140 IRE TRANSACTIONS ON CIRCUIT THEORY June

hi, 23 Y 1 hI, ’ - 1 + h,,h,, > ”

h ,h,,h,, (hllh22)2-n is a mismatch factor, since for an ideal im- - hT,l(l + h,,h,,) . (15) pedance match hllh,, = 1. (1 + h,,h,,)-’ i.s a feedback

A simple interpretation of this is possible if it is assumed factor, since h,, is a measure of the feedback in a single

that h,,h,, >> D, << h,,h,,, as is often true. Then, since transistor. The sign .must be selected by knowledge of

hh the number of phase reversals.

11 22 = h,,h,,, or l/L M hz1/hllhz2, (15) reduces to In this way a simple result is obtained for a particular

. h;, problem. The theory developed is, of course, applicable to ZIG, 23 ~

(h,,h,,)“-“(1 + h,,h,,) ‘I* (W any case. For general use, it would be very useful if there were available for this ‘and other applications table or

Since h,, is the short circuit current gain of one stage, hl, charts of U*,(X), .for integral values of n and complex would be the all-over gain in the absence of other factors. values of z. .

Frequency Transformations in Filter Design* A. PAPOULISt

INTRODUCT&N

.s

UPPOSE that A(w) is the amplitude characteristic of a network whose network function is H(p). In network synthesis it is of interest to know what are

’ the permissible functions w = cp(y)’ that will transform A(w) into the amplitude characteristic A(&/)) of another network; and how to determine the network function of the new network in terms of H(p) and p(y). In the litera- ture one finds special forms of such transformations, but the general, problem does not seem to have been dis- cussed, although it arises often and its solution is not difficult. In fact it has been stated that only the trans- formations that change reactive elements to reactive networks are useful. This conclusion is reached because the real frequency transformations are obtained through the complex frequency transformation p = p(s)l leading from H(p) to the network function H&J(S));’ however, it will be seen that this is not necessary.

In this paper we shall give a general form of the trans- formations w = q(y) and show how they can be used in

frequency transformations will be used to simplify in some cases the problem of network analysis, if the given amplitude characteristic B(y) can be written in the form A(q(y)); the study of B(y) is then reduced to the study of the simpler function A(w).

REAL FREQUENCY TRANSFORMATIONS

The amplitude characteristic A(w) of a network equals the magnitude of its network function H(p).for p = Jo; thus

AZ(W) = 1 H(jw) 12. (1)

Suppose that pi is a root3 of H(p); then -pi is a root of H( -p) and the product H(p) H( -p) is a rational function of p2, since its roots are &pi. We denote this function by h(p2)

Mp*) = H(P)H( -PI. (2)

From

1 H&I) 1’ = H(jw)H&o)

the design of filters; a method for determining the net- work function of a minimum phase-shift (MPS) network

and (1) and (2) we obtain

whose amplitude characteristiti is A(&)) will follow. A”(w) = h(4). (3) The Butterworth and Tchebycheff filters will result naturally as special forms of such transformations applied Consider the transformation

to the characteristic A(w) of a one- or two-pole network w2 = F(Y*) (4) function; it will further be shown how use of other func- tions might improve the filter characteristics. Finally where F(y2) is a rational function of y2 with real coefficients

and positive for every y.

* Manuscript received by the PGCT, February 3, 1956; revised manuscript received, February 25, 1956.

t Polytechnic Inst. of Brooklyn, N. Y.

NY? L 0 (5)

1 We shall prefer to use y and s for the real and complex trans- with s = x + jy and p = u + jw the function formed frequencies rather than use the transformations w =’ q(w), P = VP(P).

2 H. W. Bode: “Network Analysis and Feedback Amplifier De- sign.” D. Van Nostrand Co., Inc., New York, p. 208; 1945.

s By “roots” we shall mean in this paper both “zeros” and “poles” including the point at infinity.

1956 . Papoulis: Frequency Transformations in Filter Design 141

-p2 = F(4) (6)

transforms the imaginary s axis into a portion of the imaginary p axis; indeed for s = jy, pz = -F(y’) 5 0 [see (5)] hence p is pure imaginary p = =ttj~ and w2 = F(y’). We shall show that the function h(-F(y’)), ob- tained from h(-w’) through the transformation (4), is the amplitude characteristic of a network function. Indeed if the transformation (6) is applied to the function h(p’), the function h(-F(-s’)) results; we denote this function by r(s2)

h( - F( -5’)) = r(s*) (7)

and we have

W--F(+Y’)) = r(-y*). (8)

Since r(s’) is a rational function of a2 with real coefficients, its room must be of the form

ffff.ib

it can therefore be factored as in

r(2) = R(s)R(-s) (9)

where R(s) has as roots only the roots of r(s’) that lie in the left half of the s plane, and is rational with real coefficients; it is, therefore, the network function of a MPS network. The amplitude characteristic B(y) of this function can be obtained from A(w) through the trans- formation (4); indeed

B’(Y) = WY)R( 7 jy) = r( -Y’) = M -F(Y% .

