Chebyshev approximation problem of multidimensional IIR digital filters in the frequency domain

Abstract

This paper investigates the problem of simultaneous approximation of a prescribed multidimensional frequency response. The frequency responses of multidimensional IIR digital filters are used as nonlinear approximating functions. Chebyshev approximation theory and the notion of line homotopy are used to reveal the approximation properties of this set of IIR functions. A sign condition is derived to characterize a convex stable domain in this set. This sign condition can be incorporated into the optimization of the Chebyshev simultaneous approximation. The generally sufficient global Kolmogorov criterion is shown to be a necessary condition, for the characterization of best approximation, in the considered set of approximating functions. Thus, it can be incorporated, as a stopping constraint, in the design of the optimal filter. Moreover, the local Kolmogorov criterion is shown to be also necessary for the set of approximating functions. Finally, it is proved that the best approximation is a global minimum.

Appendix

1.1 Examples

Example 1

(2D circular Low Pass Filters) Two low pass filters, $H_{10}$ (Chen 1988, P. 94) of order 8 (4, 4), and $H_{11}$ (Sheng-Lu and Antoniou 1992, P. 249) of order 10 (5, 5), are considered to show that under the validity of the sign condition the homotopic sequence $h_{\lambda }$ is strictly monotone. The selected filters have circular symmetry. The denominators $D_{10}$ and $D_{11}$, with coefficients in the following matrices $A_{10}$ and $A_{11}$, were formed.

$$\begin{aligned} D_{10}= & {} {{\sum _{i=0}^{4}\sum _{j=0}^4} a_{10}(i,j){z_1}^{i} {z_2}^{j}} \\ D_{11}= & {} {{\sum _{i=0}^{5}\sum _{j=0}^5} a_{11}(i,j){z_1}^{i} {z_2}^{j}} \end{aligned}$$

The real parts $g_{10}(\omega _1,\omega _2 )$, $g_{11}(\omega _1,\omega _2 )$ and imaginary parts $u_{10} (\omega _1,\omega _2 )$, $u_{11} (\omega _1,\omega _2 )$ of $D_{10}$ and $D_{11}$ were extracted. The product $g_{10} g_{11}+ u_{10} u_{11}$ was computed and plotted in the frequency range $|\omega _1| <1$ and $|\omega _2| <1$ to check the SC, i.e., $g_{10} g_{11}+ u_{10} u_{11}>$0. Figure 1 shows that the minimum value of the product $g_{10} g_{11}+ u_{10}u_{11}$ is positive (minimum value = 0.0019) and thus the sign condition is satisfied. The filters $H_{10}$ and $H_{11}$, for which the sign condition was validated, were combined to generate the homotopic sequence:

$$\begin{aligned} |H_{10}-h_{\lambda }|=\left| \frac{\lambda {D_{11}}}{\lambda {D_{11}}+(1-\lambda )D_{11}}\right| |(H_{10}-H_{11})| \end{aligned}$$

and its derivative

$$\begin{aligned} \frac{\partial |H_{10}-h_{\lambda }|}{\partial \lambda }={\frac{|D_{10}|}{|D_{\lambda }|^3}}\left[ |D_{10}|^2+\lambda \left( g_{10}g_{11}+u_{10}u_{11}-|D_{10}|^2\right) \right] |H_{10}-H_{11}| \end{aligned}$$

The derivative was computed as a function of $\lambda$ for different ($\omega _1$, $\omega _2$ ) pairs in the considered frequency range. Results showed that the sign of the derivatives is always positive and consequently, the homotopic sequence is strictly monotone. Figure 2 shows the plot of the derivative versus $\lambda$ for $(\omega _1, \omega _2)=(-0.5566, 0.7279),(-0.3425, 0.5138),(-0.1285, 0.9957)$ and (0.0856, 0.5138). This plot shows, that under the validity of the sign condition, the derivative $|H_{10}- h_{\lambda }|$ with respect to $\lambda$, is positive for all the values of $\lambda$.

