IX.
A.
COMMUNICATION RESEARCH
MULTIPATH TRANSMISSION
H.
W.
E.
R.
Prof. L. B. Arguimbau
Dr. J. Granlund
Dr. C. A. Stutt
1.
Speech and Music.
H. Cross
C. Kinzinger
E. Manna
A. Paananen
E. M. Rizzoni
G. M. Rodgers
R. D. Stuart
Transatlantic Tests
The series of tests discussed in the last few Quarterly Progress Reports has been
concompleted. As a result of these tests, it can be said that under many ionospheric
ditions it is not practical to use the standard system of FM transmission for transmitting
speech and music over long short-wave paths.
The reason is roughly as follows.
An FM receiver will select the stronger of two
signals (possibly arriving from the same transmitter over paths differing in delay) and
reject the weaker. If three or more signals are received simultaneously, there may
not be one signal that is always stronger than the rest, that is,
stronger than the alge-
In this case, the receiver output depends
braic sum of strengths of the other signals.
on each of the signals, and serious distortion may result.
if all signals arrive from the same transmitter with very nearly the same
delay, the composite signal at the receiver is very much like any one of its components.
Using standard FM broadcast conditions, we found experimentally that if the times of
flight of all signal components lie within 100 psec, the resulting distortion is not very
Of course,
serious and can often be improved appreciably by special circuitry.
We have acquired a much better understanding of the ionosphere while performing
these tests, largely by observing narrow amplitude-modulated pulses received from the
transmitter.
The duration of the received pulse pattern or "smear" time is,
the spread in time of flight mentioned above.
of course,
The limitation on fidelity (or rate of trans-
mission of information) imposed by the smear is not peculiar to frequency modulation
and for this reason we attach major importance to pulse patterns.
Figure IX-1 is a collection of pulse patterns that is representative of our reception
during the past year.
The first nine patterns were observed in a 200-kc/sec bandwidth;
these are probably of greater interest, since it is not usually possible to obtain such
detail in pulse experiments.
The set a, b, c, d is a sequence which we interpret as reception just below the maximum usable frequency (MUF).
The MUF is increasing throughout the sequence.
For
comparison, we interpret e, f, g as scatter reception just above MUF, again with MUF
increasing throughout the sequence. Similarly, h, i, and j represent more samples of
scatter reception; note the lack of detail in j caused by the narrower receiver bandwidth.
Traces k and I are typical of nighttime reception well below MUF.
These pictures were
taken with a longer sweep length than the others.
J.
-53-
Granlund, C. A. Stutt, L. B. Arguimbau
_iiIIIII.
i!_".~.: . -_ _"
_ _. .
a
b
c
d'
e
f
9
h
k
I
U1
~
I
Fig. IX-1
Representative patterns received during pulse transmissions over transatlantic path in 1951.
a
Da.te
Time
(GMT)
Carrier
Frequency
(Me/sec)
Mar. 21
13:40
23.09
Signal Strength
(db above 1 f.l.v
on 50 -ohm feeder)
Duration of
Transmitted Pulse
(f.l.sec)
Sweep
Length
(f.l.sec)
Duration of
Pulse Pattern
(f.l.sec)
Receiver
Bandwidth
(kc/sec)
Quality of Program
Immediately after
Pulse Transmission
72
20
2000
400
200
fair to poor
20
2000
550
200
mostly poor
b
Mar. 21
14: 10
23.09
65
c
Mar. 21
15: 10
23.09
75
20
2000
650
200
fair
2000
900
200
good
d
Mar. 21
17:40
23.09
60
20
e
Mar. 20
13:40
23.09
75
20
2000
200
200
fair
f
Mar. 20
14:05
23.09
70
20
2000
300
200
fair
g
Mar. 20
14:40
23.09
50
20
2000
350
200
h
Jan. 26
14:05
26.0
50
20
2000
120
200
Feb. 1
16:05
26.0
65
20
2000
500
200
j
Apr. 16
13:45
23.09
30
20
2000
250
40
k
Oct. 6
00:15
11.6575
60
100
5000
1400
40
bad
1
Oct. 6
01: 15
11. 6575
60
100
5000
1800
40
poor
fair to poor
poor
fair
(IX.
COMMUNICATION RESEARCH)
26
2.05
/
N
5
08
n
10
100
1000
10,000
100,000
I VOLT
INPUT IN /L VOLTS
Fig. IX-5
Suppression threshold as a function of input.
Report,
July 15,
1950.
The better takeover performance at low signal input values is
especially useful.
One problem still needing attention is the rather high interstation noise levels present in the receiver output.
This is partly due to the high sensitivity of the design while
the rest may be attributed to the widebanding of the ratio detector.
That is,
these
circuits have about the same sensitivity to amplitude modulation regardless of their
bandwidth, while the FM output is inversely proportional to this quantity. Even with two
limiters the noise is not limited completely and hence the resultant noise output is
caused by both the AM and FM components of the noise, with the FM/AM ratio considerably worse than with a narrowband FM detector. Either a squelch circuit or a
mechanical tuning cut-off switch should provide a solution.
R. A. Paananen
-56-
I
+150
-I
I
+ 300
+150
-150
-300
1/2 6AL5
JA /--\
+150
IOK
I
-j
100 K
O.IM
5687
2M
-300
Fig. IX-6
Pentode-gate decoder.
(IX.
B.
COMMUNICATION RESEARCH)
STATISTICAL THEORY OF COMMUNICATION
Prof.
Prof.
Prof.
Prof.
Prof.
1.
J.
W.
R.
Y.
J.
B.
B.
M.
W.
F.
Wiesner
Davenport, Jr.
Fano
Lee
Reintjes
Dr. P. Elias
B. L. Basore
J. J. Bussgang
C. A. Desoer
L.
P.
A.
R.
M.
Dolansky
E. Green, Jr.
J. Lephakis
M. Lerner
J. Levin
Multichannel Analog Electronic Correlator
The sampling pulse generator has been redesigned and rebuilt.
Testing of the equip-
ment with a single channel is being carried out.
Y. W. Lee, J.
2.
F.
Reintjes, M.
J.
Levin
Pulse Code Magnetic Recorder
Successful experiments were conducted on a two-channel part of the pentodegate decoder and, in accordance with the results, a complete seven-channel decoder
was constructed.
The diagram is given in Fig. IX-6.
The decoder requires three
tubes less than the simplest of the former models,
although an additional mastertiming circuit is included to ensure perfect time coincidence of the leading and
trailing edges of the individually weighted decoder pulses.
To establish the over-all performance of a major part of the electronic equipment, the various units were connected as shown in Fig. IX-7. An auxiliary unit (A)
consisting of one multivibrator per channel is used to bypass temporarily the recorder
proper and the following amplifier. Sawtooth, sine-wave, and square-wave signals
were connected to the input. The experimentally obtained waveforms are shown in
Fig.
IX-8, where (a),
(b) and (c)
show the phantastron-plate
Fig. VII-24, Quarterly Progress Report, April 15,
the filter output,
voltage (i. e.
1950), the decoder output and
respectively; the corresponding input voltage (sine wave,
mately 10 kc/sec) is
included in the top part of each picture.
approxi-
For a 50-cps sine
Fig. IX-7
Block diagram of equipment used in over-all tests.