Thus functions of the form (4) satisfying (5) can be used to transform the amplitude characteristic of one network into the amplitude characteristic of another MPS network.

We shall now determine R(s) from H(p) and the given transformation. Although h(p’) and r(s2) are related through (7), there is no similar relationship between H(p) and R(s). To determine R(s) it will suffice to find the roots of r(s’); suppose that si is.such a root. Since

r(s:) = h(-F(-s:))

-F( -s:) must be a root of h(p’); .si therefore can be found by solving

-F(-s’) = p: 00)

or its equivalent

F((;)‘) = @) (10

for every root pi of H(p). Special forms of the transformations (4) are given by

w = cp(Yl* (12)

From the above follows that such transformations can be used only if [(p(y)]’ is a function of y* which is possible only if (p(y) is either an odd or an even function of y; the roots si can then be determined by solving

v; 0 = *e 3 clearly the form (4) includes (12).

Example 1

(13)

We shall take for A(w) the normahzed amplitude characteristic of a one-pole network.

1 A”(4 = l + uz (14)

the corresponding network function is given by

N(P) = .j+j

therefore [see (a)]

wp7 = &5

we apply the transformation

0 = y” (15)

of the form (12); the function

my) = & (161

results. To find the corresponding network function R(s) we solve (see (13)]

Sn 0 -7 3

=*+*j

since pi = - 1, and retain the proper roots; these roots will be the poles of R(s), since pi is a pole of H(p). The familiar Butterworth filter results.

Example 2

Suppose that A(w) is given by

A%) = &2

we have

(17)

N(P) = &j’

with a pole

pi = -1 e (18)

we apply the transformation

w = cos [n cos-1 y] = C,,(y) (19)

where n is an integer; clearly w is a polynomial in y of the form (12). The function

1 ___- my1 = 1 + cc:(y) (20)

results. To find the corresponding R(s) we solve [see (11) -and (IS)]

142 IRE TRANSACTIONS ON CIRCUIT THEORY . June

cos [n cos-1 (:)I= -$$

and retain the proper roots as poles of R(s). The Tcheby- cheff filter results.

THE LOW-PASS FILTER .

To illustrate the use of frequency transformations we shall consider the problem of designing a low-pass filter whose amplitude characteristic must satisfy the following conditions:

A(w) > 1 - e1 for w < 1

A(w)<ez . for w > 1 + 6 (21)

and A(w) decreasing in the internal (1, 1 + 6) (see Fig. 1); the constants Ed, c2, and 6 are given. If no limitation is placed on 6, then the requirements (13) can be trivially

A(w)

Fig. l-Desired low-p: ss filter characteristic.

Fig. 2-(a) One-pole amplitude characteristic A(w) (upper portion of graph). (b) The transformation function w2 = F(y2) (lower right portion of graph). (c) Resulting characteristic B(u) .(lower left

. portion of graph).

satisfied with a one-real-pole network [Fig. 2(a)]; the resulting value 6, of 6, for which (21) is true, is of course large. Suppose a function w2 = F(y’) can be found satisfy- ing the following conditions [Fig. 2(b)]

jwl<l for ‘]y]<l (22)

[wj>l+f% for jy]>1+6 e

and w increasing in the (1, 1 + 6) internal; then the func- tion B(y), obtained from A(w) through the above trans- formation WY = F(y’), will satisfy conditions (21) [Fig. 2(c)]. If in particular the extrema of F(y’) are equal in the (0, 1) and (1 + 6, a) intervals, then the correspond- ing function B(y) will have equal ripple in these intervals (dotted curves in Fig. 2). Since it is easier to satisfy conditions (22), the frequency transformation facilitates the design of our filter. For B(y) to be the amplitude characteristic of a network function, the transformation W2 = F(y’) must be of the form (4) satisfying (5).

APPLICATIONS

We shall first develop some simple relations for the one-pair pole network, that are also useful in the analysis of a symmetrical double-turned circuit. Suppose that the poles of this network are given by

P ,,2 = R e+i(n/z+e) = -a A jp, O&9$ (23)

(see Fig. 3).

Fig. .3-Two-pole geometry.

If

0 s B 5 % or the equivalent p > CX,

then ,A(w) has a maximum A, which is found when the product (AP) (BP) is maximum, where AP and BP are ; the distances from the point jw to the two poles pl and pz. . ;’ It is easily seen that this happens when the point P lies at the intersection P, of the circle K, whose diameter is

i, k

the line AB connecting the two poles, with the imaginary P axis. If the corresponding frequency is denoted by w,, we must have (see Fig. 3)

i‘,

%I ’ = fi2 - a2 = R2[cos2 8 - sin* 131 c

with A(0) the value of A for w = 0, we further have 3 -

2~43 = sin 2e. A(O) _ W3W3

A, (OA)(OB)- = R2

1956. Papodis: Frequency Transformations in Filter Design 143

The value A(0) is reached again at the point P, such that (AP,) (BP,) = R’; f rom this relationship we easily obtain

(OP,) = w,dZ:

,We thus have

To compare with the Butterworth filter of the same order we must take n = 4 in (16); in Fig. 5 the two charac- teristics are plotted.

