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PROEFSCHRIFT
ter verkrijging van de graad van doctor in de technische wetenschap aan de Technische Hogeschool te Delft, op gezag van de Rector Magnificus DL R. Kronig, hoogleraar in de Afdelirig der Technische Natuurkunde,
voor een commissie uit de Senaat te verdedigen op woensdag 21 maart 1962 des namiddags te 2 uur
door
BRUCE BARTON BARROW
geboren te Mahoning Township, Pennsylvania, U.s.A.
TE ASSEN BIJ
Dit proefschrift is goedgekeurd door de promotor, PROF. DR. IR. J. L. VAN SOEST
Ta my parents and ta Trudie
ACKNOWLEDGEMENTS
The author acknowledges with de ep gratitude his debt to the SHAPE Air Defence Technical Centre for supporting this work, thanking
particularly the members of the SADTC Board and Ir.
J.
Piket and Mr. E. C. Williams, respectively the past and present Directors of SADTC. It is a pleasure to acknowledge the constant interest and encouragement of Dr. Nic. Knudtzon, Chief of the Communications Group, who made many helpful suggestions, and to thank all those whohelped prepare the final manuscript, especially Mrs. M. Gye, Assistant Scientific Editor.
During the early stages of the research the author had many stimulating conversations with Dr. Herbert B. Voelcker, Jr., then of
the University of London, with Dr. Robert Price of the M.LT. Lincoln Laboratory, and with Mr. John N. Pierce of the U.S. Air Force Cambridge Research Laboratories. A number of the ideas in this dissertation sprang from these discussions.
SUMMARY
This study presents theoretical results on the radio transmission of information by the use of discrete signals, that is, by signals that can take only a finite number of different forms. Particular attention is given to the effects of fading in the transmission medium, and various receiving systems, including some that use diversity reception, are examined. The models used are most appropriate for scatter propa-gation, but are to some ex tent applicable also to other modes of propagation.
The transmitted signals are assumed to be coded into binary form, and error probabilities are calculated both for individual binary elements and for characters (i.e. short blocks of elements). Various methods of keying are considered, with particular emphasis on frequen-cy-shift keying. After t.ransmission the signal is corrupted by fading and by additive Gaussian noise. The assumed fading is single-path, frequency-nonselective, and slow enough so that the individual binary elements are not distorted. Rayleigh fading, or a generalization of it, is assumed for numerical work, but a number of the principal results of the study are shown to apply for more general fading distri-butions. The receiver is assumed to have knowledge of the waveforms of the two elementary signals, though the amplitudes mayor may not be known. Element-synchronous reception is assumed; both phase-coherent and nonphase-coherent reception are studied.
Chapter l is a general introduction. Chapter II discusses fading and diversity reception. It is shown that the diversity technique of equal-gain combination is nearly as effective for discrete-signal reception as maximal-ratio combination, which is the optimum method. This result holds for a number of classes of fading distributions.
Chapter IIl compares some different techniques of modulation, diversity combination, and reception. Diversity reception is shown to be a highly effective way of reducing error probability under extremely
general circumstances. A theorem is proved that relates element-error probability on a single diversity branch to the element-error probability that would be obtained aft er diversity combination.
Chapter IV discusses frequency-division-multiplexed FM trans-mission, which is commonly used with multichannel
tropospheric-scatter systems. It is shown that virtually all discrete-signal errors wiU occur while the radio-frequency carrier is below the FM threshold, and
that the beha vi or of the FM receiver and diversity combiner with weak
signals is therefore critically important. Other conclusions are (1) that
considerations involved with discrete-signal transmission do not justify the use of preëmphasis with the baseband signal; (2) that increasing
the level of the discrete-signal sub-carrier is proportionately much less effective in reducing error probability than is increasing the level of the
transmitted radiofrequency carrier; (3) that equal-gain combination of intermediate-frequency signals is theoretically superior to maxima
l-ratio combination of signals at baseband frequency; and (4) that there
is a conflict between the requirements for discrete-signal channels and those for telephone channels.
Chapter V is concerned with systems in which mark and space signals
are subjected to more or less independent fading. It is shown that the
theoretical advantage to be gained with such systems decreases with
increasing order of diversity.
Chapters VI and VII treat character-error probabilities for
short-block codes. The constant-ratio code, in particular, is analyzed. Chapter VI considers detectors th at operate on the individual received
elements. It is shown that, if the fading is slow as compared to the
length of a character, there is a considerable clustering of element
errors, so that multiple errors in a character are common. Increasing transmitter power under these circumstances, although it reduces the total number of errors, does not significantly change the ratio of multi-ple errors to single errors, which fact sharply limits the effectiveness of redundant codes. On the other hand, increasing the order of diversity greatly reduces the clustering of errors and hence the pro-portion of multiple errors. It follows th at two systems operating at the same average element-error probability, and subjected to the same fading in the propaga tion medium, ma y ha ve grea tl y different character-error probabilities.
Chapter VII gives error probabilities for systems that operate on the whole character. The ideal character receiver and a simpler, non-ideal
character receiver are considered, and both are generalized to all ow for the rejection of excessively ambiguous characters. Therefore these
receivers can be used in automatic error-correcting systems, in which
the reception of unacceptable characters initiates a request for retransmission of the information. Character-detecting receivers are superior in two ways to receivers that make independent decisions for
each element. Firstly, they make better use of the received information
and therefore show lower error probabilities. Secondly, they are more flexible, in that the criteria for acceptance of a eh ar act er can be set quite arbitrarily, so that the probability that an incorrect character will be accepted can be made as low as desired.
SAMENVATTING
Deze verhandeling behelst een theoretisch onderzoek omtrent het overbrengen langs radioverbindingen van informatie in de vorm van discrete signalen, d.w.z. signalen, die slechts een beperkt aantal vormen kunnen aannemen. De voornaamste aandacht is gewijd aan de invloed van fading bij de transmissie, waarbij verschillende ontvangstsystemen worden bestudeerd; daaronder zijn er enkele met diversity-ontvangst. In het bijzonder is de voortplanting met behulp van "scatter" voorop-gesteld, maar in zekere mate zijn de resultaten ook van toepassing op andere voortplantingsomstandigheden.
Verondersteld wordt, dat de overgebrachte signalen binair zijn ge-codeerd; de waarschijnlijkheid van optreden van een fout wordt be-rekend zowel voor individuele binaire elementen als voor lettertekens
(d.w.z. korte groepen van elementen). Verschillende technieken worden in ogenschouw genomen, waaronder in het bijzonqer het sleutelen door frequentie-variatie (FSK). Het signaal wordt na uitzending vervormd door fading en door cumulatieve ruis, welke laatste de statistiek van Gauss volgt. De fading-verschijnselen worden verondersteld betrek-king te hebben op één enkel traject, niet-selectief te zijn voor wat de frequentie betreft en voldoende traag te wijzigen om geen vervorming van individuele binaire elementen te veroorzaken. Voor het numerieke wordt fading met een Rayleigh-distributie - of een veralgemening daarvan - vooropgesteld, maar de auteur toont aan dat een aantal van de voornaamste resultaten van zijn studie van toepassing is op meer algemene verdelingen bij fading. Verondersteld wordt dat de ontvanger de golfvormen van de twee elementaire signalen kent, waar-bij de amplitudines al of niet bekend zijn. De ontvangst van de signalen wordt verondersteld gesynchroniseerd te zijn op de elementen; aandacht wordt gewijd aan fase-coherente en aan niet-coherente ontvangst.
