IX.
A.
COMMUNICATION RESEARCH
MULTIPATH TRANSMISSION
Prof. L. B. Arguimbau
Dr. J. Granlund
Dr. C. A. Stutt
1.
Speech and Music.
E. M. Rizzoni
G. M. Rodgers
R. D. Stuart
Cross
H.
W.
E.
R.
Kinzinger
Manna
Paananen
Transatlantic Tests
The present series of tests has been completed and will be discussed in some detail
in the next Quarterly Progress Report.
J. Granlund, C. A. Stutt, L. B. Arguimbau
2.
Television
Tests are being made to check the typical requirements placed on a television system
by multipath conditions.
A single-line scanner is being constructed to display test pat-
terns from local television stations and provide a measure of the relative delays and
amplitudes of the various paths.
E.
3.
M.
Rizzoni
Simplified FM Receiver
The design of the i-f amplifier has been completed,
technique outlined in Technical Report No.
using the approximation
145 by J. G. Linvill.
Seven tuned circuits are
used as opposed to the typical six circuits used in home receivers.
This design has
met the requirements broadly outlined in the Quarterly Progress Report, October 15,
1951.
Limiters have been made, using gated-beam (6BN6) tubes; they have been compared
with the crystal diode limiters discussed in earlier Quarterly Progress Reports.
The
gated-beam type has been found to saturate more fully than the crystal type and to be
superior in most respects.
In particular,
it has been found workable in conjunction
with wide-band discrimination and preserves all the accompanying advantages of low
6AI
Fig. IX-1
A simplified version of the wide-band detector.
-49-
(IX.
COMMUNICATION RESEARCH)
capture ratios.
A slight modification of the wide-band detector discussed in Technical Report No. 42,
by J. Granlund produces the circuit shown in Fig. IX-1. Notice that the cathode supplies
the bias needed for the crystals.
R. A. Paananen
The design of a tuning head for a frequency modulation broadcast receiver has been
brought to the working-model point.
Using variable capacitor tuning, and a single stage
of tuned radiofrequency amplification,
all spurious responses are down at least
80 db (for interfering signals below 0. 4 volt) and the sensitivity is comparable to usual
commercially available receivers.
At a slight sacrifice in spurious response rejection,
sensitivities can be improved by increasing the couplings in the input tuned circuits.
Noise figures as low as 6 db have been measured, in the latter case.
H. H. Cross
The effect of discriminator bandwidth on adjacent and alternate channel interference
is under investigation.
W. C. Kinzinger
-50-
(IX.
B.
COMMUNICATION RESEARCH)
STATISTICAL THEORY OF COMMUNICATION
Prof. J. B. Wiesner
Prof. W. B. Davenport,
Prof. R. M. Fano
Prof. Y. W. Lee
Prof. J. F. Reintjes
Dr. P. Elias
1.
B.
R.
J.
M.
C.
L.
B.
Jr.
E. Green, Jr.
Howland
G. Kraft
A. Basore
S. Berg
J. Bussgang
Coufleau
A. Desoer
Dolansky
M. Eisenstadt
A. J. Lephakis
R. M. Lerner
M. J. Levin
Multichannel Analog Electronic Correlator
A suitable integrating circuit has been developed. The remainder of the design work
for the correlator has been completed. Testing of the equipment is proceeding.
Y. W. Lee, J. F. Reintjes, M. J. Levin
2.
Analog Electronic Correlator for Second-Order Correlation
The second-order autocorrelation function of a random or periodic function fl(t) is
defined as
111(
T1)) = lim
T--oo
1
f fl(t) fl(t
+ T1) fl(t + T2 ) dt.
-T
Equipment is not yet available for plotting this function.
111 ( 0 , T2)
=
f (t) f(t
lim
T-oo
=
However, if T 1
+ T)
0
(2)
dt.
For this special case the second-order autocorrelation function can be obtained by
crosscorrelating f (t) with fl(t). This has been done as shown in Fig. IX-2.
LADDER NETWORK
ELECTRONIC
ANALOG
CORRELATOR
QU
A CHANNEL
CIRCUIT
B CHANNEL
f(t)
Fig. IX-2
Fig. IX-3
Arrangement for obtaining second-order
autocorrelation functions.
Squaring circuit.
-51-
(IX.
COMMUNICATION RESEARCH)
The squaring circuit(J. S. Rochefort: "Design and Construction of a Germanium-Diode
Square-Law Device, " Master's Thesis in Electrical Engineering, M.I.T. 1951) makes
use of the nonlinear properties of 1N34 germanium diodes (see Fig. IX-3). A twoterminal ladder network composed of resistors and these diodes has been constructed.
This network has a variable driving-point impedance such that the current through it
is proportional to the square of the voltage across it.
The voltage across a small
resistor in series with the network is then proportional to the square of the input voltage.
The sign of the output should be positive whether the input is positive or negative for
proper square-law action.
circuit.
This characteristic is obtained by using a push-pull driver
Since the driver circuit works into a varying impedance it was designed to
have a very low output impedance for accurate operation.
Appreciation is expressed
to Mr. Rochefort for providing the necessary information for the design of this unit.
If the function fl(t) in Eq. 1 is periodic and has the spectrum Fl(n) with the fundamental angular frequency wl,
so that
oo00
f (t) =
F 1 (n) e
(3)
n= -c
it can be shown that the second-order autocorrelation function of fl(t) is
o00
111(T2)
0m
F 1 (m) Fl(n)
=21
F 1 (m+n)
e
j(m+n)wl(Tl + T )
2
m=-oo n=-oo
For the particular case where fl(t) is an odd-harmonic function,
111 (T 1 , T 2
) vanishes
for all values of T 1 and 7 2 since Fl(m+n) is zero.
Figure IX-4b shows the autocorrelation function of the half-wave rectified sinusoid
which is illustrated in Fig. IX-4a. Figure IX-4c shows the second-order autocorrelation function of this wave. For a wave of the form shown in Fig. IX-5a, the autocorrelation function and the second-order autocorrelation function are given in
Figs. IX-5b and IX-5c respectively.
Y. W. Lee, J. F. Reintjes, M. J. Levin
3.
a.
Information Theory
Transmission of information through channels in cascade
Further investigation of this problem has indicated that the operation of the intermediate receivers is the important factor in the performance of the over-all system.
For example, in the pulse code modulation system analyzed previously the receiver
requantizes the received pulses.
This requantization process eliminates the noise in
almost all instances although errors are introduced once in a while.
-52-
(A relatively
Fig. IX-4a
Fig. IX-Sa
Half -wave rectified
sinusoid.
Triangular wave with odd
harmonic function.
Fig. IX-4b
Fig. IX-Sb
Autocorrelation of waveform
in Fig. IX-4a.
Autocorrelation of waveform
in Fig. IX-Sa.
Fig. IX-4c
Fig. IX-Sc
Second-order autocorrelation of
waveform in Fig. IX -4a.
Second -order autocorrelation of
waveform in Fig. IX-Sa.
General data pertaining to curves: AT = 5 IJ.sec; number of samples = 8000; fundamental frequency of input wave = 3.5 kc/sec except for Fig. IX-4c where it is 2 kc/sec.
-53-
Fig. IX-6
Fig. IX-7
E as in Eve.
U as in Boot.
Fig. IX-8
Fig. IX-9
S as in Hiss.
F as in Gaff.
Fig. IX-lO
Fig. IX-II
Z as in Craze.
A as in Father.
-54-
(IX.
high S/N is assumed.)
COMMUNICATION
RESEARCH)
The introduction of definite errors can be shown to result in
additional loss of information; it appears that any attempt to eliminate a fraction of
the noise necessarily involves an additional loss of information.
On the other hand,
if some of the noise contained in the signal is eliminated, the fraction of information
lost in the next transmission is
reduced.
optimum degree of noise elimination.
It appears therefore that there is
some
In the above case the over-all system is improved
by quantization, at the intermediate stations, to the original levels, although in some
cases quantization to a larger number of levels may be better.
The question of noise
elimination is the main problem under study.
C. A. Desoer, R.
b.
M.
Fano
Vocoder
The possibilities of a high quality vocoder are being investigated using, initially,
the approach of tracking the harmonics of pitch frequency (1).
This and related tech-
niques are critically dependent upon an accurate knowledge of the pitch period of voiced
sounds.
Previously used methods of obtaining the pitch period appear unsatisfactory
for our purposes.
We are currently investigating the use, in this connection,
"Ianalytic function" (2)
function of time.
of speech.
of the
The analytic function of a speech wave is a complex
Its real part is the speech wave itself, and its imaginary part is the
same speech wave in which each frequency component has been shifted in phase by 90.0
To this end, a wideband (50 cps to 15, 000 cps) phase-splitter has been constructed.
When the outputs of the phase-splitter are connected to the vertical and horizontal
deflection plates, respectively, of an oscilloscope,
patterns result which constitute
a polar representation of the analytic function.
As a by-product of the above work, such polar patterns may be useful as a new form
of visible speech.
Several patterns are shown in Figs. IX-6 through IX- 11.
to the static characteristics that may be observed in these photographs,
In addition
dynamic char-
acteristics have been observed which might play an important role in the visual identification of speech sounds.
