Wiener (least-squares) inverse filters

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Problem 9.22a

Plot cumulative energy as a function of time for wavelets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A=[1,\; -2,\; 3]} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B=[3,\; -2,\; 1]}

Background

A filter that will change a signal so as to make it as close as possible to a desired signal can be designed in a least-squares sense to minimize the sum of the squares of the “errors” (differences between a desired signal and the filtered signal). Such a filter is called a least-squares filter or Wiener filter.

Using Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle g_{t}} to denote the original signal and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h_{t}} the desired signal, the sum of the errors squared is


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} E=\mathop{\sum}\limits_{t=0}^{n} (h_{t} -f_{t} *g_{t})^{2}. \end{align} } (9.22a)

Since all quantities on the right-hand side of equation (9.22a) are fixed except the filter elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}} , we vary the coefficients Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}} to minimize Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} . We differentiate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} with respect to each of the elements Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{i}} and equate the derivatives to zero. This gives Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)} equations which can be solved for the Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (n+1)} elements of the filter. Differentiating Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle E} , we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial E}{\partial f_{i}} =0=\mathop{\sum}\limits_{t=0}^{n} (h_{t} -f_{t} *g_{t})\frac{\partial (f_{t} *g_{t})}{\partial f_{i}} \\ =\mathop{\sum}\limits_{t=0}^{n} \left(h_{t}-\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right)\frac{\partial}{\partial f_{i}} \left(\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right), \end{align} }

using equation (9.2b) to replace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t} *g_{t}} with a summation. The derivative now becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{\partial}{\partial f_{i}} \left(\mathop{\sum}\limits_{t=0}^{n} g_{k} \ f_{t-k}\right)=g_{t-i}, \end{align} }

since the only nonzero term in the differentiation is that in which Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{t}} appears, that is, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle t-k=i} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k=t-i} . Substituting this result, we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{t=0}^{n} \left(h_{t} -\mathop{\sum}\limits_{k}^{} g_{k} \ f_{t-k}\right)g_{t-i} =0=\mathop{\sum}\limits_{t=0}^{n} h_{t} \ g_{t-i}-\mathop{\sum}\limits_{t=0}^{n} \left(\mathop{\sum}\limits_{k}^{} (g_{k} \ f_{t-k})g_{t-i}\right). \end{align} }

The first term is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh}} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i} ) from equations (9.8a) and (9.8b). Interchanging the order of summation in the right-hand term, letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j=t-k} , and summing over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle j} , the equation becomes

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{j=0}^{n} \left(f_{j} \mathop{\sum}\limits_{t=0}^{n} g_{t-j} \ g_{t-i}\right)=\mathop{\sum}\limits_{j=0}^{n} f_{j} \phi _{gh} (i-j) \end{align} }

(see problem 9.21a). Thus we arrive at the normal equations


Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \mathop{\sum}\limits_{j=0}^{n} \phi_{gg} (i-j)f_{j} =\phi _{gh} (i),\qquad i=0, 1, 2, \ldots , n. \end{align} } (9.22b)

Solution

The transforms of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1-2z+3z^{2})} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (3-2z+z^{2})} , so the roots of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{1}{3} (1\pm j\sqrt{2})} and those of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1\pm j\sqrt{2})} , the magnitudes of the roots being Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/\sqrt{3}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{3}} . Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} is maximum-phase and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} is minimum-phase.

The energy of a wavelet at any instant is proportional to its amplitude squared. We thus get for the cumulative energy of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} [1, 5, 14] and for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} , [9, 13, 14], as plotted in Figure 9.22a. The cumulative energy of a minimum-phase wavelet at any instant is always greater than that of any other wavelet with the same amplitude spectrum.

Figure 9.22a.  Cumulative energy.

Problem 9.22b

Calculate three-element Wiener inverse filters assuming the desired output is (i) [1, 0, 0] and (ii) [0, 1, 0], then apply the inverse filters to wavelets Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} .

