(New page: ==Lecture Notes March 11, 2009== 1. <p><math>E(Y(n)) = \mu_x\sum_m h(m)</math>, ind of n</p>)
 
 
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==Lecture Notes March 11, 2009==
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[[Category:ECE]]
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[[Category:ECE438]]
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[[Category:signal processing]]
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[[Category:ECE438Spring2009mboutin]]
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[[Category:lecture notes]]
  
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1. <p><math>E(Y(n))  = \mu_x\sum_m h(m)</math>, ind of n</p>
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== [[ECE_438_Spring_2009_mboutin_Course_Notes|Course Notes]], March 11, 2009 ==
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[[ECE438_%28BoutinSpring2009%29|ECE438, Spring 2009]]
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----
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<p>1. <math>E(Y(n))  = \mu_x\sum_m^{} h(m)</math> ind of n</p>
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<p>2. <math>C_{xy}(m,n) = h*r_{xx}(n-m), only depends on n-m</math></p>
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<p>3. <math>r_{yy}(m,n) = h\_*C_{xy}(m-n) = h * C_{yx}(n-m)</math></p>
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<span class="sheader">because:</span>
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<math>r_{yy}(m,n) = E(Y(m)Y(n))</math><br/>
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<math> = E(Y(m)\sum_k^{}h(n-k)X(k))</math><br/>
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<math> = \sum_k^{}h(n-k)E(Y(m)X(k))</math><br/>
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<math> = \sum_k^{}h(n-k)C_{yx}(k-m)</math><br/>
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let <math>l = k-m</math><br/>
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<math> = \sum_l^{}h(n-m-l)C_{yx}(l) = h*C_{yx}(n-m)</math><br/>
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<math> = h\_*C_{xy}(m-n)</math><br/>
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<span class="sheader">observe:</span>
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<p>
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<math>r_{yy}(m,n)</math> only depends on <u>n-m</u><br/>
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<math>\Rightarrow Y(n)</math> is wss<br/>
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</p>
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<span class="sheader">Flow Diagram of autocorollation transformation</span>
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<p>
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<math>r_{xx}(n) \Rightarrow h(n) \Rightarrow C_{xy}(n) \Rightarrow time reversal \Rightarrow C_{xy}(-n) = C_{yx}(n) \Rightarrow h(n) \Rightarrow r_{yy}(n)</math>
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</p>
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<span class="sheader example">Example</span>
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<p><math>Y(n) = X(n) + X(n-1)</math><br/>
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X(n) is i.i.d Gausian 0 mean with variance <math>\sigma_x^2</math><br/>
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Lets check if its wss<br/>
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1.<math> E(X(n)) = 0, \forall_n</math><br/>
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2.<math> r_{xx}(m,n) = E(X(m)X(n)) = E(X(m)^2)</math> if m=n else <math>E(X(m))E(X(n))</math><br/>
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<math> = \sigma^2, m = n ,else = 0</math><br/><math> = \sigma^2\delta(m-n)</math>
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<br/>
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<math> \Rightarrow r_{xx}(m,n) = r_xx{n-m}</math><br/>
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Now lets compute E(Y(n))<br/>
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<math>E(y(n)) = E(X(n) + X(n-1))</math><br/>
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<math> = E(X(n)) + E(X(n-1))</math><br/>
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<math> = 0 + 0 = 0</math><br/>
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Now lets compute <math>r_{yy}(m)</math><br/>
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<math>r_{yy}(m) = h\_*C_{xy}(m)</math><br/>
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<math> = h\_*(h * r_{xx}(m))</math><br/>
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<math> = \sigma^2h\_*(h * \sigma^2\delta(m))</math><br/>
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<math> = \sigma^2(\delta(-n) + \delta(-n-1))*(\delta(n) + \delta(n-1))</math><br/>
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<math> = \sigma^2(\delta(n-1) + 2\delta(n) + \delta(n+1))</math><br/>
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...(erased too quickly, didnt get all of it)...
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</p>
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<span class="cheader">3.1.6 Estimating Correlation functions</span>
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<p>suppose X(m) is wss process to estimate.<br/>
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<math>r_{xx}(m) = E (X(n),X(n+m))</math><br/>
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<span class="note">any n would do. each n gives a sample</span>
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</p>
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--[[User:Drestes|Drestes]] 14:18, 11 March 2009 (UTC)
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==Suggestions for Wiki===
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would be kinda nice if i could use some standard css classes to have more uniform header styling across all the notes.. i.e.
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<style type="text/css" media="screen">
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.cheader {
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  font-size: 14px;
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  font-weight: bold;
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  text-decoration:underline;
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  clear:both;
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}
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.cheader {
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  font-size: 12px;
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  font-weight: bold;
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  text-decoration:none;
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  clear:both;
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}
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.note {
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  color:#0000BB;
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  padding-left:20px;
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  clear:both;
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}
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</style>

