*--'De:
O
S
T ROOM 36-412
9
*3
,,,
THE SELECTION OF NETWORK FUNCTIONS TO APPROXIMATE
PRESCRIBED FREQUENCY CHARACTERISTICS
J. G. LINVILL
I
I*.
i
j
's
i
TECHNICAL REPORT NO. 145
MARCH 14, 1950
RESEARCH LABORATORY OF ELECTRONICS
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
The research reported in this document was made possible
through support extended the Massachusetts Institute of Technology, Research Laboratory of Electronics, jointly by the Army
Signal Corps, the Navy Department (Office of Naval Research)
and the Air Force (Air Materiel Command), under Signal Corps
Contract No. W36-039-sc-32037, Project No. 102B; Department
of the Army Project No. 3-99-10-022.
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
RESEARCH LABORATORY OF ELECTRONICS
Technical Report No. 145
March 14, 1950
THE SELECTION OF NETWORK FUNCTIONS TO APPROXIMATE
PRESCRIBED FREQUENCY CHARACTERISTICS
J. G. Linvill
Abstract
A successive-approximations method is applied to the selection of network functions
having desired magnitude and phase variation with frequency.
The successive trial
functions are specified in terms of the location of their poles and zeros in the complex
frequency plane.
The first approximation is an estimate of suitable pole and zero posi-
tions which only roughly fulfills the needs of the problem.
It can be made on the basis
of known solutions to other problems or through use of a set of curves which are
presented.
The deviation of characteristics of the first estimate from the desired characteristics
is determined, and changes in pole and zero positions are evaluated which tend to erase
these deviations in the succeeding approximation.
A set of normalized functions is pre-
sented, relating changes in the magnitude and phase characteristic to changes in the pole
and zero positions.
At the outset of the adjustment procedure, rough calculations based
on the normalized functions suffice to obtain rapid improvement in the approximation.
A far more precise but also more complicated algebraic process is applied to determine
appropriate pole and zero shifts as the approximation becomes closer. This process
adjusts the magnitude and phase characteristics simultaneously.
The whole procedure
is very flexible and convenient, permitting the accommodation of practical constraints
not possible with other methods.
THE SELECTION OF NETWORK FUNCTIONS TO APPROXIMATE
PRESCRIBED FREQUENCY CHARACTERISTICS
1. 0 Introduction
The approximation problem is a component problem of almost every network design.
The typical network design is the choice of a network which exhibits prescribed response
characteristics within tolerable limits. The prescribed response characteristics are
frequently graphical plots of the phase and magnitude of the desired system function given
as functions of frequency.
Since system functions of lumped-element linear networks
are rational functions of frequency, the first problem of the designer is to select a realizable rational function which has approximately the prescribed magnitude and phase
variation with frequency.
The choice of the rational system function is the approximation
problem, and a solution to this problem is the subject of this report.
The method of solution of the approximation problem presented is one of successive
approximations and it has a number of unusual characteristics of practical value. In the
first place, one chooses the complexity of the approximating function (the number of poles
and zeros) at the outset and successive adjustments always leave the number of poles and
zeros unchanged. One can approximate to magnitude and phase characteristics simultaneously.
The approximating method is extremely flexible; one can easily impose constraints on the form of network to be obtained or on the nature of its elements.
The solution of the approximation problem is naturally divided into two parts. The
first part of the solution is the restriction of attention to the class of functions which are
realizable for a circuit of the type desired. For instance, if the system function is to
be a driving-point reactance, one's choice is limited to reactance functions, functions
having simple poles only on the imaginary axis of the complex frequency plane and having
positive real residues in those poles. Once one's attention is restricted to the class of
realizable functions, the second part of the approximation problem is to select from this
class of functions one which fulfills the requirements of phase and magnitude variation
within tolerable limits.
The difficulty of the approximation problem is dependent, among other things, upon
the constraints which are imposed on the class of functions to be considered. Increasing
the number or complexity of constraints ordinarily makes the approximation problem
more difficult. Practically, it is very desirable to be able to impose constraints to get
a network of a particular configuration, or containing elements of a prescribed nature,
etc. Clearly there is a need for flexibility in the approximation procedure so that constraints can be most effectively accommodated.
The system function of a lumped-element network, being a rational function of frequency, can be expressed as follows:
F(Xk)
a +a
p()
-b
q-
+b±
o(X
lX +..
bo + bX+..b
amm
b(n
h(k XX ol)(X
x
)- (X o)
(kX
pl(X - p2)... ( pn)
pn
-1-
(1)
The quantities Xol, . .. kom are the zeros of the function and pl, ... pn are the poles.
It is apparent that the system function is specified within a constant multiplier by specification of the poles and zeros of the function. The poles and zeros represent the most
convenient "common denominator" in terms of which requirements and constraints can
be expressed. The constraint that a network be of a specified form may ordinarily be
translated into a constraint on the pole and zero positions. For instance, lossless ladder
networks terminated in resistance have transfer impedances with zeros on the imaginary
axis of the X-plane and poles inside the left half of the X-plane. Furthermore, the behavior of the network expressed in terms of the magnitude and phase of the system function is readily related to the pole and zero positions of the system function. Accordingly,
the approximation problem is, in these terms, to determine in the first part the possible
range in the complex plane of positions of poles and zeros in which range the specified
constraints apply.
The second part is the selection, from the restricted range, of poles
and zeros which lead to the most satisfactory network characteristics. The method of
selection employed consists of a preliminary choice of pole and zero positions followed
by successive adjustments of these positions. Ordinarily, the preliminary function
chosen, or first estimate, only roughly approximates the desired characteristics. The
successive adjustments, specified in terms of shifts in pole and zero positions, are made
to reduce the deviation of the trial characteristics from the desired characteristics. The
method of successive approximations is really a systematic fitting procedure with each
successive trial producing a better fit with a function of the same complexity. The fitting
procedure is unique in that both phase and magnitude characteristics can be fitted simultane ously *.
Once the restrictions imposed by physical realizability conditions on the poles and
zeros of the approximating function have been determined, the remainder of the approximation procedure consists of two steps. The first step is the choice of the first approximation and the second is the process of successive adjustments leading to the final result. The two steps are distinct and can best be examined separately.
2. 0 The First Approximation
The first approximation can be made on the basis of known solutions to similar problems which are roughly suitable to the given problem, or it can be made purely on the
basis of a set of curves presented at a later point in this report. When a designer uses
the solution of a similar problem as a starting point, wide experience and ingenuity have
*The implicit relation between magnitude and phase characteristics is well known. If
the magnitude characteristic is completely specified over the whole frequency range from
zero to infinity, the corresponding minimum-phase characteristic is implicitly and inflexibly determined. If one specifies the magnitude characteristic over a fraction of the
spectrum only (as is ordinarily sufficient for practical purposes), or specifies it less
exactly, more freedom is obtained in choice of the phase characteristic. The approximation method described here uses the available freedom to obtain suitable approximations to both phase and magnitude characteristics over a fraction of the complete
spectrum.
-2-
a strong effect in decreasing the labor since a good first approximation reduces the
amount of work to be done in the "fitting" step.
A brief survey of previous contributions
to the approximation problem is useful in providing a stock of first estimates on which to
draw for particular problems.
2. 1 Survey of Previous Contributions
The problem which has received the most attention and yielded the simplest and most
satisfactory solution is that concerned with filters in which the characteristic desired is
that sketched in Fig. 1.
Early low-pass designs of such filters (due to Campbell and
Zobel) did not make a systematic use
-'-
rational functions.
t 12
1t
of an approximation procedure with
Cauer (1, 2) and
Bode (3, 2) introduced methods of obE
2
12 =
taining rational functions identifiable
as the reactance of the branches of
IY21j,1z12 or
IE2/E,
I
W
,,
symmetrical lattices for which lattices
-i--:
Z
12
the propagation function exhibits filtering characteristics.
W-.-
Fig. 1 Ideal low-pass characteristic and circuits to which it might apply.
Since the
quantities on which Bode and Cauer
primarily focused attention in the approximation procedure were not the
rational functions indicated in Fig. 1, but rather some related functions (image impedance and index functions), their work is more suggestive than directly useful to the problem at hand.
Darlington (4) applied an approximation procedure to rational functions,
and the results are directly useful.
tions of
X
He used Tschebyscheff polynomials in obtaining func-
(the radian frequency), which approximate the characteristic of Fig. 1.
This
approximation gives uniform tolerance in one range and monotonic behavior in the other
(Fig. 2a).
Darlington also used Jacobian elliptic functions to provide uniform-tolerance
behavior (called Tschebyscheff behavior) in both pass and stop bands
(Fig. 2b).
M
A somewhat simpler approximation
function than those used by Darlington
(a)
was suggested by Butterworth (5). The
characteristic corresponding to this
(b)
Fig. 2 Tschebyscheff behavior in filter characteristics.
function (Fig. 3) exhibits monotonic be-
havior in both pass and stop bands, the
discrimination being higher the higher the order of the function.