Hoofdstuk I geeft een algemene inleiding. Hoofdstuk II behandelt fading-verschijnselen en diversity-ontvangst. Bewezen wordt dat diversity-ontvangst waarbij alle kanalen eenzelfde transmissie-winst
hebben, voor wat de ontvangst van discrete signalen betreft, bijna even
efficiënt is als de optimale methode (d.w.z. "maximal-ratio" c om-binatie). Dit resultaat geldt voor verschillende categorieën van fading -distributies.
Hoofdstuk III behelst een onderlinge vergelijking van methodes op het gebied van modulatie, diversity-combinatie en ontvangst. Aa
n-getoond wordt dat diversity-ontvangst onder zeer algemene omstandig-heden de kans op het optreden van fouten vermindert. Een stelling
wordt bewezen die het verband legt tussen de foutenkans in één di ve
r-sity-kanaal en deze kans na diversity-combinatie.
Hoofdstuk IV behandelt de transmissie bij gebruik van een fre -quentie-multiplex-systeem met frequentie-modulatie (FDM-FM), een methode die doorgaans wordt aangewend in een meerkanalig tropo
-sferisch scatter-systeem. Aangetoond wordt dat praktisch alle fouten
in de discrete signalen zich voordoen als de hoogfrequent-draaggolf
zich onder de FM-drempel bevindt en dat het gedrag van de F
M-ontvanger en van de diversity-combiner in aanwezigheid van zwakke signalen daarom van kritisch belang is. Andere conclusies zijn, dat:
(1) overwegingen, verb,md houdend met het overbrengen van discrete signalen, het gebruik van "preëmphasis" van het signaal in de basis -frequentieband niet kunnen rechtvaardigen; (2) een verhoging van het
niveau van de hulpdraaggolf van het discrete signaal verhouding
s-gewijze veel minder nut heeft dan een verhoging van het niveau van de hoogfrequent-draaggolf voor een vermindering van de wa arschijnlijk-heid van het optreden van fouten; (3) een combinatie van midde n-frequent-signalen op basis van gelijkwaardige transmissiewinst theor
e-tisch beter is dan de optimum-combinatie van signalen in de bas is-frequentieband ; en (4) de eisen geldig voor kanalen voor het ove r-brengen van discrete signalen onverenigbaar zijn met de voorwaarden
gesteld aan telefoonkanalen.
Hoofdstuk V handelt over systemen waarvan de rust-en werkeleme n-ten aan min of meer onafhankelijke fading-verschijnselen onderhevig zijn. Aangetoond wordt dat het theoretische voordeel dat men uit dergelijke systemen zou kunnen halen, vermindert naarmate de orde van diversity in grootte toeneemt.
Hoofdstukken VI en VII zijn gewijd aan de waarschijnlijkheid van het optreden van foutieve lettertekens in korte code-combinaties.
Meer in het bijzonder wordt het codesysteem met constante verhouding
uitvoerig besproken. Hoofdstuk VI handelt over detectoren die op
element-basis werken. Aangetoond wordt dat, indien de fa ding-verschijnselen traag verlopen in verhouding tot de duur van een lett
er-teken, er een aanzienlijke concentratie van element-fouten zal optreden,
zodat veelvoudige fouten in één letterteken een normaal verschijnsel zullen zijn. In dergelijke omstandigheden leidt een verhoging van het zendervermogensniveau - hoewel dit een vermindering van het totale aantal fouten met zich meebrengt - tot geen diepgaande wijziging van
de verhouding tussen veelvoudige en enkelvoudige fouten, waardoor
het nut van redundante codesystemen sterk wordt beperkt. Anderzijds heeft een vergroting van de orde van diversity een sterke teruggang van fouten-concentraties en derhalve ook van veelvuldige fouten tot re
-sultaat. Daaruit vloeit voort, dat twee systemen met éénzelfde
ge-middelde elementfoutenkans die beide onderhevig zijn aan dezelfde
fading-verschijnselen in het voortplantingsmedium, sterk uiteenlopende
eigenschappen kunnen vertonen voor de foutenkans van een letterteken. Hoofdstuk VII behandelt de foutenkans bij systemen die op letter-teken-basis werken. Een ideale ontvanger van dit type en een meer
eenvoudige, niet-ideale ontvanger worden bestudeerd; beide ontvangers worden zo algemeen genomen dat er een mogelijkheid bestaat àl te
twijfelachtige lettertekens te verwerpen. Deze ontvangers kunnen der-halve worden aangewend in systemen met automatische foutencorrectie, waar de ontvangst van een als onaanvaardbaar erkend teken een
procedure inzet waarbij om heruitzending van de informatie wordt gevraagd. Ontvangers met letterteken-detectoren zijn in tweeërlei op-zicht te verkiezen boven ontvangers waarbij het beoordelingscriterium op ieder element afzonderlijk werkt. In de eerste plaats maken derge -lijke ontvangers beter gebruik van de binnengekomen informatie en vertonen derhalve een kleinere foutenkans. Ten tweede laat hun
ge-bruik meer soepelheid toe, in die zin dat de criteria voor het als juist
aanvaarden van een letterteken arbitrair kunnen worden vastgelegd, zodat de waarschijnlijkheid dat een verkeerd teken als juist zal worden aanvaard zo laag kan worden gehouden als gewenst is.
