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`Page 1
`
`Facebook's Exhibit No. 1019
`Page 1
`
`

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`« Jan. 6, 1970
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`Facebook's Exhibit No. 1019
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`Page 2
`
`Facebook's Exhibit No. 1019
`Page 2
`
`

`
`Jan. 6, 1970
`
`R. w. CHANG
`
`3,488,445
`ORTHOGONAL FREQUENCY MULTPLEX DATA TRANSMISSION SYSTEM
`Filed Nov. 14, 1966
`
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`Facebook's Exhibit No. 1019 %
`
`Page 3
`
`Facebook's Exhibit No. 1019
`Page 3
`
`
`

`
`United States Patent Oflice
`3,488,445
`Patented Jan. 6, 1970
`
`1
`
`2
`
`3,488,445
`ORTHOGONAL FREQUENCY MULTIPLEX
`DATA TRANSMISSION SYSTEM
`Robert W. Chang, Eatontown, N.J., assignor to Bell Tele-
`phone Laboratories, Incorporated, Murray Hill, N.J.,
`a corporation of New York
`Filed Nov. 14, 1966, Ser. No. 594,042
`Int. Cl. H04j 1/00; H041) 1/66
`U.S. Cl. 179-15
`
`10 Claims
`
`
`
`C1
`
`10
`
`ABSTRACT OF THE DISCLOSURE
`
`Apparatus and method for frequency multiplexing of
`a plurality of data signals simultaneously on a plurality of
`mutually orthogonal carrier waves such that overlapping,
`but band-limited, frequency spectra are produced without
`causing interchannel and intersymbol interference. Am-
`plitude and phase characteristics of narrow-band filters
`are specified for each channel in terms of their symmetries -
`alone. The same signal protection against channel noise is
`provided as though the signals in each channel were trans-
`mitted through an independent medium and intersymbol
`interference were eliminated by reducing the data rate.
`As the number of channels is increased, the overall data
`rate approaches the theoretrical maximum.
`
`
`as Dr
`_
`
`transmitting _
`for
`to systems
`invention relates
`This
`multiple channels of information signals over band-limited
`transmission media.
`‘
`Multiplex transmission systems employing sinusoidal
`carriers separated in frequency, or rectangular pulse car-
`riers separated in time, or combinations thereof, are well
`known. These known systems have the common charac-
`teristic that in order to avoid mutual interference among
`the channels guard bands of frequency or time are pro-
`vided between channels. These guard bands represent a
`waste of valuable and limited bandwidth.
`In digital data transmission, for example, it is common
`practice to transmit a plurality of data channels through
`a single band-limited transmission medium. In view of the
`limtation of frequency bandwidth in practical transmission
`media, the problem of maximization of the overall data
`rate and the concomitant minimization of interchannel
`and intersymbol interference arises. The general solution
`has been to center the individual channels ‘on equally
`spaced carrier frequencies and to provide a finite guard
`band between channels. This has meant limiting the usable
`bandwidth of each channel to somewhat less than the car-
`rier wave spacing in order to avoid interchannel
`inter-
`ference in the frequency domain. The overall data rate is
`therefore much less than that attainable if the guard space
`could be eliminated without causing interference.
`In the time domain, on the other hand, because the
`impulse response of band—limited transmission media is
`spread out in time,
`the signalling rate is generally held
`below the theoretical maximum in order to avoid inter-
`symbol interference.
`It is one object of this invention to define a new class
`of band-limited signals capable of being transmitted in
`parallel channels at substantally the maximum possible
`data rate without incurring either interchannel or inter-
`symbol interference.
`It is another object of this invention to so shape the
`spectra of individual signaling channels that the spectra
`of adjacent channels by virute of their orthogonality can
`overlap without producing interchannel interference.
`It is still another object of this invention to render the
`elimination of interchannel and intersymbol interference
`in frequency multiplexed parallel data signaling channels
`
`55
`
`60
`
`70
`
`independent of the phase characteristic of the transmis-
`sion medium.
