ABSTRACT

DELAY EQUALIZER DESIGN-~NUMERICAL ANALYSIS,
DIGITAL COMPUTER TECHNIQUES

by Joseph John Lang

The existing delay equalizer design techniques are not completely
satisfactory in the sense that they contain many undesirable features
such as:

(1) they are very tedious and extremely time consuming;

(2) the design is guided by trial and error process;

(3) the choice of design parameters at each stage of the trial and
error process requires skill and experience on the part of the
designer.

In recent years, the accelerating demand for higher quality of signal
transmission has led to increasingly stringent requirements on delay
equalization. As a consequence, the standard curve plotting techniques
have become increasingly inadequate and there is a very great need for
improved design methods.

In this thesis, the problem of delay equalization is first studied
with the Specific purpose of locating the drawbacks inherent in the exist-
ing delay equalizer design techniques. The results of this study have
yielded a new method of delay equalizer design, based on numerical
analysis and digital computer techniques. This new design method offers
definite improvement over the existing methods:

(1) Given a set of satisfactory initial estimates of the design para-

meters, the entire design problem is effected in a matter of

minutes of computer time;

Abstract Joseph John Lang

(2) The thesis method provides a systematic approach to the design
problem and eliminates to a large extent the need of skill and
experience;

(3) The extensive amount of the trial and error process associated
with the existing design techniques is almost eliminated in the
thesis design method. The trial and error aspect of the thesis
method is reduced to two computer applications;

(4) It appears that the thesis method is capable of producing approxi-
mations to a constant delay which are beyond the power of the
existing design techniques even though pursued with the utmost
skill, experience, and patience over an extended period of time.

The conditions on the delay equalizer network are derived and the

problem of delay equalization is defined. The problem of delay equalization
is formulated in terms of a system of non-linear algebraic equations.

A numerical method for a solution of this system of equations is presented
and a general computer program is written. The important consequence

of the new design method is demonstrated by means of several solutions to
actual equalization problems. The estimate on the number of sections of

a delay equalizer is considered and an approach to a solution of this
problem is presented. A method for obtaining the initial estimates of the
design parameters is given. Formulae for the element values of the delay

equalizer, in terms of the design parameters, are presented.

DELAY EQUALIZER DESIGN--NUMERICAL ANALYSIS,
DIGITAL COMPUTER TECHNIQUES

BY

JO 5 eph John Lang

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOC T OR OF PHILOSOPHY

Department of Electrical Engineering

1961

ACKNOWLEDGMENTS

The author expresses his indebtedness to his major
professor, Dr. Myril B. Reed, for the helpful advice and
guidance throughout the preparation of this thesis. Special
thanks are due Dr. Herman E. Koenig of the Electrical
Engineering Department, Michigan State University, Dr.
Gerard P. Weeg and Mrs. Georgia B. Reed of the Computer
Laboratory, Michigan State University, and Mr. J. L.
Garrison of the Bell Telephone Laboratories, for their many
valuable suggestions.

ii

CONTENTS

Page
LIST OF SYMBOLS ......................... iv
1. INTRODUCTION . ........ . . ........... 1
II. THE DELAY EQUALIZATION PROBLEM. . ....... 4
2. 1 Insertion Factor and Insertion Function ...... 4
Z. 2 Requirements on the Loss and Phase Function . . 5
2. 3 The Relationship Between the Insertion Functions,
EST and PSF’ and the Image Transfer Function, 6
Z 4 The Tnsertion Delay Function ........... 12
Z. 5 Requirements on the Insertion Delay Function . . 12
Z. 6 The Insertion Delay Function of Matched
Cascaded Lattice Sections. . ......... 13
2. 7 The Design Problem ..... . ........ 15
III. DELAY EQUALIZER DESIGN BY NUMERICAL METHODS 16
3. 1 The Design Problem in Terms of a Solution to a
System of Non-linear Algebraic Equations . . . 16
3. 2. A Numerical Method for a Solution of a System of
Non-linear Algebraic Equations ......... 18
3. 3 The Digital Computer Programs . ..... . . 20
IV. SOLUTIONS TO TYPICAL EQUALIZATION PROBLEMS
USING THE METHOD OF THIS THESIS ........ 24
4.1 The Commercial Problem . ............ 24
4.2 The Commercial Solution ............. Z6
4. 3 Solution by the t— Method . . ........ . . 26
4. 4 Another Example of Delay Equalization by the
t-Method....... ........ 36
4. 5 Criteria for the Choice of Input Data for
Program 1 .................... 42
V. AN ESTIMATE ON THE NUMBER OF SECTIONS. . . . . 45
VI. INITIAL ESTIMATES OF THE DESIGN PARAMETERS. . 52
6.1 Properties of the Insertion Phase Function. 52
6.2 Properties of the Insertion Delay Functiono . . . . 52
6. 3 The Choice of the Initial Estimates, xk, yk. 56
VII. THE ELEMENT VALUES OF THE LATTICE SECTION 60
VIII. CONCLUSIONS ....................... 61
BIBLIOGRAPHY .......................... . 63

iii

Symbol

ASE
SF
ST

{bib-

11>

STR

SE
SF

wwww

ST

U1

STR

LIST OF SYMBOLS

Description

Insertion Loss Function
Insertion loss function of a delay equalizer
Insertion loss function of a filter

Total insertion loss function (insertion loss function of a
filter and a delay equalizer connected in tandem)

Total insertion loss requirement (the requirement on the
insertion loss function of a filter and a delay equalizer
connected in tandem)

Insertion phase function
Insertion phase function of a delay equalizer
Insertion phase function of a filter

Total insertion phase function (the insertion phase function
of a filter and a delay equalizer connected in tandem)

Total insertion phase requirement (the requirement on the
insertion phase function of a filter and a delay equalizer
connected in tandem)

Design parameter—pair of one lattice section of a delay
equalizer

A set of n parameter-pairs for a delay equalizer of 11
sections (kzl, Z, 3, - - - , n)

Insertion factor
Frequency (in cycles per second)
Normalizing frequency

A set of Zn frequency points from which a system of Zn
equations (eq. 3.1. 6) is formed. 11 is the number of lattice
sections of the delay equalizer: (i=1, 2, 3, - - - , Zn)

Delay equalization interval on the frequency axis

iv

E
0"
O
H

smamwmeme :1
u: m "1 N

(I)
*-1

2°

.2"

SE
SER
SF

HHF-It-ir-lv-l

SFmax

TSFmin

ATSF

ST

X,y

O O
xk’ Yk

Description
Number of lattice sections of a delay equalizer
Image transfer function
Image transfer function of a delay equalizer
Image transfer function of a filter
Insertion function
Insertion function of a delay equalizer
Insertion function of a filter

Total insertion function (the insertion function of a filter
and a delay equalizer connected in tandem)

Terminating resistance at the sending (input) terminal-pair
of a network

Terminating resistance at the receiving (output) terminal-
pair of a network

Delay level of the delay equalized filter
Insertion delay function

Insertion delay function of a delay equalizer
Equalizer delay requirement (eq. 2. 5. Z)
Insertion delay function of a filter

Maximum value of the insertion delay function
of a filter in the equalization interval

Minimum value of the insertion delay function of a filter in
the equalization interval

The difference of the maximum and minimum values of the
insertion delay function of a filter in the equalization interval
(eq. 5. 8)

Total insertion delay function (the delay function of a filter
and a delay equalizer connected in tandem)

Design parameter-pair of one lattice section of a delay
equalizer

A set of n parameter-pairs for a delay equalizer of n sections
(kt-1, 2: 39 ° ° °:n)

A set of Zn initial estimates for the 11 design parameter-pairs,
xk, yk, of a delay equalizer of n sections

V

Symbol

0.)

t-method

Description
Normalized frequency with respect to the normalizing
frequency, fd

A set of Zn normalized frequency points from which a system
of Zn equations (eq. 3.1. 6) is formed. 11 is the number of
lattice sections of the delay equalizer, (i=1, 2, 3, . - ~ , 2n)

Image impedance function

Normalized image impedance function with respect to
terminating resistance (r=l, 2, 3, - - -)

Maximum allowable delay deviation of the total insertion delay
f .

unct1on, TST' from the delay level, To

Angular frequency (in radians per second)

Thesis method of delay equalizer design

vi

I. INTRODUCTION

In the early design of telephone transmission system, only the
effects of amplitude distortion in the transmission of signals were taken
into consideration. The attenuation (the insertion loss characteristic)
within the transmitting band of frequencies was carefully equalized to
approximate a constant characteristic to a specified degree of accuracy.
However, with the advance of long distance telephone systems and with
the demand for improved performance (telephotography, interconnection
of radio broadcasting stations over telephone transmission facilities,
and finally television), certain disturbing effects were noticed in the
transmitted signals. These effects were attributed to insertion phase
distortion (deviation of the insertion phase characteristic from linear
phase with frequency). The problem of insertion phase equalization
appears in literature as early as the 1920's. Thus, a new condition, in
addition to constant insertion loss, was found necessary to impose on
transmission networks, in the transmitting frequency interval, an in-
sertion phase characteristic which is linear with frequency [8, 27, 31, 36].
This condition of linear insertion phase is given by eq. 2. Z. 3 and the
corresponding condition of constant insertion delay, by eq. 2. 5. 3.

, Filter networks have delay characteristics which are substantially
constant over a wide frequency range of the pass-band. - Near the cut-off
frequency (or frequencies) the delay characteristic increases rapidly.
Figures 1. l, l. 2, and 1. 3 depict typical delay characteristics of filters
with dissipation, for the low-pass, band-pass, and high-pass types,
respectively.

