THE normalized low-pass filter magnitude specifications

Transcription

1 780 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 54, NO. 9, SEPTEMBER 2007 Optimal Use of Some Classical Approximations in Filter Design Hercules G. Dimopoulos Abstract The classical, and Elliptic (Cauer) low-pass filter approximations can be used in the design of analog and IIR digital filters in such a way as to obtain passband, stopband and transition band optimized filters at no order cost. The exact analytical relationships for such an optimal deployment of these approximations are developed and presented in this paper and their use is demonstrated through design examples. Index Terms, Cauer,, elliptic filters, equiripple filters, filter approximations, filter theory, filters. I. INTRODUCTION THE normalized low-pass filter magnitude specifications are given in terms of plain gain or logarithmic gain in decibels as shown in Fig. 1. is a reference level usually taken equal to 1 (i.e., db) out loss of generality and therefore filter requirements are determined by three numbers where or Fig. 1. Normalized LP filter specifications. (a) Plain gain. (b) Logarithmic gain in decibels The critical frequencies are given [1] by (3b) It is clear that only in the elliptic case the approximating function depends on the stopband edge frequency. The complete elliptic integral having a modulus and the Jacobi elliptic sine of modulus,, (see Appendix ) are responsible for the complexity of the mathematics of the elliptic filters. In all cases, is such that and the value, called the discrimination factor, is given by The gain specifications of the filter are approximated in the, and Elliptic cases using an approximating function in a gain expression of the form (1) Elliptic In the elliptic case, (4a) (4b) (4c) can be calculated [2] from The function is well defined in all cases Elliptic (2a) (2b) (2c) is the polynomial of order and ) is the elliptic (or ) rational function defined as for even (3a) Manuscript received March 11, This paper was recommended by Associate Editor A. B. da Silva. The author is the Department of Electronics, Technological Education Institute of Piraeus, Egaleo, Greece ( hdimop@teipir.gr). Digital Object Identifier /TCSII for even. (5) Given the specifications, the order equation for each approximation family gives, in general, a non-integer value which is rounded up to the next integer value. Traditionally, the filter of order is then designed using, a choice that ensures exact satisfaction of the passband requirement and improved stopband performance, as a trade-off of the increase of the order from to. This traditional design is shown in Fig. 2(I) and is referred to as stopband edge gain optimized. The difference allows for trade-offs in requirements so that the filter can be optimized for the given order [3]. For example, the specifications can also be satisfied in a manner shown in Fig. 2(II) minimum passband tolerance to, a value that depends on and. Furthermore, in the elliptic case, due to the dependence of the /$ IEEE

2 DIMOPOULOS: OPTIMAL USE OF SOME CLASSICAL APPROXIMATIONS 781 Fig. 3. Optimization for: (a) stopband edge gain and (b) passband gain. or equivalently (7) (8) Fig. 2. The optimization cases (I) Stopband edge gain (II) Passband tolerance (III) Transition band (IV) Stopband attenuation. approximating function on the stopband edge frequency, two more optimizations are available. The transition band and the stopband attenuation optimizations, shown in Fig. 2 (III) and (IV), respectively. In the transition band optimization of Fig. 2(III), a new stopband edge frequency, is calculated from and, that leads to stopband gain tolerance equal to the specified. In the stopband attenuation optimization of Fig. 2(IV), a new stopband edge frequency, can be calculated from and, that leads to stopband gain tolerance equal to the specified (i.e., the gain at ) but much higher attenuation (lower gain ) for. In all cases, when the optimized requirements are used in the order equation, the result will be exactly the integer, i.e., In [3], a method is developed using nomographs and the band edge selectivity (BES) concept to calculate approximately,, and for the optimized designs. In this paper the exact equations are presented for the most common types of approximation and the results of an example design are compared to those presented in [3]. For given and, (8) relates the three filter requirements that can be exactly realized order. This means that if we specify two of them the third can be calculated solving (8). A. Stopband Edge Gain Optimized Filter In this case, and are specified and solution of (8) for gives the minimum value required order Design of the filter leads to the stopband edge gain optimized filter of order [Fig. 3(a)]. This in fact is the standard procedure leading to a filter designed from which actually realizes. B. Passband Tolerance Optimized Filter In this case, and are specified and solution of (8) for gives the maximum value that can be required order. (9) II. DESIGN EQUATIONS AND OPTIMIZATION Given, the integer order can be calculated from the order equation of each approximation case. If we require full exploitation of both passband and stopband tolerances for given integer order we must satisfy the following two design equations: (6a) (6b) (10) Design of the filter leads to the passband tolerance optimized filter of order [Fig. 3(b)]. The pole positions of the transfer function are different from those of the stopband edge gain optimized filter. It should be noticed however that in the elliptic case the transmission zeros do not change since the elliptic rational function depends only on the order and the stopband edge frequency, parameters that are common in both cases.

