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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  
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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.  
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Derivation F = fourier transform
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<div style="margin-left: 3em;">
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<math>
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\begin{align}
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comb_T(x(t)) &= x(t) \times p_T(t) &= x_s(t)\\
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\end{align}
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</math> \\
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Explanation 1
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<math>
 +
\begin{align}
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X_s(f) &= F(x_s(t)) &= F(comb_T)\\
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&= X(f)*\frac{1}{T}\sum_{n = -\infty}^\infty \delta(f-\frac{n}{T})\\
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&= \frac{1}{T}X(f)*P_\frac{1}{T}(f)\\
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&= \frac{1}{T}rep_\frac{1}{T}X(f)\\
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\end{align}
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</math>
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</div>
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<font size="size"></font>
  
 
<br>  
 
<br>  

Revision as of 18:48, 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

Alumni Liaison

Correspondence Chess Grandmaster and Purdue Alumni

Prof. Dan Fleetwood