The delay of the equalized filter should approximate a constant over

a portion or over the entire pass—band of the filter. Consequently, a delay

 

 

 

 

 

 

 

 

>~
:3
v
Q
Frequency
Figure 1. 1. Typical Delay Characteristic of a
Low-Pass Filter
>.
3.“.
o
O
I Frequency \
Figure 1. 2. Typical Delay Characteristic of a
Band-Pas s - Filter
>.
(d
H
v
O
1 Frequency
Figure 1. 3. Typical Delay Characteristic of a

High-Pas 3 Filter

equalizer is a network which is to be connected in cascade with the net-
work to be equalized. This cascade arrangement is to have a delay
characteristic approximating a constant to a specified degree of accuracy,
throughout a specified interval of equalization. The requirements on the
delay equalizer are given in terms of maximum allowable delay deviation
of the total delay (the delay of the network to be equalized and the equalizer
when connected in tandem) from a constant delay level (see figs. 4. l. l
and 4. 3. Z). The stringency of the requirements on the delay equalizer
depends on the nature of the signals for which the transmission system is
intended. Less stringent requirements are imposed on delay equalization
of networks in telephone systems, for example, than on those networks

in color television systems.

This thesis presents a new method of delay equalizer design which
is based on numerical analysis and use of a digital computer. Although
both facets (numerical analysis and use of digital computer) have been
used extensively in the past, the pattern developed in this thesis combines
their use in a new way, leading to a very effective delay equalizer design

technique.

II. THE DELAY EQUALIZATION PROBLEM

2. 1 Insertion Factor and Insertion Function

The 1nsert1on factor, e s, of a two term1nal-pa1r network 13 defined
as the ratio of the load current, before insertion of the network between
source and load terminations, to the load current after insertion--see

figure 2.. 1. 1. The insertion function, PS, is defined by the equations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

R1 I'2 R1 1:2
—a- ‘— —-
o w , {—00— 1——o 0——l
+ j +.
+ +
V
E v, R; E ZT-PN vz Rz
T oo—- T oo— —0 cu
Figure 2.1.1
' '
P : IZ : v2 1 1
e S 12 V2 (2. . )
I; v'
PS = Ln (——) = Ln (-7-) (2.1. Z)
12 V2

In general the voltage and current variables in the last two equations
are complex quantities, hence both the insertion factor and the insertion
function are complex functions of frequency w. The insertion function

can be written in the rectangular form as

'PszAs + jBS (2.1.3)

where the insertion loss function, AS, is given in nepers and the insertion

phase function, BS, is expressed in radians.

2. 2 Requirements on the Loss and Phase Functions

Figure 2. 2. 1 (a) shows a two terminal-pair network (a filter) and

figure 2. Z. 1 (b) shows a tandem arrangement of a filter and an equalizer.

 

 

 

 

 

 

 

 

 

R1 I1 12
N
+ + 2 F 2 +
E V1 i> (p ) g— V2 R2
T IF
0 o o C
(a)
R1 II 12 13
F0 o—>— —->—o o —>—o
+ t N N
+ Z F 2 + 2; E Z +
E V1 —191 6;?- Vz—g (P 624— V3 Rz
(r (P11?) IE)
0 e o C o o

 

 

 

 

 

 

 

 

 

 

Figure 2. 2.1

Let PSF denote the insertion function of the network NF (the filter)

in figure 2. 2. 1 (a) and let P be the insertion function of the tandem

ST
arrangement of the two networks (the filter and the equalizer) in figure

2. 2. 1 (b). Writing the insertion functions in the rectangular form

:A P 3A

PSF SF + JBSF' ST ST + 3B (2. 2.1)

ST

The network NF is considered to be an existing filter, hence the insertion

function PSF is a known function of the radian frequency w. The equali-

zation interval (ma, cob) on the (.0 axis may be a portion or the entire pass-
band of the filter. The network NE (the phase equalizer) is to be designed

such that the total insertion loss function, A and the total insertion

ST’

 

phas e function, BST’

ment, ASTR’

to any specified degree of accuracy:

AST = ASTR : ASF + k1
- <
such that IAST ASTR I 6:4
_. : k
BST BSTR 203+ 21Tk3

h th t - <
suc a 'BST BSTRI JB

and the total insertion phase requirement, B

STR’

will approximate the total insertion loss require-

respectively,

(2.2.2)

(2.2.3)

for all to in (ma, cob). k, and k; are positive constants, preferably as small

as possible and k3 is any integer or zero [27, 31, 36, 52].

In order to obtain conditions on the properties of the phase equalizer,

NE' such that eqs. 2. 2. 2 and 2. 2. 3 hold over the interval (ma, cob), a

relationship between the functions P

E (the image transfer

function of the equalizer) must be found. A. particularly convenient

relationship would be an expression for P

and PSF'

2. 3 The Relationship Between the Insertion Functions P
and PSF and the Image Transfer Function P

as an explicit function of P

ST

In the view that the final goal of this thesis is directed towards a new

design method of phase equalizers, it is recognized that an involved

relationship between the functions might make the design too cumbersome.

A very simple relationship can be obtained if two conditions are imposed

on the equalizer network. These two conditions and the resulting relation-

ship are stated in the form of a theorem [31, 52]

Theorem: If in figure 2. 2. l (b) the following two conditions

hold for all values of (0

(2.3.1)

then for all values of w, the following relationship is true

PST = PSF + PIE (2. 3. 2)

Proof: The input and output voltage and current variables relation-

ship of the filter network NF in figure 2. 2.1 (a) is given by [33, 42]

 

 

 

 

 

 

 

 

 

 

 

 

 

I v 1 ' 211 l ' l
l .
ZIZ cosh PIF VZIizlz Sinh PIF V2
2 (2. 3. 3)
1 . Z
Il —— Sinh PIF 12 cosh PIF ‘ I;
L l H Zlizlz Zn - L
This last equation can be written in symbolic form as
FV, 1 r A B d P v 1
F F ’-
= (2. 3.4)
[I1 . L CF DF .[ 12
In addition, from figure 2. 2. 1 (a)
V2 : R212 (2. 3. 5)
V1 : E " R111 (2.3.6)

From eqs. 2.3.4, 2.3.5, and 2.3.6, after eliminating V1, I1, and V2, a

relation for 12 in terms of E is given by

 

E
12 = (2. 3. 7)
CFRle + AFR; + DFRI + BF

On the other hand, before insertion of the network NF

I E

: —— 2. 3.

12 R1 + Rz ( 8)
Therefore, the ratio which determines the insertion function, PSF' is

ePSF : Eli : CFRIRz + AFRZ + DFRI + BF (2. 3. 9)

2

 

R1+Rz

For convenience, the normalized image impedances with respect to the
terminating resistances are defined

11 I
Z Z 2 3 0
“ —_.. O . 1

 

21:

Substituting the values of AF, BF’ CF’

2. 3. 9 and eq. 2. 3. 10 into the result, the insertion factor is obtained [42]

and DF from eq. 2. 3. 3 into eq.

P VRIRZ' 1 + 2122 . Z1+ Z2
SF = ——-—— W— smh P + cosh P 2. 3.11

Next, the insertion factor, ePST, for the tandem arrangement of the
filter and equalizer in figure 2. 2. 1 (b) is derived. In addition to the

terminal equations for the filter network N , given by eq. 2. 3. 3 and in

F
addition to eqs. 2. 3. 5 and 2. 3. 6, there are the terminal equations for the

equalizer network NE, given by [33, 42]

 

 

 

 

 

 

 

' 1 t 21 . ‘ ‘ 1
V; g cosh PIE [ZI3ZI4 Sinh PIE V3
= (2. 3. 12)
1 . 21
I2 —— Sinh P 4 cosh P 13
L - LVZI3ZI4 IE ZI3 IE‘ _ .-
This last equation can be written in symbolic form as
- 7 r- q — T
V; AE BE V3
= (2. 3. 13)
L12. (CE DE._13_

 

 

 

 

 

 

From eqs. 2. 3. 3, 2.3. 5, 2. 3.6, and 2. 3.13, after eliminating V1, I1,
V2, 12, and V3, a relationship for 13 in terms of E is given by

E

2. 3. 14
ARIRZ + BR; + CR1 + D ( )

 

I3:

h :
w ere A CFAE + DFCE
B = AFAE + BFCE
C = CFBE + DFDE

D :

AFBE + BFDE

On the other hand, before insertion of the NF and NE networks

v E

: _ 2. .1
13 R1 + R; ( 3 5)
Therefore, the ratio which determines the insertion function, PST’ is
given by
V
PST ___ __I_3_ : ARIRZ + BRZ + CR1+ D (2. 3.16)

 

Again, for convenience the normalized image impedances are defined

as follows

 

Z Z Z
11 I I4

Substituting the values of A , B C D A B C and D into

 

 

 

 

F F’ F’ F' E’ E’ E’ E
eq. 2. 3.16 and eq. 2. 3.17 into the result, and insertion factor, ePST, is
obtained
'
ePST z _.1_3.__ :
I
3
VRIRZ 1+ ZIZZZ} . Z3 + 2122 .
= —-—-— h P h nh
R1+R2 [Vzlzzz3‘ 5m IF (“33 PIE + 1 + 2,222, 8‘ PIE)
+ 21+ ZggL cosh P (cosh P + 22 + ZIZL Sinh P ) (2.3.18)
VZ—IZ'Z‘ IF IE Zl + 222;, IE

The expressions for the insertion factors, ePSF and ePST, are given by

eqs. 2. 3. 11 and 2. 3. 18, re3pectively. 2 From the hypothesis of the theorem

ZI3 : ZI4 : R2 (2. 3.19)

10

Substituting eq. 2. 3. 19 into eq. 2. 3. 17, the normalized image impedances

become
211 Z12
1 R1 Zz R2 Z3 1 3 0)

In this last equation, the normalized image impedances Zl and Zz are
identical with those given by eq. 2. 3. 10. Substituting eq. 2. 3. 20 into

eq. 2. 3. 18, the expression for the insertion factor, ePST, becomes

'!

P .
e ST: @1— [L'L—ZJZJL sinh P + _Z__1:_Z_z_ cosh PIF] ePIE

R1 + R; W; 1F 2122
(2. 3. 21)
Now, from eq. 2. 3. 11 and from this last equation
ePST = ePSF-ePIE , (2.3.22)
hence 1
P = P + P 1 (2.3.23

ST SF IE

This proves the theorem.

