328
`
`PROCEEDINGS OF THE IEEE. VOL. 68, NO. 3, MARCH 1980
`
`[66] —, l"1'oward a unified theory of modulation," Proc. IEEE, vol.
`54, pp. 340-353, Mar. 1966. and pp. 735-755. May 1966.
`—, “Generation of digital signalling waveforms,” IEEE Trans.
`Commun., vol. COM-16, pp. 81-93, Feb. 1968.
`—~, “Method and apparatus for intepolation and conversion of
`signals specified by real and complex zeros." U.S. Patent 3510640,
`May 1 970.
`—, “Zero—crossing properties of angle-modulated signals,”IEEE
`Trans. Commun., vol. COM-20, pp. 307-315. June 1972.
`H. B. Voelcker and A. A. G. Requicha, “Clipping and signal
`determinkm: Two algorithms requiring validation." IEEE Mm.
`Commun.. vol. COM-21, pp. 733-?44, June 1973.
`-—.
`"Band-limited random-real-zero signals,"
`Commun..vol. COM-21. pp. 933-936. Aug. 1973.
`
`[67]
`
`[63]
`
`{69]
`
`[70]
`
`[71 ]
`
`IEEE Trans.
`
`[NI A. Walther, “The question of phase retrieval
`Acre, vol. 10, pp. 41-49, Jan. 1963.
`N. Wiener, The Fourier Integral and Certain of its Appticnrions.
`New York: Dover, 1958.
`E. Wolf, “Is a complete determination of the energy spectrum of
`light possible from measurements of the degree of coherence?"
`Proc. Phys; Soc., vol. 80. part 6, pp. 1269-1272, 1962.
`J. L. Yen, “0n nonuniform sampling of bandwidth-limited sig-
`nals,” IRE Trent. Circuit Theory, vol. (IT-3. Pp. 251-257, Dec.
`1 9 56.
`M. Zakai, “Band-limited functions and the sampling theorem,”
`Inform. Conan, vol. 3, pp. 143-153. 1965.
`A. Zygmuud, Tn‘gonometric Interpolation.
`of Chicago Press, 1950.
`
`in optics,” Opt.
`
`Chicago, IL: Univ.
`
`[73]
`
`[74]
`
`[75]
`
`[76]
`
`[T71
`
`Introduction to Spread-Spectrum Antimultipath
`
`Techniques andTheir Application to Urban
`Digital Radio
`
`GEORGE L. TURIN, FELLOW, IEEE
`
`Abstract—In a combination tutorial and research paper, spread-
`apectrum tedmlqllee for mmbating the eflecta of multipath on high-
`ntedltatranmtiaaionsviaradioareexplored. Thetutorialaapecto!
`thepaperpresents: Daheuriticoutfineoftheflrmyofrpread-
`spectrum antimuitipatll radio waivers and 2) a summary of a statistical
`model of mbanfaublu'bm multipath. The research section of the paper
`presents results of analyses and drnulationa of various candidate re-
`ceivus indicated by the theory. as they perform through urbanisati-
`urban multipath. A major result shows that megabit-mud rates
`through urban multipath (which typically lasts up to 5 pa)arequite
`feasible.
`
`1. INTRODUCTION
`
`0MB DIGITAL radio systems must operate through an
`extremely harsh multipath enrironment,
`in which the
`duration of the multipath may exceed the symbol length.l
`Two disciplines combine to shed light on receiver design for
`this environment: the modeling and simulation of multipath
`
`Manuscript received February a. 1979;reviaed October 9. 1979. This
`work was supported by the National Science Foundation under Grant
`ENG 21512 and SRI International under Advanced Research Projects
`Agency Contract MDA 903-?B-C-0'216.
`The author is with the Department of Electrical Engineering and Com-
`puter Sciences and the Electronics Research Laboratory. University of
`California, marketer, CA 94720.
`'An example is the ARPA Packet Radio network [38].
`
`channels and the theory of multipath and other diversity
`receivers.