TABLE OF CONTENTS
CHAPTER I. AN INTRODUCTION TO THE PROBLEM 1 1. The Nature of the Problem Studied 1
2. Previous Related Work. 2
3. This Study . . 5
CHAPTER 11. FADING AND DIVERSITY RECEPTION. 7
1. Introduction . 7
2. The Phenomenon of Fading. 7
3. A Mathematical Model for Simple Fading. 8 3.1 Fast Fading and Slow Fading . . . . 8 3.2 Time Intervals for Discussion of Fading. 9 4. Probability Distributions for Fading. . . . 11 4.1 Amplitude Distributions for the Path Strength. 11 4.2 Distributions for Signal-to-Noise Power Ratio 12 4.3 Notes on Applying the Distributions 14 4.4 Phase-Shift and Time-Variations . 16 5. Diversity Reception . . . 16 5.1 Background . . . 16
5.2 Diversity-Receiver ModeIs. 17
5.2.1 Conditions and assumptions. 18 5.2.2 Three Iinear diversity-combining
tech-niques . . . 18 5.2.3 Nonlinear combining techniques . 19 6. Probability Distributions af ter Combination. 20 6.1 Independent Fading . . . 20 6.1.1 General results. . . 20 6.1.2 Independent Rayleigh fading, all (Rj) equal 21 6.1.3 Independent Rayleigh fading, unequal (Rj) 23
6.1.4 Non-Rayleigh fading 28
6'1
Correlated Fading 30CHAPTER lIl. THE RECEPTION OF DISCRETE SIGNALS 32 1. Introduction . . . 32 1.1 Material to Be Treated 32 1.2 Model. . . 32 1.2.1 Transmitted signals 32 1.2.2 Fading medium 32 1.2.3 Noise 33 1.2.4 Radio receiver 33 1.2.5 Discrete-signal receivers 33
2. Error Probabilities for Discrete-Signal Receivers. 34 2.1 The Probability-Computing Receiver 34
2.2 Error Probabilities. . . 34 2.2.1 Coherent reception. no fading . . 34
2.2.2 Noncoherent reception. no fading. 36
2.2.3 General comments on reception. no fading 36
3. Discrete-Signal Reception in the Presence of Fading 38 3.1 Reception with Amplitude and Phase of Signal
Known . . . . . 38
3.1.1 Single diversity branch, m-distribution 38
3.1.2 Diversity reception . . . . . 40 3.2 Reception when Amplitude and Phase of Signal
Are Unlmown . . . 41 3.2.1 Single diversity branch . . . . . 41
3.2.2 Diversity reception. Rayleigh fading 44 3.3 Maximal-Ratio Combination with Envelope
De-tection . . . . . 44
4. CaJculations. Comparisons. and Discussion. . . . . 46
4.1 Distribution of the SNRs at which Errors Occur 46 4.2 Comparisons of Receiving Technigues. . . . . 48
4.2.1 Coherent and noncoherent diversity
re-ception. . . 48
4.2.2 Further discllssion of the theorem. 49
4.3 A Numerical Example. 50
5. ConcJuding Remarks . 52
CHAPTER IV. DISCRETE-SIGN AL TRANSMISSION
WITH MULTIPLEXED SIGN ALS ON AN FM
CARRIER 54
1. Introduction . . . 54 1.1 The Nature of the Problem 54
1.2 Method of Attack . . . . 55
1.3 Details about the Assumptions . 56 ] .3.1 Gaussian noise . . . 56 1.3.2 Models for the FM-receiver threshold . 59 2. CaJculations of Error Probability with Fading . 61
2.1 Range of Parameters in Threshold Modeis. 61
2.2 Error Probability with the Sq uare-Law Threshold 63
2.4 Non-Rayleigh Fading. . . . . 66
2.4.1 Vertical threshold . . . . . 66 2.4.2 Square-law threshold with m-distribution
fading . . . . . 66 2.4.3 Square-law threshold with Nakagami-Rice
fading . . . . . 68
2.5 Implications for System Design. . . . . 68 2.5.1 Summary of error-probability analysis. 68 2.5.2 Effect of increasing link strength. . . . 68
2.5.3 Effect of increasing level of discrete signal 69
2.5.4 Effect of position in baseband . 69
2.5.5 Effect of increasing deviation of RF carrier . . . 69 3. Systems Compared and the Design of Receiving
Equipment. . . . . 70
3.1 Error Probabilities for Various Methods of Combination . . . 70
3.1.1 Maximal~ratio IF combination . . . . . 70
3.1.2 Equal-gain IF combination . . . . . 72 3.2 Other Techniques of Transmission and Reception 72 3.2.1 Coherent FSK reception 72 3.2.2 Coherent AK reception . . . 73 3.2.3 PSK reception . . . 73
3.3 Diversity Combination of Weak Signais. 74
3.3.1 Why it is important . . . 74
3.3.2 Why it is difficult to combine weak signals 74
3.3.3 Toward a practical colution. 76 4. Concluding Remarks . . . 77
CHAPTER V. RECEPTION OF DISCRETE SIGN ALS
WHEN MARK AND SPACE SIGNALS FADE
SEPARATELY. 79
1. Introduction . 79
1.1 The Model. . 80
1.2 Notation . . 81
2. Maximum-Likelihood Detection 81
2.1 The Likelihood Ratio. . . 81
2.2 Optimum Diversity Reception 82
3. Element-Error Probabilities with Fading 85 3.1 Flat Fading and Independent Fading on Mark
and Space Channels. . . . . 85
3.2 Correlated Fading on Mark and Space Channels 86 3.2.1 One diversity branch . . . . 86 3.2.2 Dual and quadruple diversity. 87 4. Discussion and Conclusions . . . 88
CHAPTER VI. CHARACTER-ERROR PROBABILITIES
WITH CONSTANT-RATIO CODES.
1. Introduction .
91 91
1.1 Questions Examined in This Chapter . . 92 1.2 Error Relations for the Moore Code. . . 93 2. Errors with IE Fading and with COC Fading 95 2.1 Character-Error Probabilities with IE Fading 96 2.2 Character-Error Probabilities with COC Fading 96 2.2.1 Element-error probabilities. 96 2.2.2 Character-error probabilities . . . . . . 97 2.3 Discussion of Results. . . . . 98 3. Character-Error Probabilities with
Maximum-Likeli-hood Coherent Detection of Elements . . . 103 3.1 Mathematical Formulation. . . 103 3.2 Comparison of Coherent and Square-Law Methods
of Combination and Detection . . . 104 3.3 An Analysis of the Various Types of Errors . . 106 3.4 Discussion . . . 107 4. Character-Error Probabilities with Independent Mark
and Space Fading. . . 108
4.1 Maximal-Ratio Combination . 108
4.2 Equal-Gain Combination 109
4.3 Discussion . 111
5. Conclusions . . . 113 CHAPTER VII. DETECTION OF CHARACTERS AS A WHOLE 115 1. Previous Work . . . . . 115 2. Maximum-Likelihood Detection of Characters.. 116 2.1 Ideal Detection of Binary Compound SignaIs. 116 2.2 Maximum-Likelihood Reception of Binary
Com-pound Signals . . . 119 2.2.1 Non-redundant codes . . . . 119
2.2.2 Redundant codes . 119
2.3 The Likelihood Ratio as a Criterion for Signal Rejection . . . 120 2.3.1 The generalized likelihood-ratio test . . 120 2.3.2 Application of the generalized test . . . 122 2.3.3 Implementation for the constant-ratio
code. . . 123 3. Error Probabilities for the Generalized
Likelihood-Ratio Receiver . . . 125 3.1 General Expressions for Error Probabilities 125 3.2 Expressions if Noise is Gaussian . . . 127 3.3 Calculations. . . 128 4. Error Probabilities for a Constant-Gain Receiver. 130
5. Concluding Remarks . . . 136
BIBLIOGRAPHY. . . 138
LIST OF SYMBOLS AND NOTATlONS 144
LIST OF ABBREVIA TIONS . 148
CHAPTER I
AN INTRODUCTION TO THE PROBLEM
J. The Nature of the Problem Studied
This study presents theoretical results on the radio transmission of discrete messages, which are messages th at can take only a finite numb~r of different forms. The set of discrete messages available may be the twenty-six letters of the English alphabet, for example, or it may be the mark and space used in binary telegraphic transmissions. In what follows we shall usually be concerned with binary transmissions, 50 that there are only two elementary messages, i.e. two basic building blocks from
which complete transmitted messages are made.
The waveforms th at are used to transmit discrete messages through
a communications channel are called discrete signals, and it is assumed
th at they are completely specified, a different one for each of the possible
discrete messages. The problem of transmitting information in th is case
is reduced to the problem of transmitting a sequence of waveforms, each
one selected from a specified and fini te set. This contrasts with the problem of transmitting an ordinary telephone signal, for the set of possible waveforms that may be produced by the human voice is infinite and only loosely specified.