The theorem states that if the equalizer network is a symmetric,
constant resistance network, with the resistance level equal to the receiv-
ing (load) resistance, then the insertion function of the tandem arrange-
ment of a filter and an equalizer is given by the sum of the insertion
function of the filter and the image transfer function of the equalizer.

But under these restrictions on the equalizer, the insertion function of
the equalizer, defined with respect to an R2 load termination, is equal to
the image transfer function of the equalizer. Then eq. 2. 3. 23 may also
be written as

PST = PSF + PSE (2.3.24)

These three insertion functions are complex functions of frequency.

Equating the real and imaginary parts yields

11

AST : ASF + ASE (2.3.25)

BST = BSF + BSE (2. 3.26)

From eq. 2. 2. 2 and 2. 3. 25
A : é ° '
SE AIE k1 (2 3 27)
where k1 is to be made as small as possible. Theoretically, k, can be ]
made zero for all values of frequency if an additional restriction is imposed
on the equalizer network, namely, restricting the equalizer to a lossless

LC structure. Under such restriction

.3":
C
ASE = AIE = 0 for all frequenc1es (2. 3. 28)
and then
AST = ASF for all frequenc1es (2. 3. 29)

Thus, the first requirement on the equalizer, given by eq. 2. 2. 2, is
satisfied by restricting the equalizer to a class of symmetric, constant
resistance, all pass networks, with the resistance level equal to the
receiving resistance [27, 31, 36, 52].

From the second requirement on the equalizer, given by eq. 2. 2. 3,
and from eq. 2. 3. 26

E - < < . .
ESE kzw + ZTTk3 BSF wa_w_wb (2 3 30)

where k; is to be as small as possible and k3 is any integer or zero.
The relation 2. 3. 30 states that in the frequency interval (ma, oh) the
insertion phase of the equalizer must approximate a function which is
complementary to the insertion phase function of the filter with respect

to a straight line of slope k; and intercept 21rk3 [27, 31, 36, 52].

12

2.4 The Insertion Delay Function

The insertion delay function, Ts(w), used throughout this thesis is

defined

 

TS(oo) = Bs(w) (2.4.1)

where B (on) is the insertion phase function of the network in question and
s
is expressed in radians. Ts(w) is the insertion delay function, has units

of time and is expressed in seconds. §~ ‘

2. 5 Requirements on the Insertion Delay Function

The requirement on the insertion phase function, given by eq. E

2. 3. 20, can be written in terms of the delay function as follows

d ' . ‘
—d—JBSE(6) _ k2 - TSFQ»), wager: cob (2.5.1)

TSEM :

The relation 2. 5. 1 states that in the frequency interval (wa, cob) the in-
sertion delay function of the equalizer must approximate a function which
is complementary to the insertion delay function of the filter with respect
to a constant kg. The constant k; is commonly referred to as the delay
level and is denoted by To. The right side of eq. 2. 5. 1 is called the

equalizer delay requirement and is denoted by T The equalizer delay

SER'
requirement then becomes

: - < < . .2
TSER To TSF ”at—w—“b (25 )

If a given filter is to be delay equalized in the frequency interval

(ma, cub), the insertion delay of the filter, T is a known function of

frequency. The insertion delay of the filtersi‘an be measured on a test

set, if the actual filter is available, or the delay function can be calculated
on a digital computer, if only the schematic diagram and element values

of the filter network are at hand. . Hence, from eq. 2. 5. 2, given any filter,

the equalizer delay requirement, T is known within the delay level To.

SER’

13

In practice, the requirements on the total delay, T T’ (the delay

5
of the filter in tandem with the equalizer) are given in terms of the maxi-
mum allowable delay deviation, 6T’ from a constant value (from the

delay level To). Stated mathematically

I< 6‘ (2.5.3)

1T -TOI=IT T

ST ToI : 'T

SE+TSF- SE-TSER

form <to<u>
a— —

b

2. 6 The Insertion Delay Function for Matched Cascaded
Lattice Sections
From the theorem of section 2. 3 and from eq. 2. 3. 28, it follows I
that, if the class of networks considered for the delay equalizer is limited
to the class of constant resistance, all pass networks, with the resistance
level equal to the receiving resistance, R2, of the filter, then the additive
property of the insertion functions of the filter and the equalizer is pre-
served and the insertion loss of the equalizer is zero.
The fundamental building block of a delay equalizer is a lattice,
constant resistance, 360 degree, all pass section, shown in figure 2. 6.1

[27,31,34,36,52L

 

 

 

 

 

 

 

 

   

Figure 2. 6.1

The series arm and the cross arm impedances of this section are inverse

with respect to the terminating resistance, R;

14

z z =R§ (2.6.1)

_ = __ = R§ (2.6.2)

When this sectionis terminated at one port by the resistance, R2, the
input impedance at the other port is R2, for all values of frequency.

- Although the section contains eight elements, only four elements
are distinct. Two additional restrictions are imposed on the four element 1‘ J
values, L1, L2, C1, and C2: by eq. 2. 6. 2,. so that one such section con-
tains only two independent parameters. The insertion function with
respect to R2 terminations and the image transfer function are identical

and are both pure imaginary. For all values of frequency
Asks) = AI (to) = O (2.6. 3)

Bs(w) : BI (co) = 2 arccot V—t—‘l—l: 1 - w VLICI ] (2.6.4)
1

w VL1C1|

 

The insertion delay function of the section is obtained by differentiating

this last equation with respect to w

L 1
2 V—é— 1 + T—
L1 0.) LLCI

 

 

 

 

 

Ts(w) = 1 L 1 ’72 (2.6.5)
__ 1 + _7;. [ j - wv LlCl ]
leCl Ll Q) VLICI
In these last two equations, let
L3 1 l
d = and f = = . .
L. o 2.1/no; 211 V1.30; (2° 6' 6’

The insertion phase and delay functions for one section then become

 

f f
BS(f) = 2 arccot (d) (79- - 72—) (2. 6.7)
_ d _ 1+ (f /£)Z
TS(f) — Tf— - 0 (2.6.8)

2 Z
0 1 + d (f/fo - fo/f)

15

where

fo is the value of frequency at which the insertion phase function,

Bs(f), takes on the value of TI' radians; d is a dimesionless constant;

Ts(f), the insertion delay function has units of time.

If the impedance level of each section is fixed at the value of the
terminating resistance, R2, then the sections are image impedance matched.
Consequently, the insertion function of a cascade arrangement of such
sections is equal to the sum of the insertion functions of the individual
sections. Then, the insertion delay function of a delay equalizer, com— ._ “
posed of 11 sections connected in cascade, is given by

f
0k 2 . .
dk . 1~1~(—f ) [

(2.6.9)

 

 

n
TSE(f) = Z

klnfok 1+d2 f fokz
kf ”f

ok

2. 7 The Design Problem

The design problem is to approximate the equalizer delay require-

ment, T , given by eq. 2. 5. 2, by a finite sum, given in the last

SER
equation.

Stated mathematically

Given the insertion delay function of a filter, T F(f), and the

S
maximum allowable delay deviation, 6T > 0, determine the positive

constants To’ d fok’ (k=1, 2, 3, - - - , n) and the positive integer n, such

k!
that

ITSE(f) + TSF(f) - To! < cf (2.7.1)

T

with faZOand f >0

b

forall f <f<f,
a— — b

III. DELAY EQUALIZER DESIGN BY NUMERICAL METHODS

3. 1 The Design Problem in Terms of a Solution to a System of
Non—linear Algebraic Equations

If an estimate on the number of sections, 11, of the delay equalizer
is assumed, i. e. , n is taken as known, then the design problem can be
formulated in terms of a system of non-linear algebraic equations.
A solution to such a system of equations is obtained by numerical tech-
niques in conjunction with the use of a digital computer.

The insertion delay function of a delay equalizer of n sections,
given by eq. 2. 6. 9, in terms of normalized frequency, 2, with respect
to some arbitrary frequency, fd (fd 7’ 0), and in terms of a new set of

parameters, the x and y parameters, is

2
1 n zxk+yk

 

T (2).: —_ Z 2 z 2 (3.1.1)
SE 1'de k=1 2 + (z xk - yk)
where
f f
z=—f—, xzd—d— , y: d—O—- (3.1.2)
f f f
d o d

The problem is to approximate the equalizer delay requirement,

, given by eq. 2. 5. 2, by T (z), in the equalization interval (za, zb),

TSER SE

within the maximum allowable delay deviation, (IT:

TSE(z) 5: T - TSF(z) (3.1.3)

0

This last relation contains 2n+1 unknowns: To’ xk, and yk
(kzl, 2, 3- - - , n). A system of 2n+1 equations is formed from the relation
3.1. 3, one equation for each distinct value of zi (i=1, 2, 3, - - - , 2n+l),

2i 7! z.j for i 7! j, and is given by

TSE(zi’ xk, yk) - To + TSF(zi) = 0 (3.1.4)

16

17

This system of 2n+1 equations is reduced to a system of Zn equations,

in the unknowns xk and yk, by eliminating the unknown To. An explicit
expression for To is obtained from that equation of the set 3. 1. 4, corres-
ponding to zi = 1 (fi = fd) and then this expression is substituted into the

remaining set of Zn equations, yielding
TSE(zi’ xk, yk) - TSE(1,xk, yk) - TSF(1) + TSF(zi) = 0 (3.1.5)

where

ziflzj, foriylj (i=1,2,3,'--,2n) and 21}! l for all i.

In further detail, this last system of equations becomes

 

 

 

1 n ?-
11f E zlxk+yk xk+yk )1; (1)+r (z)-o
d = 2 z _ 2 " _ 2 " - ‘
k 1 z1 + (zixk yk) 1 + (fl yk) SF SF 1
(3.1.6)
which on multiplying by wfd and letting
Ad 2 TrdeSF(1), A1 = wdeSF(zi)
_ z _
1311(2th Vk' Bk‘xk+yk
(3.1.7)
: 2 .. = -
Cik zixk Yk ’ Ck xk .. yk
_ 2 _. 2
Dik—1+Cik, Dk—1+Ck
yields
B B
n o
2: (le - 55—) - Ad+Ai=O (3.1.8)
k=1 ik k
which is of the general form
Fi(X)= 0 (i=1,2,3,~--,2n) (3.1.9)

where

FI(X) : Fi(x19xZ:x3i...:xniYI’YZ’Y3’...7Y )

18

If a solution to this last system of Zn equations is obtained exactly,

then the approximation of the equalizer delay requirement, by the

TSER'
insertion delay function of the equalizer, TSE(z), is exact at 2n+1 points,

(at the Zn points, zi, and at z = 1).