The approximations mentioned above are treated in detail in the original reference
cited and in simple form elsewhere (6, 7, 8, 9).
The function, F, which has the magnitude
variation shown in Fig. 2a or 3 when expressed in the form (1) possesses no internal
-3-
zeros and an array of poles as shown in Fig. 4a or 4b respectively for functions of the
fifth order.
The geometric figure on which the poles shown in Fig. 4a lie is an ellipse.
The cut-off frequency is only slightly less than the value of o at the intersection of the
X- PLANE
X-PLANE
m OR j
X
OR jw
- jw
x- POLES OF
RATIONAL
FUNCTION F
F
n IS ORDEROF
THE FUNCTION
Re OR
Re OR (
v
n, >n1
n
2
I
I
(a)
Fig. 3 Butterworth filter characteristic.
TSCHEBYSCHEFF
(b) BUTTERWORTH
MAP
Fig. 4 Maps of poles for Tschebyscheff
and Butterworth approximations.
F=
ellipse and the axis of imaginaries.
MAP
1
(X -
pl)(k-
X)...
(
-
Xp5)
Simple relationships between the level and tolerance
of IF I of Fig. 2a and the size of the minor axis of the ellipse of Fig. 4a are given in the
For the purposes of this report it is sufficient to state that as the
references cited.
minor axis of the ellipse is decreased (the poles brought nearer the imaginary axis always along lines parallel to the real axis) the level of F in the pass-band increases as
does the tolerance.
Increasing the number of poles* increases the level of IFI for a fixed
ellipse and the number of cycles of variation of IFI in the pass band while it decreases the
tolerance.
From Fig. 4 it is apparent that the Butterworth approximation is a degenerate form
of the Tschebyscheff approximation in which the ellipse becomes a circle.
One observes
from Fig. 2a, 3 and 4 that small shifts in the position of the critical frequencies of a
function cause marked changes in the characteristics of the function.
This fact suggests
the study of the relationship between changes in pole positions and changes in the function
characteristics.
This study and a method of successive approximations arising from it
constitute the results given in this report.
The function corresponding to the characteristic of Fig. 2b has internal poles which
are distributed on a figure slightly different from an ellipse.
There are also internal
zeros distributed along the imaginary axis beyond the cut-off frequency.
That networks having band-pass characteristics can be derived from those having lowpass characteristics is well known (6,
9, 10).
It is enlightening to view the low-pass to
band-pass transformation by comparing the maps of critical frequencies for corresponding low-pass and band-pass transfer functions.
Band-pass characteristics and the map
*For odd numbers of poles, one always lies on the negative real axis; for even numbers
of poles, none lies on the real axis.
-4-
l
M
I
of critical frequencies corresponding to the
low-pass case of Fig. 2a and Fig. 4a are shown
-
Am
I
/1 I
it
(
a
I
I
I
in Fig. 5.
I1
1\
II
quency (c
) is large compared to the bandwidth
(W0) the poles in Fig. 5b fall on an ellipse just
half the size of the corresponding ellipse of
(a)
Fig. 4a (see reference 6 for proof). If the
mean frequency (c ) is not large compared to
x) Im ORjw
x
For the case in which the mean fre-
the bandwidth (0), the poles of the band-pass
transfer function no longer fall on an ellipse.
-j-
x
Those nearest the origin are warped slightly
closer both to the origin and to the imaginary
axis .
5 ZEROS
.F
a
That the magnitude and argument of a rational function are related to a potential dis-
(b)
tribution in a two-dimensional field problem is
Fig. 5 Transfer characteristics and
mapcito f
ciestr
andmap of critical frequencies for bandpass network corresponding to Fig. 2a
and Fig. 4a.
well known.
This relationship can be exploited
as an aid in solving the approximation problem.
The relationship arises through the fact that
the real and imaginary parts of the logarithm
of a rational function are conjugate potential functions. This point may be easily appreciated by considering In IF
InlFI
I
from Eq. 1.
=lnh+
The relation of Eq. 2
n
m
Zlnl-
oil-
i=l
Zlnl
-
kil
(2)
i=l
to a potential distribution is evidenced through considering an infinite conducting plane
(identified with the X-plane) to which a unit current is supplied at some point in the plane.
If the coordinates of the point at which current is injected are those of Xo, the potential
at any point X is given by C ln(l/IX - Xol). The conductivity of the conducting plane is
determined by C.
If a set of negative current sources are placed at points corresponding
to the
oi's of Eq. 2, and a set of positive current sources are placed at points corresponding to the Xpi's, the potential resulting along the imaginary axis of the -plane is an
exact measure of In IF I
Hansen and Lundstrom (11) used an electrolytic tank with probe current sources representing poles and zeros to set up two-dimensional potential distributions corresponding
to rational functions. They proposed the tank as a convenient calculating device to get
plots of the magnitude of rational functions for specified pole and zero positions. The
author (12) used an electrolytic tank to solve approximation problems. The procedure
was simply to find, by successive trials, a set of positions for the current sources leading to the desired kind of potential distribution along the imaginary axis. The method,
-5-
which is one of successive approximations, is simple and enlightening.
With a small
amount of adjustment one frequently can arrive at a good answer, and there is great
flexibility in imposing arbitrary restrictions on the positions of poles and zeros of the
function.
A difficulty with the method lies in the fact that at times adjustments must be
made to yield a compromise between conflicting requirements.
For instance, a devia-
tion from the desired characteristic at one frequency may suggest a shift in current
sources of one nature, while the deviation at another frequency suggests a different kind
of shift.
In such a situation the designer finds himself essentially trying to solve a set
of simultaneous equations by trial, and he may find the solution a slow task.
The method of solution of the approximation problem presented in this report is quite
similar to the experimental solution by successive approximations.
characteristics of successive trials are computed from curves.
However, here the
Successive shifts of
pole and zero positions are made as the result of an algebraic computation.
Each shift
is chosen to remove the deviations of the preceding trial from the desired characteristic;
appropriate compromises are made whenever it is impossible to reduce all deviations
simultaneously.
2.2 Computation of Phase and Magnitude Characteristics for Specified Pole
and Zero Positions
Normalized curves are used for the computation of the characteristics of the successive trials.
The same curves may be used as a guide in the choice of the first approx-
imation.
The process of normalization is easily understood by considering the logarithm
of F (F is defined in Eq. 1).
ln F = in FI + j Arg F
m
in
F
= n h +
n
In
X -
i-
i=l
In
- Xpil
(2)
Arg( -
pi)
(4)
i=l
m
Arg F = >,
(3)
n
Arg(k - Xoi) -
i=l
i=l
The consideration of the logarithm of F is convenient in that it makes the contribution of
each pole and zero separately evident.
One observes that the sums in Eq. Z and also in
Eq. 4 are really made up of only two kinds of terms, terms corresponding to conjugate
complex critical frequencies (poles or zeros) and terms corresponding to real critical
frequencies.
The distinction between zeros and poles is merely the distinction between
plus and minus signs associated with the components in the sum.
-6-
For the complex critical frequencies, one may write the following from Eq. 2 and
Eq. 4.
In IFcI = ln
X-
c I
-
cl
(5)
Xc)
(6)
and
Arg F
= Arg(X - Xc)(X
-
in which
Xc
jc+Wc
*
(7)
If one divides Fc by ec Eqs. 5 and 6 may be written
j+
c
S n r2 l xc
inrested
on li-in reac frquecis,
w
W
Wno
c
c
c
c
c
qs.8
nd
beom
(8)
and
Arg F
= Arg -c
)
=Arg
(9)
C
Since one is ordinarily interested only in real frequencies, Eqs. 8 and 9 become
IFI
ploIn2
_
jw
c
a'__
__
c-.
= In
c~c
c+
c
(10)
c
and
Arg F
= Arg (t
c(11)
c
c
c
c
Clearly n IFs[ and Arg F c are easily found as functions of w for any Xc if one has simply
plots of ln IFcI / 2 and Arg Fc/2cas functions of w/w for appropriate values of the parameter 6c/wc . Figure 6 gives sketches of families of curves corresponding to Eqs. 10
and 11.
An accurate set of curves of the same nature is given in the appendix**.
Through use of the curves in the appendix one can readily obtain plots of in IFcI and
Arg F c as functions of w for any complex value of Xc .
tion with the approximation problem.
Figure 6 is instructive in connec-
It is apparent that critical frequencies close to the
imaginary axis in the complex frequency plane cause much more violent changes in the
magnitude of the function for nearby real frequencies than do critical frequencies removed a considerable distance from the imaginary axis.
Accordingly, if one is choosing
a function which is to exhibit abrupt changes in magnitude at some range of frequencies,
the suggestion is definite that critical frequencies with small displacement from the
*In the following Fc will always represent the product of a pair of factors with complex
conjugate zeros, Xc and Xc.
**All logarithms in the curves are to the base 10 and all arguments are given in degrees.
In the text, log refers to the base 10 and n refers to the base E; Arg is the angle in
radians unless specifically stated otherwise.
