(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== | ==Lecture Notes March 11, 2009== | ||
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| + | <p>1. <math>E(Y(n)) = \mu_x\sum_m^{} h(m)</math> ind of n</p> | ||
| + | <p>2. <math>C_{xy}(m,n) = h*r_{xx}(n-m), only depends on n-m</math></p> | ||
| + | <p>3. <math>r_{yy}(m,n) = h\_*C_{xy}(m-n) = h * C_{yx}(n-m)</math></p> | ||
| + | |||
| + | <span class="sheader">because:</span> | ||
| + | <math>r_{yy}(m,n) = E(Y(m)Y(n))</math><br/> | ||
| + | <math> = E(Y(m)\sum_k^{}h(n-k)X(k))</math><br/> | ||
| + | <math> = \sum_k^{}h(n-k)E(Y(m)X(k))</math><br/> | ||
| + | <math> = \sum_k^{}h(n-k)C_{yx}(k-m)</math><br/> | ||
| + | let <math>l = k-m</math><br/> | ||
| + | <math> = \sum_l^{}h(n-m-l)C_{yx}(l) = h*C_{yx}(n-m)</math><br/> | ||
| + | <math> = h\_*C_{xy}(m-n)</math><br/> | ||
| + | |||
| + | |||
| + | <span class="sheader">observe:</span> | ||
| + | <p> | ||
| + | <math>r_{yy}(m,n)</math> only depends on <u>n-m</u><br/> | ||
| + | <math>\Rightarrow Y(n)</math> is wss<br/> | ||
| + | </p> | ||
| + | <span class="sheader">Flow Diagram of autocorollation transformation</span> | ||
| + | <p> | ||
| + | <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> | ||
| + | </p> | ||
| + | |||
| + | <span class="sheader example">Example</span> | ||
| + | <p><math>Y(n) = X(n) + X(n-1)</math><br/> | ||
| + | X(n) is i.i.d Gausian 0 mean with variance <math>\sigma_x^2</math><br/> | ||
| + | Lets check if its wss<br/> | ||
| + | 1.<math> E(X(n)) = 0, \forall_n</math><br/> | ||
| + | 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/> | ||
| + | <math> = \sigma^2, m = n ,else = 0</math><br/><math> = \sigma^2\delta(m-n)</math> | ||
| + | <br/> | ||
| + | <math> \Rightarrow r_{xx}(m,n) = r_xx{n-m}</math><br/> | ||
| + | Now lets compute E(Y(n))<br/> | ||
| + | <math>E(y(n)) = E(X(n) + X(n-1))</math><br/> | ||
| + | <math> = E(X(n)) + E(X(n-1))</math><br/> | ||
| + | <math> = 0 + 0 = 0</math><br/> | ||
| + | Now lets compute <math>r_{yy}(m)</math><br/> | ||
| + | <math>r_{yy}(m) = h\_*C_{xy}(m)</math><br/> | ||
| + | <math> = h\_*(h * r_{xx}(m))</math><br/> | ||
| + | <math> = \sigma^2h\_*(h * \sigma^2\delta(m))</math><br/> | ||
| + | <math> = \sigma^2(\delta(-n) + \delta(-n-1))*(\delta(n) + \delta(n-1))</math><br/> | ||
| + | <math> = \sigma^2(\delta(n-1) + 2\delta(n) + \delta(n+1))</math><br/> | ||
| + | ...(erased too quickly, didnt get all of it)... | ||
| + | </p> | ||
| + | |||
| + | <span class="cheader">3.1.6 Estimating Correlation functions</span> | ||
| + | <p>suppose X(m) is wss process to estimate.<br/> | ||
| + | <math>r_{xx}(m) = E (X(n),X(n+m))</math><br/> | ||
| + | <span class="note">any n would do. each n gives a sample</span> | ||
| + | </p> | ||
| + | |||
| + | --[[User:Drestes|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> | ||
Revision as of 09:18, 11 March 2009
Lecture Notes March 11, 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>