3. 2 A Numerical Method for a Solution of a System of
Non-linear Algebraic Equations

A number of numerical methods for a solution of a system of non-
linear algebraic equations appears in literature on numerical analysis
[22, 23, 26]. One class of numerical methods, such as the method of
steepest descent, the method of Seidel, or the method of relaxation, re-
duces the problem to that of solving a sequence of single non-linear alge-
braic equations. The class of functional iteration methods such as the
Newton-Raphson method, reduces the problem to that of solving a sequence
of systems of linear algebraic equations.

The method of Seidel is here used to obtain a solution of the system
of Zn equations, given by eq. 3. 1. 8. A brief presentation of this method
follows.

Consider a system of m non-linear algebraic equations

Fi(X)= O (i=1,2,3,°°',m) (3.2.1)
where

Fi(X) = Fi(x1, x2, x3, - - - , x )

m
Define the function M(X) as

M(X) = :2n F?(X) (3.2.2)
121 1
If the functions Fi(X) are real functions of real variables (x1, x2, x3, . - . , xm),
then the function M(X) is either positive or equal to zero, and will be
equal to zero only if X is a solution of eq. 3. 2. 1. Therefore, a solution
of eq. 3. 2. 1 is obtained by finding a value of X at which the function M(X)
has an absolute minimum. The method of Seidel is a technique which

minimizes the function M(X) by correcting one variable at a time in a

fixed order.

19

If a set of initial estimates
0 o o 0
X0 - (x1,xz,x3, ---,xm)
is sufficiently close to a solution X of eq. 3. 2. 1, then an improved value,
x], of the first variable, x10, is obtained by the Seidel method, such that
M(X1)< M(XO)’

o o o
where X1: (x[,xz,x3,-o-,xm)-

An improved value, x], for the second variable, x2, is obtained such that
M(Xz) < M(Xi).

o o
where X2 = (XL xi, x3 , . - - , xm).

When all m variables have been corrected once, the set of m cycles is
called the Seidel cycle and the process is repeated. Let the letter a denote

the number of Seidel cycles, (a=0, 1, 2, 3, - - - ). Then, an improved value,

am (a-1)m

xj , for the jth variable, xj , is obtained such that

M(X .) < M(

am+ j )

X(a-—1)m+j

The improved values of the variables are found in the following

manner. To find an improved value, x§a+1)m, for the jth variable, Xi'm,

. . . . . .th .
the function M(X) is differentiated With reSpect to the J variable, all
other variables are considered constant, given by their current estimates.
The derivative of M(X) is equated to zero and a solution of this resulting

equation for the jtlr1 variable, x:m, yields the improved value, x(a+1)m

.2— M(x .
a xam am+j

J

)=0 (3.2.3)

This last equation is ordinarily a non-linear equation in xém and
some method of successive approximation for an equation in one variable
must be applied. In the problem of this thesis, a single application of
the Newton-Raphson method was selected to obtain an approximate solu-

tion for x§a+1)m

20

 

 

 

 

3
M(X .)
3 am am+j
x§a+11m :m J (3. 2.4)
a 2
am M()(am+-')
(a )2 .1
which when written in terms of the Fi(X) functions become
2 Fil‘xamfij). 3x am Fi( am+j)
X(a~l-1)m_ am 121 ' '
j J m 3 3 97‘
Z (3 am F1(Xam+j)) + Fi(Xam+j)-__a-r‘;2_ Fi(xam+j)
i=1 x, (8x, )
J J
(3. 2. 5) vi

This last equation is written for a general system of m non-linear
algebraic equations. The corresponding equation for the Special problem
at hand is obtained by replacing m by 2n and by taking the functions Fi(X)
as defined by eqs. 3.1.9, 3.1.8, 3.1.7, and 3.1.6.

3. 3 The Digital Computer Programs

Two computer programs were written in connection with this prob-
lem of delay equalization. Both programs are general in the sense that
they can be utilized for any delay equalizer design, that is, they are
written in terms of the parameter n (the number of sections of the delay

equalizer).

Program 1

 

This program computes a solution to the system of Zn equations,
given by eq. 3. 1. 8, and employs the numerical methods of Seidel and
Newton-Raphson, as described in section 3. 2. The input data for this

program are as follows:

21

n, the number of sections of the delay equalizer.
(As here exemplified, n_<_ 10, due to limited storage capacity
of the Mistic computer);

x0, y]: (k=1, 2, 3, - - - , n), the initial estimates for a solution of

k
eq. 3. 1. 8. (These are the initial estimates for the
equalizer design parameters--see eq. 3.1. 2);
fd, the normalizing frequency;

TSF“ the value of the insertion delay of the filter at the

d),
frequency fd;
fi(i=1, 2, 3, - - - , Zn), The 2n frequencies from which the system of
Zn equations is formed, (fi 7! fd for all i);
TSF(fi), the values of the insertion delay of the filter at each
frequency fi;
JT’ the maximum allowable total delay deviation.
The units of frequency are in cycles per second and the units of delay are
in seconds. ‘
The computer prints out all xim, yém, and all Fi(xam+j) after
each Seidel cycle. The approximation to a solution of eq. 3. 1.8 is con-

side red complete when

)(< (5 for alli (3.3.1)

1
T
nle.(X T

1 am + j

and the computer automatically stops when this inequality is satisfied.
. Criteria for the choice of the input data are given in chapters V
and VI.
A simplified flow diagram of program 1 is presented in figure

3. 3. 1. 3 Each block in the diagram lists the quantities which are calcu-
lated by the various subroutines. The notation used in figure 3. 3. 1 is
consistent with the notation given in eqs. 3.1.6, 3.1. 7, 3.1.8, and 3. 2. 5.
The number of times each subroutine is entered during the calculation
corre5ponding to one Seidel cycle is indicated on the diagram in terms

of the parameter n.

22

*Program 2

 

This program computes the sum of the insertion delay of the filter
and the equalizer, at specified frequencies fp. The input data for this
program are as follows:

fp (p= 1, 2, 3, - - - , r _<_ 40), the frequencies at which the total delay

is to be calculated;

TSF(fp), the insertion delay of the filter at each frequency fp;

n, the number of sections of the delay equalizer;

xk, yk, the delay equalizer design parameters, (see eq. 3.1. 2). .

These parameters are obtained by program 1 as an approxi-
mation to a solution of eq. 3. 1. 8.
All frequency values are given in cycles per second and all delay values
in seconds.

The insertion delay of the equalizer is computed from eq. 3. 1. 1
and added to the insertion delay of the filter.

The format of the output data of program 2 is printed in two
columns. The left column with a heading "FREQ. " lists the frequencies
f and the right-hand column headed ”TOTAL DELAY" gives the values
01; the sum of the insertion delay of the filter and the equalizer at the

corresponding frequencies .

 

 

23

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3. 3.1.

 

1 2
INPUT DATA > zi, zi, Ai’ Ad
\Zn-l 1 in 2n(2n- 1)
/ \ / ‘
V 2n V4nZ
Bk’ Ck’ Dk ik’ Cik’ ik
n n
E 131/13k + A d z Bik/Dik 4. Ai
kgl =1
\ 2n 4n2
/n K \ \an
K \/ n ZnZ / 7
a n a TI
—— B /D z —— E B. /D.
3y k=1 k k \n 2n/ ayk-l 1k 1k
/ \
2 2 n a z n
Way >3 Bk/Dk 7y 2 Blk/le
k-1 k=l
/\ v n /\ V n V /\
a a
8x 2 Bk/Dk 2x 2 Bik/D'k
k=1 :1
> <
a 2 n 2 2 2 n
'23.! E Bk/Dk n n 3:2 E B k/Dik
k=1 4n-’- \/ k=l
_ a a Z 2117- 2n2 a 32
L73? F1111” 2x2 F101) 4 F101) > 7717 Fi(X). T552 Fi(X)
2 2
2n \/ . \n(2n-l) 2n \/ \2n(2n-1)
/ /
n V n l
n a n 2
2 F1 3 x F1 2 F. T F.
_ k=1 k=1 1 Y 1
n n
a z a z a 1 a z
kéllFi—a—szitlaxF.” 13111327 Ffi‘ay Fa”
‘- xam - am
1 1'3
' / V n
, J 211-]. \ n 111-]. _ l
\ \ -
1/ NO CONVERGENCE TEST IPRINTS OUT ALL
1 \ 1 < am am
<
ESTOP YES nfleil (IT 1 xj , yj , Fi(xam+j)

 

Flow Diagram of Program 1

IV. SOLUTIONS TO TYPICAL EQUALIZATION PROBLEMS
USING THE METHOD OF THIS THESIS

In this chapter, the thesis method of delay equalizer design, here-
after referred to as the t-method for brevity, is applied to a commercial
problem of very exceptionally stringent requirements. The quality of
delay equalization attained by this new method is demonstrated by means
of comparison with a commercial solution to the problem. This com-
mercial solution, obtained by a method described in chapter VI, was
considered to be a very good solution. ‘ In fact, it was agreed that a possible
improvement of the commercial delay characteristic, if any, would not

justify the extensive amount of time involved.

4. 1 The Commercial Problem

The delay characteristic of the filter to be equalized as well as the
upper and lower limits of the equalized delay are shown in figure 4. 1. 1.
The delay equalization interval, in this problem, extends from zero to
7. 2 megacycles. The requirements on the total delay (the sum of the delay
of the filter and the delay of the equalizer), in terms of the maximum

allowable delay deviation, are given by

O. 0 - 5. 0 megacycles :t 2 millimicroseconds
5. O - 5. 5 megacycles :1: 4 millimicroseconds
5. 5 - 6. O megacycles :1: 8 millimicroseconds
6. 0 - 7. 2 megacycles :1: l6 millimicroseconds

This set of requirements is drawn at an arbitrary delay level and

is shown in figure 4. 1. 1 to indicate the stringency of the requirements.