`It is yet another object of this invention to achieve an
`overall date rate in a band-limited transmission medium
`approaching the theoretical maximum rate with physically
`realizable filters having smooth amplitude rolloffs and
`arbitrary phase characteristics.
`.
`It is a further object of this invention to so shape the
`response functions of adjacent channels in a frequency
`multiplex transmission system that the distance between
`any two sets of received signals in the signal space avail-
`able defined by vectors representing all possible signals
`present at one time and which must be individually dis-
`tinguishable is the same as if the signals in each channel
`were transmitted through independent media and inter-
`symbol
`interference were eliminated by reducing the
`signaling rate. The concept of signal space is discussed
`more fully by J. R. Davey in his paper “Digital Data
`Signal Space Diagrams” published in the Bell System
`Technical Journal (vol. XLIII, No. 6, November 1964)
`at p. 2973.
`According to this invention, a plurality of data signal
`samples are orthogonally multiplexed on equally spaced
`carrier frequencies for transmission over a band-limited
`transmission medium in channels having overlapping fre-
`quency spectra. Because of the orthogonal relationships
`achieved within and between channels intersymbol and
`interchannel interferences are avoided and a theoretically
`maximum data transmission rate is attained in each
`channel.
`Orthogonality is a mathematical ‘concept derived from
`the vector representation of time-de-pendent waveforms.
`Any two vectors are orthogonal if the cosine of the angle
`between them is zero, i.e., they are perpendicular to each
`other. The test for orthogonality between vectors is that
`the product of their amplitudes (lengths) and the cosine
`of the angle formed between them when their points of
`beginning are brought
`to a common origin without
`changing their relative directions is zero. Periodic wave-
`forms, such as sine and cosine waves, are commonly
`represented by vectors. More complex waveforms can
`by the well-known methods of Fourier analysis be repre-
`sented by summations of sine and cosine terms. Both
`simple and complex waveforms can be testedfor orthog-
`onality by analogy with the vector multiplication men-
`tioned above. If the periodic waveforms to be compared
`are laid out on the time axis and the average of the in-
`tegral of the -products of pairs of values for all instants
`of time extending over their common period is taken, and
`this average is found to be zero, then the waveforms are
`said to be orthogonal. Thus, orthogonality becomes a
`broader concept than perpendicularity. In general,
`two
`time-dependent waveforms Sm(t) and Sn(t) and deemed
`to be orthogonal if
`
`51,; LTT s.,(z)s,(z)czi=o
`for 7717511 over the interval 2T,
`the common repetition
`period.
`_
`Closely allied with the orthogonality concept is that
`of symmetry. A function f(t), which can be represented
`graphically by a waveform and is defined on an interval
`centered at the origin (1:0), is said to be even if
`J‘(-l)=f(t)
`
`for :all values of t in t-he assigned interval, and odd if
`f(—-l‘)=—f(l‘)
`In graphic terms even functions are symmetric about
`vertical axis erected at the origin, i.e., the negative half
`is the mirror image of the positive half. Odd functions
`are symmetric about the origin itself, i.e., are skew sym-
`metric. From this it follows that the product of two even
`
`Faoebook's Exhibit No. 1019
`
`Page 4
`
`Facebook's Exhibit No. 1019
`Page 4
`
`

`
`3,488,445
`
`3
`functions is even whenever both functions are even or
`both are odd, and is odd whenever one of the functions
`is even and the other is odd. Summarizing,
`
`(Even) (Even): (Odd) (Odd) =Even
`(Even) (Odd) = (Odd) (Even) =Odd
`
`It can therefore be further stated from the orthogonal-
`ity integral above that whenever the function Sm(t)
`is
`of opposite parity to the function S,,(t) and both are
`centered in a common interval, they are mutually orthog-
`onal. Since the interval is common to both functions and
`both functions are periodic with respect to this interval,
`the implication is that the two functions are synchron-
`ized.
`The orthogonality concept is not limited to two func-
`tions. Any number of functions can be mutually orthog-
`onal and mutually synchronized in a common interval.
