computer program OPTAS considers the following sequence of possibilities of choosing these constraints. (i) n-i constraints are chosen as ak =-bk (ke [1, n-1], i=l,2,...,n-l) and the remaining i-1 constraints are chosen as Sj = 0 ( (sj=aj, j e[l,n] J^k) or ( Sj = bj, j e[l,n- 1], j * k ) ) from the remaining 2İ-1 variables. (ii) One constraint is chosen to be an=-bn; n-i-1 of the remaining n-2 constraints as chosen as in (i), further, the remaining i-1 constraints are chosen as zero from the remaining 2i variables. (iii) n-1 of the 2n-l free variables are constrained to be zero. The number of possibilities uses of the (n-l)th degree of freedom in the above cases can be given as below. '2i-r iv-i-i 2P i-1, and 2n-f n-1, (18) Thus the total number of possibilities related to the first block of equation (6) is the sum of the above three, which can be written as _*!/n-lV2i-f 1 wU-iA n-1 n-1 ¥ 2i ; A i-1 ^In-i-lAi-l + 2n-l. n-1 (19) In a similar manner the total number of different possibilities of setting the degrees of freedom in the choice of the variables (c,,c2,....,cn.1;d2,d3,....,dn.1) in the second block of equation can be expressed as n-2 p2=s ( n-2 Y2i-f/ £* i=l n-i-1 i-1 + i-i V n-2 V2P n-i-2jli-L '2n-3`/ n-2 (20) The overall number of possibilities of 2n-3 degrees of freedom in equation (6) is equal to the product of P, and P2, i.e. PT = P,P2 for example, PT=5, 105 and 2415 for n=2,3 and 4 respectively. The computer program OPTAS tries all these dependencies between the entries of the vectors a-b and c-d one by one and gives the network topology and related switched capacitor numbers for each case ifa solution is possible. Meanwhile, OPTAS determines the stray-insensitive network containing the rninimum number of switches and/or capacitors and the minimum capacitance area. The computer program OPTAS can find the optimum design parameters concerning the number of elements and the total capacitance area for lower-degree filters (n^5); the optimization can be achieved practically for higher degree as well. When other aspects of the filters to be designed are important in the optimization, OPTAS can be advanced to solve any problem associated with a general objective IXfunction considering the network topology, sensitivity, economic factors, switch- sharing, etc. in addition to switch and capacitor numbers and total capacitance area. Another important case is the design of the optimum SC network by means of the SFG given in Figure 4.1. The reason is that the relations between the parameters of such an SFG and the coefficients of the transfer function to be realized are linear (equation (6)). A similar signal flow graph is used in Reference 3 and 15 for the design of parasitic-insensitive switched capacitor biquads. Since the arbitrarily chosen signal flow graphs results generally in non-linear relations, they lead to difficulties with more complicated and hardly solvable equations. To illustrate the application of the proposed technique, the realization of the z-domain transfer function -2. 1451z`3 +2.03967z`2 +-6.32989z`' +2.145 1 z-3 -2.5175Z-2 +7.00134z-1 -4.18866 H(2)= --- _- - z`;_7 t,.:;. w of a third-order elliptic highpass filter is given as an example at the end of Chapter 4. The inputs n=3, a=(2.1451 -6.32989 2.03967 -2.1451) and b=(-4- 18866 7.00134 -2.5175) are given to the computer to start OPT AS and 52 different network structures satisfying the given transfer function are obtained. The stray-insensitive SC networks with the minimum capacitance area, with the minimum number of switches and with the minimum number of capacitors are listed in Table D, while the network with the minimum capacitance area is shown in Figure 4.2.a. Although the transmittance each branch of the signal flow graph shown in Figure 4.1. is realized by using one or two of basic switched capacitor components given in Table I, the number of switches can be reduced by replacing some switch pairs without affecting the charge flow in the network. The number of switches in the SC network shown in the Figure 4.2.a. is 36. This can be reduced to 26 by sharing some suitable switch pairs and this version of network is given in Figure 4.2.b.computer program OPTAS considers the following sequence of possibilities of choosing these constraints. (i) n-i constraints are chosen as ak =-bk (ke [1, n-1], i=l,2,...,n-l) and the remaining i-1 constraints are chosen as Sj = 0 ( (sj=aj, j e[l,n] J^k) or ( Sj = bj, j e[l,n- 1], j * k ) ) from the remaining 2İ-1 variables. (ii) One constraint is chosen to be an=-bn; n-i-1 of the remaining n-2 constraints as chosen as in (i), further, the remaining i-1 constraints are chosen as zero from the remaining 2i variables. (iii) n-1 of the 2n-l free variables are constrained to be zero. The number of possibilities uses of the (n-l)th degree of freedom in the above cases can be given as below. '2i-r iv-i-i 2P i-1, and 2n-f n-1, (18) Thus the total number of possibilities related to the first block of equation (6) is the sum of the above three, which can be written as _*!