Solution

To write the normal equations in equation (9.22b) in explicit form, we give Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i} the values 0, 1, and 2 in succession. The result is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi_{gg} (0)f_{0} +\phi_{gg} (-1)f_{1} +\phi_{gg} (-2)f_{2} =\phi_{gh} (0) ,\\ \phi_{gg} (1)f_{0} +\phi_{gg} (0)f_{1} +\phi_{gg} (-1)f_{2} =\phi_{gh} (1) ,\\ \phi_{gg} (2)f_{0} +\phi_{gg} (1)f_{1} +\phi_{gg} (0)f_{2} =\phi_{gh} (2) .\\ \end{align} }

Shaping wavelet A [1,−2, 3] into [1, 0, 0]

To solve these equations, we need the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (\tau)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (\tau)} . Using equations (9.8a) and (9.8e) we get

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi _{gg} (0)=1^{2} +2^{2} +3^{2} =14; \quad \phi_{gg} (1)=\phi_{gg} (-1)=-2-6=-8;\\ \phi_{gg} (2)=\phi_{gg} (-2)=3; \quad \phi_{gh} (0)=1; \quad \phi_{gh} (1)=0; \quad \phi_{gh} (2)=0. \end{align} }

Substituting these values in the normal equations gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} &=1,\\ -8f_{0} +14f_{1} -8f_{2} &=0,\\ 3f_{0} -8f_{1} +14f_{2} &=0. \end{align} }

The solution of these equations can be obtained using Cramer’s rule [see Sheriff and Geldart, 1995, equation (15.3b)]. We first calculate the following determinants:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta =\left|\begin{array}{ccc} {14} {-8} {3} \\ {-8} {14} {-8} \\ {3} {-8} {14} \end{array}\right|\\ =14(14\times 14-8\times 8)-8(-8\times 3+8\times 14)+3(8\times 8-14\times 3)=1210;\\ \Delta _{0} =\left|\begin{array}{ccc} {1} {-8} {3} \\ {0} {14} {-8} \\ {0} {-8} {14} \end{array}\right|=(14\times 14-8\times 8)=132;\\ \Delta _{1} =\left|\begin{array}{ccc} {14} {1} {3} \\ {-8} {0} {-8} \\ {3} {0} {14} \end{array}\right|=(-8\times 3+8\times 14)=88;\\ \Delta _{2} =\left|\begin{array}{ccc} {14} {-8} {1} \\ {-8} {14} {0} \\ {3} {-8} {0} \end{array}\right|=(8\times 8-14\times 3)=22. \end{align} }

Then, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =\Delta_0/\Delta =132/1210=0.1091} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =88/1210=0.0727} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =22/1210= 0.0182} . Applying this filter to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =\left[0.1091,\; 0.0727,\; 0.0182\left]*\right[1,\; -2,\; 3\right]\\ =0.1091 , 0.0727, 0.0182\\ -0.2182,\; -0.1454,\; -0.0364\\ 0.3273, 0.2181, 0, 0546\\ =0.1091 , -0.1455 , 0.2001, 0.1817, 0.0546. \end{align} }

Normalizing the wavelet to make the first element equal to +1, we get the wavelet

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left[1, -1.3336, 1.8341, 1.6654, 0.5005 \right], \end{align} }

which is far from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0,\; 0]} , the rms difference between the two wavelets being 1.43. Thus, it appears that we cannot shape the maximum-phase wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \delta(t)} .

Shaping wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B[3,\; -2,\; 1]} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0]}

Using wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} , we get the following values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (\tau)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (\tau)} :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi_{gg} (0)=14; \phi_{gg} (1)=-8, \phi _{gg} (2)=3;\phi _{gh} (0)=3;\phi _{gh} (1)=0;\phi _{gh} (2)=0. \end{align} }

The normal equations now become

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} &=3,\\ -8f_{0} +14f_{1} -8f_{2} &=0,\\ 3f_{0} -8f_{1} +14f_{2} &=0. \end{align} }

Proceeding as before,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210} from calculation for wavelet A;

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {3} & {-8} & {3} \\ {0} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=3(14\times 14-8\times 8)=396;\\ \Delta _{1} =\left|\begin{array}{ccc} {14} & {3} & {3} \\ {-8} & {0} & {-8} \\ {3} & {0} & {14} \end{array}\right| =-3(-8\times 14+8\times 3)=264;\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {3} \\ {-8} & {14} & {0} \\ {3} & {-8} & {0} \end{array}\right|=3(8\times 8-14\times 3)=66. \end{align} }

The solution is

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{0} =396/1210=0.3273 , f_{1} =264/1210=0.2182, f_{2} =66/1210=0.0545. \end{align} }

Applying the filter [0.3273, 0.2182, 0.0545] gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} &=\left[0.3273,\; 0.2182,\; 0.0545\left]*\right[3,\; -2,\; 1\right]\\ &=0.9819, 0.6546, 0.1635\\ &\qquad\qquad-0.6546,\; -0.4364,\; -0.1090\\ &\qquad\qquad\qquad\qquad0.3273, 0.2182, 0.0545\\ &0.9819, 0.,\quad 0.0544, 0.1092, 0.0545. \end{align} }

Normalizing the first value to 1 gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *g_{t} =[1 , 0, 0.056, 0.114, 0.056], \end{align} }

which is close to the desired wavelet of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [1,\; 0,\; 0]} beause Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} is minimum-phase. The rms difference from the desired output is 0.069.