`Orthogonality with respect to time within each chan-
`nel and with respect to frequency between channels is
`preserved by shaping the signals applied to each channel
`such that the integral of the mathematically transformed
`product of the squares of the shaping function applied
`to the individual channel and the channel transfer func-
`tion and the integral of the transformed products of the
`shaping functions applied to adjacent channels and the
`square of the channel
`transfer function are each zero.
`These conditions are met in practical cases by shaping
`functions Whose squares have even symmetry about the
`channel center frequencies and odd symmetry about fre-
`quencies located halfway between the channel center
`frequency and the channel band-edge frequencies. At the
`same time the phase characteristics of adjacent chan-
`nels may be arbitrary, provided only that
`their phase
`characteristics differ by ninety electrical degrees plus an
`arbitrary phase function with odd symmetry about the
`frequency midway between the channel center frequen-
`cies.
`
`The required symmetries are achievable in a half-cycle
`of the -cosine wave whose square is the raised cosine
`shaping function as one readily definable illustrative ex-
`ample.
`Preservation of orthogonality within each channel per-
`mits establishing individual channel data transmission
`rates equal
`to the channel bandwidth. This is half the
`ideal Nyqquist rate. However, due to the fact that ad-
`jacent channels are synchronized, they can be overlapped
`by 510 percent. The overall data transmission rate for the
`full channel bandwidth then becomes the ideal Nyquist
`rate times the ratio of the number of channels to the
`number of channels plus one.
`Inasmuch as the amplitudes of the shaping functions
`are proportional
`to the amplitudes of the samples by
`which they are multiplied, transmission is in no way re-
`stricted to binary digits. Multilevel symbols and symbols
`of arbitrary height derived from analog samples are equal-
`ly transmissible.
`Orthogonal signals are readily detectable by correlation
`procedures using matched filter techniques.
`A feature of this invention is that the band-limited
`shaping filters for each channel can be identical.
`Another feature of this invention is that the amplitude
`and phase characteristics of the transmitting filters can
`be synthesized independently.
`A particular advantage of the orthogonal multiplex
`transmission system of this invention is that received
`signals can be recovered by using adaptive correlators
`regardless of the phase distortion arising in the trans-
`mission medium. In addition, synchronization problems
`are minimized because stationary phase differences be-
`tween modulating and demodulating carrier waves are
`taken into account by the adaptive correlators.
`Other objects, features and advantages of this inven-
`tion will be readily appreciated from a consideration of
`the following detailed description and the accompany-
`ing drawing in. which:
`
`10
`
`20
`
`25
`
`30
`
`40
`
`50
`
`60
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`65
`
`70
`
`75
`
`4
`FIG. 1 is a block diagram of the basic orthogonal fre-
`quency-multiplex transmission system of this invention;
`FIG. 2 is a waveform diagram showing the develop-
`ment of a shaping filter characteristic satisfying the con-
`dition of orthogonality according to this invention;
`FIG.
`3 is another waveform diagram showing the
`development of a shaping filter characteristic satisfying
`the condition of orthogonality according to this invention;
`FIG. 4 is a block diagram of a representative three-
`channel orthogonal
`frequency multiplex transmitter
`according to this invention using identical shaping filters
`for all channels;
`in
`FIG. 5 is a series of waveform diagrams useful
`explaining the operation of the system of FIG. 4; and
`FIG. 6 is a block diagram of a representative correla-
`tion detection system capable of recovering the data
`signals generated in the transmitting system of FIG. 4.
`FIG. 1 is a generalized block diagram of an orthogonal
`multiplex data transmission system according to this
`invention. From data sources on the left (not shown)
`impulse samples are applied in synchronism on a plurality
`of lines such as those designated 10, 11 and 12. Each
`impulse is shaped in associated transmitting filters 15,
`16, and 17 and others not shown for additional sources.
`Line 13 symbolically indicates
`such other
`signaling
`channels. The passbands of the several transmitting filters
`are centered at equally spaced frequencies with the spac-
`ing equal to half the data rate per channel. Their outputs
`are combined on line 14 and applied to common trans-
`mission medium 18, having an impulse response h(t)‘
`and a transfer function
`
`H(f) gnu)
`
`where H()‘) and 'r](f) are respectively the amplitude and
`phase characteristics of medium 18, e is the base of
`natural logarithms and J is the imaginary number \/—l.