-7-
.
·_
uJ
z3
o
a)
bL,
.
C)
-3bU)
b
3
0
6
Zlb
j
0
t_
o~~~~~1-
*r
D
°
3
t
t
U)
'aced
a)
4a)
C) co
I-
tM a)
$-4 r.
Mn
-<3
r.e-e Cd
-
Cd
Cd
.,
0
C
C)3U
~u
_ .3
+
._
tC
~<
-
3
le.
0
4
!
,.o
-
O
ao
0
U)
zl
0
a
0
4
-8-
r C)
dH
imaginary axis be placed near the range of jo where the change is desired.
From Fig. 6
also it is clear that such critical frequencies cause rapid phase shift at the same time
they cause an abrupt change in magnitude. This situation is a manifestation of the implicit relationship between magnitude and phase.
The second kind of terms mentioned in connection with Eqs. 2 and 4 are those representing a critical frequency falling on the real axis of the complex frequency plane*.
in IFI =ln IX- ar1
Arg Fr
If one divides F r by -r- r and sets
(12)
where Tr is real.
r
= Arg(X - o-r)
= j,
(13)
the result is
F
ln
r=
r
In
jW +
(14)
r
and
F
Arg
r -a-(= Arg F if ar is negative) = Arg
r
r
r
+ 1)
(15)
r
The sketch of Fig. 7 indicates the form of variation of n IFr and Arg Fr for any real
critical frequency, arr
Corresponding accurate plots for computational purposes are
given in the appendix.
The sketches of Figs. 6 and 7 are given for critical frequencies
in the left half-plane.
The same curves are useful for critical frequencies in the right
half-plane since the logarithm of the magnitude of F c or F r is unchanged if the sign of
acc or cr is changed and only the sign of the argument of F c or F changes when the sign
r
c
r
of arcc or a rr is changed.
2.3 An Illustrative Example
To illustrate the choice of the first approximation, a function of the form
1
3
will be chosen.
k
+aX
2
1
(X -
+ aX + a0
a 1)(X - az - jw
)(
-
? + jwZ)
The characteristics to be approximated are described in Fig. 8.
(16)
It will
be required that the approximating function
LOG,, I Fl
.
-Arg F
have three poles, have approximately a
01
linear phase shift over the range 0Oo
I
.
I0_
*
1 and
have a logarithm of magnitude which de0
1.0-
Fig. 8 Partial specification of requirements for illustrative example.
creases linearly (approximately) with frequency such that IFI at
= 0.
= 1 is half of IFI at
The magnitude of the argument of F
--
*In the following Fr will always represent a factor with a single real zero, er r.
r
-9-
at
= 1 is immaterial so long as its variation with frequency is linear. The level of
log1 0 IFI is not specified. A large number of functions of the form represented in Eq. 16
will have characteristics of approximately the nature indicated in Fig. 8. Some of them
approximate the magnitude characteristic well and the phase characteristic badly; some
X
approximate the phase characteristic well and the magnitude characteristic badly; others
fall between these classes approximating both phase and magnitude reasonably well. Before one can make a final choice of approximating function, some measure of quality of
the approximation must be formulated. Such a measure of quality is given at a later
point; at this point it is necessary only to choose any function of the form of Eq. 16 which
has roughly the characteristics desired and to proceed from it as a starting point.
Knowledge of the functions used for Butterworth filter characteristics suggests that
a function with poles distributed on a semi-circle as in Fig. 4b will give roughly the kind
of characteristics desired.
Hence a first approximation is
1
F()
= (X + 1. 00)(X + 0.50 + j0. 866)(X + 0.50 - j 0. 866)
(17)
Figure 9, which is a plot of log IF I and Arg F for Eq. 17, indicates that the first approximation shows a too slow decrease in magnitude with frequency. Hence some motion of
the poles from their semi-circular distribution is desirable.
The choice of the first
LOGIFI
0.10
Fig. 9 Log FI and Arg F for the
first approximation to the prescribed characteristics of Fig. 8.
0.00
-0.10
MAP OF POLES
-0.20
_
approximation is quite arbitrary and different first approximations will lead to substantially the same end result. The closeness of the first approximation influences the length
of the adjustment process which is the next step in the procedure; good first approximations decrease the number of adjustment steps required.
3. 0 Successive Adjustments
Once the first approximation is chosen, attention is shifted to the problem of determining the changes in pole and zero positions required to effect the desired changes in
the approximating function. One observes that the number of variable quantities which
may be adjusted is equal to the total number of poles and zeros. In the illustrative example above these are three adjustable quantities, Io, 0 2 and w2. In that instance one wishes
to choose the combination of values for aaly aa 2 , and &w2 which together change the
characteristics from those of Fig. 9 to characteristics much nearer those of Fig. 8.
-10-
A study of the influence of elementary shifts of the critical frequencies, AC 1 by itself,
Ao2 by itself and A 2 by itself, is an enlightening guide to the choice of the combination.
This fact suggests the study of
a Arg
a log IFI
80- I
aC1
F
0 logIFI
acrZ
v
2oI
etc. since
AaI
A logIF
A(2
81
logF 1
0
(for small changes)
+
log
a
(W
2
8%I
0
+
d
.
lodg F1
80'T
a
a log FI d z
"'2
2
o+
o
oglFI
8
2
(18)
2
2
2 C
Similarly,
A Arg FZ (for small changes)
aAr F
01
Arg F Ad+
aC2
2
a+
Arg F A
_
(19)
2
Fortunately, the use of the logarithm of F has separated the influence of individual critical frequencies as indicated by Eq. 2 and one has, for instance
a logIFI
a,
=
-a loglI-
ol
(20)
ac1
From Eq. 20 it is apparent that the derivative functions can be normalized and presented
in curves just as was done for the logarithm and argument curves in Section 2.2.
Fur-
ther discussion of the use of the derivative curves will be given subsequent to the development of expressions for the derivative functions.
3. 1 Normalization of the Derivative Functions
From the foregoing one appreciates that the functions which it is sufficient for all
cases to consider are:
a loglFcl
a c
a logIFcl
a log IFrI
a Arg Fc
accc
r
c
(see Eqs. 5, 6, 7, 12 and 13).
At this point expressions for
a Arg F
a Arg Fr
and
c
logIFc I/ac
r
and
Arg Fc/
ac
will be developed in detail and the expressions for the other derivative functions
(which are developed by similar techniques) will be stated without proof.
in Fc(X) = n IFC(X)
+ j Arg Fc(k)
Since F c is a rational function of X with real coefficients, for X = jw,
-11-
(21)
= Fc(- X)x
F (X)
= FC(X)
F(X) Fc()=
X=jw
In =F(X)I
ln
in IFc(jw)I =1in I(jwIn I F(j')
=2
IFC
=
0c - jw)(O
in FC()
( ()cX=jw
X
- o
+
+
(
-
[
)2]
a c
2
C
[C
1
+ (
a'O2
- l
ji
(24)
-
-)
-
+ ( +
[
]
+
11c
w)
c + w
+
- j(
(25)
j)]
+
c)]
(26)
C)2 [
[c2 + (w -
-
+ j( + °c)] [° c
I ar
(7-
C
- w)
W
c
2
FC(-X)
-
wc)(
(23)
=jw
C) ]
(
a [C' + ( -
a lnF I
FC(- )I
'c) ] [-crc
ln{-ac + j( - c)] [-Cc - j(wic c n
1o 2
= IIn[C2
= FC()
x =jw
X=jw
(22)
j
+a
c )]
(
l
+
c)21
)2]
2 +(
c
[C + (
+
)2
c)]
+
(27)
IF 12
The derivative may be conveniently normalized.
r
2
a n
ao-
IFcI
2
C
2
+ 1+ ( \\2
1
C
(28)
A
C
C
C
2_
c
a In IFc I
CO
c
acr
-
c
c
c(
C
+
FCZ
'a
C
In terms of the logarithms to the base 10 one has
-12-
c+ l
c
(29)
a log FlI
Wc
ac
c
2
W
= 2.305
+Q
c1
W
O
F
(30)
2
--cC
A sketch of xc a log
IFcl/acc
fac/Oc is given in Fig. 10.
as a function of W/wc for various values of the parameter
A set of accurate charts corresponding to Fig. 10 are given
1
a log IFI
alog IFI
.
Fig. 10 Sketches of or
-~~f CA
'.
frv'- -
and aco
-i
f"i
0
C.1ii
f
c
c
in the appendix.
At
c
Figure 10 reveals that shifts in the displacement from the imaginary
axis of poles or zeros influence the magnitude of the function most in a frequency range
nearest the critical frequency being shifted.
Through use of the charts in the appendix
one can predict accurately the influence of a shift of a critical frequency.
Alternately,
one has suitable information for prescribing what kind of shift and how large a shift is
necessary to obtain a required change in the magnitude characteristic.
