Filter-Generating Systems

Transcription

1 24 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH 2000 Filter-Generating Systems Saed Samadi, Akinori Nishihara, and Hiroshi Iwakura Abstract Combining the methods of generating functions and discrete-time linear systems, we can develop multidimensional (M-D) digital systems for systematic generation of entire families of interrelated digital filters. These M-D systems, dubbed filter-generating systems, are mostly of infinite-impulse response (IIR) type and can produce the impulse response sequence of any member of the family. Temporal and spatio-temporal implementation of filter-generating systems are studied. A temporal implementation scheme is devised for causal filter-generating systems in order to implement an arbitrary filter member using (M-D) signal processing techniques. It is shown that this may result in low order recursive filtering structures that can implement arbitrarily high order members of the family. A spatio-temporal implementation scheme is developed for generation of regular signal flow-graphs for the generated family of finite-impulse response (FIR) filters. It is shown that the signal flow-graphs give rise to cellular array structures. Concrete eamples are provided for maimally flat FIR filters. Inde Terms Cellular array structures, digital filters, generating function, maimally flat magnitude response, systolic arrays, z transform. I. INTRODUCTION We develop a novel technique for generation and implementation of families of digital filters using multidimensional (M-D) systems that we call filter-generating systems. The technique utilizes the recursive relationship eisting in some classes of finite-impulse response (FIR) digital filters. It is shown that for some families of FIR digital filters, it is possible to derive infinite-impulse response (IIR) filter-generating systems of low orders that generate the whole class of the filters as their impulse-response sequence. The filter-generating system may then be used to generate any desired filter of the family, with arbitrarily high order, provided that sufficient processing time and computational resources are available. Filter-generating systems can be used for three purposes. First, they can be used as an algebraic or analytic apparatus for generation of the impulse response coefficients of the family, or designing new families of FIR filters. Alternatively, they can be used as an implementation tool to provide efficient filtering algorithms for the family. If used in such fashion, filter-generating systems can operate as variable-length filters that can produce output sequences corresponding to any desired filter in the family. As a useful by-product, in the process of generating the intended output sequence, they successively generate output sequences that are results of filtering the input signal by lower order filters of the family. The processing speed of such implementations is mainly constrained by the order of the desired filter and the order of the filter-generating system. Third, we will see that filter-generating systems are very efficient tools for systematic generation of various modular structures for the associated family of filters. These are array structures consisting of layers of identical building blocks that are interconnected in a series-parallel configuration and are suitable for high-speed systolic implementations. Filter-generating systems provide a mathematical tool for design and manipulation of such array structures. Manuscript received April 999; revised October 999. This paper was recommended by Associate Editor M. Simaan. S. Samadi and H. Iwakura are with the Department of Information and Communication Engineering, the University of Electro-Communications, Tokoyo, Japan. A. Nishihara is with the Center for Research and Development of Educational Technology, Tokyo Institute of Technology, Tokyo, Japan. Publisher Item Identifier S (00) The concept of filter-generating systems is a hybrid of the z transform [] and the method of generating functions [2], two closely related tools used in computer science, engineering, and mathematics. An early application of generating functions to digital filter design can be found in [3]. Hamming describes a method based on manipulation of a generating function for Bessel functions to design discrete-time integrators. A recent work by Schmidlin [4] may be regarded as an eample of the largely uneplored possibilities of combining the concepts of generating functions and z transforms. Our main focus here is on the application of generating function framework to implementation and modular realization of families of FIR digital filters. If the z transform polynomials corresponding to the transfer functions of a family of digital filters, or any discrete-time linear shift-invariant (LSI) system, can be put in this framework, and generating functions with rational closed forms can be derived as the result. Then, we get a multivariable function that can be viewed as the transfer function of a higher dimensional LSI system. This system is a filter-generating system. As a concrete eample, we show that at least one well-known family of FIR digital filters, the linear-phase maimally flat FIR filters, possesses very simple rational filter-generating functions. We use this function to derive array structures for modular realization of the member filters. II. FILTER-GENERATING FUNCTIONS AND SYSTEMS Consider the rational function P A i (z) i G(; z)= i=0 0 P B i(z) i () i= where A i (z) and B i (z) are polynomials in indeterminate z 0. Epanding G(; z) as a power series in ; we have G(; z)= H i(z) i (2) i=0 where H i (z) are polynomials composed of A i (z) and B i (z). The polynomials H i (z) can be viewed as transfer functions of FIR discrete-time systems. The degree of H i(z) tends to increase with the inde i,however this is not a general rule. Now consider a family of digital systems fh i (z); i =0; ; g. The family may contain a finite or infinite number of digital filters. For eample, we can think of a family of filters with increasing orders that give improved approimations to an ideal frequency response as the inde i increases. We define the formal power series G(; z) of the form (2) as the filter-generating function for the family. Note that the convergence of the defined power series is not of our concern here as is the case for other formal power series [5]. Now, if a closed-form rational epression of the form () eists for the defined generating function, then it is possible to view the polynomials A i(z) and B i(z) as the coefficients of a one-dimensional (-D) discrete-time LSI system with delay operator. This system is called the filter-generating system for the family and operates on polynomial signals in indeterminate z 0. The filter-generating system may be an FIR or IIR system, depending on the actual forms of the coefficients A i (z) and B i (z). If the filter-generating system is ecited by a polynomial impulse signal under the condition of zero initial states, the ith impulse response of the system is a polynomial equal to H i (z), i.e., /00$ IEEE