`Noise is also added at various points in the system as
`indicated symbolically by adder 19. The several signal-
`ing channels are separately detected in receiver 20. It
`is assumed for the present
`that
`the channel with the
`lowest frequency is operating at baseband. Carrier modu-
`lation and demodulation at passband can be accomplished
`by standard techniques.
`the critical element here. Let
`Channel
`shaping is
`b0, b1, b2 .
`.
`. be a sequence of m-ary signal digits
`(mil) or a sequence of analog samples to be trans-
`mitted over an arbitrary ith channel. Each of b0, b1,
`b2 .
`.
`. can be represented by an impulse with height
`proportional to that of the corresponding sample. These
`impulses are applied to the ith transmitting filter at the
`rate of one impulse every T seconds
`(data rate per
`channel equals 1/T bauds). Let a,(t) -be the impulse
`response of the associated ith transmitting filter. Then
`this filter transmits a sequence of signals as
`
`bgai(t), b1(Zj(t--T), b2ai(t—2T)
`
`.
`
`.
`
`.
`
`The received signals at the output of transmission medium
`18 are
`
`where
`
`b0llj_(t), b]_M1(t—-T), b2L!1(t—-2T)
`
`.
`
`.
`
`.
`
`u;(t) =f_: h(t—7)a;(-r)cZ-r
`
`(-r is a dummy variable of integration.)
`These received signals overlap in time, but they are
`orthogonal (noninterfering) if
`
`L" u;(t)u;(t——kT)dt=0, lc=:l:1, 42 . ..
`
`(1)
`
`Intersymbol interference in the ith channel is eliminated
`if Equation 1 is satisfied.
`Now let co, cl, C3 .
`.
`
`. be the In-ary signal digits or
`
`Facebook's Exhibit No. 1019
`
`Page 5
`
`Facebook's Exhibit No. 1019
`Page 5
`
`

`
`3,488,445
`
`5
`analog samples transmitted over an adjacent jth channel
`which has a transmitting filter impulse response of aJ~( 1).
`Since all signaling channels are assumed to be synchro-
`nized, the jth transmitting filter transmits a sequence of
`signals
`
`.
`.
`.
`C0(Ij(t), C1aj(f-2T), C2(lj(t—2T)
`The received signals at the output of medium 18 are now
`
`.
`.
`.
`C0llj(t), C1l[j(t--T), C2l[j(f--2T)
`Although these signals overlap those of the ith channel
`in both time and frequency,
`they are nevertheless
`mutually orthogonal if
`
`(2)
`f_°° u;(t)u,»(t—IcT)dt=0, k=0, i1, i2 . ..
`Intersymbol and interchannel
`interference can be
`simultaneously eliminated if Equation 1
`is satisfied for
`all z’ and Equation 2, for all i and j (z’#]').
`By well known principles of Fourier transform analysis
`Equations 1 and 2 can be transformed into the frequency
`domain, such that Equation 1 becomes
`
`fi]wAi2(f)H2(f) cos 27rflcTdf=O
`(3)
`fork=1, 2,3 .. .,i=-1,2,3
`N; and Equation 2
`becomes
`
`La Aa(f)A;(f)H”(f) 005 [ozs(f)-a;(f)] cos 27rfkTdf=0
`(real part)
`(4)
`
`and
`
`L” Aa(DAr(f)H2(f) Sin [on (f) -06; (M Sin 2vrfkTdf=0
`(imaginary part)
`(5)
`
`for
`
`k=0, 1, 2 .
`
`.
`
`.
`
`i, ]'=1, 2 .
`
`. .N,
`
`i%]'
`
`is the amplitude char-
`In Equations 3, 4, and 5 A1()‘)
`acteristic and oq(f) is the phase characteristic of the ith
`transmitting filter. A30‘) and ea,-(f) for the jth transmitting
`filter are similarly defined. H0‘) is the amplitude char-
`acteristic of medium 18.