Since the set of possible discrete messages is by definition finite, the
problem of reception of discrete signals may be expressed as a problem
in statistical decision theory - given certain experimental data (a
re-ceived signal waveform), is a particular hypo thesis likely to be true (for
example, the hypothesis that the discrete message transmitted was the number 7). Since a decision is either right or wrong, a discrete-signal receiver is ordinarily judged in terms of error probability - how of ten is an incorrect message chosen from the set of possible transmitted messages. A telephone receiver, by contrast, is ordinarily judged in terms of the noise or distortion that corrupts the voice signal that passes through it.
In what follows we shall generally assume th at the receiver has nearly full knowledge about each of the possible transmitted discrete signais.
The wa veform corresponding to each discrete signal is assumed to be known, thougr its amplitude mayor may not be known. The
signiji-cant instants, i.e. the instants that separate the elementary signals from each other, are assumed known at the receiver, although the re-ceiver mayor may not know the radiofrequency phase of the received signal. Thus the problems considered include element-synchronous reception at all speeds, and there is in principle no difference between data transmission and telegraphic traffic. Occasionally, of course, a distinction will be made in numerical examples.
The discrete signals that are transmitted are assumed to be corrupted in the transmission medium by a multiplicative disturbance (fading) that introduces random variations in amplitude and RF phase. They are further corrupted by additive Gaussian noise.
2. Previous Related Work
The first modern work on discrete-signal reception was th at of Kotel'nikov,l who defined the "ideal" discrete-signal receiver as the receiver th at would on the average make the fewest incorrect decisions as to which discrete signal had been transmitted. He then showed th at that receiver is ideal which chooses the signal that, on the basis of the information available to it, is the one most probably transmitted. The mathematical formulation of th is idea is very direct, and proceeds as follows. Let there be K possible discrete messages, and denote the K discrete signals corresponding to them by
One of these signals may be identically zero. Denote the received signal, which is a corrupted vers ion of one of the discrete signais, by y(t). From the product law of probabilities we can write
(1-1) where P(y, Xk) is the joint probability th at Xk will be transmitted and y will be received, P(Xk) is the a priori probability th at Xk win be trans-mitted, P(y J Xk) is the conditional probability th at y will be received if Xk is transmitted, P(Xk Jy) is the a posteriori probability that if y is re-ceived Xk was the signal th at produced it, and P(y) is the total probability that y will be received, given by ~kP(Xk)P(yJXk). Since 1 V. A. Kotel'nikov, The Theory of Optimum Noise Immunity, trans. by R. A. Silverman (New Vork: McGraw-Hill, 1959). This book was published in Russia
P(y) is independent of k it is of no importance in deciding which signal was transmitted. Equation 1-1 may be rewritten
(1-2)
which is a form of Bayes's theorem. Kotel'nikov's ideal receiver chooses the message corresponding to the signal that was most likely transmittéd, i.e. to the signal that maximizes P(Xk Iy) or, what is the same thing, tha t maximizes
(1-3) The set of a priori probabilities given by the P(Xk) are assumed to be part of the information available to the receiver. Thus the problem of ideal reception reduces to the problem of calculating P(y
I
Xk) for the given y and for each of the set of Xk. This formulation is quite general. For the special case of additive-noise interference, P(yI
Xk) is exactly the probability of occurrence of a noise waveform given by y-Xk, and Kotel'nikov proceeded to evaluate this probability, assuming white Gaussian noise of a given power density.2When y is specified and the quantity P(y
I
Xk) is considered as a function of k, it is then not a probability distribution at all, and is given the name likelihood junction. A maximum-likelihood receiver is one that chooses the message corresponding to thesignal thatmaximizesP(y IXk).In general a maximum-likelihood receiver is not an ideal receiver, but in the important practical case in which the P(Xk) are all equal, e.g. telegraphy with marks and spaces equally common, the maximum-likelihood receiver maximizes expression 1-3 and is therefore ideal in Kotel'nikov's sense.
It is only recently that Kotel'nikov's work has become known in the West, and work here has developed instead from the very similar conclusions on ideal reception reached by Woodward and Davies.3
Kotel'nikov treated telegraph reception in detail, whereas Woodward 2 Strictly speaking, P(y) and P(Y!Xk) in Eq. 1-2 are probability densities, for the
probability that a specific waveform Y will occur is zero. The difficulty may be
handled by sampling the waveforms at, say, L points. Then P(y) and P(Y!XkJ may be expressed as probability-density functions in L dimensions.
3 P. M. Woodward and 1. L. Davies, "Information theory and inverse probability in telecommunication", Proe.lEE, vol. 99, part. 111, pp. 37-44 (March 1952). See also P. M. Woodward, Probability and lnformation Theory with Applieations to Radar (London: Pergamon Press Ltd., 1953).
and Davies, although they formulated the problem of discrete-signal reception, were primarily interested in radar reception. The problem of radar reception differs in several important ways from th at of dis-crete-signal reception. For example, with radar one may not have a very good idea whether or not a signal is in fact present, and the precise shape and location in time of the signal waveform are usually not known.
The work of Woodward and Davies was soon extended by Helstrom,4 who considered the pro blem ofresol ving discrete signals in Gaussian noise, and who studied the binary case in detail. Helstrom considered both the case in which the RF phase of the signal is known and that in which it is unknown, showed how the two discrete signals should be made dif-ferent from each other, and computed errorprobabilities formany cases.
All the work cited above concerns non-fading signals. Very general work on the reception of fading signals has been done by Price5 and Kailath,6 who have considered a general type of fading th at includes various classes of multipath and frequency-selective disturbances, and who have succeeded in specifying forms for the ideal receiver of signals corrupted by such fading and additive Gaussian noise. The work th at follows in this study is based on a simpIer fading model and is largely concerned with calculations of error probabilities for specific types of recept ion and with the assessment of non-ideal methods of reception. In some respects it extends published work of Law7 and Turin.8 Law considered ideal diversity reception and Rayleiglj. fading, with the phase
4 C. W. Helstrom, "The resolution of signals in white, Gaussian noise", Proc. I RE, voi. 43, pp. 1111-1118 (Sept. 1955).
5 Robert Price, "The detection of signals perturbed by scatter and noise", Trans. of the lRE Professional Group on lnformation Theory, no. PGIT-4, pp.163-170 (Sept. 1954); "Optimum detection of random signals in noise, with applications
to scatter-multipath communication, I", lRE Trans. on lnformation Theory, vol. IT-2, pp. 125-135 (Dec. 1956); "Error probabilities for the ideal detection of signals perturbed by scatter and noise", Group Report 34-40, M. 1. T. Lincoln Laboratory, Lexington, Mass. (October 1955).
6 Thomas Kailath, "Optimum receivers for randomly varying channels", in lnformation Theory, ed. by Colin Cherry (London: Butterworths, 1961).
7 H. B. Law, "The detectability of fading radiotelegraph signals in noise", Proc. lEE, vol. 104, part B, pp. 130-140 (March 1957).