58-
V
a
b
c
d
e
f
9
h
Fig. IX-8
Waveforms obtained experimentally at the phantastron plate, the decoder output,
and the filter output, related to the input waveforms.
-59-
(IX.
COMMUNICATION RESEARCH)
wave the corresponding waveforms are represented
by (d),
(e)
and (f).
The input
and output waveforms for a 300-cps square wave are shown in (g); a 360-cps sawtooth in (h). The performance for sawtooth waveforms was used during an approximate visual adjustment of the plate resistors in the individual decoder stages as
indicated by (i).
The lower part of (i) represents the plate waveform in the phantastron stage of the coder; the upper part represents the decoder output. By changing
the values of the individual plate resistors,
is
adjusted to approximately a straight line.
the lower edge of the decoder output
It
should be pointed out that a cer-
tain delay between the upper and lower figures is
present in all the pictures. This
is more noticeable for the sawtooth waveform (see (g)).
Part of this delay is due
to the oscilloscope; another part is caused by the fact that the coder output lags
behind the sampling time; and, finally, for some frequencies at least, additional delay
is caused by the output filter.
To ensure that no frequencies above 10 kc/sec enter the coder, a filter similar to the
output filter (Fig. VIII-9, Quarterly Progress Report, April 15, 1951) was constructed
and is being tested.
J.
3.
a.
B. Wiesner, L. Dolansky
Information Theory
Transmission of information through channels in cascade
The problem of quantization to a larger number of levels, as described in the
Quarterly Progress Report, January 15, 1952, has been the problem of chief interest
during the last quarterly period.
Consider a communication system consisting of two channels in cascade. The
transmitter sends pulses of amplitude
l1.
The intermediate station may operate
according to the following rules: if the received pulse is positive (or negative) the
intermediate transmitter retransmits a pulse of amplitude + 1 (or - 1 respectively). The
output of the second receiver is a pulse of amplitude + 1 (or - 1) if the pulse received
through the second channel is positive (or negative, respectively).
The capacity of such
a communication system is easily obtained.
It has been shown, however, that if the average power of the intermediate transmitter is left constant, the amount of information received increases as long as the
intermediate station does not retransmit anything when the received pulse belongs to
the interval (-a, +a). This saving of energy permits a larger pulse amplitude to be used
by the intermediate station.
The optimum a increases the information received by
17 percent when S/N = 1 and only 1. 1 percent when S/N = 3.
The amount of information
received can be further increased if the second receiver also introduces a rejection
interval (-p, +p) so that the output of the system now consists of the three-valued pulse
-60-
(IX.
+ 1, 0,
- 1.
P are not given by explicit formulas and could
The optimum values of a and
be found by trial and error only.
COMMUNICATION RESEARCH)
For S/N = 1, the relative increase of information
received (with respect to the case a = p = 0) is 33 percent.
For S/N = 4, the relative
increase is 5 percent.
These results indicate that the type of detection used by the second receiver is of
importance, and consequently it would be highly desirable to be able to compute the
amount of information actually contained in the received pulse.
Such a formulation
would give results which depend only on the operation of the intermediate station.
This
formulation involves the difficulty of integrating expressions of the type
+oo
f
p(x) log p(x) dx
- 00
where p(x) is a weighted sum of gaussian distributions.
The amount of information
received in the case of a binary channel (samples ± 1) was computed by the combined use
of power series and asymptotic expansions.
It is expected that the same procedure will
be applicable to the case of channels in cascade,
even when the intermediate
station
requantizes to a number of levels larger than two.
C. A. Desoer, R. M. Fano
b.
Vocoder
Equipment for operating on the magnitude of the analytic function of speech to obtain
the fundamental pitch period is under test.
Investigations continue in the application of
polar representation to speech and electroencephelography.
A method for designing
wideband phase-splitting structures has been developed and is being written up.
R. M. Lerner
-61-
(IX.
C.
COMMUNICATION
RESEARCH)
HUMAN COMMUNICATION SYSTEMS
1.
D. G. Senft
P. F. Thorlakson
J. B. Flannery
J. Macy, Jr.
E. S. Palmer
Dr. L. S. Christie
Dr. R. D. Luce
Technical Report
Much of the time of the staff members was devoted to continuing the writing of the
comprehensive report mentioned in the Quarterly Progress Report, January 15,
1952.
Data analysis for this report consumed well over 50 percent of the time of our technicians.
At present the report is nearly completed.
We shall not attempt to present a
resume here of the empirical and theoretical results to be presented in detail in that
study, but we will mention the four main areas of analysis:
A) The development of learning within the group as a function of the communication
This group-learning is interpreted as a combinatorial resultant of
network is studied.
individual learning.
B)
The temporal characteristics of group communication are analyzed and shown
to be readily explained by comparatively simple assumptions as to individual behavior.
Again the group behavior is a fairly simple, mathematical resultant of the individual
behavior.
C)
An analysis of the effects of encoding-decoding noise on the communication pro-
cess is examined.
It is shown that there is considerable interaction between the noise
present and the feedback nature of the network.
Notions from Information Theory play
an important role in this analysis.
D)
From questionnaire data, various correlations are shown to exist between a
person's role in the group and his emotional reaction to the situation.
in all cases what would be expected,
These are not
and it is felt that only a very partial understanding
of this area yet exists.
2.
Experiment on the Effect of Change in Communication Network
Experimental work has begun on the problem of determining the influence of the
group organization developed in one network upon subsequent performance of the same
subjects in a different network.
given previously (1).
A preliminary description of this experiment has been
It has been found desirable, on the basis of preliminary tests, to
make certain modifications in that plan.
The final test network will be "diablo" in every case.
(See Fig. IX-A. ) In this net-
work the node A has the most critical effect on group performance,
since information
originating at B or C can get to D and E only by way of A, and conversely.
node A the "center man."
Call
The 6 networks for the conditioning trials have been selected
-62-
(IX.
0B
Do
COMMUNICATION RESEARCH)
to favor to various degrees the development of organization
about a center man. After having run the groups on these
Spatterns
Eo
o C
Fig. IX-A
Diablo.
in which centralized organization develops, it is
possible to change to the test network diablo in two ways,
i. e. with the same man in the center and with a different
man in the center. Taking these possibilities into account,
we plan to run 10 groups of 5 military subjects, each on the
following network sequences (C = circle, D = diablo, P =
pinwheel, S = star; a prime denotes change of center man):
CD.
DD, DD', SD,
SD',
PD,
Fifteen trials will be run in each phase of the experiment, preceded by 5 trials
on network P to be used for adjustment to the apparatus.
Our previous experiments have been run in such a way that group trial time was
composed of individual action times, either with no temporal constraint on the individuals, or with each individual constrained to act only when every member of the group
has signaled his readiness to act.
In the former case, the complexity of the relations
between one group member's and another's action make it impossible to analyze the
group process in
a way which explains how group action is
composed of individual
The latter case does permit this reconstruction, but it has the disadvantages
actions.
of considerably lengthening the experiment and seeming highly artificial to the subjects.
In the present experiment we have adopted a compromise between the two extremes
above.
A "sending signal"'
We now quantize the time scale rather than the group action.
is sounded at regular intervals and the subjects are told that they may send messages,
if they desire, when they hear the signal but at no other times.