`. N) denote the equally spaced
`Let f1(i=1, 2, 3 .
`.
`center frequencies of the N independent signaling chan-
`nels. Let the lowest channel center frequency be
`
`(6)
`f1=(h+§)f.
`where h is zero or any positive integer and f5 is the dif-
`ference between the center
`frequencies of
`adjacent
`channels. Thus, the center frequency of the ith channel IS
`
`.
`_
`pl
`(7)
`fa=f1+(z—1)fs—(h+¢ 2):
`Each amplitude-modulated data channel is assumed to
`transmit at Zfs bauds (symbols per second). Hence
`1
`T‘???
`
`(8)
`
`there is
`Since the ‘bandwidth of each channel is 2fs,
`no inherent difliculty in transmitting a.t 2fs bauds for an
`arbitrary channel shaping.
`For a given amplitude characteristic H0‘) of transmis-
`sion medium 18, band-limited transmitting filters (15, 16,
`17) can be devised to satisfy Equations 3, 4, 5 and 8
`simultaneously and thereby eliminate both intersymbol
`and interchannel interference for a data rate of Zfs per
`channel. At the same time the objects of this invention
`will be met.
`I propose a general method of designing the required
`transmitting filters in the form of a theorem.
`For a given characteristic H0‘) of a transmission me-
`
`5
`
`10
`
`20
`
`30
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`35
`
`40
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`45
`
`50
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`60
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`65
`
`70
`
`6
`dium, a channel amplitude characteristic A,(f) (i=1, 2
`.
`.
`.
`, N) exists such that
`
`Ai2(f)H2(f)=Cx+Qi(J‘)
`
`C1+Q1(f) is greater than zero for all f in the range
`f,:L-fs and zero outside this range. C, is an arbitrary con-
`stant and Q1(f) is a shaping function having odd symme-
`tries about fiifs/2. Furthermore,
`the products of the
`shaping functions for adjacent channels
`[C1+Q1 (f)]
`[C1+1+Q-1+1(f)] are even functions about the frequency
`(f1-f-fs/2) midway between adjacent channel center fre-
`quencies fi and fi+1.
`As a further part of my theorem the channel phase
`characteristic a1(f)
`(i=1, 2 .
`.
`.
`, N) can be shaped
`such that
`
`(9)
`aa(f)-aa+1(f)=i%+w(f)
`in the frequency range between channel center frequen-
`cies. 'y1(f) is an arbitrary phase function having odd sym-
`metry about the frequency (f,+f5/2) midway between
`adjacent channel center frequencies f, and f1+1.
`If A10‘) and oc1(f) are -shaped in accordance with my
`theorem and fl is chosen according to Equation 6, then
`Equations 3, 4, 5, and 8 are simultaneously satisfied.
`There is then no intersymbol and interchannel interfer-
`ence for a synchronous data rate of Zfs bauds per channel.
`Furthermore, the transmitting filters have gradual rolloifs,
`the overall data rate is maximized, the transmitting fil-
`ters are matched to the transmission medium, and for
`band-limited Gaussian noise the receiver receives each
`of the overlapping signals with the same probability of
`error as if only that signal were transmitted.
`Detailed proofs of my theorem are omitted here, but
`its practical consequences will be dealt with hereinafter.
`These proofs are set forth in the appendices A and B
`of my paper “Synthesis of Band-Limited Orthogonal Sig-
`nals for Multichannel Data Transmission” published in
`the Bell System Technical Journal (vol. XLV, No. 10,
`December 1966) on pp. 1790 to 1794.
`The first part of my theorem can be readily satisfied
`by any number om symmetrical waveshapes. The second
`part, relating to the product of the shaping functions of
`adjacent channels, can be satisfied according to the fol-
`lowing corollary 1 to my theorem.
`Under the simplifying conditions that C, is chosen the
`same (Co) for all i and all Qi(f) (i=1, 2 .
`.
`.