Through a procedure similar to that presented from Eqs. 21 to 30 one can obtain
c
(31)
c
c
Equation 31 and the corresponding sketch in Fig. 10 indicates the influence on the
-13-
magnitude characteristic of a shift parallel to the imaginary axis of a conjugate pair of
critical frequencies.
The starting point to evaluate a Arg Fc/aCrc is again Eq. 21.
in Fc(k)
=
ln IFc()l + j Arg Fc(k)
(32)
If X = jo,
in Fc(-x) = in IFc(Xk)I - j Arg Fc(X)
,
(33)
and
=
n
-ln IFc()I + j Arg F (X)
F(X)
1
Arg Fc(
a
Arg F-c
c
ac
c
1c (-jw = 1 in (j
)
c
c
- j)(-jw W i)(W
+
c
(34)
C(3)
+ jc)
+
c)
-c
c
c
c
After a straightforward manipulation one has
a Arg F
2 jc
=0g~c
iw
2
2
( c c + oc -
2
w+
2
)
+
2
](38)
o.2
2 2
+ 4a c o
Through normalizing the frequency with respect to wc and multiplying by 57. 3 so that
the units of Arg F are degrees for the curves, one has
ec
2 5
57.3X 2 C
8 Arg Fc(degrees)
a(T
oa
2
c
2
c
3c
(39)
c
Through a similar procedure
a Arg Fc(degrees)
c
awc
57.3 X4
F
2
c
-14-
c
Wc WcC
(40)
c(a Arg Fc/a c) and oc(a Arg Fc/
Figure 11 shows sketches of
c) for various values
of the parameter c/
c . Figures 10 and 11 give complete information on the influence
on magnitude and phase characteristics for any shifts of a conjugate pair of complex
critical frequencies.
Fig. 11
a Arg F c
a
c
Sketches of
Sac
I
and wc
I
a Arg Fc
as functions of '
W
c
c
-0.1
oC
for various values of
c .
c
-WcbAn
be
6
w
In an analogous manner to that just presented for conjugate critical frequencies one
can determine the corresponding derivative functions for real critical frequencies.
,
8 log IjrI-
a Cr
-1
/ \)2 1
--
2.305
rr
1+-
(41)
2')
o
a
r
Arg F
57.3
r
1
3 r
-
(
rr
r2
(42)
r~~~
Accurate plots
Arg Fr/a or) and -crr(3 log IFrl /a r)
Figure 12 presents sketches of -Cr(
for computational purposes are given in the appendix for functions corresponding to
Fig. 11 and Fig. 12.
-15-
~~~~~~~~
8 logFr
Fig. 12 Sketches of -or
and
a
r
a Arg F
-o
as functions of
r
r
-a, b ArgF
bar
3.2 Illustration of the First Stage of the Adjustment Procedure
The preliminary step to stating what changes should be made in the pole and zero
positions is the sketching of the derivative functions for the case at hand.
medium to illustrate the process is the illustrative example of Section 2. 3.
A convenient
One observes
that the deviation of the magnitude characteristic from the desired is much more serious
than that of the phase characteristic.
Accordingly it is appropriate for the first adjust-
ment to base the choice of changes on the deviation of the magnitude characteristic alone.
Figure 13 shows plots of the desired change in the magnitude characteristic and the derivative functions.
bLOGIFI
On the basis of Fig. 13 one estimates the size changes to be used to
bLOGIFI
bC2
0.4
-0.8t
-
bLOG IFI
)bw,
oOs
W-
Fig. 13 Derivative curves and desired change in magnitude characteristic for the example of Fig. 9.
W'-
DESIRED A LOG Fi
(LEVEL OF A LOG F
IS ARBITRARY)
-16-
give the desired changes in the magnitude characteristic.
any positive increment in
1
will cause an improvement since the first estimate gives
insufficient amplification in the low range.
0.2,
A
2
= -0.2 and Aw2 = -0.1.
ing to these changes.
Rough calculations suggest the values
C1 =
Figure 14 shows the second approximation correspond-
As long as it is obvious without a more precise method how to
LOGF
-A
F
---DESIRED CHARACTERISTICS
X APPROXIMATING FUNCTION
0.150
lO
For instance, it is clear that
'".10
_
*I
100
I
.
-.
'L.Nt
.
N1
V.
0.00
,X
N'.
11N x
X
I,
"I
",
/J
-I
"I
-0.10-o
50
-0.25U'
'1 N/~/~
x...
I e'W/
N
'~~
j~~
~^
MAP OF POLES
"
-0.25
0
O
0.2
O.
O.
04
0.6
I
I
0.8
,-
Fig. 14 Second approximation to the characteristics of Fig. 8.
obtain improvement in the characteristics, one continues to use this simple method of
adjustment.
Fortunately, for many problems, specifications allow considerable toler-
ance and one arrives at sufficiently close approximations without going to a more accurate method.
Actually, as the present example illustrates, the simple procedure care-
fully applied leads to a very nice approximation.
After a few more adjustments of the
nature of that illustrated above, one arrives at the characteristics of Fig. 15.
-Ara F
LOGIFI
150
---- DESIRED CHARACTERISTICS
X APPROXIMATING FUNCTION
,,/x
0.30
~__~,~X-PLANE
/~~/
:h
'-.
X," ..
x
-045,1.066
100
0.20
N
0.10
-x
K
50
-04,0
0.00
I
0
0.2
04
,' I
IN~~~~~
I
aMAP
IN~
0.6
0.8
1.0
OF POLES
Fig. 15 Approximation to characteristics of Fig. 8
obtained by the first stage of the adjustment procedure.
The method of adjustment illustrated in the foregoing paragraphs becomes ineffective
in complicated situations.
For instance, the deviation of the characteristics at one fre-
quency may indicate one shift of a pole while the deviation at a different frequency may
indicate an entirely different shift.
In such a situation it is difficult to prescribe appro-
priate shifts of the critical frequencies.
If one tries to prescribe the shifts by the simple
procedure illustrated, he finds himself in the predicament of one trying to solve a set of
simultaneous equations one at a time.
Incidentally, this same difficulty is encountered
in solving the approximation problem with an electrolytic tank.
-17-
A further defect of the
simple method of adjustment, and with the electrolytic tank as well, lies in the fact that
one stops the procedure when he cannot obtain further improvement.
He may be uncer-
tain as to whether or not someone else can obtain a better approximation by stumbling
upon a more appropriate combination of critical-frequency shifts.
At this point a more precise method of adjustment is illustrated which will handle
the difficult cases.
Moreover,
it gives the designer a very strong signal when he has
arrived at the "best" approximation (he will have had to define a criterion for "best",
however).
Finally, it will tell him whether he can substantially improve the approxima-
tion by asking for a different level of magnitude or an added increment of phase shift
which is linear with frequency (different time delay).
3. 3
Final Adjustments of the Positions of Poles and Zeros
The problem which must be solved more adequately in the final adjustment of the
position of poles and zeros is the choice of Ac's and A's
in Eqs. 18 and 19.
The means
employed to prescribe the shifts must account for the effect of all shifts simultaneously.
At this point it is helpful to consider the derivative curves and to observe that changes
of 10 to 20 percent in the pole positions cause rather small changes in the derivative.
This fact indicates that in Eqs.
small changes.
18 and 19 the derivatives may be considered constant for
Equations 18 and 19
, 1og IFl " a logIF,
AlogF
Fy
=
a 1ogll
.1
1
• Arg F
aLOGIFI
Arg F
a log IFI
A2 +
a
w
A61
a Arg F A
~ 2~22
1
+
a
Arg F Aw
ArgF
do,
i_
aat
0
0.5
1.0
-100
'
a LOGIFI
a Arg F
.¾_
a,
I a-,
lb
Ketcnes
erlvatilve unctions for the pole positions of Fig. 15.
1.0
0
3 LOGIFI
0
-O
U
-11,
Vig.
0.5
0.5
0.5
I
1.0
w-_
(b)
Arg F
ao ,
, §
1.0
I-
0-
(c)
°
(18)
2
-.-
..-
0.Z
1.0
w-..
-18-
(19)
suggest a mathematically convenient formulation of the problem of final adjustment which is
ArgFDESIRE
D
,o0
o0.5s
-°"
-IO
to.
-
'u-
most simply understood by reference to the
problem of Fig. 15. In Fig. 16 are shown plots
Fig. 17 Desired changes in magnitude
of the derivative functions corresponding to the
and phase from Fig. 15.
critical frequency positions of Fig. 15. Figure
17 shows plots of the desired changes in the magnitude and phase from that illustrated in
Fig. 15. (Note that a constant change in level or a linear change in argument may be
Aa(l,
prescribed in addition to the changes of Fig. 17.) The problem of the choice of
A2
and A Z is the choice of these quantities such that, when the functions of (a) of Fig. 16
are multiplied by the proper Aai, those of (b) are multiplied by the proper Aa 2 and those
of (c) are multiplied by the proper
2 and the sums formed in accordance with Eqs.
