Difference between revisions of "Text slecture" - Rhea

Revision as of 11:01, 7 October 2014

Text slecture

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-[[Category:slecture]]
+Text slecture
-[[Category:ECE438Fall2014Boutin]]
-[[Category:ECE]]
-[[Category:ECE438]]
-[[Category:signal processing]]
-<center><font size= 4>
-Frequency domain view of the relationship between a signal and a sampling of that signal
-</font size>
-A [https://www.projectrhea.org/learning/slectures.php slecture] by [[ECE]] student Yerkebulan Yeshmukhanbetov
-Partly based on the [[2014_Fall_ECE_438_Boutin|ECE438 Fall 2014 lecture]] material of [[user:mboutin|Prof. Mireille Boutin]].
-</center>
-----
-<font size = 3>
-==Outline==
-#Introduction
-#Derivation
-#Example
-#Conclusion
-----
-==Introduction==
-In this slecture I will discuss about the relations between the original signal <math> X(f) </math> (the CTFT of <math> x(t) </math>  ), sampling continuous time signal <math> X_s(f) </math> (the CTFT of <math> x_s(t) </math> ) and sampling discrete time signal <math> X_d(\omega) </math> (the DTFT of <math> x_d[n] </math> )  in frequency domain and give a specific example showing the relations.
-----
-==Derivation==
-The first thing which need to be clarified is that there two different types of sampling signal: <math> x_s(t) </math> and <math> x_d[n] </math>. <math> x_s(t) </math>  is created by multiplying a impulse train <math> P_T(t) </math> with the original signal <math> x(t) </math> and actually <math> x_s(t) </math>  is  <math> comb_T(x(t)) </math> where T is the sampling period.
-However the <math> x_d[n] </math> is <math> x(nT) </math> where T is the sampling period.
-Now we first concentrate on the relationship between <math> X(f) </math> and <math> X_s(f) </math>.
-We know that <math> x_s(t) = x(t) \times P_T(t) </math>, we can derive the relationship between <math> x_s(t) </math> and <math> x(t) </math> in the following way:
-<div style="margin-left: 3em;">
-<math>
-\begin{align}
-F(comb_T(x(t)) &= F(x(t) \times 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}
-</math>
-</div>
-<font size>
-Show this relationship in graph below:
-----
-==example==
-[[Image:Xfcbt.png]]
-[[Image:xsfcbt.png]]
-----
-==Derivation==
-Then we are going to find the relation between <math> X_s(f) </math> and <math> X_d(\omega) </math>
-We know another way to express CTFT of <math> x_s(t) </math>:
-<div style="margin-left: 3em;">
-<math>
-\begin{align}
-X_s(f) &= F(\sum_{n = -\infty}^\infty x(nT)\delta(t-nT))\\
-&= \sum_{n = -\infty}^\infty x(nT)F(\delta(t-nT))\\
-&= \sum_{n = -\infty}^\infty x(nT)e^{-j2\pi fnT}\\
-\end{align}
-</math>
-</div>
-<font size>
-compare it with DTFT of <math> x_d[n] </math>:
-<div style="margin-left: 3em;">
-<math>
-\begin{align}
-X_d(\omega) &= \sum_{n = -\infty}^\infty x_d[n]e^{-j\omega n}\\
-&= \sum_{n = -\infty}^\infty x(nT)e^{-j\omega n}\\
-\end{align}
-</math>
-</div>
-<font size>
-we can find that:
-<div style="margin-left: 3em;">
-<math>
-\begin{align}
-X_d(2\pi Tf) &= X_s(f)\\
-\end{align}
-</math>
-</div>
-<font size>
-if <math> f = \frac{1}{T} </math>
-we have that:
-<div style="margin-left: 3em;">
-<math>
-\begin{align}
-X_d(2\pi ) &= X_s(\frac{1}{T})\\
-\end{align}
-</math>
-</div>
-<font size>
-from this equation, we can know the relationship between <math> X_s(f) </math> and <math> X_d(\omega) </math> and the relationship is showed in graph as below:
-----
-==example==
-[[Image:xsfcbt.png]]
-[[Image:xdwcbt.png]]
-----
-==conclusion==
-So the relationship between <math> X(f) </math> and <math> X_s(f) </math> is that <math> X_s(f) </math> is a a rep of <math> X(f) </math> in frequency domain with period of <math> \frac{1}{T} </math> and magnitude scaled by <math> \frac{1}{T} </math>.
-the relationship between <math> X(f) </math> and <math> X_d(\omega) </math> is that <math> X_d(\omega) </math> is also a a rep of <math> X(f) </math> in frequency domain with period <math> 2\pi </math> and magnitude is also scaled by <math> \frac{1}{T} </math>, but the frequency is scaled by <math> 2\pi T </math>
-----

Difference between revisions of "Text slecture" - Rhea

Revision as of 11:01, 7 October 2014

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