Possible uses for this method of representation are being investigated.
R. M. Lerner, R. M. Fano
References
1. R. M. Fano:
The Information Point of View in Speech Communication,
J. Acous. Soc. Am. 22, No. 6, 694-695, 1950
2.
D. Gabor:
Theory of Information, J.
Inst. Elec. Eng. 93,
-55-
Part III, 429-431,
1946
(IX.
C.
COMMUNICATION RESEARCH)
HUMAN COMMUNICATION SYSTEMS
Dr. L. S. Christie
Dr. R. D. Luce
F. D. Barrett
1.
J.
J.
B. Flannery
Macy, Jr.
D. G. Senft
A. G. Simmel
P. F. Thorlakson
Technical Report
Our major project during the past quarter has been the preparation of a technical
report summarizing the results and developments of the past two years.
The report
will describe the 1949-1950 experiments of Professor A. Bavelas and associates,
first explored the problem area of task-oriented groups.
who
The work of the past year,
involving a more intensive study of a restricted part of the problem area and employing
more refined experimental techniques, will also be described.
The principal experimental program (1) has been concluded.
The primary purpose
of this program was to examine learning within a task-oriented group.
light on earlier work of Leavitt and Smith.
complex data analysis required.
The results shed
Considerable time has been spent on the
This analysis will be concluded during the next quarter.
The principal development in experimental technique,
"Octopus",
is reported briefly
below and will be discussed in full in the forthcoming report.
F.
2.
D.
Barrett, L.
A Problem in Data Analysis:
S. Christie, R. D. Luce, J.
Macy, Jr.
Programming for Whirlwind I
For the analysis of the data from experiments on network patterns and group
learning (1) it is necessary to have the distributions of the number of time units, i. e.
opportunities to send a message,
required to distribute all the relevant information to
all the members of the group,
assuming equilikelihood for the use of the different
channels going out from any individual.
For many networks this problem appears sufficiently complicated mathematically to
warrant a quasi-empirical approach by means of random numbers and the use of highspeed computing machinery.
Whirlwind I was made available for this work, and a code
has been developed which is of sufficient generality to permit adjustment to different
networks by very simple modifications.
If it should become of interest to investigate
n-man groups, n 4 10, or to make some assumptions other than equilikelihood for the
use of the different channels going out from the individuals in the group, the completed
code may serve as a core around which such modifications can be constructed.
We define matrices S such that s.. = 1 or 0 according as individual j has or has not
the information originally possessed by individual i,
0 according as individual i is
and matrices T such that tij = 1 or
sending all the information at his disposal to individual
j or to someone else, letting the diagonal terms be unity in the matrices of both types.
-56-
(IX.
If we further let I be the identity matrix
COMMUNICATION RESEARCH)
having e.. = 1 for all i and j, we may state our problem as follows.
S
n
Ieij
e
I,
and E the universal matrix
I 18ij I
Given
S
=I
(1)
= S
n-l T n
(2)
where the product in Eq. 2 is the Boolean matrix product defined in strict analogy to the
ordinary matrix product by
S' = ST
if and only if
n
=SU
) tkj)
(ik
k=l
We want to know the distribution of N where SN = E and for all n < N, S
E.
The Whirlwind I program constructs the T-matrices as needed by means of random
digits fed from a tape into the high-speed storage in blocks of 375, guided by a set of
5 control numbers peculiar to the network under investigation.
The distribution of the
values of N < 10 will be typed out.
Testing of the program is under way.
A. G. Simmel
3.
"Octopus"
The electrical experimental device (2),
experimental groups of human subjects,
nicknamed "Octopus, " for running controlled
has been partially completed.
The individual
stations for the subjects and the central control unit have been completed and tested,
and preliminary trials using military subjects have been successfully completed.
Con-
struction of the component parts of the automatic data-recording and analyzing unit has
been finished, and final assembly and testing of this unit is nearing completion.
The
tape-recording device, using binary sequences time-multiplexed on punched magnetic
recording tape, has been completed.
Future experiments employing Octopus are in the
planning stage.
J.
4.
Macy, Jr.
Experiment on the Persistence of Organization
In conjunction with Harvey Hay, graduate student in the Department of Economics,
M. I. T.,
an experiment has been designed to detect the persistence of inappropriate
voluntary organization within a task-oriented group.
The question being asked is:
Given a group having a simple task to perform and communication by written messages
subject only to network constraints, what is the effect of shifting the group from one
-57-
(IX.
COMMUNICATION RESEARCH)
network to another ? It has been found in earlier experiments of a similar type that
under certain networks, groups develop an internal organization which is appropriate
to that network. Does such an informal organization persist in a new network, or does
it decay as rapidly as it is formed, when it is no longer appropriate? Further, does
the experience and consequent learning of the first network tend to affect adversely
learning in the second network?
For any group of subjects the experiment will have three phases. Five trials will
be performed on a network which we shall call P. Following these, a questionnaire
will be administered. After a break for lunch, two sets of 15 trials each will be run
on different networks, each followed by the same questionnaire as used after P. Our
interest is only in the last two phases. The network P is run in order to eliminate a
transient phase of confusion and adjustment to the apparatus.
P was chosen to be a
network within which very little organization can occur in five trials.
The questionnaire
is also used after P so that the subjects will have the same knowledge as to the content
of the questionnaire when entering phase two as when entering phase three. In addition
to P, there will be three networks used in the afternoon sessions: circle C, star S,
and diablo D. The combinations to be run are PPC, PPD, PSC, PSD, PDC, PCD.
10 groups will be run in each combination.
The apparatus to be used for this experiment is a modification of that described in
reference 1; and, in particular, includes the important feature that all communication
occurs in a quantized time scale.
This, or something equivalent to it, is necessary for
the determination of the time order of the messages sent.
The subjects to be used are enlisted Army personnel.
L. S. Christie, R. D. Luce, J. Macy, Jr.
References
1.
Quarterly Progress Report, Research Laboratory of Electronics, M. I. T. Jan. 15,
1951, pp. 79-80
2.
Quarterly Progress Report, Research Laboratory of Electronics, M. I. T. April 15,
1951, p. 52
-58-
COMMUNICATION RESEARCH)
(IX.
D.
REPLACEMENT OF VISUAL SENSE IN TASK OF OBSTACLE AVOIDANCE
Dr. C. M. Witcher
E. Ruiz de Luzuriaga
A small project has been set up in the Laboratory;
its object is to extend the field
of sensory replacement to the visual domain as specifically applied to the problem of
independent travel by the blind. Analysis of the performance of all obstacle avoidance
devices developed to date strongly suggests that their major failing has been a result of
inadequate means for transmission of information from device to user. In all schemes
thus far employed the information was transmitted to the user as a time series of data;
each element provided information as to the presence or absence of an obstacle in some
small specified portion of the environment.
Thus the necessary integration always had
to be done within the brain of the user.
The present aim of our project is to devise a method of transmission of information
from guidance device to user in which part of the integration process can be taken over
by the device, enabling a much needed increase of speed in the process and affording a
decrease in the mental effort which must be expended by the user.
We have started from the obstacle avoidance device developed a few years ago by
the Signal Corps, and which appears to be the most reliable and satisfactory type
available at present.
Our proposed solution to the problem of providing integrated
information consists of two modifications of this device:
1) addition of an automatic
optical scanning system, and 2) presentation of the information through the presence or
absence of projecting points, or, more precisely, round-headed pins, at various positions on a signal presentation plate.
The positions of the pins, which at any moment
project above the surface of the signal presentation plate, can be quickly surveyed by
very slight movements of the index finger of the blind user, and he can thus obtain an
almost instantaneous, if rather crude, picture of the obstacles in his immediate environment.
The positions (radial and angular) at which pins appear on the plate will
correspond roughly to the positions of the obstacles in range and azimuth, much like
the situation represented by a PPI radar presentation.
The moving pins will be actuated
by relays fed from the output of the small hearing-aid amplifier of the device.
mechanical design of the system is now fairly complete,
The
and the circuits for relay
operation have been checked experimentally.
In addition to this system for presentation of information, a tentative design for a
step-down indicator,
an element which has long been recognized as a necessity for safe
travel, has been completed.
-59-
(IX.
E.
COMMUNICATION RESEARCH)
COMMUNICATIONS BIOPHYSICS
Prof. W. A. Rosenblith
K. Putter
1.
Interaction of Cortical Activity and Evoked Potentials
Sizable responses to clicks are recorded from the auditory area of the cerebral
cortex of anesthetized animals. The present study proposes to investigate systematically the extent to which these evoked responses modify cortical activity and are in
turn affected by this activity.
Preliminary experimentation has been started at the
Massachusetts General Hospital.
The use of correlational techniques for purposes of
data analysis is projected.
W. A. Rosenblith with Dr. M.
2.
A. B. Brazier (Massachusetts General Hospital)
Variability of Cortical Responses to Acoustic Clicks
Responses to clicks recorded by fine wire electrodes from the auditory area of the
cortex of anesthetized animals are characterized by large variability. If a theoretical
useful quantitative description of such responses is to be given, reliable data on cortical
variability are essential.
