + | |||
| + | Conclusion | ||
Revision as of 19:08, 6 October 2014
Outline
Introduction
Derivation
Example
Conclusion
Introduction Hello! My name is Ryan Johnson! You might be wondering was 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} $ Explanation 1 $ \begin{align} X_s(f) &= F(x_s(t)) = F(comb_T)\\ &= F(x(t)p_T(t))\\ &= X(f) * F(p_T(t)\\ &= 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} $
Example
Conclusion
