Aktif tüm-geçiren anahtarlı kapasite filtrelerinin kaskad kafes yapılarla gerçeklenmesi

Abstract

Sayısal tüm geçiren filtreler, genellikle» gecikme elemanı yada faz dengeleyici olarak kullanılmaktadır. Bu tür filtrelerin sayısal elemanlarla (çarpma, toplama ve gecikme) sentezi ve tasarımında çeşitli yöntemler izlenmektedir. Bu çalışmada, ilgili makaleler [1], [2], [3] ve lineer sistemlerin kararlılığını inceleyen Schur-Cohn test yön temi yardımıyla, kafes yapıdaki sayısal tüm geçiren filtrelerin, aktif anahtarlı kapasite devreleri ile gerçeklenmesi yapılmış ve devreler duyarlık, eleman dağılımı (element spread), toplam kapasite, eleman sayısı, dinamiklik (dynamic range), kutuplama gibi yönlerden incelenmiştir. Bu yöntem, n'inci dereceden sayısal tüm geçiren filt relerin, l'inci dereceden iki-çift ( two-pair ) yapısında temel kafes hücrelerin ard-arda (kaskad) bağlanarak gerçeklenmesine dayanmaktadır. Modüler olan bu yapı, tümdevre yapısına uygun olan aktif anahtarlı kapasite devreleriyle gerçeklenmiştir. Aktif anahtarlı kapasite filtrelerinin gerçeklenmesi sırasında ortaya çıkan temel problemler, kısaca belirtilerek, giderilmeye çalışılmış ve bunlara ilişkin devreler incelenmişti r. Temel kafes hücre kullanılarak, tüm geçiren transfer fonksiyonunun toplam ve farkından elde edilen, bant geçiren ve bant söndüren transfer fonksiyonları incelenmiştir.