18 and 19, these sums approximate the functions of Fig. 17 in the best possible manner.
Clearly "the best possible manner" is in general not a perfect fit. One criterion of "the
best possible manner" is that the integral of the square of the deviation subsequent to the
chosen shifts be a minimum. The method of choosing A1, L 2 and /A 2 for such a criterion is well known. One forms a set of normal orthogonal functions from linear combinations of the derivative functions and evaluates the optimum Aal, Az 2 and Ad
by the
Z
evaluation of appropriate integrals. However, the formation of the normal orthogonal
functions is so laborious because of the evaluaaLOGiI
tion of the integrals that the method is entirely
1a
OF
oo
0
T
T
An alternate method which is based on the
same kind of approach but is far simpler is that
of approximating at a set of points rather than at
,ool
T
cu
impractical.
(a)
6LOGIFI
bArgF
as,
as-
every point from
. .
o-
o
0
dafw,
A
_
= 1.
The labor in-
volved in forming the normal orthogonal functions
arises because of the fact that one uses an infinite set of samples to get an average.
It is easy
to see by consideration of Figs. 16 and 17 that,
if one finds the sums of Eqs. 18 and 19 matching
a ArgF
aLOGIFI
o
T
= 0 to
a,
0
well with Fig. 17 at five or six points on the
IC?
00
-T T
1.0
[-
= 0 to w = 1, the match will be good
range from
elsewhere as the derivative functions and the desired deviations are smooth.
0
-0.01
in matching at eleven points (six on the magnitude
~AArg
F
L0GiFI
ALOGFDESIRED
10
o
1
curve and five on the phase curve) is very much
DESIRED
T
The labor involved
Io
(d)
Fig. 18 Values of functions of Figs.
16 and 17 at selected matching points.
reduced. To formulate the procedure of solution
suppose a set of points have been chosen from
the range
= 0 to 1 and the values of the functions of Figs. 16 and 17 are indicated at those
-19-
points as shown in Fig. 18.
One is under no compulsion to choose the same matching
frequencies for the magnitude and argument curves though he is choosing Aal, Ar 2 and
Ordinarily it is expedient to choose matching
a,2 which apply to both at the same time.
frequencies where the worst deviations occur or where the derivative curves are large.
In any event one guides the choice of matching frequencies to fit the needs of the particular case.
The ordinates of Fig. 18 are of two different kinds; some are in terms of logarithms
to the base 10 and some are in terms of degrees. Before proceeding one must put all of
the ordinates on a common basis.
In the problem at hand a deviation of 0. 05 in logIF
will be said to be as serious as a deviation of 10 degrees in the argument.
[
Accordingly
all quantities referring to arguments should be divided by 200 to have the quantities on a
uniform basis.
One appreciates at this point that if the magnitude or phase at a particu-
lar frequency is more important than others it may be given a heavier weight.
18 the phase shift at
If in Fig.
= 1 is to be more closely controlled than at other frequencies the
ordinates of argument curves there might be divided by 100 instead of 200, for instance.
A compact notation for the functions of Fig. 18 (after weighting) is vector notation.
The ordinates of the functions define vectors of eleven dimensions. Call FD a vector in
1 1-space associated with the weighted ordinates of (d), Fi
ciated with the ordinates of (a), F
ciated with (c).
2
a vector in 1 1-space asso-
a vector associated with (b) and F 2 a vector asso-
For the situation illustrated in Fig. 18, one has the following vectors.
-0.006, -0.006, +0.030, +0.024, -0.004,
-0.027, -0.027)
FD = +0.010,
-0.010, +0.004, +0.008,
F
= +1.08,
+0.87,
+0.54,
+0.33,
+0.22,
+0.15,
-0.29,
-0.37,
-0.33,
-0. 29,
F
= +0.29,
+0.33,
+0.40,
+0.56,
+0.76,
+0.98,
+0.059, +0.11,
+0.17,
+0.13,
-0.25
-0.25 ~
-0.055
F
= -0.69,
-0.69,
-0.71,
-0.73,
-0.64,
-0.33,
+0.065, +0.15,
+0.26,
+0.44,
+0.60
The problem to be solved, expressed in vector terminology, is to select Ay1,
Ar
(43)
)
2
and
Aw2 such that
FD
AC1 F
1
+
2
F62 + A
Z
F 2
*
(44)
The a's found to solve the problem stated in vector terms are appropriate to be used as
shifts of pole positions. Clearly the approximation can not be made exact in the general
case when the number of dimensions is greater than the number of adjustable quantities
as is the case here, eleven being larger than three. Accordingly, one must establish a
measure of approximation and on the basis of it choose the A's.
A mathematically con-
venient and physically practical approximation is that in which the sum of squares of
deviations at the points considered is minimized. In vector terms the A's are chosen
such that
[FD
^-alF1 + AC2Fa2 + At
2
F021
[Same] is minimized.
(45)
The solution for the A's of Eq. 44 is most conveniently done by choosing a set of
normal orthogonal vectors which are linearly dependent with Fr.1 , F2 and F2 and first
-20-
Finally one is led to the
approximating FD with them.
A simple set of
's of Eq. 44.
normal orthogonal vectors is:
Fnl = all F0l
Fn2 =a
21
Fn 3 =a
31
Fcy1 + a22 FC2
(46)
+ a 3 2 F(2 + a 3 3 Fo2
The a's are chosen to fulfill the following relationships:
Fnl .
Fnl = 1
=
Fnl .F n
Fnl . Fn3 =0
Fn2
Fn 2 =1
Fn2
Fn3 =0
(47)
.
Fn3 . Fn3
1
The solution of the equations above, or similar equations for a case of any number
of poles and zeros, is fortunately simple.
and one solves for all first.
volves a 2 1 and a22.
The first row of Eqs. 47 involves only al 1
The second row (using the numerical values of all) in-
The third row involves (using numerical values for a, 1 1' a 2 1 and
a22) a 3 1 , a 3 2 and a 3 3 .
It is simplest to solve the equations from top to bottom.
If one
considers any row of equations of Eqs. 47, he finds that only the last equation is quadratic.
The equations preceding it in the row are linear.
a time.
Further they can be solved one at
For the third row for instance, the first and second equations are of the form
(48)
m2
2
a3
2
+ m 23 a3
2
0
Hence one can solve for a32 in terms of a33 and then for a31 in terms of a 3 3 .
the third equation in the row is quadratic but is in terms of a33 only.
Finally
By considering the
equations in the proper order one completely avoids the solution of general simultaneous
equations.
The simplicity indicated above applies to the determination of orthogonal vectors regardless. of the number.
The degeneracy of the equations of the form of Eq. 48 arises
from the special manner in which the orthogonal vectors are obtained.
For the illustrative example at hand, one has the following:
al
=
0. 5902
a 2 1 = -0. 3608
a 2 2 =0.748
a31 =
a 3 2 =0.6356
1.063
(49)
a 3 3 = 1.463
-21-
__.
_
_
_
F
ni
+0.6374,
+0.5135, +0.3187, +0.1948,
+0. 1298, +0.0885,
-0.1712,
-0.2184,
-0.1948,
-0.1712,
-0.1476
= -0.1144,
-0.0200,
+0.1335, +0.3177,
+0.5010, +0.6870, +0.1331,
+0.1958, +0.2248, +0.1862, +0.0355
Fn3 = +0.3228,
+0.1250,
-0.2103,
-0.2191,
-0.1038,
Fn2
-0.3609,
+0.2997,
-0.1756,
+0.1376, +0.4180,
(50)
+0.5768
Once the normal orthogonal vectors are selected, one approximates FD in terms of
them.
FD-
1 Fnl
C2 Fn2 +
Fn3
3
(51)
This approximation involves choice of the c's such that
[
is a minimum.
-
(c1 Fn + C2 Fn2 + c 3
[Same]
n3]
(52)
This problem is particularly easy to solve since the vectors used are
an orthogonal set.
Once the c's of Eq. 51 are selected one can obtain directly to the
Considering Quantity 52 and observing the ortho-
Co 2 and Aw2
optimum values for A l, A
gonality of the vectors one has
[
]=FD
F -2
2cF
- 23Fn3
n3
Fl
FD + c - 2c F
2
F+c
FD
FD
c2
(53)
c3
To minimize Quantity 52 one determines the c's by setting the partial derivatives with
respect to ral,
I2 and w2 of Eq. 53 equal to zero.
a1
2cl - 2Fnl
][
i2c
a
ac
ac
This step gives
2
3
[
]3
2c 3
n2
-
2F
3
F
O
F
0
D
FD
(54)
0
One observes that the second derivatives of Eqs. 54 with respect to the c's give 2 and
accordingly c's chosen to satisfy Eqs. 54 give a minimum of Quantity 52. Consequently,
C1 = Fnl
FD
c2
Fn2
FD
c3
Fn3
FD
Finally, Eqs. 55 lead to the optimum pole shifts indicated in Eq. 56.