A large number of responses have been recorded simultaneously from two locations
of a cat's auditory cortex; the rate of stimulation was varied from 2 clicks per second to
1 click every 10 seconds. The data are being analyzed by different statistical techniques
in order to determine the character of the observed variability. If the variability is nonrandom, an attempt will be made to differentiate between response-induced nonrandomness and nonrandomness due to physiological periodicities (e. g. cortical rhythms).
The experiments were carried out at the Psycho-Acoustic Laboratory, Harvard
University.
W. A. Rosenblith, K. Putter, with K. Safford (Harvard Psycho-Acoustic Laboratory)
3.
Instrumentation
A useful device for the analysis and presentation of electrophysiological data
recorded in response to discrete stimuli might have the following properties: a) it
would take "peeks" of adjustable duration at preset intervals after the occurrence of the
stimulus; b) it would quantize the evoked response at the highest level reached during
the "peeking-interval; " and c) it would present the central tendency and the variance of
the quantized data and also record the data in sequential form.
A preliminary model of such a time-gated amplitude quantizer having most of the
enumerated features is now in the design stage.
W. A. Rosenblith, K. Putter
-60-
(IX.
F.
COMMUNICATION RESEARCH)
ELECTRONEUROPHYSIOLOGY
J. Y. Lettvin
W. Pitts
B. Howland
With the design and building of two constant-current stimulators, a machine that
manufactures the numerous microelectrodes used in our experiments and an elaborate
control device for programming whole experiments, an electrophysiological laboratory
is being set up. The shop is now working on the design and construction of a new stereotactic instrument.
We have been applying a constant input-volley to the spinal cord, and plotting the
potential field and its changes with time at as dense a network of recording points within
the cord as is practicable.
Since the cord is narrow and a good conductor, the potential
at each point is determined by an integration over all the sources and sinks of current
in the entire cross section; and the more remote suffer only a small decrement with
distance.
These sources and sinks represent the activity of cells and fibers where they
are, for flow of current from the inside of a cell outward makes a source in the external medium in which we measure; current inward makes a sink.
The density of these
sources and sinks can be calculated by taking the Laplacian of the potential; the computing room has been doing this for us numerically from the potential maps derived from
earlier experiments.
The result is a series of maps representing the successive acti-
vity of different groups of cells or fibers.
From these maps we hope to find out how the
groups of cells combine to produce the complicated structure of the spinal reflex, and
how the transmission of information to them along pathways descending from the brain
so modifies the structure as to control movement.
To this end we plan first to complete
our analysis of the sequence of maps of sources and sinks produced by various inputs to
the isolated cord,
then to note the differences when certain of the most important
descending systems are stimulated concurrently.
Somewhat aside from this, we expect
these maps to contribute crucial evidence for or against our earlier theory of inhibition
at the synapse.
// We plan to apply the same methods, combined with various forms of statistical analysis, to a study of some of the higher sensory systems, such as the visual cortex and
lateral geniculate, from the point of view of communication theory.
To record the
potentials from one or a few points on the surface of the cortex, as has been usually
done, does not, in our opinion, furnish suitable material for such an analysis; for we
This section has been supported in part by the Department of the Navy, Office of Naval
Research, under Contract No. N5 ori-07868. Authority: NR 113-004/9-6-51, Biological Sciences Division.
-61-
(IX.
COMMUNICATION RESEARCH)
cannot ordinarily discover which of the numerous groups of cells and fibers, distinct in
function and connections, are responsible for the potentials recorded. But the two
methods: that of measuring potential histories at enough points to compute sources and
sinks, localizing the generators of the potentials; and the statistical methods of communication theory, should provide more information about the system, and better means
of analyzing it.
-62-
(IX.
G.
PARALLEL
COMMUNICATION RESEARCH)
CHAIN AMPLIFIER
The behavior of the high frequency amplifier chain of a parallel chain amplifier
has been reported in the Quarterly Progress Report, October 15,
An average
1951.
gain of about 5 db per stage was obtained over a band from 130 to 260 Mc/sec
using 6AK5's with double-tuned interstages.
The results are mentioned here only
An input circuit for this chain has been tried.
as they give an idea of the problem of network complexity when dealing with small shunt
capacitances.
It will be shown that the use of this input circuit should have considerably
Instead it was found that the gain was slightly reduced.
improved the gain of the chain.
The poor results are probably due to stray capacitances disturbing the network structure.
The input capacitance of the first stage is
terminated 50-ohm cable, which provides a 25-ohm source.
connected directly to the grid as shown in Fig. IX-12.
in Fig. IX-13,
where the source is
R = 25 ohms and C 2 = 8
pif,
The input is
about 8 p.aif.
fed from a
Originally, the cable was
The input circuit tried is shown
represented by its Thevenin equivalent.
and the products LIC
1
Here
and L2C 2 are equated to make this
circuit the bandpass equivalent of a simple series LC circuit.
If R is temporarily assumed zero, the poles of the transfer ratio, E 2 /E
occur on the jw axis, say at wI and w2 .
1,
will
Then, if we solve for the element values we
find
1
21
C
1
= C
12
2
(3)
1lW2
Returning to the lossy case, R * 0, we may conveniently express the transfer ratio
in terms of the critical frequencies of the lossless case, w1 and w
2
E2
1
=4
+ RC 2(o
2
-
1
Z 3
X + (2
(2
2
,1 2 2
2
+ RC 2 W12
1)
and also R and C 2
,
)
2
1
1+X
1
.
(4)
We may now quickly evaluate the contribution of the input network over the band for a
simple but representative case.
-63-
6AK5
50- OHM CABLE
Fig. IX-12
Fig. IX-13
Direct connected input.
Input circuit.
-
0X
*
-------.. .
L ..
jw X-PLANE
X
INPUT CIRCUIT
OF FIG. I- 13
WITHOUT INPUT
CIRCUIT (SEE FIG.I-
a,
-
130
Fig. IX-14
/
260
FREOUENCY IN Mc/sec
Fig. IX-15
Assignment of poles. The extreme
pair is realized by the input network;
other pairs are realized by interstages.
Transfer function of input circuit.
.T
T.
Fig. IX-16
An interstage used in the
low frequency chain.
-64-
12)
(IX.
COMMUNICATION RESEARCH)
1) 2 .
First we notice that the multiplying factor of the transfer ratio is (W -
To aid in selecting two poles from the whole array of poles of the over-all transfer
function, we also consider the multiplying factors of the interstage networks used in
this chain.
These multiplying factors are equal to ( b
-
wo)/2C where the w's refer to
the critical frequencies of a particular interstage and C is the shunt capacitance
imposed at each terminal pair.
We maximize the over-all multiplying factor, which is
just the product of the individual multiplying factors, by pairing the poles as shown in
Fig. IX-14.
This maximization gives the greatest spread for the poles of the input
network and equal spread for the poles of the interstages.
turns out to be about 5 p.Lff.
For this arrangement C 1
For expediency, let us set wl and w2 equal to the upper and lower edges of our band
(corresponding to 130 and 260 Mc/sec).
edges is,
from Eq. 4, equal to 1/[RC 2(
The magnitude of the transfer ratio at the band
2
- w1)
= 6. 1 (ratio) or 15. 7 db.
the geometrical band center, this ratio is identically one or zero db.
Similarly, at
In Fig. IX-15
these values are used to sketch the approximate transfer ratio over the band.
transfer ratio of Fig. IX-12 is also sketched for comparison.
The
(Of course, with the
addition of the input network, the interstages are appropriately realigned. ) Thus we
see that the input network should result in an over-all improvement of gain.
The low frequency chain has been built, but the alignment has not been completed.
After some delay, a setup was obtained to measure the gain over this band
(0-130 Mc/sec).
Also, a generator to sweep the entire band (0-260 Mc/sec) for align-
ment purposes has been built, using a thermally-tuned klystron.
After the gain measuring equipment was set up and a calibration run taken, a preliminary run was made on the gain.
120 Mc/sec and 130 Mc/sec.
This run showed serious attenuation between
The trouble was traced to a self-resonance of series
coils in two circuits of the type shown in Fig. IX-16.
Although some care had been
taken to keep the distributed capacitance of these coils small, they were found to resonate in the affected region, producing zeros there.
The coils are being rewound, and
it is expected that the new coils will remedy this trouble.
R. K. Bennett
H.
A METHOD OF WIENER IN A NONLINEAR CIRCUIT
Technical Report No. 217 has been prepared and has been scheduled for publication.
S. Ikehara
-65-
COMMUNICATION RESEARCH)
(IX.
I.
TRANSIENT PROBLEMS
Prof. E. A. Guillemin
Dr. M. V. Cerrillo
F. Reza
1.
Network Synthesis for Prescribed Transient Response
A given transient time function f(t) is to be the unit impulse response of a finite
passive lumped-parameter
network.
The problem is to find the pertinent system
function h(s) of this network.
If,
for the moment, one were to consider a periodic time function f(t), the desired
system function could be found at once from a Fourier series representation for f(t);
and the requirement that the system function be rational could be met through being
content with the approximation to f(t) afforded by a partial sum.
