1958
`
`PROCEEDINGS OF THE IRE
`
`555
`
`A Communication Technique for Multipath Channels*
`
`R. PRICET, ASSOCIATE MEMBER, IRE AND P. E. GREEN, JR.T, SENIOR MEMBER,
`
`IRE
`
`Summary—Application of principles of statistical communication
`theory has led to .a new communication system, called Rake, de-
`signed expressly to work against the combination of random multi-
`path and additive noise disturbances. By coding the Mark-Space
`sequence of symbols to be transmitted into a wide-band signal, it
`becomes possible at the receiver to isolate those portions of the
`transmitted signal arriving with different delays, using correlation
`detection techniques. Before being recombined by addition, these
`separated signals are continuously and automatically processed so
`as to 1) apply to each an optimum weighting coefficient, derived from
`a measurement of the ionosphere response, and 2) introduce in each
`an appropriate delay such that they are all brought back into time
`coincidence.
`After a brief introduction, a functional description of the system
`is presented. There follows a review of the communication theory
`studies, which indicate that such systems have certain optimal prop-
`erties. Details of design of an experimental prototype Rake system
`are followed by the results of limited field tests of this prototype.
`Conclusions and recommendations for future work are given.
`
`I.
`
`INTRODUCTION
`
` ULTIPATH is a troublesome condition in many
`
`communication channels; the signal proceeds
`to the receiver along not one, but many paths,
`so that the receiver hears many echoes having, in gen-
`eral, different and randomly varying delays and ampli-
`tudes. The most familiar example occurs in high—fre-
`quency communication via the ionosphere [1, 2].
`Multipath has been a problem from the early days of
`short-wave radio communication.
`Its influence on a
`
`communication system is usually described in terms of
`two effects—selective fading, and intersymbol interfer-
`ence—of which one or the other may be of predominant
`importance in the particular communication system
`being discussed
`Selective fading has to do with the relative rf phases
`of the signals delivered to the receiving antenna via the
`various paths. At any one frequency, the total received
`signal is a vector sum of individually delayed signals,
`their relative phase angles depending on the frequency
`and the echo amplitudes and delays. Therefore, since
`the echo amplitudes and delays are time varying, one
`observes large variations of the received signal strength
`at a single frequency as a function of time, or of the
`strength at a given time as a function of frequency; the
`latter is termed “selective fading.”
`Intersymbol
`interference is associated simply with
`the time delay between first and last significantly large
`echoes. If the modulation is rapid enough, the echoes
`appearing in this modulation will result in a jumbling
`
`* Original manuscript received by the IRE, August 8, 1957;
`revised manuscript received, November 29, 1957. The research in
`this document was supported jointly by the Army, Navy, and Air
`Force under contract with Mass. Inst. Tech., Cambridge, Mass.
`‘(Lincoln Lab., M. I. T., Lexington, Mass.
`
`or smearing of the intelligence, regardless of the form
`of signal used.
`Previous system design for effectively combating
`multipath disturbances might be considered “passive”;
`that is,
`the effort has mainly been directed toward
`minimizing the undesired effects without having actual
`knowledge of the multipath characteristic. The use of
`single-sideband transmission (with or without exalted-
`carrier reception)
`[3], synchronous AM reception [4],
`and various forms of diversity reception [5] as anti-
`selective fading measures represent
`this type of ap-
`
`rfroach. Fading has also been attacked by using coding
`
`6].
`The intersymbol interference aspect of multipath was
`long ago recognized to place a limit on the rate at which
`digital information could be communicated with time-
`division schemes. The use of multiple subcarriers (“fre-
`quency division”) each having long symbol-waveforms
`to carry a fraction of the total information rate over
`multipath has been standard for many years, and has
`recently received additional
`impetus from new tech-
`niques which yield considerably greater efficiency of
`frequency spectrum utilization [7—8]. A method of ex-
`tending frequency—diVision approach to transmission
`of analog information has also been proposed [10].
