For the regular Farrow structure in Fig. 6, together with the realization in Fig. 9 a total of 33 adders are required for the multiplier block using the RAG-n algorithm in [45]. This is an optimal result since there are 33 different (odd) coefficients as discussed in [58], and, hence, there is no need to apply the slightly more efficient algorithms in [47–49]. Furthermore, 80 structural adders and 118 registers are required for the FIR subfilters. Further, five general multipliers and four additional adders are required (for both the regular Farrow and two-rate based filters). Alternatively, using the proposed structure in Fig. 10 the constant matrix multiplication can be realized using 96 adders using the algorithm in [53]. In addition, 26 structural adders are required. One observation is that separating the symmetric and anti-symmetric subfilters may reduce the complexity as some algorithms work better for fewer columns. If this is utilized, the number of adders can be reduced to 107 by applying the algorithm in [52] to the resulting two matrices and adding and subtracting the results. The number of registers is now decreased to 30, whereas the number of general multipliers and additional adders is constant.

For the two-rate based structure in Fig. 7, the HB filter requires 43 structural adders and 29 registers. The sum of products are realized by computing the corresponding multiplier

10.5772/52198

209

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[5] T. Saramäki, Y. C. Lim, and R. Yang. The synthesis of half-band filter using frequency-response masking technique. *IEEE Trans. Circuits Syst. II*, 42(1):58–60, Jan.

Two-Rate Based Structures for Computationally Efficient Wide-Band FIR Systems

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1995.

2004.

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**Table 5.** Results of Example 5.

block with RAG-n and transposing it. For the constant matrix multiplication, 38 adders are required. This number is not reduced by separating the symmetric and anti-symmetric subfilters. A total of six additional registers are required as well as the five general multipliers and four additional adders.

The results are summarized in Table 5. It is seen that the two-rate based structure still has the lowest complexity for most implementation technologies as five registers will typically be less complex to implement than 26 adders. Furthermore, whereas the use of transposed direct form subfilters for the Farrow filter, as proposed in [50, 57], reduces the number of adders related to the multiplication, it is for the two-rate approach still more efficient to use direct form subfilters.

## **6. Conclusion**

This chapter has reviewed recent two-rate based structures and their design for obtaining efficient wide-band FIR systems. Left-band, right-band, and mid-band systems, as well as single-function and multi-function systems, were covered. Several design examples were given, for differentiators and VFD filters (a special case of multi-function systems), revealing dramatic complexity savings for wide-band specifications. More details can be found in [12–19].

## **Author details**

Håkan Johansson and Oscar Gustafsson

Division of Electronics Systems, Department of Electrical Engineering, Linköping University, Sweden

## **References**

[5] T. Saramäki, Y. C. Lim, and R. Yang. The synthesis of half-band filter using frequency-response masking technique. *IEEE Trans. Circuits Syst. II*, 42(1):58–60, Jan. 1995.

20 Digital Filters and Signal Processing

**Table 5.** Results of Example 5.

direct form subfilters.

**6. Conclusion**

**Author details**

Håkan Johansson and Oscar Gustafsson

*Relat. Technol.*, 1(2):103–109, Mar.–Apr. 1990.

[12–19].

Sweden

**References**

and four additional adders.

**Filter structure Mult Add Registers**

Farrow with transposed direct-form subfilters 5 117 118 Farrow with direct-form subfilters 5 111 30 Proposed two-rate based structure 5 85 35

block with RAG-n and transposing it. For the constant matrix multiplication, 38 adders are required. This number is not reduced by separating the symmetric and anti-symmetric subfilters. A total of six additional registers are required as well as the five general multipliers

The results are summarized in Table 5. It is seen that the two-rate based structure still has the lowest complexity for most implementation technologies as five registers will typically be less complex to implement than 26 adders. Furthermore, whereas the use of transposed direct form subfilters for the Farrow filter, as proposed in [50, 57], reduces the number of adders related to the multiplication, it is for the two-rate approach still more efficient to use

This chapter has reviewed recent two-rate based structures and their design for obtaining efficient wide-band FIR systems. Left-band, right-band, and mid-band systems, as well as single-function and multi-function systems, were covered. Several design examples were given, for differentiators and VFD filters (a special case of multi-function systems), revealing dramatic complexity savings for wide-band specifications. More details can be found in

Division of Electronics Systems, Department of Electrical Engineering, Linköping University,

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**Chapter 9**

**Provisional chapter**

**Analytical Approach for Synthesis of Minimum**

**Analytical Approach for Synthesis of Minimum**

*L*2**-Sensitivity Realizations for State-Space Digital**

Shunsuke Yamaki, Masahide Abe and

Masahide Abe and Masayuki Kawamata

Additional information is available at the end of the chapter

Additional information is available at the end of the chapter

Masayuki Kawamata

Shunsuke Yamaki,

10.5772/52194

**1. Introduction**

the minimum *L*2-sensitivity.

**Filters**

http://dx.doi.org/10.5772/52194

*L2***-Sensitivity Realizations for State-Space Digital Filters**

On the fixed-point implementation of digital filters, undesirable finite-word-length (FWL) effects arise due to the coefficient truncation and arithmetic roundoff. These FWL effects must be reduced as small as possible because such effects may cause serious degradation of characteristic of digital filters. *L*2-sensitivity is one of the evaluation functions which evaluate the coefficient quantization effects of state-space digital filters [1–14]. The *L*2-sensitivity minimization is quite beneficial technique for the synthesis of high-accuracy digital filter

To the *L*2-sensitivity minimization problem, Yan *et al.* [1] and Hinamoto *et al.* [2] proposed solutions using iterative calculations. Both of the solutions in [1] and [2] try to solve nonlinear equations by successive approximation. Their solutions do not guarantee that the *L*2-sensitivity surely converges to the minimum *L*2-sensitivity since their solutions are not analytical solutions. It is necessary to derive some analytical solutions to the *L*2-sensitivity minimization problem in order to guarantee that their conventional solutions surely derive

This chapter presents analytical approach for synthesis of the minimum *L*2-sensitivity realizations for state-space digital filters. In Section 3, we derive closed form solutions to the *L*2-sensitivity minimization problem for second-order digital filters [12, 13]. This problem can be converted into the problem to find the solution to fourth-degree polynomial equation of constant coefficients, which can be algebraically solved in closed form. Next, we reveal that the *L*2-sensitivity minimization problem can be solved analytically for arbitrary

> ©2012 Yamaki et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Yamaki et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Yamaki et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

structures, which achieves quite low-coefficient quantization error.

**Provisional chapter**

## **Analytical Approach for Synthesis of Minimum** *L2***-Sensitivity Realizations for State-Space Digital Filters Analytical Approach for Synthesis of Minimum** *L*2**-Sensitivity Realizations for State-Space Digital Filters**

Shunsuke Yamaki, Masahide Abe and Masayuki Kawamata Shunsuke Yamaki, Masahide Abe and Masayuki Kawamata

Additional information is available at the end of the chapter Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/52194 10.5772/52194

## **1. Introduction**

24 Digital Filters and Signal Processing

212 Digital Filters and Signal Processing

1–4, 2003.

40(11):651–652, May 2004.

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53–56, Espoo, Finland, June 9–11, 2004.

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[51] O. Gustafsson and H. Johansson. Efficient implementation of FIR filter based rational sampling rate converters using constant matrix multiplication. In *Proc. Fortieth Asilomar*

[52] A. G. Dempster, O. Gustafsson, and J. O. Coleman. Towards an algorithm for matrix multiplier blocks. In *Proc. European Conf. Circuit Theory Design*, Krakow, Poland, Sept.

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[54] N. Boullis and A. Tisserand. Some optimizations of hardware multiplication by

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[56] A. G. Dempster and N. P. Murphy. Efficient interpolators and filter banks using

[57] M. Abbas, O. Gustafsson, and H. Johansson. On the fixed-point implementation of fractional-delay filters based on the farrow structure. *IEEE Trans. Circuits Syst I: Regular*

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*Conf. Signals, Systems and Computers ACSSC '06*, pages 888–891, 2006.

constant matrices. *IEEE Trans. Computers*, 54(10):1271–1282, Oct. 2005.

multiplier blocks. *IEEE Trans. Signal Processing*, 48(1):257–261, Jan. 2000.

On the fixed-point implementation of digital filters, undesirable finite-word-length (FWL) effects arise due to the coefficient truncation and arithmetic roundoff. These FWL effects must be reduced as small as possible because such effects may cause serious degradation of characteristic of digital filters. *L*2-sensitivity is one of the evaluation functions which evaluate the coefficient quantization effects of state-space digital filters [1–14]. The *L*2-sensitivity minimization is quite beneficial technique for the synthesis of high-accuracy digital filter structures, which achieves quite low-coefficient quantization error.

To the *L*2-sensitivity minimization problem, Yan *et al.* [1] and Hinamoto *et al.* [2] proposed solutions using iterative calculations. Both of the solutions in [1] and [2] try to solve nonlinear equations by successive approximation. Their solutions do not guarantee that the *L*2-sensitivity surely converges to the minimum *L*2-sensitivity since their solutions are not analytical solutions. It is necessary to derive some analytical solutions to the *L*2-sensitivity minimization problem in order to guarantee that their conventional solutions surely derive the minimum *L*2-sensitivity.

This chapter presents analytical approach for synthesis of the minimum *L*2-sensitivity realizations for state-space digital filters. In Section 3, we derive closed form solutions to the *L*2-sensitivity minimization problem for second-order digital filters [12, 13]. This problem can be converted into the problem to find the solution to fourth-degree polynomial equation of constant coefficients, which can be algebraically solved in closed form. Next, we reveal that the *L*2-sensitivity minimization problem can be solved analytically for arbitrary

Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Yamaki et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2013 Yamaki et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

©2012 Yamaki et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative

**Figure 1.** Block diagram of a state-space digital filter.

filter order if second-order modes are all equal [14] in Section 4. We derive a general expression of the transfer function of digital filters with all second-order modes equal. We show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter. Furthermore, we show the absence of limit cycles of the minimum *L*2-sensitivity realizations [11] in Section 5. The minimum *L*2-sensitivity realization without limit cycles can be synthesized by selecting an appropriate orthogonal matrix in the coordinate transformation matrix.

## **2. Preliminaries**

This section gives the preliminaries in order to lay groundwork for the main topics of this chapter, which appear in later sections. In Subsection 2.1, we begin with introduction of state-space digital filters. Subsection 2.2 provides the introduction of *L*2-sensitivity. Subsection 2.3 explains coordinate transformations, the operation for changing the structures of state-space digital filters under the transfer function invariant. Subsection 2.4 formulates the *L*2-sensitivity minimization problem.

#### **2.1. State-space digital filters**

It is beneficial to introduce the state-space representation for the synthesis of high accuracy digital filters. For a given *N*th-order transfer function *H*(*z*), a state-space digital filter can be described by the following state-space equations:

$$\mathbf{x}(n+1) = \mathbf{A}\mathbf{x}(n) + \mathbf{b}u(n) \tag{1}$$

$$y(n) = \mathfrak{cx}(n) + du(n) \tag{2}$$

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*<sup>H</sup>*(*z*) = *<sup>c</sup>*(*z<sup>I</sup>* − *<sup>A</sup>*)−1*<sup>b</sup>* + *<sup>d</sup>*. (3)

*b* + *d*. (4)

In this chapter, the state-space representation (*A*, *b*, *c*, *d*) is assumed to be a minimal realization of *H*(*z*), that is, the state-space representation (*A*, *b*, *c*, *d*) is controllable and

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

adj(*zI* − *A*) det(*zI* − *A*)

We know from the above equation that the poles of *H*(*z*) are the solutions of the characteristic equation det(*zI* − *A*) = 0, that is, the eigenvalues of the coefficient matrix *A*. Since we assume that the transfer function *H*(*z*) is stable, all absolute values of the eigenvalues of the coefficient matrix *A* are less than unity. It follows from Eq. (4) that the absolute values of

The *L*2-sensitivity is one of the measurements which evaluate coefficient quantization errors of digital filters. The *L*2-sensitivity of the filter *H*(*z*) with respect to the realization (*A*, *b*, *c*, *d*)

>

  2 *dω* +

*N* ∑ *l*=1

>

2

2

*∂H*(*z*) *∂c*

*<sup>∂</sup><sup>b</sup>* <sup>=</sup> *<sup>G</sup>T*(*z*), *<sup>∂</sup>H*(*z*)

1 2*π*  <sup>2</sup>*<sup>π</sup>* 0

*<sup>F</sup>*(*z*)=(*z<sup>I</sup>* − *<sup>A</sup>*)−1*<sup>b</sup>* (7) *<sup>G</sup>*(*z*) = *<sup>c</sup>*(*z<sup>I</sup>* − *<sup>A</sup>*)−<sup>1</sup> (8)

  *∂H*(*ejω*) *∂cl*

*<sup>∂</sup><sup>c</sup>* <sup>=</sup> *<sup>F</sup>T*(*z*) (6)

2 *dω*

(5)

2 *dω*

observable. The transfer function *H*(*z*) in Eq. (3) can be rewritten as

*H*(*z*) = *c*

coefficient matrices (*A*, *b*, *c*, *d*) as

poles of *H*(*z*) are less than unity.

*S*(*A*, *b*, *c*)=

*N* ∑ *k*=1

+ *N* ∑ *k*=1

=

to the coefficient matrices are described by

*∂H*(*z*)

where *F*(*z*) and *G*(*z*) are defined by

*N* ∑ *l*=1

*∂H*(*z*) *∂A*

1 2*π*

1 2*π*

>

2

2 +

*<sup>∂</sup><sup>A</sup>* <sup>=</sup> *<sup>G</sup>T*(*z*)*FT*(*z*), *<sup>∂</sup>H*(*z*)

 <sup>2</sup>*<sup>π</sup>* 0

 <sup>2</sup>*<sup>π</sup>* 0

  *∂H*(*ejω*) *∂akl*

*∂H*(*ejω*) *∂bk*

>

2

2 +

where �·�<sup>2</sup> denotes the *L*2-norm. The derivatives of the transfer function *H*(*z*) with respect

*∂H*(*z*) *∂b*

**2.2.** *L*2**-sensitivity**

is defined by

where *x*(*n*) ∈ ℜ*N*×<sup>1</sup> is a state variable vector, *u*(*n*) ∈ ℜ is a scalar input, *y*(*n*) ∈ ℜ is a scalar output, and *A* ∈ ℜ*N*×*N*, *b* ∈ ℜ*N*<sup>×</sup>1, *c* ∈ ℜ1×*N*, *d* ∈ ℜ are real constant matrices called coefficient matrices. The block diagram of the state-space digital filter (*A*, *b*, *c*, *d*) is shown in Fig. 1(a). In case of second-order digital filters, the block diagram in Fig. 1(a) can be rewritten as shown in Fig. 1(b). The transfer function *H*(*z*) is described in terms of the <sup>214</sup> Digital Filters and Signal Processing Analytical Approach for Synthesis of Minimum *<sup>L</sup>*2-Sensitivity Realizations for State-Space Digital Filters 3 10.5772/52194 Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters http://dx.doi.org/10.5772/52194 215

coefficient matrices (*A*, *b*, *c*, *d*) as

2 Digital Filters and Signal Processing

A

z−<sup>1</sup>

a<sup>22</sup>

*x*(*n* + 1)= *Ax*(*n*) + *bu*(*n*) (1) *y*(*n*)=*cx*(*n*) + *du*(*n*) (2)

u(n) y(n)

z−<sup>1</sup>

a<sup>11</sup>

a<sup>12</sup> a<sup>21</sup>

b1

b2

(a) *N*th-order digital filter (b) Second-order digital filter

filter order if second-order modes are all equal [14] in Section 4. We derive a general expression of the transfer function of digital filters with all second-order modes equal. We show that the general expression is obtained by a frequency transformation on a first-order prototype FIR digital filter. Furthermore, we show the absence of limit cycles of the minimum *L*2-sensitivity realizations [11] in Section 5. The minimum *L*2-sensitivity realization without limit cycles can be synthesized by selecting an appropriate orthogonal matrix in the

This section gives the preliminaries in order to lay groundwork for the main topics of this chapter, which appear in later sections. In Subsection 2.1, we begin with introduction of state-space digital filters. Subsection 2.2 provides the introduction of *L*2-sensitivity. Subsection 2.3 explains coordinate transformations, the operation for changing the structures of state-space digital filters under the transfer function invariant. Subsection 2.4 formulates

It is beneficial to introduce the state-space representation for the synthesis of high accuracy digital filters. For a given *N*th-order transfer function *H*(*z*), a state-space digital filter can be

where *x*(*n*) ∈ ℜ*N*×<sup>1</sup> is a state variable vector, *u*(*n*) ∈ ℜ is a scalar input, *y*(*n*) ∈ ℜ is a scalar output, and *A* ∈ ℜ*N*×*N*, *b* ∈ ℜ*N*<sup>×</sup>1, *c* ∈ ℜ1×*N*, *d* ∈ ℜ are real constant matrices called coefficient matrices. The block diagram of the state-space digital filter (*A*, *b*, *c*, *d*) is shown in Fig. 1(a). In case of second-order digital filters, the block diagram in Fig. 1(a) can be rewritten as shown in Fig. 1(b). The transfer function *H*(*z*) is described in terms of the

x1(n)

c1

d

x2(n)

c2

b c

**Figure 1.** Block diagram of a state-space digital filter.

coordinate transformation matrix.

the *L*2-sensitivity minimization problem.

described by the following state-space equations:

**2.1. State-space digital filters**

**2. Preliminaries**

d

u(n) x(n) y(n) z−<sup>1</sup>

$$H(z) = \mathfrak{c}(zI - A)^{-1}\mathfrak{b} + d.\tag{3}$$

In this chapter, the state-space representation (*A*, *b*, *c*, *d*) is assumed to be a minimal realization of *H*(*z*), that is, the state-space representation (*A*, *b*, *c*, *d*) is controllable and observable. The transfer function *H*(*z*) in Eq. (3) can be rewritten as

$$H(z) = c \frac{\text{adj}(zI - A)}{\det(zI - A)} b + d. \tag{4}$$

We know from the above equation that the poles of *H*(*z*) are the solutions of the characteristic equation det(*zI* − *A*) = 0, that is, the eigenvalues of the coefficient matrix *A*. Since we assume that the transfer function *H*(*z*) is stable, all absolute values of the eigenvalues of the coefficient matrix *A* are less than unity. It follows from Eq. (4) that the absolute values of poles of *H*(*z*) are less than unity.