`, N) are
`identically shaped,
`the product
`function [C1+Q1(f)]
`[Ci+1+Q1+1(f)] is an even function about the frequency
`midway between adjacent channel center
`frequencies
`(fi—|—fS/2) provided Qi(f)
`is an odd function about
`(f,+fs/2) and is an even function about the channel
`center frequency fi. This corollary follows directly from
`the theorem and needs no proof. The product of two
`odd functions is always an even function.
`Practical examples of representative shaping functions
`that satisfy the requirements of my theorem and corollary
`1 are shown in FIGS. 2 and 3.
`In FIG. 2(A) waveform 21 is
`
`Qs(f)=—;- cos‘2arf—'f‘
`.
`Zfs
`where 1‘ lies between f1—--f5 and f1+f5 and z‘ is any positive
`integer.
`Choose
`
`1
`
`and FIG. 2(B) is identical to FIG 2(A) with the zero
`intercept changed to coincide with the minimum value of
`the -function. Then waveform 22 is the raised cosine
`function
`
`A.2mH2<r> =o.+c2.<n =§+§- cos 2-ri2—‘f:i*
`
`Facebook's Exhibit No. 1019
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`

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`3,488,445
`
`7
`From trigonometric identities the square root of this
`equation is
`
`=GOS
`
`2]:-3
`
`Waveform 23 in FIG. 2(C) is seen to be the positive
`half-cycle of a cosine wave, having zero transmission
`beyond the band-edge frequencies f1:L-fs and maximum
`transmission at the center frequency f1.
`Waveforms 21 and 22 meet the symmetry properties
`about
`
`postulated above. Adjacent overlapping channels spaced
`by a -frequency fs and identically shaped in this manner
`are readily seen to satisfy corollary 1 also.
`A second example of a shaping function satisfying
`Equation 3 is shown in FIG. 3. Waveforms 31 and 32 are
`identically shaped functions similar to that of a multiple
`tuned circuit. The waveforms of FIGS. 3(A) and 3(B)
`differ only in the value of the ordinate. Waveform 33
`of FIG. 3(C) is the square root of waveform 32.
`the
`It may be observed from these waveforms that
`center frequency need not be the frequency of maximum
`response. There are dual maxima symmetrical about the
`channel center frequency as shown. The waveforms of
`FIG. 3 are not as readily characterized mathematically as
`those of FIG. 2, but are nevertheless practically attainable.
`Reference is made to such standard texts as E. A.
`Guillemin’s Synthesis of Passive Networks (John Wiley
`and Sons, Inc., New York, 1957) for filter design methods.
`It can be seen from these two examples that a great
`deal of freedom is allowed in choosing the shaping func-
`tion Q;(f). Consequently A1(f)H(f) can also assume
`various forms. If H(f) is flat over the narrow frequency
`band of the individual channel, A,(f) may have the
`same shape as A;(f)H(f). If H0‘) is not flat in the indi-
`vidual channel band, A,(f) can be obtained from a
`division of the product A1(f)H(f) by H(f).
`My theorem also places constraints on the phase char-
`acteristic u,(f) of the transmitting filters. It is only re-
`quired that Equation 9 be satisfied in order to insure
`orthogonality between adjacent channels. However,
`if
`it is desired to have identically shaped transmitting filter
`characteristics for all channels, I propose the following
`corollary 2.
`Under the simplifying condition that all transmitting
`filter phase characteristics a1(f)(1=1, 2 .
`.
`. N) be
`identically shaped, Equation 9 holds if
`
`0‘i(f)=hgf;]fi+%<00
`+>:,.;,,. cos m27rf‘+>:¢,, sin n27rfi
`
`in the range
`.
`.
`. and n=2, 4, 6 .
`.
`for m'=1, 2, 3 .
`fii-fs, where h is an arbitrary odd integer and (p0, gum, 30,,
`are all arbitrarily chosen.
`The first term of this equation is a linear term. The
`second term is an intercept term which may conveniently
`be zero. The last two terms are ripple terms having odd
`symmetry about the frequencies flifs/2. The only real
`constraint is that n be even. If n were allowed to assume
`odd as well as even values, the form of oq(f) would be
`completely arbitrary.