Shaping wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A[1,\; -2,\; 3]} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [0,\; 1,\; 0]}

We use the values of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gg}(\tau)} from the previous calculations: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gg} (0)=14} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gg} (1)=-8,\,\phi_{gg} (2)=3} ; and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (0)=-2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi_{gh} (1)=1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi _{gh} (2)=0} . The normal equations thus become

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} =-2,\\ -8f_{0} +14f_{1} -8f_{2} =1,\\ 3f_{0} -8f_{1} +14f_{2} =0. \end{align} }

We next calculate the determinants:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210} as before,

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {-2} & {-8} & {3} \\ {1} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=-176,\\ \Delta _{1} =\left|\begin{array}{ccc} {14} & {-2} & {3} \\ {-8} & {1} & {-8} \\ {3} & {0} & {14} \end{array}\right|=11\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {-2} \\ {-8} & {14} & {1} \\ {3} & {-8} & {0} \end{array}\right|=44. \end{align} }

Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =-0.1455} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =0.0091} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =0.0364} and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =\left[-0.1455,\; 0.0091,\; 0.0364\left]*\right[1,\; -2,\; 3\right]= \end{align} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} -0.1455 , 0.0091, 0.0364\\ 0.2910, -0.0182, -0.0728\\ -0.4365 , 0.0273 0.1092\\ -0.1455 , 0.3001, -0.4183, -0.0455 , 0.1092. \end{align} }

Normalizing the second value to 1 gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *A_{t} =[-0.485,\; 1,\; -1.394,\; -0.152,\; 0.364], \end{align} }

and the rms difference from the desired output is 0.764. The result is poor because the input wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} is maximum-phase, but it is much better than when we tried to make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} into [1,0,0].

Shaping wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B[3,\; -2,\; 1]} into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [0,\; 1,\; 0]}

We have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \phi _{gg} (0)=14,\; \phi _{gg} (1)=-8,\; \phi _{gg} (2)=3;\phi _{gh} (0)=-2,\\ \phi _{gh} (1)=3,\; \phi _{gh} (2)=0. \end{align} }

The normal equations are

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} 14\; f_{0} -8f_{1} +3f_{2} =-2,\\ -8f_{0} +14f_{1} -8f_{2} =3,\\ 3f_{0} -8f_{1} +14f_{2} =0. \end{align} }

We next calculate the following determinants:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Delta =1210} as before;

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} \Delta _{0} =\left|\begin{array}{ccc} {-2} & {-8} & {3} \\ {3} & {14} & {-8} \\ {0} & {-8} & {14} \end{array}\right|=0,\\ \Delta _{1} = \left|\begin{array}{ccc} {14} & {-2} & {3} \\ {-8} & {3} & {-8} \\ {3} & {0} & {14} \end{array}\right|=385,\\ \Delta _{2} =\left|\begin{array}{ccc} {14} & {-8} & {-2} \\ {-8} & {14} & {3} \\ {3} & {-8} & {0} \end{array}\right|=220. \end{align} }

We now find that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{0} =0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{1} =-0.3182} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_{2} =0.1818} , and

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} =[0, 0.3182, 0.18181, 0.0182]*[3,\; -2,\; 1]\\ =0, 0.9546, 0.5454\\ 0,\; -0.6364,\; -0.3636,\\ 0, 0.3182, 0.1818\\ =0, 0.9546, -0.0910, -0.0454, 0.1818. \end{align} }

Normalizing the second value to 1 gives

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \begin{align} f_{t} *B_{t} =[0,\; 1,\; -0.0964,\; -0.0476,\; 0.1904], \end{align} }

and the rms difference from the desired output is 0.109. The result is not as good as for the previous case, but it is much better than for the wavelet Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} . Wiener filtering works best when input and desired wavelets have the same phase.

Figure 9.22b.  Original and Wiener-filtered waveforms.; first value = 1 Top row, waveform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} ; bottom row, waveform Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} .

The original and filtered wavelets are shown in Figure 9.22b. Reviewing the errors:

To make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} into [1, 0, 0]: rms error = 1.426;

To make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} into [1, 0, 0]: rms error = 0.069;

To make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A} into [0, 1, 0]: rms error = 0.764;

To make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle B} into [0, 1, 0]: rms error = 0.109.

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