#### **2.2.** *L*2**-sensitivity**

The *L*2-sensitivity is one of the measurements which evaluate coefficient quantization errors of digital filters. The *L*2-sensitivity of the filter *H*(*z*) with respect to the realization (*A*, *b*, *c*, *d*) is defined by

$$\begin{split} S(\mathbf{A}, \mathbf{b}, \mathbf{c}) &= \sum\_{k=1}^{N} \sum\_{l=1}^{N} \frac{1}{2\pi} \int\_{0}^{2\pi} \left| \frac{\partial H(e^{j\omega})}{\partial a\_{kl}} \right|^{2} d\omega \\ &+ \sum\_{k=1}^{N} \frac{1}{2\pi} \int\_{0}^{2\pi} \left| \frac{\partial H(e^{j\omega})}{\partial b\_{k}} \right|^{2} d\omega + \sum\_{l=1}^{N} \frac{1}{2\pi} \int\_{0}^{2\pi} \left| \frac{\partial H(e^{j\omega})}{\partial c\_{l}} \right|^{2} d\omega \\ &= \left\| \frac{\partial H(z)}{\partial \mathbf{A}} \right\|\_{2}^{2} + \left\| \frac{\partial H(z)}{\partial \mathbf{b}} \right\|\_{2}^{2} + \left\| \frac{\partial H(z)}{\partial c} \right\|\_{2}^{2} \end{split} \tag{5}$$

where �·�<sup>2</sup> denotes the *L*2-norm. The derivatives of the transfer function *H*(*z*) with respect to the coefficient matrices are described by

$$\frac{\partial H(z)}{\partial \mathbf{A}} = \mathbf{G}^T(z)\mathbf{F}^T(z), \ \frac{\partial H(z)}{\partial \mathbf{b}} = \mathbf{G}^T(z), \ \frac{\partial H(z)}{\partial \mathbf{c}} = \mathbf{F}^T(z) \tag{6}$$

where *F*(*z*) and *G*(*z*) are defined by

$$\mathbf{F}(z) = (z\mathbf{I} - \mathbf{A})^{-1}\mathbf{b} \tag{7}$$

$$\mathbf{G}(z) = \mathbf{c}(z\mathbf{I} - \mathbf{A})^{-1} \tag{8}$$

respectively. Substituting Eqs. (6) into Eq. (5), the *L*2-sensitivity can be rewritten as

$$\begin{split} S(\mathbf{A}, \mathbf{b}, \mathbf{c}) &= \left\| \frac{\partial H(\mathbf{z})}{\partial \mathbf{A}} \right\|\_{2}^{2} + \left\| \frac{\partial H(\mathbf{z})}{\partial \mathbf{b}} \right\|\_{2}^{2} + \left\| \frac{\partial H(\mathbf{z})}{\partial \mathbf{c}} \right\|\_{2}^{2} \\ &= \left\| \mathbf{G}^{T}(\mathbf{z}) \mathbf{F}^{T}(\mathbf{z}) \right\|\_{2}^{2} + \left\| \mathbf{G}^{T}(\mathbf{z}) \right\|\_{2}^{2} + \left\| \mathbf{F}^{T}(\mathbf{z}) \right\|\_{2}^{2}. \end{split} \tag{9}$$

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Second-order modes are defined by the square roots of the eigenvalues *θi*'s as follows [3, 5]:

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

In the field of digital signal processing, the controllability and observability Gramians are

It is well known that the number of state-space realizations of a transfer function is infinite since choice of the state variable vector *x*(*n*) is not unique. We can change the filter structures of state-space digital filters by the operation called *coordinate transformation* under the transfer

is applied to a filter structure (*A*, *b*, *c*, *d*), we obtain a new filter structure which has the

It should be noted that the coordinate transformation does not affect the transfer function

*<sup>H</sup>*(*z*)=*c*(*z<sup>I</sup>* − *<sup>A</sup>*)−1*<sup>b</sup>* + *<sup>d</sup>* =*c*(*z<sup>I</sup>* − *<sup>A</sup>*)−1*<sup>b</sup>* + *<sup>d</sup>*

It implies that there exist infinite filter structures for a given transfer function *H*(*z*) since nonsingular *N* × *N* matrices exist infinitely. Therefore, one can synthesize infinite filter structures by the coordinate transformation with keeping the transfer function invariant. Under the coordinate transformation by the nonsingular matrix *T*, the general controllability Gramian *K<sup>i</sup>* and the general observability Gramian *W<sup>i</sup>* are transformed into *K<sup>i</sup>* and *W<sup>i</sup>* given

Let *T* be a nonsingular *N* × *N* real matrix. If a coordinate transformation defined by

also called the covariance and noise matrices of the filter (*A*, *b*, *c*, *d*), respectively.

**2.3. Coordinate transformations**

function invariant.

*H*(*z*), that is,

by

following coefficient matrices:

respectively. Letting *i* = 0 in Eqs. (20) yields

(*θ*1, ··· , *θN*) = Eigenvalues of *K*0*W*0. (16)

*<sup>x</sup>*¯(*n*) = *<sup>T</sup>*−1*x*(*n*) (17)

= *H*(*z*). (19)

(*Ki*,*Wi*)=(*T*−1*KiT*−*T*, *<sup>T</sup>TWiT*) (20)

(*A*, *<sup>b</sup>*, *<sup>c</sup>*, *<sup>d</sup>*)=(*T*−1*AT*, *<sup>T</sup>*−1*b*, *cT*, *<sup>d</sup>*). (18)

We can express the *L*2-sensitivity *S*(*A*, *b*, *c*) by using complex integral as [1]

$$\begin{split} S(\mathbf{A}, \mathbf{b}, \mathbf{c}) &= \text{tr}\left[\frac{1}{2\pi i} \oint\_{|z|=1} \mathbf{F}(z) \mathbf{G}(z) (\mathbf{F}(z) \mathbf{G}(z))^\dagger \frac{dz}{z}\right] \\ &+ \text{tr}\left[\frac{1}{2\pi j} \oint\_{|z|=1} \mathbf{G}^\dagger(z) \mathbf{G}(z) \frac{dz}{z}\right] + \text{tr}\left[\frac{1}{2\pi j} \oint\_{|z|=1} \mathbf{F}(z) \mathbf{F}^\dagger(z) \frac{dz}{z}\right]. \end{split} \tag{10}$$

Applying Parseval's relation to Eq. (10), Hinamoto *et al.* expressed the *L*2-sensitivity in terms of the general Gramians such as [2]

$$S(\mathbf{A}, \mathbf{b}, \mathbf{c}) = \text{tr}(\mathbf{W}\_0)\text{tr}(\mathbf{K}\_0) + \text{tr}(\mathbf{W}\_0) + \text{tr}(\mathbf{K}\_0) + 2\sum\_{i=1}^{\infty} \text{tr}(\mathbf{W}\_i)\text{tr}(\mathbf{K}\_i). \tag{11}$$

The general controllability Gramian *K<sup>i</sup>* and the general observability Gramian *W<sup>i</sup>* in Eq. (11) are defined as the solutions to the following Lyapunov equations:

$$\mathbf{K}\_{i} = \mathbf{A}\mathbf{K}\_{i}\mathbf{A}^{T} + \frac{1}{2} \left(\mathbf{A}^{i}\mathbf{b}\mathbf{b}^{T} + \mathbf{b}\mathbf{b}^{T}(\mathbf{A}^{T})^{i}\right) \tag{12}$$

$$\mathbf{W}\_{l} = \mathbf{A}^{T}\mathbf{W}\_{l}\mathbf{A} + \frac{1}{2}\left(\mathbf{c}^{T}\mathbf{c}\mathbf{A}^{i} + (\mathbf{A}^{T})^{i}\mathbf{c}^{T}\mathbf{c}\right) \tag{13}$$

for *i* = 0, 1, 2, ··· , respectively. The general controllability and observability Gramians are natural expansions of the controllability and observability Gramians, respectively. Letting *i* = 0 in Eqs. (12) and (13), we have the Lyapunov equations for the controllability Gramian *K*<sup>0</sup> and the observability Gramian *W*<sup>0</sup> as follows:

$$\mathbf{K}\_0 = \mathbf{A}\mathbf{K}\_0\mathbf{A}^T + \mathbf{b}\mathbf{b}^T \tag{14}$$

$$\mathbf{W}\_0 = \mathbf{A}^T \mathbf{W}\_0 \mathbf{A} + \mathbf{c}^T \mathbf{c}.\tag{15}$$

The controllability Gramian *K*<sup>0</sup> and the observability Gramian *W*<sup>0</sup> are positive definite symmetric, and the eigenvalues *θ*<sup>2</sup> *<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>N</sup>*) of the matrix product *<sup>K</sup>*0*W*<sup>0</sup> are all positive. Second-order modes are defined by the square roots of the eigenvalues *θi*'s as follows [3, 5]:

$$(\theta\_1, \dots, \theta\_N) = \sqrt{\text{Eigenvalues of } \mathbf{K}\_0 \mathbf{W}\_0}. \tag{16}$$

In the field of digital signal processing, the controllability and observability Gramians are also called the covariance and noise matrices of the filter (*A*, *b*, *c*, *d*), respectively.

#### **2.3. Coordinate transformations**

4 Digital Filters and Signal Processing

*S*(*A*, *b*, *c*)=tr

of the general Gramians such as [2]

*S*(*A*, *b*, *c*)=

 1 2*πj* |*z*|=1

+tr 1 2*πj* |*z*|=1

=  *∂H*(*z*) *∂A*

respectively. Substituting Eqs. (6) into Eq. (5), the *L*2-sensitivity can be rewritten as

*GT*(*z*)*FT*(*z*)

We can express the *L*2-sensitivity *S*(*A*, *b*, *c*) by using complex integral as [1]

*S*(*A*, *b*, *c*) = tr(*W*0)tr(*K*0) + tr(*W*0) + tr(*K*0) + 2

are defined as the solutions to the following Lyapunov equations:

*K<sup>i</sup>* = *AKiA<sup>T</sup>* +

*W<sup>i</sup>* = *ATWiA* +

*K*<sup>0</sup> and the observability Gramian *W*<sup>0</sup> as follows:

symmetric, and the eigenvalues *θ*<sup>2</sup>

2

*∂H*(*z*) *∂b*

  2

*∂H*(*z*) *∂c*

  2

2

*F*(*z*)*F*†(*z*)

*dz z*

tr(*Wi*)tr(*Ki*). (11)

. (10)

(12)

(13)

. (9)

2 +

> *z*

*bb<sup>T</sup>* + *bbT*(*AT*)*<sup>i</sup>*

*cTcA<sup>i</sup>* + (*AT*)*<sup>i</sup>*

∞ ∑ *i*=1

*K*<sup>0</sup> = *AK*0*A<sup>T</sup>* + *bb<sup>T</sup>* (14) *W*<sup>0</sup> = *ATW*0*A* + *cTc*. (15)

*<sup>i</sup>* (*<sup>i</sup>* <sup>=</sup> 1, ··· , *<sup>N</sup>*) of the matrix product *<sup>K</sup>*0*W*<sup>0</sup> are all positive.

*cTc*

2 +

> 2 2 + *GT*(*z*) 2 2 + *FT*(*z*) 2 2

*<sup>F</sup>*(*z*)*G*(*z*)(*F*(*z*)*G*(*z*))† *dz*

Applying Parseval's relation to Eq. (10), Hinamoto *et al.* expressed the *L*2-sensitivity in terms

The general controllability Gramian *K<sup>i</sup>* and the general observability Gramian *W<sup>i</sup>* in Eq. (11)

1 2 *Ai*

> 1 2

for *i* = 0, 1, 2, ··· , respectively. The general controllability and observability Gramians are natural expansions of the controllability and observability Gramians, respectively. Letting *i* = 0 in Eqs. (12) and (13), we have the Lyapunov equations for the controllability Gramian

The controllability Gramian *K*<sup>0</sup> and the observability Gramian *W*<sup>0</sup> are positive definite

*dz z* + tr 1 2*πj* |*z*|=1

*G*†(*z*)*G*(*z*)

It is well known that the number of state-space realizations of a transfer function is infinite since choice of the state variable vector *x*(*n*) is not unique. We can change the filter structures of state-space digital filters by the operation called *coordinate transformation* under the transfer function invariant.

Let *T* be a nonsingular *N* × *N* real matrix. If a coordinate transformation defined by

$$\bar{\mathfrak{x}}(n) = T^{-1}\mathfrak{x}(n) \tag{17}$$

is applied to a filter structure (*A*, *b*, *c*, *d*), we obtain a new filter structure which has the following coefficient matrices:

$$(\overline{A}, \overline{b}, \overline{c}, \overline{d}) = (T^{-1} A \mathbf{T}, T^{-1} b, \mathbf{c} \mathbf{T}, d). \tag{18}$$

It should be noted that the coordinate transformation does not affect the transfer function *H*(*z*), that is,

$$\begin{split} \overline{H}(z) &= \overline{c}(zI - \overline{A})^{-1}\overline{b} + \overline{d} \\ &= c(zI - A)^{-1}b + d \\ &= H(z). \end{split} \tag{19}$$

It implies that there exist infinite filter structures for a given transfer function *H*(*z*) since nonsingular *N* × *N* matrices exist infinitely. Therefore, one can synthesize infinite filter structures by the coordinate transformation with keeping the transfer function invariant.

Under the coordinate transformation by the nonsingular matrix *T*, the general controllability Gramian *K<sup>i</sup>* and the general observability Gramian *W<sup>i</sup>* are transformed into *K<sup>i</sup>* and *W<sup>i</sup>* given by

$$(\overline{\mathbf{K}}\_{l}, \overline{\mathbf{W}}\_{l}) = (\mathbf{T}^{-1}\mathbf{K}\_{l}\mathbf{T}^{-T}, \mathbf{T}^{T}\mathbf{W}\_{l}\mathbf{T}) \tag{20}$$

respectively. Letting *i* = 0 in Eqs. (20) yields

$$(\overline{\mathbf{K}}\_0, \overline{\mathbf{W}}\_0) = (T^{-1}\mathbf{K}\_0 T^{-T}, T^T \mathbf{W}\_0 \mathbf{T}) \tag{21}$$

From Eqs. (21), we have

$$\overline{\mathbf{K}}\_0 \overline{\mathbf{W}}\_0 = T^{-1} \mathbf{K}\_0 \mathbf{W}\_0 T \tag{22}$$

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*<sup>S</sup>*(*P*)=tr(*W*0*P*)tr(*K*0*P*−1) + tr(*W*0*P*) + tr(*K*0*P*−1)

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

where *P* = *TTT*. The problem we consider here is to derive *the optimal positive definite*

If one can obtain the optimal positive definite symmetric matrix *P*opt, the optimal coordinate

where *U* is an *arbitrary* orthogonal matrix. It implies that the minimum *L*2-sensitivity realizations exist infinitely for a given digital filter *H*(*z*). The minimum *L*2-sensitivity

This section proposes analytical synthesis of the minimum *L*2-sensitivity realizations for second-order digital filters. We propose closed form solutions to the *L*2-sensitivity minimization problem of second-order state-space digital filters. The proposed closed form solutions can greatly save the computation time and guarantee that the *L*2-sensitivity obtained by the iterative algorithm of the conventional method surely converges to the theoretical minimum. We show that the *L*2-sensitivity is expressed by a simple linear combination of exponential functions, and we can obtain the minimum *L*2-sensitivity realization by solving a fourth degree polynomial equation of constant coefficients in closed

We adopt the balanced realization (*A*b, *b*b, *c*b, *d*b) as the initial realization to synthesize the minimum *L*2-sensitivity realization. The coefficient matrices (*A*b, *b*b, *c*b, *d*b) satisfy symmetric

<sup>b</sup> <sup>=</sup> **<sup>Σ</sup>***A*b**Σ**, *<sup>c</sup><sup>T</sup>*

*AT*

1 2

*T*opt = *P*

**3. Analytical solutions to the** *L*2**-sensitivity minimization problem for**

*symmetric matrix P*opt, which gives the global minimum of *S*(*P*) as follows:

*∂S*(*P*) *∂P P*=*P*opt

tr(*WiP*)tr(*KiP*−1) (26)

= **0**. (27)

opt*<sup>U</sup>* (28)

<sup>b</sup> <sup>=</sup> **<sup>Σ</sup>***b*<sup>b</sup> (29)

+2 ∞ ∑ *i*=1

realizations have freedom for orthogonal transformations.

transformation matrix *T*opt is given by

**second-order digital filters**

form without iterative calculations [12, 13].

