Realizability-Preserving Transformations for Digital and Analog Filters

Realizability-Preserving Transformations for Digital and Analog Filters by

ARTICE

Department California

M. DAVIS

Engineering,

of Electrical

95192,

San Jose State

University,

San

Jose,

U.S.A.

The theoretical basis for the design of analog and digital filters by prototype transformation is studied. Necessary and suficient conditions are developed for a transformation to preserue realizibility as well as the frequency response. The attendant structural properties of such transformations are developed and compared with the reactance transformations of classical analog filter theory. The superiority of direct analog-todigital transformation to the Constantinides approach is proven.

ABSTRACT:

and

I. Introduction The importance of frequency transformations in the realization of filters from a prototype has long been recognized (1,2,3). However, no effort has been directed toward examining the subject from a unifying point of view. Previous work has merely derived special cases which are justified simply on the basis that “they work”. To cite two important examples, there are the reactance transformations of classical analog filter theory (1) and the all-pass digital filter transformations of Constantinides (2). This paper pursues a more general path, investigating the following question: What is the most general class of frequency transformations which preserves realizability, that is which map the class of realizable jilters into realizable filters? In this context, a transfer function is defined to be realizable if it is a stable, rational function of s or z with real coefficients. Another important criterion is usually added: the sinusoidal steady-state frequency response contour in the prototype domain should map into the corresponding contour in the object domain. The class of transformations having this important property will be delineated as sub-classes of the above realizability preserving maps. The remainder of the paper will consider the above questions for mappings from analog and digital prototypes to both types of object filters. II. Analog-to-Analog

Transformations

Although these transformations are well known (see, for example, (1) and (4)) they have not, to the best of the author’s knowledge, been derived from the general constraints of realizability. For completeness, then, as well as to establish the tenor for the succeeding work, such a development will be presented. @The

Franklin Institute 0016-0032/81/0201-0111$02.00/00

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A. M. Davis Consider the transformation p =f(s), where p is the prototype frequency variable and s is the object filter variable. Since a rational function of any order in p must become a rational function in s under the above transformation, it is clear that f must itself be a rational function. Furthermore, the coefficients of the object function are real linear combinations of the coefficients of f. Since the coefficients of the object function must be real, it is clear that the coefficients of f must be real also. In order for f to map stable filters into stable filters, one must demand that Re(p)
(1)

= -f(s).

Combining

this result with Theorem I, there results Theorem II. p = f(s) maps the class of realizable analog filters into itself with the frequency response preserved if and only if f is an odd positive real function (that is, a reactance function). Example. Again, let H(p) = l/(p+ 1) be a realizable prototype. Let f(s) = (s*+o~)/Bs. Clearly f is a reactance function, and

Theorem II. The transformation

H’(s)=H~~(s)I=~~+BB~S+~~ 0

is a realizable second order band-pass simultaneously with s.

ILI. Analog-to-Digital

filter. Note that p is purely imaginary

Transformations

An important technique in digital filter design is the transformation of existing analog filters into their digital counterparts. In this case, the required mapping is given by s =f(z). Again, it is clear that f(z) must be a rational function with real coefficients. For realizability, Re(s) < 0 must map into IzI < 1. The logical contrapositive states that (z[ 2 1 must map into Re(s) 20. The following definition and theorem express these ideas in a concise fashion.

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Transformations for Digital and Analog Filters Definition. A function f which maps )z) 2 1 into Re(s) 2 0 is called a positive exterior function (or a P.E. function). Theorem 111. The transformation s = f(z) maps the class of realizable analog filters into the class of realizable digital filters if and only if f(z) is a real rational positive exterior function. P.E. and P.R. functions have similar properties, as is illustrated by the next theorem. Theorem IV. A positive exterior function can have no poles outside the unit circle. Any poles on the unit circle are simple. Proof: The proof proceeds in a fashion identical to that for a pr function. Assume that z,, is a pole of a positive exterior function f(z). Then, in a sufficiently small neighborhood of zO, the following approximation holds: k f(z)-(Z_zO)n

