Wiener Filter - Finite Impulse Response Wiener Filter For Discrete Series

Finite Impulse Response Wiener Filter For Discrete Series

The causal finite impulse response (FIR) Wiener filter, instead of using some given data matrix X and output vector Y, finds optimal tap weights by using the statistics of the input and output signals. It populates the input matrix X with estimates of the auto-correlation of the input signal (T) and populates the output vector Y with estimates of the cross-correlation between the output and input signals (V).

In order to derive the coefficients of the Wiener filter, consider the signal w being fed to a Wiener filter of order N and with coefficients, . The output of the filter is denoted x which is given by the expression

The residual error is denoted e and is defined as e = x − s (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:

where denotes the expectation operator. In the general case, the coefficients may be complex and may be derived for the case where w and s are complex as well. With a complex signal, the matrix to be solved is a Hermitian Toeplitz matrix, rather than symmetric Toeplitz matrix. For simplicity, the following considers only the case where all these quantities are real. The mean square error (MSE) may be rewritten as:

\begin{array}{rcl} E\{e^2\} &=& E\{(x-s)^2\}\\ &=& E\{x^2\} + E\{s^2\} - 2E\{xs\}\\ &=& E\{\big( \sum_{i=0}^N a_i w \big)^2\} + E\{s^2\} - 2E\{\sum_{i=0}^N a_i ws\} . \end{array}

To find the vector which minimizes the expression above, calculate its derivative with respect to

\begin{array}{rcl} \frac{\partial}{\partial a_i} E\{e^2\} &=& 2E\{ \big( \sum_{j=0}^N a_j w \big) w \} - 2E\{sw\} \quad i=0,\, \ldots,\, N\\ &=& 2 \sum_{j=0}^N E\{ww\} a_j - 2E\{ ws\} . \end{array}

Assuming that w and s are each stationary and jointly stationary, the sequences and known respectively as the autocorrelation of w and the cross-correlation between w and s can be defined as follows:

\begin{align} R_w =& E\{ww\} \\ R_{ws} =& E\{ws\} . \end{align}

The derivative of the MSE may therefore be rewritten as (notice that )

Letting the derivative be equal to zero results in

which can be rewritten in matrix form

\begin{align} &\mathbf{T}\mathbf{a} = \mathbf{v}\\ \Rightarrow &\begin{bmatrix} R_w & R_w & \cdots & R_w \\ R_w & R_w & \cdots & R_w \\ \vdots & \vdots & \ddots & \vdots \\ R_w & R_w & \cdots & R_w \end{bmatrix} \begin{bmatrix} a_0 \\ a_1 \\ \vdots \\ a_N \end{bmatrix} = \begin{bmatrix} R_{sw} \\R_{sw} \\ \vdots \\ R_{sw} \end{bmatrix} \end{align}

These equations are known as the Wiener–Hopf equations. The matrix T appearing in the equation is a symmetric Toeplitz matrix. These matrices are known to be positive definite and therefore non-singular yielding a unique solution to the determination of the Wiener filter coefficient vector, . Furthermore, there exists an efficient algorithm to solve such Wiener–Hopf equations known as the Levinson-Durbin algorithm so an explicit inversion of is not required.

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