A General Class of Optimal Polynomials Related to Electric Filters

Transcription

1 J. Inst. Maths Applies (1979) 24, A General Class of Optimal Polynomials Related to Electric Filters E. COHEN Department of Electrical Engineering, New Jersey Institute of Technology, Newark, New Jersey [Received 28 November 1977] A class of optimal polynomials is obtained satisfying general constraints imposed on the pass and stop bands of low-pass and band-pass electric filters. Mathematically speaking, the x-axis is divided into intervals called bands. One of them, the pass band, is divided into sub-intervals with various upper-bound constraints on the value of the polynomial. Others, the stop bands, are divided into sub-intervals with lower-bound constraints. The pass and stop bands are separated by transition bands whose widths are minimized by the optimal polynomials. Algorithms are presented to obtain the optimal polynomials and they are proved to converge. 1. Introduction FUNCTIONS USED in the design of electric-wave filters have often been characterized as approximations to certain ideal curves known to be irrealizable in practice (Vlach, 1969 and Daniels, 1974). A proper characterization, however, would have been the selection of a function from a collection of allowable ones, fitting within certain bounds and resulting in a physical electric circuit of minimal complexity. For example, if the function is to be a polynomial, its degree should be minimized. Papoulis recognized this (Papoulis, 1957), but dealt with the standard filters only, namely, the Chebyshev low-pass type with a single bound in each of the pass and stop bands. Another aspect of approximation concerns band-pass filters. The suitable functions are obtained from the optimal low-pass functions through a frequency transformation. This restricts the selection of the optimum to the set of functions obtainable through such a transformation. The purpose of this paper is to generalize the constraints on the functions and to avoid any frequency transformation. This is done by considering any finite number of suitable bounds in the pass and stop bands, for a band-pass (BPF) as well as a lowpass filter (LPF). The functions, however, are restricted to being polynomials. They may be obtained by numerical methods. 2. Statement of the Problem Let the "ceiling function" C L (x) of a LPF be defined as follows (see Fig. 1): C L (x) = B, for x,_, < x < x h i = 1, 2,..., m, (la) /79/ $02.00/ Academic Press Inc. (London) Limited

2 198 E - COHEN 0 = x 0 <Xi <... <x m _, <x m = 1, 0<B ( < 1, B t = 1 for some i; C L (0) = B,; (lb) C t (l) = B m ; (lc) C L (x I ) = inf(b 1,B 1+1 ), i=l,2,...,m-l. (Id) The ceiling function C B (x) of a BPF is defined as follows (see Fig. 2): C B (x) = Bi forx,.! <x<x,, i= 1,2,..., m, (2a) x_ = x 0 < X, <... < x m _, < x m = x +, 0 < x_ < 1 < x+, 0<B,^ 1, B, = 1 for some i; C B (x_) = B,; C B (x + ) = B m ; C B (x,) = inf(b 1. ) B I+1 ), i=l,2,...,m-l. The "floor functions" F_(x) and F + (x) are defined as follows: F_(x) = N ; for x_ s _, <x <x_ s _ 1+1, i=l,2,... t, 0 = x_ s _, <x_ s _, + 1 <... <x_ s+1 <x_ s <x_, N,>1; F-{0)=N,; F.(x. M ) = N l ; F_(x_,_,)=sup(N,,N, + I ), i= 1,2,...,t-l. The function F_ is defined only for a BPF. (2b) (2c) (2d) (3a) (3b) (3c) (3d) F + (x) = M t for x s+i _i <x<x s+i, i=l,2,...,r, (4a) Kx s <x s + 1 <...<x s + r =co x + < x s < x s+,<...< x s+r = 00 M, > 1; F + (x s ) = M,; F + (x s+i )=sup(m I.,M I. + 1 ), i= 1,2,..., r-1. foralpf, for a BPF, (4b) Based on electric filter design common criteria, the problem is now stated as follows. For a LPF let S L be the set of polynomials f(x) satisfying the following constraints (see Fig. 1): (4c) 0 < f{x) s: C L (x) for 0 ^ x ^ 1; (5a)

3 OPTIMAL POLYNOMIALS RELATED TO ELECTRIC FILTERS 199 *, x z I x. stop 0 x_ s _ "s+l X s+2 FIG. 1. A low-pass filter polynomial. 's "*s+l FIG. 2. A band-pass filter polynomial. forx$sx s. stop For a BPF let S B be the set of polynomials f(x) such that (see Fig. 2): 0 < /(x) ^ C B (x) for x_ < x < x + ; A s+z (5b) (6a) (6b) (6c) In electric filter parlance the intervals [0, 1], (1, x s ) and [x,, oo] are called the pass band, the transition band, and the stop band of a LPF, respectively. Likewise, for a BPF [0, x_ J and [x s, oo] are stop bands, [x_, x + ] is a pass band, while (x_ s, x_) and (x +, x s ) are transition bands. The set S L is not empty. In fact, there exists a polynomial degree n 0 such that, for all n > «0, x" (min B,) e S L.