**3.1. Problem formulation**

properties as follows:

which shows that *K*0*W*<sup>0</sup> has the same eigenvalues of *K*0*W*0. Thus, the second order modes defined by Eq. (16) are invariant under the coordinate transformations. It implies that second-order modes depends on only the transfer function.

## **2.4.** *L*2**-sensitivity minimization problem**

The value of *L*2-sensitivity depends on not only the transfer function *H*(*z*) but also the coordinate transformation matrix *<sup>T</sup>*. The *<sup>L</sup>*2-sensitivity of the filter (*T*−1*AT*, *<sup>T</sup>*−1*b*, *cT*, *<sup>d</sup>*) can be expressed in terms of the complex integral as

$$\begin{split} &S(T^{-1}AT, T^{-1}b, cT) \\ &= \text{tr}\left[\frac{1}{2\pi i}\oint\_{|z|=1} T^{-1}F(z)\mathbf{G}(z)\mathbf{T}T^{T}(F(z)\mathbf{G}(z))^{\dagger}T^{-T}\frac{dz}{z}\right] \\ &+ \text{tr}\left[\frac{1}{2\pi i}\oint\_{|z|=1} \mathbf{T}^{T}\mathbf{G}^{\dagger}(z)\mathbf{G}(z)\mathbf{T}\frac{dz}{z}\right] + \text{tr}\left[\frac{1}{2\pi i}\oint\_{|z|=1} \mathbf{T}^{-1}F(z)\mathbf{F}^{\dagger}(z)\mathbf{T}^{-T}\frac{dz}{z}\right] \tag{23} \end{split} \tag{23}$$

or in terms of the general Gramians as

$$\begin{split} &S(T^{-1}AT, T^{-1}b, cT) \\ &= \text{tr}(T^{T}\mathbf{W}\_{0}T)\text{tr}(T^{-1}\mathbf{K}\_{0}T^{-T}) + \text{tr}(T^{T}\mathbf{W}\_{0}T) + \text{tr}(T^{-1}\mathbf{K}\_{0}T^{-T}) \\ &+ 2\sum\_{i=1}^{\infty} \text{tr}(T^{T}\mathbf{W}\_{i}T)\text{tr}(T^{-1}\mathbf{K}\_{i}T^{-T}). \tag{24} \end{split} \tag{24}$$

The *L*2-sensitivity (23) can be expressed as the function of the positive definite symmetric matrix *P* as follows [1]:

$$\begin{split} S(\mathbf{P}) &= \text{tr}\left[\frac{1}{2\pi j}\oint\_{|z|=1} F(z)\mathbf{G}(z)\mathbf{P}(\mathbf{F}(z)\mathbf{G}(z))^\dagger \mathbf{P}^{-1}\frac{dz}{z}\right] \\ &+ \text{tr}\left[\frac{1}{2\pi j}\oint\_{|z|=1} \mathbf{G}^\dagger(z)\mathbf{G}(z)\mathbf{P}\frac{dz}{z}\right] + \text{tr}\left[\frac{1}{2\pi j}\oint\_{|z|=1} \mathbf{F}(z)\mathbf{F}^\dagger(z)\mathbf{P}^{-1}\frac{dz}{z}\right] \end{split} \tag{25}$$

where *P* = *TTT*. Similarly, the *L*2-sensitivity (24) can be expressed as the function of the positive definite symmetric matrix *P* as follows [2]:

$$\begin{split} S(\mathcal{P}) &= \text{tr}(\mathbf{W}\_0 \mathbf{P}) \text{tr}(\mathbf{K}\_0 \mathbf{P}^{-1}) + \text{tr}(\mathbf{W}\_0 \mathbf{P}) + \text{tr}(\mathbf{K}\_0 \mathbf{P}^{-1}) \\ &+ 2 \sum\_{i=1}^{\infty} \text{tr}(\mathbf{W}\_i \mathbf{P}) \text{tr}(\mathbf{K}\_i \mathbf{P}^{-1}) \end{split} \tag{26}$$

where *P* = *TTT*. The problem we consider here is to derive *the optimal positive definite symmetric matrix P*opt, which gives the global minimum of *S*(*P*) as follows:

$$\left. \frac{\partial \mathcal{S}(\mathcal{P})}{\partial \mathcal{P}} \right|\_{\mathcal{P} = \mathcal{P}\_{\text{opt}}} = \mathbf{0}.\tag{27}$$

If one can obtain the optimal positive definite symmetric matrix *P*opt, the optimal coordinate transformation matrix *T*opt is given by

$$T\_{\rm opt} = P\_{\rm opt}^{\frac{1}{2}} \mathcal{U} \tag{28}$$

where *U* is an *arbitrary* orthogonal matrix. It implies that the minimum *L*2-sensitivity realizations exist infinitely for a given digital filter *H*(*z*). The minimum *L*2-sensitivity realizations have freedom for orthogonal transformations.

## **3. Analytical solutions to the** *L*2**-sensitivity minimization problem for second-order digital filters**

This section proposes analytical synthesis of the minimum *L*2-sensitivity realizations for second-order digital filters. We propose closed form solutions to the *L*2-sensitivity minimization problem of second-order state-space digital filters. The proposed closed form solutions can greatly save the computation time and guarantee that the *L*2-sensitivity obtained by the iterative algorithm of the conventional method surely converges to the theoretical minimum. We show that the *L*2-sensitivity is expressed by a simple linear combination of exponential functions, and we can obtain the minimum *L*2-sensitivity realization by solving a fourth degree polynomial equation of constant coefficients in closed form without iterative calculations [12, 13].

#### **3.1. Problem formulation**

6 Digital Filters and Signal Processing

From Eqs. (21), we have

*<sup>S</sup>*(*T*−1*AT*, *<sup>T</sup>*−1*b*, *cT*)

=tr 1 2*πj* |*z*|=1

> +tr 1 2*πj* |*z*|=1

matrix *P* as follows [1]:

*S*(*P*)=tr

or in terms of the general Gramians as

*<sup>S</sup>*(*T*−1*AT*, *<sup>T</sup>*−1*b*, *cT*)

+2 ∞ ∑ *i*=1

 1 2*πj* |*z*|=1

positive definite symmetric matrix *P* as follows [2]:

+tr 1 2*πj* |*z*|=1

(*K*0,*W*0)=(*T*−1*K*0*T*−*T*, *<sup>T</sup>TW*0*T*) (21)

*z*

tr(*TTWiT*)tr(*T*−1*KiT*−*T*). (24)

*z*  *<sup>T</sup>*−1*F*(*z*)*F*†(*z*)*T*−*<sup>T</sup> dz*

*<sup>F</sup>*(*z*)*F*†(*z*)*P*−<sup>1</sup> *dz*

*z*

*z*

(23)

(25)

which shows that *K*0*W*<sup>0</sup> has the same eigenvalues of *K*0*W*0. Thus, the second order modes defined by Eq. (16) are invariant under the coordinate transformations. It implies that

The value of *L*2-sensitivity depends on not only the transfer function *H*(*z*) but also the coordinate transformation matrix *<sup>T</sup>*. The *<sup>L</sup>*2-sensitivity of the filter (*T*−1*AT*, *<sup>T</sup>*−1*b*, *cT*, *<sup>d</sup>*) can

*<sup>T</sup>*−1*F*(*z*)*G*(*z*)*TTT*(*F*(*z*)*G*(*z*))†*T*−*<sup>T</sup> dz*

*z* + tr 1 2*πj* |*z*|=1

<sup>=</sup>tr(*TTW*0*T*)tr(*T*−1*K*0*T*−*T*) + tr(*TTW*0*T*) + tr(*T*−1*K*0*T*−*T*)

The *L*2-sensitivity (23) can be expressed as the function of the positive definite symmetric

*<sup>F</sup>*(*z*)*G*(*z*)*P*(*F*(*z*)*G*(*z*))†*P*−<sup>1</sup> *dz*

*z* + tr 1 2*πj* |*z*|=1

where *P* = *TTT*. Similarly, the *L*2-sensitivity (24) can be expressed as the function of the

*<sup>G</sup>*†(*z*)*G*(*z*)*Pdz*

*<sup>T</sup>TG*†(*z*)*G*(*z*)*<sup>T</sup> dz*

second-order modes depends on only the transfer function.

**2.4.** *L*2**-sensitivity minimization problem**

be expressed in terms of the complex integral as

*<sup>K</sup>*0*W*<sup>0</sup> <sup>=</sup> *<sup>T</sup>*−1*K*0*W*0*<sup>T</sup>* (22)

We adopt the balanced realization (*A*b, *b*b, *c*b, *d*b) as the initial realization to synthesize the minimum *L*2-sensitivity realization. The coefficient matrices (*A*b, *b*b, *c*b, *d*b) satisfy symmetric properties as follows:

$$A\_{\mathbf{b}}^{T} = \Sigma A\_{\mathbf{b}} \Sigma \,\,\mathbf{c}\_{\mathbf{b}}^{T} = \Sigma b\_{\mathbf{b}} \tag{29}$$

where **Σ** is a signature matrix defined as follows:

$$\Delta = \text{diag}(\sigma\_1 \cdot \cdots \cdot \sigma\_N), \ \sigma\_i = \pm 1 \ (i = 1, \cdots, N). \tag{30}$$

where

Gramian *<sup>W</sup>*(b)

matrix *P*opt [1].

*definite symmetric matrix P*opt *satisfying*

*then the positive definite symmetric matrix* **<sup>Σ</sup>***P*−<sup>1</sup>

*λ* = *λ*<sup>r</sup> + *jλ*i, *α* = *α*<sup>r</sup> + *jα*i, *κ* =

<sup>2</sup> , *<sup>µ</sup>*<sup>2</sup> <sup>=</sup>

*P*<sup>2</sup> − *Q*<sup>2</sup> + *R*,

In this subsection, we consider the property of the positive definite symmetric matrix *P*. The following two theorems lead a symmetric property of the optimal positive definite symmetric

**Theorem 1.** [9] *L*2*-sensitivity S*(*P*) *has the unique global minimum, which is achieved by a positive*

*∂S*(*P*) *∂P* � � � � *P*=*P*opt

*∂S*(*P*) *∂P* � � � � *P*=*P*opt

*∂S*(*P*) *∂P* � � � � *P*=**Σ***P*<sup>−</sup><sup>1</sup>

opt**<sup>Σ</sup>** *also satisfies*

*for the signature matrix* **Σ** *which satisfies Eq. (29).* ✷

�*κ*(|*α*| − *α*i)

Using the parameters *<sup>P</sup>*, *<sup>Q</sup>*, and *<sup>R</sup>*, the controllability Gramian *<sup>K</sup>*(b)

**Θ**=diag(*θ*1, *θ*2)

�

=diag(

**3.3. Property of the positive definite symmetric matrix** *P*

**Theorem 2.** [1] *If a positive definite symmetric matrix P*opt *satisfies*

� *P* + *Q <sup>P</sup>* <sup>−</sup> *<sup>Q</sup>*,

<sup>2</sup>*<sup>κ</sup>* sign(*α*r).

<sup>0</sup> <sup>=</sup> **<sup>Θ</sup>** (38)

�|*α*| + *α*<sup>i</sup>

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

<sup>0</sup> of the balanced realization (*A*b, *<sup>b</sup>*b, *<sup>c</sup>*b, *<sup>d</sup>*b) can be expressed as follows:

�

*µ*<sup>1</sup> =

*K*(b) <sup>0</sup> <sup>=</sup>*W*(b) 10.5772/52194

<sup>0</sup> and the observability

http://dx.doi.org/10.5772/52194

*P*<sup>2</sup> − *Q*<sup>2</sup> − *R*). (39)

= **0**. (40)

= **0** (41)

opt**<sup>Σ</sup>** <sup>=</sup> **<sup>0</sup>** (42)

(37)

221

✷

We exploit the symmetric properties of the balanced realization in order to simplify the *L*2-sensitivity formulation and minimization in the following discussion. Under this condition, the *L*2-sensitivity *S*(*P*) in Eq. (26) is rewritten as

$$\begin{split} S(\mathbf{P}) &= \text{tr}(\mathbf{W}\_0^{(\mathbf{b})} \mathbf{P}) \text{tr}(\mathbf{K}\_0^{(\mathbf{b})} \mathbf{P}^{-1}) + \text{tr}(\mathbf{W}\_0^{(\mathbf{b})} \mathbf{P}) + \text{tr}(\mathbf{K}\_0^{(\mathbf{b})} \mathbf{P}^{-1}) \\ &+ 2 \sum\_{i=1}^{\infty} \text{tr}(\mathbf{W}\_i^{(\mathbf{b})} \mathbf{P}) \text{tr}(\mathbf{K}\_i^{(\mathbf{b})} \mathbf{P}^{-1}) \end{split} \tag{31}$$

and thus, the *L*2-sensitivity minimization problem is formulated as follows:

$$\min\_{\mathbf{P}} S(\mathbf{P}) \text{ in Eq. (31)}\tag{32}$$

where *P* is an arbitrary positive definite symmetric matrix.

We derive the optimal positive definite symmetric matrix *P*opt which gives the global minimum of the *L*2-sensitivity *S*(*P*) in Eq. (31).

#### **3.2. Second-order digital filters**

Consider second-order digital filters with complex conjugate poles given by

$$H(z) = \frac{a}{z - \lambda} + \frac{a^\*}{z - \lambda^\*} + d \tag{33}$$

where (*λ*, *λ*∗) are *complex conjugate* poles, *α* is a *complex* scalar, and *d* is a *real* scalar 1. We define scalar parameters *P*, *Q*, and *R* as follows:

$$P = \frac{|\alpha|}{1 - |\lambda|^2} \tag{34}$$

$$R + jQ = \frac{\alpha}{1 - \lambda^2} \tag{35}$$

which can be calculated directly from the transfer function *H*(*z*). The closed form expression of the balanced realization of the filter *H*(*z*) is given as follows [15]:

$$
\begin{bmatrix}
\frac{A\_{\mathtt{b}}\mid\mathsf{b}\_{\mathtt{b}}}{\mathsf{c}\_{\mathtt{b}}\mid\mathsf{d}\_{\mathtt{b}}}
\end{bmatrix} = \begin{bmatrix}
\lambda\_{\mathtt{r}} - \frac{\mathsf{x} - \mathsf{x}^{-1}}{2}\lambda\_{\mathtt{i}} & \frac{\mathsf{x} + \mathsf{x}^{-1}}{2}\lambda\_{\mathtt{i}} & \left|\boldsymbol{\mu}\_{1} + \boldsymbol{\mu}\_{2}\right| \\
\hline
\boldsymbol{\mu}\_{1} + \boldsymbol{\mu}\_{2} & -(\boldsymbol{\mu}\_{1} - \boldsymbol{\mu}\_{2}) & d
\end{bmatrix} \tag{36}
$$

<sup>1</sup> For also second-order digital filters, we can derive analytical solutions to the *L*2-sensitivity minimization problem [13].

where

8 Digital Filters and Signal Processing

where **Σ** is a signature matrix defined as follows:

*<sup>S</sup>*(*P*)=tr(*W*(b)

min *P*

**3.2. Second-order digital filters**

minimum of the *L*2-sensitivity *S*(*P*) in Eq. (31).

define scalar parameters *P*, *Q*, and *R* as follows:

� *A*<sup>b</sup> *b*<sup>b</sup> *c*<sup>b</sup> *d*<sup>b</sup>

[13].

� =

+2 ∞ ∑ *i*=1

condition, the *L*2-sensitivity *S*(*P*) in Eq. (26) is rewritten as

<sup>0</sup> *<sup>P</sup>*)tr(*K*(b)

tr(*W*(b)

and thus, the *L*2-sensitivity minimization problem is formulated as follows:

Consider second-order digital filters with complex conjugate poles given by

*<sup>H</sup>*(*z*) = *<sup>α</sup>*

of the balanced realization of the filter *H*(*z*) is given as follows [15]:

*<sup>λ</sup>*<sup>r</sup> <sup>−</sup> *<sup>κ</sup>* <sup>−</sup> *<sup>κ</sup>*−<sup>1</sup>

<sup>−</sup>*<sup>κ</sup>* <sup>+</sup> *<sup>κ</sup>*−<sup>1</sup>

**Σ** = diag(*σ*1, ··· , *σN*), *σ<sup>i</sup>* = ±1 (*i* = 1, ··· , *N*). (30)

*S*(*P*) in Eq. (31) (32)

<sup>0</sup> *<sup>P</sup>*) + tr(*K*(b)

<sup>0</sup> *<sup>P</sup>*−1)

*<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>*<sup>∗</sup> <sup>+</sup> *<sup>d</sup>* (33)

<sup>1</sup> − |*λ*|<sup>2</sup> (34)

<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*<sup>2</sup> (35)

(36)

*<sup>i</sup> <sup>P</sup>*−1) (31)

We exploit the symmetric properties of the balanced realization in order to simplify the *L*2-sensitivity formulation and minimization in the following discussion. Under this

<sup>0</sup> *<sup>P</sup>*−1) + tr(*W*(b)

where *P* is an arbitrary positive definite symmetric matrix.