$$\begin{aligned} A_{10}= & {} \left[ \begin{array}{ccccc} 1.0000 &{}\quad -4.1440 &{}\quad 6.1910 &{}\quad -4.9780 &{}\quad -0.0160\\ -4.1440&{}\quad 17.7000 &{}\quad -24.4800 &{}\quad 21.5000 &{}\quad -0.0177\\ 6.1910 &{}\quad -24.4800&{}\quad 35.5800&{}\quad -30.5800 &{}\quad -0.0047\\ -4.9780 &{}\quad 21.5000&{}\quad -30.5800 &{}\quad 27.7300 &{}\quad 0.0040\\ -0.0160 &{}\quad 0.0177 &{}\quad -0.0047 &{}\quad 0.0040 &{}\quad 0.0172 \end{array} \right] \\ A_{11}= & {} \left[ \begin{array}{cccccc} 0.0652 &{}\quad -0.6450&{}\quad 3.3632 &{}\quad -4.8317&{}\quad 0.3218 &{}\quad -0.1645\\ -0.7930 &{}\quad 7.8851 &{}\quad -25.8710 &{}\quad 23.8380 &{}\quad 3.4048 &{}\quad 3.4667\\ 4.2941 &{}\quad -28.7340 &{}\quad 61.5510 &{}\quad -29.3020&{}\quad -13.2490 &{}\quad 25.5190\\ -6.3054 &{}\quad 28.7070 &{}\quad 33.4870 &{}\quad -7.2275 &{}\quad 22.7050 &{}\quad 83.0110\\ 0.7907 &{}\quad 1.4820 &{}\quad -7.4214 &{}\quad -33.3130 &{}\quad 136.7600 &{}\quad -128.4300\\ -0.4134 &{}\quad 6.0739 &{}\quad -36.0290 &{}\quad 101.4700 &{}\quad 140.2000 &{}\quad 78.4280 \end{array}\right] . \end{aligned}$$

Fig. 2

Example 1: the derivative of the homotopic sequence $|H_{10}-h_{\lambda }|$ with respect to $\lambda$ for different values of $\omega _1$ and $\omega _2$

Fig. 3

Example 2: plot of $G = g_{20}g_{21} + u_{20}u_{21}$, versus ($\omega _1, \omega _2$) and different values of $\omega _3$, for the combination of the two 3D circular filters $H_{20}$ and $H_{21}$

Example 2

(3D circular Low Pass Filters) In this example, we considered two 3D circular LP filters, $H_{20}$ of order 5 (1, 2, 2) and $H_{21}$ of order 6 (2, 2, 2) (Chen 1988, pp. 154, 156), to show that under the validity of the sign condition the homotopic sequence $h_{\lambda }$ is strictly monotone. As expected, the computation cost increased significantly with respect to the two-dimensional case (Example 1). The denominators $D_{20}$ and $D_{21}$, with coefficients in the following two 3D matrices $A_{20}$ and $A_{21}$, respectively, were formed.

$$\begin{aligned} D_{20}= & {} {{\sum _{i=0}^{1}\sum _{j=0}^2}\sum _{j=0}^2} a_{20}(i,j,k){z_1}^{i} {z_2}^{j}{z_3}^{k } \\ D_{21}= & {} {{\sum _{i=0}^{2}\sum _{j=0}^2}\sum _{j=0}^2} a_{21}(i,j,k){z_1}^{i} {z_2}^{j}{z_3}^{k} \end{aligned}$$

The real parts $g_{20}(\omega _1,\omega _2,\omega _3 )$, $g_{21}(\omega _1,\omega _2, \omega _3 )$ and imaginary parts $u_{20}(\omega _1,\omega _2, \omega _3 )$, $u_{21} (\omega _1,\omega _2, \omega _3 )$ of $D_{20}$ and $D_{21}$ were extracted. The product $g_{20}g_{21}+ u_{20}u_{21}$ was computed and plotted in the frequency range $|\omega _1| <0.4$, $|\omega _2| <0.4$, and $|\omega _3| <0.4$ to check the SC in the 3D case, i.e. $g_{20}g_{21}+u_{20}u_{21}>$0. Figure 3 shows that the minimum value of the product $g_{20}g_{21}$+$u_{20}u_{21}$ is positive (minimum value $=$ 0.1145) and thus the sign condition is satisfied. The filters $H_{20}$ and $H_{21}$, that satisfy the SC, were combined to generate the homotopic sequence:

$$\begin{aligned} |H_{20}-h_{\lambda }|=\left| \frac{\lambda {D_{21}}}{\lambda {D_{21}}+(1-\lambda )D_{21}}\right| .|(H_{20}-H_{21})| \end{aligned}$$

and its derivative

$$\begin{aligned} \frac{\partial |H_{20}-h_{\lambda }|}{\partial \lambda }={\frac{|D_{20}|}{|D_{\lambda }|^3}}\left[ |D_{20}|^2+\lambda \left( g_{20}g_{21}+u_{20}u_{21}-|D_{20}|^2\right) \right] |H_{20}-H_{21}| \end{aligned}$$

Fig. 4

Example 2: the derivative of the homotopic sequence $|H_{20}-h_{\lambda }|$ with respect to $\lambda$ for different values of $\omega _1$, $\omega _2$, and $\omega _3$