`
`In this paper, we first present a tutorial review of pertinent
`aspects of both underlying disciplines, particularly in the con-
`text of spread-spectrum2 systems. We then carry out rough
`analyses of the performances of two promising binary spread-
`spectrum antimultipath systems. Finally, since the analyses
`contain a number of oversimplifications that make them
`heuristic rather than definitive, we present results of computer
`simulations of the two proposed configurations and others, as
`they operate through simulated urbanlsuburban multipath.
`The simulation results highlight the importance of using realis-
`tic simulations of complex charmels rather than simplified
`analyses, or they show that the analytic results, although based
`on standard assumptions, are unduly optimistic.
`
`II. MODELING MULTIPATH PROPAGATION
`
`Ultimately, a reliable multipath model must be based on
`empirical data rather than on mathematical axioms. Two types
`
`“In a spread-spectrum system, the bandwidth W of the transmitted
`signals is much larger than UT, the reciprocal of the duration of the
`fundamental signalling interval. so Tilt" >> 1 . The transmitted spectrum
`is said to be "spread” since a signal lasting T seconds need not occupy
`more than the order of W! ”T H: of bandwidtth whichcaseTWi 1.
`See [6] for references on the spread-spectrum concept.
`
`0018-9219f80f0300-032330035 © 1980 IEEE
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 1
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 1
`
`

`

`TURIN: SPREAD SPECTRUM ANTIMULTIPATH TECHNIQUES
`
`329
`
`
`
`i230 MHz— DENSE H!GH RISE
`
`3 I00
`I
`
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`
`3g
`
`
`
`
`
`“bursty” transmissions, measurement of sequences of “im-
`pulse responses” of the propagation medium suffices.
`The simulations of data reception that are presented in a
`later section are based on the “impulse response" approach,
`and it is to this type of model that we restrict ourselves.
`In
`order that the model and the simulations themselves be fully
`understood, we shall review here the experiments underlying
`the model. These were performed in urbanfsuburban areas
`[321,l33].
`
`A. The Underlying Experiments
`
`Pulse transmitters were placed at fixed, elevated sites in the
`San Francisco Bay Area. Once per second, these would simul-
`taneously send out IOO-ns pulses of carrier at 488, 1280, and
`2920 MHz. The pulses were received in a mobile van that
`moved through typical urbanfsuburban areas, recording on a
`multitrace oscilloscope the logarithmically scaled output of
`the receiver’s envelope detectors (see Fig. 1).
`Since the
`oscilloscope was triggered by a rubidium frequency standard
`that was synchronized with a similar unit at the transmitters
`prior to each experimental run, absolute propagation delays
`could be measured within experimental accuracies of better
`than 20 ns.
`Four se1ies of experiments were performed, in the following
`typical urbanfsuburban areas:
`
`A) dense high-rise—San Francisco financial district,
`B) sparse high-rise—downtown Oakland,
`C) low rise—downtown Berkeley,
`D) suburban—residential Berkeley.
`
`regions of dimensions roughly 500-1000 ft
`In each area,
`(along the transmitter—receiver line of sight) by 2500-4000 ft
`(tangential to line of sight) were exhaustively canvassed, with
`care taken to include proper topographic cross sections: inter-
`sections, midblocks, points at which the transmitter site was
`visible or occluded, etc. About 1000 frames of data of the
`type shown in Fig. 1(a) were obtained in each area.
`
`B. A Fundamental Model
`
`The model upon which data reduction was based was one
`first posed in [27}.
`In this model, it is assumed that a trans-
`mission of the form
`
`:0) = Re [0(t) exo {fund}
`
`will be received as
`
`where
`
`r0) = Re [9(1) exp (it-101)] + M!)
`
`[(-1
`
`90‘) = 2 Real! ‘ it) eprflk).
`k-o
`
`(1)
`
`(2)
`
`(3)
`
`In (ll-(3), 0(r) is the complex envelope of the transmission, i.e .,
`|a(r)l is its amplitude modulation and tan"1 [lm o(t){Re 00)]
`is its phase modulation. The transmission is received via K
`paths, where K is a random number that may vary from trans-
`mission to transmission. The kth path is characterized by
`three variables: its strength ck, its modulation delay tk, and
`its carrier phase shift Bk. The waveform Mr) is an additive
`noise component.