A study of the nature
of this approximation could be dealt with in the time domain according to familiar
techniques; and since no further approximations in the frequency domain were called
for, the ultimate response would be assured of having the approximation properties of
the partial sum.
One way of dealing with a given transient function f(t) of finite duration is to consider its periodic repetition in several ways such that an appropriate combination of
the resulting periodic functions cancels everywhere except over the first period.
A
simple pattern accomplishing this end is described in the following.
Suppose a desired f(t) is the transient pulse shown in Fig. IX-17.
We consider the
two periodic repetitions of this function as shown in Fig. IX- 18 and observe that the
corresponding transforms hl(s) and h 2 (s) are readily constructible from the appropriate
Fourier series for fl(t) and f 2 (t).
If we could synthesize a pair of two-terminal net-
works N 1 and N 2 having the driving-point functions hl(s) and h 2 (s), it is clear that their
unit impulse responses would be fl(t) and f 2 (t) respectively.
Or we can say that if we
were to apply an excitation fl(t) to N 1 or an excitation f 2 (t) to N 2 , the response in each
case would be a unit impulse.
If we now observe that
and
f(t) = f2(t)
t -
)
(2)
we can conclude that if we apply an excitation f(t) to N 1 the response is a unit impulse
at t = 0 followed by a negative unit impulse at t = T/2; while if we apply an excitation f(t) to N2 the response is
a unit impulse at t = 0 followed by another at t = T/2.
-66-
COMMUNICATION RESEARCH)
(IX.
Therefore,
f(t) applied to N 1 and N 2 in series produces simply an impulse at t = 0,
It remains to fill in the details
or the latter produces f(t), which is what we wanted.
with appropriate mathematical symbols.
We begin by considering the periodic function f p(t) shown in Fig. IX- 19,
Writing the Fourier series
of a repetition of f(t) at half-period intervals.
f (t) = E
consisting
akejkt
k=- oo
we have for its transform
p(s)
q( "
ak
h (s)
s - jkw
hp(S)
k=-oo
(4)
The transforms hl(s) and h 2 (s) are readily constructible from h (s).
p
oo
ak(1 + e-Jk)
hl(s) = hp (s)
s - jkw
(5)
and
00
I-.a
h 2 (s) = h (s) (1
- e
-
a_(1 - e
- jkw
K'
ST/)
-00
Noting that the factor (1 + e
- jk
w) equals 2 for k even and zero for k odd, while the
factor (1 - e - jk w ) equals 2 for k odd and zero for k even, we have more simply
ak
h l (s) = 2
s-jk
k=0+(2,4, ..
P 1(s)
ql(s)
(7)
)
and
oo
ak
-jk
h 2 (s) = 2
P 2 (s)
S
(8)
k=+(1, 3 ... .)
That is to say, hl(s) is twice the sum of the
even order terms in hp(s), given by Eq. 4,
while h 2 (s) is twice the sum of the odd order
terms.
As long as the sums in the expressions for
hp(s), hi(s) and h 2 (s) are regarded as having
Fig. IX-17
The desired transient time function.
an infinite number of terms, these transforms
are, of course, not rational and the polynomials
-67-
Fig. IX-18
The auxiliary periodic repetitions of f(t).
i fp(t)
II
-
r\
%
#
r/2
T
T/2
-
./2
Fig. IX-19
The basic periodic repetition of f(t) from
which the auxiliary functions of Fig. IX-18
are derived.
fp*(t)
/'""
O
rj
-
I
I
1
f*(t-r/2)+A f(t)
af(t)
III
,.~~~-"1.
A
I
f
I
v.
.
I
-
I
Fig. IX-20
Approximation to the periodic function of
Fig. IX-19 afforded by a partial sum of its
Fourier series, and the imperfectly delayed
version of this function.
-68-
COMMUNICATION RESEARCH)
(IX.
p(s), q(s), p 1 (s),
Since we anticipate replacing
q 1(s), pZ(s) and q 2 (s) are not finite.
these expressions by their partial sums, it is appropriate even at this stage to think of
the polynomials as though they were finite. They obviously have real coefficients even
though the Fourier coefficients ak (which are residues of the h functions in their j axis
poles) are complex.
If we think of networks N 1 and N 2 as having the driving-point functions hl(s) and
h 2 (s), then a unit impulse u (t) applied to N 1 produces fl(t) and if applied to N 2 , it
produces f 2 (t). If we now interchange the roles of excitation and response, and say that
fl(t) applied to N 1 or f 2 (t) applied to N 2 produces u (t), then the system functions for
these networks are the reciprocals 1/hl(s) and 1/h 2 (s). Noting Eqs. 1 and 2 we may
then state that f(t) applied to N 1 having the system function 1/hl(s) produces the
response
(9)
uo(t) - u 0 (t -)
while f(t) applied to N2 having the system function 1/h
(1/hl) + (1/h
(s) produces the response
).
u (t) + u0 (t Wherefore,
2
(10)
f(t) applied to N 1 and N 2 in series with the resultant system function
2
) produces the response 2u o (t); or if we again interchange the roles of
excitation and response and consider the series combination of N
1
and N2 as having the
system function
1
1
2
then its unit impulse response becomes (1/2) f(t), as may readily be verified through
substitution of Eqs. 5 and 6 into Eq. 11, which yields
h(s) =
1
h p(s) (1 - e-s
S
P(s) " P 2 (s)
2p(s)
"
(12)
Since the time-domain equivalent of this equation reads
f(t) =
(f
(t)
- f(t-T))
(13)
the above statement is seen to be true.
Neglecting second-order effects, it is easy to show that when one replaces the above
infinite series by their partial sums, the resulting transient response is as good an
approximation to f(t) as the partial sum of the Fourier series given in Eq. 3 is to f p(t)
over any one of its periods.
Thus, if we indicate any approximate function through
attaching an asterisk to the corresponding exact function, we may write
-69-
(IX.
COMMUNICATION RESEARCH)
n
f (t) = )
akeJt
(14)
k=-n
n
h(s)
hp (s)
a
=k
(15)
s - jkw
k=-n
and
-jk*
n
h (s) e-ST/2*
p
ake
k=-n
(16)
s - jko
(16)
where n is a finite integer.
Figure IX-20 shows how the approximation to f p(t) given by the partial sum (Eq. 14)
might look, and correspondingly how the inverse transform of the frequency function
(Eq. 16) will appear. It is important to note that this time function may be regarded
as an imperfect version of f (t) delayed by a half period. Thus, the perfect version of
f (t) delayed by a half period differs from the function shown in the bottom half of
Fig. IX-20 in that it is exactly zero throughout the first half period. The imperfect
version, by contrast, has the same nonzero ripply variations here that it has in any of
the corresponding subsequent half-period intervals. Observe, however, that the imperfect and perfect versions are exactly alike everywhere except during the first half period
where they differ by the residual ripple which in Fig. IX-20 is denoted by Af(t). That
is to say, the net difference Af(t) between the imperfectly and the perfectly delayed
functions f (t) is nonzero only for the interval 0 < t < 7/2!
p
In the frequency domain we can express this result through writing
h (s) e-/
p
2
h (s)
p
e- r/2+
Ah(s)
(17)
and interpreting Ah(s) as a frequency function whose inverse transform is Af(t) and
hence is nonzero only for 0 < t < 7/2 where it amounts to a small ripple. From Eq. 17
we may now form
- s r/ 2
- s
e
e
/ 2
h(18)
+(18)
h
p
Through squaring and neglecting the term involving (Ah)
2,
we have
- ST
ST*= e --S-+ 2h
2Ah eT/
- sr/2
h
p
whereupon the pertinent relation (Eq. 12) becomes
h *(s) =
h (s) (1 - e7
*) =
p
(s) (1 - e-T) - Ahe
7 h P20
70-
(19)
- s r / 2
(20)
(IX.
COMMUNICATION RESEARCH)
and the corresponding time function is
f(t) =
(fp(t) - fp(t-)
-
Af(t -
(21)
.
for the
The first term in this last expression equals one-half the function f*(t)
p
*
interval 0 < t < T and is zero otherwise. The second term is f (t) for 7/2 < t < T with
P
The result (except for a factor of 1/2) equals f (t) for 0 < t < '/2,
its sign changed.
-f (t) for T/2 < t < T, and is zero otherwise.
p
The second-order effects, which have been neglected, will slightly alter the form
of f*(t) during the interval 0 < t <
T,
and will prolong the error in an asymptotically
decreasing fashion thereafter.
The method of finding a system function appropriate to a desired transient pulse of
arbitrary shape as discussed in the above paragraphs is the logical generalization of
a method used during the war for designing networks to produce rectangular radar
pulses. It took the writer approximately ten years to stumble upon this generalization.
E. A. Guillemin
2.
Basic Existence Theorems
Section 1
A New Integral Representation of Laplace Transforms
A set of new integral expressions for the representation of direct and inverse
Laplace transformations will be presented first. These integral expressions are the
1. 0
basic tool for proving a set of existence theorems for fundamental problems in the
theory of network synthesis. Several existence theorems are presented in this report.
1.1
Let
f(t) =
f 0
0
for t > 0, and such that
(1)
for t < 0
where M and c are finite positive constants.