`In FIG. 4 to be discussed more fully later the phase
`function oq(f) is sketched as identical curves 56, 58, and
`60. For this particular choice I2 is set to -1, <p0, and (pm
`equal zero, m equals 1, n equals 2 and 1//2:1. The linear
`term is thus -——7r/2 and a sine function with odd sym-
`metry about fiifs/2 is superimposed thereon.
`
`CI
`
`10
`
`25
`
`30
`
`CO 01
`
`40
`
`50
`
`55
`
`60
`
`65
`
`70
`
`76
`
`8
`It is readily appreciated that the phase characteristic
`oc,(f) is independent of the amplitude characteristic A,(f).
`Further, the phase function of the channel is absent from
`both corollaries 1 and 2. Hence, the amplitude and phase
`characteristics of
`the transmitting filters can be syn-
`thesized independently of each other and of the phase
`characteristic of the transmission medium.
`Variations in the amplitude characteristic H()‘) can
`be taken into account for each individual channel. How-
`ever, it may be more convenient to use a single com-
`pensating network for the entire bandwidth of the trans-
`mission medium. For convenience in implementation the
`amplitude offset C, and shaping functions Q,(f) can be
`chosen in an identical manner for all channels, according
`to my corollaries 1 and 2. Then A1(f)H(f) will be identi-
`cal
`(except for a shift
`in center frequencies)
`for all
`channels. This permits the use of identical shaping filters
`for all channels coupled with frequency translation to
`the equally spaced center channel frequencies.
`FIG. 4 illustrates in block diagram form a three-chan-
`nel system using identical channel shaping filters plus
`frequency translations. FIG. 5 is a waveform diagram
`useful in explaining the transmitter of FIG. 4.
`In FIG. 4 data sources a, b, and c (not shown) de-
`liver synchronized impulse samples to lines 41, 42, and
`43 which in turn are connected to identical shaping
`filters 44 having an amplitude characteristic H1()‘) and a
`phase characteristic oc1(f) as shown in FIG. 5. Char-
`acteristics H1(f) and o¢1(f) have the properties described
`in corollaries 1 and 2. Filters 44 are bandpass filters of
`Zfs bandwidth centered on a frequency lying outside the
`transmission band of the transmission medium. Here this
`center frequency is chosen for convenience as (k+0.5)fs,
`k being an arbitrary odd integer.
`The waveforms of FIG. 5 use frequency as the abscissa
`and amplitude and phase as the ordinate. In FIG. 5 line
`(D) vertical line 61 on the right side indicates the center
`frequency of filters 44 at the frequency (k+0.5)f5. On
`lines (A), (B), and (C) of FIG. 5 the identical ampli-
`tude characteristics 55, 57, and 59, shown here as the half
`cycle of a cosine wave, are centered on the frequency
`(k+0.5)fs. The phase characteristics 0:10‘)
`in broken
`line form are superimposed on the amplitude character-
`istics as identical waveforms ‘S6, 58, and 60. The average
`slope is linear and the difference in slope between chan-
`nels is equal to .—1r/2. A sinusoidal phase ripple is also
`present. Since all three waveforms 55, 57 and 59 are
`derived from identical filters, their amplitude, as well as
`their phase, characteristics 56, 58 and 60, are also
`identical.
`The shaped outputs of filters 44 are modulated by
`equally spaced frequencies f1, f2, and f3 in modulators
`45. The frequency fl is chosen equal to (k—-1))‘, to form
`a lower sideband centered on a frequency of 1.5fs as
`shown by waveform 51 on line (A) of FIG. 5. This
`waveform has a bandwidth extending from 0.5;‘, to 2.5fs.
`Similarly, frequencies f2 and f3 are chosen respectively
`to be (k——2)f5 and (k—3)f,
`to form lower sidebands
`52 and 53 on lines (B) and (C) of FIG. 5. The new cen-
`ter frequencies are 2.5fs and 3.5fs. The center frequency
`spacing is clearly fs.