3 782 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 54, NO. 9, SEPTEMBER 2007 Fig. 4. Elliptic transition band optimized filter. C. Transition Band Optimized Filter If and are specified, solution of (8) for gives the minimum value of the stopband edge frequency that can be used order to exactly satisfy the gain requirements and.in this case, only can be directly calculated from (8) where and (11) Fig. 5. (a) Typical. (b) Stopband attenuation optimized. D. Stopband Attenuation Optimized Filter In the elliptic case more space for optimization exists due to the dependence of the approximating elliptic rational function on the stopband edge frequency and a fourth design is therefore of special interest. Fig. 5(a) shows the gain response of an elliptic filter designed. According to (9), the gain at the stopband edge frequency is. It is obvious that increasing the value of the stopband edge frequency, a value exists at which as shown in Fig. 5(b). Referring to Fig. 5, and are related (11) as follows: In fact (11) gives the relationship of and the gain requirements which can be exactly satisfied. Calculation of from is straightforward for the and cases using (4a) and (4b) respectively. Since is the minimum value of the stopband edge frequency that satisfies exactly the gain requirements, is denoted as From which we get (14a) (14b) (12a) The following equation yields : (12b) For the elliptic case, Huber in [2] has shown that since the relation of and (when ), is given by (11) as, is given in terms of by (13) or equivalently equation (15a) Since the elliptic rational function depends on and so do the poles and zeros of the transfer function [4], requiring in the specifications instead of. This leads to a completely different filter of the same order that exactly satisfies the stopband and passband specifications and. This filter is referred to as the transition band optimized filter [Figs. 2 (III) and 4]. In the and cases, the poles of the transfer function do not depend on the stopband edge frequency [4], [5]. Thus, requiring instead of in the design procedure does not produce a different filter as in the elliptic case, thereby making simply the frequency at which. (15b) This means that the elliptic rational function must be determined for the unknown stopband edge frequency the additional condition that its value for must be. This could not been done analytically, given the inherent difficulty in determining even for known and an algorithmic approach is inevitable. This is embodied in the software in [6]. The algorithms calculate the elliptic rational function for and any stopband edge frequency. The gain specified

4 DIMOPOULOS: OPTIMAL USE OF SOME CLASSICAL APPROXIMATIONS 783 Fig. 7. Passband responses of the example design. Fig. 6. example (N =5). is calculated for stopband edge frequencies from upwards, until it becomes equal to the specified. III. DESIGN EXAMPLES For a filter specifications,, and, the order was found. From (4b) we find and from (9), (10) and (12b) respectively,, and. The transfer function [5] of the typical and the passband tolerance optimized designs are Fig. 8. Stopband responses of the example design. B. Passband Tolerance Optimized Filter The discrimination factor is again. From (10) we find or db ( db in [3]). The parameters of the transfer function in this case will be and. Fig. 6 shows the responses of the typical design (stopband edge gain optimized filter) and of the passband tolerance optimized filter. As mentioned, in this case is simply the frequency at which the gain becomes equal to. As a second example, an elliptic filter transfer function specifications will be designed and optimized. The order of the filter is calculated from (19):. The transfer function of the fourth-order elliptic filter will be of the form and parameters and are calculated for all for optimization cases. A. Stopband Edge Gain Optimized Filter From (5) we find. From (9) we get or db. The corresponding transfer function can be calculated [6], where db db C. Transition Band Optimized Filter From (13) in combination (5) we find ( in [3] ). From (5),. The parameters of the transfer function in this case are D. Stopband Attenuation Optimized Filter Using our algorithm as embodied in [6] we find ( in [3] ). Using (5) we find and from (14a) we find or db ( db in [1] ). The parameters of the transfer function are Figs. 7 and 8 show the passband and stopband details of the four designs. IV. CONCLUSION The classical approximations are traditionally used in analog and IIR digital filter design in such a way as to satisfy the specified passband tolerance in an exact way minimum gain at the stopband edge frequency [Fig. 2(I)]. It is shown in this paper that filters, and elliptic response can be designed optimum passband behaviour [i.e., lower gain tolerance, Fig. 2 (II)] at no order cost and that the dependence of the approximating elliptic rational function, on the stopband edge frequency, allows two more designs: one optimized transition band [Fig. 2(III)] and one optimized stopband attenuation [Fig. 2(IV)]. In contrast to other relevant