The derivative was computed as a function of $\lambda$ for different ($\omega _1,\omega _2,\omega _3$) in the filter frequency range. Results showed that the sign of the derivative is always positive and consequently, the homotopic sequence is strictly monotone. Figure 4 shows the plot of the derivative versus $\lambda$ for ($\omega _1,\omega _2,\omega _3) =(0.2042,-0.2356,-0.0666),(-0.0471,-0.0471, -0.2444),(-0.1728,-0.0471,-0.2444)$ and $(0.3299,0.0157,-0.1555)$. The plot shows, that under the validity of the sign condition, the derivative $|H_{20}- h_{\lambda }|$ with respect to $\lambda$, is positive for all the values of $\lambda$.

$$\begin{aligned} A_{20}(:,:,1)= & {} \left[ \begin{array}{ccccc} 1.0000 &{}\quad -2.0672 &{}\quad 1.9123 &{}\quad -0.9076 &{}\quad 0.1817\\ -2.0672 &{}\quad 4.2732 &{}\quad -3.9529 &{}\quad 1.8761 &{}\quad -0.3756\\ 1.9123 &{}\quad -3.9529 &{}\quad 3.6567 &{}\quad -1.7355 &{}\quad 0.3475\\ -0.9076 &{}\quad 1.8761 &{}\quad -1.7355 &{}\quad 0.8237 &{}\quad -0.1649\\ 0.1817 &{}\quad -0.3756 &{}\quad 0.3475 &{}\quad -0.1649 &{}\quad 0.0330\\ \end{array} \right] \\ A_{20}(:,:,2)= & {} \left[ \begin{array}{ccccc} -2.0672 &{}\quad 4.2732 &{}\quad -3.9529 &{}\quad 1.8761 &{}\quad -0.3756\\ 4.2732 &{}\quad -8.8333 &{}\quad 8.1714 &{}\quad -3.8782 &{}\quad 0.7765\\ -3.9529 &{}\quad 8.1714 &{}\quad -7.5590 &{}\quad 3.5875 &{}\quad -0.7183\\ 1.8761 &{}\quad -3.8782 &{}\quad 3.5875 &{}\quad -1.7027 &{}\quad 0.3409\\ -0.3756 &{}\quad 0.7765 &{}\quad -0.7183 &{}\quad 0.3409 &{}\quad -0.0683\\ \end{array} \right] \\ A_{20}(:,:,3)= & {} \left[ \begin{array}{ccccc} 1.9123 &{}\quad -3.9529 &{}\quad 3.6567 &{}\quad -1.7355 &{}\quad 0.3475\\ -3.9529 &{}\quad 8.1714 &{}\quad -7.5590 &{}\quad 3.5875 &{}\quad -0.7183\\ 3.6567 &{}\quad -7.5590 &{}\quad 6.9926 &{}\quad -3.3187 &{}\quad 0.6645\\ -1.7355 &{}\quad 3.5875 &{}\quad -3.3187 &{}\quad 1.5751 &{}\quad -0.3154\\ 0.3475 &{}\quad -0.7183 &{}\quad 0.6645 &{}\quad -0.3154 &{}\quad 0.0631\\ \end{array} \right] \\ A_{20}(:,:,4)= & {} \left[ \begin{array}{ccccc} -0.9076 &{}\quad 1.8761 &{}\quad -1.7355 &{}\quad -1.7355 &{}\quad 0.3475\\ 1.8761 &{}\quad -3.8782 &{}\quad 3.5875 &{}\quad -1.7027 &{}\quad 0.3409\\ -1.7355 &{}\quad 3.5875 &{}\quad -3.3187 &{}\quad 1.5751 &{}\quad -0.3154\\ -1.7355 &{}\quad -1.7027 &{}\quad 1.5751 &{}\quad -0.7475 &{}\quad 0.1497\\ 0.3475 &{}\quad 0.3409 &{}\quad -0.3154 &{}\quad 0.1497 &{}\quad -0.0300\\ \end{array} \right] \\ A_{20}(:,:,5)= & {} \left[ \begin{array}{ccccc} 0.1817 &{}\quad -0.3756 &{}\quad 0.3475 &{}\quad -0.1343 &{}\quad 0.0330\\ -0.3756 &{}\quad 0.7765 &{}\quad -0.7183 &{}\quad 0.3409 &{}\quad -0.0683\\ 0.3475 &{}\quad -0.7183 &{}\quad 0.6645 &{}\quad -0.3154 &{}\quad 0.0631\\ -0.1649 &{}\quad 0.3409 &{}\quad -0.3154 &{}\quad 0.1497 &{}\quad -0.0300\\ 0.0330 &{}\quad -0.0683 &{}\quad 0.0631 &{}\quad -0.0300 &{}\quad 0.0060\\ \end{array} \right] \\ A_{21}(:,:,1)= & {} \left[ \begin{array}{ccccc} 1.0000 &{}\quad -1.6218 &{}\quad 1.4840 &{}\quad -0.7392 &{}\quad 0.1514\\ -1.6218 &{}\quad 2.6303 &{}\quad -2.4068 &{}\quad 1.1988 &{}\quad -0.2455\\ 1.4840 &{}\quad -2.4068 &{}\quad 2.2024 &{}\quad -1.0970 &{}\quad 0.2246\\ -0.7392 &{}\quad 1.1988 &{}\quad -1.0970 &{}\quad 0.5464 &{}\quad -0.1119\\ 0.1514 &{}\quad -0.2455 &{}\quad 0.2246 &{}\quad -0.1119 &{}\quad 0.0229\\ \end{array} \right] \end{aligned}$$