`In the context of the spread-spectrum systems on which we
`shall concentrate, it is desirable to assume that all paths are
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 2
`
`on
`L35
`
`s
`a
`IN HICROSECDNDS
`
`s
`
`55
`
`a
`_2
`I
`DELA‘I BEYOND LINE OF SIGHT [LOSI
`(b)
`Fig. 1. Example of measured multipath profiles for a dense high-rise
`topography.
`(a) Top to bottom: 2920, 1280, 488 MHz. Vertical
`scale: 35 dnfcm. Horizontal scale: 1 usfcm. Different apparent L03
`delays are due to difference in equipment delays. (b) Middle trace of
`(a) on a linear scale.
`
`of data are available. The more common type give the results
`of narrow-band or CW measurements, in which only a single
`fluctuating variable, a resultant signal strength, is measured
`[ 11]. Although the fluctuation of this strength variable de-
`pends on reception via multiple paths, these paths are not re-
`solved by the measurements. We shall denote the results of
`such measurements as “fading" data rather than multipath
`data, because they determine a fading distribution of the
`single strength variable, e.g., Rayleigh, log-normal, Rice, etc.
`Wide-band experimental data that characterize individual
`paths are less common [51,[10], [19}, [33], [37].
`In order
`to resolve two paths in such measurements, the sounding sig-
`nal’s bandwidth must be larger than the reciprocal of the dif-
`ference between the paths’ delays. Although bandwidths of
`100 MHz or more have been used in exceptional circumstances
`to resolve path delay differences of less than 10 ns [10], the
`bulk of available data derives from IO—MH: bandwidths or less
`[5],
`[19],
`[33], [37].
`In the latter measurements, paths
`separated by delays of more than 100 ns are resolved; multiple
`paths with smaller separations are seen as single paths.
`The nature of the multipath measurements depends somewhat
`on the use envisioned for them. If understanding of the effect
`of the multipath channel on CW transmissions is required,
`measurements that show Doppler effects may be important
`{5-}, and these are reasonably related to a scattering-medium
`model of the channel [1], [12]. For high-rate packetized-
`data transmission, for vehicle-location sensing, and for other
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 2
`
`

`

`330
`
`PROCEEDINGS OF THE IEEE, VOL. 68, NO. 3, MARCH 1980
`
`sms
`
`
`
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`(a) Division of excuses delay axis into
`Fig. 2. Discrete-time model.
`seventy one lilo-n3 bins. (b) A typical multipath profile.
`(c) Discrete—
`time path- delay indicator string, {7;}Foo.
`
`resolvable, i.e., that
`
`Irk-r;|>.l/W,
`
`forallkrfil
`
`(4)
`
`holdS, where W is the transmission bandwidth. Distinct paths
`in the physical medium that violate this resolvability condition
`are not counted separately, since they cannot be distinguished
`by a measurement using bandwidth W.
`Instead, any two
`paths—call
`them In and kg —for which “'51 - r,2|<1;w are
`considered as a single path in (3),.with a common delay :3, E
`‘k, a ‘3‘: and a strengthiphase combination given by
`at eprfii) gar, exp (ism) + as, exp 091;).
`
`It is the triplet {th at, 63;} that is to be determined for each
`“resolvable" path. To be sure, if a continuum of paths existed,
`it Would be difficult uniquely to cluster the “subpaths” into
`paths. But many media, including the urbanfsuburban one,
`have a natural clustering, e.g., groups of facades on buildings,
`that make the model feasible.
`
`C. A Discrete-Time Approximation to the Model
`
`hi addition to the additive random noise 310) in (2), the re-
`ceived signal rfr) is therefore characterized by the random
`variables {rk}ox'l, {ak}§‘1, {fig-1 and K. The purpose
`of data reduction from the “multipath profiles" exemplified
`by Fig.
`1 was to obtain statistics of these random variables
`upon which to base a simulation program. A number of gen-
`erations of statistical models—based both_on the data and on
`physical reasoning when the data were insufficient or unde-
`cisive—ensued [8], I9], [25], [26], [32]. The following final
`version emerged.