If the interval is very
short, this regime approximates the unconstrained situation; if it is
approximates the action-quantized scheme.
very long, it
It appears that an interval of 15 seconds
will accomplish the objective of making the data amenable to analysis and, at the same
time, remove much of the artificiality.
Developments in the apparatus are described in section 3 below and in data analysis
in section 4 below.
3.
Experimental Apparatus
In order to implement the experimental scheme described above,
and at the same
time to permit a complete reconstruction of the actions of the group, several modifications to the apparatus have been necessary.
The table previously described (2) is being used for this experiment, with certain
changes.
The "ready" buttons have been removed from the subjects' compartments,
and a new centerpiece has been constructed.
the original one, with send and receive
This centerpiece follows the scheme of
slots between each man.
The slots have
been enlarged to accommodate IBM cards, and each slot has been equipped with a
-63-
(IX.
COMMUNICATION
RESEARCH)
spring-loaded finger which detects the passage of a card.
These fingers break a circuit
every time a card is passed through the given slot, and these circuits are each connected
to a different pen in an Esterline-Angus operations recorder.
Since the transmission
time of the messages is quantized as mentioned in part 2 above, the correlation of this
record with the possible transmission times makes possible a complete reconstruction
of the location of all information in the net at any instant.
In addition, the instant at which each man gets the answer is recorded on another
20-pen recorder, and also a series of marks representing the possible transmission
times.
All the information derived from the recorders mentioned above will subsequently
be punched on the message cards themselves,
described below.
and these will be used for analysis, as
These modifications to the experimental apparatus have been com-
pleted and tested, and are now in use.
4.
IBM Analysis of Data
During the time that the equipment conversion for the Network Change experiment
was being planned and carried out (see sec.
3 above), an extensive analysis of the data
from Network Pattern and Group Learning experiment (2) was under way.
This analysis
ultimately assumed the form of determining conditional probabilities for the behavior
of individuals in the groups studied.
It was found that except for comparatively simple
conditions, which meant also comparatively simple networks, it is practically impossible
to carry out the necessary computations by hand, even in the highly simplified action
quantized case.
Since the Network Change experiment,
as described above, is not
action quantized, the calculation of these probabilities would be even more difficult.
These considerations lead us to investigate the possibility of computational aid from
the Center of Analysis, using IBM equipment.
It became apparent that this would be
satisfactory for our purposes, and that in payment for some labor in the preparation of
the IBM message cards, the computation of the conditional probabilities desired could
be reduced to a purely machine operation.
It is felt that this is an appreciable improvement in technique,
for we will be able to
obtain somewhat more complex probability conditions with less labor.
L.
S.
Christie, R. D. Luce, J.
Macy, Jr.
References
I.
Quarterly Progress Report, Research Laboratory of Electronics, M.I.T. Jan. 15,
1952, pp. 57-58
2.
Quarterly Progress Report, Research Laboratory of Electronics, M.I.T. Jan. 15,
1952, pp. 79-80
-64-
(IX.
D.
COMMUNICATION RESEARCH)
REPLACEMENT OF VISUAL SENSE IN TASK OF OBSTACLE AVOIDANCE
Dr. C. M. Witcher
E. Ruiz de Luzuriaga
L. Washington, Jr.
Work on this project has proceeded along two principal lines:
continuation of the
program for modification of the Signal Corps sensory aid as outlined in the Quarterly
Progress Report, January 15,
1952; and work on the development of a device for the
reliable detection of step-downs or other sudden drops.
Work on the modification of the Signal Corps obstacle detector has thus far consisted
of the completion of the necessary electronic modifications to yield an output signal
suitable for operation of relays to actuate the signal presentation pins discussed in the
Quarterly Progress Report, January 15,
1952.
Rough estimates based on previous
experience with tactile stimulators of this kind have indicated that the signal presentation pins should be actuated by a force whose minimum value is not less than about
15 gm, and that the length of stroke for these pins should be of the order of 0. 025 inch.
Measurements indicate that the signals from our modified Signal Corps obstacle
detector, when applied to the relays which we have selected (Sigma 4F), will yield armature motions which will more than meet these requirements.
The next steps in the work
with this device will consist of the development of the necessary mechanical and optical
systems to facilitate automatic scanning,
and the modification of the scheme by which
information received by the device is coded so as to yield a signal presentation of the
form outlined in the Quarterly Progress Report, January 15,
1952.
It was long ago realized by all of those familiar with the problem of independent
travel by the blind that their greatest dangers lay in their encounters with unexpected
step-downs or other sudden drops.
It was therefore felt that an attack on this aspect
of the problem should be undertaken as soon as possible.
During the past three months
a tentative solution has been developed, and a model embodying it is now under construction.
The requirements for an adequate step-down detector were fairly well estab-
lished about three years ago by a group of experienced blind travelers at a meeting of
the Technical Research Council in New York.
They are:
1) the device should provide
definite indication of a step-down or other drop at distances of 5 to 7 feet in front of it;
2) the device should (except for possible false alarms) give no signal except when a
step-down is encountered;
3) the step-down signal should be very distinct from that
given by any obstacle-detecting equipment with which the step-down device is to be combined.
It is generally conceded that some false alarms (perhaps even 20 percent of the
total signals) can be tolerated, provided the detection of step-downs of height greater
than 3 inches is absolutely reliable.
-65-
(IX.
COMMUNICATION RESEARCH)
Fig. IX-9
Fig. IX-10
Step-down detector.
Approximate pulse shape when the beam of
light crosses the step-down.
Our proposed solution to this problem can be understood by reference to Fig. IX-9.
Light from a flashlight bulb and reflector (a) passes downward through a rectangular
collimating tube (b) to fall on a small rotating mirror (c).
As the mirror turns, the
light beam will be periodically reflected so as to pass out through the rectangular
opening (d) in the bottom of the housing for the device.
The dimensions of the system
have been so chosen that the beam will initially strike the ground at a distance of about
5 feet in front of the device and sweep along the ground to a point about 7 feet in front
of it when the device is held horizontally with its bottom surface about 2 feet above the
ground.
A small amount of the light scattered from the spot where the beam strikes
the ground will enter the receiving lens (e) on the front of the device and will, after
reflection from the mirror (f) come to a focus near the front surface of the photo cell
(g).
The window of this photo cell is masked except for a narrow horizontal slit across
its center.
The position of the photo cell can be adjusted so that the image formed by
the lens will pass across the slit on the cell mask when the light beam is falling on the
ground at any desired spot along its path.
to send the beam out of the device,
Thus, each time the mirror (c) turns so as
a light pulse will be registered by the photo cell and
amplified by the amplifier (h).
When the surface of the ground is flat or merely sloping along the path of the beam,
the pulses received by the photo cell will be approximately rectangular or trapezoidal
in shape.
However, if at the moment the image is crossing the slit of the cell,
the beam
is passing over the edge of a step-down, the image will consist of two bright bands
separated by a relatively narrow dark line,
the shadow of the step-down.
image will, accordingly, give rise to a pulse of the approximate
-66-
This
shape shown in
(IX.
Fig. IX-10.
COMMUNICATION RESEARCH)
Although the design of the circuits has not been completed,
it is
fairly
certain that these pulses can be electronically distinguished from those produced
by level or sloping ground.
Except for the pulse-discriminating
now complete,
and it
is
circuits, the construction of the device is
to be tested by oscilloscopic
-67-
observation.