$$\begin{aligned} A_{21}(:,:,2)= & {} \left[ \begin{array}{ccccc} -1.6218 &{}\quad 2.6303 &{}\quad -2.4068 &{}\quad 1.1988 &{}\quad -0.2455\\ 2.6303 &{}\quad -4.2659 &{}\quad 3.9035 &{}\quad -1.9443 &{}\quad 0.3981\\ -2.4068 &{}\quad 3.9035 &{}\quad -3.5718 &{}\quad 1.7791 &{}\quad -0.3643\\ 1.1988 &{}\quad -1.9443 &{}\quad 1.7791 &{}\quad -0.8861 &{}\quad 0.1815\\ -0.2455 &{}\quad 0.3981 &{}\quad -0.3643 &{}\quad 0.1815 &{}\quad -0.0372\\ \end{array} \right] \\ A_{21}(:,:,3)= & {} \left[ \begin{array}{ccccc} 1.4840 &{}\quad -2.4068 &{}\quad 2.2024 &{}\quad -1.0970 &{}\quad 0.2246\\ -2.4068 &{}\quad 3.9035 &{}\quad -3.5718 &{}\quad 1.7791 &{}\quad -0.3643\\ 2.2024 &{}\quad -3.5718 &{}\quad 3.2684 &{}\quad -1.6279 &{}\quad 0.3334\\ -1.0970 &{}\quad 1.7791 &{}\quad -1.6279 &{}\quad 0.8108 &{}\quad -0.1660\\ 0.2246 &{}\quad -0.3643 &{}\quad 0.3334 &{}\quad -0.1660 &{}\quad 0.0340\\ \end{array} \right] \\ A_{21}(:,:,4)= & {} \left[ \begin{array}{ccccc} -0.7392 &{}\quad 1.1988 &{}\quad -1.0970 &{}\quad -1.0970 &{}\quad 0.2246\\ 1.1988 &{}\quad -1.9443 &{}\quad 1.7791 &{}\quad -0.8861 &{}\quad 0.1815\\ -1.0970 &{}\quad 1.7791 &{}\quad -1.6279 &{}\quad 0.8108 &{}\quad -0.1660\\ -1.0970 &{}\quad -0.8861 &{}\quad 0.8108 &{}\quad -0.4039 &{}\quad 0.0827\\ 0.2246 &{}\quad 0.1815 &{}\quad -0.1660 &{}\quad 0.0827 &{}\quad -0.0169\\ \end{array} \right] \\ A_{21}(:,:,5)= & {} \left[ \begin{array}{ccccc} 0.1514 &{}\quad -0.2455 &{}\quad 0.2246 &{}\quad 0.1514 &{}\quad 0.0229\\ -0.2455 &{}\quad 0.3981 &{}\quad -0.3643 &{}\quad 0.1815 &{}\quad -0.0372\\ 0.2246 &{}\quad -0.3643 &{}\quad 0.3334 &{}\quad - 0.1660 &{}\quad 0.0340\\ 0.1514 &{}\quad 0.1815 &{}\quad -0.1660 &{}\quad 0.0827 &{}\quad -0.0169\\ 0.0229 &{}\quad -0.0372 &{}\quad 0.0340 &{}\quad -0.0169 &{}\quad 0.0035\\ \end{array}\right] . \end{aligned}$$

1.2 Proofs

Proof of Theorem 4

Sufficiency is a direct result from Dunham (1975), Meinardus and Schwedt (1964) and Meinardus (1967).