24

 

26

4. 2 The Commercial Solution

The commercial solution to this design problem is given in figure
4. 2. 1, in terms of the total delay characteristic (the sum of the delay
characteristics of the filter and the commercially designed equalizer) .
This commercially designed equalizer consists of six lattice sections
of the form shown in figure 2. 6. 1, connected in cascade. From figure
4. 2. 1, it is evident that the commercial solution fails to satisfy the
requirements on the total delay in the neighborhoods of 5. 5 and 5. 75 mega-
cycles. Because of the difficulty and time consuming nature of the
commercial delay equalizer design technique, this failure to meet the

requirements was accepted as the best that could be done.

4. 3 Solution by the t-Method

In this section, several solutions to the design problem as obtained
by the t-method are presented and the procedure followed in each case
is described. Program 1 (section 3. 3) was used to calculate the equalizer

design parameters, x , yk, given by eq. 3.1. 2, and program 2 (section

k
3. 3) was employed to calculate the total delay characteristic.

Solution 1

 

The first test for the t-method was an investigation of the possibility
of improving the commercial solution.

In this case, the input data for program 1 were taken as follows:

The number of sections, n, of the delay equalizer was taken the
same as for the commercial design (11 =‘ 6).

The initial estimates, xi, yi, for the design parameters of the
equalizer were taken as those of the commercial design.

The normalizing frequency, f , was chosen as 2 megacycles. The

d
normalizing frequency is taken as any convenient number in the frequency

interval where the delay of the filter, T , undergoes small variation.

SF

r! in? .lL

 

28

The 2n frequencies, fi, (12 in this case) were selected as follows:
In the interval 1. O to 5. 5 megacycles, nine frequency points were chosen
at uniform intervals of O. 5 megacycles. (Note that the 2. O megacycle
point has already been selected by the choice of the normalizing frequency,
fd). In addition, the 0. 25 megacycle point was chosen. The remaining
two frequency points were selected on the basis of the behavior of the
commercial delay characteristic (fig. 4. 2. l) in the 5. 75 to 7. 2 megacycle
interval. The points 5. 75 and 7. 1 megacycles were chosen in expectation
of improving the delay characteristic at these points.

In this trial, it was decided to apply the Mistic computer to program
1 for about 30 minutes. In order to prevent the computer from stopping
short of this time interval, a very small maximum allowable delay devi—
ation, J , was chosen for the stop order criterion in program 1 (see eq.

T
3. 3. 1), namely: (1 = O. 25 millimicroseconds. The actual computer

time was 25 minutes? in which period 22 Seidel cycles were completed.
Program 2 was utilized to compute the total delay characteristic, at 0. 25
megacycle intervals, on the basis of the design parameters of the 22nd
Seidel cycle. The result is shown plotted in figure 4. 3. 1. The points at
which the delay characteristic was calculated are indicated on the plot by
circles. The delay values corresponding to the frequency points fi and

the normalizing frequency f are identified by double circles.

d
A comparison of figures 4. 2.1 and 4. 3.1 clearly reveals that the

application of the t-method produced a considerable improvement over

the commercial delay characteristic. The new delay characteristic not

only satisfies the original stringent requirements, but actually satisfies a

set of more stringent requirements, given by

O. 00 - 5. 75 megacycles :t 1. 70 millimicroseconds
5. 75 - 6. 00 megacycles :l: 4. 00 millimicroseconds
6. 00 - 6. 50 megacycles :1: 10. 00 millimicroseconds

6. 50 — 7. 20 megacycles :1: 12. 00 millimicroseconds

.‘u..L_

 

30

Solution 2

 

In this solution, an improvement over the delay characteristic
obtained by solution 1 (fig. 4. 3. 1) is sought by selecting a different set of
the fi frequency points. ¢The following basis is used for the choice of the
new set of points from the set of points of the previous solution: Those fi
points at which the total delay characteristic has neither a relative
maximum or minimum are deleted. Those frequency points at which the
total delay characteristic has a relative maximum or minimum or at which
a delay improvement is desired are included in the new set of f1 points.
Since the number of the fi frequency points is fixed, for a fixed 11, the
number of added points is equal to the number of deleted points.

From the total delay characteristic of solution 1, shown in figure
4. 3. 1, and on the basis of the criterion, given in the last paragraph, a
new set of the fi frequency points was obtained by deleting the points at
2. 5 and 3. 5 megacycles, from the set of the fi frequency points of solution
1, and by adding the points at 4. 25 and 6. O megacycles.

The initial estimates, xi, y]: were taken as those obtained by
solution 1.

All other input data are as those used in solution 1.

The computer was applied to program 1 for about one hour, in which
time 52 Seidel cycles were completed. On the basis of the design para-
meters of the 52nd Seidel cycle, the total delay characteristic was calcu-
lated by program 2 and is shown in figure 4. 3. Z.

‘ An improvement of this total delay characteristic (fig. 4. 3. 2),
particularly at the 6. 0 megacycle point, over that of solution 1 (fig. 4. 3. 1)
should be noted. Solution 2 (fig. 4. 3. 2) is also well within the original

requirements and is actually confined within a set of requirements given by

O. 00 - 5. 50 megacycles :h 1. 5 millimicroseconds
5. 50 - 6. 00 megacycles i 3. O millimicroseconds
6. 00 — 6. 75 megacycles :I: 8. 5 millimicroseconds
6. 75 - 7. 20 megacycles i 13. 5 millimicroseconds

 

32

which is a better overall delay characteristic than solution 1 of figure

4.3.1.

Solution 3

 

The success of solutions 1 and 2 lead into an investigation of the
possibility to satisfy the original requirements with a delay equalizer

consisting of five sections rather than six.

 

In this case, the input data for program 1 were taken as follows:
The number of sections, 11, was taken as five (n = 5). j.
The initial estimates, xi, y}: were obtained by the method described .
in chapter VI. The total delay based on these initial estimates was calcu- . 4
lated by program 2 and is shown in figure 4. 3. 3.
The frequency points fi were chosen in accordance with criterion
4. 5. 2, given in section 5 of this chapter, and are indicated in figure 4. 3. 4.
The computer was applied to program 1 to compute 6O Seidel cycles.
The total delay characteristic as calculated by program 2, on the basis of
the design parameters of the 60th Seidel cycle, is shown in figure 4. 3.4.
InSpection of this figure reveals that the delay characteristic does not
satisfy the original requirements in the neighborhoods of 6. O and 6. 5
megacycles. At the same time, it is noticed that none of the fi frequency
points were chosen in the interval 5. 5 to 7. 1 megacycles. To improve
this delay characteristic, a new set of the fi frequency points was chosen
according to criterion 4. 5. 2 and on the basis of the obtained delay
characteristic (fig. 4. 3. 4). The design parameters of the 60th Seidel
cycle were taken for the initial estimates of this run. After the application
of one Seidel cycle, the total delay was calculated and is shown in figure
4. 3. 5. From this figure it is seen that the original stringent requirements
are satisfied by a delay equalizer of only five sections.
Solutions 1 and 2 (figs. 4. 3. 1 and 4. 3. 2) clearly demonstrate that
for a delay equalizer of the same number of sections, the original delay

requirements are fully met. The quality of delay equalization obtained by

 

a...

1. tan
, .

34

 

 

36

the t-method exceeds that of the accepted commercial design (fig. 4. 2. l),
which actually did not fully meet the original requirements. Solution 3
(fig. 4. 3. 5) shows that the original requirements can indeed‘be met not
only with an equalizer of six sections, but with an equalizer of one less
section than the commercial one, i. e. , with five sections.

Another important feature of the t-method should be noted. The total
delay characteristic (fig. 4. 3. 3), based on the initial estimates, xi, yi,
varies from 540 to 586 millimicroseconds, in the frequency interval zero
to five megacycles. This corresponds to a delay deviation of about i 23
millimicroseconds. In the same frequency interval, the final result (fig.
4. 3. 5) is confined within :h 2 millimicroseconds. This example demon-
strates that the t-method converged, even though the initial estimates were

poorly chosen.

4. 4 Another Example of Delay Equalization by the t-Method

The original stringent requirements, given in section 4. l, are for
delay equalization of a loss filter in a very special, very high quality
transmission system. A set of delay requirements for delay equalization
of the same loss filter in a transmission system for color television is

the very much relaxed requirement schedule:

0. 0 - 5. O megacycles :1: 2 millimicroseconds
5. 0 - 5. 5 megacycles d: 4 millimicroseconds
5. 5 - 6. 0 megacycles :l: 8 millimicroseconds

with no requirements on the total delay for frequencies greater than six

megacycles .

Solution A

 

In this solution, the input data for program. 1 were taken as follows:
The number of sections, n, was taken as four (11 = 4). 1 No attempt
was made here to arrive at an estimate on the number of sections. Since

an equalizer of five sections satisfied the requirements in the interval

37

zero to 7. 2 megacycles (solution 3, fig. 4. 3. 5), an equalizer of fewer
sections can be expected to effect the equalization in the interval zero

to six megacycles. A few minutes of computer time determines whether
the number of equalizer sections can be reduced still further.

The initial estimates, 12:, y}: were obtained by the method given
in chapter VI. The total delay on the basis of these initial estimates,
as calculated by program 2, is shown in figure 4. 4. 1.

The frequency points fi were chosen according to criterion 4. 5. 2
and are indicated in figure 4. 4. 2.

‘ In 30 minutes of computer time, 68 Seidel cycles were completed.
The total delay characteristic on the basis of the design parameters of
the 68th Seidel cycle is shown in figure 4.4. 2. InSpection of this figure
shows that this solution satisfies a set of considerably more stringent
requirements than those for color television, namely, a set given by

0. O - 6. 0 megacycles :1: 1.1 millimicroseconds

Solution B

 

The outcome of the previous solution (fig. 4.4. 2) suggests that an
equalizer of three sections might meet the requirements for color tele-
vision.