8 George L. Turin, "Communication through noisy, random-multipath channels", 1956 lRE Convention Record, Part 4, pp. 154-166; "Error probabilities for binary sy,mmetric ideal reception through nonselective slow fading and noise",
Proc. lRE, vol. 46, pp. 1603-1619 (Sept. 1958); "Some computations of error
rates for selectively fading multipath channels", Proc. of the No.tl. Electronics Conference, vol. 15, pp. 1-10 (1959).
of the signal known exactly at the receiver. Turin studied reception over a single channel with a more general type of fading, and with the mean value of the phase of the signal assumed to be known at the re-ceiver. Many other authors who have worked on related aspects of the problem of discrete-signal reception are cited in the chapters th at follow and in the Bibliography.
3. This Study
The work in th is study is based for the most part on a very simple type of fading - single-path, frequency-nonselective, and sufficiently slow that the individual discrete signals are not distorted. Such a model is most appropriate for the fading th at is met with tropospheric-scatter and ionospheric-scatter propagation. It is also useful for conventional ionospheric propagation when only one path is effective, and for de-scribing fading on line-of-sight paths.
Much of the work that follows, but not all of it, is concerned with binary transmission. The emphasis is not on ideal recept ion as such, although a few new results on ideal reception are presented. The emphasis is rather on comparison of various practical techniques of reception, especially diversity techniques. Many special examples are calculated, both for ideal receivers and for non-ideal receivers th at may be easier to build, in order to get an idea of how much might be gained by various refinements of technique.
A number of results are given that apply whatever the distribution of signal amplitudes produced by the fading, or th at depend only weakly upon the fading distribution, it being assumed th at the restrictions noted in the above paragraph are met. These results are among the most important ones in the dissertation. The author knows of no other work on discrete-signal reception that gives general results except for Rayleigh fading or a mathematical generalization of Rayleigh fading, even though experimental evidence indicates th at non-Rayleigh fading is common.
The dissertation is organized as follows. Chapter Il contains a dis-cussion of fading and diversity reception, and is mainly intended to lay a foundation for the work in the succeeding chapters. Most of the results in Chapter Il are not new, but it contains at least one extension of earlier work on diversity-combining techniques.
Chapter III discusses discrete-signal reception in the presence of fading for some simple cases. Error probabilities for two important classes of discrete-signal receivers are discussed - those for which the
RF phaseof the signal is known, and se ver al types for which it is un-known. A number of results are presented for comparison of different techniques of modulation, diversity combination, and reception. A theorem is proved which, for aspecific class of systems, relates the error
rate for individual fading signals to the error rate th at would be
obtain-able aft er diversity combination of these signals. This theorem holds for any fading distribution.
Chapter IV contains an analysis of a common system th at uses double modulation - the FDM-FM system frequently used with tropospheric scatter, in which a nu mb er of frequency-division-multiplexed channels are used to frequency-modulate an RF carrier.
The FM threshold implied in reception of such a signal is extremely
important for discrete-signal channels, because nearly all errors occur
while the RF carrier is below threshold. The effect of this fact upon
system design is studied in detail.
In Chapter Va study is made of one problem in HF transmission
via the ionosphere - what is the effect of the independent or partially-correlated fading observed on the mark and space channels when these
are separated (in frequency-shift-keyed systems) by several kilocycles per second? Law's results on this problem are extended and expressed
in closed form, and partial correlation is treated.
Chapter VI treats character-error probabilities for short-block codes,
and constant-ratio codes are given particular attention.9 Various factors that influence the clustering of element errors within a
char-acter are discussed.
Chapter VII treats ideal reception of characters from a constant-ratio
code, and extends Kotel'nikov's ideas of ideal reception to a receiver that is allowed to reject signals th at are too ambiguous. Such a tech-nique is appropriate, for example, in those systems in which it is possible for the receiver. to request repetition of an ambiguous signal. The criterion for ambiguity may be defined in terms of the likelihood function, and when this is done for a constant-ratio code, the mathe-matical formulation of the generalized ideal receiver becomes quite
simple. Computed results are presented for the generalized ideal
re-ceiver and for a non -ideal modified version tha t would be simpler to build. Details about the models studied and about the assumptions made
are given at the beginning of each chapter, so th at the models may be developed as the work proceeds. Conclusions are summarized at the
end of each chapter.
9 The characters in constant-ratio codes are composed of binary elements, and all characters have the same number of marks and the same number of spaces.
CHAPTER II
FADING AND DIVERSITY RECEPTION
1. Introduction
The primary purpose of this chapter is to bring together the principal results on fading and diversity reception that are used in this study. It also serves to fix terminology and notation th at will be used throughout the later chapters. The first sections briefly discuss fading
as a propagation phenomenon and present some of the probability distributions commonly used in the statistical description of fading. The later sections discuss diversity reception as a technique for combatting fading.
The excellent summary papers by Brennan1 and Stein2 give a thorough introduction to the subjects of fading and diversity reception, and much of what follows is drawn from them.
2. The Phenomenon of Fading
When an unmodulated radiofrequency carrier is transmitted to a receiver th at is some distance away, the received carrier is observed to fluctuate randomly in both amplitude and phase. This phenomenon is termed lading, and it appears quite generally. Special techniques
were early developed to combat fading on the shortwave bands, including automatic gain control, diversity reception, and (for reliable discrete-signal reception) transmission using error-detecting codes. These techniques are also used for so-called line-of-sight transmission, where fading must be planned for even when the path between trans-mitter and receiver is apparently free of obstructions.
In recent years two new developments have stimulated much new work on fading and on all of the special techniques listed above. The first was the discovery of the various scatter modes of propagation, for which fading is much more severe than that met formerly. The second 1 D. G. Brennan, "Linear diversity combining techniques", Proc.IRE, vol. 47, pp. 1075-1102 (June 1959).
2 S. Stein, "Clarification of diversity statistics in scatter propagation", pp. 274-295 in Statistical Methods in Radio Wave Propagation, ed. by W. C. Hoffman (London: Pergamon Press, 1960).
is the greatly increased de mand for circuits to transmit digital infor-mation, which has led to the proposal of many new techniques in modulation and coding that cannot properly be evaluated until more is known about fading.
3. A Mathematical Model for Simple Fading
The effects of simple fading on a radio path may be described mathematically by introducing a complex factor
Y(t)ei<p(t)
by which the transmitted carrier is multiplied. Y, sometimes called the path strength, describes the amplitude variation of the received carrier caused by fading, while cp, the carrier phase-shijt of the path, describes the phase variation. Y and cp are usually random variables
in time, i.e. s'~ochastic processes. The fading must therefore be described by specifying them in statistical terms.
For the model of the above expression to be useful, the transmitted
signal must be narrow-band. Furthermore, the expression implies th at Y and cp remain essentially constant over many cycles of the carrier
frequency. An additional restriction is that it cannot be used as it stands
to describe multipath phenomena or the effects of a frequency-selective
or dispersive medium. A generalization can, however, be used to
de-scribe a selectively fading multipath channel. 3
As already noted, this model for fading is most useful with the various scatter modes of propagation, but it can also be applied in some cases
to conventional ionospheric propagation and to line-of-sight pat:b.s. 3.1 Fast Fading and Slow Fading
The earliest ob servers of fading phenomena made a distinction
between two types of fading, fast and slow, simplyon the basis of what
they observed. The distinction remains important, not only for an
understanding of the physical causes of fading, but also for the mathe-matical description of fading and for the design of communications systems.