2 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH Fig.. Direct-form recursive structure for rational filter-generating system. Transfer functions of the associated family of FIR filters appear as the impulse-response sequence. g(n )j n =i = H i (z), where g(n ) denotes the n th (n =0; ; ) impulse response coefficient of the -D system (with respect to ) G(; z). In this way, we can generate the transfer functions of the entire members of the family fh i (z)g. In the above argument, G(; z) is considered as a -D function in whose coefficients are polynomials in z 0. If the generated family of FIR filters contains an infinite number of elements, the associated filter-generating system should be an IIR system, i.e., there should eist at least one B i(z) 6=0. Using a well-known recursive structure for realization of IIR systems, the filter-generating system can be realized as a special recursive -D system as shown in Fig.. The system of Fig. operates on polynomial signals by performing polynomial additions, polynomial multiplications and polynomial delays. As mentioned above, the system can be used to compute the transfer functions of the filters H i (z) in a successive manner by applying the polynomial impulse sequence (n )= ; n =0 0; n =; 2; which is identical to the numerical discrete-time impulse sequence, and then taking the output polynomial signal at the ith sample. If the members of a family of digital filters are doubly indeed as H i; j (z), the definition of filter-generating function can be etended as G(; y; z)= H i; j (z) i y j : (3) with polynomial coefficients A i; j(z) and B i; j(z) is called the filtergenerating system of the family. Like the -D case, the transfer functions of the members H i; j (z) can be computed by applying a 2-D polynomial impulse signal of the form (n ;n y )= ; n = n y =0 0; n ;n y =; 2; and taking the output at the appropriate instance. The discrete variables n and n y correspond to indeterminates and y of the generating function, respectively. The application of the filter-generating system is not limited to the computation of transfer functions. Net, we devise a method to use the system as a filtering structure that can process conventional numerical signals. III. TEMPORAL IMPLEMENTATION OF FILTER GENERATING SYSTEMS Let u(n z ) be the -D input signal we wish to convolve with the ith member of the family fh i(z)g. The result of convolution is denoted by u 3 h i. The discrete-time variable n z takes on nonnegative integer values. The use of suffi z emphasizes the correspondence of the variable to the summation counter in the z transform of the input signal. In the preceding section, we made a similar usage of the suffies and y in n and n y, respectively. Those suffies correspond to the location of the member filters in the filter-generating system. Henceforth, we will use the three suffies, y, and z to distinguish between the discrete variables. Now consider the 2-D impulse response sequence g(n ;n z) of the filter-generating function G(; z) () that is now looked upon as a 2-D system. The impulse response coefficients of the filter H i (z) appear at the ith row of the 2-D impulse response, i.e., Again if a closed-form rational epression can be obtained for the above formal power series, the two-dimensional (2-D) LSI system (in and y) P Q A i; j (z) i y j G(; y; z)= (4) P Q 0 B i; j(z) i y j g(n ;n z )j n =i = h i (n z ) (5) where h i(n z) denotes the n zth impulse response coefficient of the ith filter of the family, i.e., H i (z)= n =0 h i(n z )z 0n. In other words, the filter-generating system can be viewed as a 2-D causal IIR system that, when ecited by the 2-D impulse signal, generates the impulse response of the ith member of the family at the ith row of its 2-D impulse response. Now we convert the input signal u(n z ) to the 2-D signal u(n ;n z)= u(n z); n =0 0; otherwise (6)