Filters, whether theoretically or practically, have been a great and significant part of engineering for seve ral decades. Because, they take part almost in every elec tronic circuit, they also take part in most of the research areas. All-pass filters form a sub-section of this area. Like other types of filters, all-pass filters can be reali zed with active or passive components in continuous or dis crete-time domain. Studies to obtain optimum conditions for filter design are carried on and new methods are brought out. Because all -pass filters provide constant magnitute of unity, they can be used as delay elements or phase shif ters. Properties of m-th order all-pass filters ( usually shown as G (z) ) are as follows [4] : m - vn ** * 1. G (z) = -^D-(1Z*_>_ D(z) where "*" denotes complex conjugate, 2. |G (eJV) I = 1 for all w. 1 m ? 3. If G (z) has a pole (say r. ) at z=j"., then it has m k k a pole z=Tk. 4. The poles and zeros of G (z) are complex conjugate. The all-pass filter is a computationally efficient signal processing building block which is quite useful in many signal processing applications. In practice, a digi tal filter is implementated either on a general -purpose com puter or by using special-purpose hardware.lt is different from its idealized design due to the finite word length available to present the signal variables and the multip lier coefficients. One of the practical issues, among ot hers, is sensitivity of filter performance to minor -v- variations of the multiplier coefficients. The importance of low-sensitivity structures arises out of the fact that the characteristics of such structure implemented with qu antized multiplier coefficients can be presented by fewer bits, the implementation could operate at a faster speed and/or be less expensive. Moreover, there exist certain structures in which, if the multiplier coefficients are restricted to a certain range, the structures are neces sarily stable. Thus for such structures, parameter quan tization can be done in such a way that stability is not impared. The concept of losslessness and maximum available po wer are basic to low-sensitivity properties of doubly ter minated lossless networks of the continuous-time domain. Based on similar concepts, a theory is developed for low sensitivity digital filter structures [5]. The mathemati cal set up for the development is the bounded real proper ty. Most of the structures generated by this approach are interconnections of a basic building block, called digital "two pair", shown in figure 1, each two-pair is characte rized by a chain matrix or transfer matrix. ? Y. Figure 1. Digital two-pair Input variables are Xi, Xz and output variables are Yi, Y2 of digital two-pair. Relations between input and output variables, defined by chain matrix and transfer matrix, are as follows: The chain matrix : A C B D (1) The transfer matrix : 11 12 T T 21 22 (2) -vi- There is evidence that in addition to standard latti ce digital filter forms such as the direct, parallel and cascade forms, digital lattice and ladder filters may play an important role in finite word length implementation problems. A method is developed by Gray-Markel [1] for ef ficiently synthesizing digital lattice and ladder filters from any stable direct form. In this study, the aim is to realize all-pass switched capacitor filters which have digital lattice structers by using signal-flow graph theory. Switched capacitor circuits are sampled-data analogy systems and they occupy an intermediate position between fully analog (continuous-time) and fully digital (discre te-time) systems. In analog signal processing, especially filtering apllications, they have so far offered several advantages over fully analog circuits. While the latter up to now have been fabricated mostly in discrete or hybrid forms, switched capacitor circuits are realized as integra ted circuits and hence are compact, reliable and (for lar ge-volume applications) inexpensive. Their frequency res ponses are controlled by clock signals, hence these fil ters can be synchronized multiplexed and programmed due to their sample data character, however, switched capacitor circuits mostly are proceeded and followed by fully analog filters. Compared to an equivalent digital filter, the switched capacitor realization usually requires a less complicated structure and hence often much less chip area on an integ rated circuit. On the other hand, its accuracy is limited approximately ten bits, which may proclude the use of switched capacitor filters in critical applications. Hence switched capacitor and digital filters tend to have comple mentary applications and are usually not directly compati- tive in any one situation. Nowadays, with Large Scale Integration (LSI) technics, millions of transistors can be placed on a single chip. As the number öf elements integrated in one chip increases, cost of complex naturated integrated circuits continuously decrease. Switched capacitor circuits, with the qualificati on of elements they contain, can be realized as MOS integ rated circuits. They appear to be totaly suitable for in tegrated circuit production. For this reason, lately switched capacitor circuits have a special importance and have been a subject of intensive research study. With broad explanation, circuits composed of periodi cally functioning switches, capacitors and active elements -vii- are called switched capacitor circuits. In this study especially, switched capacitor circuits composed of two- phase periodically functioning switches, capacitors and operational amplifiers are used. The coefficients of the numerator and denominator polynomials of a transfer function belonging to a switched capacitor filter, can be set by highly stable sampling fre quency and by very low capacitor ratios. With approprite synthesis technics, the tolerance level can be less than 0,2% and capacitor levels as low as 8,5 pF can be used. While different capacitors of the same circuit, integrated simultaneously, their proportion is a geometric dimension scale. Also, heat sensibility of a MOS capacitor can be less than 25 ppm/ C. Again, with approprite topology cho ice, parasitic capacitor effects can be reduced and this makes use of low value capacitors. So, this decreases the surface, occupied by the capacitor, and more complex switched capacitor circuits can be fitted on a single chip. The switches used in switched capacitor circuits can be realized with MOS transistor arrays. These elements be have as a small resistor (about 80 O), when its "on" posi tion, otherwise behave as a large resistor (about to n >. There exist some parasitic capacitances while realizing it, These capacitances occurs between drain, source and ground or between drain, source and gate. Their values are about 0,02 pF -0,005 pF. Several technics are developed for in sensitive structure to parasitic capacitances. In this study, the switches controlled by two-phase, non-overlap ping clock frequency of f=1/(2.T) (T:1/2 of sampling peri od) and assumed to have 50% duty cycle with equal on and off time periods. Recently, above explained advantages of switched capacitor circuits, motivated most researchers to work on this area. This study presents a method for realizing all-pass active switched capacitor filters and a method for obtain ing an algorithm to realize digital transfer functions. The method is based on the realization using two-pairs in a cascaded lattice form [1]. The circuit realized with this method, consists of basic cells connected in cascade whereas every element has the same value with the expecti- on of a few elements which have to be calculated separetly for every basic cell. The main idea of the circuit is to obtain another all- pass function which is one degree lower in every basic cell. The reduction process goes up to zero-th order. The all- pass function of zero-th order is 1 (or -1 according to the sign of given function). Therefore, the output terminals of the last lattice cell presenting the first order are -viii- connected together. As a result, a given all-pass trans fer function of m-th order is realized with m lattice cells connected in cascade and the output terminals of the last cell connected together. So, the input and the output variables are the input and output sampled voltages at the input terminals of the first lattice cell. The second section of this study includes the theore tical concepts about cascaded lattice realization on which the given method based and gives the building blocks to realize the all-pass functions. The third section gives some basic principles of switched capacitor circuits, presents some types of circu its which are used in design procedure and explains how the given building blocks can be realized with these cir cuits. Some problems which exist in practical realizati ons and their solutions are given. Moreover, sensitivity, element spread, total capacity, number of elements, bia sing and dynamic range of circuits are investigated. In the fourth section, sensitivity considerations of the circuits is presented and convinience of the circuits from this point of view is examined. In the last section, applications on filter functi ons either with general or numerical coefficients are gi ven in order to make clear how the method works. It also explains how band-pass or band-reject filters can be obta ined from all -pass filters.