-22-
(55)
C 1 = clall
+ C2a
AC2 =
+ c3a31
21
(56)
ca22 + C3a32
2
32
Inconnection
with
example
theillustrative
being 3c3a3
In connection with the illustrative example being considered one has
c
= 0. 00175,
asc 1 = -0.039,
'1 = -0. 439
C2 = 0.00086,
Ao - = -0. 024,
2
c 3 = 0. 0374,
A
2
1
2 = -0. 055,
A plot of the characteristics of the
LOGIFI
Fig. 19.
-Arg F
tics.
\
\
APPROXIMATED
BY a'S
/
0.250
0.150
has been quite small; this was to be ex-
PONO CHAR. CORRESI
ING TO EQS.5
pected since the example is simple and
the first stage of adjustment is very ef-
+0.100
A CHAR.CORRESI
PONDING TO EOS.72
//\//
__-
_
\x
\
A/
\/
Iy
)OTTED LINES ARE C
CHARACTERISTICS BEING
APPROXIMATED.
\
X\ //
_60
----
one
now can state definitely that it is impossible to obtain a better approximation to
It is neces-
sary to recall that the measure of quali-
20
_
However,
further shifts of the poles.
+0.050 _-
+0.000
fective in simple cases.
the "o" characteristics of Fig. 19 by
xX
A(
iv _
-0.025
In this case the improvement
x CHAR.OF FIG.I15
100 _-
_
80
Quantity 52 changed from 0. 0033
to 0. 0021.
//I
_
12o )
0.200
One can see by comparison
adjustment has changed the characteris-
-
140
0.300
1. 01
of the 0's and the x's how the final
-
160
0350
(57)
474}
mating function given by Eq. 57 is a part of
_.
0.400
= -0
ty of the approximations has been chosen
Li
I
0
0.2
0.4
I
0.6
I
0.8
"
and the approximation arrived at is best
1.0
O,-_
in the sense of the chosen measure.
Fig. 19 Characteristics of approximating
functions.
Dif-
ferent weightings of magnitude and phase
characteristics would have resulted in
somewhat different approximations.
In cases where the first stage of approximation gives a far less satisfactory result
than that of the accompanying illustrative example, it may be necessary to use more than
one adjustment in the final state.
This will be necessary only if the shifts required are
so large that the derivative functions change a large amount. A simple method is given
presently to compensate for changes in the derivative functions for large shifts. However,
the compensation is not exact and one may wish to carry through the final stage of the
adjustment procedure again with the correct values of the derivative functions to obtain
the optimum result.
-23-
_______
__._
3.4 Further Topics in Connection with the Final Adjustment Procedure
Further useful conclusions can be drawn from the results presented above.
measure of seriousness of the deviations is FD . FD (or Quantity 52).
The
As FD . FD (or
Quantity 52) is decreased the quality of the approximation is increased.
For the optimum
choice of c's according to Eq. 55 one sees from Eq. 53 that the corresponding sum of
deviations squared is reduced by the sum of the c's squared.
At times it may not be desirable to make the total shift which is specified by the c's.
In the event that a shift is used which is k times the optimum shift one has the following
sum of squares of deviations.
(See Eqs. 53 and 55.)
clf = kcl, c 2 f
F
FD
D
2k
.D
+ k2c
c
2
k c 2 , etc.
k2
-2k
cl
2
2
(58)
2k
2c
+ k22
2c 3 k
3
3
FD
(2k - k2)
.D
c2
(59)
i=l
The second part of the last equality in Eq. 59 indicates the change in the square of deviations caused by the shift in poles.
Consideration of Eq. 59 reveals that the maximum
improvement occurs for k = 1 as it should.
in improvement.
Any value of k between zero and two results
Larger values of k increase the square of deviations.
3.41 Compensation for Changes in the Derivative Functions
When one chooses the c's of Eq. 51, he assumes the F's are constant vectors.
How-
ever when the c's lead to large changes in the pole or zero positions, one may find that
the change in the approximation is not represented by the vector
C1 F
nl
+
F n2
+
..
Cm F n m
but is instead represented by
C1(Fn 1 + 'nl
+ c 2 (Fn2 +
Fn2 ) + ...
cm(Fnm + AFnm)
since the value of the derivative functions has changed.
If this is the case, the c's
chosen do not lead to the best approximation and one needs to choose slightly different
values. The problem then is to change the values of c to compensate for the change in
the values of the F n's. This means that one should consider
(c 1 + AC)(Fn
)
+ (
+
c 2 )(Fn+Fn
+
(cm + ACm)(Fnm
+AFnm) FD
(60)
-24-
it
In Eq. 60 Fni + AFni is the mean value of the vector Fni as the poles are shifted to the
positions indicated by c i 's.
Appropriate values of the
ci's may be obtained by neglecting
the second order terms in Eq. 60. Accordingly,
m
±
AclF1
AC 2Fn +
...
AC Fnm
m
D-
F
i=l
Cni-ciaFni
.
(61)
i=l
Because of the orthogonality of the F 's one can write immediately the appropriate values
of Acj,
Acj=
=
Fnj.
(62)
i
One recognizes that
m
i
AFni
i=l
is the difference between the adjustment in characteristic actually obtained and that which
would be obtained if the derivatives were constant.
In difficult cases the compensation
procedure just outlined reduces the number of stages of the final adjustment procedure
and proportionately reduces the computational labor.
3. 42 Choice of Level of Magnitude and Total Phase Shift for
Best Approximation
In the choice of the desired changes in the magnitude and phase indicated in Fig. 17
an additional constant increase or decrease in level of magnitude or an added change in
the phase shift which is linear with frequency may be applied without violating the requirements being approximated (Fig. 8).
To approximate a different level of magnitude
or a different linear phase shift is appropriate if these changed characteristics can be
better approximated.
this situation.
The implicit relation between magnitude and phase actually enters
For a prescribed level of magnitude (Fig. 8) over the frequency range of
interest there is a most compatible linear phase shift over that range which can be best
approximated.
To investigate the question of added magnitude it is appropriate to define Fm, a vector which added to FD gives a unit added constant change in magnitude.
For the illustra-
tive example,
F m = 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0.
(63)
Fp is a vector which added to FD gives a unit added linear phase shift.
Fp
p
0, 0, 0, 0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1.0
.(64)
-25-
-
~
The problem is to determine how much magnitude and how much phase shift to add.
One
should ask for the vector change FD + kmFm + kpFp in which the k's are chosen constants to make the best approximation possible.
To study this problem it is first desir-
able to recognize that F m and Fp are best approximated by
Fm
mlFnl
+m
2 Fn2
+ m 3 Fn 3
(65)
P 3 Fn3
(66)
and
+
Fp - PlFnl
P 2 Fn2
+
where
m
=
Fnl ' F
= Fnl
Fm
P1
m2 =Fn2
Fm
P2 = Fn2
m 3 =Fn3
Fm
P3
=
F
(67)
Fn3 ' F P)
For the present illustrative example
m
= +1.88
1
P1 = -0. 523
m 2 = +1. 50
P2
m 3 = -0. 0428
P 3 = +0. 917
It is simple to verify that FD + kF
Dm
3
z
=
(68)
+0.427
+ k pFp is best approximated by
mp
p
(ci + kmmi + kpPi) Fni
i=l
For any choice of k m and kp the sum of deviations squared (Ds) is
D
s
=
kmF m
+
k mi
+kpFp -
k
(ci
+
kmml +kpPi) Fni
(69)
=
The variables in the above expression are km and k.
of information.
Same
Equation 69 contains a great deal
One can say, by considering it, how changes in magnitude and phase shift
influence the quality of the approximation.
For the illustrative example,
aD
7TI m
= 0.375 k
m
+ 0.782 k
p
- 0. 00722
(70)
+ 0.0106
(71)
aDs
= 0.782 k
PP km
+ 1.806 k
P
By setting Eqs. 70 and 71 to zero and solving one finds the optimum km and kp.
m p
-26-
This
gives km = 0. 33 and kp = -0. 15.
However, these changes lead to shifts of the pole posi-
tions which are too great for the derivative functions to be approximately constant. If
one used these shifts he would find the errors introduced by the changes in the derivative
functions would consume the advantage indicated and it is necessary to proceed to the optimum by smaller steps. Less drastic changes may be made on the basis of Eqs. 70 and
71. One can find the most desirable changes in km for any specified change in kp. As
an example if one sets k
= -0. 025, he finds that the best k is 0. 0715.
p
m
sponds to the choice of poles indicated below.
This corre-
= -0. 414
2
= -0.418
2
= +0. 973
(72)
)
By integration of Eqs. 70 and 71 over the appropriate ranges one would predict the improvement in D s to be about 0. 0006. The characteristic corresponding to Eq. 72 is
shown by A's in Fig. 19. The sum of squares of deviations for this choice is 0. 0018 according to the calculation of the characteristics of Fig. 19 from the normalized curves
in the appendix. The correlation between predicted and calculated deviations cannot be
too great since small inaccuracies in reading the curves substantially influence the deviations which are themselves small.