3 26 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH 2000 Fig. 2. Filtering an infinite length -D signal using 2-D IIR filter-generating system; a temporal scheme. and feed it to the filter-generating system. We get a 2-D output sequence of the form v(n ;n z )= u(i; j)g(n 0 i; n z 0 j): (7) Using (5) and (6) we can rewrite this as 0 v(n ;n z )= u(0; j)g(n ;n z 0 j) = u(j)h n (n z 0 j) =u 3 h n : (8) j=0 The above relations show that the filter-generating system G(; z) can be used to convolve the -D signal u(n z ) and the sequence h i (n z ), the impulse response of the ith member of the family of generated FIR filters, using the simple scheme illustrated in Fig. 2. In the above description, we considered a 2-D filter-generating system implemented as a recursive IIR system. A main concern with IIR systems is the stability of the system, which is an asymptotic concept. However, stability issues do not arise in the temporal implementation of the filter-generating system, for the subject of operation undergoes, in effect, FIR filtering in one direction (z direction), and for the other direction ( direction), the operation is terminated after a finite number of iterations. Temporal implementation of filtering structures using 3-D generating systems follows from the 3-D etension of the method proposed for the 2-D case. IV. SPATIO-TEMPORAL IMPLEMENTATION OF FILTER-GENERATING SYSTEMS Another method for filtering digital signals through the filter-generating system is a spatio-temporal scheme that generates modular array structures for the family of filters. In this scheme, the indeterminate z 0 is regarded as a delay operator in time while is regarded as a spatial delay operator, a delay operator that acts on the nodes of system s signal flow-graph. We consider three types of rational filter-generating systems G(; z), namely: ) nonrecursive; 2) all-pole; and 3) general recursive filter-generating systems. We provide a simple and systematic method for deriving modular array structures for members of the family generated by G(; z). A. Nonrecursive Filter-Generating Systems Consider the second order filter-generating system G(; z) =A0(z) +A(z) + A2(z) 2 : (9) This system can be associated with the -D difference equation v(n )= A0(z)u(n )+A(z)u(n 0 ) + A2(z)u(n 0 2) n =0; ; (0) where v(n ) and u(n ) denote the spatial (as opposed to temporal) input and output signals, respectively. These signals can also be associated with the nodes of a signal flow-graph. Hence, eecution of the recurrence under the assumption of zero initial values can be performed over a signal flow-graph as shown in Fig. 3(a). Higher order generating systems can be built in a similar manner. The number of branches incident to each computing node increases with the order of the generating system. If only local connections are desired, the order of the generating system should be at most in. Plugging the spatial impulse signal (n ) into the above recurrence is equivalent to allocating values or 0 to the source nodes of the signal flow-graph. We then get the transfer functions fh i (z); i=0; ; g of the associated family of digital filters as the impulse response at sink nodes. Thus, the signal flow-graph gives rise to a modular structure for realization of the digital filters in the family. Evidently, there are only three non-zero transfer functions in the family generated by the second order generating system, namely, H i (z) =A i (z), i =0; ; 2. Later, we will give eamples where nonrecursive generating systems appear in cascade with all-pole generating systems. B. All-Pole Filter-Generating Systems For the second order all-pole filter-generating system G(; z) = we get a spatial difference equation of the form 0 B(z) 0 B2(z) 2 () v(n )= u(n )+B(z)v(n 0 ) + B2(z)v(n 0 2); n =0; ; (2) which, under zero initial conditions, gives rise to the signal flow-graph of Fig. 3(b). Again to restrict the connections to a local form, the generating system should be of order at most in. When driven with the impulse sequence, the signal flow-graph generates digital filters H i (z), i =0; ;. An infinite number of filters can be generated due to the recursive form of generating function. However, in practice, the signal flow-graph is terminated at the sink node associated with the desired member of the family. The modularity of the resulting filter structure is evident.

4 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH (a) (b) (c) Fig. 3. Spatio-temporal implementation of: (a) nonrecursive; (b) all-pole; and (c) general recursive generating systems of order 2. Applying the -D impulse signal u(n )= (n ), the members of the family will be generated at sink nodes v(n ). C. General Recursive Filter-Generating Systems A general rational generating system G(; z) can be decomposed to the cascade of an FIR system G (; z) and an all-pole IIR system G 2 (; z). For a second-order system, we have G (; z) =A 0(z) +A (z) + A 2(z) 2 G 2(; z) = B 0 (z) 0 B (z) 0 B 2 (z) 2 : (3) Since the scope of this brief is limited to generation of FIR filters, we are concerned with the special case of B 0 (z) =. We can write the spatial difference equations for the cascaded structure as v (n )=A0(z)u(n )+A(z)u(n 0 ) + A2(z)u(n 0 2) v 2 (n )=v (n )+B (z)v 2 (n 0 ) + B 2 (z)v 2 (n 0 2); n =0; ; : (4) The recurrence can be carried out by the signal flow-graph shown in Fig. 3(c). Recall that for the impulse input signal, the output sequence v (n ) takes on zero values for n 3. Therefore, we need to carry out the eecution of the associated recurrence only for n =0; ; 2. On the other hand, the system G 2 (; z) is an all-pole IIR system and the difference equation produces an infinite-length sequence of nonzero output values. We need to carry out the recurrence up to n = i to realize the filter H i (z). This means that for the spatial impulse signal, the upper layer of the signal flow-graph may be simplified to three nodes while the lower layer needs all i nodes. The simplified signal flow-graph is given in Fig. 4(a). The overall modular filter structure generated by the system is illustrated in Fig. 4(b). It consists of a conditioning part for the first 3 modules, generated by the numerator of the generating function, and a modular part generated due to the recursive structure of the generating system. The generated -D filter is implemented by feeding the input signal from the source node that is labeled with a in Fig. 4(a). The source nodes labeled with a 0 are fed with the zero sequence. The output taken at the ith port is the signal filtered by H i (z). In general, if the generating system G(; z) can be decomposed to the cascade of K subsystems, the resulting family of filters can be real-