A(O) R’ cos 28 = w:, B = sin 28, A(0) = A(w,v$. (24) m

If 0 > n/4 (p < a), the circle K does not intersect the w axis hence A(w) does not have a maximum; it is a decreasing function of W. The above two cases correspond to the over- and undercoupled conditions of the symmetrical doubled-tuned circuit if one displaces the poles vertically.

On the Butterworth Filter

Suppose that we apply the transformation (15) of the Butterworth filter not to the function in (14) but to another function with better cutoff characteristics in some sense; then a more favorable characteristic B2(y) will result which will also be flat at the origin (zero deriva- tives of order up to. 2n - 1). The degl’ee n however should be smaller than the corresponding degree of the Butter- worth filter so that the two networks will be of the same order. To illustrate we shall take for A(W) the character- istic of a one-pair pole network, with pass band equal to one (A(0) = A(1)) and the same 45 ratio between maximum and minimum in the (0, 1) interval, as in the function (14). We thus have [see (24)]

A@) L - = dz = sin’28, A,

w,v?ii = 1, R2 cos 20 = w;

from which we obtain

e=i, .R =& (25)

the corresponding normalized characteristic is given by

A’(w) = ;z--- ;z + L 2

we apply the transformation

to the above function; the result is given by

B'(y) = * gxy4+*'

To determine the poles of the corresponding function R(s) we solve [see (13)]

si 2 (-) j

= *I+ 3

/ Fig. 4-+ : Pole location of Butterworth filter with n = 4. 0: Pole

location of modified filter.

B(Y)

t fi-------- om*

Fia. 5-Amplitude characteristics of Butterworth filter and modified filter. Curve I: B (y) = l/d/2/8 Curve 2: B (y) 4\/4jys - yd + f.

On the Tchebychef Filter

where pi are the roots p,,, as given by (23) with 0 and R as in (25). Ret’aining the proper roots we obtain

(see Fig. 4).

W2 = ay4 + by2 + c.

The normalization condition gives a + b + c = 1; from (22) we further obtain 0 5 c 5 1 and

Suppose that we want a polynomial transformation satisfying requirements (22), with maximum cutoff rate at the point y = 1. Let us consider a fourth degree poly- nomial; since it must be of the form (4), it can be written as

144 IRE TRANSACTIONS ON CIRCUIT THEORY June

1>4ac--b2>0 4a _’

From the above we obtain

b as’c+1+2&.

To maximize the slope at y = 1 we must make 2a + b = a - c + 1 maximum; therefore we must have c = 1, a = 4, b = -4 and the corresponding polynomial is given by

or

m2 z 4y4-4yj2+l=(2y2-l)2

w = 2y2 - 1 (27)

which is the Tchebycheff polynomial of order 2. Thus by utilizing completely the available variation in the band- pass we made the slope maximum at the cutoff point and the Tchebycheff filter resulted; this seems to be true in general but tie don’t readily see a proof at the moment. Eq. (27) has also the equal-ripple property but this fol- lowed and was not required in its derivation.

One is next tempted to apply the Tchebycheff trans- formation not to (17) but to a two-pole characteristic which has better cutoff properties; the general form of this characteristic is given by

A?(U) = 1

cd4 - w”+K’ K>4

if we assume A(0) = A(1) and

2L=4K. A 0 A(O) 4K - 1

Suppose we apply the transformation (19) to the above function; we obtain

1 B2(Y) = c;t _ (7; + K

but

c2 = cat + 1 n 2

therefore (28) can be written in the form

(29)

and nothing is gained since (29) is of the form (20) with

4 ‘=m-

On.Network Analysis

The above discussion suggests a use of frequency trans- formations in network analysis. Indeed suppose that we want to investigate the amplitude characteristic of a two-pair pole network whose poles si are symr,netrical about the +3?r/4 lines.

It can readily be seen that this network can be obtained from a one-pair pole network whose poles are given by

pi = r2 exp [*jk + 2~)) I \Y

through the transformation (26). But the latter network are known; if

the properties of

. 0s Q$

then it is overcoupled with

A(O) A,

= sin 4~

and the maximum occurs at

w-i = T4 COS 4Q

.[see (24)]. Therefore the corresponding B(y) will have a maximum B, at

further

y=ym= ri%zQ

B(O) i sin qQ B,

and the value B(0) is again reached at

.y = i@/m.

If

g<Q$

then the corresponding one-pair pole

network function

the point

network w{ll be updercoupled and B(y) will be a decreasing function of y.

The above analysis can be applied to amplitude charac- teristics B(y) that can be written as functions of F(y’), if F(y’) is of the form (4) satisfying (5).