`Two other techniques, of quite a different nature
`from those mentioned previously, are directed toward
`the actual or effective suppression of all but one domi-
`nant path. The first employs a complex, steerable an-
`tenna array [50], while the second uses frequency
`modulation [12]. The latter method appears to work
`under only certain conditions which, unfortunately, are
`seldom met in practice [11
`In contrast to the philosophy of the systems just
`enumerated,
`the system which is the subject of this
`paper performs a continuous, detailed measurement of
`the multipath characteristic. This knowledge is then
`actively exploited to combat the multipath effectively.
`Simply stated, selective fading is opposed by detecting
`the echo signals individually, using a correlation method,
`and adding them algebraically (with the same sign)
`rather than vectorially, and intersymbol
`interference
`is dealt with by reinserting different delays into the
`various detected echoes so that they fall into step again.
`For reasons that will be seen shortly, this approach has
`been dubbed the “Rake” system.
`This system evolved from and is largely justified on
`the basis of the application of methods of statistical
`communication theory to the problem of communica-
`tion through multipath disturbances. Soon after the
`idea was thus established, as often happens, a heuristic
`
`PETITIONERS 1010-0001
`
`

`
`556
`
`PROCEEDINGS OF THE IRE
`
`March
`
`MARK
`OSCILLATOR
`4,,-A
`
`MIXER —-.
`$4
`
`.?
`
`FILTER
`TUNED
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`MIXER
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`OSCILLATOR
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`......._.
`
`ENVELOPE
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`
`RECEIVER
`
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`II
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`
`IIIIIIIII IIIIIIIIIIIII
`
`
`
`CHANNEL
`
`BINARY INPUT ~'<«
`
`
`
`
`
`MARK
`OSCILLATOR
`
`
`
`SPACE
`OSCILLATOR
`'5
`
`IIIAIISIII IIEII
`
`Fig. 1—~Simple fsk system.
`
`ence signals, with which the received signal is compared.
`The comparison takes the form of determining (in the
`decision circuit) whether it is the envelope of the differ-
`ence-frequency tone from the Mark filter or the Space
`filter that is the larger. The decision then represents one
`binary element of the output teletype sequence. Mixing
`(multiplying) the incoming and reference signals, and
`passing the difference-frequency result through a nar-
`row-band filter is equivalent to the operation of cross
`correlating the two [16, 17] (that is, multiplying and
`then integrating)? Since the decision is based on which
`symbol yields the larger correlation, it is apparent that
`for best performance the frequency-shift
`IfM -
`should be sufficient
`to yield uncorrelated Mark and
`Space waveforms [18].
`A plot of the output of the filter as a function of
`relative delay between the reference signal and a copy
`of it arriving from the transmitter is the autocorrelation
`function of the reference (the Fourier transform of its
`power density spectrum) [17 In the case of a reference
`having a continuous spectrum of nominal bandwidth
`W,
`the autocorrelation function has a single peak of
`width about 1/ W seconds, centered at the origin, and
`disappearing toward zero elsewhere. Portions of
`the
`transmitted signal arriving with various delays relative
`to the appropriate reference appear in the output of the
`corresponding integrating filter as sine waves of ampli-
`tudes given by the values of the autocorrelation func-
`tion of the reference at the respective delays. With fsk,
`W is small, the central correlation peak is very wide,
`and path contributions will appear in the filter outputs
`for a wide variety of delays. We have already discussed
`the destructive interference (selective-fading) that can
`result when contributions add with random phases.
`When the receiver of Fig. 1 is modified by the substi-
`tution of wide-band transmitted and reference wave-
`
`forms, the correlation function narrows very greatly, so
`that the correlator will make use of only those echoes
`which arrive within 1/ W of synchronization with the
`references. Thus we need only make W sufficiently
`
`3 A more detailed discussion of the relationship of this mix-and-
`filter scheme of correlation (“difference-frequency correlation” or
`“band—pass correlation”)
`to the true mathematical operation of
`multiplying and integrating is given in [17].
