A. MULTIPATH TRANSMISSION
Prof. L. B. Arguimbau H. H. Cross E. M. Rizzoni Dr. J. Granlund W. C. Kinzinger G. M. Rodgers
Dr. C. A. Stutt E. E. Manna R. D. Stuart
R. A. Paananen
1. Speech and Music. Transatlantic Tests
The series of tests discussed in the last few Quarterly Progress Reports has been completed. As a result of these tests, it can be said that under many ionospheric con-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-braic sum of strengths of the other signals. In this case, the receiver output depends on each of the signals, and serious distortion may result.
Of course, 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 veryserious 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, of course, the spread in time of flight mentioned above. 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 maxi-mum 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.
a
e
b
fc
9
k
d'
h
I U1 ~ I Fig. IX-1Representative patterns received during pulse transmissions over transatlantic path in 1951.
Carrier Signal Strength Duration of Sweep Duration of Receiver Quality of Program Time Frequency (db above 1 f.l.v Transmitted Pulse Length Pulse Pattern Bandwidth Immediately after
Da.te (GMT) (Me/sec) on 50 -ohm feeder) (f.l.sec) (f.l.sec) (f.l.sec) (kc/sec) Pulse Transmission
a Mar. 21 13:40 23.09 72 20 2000 400 200 fair to poor
b Mar. 21 14: 10 23.09 65 20 2000 550 200 mostly poor
c Mar. 21 15: 1 0 23.09 75 20 2000 650 200 fair
d Mar. 21 17:40 23.09 60 20 2000 900 200 good
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 fair to poor
h Jan. 26 14:05 26.0 50 20 2000 120 200 poor
26 2.05 5 08 n 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 pres-ent 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 con-siderably worse than with a narrowband FM detector. Either a squelch circuit or a mechanical tuning cut-off switch should provide a solution.
-I 1/2 6AL5 +150 JA /--\ IOK I --j O.IM 2M -300
Fig. IX-6
Pentode-gate decoder.
+ 300 +150 -150 -300 100 K 5687 IB. STATISTICAL THEORY OF COMMUNICATION
Prof. J. B. Wiesner Dr. P. Elias L. Dolansky Prof. W. B. Davenport, Jr. B. L. Basore P. E. Green, Jr. Prof. R. M. Fano J. J. Bussgang A. J. Lephakis
Prof. Y. W. Lee C. A. Desoer R. M. Lerner
Prof. J. F. Reintjes M. J. Levin
1. 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. F. Reintjes, M. J. Levin
2. Pulse Code Magnetic Recorder
Successful experiments were conducted on a two-channel part of the pentode-gate 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 master-timing 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 equip-ment, 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 voltage (i. e. V
Fig. VII-24, Quarterly Progress Report, April 15, 1950), the decoder output and the filter output, respectively; the corresponding input voltage (sine wave, approxi-mately 10 kc/sec) is included in the top part of each picture. For a 50-cps sine
Fig. IX-7
d
9
e
h
f
Fig. IX-8
Waveforms obtained experimentally at the phantastron plate, the decoder output,
and the filter output, related to the input waveforms.
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 saw-tooth in (h). The performance for sawtooth waveforms was used during an approxi-mate 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 phan-tastron stage of the coder; the upper part represents the decoder output. By changing the values of the individual plate resistors, the lower edge of the decoder output is adjusted to approximately a straight line. 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. B. Wiesner, L. Dolansky
3. Information Theory
a. 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 trans-mitter 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
+ 1, 0, - 1. The optimum values of a and
P
are not given by explicit formulas and could be found by trial and error only. 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 useof 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 stationrequantizes to a number of levels larger than two.
C. A. Desoer, R. M. Fano
b. VocoderEquipment 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 ofpolar representation to speech and electroencephelography.