Necessity depends upon the existence of the homotopy, $h_{\lambda }$, and on general concepts of approximation theory (Brosowski 1968; Dunham 1975; Siam et al. 2014).

Necessity Let $H^*$ be not a best approximation and suppose there exists $H_1$ such that $||H_d-H_1||<||H_d-H^*||~ \forall ~\omega \in {M^*}$. Since the set $M^*\subset {\varOmega }$ is a compact set, there exists $\eta {>}0$ such that $S(\omega )=Re[\overline{(H_d-H^*)}(H_1-H^*)]\ge \eta$ for $\omega \in {M^*}$. By the continuity of $(H_d-H^*)~and~(H_1-H^*)$ there exists an open set W that covers $M^*$, $W=\{\omega \in {\varOmega }|S(\omega )~>{\eta \over {2}}\}$, i.e., $\overline{W}$ includes $M^*$. By Theorem 3 there exists an element $h_{\lambda }$ between $H_1$ and $H^*$ such that $||H_d-h_{\lambda }||<||H_d-H^*||$ and therefore

$$\begin{aligned} 2Re[\overline{(H_d-H^*)}(h_{\lambda }-H^*)]>|h_{\lambda }-H^*|^2,\omega \in W \end{aligned}$$

(31)

and by Theorem 3

$$\begin{aligned} ||h_{\lambda }-H^*||\rightarrow ~0 \end{aligned}$$

(32)

and also the following estimation on W

$$\begin{aligned} |H_d-h_{\lambda }|^2= & {} |H_d-H^*|^2-2Re[\overline{(H_d-H^*)}(h_{\lambda }-H^*)]\nonumber \\&+\,|h_{\lambda }-H^*|^2, \omega \in W\nonumber \\= & {} |H_d-H^*|^2-2Re\bigg \{\overline{(H_d-H^*)}(H_1-H^*)\nonumber \\&\times \,\frac{\lambda {D_1}}{\lambda {D_1}+(1-\lambda )D_0}\bigg \}+|\frac{\lambda {D_1}}{\lambda {D_1}+(1-\lambda )D_0)}|^{2}|H1-H^*|^2\nonumber \\\le & {} |H_d-H^*|^2 \nonumber \\&-\,2Re\left\{ (H_d-H^*)(H_1-H^*)\frac{\lambda {D_1}}{\lambda {D_1}(1-\lambda )D_0}\right\} \nonumber \\&\left. +\,|\frac{\lambda {D_1}}{\lambda {D_1}+(1-\lambda )D_0}|^2\times ||H_1-H^*||^2\right] \nonumber \\&<\,||H_d-H^*||^2 \end{aligned}$$

(33)

A second estimation is performed on V, $V=\varOmega {\setminus }{W}$. Consider the expression, $\max _{V}|H_d-H^*|$. As V is a compact set and $M^*\cap {V}=\phi$, then $\max _{V}|H_d-H^*|$ and $||H_d-H^*||$ cannot be on $M^*\cap {V}$. Let

$$\begin{aligned} \max _{M^*}|H_d-H^*| -\{\max _{V}|H_d-H^*|\}=\mu >{0} \end{aligned}$$

By Theorem 3, $||h_{\lambda }-H^*||$ converges uniformly to 0. Choosing $\lambda$ such that

$$\begin{aligned} ||H^*-h_{\lambda }||<{\mu >0} \end{aligned}$$

and

$$\begin{aligned} |H_d-h_\lambda |= & {} |H_d-H^*-(h_{\lambda }-H^*)| \nonumber \\\le & {} |H_d-H^*| +|H^*-h_{\lambda }|\nonumber \\\le & {} |H_d-H^*|-\mu +\mu < \nonumber \\&|H_d-H^*| \end{aligned}$$

(34)

The two estimations (33, 34) on $W\cup {V}$ contradict the optimality of $H^*$. $\square$

Proof of Theorem 5

Let $H^*$ be not a global minimum. Then there exists $H_1$ such that $||H_d-H_1||<||H_d-H^*||$, implying $Re[\overline{(H_d-H^*)}(H_1-H^*)]>{0} \forall \omega \in {M^*}$. By theorem (3) there exists a sequence, $\{h_{\lambda }\}\subset \mathcal {H}$ which converges uniformly to $H^*$ such that $||H_d-h_{\lambda }||< ||H_d-H^*||$, implying $Re\{\overline{(H_d-H^*)}(h_{\lambda }-H^*)>{0}$ for $\omega \in ~M^*$. Hence by Lemma 1 and the expression of Theorem (4). $H^*$ is not a local minimum. $\square$