Let c o , the abscissa of convergence,
the lowest bound of c which satisfies this inequality.
Laplace transformable.
If(t)I < Mect
be
Then, as is well known, f(t) is
Its transform F(s) is analytic in the right half of the s plane
beyond the line parallel to the w axis and at a distance c o from it.
The direct and inverse Laplace transformations are given,
integrals
-71-
respectively, by the
(IX.
COMMUNICATION RESEARCH)
00
f(t) e - s t dt
F(s) =
s
+ iw
0
f(t) = I
JF(s)
est ds
fT171d
where r is a line going from -ioo to ioo such that all the singularities of F(s) lie to the
left of F.
1.2
For the purpose of our representation we shall select F as a symmetric curve,
having the branches F and F'. F' is the image of F with respect to the real axis, as
shown in Fig. IX-21. An arbitrary point of the s plane will be denoted by
s =
For points on the contour
cr + iW
(3)
Ewe shall use, for convenience, the notation
S = y + iX
(4)
Finally, the analytic expression for 1 is
(y,
)= 0
(5)
A simple algebraic process shows that the second integral in (2) can be expressed
by an integral along F only,
f(t)
=1
f
F(S) eSt dS - F(S) et
Zi
dS
from which we obtain the form
f(t) =
±
eyt
U(y,k) [sin Xt dy + cos Xt dX
F-
+ V(y,k) [cos Xt dy - sin Xt dX]
(7)
in which
F(s) = U(o-,
) + iV(cr, w)
F(S) = U(y,X)+
(8)
iV(y, X)
1.3
Let us indicate by I the distance from the point P to yo along the curve T as
indicated in Fig. IX-22.
From Eq. (5) we get
-ad
dy + a
-
d X=
d
= 0
and we shall introduce the well-known relations
-72-
(9)
Fig. IX-21
The contour P
is defined by the function €(y, X) = 0.
Fig. IX-22
The contour I, the coordinate I and the direction cosines of d.
s PLANE
Fig. IX-23
The contour r.
-73-
0
(IX.
COMMUNICATION RESEARCH)
a
cos 0(y,X) =
dy
)2
2~
(10)
2a
sin
()D
(y, x) =
A few algebraic manipulations lead to
f(t) =
f e
IF(y, k)
sin [Xt +
(y, ) +
(y, )] dcU
(11)
in which
'
F(S) = IF(y,X) I ei(y, X)
1.4
(12)
We shall now obtain the corresponding expression for F(s).
Since
f(t) est dt
F(s) =
0
then one can multiply (7) by (e s t dt) and integrate between zero and infinity. By
reversing the order of integration (which is justified here because r is to the right of
the singularities of F(s)), and considering the integrals
X
(s-y) 2 + 2
e-st e
sin
t
(13)
s-y)
(s-y)
+x
one gets
F(s) = IfU(y,X)
[Xdy + (s-y) d
+ V(y,k
)
[(s-y) dy -
1
dk]
(s-y)
+X
(14)
By introducing the functions defined by
s - -
(s-y)
(s-y)
=
cos K(s, y, k)
+
+ x
sin K(s, y, X)
-74-
(15)
COMMUNICATION RESEARCH)
(IX.
one gets
F(y,X) sin
F(s) =
1.5
K(s,, y,)
(16)
d
(y)
+(y,)+
The expressions for f(t) and F(s) of a strategic contour are very useful in proving
the final forms at which we are aiming.
They also play a basic role in later theorems.
The new contour is the semi-infinite line obtained by setting y = -o = constant > c .
See Fig. IX-23.
This particular contour will be designated by 0
The corresponding integrals follow by setting dy = 0, or 0 = w/2, and dX = dl.
They
are
yt
f(t) = e
1
U(Y o
,
) cos Xt - V(Yo,
Xt
) sin
dX
o
(17)
IF(yO, X ) I cos [Xt +
= e-
(yo,k)
dX
(s -
P
) +X
(18)
=
IF(
,X)I cos [K(so,
k) + P(,)
dX
ro
1. 6
Our aim is to produce two final sets of basic integral representations.
The first
of these sets reads
f (t) =-
2 r vt..
e' U(y,
r
=2f eYtU(y,X)
7
r
) Lcos Xt dX + sin Xt dYjI
[sinXt + 0(y ,)]
Cd
(19)
F(s) =
j
Xdy + (s-y) d)
S(s-y)
=.
sin
)
U
2
+
X
K(s, y, ) 6(y, )1 U(y, k) d~i
Iwhich
both
permits f(t)
usto express
and F(s) in terms of the real part of F(s)
which permits us to express both f(t) and F(s) in terms of the real part of F(s)
-75-
(IX.
COMMUNICATION RESEARCH)
along the contour F.
A direct proof of (19) is rather long for this report.
Instead, we shall first furnish
a simplified (and correct) proof, valid for the particular contour F
F to F is not difficult.
o .
The passage from
0
1.7
The proof for
example,
E.
Fo
follows with ease by introducing Hilbert transforms. (See,
A. Guillemin: "'The Mathematics of Circuit Analysis, " John Wiley,
New York 1949,
for
pp. 330-349.)
Let f(z) = U(x,y) + iV(x, y), z = x + iy.
The transforms are given by
+00
U(x,y) = -f
xU(o,
Tx
d
0) x 2+ (y-n)
(20)
+00
V(x,y) = -f
(y-
) U(0,
d
- 00
To apply them to our case we set x = y - yo,
F0
+ F'.
0
y = X, and designate by
0 the contour
We obtain
U(y,X)
-
=
Y)
U(Y,
S(V -Vo)
d
+ (X-)
Fo
(21)
V(y,)
(_
= -
-YO)2
'T
- (y - y 0
(
+
(X-Tl)
10
1
2 d_)
/
where yo > c o . Substituting these expressions in (7) and inverting the order of integration (justified by the condition yo > c ) one gets
f(t)=
1
F
U(yo,1) d
f et
r0
I
U(y,X) (sin Xt dy + cos Xt dX)
(21a)
+ V 1 (Y, X) (cos Xt dy - sin Xt dk
where
U(Y,X
) =
Y - ¥o
V
(V - YO)
2
+(X-q)
(22)
V(Y,
X)
=
-
)
(V- ( ,)2 (X+ (×-hi
n
-76-
COMMUNICATION RESEARCH)
(IX.
respectively,
One recognizes immediately that expressions (22) are,
the real and
imaginary parts of the function
(23)
F 1(S) = s - (y
0 + in)
=
The corresponding time function, say fl(t), is
S
t
0
for t < 0
(24)
fl(t) =
e+(o +i)t
Now,
reveals immediately that the bracket parenthesis of (21a) is
expression (7)
Consequently, Eq. (21a) becomes
exactly the function fl(t).
U(y 0 ,
)
for t > 0
e
(y +iq)t
U(y , 0 )
dq = e T
cosi
t +
i sin
t]
d'
for t >, 0
.
L
(25)
f(t) =
for t < 0
Finally, recalling that U(y 0 , 1) (in Laplace transforms) is necessarily an even function,
one gets, by changing 9 to X
U(y
S2e
,
k) cosX t d
for t > 0
r
(26)
f(t) =
fort <0
0
where yo > c .
By the direct Laplace transformation of (26) we get
F(s) =
2(s -Yo)
IT
U(o, X) dX
for yo > co
(s -
-77-
o) +X
(27)
COMMUNICATION RESEARCH)
(IX.
Therefore,
1.8
expressions (26) and (27) show the veracity of (19),
at least for
=
0o
Expressions (26) and (27) produce both f(t) and F(s) in terms of a generic function
U(Y0o ,
) for F = o .
These expressions can be solved for U(a, o).
From the first integral in (2) one gets
F(s) = U(cr,
f(t) e - 3 t (cos ot - i sin ot) dt
) + iV(a,wo)=
(28)
from which
U(cr, w) =
Je
-
t
cos wt f(t) dt
0
Hence
(28a)
0o
U(Yo, X) = e
f(t) cos Xt dt
o
0
which inverts (26).
The inversion of (27) is obviously given by
= Real (F(s))
U(o-,o)
U(o ,
) = Real (F(s)s
=S
for S = 'o
+
ii
For convenience we shall put these integral representations together for
will be very helpful in a subsequent discussion.
f(t) =
2e
(29)
I
= po .
0
They
ytt
U(y ,X) cos Xt dX
o
F(s) =
2(s - yo)
U(y0 ,X) dX
(s - y
U( 0 ,kX) = e
o
U(o, X) = Real
)2
+ X2
(30)
'f
f(t) cos Xt dt
F(s)4s = S
-78-
for S = -y + iX
COMMUNICATION RESEARCH)
(IX.
where yo > c 0
The reader may note the apparent simplicity of these expressions and
the interchangeability of the functions U(-
r=r
o
f(t) e
, X) and (
t) along a contour
o
1.9 We must now prove the correctness of (19) for any F contour (for which all the
singularities of F(s) lie to the left of F). A simple (heuristic) reasoning renders a
lucid proof. It is based on the following well-known facts of the Laplace transforms.
Let f(t) have F(s) = U(cr, w) + iV(or, o) as Laplace transforms.
