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