The result obtained in Fig. 19 is not the absolute optimum. One can proceed to the
optimum by further steps of the kind just completed, using the accurate values of the
derivative functions for the pole positions of Eq. 72. In a practical problem one must
balance the need for precision in the result against the labor involved in obtaining the result. If one is unable to obtain good elements whose behavior is close to ideal, it is as
needless to carry the approximation problem to the finest point as it is to specify the
dimensions of a house to thousandths of inches.
As has been pointed out previously, the
first stage of the adjustment procedure is sufficient for most problems; however, when
one needs to obtain precise results, the method given is applicable as far as one needs
to go.
-27-
___
References
1.
W. Cauer: Die Siebschaltungen der Fernmeldtechnik, Zeit. f. angew. Math u. Mech.
10, 425-33 (1930).
2.
E. A. Guillemin:
Communication Networks, 2, Chapter 10 (John Wiley and Sons,
1935).
3.
H. W. Bode: A General Theory of Electric Wave Filters, Jour. Math and Phys. 13,
275-362 (1934).
4.
S. Darlington: Synthesis of Reactance Four Poles, Jour. Math. and Phys. 18, 257353 (1935).
5.
S. Butterworth: On the Theory of Filter Amplifiers, Exp. Wireless and the Wireless
Eng. 7, 536-41 (1930).
6.
J. G. Linvill: Amplifiers with Arbitrary Amplification-Bandwidth Product and Controlled Frequency Characteristics, Sc. D. Thesis, M.I.T. (1949).
7.
R. M. Fano: A Note on the Solution of Certain Approximation Problems in Network
Synthesis, Technical Report No. 62, Research Laboratory of Electronics, M.I.T.
(April 16, 1948).
8.
R. F. Baum: A Contribution to the Approximation Problem, Proc. I.R.E. 36,
863-9 (1948).
9.
10.
G. L. Ragan: Microwave Transmission Circuits, Chapter 9 (McGraw-Hill,
1948).
H. W. Bode: Network Analysis and Feedback Amplifier Design, p. 209 (Van
Nostrand Company, 1945).
11.
W. W. Hansen, O. C. Lundstrom:
Experimental Determination of Impedance Func-
tions by Use of an Electrolytic Tank, Proc. I.R.E. 33, 528-34 (1945).
12.
J. G. Linvill: An Experimental Approach to the Approximation Problem for DrivingPoint and Transfer Functions, S. M. Thesis, M.I.T. (1945).
-28-
_
__._
I
_1
__
___
Appendix
Two rational functions are considered.
(a)
(b)
The functions log
F
(X -Xc)(k
-
Xc
Cc + j~c
Fr =X-
c)
o
logl0
1Fc, Frl, Arg F c , and Arg F
(normal-
ized) are shown in Set 3 and Set 4 plotted as a function of X = j.
In
connection with the function F c , families of curves are given for a
range of values of the parameter -c/wc.
All logarithms are to the
base 10, and all arguments are expressed in degrees.
The rates of change of the functions of Set 3 and Set 4 with shifts
in the zero positions of F c or Fr are given in Set 1 and Set 2.
-29-
__
Set 1
a lOglo IFI
8 loglo IFCI
-W
c
c
acc 0
and Xc -
-a-
of the parameter.
-
r
a logl 0
c
IF
c
as functions of
c
a
as a function of
-
c
c
r
for a range of values
-10
-9~~~~~~~J-
A
7
F
|
|
+
-',
I-t
t t
i
t -t,
~~ ~ ~ ~
.
4i |
+i t i
T
f t!
.t,~-
;
t
.,i
1
F i < i4t I
4
:
i
LI
;
<] ti ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~1--r1:1
11i.41,<+,,,
44
' t
-30E
-3~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~oi
';0 lLLI<
t :tI4i11t : : , ~
!t~ ~t
'. i1
- Isf1t~! t-.':'1/~
1 K';:1:,i ,.fi :i ;t. ~ !
I !6~~~~~~~~~I I
tI Illl
,] ;
' t ! ! ' ',_J
0l ~ ~~t
~_M:__ t tlG
!t;::i
nt;\\ |*l
t::
f
'/!
XI|
i i]
1.3
1L4
! 0 1T
F
~'+'
0
Q-l
0.2
Q3
04
05
0.6
0
Q7
0.8
0Q9
1.
1
.2
1.5
we
-I~~~~~~~~~~~~~~~~~~o
~~
~
-0.8
~
3
W
C
08ki~:~~~~~~~~~~_
082
; l:I-.·
-0.8
||
'f2-f
i 0f l Tl 11I-ilr l-f ii :i !i i 161i 1i i Ih-01 -11' t;i
44~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.
!
p-Tt;J40.:,--:,4,'il:
i
i~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
-07 -4_
.: . ! .
!
os
Y
-0.4
4:~~~~~~~~~~~~~~~-
-02C
o~~~~~~ll
1rl 1;1-~~
4:01 i !~01
.l|A,~ ;! I 'tI.
1 <
l li::all 1
0.I1
U
0~~~~~~~~~~~~~~~~~~~~~~~~~.
^
U.I
UL
U.O
n
S
UD
.
s
U. 10
t
A
it
til-X
II:t
-w
,tA-t.
i7i l I i li: - H1I1-Ll--l
1 i IL '1
1 ~!..71~~-!s-i,'
l <
L
0{
;|W LL
7
U
o
a
n
U
U..
¢
WC
----
z
z.
I.U
t
1.I
I.Y
I'>
1.3
1.4
1.0
--
I~~~~~~~~~~
I
4R
T-F
Ittl
lit~~~~~~i
I
1TI
1: IIIIIt
_T FF
14~1r I
I
~I/Iv
itI 1 TT-1
fl~~~~~~ C
IH
1
4
:CI i I =
'~ I j ,
t ~~
'
4-~ 4f
r.
"
'..,...
.........
-VII~
VI
_
4
d6000
I
_ _
-
t {
f/-:t
//-
-
I-
_
,
o
.
000
:'
I
leQ
C4
D
Y)~~~~~~~~~~~~~~~~~~~~~~~~t
i:,!t,!
{
-
-
'
S'
~
-
II
t
/ t.~flA
I.....,",..P,-:.":,tY_
~ ±
lI~f7A1.0~~~~~~~~~~i
-
-
'
-
w
4
N0
M ___
0
-
-
-
I~ I ;I D
-7-
Q
Q
0
~
Q
0
(
Q
0
v
rT1
II
..
_._._..._
IMi#t
.
IIt
N
T#It fUffiMv
MTlig
Cp
Tly'klitv
fi
fit
4 1-
mix"
o
al
i
MIi4,ti'
4
i 4 MITM
t Xf t~~~~~t
t
ATlf~
1
tt~~ r,
tf+4
i f9
iffie
W
~~
E on
;~~~~~~~~~~~~~~~~~~C
2
Wt[ 0r
0t4f~
t4
CD
f
ci
01
(i,
o
110
,
li
|
l
40
0 0
iii| 0
5
NJ
0
rm l|
0
W 0g
| @k SW 4
N
944~~~~~~~~~~~~~~
|
0 MM| W
WX
gett i tt
W
o~~~~~~~~atiP~
ss!
~~~~~~~~~~~~~~~~~~~~~~li
ti~~~~~ll
-1froAM".
t
4
Pi
ii~~~~N
Ff T Z v iT ;P'&F
i-TT~~i
f4H4Whi
V.Tt
Hditor'"
_T
f
;F
10
I
t
v
i&
h_l
Pil TI i~~~~f
M'111~~~~~~_
1ii-;, vJ,:m 5 II
_P
Tit;
if
Ir'l:
i
4i
W
Pl_if
"i
W
to q
IQ
0
R
0
Q
.?
-----
Q
i v
W?
I
q
7
*
°
f
-IJ.C.v
} E E E Wm ~~~~gig
#y41#i
01r4
a, MR 4 OR
4
RM
M
t
M
99
iw
IR
Mfi5WW
XXgSXXW
T
mgW
ORM
!9Q MT4 i#
WI
t Mat
3
4i
9
i
MUM49
R
gf
V,
EmEgg.E
gol
9
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X~~~~~~~~~~M
MMV
~~~~~~~~AMMO
0~~~~~~~~~~R
m m"
~~~~~~~~~~~~VONM
-030
'-a
41"
Lif
-0.35
if
w
m~~~~~
IM
0
M
nm
HI41
MZ99
IM
TM
MIng
M
-
AltM_
MJ
44 ffl
mm fw
MM AM Y
-9
mm
n
q
RmggU
M
4
gJ
m ft la
a
W@ $0
g
OR kW 0MAwga MMI
m
-043
-040)
ih
uXW
W
Mltm
-
99~~~~~~~~~~~~~~~
t
oa4
0.2
0.3
Q4
-a
A.
OR
,
0.7
Q8
A
no
....