We derive the optimal positive definite symmetric matrix *P*opt which gives the global

*<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>* <sup>+</sup>

where (*λ*, *λ*∗) are *complex conjugate* poles, *α* is a *complex* scalar, and *d* is a *real* scalar 1. We

*<sup>P</sup>*<sup>=</sup> <sup>|</sup>*α*<sup>|</sup>

*<sup>R</sup>* <sup>+</sup> *jQ*<sup>=</sup> *<sup>α</sup>*

<sup>2</sup> *<sup>λ</sup>*<sup>i</sup>

<sup>2</sup> *<sup>λ</sup>*<sup>i</sup> *<sup>λ</sup>*<sup>r</sup> <sup>+</sup>

<sup>1</sup> For also second-order digital filters, we can derive analytical solutions to the *L*2-sensitivity minimization problem

which can be calculated directly from the transfer function *H*(*z*). The closed form expression

*κ* + *κ*−<sup>1</sup>

*µ*<sup>1</sup> + *µ*<sup>2</sup> −(*µ*<sup>1</sup> − *µ*2) *d*

*κ* − *κ*−<sup>1</sup>

<sup>2</sup> *<sup>λ</sup>*<sup>i</sup> *<sup>µ</sup>*<sup>1</sup> <sup>+</sup> *<sup>µ</sup>*<sup>2</sup>

<sup>2</sup> *<sup>λ</sup>*<sup>i</sup> *<sup>µ</sup>*<sup>1</sup> <sup>−</sup> *<sup>µ</sup>*<sup>2</sup>

*α*∗

*<sup>i</sup> <sup>P</sup>*)tr(*K*(b)

$$\begin{cases} \lambda = \lambda\_\mathbf{r} + j\lambda\_\mathbf{i}, \; \alpha = \mathbf{a}\_\mathbf{r} + j\mathbf{a}\_\mathbf{i}, \; \kappa = \sqrt{\frac{P+Q}{P-Q}},\\ \mu\_1 = \sqrt{\frac{\kappa(|\boldsymbol{\alpha}| - \boldsymbol{a}\_\mathbf{i})}{2}}, \; \mu\_2 = \sqrt{\frac{|\boldsymbol{\alpha}| + \boldsymbol{a}\_\mathbf{i}}{2\kappa}} \operatorname{sign}(\boldsymbol{a}\_\mathbf{r}). \end{cases} \tag{37}$$

Using the parameters *<sup>P</sup>*, *<sup>Q</sup>*, and *<sup>R</sup>*, the controllability Gramian *<sup>K</sup>*(b) <sup>0</sup> and the observability Gramian *<sup>W</sup>*(b) <sup>0</sup> of the balanced realization (*A*b, *<sup>b</sup>*b, *<sup>c</sup>*b, *<sup>d</sup>*b) can be expressed as follows:

$$\begin{aligned} \mathbf{K}\_0^{(\mathbf{b})} &= \mathbf{W}\_0^{(\mathbf{b})} = \boldsymbol{\Theta} \\ \boldsymbol{\Theta} &= \text{diag}(\theta\_1, \theta\_2) \end{aligned} \tag{38}$$

$$\mathbf{p} = \text{diag}(\sqrt{P^2 - Q^2} + \text{R}, \sqrt{P^2 - Q^2} - \text{R}). \tag{39}$$

## **3.3. Property of the positive definite symmetric matrix** *P*

In this subsection, we consider the property of the positive definite symmetric matrix *P*. The following two theorems lead a symmetric property of the optimal positive definite symmetric matrix *P*opt [1].

**Theorem 1.** [9] *L*2*-sensitivity S*(*P*) *has the unique global minimum, which is achieved by a positive definite symmetric matrix P*opt *satisfying*

$$\left. \frac{\partial \mathcal{S}(\mathcal{P})}{\partial \mathcal{P}} \right|\_{\mathcal{P} = \mathcal{P}\_{\text{opt}}} = \mathbf{0}. \tag{40}$$

✷

**Theorem 2.** [1] *If a positive definite symmetric matrix P*opt *satisfies*

$$\left. \frac{\partial \mathcal{S}(\mathcal{P})}{\partial \mathcal{P}} \right|\_{\mathcal{P} = \mathcal{P}\_{\text{opt}}} = \mathbf{0} \tag{41}$$

*then the positive definite symmetric matrix* **<sup>Σ</sup>***P*−<sup>1</sup> opt**<sup>Σ</sup>** *also satisfies*

$$\frac{\partial S(\mathcal{P})}{\partial \mathcal{P}}\Big|\_{\mathcal{P} = \boldsymbol{\Sigma}\boldsymbol{P}\_{\mathrm{opt}}^{-1}\boldsymbol{\Sigma}} = \mathbf{0} \tag{42}$$

*for the signature matrix* **Σ** *which satisfies Eq. (29).* ✷

The derivative *∂S*(*P*)/*∂P* is given by differentiating *S*(*P*) in Eq. (31) with respect to *P* as

$$\frac{\partial \mathcal{S}(\mathbf{P})}{\partial \mathbf{P}} = (1 + \text{tr}(\mathbf{K}\_0^{(\mathbf{b})} \mathbf{P}^{-1})) \mathbf{W}\_0^{(\mathbf{b})} + 2 \sum\_{i=1}^{\infty} \text{tr}(\mathbf{K}\_i^{(\mathbf{b})} \mathbf{P}^{-1}) \mathbf{W}\_i^{(\mathbf{b})}$$

$$-\mathbf{P}^{-1} \left( (1 + \text{tr}(\mathbf{W}\_0^{(\mathbf{b})} \mathbf{P})) \mathbf{K}\_0^{(\mathbf{b})} + 2 \sum\_{i=1}^{\infty} \text{tr}(\mathbf{W}\_i^{(\mathbf{b})} \mathbf{P}) \mathbf{K}\_i^{(\mathbf{b})} \right) \mathbf{P}^{-1}. \tag{43}$$

From Theorem 1 and Theorem 2, it is proved that the optimal positive definite symmetric matrix *P*opt has the following symmetric property [1]:

$$P\_{\rm opt} = \Sigma P\_{\rm opt}^{-1} \Sigma \tag{44}$$

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223

(49)

. (50)

b*P*)tr(**Θ**(*A<sup>T</sup>*

*np*. (53)

b )*i*

. (52)

*<sup>P</sup>*−1). (51)

On the other hand, in case of **Σ** = ±diag(1, −1), the authors have derived the closed form expression of a positive definite symmetric matrix *P* which satisfies Eq. (45) as follows:

In this subsection, the closed form expression of the *L*2-sensitivity of second-order digital filters is given. We give the closed form expression of the *L*2-sensitivity *S*(*P*) in Eq. (31). We first express the general Gramians of the balanced realization (*A*b, *b*b, *c*b, *d*b) as follows:

<sup>0</sup> <sup>+</sup> *<sup>K</sup>*(b)

<sup>b</sup> + (*A<sup>T</sup>* b )*i W*(b) 0

<sup>b</sup>**<sup>Θ</sup>** <sup>+</sup> **<sup>Θ</sup>**(*A<sup>T</sup>*

<sup>b</sup> + (*A<sup>T</sup>* b )*i* **Θ**

We express the *L*2-sensitivity *S*(*P*) by substituting Eqs. (49) and (50) into Eq. (31) as follows:

The *L*2-sensitivity *S*(*P*) in Eq. (51) can be expressed more simply. Exploiting the symmetric property of coefficient matrix *A*<sup>b</sup> and *P* given in Eqs. (29) and (45) respectively, we can

In order to give the closed form expression of the *L*2-sensitivity *S*(*P*) in Eq. (52), it is necessary to derive the closed form expressions of matrices *A*b, **Θ**, and *P*. The closed form expressions of matrices *A*<sup>b</sup> and **Θ** are given in Eqs. (36) and (39), respectively. The closed form expression of the positive definite symmetric matrix *P* is given in Eq. (48). Substituting the closed form expressions of matrices *A*b, **Θ**, and *P* into Eq. (52) gives the closed form

*S*(*P*) = *S*(*p*) =

<sup>2</sup> + 2

2 ∑ *n*=−2

*sne*

∞ ∑ *i*=0 tr(**Θ***A<sup>i</sup>* <sup>b</sup>*P*) 2

<sup>0</sup> (*A<sup>T</sup>* b )*i*

> ∞ ∑ *i*=1

tr(**Θ***A<sup>i</sup>*

b )*i*

cosh(*p*) sinh(*p*) sinh(*p*) cosh(*p*) Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

*P* =

**3.5. Closed form expression of the** *L*2**-sensitivity** *S*(*P*)

*K*(b) *<sup>i</sup>* <sup>=</sup> <sup>1</sup> 2 *Ai* b*K*(b)

*W*(b) *<sup>i</sup>* <sup>=</sup> <sup>1</sup> 2 *W*(b) <sup>0</sup> *<sup>A</sup><sup>i</sup>*

*<sup>S</sup>*(*P*) = tr(**Θ***P*)tr(**Θ***P*−1) + tr(**Θ***P*) + tr(**Θ***P*−1) + <sup>2</sup>

*S*(*P*)=2tr(**Θ***P*) − (tr(**Θ***P*))

rewrite the *L*2-sensitivity *S*(*P*) as

expression of the *L*2-sensitivity *S*(*p*) as

= 1 2 *Ai*

= 1 2 **Θ***Ai*

where *p* is a real scalar variable [12, 13].

for the signature matrix **Σ** which satisfies Eq. (29). We will thus search the optimal positive definite symmetric matrix *P*opt among the positive definite symmetric matrices *P* which satisfy

$$P = \Sigma P^{-1} \Sigma.\tag{45}$$

## **3.4. Closed form expression of the positive definite symmetric matrix** *P*

In this subsection, we consider the case of second-order digital filters and give the closed form expression of the positive definite symmetric matrix *P*. When we restrict ourselves to the second-order case of state-space digital filters, we can give an closed form expression of the positive definite symmetric matrix *P* which satisfies Eq. (45), considering the form of the signature matrix **Σ** classified into the following two cases:

$$\begin{cases} \mathbf{\dot{\Sigma}} = \pm \text{diag}(1, 1) = \pm I \\ \mathbf{\dot{\Sigma}} = \pm \text{diag}(1, -1) . \end{cases} \tag{46}$$

For each case, we next consider the closed form expression of the positive definite symmetric matrix *P* which satisfies Eq. (45).

In case of **<sup>Σ</sup>** <sup>=</sup> <sup>±</sup>*I*, Eq. (44) yields *<sup>P</sup>*opt <sup>=</sup> *<sup>P</sup>*−<sup>1</sup> opt, that is, *<sup>P</sup>*opt <sup>=</sup> *<sup>I</sup>*. It means that the minimum *L*2-sensitivity realization can be synthesized without any coordinate transformation to the balanced realization, that is, the initial realization. In other words, the minimum *L*2-sensitivity realization is the balanced realization as follows:

$$(A\_{\rm opt}, \mathbf{b}\_{\rm opt}, \mathbf{c}\_{\rm opt}, d\_{\rm opt}) = (A\_{\mathbf{b}\prime} \mathbf{b}\_{\mathbf{b}\prime} \mathbf{c}\_{\mathbf{b}\prime} d\_{\mathbf{b}}).\tag{47}$$

Thus, we need no more discussion on this case since the minimum *L*2-sensitivity realization is already achieved as the balanced realization.

On the other hand, in case of **Σ** = ±diag(1, −1), the authors have derived the closed form expression of a positive definite symmetric matrix *P* which satisfies Eq. (45) as follows:

$$P = \begin{bmatrix} \cosh(p)\sinh(p) \\ \sinh(p)\cosh(p) \end{bmatrix} \tag{48}$$

where *p* is a real scalar variable [12, 13].

10 Digital Filters and Signal Processing

satisfy

*∂S*(*P*)

*<sup>∂</sup><sup>P</sup>* = (<sup>1</sup> <sup>+</sup> tr(*K*(b)

−*P*−<sup>1</sup>

matrix *P*opt has the following symmetric property [1]:

signature matrix **Σ** classified into the following two cases:

matrix *P* which satisfies Eq. (45).

In case of **<sup>Σ</sup>** <sup>=</sup> <sup>±</sup>*I*, Eq. (44) yields *<sup>P</sup>*opt <sup>=</sup> *<sup>P</sup>*−<sup>1</sup>

is already achieved as the balanced realization.

*L*2-sensitivity realization is the balanced realization as follows:

The derivative *∂S*(*P*)/*∂P* is given by differentiating *S*(*P*) in Eq. (31) with respect to *P* as

<sup>0</sup> + 2

<sup>0</sup> *<sup>P</sup>*))*K*(b)

From Theorem 1 and Theorem 2, it is proved that the optimal positive definite symmetric

*<sup>P</sup>*opt <sup>=</sup> **<sup>Σ</sup>***P*−<sup>1</sup>

for the signature matrix **Σ** which satisfies Eq. (29). We will thus search the optimal positive definite symmetric matrix *P*opt among the positive definite symmetric matrices *P* which

In this subsection, we consider the case of second-order digital filters and give the closed form expression of the positive definite symmetric matrix *P*. When we restrict ourselves to the second-order case of state-space digital filters, we can give an closed form expression of the positive definite symmetric matrix *P* which satisfies Eq. (45), considering the form of the

**Σ** = ±diag(1, 1) = ±*I*

For each case, we next consider the closed form expression of the positive definite symmetric

*L*2-sensitivity realization can be synthesized without any coordinate transformation to the balanced realization, that is, the initial realization. In other words, the minimum

Thus, we need no more discussion on this case since the minimum *L*2-sensitivity realization

**3.4. Closed form expression of the positive definite symmetric matrix** *P*

∞ ∑ *i*=1

<sup>0</sup> + 2

tr(*K*(b)

∞ ∑ *i*=1

*<sup>i</sup> <sup>P</sup>*−1)*W*(b) *i*

tr(*W*(b)

*<sup>i</sup> <sup>P</sup>*)*K*(b) *i*

opt**<sup>Σ</sup>** (44)

*<sup>P</sup>* = **<sup>Σ</sup>***P*−1**Σ**. (45)

**<sup>Σ</sup>** <sup>=</sup> <sup>±</sup>diag(1, <sup>−</sup>1). (46)

(*A*opt, *b*opt, *c*opt, *d*opt)=(*A*b, *b*b, *c*b, *d*b). (47)

opt, that is, *<sup>P</sup>*opt <sup>=</sup> *<sup>I</sup>*. It means that the minimum

*<sup>P</sup>*<sup>−</sup>1. (43)

<sup>0</sup> *<sup>P</sup>*−1))*W*(b)

(<sup>1</sup> + tr(*W*(b)

## **3.5. Closed form expression of the** *L*2**-sensitivity** *S*(*P*)

In this subsection, the closed form expression of the *L*2-sensitivity of second-order digital filters is given. We give the closed form expression of the *L*2-sensitivity *S*(*P*) in Eq. (31). We first express the general Gramians of the balanced realization (*A*b, *b*b, *c*b, *d*b) as follows:

$$\begin{split} \mathbf{K}\_{i}^{(\mathbf{b})} &= \frac{1}{2} \left( \mathbf{A}\_{\mathbf{b}}^{i} \mathbf{K}\_{0}^{(\mathbf{b})} + \mathbf{K}\_{0}^{(\mathbf{b})} (\mathbf{A}\_{\mathbf{b}}^{T})^{i} \right) \\ &= \frac{1}{2} \left( \mathbf{A}\_{\mathbf{b}}^{i} \boldsymbol{\Theta} + \boldsymbol{\Theta} (\mathbf{A}\_{\mathbf{b}}^{T})^{i} \right) \\ \mathbf{W}\_{i}^{(\mathbf{b})} &= \frac{1}{2} \left( \mathbf{W}\_{0}^{(\mathbf{b})} \mathbf{A}\_{\mathbf{b}}^{i} + (\mathbf{A}\_{\mathbf{b}}^{T})^{i} \mathbf{W}\_{0}^{(\mathbf{b})} \right) \\ &= \frac{1}{2} \left( \boldsymbol{\Theta} \mathbf{A}\_{\mathbf{b}}^{i} + (\mathbf{A}\_{\mathbf{b}}^{T})^{i} \boldsymbol{\Theta} \right) . \end{split} \tag{50}$$

We express the *L*2-sensitivity *S*(*P*) by substituting Eqs. (49) and (50) into Eq. (31) as follows:

$$S(\mathbf{P}) = \text{tr}(\boldsymbol{\Theta}\boldsymbol{P})\text{tr}(\boldsymbol{\Theta}\boldsymbol{P}^{-1}) + \text{tr}(\boldsymbol{\Theta}\boldsymbol{P}) + \text{tr}(\boldsymbol{\Theta}\boldsymbol{P}^{-1}) + 2\sum\_{i=1}^{\infty} \text{tr}(\boldsymbol{\Theta}\boldsymbol{A}\_{\mathbf{b}}^{i}\boldsymbol{P})\text{tr}(\boldsymbol{\Theta}(\boldsymbol{A}\_{\mathbf{b}}^{T})^{i}\boldsymbol{P}^{-1}).\tag{51}$$

The *L*2-sensitivity *S*(*P*) in Eq. (51) can be expressed more simply. Exploiting the symmetric property of coefficient matrix *A*<sup>b</sup> and *P* given in Eqs. (29) and (45) respectively, we can rewrite the *L*2-sensitivity *S*(*P*) as

$$S(\mathsf{P}) = 2\mathrm{tr}(\boldsymbol{\Theta}\boldsymbol{\mathsf{P}}) - \left(\mathrm{tr}(\boldsymbol{\Theta}\boldsymbol{\mathsf{P}})\right)^2 + 2\sum\_{i=0}^{\infty} \left(\mathrm{tr}(\boldsymbol{\Theta}\boldsymbol{A}\_{\mathsf{b}}^i\boldsymbol{\mathsf{P}})\right)^2. \tag{52}$$

In order to give the closed form expression of the *L*2-sensitivity *S*(*P*) in Eq. (52), it is necessary to derive the closed form expressions of matrices *A*b, **Θ**, and *P*. The closed form expressions of matrices *A*<sup>b</sup> and **Θ** are given in Eqs. (36) and (39), respectively. The closed form expression of the positive definite symmetric matrix *P* is given in Eq. (48). Substituting the closed form expressions of matrices *A*b, **Θ**, and *P* into Eq. (52) gives the closed form expression of the *L*2-sensitivity *S*(*p*) as

$$S(\mathcal{P}) = S(p) = \sum\_{n=-2}^{2} s\_n e^{np}.\tag{53}$$

It is remarkable that Eq. (53) is a simple linear combination of exponential functions which does not contain infinite summations. These coefficients *sn*'s are easily computed directly from the transfer function *H*(*z*) [12, 13].

