CourseNotes20 (BoutinSpring2009) - Rhea

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)

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