(IX.
E.
COMMUNICATION RESEARCH)
COMMUNICATIONS BIOPHYSICS
Prof. W. A. Rosenblith
K. Putter
1.
Interaction of Cortical Activity and Evoked Potentials
Preliminary experimentation on this topic continues at the Massachusetts General
Hospital.
Observations have been made concerning such phenomena as masking of
clicks by means of noise and cortical driving and also on the influence of metrazol upon
evoked responses.
More systematic experimentation will be carried out when satis-
factory facilities and equipment become available.
In the meantime, the construction
of an electronic correlator having the appropriate characteristics for this type of work
will be undertaken.
W. A. Rosenblith with Dr. M.
2.
A. B. Brazier (Massachusetts General Hospital)
Variability of Cortical Responses to Acoustic Clicks
Research on this topic and statistical analysis of the data continues to be carried
out at the Psycho-Acoustic Laboratory, Harvard University.
W. A. Rosenblith with K. Safford (Harvard Psycho-Acoustic Laboratory)
3.
Instrumentation
The time-gated amplitude quantizer is in the experimental stage.
The unit has at
present a single-channel time gate, and the quantizing process is done electromechanically.
The design characteristics call for a quantizing rate of 10/sec.
K. Putter, W. A. Rosenblith
-68-
(IX.
F.
COMMUNICATION RESEARCH)
NEUROPHYSIOLOGY
W. H. Pitts
J. Y. Lettvin
P. D. Wall
B. Howland
Our stereotaxic instrument is completed and in use.
It is to be on display at the
American Physiological Society meeting in April.
The source-sink maps of the spinal cord following dorsal and ventral root stimulation are now complete for two experiments, and they agree beyond our expectations.
The anatomical match of the two series at corresponding times is
good enough to
encourage us in attacking the more difficult mapping of inputs from separate modalities
without any further gross justification of the technique.
The data will appear in a tech-
nical report now in preparation.
Because of the tedium of computation and the length of time required for calculating
one experiment
(two months),
we are designing an analog computer for finding the
"smoothest" distribution of sources and sinks that will reproduce an irregularly spaced
matrix of potentials in a bounded volume conductor.
The details of this computer will
be included in the technical report.
In the experiments of the past three months we have tried to map the potentials
arising from bulbar reticular inhibition in the cord.
Twelve attempts have failed to
produce a single set of data that is satisfactory for computation.
We had not hoped for
earlier success, principally because the inhibitory system is so ill-defined in the bulb
that, although repeated stimulation anywhere around it will nicely suppress tone and
reflex, single stimuli, save with luck in placing the electrode,
rarely do so.
But the
map must be made; for unless we can exhibit an example of pure inhibition, we cannot
argue with certainty about its nature.
In December J.
C. Eccles put an important problem to us.
Recording within cells,
he has found a twenty-fold amplification and inversion of certain potentials relative to
their values just outside the cell-body.
Fortunately, under simplifying assumptions
about the geometry, this problem is susceptible of analytic treatment.
One can derive
relations between the Fourier transforms of the potentials inside and outside cells which
are quite different in the two cases:
the one where the cause is in the external medium
outside the cell, the other where it lies in the cell membrane.
It is a remarkable fact
that both relations are independent of the details of the processes in the membrane.
We are now using the Integral Transform Computer developed by J.
M. Ham to convert
these relations into a form expressing the internal potential as a weighted average (a
convolution integral) over the external, and vice versa, in the two cases.
computations are finished,
When the
we hope to use them to determine the causes of various
measured potentials by comparing their magnitudes inside and outside of cells.
-69-
COMMUNICATION RESEARCH)
(IX.
Another difficult problem is the fate of an impulse arriving at a branch in a single
axon.
With the kind cooperation of Dr. Aldo Truant of Tufts we shall try to get some
experimental data on squid axon this summer at Woods Hole.
Such data are necessary
to any theory of transmission in the central nervous system, since axons there almost
always arborize where they terminate,
certainly where they end on the motor neurons in
the spinal cord.
Mr. Pitts presented our work on inhibition at the Josiah Macy Jr. Foundation
Conference on Cybernetics on March 20, and we are submitting an abbreviated version
of the technical report to the Journal of Comparative Neurology.
-70-
(IX.
G.
COMMUNICATION RESEARCH)
MECHANICAL TRANSLATION
Much valuable time is now being spent by highly trained people on language trans-
lation.
Probably the amount of material that has to be translated will increase even
more in the future.
Plausible estimates indicate that the scarcity of expert translators
causes a log jam in scientific translation that alone costs American research hundreds
of thousands of dollars yearly.
In addition to the need for high-accuracy translation in
science, finance, and diplomacy, there also exists an urgent need for high-speed translation of the huge printed output of actual or potential enemies, i. e. newpapers, journals,
propaganda leaflets and the like.
Some preliminary research for the use of electronic computer-like machinery as an
aid in translation has been going on in various places these last few years, but the proThe author has made a survey of the various
gress made so far has been rather slight.
possible machine-man partnerships in translation.
For the time being, no completely
mechanical translation seems to be possible with respect to an actual language,
there is
since
no way in sight by which a machine with its limited memory-capacity
overcome semantical ambiguities.
for instance,
could
The number of possible word-digrams in German,
is probably more than a billion.
Therefore, the method by which a human
being reduces ambiguity (i. e. through consideration of the context of the ambiguous
word) is out of reach for present machines.
In this discussion the following terminology and abbreviations will be used:
MT:
mechanical translation
FL:
foreign language
TL:
target language
Pre-editor:
Post-editor:
the human partner who knows the FL alone
the human partner who knows the TL alone
Bilingual editor:
the human partner who knows both FL and TL.
The major practical problem is to reduce the required services of a bilingual editor
as much as possible and, if possible, to eliminate him completely.
nerships were studied:
Accordingly four part-
MT with pre-editor, MT with post-editor, MT with pre-editor
and post-editor, and MT with limited availability of a bilingual editor.
It seems that a pre-editor would be most efficient in eliminating morphological and
syntactical ambiguities and in rearranging the FL text in an order corresponding to that
of some standard order in the TL.
And there also exists a method by which the pre-
editor could eliminate semantical ambiguities without any knowledge of the TL.
It is
plausible that this method would be more time-consuming than having the semantical
ambiguities eliminated by the post-editor.
It has been shown experimentally that a post-
editor is able to pick out from the millions of suitably arranged correlates of an average
FL-sentence just one,
up to synonymy; and, in general, is able to do it without difficulty.
-71-
(IX.
COMMUNICATION RESEARCH)
If the average number of possible translations of an FL-word within a 25-letter sentence
is 2, the number of possible correlates of this sentence is 2 25, i. e. almost 40, 000, 000.
A satisfactory explanation of this rather astonishing fact can be given on the basis of
certain other experimental results achieved by Abraham Kaplan with respect to the
reduction of ambiguity through context.
The main problem is,
therefore, to have the machine do the preliminary syntactical
analysis of the FL material.
This could be achieved only if an operational syntax for
this FL were available which would take the form of a sequential program as used for
electronic computers, though it would be incomparably more complicated.
operational syntax exists so far for any natural languages,
been clearly seen until now.
No such
and the need for one has not
No insuperable difficulties for its preparation can be seen.
Another major obstacle lies on the hardware level.