5 28 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH 2000 (a) (b) Fig. 4. Spatio-temporal implementation of a recursive filter-generating system of order 2. The scheme results in a modular array for the desired filter that simultaneously generates all filters of lower orders. (a) Overall signal flow-graph driven by the impulse sequence, after simplification. (b) Modular filter structure. ized by interconnecting the corresponding sink and source nodes of the signal flow-graphs generated by subsystem difference equations. This is shown schematically in Fig. 5, where a layered signal-flow graph is formed. The impulse sequence is applied at the source nodes of the first subsystem and all initial conditions are set to zero. Since the degree of each subsystem depends on the actual forms of the overall coefficient polynomials A i(z) and Bi(z), the number of interlayer (horizontal) and intralayer (vertical) interconnections is not specified. The branch transmittances are not shown either. V. FILTER-GENERATING SYSTEMS FOR MAXIMALLY FLAT FIR FILTERS A. Basic Formulas In this section, we provide a concrete eample of a family of filters with a simple rational generating function. We give details of temporal and spatio-temporal implementations of the family using its filter-generating system. The maimally flat linear-phase FIR filters construct a family of digital filters whose members are denoted by H i; j (z). The suffies i and j represent the degrees of flatness of the 2(i + j 0 )th order maimally flat filter H i; j (z) at frequencies! =0and! =, respectively. Specifically, the zero-phase response of the filters has 2i 0 vanishing derivatives at! = 0and 2j 0 vanishing derivatives at! =. A number of eplicit epressions have been derived in the literature for H i; j(z). In [7] it is shown that using the Bernstein approimation, the transfer function of low-pass maimally flat filters can be written as H i; j(z) = i0 p=0 i + j 0 p (Q(z)) p (P (z)) i+j00p (5) where the two kernel filters P (z) and Q(z) are given as P (z) = +z0 2 2 ; Q(z) =0 0 z0 2 2 : (6)

6 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH Fig. 5. General connection patterns in signal flow-graphs generated by spatio-temporal implementation of the cascade of K filter-generating systems. (a) (b) Fig. 6. (a) Filter-generating system for maimally flat FIR filters. (b) Using the 3-D etended scheme of Section III, the system can be used to implement any desired FIR maimally flat filter. This formula was initially derived in [6] via a different method. The Bernstein representation of the transfer function leads to a recursive decomposition of H i; j (z) in terms of filters of lower orders. The recursive representation and initial conditions are given as H i; 0(z) =z 0(i0) ; i =; 2; H 0;j (z) =0; j =0; ; H i; j (z) =P (z)h i; j0(z) +Q(z)H i0;j(z); i; j : (7) B. Filter-Generating System for Entire Family The generating function for fh i; j (z); i 0; j 0g defined by (5) can be written as G(; y; z) = H i; j (z) i y j : (8) Using the relations given in (7), we have G(; y; z) = = i= z 0(i0) i + 0 z0 + y i= j= H i; j(z) i y j (P (z)h i+; j(z) + Q(z)H i; j+(z)) i y j : (9) The right-hand side of the last relation above can be epressed as G(; y; z) = 0 z0 + yp (z)g(; y; z) +Q(z) G(; y; z) 0 0 z0 : (20)