4 200 E. COHEN The set S B is not empty either. In fact, for all n even, larger than some degree n 0, (min B,)[x-Affix+ -AfeS B, A = (x + + x_)/2. Likewise, for all n odd, larger than some n 0, The optimality criterion (based on realistic practice) is now enunciated as follows: An optimal polynomial for a LPF is a polynomial which, among the polynomials of lowest degree in S L, maximizes /(x s ). An optimal polynomial for a BPF is a polynomial which, among the polynomials of lowest degree in S B, maximizes /(x_ s ) or /(*,), or both. The optimal polynomial (o.p.) will now be described. The proof of its existence and optimality will be treated later. Figure 3 provides some illustrations. I *s I X, 0 x_s x. 0 x.s x. FIG. 3. Illustrating examples of optimal polynomials: (a) LPF, n = 6; (b) LPF, n = 7; (c) BPF, n = 6; (d) BPF, n = 7. Even Degree For a LPF the optimal polynomial f*(x) of even degree n possesses n/2 distinct vanishing minima in (0, 1). The points of location of these minima divide [0, 1] into n/2 +1 closed intervals in each of which /*(x) = C L (x) at some point; that is, fix) = a;(x- Zl ) 2 (x-z 2 ) 2... (x-z n/2 ) 2,

5 OPTIMAL POLYNOMIALS RELATED TO ELECTRIC FILTERS 201 and such that at some point x,, 0 = z 0 <z, <z 2 <... <z n/2 <z n/2 + l = 1, / (*,) = CJx,) z ; < x, ^ z,+! for i = 0, 1, 2,..., n/2. For a BPF the o.p. of even degree behaves in the same manner in [x_,x + ] in regard to C B (x), except that z 0 = x_ and z n/2 + l = x+. As for a*, it must be positive in both cases (see 3a and 4a). Odd Degree For a LPF the optimal polynomial is /*(x) = an*x(x-z,) 2 (x-z 2 ) 2... (x-z (n _ 1)/2 ) 2, 0 = Z 0 < Z, < Z 2 <... < Z< n -i )/2 < Z( B +l)/2 = J. and such that /*(*,.) = C L (x,) at some point x,-, z,- ^ Xj ^ z j+, for i = 0,1, 2,..., (n-1)/2. The coefficient a* is positive. For a BPF the optimal polynomial f*{x) = a*(x-z_ 1 )(x-z 1 ) 2 (x-z 2 ) 2... (x-z (n _ 1)/2 ) 2, Z_l <0<X_ =Z 0 <Z, <Z 2 <... <Z ((I -i)/2 <Z (n+ i )/2 = X +, and such that f*(5t,) = C B (x,) at some point x,, z, ^ x, ^ z i+, for i = 0,1, 2,..., (n- l)/2, and / (x) = F_(x) at some point in the left stop band. The coefficient a* is positive. As will be shown later, the BPF o.p. maximizes both /(x_ s ) and /(x s ) simultaneously if it is of even degree. 3. Algorithms To obtain the o.p. of any degree n the following algorithms akin to the Remez' are suggested.

6 202 E. COHEN Even Degree The same algorithm applies to both the LPF and BPF. It will be described here only for the BPF case. At the feth iteration one has n +1 different values of x, namely, x'o*', x\ k \..., x ik) such that x_ ^ x'o 10 < x*/" < x ( 2 4) <... < x*,*' < x +. An interpolating polynomial of the nth degree is then found such that i = 0 ) = 0, i= 1,2 n/2, f k) (x%) = C B (x >), i = 0, 1,... n/2. The location of the minima of /**> are next obtained, namely, y ik), y (k),..., yj*2 1. Let y { o } and y (k) be the points in [x-.y'j* 1 '] and [y (k -i,x+], respectively, at which / <k) (x)-c B (x)is maximized. Lety be the points in [yi-'um+i].'= 2 > 4 >- -,"-2at which f ik) (x) C B (x) is maximized. A new set of interpolating points is then determined as x<.' [+1 )= v (t> i = n The two sets {x ( i ll+1) } and {x ik) } are next compared. If they are equal, then/" 0 is the optimal polynomial. If they differ, the k+ lth iteration is performed. For the starting solution it is convenient to take equally spaced points on x - u = x_ i(x+ x_)/n, i = 0, 1,..., n. It is to be noted that y (, i = 0, 1,..., n/2 are relatively easy to get. Each must be one of the break points of the pass band, i.e. x_, x 1( x 2,..., x m, x+, or else a maximum of/""(x) as illustrated in Fig. 4. FIG. 4. Discrepancies (overshoots) resulting at the fcth iteration of an algorithm: (a) at minima and maximum of interpolation polynomial p k) (x); (b) at minima and break point. (b)

