http://i709.photobucket.com/albums/w.../Untitled2.jpgcan someone please help me with this.....

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- May 6th 2009, 09:24 PMkrispiekreamLinear Prediction Filter
http://i709.photobucket.com/albums/w.../Untitled2.jpgcan someone please help me with this.....

- May 7th 2009, 07:04 AMcl85
Let $\displaystyle \hat{s} = h_1u(n-1) + h_2u(n-2) + h_3u(n-3)$ where $\displaystyle h_i, i=1,2,3$ are the coefficients to be determined.

Then apply the orthogonality principle

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

Solve for the three coefficients using the 3 equations.

After computing the coefficients, compute

$\displaystyle E[(s(n)-\hat{s})^2]$

for the MMSE. - May 7th 2009, 04:16 PMkrispiekream
what is the the orthogonality principle?

__I couldnt get__the 3 equations you mentioned. please give me more details? - May 7th 2009, 07:54 PMcl85
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 $\displaystyle <\bar{x} , \bar{y}> = 0 $.

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

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

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

Now apply the orthogonality principle, and we have

$\displaystyle E[(s(n)-\hat{s}) A] = 0$ where $\displaystyle A$ is any possible estimator.

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

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

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

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

Substitute in the expression for $\displaystyle \hat{s}$, expand out, plug in the values, and solve for the 3 unknown coefficients and you're done. - May 7th 2009, 08:00 PMkrispiekream
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..