`Each multipath profile starts with the line-of-sight (L05)
`delay, which is chosen as the delay origin. Since the resolu-
`tion of the original experiment is 100 ns, the delay axis is
`made discrete by dividing it into lOO-ns bins, numbered from
`0 to 70. Bin 0 is centered on LOS delay, subsequent bins
`being centered on multiples of 100 us. The delay of any
`physical path lying in bin I is quantized to 1001 ns, the delay
`of the bin’s center. Fig. 2 shows the bin structure, a multi-
`path profile, and the resulting discrete-time path-delay struc-
`ture. Notice that only paths with delays lessr than 7.05 its
`beyond L05 delay are encompassed in this model: experi-
`mental evidence shows that significant paths with larger de-
`lays are highly improbable.
`The path-delay sequence {:kE,‘ '1 is approximated by a
`string {r;}3° of 0’s and 1’s, as shown in Fig. 2(c).
`If a path
`
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`(a) A vehicle’s track,
`Fig. 3. Spatial variation of the multipath profile.
`with spatial sample points.
`(b) The sequence of multipath profiles
`at the sample points on the track.
`
`In the sample string
`exists in bin 1, r1= 1; otherwise 1'; = 0.
`in Fig. 2(c}, only To, 1'3, 1'4, rs, '
`'
`'
`, 1'51, 1'51 are nonzero, cor-
`responding to the quantized path delays to = 0, f1 = 300 ns,
`t; =400 ns,
`’13 = 600 ns, -
`- - Err-1 = 6200 ns, ?K_1 = 6700
`ns.
`(33], is the value of It, as quantized to the nearest 100 ns.)
`Associated with each nonzero r; is the corresponding (ck, Bk)
`pair Thus the discrete-time model is completed by appending
`to the r; string a set of strength-phase pairs {(ak, Bk) £01,
`where the index k refers to the kth nonzero entry in the r,
`string. This is shown in Fig. 2(b).
`The discrete-time model of Fig. 2 pictures the multipath
`profile at a single point in space. A sequence of such profiles
`is needed to depict
`the progression of multipath responses
`that would be encountered by a vehicle following a track such
`as shown in Fig. 3(a). One imagines points 1, 2,
`,n, ' ' '
`arbitrarily placed on the track, at each of which a multipath
`response is seen. The discrete-time versions of these responses
`are arrayed in Fig. 3(b), where an additional spatial index n
`has been superscribed on all variables.
`One begins to recognize the complexity of the model and of
`the required reduction of experimental data on realizing that
`in addition to the need for first- order statistics of the random
`variables 7}”) at"), 35,“) and K00 (where 0 $1 a 70- o a k s
`K00 - l; l fin (“0), there are two dimensions along which
`at
`least second-order statistics are necessary:
`temporal and
`spatial. For each profile (fixed n), there are temporal cor-
`relations of the delays, strengths and phases of the several
`paths;
`in addition,
`there are spatial correlations of these
`variables at neighboring geographical points.
`[33] and
`The reduction of experimental data [9], [25],
`physical reasoning led to the following model, which was the
`basis for simulation.
`1) The {7,00} string of the nth profile is a modified Ber-
`noulli sequence, in which the probability of a l
`in the 1th
`place depends on: a) the value of l; b) whether a l or a 0 oc-
`curred in the (I - l)th place of the same profile; c) whethera
`1 or a 0 occurred in the 1th place of the (a - Uth profile.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 3
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 3
`
`

`

`QRGEASILETY—GF—GCCUFANCY iii—2‘1
`
`331
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`Fig. 4. Comparison of empirical statistics with statistics of simulation
`runs: sparse high-rise, 1280 MHz; 3000 simulation couples at l-ft
`spacings.
`Solid curves: empirical; broken curves: simulation.
`(is)
`Ordinate is probability that a path occurs within 150 us of abscissa
`value.
`(b) Ordinate is probability that there are the number of paths
`given by the abscissa within the first N hm (c) Ordinate la the prob-
`ability that the strength of a path in the indicated delay interval is
`less than the abscissa value.
`
`ble to, say, downtown New York City or Chicago as to down-
`town San Francisco.
`This initial success encouraged the
`development of the more elaborate simulation capability just
`described.