Existing memory capacities of
electronic computers are still unable to handle with an appropriate speed the task of
picking out one word from approximately a million.
with the mechanical word-analyzer,
This is one of the steps to be taken
that gives for each running word its various
divisions into stem or morphological category.
A high reduction of the memory-capacity can be achieved through the restricted use
of a bilingual editor.
With a storage of a few tens of thousands of words, the analysis
and translation of perhaps 90 percent of the words of a foreign text should be possible.
The bilingual editor would be required to deal only with the remaining 10 percent.
Various methods of dealing with so-called idioms have been developed.
All the methods treated so far deal with MT from one specific language into another
specific one.
There are practical situations in which multiple translation into various
languages is of importance.
than that of specific MT.
The task of general MT is incomparably more complex
The only reasonable method of handling the problem of general
MT would be to devise a universal syntax.
All attempts undertaken until recently in this
direction failed, mainly because they were based on Aristotelian logic,
physical prejudices,
and lack of acquaintance with exotic languages.
certain meta-
It is possible that
a combination of the best methods developed by modern structural linguists and by mathematical logicians might lead to better results (see sec. IX-H).
The problem of MT is much less formidable when applied to certain artificial or
semi-artificial languages with restricted vocabularies and regular grammars, such as
Interlingua or Basic English.
These languages are therefore most suitable for future
experimental work to be undertaken for finding out the optimum combination of man and
machine in translation or the optimal combinations.
Other types of restricted languages that lend themselves easily to MT are those in
which the syntactical structure of a natural language is left completely unchanged, but
there the field of relevant messages is reduced either by the nature of the field or by
artificial convention.
It seems that this is the situation with respect to pilots' language
-72-
(IX.
COMMUNICATION RESEARCH)
or the meteorologists' code.
Some investigations were made into the degree of distortion introduced by MT in
comparison with purely human translations.
It appears that under suitable conditions
a man-machine partnership would do, on the average, not worse than a human translator
alone, at least with regard to material where the emotive component is of relatively
small importance.
A more extensive interim report, "The Present State of Research on Mechanical
Translation," containing summaries and evaluations of the work done by other groups,
will appear in American Documentation, Vol. II, No. 4.
Y.
-73-
Bar-Hillel
(IX.
H.
COMMUNICATION RESEARCH)
A NEW METHOD OF SYNTACTICAL DESCRIPTION
A method proposed by the Polish logician, K. Ajdukiewicz (1),
for syntactical des-
cription has apparently exerted no influence on American linguistics.
ever, that if some of its shortcomings are removed,
opments among American structural linguists (2),
and efficient.
It seems, how-
mainly on the basis of recent devel-
it should turn out to be quite powerful
Its main importance lies in its quasi-mathematical character,
which
enables a purely formal and completely mechanical analysis of the structure of any sentence of a given language to be made on the basis of the morphological-syntactical
gories to which the words of the sentence belong.
cate-
It should be, therefore, of special
value in those cases where a mechanical analysis is indispensable,
as in mechanical
translation (see sec. IX-G).
Preliminary results show that this new method is very efficient and perspicuous
with regard to artificial or semi-artificial languages.
however,
With respect to natural languages,
its efficiency is hampered by their enormous syntactical complexity and phe-
nomena like homonymity, idioms,
etc.
Since complete exposition of this method cannot
be given here, let us simply say that it consists of assigning to appropriate linguistic
units (generally words or morphemes) such symbols designating their morphologicalsyntactical categories as are easily amenable to a formal manipulation.
This manipula-
tion is very similar to the multiplication of fractions and is governed by a few, very
simple rules.
All of customary syntax is embodied in this categorical symbolism and
in its rules of operation.
Y. Bar-Hillel
References
1.
K. Ajdukiewicz: Die Syntaktische Konnexitaet, Studia Philosophica 1,
(Mimeographed translation by University of Chicago, 1951.)1935.
2.
Z. S.
3.
Y.
Harris:
Bar-Hillel:
1-27, Lwow,
Methods in Structural Linguistics, University of Chicago Press, 1951
On Syntactical Categories, J.
-74-
Symbolic Logic 15,
1-16,
1950
(IX.
I.
COMMUNICATION
RESEARCH)
SEMANTIC INFORMATION AND ITS MEASURES
In existing Theory of Information the semantic aspects of communication have been
deliberately ignored. It is the purpose of the present inquiry, undertaken in collaboration with Professor Rudolf Carnap, Department of Philosophy, University of Chicago,
to investigate just these aspects and to arrive at an explication of the concept of information as used on this level, and of its measures.
treated as a function whose arguments are
Information in its semantic aspect is
statements.
For the time being, only languages of a very limited complexity of structure
are treated, essentially those designated by Carnap (1) by "LN", i. e. languages containing
a finite number N of individual symbols and a finite number, Ir,of one-place predicates,
in addition to the customary symbols of a lower functional calculus with identity.
Abbreviating the presystematic "the semantic information carried by the sentence i"
to "In(i)",
R1:
has to fulfill the requirement
any adequate explication of "In"
In(i) includes In(j), if and only if i L-implies j.
Various possible explications which fulfill this condition are treated. The most
fruitful one seems to be that in which In(i) is identified with Cont(i) (read: the content
of i), defined as the class of all content-elements of i.
It can be shown that Cont is indeed an adequate explicatum of the presystematic concept of semantic information. On the basis of Cont(i) (the absolute information carried
by i) Cont(j/i) (the information carried by j relative or in excess to i) is defined by
Cont(j/i) =Df Cont(i. j) - Cont(i).
It can be shown that for any L-true statement t,
Cont(j/t) = Cont(j), so that we could
have also defined Cont(j) on the basis of Cont(j/i).
From any adequate explication of the concept of amount of semantic information, the
fulfillment of the following requirements is stipulated:
Abbreviating "the amount of semantic information carried by i" to "in(i)",
in(i) >, in(j) if (but not only if) i L-implies j
in(i) = 0 if i is L-true
in(i) > 0 if i is not L-true.
In addition to these three requirements,
cated.
It is,
however,
under certain conditions additivity is indi-
not easy to specify at this stage the exact nature of the conditions,
and it turns out that these conditions are different for various plausible explicata.
To explicate the function in(i),
has to be defined. In [Prob.],
a measure-function ranging over the content-elements
a related problem, namely, that of defining measure-
functions ranging over negations of content-elements (the state-descriptions) has been
treated at great length.
With respect to any measure-function defined over state-
descriptions, a so-called m-function, we take as our first explicatum the function cont(i),
-75-
(IX.
RESEARCH)
COMMUNICATION
defined as equal to m(-i) and called content-measure.
The range of cont is then extended
to include not only content-elements but any sentences whatsoever.
The most important
properties of cont are
cont(i) = 1 - m(i)
0 < cont(i) < 1
cont(i.j) = cont(i) + cont(j), if iVj is L-true.
The last property shows that this measure of information is additive with respect to
L-disjunct sentences.
It can easily be shown that cont fulfills the above requirements.
The relative content-measure of j with respect to i,
cont(j/i), is defined as cont(i. j)
- cont(j).
The most interesting property of cont(j/i) is
perhaps its being equal to
cont(iDj).
Interpreted in terms of an "ideal" receiver,
this means that for such a
receiver with the previous knowledge i, the acquisition of the additional information j
increases his information as measured by cont by exactly the same amount as it would
have been increased by acquiring the "weaker" statement i = j.
