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I have this question: "Show that for the optimal filter coefficients the error $e[n]$ is orthogonal to the input signal $w[n]$." anybody have an idea?

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For conveniece I'm going to drop the time indexes. Let $e=d-\mathbf{a}^T\mathbf{x}$, where $d$ is the desired signal and $\mathbf{a}$ weights the elements of the observations/input vector $\mathbf{x}$. The goal is to minimize expected squared error with respect to the vector $\mathbf{a}$.

$$\min J=\min E[e^2],$$

where $E$ is the expectation operator. To find the minimum we take the derivative of the objective function w.r.t. $\mathbf{a}$ and set it equal to zero. Therefore $$ \mathbf{g}=\frac{\partial E[e^2]}{\partial \mathbf{a}}=E[2e\frac{\partial e}{\partial \mathbf{a}}]=0,$$

but from the defintion of $e$ earlier, we have $$ \frac{\partial e}{\partial \mathbf{a}}= -\mathbf{x}$$ Therefore $$ E[2e\frac{\partial e}{\partial \mathbf{a}}]= -E[2e\mathbf{x}]=0. $$ The factor 2 can be removed and thus $E[e\mathbf{x}]=0$, which means the error $e$ is orthogonal to the input signal $\mathbf{x}$.

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In other words,you should show that the expected value of the error term multiplied by input signal is zero.

If you ask about the reason, the linear estimator of the signal that minimizes the error(between the estimator and the signal) is given by the projection theorem and the projection is computed by directly minimizing the mean square error or by applying the orthogonality between error and the input signal.

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