1. OPTIMUM FILTERING

2. WIENER FILTER
This is an optimum filter that is based on the minimization of mean square error between the filter output and the a desired signal
x(n)
v(n)
h(n)
y(n)
Interference due to noise
Ideally y(n) should be equal to xd(n)
Filter impulse response
sd(n)
Desired sequence
e(n)
Error signal
s(n)
Assuming that h(n) is of length N, the mean square error is defined as
By taking the derivative of the mean square error and let it equal to
zero, h(n) is solved by

3. WIENER FILTER
When expressed in terms of the autocorrelation function and
crosscorrelation function is
Since the x(n) is a summation of the true signal s(n) and noise w(n), and sd(n) is not correlated with the noise v(n), then

4. WIENER FILTER The matrix representation is
The h(n) can be obtained by taking the inverse of the autocorrelation matrix and multiply with crosscorrelation matrix.
Minimum mean square error is

5. WIENER FILTER EXAMPLE A signal is defined as follows
where v(n) is additive white Gaussian noise with zero mean and variance 0.1.
The Wiener filter length is 4.

6. WIENER FILTER EXAMPLE The matrix representation is
By taking inverse matrix, the filter coefficients are
Minimum means square error is

7. WIENER FILTER CONFIGURATION
The various configuration of the Wiener filter is often referred as the linear estimation problem.
sd(n)=s(n) filtering
sd(n)=s(n+D) D>0 signal prediction
sd(n)=s(n-D) D>0 signal smoothing
The materials presented will focus only on filtering and prediction.

8. WOLD REPRESENTATION H(z) V(n) X(n) White noise Random process
H(z) - all pole, AR process
- all zero, MA process
- pole-zero, ARMA
1/H(z)
V(n)
X(n)
White noise
Random process

9. AUTOREGRESSIVE PROCESS
Difference equation for M th order AR process is
where V(n) is white noise with zero mean and variance sv2. The autocorrelation function is
Since RVV(m)= sv2d(m), the autocorrelation function for AR process

10. AUTOREGRESSIVE PROCESS
Expanding the autocorrelation function results in
The matrix representation
The results is known as the Yule-Walker equations.

11. LINEAR PREDICTION
The linear predictive filter is used to predict the model of the underlying random process.
Desired sequence
h(n-1)
X(n)
Ideally Y(n) should be equal to Xd(n)
Filter impulse response
Xd(n)
e(n)
Error signal
Assuming that h(n) is of length P, then one step forward predictor filter is defined as

12. LINEAR PREDICTION The forward prediction error is
The mean square prediction error is
By taking the derivative of the mean square error and let it equal to
zero, ap(n) is solved by

13. LINEAR PREDICTION
When expressed in terms of the autocorrelation function is
The minimum mean square error predictor error is
Combining the above two equation results in the augmented normal equations where the solution for the coefficients are derived

14. LINEAR PREDICTION
The matrix representation for the augmented normal equation is
where ap(0)=1. For sample functions, the time averaged autocorrelation function is used. The solution is calculated by taking the inverse matrix.

15. LINEAR PREDICTION EXAMPLE
A sample function x(n) is defined by as follows
x(n)=[7.0718, , , , , 2.011, , 1.001, , ]
The biased time-average autocorrelation function is
Rxx(l)=[24.39, , , 1.662, , , , 1.617, , ]
For 4 th order predictor, the augmented normal equation is

16. LINEAR PREDICTION EXAMPLE
The solution for the predictor filter is
ap(n)=[0.1104, , , ]
The normalized solution is
ap(l)=[1, , , ]