IIR/FIR filtrelerinin SC türev alıcılarla gerçeklenmesi

Abstract

Bu çalışmada yeni bir SC türev alıcı devre ele alınmış tır. Daha sonra, ele alınan SC türev alıcı devre blokları ve sentetik bölme tekniği kullanılarak, IIR ve FIR biçimindeki ayrık zaman transfer fonksiyonlarının doğrudan gerçeklenme- sine ilişkin bir tasarım yöntemi gösterilmiştir. SC integral alıcı devreler ile gerçeklenen ikinci dereceden kararlı yapı bloklarını kaskad bağlayarak elde edilen IIR filtrelere göre, önerilen yöntem ile gerçeklenen IIR filtre tasarımı daha esnek bir yapıya sahiptir. önerilen yöntem ile IIR filtre fonksiyonlarının doğrudan gerçeklenme- sinde kanonik ve basamaklı yapılar kullanılmıştır. Bu yapılardan yola çıkılarak gerçeklenen IIR filtre transfer fonksiyonlarının, yine sentetik bölme tekniğine dayanan fakat SC integral alıcı devre blokları kullanılarak gerçeklenen yapılara göre daha düşük bir duyarlık verdikleri görülmüştür. Ayrıca SC türev alıcı devre blokları kullanılarak gerçeklenen yapılara göre de duyarlık açısından üstünlük göstermektedir. önerilen yöntem ile, SC integral alıcı devreler kullanılarak doğrudan gerçekl enemeyen FIR filtre fonksiyonlarının SC türev alıcı devre blokları kullanılarak doğrudan gerçek- lenebildigi görülmektedir. Bu şekilde gerçeklenen FIR filtreler basit ve toplu elemanlı bir yapıya sahip olmakla birlikte, diğer gerçeklemelere göre daha düşük bir katsayı ve eleman duyarlığı göstermektedirler. SC türev alıcı devre ile doğrudan z domeni karakteris tiklerinden gerçeklenen IIR ve FIR filtrelerin tasarım yön temlerinin oldukça kolay olduğu görülmektedir. Ayrıca bu şekilde gerçeklenen tüm filtreler SC türev alıcı devrenin basit yapısı, parazitik kapasite etkilerine karşı duyarlıksız olması gibi özellikleri de taşımaktadırlar.

Switched - Capacitor ( SC ) circuits in MOS technology have been widely applied to analog signal processing with a good accuracy and less chip area. Most SC circuits are based on the SC integretors, - although it has certain limits in realizing analog functions. In this thesis, a new SC differentiator is introduced. The proposed SC differentiator is a simple structure and stray insensitive. In addition, it is compatable in both fabrication technology and operation with conventional SC integrators. The proposed SC differentiator can be- applied to design useful SC circuits and systems. Fig. 1 shows the structure of the proposed SC differen tiator. Where two nan _ overlapping clocks 'e' and 'o' are required to operate the differentiator. ''- Vout (a) & Yfc %.V$e T/a --t T/2. (b) Fig-1 (a) The SC differentiator circuit (b) The clock waveforms If the nodal-charge equations of the circuit i; for a clock period T ( when e is closed ), the following z - domein equation is found. writtene e -i e Cı.Vin(z)-C.Vout(z)-z.q(-Vin(z)) = 0 (D From (1), the transfer function H(z) of the SC differentiator can be written as m= Vout(z) Vintz) ü-.(r1-i) «* When sT << 1 = f 1 / T ).( 1, the backward - difference mapping - z ') can be applied to (2). The result i< C1 (3) As may be seen from ( 3 ), the time constant of the differentiator is equal to the capacitance ratio ( CI / C ) times the clock period T, which can be precisely controlled in integreted circuit technology. To invert a signal, the SC inverter shown in fig. 2 can be used. This circuit can be conveniently incorporeted with SC differentiator circuits to perform signal inversations. The transfer function H(z) of this circuit can be expressed as H&e). Vout(z) Vin(z) C2 (4) out fig.2 A SC inverter There is an interesting application of SC circuits which is studied in this thesis. This is the realization of discrete-time transfer functions in both infinite - impulse response ( IIR ) and finite - impulse - response ( FIR ) forms. There Are two methods C 2 ], [ 3 ] to implement an IIR transfer function by using SC integrators. In the first method [ 2 3, the SC circuit is first realized from the s - domain transfer functions. Then the z - domain transfer function of the circuit is generated and compared to the specified one. Finally, capacitor values of the SC circuit can be determined to satisfy the specified z - domain VItransfer function. It is difficult. however, to use the proposed SC structure to implement a high order IIR filter (n> 2 ) [7]. The second method proposed by Davis and Smith C 3 ] relies on a polinomial transformation method so called Synthetic Division Technique. It is known that a z -domain IIR transfer function can be written as m= - (5) Appliying the synthetic division technique, ( 5 ) can be transformed into -1 n 2 Mz-1) n=D m = _ (B) -1 n N 1- S an-U - 1) n=1 If nominator and denominator of the transfer function in ( 6 ) is divided by _ aN.( z~\_ 1 )**, the following form z - domain transfer function is found. N.i. -1,x-n m= N,.,," (?) 1- Z «"(»-!) n-1 Then, the IIR transfer function in ( 7 ) can be directly realized by using SC integrators with canonical stractures. By using this method, SC circuits can be constructed efficiently and directly from z - domain specifications. This method is a good design skill for SC circuits because the design procedure is clear and easy. However, the proposed structure has a high component sensitivity [ 3 3. In this thesis, canonical and ladder stractures are proposed for the realization of IIR transfer function of (6) by using SC differentiator with SC inverter. This method is based on synthetic division technique again. In the proposed VIIrealization method, the z _ 1 SC differentiator circuit is used instead of the z~i digital delay element. The resultant SC IIR filters have shown the superiority over the conventional structures using z"1 in component sensitivity. They can also retain the adventages of SC differentiators, such as simple structure, good noise performans at low frequency and stray insensitive. As mentioned above, in the proposed structures, z~'_ 1 is the basic element. The design procedures in conventional digital structures with z ~1 as the basic delay element can be applied without modification. Thus the network function of (6 ) can be generated with adders, multipliers and the differentiator type elements z~1_ 1. It can be shown that the realized IIR filters with SC differentiators have much lower component sensitivities than the SC integrator based IIR filters. This is because the values of z-__ 1 in SC differentiator based structures is much lower than those of 1 / z_{_ 1 in SC integretor based structures [ 1 ]. Although conventional SC integrator can be used to realize IIR filters, they can not be applied to realize FIR filters directly using the above mentioned design technique. But the proposed SC differentiator can be applied directly to realize FIR filter transfer functions. A FIR filter transfer function can be characterized by the z - domain transfer function : N HB- Zhnzn (8) n=0 To realize FIR filter using the proposed SC differentia tors, the transfer function of ( 8 ), must be transformed into the following form appliying synthetic division technique. N hKz)=Sbn.(zl 1)" (9) Then, the transfer function in ( ? ) can be realized by the block diagram shown in Fig. 3. In this block diagram, all the z~^_ 1 blocks with their coefficients can be implemented by SC differentiators whereas the constant coefficient b by a SC inverter. Where the absolute values of bn filter coefficients are implemented by capacitor ratios in SC differentiators and SC inverter.