## **3.6. Synthesis of minimum** *L*2**-sensitivity realizations**

The parameter *p* which minimizes *S*(*p*) in Eq. (53) can be derived by solving the following equation with respect to *p*:

$$\frac{\partial S(p)}{\partial p} = \sum\_{n=-2}^{2} n s\_n e^{np} = 0. \tag{54}$$

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where *U* is an arbitrary orthogonal matrix. We have to note that the optimal coordinate transformation matrix *T*opt is not unique because of the non-uniqueness of matrix *U*. By letting *U* = *I* in Eq. (58), for instance, one of the optimal coordinate transformation matrices

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

� *β* 1 2 opt 0 0 *β* − <sup>1</sup> 2 opt

> − <sup>1</sup> 2 opt *<sup>β</sup>* 1 2 opt <sup>−</sup> *<sup>β</sup>*

> − <sup>1</sup> 2 opt *<sup>β</sup>* 1 2 opt <sup>+</sup> *<sup>β</sup>*

minimum *L*2-sensitivity realization (*A*opt, *b*opt, *c*opt, *d*opt) is synthesized as follows:

= � *T*−<sup>1</sup>

*S*min =

We can give a geometrical interpretation of the above optimal coordinate transformation matrix *T*opt. In Eq. (59), *R* is an orthogonal matrix, which means *π*/4[rad] rotation of the coordinate axes, *B*opt is a positive definite diagonal matrix, which means a simple scaling of each coordinate axis. Eq. (59) shows that the minimum *L*2-sensitivity realization is obtained by the following operations for the coordinate axes of the initial balanced realization; −*π*/4[rad] rotation, simple scaling, and +*π*/4[rad] rotation. Finally, the

and the minimum *L*2-sensitivity *S*min, which is achieved by *p*opt = log(*β*opt), is expressed by

2 ∑ *n*=2

We present numerical examples to demonstrate the validity of the proposed method. Consider a second-order digital filter *H*(*z*) with complex conjugate poles, of which transfer

> *α*∗ *<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>*<sup>∗</sup> <sup>+</sup> *<sup>d</sup>* <sup>=</sup> 0.0396 <sup>+</sup> 0.0793*z*−<sup>1</sup> <sup>+</sup> 0.0396*z*−<sup>2</sup>

*snβ<sup>n</sup>*

− <sup>1</sup> 2 opt

− <sup>1</sup> 2 opt

opt*A*b*T*opt *<sup>T</sup>*−<sup>1</sup>

*c*b*T*<sup>b</sup> *d*<sup>b</sup>

opt*b*<sup>b</sup>

�

<sup>1</sup> <sup>−</sup> 1.3315*z*−<sup>1</sup> <sup>+</sup> 0.49*z*−<sup>2</sup> (62)

opt. (61)

� √ 1 <sup>2</sup> <sup>√</sup> 1 2

− √ 1 <sup>2</sup> <sup>√</sup> 1 2 �

. (59)

*T*opt = *P*1/2

opt is given by

*T*opt =*P*

= � √ 1 <sup>2</sup> <sup>−</sup> <sup>√</sup> 1 2

= 1 2

�

substituting *p*opt into Eq. (53) as

**3.7. Numerical examples**

function is given by

*A*opt *b*opt *<sup>c</sup>*opt *<sup>d</sup>*opt �

*<sup>H</sup>*(*z*)= *<sup>α</sup>*

*<sup>z</sup>* <sup>−</sup> *<sup>λ</sup>* <sup>+</sup>

1 2 opt =*RTB* 1 2 opt*<sup>R</sup>*

> √ 1 <sup>2</sup> <sup>√</sup> 1 2

 *β* 1 2 opt <sup>+</sup> *<sup>β</sup>*

*β* 1 2 opt <sup>−</sup> *<sup>β</sup>*

Letting *β* = *e<sup>p</sup>* gives

$$\sum\_{n=-2}^{2} n s\_n \beta^n = 0.\tag{55}$$

The above equation is a fourth-degree polynomial equation with respect to *β* of constant coefficients. In 1545, G. Cardano states in his book entitled *Ars Magna* (its translated edition is [16]) that there exists the formula of solutions for fourth-degree polynomial equations. Therefore, Eq. (55) can be solved analytically. Eq. (55) has four solutions, from which the positive real solution *<sup>β</sup>*opt <sup>=</sup> *<sup>e</sup>p*opt is adopted to derive the optimal positive definite symmetric matrix *P*opt as

$$\begin{split} \mathbf{P}\_{\text{opt}} &= \begin{bmatrix} \cosh(p\_{\text{opt}}) \sinh(p\_{\text{opt}}) \\ \sinh(p\_{\text{opt}}) \cosh(p\_{\text{opt}}) \end{bmatrix} \\ &= \frac{1}{2} \begin{bmatrix} \beta\_{\text{opt}} + \beta\_{\text{opt}}^{-1} \beta\_{\text{opt}} - \beta\_{\text{opt}}^{-1} \\ \beta\_{\text{opt}} - \beta\_{\text{opt}}^{-1} \beta\_{\text{opt}} + \beta\_{\text{opt}}^{-1} \end{bmatrix} . \end{split} \tag{56}$$

The diagonalization of the optimal positive definite symmetric matrix *<sup>P</sup>*opt <sup>=</sup> *<sup>T</sup>*opt*T<sup>T</sup>* opt is given by

$$\begin{split} \mathbf{P}\_{\mathrm{opt}} &= \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} \boldsymbol{\beta}\_{\mathrm{opt}} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{\beta}\_{\mathrm{opt}}^{-1} \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \\ &\equiv \mathbf{R}^{T} \mathbf{B}\_{\mathrm{opt}} \mathbf{R}. \end{split} \tag{57}$$

Once the optimal positive definite symmetric matrix *P*opt is derived, the optimal coordinate transformation matrix *T*opt is calculated as

$$T\_{\rm opt} = P\_{\rm opt}^{\frac{1}{2}} \mathcal{U} \tag{58}$$

where *U* is an arbitrary orthogonal matrix. We have to note that the optimal coordinate transformation matrix *T*opt is not unique because of the non-uniqueness of matrix *U*. By letting *U* = *I* in Eq. (58), for instance, one of the optimal coordinate transformation matrices *T*opt = *P*1/2 opt is given by

$$\begin{split} T\_{\rm opt} &= \mathbf{P}^{\frac{1}{2}}\_{\rm opt} \\ &= \mathbf{R}^{T} \mathbf{B}^{\frac{1}{2}}\_{\rm opt} \mathbf{R} \\ &= \begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \begin{bmatrix} \boldsymbol{\beta}^{\frac{1}{2}}\_{\rm opt} & 0 \\ 0 & \boldsymbol{\beta}^{-\frac{1}{2}}\_{\rm opt} \end{bmatrix} \begin{bmatrix} \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \\ -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \\ &= \frac{1}{2} \begin{bmatrix} \boldsymbol{\beta}^{\frac{1}{2}}\_{\rm opt} + \boldsymbol{\beta}^{-\frac{1}{2}}\_{\rm opt} \boldsymbol{\beta}^{\frac{1}{2}}\_{\rm opt} - \boldsymbol{\beta}^{-\frac{1}{2}}\_{\rm opt} \\ \boldsymbol{\beta}^{\frac{1}{2}}\_{\rm opt} - \boldsymbol{\beta}^{-\frac{1}{2}}\_{\rm opt} \boldsymbol{\beta}^{\frac{2}{2}}\_{\rm opt} + \boldsymbol{\beta}^{-\frac{1}{2}}\_{\rm opt} \end{bmatrix}. \end{split} \tag{59}$$

We can give a geometrical interpretation of the above optimal coordinate transformation matrix *T*opt. In Eq. (59), *R* is an orthogonal matrix, which means *π*/4[rad] rotation of the coordinate axes, *B*opt is a positive definite diagonal matrix, which means a simple scaling of each coordinate axis. Eq. (59) shows that the minimum *L*2-sensitivity realization is obtained by the following operations for the coordinate axes of the initial balanced realization; −*π*/4[rad] rotation, simple scaling, and +*π*/4[rad] rotation. Finally, the minimum *L*2-sensitivity realization (*A*opt, *b*opt, *c*opt, *d*opt) is synthesized as follows:

$$
\begin{bmatrix}
\frac{A\_{\rm opt}}{c\_{\rm opt}}
\frac{|b\_{\rm opt}|}{d\_{\rm opt}}
\end{bmatrix} = \begin{bmatrix}
T\_{\rm opt}^{-1} A\_{\rm b} T\_{\rm opt} \left|
\boldsymbol{T}\_{\rm opt}^{-1} \boldsymbol{b}\_{\rm b}
\right| \\
\hline
 c\_{\rm b} T\_{\rm b} \quad \left|
\quad d\_{\rm b}
\right.
\end{bmatrix} \tag{60}
$$

and the minimum *L*2-sensitivity *S*min, which is achieved by *p*opt = log(*β*opt), is expressed by substituting *p*opt into Eq. (53) as

$$S\_{\rm min} = \sum\_{n=2}^{2} s\_n \mathcal{J}\_{\rm opt}^n. \tag{61}$$

#### **3.7. Numerical examples**

12 Digital Filters and Signal Processing

equation with respect to *p*:

Letting *β* = *e<sup>p</sup>* gives

matrix *P*opt as

given by

from the transfer function *H*(*z*) [12, 13].

**3.6. Synthesis of minimum** *L*2**-sensitivity realizations**

*P*opt =

*P*opt =

transformation matrix *T*opt is calculated as

 √ 1 <sup>2</sup> <sup>−</sup> <sup>√</sup> 1 2

√ 1 <sup>2</sup> <sup>√</sup> 1 2

= 1 2

*∂S*(*p*) *<sup>∂</sup><sup>p</sup>* <sup>=</sup>

> 2 ∑ *n*=−2

It is remarkable that Eq. (53) is a simple linear combination of exponential functions which does not contain infinite summations. These coefficients *sn*'s are easily computed directly

The parameter *p* which minimizes *S*(*p*) in Eq. (53) can be derived by solving the following

2 ∑ *n*=−2

The above equation is a fourth-degree polynomial equation with respect to *β* of constant coefficients. In 1545, G. Cardano states in his book entitled *Ars Magna* (its translated edition is [16]) that there exists the formula of solutions for fourth-degree polynomial equations. Therefore, Eq. (55) can be solved analytically. Eq. (55) has four solutions, from which the positive real solution *<sup>β</sup>*opt <sup>=</sup> *<sup>e</sup>p*opt is adopted to derive the optimal positive definite symmetric

> cosh(*p*opt) sinh(*p*opt) sinh(*p*opt) cosh(*p*opt)

> > *<sup>β</sup>*opt <sup>+</sup> *<sup>β</sup>*−<sup>1</sup>

*<sup>β</sup>*opt <sup>−</sup> *<sup>β</sup>*−<sup>1</sup>

The diagonalization of the optimal positive definite symmetric matrix *<sup>P</sup>*opt <sup>=</sup> *<sup>T</sup>*opt*T<sup>T</sup>*

 *β*opt 0 <sup>0</sup> *<sup>β</sup>*−<sup>1</sup> opt

Once the optimal positive definite symmetric matrix *P*opt is derived, the optimal coordinate

1 2

*T*opt = *P*

opt

. (56)

opt is

opt

 √ 1 <sup>2</sup> <sup>√</sup> 1 2

− √ 1 <sup>2</sup> <sup>√</sup> 1 2

<sup>≡</sup>*RTB*opt*R*. (57)

opt*<sup>U</sup>* (58)

opt *<sup>β</sup>*opt <sup>−</sup> *<sup>β</sup>*−<sup>1</sup>

opt *<sup>β</sup>*opt <sup>+</sup> *<sup>β</sup>*−<sup>1</sup>

*nsne*

*np* = 0. (54)

*nsnβ<sup>n</sup>* = 0. (55)

We present numerical examples to demonstrate the validity of the proposed method. Consider a second-order digital filter *H*(*z*) with complex conjugate poles, of which transfer function is given by

$$\begin{split}H(z) &= \frac{a}{z - \lambda} + \frac{a^\*}{z - \lambda^\*} + d\\ &= \frac{0.0396 + 0.0793z^{-1} + 0.0396z^{-2}}{1 - 1.3315z^{-1} + 0.49z^{-2}}\end{split} \tag{62}$$

**Figure 2.** Frequency responses and zero-pole configurations of *H*(*z*).

where *λ* = 0.7 exp(*j*0.1*π*), *α* = 1.6657 − *j*6.3055, and *d* = 1. The frequency responses and zero-pole configurations of digital filter *H*(*z*) are shown in Fig. 2. From the transfer function *H*(*z*), parameters *P*, *Q*, and *R* are calculated as

$$P = 0.5068, \ Q = -0.2947, \ R = 0.25. \tag{63}$$

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0 2 4 6 8 10

Minimum L2−sensitivity

opt = 3.6070. (68)

Method in [1] Method in [2]

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

Iteration k

**Figure 3.** Minimum *L*2-sensitivity and convergence behaviors of *L*2-sensitivity in conventional methods [1] and [2].

2 ∑ *n*=−2

*snβ<sup>n</sup>*

Figure 3 shows the comparison of our proposed method with the iterative methods reported in [1] and [2], where the initial realization is the balanced realization. Our proposed method achieves the minimum *L*2-sensitivity by only solving a fourth-degree polynomial equation without iterative calculations, while both of the methods in [1] and [2] require many iterative calculations to achieve the minimum *L*2-sensitivity. Furthermore, our proposed method can guarantee that the *L*2-sensitivity surely converges to the theoretical minimum when using

**4. Analytical solutions to the** *L*2**-sensitivity minimization problem for**

This section reveals that the *L*2-sensitivity minimization problem can be solved analytically if second-order modes are all equal. Furthermore, we clarify the general expression of the

We have discovered that there exist some digital filters whose minimum *L*2-sensitivity realization is equal to the balanced realization. Such digital filters satisfy a sufficient

**Theorem 3.** *If all the second-order modes θi*(*i* = 1, ··· , *N*) *of a digital filter H*(*z*) *are equal, then*

*that is, the minimum L*2*-sensitivity realization is equal to the balanced realization.* ✷

(*A*opt, *b*opt, *c*opt, *d*opt)=(*A*b, *b*b, *c*b, *d*b) (69)

*S*min =

**digital filters with all second-order modes equal**

condition summarized in the following theorem:

transfer functions of digital filters with all second-order modes equal [14].

**4.1. Analytical synthesis of the minimum** *L*2**-sensitivity realizations**

3.6

3.62

3.64

3.66

L2−sensitivity S(Pk

of which *L*2-sensitivity *S*min is

conventional methods in [1] and [2].

)

3.68

3.7

The coefficients *sn*'s are computed as follows:

$$(s\_{-2}, s\_{-1}, s\_0, s\_1, s\_2) = (0.3345, 0.8246, 0.8987, 0.8246, 0.7951). \tag{64}$$

We solve the following fourth-degree polynomial equation to derive the optimal solution *β*opt:

$$\sum\_{n=-2}^{2} n s\_n \mathfrak{E}^n = 0.\tag{65}$$

The fourth-degree polynomial equation above has the following four solutions:

$$
\beta = 0.8568, -0.6960, -0.3396 \pm j0.7682.\tag{66}
$$

We adopt *β*opt = 0.8568, which is a positive real scalar, to derive the optimal positive definite symmetric matrix *P*opt. We can derive the minimum *L*2-sensitivity realization (*A*opt, *b*opt, *c*opt, *d*opt) in closed form as follows:

$$
\begin{bmatrix}
\frac{\mathbf{A}\_{\rm opt}}{\mathbf{c}\_{\rm opt}}
\begin{vmatrix}
\mathbf{b}\_{\rm opt} \\
\mathbf{c}\_{\rm opt}
\end{vmatrix}
\end{bmatrix} = \begin{bmatrix}
0.7810 & 0.2451 \begin{vmatrix}
0.4751 \\
0.5505 \begin{vmatrix}
0.3061 \\
0.3061 \end{vmatrix}
\end{bmatrix} \tag{67}
$$

**Figure 3.** Minimum *L*2-sensitivity and convergence behaviors of *L*2-sensitivity in conventional methods [1] and [2].

of which *L*2-sensitivity *S*min is

14 Digital Filters and Signal Processing

**Figure 2.** Frequency responses and zero-pole configurations of *H*(*z*).

*H*(*z*), parameters *P*, *Q*, and *R* are calculated as

The coefficients *sn*'s are computed as follows:

(*A*opt, *b*opt, *c*opt, *d*opt) in closed form as follows:

�

*A*opt *b*opt *c*opt *d*opt � =

*β*opt:

where *λ* = 0.7 exp(*j*0.1*π*), *α* = 1.6657 − *j*6.3055, and *d* = 1. The frequency responses and zero-pole configurations of digital filter *H*(*z*) are shown in Fig. 2. From the transfer function

We solve the following fourth-degree polynomial equation to derive the optimal solution

We adopt *β*opt = 0.8568, which is a positive real scalar, to derive the optimal positive definite symmetric matrix *P*opt. We can derive the minimum *L*2-sensitivity realization

2 ∑ *n*=−2

The fourth-degree polynomial equation above has the following four solutions:

*P* = 0.5068, *Q* = −0.2947, *R* = 0.25. (63)

*β* = 0.8568, −0.6960, −0.3396 ± *j*0.7682. (66)

0.7810 0.2451 0.4751 −0.2451 0.5505 0.3061 0.4751 −0.3061 0.0396

*nsnβ<sup>n</sup>* = 0. (65)

(67)

(*s*−2,*s*−1,*s*0,*s*1,*s*2)=(0.3345, 0.8246, 0.8987, 0.8246, 0.7951). (64)

$$S\_{\rm min} = \sum\_{n=-2}^{2} s\_n \beta\_{\rm opt}^n = 3.6070. \tag{68}$$

Figure 3 shows the comparison of our proposed method with the iterative methods reported in [1] and [2], where the initial realization is the balanced realization. Our proposed method achieves the minimum *L*2-sensitivity by only solving a fourth-degree polynomial equation without iterative calculations, while both of the methods in [1] and [2] require many iterative calculations to achieve the minimum *L*2-sensitivity. Furthermore, our proposed method can guarantee that the *L*2-sensitivity surely converges to the theoretical minimum when using conventional methods in [1] and [2].