7 OPTIMAL POLYNOMIALS RELATED TO ELECTRIC FILTERS 203 Odd-degree LPF At the fcth iteration one has n +1 interpolation points, namely, x$\ x'/ 0, x^',..., xj,*', 0 = x ( 0*> < xv> < x ( 2*> <... < x<*> ^ 1. The interpolating polynomial /* k) satisfies the following: /<V 2 Y) = 0, «= 0,l,...,(n-l)/2, f k) (A%,) = C Jx< 2 *>_1), i = 1, 2,..., (n +1 )/2. Let the minima of /"" be located at y^\yt\ -,y ( n-i ' et y {k) be the point in Lvli*-!.!] at which /""(x) C L (x) is maximized. Also, let y<*> be the points in [yi-i.^i+i]. i = 1,3,..., n-2, at which f k) (x)-c L {x) is maximized, y^* 1 = 0. The new set of interpolation points becomes: Odd-degree BPF (*+!> = v(*), = 012 n At the /cth iteration one has n +1 interpolation points, namely, x'o", x'*',..., xj,*', anql x< 0 *» < x_ < x 1," < x ( 2 k) <... < JCJ,*' < x +. The interpolating polynomial / *' is then determined on the following basis: /""(4V-i) = C B (4V-i), '=1,2,..., Z 1 *^) = 0, i= 1,2 («- Let the minima of/ (t> be located at y^\ yfk, yfl i; let y*' and yj,*' be the points in [x_, yj**] and [yj, k 2!, x + ] at which /""(x) C B (x) is maximized; let y ( o k) be the point in [0, x_j at which/" I) (x) F_(x) is minimized (a break point); and, finally, let yj*' be the points in [yj-l'i,y'+'i], i = 3, 5,..., w-2 at which / (([> (x)-c B (x) is maximized. The new set of interpolation points becomes x * +1( = yj*', i = 0, 1, 2,..., n. A starting solution may be: x{" = x_ + i(x + -x_)/(n-l), i = 0, 1,2,...,II-1. The above algorithm obtains the o.p. which maximizes /(x s ). The o.p. which maximizes /(x_ s ) is obtained by a similar procedure, the difference lying in the position of X'Q*' which is now required to be greater than x +. The above algorithm is then changed only to the extent of replacing F_ by F +> and x_, by x,. 4. Existence of the Optimal Polynomial The successive polynomials obtained in the course of execution of the algorithms described above will now be shown to converge to the optimal polynomial. The evendegree BPF case alone will be tackled here.

8 204 E - COHEN Let a ( * +1) be the vector of order n+ 1 whose ;th element a'-lv' is the coefficient of x J ~ x of the interpolation polynomial f ik+l) (x). Let A- ( * +1) be the square matrix of order n + l whose y-element is [x _ + 1 1) ]-'~ 1, xfj 1, 1 ' is the (i l)th interpolation point at the (/c + l)th iteration. The determinant of A r((t+1), say M (<1+1), as well as all the minors M * n Vi of its (n + l )th column, is Vandermonde and positive. Let c <k+1) be the vector of order n+l whose ith element is equal to the ceiling function Cj^x^V') for i odd and is equal to zero for i even. Obviously, The highest-order coefficient of/ ( * +1) (x) is thus given by: i= 1 i odd Since n is even, a ik+1) is therefore strictly positive. Thus, every iteration of the algorithm must result in a polynomial of degree n. It will next be shown that aj, k) is lower-bounded away from zero. A polynomial /(x) is said to "link C B (x) over a closed sub-interval [x!,x 2 ] of [x_, x + ]" if either condition (a) or condition (b) is satisfied: (a) /(x,) = 0, /(x 2 ) = C B (x 2 ), 0</(x)<C B (x) forx 1< x<x 2 ; (b) /(x 1 ) = C B (x 1 ), /(x 2 ) = 0, 0</(x)<C B (x) forx 1 <x<x 2. Two distinct closed intervals over which/links C B (x) cannot overlap except possibly at one of their boundary points. Clearly, the interpolating polynomial /" (x), for any k, links C B (x) over n closed subintervals of [x_,x + ]. Let us consider the polynomial On the interval [x_, x + ], 0 < g(x) < C B (x). Hence, g{x) intersects/""'(x) at least once in every interval over which /" I) (x) links C B (x). Since both polynomials are of degree n and there are n such intervals, there is exactly one point of intersection in each of these intervals and there is none outside [x_,x + ]. Consider the rightmost subinterval of [x_,x + ] over which /""'(x) links C B (x), namely, [a,fc].as illustrated in Fig. 5,f k) (x) and #(x) increase monotonically to the right of point a and do not intersect past x +. If the highest-order coefficient olg(x) is then a n = (min B,)/2"- 1 (x + -x_f, Therefore, a {k) is lower-bounded above zero. Consider now the vector f" of order n + l, the ith element is f k) (x l /L\ u ). a<*> =f<*>. (8)