-oTr
I
t
-0.04
_~~~~~~~~
ma
-o.oe
-012
-O.16
.0.20
b4IA
-0.24
MI
4
44A
-0.32
-0362as
~ ~
8
~
10
~
12@
-
4
16
18OM22
2
Set 2
8aArg F
ic -~aC
, and
c
C-c
C
8 Arg F
c
a
as functions of
C
-(r
of the parameter
-.
C
8 Arg F
- (ras
r
a function of
a%
-_r
-'
C
for a range of values
0
17 7- -1-I -I I I I FI -I
-I
I -…-I
f I I I I I 1 -"- I I I " I 0 1, I 1 i j I ; I 1 " 1 1 'l I I i i 1 I I -I I I I I I
-I
I
I
I
1
i
[
I
I
I
I
__-
_
I
I t I_,
i '1f
fir
I
_
I 1
!M
I II~t
I 1
I
1
1
i-
f
iI i
H-
I_
IIIItIIL
-
I
I J..
1
I
I
oD~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
I
-I--- I--___
-:
Ii
I
I
L
i' 11"'t'I
~~~~I
i' tIi : Ii. I. , .I. I- . .
t I, I I i [ i t. I l i ' lJ:,,! L- I
!!~
, :I ,![:'
;
-V
-
-
i
i
1
U I
I-1
-LL
I ----
--I
L....I
I.
L
c
I'<
i
i
-L~
I
11
_J
1,: tI t] ,11: -::tI:11 I
I
I
I
;~~~~T
I
'If
~~~~~~~~
I
I
I
II~~~~~II
~-II~C
t--~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i''
-~~~~~~~~~~~~~~~~~~~~~
I~~~~M
~~~~~~~~~~~~~~~~~
I
o
~:
1
[
'
I
I
'
I-
I
:I
I
0
.)
I
·-
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M
;
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~ ~ ~ ~[ ~~~~~
:
1
I
I
I
ii
I~II
I
:~
I
I
i:l:--i-
'"l
1J-
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:'2
'
1:
LLII.I-;J.r'
l
'
O
21:":?:'
,
b,;
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m'W
i t i l
0
I,-
0O
D
0u
o
0
v
I J f^l
oo
0
10
0
OJ
: t l
0
0
i l i J i t i~:J [+~,:l't
0
0
(3
~f
0
0
I
I
:1 ~:t [::t 'i l t:-J J0
0
I
0
0
0
I
0
| ^ . | 0Mill. ON
CD
SlD
~
'
.1
re4
.S
eD
I
W~~~~X
S~~~~~~~~~~~~~
B
.f
C~I
44_
1
f~
t~J
C
#ti_
m~~~~~~~~~~g
4 . 4 JU oc~~~~~~Nf
fl_
C
n
0
c;
t
i t
[ ..g
E 4_g0g W~~~~~~~~~fil
oIl
10M. !
1g
0
4v
B
Cli
g~~~~~~~~~~~O
g
E10
~~~~ ~ ~ ~ ~ ~ ~ ~
W
1
i
|
SD 10 It
l
cl
cl
cl
C
cl
i-
I
[~~~~~~~~~~~~~~~ H
iiii
,~~~~~~~
.
40
Wu
04
C4
CY
,l,~~~~~~~~~~~~~~~~l
l
cu
C-i
OD,
LOJ
m
W~~~~~~~
ala
m V-m
q
u 11
v
u
u
g
d
O
bARG F
-W
bW 0
I
z20
7-77,77. 3
I
le
I
--
-,jl 1
1jI i 1'
_=
16
1
I
I4
X
1'
:1 3 1
-f I
i1
|t
141
7
r-ti:l .
:
W 4tl-4 4ll,
h.Hi
%:I
E .TF
ia'
Ii
:1 -11
1,:
-F-
-- _-1.
'41
_'1
_
-I
*L-!
it
T ,
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i
1:I
1W
I 'A
; !: 1-X - i
it',
i"I
,
T
. !he
X4ltr
t-i
7~XtHOr
-40:'~
4-.
71-
,j ,,
74
i
+I j
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LLL
I
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_
2
-i
4
6
8
10
12
14
WC
16
18
r
i
1--1 8
_
I
i:il
-t
ill
X
,
-, I, I I -,".
- 1,
, 1 ; T;, -:
20
:.-
':
22
HI~f
.. ;
' '- , 24
f-
I
ate
MRS"
a
R
-m T It Tiff
iTit
ffttWE
r:
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f- if!
t
TaffTf i
IIfIs It
TOM
fi M I
is WON
4 IN
lis
am
14 VI' at XIMM IWAR"M
0
q ilt
no
mile
9 _U
M
Mai
qa
.... ....
M lit
S.i4
Q6
Q8
10
1.2
L4
t 4TR 1 Ti TFF
1.6
1.8
2.0
2.2
2.4
vt~~l
.mi. , .t E . .X ^
~
Hii i
26
#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Fir
14 f i +.. i
24
22
:
h
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I
t, ft Ail
2C
It
i
14
_d~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~m
"i i
f11
S~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~N
R Sm
iII
IV
9
IR ff
~~~~~~~~~~~~~~~~~~~~~~~~~M
i i
tt
ts I
$t 14g
ti
t iit
t : t X i i Nii~~~~~~~~~~~~~~~~~~~~~~~ti
IC
~~~~~~~~~~~~~~~~~~~~1
1
14
16
4
c
-ii
-
--
--
la
20
22
Set 3
Logo
10
IFcl
w
2- as a function of -
Wo
for a range of values of the parameter
C
c
.
C
F
Loglo
0T
r
c
as a function of -0r
r
___I_
I_
0.6(
T
0. 5
7
Teeel
tfi
-'f
r`A:tttrta:ct 1
0.5(
tF
:
rt~~~~t~
m
7
bt~~~~~~~_7
atl-lfer-;
_
t~~tt'tttlL~~~lt
:r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~L
I f f~:
(1.6)
tt37=
>7
aA
X
L
if3
tt
4 7t t ,
ss
0.5)
4
riil~i'
~r~~
4
-
;W
7~~t~
g
v4 Tl'-GWig
# 0 ; tt~:7itet;4;0lll f;1;if 1 5~~~f:
X W W~~~~~~~g
i9 i i
044
(1.4)
(1.3)
0.41
(1.2)
t+-F
I+7t$It
t'-T8
Lq
t#
q' Tt~t 4 j
-4i;
rItt:
4i 1.t'1.
TSET
i+F11;
i.it-.+
.t '
:·l.L~~~ ;
3
L:-~~~Et
':l~~~t"-tt~~-It :7
(I 1)
(110)
0.'31
X11l
ea i
;Wt4
tt
15i~t~
tI-ii4
i
3
2mfi
vri-itF
if~~~~~~~~~~~~~~~~~~~~~~~~~~~~i
n z i!
m~t·
::
e
-fill
ttintl-<etf~rti~14fi
7I
rettsI] t-;Z~t|
.'i~i~LIi~r~kr
1fTlh
ti 41 + ttllI14
"It
E
1X~~~~~~F
_
0.2(
4-iT iI-, t~
jt`imIf t
tt
,C~~~~~~~~~~~~~~~~~~~~~9
+1
0.2
Em
Fm f-Ft
I Et
~--~lil::t tt
jfri~fi%~
~~ili-i~··
E
4
TTta 77 ;l$;:t~wS
g
w
(0.9)
l
- 1;
(0.8))
E
(0.7)
0.1
Xiji4~t~~~,r
~~~~~~~~i-tf
WiitiilWf~;f~fil~·~f t X
0.1
,.t EEES
,4
0.0
9~~~~~~~~~~
tt
)
A
i tf:~ttflr S~ i
(0.6)
jSXX
SX
w
ffiffi~~~~~~~t
¢E0at
~~~~~~~~~~~~~~~~I
~
~
nt
ta
H1 IF4 Im
;i1:~
Cftr
~l
nC-Igl--~-
F
.
na
nx
n
(0.5)
j gz
) ) R= x~~~~~~~~~~~~~~~~~~~~ij
Mm~~~~~~~~~~~~~~~~
(0.4)
-0.0
(0.3)
iStgw
s
(O2) )
-0.1
(0-1)
(0.0:)
-0.
-_l
v
-A
Jv.
co
'IC
u.4
v.
5
-0.02
-004
-0.06
il04)
-- 0.081Sgig
-0.10
-0.15
t
2l
4)
-O.3
-0.40
-0.65
-0.c 0. 1W
-C5
1 'WX
-OL6 t
-0.470
[Ill~l~ilgllITIIllStilT
-.-----ITl M
X
-0.70
-0.75
tXt
(I)
O~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
0.6
0.
Q5
05
12t
x111$11N1111x1|1111
t
T 0.6
0.
gwIIILRIillLIIIIII~iiillllllllliiiiilllliiiiilllull~hxlluiliiltWC~il~ii
07 0.8
0.
____11
0.
I.3............fXff LOIXLINIIIbLIIIIMNIIIL
....... ..
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