*

If one now lets z - z0 = ae’* and k = (kleib one obtains

and the real part is ReCf(z>] = lkl a-” cos (b - n4). Clearly, Re[f(z)] changes sign for any value of n. Thus z0 cannot be exterior to the unit circle. In order to prove simplicity of unit circle poles, assume that )zO]= 1. Then Relf(z)] can only change signs once. By making a sufficiently small, the unit circle approximates its tangent at zO. Thus, by selecting k appropriately (not necessarily real), it is easy to see that Rev(z)] can be made positive outside the unit circle if n = 1. Example. Consider the transfer function of the backward difference operator f(z) = 1 - z-’ (5). It is easy to see that f is a P.E. function. Writing z = aeim, one obtains Rev(z)] = 1 -(l/a) cos 4, which is non-negative if a is at least unity. Once more letting the analog prototype transfer function be H(s) = l/(s + l), one obtains H’(z) = HLf(z)] = z/(22 - 1). H’( z ) is clearly a realizable digital transfer function. Note, however, that f-‘(z) maps the imaginary axis into a circle strictly contained within the unit circle. Thus, this transformation does not preserve the frequency response. The backward difference transformation is of first order. Thus, object filters will have the same order as their prototypes. Although this is a desirable property in some instances, in others it is not, This will be illustrated later. If one now adds the requirement that the frequency response be preserved, one can write io’ = f(eiwT). Here o’ is the analog frequency, and o is the digital frequency. Thus, f(e-‘“‘) = f*(eioT) = -io’ = -f(e’“‘). From this, it follows that f(z-‘> = -f(z). Vol. 311, No. 2, February 1981 F’rinted in Northern Ireland

(2) 113

A. M. Davis This observation prompts the following definition. Definition. f(z) is said to be (1) symmetic if and only if f(z-‘) = f(z). (2) antisymmetric if and only if f(z-‘> = -f(z). The following theorem is an immediate result of the above observations. Theorem V. The transformation s = f(z) maps the class of realizable analog filters into the class of realizable digital filters with the frequency response preserved if and only if f(z) is a real rational antisymmetric P.E. function. Note that this theorem characterizes a class of functions which are the digital analogs of reactance functions. Example. Consider the bilinear transformation f(z) = (z - l)/(z + 1). It is straightforward to show that f(z) is P.E. Indeed, writing z = u + jv results in

which is certainly non-negative if z 2 1. f(z) is also antisymmetric. Hence it transforms realizable analog filters into realizable digital filters while preserving the frequency response. IV. The Structure of AnaZog-to-Digital

Transformations

It is possible to develop additional properties of those transformations which preserve the frequency response. In order to render the exposition more concise, a few more definitions are in order. Definition. P(z) is said to be a mirror image polynomial (MIP) if and only if there exists an integer k such that zkP(zV1) = P(z). Definition. P(z) is said to be an antimirror image polynomial (AMIP) if and only if there exists an integer k such that zkP(z-‘) = -P(z). Lemma.* If P(z) is an MIP or an AMIP of order N, then the integer k in the above definitions is unique and has the value k = N+ m, where m is the number of zeros of P(z) at z = 0. Proof. Assume for concreteness that P(z) is an MIP of order N. Then P(z) = zkP(z-‘). Assume that P(z) has zero at z = 0 of order m. Then P(z)=

i

a,z’

i=m so

2

zkp(z-l)=

a&-‘.

i=m

Hence,

k

r N. Now write the above as k-N zkp(z-‘)

=

c l=k-m

k-m ak_fzf

=

c

ak-fzf

f=k-N

* An additional structural property of these polynomials is as follows: An AMIP must have a zero of odd order at z = 1 and-if k is even-it must have a zero of odd order at z = -1. An MIP must have a zero of odd order at z = -1 if k is odd. The proof is simple; hence it is omitted.

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Transformations for Digital and Analog Filters

which is equal to P(z) if and only if k =iV+ m and aN+m-l = al; l= m,m+l,..., N. Note that the definitions of MIP and AMIP are slightly at variance with common usage, for what is usually termed (for example) a mirror image polynomial is not a polynomial at all. As defined here, it is a symmetric function, albeit a rational one with all its poles at z = 0. With the above definitions at hand, one can state and prove the following structural facts. Theorem VI. Let f(z) be a real rational function. Then f(z) is antisymmetric if and only if it can be written as the quotient of two polynomials of equal order, one of which is an MIP and the other an AMIP, neither having zeros at z =o. Proof. Assume first that f(z>=P(z)/Q( 2 ) is an antisymmetric rational function, where P and Q are coprime polynomials. Furthermore, assume that the degree of P is N. Then, since f(z) is antisymmetric, f(z-‘) = P(z-‘)/Q(z-‘) = -f(z) = -P(z)/Q(z). Rearranging, one obtains P(z-1) = --+$

Z

P(z).