`Thus the seqmnce generated by SURP, described above,
`provide a data base with which to perform accurate experi-
`ments with urbanfsuburban radio systems, and the results of
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 4
`
`TURIN: SPREAD SPECTRUM ANTIMULTIPATH TECHNIQUES
`
`2) The strength a?) of the kth path of the nth profile is
`conditionally log-normally distributed? the conditions being
`the values of strengths of the (k - llth path of the nth profile
`and of the path with the closest delay in the (n - 1)th profile;
`appropriate empirically determined correlation coefficients
`govern the influence these conditions exert on cg"). The mean
`and variance of the distribution of cg") are also random vari-
`ables, drawn from a spatial random process that reflects large-
`scale inhomogeneities in the multipath profile as the vehicle
`moves over large 3.1'835.
`3) The phase 6%") of the kth path of the nth profile is in-
`dependent of phases of other paths in the same profile, but
`has a distribution depending on the phase of a path with the
`same delay in profile (n - 1), if there is such a path; if no
`such path exists at") is uniformly distributed over [0, 2n).
`4) The spatial correlation distances of the variables just
`described vary considerably, ranging from less than a wave-
`length for the Gk’s, through tens of wavelengths for the ak’s
`and ry’s, to hundreds of wavelengths for the means and vari-
`ances of the ak’s.
`These statistics are more fully explained in [8]. [9}.
`
`D. The Simulation Program
`
`Hashemi’s simulation program SURP, based on the statistics
`just outlined, generates sequences of multipath profiles, as
`depicted in Fig. 3.
`If one were to examine a sequence of such
`profiles, he would see paths appearing and disappearing at a
`rate depending on the spacing of points on the vehicle's track
`(Fig. 3(a)).
`Profiles at only slightly separated geographical
`points would look very similar, with high correlations of path
`delays and strengths (and, for very close points, phases). Pro-
`files at greatly separated points would not only have grossly
`dissimilar {n}, {“k: 61,} strings, but the gross strength statis-
`tics of these strings {e.g., the average strength of the paths in a
`string) would be dissimilar, reflecting the spatial inhomogeneity
`incorporated into the model. The “motion picture” of simu-
`lated profiles just described is in fact very much like experi-
`mental data [37].
`The simulation program can be run, using empirically deter-
`mined parameters, for each of the three frequencies and four
`areas of the original experiment. Long sequences of strings
`were in fact generated for each of the twelve frequencylarea
`combinations, assuming that the points on the vehicle track
`are uniformly spaced by distance d. An example of such se-
`quences is given later in Fig. 25. For various values of d, the
`statistics of the simulated sequences were then compared with
`the original empirical statistics. Excellent agreement was ob-
`tained [8], [9].
`(See Fig. 4 for examples.)
`It should be noted that initial simulation experiments on
`urbanfsuburban radio ranging and location systems, using a
`rudimentary propagation simulation program preceding SURP,
`gave results which compared extraordinarily accurately with
`actual hardware experiments
`[34}.
`In particular,
`it was
`verified that although the data upon which the simulation
`program is based were taken in the San Francisco Bay area,
`one can expect simulation results that are not correspondingly
`restricted geographically. For example, use of the Area-A
`parameters in the program led to results that are as applica-
`
`‘Actually, Suzuki [25] showed that paths with small delays (beyond
`L05 delay) were better modeled by Nakagamr’ distributions,but Haaherni
`[9] was forced to approximate these log-normally because of the com-
`plexity of the simulation program.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 4
`
`

`

`332
`
`PROCEEDINGS OF THE IEEE, VOL. 68, N0. 3, MARCH 1980
`
`these experiments can be expected to have wide applicability
`to typical urbanfsuburban topographies.
`
`III. DESIGN or MULTIPATH RECEIVERS
`
`Multipath reception is one form of diversity reception, in
`which information flows from transmitter to receiver .via the
`
`natural diversity of multiple paths rather than via the planned
`diversity of multiple frequency channels, multiple antennas,
`multiple time slots, etc. Thus instead of regarding the multi-
`path phenomenon as a nuisance disturbance whose effects are
`to be suppressed, it should be regarded as an opportunity to
`improve system performance.