Among the various m-functions, that one which assigns equal values to each statedescription, mD (called "mt" in [Prob.] ) is especially simple but unsuitable for purposes
of inductive logic, mainly because it does not fulfill the requirement of instantial relevance.
In the interest of deductive logic, however,
cont-function based upon mD, namely,
contD.
it is also worthwhile to study the
Many theorems referring to contD have
indeed an especially simple form.
Among the other m-functions, those suitable for inductive logic are denoted by "m" 1
and the cont-functions based on them by " conti."
Though the cont-functions fulfill the requirements stated for explicata of amount of
information and have many plausible properties (for example,
additivity with respect
to L-disjunct sentences) they also have other properties which may look less plausible.
One of these is the following:
If several basic sentences with distinct primitive predi-
cates are received as information, the cont-value for the first is one-half, but the relative cont-value for the second is only one-fourth; and, in general,
every new basic
sentence adds to the content-measure only half as much as its immediate predecessor.
This is disturbing,
since basic sentences of this type are not only deductively independ-
ent, but even inductively independent, if we take inductive independence to be identical
with what has been called in [Prob.] initial irrelevance.
We might, therefore, be inter-
ested to have at our disposal another measure of information which assigns the same
value to basic sentences of the type considered,
by other basic sentences.
If,
whether or not they have been preceded
in addition (for the sake of normalization) we require that
the information-value for each such sentence be 1, it can be shown that these requirements are most simply fulfilled by defining the new concept of measure of information,
denoted by "inf,"
as follows:
-76-
(IX.
COMMUNICATION RESEARCH)
1
inf(i) =Df Log 1 cont(i)
where "Log" is short for "log 2 .'
Among the more interesting theorems we have
1
inf(i) = Log m(i) = -Log m(i)
which recalls, of course, the well-known formula of Hartley.
0 . inf(i) . oo
inf(i V j)
inf(i) . inf(i.j)
inf(i.j) = inf(i) + inf(j), if and only if i is initially irrelevant to j with respect to that m-function on which inf is based.
The last theorem states that inf is additive with respect to inductively independent
sentences.
Against this plausible result, we get another result which looks strange.
Whereas cont(i. j) is,
at most, equal to the sum of cont(i) and cont(j), inf(i. j) may, for
suitable arguments, be greater than inf(i) + inf(j).
The relative measure of information can now be defined in the familiar pattern, and
among the various measures of information the deductive infD and the inductive inf
been specially treated.
theorem holds:
I
have
With respect to infD, for instance, a much stronger additivity
If i and j are molecular sentences such that no atomic sentence occurs
simultaneously in both, then infD(i.j) = infD(i) + infD(j).
The fact that each of the proposed explicata for amount of information has plausible
as well as implausible properties seems to indicate that we have more than one explicandum in mind, and that we require incompatible properties because we are confusing
different, though related, concepts.
its field of application.
It seems that each explicatum has its value and
To make this statement more definite would require additional
studies.
One of the most promising m-functions is m
.
Therefore, the properties of cont
and inf , based upon m*, have been studied with special care, and many numerical examples have been worked out.
Y. Bar-Hillel
Reference
1.
R. Carnap: Logical Foundations of Probability [Prob.] , University of Chicago
Press, Chicago 1950. The terminology of this book is used in this discussion.
-77-
(IX.
J.
COMMUNICATION RESEARCH)
PARALLEL CHAIN AMPLIFIER
The coils mentioned in the Quarterly Progress Report, january 15, 1952, have been
rewound with a sufficiently small distributed capacitance (large self-resonance) as to
make them appear to be satisfactory.
Progress has been hampered by the cumbersomeness of the present measurement
technique. The difficulty lies in the very wide frequency range involved. (We are interested in the amplitude response from approximately 1 Mc/sec to 270 Mc/sec.)
We are
working on some ideas to ease this situation.
R. K. Bennett
-78-
(IX.
K.
CORRELATION FUNCTIONS OF AMPLITUDE
COMMUNICATION RESEARCH)
DISTORTED NOISE
Correlation functions of stationary gaussian noise passed through nonlinear circuits
can be computed by probability techniques.
derived:
The following general property has been
The crosscorrelation function of two coherent gaussian signals, taken after
one of them has undergone amplitude distortion, is identical, except for a factor of
proportionality,
to the crosscorrelation function taken on undistorted signals.
A technical report (No. 216) has been prepared in which this relation is demonstrated, and its possible practical applications discussed.
The report includes a number
of correlation functions computed for some of the more common types of nonlinear distortion.
J.
-79-
J.
Bussgang
(IX.
COMMUNICATION RESEARCH)
L.
TRANSIENT PROBLEMS
Prof. E. A. Guillemin
Dr. M. V. Cerrillo
I.
E. F. Bolinder
Dr. F. M. Reza
Basic Existence Theorems (continued from Quarterly Progress Report,
Jan. 15, 1952)
Synopsis of Previous Report
A new set of integral representations of direct and inverse Laplace transformations
was given.
The main feature of this representation is that both f(t) and F(s) can be
simultaneously expressed in terms of integrals containing the real part of F(s) along
certain semi-infinite contours.
When the contour runs over one or several singularities
of F(s), but leaves the rest to the right, then the integral must be taken in the Stieltjes
sense.
When the contour leaves some of the singularities to its right, the corresponding
Stieltjes integral representation is also produced.
With the aid of this integral representation, a set of theorems was obtained,
particular, the following:
t < 0,
in
first, let f(t) be a bounded function for t > 0 and be zero for
then F(s) is necessarily a transfer function;
second, the necessary and sufficient
condition for F(s) to be a transfer function is that F(s) satisfy the integral representation.
The theorems are of a constructive nature.
The corresponding electrical network
for a certain transfer function can be obtained.
Part II
Introduction
The aim of this report is to push the previously reported results somewhat further
toward the construction of methods on network synthesis, both in the frequency and time
domains. The material given here serves as a preparatory introduction to the basic
future results.
Because of lack of space and for clarity in the explanation, the presentation of the subject follows heuristic and suggestive lines. The formal mathematical
aspect is then not hard to derive.
Repetitions of equations already given will be omitted
here, so that the reader must continually consult the Quarterly
January 15, 1952.
-80-
Progress Report,
(IX.
COMMUNICATION
RESEARCH)
Section 4
Introductory Ideas Concerning New Strategic Integral
Representations on Transfer Functions
II-4. 0
This section deals with the first steps tending to develop new integral repre-
sentations of transfer functions which lately have become quite suitable in founding the
theories of constructions of methods for network synthesis.
11-4. 1
In section I, particularly in Eqs.
19,
30, 43,
44, 45,
52,
53,
(I-l. 14) we have given sets of representations of Laplace transforms
to two basic positions, with respect to the singularities of F(s),
54 and theorem
corresponding
of the semi-infinite
contour of integration 2F (see ref. 1).
Our next move is to deform the contour
idea of deforming the contour r
F
in certain strategic configurations.
This
plays a basic role in the present and in future discus-
sion.
Let f(t) be a bounded function as described
in theorem (2-2. 1) (ref. 1, sec. 1).
Then
F(s) is a transfer function whose singularities
lie on or to the left of the imaginary axis in
the s plane.