Multiplying by zNIk yields zN+kp(z-l)

= _ZN+k

_Q(z-'1

Q(z)

P(z)

where zN+kP(z-‘) is a polynomial of degree N. The assumption is that P(z) has a zero at z = 0 of order k. From this it follows that Z N+kQ(~-l)

= *Q(z)

(5)

= rP(z).

(6)

and Z N+kP(Z--l)

But k = 0 since P and Q were assumed coprime, so the theorem has been proved. The converse is immediate. Theorem VI states necessary and sufficient conditions for a mapping f(z) to be a frequency response preserving transformation. In order to insure that f(z) is a P.E. function, however, more work is necessary. A sufficient condition is given in the following theorem. Theorem VII. Let f(z) = P(z)/Q(z), where P and Q are real polynomials of equal order with no zeros at the origin, one of which is an MIP and the other an AMIP. Then f(z) is a P.E. function if and only if none of the zeros of g(z) = P(z) + Q(z) are exterior to the unit circle. Proof. Suppose none of the zeros of g(z) are exterior to the unit circle. Then define the following auxiliary function: (7) Vol. 311, No. 2, February 1981 Printed in Northern Ireland

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A. M. Davis It is easy to show that lZ\s 1 if and only if Re[f(z)]$$O. show that Jz( 2 1 maps into lZ( 2 1. Thus, write z

=

The objective

is to

P(z) + Q(z) P(z) - Q(z) .

Assuming that P is an MIP and Q an AMIP, one can write

z=

P(z) + Q(z) z"[P(z-')+

Q(z-')I

g(z) = zNg(z-‘) ’

If g(z) has zeroes at zi, then

g(z>=k * ff (z-Zi). i=l

where k is a multiplicative

constant. Thus

Clearly, if Iz( 2 1, lz( =

lg(z)l (zNg(z-l)J Z l*

The converse is proved simply by noting that the argument proceeds equally well in the reverse order. This paper has not developed the structure of analog-to-analog transformations since the properties of reactance functions are well known (1). Comparing the above work with the known results on reactance functions, however, indicates that the analog-to-digital transformations are the digital equivalents, structurally speaking, of reactance functions. Example. The general transformations developed above are of practical interest in that they permit the mapping of a low-pass prototype analog filter directly into (for example) a band-pass digital filter. This procedure is more efficient than that of Constantinides, which will be discussed in the next section. In that case, one must first map the analog low-pass prototype into a digital low-pass filter. One then applies a Constantinides transformation. As an example of the efficiency of the present procedure, the low-pass analog to band-pass digital transformation will next be derived. Since the bi-linear transformation preserves order, it is clear that one must select a transformation of at least second order to map an analog low-pass into a digital band-pass filter. Thus, it will be assumed that az*+bz+c

f(z) = z’+dz+e

*

Since f must be the quotient of an MIP and an AMIP, one must select c = *a and e = ~1. Suppose that c = a and e = -1. One now imposes the following conditions: (1) s = 0 must map into the center frequency o0 (z = eiOoT).

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Transformations for Digital and Analog Filters

0=B/2 b=JZ.S

FIG. l.RootLocusof(z*-l)+A(z’--Bz+l). (2) s = 00 must map into 0 = O,m (2 = *l) (3) s = j (the analog cut-off frequency) must map into the digital cut-off w1 > o0 (2 = ei”lT) Now condition (1) implies that b = -2a cos (o,T), (2) results in d = 0, while (3) leads to sin (wl T) a=cos(o,T)-cos(w,T). Hence,

the required transformation

f(z) = cos

is given by

sin (w,T) (o,T) -cos

.z*-2cos(w,T)2+1 2*-l

(o,T)

(9)

The transformation (9) has been derived by imposing necessary conditions.* One must, however, ascertain whether or not this f(z) satisfies the requirements of a realizability and frequency response preserving transformation. A convenient tool for this is Theorem VII. Writing f(z)=A

2*-Bz+l zz_l

,

(10)

one can note from (9) that 0
(z*-l)+A(z*-Bz+l)=O.

(11)

The root locus of (11) as a function of A is shown in Fig. 1. Note that the zeros of z*-BZ + 1 are always complex and have unit magnitude. Since the root locus is contained within the unit circle, response preserving transformation.