Now, let us construct a new function
G(s) = U(a-, o) - iV(o-, o)
Then
for t> 0
0
(31)
for
L-1(G(s)) = g(t)
for t < 0
f(t)
(The proof follows by setting
f(t) = f 2 (t) + fl(t)
0 = f 2 (t) - f (t)
1
where f 2 (t) is an even function of (t), and f 1 (t) is an odd function of t.)
To prove our expression (19), we use (7) and set
g(t) =
r
sin Xt dy + cos Xt d] - V(y,
eYt (U(y,)
) [cos Xt dy - sin Xt dXk
(32)
Therefore, as a consequence of (31) for t > 0,
f(t)+ g(t) = f(t)
f
et
U(y,
) [sint
dy + cos Xt dX}
fort> 0
(33)
By direct integration of (33) one gets
F(s) = 2 f
F(s)
dy + (s-y) dk}
II(s-y)
U(
X)
+ X
(34)
Therefore, the representation (19) is correct for every contour F as prescribed before.
Similarly, as in Eq. (30), Eqs. (33) and (34) can be solved for U(y,X). Take (28)
and set s = S.
Then
-79-
(IX.
COMMUNICATION RESEARCH)
Yt
f(t) e
U(y, X) =
(cos Xt) dt
0
(35)
X(y,
) =0
1. 10
The integral representations developed in the previous pages have been found by
using the two basic assumptions:
c t
f(t)
(a)
(b)
4 Me o
The contour
for every value of (t)
P
must always stay to the right of every singularity of F(s).
Under these conditions all of the integrals above exist in a Riemann sense.
Our analysis
breaks down if we allow the contour of integration to go through one or several singularities of F(s), keeping the rest of the singularities to the left of F.
for
=
0 this situation arises when yo = c .
integral fails to exist.
impulse at t = t o
for t = t
.
.
For example,
Under this assumption the Riemann
Stipulation (a) excludes a very important time function:
Clearly condition (a) does not hold for every value of t,
However,
the unit
in particular
U(y,X) exists in this case since
U(y, X) I
dt < j
ff(t) e
0
f(t)
(36)
dt = 1
0
A set of important existence theorems in network synthesis can be obtained with
' to run through several or all singularities of F(s),
always keeping the remaining singularites to the left of r.
It is also important to
ease when we allow the contour
admit time functions formed by a time distribution of impulses whose total area is finite.
This new situation can be handled by the introduction of a new Stieltjes integral
representation which contains, as particular cases, the Riemann representations derived
above.
For the purpose of this report, it is enough to produce the Stieltjes integral
representation for r =
Fo valid now for yo > co (abscissa of convergence).
progress reports we shall consider the general case
.)
(In future
The briefness of a progress
report forces the assumption that the reader is acquainted with at least a few elementary properties of Stieltjes integrals.
1. 11
We will introduce directly,
without further elaboration,
Stieltjes integral representation for
P=
I
o
, -y >Co .
the corresponding
Let us introduce the following
functions, which we will assume exist and are of bounded variation;
The reader may immediately produce the representation for any P with relative ease,
particularly when r possesses continuous first derivatives for all values of 1.
-80-
COMMUNICATION RESEARCH)
(IX.
(yo,
along F
U(yo, 1 1 ) d
=
)
(37)
t
"(y , t) =
(38)
e
f()
0
and an alternate function
t
T(t) =
f()
(39)
d.
The definition of "bounded variation" implies the existence of the integral inequalities
IU(y,X)
If(t) e
I
-y t
dX <
(40)
oc
dt <
(41)
00
and alternatively
If(t)l dt < 00
(42)
0
Under these stipulations, the corresponding Stieltjes integral representations (which
contain (30) as a particular case) exist:
f(t) = 2e
yt
cos Xt dp(Yo,X)
(43)
Fw)=2(
F(s) =
2(s - yo)
1T
d((y , X)
(s- yo
U(y0 , ) = j cos Xt
0
and alternatively
-81-
)
+X
dT(yo, t
)
(44)
(45)
(IX.
COMMUNICATION RESEARCH)
00
k) =
U(Yo,
(46)
e Yot cos Xt dT(t)
0
if (42) holds.
With these integrals, as will be shown later, we can set yo > co.
The equality sign
is unacceptable in the Riemann integral representation.
For yo > co the representations above coincide with (30). Take (43) and (44), which
depend on (Yo0 , X). A well-known theorem of the Laplace transform of a time function
c t
f(t), If(t)I 4 Me o , states that F(s) is analytic for every point s of the right half-plane
defined by the line s = c + iw. The selected contour lies completely in points of analyticity of F(s), since yo > c .
The corresponding continuity and finiteness of U(y
allow us to differentiate the integral (37) with respect to X. Hence
d4(yo,X) = U(yo,X) dX
o ,r)
(47)
which justifies the assertion.
Examples of the application of these integrals will be found later.
1. 12
Equations (43) to (46), particularly (44), will be written in a canonical form which
is basic in the existence theorem of transfer functions.
we will refer directly to the integrals (43) and (44).
For briefness in presentation
The corresponding extension to
(45) and (46) is obvious.
From (37) and (40)
IYo ,
< i IU(Y, X)) dX < 00
(48)
0
indicating that
(Yo, X) is a function of "bounded variation" in the interval (0, oo).
The following theorem (well-known) will be used.
Theorem (1-1.12).
(a, b).
Let
c(X)
be a real function of "bounded variation" in the interval
Then, there exist two functions c(+)(X) and
()()
which in (a, b) are:
i) non-
negative; ii) nondecreasing; iii) bounded, and vanishing at X = a; iv) discontinuous at
the same points as p(X),
such that
p(X) -
(a)
(X) -
(_()(X)1
(49)
Va(X)
where Va(k) is the variation of
=(+)()
+ ()()
(X) in the interval (a,
Although the proof is simple, we omit it here.
functions Va(X)
4(+)(),
(_)().
The function Va(X) is given by
,
-82-
k).
Our concern is to construct the
(IX.
Va(Yo,
) 1dT
0
IU(-o,
) =
COMMUNICATION RESEARCH)
(+)(Yo, ) -= U(1)(Yo,
(50)
) di
0
,( )(yo,
X)=
where
U(1)(Yo, K) =
IU(Yo, X)
for U(Yo ,X) > 0
0
for U(Yo ,X)< 0
0
for U(yo,
(51)
U ()(o,
)=
)
> 0
(YX)
)l for U(Y o ,X) <
=I|U(vo,
0
Figure IX-24 provides a simple graphical illustration of the process of the extraction
of U( 1 )(Yo, X) and U( 2 )(Yo, k) from U(y 0 , X). Figure IX-25 produces the corresponding
graphs of Va(X),
(+)(), (_)().
The above theorem and the conditions of boundedness imposed on the functions
(43) to (46) as the difference
(y', X), T(y o , X), T(y o , X) justify the writing of integrals
always real, non-negative,
T
(_)are
T +),
(_), T(+), T(_),
of integrals, where (+),
nondecreasing, etc., functions.
Hence
os Xt
f(t) = f( 1)(t) - f( 2 )(t) =
d()(y,
)
cos Xt d(_)(
-e
F(s) = F(1)(s)-
F(2)(s)
d (+ )(Y0 ,X)
2(s - y )
I
(s - yo)
2
2(s-
z
-y
)
I
dP(
(s -
T+
)(Y
(53)
o ) + X(53)
1o
ro
U(,
(52)
To
ro
=
,
0 X)
X)= U( 1 )(Yo,X) - U( 2 )(Y0 , X)= f cos Xt
dT
)(,
t) -
Scos
0
0
or alternatively
-83-
t
dTr)(y0 ,t)
(54)
(IX.
COMMUNICATION RESEARCH)
U(o,
k)
) = U( 1 )(Yo,X) - U( 2 )(y,
o-yY t
e
cos Xt dT(+)(t) -
=
0
e
-y t
cos Xt dT _ (t)
0
(55)
which are the forms we wanted to introduce. The following properties are important.
It can be noted that when U(y o , X) along 'o , (Yo > C ),is either a non-negative definite
or a nonpositive definite function of X, then only the (1) system or the (2) system,
respectively,
exists.
Several other theorems are omitted for brevity.
1.13
A series of basic existence theorems on transfer functions will be deduced, in
section 2, from the integral representation given above. We are going to show that the
functions Fl(s) and FZ(s) are positive real, (p, r) when c , the abscissa of convergence,
is equal to or less than zero.
In the light of the result given above,
F(s) can immediately be recognized as a
Now, all real, single-valued, bounded functions of time
transfer function if co < 0.
have an abscissa of convergence in the Laplace transform c o which is equal to or
less than zero. Besides, the Laplace transform is a transfer function which can
degenerate into a single (p, r) function if U(O, X) >, 0 for every point in the interval
(0o< X< oo).
The transfer character of F(s) immediately suggests the existence of a certain
discrete network which realizes this transfer function. We will show that for Co
0
0
we can find a four-terminal passive network which realizes this F(s). For c > 0
we enter the realm of active networks. Section 2 presents a series of theorems concerning the situation indicated in this subsection.