`Two bodies of work in the literature are concerned with
`
`multipath receiver design. The older (see, e.g., [l], [4}, [22],
`[23], [27]) concentrates on the explicit diversity structure of
`resolvable paths; its thrust is to take advantage of this struc-
`ture by optimally combining the contributions of different
`paths.
`In its simplest form, this approach ignores the inter-
`symbol interference that can be caused when the multipath
`medium delays a response from a transmitted symbol into
`intervals occupied by subsequent symbols, an approach that
`is justified only when the duration of the transmitted symbol
`is large compared with the duration of the multipath profile.
`More recently [15] —[ 18], equah‘Zation techniques that were
`developed for data transmission over telephone lines [14] have
`been applied to the radio multipath problem. Here, receiver
`design concentrates on reduction of the effects of intersymbol
`interference, and the diversity-combining properties of the re-
`ceiver are only implicit. This approach appears most suitable
`when the paths are not resolvable and when the symbol dura-
`tion is much smaller than the multipath profile’s “spread.”
`A melding of the two approaches is currently being worked
`on by L-F. Wei of ERL, UC Berkeley. Since we are concerned
`here with the case in which resolvability condition (4) is satis-
`fied, we shall in this paper pursue only the former diversity-
`oriented approach, as modified to take into account
`the
`deleterious effects of intersyrnboi interference-
`Instead of
`indulging in general and complex derivations, however, we
`shall present results using a tutorial “building block" approach,
`employing intuitive arguments that are justified by references
`to more formal developments in the literature.
`
`A. The Optimal Single-Path Receiver
`
`We start with the simple case in which the channel com-
`prises only one path: K = l
`in (3). We assume initially that
`the path strength so and delay to are known (to =0 for
`simplicity), but that the carrier phase 30 is unknown, being
`a random variable, uniformly distributed over [0, 2n).4
`Since the absence of multipath implies the absence of inter-
`syrnbol interference (a point we discuss more fully later), we
`can concentrate on the mception of a single symbol, say over
`the interval 0 ‘E r < T. Knowledge of this interval of course
`implies. some sort of synchronization procedure at the re-
`ceiver, a question discussed below.
`Suppose the received signal r(r) is as in {2), with 0 < t < T,
`and where, in (3), K = l, to = 0, on is known, and 90 is ran-
`dom as described above. The transmitted signal rift) of (l)
`
`‘ Random path phases are assumed throughout this paper, since these
`generally change too rapidly in the mobile environment to make use of
`coherent-receiver techniques.
`
`FILTER
`MRTCHED
`TO 5"”
`FILTER
`“‘TCHED
`TO Salli
`
`streets
`FILTER
`to sin)
`
`FILTER
`IATCHED
`
`| sacrum:
`'llVE F0 Ru
`[{1}
`
`ENVELOPE 0%.
`21' DECISION
`CIRCUIT
`DETECTOR
`ENVELGFE QM ””PLE
`flT
`O
`E
`I: T
`[MID
`OUTPUT
`INDEX
`or
`Lanstsr
`SAMPLE
`
`ENVELOPE
`ctrtcrcs
`
`D IGI TAL
`0U TPUT
`
`ENVELOPE
`ctrscron
`
`To sum
`
`Fig. 5. Optimal noncoherent-phase receiver for single-path channel.
`(Equlprobable, equienergy signals, Gaussian noise.)
`
`can be any of M possible waveforms
`
`sifrl= Re [0:(0 6X13 (lo->00].
`
`i=1."',M-
`
`(5)
`
`We assume that the transmitter has chosen among the r; at
`random with equal probability and that the a; have equal
`energy:
`
`T
`
`f sf(r)d¢=a,
`
`O
`
`forallr‘.
`
`(6)
`
`The additive noise at!) of (2) is for simplicity assumed to be
`white and Gaussian, although non-Gaussian noise is also com-
`mon in urban radio communication.
`(See [39]
`for a com-
`prehensive survey of urban noise.)
`It is well known that the optimal receiver—i.e., the receiver
`that decides which if was sent with minimum probability of
`error—has the form depicted in Fig. 5 (see, e.g., [36]). As shown
`there, 31:) is passed through a bank of M filters, “matched”
`respectively to 5:0“), i= 1,“ '
`' ,M,
`i.e., having impulse re-
`sponses r,(T- t), 0 €t< T [29]. The filter outputs are en-
`velope detected and the envelopes sampled at r = T and com-
`pared. The index r‘= l, -
`-
`- , M of the largest sample is the
`receiver's output.