A method for designing
wideband phase-splitting structures has been developed and is being written up.C. HUMAN COMMUNICATION SYSTEMS
Dr. L. S. Christie J. B. Flannery D. G. Senft
Dr. R. D. Luce J. Macy, Jr. P. F. Thorlakson E. S. Palmer
1. 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 techni-cians. 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 network is studied. This group-learning is interpreted as a combinatorial resultant of 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. These are not in all cases what would be expected, 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. A preliminary description of this experiment has been given previously (1). 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. Call node A the "center man." The 6 networks for the conditioning trials have been selected
Do 0B
to
favor to
various degrees the development of
organization
about a center man. After having run the groups on theseSpatterns
in which centralized organization develops, it isE o o C possible to change to the test network diablo in two ways, i. e. with the same man in the center and with a different Fig. IX-A man in the center. Taking these possibilities into account,
Diablo. 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): DD, DD', SD, SD', PD, CD. 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 individ-uals, 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 actions. The latter case does permit this reconstruction, but it has the disadvantages 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. We now quantize the time scale rather than the group action. A "sending signal"' 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 very long, it approximates the action-quantized scheme. 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 modifica-tions 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. This centerpiece follows the scheme of the original one, with send and receive slots between each man. The slots have been enlarged to accommodate IBM cards, and each slot has been equipped with a
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, and these will be used for analysis, as described below. 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
D. 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 presen-tation 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 arma-ture 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
con-struction. 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 com-bined. 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.
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. Thus, each time the mirror (c) turns so as to send the beam out of the device, 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. This image will, accordingly, give rise to a pulse of the approximate shape shown in
Fig. IX-10. 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 circuits, the construction of the device is now complete, and it is to be tested by oscilloscopic observation.
E. 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. A. B. Brazier (Massachusetts General Hospital)
2. 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 electromechan-ically. The design characteristics call for a quantizing rate of 10/sec.
F. NEUROPHYSIOLOGY
W. H. Pitts P. D. Wall
J. Y. Lettvin 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 stimu-lation 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. When the computations are finished, we hope to use them to determine the causes of various measured potentials by comparing their magnitudes inside and outside of cells.
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.
G. 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 trans-lation 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 pro-gress made so far has been rather slight. The author has made a survey of the various possible machine-man partnerships in translation. For the time being, no completely mechanical translation seems to be possible with respect to an actual language, since there is no way in sight by which a machine with its limited memory-capacity could overcome semantical ambiguities. The number of possible word-digrams in German, for instance, 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: the human partner who knows the FL alone Post-editor: 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. Accordingly four part-nerships were studied: 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.
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. No such operational syntax exists so far for any natural languages, and the need for one has not been clearly seen until now. 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. This is one of the steps to be taken with the mechanical word-analyzer, 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. The task of general MT is incomparably more complex than that of specific MT. 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, certain meta-physical prejudices, and lack of acquaintance with exotic languages. It is possible that a combination of the best methods developed by modern structural linguists and by math-ematical 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
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.
H. 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. It seems, how-ever, that if some of its shortcomings are removed, mainly on the basis of recent devel-opments among American structural linguists (2), it should turn out to be quite powerful
and efficient. Its main importance lies in its quasi-mathematical character, which enables a purely formal and completely mechanical analysis of the structure of any sen-tence of a given language to be made on the basis of the morphological-syntactical
cate-gories to which the words of the sentence belong. 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. With respect to natural languages, however, 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
morphological-syntactical categories as are easily amenable to a formal manipulation. This manipula-tion is very similar to the multiplicamanipula-tion of fracmanipula-tions 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, 1-27, Lwow, 1935. (Mimeographed translation by University of Chicago,
1951.)-2. Z. S. Harris: Methods in Structural Linguistics, University of Chicago Press, 1951 3. Y. Bar-Hillel: On Syntactical Categories, J. Symbolic Logic 15, 1-16, 1950
I. 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 collabora-tion 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
infor-mation as used on this level, and of its measures.
Information in its semantic aspect is treated as a function whose arguments are 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)", any adequate explication of "In" has to fulfill the requirement
R1: 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 con-cept 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, under certain conditions additivity is indi-cated. It is, however, 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), a measure-function ranging over the content-elements has to be defined. In [Prob.], 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),
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 state-description, 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 rele-vance. In the interest of deductive logic, however, it is also worthwhile to study the cont-function based upon mD, namely, contD. 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 rela-tive 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 independindepend-ent, 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, whether or not they have been preceded by other basic sentences. If, 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 require-ments are most simply fulfilled by defining the new concept of measure of information,
1 inf(i) =Df Log 1 - cont(i)
where "Log" is short for "log2.'