9 OPTIMAL POLYNOMIALS RELATED TO ELECTRIC FILTERS 205 FIG. 5. Right-most intersection of/"''(.x) and g(x). Expressions (7) and (8) give: A r< * +U [a <l ' +1> a'*'] = c<*+i>_f<*) which gives rise to the overshoot vector a" 0 whose ith element a'*' is the absolute value of the ith element of c lt + 1) -f* ) (see Fig. 4). Thus, n+ 1 Therefore, the sequence {a[ k), k = 1, 2,...} converges to some positive number a* n > 0. A positive lower bound for the distance between adjacent interpolation points of P k) (x) for all k (Markov's inequality) leads to a positive lower bound for the values of the Vandermonde determinants in expression (9). Thus, there exists a positive constant 6 n+l such that Hence, the sequence _(k+l)_ +1 (It) converges to zero, i.e. all the overshoots tend individually to zero. As in (9), one can write: for 7 = 0, 1, 2,..., n 1 dj is an upper bound on the values of IMjf^V, 1! over ' and k. Hence, all the sequences {af\ k = 1, 2,...}, for ally, converge. Consequently, the sequence of interpolating polynomials I/ 110, k= 1,2,...} converges uniformly to a polynomial having all the attributes characterizing the optimal polynomial. This completes the existence proof of the polynomial called optimal in the foregoing treatment. The proof of optimality follows next. n+l <*) n+l 1=1

10 206 E. COHEN I > <]/ ' v 1 ' * 1' (o) ;b) */+ (c) (d) A. ' \ \ \ \ \ \ \ i / \ / v / v \ i \ \ (e) \ \ \ x- z, (g) FIG. 6. Possible intersections of the optimal polynomial ( ( ) satisfying the filter constraints. 5. Optimality and Uniqueness (f) ) with a polynomial of the same degree Let/*(x) be the optimal polynomial of the nth degree as described above. Referring to Fig. 6, it can be shown that/*(x) intersects/(x) at at least two points in the closed interval defined by two adjacent minima of/*(x), f(x) e S L (S B ). This is also true of the interval [0, z,] for n odd, z, is the left-most minimum off*{x), as evidenced from Figs. 6(f) and (g). It is to be noted that a point of tangency of/*(x) and/(x) is considered as two points of intersection. If the point of tangency falls inside the interval [z,-, z I+1 ], two points of intersection are considered to occur in the interval. If the point of tangency occurs at a boundary point of the interval [z,, z i+1 ],