`The translated outputs of modulators 45 are combined
`on line 46 and result in the overlapping spectra 51, 52,
`and 53 on line (D) of FIG. 5. In adder 47 connected
`to line 46 a component at the frequency fs is inserted to
`facilitate demodulation at a receiver. To eliminate the
`upper sidebands in the outputs of modulators 45 and to
`confine the transmitted spectrum to the bandwidth of
`the transmission medium the signal from adder 47 is
`applied to low—pass filter 48 having the flat amplitude
`characteristic H2(f) out to the frequency 4.5f5 shown
`in waveform 62 on line (D) of FIG. 5. The composite
`signal in the output of filter 48 is translated in modula-
`tor '50 on a carrier frequency fc and then appears on line
`49 for application to a connected transmission medium.
`
`Facebook's Exhibit No. 1019
`
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`Facebook's Exhibit No. 1019
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`

`
`9
`The passband of the transmission medium is assumed to
`be centered on the frequency fc.
`is Zfs
`Since the transmission rate in each channel
`bands, the total
`transmission rate for three channels is
`6fs bauds in a transmission band of 4fS. This is a rate
`of 1.5 bauds per cycle of bandwidth, 50‘ percent greater
`than that possible by use of conventional nonoverlap-
`ping frequency spectra. By extension of the principle of
`this invention it is apparent that the more channels used,
`the closer is the approach to the theoretical maximum
`of 2 bauds per cycle of bandwidth. In general
`
`_N_
`N + 1
`
`times 2 bands per cycle of bandwidth is obtained, where
`N is the number of channels used.
`The data in individual channels of a composite signal
`as shown on line (D) of FIG. 5 can be demodulated
`and detected by the use of adaptive correlation techniques
`as shown in the block diagram of FIG. 6. The composite
`signal arriving on line 65 after having traversed the trans-
`mission medium has the reference frequency fs removed
`in pickolf device 70. Pickolf 70‘ may comprise a narrow-
`band filter and frequency multipliers by means of which
`the sampling frequency Zfs and the several demodulating
`carriers are derived for application to conductor 66.
`Pickolf 70 can alternatively be placed to the right of
`upper modulator 74, if desired. The received signal is next
`applied to modulators 74, having deunodulating fre-
`quencies chosen to translate the respective channel band-
`widths to a common frequency range. This frequency
`range is defined by the characteristic H3(f) of low-pass
`filters 76. The characteristics of filters 76 are identical
`as shown in waveform 75. The characteristic is flat to 2.5f5
`cycles and falls off to zero beyond that frequency.
`On the top line channel 1 is translated back to its base-
`band position centered on the frequency l.5fs by de-
`modulation with a frequency fc,
`the same carrier fre-
`quency used at the transmitter. In passing through filter
`76 channel 3 is severely attenuated and channel 2 to a
`lesser extent as shown in waveform 83. Only channel 1
`produces a full response, however. On the middle line
`channel 2 is translated to the baseband position centered
`on-a frequency of 1.5fs by demodulation with a fre-
`quency of fc+fs. Filter 76 has an output as shown in
`waveform 84. Finally on the bottom line channel 3 is
`translated to the baseband position centered on a fre-
`quency of 1.5fs by demodulation with a frequency fc—{-Sfs.
`The respective channels now appear in reverse order as
`shown in waveform 85. All three chanels have been trans-
`lated into a position in the frequency spectrum which
`satisfies Equation 7. The signals in each individual channel
`remain orthogonal
`in time. The overlapping frequency
`spectra occur only between pairs of channels and the
`phase differences are unchanged. The signals in these
`channels thus remain mutually orthogonal in frequency.
`The remaining channel on each line does not overlap the
`desired channel in the baseband position and can there-
`fore produce no interference.
`The outputs of filters 76 are in turn applied to matched
`filters 78, which function as correlators. A matched filter
`is a linear system whose impulse response is the time
`inverse or complex conjugate of the waveform of the
`signal to which it is being matched. A matched filter is
`usually implemented by a tapped delay line with weight-
`ing resistors between each tap and a summing circuit.
`The weighting resistors are set according to samples at
`the corresponding taps when the