## **4. Analytical solutions to the** *L*2**-sensitivity minimization problem for digital filters with all second-order modes equal**

This section reveals that the *L*2-sensitivity minimization problem can be solved analytically if second-order modes are all equal. Furthermore, we clarify the general expression of the transfer functions of digital filters with all second-order modes equal [14].

## **4.1. Analytical synthesis of the minimum** *L*2**-sensitivity realizations**

We have discovered that there exist some digital filters whose minimum *L*2-sensitivity realization is equal to the balanced realization. Such digital filters satisfy a sufficient condition summarized in the following theorem:

**Theorem 3.** *If all the second-order modes θi*(*i* = 1, ··· , *N*) *of a digital filter H*(*z*) *are equal, then*

$$(A\_{\rm opt}, b\_{\rm opt}, c\_{\rm opt}, d\_{\rm opt}) = (A\_{\rm b}, b\_{\rm b}, c\_{\rm b}, d\_{\rm b}) \tag{69}$$

*that is, the minimum L*2*-sensitivity realization is equal to the balanced realization.* ✷

*Proof:* The general Gramians of the balanced realization are given by Eqs. (49) and (50), respectively. If all the second-order modes *θi*(*i* = 1, ··· , *N*) satisfies

$$\theta\_{\bar{l}} = \theta \text{ ( $i = 1, \dots$ ,  $N$ )}\tag{70}$$

*4.2.1. General expression*

*a nonzero real scalar.*

*can be expressed as the following form:*

*4.2.2. Frequency transformation*

all-pass digital filter.

*such as*

*equal such as*

digital filters with all second-order modes equal.

*The second-order modes of the digital filter H*(*z*) *are given by*

digital filters with all second-order modes equal.

In Ref. [14], we have newly derived a general expression of the transfer function of *N*th-order

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

**Corollary 1.** *Let the second-order modes of an Nth-order digital filter H*(*z*) *be θi*(*i* = 1, ··· , *N*)*. The second-order modes of the transfer function θH*(*z*) *are given by* |*θ*|*θi*(*i* = 1, ··· , *N*)*, where θ is*

**Theorem 4.** *The transfer function of an Nth-order digital filter H*(*z*) *with all second-order modes*

*where θ is a nonzero real scalar, ρ is a real scalar, and HAP*(*z*) *is an Nth-order all-pass digital filter.*

Furthermore, it is remarkable that the transfer function *H*(*z*) in Eq. (77) is generally obtained by the frequency transformation on a first-order FIR prototype filter using an *N*th-order

**Remark 1.** *An Nth-order digital filter H*(*z*) *in Eq. (77) is obtained by the frequency transformation*

*where HP*(*z*) = *<sup>θ</sup>z*−<sup>1</sup> <sup>+</sup> *<sup>ρ</sup> is a first-order prototype FIR digital filter and HAP*(*z*) *is an Nth-order all-pass digital filter.* ✷

The variable substitution in Eq. (79) represents the frequency transformation, where *H*P(*z*) is the prototype digital filter. Block diagrams of the prototype digital filter *H*P(*z*) and transformed digital filter *H*(*z*) are shown in Fig. 4. These figures show that a block diagram of the transformed digital filter *H*(*z*) can be obtained by simple substitution of the all-pass digital filter *<sup>H</sup>*AP(*z*) into the unit delay *<sup>z</sup>*<sup>−</sup>1. Theorem 4 and Remark 1 give us the class of

*θ*<sup>1</sup> = *θ*<sup>2</sup> = ··· = *θN*−<sup>1</sup> = *θ<sup>N</sup>* (76)

*H*(*z*) = *θHAP*(*z*) + *ρ* (77)

*θ<sup>i</sup>* = |*θ*| (*i* = 1, ··· , *N*). (78)

*<sup>H</sup>*(*z*) = *HP*(*z*)|*z*−1←*HAP*(*z*) (79)

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✷

the controllability and observability Gramians are expressed as

$$\mathbf{K}\_0^{(\mathbf{b})} = \mathbf{W}\_0^{(\mathbf{b})} = \text{diag}(\theta\_\prime \cdot \cdots \prime, \theta) = \theta \mathbf{I}. \tag{71}$$

Substituting Eq. (71) into Eqs. (49) and (50), the general Gramians are given by

$$\mathbf{K}\_{i}^{(\mathbf{b})} = \mathbf{W}\_{i}^{(\mathbf{b})} = \frac{1}{2}\theta(\mathbf{A}\_{\mathbf{b}}^{i} + (\mathbf{A}\_{\mathbf{b}}^{T})^{i}).\tag{72}$$

We can express the general Gramians as *<sup>K</sup>*(b) *<sup>i</sup>* <sup>=</sup> *<sup>W</sup>*(b) *<sup>i</sup>* = **Θ***i*, which is defined by

$$\Theta\_{\dot{i}} = \frac{1}{2}\theta(A\_{\mathbf{b}}^{\dot{i}} + (A\_{\mathbf{b}}^{T})^{\dot{i}}) \ (\dot{\imath} = 0, 1, \cdots). \tag{73}$$

Substituting *<sup>K</sup>*(b) *<sup>i</sup>* <sup>=</sup> *<sup>W</sup>*(b) *<sup>i</sup>* = **Θ***<sup>i</sup>* into Eq. (43) yields

$$\frac{\partial \mathcal{S}(\mathbf{P})}{\partial \mathbf{P}} = (1 + \text{tr}(\boldsymbol{\Theta}\_0 \mathbf{P}^{-1})) \boldsymbol{\Theta}\_0 + 2 \sum\_{i=1}^{\infty} \text{tr}(\boldsymbol{\Theta}\_i \mathbf{P}^{-1}) \boldsymbol{\Theta}\_i$$

$$-\mathbf{P}^{-1} \left( (1 + \text{tr}(\boldsymbol{\Theta}\_0 \mathbf{P})) \boldsymbol{\Theta}\_0 + 2 \sum\_{i=1}^{\infty} \text{tr}(\boldsymbol{\Theta}\_i \mathbf{P}) \boldsymbol{\Theta}\_i \right) \mathbf{P}^{-1}. \tag{74}$$

It is obvious that

$$\left. \frac{\partial S(P)}{\partial P} \right|\_{P=I} = \mathbf{0} \tag{75}$$

which means that the minimum *L*2-sensitivity realization can be synthesized without any coordinate transformation to the balanced realization, that is, the initial realization. Therefore, it is proved that the minimum *L*2-sensitivity is equal to the balanced realization. ✷

#### **4.2. Class of digital filters with all second-order modes equal**

In the previous subsection, we revealed that the *L*2-sensitivity minimization problem can be solved analytically if second-order modes are all equal. We next clarify the class of digital filters with all second-order modes equal. We have newly derived a general expression of the transfer function of *N*th-order digital filters with all second-order modes equal.

#### *4.2.1. General expression*

16 Digital Filters and Signal Processing

*Proof:* The general Gramians of the balanced realization are given by Eqs. (49) and (50),

*θ<sup>i</sup>* = *θ* (*i* = 1, ··· , *N*) (70)

<sup>0</sup> <sup>=</sup> diag(*θ*, ··· , *<sup>θ</sup>*) = *<sup>θ</sup>I*. (71)

*<sup>i</sup>* = **Θ***i*, which is defined by

) (*i* = 0, 1, ···). (73)

= **0** (75)

*<sup>P</sup>*<sup>−</sup>1. (74)

). (72)

respectively. If all the second-order modes *θi*(*i* = 1, ··· , *N*) satisfies

the controllability and observability Gramians are expressed as

*K*(b)

**<sup>Θ</sup>***<sup>i</sup>* <sup>=</sup> <sup>1</sup> 2 *<sup>θ</sup>*(*A<sup>i</sup>*

We can express the general Gramians as *<sup>K</sup>*(b)

*<sup>i</sup>* <sup>=</sup> *<sup>W</sup>*(b)

*∂S*(*P*)

Substituting *<sup>K</sup>*(b)

It is obvious that

✷

<sup>0</sup> <sup>=</sup> *<sup>W</sup>*(b)

Substituting Eq. (71) into Eqs. (49) and (50), the general Gramians are given by

*<sup>i</sup>* <sup>=</sup> <sup>1</sup> 2 *<sup>θ</sup>*(*A<sup>i</sup>*

<sup>b</sup> + (*A<sup>T</sup>* b )*i*

(1 + tr(**Θ**0*P*))**Θ**<sup>0</sup> + 2

which means that the minimum *L*2-sensitivity realization can be synthesized without any coordinate transformation to the balanced realization, that is, the initial realization. Therefore, it is proved that the minimum *L*2-sensitivity is equal to the balanced realization.

In the previous subsection, we revealed that the *L*2-sensitivity minimization problem can be solved analytically if second-order modes are all equal. We next clarify the class of digital filters with all second-order modes equal. We have newly derived a general expression of

the transfer function of *N*th-order digital filters with all second-order modes equal.

*∂S*(*P*) *∂P P*=*I*

*<sup>i</sup>* <sup>=</sup> *<sup>W</sup>*(b)

∞ ∑ *i*=1

tr(**Θ***iP*−1)**Θ***<sup>i</sup>*

tr(**Θ***iP*)**Θ***<sup>i</sup>*

∞ ∑ *i*=1

<sup>b</sup> + (*A<sup>T</sup>* b )*i*

*<sup>i</sup>* <sup>=</sup> *<sup>W</sup>*(b)

*<sup>i</sup>* = **Θ***<sup>i</sup>* into Eq. (43) yields

*<sup>∂</sup><sup>P</sup>* = (<sup>1</sup> <sup>+</sup> tr(**Θ**0*P*−1))**Θ**<sup>0</sup> <sup>+</sup> <sup>2</sup>

**4.2. Class of digital filters with all second-order modes equal**

−*P*−<sup>1</sup>

*K*(b)

In Ref. [14], we have newly derived a general expression of the transfer function of *N*th-order digital filters with all second-order modes equal.

**Corollary 1.** *Let the second-order modes of an Nth-order digital filter H*(*z*) *be θi*(*i* = 1, ··· , *N*)*. The second-order modes of the transfer function θH*(*z*) *are given by* |*θ*|*θi*(*i* = 1, ··· , *N*)*, where θ is a nonzero real scalar.*

**Theorem 4.** *The transfer function of an Nth-order digital filter H*(*z*) *with all second-order modes equal such as*

$$
\theta\_1 = \theta\_2 = \dots = \theta\_{N-1} = \theta\_N \tag{76}
$$

*can be expressed as the following form:*

$$H(z) = \theta H\_{\text{AP}}(z) + \rho \tag{77}$$

*where θ is a nonzero real scalar, ρ is a real scalar, and HAP*(*z*) *is an Nth-order all-pass digital filter. The second-order modes of the digital filter H*(*z*) *are given by*

$$\theta\_{\hat{i}} = |\theta| \ (\hat{\imath} = 1, \hat{\imath} \cdots, \hat{\imath} \text{ N}). \tag{78}$$

✷

#### *4.2.2. Frequency transformation*

Furthermore, it is remarkable that the transfer function *H*(*z*) in Eq. (77) is generally obtained by the frequency transformation on a first-order FIR prototype filter using an *N*th-order all-pass digital filter.

**Remark 1.** *An Nth-order digital filter H*(*z*) *in Eq. (77) is obtained by the frequency transformation such as*

$$H(z) = H\_P(z)|\_{z^{-1} \leftarrow H\_{AP}(z)}\tag{79}$$

*where HP*(*z*) = *<sup>θ</sup>z*−<sup>1</sup> <sup>+</sup> *<sup>ρ</sup> is a first-order prototype FIR digital filter and HAP*(*z*) *is an Nth-order all-pass digital filter.* ✷

The variable substitution in Eq. (79) represents the frequency transformation, where *H*P(*z*) is the prototype digital filter. Block diagrams of the prototype digital filter *H*P(*z*) and transformed digital filter *H*(*z*) are shown in Fig. 4. These figures show that a block diagram of the transformed digital filter *H*(*z*) can be obtained by simple substitution of the all-pass digital filter *<sup>H</sup>*AP(*z*) into the unit delay *<sup>z</sup>*<sup>−</sup>1. Theorem 4 and Remark 1 give us the class of digital filters with all second-order modes equal.

**Figure 4.** Block diagrams of prototype digital filter *H*P(*z*) and transformed digital filter *H*(*z*).

#### **4.3. Examples of digital filters with all second-order modes equal**

There are many types of digital filters with all second-order modes equal. We can design various digital filters by setting variables *θ*, *ρ*, and *H*AP(*z*).

#### *4.3.1. The unit delay*

The unit delay is the simplest example for digital filter with all second-order modes equal. It is obvious that letting *<sup>θ</sup>* <sup>=</sup> 1, *<sup>ρ</sup>* <sup>=</sup> 0, and *<sup>H</sup>*AP(*z*) = *<sup>z</sup>*−<sup>1</sup> in Eq. (77) yields the unit delay *<sup>z</sup>*<sup>−</sup>1.

#### *4.3.2. First-order digital filters*

Any first-order digital filter can be expressed in the form of Eq. (77). Consider a first-order IIR digital filter given by

$$H\_{IIR}(z) = \frac{b\_0 + b\_1 z^{-1}}{1 + a\_1 z^{-1}} \tag{80}$$

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(85)

*<sup>H</sup>*MN(*z*) = <sup>1</sup> <sup>+</sup> *<sup>α</sup>*

, *<sup>ρ</sup>* <sup>=</sup> <sup>1</sup> 2

that Eq. (82) can be rewritten as the form of Eq. (77) where

Consider a first-order FIR digital filter *H*FIR(*z*) given by

**4.4. Numerical examples**

*4.4.1. First-order FIR digital filters*

and controllability Gramian *<sup>K</sup>*(b)

*4.4.2. First-order IIR digital filters*

*<sup>θ</sup>* <sup>=</sup> <sup>1</sup> 2 2

This filter has *N* notches at the frequency 2*πk*/*N*[rad] for *k* = 1, ··· , *N*. One can easily show

This subsection gives numerical examples of synthesis of the minimum *L*2-sensitivity

of which frequency magnitude and phase responses are shown in Fig. 5 (a). The second-order mode of the digital filter *H*FIR(*z*) is *θ* = 0.5. The balanced realization (*A*b, *b*b, *c*b, *d*b), which

> 0 0.7071 0.7071 0.5

<sup>0</sup> and observability Gramians *<sup>W</sup>*(b)

realizations for various types of digital filters with all second-order modes equal.

is equal to the minimum *L*2-sensitivity realization, of *H*FIR(*z*) is derived as

 =

*K*(b)

<sup>0</sup> <sup>=</sup> *<sup>W</sup>*(b)

*<sup>H</sup>*IIR(*z*) = 0.25 <sup>+</sup> 0.25*z*−<sup>1</sup>

of which frequency magnitude and phase responses are shown in Fig. 5 (b). The second-order mode of the digital filter *H*IIR(*z*) is *θ* = 0.5. The balanced realization

 *A*<sup>b</sup> *b*<sup>b</sup> *c*<sup>b</sup> *d*<sup>b</sup>

Consider a first-order IIR digital filter *H*IIR(*z*) given by

1 − *z*−*<sup>N</sup>*

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

, *<sup>H</sup>*AP(*z*) = *<sup>α</sup>* <sup>−</sup> *<sup>z</sup>*−<sup>1</sup>

<sup>1</sup> <sup>−</sup> *<sup>α</sup>z*−*<sup>N</sup>* . (82)

<sup>1</sup> <sup>−</sup> *<sup>α</sup>z*−<sup>1</sup> . (83)

<sup>0</sup> are calculated as

<sup>0</sup> = 0.5. (86)

<sup>1</sup> <sup>−</sup> 0.5*z*−<sup>1</sup> (87)

*<sup>H</sup>*FIR(*z*) = 0.5 <sup>+</sup> 0.5*z*−<sup>1</sup> (84)

where *b*<sup>0</sup> and *b*<sup>1</sup> are numerator coefficients, *a*<sup>1</sup> is a denominator coefficient. One can easily show that Eq. (80) can be rewritten as the form of Eq. (77) where

$$\theta = \frac{b\_1 - a\_1 b\_0}{1 - a\_1^2}, \; \rho = \frac{b\_0 - a\_1 b\_1}{1 - a\_1^2}, \; H\_{\rm AP}(z) = \frac{a\_1 + z^{-1}}{1 + a\_1 z^{-1}}.\tag{81}$$

#### *4.3.3. All-pass digital filters*

It is obvious that all-pass digital filters are included in the class of digital filters expressed as Eq. (77). The transfer function *H*(*z*) in Eq. (77) is an all-pass digital filter when we let *θ* = 1 and *ρ* = 0.