`
`PETITIONERS 1010-0002
`
`physical interpretation of the system became apparent.
`We shall, in the interest of clarity, present the latter
`straightforward,
`functional explanation of
`the Rake
`system first, in Section II. Without detracting from its
`significance, we defer until Section III the more rigor-
`ous derivation, which indicates that such a system, in
`addition to making intuitive sense, has certain optimal
`properties. In Section IV, the details of the construction
`of an experimental Rake system are given, and in Sec-
`tion V some results obtained in tests of the system over
`a transcontinental circuit are presented. In the con-
`cluding section the advantages and present drawbacks
`of the Rake system in relation to conventional systems
`are discussed, and suggestions are made for future im-
`provement and extension.
`
`II. FUNCTIONAL DESCRIPTION 01+‘ THE
`RAKE SYSTEM
`
`A. General Principles of Design
`
`Suppose we have decided to build a radioteletype
`system that will detect separately, then add up, each
`of the multiplicity of delayed signals arriving as a result
`of multipath. The transmission will then have to be
`wide-band,‘ for otherwise its time waveform cannot
`possess sufficient detail to permit the waveform at one
`instant of time to be distinguished unambiguously from
`that at another. Naturally, the wider the bandwidth,
`the finer will be the time resolution.
`
`In addition to the requirement of providing multi-
`path resolution,
`the transmission must of course be
`capable of carrying information. In a teletype system,
`this is accomplished by transmitting at will two dis-
`tinguishable waveforms, one representing the Mark
`baud (or signalling element)? and the other Space. A
`commonly-used system is frequency-shift keying (fsk),
`where two sine waves of slightly different frequency are
`employed [13]. By analogy, we shall employ two differ-
`ent wide-band waveforms for the Rake transmission.
`
`Then by proper treatment of the received signal we can
`isolate a narrow region of delay and select for demodula-
`tion only the signal lying within it.
`The principle of the Rake technique can be explained
`by starting with a simple fsk system shown in Fig. 1,
`which will shortly be reinterpreted to permit the use
`of a wide-band transmission and the attainment of the
`
`desired delay—isolation. The Mark and Space local os-
`cillators shown in the receiver may be viewed as refer-
`
`1 The inherent advantage of using a wide-band transmission in a
`multipath environment was first suggested by R. M. Fano in 1952
`(private communication), who reasoned that it averaged out the
`selective fading. Subsequent channel-capacity analyses carried out
`on the fading justified this notion from the information—theory point
`of View [14]. An informal note by Fano [15] proposes another method
`of employing wide-band signals against multipath. Kharkevich [52]
`has independently suggested the use of wide-band signals as a measure
`against selective fading, but his receiver employs completely in-
`coherent detection and this is comparatively ineflicient.
`"’ The word “baud,” taken from teletype usage, will be employed
`frequently in this paper instead of “signalling element,” or “hit.”
`
`

`
`1958
`
`Price and Green: A Communication Technique for Multipath Channel:
`
`MARK BUS
`A2
`‘Mm »~—-*~4
`
`MARK
`REFERENCE
`
`I
`1
`
`4*:
`
`42
`
`$3
`
`SPACE
`REFERENCE
`
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`wide to separate out the various echoes (this also corre-
`sponds to making W wide enough to “average out” the
`selective fading). Finally, we can be assured of making
`use of all the signals delivered to the receiver by the
`entire multipath structure, regardless of whether it con-
`sists of discrete paths or is a continuum, if we use a se-
`ries of such correlators, each synchronized at successive
`delay increments of roughly 1/ W. Enough of them must
`be employed to span a region of delay sufficiently wide
`to encompass all echoes that are likely to appear. The
`name “Rake” seems an appropriate designation for such
`a scheme. Making each correlator synchronize at its
`assigned value of delay can be done by inserting the
`right amount of delay in either the reference or received
`signals. The latter has been chosen for several impor-
`tant reasons to be dealt with later.