(9) is a realizability

and frequency

‘* Rader and Gold (6) have presented this particular transformation. It was developed from empirical considerations, however, and no proof of realizability or frequency response preservation was given. Vol. 311. No. 2, February 1981 Printed in Northern Ireland

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A. M. Davis The same procedure can be followed for high-pass, pass-band, or any other type of filter desired.

V. Digital-to-Digital

band-reject,

multiple

Transformations (2)

The digital-to-digital transformation must take the form z’ = f(z), where z’ is the prototype variable and z is the object variable. Again, f must be a real, rational function. It is clear that lz’l< 1 must map into Iz) < 1 for the maintenance of stability. The contrapositive states that IzI 2 1 must map into If(z)I 2 1. This is formalized in the following definition. Definition. If f(z) has the property that If(z)1 2 1 whenever )z ( L 1, it will be termed an exterior function. The next theorem summarizes the statements above. ‘Theorem VIII. The transformation z’=f(z) maps the class of realizable digital filters into itself if and only if f is a real rational exterior function. Example. Let f(z) = 1-f 22. Then it is straightforward to show that f is an exterior function. The unit circle in the prototype domain, however, maps into )z ++I =$. Thus, f p reserves stability, but not the frequency response. If one adds the criterion that the frequency response be preserved, one can write eio’T -f(e’“‘>. Here, o’ is the prototype frequency, and o is that of the o$ct filter, Hence, f(eeioT) = f*(ejoT) = e- j”‘* = l/f(ejoT). From this, it follows f(z) * f(z-‘)

= 1.

(12)

Definition: If f(z-‘) +f(z) = 1, f(z) will be termed reciprocal; and if f(z-‘) +f(z) = -1, f(z) will be called antireciprocal. In either case, f(z) will also be called a digital all-pass function. The next theorem characterizes those mappings which preserve the frequency response. Theorem IX. The transformation z’ = f(z) maps the class of realizable digital filters into itself with the frequency response preserved if and only if f(z) is a real reciprocal exterior function. Note that by defining z’ = (s + l)/(s - l), it is easy to see that Theorem V implies Theorem IX directly. In order to characterize the structure of digital-to-digital transformations more thoroughly, one can write the rational function f(z) as f(z) = P(z)/Q(z), where P(z) and Q(z) are coprime. Then applying the condition that f is recinrocal leads to f(z)

. f(z-l)

=

p(z)p(z-‘)=

1.

Qtz)Q(z-') Rearranging,

one has P(z) - P(z-‘)

= Q(z) . Q(z-‘).

(13)

Clearly P(z) and Q(z) have the same order, say N. Hence P(z) 1 zNP(z-‘)

= Q(z) . z”Q(z-‘)

(14)

where each side is a product of two polynomials.

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Transformations

for Digital

and Analog

Filters

Equation (14) indicates that the set of zeros of the polynomial P(z) * zNP(fl) is identical to that of Q(z) * zNQ(z-‘). This common set of zeros can be further sorted into the set of zeros interior to the unit circle and the set exterior to it, those in the exterior being the reciprocals of those in the interior. Since f(z) is an exterior function, all of the zeros produced by (14) which are exterior to the unit circle belong to Q(z); all those interior to it must therefore belong to P(z). Since P(z) and Q(z) are coprime, (14) indicates that neither P(z) nor Q(z) can have zeros on the unit circle. This leads to the following result. Theorem X. The transformation z’ = f(z) preserves realizability and the frequency response if and only if f(z) = P(z)/zNP(z-‘); where N is the order of P(z), and P(z) has all its roots strictly contained within the unit circle. Note that f(z) is a digital all-pass transformation, as is f(z-‘). Furthermore, f(z-‘) must be stable (2). It should be evident at this juncture that any analog-to-digital transformation can be represented (albeit in a non-unique manner) as the composition of an analog-to-digital transformation and a digital-to-digital transformation. Indeed, it is straightforward to show that if s = f(z’) is an analog-to-digital transformation and z’ = g(z) is a digital-to-digital transformation, both of which preserve realizability and the frequency response, then s =f[g(z)] is a transformation of the former type. Example (2). The low-pass to band-pass transformation tinides has the form z2_-z+k-l

derived by Constan-

2ak

f(z)=k_lk+12akk+1 -z*--z+l k+l k+l

(1%

where

a=coso,T=

and k =cot ((w2--q)/2)T * tan @T/2). Here, wO is the band-pass center frequency, o2 is the upper cutoff, and o1 is the lower cutoff. p is the cut-off frequency of the low-pass prototype. By Theorem X, f(z) is a realizability and frequency response preserving transformation.