Suppose that F(s) possesses a
group of singularities, say poles, branchpoints, etc. in certain domains, say D v ,
v = 1,2,... n, of the s plane. Figure IX-11
illustrates a possible situation.
Now let us deform the contour of integration F continuously in such a way that the
singularities and branch-cuts are surrounded,
but leave all of them always to the left of the
deformed contour when traveling from -ioo
to +ioo,
as indicated by Fig. IX-11.
Note
that our present aim is to surround regions
containing groups of singularities but not
necessarily isolating individually each indiFig. IX-11
Deformation of the contour of integration F around the domains D containing
the singularities of F(s). Branch-cuts
must not necessarily be straight-line
segments, except along the real axis.
vidual singularity.
Since we have shown in (1)
that the transfer function belongs to Laplace
transforms, then such deformation is permitted and the functions F(s) and f(t) both
remain invariant to such deformations.
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(IX.
COMMUNICATION RESEARCH)
We want to develop the corresponding integral representation of F(s) and f(t) under
such deformations of the semi-infinite contour F.
Such representation is not difficult
to obtain from Eqs. 19, 30, 43, etc., but it is not so obvious as it may seem at first sight.
Due care must be taken at certain steps of the derivations to avoid fatal violations of the
conditions under which the existence of the integral representation is valid.
11-4. 2 To facilitate for the reader the proof of the integral representation now reported,
we will first perform a simple operation which takes the function F(s) into the wellknown classical Cauchy integral representation.
Let us call
This step is convenient but not needed.
Fd the deformed contour equivalent to F , already described.
thesis F d does not touch any singularity of F(s).
By hypo-
It is well-known that the fundamental
basic expression for f(t) may be written as
f(t)
F(S) e
=-
- St
dS
1,(II-4.2)
Fd
where S indicates a point on
By multiplying f(t) by e
d"
dt, integrating from 0 to oc,
and inverting the order of
integration (permitted because Fd does not touch singularities of F(s)), one gets
00cc
F(s)
f(t) e -
=
st
dt =
0
(S-s)
(S-s)
2, (II-4.2)
rd
where f
is taken along the closed contour formed by Fd plus the infinite circle in the
right-half s plane. The integral vanishes along such semicircles. The function F(s)
is represented in the inside of the above closed region, where F(s) is analytic. It must
be recalled that if F d is allowed to form a "pocket" which contains no singularities of
F(s) inside, then the integral 2, (1I-4. 2) produces a zero contribution to F(s). (The integral in this pocket represents an analytic function inside the pocket and zero outside.)
11-4. 3
For clarity in the explanation we will assume that F(s) is decomposed into a sum
of function components, say KV(s), which are each generated only by the singularities
contained inside the respective pockets or branch-point circuits.
This function decom-
position is not a necessity in the final theorems.
Assume first that F(s) contains singularities in the inside of only two pockets mirrorsymmetrical with respect to the real axis, or in one symmetrical pocket (if the first two
pockets have a non-empty intersection).
The singularities need not be confined to poles.
If there are branch-points inside, let us assume that the cuts are formed by points all
contained inside the respective pockets.
-82-
COMMUNICATION RESEARCH)
(IX.
The corresponding integral representation for the function F(s) for such pockets,
say D 1 and D 2 , is given by
F( s) =72
= f
F
Xdy + (s-y)dXk U(Y,
2 U(,
(s-)2
)
+
(Riemann's sense)
2
=-f
7rf
Xdy + (s-y)dk U,
1(Y,
2
(K, 1)( ,
(s- )2 + X
where (y, k) are the coordinates of points of the contour around the pocket.
U(K,
1 )(y,
1, (II-4. 3)
U(y, k) and
K) are respectively the real part of F(s) or Kl(s) on the pocket contour.
The
They may have a quite different
reader must not confuse the function F(s) and Kl(s).
character, because F(s) is necessarily a transfer function but Kl(s) is not necessarily
so.
For example, let the singularities inside the pockets D 1 and D 2 be a simple pole
of residue one located at s = -o-
+ iw .
Then, the corresponding transfer function is
F(s)
2(s + O- )
2
(
+
(s+ o)Z
1
o
s-s
+
S
o
1
s-s
-
I
o
2, (11-4.3)
= K (s) + K 2 (s);
s
: - o-
+
i=
Alternate forms to 1, (II-4. 3) are
where neither K 1 (s) nor K 2 (s) are tr ansfer functions.
given by
[X + (s-y)g'(y)]
(s-y)2 +2
+ g,()
3, (11-4. 3)
2/
0D
U(y, k)di
[x + (s-y)g'()lU(K,
(s-y)
2
+
1 )(Y-, )dl
1i + g'(y)2
1
where g'(y) indicates the derivative at a point (y, k) on the contour of integration.
The
above expression presupposes that the contour of integration is a closed curve having
a continuous turning tangent.
(The corresponding expression when the above closed
contour possesses corners will be given later.)
For the particular case of a circular region D 1 , centered at oR, the above expressions reduce to
-83-
+ iw
and with radius
(IX.
COMMUNICATION RESEARCH)
+T-
F(s) =
-TT
s -
-)2
0
0
(s -
+w 2 + R2]
- 2R(-w sin 0 + (s 0
0
u(R,0)
) cos 0 - R
[-w 0sin
- ) cos 0)
0O
UK
1 (R,
0)
RdO
4, (11-4. 3)
It is convenient to illustrate the correctness of 4, (11-4. 3) by means of a few simple
examples:
Take first F(s) as given by 2, (11-4. 3) and set the pole s at the center of the circle.
R is arbitrary, and for simplicity let us make RK (S)
1
1
s
s - s
.
Here
0.
cos 0
cos
R
(R, 6)
U,
K
OR
Hence
5, (1I-4. 3)
[- osin 0 + (s + - ) cos
2
F(s) --4-
+
(s+o)
-W
as it should be.
O] cos 0 dO
2 +
O
+0
2(s + o)
(s+a-)
0
O
2
+w
2
0
The important feature of this new representation comes from the fact
that F(s) is here generated by a K (s) function.
Of course, computing the above integral
directly with the complete real part of F(s) on the circle, one obtains exactly the same
result.
Suppose now that the "pocket" D 1 contains a finite number n of simple poles, say
at sk' k = 1, 2, ...
n, each having a positive residue a k
.
The corresponding F(s) follows
with ease from 3, (11-4. 3) or 4, (11-4. 3) since the domain D1 can be broken in n circular
nonoverlapping small subdomains,
each surrounding one pole.
(This is
justified by
2, (11-4. 2).)
By applying 5, (II-4. 3) to each subdomain, we finally get
F(s)
k=n
(s +
k)
6, (11-4. 3)
k=l
11-4. 4
(s + ok
k
In this subsection, let us consider a pocket which is symmetrically bisected by
the real axis, such as D 0 in Fig. IX-11.
The corresponding integral representation of
the transfer function F(s) which is generated by the singularities in such a pocket has
the following integral representation
F(s) = 2
[X + (s-y)g'(y)
(s-y) + X2
fU(y,X)
UK, o (-Y,k
dI
1, (II-4.4)
1 + g'(N)2
The reader must note that the integration is to be taken only along the upper contour of
-84-
(IX.
COMMUNICATION RESEARCH)
D . The representation 4, (11-4. 3) can also be used here by taking the integral between
o
0 and Tr only.