5 784 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II: EXPRESS BRIEFS, VOL. 54, NO. 9, SEPTEMBER 2007 work, for the first three optimized designs, exact analytic relationships of all necessary parameters are given and the results of a design example were found much more accurate as compared to those of the nomographic technique proposed in [3]. The analytic solution of (15b) of the stopband attenuation optimized elliptic filter is a subject for further research and it is shown that can be calculated using a small computer program [6]. The proposed method, in conjunction to the computationally simplified expressions of the involved elliptic functions given in the Appendix, can be easily programmed in any computer or programmable calculator language, even in those out inherent advanced mathematical features. APPENDIX Complete Elliptic Integral and the AGM: If the arithmetic and the geometric mean values of two positive numbers and are calculated and then proceed to find the arithmetic and geometric means of these two values and so on, then both sequences of mean values converge very rapidly to the same value defined as the arithmetic geometric mean (AGM) of and [7]. The AGM algorithm can be programmed very easily and converges uniformly quadratically [7]. In fact for the range of and encountered in elliptic filter design, less than ten iterations are required for very high precision. The complete elliptic integral of the first kind modulus,defined as it is necessary to calculate the elliptic parameters modular constants or Jacobi s Nome,defined as, called (20) Elliptic Sine and Theta Functions: The elliptic sine, can be calculated via the Jacobi theta functions and [1], [7] as (21) The theta functions are expressed as fast converging series [1], [7] calculated high accuracy using a few terms (22) Calculation of the Discrimination Factor: Combining (5), (20) (22), we find (16) is connected to the easily calculated arithmetic-geometric mean by a straightforward relationship [7] when (17) Under (17), the awkward order equation for elliptic filters [1], [8] given by (18) is transformed into an easily computed expression requiring only calculation of four arithmetic geometric means (19) This is of course possible because the moduli of all involved in elliptic filter design,. In elliptic filter design, (23) for even This is an expression that can be easily programmed in any computer or programmable calculator language. In fact a function to calculate the AGM of two numbers is required and a function to calculate the theta series. Even a few terms (e.g., 9 terms) very high accuracy is achieved. REFERENCES [1] M. D. Lutovac, Tosic, V. Dejan, and B. L. Evans, Filter Design for Signal Processing using MATLAB and Mathematica. Upper Saddle River, NJ: Prentice-Hall, 2001, pp [2] K. Huber, On the design of Cauer filters, AEÜ Int. J. Electron. Commun., vol. 58, no. 3, pp , May [3] C. A. Corral and C. S. Lindquist, Design for optimum classical filters, Proc. IEE Circuits Devices Syst., vol. 149, no. 5/6, pp , [4] S. Tantaratana, S. S. Lawson, and Y. C. Lim, IIR filters, in The Circuits and Filters Handbook, W.-K. Chen, Ed. Boca Raton, FL: CRC, 1995, ch. 83, pp [5] R. Schaumann and M. E. Van Valkenburg, Design of Analog Filters. Oxford, U.K.: Oxford University Press, 2001, p and [6] H. G. Dimopoulos, A small program to calculate optimized elliptic filters Technological Education Institute of Piraeus, Egaleo, Greece [Online]. Available: [7] J. M. Borwein and P. B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts. New York: Wiley, 1987, vol. 4, ch. 1 and 2. [8] P.-M. Lin, Single curve for determining the order of an elliptic filter, IEEE Trans. Circuits Syst., vol. 37, pp , 1990.