7 220 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH 2000 (a) (b) Fig. 7. Array realization of maimally flat FIR filters using filter-generating system. (a) Signal flow-graph. (b) Simplified signal flow-graph. Solving (20) for G(; y; z) we have G(; y; z)= ( 0 Q(z)) ( 0 z 0 )( 0 Q(z) 0 P(z)y) : (2) This is an eplicit formula for filter-generating function of the entire family of linear-phase maimally flat filters fh i; j (z)g. Note that the filter-generating function may take other forms depending on the way we number the filters. Thus, (2) serves as the generating function for the filters numbered according to our convention. Fig. 6 shows a recursive temporal structure for the three-dimensional (3-D) transfer function G(; y; z) and a method for implementing the generated -D FIR filters. This method is a straightforward 3-D etension of the scheme developed in Section III for temporal implementation of 2-D filter-generating systems. Note that the recursive computation is terminated after a prespecified number of iterations in and y directions while for the z direction it continues as long as input samples are fed to the system. It is intriguing that the whole family of maimally flat filters can be implemented via a simple recursive LSI structure of order 2 that requires no multiplier coefficients ecept simple powers of /2.

8 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: ANALOG AND DIGITAL SIGNAL PROCESSING, VOL. 47, NO. 3, MARCH (c) Fig. 7. (Continued.) Array realization of maimally flat FIR filters using filter-generating system. (c) Cellular mesh array structure. Net, let us consider the spatio-temporal implementation of the system (2). The system can be decomposed as G(; y; z) =G (; z)g 2(; z)g 3(; y; z), where G (; z) = 0 z 0 G 2(; z) =0 Q(z) G 3(; y; z) = 0 Q(z) 0 P (z)y : (22) For the overall cascaded system, we can write difference equations of the forms v (n )=u(n 0 ) + z 0 v (n 0 ) v 2 (n )=v (n ) 0 Q(z)v (n 0 ) v 3(n ;n y)=v 2(n ;n y)+q(z)v 3(n 0 ; n y) where + P (z)v 3 (n ;n y 0 ); n ;n y =0; ; v 2(n ;n y)= v2(n); ny =0 0; n y =; 2;. (23) (24) Implementation of the above recurrences over a signal flow-graph for the -D impulse input signal u (n )=(n ) is illustrated in Fig. 7(a). It can be seen that many nodes and branches of the first and second recurrences contribute to redundant computations and hence can be removed. The simplified signal flow-graph is given in Fig. 7(b). Finally, a modular structure for the nontrivial filters in the family is given in Fig. 7(c). VI. CONCLUSION Filter-generating systems contain, in a very compact form, all information needed to synthesize and implement an entire family of digital systems using the most basic properties underlying each member of the family. They can be thought of as a compressed form of the transfer functions of the members of the family that may be unfolded to yield any member of the family whenever necessary. An equivalently important point of view emerges from the possibility of implementing any filter member of the family, of arbitrarily high order, using low-order M-D recursive filter-generating structures. Filter-generating systems also provide a systematic method to develop arrays of regularly interconnected signal flow-graph nodes with simple and in some cases almost identical branch transmittances for the members of the family. This gives rise to a cellular realization of the filter members through regularly interconnected identical cells of low orders. Such cellular structures are interesting from both theoretical and practical considerations. They may yield high-speed systolic architectures suitable for VLSI digital circuits that inherit the desirable property of scalability. Note that there is a connection between the overall dimension M (including z) of the generating system and that of the array realization of the family. An M -dimensional generating system for a family of -D filters has an M 0 -dimensional array realization. For eample, a 3-D generating system has a 2-D mesh array realization as shown in Section V. The application of the proposed techniques need not be limited to frequency-selective maimally flat systems; any family of LSI discrete-time systems with recursive relation between the members of the family may be a suitable candidate. If the concept of filter-generating systems is to be used for designing new families of digital filters, we need a method to connect the properties of the M-D generating system with those of the family of generated digital filters. For instance, to design novel families of maimally flat filters, we should look for a 2-D or 3-D property that corresponds to the -D maimal flatness of the resulted family. This issue will be addressed in a future work. REFERENCES [] E. I. Jury, Theory and Application of the Z-Transform Method. Huntington, NY: R. E. Krieger, 973. [2] H. S. Wilf, Generatingfunctionology, 2nd ed. San Diego, CA: Academic, 994. [3] R. W. Hamming, Digital Filters. Englewood Cliffs, NJ: Prentice-Hall, 977. [4] D. J. Schmidlin, Realization of irrational transfer functions, IEEE Trans. Circuits Syst. I, vol. 43, pp , July 996. [5] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete Mathematics. Reading, MA: Addison-Wesley, 989. [6] J. A. Miller, Maimally flat nonrecursive digital filters, Electron. Lett., vol. 8, pp , 972. [7] L. R. Rajagopal and S. C. Dutta Roy, Design of maimally-flat FIR filters using the Bernstein polynomial, IEEE Trans. Circuits Syst., vol. CAS-34, pp , Dec. 987.