VI. Digital- to-Analog

Transformations

The appropriate transformation here is z = f(s), where f is a real rational function. In this case, it is desired that ]z]< 1 map into Re(s)
119

A. M. Davis to pursue the tack alluded to in the previous section. Performing mapping, one has

+*+1w+1 -= 2 -

1

f(s)-

1

an ancillary

(16)

F(s).

Since Re(s’) 50 if and only if Iz 1% 1, one immediately has the following result. Theorem XI. The real rational transformation z = f(s) maps the class of realizable digital filters into realizable analog filters if and only if F(s) = V(s) + l)/(f(s) - 1) is a positive real function. Once again stipulating that the frequency response should be preserved leads to eio’T = f(jw), wh ere U’ is the digital frequency and w is the frequency of the analog filter. It then follows that f( - jo) = f*(j~) = e--io’T = l/f(jw), leading to f(s) * f(-s)

= 1.

(17)

In order to clearly distinguish analog symmetry properties from similar digital properties, one is led to the following definitions. Dejinitions. A function f(s) is said to be (1) even reciprocal if and only if f(s) * f(-s) = 1. (2) odd reciprocal if and only if f(s) * f(-s) = -1. In either case, f(s) is said to be an analog all-pass function. The following theorem has thus been proved. Theorem XII. The real rational transformation z = f(s) maps the class of realizable digital filters into realizable analog filters with the frequency response preserved if and only if F(s) = (f(s) + l)/(f(s) - 1) is a positive real odd function (a reactance function). Additional structural properties can be obtained by considering the symmetries of analog all-pass functions. Since f(s) is rational, one has f(s) = P(s)/Q(s). P(s) and Q(s) can, of course, be assumed coprime. If, in addition, f(s) is even reciprocal, one can write f(s) * H-s)

P(s) * P(-s) = Q(s) . Q(_s) = 1

(18)

from which it follows that P(s) - P(-s) and Q(s) .Q(-s)have the same zeros. Furthermore, P(s) and Q(s) have the same order. Also, since f(s) must map Re(s) 20 into If(s)1 5: 1, P( s ) must have all its zeros strictly in the left half plane. The zeros of Q(s) are the negatives of those of P(s). This is summarized in the following theorem. Theorem XIII. The real rational function z = f(s) maps the class of realizable digital filters into the class of realizable analog filters with the frequency response preserved if and only if it can be written as f(s) = *(P(s>/P(-s)), where all the zeros of P(s) are in the open left half plane. Note that l/f(s) must be a strictly stable analog all-pass transformation.

VIZ. Conclusion This paper has developed the theory of realizability-preserving filter transformations. All types of transformations between analog and digital prototypes

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and object filters have been investigated. The result is a clear delineation of necessary and sufficient conditions for realizability and frequency response preservation. The design of filters by prototype and transformation has therefore been placed upon a solid theoretical foundation. The intent here was not to derive new transformations but to show the efficiency of direct analog low-pass prototype to general digital filter transformations vis-a-vis the more laborious method of Constantinides. In addition to the foregoing items, the structures of the various transformations were developed and compared to the classical reactance transformations of analog filter theory. References (1)E. A. Guillemin,

“Synthesis of Passive Networks”, John Wiley, New York, 1957. (2) A. G. Constantinides, “Spectral Transformations for Digital Filters”, Proc. Institute of Electrical Engineers, Vol. 117, pp. 1585-1590, Aug. 1970. (3) L. R. Rabiner and B. Gold, “Theory and Application of Digital Signal Processing,” Prentice-Hall, Englewood Cliffs, N.J., 1975. (4) G. C. Temes and J. W. LaPatra, “Introduction to Circuit Synthesis and Design,” McGraw-Hill, New York, 1977. (5) A. V. Oppenheim and R. W. Schafer, “Digital Signal Processing.” Prentice-Hall, Englewood Cliffs, N.J., pp. 204-205, 1975. (6) C. M. Rader and B. Gold, “Digital Filter Design Techniques in the Frequency Domain”, Proc. IEEE, Vol. 55, pp. 149-171, Feb. 1967.

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