In this solution, the input data for program 1 are as follows:

The number of sections, n, was taken as three (n == 3).

. . . . 0 . .
The initial estimates, x were obtained by the method described

0
k9 yk!
in chapter VI. The total delay characteristic in terms of these initial
estimates, as calculated by program 2, is shown in figure 4. 4. 3.

The frequency points f, were chosen according to criterion 4. 5. 2

i

and are indicated in figure 4.4.4.

In this case, 110 Seidel cycles were completed in about 30 minutes.
The total delay characteristic on the basis of the design parameters of

the 110th Seidel cycle is shown in figure 4.4.4. It is seen from this

figure that the color television requirements are indeed satisfied by a

 

 

 

 

42

delay equalizer of three sections. A selection of a new set of the fi fre-
quency points (see criterion 4. 5. 2) and an application of a few additional
Seidel cycles might still further improve this delay characteristic. This
step was not carried out in this particular example.

The delay equalization of the given filter, satisfying the exceptionally
stringent requirements (section 4. 1) was effected by a delay equalizer of

five sections (fig. 4. 3. 5), while the color television requirements (section

 

4.4) were met by an equalizer of three sections (fig. 4.4.4). This is to be

 

contrasted with the commercially designed delay equalizer, consisting of

six sections (fig. 4. 2. 1), which was accepted as a solution to the delay

 

equalization problem for both sets of requirements, in Spite of the fact

that this commercial design did not fully meet either set of the requirements.

4. 5 Criteria for the Choice of Input Data for Program 1

In this section, a discussion on the choice of input data for program 1
is presented and the discussions are summarized in forms of criteria. The
estimate on the number of sections, n, of a delay equalizer is treated in
chapter V and a method for the selection of the initial estimates, x12, y]:

for the design parameters is presented in chapter VI.

Criterion 4. 5. l. The normalizing frequency, fd, is chosen as any

convenient number in that frequency interval where the delay of the filter,

 

TSF’ undergoes small variation.

Criterion 4. 5. 2. The 2n frequency points, fi (fi ;/ fd for all i), are

 

chosen as follows:
(a) For the first application of the t-method, the fi frequency points

are selected such that the set of the Zn points fi and the point fd are

uniformly spaced in the equalization interval. (Although this choice is
always theoretically possible, the followingpractical difficulties arise:
The t-method requires the functional values of the delay of the filter,

T If the delay of the filter was measured

SF’ at the frequenc1es fi and fd.

43

or calculated beforehand, at a set of frequencies, fq, the choice of the
frequencies f1 and the frequency fd must be made to coincide with a subset
of the fq frequencies in order to avoid measurement or calculation of the
delay of the filter at additional frequencies. Then, in the case where

uniform spacing of the fi and f frequencies would lead to a set of points

d
not coincident with a subset of the fq points, smaller subintervals are
selected in that frequency region where the delay requirements are most
stringent).

(b) An improvement over the total delay characteristic obtained by
the first application of the t-method is often effected by a choice of a new
set of the fi frequencies. This new set of the fi frequency points is chosen
on the basis of the previous set of the fi points and from the behavior of
the total delay characteristic of the first application of the t-method, as
follows: Those fi points in which neighborhoods the total delay character-
istic has neither a relative maximum nor minimum are deleted. Those
fi frequency points in which neighborhoods the total delay characteristic
has a relative maximum or minimum, or at which point a delay improve-
ment is desired, are included in the new set of the fi points. Since the
number of the fi frequencies is fixed, for a fixed 11, the number of added
points is equal to the number of deleted points.

The t-method of delay equalizer design was applied to a number of
additional problems over those presented in this chapter, in an attempt
to arrive at some convergence criterion in terms of the initial estimates,
ii, Vi. It was found that in all cases considered, an improvement of the
delay characteristic over that of the characteristic based on the initial
estimates was effected. However, the success of the t-method depends
not only on the improvement of the delay characteristic, but one additional
condition must be met, namely, that all of the final design parameters,
xk, yk, must be positive numbers. This is a necessary condition for the
realizability of the equalizer network (see chapter VII). Although positive

. . . . 0
initial estimates, x

0 . . .
k’ yk, were used in all cases, it was noted in a few

44

instances that if after the first Seidel cycle one or more of the corrected
design parameters took on negative values, invariably these parameters
remained negative for all subsequent Seidel cycles. On the basis of this

investigation, the following criterion is stated:

Criterion 4. 5. 3. If after the application of one or a few Seidel

 

cycles, at least one of the design parameters, xk, yk’ takes on a negative
value, a new set of the initial estimates, xi, yi, should be chosen. In such
case, program 2 is utilized to calculate the total delay characteristic in
terms of the old set of the initial estimates. From the behavior of this
delay characteristic and on the basis of the method for the choice of the

initial estimates (chapter VI), a new set of initial estimates is obtained.

V. AN ESTIMATE ON THE NUMBER OF SECTIONS

The problem of estimating the number of sections, 11, of a delay
equalizer is indeed a difficult task. If the estimate is to be of value to
the designer, it is necessary to obtain an expression for n in terms of
the known quantities, i. e. , in terms of

(l) the delay characteristic of the filter, TSF;

(2) the equalization interval, (fa, f5);

(3) the maximum allowable total delay deviation, CST.

In functional notation:

,f,£,cl

F a b T) (5.1)

n'-'F(Ts

From section 2. 7, the delay equalization problem is to approximate

the delay equalizer requirement, T , (eq. 2. 5. 2) by the delay equalizer

SER

function, T (eq. 2. 6. 9), throughout the equalization interval, (fa, fb),

. SE’
to within a Specified degree of accuracy:

T (f)é T -'I‘ (f) forf <f<f (5.2)

SF b
Integrating this last equation with respect to frequency, f, over the

equalization interval, yields
f f

If: TSE(f)df é To(fb-fa) - ff: TSF(f)df (5. 3)
The integral on the left side of this last expression can be related to the
number of sections, 11, of the delay equalizer as follows. From the
definition of the insertion delay function (eq. 2.4. l) and from eqs. 2. 6. 7,

2. 6.8, and 2. 6. 9, for an equalizer of n sections

[CDT (f)df= —-1— fm—L B (f)df=3— (211 - O)=n (5 4)
0 SE 211 0 df SE 211 '
then f f m
.f a T (f)df = n - f a T (f)df — f T (f)df (5.5)
f SE 0 SE f SE

b b

45

46

Substituting this last equation into eq. 5. 3, yields

fb fa
s: - _ d
1‘ T0(fb fa) ff TSF 1+ [0 TSE

co
df+f T df (5.6)
f
a b

SE

Before proceeding further, the three types of equalization intervals are
considered separately, 1. e. ,

(l) equalization of low-pass loss filter: (0, fb);

(2) equalization of band-pass loss filter: (fa, fb);

(3) equalization of high-pass loss filter: (fa, 00 ).

In the case of low-pass equalization, the interval is (O, fb), hence
fa=0 and eq. 5.6 reduces to

fb

w
0 TSF(f)df+ ff TSE(f)df (5.7)

n Tofb f b
The delay functions TSF(f), TSE(f), and the delay level To corresponding
to the delay equalization solutions as obtained in the previous chapter
(figs. 4.4.4 and 4. 3. 5) are plotted in figures 5. 1 and 5. 2, for the cases
n=3 and n25, respectively. For any low-pass delay equalization problem,

the delay functions, TSF(f) and T (f), have typical delay characteristics

E
similar in form to those shown insthese two figures. An estimate for
each of the two integrals in eq. 5. 7 is made on the basis of the shapes of
these typical delay characteristics as follows:

First, the following quantities are defined.
represents the maximum value of the function T

SF
(the delay of the filter), in the equalization interval;

SFmax:

TSFmin: represents the minimum value of the function, TSF’
in the equalization interval;
Also, let AT - T - T (5.8)

SF ‘ SFmax SFmin

The integral corresponding to the second term on the right-hand
side of eq. 5. 7 is approximated by the sum of the areas of the rectangle

and the triangle, shown in figures 5. l and 5. 2. The rectangle has width

 

4.

3.14.1- ‘. Q _ .

 

49

fb and height TSFmin

ATSF. The constant, k (0< k <1), is to be determined on the basis of

the particular filter delay function, TSF’ so that a good approximation

, while the triangle has base (l-k)fb and height

of the integral results. Then

f

fob TSF(f)df s [T .n + .5-(1-k)/_\.T (5.9)

SFmi SF]fb

The last integral on the right-hand Side of eq. 5. 7 is approximated
by the area of the triangle shown in figures 5. 1 and 5. 2. It should be
remembered, however, that the function, TSE(f) (the equalizer delay), is
not known prior to completion of the equalization problem, hence, it is
necessary to obtain empirical expressions for the height and the base of
this triangle in terms of known quantities, namely, in terms of the
AT f and k, associated with the filter

SFmin’ TSFmax’ SF’ b’
delay function, T Such empirical relations Should be arrived at from

quantities, T
SF'
a large number of delay equalization solutions, all of comparable quality
of equalization (for example, the solutions could be selected such that
in each solution the percent delay deviation is less than x% over more than
y% of the delay equalization interval). The estimate on the number of
sections, n, so obtained is valid for delay equalizer networks of corres-
ponding delay equalization quality. For an equalizer requiring higher
quality of equalization a larger number than that obtained by this estimate
would be taken.

To illustrate the above procedure, the integral corresponding to
the last term on the right-hand side of eq. 5. 7 might be approximated by
a triangle as follows: The height of the triangle is taken as TSE(fb).

From eq. 5. 2

TSE(fb) = To ' Tsrub) 2 To ' TSFmax (5'10)

Inspection of the delay characteristics in figures 5. l and 5. 2 suggests

that the base of this approximating triangle is directly prOportional to the

50

factor (l-k). Let the constant of proportionality be denoted by k1. The
numerical value of this constant k1 must be determined from a large
number of solutions, as mentioned above. The approximation of the

integral then becomes

[:0 TSE(f)df 5 7;- k1(l-k) (TO — T ) (5.11)

b SFmax

Substituting eqs. 5. 9 and 5.11 into eq. 5. 7, yields

n5: [T -T

o SFm ]f

jn-%U-MATS +%kfiLk)flde )(SJZ)

F b SFmax

This last equation gives an estimate on the number of sections, n, of a
delay equalizer. However, the quantity To is unknown and an estimate
for it in terms of the known quantities must be obtained empirically
from a large number of delay equalization solutions.

