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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 continuous time sampling of that original signal. We will also take a look at how this relationship translates to the frequency domain. Derivation | + | 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 continuous time 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;"> | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\ | ||
| + | \end{align} | ||
| + | </math> \\ | ||
| + | Explanation 1 | ||
| + | <math> | ||
| + | \begin{align} | ||
| + | X_s(f) &= F(x_s(t)) &= F(comb_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} | ||
| + | </math> | ||
| + | </div> | ||
| + | <font size="size"></font> | ||
<br> | <br> | ||
Revision as of 17:48, 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 continuous time 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)\\ &= 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 So the relationship between X(f) and Xs(f) is that Xs(f) is a a rep of X(f) in frequency domain with period of and magnitude scaled by . the relationship between X(f) and Xd(ω) is that Xd(ω) is also a a rep of X(f) in frequency domain with period 2π and magnitude is also scaled by , but the frequency is scaled by 2πT