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 irrel-evant 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 infI have been specially treated. With respect to infD, for instance, a much stronger additivity theorem holds: 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 expli-candum in mind, and that we require incompatible properties because we are confusing different, though related, concepts. It seems that each explicatum has its value and its field of application. 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 exam-ples 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.
J. 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 inter-ested in the amplitude response from approximately 1 Mc/sec to 270 Mc/sec.) We are working on some ideas to ease this situation.
K. CORRELATION FUNCTIONS OF AMPLITUDE DISTORTED NOISE
Correlation functions of stationary gaussian noise passed through nonlinear circuits can be computed by probability techniques. The following general property has been derived: 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 demon-strated, and its possible practical applications discussed. The report includes a number of correlation functions computed for some of the more common types of nonlinear dis-tortion.
L. TRANSIENT PROBLEMS
Prof. E. A. Guillemin E. F. Bolinder Dr. M. V. Cerrillo Dr. F. M. Reza
I. 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, in particular, the following: first, let f(t) be a bounded function for t > 0 and be zero for t < 0, 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 representa-tion.
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 presen-tation 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 Progress Report, January 15, 1952.
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, 54 and theorem (I-l. 14) we have given sets of representations of Laplace transforms corresponding to two basic positions, with respect to the singularities of F(s), of the semi-infinite
contour of integration 2F (see ref. 1).
Our next move is to deform the contour
F
in certain strategic configurations. This idea of deforming the contour r plays a basic role in the present and in future discus-sion.Fig. IX-11
Deformation of the contour of
integra-tion
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.
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,
branch-points, 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
integra-tion 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
indi-vidual singularity.
Since we have shown in (1)
that the transfer function belongs to Laplace
transforms, then such deformation is
per-mitted and the functions F(s) and f(t) both
remain invariant to such deformations.
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 well-known classical Cauchy integral representation. This step is convenient but not needed.
Let us call
Fd
the deformed contour equivalent toF
, already described. By hypo-thesisF
d does not touch any singularity of F(s). It is well-known that the fundamental basic expression for f(t) may be written asf(t) =- F(S) e- S t dS 1,(II-4.2)
Fd
where S indicates a point on d"
By multiplying f(t) by e 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- s t dt=
(S-s)
(S-s)
2,
(II-4.2) 0rd
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 inte-gral 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 mirror-symmetrical with respect to the real axis, or in one mirror-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.
The corresponding integral representation for the function F(s) for such pockets, say D1 and D2, is given by
2 = f Xdy + (s-y)dX U(Y, k )
F( s) =7
2 U(,
F
(s-)
2 +(Riemann's sense)
2
=-f7rf 1, (II-4. 3) Xdy + (s-y)dk U, 1(Y,(s- )2 + X2 (K, 1)( ,
where (y, k) are the coordinates of points of the contour around the pocket. U(y, k) and U(K, 1)(y, K) are respectively the real part of F(s) or Kl(s) on the pocket contour. The reader must not confuse the function F(s) and Kl(s). They may have a quite different character, because F(s) is necessarily a transfer function but Kl(s) is not necessarily so. For example, let the singularities inside the pockets D1 and D2 be a simple pole of residue one located at s = -o- + iw .
Then, the corresponding transfer function is
2(s + O-
) F(s) ( 2+
(s+ o)Z
o
where neither K1(s) given by nor K2(s) are tr 1 1 S +-s-s
s-s
I o o 2, (11-4.3) = K (s) + K2 (s); s : - o- + i=ansfer functions. Alternate forms to 1, (II-4. 3) are
2/
0D
1
[X + (s-y)g'(y)] U(y, k)di
(s-y)2 + 2 + g,()
[x
+ (s-y)g'()l U(K,
1)(Y-, )dl
(s-y)2 + 1i + g'(y)2
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 D1, centered at o- + iw and with radius R, the above expressions reduce to
+T-F(s) =
-TT
[-w 0sin
(s
-0
)cos 0
-R
u(R,0)
s
--)2
+w 2 + R2] - 2R(-w sin0
+ (s - - ) cos0)
UK 1(R, 0)0 0 0 0O
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 R- 0. Here
1 cos
K (S) s . U, (R, 6) cos
1 s - s K R
OR
Hence
2 [-
osin 0
+ (s +-
) cosO]
cos 0 dOF(s) --4-
+
2 +
(s+o) +0 -W O O 0 5, (1I-4. 3) 2(s + -o)2
2
(s+a-) +w 0 0as it should be. 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" D1 contains a finite number n of simple poles, say at sk' k = 1, 2, ... n, each having a positive residue ak . 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)
k=l (s + ok k
6, (11-4. 3)
11-4. 4
In this subsection, let us consider a pocket which is symmetrically bisected by
the real axis, such as D in Fig. IX-11.