The estimate on the number of sections, 11, as given by eq. 5. 12
is computed for the two cases, n=3 and n=5, shown in figures 5.1 and
5. 2, respectively. The numerical values of the quantities used in the

calculations as well as the results are listed in table 5. 1.

Table 5. 1: Calculation of Estimates on the Number of Sections, 11,
from eq. 5.12.

 

o TSFmin TSFmax SF b

 

3 .345-10'6 510-6 .14-10-6 .09-10'6 6.0-106

 

5 .571-10-6 510"6 .32-10-6 .27-10‘6 7.2-106

 

 

 

 

 

 

 

 

 

 

n k k Calculated
1 Estimate 3 of n:
3 .567 2.107 2.74

 

5 .653 2.107 4.35

 

 

 

 

 

 

51

The constant of proportionality, k1, was chosen on the basis of the delay
characteristics of figures 5. l and 5. 2. It is seen from this table (table
5. 1) that very good estimates on the number of sections are obtained from
eq. 5. 12, at least for the two cases considered. The number of sections
for the delay equalization problem would be taken as the next higher integer
with respect to the number obtained from eq. 5. 12.

The estimate on the number of sections, n, given by eq. 5.12, applies
for delay equalization in the low-pass band. ' Starting with eq. 5. 6, Similar
procedures would lead to expressions for estimates on the number of

sections, 11, for the band-pass and high-pass cases.

VI. INITIAL ESTIMATES OF THE DESIGN PARAMETERS

In this chapter, a method for the choice of the initial estimates,
0 o . . . .
xk, yk’ for the deSign parameters is presented. First, some important
prOperties of the insertion phase function, Bs(f), and the insertion delay

function, Ts(f), of one lattice section (fig. 2.6. l) are listed.

6. 1 Properties of the Insertion Phase Function

From eq. 2. 6. 7, the insertion phase function, BS(f), of one lattice

section, in terms of the two independent parameters (1 and f0, is given by

f
_ o __f_
BS(f) — 2 arccot (d)(-?- - f > (6.1.1)
0
From this last equation
BS(0) = O
B (f = 17
S 0
lim B (f) = 211 (6.1.2)
f-pm S

38(1) >0 forO<f<oo ,d>0,£o>0

The insertion phase function, Bs(f), is a monotonically increasing function
of frequency, f. It has a value of zero at zero frequency and approaches
the value of 271 radians asymptotically as frequency increases without

bound.

6. 2 PrOpertieS of the Insertion Delay Function

From eq. 2. 6. 8, the insertion delay function, TS(f), of one lattice

section, in terms of the parameters d and f0, is given by

d 1 + (fO/f)z
17 f 2 _ 2 '
O 1 + d (f/fo 10/1)

 

 

Ts(f) = (6.2.1

From this last equation

52

53

 

 

l
Tsw)- wfd
o
Ts<f>= .de
O 0 (6.2.2)

lim T (f) = O
S
fem

“Ts(f) >0 for 0<£<00,d>0,fo>0

The insertion delay function, Ts(f), has finite and non-zero value at zero
frequency (for finite and non-zero values of d and f0) and approaches zero
value asymptotically as frequency approaches infinity.

The insertion delay, Ts(f), is a function of frequency, f, and the two

parameters, d and f0. If the product function, Tp, is considered, given by

 

d . 1+ (f /£)-’-
Tp: Ts(f)-f0= T H dszffO _ 16m: (6.2.3)

then Tp can be viewed as a function of f/fO and d [34]. It is now possible
to plot a family of curves of the product function, Tp’ versus f/fo, each
curve for a different value of the d parameter. This family of curves is
classified into two broad classes, depending on the value of the d para-
meter, according to whether the product function takes on a maximum
value at zero frequency (class A, fig. 6. 2. 1) or at some finite and non-
zero frequency (class B, fig. 6. 2. 2).

The maximum value of the insertion delay function, TS(f), of one

lattice section, occurs at

f = o, for d3: 1/3, class A (6.2.4)

 

f = f V\/4-1/d2'- l, for dz > 1/3, class B (6.2.5)
0

Since the radical in this last equation is always less than unity, for
dz > 1/3 and finite, it follows that the insertion delay function takes on

a maximum value at some frequency less than £0.

 

 

56

Figure 6. 2. 3 illustrates the change in the insertion delay function,
Ts(f), due to a change in £0, for one value of the d parameter. The change
in TS(f), due to a change in d, for one value of the fO parameter, is shown
in figure 6. 2. 4.

A summary of the properties of the insertion delay function, Ts(f),
of one lattice section, is given below.

(1) The behavior of the Ts(f) function is illustrated in figures

6. 2.1 (class A) and 6. 2. 2 (class B);

(a) Class A consists of monotonically decreasing curves.
They all have maximum value at zero frequency
(eq. 6. 2. 4);

(b) Class B consists of curves having maximum values at
finite and non-zero frequencies, given by eq. 6. 2. 5;

(2) In the case of the class B curves

(a) for a fixed value of fo, the d parameter determines the
sharpness of the peak, as is seen from figures 6. 2. 2
and 6. 2. 4;

(b) The parameter f0 is dominant in determining the location
of the maximum value of the delay characteristic, as is

seen from figure 6. 2. 3 and eq. 6. 2. 5.

6. 3 The Choice of the Initial Estimates, x:, y:

This section presents the steps employed in the process of selection
of the initial estimates, xi, yi, for the design parameters of a delay
equalizer. In this procedure, the estimate on the number of sections, 11,
of the delay equalizer is assumed to be known.

(1) The delay characteristic, TSF(f), of the filter to be equalized

is plotted versus frequency;

(2) The equalizer delay requirements, TSERH) (eq. 2. 5. 2), is

obtained graphically from the plot of step (1). Here, the delay

level, To’ is chosen arbitrarily, (a good value is to take

Delay, Ts(f)

Delay, TS(f)

57

d, £01

d: f02

f02 > fo1

 

 

Frequency, f

Figure 6. 2. 3. Change in TS(f) due to a change in f0.

 

 

 

Frequency, f

Figure 6. 2. 4. ~ Change in Ts(f) due to a change in d.

58

To: ZTSFmax)’ since only relative values of the equalizer
delay requirement are of interest;
(3) On the basis of the behavior of the insertion delay function,

Ts(f), of one section (figs. 6. 2.1 and 6. 2. 2) a set of 11 para-

meter pairs, dk’ f k (k=1, 2, 3, ° ' ' , n), is chosen such that the
0

sum of these n functions, Tsk(f), when plotted, approximates

the Shape of the equalizer delay requirement, TSER(f)' This

step is a curve fitting problem and the degree of approximation
attained depends largely on the skill and experience of the
designer. For the initiated, steps (4) and (5) of this procedure
might well be omitted;
(4) Program 2 is used to calculate the total delay, TST(f), (the
sum of the delay of the filter and the equalizer, based on the
parameters of step 4);
(5) On the basis of the behavior of this total delay characteristic,
(f), a new set of the n parameter pairs, dk’ fok’ is obtained
(f), is to approximate a

TST
as follows: Since the total delay, TST
constant value in the interval of equalization, the d'k parameters
of those sections having a delay characteristic which peaks in

a neighborhood of a relative maximum of T T(f) are decreased

and the dk parameters of those sections hafing a delay char-
acteristic which peaks in a neighborhood of a relative minimum
of TST(f) are increased. The fok parameters are used to relo-
cate the peaks of the delay characteristics of some sections, if
necessary;

(6) Upon the selection of the normalizing frequency, fd (see
criterion 4. 5. 1), the initial estimates, xi, yi, are calculated
in terms of the dk and f0k parameters form eq. 3.1. 2.

It Should be pointed out that the steps (1), (2), and (3), of the above

procedure, in essence describe the commercial method of delay equalizer

design. In the actual design, after each choice of the n parameter pairs,

59

dk, fok’ the 11 functions, Tsk(f) (k=1, 2, 3, - - - , n), are plotted and added

graphically, giving the equalizer delay characteristic, TSE(f). This
equalizer delay characteristic is compared with the shape of the equalizer
delay requirement, TSER(f), and a new set of the parameters, dk’ fok’ is

selected on the basis of this comparison. This repetitive trial and error

process is continued until such equalizer delay characteristic, TSE(f), is

(0.

obtained, which approximates the equalizer delay requirement, TSER

to the Specified degree of accuracy [34].

VII. THE ELEMENT VALUES OF THE LATTICE SECTION

A schematic diagram of one lattice section of a delay equalizer is
shown in figure 2. 6. 1. The element values of this section in terms of
the d and fo design parameters, as obtained from eqs. 2. 6. 2 and 2. 6. 6,

are given by

R2

L1: m— : REC; (7.1)
O
R (1

L2 = m;— = Rgcl (7.2)
0

where R; is terminating resistance of the filter to be equalized and also
the resistance level of the lattice section (eq. 2. 6. l).
The element values in terms of the x and y design parameters and

in terms of the normalizing frequency, fd, as obtained from eqs. 3. 1. 2,

7.1, and 7.2, become

R2

__..__._ : Rgcz (7. 3)
ZTrfdy

L1:

XRZ

and

L2 : RECI (7'4)

A number of bridged-tee and twin-tee equivalent networks for the
lattice section can be derived. Such equivalent networks appear in

literature and are not included here [1, 33].

60

VIII. CONCLUSIONS

The usefulness of the thesis method of delay equalizer design and
its advantages over the commercial delay equalizer design technique
(section 6. 3) are discussed in the following paragraphs.

