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In order to derive the coefficients of the Wiener filter, we consider a signal w[n] being fed to a Wiener filter of order N and with coefficients 
. The output of the filter is denoted x[n] which is given by the expression
![x[n] = \sum_{i=0}^N a_i w[n-i]](../../../math/3/4/c/34caa890e9ecab29abf6192eb528e0be.png)
The residual error is denoted e[n] and is defined as e[n]=x[n]-s[n] (See the corresponding block diagram). The Wiener filter is designed so as to minimize the mean square error (MMSE criteria) which can be stated concisely as follows:
![a_i = \arg.\min ~E\{\sum_{n=0}^N e^2[n]\}](../../../math/4/8/7/48707c3b86735fbf8ddfb6d4b3fe1c66.png)
where E{.} denote the expectation operator. In the general case, the coefficients ai may be complex and may be derived for the case where w[n] and s[n] are complex as well. For simplicity, we will only consider the case where all these quantities are real. The mean square error may be rewritten as:
![\begin{array}{rcl} E\{e^2[n]\} &=& E\{(x[n]-s[n])^2\} \\ &=& E\{x^2[n]\} + E\{s^2[n]\} - 2E\{x[n]s[n]\}\\ &=& E\{\big( \sum_{i=0}^N a_i w[n-i] \big)^2\} + E\{w^2[n]\} -2 E\{ \sum_{i=0}^N a_i w[n-i]s[n]\} \end{array}](../../../math/d/9/5/d95ee7dcbe0c38b048c79aa4412590af.png)
To find the vector ![[a_0,\ldots,a_N]](../../../math/c/3/3/c335f950651d0ed4d42cf1082ba72ed7.png)
which minimizes the expression above, let us now calculate its derivative w.r.t ai
![\begin{array}{rcl} \frac{\partial}{\partial a_i} E\{e^2[n]\} &=& 2E\{ \big( \sum_{j=0}^N a_j w[n-j] \big) w[n-i] \} - 2E\{s[n]w[n-i]\} \quad i=0, \ldots ,N \\ &=& 2 \sum_{j=0}^N E\{w[n-j]w[n-i]\} a_j - 2 E\{ w[n-i]s[n] \} \end{array}](../../../math/d/e/0/de06855b73a3d8f522cf67555284eb67.png)
If we suppose that w[n] and s[n]are stationary, we can introduce the following sequences ![R_w[m] ~\textit{ and }~ R_{ws}[m]](../../../math/5/3/4/534d120823ed6fe4e31d60895c4a5803.png)
known respectively as the autocorrelation of w[n] and the cross-correlation between w[n] and s[n] defined as follows
![R_w[m] = E\{w[n]w[n+m]\} \quad \textit{ and } \quad R_{ws}[m] = E\{w[n]s[n+m]\}](../../../math/7/6/3/763678d84d6e2a7a8899529bebe99906.png)
The derivative of the MSE may therefore be rewritten as (notice that Rws[ − i] = Rsw[i])
![\frac{\partial}{\partial a_i} E\{e^2[n]\} = 2 \sum_{j=0}^{N} R_w[j-i] a_j - 2 R_{sw}[i] \quad i=0, \ldots ,N](../../../math/4/6/f/46f14653fc034300949523a734f66f71.png)
Letting the derivative be equal to zero, we obtain
![\sum_{j=0}^N R_w[j-i] a_j = R_{sw}[i] \quad i=0, \ldots ,N](../../../math/b/a/c/bac26d7be7e6f7dd1ab65722ba2524a4.png)
which can be rewritten in matrix form
![\begin{bmatrix} R_w[0] & R_w[1] & \cdots & R_w[N] \\ R_w[1] & R_w[0] & \cdots & R_w[N-1] \\ \vdots & \vdots & \ddots & \vdots \\ R_w[N] & R_w[N-1] & \cdots & R_w[0] \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_N \end{bmatrix} = \begin{bmatrix} R_{sw}[0] \\R_{sw}[1] \\ \vdots \\ R_{sw}[N] \end{bmatrix}](../../../math/7/3/2/732cda0b243d1864fae92af4172218d5.png)
These equations are known as the Wiener-Hopf equations. The matrix appearing in the equation is a symmetric Toeplitz matrix. These matrices are known to be positive definite and therefore non-singular yielding a single solution to the determination of the Wiener filter coefficients. Furthermore, there exists an efficient algorithm to solve the Wiener-Hopf equations known as the Levinson-Durbin algorithm.