`
`With the above ideas in mind, it is now possible to set
`down almost completely an elementary block diagram
`of such a Rake receiver. This is done in Fig. 2. The
`reference sources, emitting the same Mark and Space
`waveforms that the transmitter uses (except for a fre-
`quency displacement of A1), feed a series of correlators
`arranged along a delay line Whose input is the received
`signal. Each correlator consists of a multiplier and an
`integrator, but since the correlator outputs should be
`added together, in order to make full use of each echo,
`we may either combine after separate integrations, or,
`what is equivalent, use a common integrator. The latter
`is obviously simpler, so, as the figure shows, the output
`of each Mark multiplier is added to that of the others
`on a “bus,” a11d the sum is then passed into a common
`integrator. The same is done with the Space multi-
`pliers. A decision on whether Mark or Space was sent
`from the transmitter is made according to which of the
`two sums of correlations is the larger.
`In order to be sure that the difference-frequency sig-
`-nals appearing in the buses all add constructively, their
`phases must be brought into common agreement. Fur-
`thermore, those multipliers responding to large paths
`should have their contributions to the buses accentu-
`
`ated, while those not synchronizing with any significant
`path, and thus being affected mainly by the channel
`noise, should be greatly suppressed, in order to increase
`the snr and consequently decrease the probability of
`making a decision error. More precisely, we may invoke
`the well-known result [19] that the maximum snr of a
`weighted sum, of which each term is the combination of
`a signal and an additive,
`independent noise of fixed
`power, is achieved when the amplitude weighting is done
`in proportion to the signal strength (in voltage). Thus
`the weighting coelficients a,~ in Fig. 2 should be propor-
`tional to as good a measurement as can be made of path
`strengths at the corresponding delays. (In Section III—A,
`the same result will be obtained from a more general
`point of view.) The phase corrections are indicated by <15.
`Provided that W is large enough to isolate a number
`of independently-fading echoes, the deleterious effects
`of selective fading are largely eliminated by such a
`
`557
`
`BINAFH
`OUTPUT
`
`MARK
`|NT_EGRATlNG
`FILTER
`(iuned to A2)
`_l_ENV ELOPE
`DETECTOR
`
`1
`SAMPLER
`AND
`DECISION
`CIRCUIT
`
`ENVELOPE
`DETECTOR
`
`SPACE
`
`
`
`T
`
`ALL %I|E_iNALSBANDWIDTH W
`
`‘\
`
`TAP CIRCUIT
`140.!
`
`___
`SPACE Bus
`® DENOTES A MULTIPLIER
`
`A2
`
`mreennmc
`
`“HER(luned to A2)
`
`Fig. 2--Simplified block diagram of Rake receiver. The f's and A's
`refer to center frequencies. The a’s and ¢>'s are weightings and
`phase corrections, respectively based on path structure measure-
`ments.
`
`scheme, since the path contributions are added alge-
`braically, not vectorially. It happens that intersymbol
`interference is also eliminated by this Rake scheme, al-
`though a few words of explanation will be required to
`show that this is so. Consider the situation when the
`
`transmitter, which has been sending Mark, begins trans-
`mitting Space. Any particular echo delivers this Mark-
`to-Space transition to the receiver input at a different
`delay from the other echoes. The transition supplied
`by this echo, however, will be detected only at that de-
`lay line tap which corresponds to the delay of the echo,
`since at all other taps the echo contribution and the
`reference signals are uncorrelated. With the timing of
`the reference signals set properly so that they correlate
`with the last arriving echo, successively earlier echoes
`will correlate at points on the delay line correspondingly
`further delayed from its input. Thus the total propaga-
`tion time from transmitter to receiver bus outputs is the
`time from transmitter to receiver input plus the line
`length to the appropriate tap, and this sum is the same
`for all paths. Hence only a single Mark-to-Space transi-
`tion appears at the receiver output and intersymbol
`interference is eliminated.
