Math Help - Linear Prediction Filter

1. Linear Prediction Filter

can someone please help me with this.....

2. Let $\hat{s} = h_1u(n-1) + h_2u(n-2) + h_3u(n-3)$ where $h_i, i=1,2,3$ are the coefficients to be determined.

Then apply the orthogonality principle
$E[(s(n)-\hat{s})u(n-i)] = 0, i = 1,2,3$
Solve for the three coefficients using the 3 equations.

After computing the coefficients, compute
$E[(s(n)-\hat{s})^2]$
for the MMSE.

3. what is the the orthogonality principle?
I couldnt get the 3 equations you mentioned. please give me more details?

4. You didn't learn about it in class? Do you have a formula involving matrix inversion?

In its simplest form, the Orthogonality Principle states that "the error vector of the optimum estimator is orthogonal to any other possible estimator."

I assumed that you have had some linear algebra and understood the concept of inner product. If two vectors are orthogonal, then its inner product is zero, ie $<\bar{x} , \bar{y}> = 0$.

Here, the inner product of two random variables $X$ and $Y$ is defined to be $ = E[XY]$.

Now let the optimum estimator be $\hat{s} = h_1u(n-1) + h_2u(n-2) + h_3u(n-3)$ where $h_i, i= 1, 2, 3$ are the coefficients to be determined. You can think of $u(n-1), u(n-2), u(n-3)$ as three linearly independent vectors(like the cartesian basis $\hat{i},\hat{j},\hat{k}$, although I should add that the u's are not orthogonal like the cartesian basis)

The error of the optimum estimator is thus $s(n) - \hat{s}$.

Now apply the orthogonality principle, and we have
$E[(s(n)-\hat{s}) A] = 0$ where $A$ is any possible estimator.

We have three unknowns to solve for so we need three linearly independent equations. They can be obtained by substituting $A$ with $u(n-1), u(n-2), u(n-3)$ (each of the u's is a possible estimator by itself). So the three equations are

$E[(s(n)-\hat{s})u(n-1)] = 0$
$E[(s(n)-\hat{s})u(n-2)] = 0$
$E[(s(n)-\hat{s})u(n-3)] = 0$

Substitute in the expression for $\hat{s}$, expand out, plug in the values, and solve for the 3 unknown coefficients and you're done.

5. thank you for your help.
i would have to say that my teacher didnt go through the problems well. all theory and no examples so that we can work from.
i am totally lost..