11 OPTIMAL POLYNOMIALS RELATED TO ELECTRIC FILTERS 207 only one point of intersection is considered to belong to that interval, the other point being considered as part of the neighbouring interval. Let us now consider the case of n even. Both the LPF and BPF have n/2 minima in the pass band and, hence, n/2 1 such closed intervals as described above. Let Zj and z m be the locations of the left-most and right-most minima, respectively, m = n/2. Thus,/*(x) and/(x) have at least 2(n/2 1) = n 2 points of intersection in [Zi,z m ~]. Let fj and z m be the locations at which the o.p. is equal to the ceiling function in the interval [0, z,] for a LPF or [x_, z,] for a BPF, and in the interval [z m) 1] for a LPF or [z m, x + ] for BPF. Since f*(x) links the ceiling function on both the intervals [fj, z t ] and [z m, z m ], then it must intersect/(x) at each of these intervals. Thus,/*(x) and /(x) intersect exactly once in each of these intervals and cannot intersect at all outside the interval [z,, 2J. If /(zj = /*(z m ), then /(x) </*(x) for all x > z m. If /(* m ) >f*(zj, then f(z m ) *S/*(zJ and /(x) </*(x) for all x > z m. Therefore, there exists no polynomial/(x) S L (S B ) such that/(x) ^/*(x) at any point x > z m. Thus, /*(x s ) is maximized. It can similarly be shown for the BPF that the same polynomial also maximizes/*(x_ s ). The foregoing proof also shows that/*(x) is unique. Let us now turn to the case of n odd. Both the LPF and BPF have (n 1)/2 minima in the pass band and, hence, 2((n- l)/2-1) = n-3 points of intersection in [z u z m ], m = (n l)/2. Since there are two points of intersection in [0, z{], there is exactly one point of intersection in [z m, z m ] and none in x > z m. As expounded above, this leads to the optimality and uniqueness of/*(x) in maximizing/(x s ). However, /*(x_ s ) of a BPF is not maximized when /*(x) > F+(x) for x ^ x s, as will next be shown. The horizontal line \ min B ; intersects/*(x) at the points b l,b 2,...,b n _ u the minima of/*(x) lie between b { and b i+l for i odd. The nth degree polynomial r(x) = -(x-x_)(x-x,)(x-x 2 )... (x-x n _!) is negative for x > x + and positive for x < x_. There exists a small positive constant d such that the polynomial is acceptable, i.e. g e S B. Thus, g(x)=f* i.e./(x_ s ) is increased at the expense of/(x s ). Hence,/*(x) cannot maximize/(x_ s ) if it maximizes/(x s ) while exceeding the right floor function F +. Consider now the optimal polynomials/ n *(x) and/ n *_,(x) of degree n and n 1, respectively, n is odd and / * maximizes /(x s ). Using arguments analogous to the foregoing ones, it can be shown that f*(x) and f*- x (x) have n points of intersection in [x_,x + ], that/ n *(x) >/ *_,(x) for x > x+, and that/ n *(x) <ff- 1 (x) for x < x_. Obviously then,/ n *(x) does not exist if/ n *_,(x) = F_(x) at some point x in [0,x_J. Bearing in mind that the even-degree o.p. is completely characterized by its behaviour in the pass band, the following scheme is therefore suggested for searching the o.p. of a BPF, subject to the constraints (6). An upper bound on the degree n is the

12 208 E - COHEN lowest even-degree n 2 of the shifted and scaled Chebyshev polynomial x + +* satisfying the constraints (6). A lower bound on n is the lowest even-degree n, of the shifted and scaled Chebyshev polynomial satisfying constraints (6b) and (6c). Starting with the degree n l one would obtain successively the prospective even-degree optimal polynomials satisfying the pass band constraints (6a) and stop at thefirstwhich satisfies the stop band constraints (6b) and (6c). Let its degree by 2/c < n 2. This is the sought o.p. if the prospective o.p. of degree 2/c 2 either (i) does not satisfy constraint (6b), or (ii) satisfies (6b), violates (6c), and equals F_ at some point in [0, x_j. If the prospective o.p. of degree 2/c 2 strictly exceeds F_ in [0, x_ s ] and the o.p. of degree 2/c exceeds F + in [0, x s ], then the o.p. of odd degree 2k 1 is sought to optimize either /(x_ s ) or /(x s ). If this odd-degree polynomial is found to satisfy constraints (6), it is the optimal polynomial; otherwise, the o.p. of degree 2k is the sought polynomial and the search is over. For a LPF the following scheme is recommended. An upper bound on the degree n is the lowest even-degree n 2 of the shifted and scaled Chebyshev polynomial satisfying the constraints (5). A lower bound on n is the lowest even-degree n l of the shifted and scaled Chebyshev polynomial HiC ni (2(x-i)) satisfying constraints (5). Starting with n t one obtains successively prospective evendegrees optimal polynomials and stop at the first which satisfies the stop band constraints (5b). Let its degree be 2/c ^ n 2. The o.p. of degree 2/c-1 is then obtained. If it satisfies the constraints (5), it is the optimal polynomial; otherwise, the o.p. of degree 2/c is the sought polynomial and the search is over. REFERENCES DANIELS, R. W Approximation Methods for Electronic Filter Design. New York: McGraw-Hill. PAPOULIS, A On the approximation problem in filter design. IRE National Convention Record 5, {2), VLACH, J Computerized Approximation and Synthesis of Linear Networks. New York: John Wiley.