`
`B. Measurement, Weighting, and Phase Correction
`
`The required weighting and phase correction is per-
`formed by the measurement function of the Rake re-
`ceiver, which determines the path strengths and phases
`(including in the latter possible small variations in the
`positioning of the taps on the delay line). This measure-
`ment is accomplished again by the use of correlation}
`since, as already shown, the use of a sufficiently wide-
`band signal enables the various paths to be isolated.
`The integrating filters in this case, however, have a very
`long integration time, in order for the measurement to
`be as noise-free as possible. The integration performed
`
`4 This method is practically identical to that proposed by J. B.
`Wiesner and Y. VV. Lee [20, 21].
`
`PETITIONERS 1010-0003
`
`

`
`558
`
`PROCEEDINGS OF THE IRE
`
`March
`
`MARK
`REFERENCE
`
`T0 MARK
`nus
`A2
`M ULTIPLIER
`
`
`D
`
`
`
`
`INJECTION FREQUENCY
`(common in all
`My circuits)
`RECEIVED SIGNAL
`AFTER FREOUENCV
`4:
`
`CONVERSION
`AND DELAY
`NARROW.
`ArA2 MEASURING FILTER AFA2
`(tuned Io Al -A2)
`c
` «,4, T0
`COMMUTATOR ron
`
`PATH smucrunz
`osssnv/mow
`
`
`
`Fig. 3~Block diagram of tap circuit. (Dashed portion of Fig. 2.)
`
`for the receiver to be able to resolve the multipath and
`isolate and constructively utilize the contributions of
`the various arriving paths. But given this requirement
`of wide-bandedness, what particular sort of waveforms
`should these signals have, out of the great number of
`possibilities? The choice is narrowed considerably when
`the following additional factors are considered:
`1) It must be possible to store the Mark and Space
`signals separately at transmitter and receiver.
`2) It must be possible to keep the timing of the
`reference signals at the receiver from drifting ap-
`preciably with respect to that at the transmitter,
`and to align the time bases to allow for the lag
`given by the time of flight along the longest path.
`3) If repetitive wide-band signals are used,
`they
`must have a repeat time at least as great as the
`multipath duration. If the repeat time is less, paths
`of separation equal to a repeat time or its multiples
`will be indistinguishable. This condition is later in-
`terpreted in terms of sampling theory.“
`4) Each signal, Mark and Space, should have an
`autocorrelation function as near zero as possible
`within the first repeat time away from the central
`peak (which is approximately 1/W wide).
`5) The spectral components outside the assigned
`frequency band should be small enough not
`to
`cause interference to other services. Also, as will
`be seen in the next section, the spectrum should
`be reasonably flat inside the band.
`6) Because high—frequency transmitters tend to be
`limited by peak power rather than average power,
`it is desired that the transmitted signal envelope
`be reasonably uniform,
`in order that a given
`transmitter may supply as much energy as pos-
`sible to each baud.
`
`As we have seen, the Rake system requires Mark and
`Space reference signals of wide bandwidth W in order
`
`5 The frequency translations of A; and A2 allow mixers rather than
`ideal multipliers to be used for A, B, D, and E. In addition to simpli-
`fying the circuitry, the translation A; has the effect of broadening the
`tolerance on reference waveform synchronization [17] in elements A
`and B.
`
`to be continuous, whereas the shift-
`“ We shall assume S(w)
`register transmission is actually a line spectrum. There is no real
`difficulty here, however, for by the sampling theorem [33] so long as
`the (complex) frequency samples are taken no farther apart than
`1/TM, where TM is the multipath spread, the multipath impulse
`response can be completely determined. The appropriate finite-
`interval, orthogonal interpolation functions are of the (sin nt/sin 1.‘)
`Variety. See p. 13 of [14].