It is generally assumed th at fast fading arises from partialor com-plete scattering of the transmitted wave, so th at the received wave may be thought of as made up of a number of sm all components.4 The
3 Turin, "Communication through noisy, random-multipath channels", p. 154.
4 J. A. Ratcliffe, "Diffraction from the ionosphere and the fading of radio waves", Nature, vol. 162, pp. 9-11 (July 3,1948).
precise nature of the ionospheric and tropospheric inhomogeneities th at produce the scattering is still under intensive investigation. For our
purposes it is sufficient to postulate that such inhomogeneities exist, and th at the inhomogeneous structure moves in some way so that the
small component "sub-waves" of the received wave randomly rein force
and cancel each other as time passes, thus producing the fast fading. The term "slow fading" is applied to a generally slower phenomenon.
This type of fading results from large-scale changes in the propagation
medium - variations in humidity or temperature in the troposphere,
or in electron density in the ionosphere, for example.
For our purposes the important distinction between fast and slow
fading, however, has to do with diversity reception. Because fast fading has its origin in tiny path-delay differences among the various
com-ponents of the received wave, it tends to be different for signals that are received on two antennas spaced some distance apart, or for signaIs
on two different carrier frequencies that are received on the same antenna, or for signals that are separated by a few seconds in time. This "diversity effect" implies th at the probability that two such signals
will simultaneously be affected by de ep fades is much less than the
probability th at either one will be so affected. Diversity-reception techniques exploit this facto Slow fading, on the other hand, since it
springs from large-scale changes in the propagation medium, does not
produce a diversity effect.
The two types of fading are superimposed in the radiofrequency wave reaching the receiver. Furthermore, although one type tends to be faster than the other, the two cannot easily be separated by that property alone. Therefore, before taking or interpreting fading data,
and before setting up theoretical modeis, it is necessary to define more
carefully the statistical structure required. Brennan5 and Stein6 have
discussed th is point at length, and it is only necessary here to review the argument.
3.2 Time Intervals for Discussion of Fading
Brennan discusses fading in terms of observations at a receiver over
time intervals of three different durations. The short est of these, T, is taken so short that the fading parameters Y and cp may be regarded as constant within any interval T. On the other hand, T must be long 5 Brennan, op. cito Sections II and XIII are particularly relevant here.
compared to the period of the radiofrequency carrier. When these
conditions are met, which will commonly mean a T of the order of a
millisecond, we can define "local" statistics by averaging over an inter-val T, and we can thus obtain alocal value for the root-mean-square (rms) noise in the receiver, or alocal value for the RF-carrier power.
The local values of signal power and noise power will in general vary randomly in time, and hence these local statistics will have distribution functions, median values, and the like, when observations are taken
over a time T 1 which is chosen much greater than T.
Similarly, observed values of T 1-medians will change from one T 1-interval to another, and will have distribution functions over some very
long time interval T 2, which is chosen much greater than T 1.
The interval T2 , which is frequently one year but may be longer or
shorter, is to be taken long enough to encompass all of the slow-fading effects of interest. Measured or computed values over the T 2-interval are assumed to be the same in other T 2-intervals still to come, and on
this assumption communications systems are designed.
Thus far we have said nothing about how long T 1 should be, except
th at it must be much longer than T, and this is required only so th at
there are enough T-intervals in T 1 to give a sample of reasonable
statistical size. If T 1 is too long, however, the effe cts of slow fading
contaminate the T 1-interval distribution functions, so that results are obtained that are at variance with what would have been expected for fast fading alone.
The whole justification of this procedure is that, for many fading circuits and for much of the time, an interval T 1 can be found that works, i.e. that permits areasonabIe separation of fast and slow fading and th at therefore allows one to predict the performance of a diversity receiving system. Values of T 1 of from 1 to 15 minutes are usually best
with scatter modes of propagation, and somewhat longer intervals are
useful for normal ion-Iayer propagation.
All of the results th at follow and that are concerned with fading are presented in terms of mean values of signal-to-noise power ratio or of signal strength that are taken over a T1-interval. When planning a system, one must take into account the distribution of these T 1-means (more often T 1-medians are used in practice) over all or part of the T 2-interval. There are various ways of doing this, and the point is discussed at length by Brennan. Probably the most appropriate
method, so far as constant-rate transmission of discrete signals is
concerned, is one which pays particular attention to the worst month
un-favorable season. This is because discrete-signal transmission systems, especially when used with diversity reception, tend to show a threshold, so that when slow fading brings the T1-medians below a certain point,
the systems become unusable. Yet when the T l-median is only six or eight decibels higher, by no means an unusual range of variation, the same systems would operate very weU. For this reason it is of little value to try to ave rage error rat es over the whole T 2-interval. One is
ordinarily far more interested in the number of hours during the year
that the error rate wiU exceed a specified limit, and one mayalso wish
to predict how many of these hours may faU within the worst single
month. Data such as these may be estimated using the results of this
study in conjunction with measured or assumed slow-fading
distri-butions (i.e. distributions of T l-medians).
4. Probability Distributions for Fading 4.1 Amplitude Distributions for the Path Strength
The simplest fading models involve only the path strength, Y, and at least three probability distributions have proved to be useful in
various circumstances as analytic expressions for the random variation
in Y. The most commonly used distribution in fading studies is the Rayleigh distribution, which has the probability-density function
(2-1)
and the distribution function
Prob{Y<y} == Wr(y) = 1-exp(-y2/(Y2»). (2-2) This distribution applies when the received waveform results entirely from Rayleigh scattering.
Here a word about notation is in order. The subscript r is used con-sistently to refer to the Rayleigh distribution, and y is a dummy
variabIe associated with the random variabIe Y. Since Y is the modulus of a complex number and is to be identified with the envelope of an RF
carrier, the probability-density function of Eq. 2-1 is obviously zero for negative values of the argument, and we rely on the context, here and elsewhere, to make th is clear. The angle brackets, ( ), denote the average of the random variabIe they enclose, so th at (Y2) denotes the mean-square value of Y, taken over aT l-interval of course.
When the received waveform consists of a constant-amplitude
component plus a Rayleigh-scattered one, Y will have the probability-density function
Wn(Y) = 2ycl exp[ - (y2
+
Yo2) /cJ
Io(2yYo/c), (2-3) where Y 0 is the rms value of the constant-amplitude component, c isthe mean-square value th at the scattered component alone would have,
and 10 is the modified Bessel function. The corresponding distribution function, which is obtained by integrating Eq. 2-3, may be expressed in terms of Marcum's Q-function, which has been tabulated.7
The probability-density function of Eq.2-3 was derived inde-pendently by Nakagami8 and Rice,9 and we shall identify it as the
Nakagami-Rice distribution, using the subscript n to distinguish it. The Nakagami-Rice distribution is one generalization of the Rayleigh distribution, to which it reduces when Y 0 = O.
The third distribution, which has been applied to fading studies by Nakagami, has a probability-density function given by
(2-4)
We shall call this distribution the m-distribition, using a subscript m to distinguish it from the others. The m-distribution is, like the Nakagami-Rice distribution, a generalization of the Rayleigh distri-bution, to which it reduces when m
=
1.4.2 Distributions for Signal-to-Noise Power Ratio
The signal-to-noise power ratio, R, will frequently be of interest. If we assume that R is proportional to Y2, which is ordinarily avalid assumption for any Tl-interval since receiver noise will normally be constant over a Tl-interval, th en we can obtain probability distributions for R by introducing a change of variables in the expressions presented in § 4.1.