`In Fig. 5, we have shown the outputs of the envelope de-
`tectors when .130“) is the transmitted signal and when the re-
`ceived noise component n(r) is negligible, assuming that the
`a; have been “well chosen.” This latter assumption means
`that if we define complex correlation function
`
`r
`
`mfllé I ai'irwsir- 0dr,
`
`0
`
`1',k=1. HM,
`
`IIIQT (7)
`
`then[27]
`
`[7,,(r)i<<2s,
`
`allkefir‘,
`
`l7,,(r)l<<2ll,
`
`forltl>ifi
`
`alli
`
`alli
`
`(8a)
`
`(8b)
`
`where W is the bandwidth shared by all 3,, and, optimally but
`not necessarily,
`
`7,,(0) = 0,
`
`all k as s,
`
`all 3.
`
`(So)
`
`In sketching the envelope detector outputs, we have also as-
`sumed that TV >> 1,
`i.e.,
`the signals are of the so—called
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 5
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 5
`
`

`

`TURIN: SPREAD SPECTRUM ANTIMULTIPATH TECHNIQUES
`
`333
`
`(D
`
`o
`
`7
`
`2T
`
`3r 2T+A
`
`CD
`
`In
`
`0
`
`I
`
`c
`
`T f .921 .57: arena
`r-
`I:
`I: °"
`3
`
`I
`
`T
`
`27
`
`ST and
`
`Fig. 6. Envelope detector output waveforms (small-noise case) for-the
`receiver of Fig. 5: four-path channel.
`
`spread-spectrum type [61. None of the foregoing assump-
`tions about the structure of the signal set {5,-(r)}‘,5§1 is neces-
`sary for the optimality of the single-path receiver of Fig. 5
`to hold; but we shall invoke them when discussing multipath
`receivers later, as they become necessary or desirable.
`The noisefree waveforms sketched in Fig. 5 are in fact given
`by [27]
`
`e,(r)é%|7fl(r- r)|,
`
`l=l,---,M,
`
`0€r<2T.
`
`(9)
`
`Conditions (8a, b) and TW >>l assure that the jth output
`
`envelope ef(r) consists of a sharp “mainlobe” peak surrounded
`by low-level “sidelobes,” while all other outputs have only
`low-level sidelobes. By careful signal selection, (83, b) can be
`satisfied with the maximum sidelobe level in all these wave-
`
`forms at a factor of about Zh/TW down from the mainlobe.
`Typically, for TW = 100, this means that the maximum side-
`lobe is about 17 dB or more down from the mainlobe.
`
`If condition (8c) is also satisfied, the values of £10") for
`ME)? are zero at
`the sampling instant r= T, so that—in the
`absence of received noise~—the receiver will not make an
`error.
`If the received noise is nonzero, the probability that
`the lth output exceeds the ,r'th at r = T for some (ski is also
`nonzero, and it
`is this probability (of erroneous decision)
`that characterizes the receiver's performance.
`A final feature of Fig. 5 is important. There, we have de-
`picted the output waveforms when a single isolated symbol
`is sent during 0 £1“ < T.
`If another symbol, say 5,0), were
`sent immediately afterward, in Te; 1‘ < 21", it is clear that the
`response to it would occur over the interval Té t < 3T. The
`mainlobe peak in the ith output would be centered exactly
`at r= 2T, precisely when all responses from the first symbol
`have died out. Thus on sampling the outputs at t= 23", one
`would be able to make a decision based on the response to
`the second symbol alone, whence our previous statement that
`no intersyrnbol interference occurs in this single-path case.
`
`B. The Optimal Mulrfpath Receiver: Known Delays
`
`If we should attempt to use the receiver of Fig. 5 when
`many paths are present (K > 1 in (3)), we would expect from
`the linearity of the medium and of the matched filters that
`the envelope detector output waveforms will look something
`like those in Fig. 6. Here, we have shown a four-path-situa-
`tion (K = 4).