`
`PETITIONERS 1010-0004
`
`
`
`
`SPACE
`REFERENCE
`
`T0 space
`sus
`
`MULTIPLIER
`E
`
`
`
`in the Mark and Space integrating filters is, by defini-
`tion, performed over the duration of a signalling ele-
`ment or baud. But the time constant of each of the
`
`measurement filters, as they are called, should be as long
`as the rate of change of the multipath structure will
`permit. In order to assure uniform measurement regard-
`less of the proportion of Marks and Spaces in the trans-
`mission, the measurement is provided by the sum of the
`Mark and Space correlations.
`The method by which the correlation measurements
`are performed and applied is shown in Fig. 3, where a
`typical tap circuit, indicated by the dashed enclosure of
`Fig. 2,
`is depicted in greater detail. The same pair of
`multipliers (A and B) is used to obtain both the baud-
`correlations and the measurement—correlation (that is,
`the tones to be fed to the integrating filters and the
`measurement filter, respectively). The integration of the
`latter is performed by narrow—band filtering of the com-
`bined Mark-Space multiplier output. The amplitude of
`the sine wave out of this measuring filter (C) is propor-
`tional to the strength of the path to which the tap cir-
`cuit responds, while the phase reflects the combination
`of the path and tap phasing. In the final multiplier
`tubes (D and E) this sine wave is mixed with the original
`outputs of A and B, thus performing the desired multi-
`plication of these latter signals by the proper weighting
`function.5
`
`The way in which the phase correction is applied
`warrants more detailed discussion. To begin with, we
`require that the Mark and Space references at the re-
`ceiver be in the same phase relationship to each other as
`at the transmitter, in order that the l\/Iark-Space modu-
`lation produce no undesirable phase discontinuities
`through the measuring filter. When this is done the
`tones of frequency A1 out of the first two multipliers,
`A and B, are phase-coherent with each other (although
`only one is on at a time). Let us call this phase angle 01.
`If 02 is the phase of the injection frequency A2, then the
`phase of the tone of frequency (A1—A2) reaching the
`second multiplier is (61—02—l—0o). 00 represents any addi-
`tional phase shift added during filtering, and the tap
`circuits are aligned to have identical 00. In the second
`multiplier the original phase 01, whatever it was, cancels
`itself out, and only (02-00) remains. But 02 is the con-
`stant phase of an oscillator acting as a common injec-
`tion for all taps circuits. Therefore, all such outputs at
`frequency A2 add with the same phase angle. (Recently
`we have learned that a practically identical scheme of
`phase alignment was invented by Earp [53] for prede-
`tection combining of diversity receivers.)
`
`C. Reference Signals
`
`

`
`1958
`
`Price and Green: A Communication Technique for Multipath Channels
`
`559
`
`I
`I
`1
`
`PULSER
`
`SHIFT REGISTER
`
`SHIFT Puféizs
`“”
`
`I
`I
`|
`I
`
`III
`
`CRYSTAL
`°$°"—W'°“
`
`I
`
`
`POSITIVE AND NEGATIVE PULSE5
`
`FREQUENCY
`MARK
`
`
`TRANSLATION
` TO ro .
`
`sscono
`LIMITER
`BANDPASS
`BANDPASS
`
`
`FILTER
`
`FILTER
`rncouancv
`
` rnnusumou
`
`
`
`
`T0 fo : I/T
`
`
`(1 an integer)
`
`SPACE
`
`
`
`Fig. 4——A method of generating suitable reference signals.
`
`register is large enough compared to W to yield approxi-
`mately Gaussian noise at the filter output [25]). Accord-
`ingly, a second filter, identical to the first, follows the
`limiter to restore the nearly rectangular power spectrum
`without producing as severe envelope fluctuations as are
`present in the output of the first filter.