U(yr,X)
0
v a (X)
I
x
i~--CD
I
I
I
I
I
UI
L
K
-
N
--
J
I
I
I
I
~-*-
I
U(2 0)( 0 ,X
t0
H-
M
I
I
Fig. IX-24
The functions U(1)(Yo, X), U( 2 )(Yo 0 ,).
I
I
I
I
Fig. IX-25
The functions Va(k),
)()
-84-
I
I
I
,
(_)().
(IX.
COMMUNICATION RESEARCH)
A final form of integral representation is given in this subsection. We have
assumed that the contour F runs to the right of all the singularities of F(s). The new
representation is valid when the contour P leaves some singularities to the right and
1. 14
This representation is important because it reveals the character
of the singularities of F(s). In fact, it shows that F(s) has a meromorphic behavior in
.
the half-plane defined to the right of
others to the left.
The following form is given with regard to r
o
, but y > co.
Let f(t) be bounded as in the previous theorems, then
Theorem (1-1.14).
(s-
exp
F(s) =
+
(s -y)
o)
ro
(X) is a function of "bounded variation. " 1T 1 and Tr2 are Blaschke products.
x
0
Note: The Blaschke products are constructed in our case as follows. Let s' , s'
where
be respectively the poles and zeros of F(s) to the right of
I
Z'
S'
Z
s-
o
. Then
cos pP + s'
s'I
r
I
TT-
+
s ' l
'
c os
+ s,
FT
1-i.
S
2
-
v,
I'
+ s's'I os P + s'
IJ
s ' I cos
x
P
+ s '2
s' = (s - y0 )
Section 2
On Transfer and Impedance (or Admittance) Functions
2.0
Two-terminal impedances, or admittances, of a discrete or a continuous passive
network belong to the class of positive real, (p, r) functions.
In general, positive (p) and positive real (p, r) are defined as follows:
Let Z ((s),
s = a- + io,
be
(a)
single-valued
(b)
analytic for all points a- > 0
(c)
such that, if we write Z = U(o-, w) + iV(C-,
0), then U(-,w) > 0 for - > 0.
If these conditions are satisfied, then Z (p(s) is called a positive (p) function (of the
right half-plane).
If,
in addition,
Zp (s) also satisfies the fourth condition
-85-
(IX.
COMMUNICATION RESEARCH)
(d) that Z(p)(s) is pure real for s real
then Z p)(s) is called a "positive real function, " (p,r) (of the right half-plane).
For convenience we will introduce the notation Z(s), composed of Y and Z, for (p,r)
functions.
We assume that the reader is acquainted with the following theorem (HerglotzCauer).
Theorem (1-2.0).
The necessary and sufficient condition for a function H(s) to be
analytic and have a positive real part for a-> 0 (s = a- + iw) and to be real for s real,
(in other words, to be (p, r)), is that it can be expressed by the Stieltjes integral
o00
H(s) = (Z(s)) =
+ sC
(56)
where p(X) is a nondecreasing, non-negative, real function of "bounded variation," and
C is a constant (equal to lim sZ(s)).
S-0o0
2. 1 With this theorem as a basis, we can produce the following theorems.
Theorem (1-2. 1).
If co 4 0, then the functions F( 1 )(s) and F( 2 )(s) are both (p,r).
For
this reason we will use the suggestive notation F(1 = Z(l and F(2
)
),
) = Z(2)
Theorem (2-2. 1).
is zero for t < 0.
Let f(t) be a real, single-valued, bounded function of time, which
f(t) may possess a denumerable set of isolated points of simple dis-
continuity and a denumerable set of isolated points where f(t) shows an impulse of finite
area, such that the sum of the areas is finite; then the associated functions F(1)(s) and
F(2 )(s) are both (p,r) functions.
It is enough to show that co
.< 0.
For if f(t) is bounded and has a bound M almost
everywhere then the condition of Eq. (1),
If(t)I < Mect, is satisfied with c = 0.
The
impulses are taken care of because they satisfy (42).
2.2
Now, let us produce two basic theorems.
If, by extension of the ideas in the theory of networks,
we define a "transfer
function" as the difference of two (p, r) functions (in section 3 there is a justification of
this extension), then we can produce the fundamental theorems of existence of transfer
functions.
Theorem (1-2.2).
The necessary and sufficient condition for a function H(s) to be a
transfer function, is that it can be represented by the Stieltjes integral, (57),
alternate forms (58), (59):
-86-
or its
COMMUNICATION RESEARCH)
(IX.
sin K(s, y,X) + 0(y,X)] d(I)
F(s) =
2
F(s) =- T (s -
<
0O s
for every continuous contour
0)
j
F
d(y , X)
(s- yo)
(57)
(58)
+
+X
+
F which shall be made to coincide with the upper
imaginary axis (as in (59)), where 4(I), 4(y
0
, X) and p(O,X),
respectively, are functions
of "bounded variation" along F.
The results are independent of f.
The condition c o
0 implies that F can coincide
<
wholly with the positive imaginary axis.
Theorem (2-2.2).
Then, its Laplace trans-
Let f(t) be as defined in theorem (2-2.1).
form is necessarily a transfer function.
Theorem (2-2.3).
Let F(s) be a general transfer function as defined above.
Then its
inverse Laplace transform is necessarily a function f(t) as defined in theorem (2-2.1).
The fundamental character of theorems (2-2.2) and (2-2.3) in regard to network
theory is quite obvious for they define the open field of network synthesis possibilities.
A group of related theorems was omitted
The network aspect is discussed in section 3.
for lack of space.
2.4
For convenience we shall illustrate with simple examples the application of the
theorems given above.
Example 1.
The function e-s is a transfer function which represents the unit delay.
Its inverse Laplace transform is
a unit impulse delayed one unit of time.
From e
theorem (1-2.2) and use, for example, (58).
-
U(y , X) = e
Direct theorem.
integral.
V0
o
- s
one gets
os X
can be taken as a Riemann
Let us consider yo > c , so that (58)
Then
Consider
00
2(s - Y0 ) -yo
F(s)
cos X dX
e
-
0
(s-g
2
)
+
+
2
The value of the integral in (60) can be obtained from the well-known result
-87-
(60)
(
(IX.
COMMUNICATION RESEARCH)
W
cos x
-a
a +x
Hence
2(sF(s) =
Example 2.
a > 0, b > 0.
yo)
Direct theorem.
-Yo
-_0
e
S
(
2(S - Ylo
e
-(s -
o)
-s
=e
Let us consider the transfer function F(s) = (s-a)/(s+b),
The abscissa of convergence is s = -b.
Then, for simplicity, we can set
o = 0 (Riemann integral).
By direct computation
2
2
L
X2 - ab
X - ab
1
L
U(0, X)=2
'
2
2+2
2
22
2X + b s +X 2X 2 + b
X2 + b
L
2
2
X2 + s
where
b+ a
L
s
2
s
2
-b
2
- ab
s
b
2
By direct substitution in (60) and using the well-known integral
+ x2 =
a
oo
F(s)
-j
0
Example 3.
22
+b
s
=
22
+X
-
7
Illustrations of the inverse theorem-Stieltjes integral.
Now we can
arbitrarily choose U(y 0 , k), except that it must be of "bounded variation." For simplicity set yo = 0 and choose U(0,X) as in Fig. IX-26a. Here, the Riemann integral evidently fails to exist. The Stieltjes integral exists and for the particular selection of
U(0,
k) it reduces to a sum of finite terms.
oo
F(s)= 2s
S
d(k)
2
2
0 s+
s
Ia
z
l
+x
= z(1)(s)-
a3
2 +2
1
s
2 +2
+1X s
a2
a4
+X2
2
a5
2
s
2
Z( 2 )(s)
from which the (p,r) character of Z( 1 )(s) , Z( 2 )(s) and the transfer character of F(s) are
evident.
-88-
n II
,
MIma
CJ
A2
0
O
1
I
X3
XI
+ C
+IN
4c.I
a,
dH
X4
Sn
(a)
(b)
X4 X5
A
(C)
(d)
Fig. IX-26
(a) The selected function U(O,
); (b)
;=
U(O,
) dk;
0
(c) the function
(
); (d) the function
-89-
(_)(k).
(IX.
COMMUNICATION RESEARCH)
Example 4.
Here, we will illustrate the use of integral (43) in computing the corresponding time response associated with U(0, X) as defined in example 3.
Setting y
= 0 in (43), the Stieltjes integral reduces to a finite sum of terms.
immediately gets
f(t) =a
One
cos Xt - a 2 cos X2t + a 3 cos X3t + a 4 cos X4t - a 5 cos X t
2.5
We close section 2 by pointing out three basic features of our integral representation:
(1) The functions f(t) and F(s) are uniquely determined in terms of the function
U(y,k) along an arbitrary contour F .
(2)
The function U(y,k) and the contour
restrictions already given.
(3)
F can be arbitrarily chosen, except for the
Then, for any selection, the above integrals allow
us to generate a transfer F(s) and its inverse Laplace transform.
Conversely, given f(t), or F(s) we can find the corresponding value of U(o-, W)
along a prescribed contour F .