0The 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
The reader must note that the integration is to be taken only along the upper contour of
RdO
4, (11-4. 3)
fU(y,X)
d I
UK, o (-Y,k 1 + g'(N)2
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 made with mirror symmetry with respect to the real axis of the s plane. Otherwise the representation given in Part I would be invalidated.
An example of the resulting violation is quite convenient. Let us consider the function 1/v,' (the Laplace transform of I/ fft). The branch-points are at s = 0, o0 . 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 and
fII as follows
jT.
-2Tr+ O < < 01 0
I*
0
<0<
0 +27rThe corresponding sign distribution of the real and imaginary parts of 1/,/ in each
0
_4f ++++++ SIGN OF REAL 1rs SIGN OF Im., --++SGN OF REAL SIGN OF REAL 1 +1 SF + , + SIGN OF Im. ,rSFig. IX-12
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 sheetI,
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 = sa , s = sb , (as for example, (s - sa)(s - Sb
It can immediately be shown that F(s) is represented by
F(s) = 2 [kdy + (s-y)dk]U( )2 2 U(Y'
X)
(s-y) + sb a 1, (II-4.5) iT ,--s b con -a aJump U (y, X) (Xd2 + (s-y)dk) + 2
2 [(s-y)+
2]
tour
where the last two integrals of the third member must be taken around each
branch-point.
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
[Xdy + (s-y)dX] 2
(s- v) + X)
[kdy + (s-)dA Jump U(y, X) 2
(-2 + /contour F(s) = 2 _ .real axis
2, (1I-4. 5)
sb
Finally, if the cut is made along the real axis, we can write
F(s) =
Tr V(Y,,O) dy +-2 +J
~~f-(s-
)
TrTr
s a sbbwhere the explicit expression for the last two integrals is given by
2 f
(s-y)dy - XdX
V(y,
2)
(s-y) + XZ
(s-y)
+ X
3, (11-4. 5) 4, (11-4.5) y = R cos0
S= R sin0
R = constant}See Eq. 7, reference 1.
Note: The reader should observe that in the last two representations we have used V(y,X) instead of U(y, k). This anomaly will be properly straightened out later. This sub-section will be closed with two illustrative examples.
Example I:
Take F(s) = 1/ s + 1, with branch points at s = ±i. Let us cut the s plane by the straight segment from +i to -i. The real part along the cut, sheet I, is given by
1/ 1
7
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, one getsF(s)= -2 s Os 2 dk
+ X2 2 5, (11-4. 5)
since the integral around +i vanishes.
To evaluate the integral we may use reference 2, page 72, entry No. 387, which gives
dx = 1 tan-1 x4(ag - bf);
(ax
2+ b) (fx
2+ g)
b(ag - bf)
b(fx
2+ g)
ag > bf
By direct application we find
1
S1
dk
s2 2 0 + 1 1 Ws
2
12
S 1+
~1
and consequently F(s) = s2 +1 as it should be. Example II: 1 -1 F(s)=; V(y, 0) s = -oo _s -a b
By direct application of 3, (11-4. 5), and since the integration 4, (1I-4. 5) around s = 0 vanishes, one gets
-00
F(s)=L2r-1 dy
Tr
J-
-
s +
y
0
Using reference 2, page 197, entry No. 856. 3, one gets
00 00
1
d
dx
1
0ys
+ y 0 T(i+t) Hence1
F(s) -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, when the pocket is symmetrically bisected by the real axis of the s plane. 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. Branch-cuts need not be straight-line segments.