`To satisfy requirement 7, we seek orthogonal, wide-
`band Mark and Space waveforms having about the
`same energy. If one reference signal, say Mark, has only
`slight envelope fluctuation (as described in the preced-
`ing paragraph), a frequency shift of only (l/ T) is ade-
`quate to produce another nearly-orthogonal waveform
`that we can use for Space, even though the Mark and
`Space spectra may overlap considerably. Here l is any
`nonzero integer and T is the baud length.” This fre-
`quency shift is the final operation shown in Fig. 4. The
`long-term zero cross-correlation function stipulated by
`requirement 8 can be attained only if the long-term
`power spectra of the Mark and Space signals are dis-
`joint. This will be assured so long as the frequency shift
`is not equal to or near a multiple of the reciprocal of
`the shift-register repeat time.
`
`D. Integration and Decision
`
`The integrations upon which each Mark-Space deci-
`sion are based are required to be performed over the
`corresponding baud interval of length T. A filter of
`very high Q (time constant several times T), tuned to
`the bus frequency A2, will faithfully perform the integra-
`tion of the tap circuit outputs. It must be “quenched”
`or “dumped,” however, after each decision to prevent
`ringing from the previous baud from carrying over into
`the succeeding baud intervals [8, 26].
`Just previous to quenching, samples are taken of the
`
`9 Proof: Let the Mark baud-Waveform be represented as R(t) cos
`[wot+¢(t)]; the frequency-shifted (Space) waveform is then R(t) cos
`[(wo-I-27rl/T)t-I-qS(t)]. Neglecting terms near Zwot, the cross-correla-
`tion envelope at the time of sampling is
`
`mgx f T1220) cos (2.4:/T + out w 0
`0
`for R(t) approximately constant, and tends more toward zero as l
`increases. See also [8].
`
`PETITIONERS 1010-0005
`
`7) For good discrimination in the presence of noise,
`Mark and Space signals should have a zero-shift
`cross correlation as near zero as possible.
`(The
`integration time here is the baud length T.)
`8) Satisfactory performance of
`the Rake receiver
`measurement function requires that the signals
`have a long-term cross-correlation function (that
`is, integrated over the “ring time” of the measur-
`ing filters) that is zero for all 7' shifts.
`
`The study of maximal-length (or null-sequency) bi-
`nary shift register sequences
`[22—24, 5.1] has revealed
`one way of constructing waveforms having the desired
`properties.7 The complete scheme for generating Mark
`and Space signals, which is duplicated at transmitter
`and receiver, is shown in Fig. 4.
`The repetitive sequence is generated by a self-driven
`binary shift register in which the modulo-two sum of the
`digits appearing in the output and certain of the inter-
`mediate stages is fed back to the input. For an n-stage
`shift register, the maximal-length binary sequence has
`a period rn=2"——1, and is generated through suitable
`choice of which intermediate stages are fed back to the
`input. One such shift register can easily be set up at the
`transmitter, and another at the receiver (requirement
`1). Each can be driven by pulses derived from crystal
`oscillators (requirement 2) with a period Wt made as long
`as necessary by choosing a sufliciently large number of
`stages n (requirement 3).
`Maximal-length shift register sequences were chosen
`in preference to other classes of binary sequences be-
`cause of a peculiar property that is useful in satisfying
`requirement 4: if we let positive and negative impulses
`represent the binary variable, the shift-register output
`is, of course, a sequence of such pulses repeating every
`in pulse. The autocorrelation function of this maximal-
`length sequence is a sequence of impulses, each of area
`(-1) except at the origin and integral multiples of the
`period m, where there are impulses of area in. This auto-
`correlation function is, for large in, very nearly that of
`a repeated impulse, and hence the power density spec-
`trum exhibits a nearly uniform comb of spectral lines.3
`To limit the frequency spectrum (requirement 5) the
`shift register output sequence is passed through a rec-
`tangular, band-pass filter of width Wcycles per second.