Section 3
The Electrical Passive Network Associated with the General Transfer Function F(s)
The functions F(s) generated by an arbitrary selection of U(y,X) and F are not
necessarily rational functions. Therefore, the existence of the electrical networks
3.0
which synthesize them is not evident.
The purpose of this section 3 is to show the
existence and construction of such networks.
Theorem (1-3.0).
Discrete networks.
The main theorem of this section is:
Let f(t) be a real, single-valued, bounded function
of time, which is zero for t < 0. f(t) may possess denumerable sets of points of isolated
simple discontinuities, as well as a denumerable set of points at which f(t) possesses
impulses of finite area whose sum is of "bounded variation. "* Then there exists a
passive linear, finite network, having finite element values such that its transfer
function F (s
approaches F(s) uniformly as n -o for every point s, C- > 0. Let f (t)
be the time response corresponding to F (s). Then, f (t) is zero for t < 0 and any n,
and fn(t) tends to f(t) as n-oo at almost every point 0
t< oo. (The exceptional points
)
are the sets of discontinuity and the impulses of f(t).)
The proof consists in finding a sequence of functions Un(X,y) such that Un(Xk,) -U(X,y),
as n oo, and each Un(X,y) possesses an Fn(s) which is necessarily a rational transfer
function. Instead of formal proof we use a simple and illuminating heuristic approach.
The theorem is true for a more general class of functions f(t).
select f(t) as defined by theorem (1-3.0).
-90-
For simplicity, we
COMMUNICATION RESEARCH)
(IX.
Theorem (1-3.0) has an alternative form associated with "continuous" networks.
3.1
Heuristically, we can see with ease that theorem (1-3.0) (and its alternative form
The conditions imposed on f(t) imply that co < 0.
for continuous networks) is correct.
(if the singularities stay on or to
Since the results of our integral are independent of P
the left of
For simplicity, let us start by taking an arbi-
F), then one can use -o = 0.
From U(0, X) we
trary function U(0, X) which satisfies the requirements of our integral.
construct the function
(0,
X)
U(0,i) dr
=
0
(+)(o0, ) and
and from it find
negative,
(_)(0, X) which are both of "bounded variation, " non-
nondecreasing and which have the same points of discontinuity as 4(0, X).
First we find the two-terminal impedance or admittance networks which correspond
We will discuss the procedure corresponding to Z( 1 )(s).
to Z( 1 )(s) or Z( 2 )(s).
The
procedure is the same for Z( 2 )(s).
(+)()has a graph as in Fig. IX-27a. We can split
Suppose that
(+)d(k) . See Figs. IX-27b and c.
continuous '(+)c(X) and the discontinuous
m2s
d(+)c(X)
2s
d(+)d(X)
2
1 2 +
s'+
2s
+
SZ(1)(s)
X2
s
k=0s
2
we introduce a sequence of functions
Let
them.
-
+s
d (+)c(X)
2T
s2
sJ+s
k
because of an elementary property of Stieltjes integrals.
manner.
2
0
0
Then
oo
00
o0
(+)(X) into the
To handle the second integral
n(+)c(k) which approaches
p(+)c(X)
in a stair-like
See Fig. IX-28.
AJ be a set of disjointed intervals which cover 0
t < o00.Consider one of
Without loss of generality we use A o , where there are, say, n o jumps.
By using
the same elementary property of the Stieltjes integral and introducing self-explanatory
notation, one gets
n
Z(1)c, n (s)=
Z( (s
ak, n
+)X
J
0
n =0
k= 0 s
+
By geometrical intuition the reader can infer that
Z(1)c,n(S)
-
Z( 1 )c(s)
uniformly
n
0
-91-
-
00
J
k, n
BOUND OF
(4)()
W
,
-N POINTS OF
DISCONTINUITY
I
n POINTS OF
I
DISCONTINUITY
___
I
m POINTS OF DERIVATIVE
DISCONTINUITY
2
S
I
--
12
I
i
i
2
(c)
Fig. IX-27
Example of ,+)(X) discontinuous.
THERE ARE no JUMPS
IN Ao
(+)c
THE AUXILIARY FUNCTION
eAo
U
o
e
AI
Fig. IX-28
The function c(X).
Zo(s)
UNIT
t)
Fig. IX-29
Example of open lattice realization of F(s) for arbitrary U(O, X).
-92-
COMMUNICATION RESEARCH)
(IX.
we have found a set of rational functions which are (p, r) and which approximate
Foster
Z( 1 )(s) as prescribed before. Then, the above procedure always leads to a
canonical two-terminal structure which approaches Z( 1 )(s) as prescribed before.
Hence,
We obtain an approximation to Z( 2 )(s) by a similar procedure.
Consequently, the
Z(2)n
corresponding transfer function F(s) is approached by Fn (s) = Z(1)n
the lattice structure as given in Fig. IX-29, for example, one gets F n
Taking
o
The associated time function f(t) corresponding to the arbitrary function 4(X),
given in Fig. IX-27a, can be obtained at once by means of the Stieltjes integral
3.2
as
00
cos Xt dp(X)
f(t) =.
0
when we use for
(Xk) the auxiliary functions as described above.
Consequently, the
time response has the general form
n
cos X t -
fn (t)=Za
0
where a
and b
n
o
o
EV
b
cos
V
t
v=O
v=O
represent the corresponding steps of the component functions.
The
existence of the limiting function is clear.
The elementary network interpretation of these processes is that one can find a
four-terminal network such that f(t) is the response when it is excited with the unit
impulse.
3.3
The alert reader may claim that there is a fallacy in the above reasoning.
For
we may say that, as n increases without limit, it is necessary that every step irak/2
Therefore, as n- oo, the network
tend to zero in the continuous interval of 4(k).
element value goes to zero.
Consequently there is no such "discrete" network.
Besides,
as n - oo, the poles become everywhere dense in those intervals of the imaginary axis
along which 4(X) is continuous. This continuous array of poles may produce confusion
at first sight.
In spite of the above consideration, there is no basic flaw in the heuristic approach
of subsection 3.2. The real meaning is that this procedure takes us into the realm of
a distributed system having a continuous spectrum of natural frequencies. The elementary theorems of separation of poles and zeros tell us that as the poles become
denser and denser, the zeros of each impedence function Z( 1 )(s), or Z( 2 )(s), also
become denser and denser.
The interpretation is that F(s) possesses a branch cut
along the segments of the imaginary axis where the continuous arrays of poles and zeros
appear.
-93-
(IX.
COMMUNICATION
RESEARCH)
A heuristic approach which produces finite element value is illustrated in a
particular, but illuminating, case in the next subsection.
3.4
We may proceed as follows. Suppose that {Ak} is a set of finite disjointed
intervals which cover 0 < X < oo. In the new approach we assume that
oo
F(s) =d(X)
2
r
k=0 A
2
+
Here we assume that the denominator (s 2 + X2 ) changes slowly with respect to X in
each interval Ak, assuming an average value (s + Xk), kkEAk. Then
F(s) =T
2
2
k=0 s +
d() = s
kA
2
k=0 s
k
2
+ k
where
dc()
Ak =
Ak
For example take U(O,X) = cos X.
We have found that F(s) = e - s . The U( 1 )(0,
) consists
of positive parts of cos X and is zero elsewhere. The U Z)(0, X) is formed by reversed
negative parts of cos X and is zero elsewhere.
We take all Ak equal to the area of
each spike times Z/r, and take Xk in the middle of each spike. Hence
F(s)=
s+
2 +
1
s
2
2
+i
2
2
1
+
2
2
sr + r
)2
2
s
1
2
+
+ (3T)
Evidently, the element value does not go to zero as n increases.
heuristic approach for discrete networks.
3.5
This finishes the
For mathematically-minded readers the theorem (1-3.0) can be proved rigorously
by a well-known theorem of analysis. It is enough to prove the realizability of Z(1)(s
).
Consider the function defined below.
Let {gp
and {rp} be two sets of real numbers such that o gp
1; -oo < r < + co,
p = 0, 1, 2 .....
Construct the functions H and H as follows
-94-
(IX.
go
COMMUNICATION RESEARCH)
s
H=
2
gl(s
1 +ir
- 1)
s+
2
- 1)
(1 - gl) g 2 (s
o
1 + ir
s +
(1 - g)
1 + ir
s +
g 3 (s 2 - 1)
1 + ir3s +
(61)
gos
H=
1 - ir
s +
g 1(s -1)
(1 - gl) g 2 (s
1 - ir
- 1)
s +
(1 - g 2 ) g 3 (s
- 1)
1 - ir s +
1 - ir3s 3+
and write F(s) = H(s) + H(s).
Theorem (1-3. 5-condensely stated).
A necessary and sufficient condition for a
function to be positive real in the s plane is that it have a continued fraction expansion
of the form
F(s) = H(s) + H(s)
See, for example,
H. S. Wall:
Continued Fractions,
D. Van Nostrand, New York 1948.
The process of computation of the element values of the sequences {rp} and {gp} is
given in the same book.
It can be shown that any approximant of H(s) + H(s) (but not of H(s) alone) is a (p, r)
function.
3. 6 The rigorous proof of the existence of the rational fraction approximation of F(s)
follows immediately from the integral representations given in subsection 1. 14.
M. V. Cerrillo
-95-