Latest revision as of 05:45, 16 September 2013


Course Notes, March 11, 2009

ECE438, Spring 2009


1. $ E(Y(n)) = \mu_x\sum_m^{} h(m) $ ind of n

2. $ C_{xy}(m,n) = h*r_{xx}(n-m), only depends on n-m $

3. $ r_{yy}(m,n) = h\_*C_{xy}(m-n) = h * C_{yx}(n-m) $

because: $ r_{yy}(m,n) = E(Y(m)Y(n)) $
$ = E(Y(m)\sum_k^{}h(n-k)X(k)) $
$ = \sum_k^{}h(n-k)E(Y(m)X(k)) $
$ = \sum_k^{}h(n-k)C_{yx}(k-m) $
let $ l = k-m $
$ = \sum_l^{}h(n-m-l)C_{yx}(l) = h*C_{yx}(n-m) $
$ = h\_*C_{xy}(m-n) $


observe:

$ r_{yy}(m,n) $ only depends on n-m
$ \Rightarrow Y(n) $ is wss

Flow Diagram of autocorollation transformation

$ r_{xx}(n) \Rightarrow h(n) \Rightarrow C_{xy}(n) \Rightarrow time reversal \Rightarrow C_{xy}(-n) = C_{yx}(n) \Rightarrow h(n) \Rightarrow r_{yy}(n) $

Example

$ Y(n) = X(n) + X(n-1) $
X(n) is i.i.d Gausian 0 mean with variance $ \sigma_x^2 $
Lets check if its wss
1.$ E(X(n)) = 0, \forall_n $
2.$ r_{xx}(m,n) = E(X(m)X(n)) = E(X(m)^2) $ if m=n else $ E(X(m))E(X(n)) $
$ = \sigma^2, m = n ,else = 0 $
$ = \sigma^2\delta(m-n) $
$ \Rightarrow r_{xx}(m,n) = r_xx{n-m} $
Now lets compute E(Y(n))
$ E(y(n)) = E(X(n) + X(n-1)) $
$ = E(X(n)) + E(X(n-1)) $
$ = 0 + 0 = 0 $
Now lets compute $ r_{yy}(m) $
$ r_{yy}(m) = h\_*C_{xy}(m) $
$ = h\_*(h * r_{xx}(m)) $
$ = \sigma^2h\_*(h * \sigma^2\delta(m)) $
$ = \sigma^2(\delta(-n) + \delta(-n-1))*(\delta(n) + \delta(n-1)) $
$ = \sigma^2(\delta(n-1) + 2\delta(n) + \delta(n+1)) $
...(erased too quickly, didnt get all of it)...

3.1.6 Estimating Correlation functions

suppose X(m) is wss process to estimate.
$ r_{xx}(m) = E (X(n),X(n+m)) $
any n would do. each n gives a sample

--Drestes 14:18, 11 March 2009 (UTC)

Suggestions for Wiki=

would be kinda nice if i could use some standard css classes to have more uniform header styling across all the notes.. i.e. <style type="text/css" media="screen"> .cheader { 

  font-size: 14px;
  font-weight: bold;
  text-decoration:underline;
  clear:both;

}

.cheader { 

  font-size: 12px;
  font-weight: bold;
  text-decoration:none;
  clear:both;

} .note { 

  color:#0000BB;
  padding-left:20px;
  clear:both;

}

</style>

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