`The envelope of the resulting signal fluctuates consider-
`ably, contrary to requirement 6, but by following the
`filter with a sharply limiting device this fluctuation can
`be suppressed. Such a nonlinear operation of course
`spreads the spectrum somewhat (resulting, for example,
`in a spilling of 10 per cent of the power outside the orig-
`inal band when the output pulse rate of
`the shift
`
`7 The use of short pulses would fit naturally into the Rake con-
`cept, if it were not for the limitation of requirement 6.
`3 The effect of the small negative impulses between the large
`positive ones in the autocorrelation function is to suppress almost
`completely every rnth tooth of the frequency comb, beginning at
`zero frequency.
`
`

`
`560
`
`PROCEEDINGS OF THE IRE
`
`March
`
`envelopes” of both integrating filter outputs, and deci-
`sion is based on whether the Mark or Space sample is the
`larger. The timing required to make the sampling and
`quenching operations occur exactly at the end of each
`received baud implies knowledge of
`the transmitter
`modulation timing. This is readily available, because
`of the degree of the transmitter—receiver synchroniza-
`tion already required of the reference sources.
`
`III. THEORETICAL FOUNDATIONS
`
`In this section we present the communication—theo-
`retical arguments that led to the Rake receiver design.
`Two rather distinct mathematical treatments have been
`
`pursued, which between them encompass a fairly real-
`istic and complete set of assumptions. As yet, no unified
`theory has been forthcoming which includes the entire
`set of assumptions.
`Both analyses are based on statistical decision theory
`[27, 28] and both lead, with some intuitive extension,
`to the Rake receiver previously described. VVe shall first
`discuss the simpler and more physical argument, and
`later briefly touch upon the more abstract analysis that
`actually first suggested the Rake configuration.
`In the application of statistical decision theory, it is
`customarily assumed that the transmitter, its signals,
`and the channel are specified, at least on a statistical
`basis. Here, we shall assume that the receiver has knowl-
`edge of the possible transmitted waveforms and their
`a priori probabilities, and complete statistical knowl-
`edge of all random elements in the transmitter and
`channel. We seek the best such receiver. Generally the
`“best” receiver is taken to be that which achieves the
`
`minimum average “cost,” suitably defined, of wrong de-
`cisions made by the receiver [27]. In the communica-
`tion problem under study it appears that the appropri-
`ate cost function is simply the probability of error-
`that is, a Mark being mistaken for a Space is no worse
`nor better than the inverse error. This assumption be-
`comes even more reasonable when we add the additional
`
`condition that the a priori probabilities of transmitting
`a Mark or a Space are to be equal. It has been shown
`that the corresponding optimum receiver, called by
`Siegert the Ideal Observer [29] is one that, from the re-
`ceived waveform, computes the a posteriori probabilities
`that a Mark, or a Space, was transmitted, and decides
`in favor of the symbol having the larger probability.
`
`A. Analysis for Multipath Assumed to .Be Perfectly
`Measurable
`
`solely by additive white Gaussian (thermal or shot-
`effect) noise [30, 31]. Assuming that the Mark and
`Space waveforms have equal energy, the Ideal receiver
`in this case simply cross correlates the received wave-
`form with the two stored references, and bases its deci-
`sion on the symbol yielding the larger correlation.
`(Thus the simple correlation receiving system of Fig. 1,
`used previously for explanatory purposes, is seen to be
`an optimum system against white noise under the fore-
`going assumptions.)
`The introduction of an arbitrary linear or nonlinear
`filter into the channel, preceding the noise, can be ac-
`commodated by a simple extension of this result, pro-
`vided that the receiver knows or can measure the filter
`
`characteristic exactly. It is apparent that the optimum
`receiver in this case first passes its local reference wave-
`forms through filters identical to the one in the channel
`and then performs cross correlations as before. (We as-
`sume that there is a negligiblemdifference in the energies
`of the filtered reference wave forms.)
`The above requirement that the receiver have com-
`plete knowledge of the channel filter is rather unrealistic
`for an io