/n-lV2i-f 1 wU-iA n-1 n-1 ¥ 2i ; A i-1 ^In-i-lAi-l + 2n-l. n-1 (19) In a similar manner the total number of different possibilities of setting the degrees of freedom in the choice of the variables (c,,c2,....,cn.1;d2,d3,....,dn.1) in the second block of equation can be expressed as n-2 p2=s ( n-2 Y2i-f/ £* i=l n-i-1 i-1 + i-i V n-2 V2P n-i-2jli-L '2n-3`/ n-2 (20) The overall number of possibilities of 2n-3 degrees of freedom in equation (6) is equal to the product of P, and P2, i.e. PT = P,P2 for example, PT=5, 105 and 2415 for n=2,3 and 4 respectively. The computer program OPTAS tries all these dependencies between the entries of the vectors a-b and c-d one by one and gives the network topology and related switched capacitor numbers for each case ifa solution is possible. Meanwhile, OPTAS determines the stray-insensitive network containing the rninimum number of switches and/or capacitors and the minimum capacitance area. The computer program OPTAS can find the optimum design parameters concerning the number of elements and the total capacitance area for lower-degree filters (n^5); the optimization can be achieved practically for higher degree as well. When other aspects of the filters to be designed are important in the optimization, OPTAS can be advanced to solve any problem associated with a general objective IXfunction considering the network topology, sensitivity, economic factors, switch- sharing, etc. in addition to switch and capacitor numbers and total capacitance area. Another important case is the design of the optimum SC network by means of the SFG given in Figure 4.1. The reason is that the relations between the parameters of such an SFG and the coefficients of the transfer function to be realized are linear (equation (6)). A similar signal flow graph is used in Reference 3 and 15 for the design of parasitic-insensitive switched capacitor biquads. Since the arbitrarily chosen signal flow graphs results generally in non-linear relations, they lead to difficulties with more complicated and hardly solvable equations. To illustrate the application of the proposed technique, the realization of the z-domain transfer function -2. 1451z`3 +2.03967z`2 +-6.32989z`' +2.145 1 z-3 -2.5175Z-2 +7.00134z-1 -4.18866 H(2)= --- _- - z`;_7 t,.:;. w of a third-order elliptic highpass filter is given as an example at the end of Chapter 4. The inputs n=3, a=(2.1451 -6.32989 2.03967 -2.1451) and b=(-4- 18866 7.00134 -2.5175) are given to the computer to start OPT AS and 52 different network structures satisfying the given transfer function are obtained. The stray-insensitive SC networks with the minimum capacitance area, with the minimum number of switches and with the minimum number of capacitors are listed in Table D, while the network with the minimum capacitance area is shown in Figure 4.2.a. Although the transmittance each branch of the signal flow graph shown in Figure 4.1. is realized by using one or two of basic switched capacitor components given in Table I, the number of switches can be reduced by replacing some switch pairs without affecting the charge flow in the network. The number of switches in the SC network shown in the Figure 4.2.a. is 36. This can be reduced to 26 by sharing some suitable switch pairs and this version of network is given in Figure 4.2.b.computer program OPTAS considers the following sequence of possibilities of choosing these constraints. (i) n-i constraints are chosen as ak =-bk (ke [1, n-1], i=l,2,...,n-l) and the remaining i-1 constraints are chosen as Sj = 0 ( (sj=aj, j e[l,n] J^k) or ( Sj = bj, j e[l,n- 1], j * k ) ) from the remaining 2İ-1 variables. (ii) One constraint is chosen to be an=-bn; n-i-1 of the remaining n-2 constraints as chosen as in (i), further, the remaining i-1 constraints are chosen as zero from the remaining 2i variables. (iii) n-1 of the 2n-l free variables are constrained to be zero. The number of possibilities uses of the (n-l)th degree of freedom in the above cases can be given as below. '2i-r iv-i-i 2P i-1, and 2n-f n-1, (18) Thus the total number of possibilities related to the first block of equation (6) is the sum of the above three, which can be written as _*!