. (l) The commercial method of delay equalizer design, as described
in section 6. 3, is a trial and error procedure, which iS very tedious and
extremely time consuming. One of the main advantages of the t-method
lies in its time-saving nature. Reference to the five t-method solutions,
presented in chapter IV, demonstrates that given a set of satisfactory
initial estimates, the entire design problem is effected in a matter of
minutes of computer time. .More Specifically, the average computer time
for these five solutions is approximately 35 minutes. Several weeks for
one man would be required to effect one such equalization problem by
the commercial method.

(2) The commercial design method depends largely on the selection
of design parameters, which requires Skill and experience on the part
of the designer. The t-method provides a systematic approach to the
design problem and eliminates to a large extent the need of skill and
experience.

- (3) The extensive amount of the trial and error process associated
with the commercial method is almost eliminated in the t-method.
Although no criterion for the choice of the fi frequency points leading to
the best approximation of constant delay is given, criterion 4. 5. 2 reduces
the trial and error asPect of the t-method to two computer applications.

(4) On the basis of the exploration with the t-method, as in chapter
IV, it appears that this method is capable of producing approximations
to a constant delay which are beyond the power of the commercial method
even though pursued with the utmost skill, experience, and patience over

an extended period of time.

61

62

The five solutions of delay equalization problems, presented in
chapter IV, demonstrate the effectiveness of the thesis method of delay
equalizer design. It is not meant, however, to imply that the t-method
is the final answer to the delay equalization problem, but it definitely
indicates that the combined use of numerical analysis and a digital
computer has promise in equalizer design.

There is essentially no discussion in the technical literature on
the problem of estimating the number of sections of a delay equalizer.
One approach to the solution of this problem is presented in chapter V.

This approach should be investigated further.

BIBLIOGRAPHY

l) Balabanian, N. Network Synthesis, New York: Prentice-Hall, Inc.,
1958.

2) Bennett, B. J. "Synthesis of Electric Filters with“ Arbitrary Phase

Characteristics, " Convention Record of the I. R. E. , 1953, Part 5,
Circuit Theory, pp. 19-26.

3) Blewett, J. P. and Rubel, J. H. ”Video Delay Lines, " Proc. I.R.E.,
December, 1947, pp. 1580-1584.

4) Bode, H. W. "A General Theory of Electric Wave Filters, " B. S. T.J.,
April, 1935, p. 211.

5) Bode, H. W. Network Analysis and Feedback Amplifier Design,
Princeton, N. J.: D. Van Nostrand Co., Inc., 1945.

6) Bradley, W. E. "Design of a Simple Bandpass Amplifier with
Approximate Ideal Frequency Characteristics, " I. R. E. Trans. ,
P. G. C. T. 2, December, 1953, pp. 30-38.

7) Brain, A. E. "The Compensation for Phase Errors in Wide-band
Video Amplifiers, " Proc. 1. R.E. , July, 1950, pp. 243—251.

8) Brand, S. and Nyquist, H. "Measurement of Phase Distortion, "
B.S.T.J., July, 1930, pp. 522-543.

9) Bresler, A. D. . "On the Approximation Problem in Network Synthesis, "
Proc. I.R.E., December, 1952, p. 1724.

10) Cannon, W. D. "Delay Distortion Correction, " Trans. A.I. E. E. ,
March, 1956, pp. 55-61.

11) Cauer, W. Synthesis'of Linear Communication Networks, New York:
McGraw—Hill, Inc., 1958.

12) Cherry, E. C. "Application of Electrolytic Tank Techniques to
' Network Synthesis, " Proc. of the Symp. on Modern Network Synthesis,

Polytech.‘ Inst. of Brooklyn, Microwave Research Institute, New York,
‘April, 1952, p. 140.

63

64

13) Corrington, M. S. and Sonnenfeldt, R. W. "Synthesis of Constant-
Time Delay Networks, " R.C.A. Review, Vol. 15-2, June, 1954,
pp. 163-186.

14) Dagnall, C. H. and Rounds, P. W. "Delay Equalization of Eight-
Kilocycle Carrier Program Circuits, " B.S.T.J. , Vol. 28, April,
1949. PP. 165-220.

15) Darlington, S. "Realization of a Constant Phase Difference, "
B. S. T. J. , January, 1950, pp. 94-104.

16) Darlington, S. . "The Potential Analoque Method of Network Synthesis, "
B.S. T.J. , Vol. 30, April, 1951, pp- 315—365.

17) Deutsch, S. "Maximally-Flat Time Delay Ladders, " Trans. 1. R. E. ,
Circuit Theory, June, 1959, pp. 214-218.

18) Di Toro, M. J. "Phase and Amplitude Distortion in Linear Networks, "
Proc. I.R.E., January, 1948, pp. 24-36.

19) Floyd, C. F., Corke, R. L. , and Lewis, H. "The Design of Linear-
Phase Low-pass Filters, " Inst. Elect. Engrs. , Paper 1295, 11 pp.
(Television Convention, 1952).

20) Golay, M. J. E. ”The Direct Method of Filter and Delay Line
Synthesis, " Proc. I.R.E., March, 1954, pp. 585-588.

21) Hebb, M. H. and Hotton, C. W. "On the Design of Networks for a
Constant Time Delay, " J. Appl. Phys., June, 1949, p. 616.

22) Hildebrand, F. B. Introduction to Numerical Analysis, New York'
McGraw-Hill, Inc., 1956.

 

23) Householder, Alston S. Principles of Numerical Analysis, New York:
McGraw-Hill, Inc., 1953.

 

24) Kallmann, H. E. "Equalized Delay Lines, " Proc. I. R. E. ,
September, 1946, pp. 646-657.

25) Kuh,‘ E. S. "Synthesis of Lumped Parameter Precision Delay Line, "
I.R.E. Nat. Convention Record, 1957, Circuit Theory, Part 2,
pp. 160-170.

26) Kunz, K. S. -Numerical Analysis, New York: McGraw-Hill, Inc.,
1957.

 

65

27) Lane, C. E. "Phase Distortion in Telephone Apparatus, " B. T. S. J. ,
July 1930, pp. 493-521.

28) Laurent, T. "Basic Theory of Delay and Build-up Times in Electric
Filters with‘Phase Distortion, " Arch. Elekt. Uebertragung, March,
1952, pp. 91-98.

29) Linvill, J. G. , "The Approximation with Rational Function of
Prescribed Magnitude and Phase, " Proc. I.R‘. E. , June, 1952, p. 711.

30) MacCall, L. A. , "An Analysis of the Distortion of Signals by Linear
Systems Having Amplitude and Phase Characteristics of Certain Type, "
Proc. I.R. E. , November, 1931.

31) Mead, S. P. "Phase Distortion and Phase Distortion Correction, "
B.S.T.J. , April, 1928, pp. 195-224.

32) Nuttall, T. C. "Some Aspects of Television Circuit Technique: Phase
Correction and Gamma Correction, " Bull. Assn. Suisse Electr. ,
August, 1949, pp. 615-622.

33) Reed, M. B. Electric Network Synthesis, New York: Prentice-Hall,
Inc., 1955.

 

34) Rendall, A. R. "Phase Compensation (II), Design of Phase Compen-
sation Networks, "Elect. Communication, Vol. 7, 1928-1929, PP-
316-327.

35) Scott, R. E. "Potential Analogs in Network Synthesis, " Convention
Record of the I. R. E. , 1955, Part 2, Circuit Theory, p. 2.

36) Steinberg, J. C. "Effects of Phase Distortion on Telephone Quality, "
B.S.T.J., July, 1930, pp. 550-566.

37) Storch, L. "An Application of Moderanetwork Synthesis to the Design
of Constant-Time Delay Networks with Low-Q Elements, " Convention
Record of the LR. E. , 1954, Part 2, pp. 105-117.

38) Storch, L. "Synthesis of Constant-Time Delay Ladder‘Networks Using
Bessel Polynomials, " Proc. I. R. E. , Vol. 32, November, 1954,
pp. 1666-1675.

39) Storer, J. E. Passive Network Synthesis, New York: McGraw-Hill,
Inc., 1957.

 

66

40) Thomson, W. E. "Delay Networks Having Maximally Flat Frequency
Characteristics, " Proc. I.R.E., Vol. 96, 1949, pp. 487-490.

41) Thomson, W. E. "Networks with Maximally-Flat Delay, " Wireless
Engr., Vol. 29, October, 1952, pp. 256-263.

42) Tokad, Y. "Some Improvements of the Image-Parameter Method for
the Design of L-C Filters, " Ph. D. Thesis, Michigan State University,
1959.

43) Turner, A. H. "Artificial Lines for Video Distribution and Delay, "
R.C.A. Review, December, 1949, pp. 477-489.

44) Valley, G. E., Jr. and Wallman, H. Vacuum-Tube Amplifiers,
New York: McGraw-Hill, Inc., 1948, pp. 84-92.

 

45) Wallis, C. M. "Design of Low-Frequency Constant Time Delay Lines, "
A.I. E. E. Trans. , Part 1, Communications and Electronics, April,
1952, pp. 135-140.

46) Weinberg, L. "Constant-Resistance All Pass Networks with Maximally
Flat Time Delay, " Journal of Franklin Institute, January, 1959,
pp. 35-53.

47) Wheeler, H. A. "The Interpretation of Amplitude and Phase Distortion
in Terms of Paired Echos, " Proc. I. R. E. , June, 1939, pp. 359-385.

48) Wheeler, H. A. "Wideband Amplifiers for Television, " Proc. I.R. E. ,
June, 1939, p. 429.

49) Winkler, S. "The Approximation Problem of Network Synthesis, "
Trans. I. R. E. , P. G. C. T. , September, 1954.

50) Wood, I. E. , ”Note on Maximally Flat Delay Networks, " Trans. 1. R. E. ,
P.G.C.T. , December, 1958, pp. 363-364.

51) Zobel, O.J. "Transmission Characteristics of Electric Wave-Filters, ”
B. S. T. J. , October, 1924.

52) Zobel, O.J. "Distortion Correction in Electrical Circuits, " B. S. T. J. ,
July, 1928, pp. 438-534.

" A

A

I

'K n“ IA.
. .

 

011

M I'll.
R H
m I'll.
L I"
v m
Tl. "III
B m5
E ”8|

1111)),

9

3123

"WWW"!!!