`The lth response in Fig. 6 is the envelope of the superposi-
`tion of the several paths’ contributions, and, when noise is
`absent, can be shown from (3) and (7) to be of the form
`[27]
`
`K—1
`
`a l.
`€r(f)=‘
`
`2 at: EXP 09k)
`2 k=0
`
`T
`
`0
`
`0,?(7) 91(1' + T'l' It ' ‘1 d7 !
`
`l= l,-'-,M,
`
`0<t<2T+A.
`
`(10)
`
`Under resolvability condition (4), the mainlobe peaks in the
`jth output 2,0) are distinct, and occur as shown at to =0,
`t1, :2, and r3 =A.5 The heights of these peaks are propor-
`
`tional to the path strengths as. The sidelobes, both of ej(r)
`and of the other outputs (none of the latter having mainlobe
`peaks), are mixtures of sidelobes due to the several paths.
`We stress that Fig. 6 is drawn for the isolated transmission
`
`of a single waveform 3,10), 0 Q r < T.
`The waveforms of Fig. 6 differ from those of Fig. 5 in
`several important respects.
`
`If
`1) Strong peaks are available in €j(f) at multiple times.
`the decision circuit of Fig. S knows the values of the path
`delays to, ' -' ,rK_1, it can sample the contributions of all
`paths and combine them, affording the receiver the advantages
`of diversity reception, as discussed earlier. The ability to re-
`solve the paths in Fig. 6 is the essence of the spread-spectrum
`approach.
`If we instead had TW55 1,
`the peaks in Fig. 6
`would merge, and explicit diversity combination would no
`longer be available.
`2) The sidelobe levels of all outputs is increased, since
`(10) shows the addition of multipath contributions.
`3‘ < T now
`3) The responses to the symbol sent during 0 €
`extends beyond t= 2T, thus overlapping with the responses
`to the next symbol, which is sent during T€r< 2T. That
`is, we now have intersymbol
`interference, caused by the
`multipath.
`Effects 2) and 3) are deleterious, while 1) is favorable. As
`we shall see, however, the benefits of 1) usually far outweigh
`the deterioration caused by 2) and 3).
`For the time being, we shall ignore the effects of intersym-
`bol interference, and inquire into the structure of the op-
`timum receiver for
`reception of a single symbol
`through
`multipath, assuming first that the path delays “all; '1 are
`known. However, we again assume random phases {emf '1,
`independently and uniformly distributed over [0, 211')“, we also
`assume that the path strengths {ak}§" are random, perhaps
`having different distributions.
`Intuitively, 'one might expect under these conditions that
`the optimal receiver is still of the form of Fig. 5, but what
`
`5The maximum excess delay anticipated in the channel—l.e., maer_1
`- min,o(rx_1 - to) 2 A—is called the multipath spread; it is by this
`amount that the waveforms of Fig. 6 can spread beyond those of Fig. 5.
`In the four-path example of Fig. 6, it is assumed that :3 - to achieves
`this maximum.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 6
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 6
`
`

`

`334
`
`PROCEEDINGS OF THE IEEE, VOL. 65, NO. 3, MARCH 1930
`
`the decision circuit now samples each of its inputs at multiple
`times T'l- it, k = D, '
`'
`' ,K - l, combines these samples for
`each input, and compares the resulting combined values; the
`decision would be the index of the largest combined value.
`Indeed this is the case, at least when (4) and (81:) hold so that
`the pulses in outputf of Fig. 6 are distinct [27]. However,
`the optimal combining law is sometimes Complicated, and de-
`pends on the statistics of the path strengths.
`Suppose that the sample of the ith output envelope at time
`T+tk is xxx.
`(In the absence of noise x”, =e;(rk) as given
`by (10).) Then, if all path strengths ck are known, the op-
`tirnal‘s combining of the samples is given by [27}
`
`occur is small if there are no intervals of length fill ,i W in which
`substantial probability is concentrated; so put) must be
`“diffuse,” without high peaks and with WA >> 1.8 Second,
`the method of generation of r, strings discussed in Section 11
`(see Fig. 2) incorporates dependences among n's for neighbor-
`ing i's, which implies corresponding dependences in the as
`sociated delays in, that are not incorporated in the simplified
`model just