/n-lV2i-f 1 wU-iA n-1 n-1 ¥ 2i ; A i-1 ^In-i-lAi-l + 2n-l. n-1 (19) In a similar manner the total number of different possibilities of setting the degrees of freedom in the choice of the variables (c,,c2,....,cn.1;d2,d3,....,dn.1) in the second block of equation can be expressed as n-2 p2=s ( n-2 Y2i-f/ £* i=l n-i-1 i-1 + i-i V n-2 V2P n-i-2jli-L '2n-3`/ n-2 (20) The overall number of possibilities of 2n-3 degrees of freedom in equation (6) is equal to the product of P, and P2, i.e. PT = P,P2 for example, PT=5, 105 and 2415 for n=2,3 and 4 respectively. The computer program OPTAS tries all these dependencies between the entries of the vectors a-b and c-d one by one and gives the network topology and related switched capacitor numbers for each case ifa solution is possible. Meanwhile, OPTAS determines the stray-insensitive network containing the rninimum number of switches and/or capacitors and the minimum capacitance area. The computer program OPTAS can find the optimum design parameters concerning the number of elements and the total capacitance area for lower-degree filters (n^5); the optimization can be achieved practically for higher degree as well. When other aspects of the filters to be designed are important in the optimization, OPTAS can be advanced to solve any problem associated with a general objective IXfunction considering the network topology, sensitivity, economic factors, switch- sharing, etc. in addition to switch and capacitor numbers and total capacitance area. Another important case is the design of the optimum SC network by means of the SFG given in Figure 4.1. The reason is that the relations between the parameters of such an SFG and the coefficients of the transfer function to be realized are linear (equation (6)). A similar signal flow graph is used in Reference 3 and 15 for the design of parasitic-insensitive switched capacitor biquads. Since the arbitrarily chosen signal flow graphs results generally in non-linear relations, they lead to difficulties with more complicated and hardly solvable equations. To illustrate the application of the proposed technique, the realization of the z-domain transfer function -2. 1451z`3 +2.03967z`2 +-6.32989z`' +2.145 1 z-3 -2.5175Z-2 +7.00134z-1 -4.18866 H(2)= --- _- - z`;_7 t,.:;. w of a third-order elliptic highpass filter is given as an example at the end of Chapter 4. The inputs n=3, a=(2.1451 -6.32989 2.03967 -2.1451) and b=(-4- 18866 7.00134 -2.5175) are given to the computer to start OPT AS and 52 different network structures satisfying the given transfer function are obtained. The stray-insensitive SC networks with the minimum capacitance area, with the minimum number of switches and with the minimum number of capacitors are listed in Table D, while the network with the minimum capacitance area is shown in Figure 4.2.a. Although the transmittance each branch of the signal flow graph shown in Figure 4.1. is realized by using one or two of basic switched capacitor components given in Table I, the number of switches can be reduced by replacing some switch pairs without affecting the charge flow in the network. The number of switches in the SC network shown in the Figure 4.2.a. is 36. This can be reduced to 26 by sharing some suitable switch pairs and this version of network is given in Figure 4.2.b.

ÖZET Bu çalışmada; özel bir işaret-akış diyagram modeli kullanarak, ele alınan herhangi bir gerilim transfer fonksiyonunun değişik aktif SC-devreleri ile gerçekleştirilmesi ve bu devrelerden en uygun olanının seçilmesi incelenmiştir. Bu amaçla, H(z) transfer fonksiyonunun pay ve payda polinomlarının katsayılarıyla, derecesini girdi olarak alan ve verilen transfer fonksiyonunu ele alınan işaret-akış diyagramı uyarınca gerçekleyen değişik tüm SC-devre yapılarını veren, OPTAS isimli bir bilgisayar programı hazırlanmıştır. Program, elde ettiği her devre yapısıyla beraber, bu devre yapılarına ilişkin anahtar ve kapasite sayılan ile kapasite alanı değerlerini de belirtmektedir. Ayrıca minimum kapasite, minimum anahtar sayısı ve minimum kapasite alanı için elde edilen devre yapılarını belirleyip, en uygun devrenin seçimine olanak sağlamaktadır. OPTAS, turbo pascal proglama dilinde yazılmış, herhangi bir özel gereksinime ihtiyaç duymayan, bellekte 65kbyte yer kaplayan ve istenilirse sonuçlan çıktı olarak verebilen bir bilgisayar programıdır.

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