7 J. 1. Marcum, "Table of Q functions", Research Memorandum RM-339, USAF Project RAND, Jan. 1950.
8 Minoru Nakagami, "Study on the resultant amplitude of many vibrations whose phases and amplitudes are at random", NiPpon Eleärieal Communieation Engineering, no. 22, pp. 69-92 (Oct. 1940); Eq. 118.
9 S. O. Rice, "Mathematical analysis of random noise", Belt System Teek. J., vol. 23, pp. 282-332 (July 1944), and vol. 24, pp. 46-156 (Jan. 1945).
For a carrier subject to Rayleigh fading, the probability-density
function of R will be
(2-5) and the distribution function will be
Vr(r) = 1 - e-r/(R>, (2-6) where r is a dummy variabie corresponding to R.
For the Nakagami-Rice distribution of Eq. 2-3 the
probability-density function of R is
vn(r) = k-1 exp[-(r
+
Ro)JkJIo[2(rRo)iJkJ, (2-7)where Ro is the signal-to-noise power ratio th at would be produced by the constant-amplitude component of the received carrier alone, and k
is the mean signal-to-noise power ratio that would be produced by the scattered component alone.
If Y fades according to the m-distribution of Eq. 2-4, the
probability-density function of R is
Vm(r) =
r(m)(R)m (2-8)
The corresponding distribution function may be expressed in terms
of the incomplete r-function, which has been tabulated:IO y(m, mrJ(R»)
V m(r)
=
.
r(m) (2-9)
Because the incomplete r-function appears frequently in various forms in wh at follows, we make a slight digression at this point and present
together some of the various relations that wi11 be used.
The incomPlete r-junction. The function th at appears in Eq. 2-9 is defined as follows:
y(a, x)
=
fX
ta-Ie-tdt. • 0(2-10)
10 Karl Pearson, ed., Tables of the Incomplete T-Function (Cambridge, England : The University Press, 1957).
There is a second form, given by
T(a, x)
=
!'ta-Ie-tdt
=
T(a) - y(a, x). (2-11)If a is a positive integer we have
I
OO xi = (a-I)' e-.
X -.,.
. J. ,~a (2-12)The error junction. A related function is the error function, defined
by
(2-13)
The complementary error function is given by
(2-14)
(2-15) The error function is monotone increasing and odd. The complementary error function is monotone decreasing, and
Erfc (-x) = 2 - Erfc x. (2-16)
4.3 Notes on Applying the Distributions
The Rayleigh distribution is completely described when (R) is specified. (R) is directly proportional to transmitter power and antenna gain and inversely proportional to receiver noise figure ; it will of course vary over a T 2-interval.
The Nakagami-Rice distribution is a two-parameter distribution, and
Ro and k are required to specify it. Both Ro and k are directly pro-portional to transmitter power, etc.; over a T 2-interval they will vary more or less independently of each other, which means th at the shape of the distribution will change from one Tl-interval to another.
and (R). m is not related to transmitter power, but both mand (R) will vary more or less independently of each other over a T2-interval.
It is not yet clear which of the three distributions is the most useful in describing actual fading. This is especially true where discrete-signal reception is concerned, for it will be seen in later chapters that errors occur in fading channels only when the received signal-to-noise ratio is rat her low. This implies that the tails of the distribution curves th at correspond to small values of r are of most interest, but few experiments have been made to investigate these tails.
The Rayleigh distribution is the most commonly used, partly because it is the simplest to work with, and partly because it frequently fits observed data rather well, especiaUy for tropospheric scatter. On the other hand, Brennan stat es flatlyll th at "the Rayleigh distribution does not provide an accurate model for actual fading distributions outside of the 0.1 per cent to 99.9 per cent range". Some experiments on discrete-signal reception support the assumed Rayleigh distribution rather weU, but in at least one case the authors have suggested that non-Rayleigh fading might account for the discrepancy between their calcu-lations and their observed data.l2
Most of the work with the m-distribution has been carried out by Nakagami,13 who reports success in describing tast fading, at frequen-cies up to 4 Gc/s, with the m-distribution. In attempting to fit this distribution to observed data he has worked with the centers of the distributions rather than with the tails, so that some of his results are not appropriate for discrete-signal reception. The m-distribution itself may prove very useful, for with an appropriate choice of parameters, Eq. 2-8 could be made to fit the tails of many types of observed distributions.
In what follows the basic results are given, where possible, for all three theoretical distributions. This makes it possible to estimate in many cases whether a change in the form of the assumed fading distribution would have a large or small effect upon a particular system. With the present state of our knowledge about fading it is certainly true th at theoretical calculations are of much more value in predicting the
11 "Linear diversity combining techniques", p. 1088.
12 F. E. Willson and Vii. A. Runge, "Data transmission tests on tropospheric
beyond-the-horizon radio system", IRE Trans. on Comm. Systems, vol. CS-8,
pp.40-43 (March 1960).
13 M. Nakagami, "The m-distribution - a general formula of intensity
distri-bution of rapid fading", pp. 3-36 in Statistical M ethods in Radio Wave Propagation,
comparative performance of similar systems than lil predicting the absolute performance of any one system.
4.4 Phase-Shift and Time Variations
Thus far we have considered only probability distributions of the path strength, Y. Yet time variations of the carrier phase-shift, qJ,
as well as of Y are important, for with discrete-signal transmission such variations in time affect both the probabilities of individu al errors and the grouping of errors in time.
Attempts to arrive at a more detailed theoretical description of fading have been made by Rice,I4 Bremmer,I5 and others. Their models imply either Rayleigh or Nakagami-Rice fading, which somewhat limits their applicability.
The work that follows assumes fading slow enough (or transmission rate fast enough) so th at Y and qJ may be taken to be constant over the time interval occupied by the signalof interest. Occasionally we shall refer, however, to work of other authors that has been based on the more detailed model.
5. Diversity Reception 5.1 Background
As noted above, fast fading is a re sult of small-scale irregularities in the propagation medium. Therefore it has a different effect on signals th at are transmitted on sufficiently separated frequencies, or over sufficiently separated paths, or at sufficiently separated times. The method by which the diversity effect is achieved is of little interest here; the important point is that when it is present one is able to obtain a number of copies of the information-bearing signal, copies that are not identically corrupted by noise and fading. One can then build a diversity receiver to obtain a resultant signal that, at least on the average, is better than any of the individual signals.
The first paper published on diversity reception seems to have been that of A. de Haas,I6 which appeared in 1927 and 1928, several years J4
s
.
o.
Rice, "Distribution of the duration of fades in radio transmission : Gaussian noise model", Belt System Teek. ]., vol. 37, pp. 581-635 (May 1958). 15 H. Bremmer, "Some theoretical investigations on fading phenomena",Sta-tistieal Metkods in Radio Wave Propagation, ed. by W. C. Hoffman (London: Pergamon Press, 1960), pp. 37-39.
ahead of the better known papers of Beverage, Peterson, and Moore.17 ,18 De Haas made his experiments and developed his diversity receiver while working in Java for the PTT of the Netherlands East Indies.