#### *4.3.4. Multi-notch comb digital filters*

The transfer function of an *N*th-order multi-notch comb digital filter is given by

$$H\_{\rm MN}(z) = \frac{1+a}{2} \frac{1-z^{-N}}{1-az^{-N}}.\tag{82}$$

This filter has *N* notches at the frequency 2*πk*/*N*[rad] for *k* = 1, ··· , *N*. One can easily show that Eq. (82) can be rewritten as the form of Eq. (77) where

$$\theta = \frac{1}{2}, \; \rho = \frac{1}{2}, \; H\_{\rm AP}(z) = \frac{a - z^{-1}}{1 - az^{-1}}. \tag{83}$$

#### **4.4. Numerical examples**

18 Digital Filters and Signal Processing

*4.3.1. The unit delay*

*4.3.2. First-order digital filters*

IIR digital filter given by

*4.3.3. All-pass digital filters*

*4.3.4. Multi-notch comb digital filters*

and *ρ* = 0.

u(n) y(n)

(a) First-order prototype digital filter *H*P(*z*)

z−<sup>1</sup>

θ

various digital filters by setting variables *θ*, *ρ*, and *H*AP(*z*).

show that Eq. (80) can be rewritten as the form of Eq. (77) where

*<sup>θ</sup>* <sup>=</sup> *<sup>b</sup>*<sup>1</sup> <sup>−</sup> *<sup>a</sup>*1*b*<sup>0</sup> 1 − *a*<sup>2</sup> 1

**Figure 4.** Block diagrams of prototype digital filter *H*P(*z*) and transformed digital filter *H*(*z*).

**4.3. Examples of digital filters with all second-order modes equal**

There are many types of digital filters with all second-order modes equal. We can design

The unit delay is the simplest example for digital filter with all second-order modes equal. It is obvious that letting *<sup>θ</sup>* <sup>=</sup> 1, *<sup>ρ</sup>* <sup>=</sup> 0, and *<sup>H</sup>*AP(*z*) = *<sup>z</sup>*−<sup>1</sup> in Eq. (77) yields the unit delay *<sup>z</sup>*<sup>−</sup>1.

Any first-order digital filter can be expressed in the form of Eq. (77). Consider a first-order

*HIIR*(*z*) = *<sup>b</sup>*<sup>0</sup> <sup>+</sup> *<sup>b</sup>*1*z*−<sup>1</sup>

where *b*<sup>0</sup> and *b*<sup>1</sup> are numerator coefficients, *a*<sup>1</sup> is a denominator coefficient. One can easily

It is obvious that all-pass digital filters are included in the class of digital filters expressed as Eq. (77). The transfer function *H*(*z*) in Eq. (77) is an all-pass digital filter when we let *θ* = 1

, *<sup>H</sup>*AP(*z*)= *<sup>a</sup>*<sup>1</sup> <sup>+</sup> *<sup>z</sup>*−<sup>1</sup>

, *<sup>ρ</sup>* <sup>=</sup> *<sup>b</sup>*<sup>0</sup> <sup>−</sup> *<sup>a</sup>*1*b*<sup>1</sup> 1 − *a*<sup>2</sup> 1

The transfer function of an *N*th-order multi-notch comb digital filter is given by

u(n) y(n)

(b) *N*th-order transformed digital filter *H*(*z*)

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*1*z*−<sup>1</sup> (80)

<sup>1</sup> <sup>+</sup> *<sup>a</sup>*1*z*−<sup>1</sup> . (81)

HAP(z)

θ

ρ

ρ

This subsection gives numerical examples of synthesis of the minimum *L*2-sensitivity realizations for various types of digital filters with all second-order modes equal.

#### *4.4.1. First-order FIR digital filters*

Consider a first-order FIR digital filter *H*FIR(*z*) given by

$$H\_{\rm FIR}(z) = 0.5 + 0.5z^{-1} \tag{84}$$

of which frequency magnitude and phase responses are shown in Fig. 5 (a). The second-order mode of the digital filter *H*FIR(*z*) is *θ* = 0.5. The balanced realization (*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity realization, of *H*FIR(*z*) is derived as

$$
\begin{bmatrix}
\frac{A\_{\rm b}}{\mathbf{c}\_{\rm b}}
\frac{|b\_{\rm b}|}{d\_{\rm b}}
\end{bmatrix} = \begin{bmatrix}
0 & \left|
\frac{0.7071}{0.7071}
\right| \\
\hline
0.7071
\end{bmatrix} \tag{85}
$$

and controllability Gramian *<sup>K</sup>*(b) <sup>0</sup> and observability Gramians *<sup>W</sup>*(b) <sup>0</sup> are calculated as

$$\mathbf{K}\_0^{(\mathbf{b})} = \mathbf{W}\_0^{(\mathbf{b})} = \mathbf{0}.5. \tag{86}$$

#### *4.4.2. First-order IIR digital filters*

Consider a first-order IIR digital filter *H*IIR(*z*) given by

$$H\_{\rm IIR}(z) = \frac{0.25 + 0.25z^{-1}}{1 - 0.5z^{-1}}\tag{87}$$

of which frequency magnitude and phase responses are shown in Fig. 5 (b). The second-order mode of the digital filter *H*IIR(*z*) is *θ* = 0.5. The balanced realization

**Figure 5.** Frequency magnitude and phase responses of digital filters with all second-order modes equal.

(*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity realization, of *H*IIR(*z*) is derived as

$$
\begin{bmatrix}
\frac{A\_{\rm b}|b\_{\rm b}}{c\_{\rm b}|d\_{\rm b}}
\end{bmatrix} = \begin{bmatrix}
\frac{0.5}{0.6124}
\begin{vmatrix}
0.6124\\
0.25
\end{vmatrix}
\end{bmatrix} \tag{88}
$$

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*4.4.3. All-pass digital filters*

Consider a fourth-order all-pass digital filter *H*AP(*z*) given by

of which poles *λp*(*p* = 1, 2, 3, 4) are given by

�

realization, of *H*AP(*z*) is derived as

� *A*<sup>b</sup> *b*<sup>b</sup> *c*<sup>b</sup> *d*<sup>b</sup>

and controllability Gramian *<sup>K</sup>*(b)

*4.4.4. Multi-notch comb digital filters*

of which poles *λp*(*p* = 1, 2, 3, 4) are given by

� =

*K*(b)

�

<sup>0</sup> <sup>=</sup> *<sup>W</sup>*(b)

Consider a fourth-order multi-notch comb digital filter *H*MN(*z*) given by

*<sup>H</sup>*AP(*z*) = 0.5184 <sup>−</sup> 1.9805*z*−<sup>1</sup> <sup>+</sup> 3.3350*z*−<sup>2</sup> <sup>−</sup> 2.7507*z*−<sup>3</sup> <sup>+</sup> *<sup>z</sup>*−<sup>4</sup>

*λ*<sup>1</sup> = 0.9 exp(*j*0.2*π*), *λ*<sup>2</sup> = 0.9 exp(−*j*0.2*π*),

and of which frequency magnitude and phase responses are shown in Fig. 5 (c). The

The balanced realization (*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity

<sup>0</sup> and observability Gramians *<sup>W</sup>*(b)

*<sup>H</sup>*MN(*z*) = 0.9073 <sup>−</sup> 0.9073*z*−<sup>4</sup>

*λ*<sup>1</sup> = 0.95, *λ*<sup>2</sup> = −0.95,

0.8144 −0.1106 0.2499 −0.5039 −0.0903 0.1599 0.4698 −0.7114 −0.1105 −0.4853 −0.1616 0.2997 0.6211 0.1062 −0.6978 0.5318 0.0448 0.0006 0.8453 0.0252 0.0481 0.8217 0.2137 −0.0894 0.5184

second-order modes *θ<sup>i</sup>* (*i* = 1, 2, 3, 4) of the all-pass digital filter *H*AP(*z*) are given by

<sup>1</sup> <sup>−</sup> 2.7507*z*−<sup>1</sup> <sup>+</sup> 3.3350*z*−<sup>2</sup> <sup>−</sup> 1.9805*z*−<sup>3</sup> <sup>+</sup> 0.5184*z*−<sup>4</sup> (90)

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

*<sup>λ</sup>*<sup>3</sup> <sup>=</sup> 0.8 exp(*j*0.2*π*), *<sup>λ</sup>*<sup>4</sup> <sup>=</sup> 0.8 exp(−*j*0.2*π*) (91)

(*θ*1, *θ*2, *θ*3, *θ*4)=(1, 1, 1, 1). (92)

<sup>0</sup> are calculated as

<sup>0</sup> = diag(1, 1, 1, 1). (94)

<sup>1</sup> <sup>−</sup> 0.8145*z*−<sup>4</sup> (95)

*<sup>λ</sup>*<sup>3</sup> <sup>=</sup> *<sup>j</sup>*0.95, *<sup>λ</sup>*<sup>4</sup> <sup>=</sup> <sup>−</sup>*j*0.95 (96)

(93)

and controllability Gramian *<sup>K</sup>*(b) <sup>0</sup> and observability Gramians *<sup>W</sup>*(b) <sup>0</sup> are calculated as

$$\mathbf{K}\_0^{(\mathbf{b})} = \mathbf{W}\_0^{(\mathbf{b})} = \mathbf{0}.5. \tag{89}$$

#### *4.4.3. All-pass digital filters*

20 Digital Filters and Signal Processing

0.5

−0.5

0.5

−3 −2 −1 0

and controllability Gramian *<sup>K</sup>*(b)

∠ HAP(ejω)/π

as

1

∠ HFIR(ejω)/π 0

1

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

Frequency ω/π

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

Frequency ω/π

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −4

(d) fourth-order multi-notch comb digital filter *H*MN(*z*)

Frequency ω/π

<sup>0</sup> are calculated as

<sup>0</sup> = 0.5. (89)

(88)

(b) first-order IIR digital filter *H*IIR(*z*)

Frequency ω/π

0.5

−0.5

0.5

−3 −2 −1 0

∠ HMN(ejω)/π

1

∠ HIIR(ejω)/π 0

1

Frequency ω/π

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −1

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> <sup>0</sup>

Frequency ω/π

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> −4

(c) fourth-order all-pass digital filter *H*AP(*z*)

Frequency ω/π

 *A*<sup>b</sup> *b*<sup>b</sup> *c*<sup>b</sup> *d*<sup>b</sup>

**Figure 5.** Frequency magnitude and phase responses of digital filters with all second-order modes equal.

 =

*K*(b)

<sup>0</sup> <sup>=</sup> *<sup>W</sup>*(b)

(*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity realization, of *H*IIR(*z*) is derived

 0.5 0.6124 0.6124 0.25

<sup>0</sup> and observability Gramians *<sup>W</sup>*(b)

(a) first-order FIR digital filter *H*FIR(*z*)

Frequency ω/π

Consider a fourth-order all-pass digital filter *H*AP(*z*) given by

$$H\_{\rm AP}(z) = \frac{0.5184 - 1.9805z^{-1} + 3.3350z^{-2} - 2.7507z^{-3} + z^{-4}}{1 - 2.7507z^{-1} + 3.3350z^{-2} - 1.9805z^{-3} + 0.5184z^{-4}}\tag{90}$$

of which poles *λp*(*p* = 1, 2, 3, 4) are given by

$$\begin{cases} \lambda\_1 = 0.9 \exp(j0.2\pi), \ \lambda\_2 = 0.9 \exp(-j0.2\pi), \\ \lambda\_3 = 0.8 \exp(j0.2\pi), \ \lambda\_4 = 0.8 \exp(-j0.2\pi) \end{cases} \tag{91}$$

and of which frequency magnitude and phase responses are shown in Fig. 5 (c). The second-order modes *θ<sup>i</sup>* (*i* = 1, 2, 3, 4) of the all-pass digital filter *H*AP(*z*) are given by

$$(\theta\_1, \theta\_2, \theta\_3, \theta\_4) = (1, 1, 1, 1). \tag{92}$$

The balanced realization (*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity realization, of *H*AP(*z*) is derived as

$$
\begin{bmatrix} A\_{\mathtt{b}} \begin{vmatrix} b\_{\mathtt{b}} \\ c\_{\mathtt{b}} \end{vmatrix} = \begin{bmatrix} 0.8144 & -0.1106 & 0.2499 & -0.5039 \end{bmatrix} - 0.0903 \\\ \begin{bmatrix} 0.1599 & 0.4698 & -0.7114 & -0.1105 \end{bmatrix} - 0.4853 \\\ \begin{bmatrix} -0.1616 & 0.2997 & 0.6211 & 0.1062 \end{bmatrix} - 0.6978 \\\ \begin{bmatrix} 0.5318 & 0.0448 & 0.0006 & 0.8453 & 0.0252 \\\ \hline 0.0481 & 0.8217 & 0.2137 & -0.0894 & 0.5184 \end{bmatrix} \end{bmatrix} \tag{93}
$$

and controllability Gramian *<sup>K</sup>*(b) <sup>0</sup> and observability Gramians *<sup>W</sup>*(b) <sup>0</sup> are calculated as

$$\mathbf{K}\_0^{(\mathbf{b})} = \mathbf{W}\_0^{(\mathbf{b})} = \text{diag}(1, 1, 1, 1). \tag{94}$$

#### *4.4.4. Multi-notch comb digital filters*

�

Consider a fourth-order multi-notch comb digital filter *H*MN(*z*) given by

$$H\_{\rm MN}(z) = \frac{0.9073 - 0.9073z^{-4}}{1 - 0.8145z^{-4}}\tag{95}$$

of which poles *λp*(*p* = 1, 2, 3, 4) are given by

$$\begin{cases} \lambda\_1 = 0.95, \, \lambda\_2 = -0.95, \\ \lambda\_3 = j0.95, \, \lambda\_4 = -j0.95 \end{cases} \tag{96}$$

and of which frequency magnitude and phase responses are shown in Fig. 5 (d). The second-order modes *θ<sup>i</sup>* (*i* = 1, 2, 3, 4) of the multi-notch comb digital filter *H*MN(*z*) are given by

$$(\theta\_1, \theta\_2, \theta\_3, \theta\_4) = (0.5, 0.5, 0.5, 0.5). \tag{97}$$

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*P*opt = *RTB*opt*R*. (100)

opt*RU*. (101)

opt. (102)

opt*b*b, *<sup>c</sup>*b*T*˜ opt, *<sup>d</sup>*b) (103)

opt (104)

opt (105)

<sup>0</sup> of the minimum *L*2-sensitivity realization

We can give the diagonalization of the matrix *P*opt as follows:

transformation matrix *T*opt is given by

matrix as *U* = *RT*, which yields

*coordinate transformation by T*˜ *opt such as*

(*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) are expressed as

<sup>0</sup> and the observability Gramian *<sup>W</sup>*˜ (opt)

the controllability and observability Gramians as follows:

*K*˜ (opt)

(*A*˜ opt, *<sup>b</sup>*˜ opt, ˜*c*opt, ˜*d*opt)=(*T*˜ <sup>−</sup><sup>1</sup>

*K*˜ (opt) <sup>0</sup> <sup>=</sup>*T*˜ <sup>−</sup><sup>1</sup>

*W*˜ (opt) <sup>0</sup> <sup>=</sup>*T*˜ *<sup>T</sup>*

Since the matrix *P*opt is positive definite symmetric, it can be diagonalized by an orthogonal matrix *R*, and *B*opt is a positive definite diagonal matrix. The optimal coordinate

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

1 2 opt*<sup>U</sup>*

=*RTB* 1 2

In the above expression, *U* is an arbitrary orthogonal matrix. It means that the minimum *L*2-sensitivity realizations exist infinitely for a given digital filter *H*(*z*). The minimum *L*2-sensitivity realizations have freedom for orthogonal transformations. We show that the minimum *L*2-sensitivity realization does not generate limit cycles if we specify the orthogonal

*T*˜ opt = *RTB*

**Theorem 5.** *The minimum L*2*-sensitivity realization* (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt)*, obtained by the*

*does not generate limit cycles.* ✷

*Proof:* Under the coordinate transformation by *T*˜ opt in Eq. (102), the controllability Gramian

opt*K*(b) <sup>0</sup> *<sup>T</sup>*˜ <sup>−</sup>*<sup>T</sup>* opt

opt*W*(b)

where **Θ** = diag(*θ*1, ··· , *θN*). From Eqs. (104) and (105), we can derive the relation between

opt*R***Θ***RT<sup>B</sup>*

<sup>0</sup> *<sup>T</sup>*˜ opt

2

1 2

<sup>=</sup>*B*<sup>−</sup> <sup>1</sup> 2 opt*R***Θ***RTB*<sup>−</sup> <sup>1</sup>

=*B* 1 2 1 2

opt*A*b*T*˜ opt, *<sup>T</sup>*˜ <sup>−</sup><sup>1</sup>

*T*opt =*P*

The balanced realization (*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity realization, of *H*MN(*z*) is derived as

$$
\begin{bmatrix}
\frac{A\_{\mathbf{b}}}{\mathbf{c}\_{\mathbf{b}}}
\frac{b\_{\mathbf{b}}}{d\_{\mathbf{b}}}
\end{bmatrix} = \begin{bmatrix}
0 \ 0 \ 0 \ 0.8145 & 0.4102 \\
1 \ 0 \ 0 \ 0 & 0 \\
0 \ 1 \ 0 & 0 \\
0 \ 0 \ 1 & 0 \\
\hline
0 \ 0 \ 0 & -0.4102 \boxed{0.9073}
\end{bmatrix} \tag{98}
$$

and controllability Gramian *<sup>K</sup>*(b) <sup>0</sup> and observability Gramians *<sup>W</sup>*(b) <sup>0</sup> are calculated as

$$\mathbf{K}\_0^{(\mathbf{b})} = \mathbf{W}\_0^{(\mathbf{b})} = \text{diag}(0.5, 0.5, 0.5, 0.5). \tag{99}$$

## **5. Absence of limit cycles in the minimum** *L*2**-sensitivity realizations**

This section proves the absence of limit cycles of the minimum *L*2-sensitivity realization from the viewpoint of the controllability and observability Gramians. The minimum *L*2-sensitivity realizations have freedom for orthogonal transformations. In other words, minimum *L*2-sensitivity realizations are not unique. We select the minimum *L*2-sensitivity realization without limit cycles among these minimum *L*2-sensitivity realizations. The controllability and observability Gramians of the selected minimum *L*2-sensitivity realization satisfy a sufficient condition for the absence of limit cycles [11].