II-4. 5
Let us consider the integral along the banks of the branch-cuts.
In order to
fulfill the symmetry condition imposed on the contour f , the branch-cutting must be
Otherwise
made with mirror symmetry with respect to the real axis of the s plane.
the representation given in Part I would be invalidated.
Let us consider the function
An example of the resulting violation is quite convenient.
1/v,'
The branch-points are at s = 0, o0 .
(the Laplace transform of I/ fft).
Suppose we
cut the s plane along the straight line inclined at an angle 0o to the real axis (see
Fig. IX-12a).
Let us define the corresponding Riemannian sheets f
jT. -2Tr+
O
1
I*
0
<
0
<
0< 0
and
fII
< 0
+27r
The corresponding sign distribution of the real and imaginary parts of 1/,/
0
_4f
++++++
SIGN OF REAL
SIGN OF Im.,
--
1rs
+1
,
SF
+
+
++SGNOF REAL
SIGNOF REAL
1
SIGN OF Im.
,rS
Fig. IX-12
Resulting asymmetry of Real (1/fs)
-85-
as follows
with respect to the real axis.
in each
(IX.
COMMUNICATION RESEARCH)
sheet is indicated in Fig. IX-12b and IX-12c.
These plots put in evidence that Real (1/f-) does not have even symmetry and Im(l/f-s)
does not have odd symmetry with regard to the real axis when 60 ± w (0o = 0 violates the
analyticity of F(s) in the right-hand side of the s plane). Hence, 6o must be set equal
to n. If 1/Tis
to represent a p. r. function, then the sheet
I, o00= w, is the only
solution.
Consider first a branch-cut in the upper part of the plane (not along the real axis)
as illustrated in Fig. IX-12a. The cut need not be a straight-line segment. Let us
assume, for simplicity in the explanation, that F(s) is a function whose singularities
are two branch-points, s = s a
s = sb
(s - sa)(s - Sb
, (as for example,
It can immediately be shown that F(s) is represented by
,
[kdy + (s-y)dk]U(
U(Y' X)
2
)2
(s-y) +
F(s) = 2
sb
a
1, (II-4.5)
Jump U (y, X) (Xd2 + (s-y)dk) + 2
2 [(s-y)+
iT
2]
,--sb
sb
con tour
-a
a
where the last two integrals of the third member must be taken around each branchpoint. These last integrals have the form 4, (11-4. 3) when R- 0.
If the branch-cut cuts the real axis, then the needed integration is similar to
1, (11-4. 5) except that we need only go around the upper part of the branch-cut. That is
2
F(s) =
[Xdy + (s-y)dX]
(s-
_ .real
axis
v)
+
2
[kdy + (s-)dA
X)
(-2
/contour
-86-
+
Jump U(y, X)
2
2, (1I-4. 5)
COMMUNICATION
(IX.
RESEARCH)
Finally, if the cut is made along the real axis, we can write
F(s) =
V(Y,,O) dy +-2
Tr
(s- )
Tr
+
3, (11-4. 5)
~~fTr
J
sa
sb
where the explicit expression for the last two integrals is given by
2 f
+ XZ
(s-y)
2)
V(y,
(s-y)dy - XdX
4, (11-4.5)
+ X
(s-y)
y = R cos 0
R = constant}
S= R sin 0
reference 1.
See Eq. 7,
The reader should observe that in the last two representations we have used V(y,X)
This anomaly will be properly straightened out later. This subinstead of U(y, k).
Note:
section will be closed with two illustrative examples.
Example I:
Take F(s) = 1/
s
+ 1,
straight segment from +i to -i.
1/ 1
7
Let us cut the s plane by the
with branch points at s = ±i.
The real part along the cut, sheet I, is
given by
at the right bank, and by -1/1 - K at the left bank of the cut.
By direct application of 2, (1I-4. 5) and setting y = 0,
-2s
F(s)=
one gets
dk
Os
2
5, (11-4. 5)
+ X2
2
since the integral around +i vanishes.
To evaluate the integral we may use reference 2, page 72,
entry No. 387, which
gives
dx
(ax 2 + b)
=
1
(fx 2 + g)
tan-1 x4(ag - bf);
b(fx 2 + g)
b(ag - bf)
By direct application we find
1
S
1
s2
0
+
dk
1
W
2
s
1
S 1
-87-
2
+
12
~1
ag > bf
(IX.
COMMUNICATION RESEARCH)
and consequently
F(s) =
s2 +1
as it should be.
Example II:
1
F(s)=;
-1
V(y, 0)
By direct application of 3, (11-4. 5),
= -oo
s
-a
_s
b
and since the integration 4, (1I-4. 5) around s = 0
vanishes, one gets
-00
F(s)=L2r-1
Tr
J-
dy
- s +y
0
Using reference 2, page 197,
entry No.
856. 3,
one gets
00
00
1 d
0 ys +
dx
y
0
1
T(i+t)
Hence
F(s) -
1
as it should be.
11-4. 6
From these particular cases, one can write down with ease the corresponding
integral representation in the general case.
To facilitate the writing, we will use a
suggestive notation as follows:
CD
indicates integration around a pocket which contains no common points with the
real axis of the s plane.
The singularities of each pocket lie inside the pocket contour.
Pockets need not be circles.
r{
Dk indicates integration around the upper contour of a pocket,
symmetrically bisected by the real axis of the s plane.
when the pocket is
The singularities of the pocket
do not lie on the pocket contour.
(~Cn indicates integration around the banks of a branch-cut which contains no points
in common with the real axis.
gPC
Branch-cuts need not be straight-line segments.
indicates integration around the upper branches of a branch-cut which is
-88-
(IX.
COMMUNICATION
RESEARCH)
symmetrically bisected by the real axis of the s plane.
(
C. indicates integration around the upper part of a straight-line branch-cut
which lies along the real axis of the s plane.
One must remember that all pockets
and branch-cuts of the s plane must have
mirror symmetry with respect to the real axis.
With this notation, the corresponding integral representation (Riemann's sense) of
F(s) is given by the condensed notation
F(s) =
{integrals
+
~
of type 3, (II-4. 3) +
D
integral7Dk of type 1, (II-4. 4)} +
k
{integral
+
+
integrals
Cg of type 1, (11-4. 5)
+
Ch of type 2, (11-4. 5)
+
1, (II-4.6)
h
+
{integrals
---
'
of type 3, (II-4. 5)}
j
This expression is not written in full; our particular aim now is to put in evidence
the structural formation of the function F(s).
In section 5 we shall study some funda-
mental structural properties of transfer functions which are of basic importance in the
theory of network synthesis.
This subject will be continued in the Quarterly Progress Report for July 15,
M. V.
1952.
Cerrillo
References
1.
Quarterly Progress Report,
1952
2.
H.
B. Dwight:
Research Laboratory of Electronics,
Table of Integrals,
MacMillan,
-89-
New York 1947
M. I. T., Jan.
15,
(IX.
COMMUNICATION
2.
RESEARCH)
Conformal Mapping of Physically Realizable Passive Networks
Certain studies have been made on the conformal mapping of physically realizable
impedance functions, with particular emphasis on the geometry of the network.
These
studies are aimed toward possible improvements of existing synthesis procedures.
F.
-90-
M.
Reza