Optimum Wiener filters

Return to the desired output — the zero-delay spike (1, 0, 0), that was considered when studying inverse and least-squares filters. Rewrite equation (16), which we solved to obtain the least-squares inverse filter, as follows:

${\begin{pmatrix}{5}/{2}&-1\\-1&{5}/{2}\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}2\\0\end{pmatrix}}.$

(16)

$2{\begin{pmatrix}5/4&-1/2\\-1/2&5/4\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}2\\0\end{pmatrix}}.$

(26)

Divide both sides by 2 to obtain

${\begin{pmatrix}5/4&-1/2\\-1/2&5/4\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}1\\0\end{pmatrix}}.$

(27)

The autocorrelation of the input wavelet (1, -  1/2) is shown in Table 2-17. Note that the autocorrelation lags are the same as the first column of the 2 × 2 matrix on the left side of equation (27).

Now compute the crosscorrelation of the desired output (1, 0, 0) with the input wavelet (1, -  1/2) (Table 2-18). The crosscorrelation lags are the same as the column matrix on the right side of equation (27).

Table 2-17. Autocorrelation lags of input wavelet (1, -  1/2).

	1	-  1/2		Output
	1	-  1/2		5/4
		1	-  1/2	-  1/2

Table 2-18. Crosscorrelation lags of desired output (1, 0, 0) with input wavelet (1, -  1/2).

	1	0	0	Output
	1	-  1/2		1
		1	-  1/2	0

In general, the elements of the matrix on the left side of equation (27) are the lags of the autocorrelation of the input wavelet, while the elements of the column matrix on the right side are the lags of the crosscorrelation of the desired output with the input wavelet.

Now perform similar operations for wavelet (-  1/2, 1). By rewriting the matrix equation (19), we obtain

${\begin{pmatrix}5/2&-1\\-1&5/2\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1\\0\end{pmatrix}}.$

(19)

$2{\begin{pmatrix}5/4&-1/2\\-1/2&5/4\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1\\0\end{pmatrix}}.$

(28)

Divide both sides by 2 to obtain

${\begin{pmatrix}5/4&-1/2\\-1/2&5/4\end{pmatrix}}{\begin{pmatrix}a\\b\end{pmatrix}}={\begin{pmatrix}-1/2\\0\end{pmatrix}}.$

(29)

The autocorrelation of wavelet (-  1/2, 1) is given in Table 2-19. The elements of the matrix on the left side of equation (29) are the autocorrelation lags of the input wavelet. Note that autocorrelation of wavelet (-  1/2, 1) is identical to that of wavelet (1, -  1/2) (Table 2-17). As discussed in inverse filtering, an important property of a group of wavelets with the same amplitude spectrum is that they also have the same autocorrelation.

The crosscorrelation of the desired output (1, 0, 0) with input wavelet (-  1/2, 1) is given in Table 2-20. Note that the right side of equation (29) is the same as the crosscorrelation lags.

Table 2-19. Autocorrelation lags of input wavelet (-  1/2, 1).

	-  1/2	1		Output
	-  1/2	1		5/4
		-  1/2	1	-  1/2

Table 2-20. Crosscorrelation lags of desired output (1, 0, 0) with input wavelet (-  1/2, 1).

	1	0	0	Output
	-  1/2	1		-  1/2
		-  1/2	1	0

Matrix equations (27) and (29) were used to derive the least-squares inverse filters. These filters then were applied to the input wavelets to compress them to zero-lag spike. The matrices on the left in equations (27) and (29) are made up of the autocorrelation lags of the input wavelets. Additionally, the column matrices on the right are made up of lags of the crosscorrelation of the desired output — a zero-lag spike, with the input wavelets. These observations were generalized by Wiener to derive filters that convert the input to any desired output .

The general form of the matrix equation such as equation (29) for a filter of length n is (optimum Wiener filters):

${\begin{pmatrix}r_{0}&r_{1}&r_{2}&\cdots &r_{n-1}\\r_{1}&r_{0}&r_{1}&\cdots &r_{n-2}\\r_{2}&r_{1}&r_{0}&\cdots &r_{n-3}\\\vdots &\vdots &\vdots &\ddots &\vdots \\r_{n-1}&r_{n-2}&r_{n-3}&\cdots &r_{0}\end{pmatrix}}{\begin{pmatrix}a_{0}\\a_{1}\\a_{2}\\\vdots \\a_{n-1}\\\end{pmatrix}}={\begin{pmatrix}g_{0}\\g_{1}\\g_{2}\\\vdots \\g_{n-1}\end{pmatrix}}$

(30)

Here r_i, a_i, and g_i, i = 0, 1, 2, …, n − 1 are the autocorrelation lags of the input wavelet, the Wiener filter coefficients, and the crosscorrelation lags of the desired output with the input wavelet, respectively.

The optimum Wiener filter (a₀, a₁, a₂, …, a_n−1) is optimum in that the least-squares error between the actual and desired outputs is minimum. When the desired output is the zero-lag spike (1, 0, 0, …, 0), then the Wiener filter is identical to the least-squares inverse filter. In other words, the least-squares inverse filter really is a special case of the Wiener filter.

The Wiener filter applies to a large class of problems in which any desired output can be considered, not just the zero-lag spike. Five choices for the desired output are:

Type 1: Zero-lag spike,

Type 2: Spike at arbitrary lag,

Type 3: Time-advanced form of input series,

Type 4: Zero-phase wavelet,

Type 5: Any desired arbitrary shape.

Figure 2.3-1  A flowchart for Wiener filter design and application.

These desired output forms will be discussed in the following sections.

The general form of the normal equations (30) was arrived at through numerical examples for the special case where the desired output was a zero-lag spike. Optimum Wiener filters provides a concise mathematical treatment of the optimum Wiener filters. Figure 2.3-1 outlines the design and application of a Wiener filter.

Determination of the Wiener filter coefficients requires solution of the so-called normal equations (30). From equation (30), note that the autocorrelation matrix is symmetric. This special matrix, called the Toeplitz matrix, can be solved by Levinson recursion, a computationally efficient scheme (spiking deconvolution). To do this, compute a two-point filter, derive from it a three-point filter, and so on, until the n-point filter is derived . In practice, filtering algorithms based on the optimum Wiener filter theory are known as Wiener-Levinson algorithms.

See also

• Spiking deconvolution
• Prewhitening
• Wavelet processing by shaping filters
• Predictive deconvolution
• Exercises
• Mathematical foundation of deconvolution

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