In recent years the exploitation of troposcatter and ionoscatter modes of transmission has stimulated considerable effort toward the design of improved diversity receivers. Kahn19 was the first to suggest, in the context of communication engineering, that signals might be combined to achieve alocal signal-to-noise ratio (SNR) greater than the SNR of any of the component signaIs. Brennan20 extended this result and justified it rigorously.
5.2 Diversity-Receiver Models
Before going deeper into a discussion of diversity reception, it is necessary to define certain terms and to fix certain features of the diversity-receiver models that are considered in what follows. To begin, we define a diversity branch as the communication path between a single transmitter and a single receiving antenna and associated re-ceiver, and a branch receiver as the receiving equipment associated with a single diversity branch.
We shall also find it convenient at times to classify combining techniques according to the stage of the demodulation process at which combination occurs. For a radio link that carries a number of data or telegraph channels we can consider I F combination (in-phase combi-nation of the IF signals of the various diversity branches) and baseband
combination (combination of the baseband signaIs). Brennan uses the terms "predetection combination" and "postdetection combination" , but this terminology is unsatisfactory for a system with several stages of modulation.
pp. 357-364 (December 1927); vol. 11, pp. 80-88 (February 1928). A trans lat ion into English of large portions of this paper appeared recently in Proc. IRE, vol. 49, pp. 367-369 (Jan. 1961).
17 H. H. Beverage and H. O. Peters on, "Diversity receiving system of R C. A. Communications, Inc., for radiotelegraphy", Proc. IRE, vol. 19, pp. 531-561 (April 1931).
18 H. O. Peterson, H. H. Beverage, and J. B. Moore, "Diversity telephone
receiving system of RC.A. Communications, Inc.", Proc. IRE, vol. 19, pp. 562-584 (April 1931).
19 L. R Kahn, "Ratio squarer", Proc. IRE, vol. 42, p. 1704 (Nov. 1954). 20 D. G. Brennan, "On the maximal signal-to-noise ratio realizable from several
Finally, combination of discrete signals becomes with some schemes an integral part of the process of deciding which signal was transmitted. 5.2.I Conditions and Assumptions. Brennan discussed2I in detail the various conditions and assumptions required for diversity reception to work, and it is therefore sufficient to give only a brief outIine here. We shall assume that the received signal in the jth diversity branch, Zj(t), can be represented by
(2-17) where x(t) is the transmitted signal (referred to the appropriate stage in the modulation-demodulation process), ni(t) is additive noise of zero mean, independent of x(t), and Ai is the amplitude disturbance caused by fading. Ai will in general be a function of the path strength, Y. We shall generally assume that the nj(t) are uncorrelated, and that the local noise powers (nj2) are slowly varying or constant. Furthermore we shall usually consider the Aj, taken as random variables over the TI-interval, to be independent. That is, we shall normally be concerned with independent fading on the various diversity branches. In Chapter V, however, we treat some cases of correlated fading.
5.2.2 Three linear diversity-combining techniques. Linear diversity techniques are those that involve a sum, in general weighted, of the Zj defined in Eq. 2-17. The combined signal, Ze, is thus
M
ze(t) =
I
Bjzj(t) , i =1(2-18)
where the Bi are the weights, and M is the nUI?ber of diversity branches, i.e. the order ot diversity. It is understood th at the zi are to be put in phase with each other, or nearly so, before combination. The Bj are not shown as functions of time because they are related to the path strengths and thus are assumed to remain constant during the length of a signal element.
An old technique, and a common one, is selection diversity, in which the best single Zj is chosen, usually on the basis of SNR. With selection diversity all of the Bj except one are equal to zero.
The second technique studied here is maximal-ratio diversity, in which 21 "Linear diversity combining techniques", Part Il.
each Bi is made directly proportional to the associated Ai (signal strength) and inversely proportional to (ni)2 (the local noise power). Brennan has shown22 that this technique gives the large st SNR of any linear diversity-combining technique, the signal-to-noise power ratio of the combined signal being exactly equal to the sum of the signal-to-noise power ratios in the various diversity branches. The principal disadvantage of maxim al-ratio diversity lies in the complexity of the instrumentation required - the combiner requires knowledge of receive:d signal streng th and noise power, which th en must be used to operate variable-gain devices in each diversity branch.
The third technique, and probably the oldest of all,23 is equal-gain
diversity, in which all of the Bi are made equal. For this combiner to work well, local noise powers should be approximately equal in the various diversity branches. Equal-gain diversity has recently received much attention sin ce modern techniques of phase con trol have made it possible to combine intermediate-frequency signals. A modification of equal-gain diversity, called constant-gain diversity, in which the Bi are constant but not necessarily equal, is referred to briefly later in this chapter. Constant-gain diversity is an appropriate technique when the noise powers in the various diversity branches are not nearly equal.
5.2.3 Nonlinear combining techniques. As was mentioned above, the processes of combination and decision may be considered together in discrete-signal systems. This was done by Law,24 and for the case he considered, in which full knowledge about the signals was available to the receiver, the ideal receiving system reduced to a maximal-ratio combiner followed by a matched-filter receiver. In other cases, how-ever, this approach leads to a receiving system that involves nonlinear combination. This was suggested by Masonson,25 who discussed differ-ent ways of weighting the envelopes of the two signals available in a dual-diversity system. He showed that for on-oft keying, decisions against a fixed threshold, and Rayleigh fading, the envelope amplitudes should be squared and added. In other words, square-law detection should be used.
22 "On the maximal signal-to-noise ratio".
23 De Haas's combiner was an equal-gain postdetection combiner. Beverage, Peterson, and Moore, on the other hand, worked toward a selection-diversity system.
24 Op. cito
25 M. Masonson, "Binary transmission through noise and fading", 1957 IRE
A similar result was obtained and justified in a much broader context by Pierce,26 who considered arbitrary orders of diversity with frequency-shift keying. Pier ce showed that, with independent Rayleigh fading, square-Iaw combination was the optimum method of deciding between mark and space if the receiver has no knowledge of ei th er the amplitude or the phase of the received signal. Pierce also gave calculations of the probability of error in the received signal.
6. Probability Distributions after Combination
6.1 Independent Fading
6.1.1 Genera} results. We assume the conditions discussed in § 5.2.1. If there are M diversity branches, and if the distribution functions of the signal-to-noise power ratios on these M branches are given by
V M(r),
then, for selection diversity the distribution function of the signal-to-noise power ratio aft er combination is given by
M V Mse(r)
=
IJ
Vj(r).i~l
(2-19)
Here the subscript M denotes Mth-order diversity, and the subscript se denotes selection diversity.
Equation 2-19 is very easy to understand physically. The probability that the power ratio aft er combination will be Ie ss than r is precisely the probability that all of the power ratios in the individual diversity branches are less than r. If these probabilities are independent, they may be multiplied to give the probability of the composite event. There is no requirement that all of the Vj be the same.
For maximal-ratio diversity the signal-to-noise power ratio aft er combination is, by Brennan's theorem, the sum of the power ratios available on the individu al diversity branches. For equal-gain diversity the local signal amplitude af ter combination is the sum of the signal amplitudes available on the individual diversity branches, and the noise power aft er combination is the sum of the noise powers on the
26 ]ohn N. Pierce, "Theoretical diversity improvement in frequency-shift keying", Proc.IRE, vol. 46, pp. 903-910 (May 1958).