## **5.1. Theoretical proof of the absence of limit cycles**

For high-order digital filters, we synthesize the minimum *L*2-sensitivity realization by the successive approximation methods in [1] or [2], for examples. For second-order digital filters, we can synthesize the minimum *L*2-sensitivity realization by the closed form solutions proposed in Section 3. For both cases, we can construct the minimum *L*2-sensitivity realization *without limit cycles*.

We begin by reviewing the procedure to synthesize the minimum *L*2-sensitivity. We solve the *L*2-sensitivity minimization problem in (32) adopting the balanced realization (*A*b, *b*b, *c*b, *d*b) as an initial realization. We obtain the optimal positive definite symmetric matrix *P*opt. In case of high-order digital filters, we can derive the optimal positive definite symmetric matrix *P*opt by successive approximation method in [1] or [2], for example. In case of second-order digital filters, we can derive the optimal positive definite symmetric matrix *P*opt analytically as proposed in Section 3.

We can give the diagonalization of the matrix *P*opt as follows:

$$P\_{\rm opt} = \mathbf{R}^T \mathbf{B}\_{\rm opt} \mathbf{R}.\tag{100}$$

Since the matrix *P*opt is positive definite symmetric, it can be diagonalized by an orthogonal matrix *R*, and *B*opt is a positive definite diagonal matrix. The optimal coordinate transformation matrix *T*opt is given by

$$T\_{\rm opt} = \mathbf{P}\_{\rm opt}^{\frac{1}{2}} \mathbf{U}$$

$$= \mathbf{R}^T \mathbf{B}\_{\rm opt}^{\frac{1}{2}} \mathbf{R} \mathbf{U}. \tag{101}$$

In the above expression, *U* is an arbitrary orthogonal matrix. It means that the minimum *L*2-sensitivity realizations exist infinitely for a given digital filter *H*(*z*). The minimum *L*2-sensitivity realizations have freedom for orthogonal transformations. We show that the minimum *L*2-sensitivity realization does not generate limit cycles if we specify the orthogonal matrix as *U* = *RT*, which yields

$$
\tilde{T}\_{\rm opt} = \mathbf{R}^T \mathbf{B}\_{\rm opt}^{\frac{1}{2}}.\tag{102}
$$

**Theorem 5.** *The minimum L*2*-sensitivity realization* (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt)*, obtained by the coordinate transformation by T*˜ *opt such as*

$$(\tilde{\mathbf{A}}\_{\mathrm{opt}} \tilde{\mathbf{b}}\_{\mathrm{opt}} \tilde{\mathbf{c}}\_{\mathrm{opt}} \tilde{\mathbf{d}}\_{\mathrm{opt}}) = (\tilde{T}\_{\mathrm{opt}}^{-1} \mathbf{A}\_{\mathrm{b}} \tilde{T}\_{\mathrm{opt}} \tilde{T}\_{\mathrm{opt}}^{-1} \mathbf{b}\_{\mathrm{b}\prime} \mathbf{c}\_{\mathrm{b}} \tilde{T}\_{\mathrm{opt}} \boldsymbol{d}\_{\mathrm{b}}) \tag{103}$$

*does not generate limit cycles.* ✷

22 Digital Filters and Signal Processing

realization, of *H*MN(*z*) is derived as

and controllability Gramian *<sup>K</sup>*(b)

realization *without limit cycles*.

as proposed in Section 3.

� *A*<sup>b</sup> *b*<sup>b</sup> *c*<sup>b</sup> *d*<sup>b</sup>

*K*(b)

satisfy a sufficient condition for the absence of limit cycles [11].

**5.1. Theoretical proof of the absence of limit cycles**

<sup>0</sup> <sup>=</sup> *<sup>W</sup>*(b)

� =

by

and of which frequency magnitude and phase responses are shown in Fig. 5 (d). The second-order modes *θ<sup>i</sup>* (*i* = 1, 2, 3, 4) of the multi-notch comb digital filter *H*MN(*z*) are given

The balanced realization (*A*b, *b*b, *c*b, *d*b), which is equal to the minimum *L*2-sensitivity

0 0 0 0.8145 0.4102 100 0 0 010 0 0 001 0 0 000 −0.4102 0.9073

<sup>0</sup> and observability Gramians *<sup>W</sup>*(b)

**5. Absence of limit cycles in the minimum** *L*2**-sensitivity realizations**

This section proves the absence of limit cycles of the minimum *L*2-sensitivity realization from the viewpoint of the controllability and observability Gramians. The minimum *L*2-sensitivity realizations have freedom for orthogonal transformations. In other words, minimum *L*2-sensitivity realizations are not unique. We select the minimum *L*2-sensitivity realization without limit cycles among these minimum *L*2-sensitivity realizations. The controllability and observability Gramians of the selected minimum *L*2-sensitivity realization

For high-order digital filters, we synthesize the minimum *L*2-sensitivity realization by the successive approximation methods in [1] or [2], for examples. For second-order digital filters, we can synthesize the minimum *L*2-sensitivity realization by the closed form solutions proposed in Section 3. For both cases, we can construct the minimum *L*2-sensitivity

We begin by reviewing the procedure to synthesize the minimum *L*2-sensitivity. We solve the *L*2-sensitivity minimization problem in (32) adopting the balanced realization (*A*b, *b*b, *c*b, *d*b) as an initial realization. We obtain the optimal positive definite symmetric matrix *P*opt. In case of high-order digital filters, we can derive the optimal positive definite symmetric matrix *P*opt by successive approximation method in [1] or [2], for example. In case of second-order digital filters, we can derive the optimal positive definite symmetric matrix *P*opt analytically

(*θ*1, *θ*2, *θ*3, *θ*4)=(0.5, 0.5, 0.5, 0.5). (97)

<sup>0</sup> = diag(0.5, 0.5, 0.5, 0.5). (99)

<sup>0</sup> are calculated as

(98)

*Proof:* Under the coordinate transformation by *T*˜ opt in Eq. (102), the controllability Gramian *K*˜ (opt) <sup>0</sup> and the observability Gramian *<sup>W</sup>*˜ (opt) <sup>0</sup> of the minimum *L*2-sensitivity realization (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) are expressed as

$$\begin{split} \tilde{\mathbf{K}}\_{0}^{\text{(opt)}} &= \tilde{T}\_{\text{opt}}^{-1} \mathbf{K}\_{0}^{\text{(b)}} \tilde{T}\_{\text{opt}}^{-T} \\ &= \mathbf{B}\_{\text{opt}}^{-\frac{1}{2}} \mathbf{R} \boldsymbol{\Theta} \mathbf{R}^{T} \mathbf{B}\_{\text{opt}}^{-\frac{1}{2}} \end{split} \tag{104}$$

$$\begin{split} \tilde{\mathbf{W}}\_{0}^{(\text{opt})} &= \tilde{\mathbf{T}}\_{\text{opt}}^{T} \mathbf{W}\_{0}^{(\text{b})} \tilde{\mathbf{T}}\_{\text{opt}} \\ &= \mathbf{B}\_{\text{opt}}^{\frac{1}{2}} \mathbf{R} \boldsymbol{\Theta} \mathbf{R}^{T} \mathbf{B}\_{\text{opt}}^{\frac{1}{2}} \end{split} \tag{105}$$

where **Θ** = diag(*θ*1, ··· , *θN*). From Eqs. (104) and (105), we can derive the relation between the controllability and observability Gramians as follows:

**Figure 6.** Synthesis of the minimum *L*2-sensitivity realization which does not generate limit cycles.

$$\mathbf{W}\_0^{(\text{opt})} = \mathbf{B}\_{\text{opt}} \mathbf{K}\_0^{(\text{opt})} \mathbf{B}\_{\text{opt}}.\tag{106}$$

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x 1 (n) x 2 (n)

0 0.2 0.4 0.6 0.8 1

Analytical Approach for Synthesis of Minimum *L2*-Sensitivity Realizations for State-Space Digital Filters

Frequency ω/π

−0.5

x

1(n), x2(n)

0

0.5

1

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> −1

<sup>1</sup> <sup>−</sup> 1.4562*z*−<sup>1</sup> <sup>+</sup> 0.81*z*−<sup>2</sup> . (108)

Time n

(b) Direct form II

x 1 (n) x 2 (n)

Consider a second-order narrow-band band-pass digital filter *H*(*z*) given by

*<sup>H</sup>*(*z*) = 0.0316 <sup>+</sup> 0.0602*z*−<sup>1</sup> <sup>+</sup> 0.0316*z*−<sup>2</sup>

The poles of the transfer function *H*(*z*) in Eq. (108) are 0.9 exp(±*j*0.2*π*), which are very close to the unit circle. The frequency response of the digital filter *H*(*z*) in Eq. (108) is shown in Fig. 7. The coefficient matrices of the minimum *L*2-sensitivity realization which is free of

0

0.2

0.4

0.6

**Figure 7.** Frequency Response of digital filter *H*(*z*) in Eq. (108).

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> −1

**Figure 8.** Zero-input responses of *H*(*z*) in Eq. (108).

*5.2.1. Second-order digital filters*

Time n

limit cycle is derived by Eq. (107) as follows:

(a) Minimum *L*2-sensitivity realization

−0.5

0

x1(n), x2(n)

0.5

1

0.8

1

Eq. (106) is equivalent to a sufficient condition for the absence of limit cycles proposed in Ref. [6]. Therefore, the minimum *L*2-sensitivity realization (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) does not generate limit cycles. ✷

Theorem 5 shows that we can synthesize the minimum *L*2-sensitivity realization without limit cycles by choosing appropriate orthogonal matrix *U*. Fig. 6 shows the synthesis procedure of the minimum *L*2-sensitivity realization which does not generate limit cycles. The coefficient matrices of the minimum *L*2-sensitivity realization without limit cycles (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) are given by

$$
\left[\frac{\tilde{\mathbf{A}}\_{\mathrm{opt}} \Big| \tilde{\mathbf{b}}\_{\mathrm{opt}}}{\tilde{\mathbf{c}}\_{\mathrm{opt}} \Big| \tilde{d}\_{\mathrm{opt}}}\right] = \left[\frac{\mathbf{B}\_{\mathrm{opt}}^{-\frac{1}{2}} \mathbf{R} \mathbf{A}\_{\mathrm{b}} \mathbf{R}^{T} \mathbf{B}\_{\mathrm{opt}}^{\frac{1}{2}} \mathbf{B}\_{\mathrm{opt}}^{-\frac{1}{2}} \mathbf{B} \mathbf{b}\_{\mathrm{b}}}{\mathbf{c}\_{\mathrm{b}} \mathbf{R}^{T} \mathbf{B}\_{\mathrm{opt}}^{\frac{1}{2}} \Big| \quad d}\right].\tag{107}
$$

#### **5.2. Numerical examples**

We present numerical examples to demonstrate the validity of our proposed method. We synthesize the minimum *L*2-sensitivity realizations of second-order and fourth-order digital filters which do not generate limit cycles.

**Figure 7.** Frequency Response of digital filter *H*(*z*) in Eq. (108).

24 Digital Filters and Signal Processing

� Aopt bopt copt dopt

(*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) are given by

**5.2. Numerical examples**

filters which do not generate limit cycles.

� *A*˜ opt *b*˜ opt *c*˜opt ˜*d*opt

� =  *B*<sup>−</sup> <sup>1</sup> 2 opt*RA*b*RT<sup>B</sup>*

> *<sup>c</sup>*b*RT<sup>B</sup>* 1 2

We present numerical examples to demonstrate the validity of our proposed method. We synthesize the minimum *L*2-sensitivity realizations of second-order and fourth-order digital

1 2 opt *<sup>B</sup>*<sup>−</sup> <sup>1</sup> 2 opt*Rb*<sup>b</sup>

opt *<sup>d</sup>*

�

1

**Figure 6.** Synthesis of the minimum *L*2-sensitivity realization which does not generate limit cycles.

*W*˜ (opt)

� A<sup>b</sup> b<sup>b</sup> c<sup>b</sup> d<sup>b</sup>

� Aopt bopt copt dopt

Balanced realization

Minimum L2-sensitivity realizations without L2-scaling constraints

<sup>0</sup> <sup>=</sup>*B*opt*K*˜ (opt)

Eq. (106) is equivalent to a sufficient condition for the absence of limit cycles proposed in Ref. [6]. Therefore, the minimum *L*2-sensitivity realization (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) does not generate limit cycles. ✷ Theorem 5 shows that we can synthesize the minimum *L*2-sensitivity realization without limit cycles by choosing appropriate orthogonal matrix *U*. Fig. 6 shows the synthesis procedure of the minimum *L*2-sensitivity realization which does not generate limit cycles. The coefficient matrices of the minimum *L*2-sensitivity realization without limit cycles

Limit Cycle Free

c˜opt ˜ dopt

� A˜ opt ˜

�

R<sup>T</sup> B <sup>1</sup> <sup>2</sup> R

U<sup>1</sup> U<sup>M</sup>

�

bopt

R<sup>T</sup>

�

� Aopt bopt copt dopt � M

<sup>0</sup> *<sup>B</sup>*opt. (106)

. (107)

**Figure 8.** Zero-input responses of *H*(*z*) in Eq. (108).

#### *5.2.1. Second-order digital filters*

Consider a second-order narrow-band band-pass digital filter *H*(*z*) given by

$$H(z) = \frac{0.0316 + 0.0602z^{-1} + 0.0316z^{-2}}{1 - 1.4562z^{-1} + 0.81z^{-2}}.\tag{108}$$

The poles of the transfer function *H*(*z*) in Eq. (108) are 0.9 exp(±*j*0.2*π*), which are very close to the unit circle. The frequency response of the digital filter *H*(*z*) in Eq. (108) is shown in Fig. 7. The coefficient matrices of the minimum *L*2-sensitivity realization which is free of limit cycle is derived by Eq. (107) as follows:

$$
\begin{bmatrix}
\frac{\mathbf{A}\_{\rm opt}}{\mathbf{c}\_{\rm opt}}
\begin{bmatrix}
\dot{\mathbf{b}}\_{\rm opt}
\end{bmatrix}
\end{bmatrix} = \begin{bmatrix}
0.7281 & 0.5229 \\
\hline
0.1282 & -0.4146 \\
\end{bmatrix} \begin{bmatrix}
0.4146 \\
0.0316
\end{bmatrix} \,. \tag{109}
$$

The controllability Gramian *<sup>K</sup>*(opt) <sup>0</sup> and the observability Gramian *<sup>W</sup>*(opt) <sup>0</sup> are given as follows:

$$
\tilde{\mathbf{K}}\_0^{(\text{opt})} = \begin{bmatrix} 0.5100 \ -0.0870 \\ -0.0870 & 0.4901 \end{bmatrix} \tag{110}
$$

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. (114)

(115)

0

<sup>0</sup> are given as follows:

. (116)

<sup>0</sup> and the observability Gramian *<sup>W</sup>*˜ (opt)

� *A*˜ opt *b*˜ opt *<sup>c</sup>*˜opt ˜*d*opt �

The controllability Gramian *<sup>K</sup>*˜ (opt)

cycles.

form II.

[0.9 0.9 0.9 0.9]

=

*K*˜ (opt) <sup>0</sup> =

*W*˜ (opt) <sup>0</sup> =

We have to note that the controllability Gramian *<sup>K</sup>*˜ (opt)

satisfy the sufficient condition of the absence of limit cycles given in Eq. (106) with

Therefore, (*A*˜ opt, *b*˜ opt, ˜*c*opt, ˜*d*opt) is the minimum *L*2-sensitivity realization without limit

We demonstrate the absence of limit cycles in the minimum *L*2-sensitivity realization by observing its zero-input response. We calculate the zero-input responses of the minimum *L*2-sensitivity realization and the direct form II, setting the initial state as *x*(0) =

complement as the overflow characteristic. The zero-input responses are shown in Fig. 10(a) and 10(b). We assume that each filter coefficient and signal have 16[bits] fixed-point representation, of which lower 13[bits] are fractional bits. In this numerical example, the overflow of the state variables occurs in both cases. Also in this case, we can confirm that the minimum *L*2-sensitivity realization does not generate limit cycles. On the other hand, a large-amplitude autonomous oscillation is observed in the zero-input response of the direct