ORTHOGONAL FREQUENCY MULTPLEX .DATA TRANSMISSION SYSTEM
`
`Facebook's Exhibit No. 1013
`Page 1
`
`

`

`« Jan. 6, 1970
`
``
`
`3,445
`oRTHoGoNAL FREQUENCY MULTPLEX DATA TRANSMISSION SYSTEM
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`Filed Nav. 14. 1966
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`Facebook's Exhibit No. 1013
`Page 2
`
`

`

`Jan. 6, 1970
`
`94v45
`R. w. @HANG
`ORTHOGONAL FREQUENCY MULTPLEX DATA TRANSMISSION SYSTEM
`Filed Nav. 14, 1966
`
`s sheets~sheet s
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`Facebook's Exhibit No. 1013
`Page 3
`
`

`

`United States Patent O ”
`ICC
`
`3,488,445
`Patented Jan. 6, 1970
`
`1
`
`3,488,445
`ORTHOGONAL FREQUENCY MULTIPLEX
`DATA TRANSMISSION SYSTEM
`Robert W. Chang, Eatontown, NJ., assignor to Bell Tele
`phone Laboratories, Incorporated, Murray Hill, NJ.,
`a corporation of New York
`Filed Nov. 14, 1966, Ser. No. 594,042
`Int. Cl. H043' 1/00; H04b 1/66
`U.S. Cl. 179-15
`
`10 Claims
`
`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. Arn
`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.
`
`2
`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 t-wo 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-dependent 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 tested'for 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
`
`for meen over the interval 2T, the common repetition
`period.
`_
`Closely allied with the orthogonality concept is that
`of symmetry. A function f(l), 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
`
`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
`
`This invention relates to systems for transmitting _
`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, 0r 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
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`60
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`70
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`Facebook's Exhibit No. 1013
`Page 4
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`

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`3,488,445
`4
`FIG. l 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;
`FIG. 5 is a series of waveform diagrams useful in
`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 11(1)
`and a transfer function
`
`3
`functions is even Whenever both functions are even or
`both are odd, and is odd whenever one of the functions
`1s 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 Sn(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 lthat 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 ybecause 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
`
`HU) entf)
`
`where HU) and 17(1‘) are respectively the amplitude and
`phase characteristics of medium 18, e is the base of
`natural logarithms and I is the imaginary number \/(cid:173)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.
`Channel shaping is the critical element here. Let
`bo, b1, b2 .
`.
`. be a Sequence of m-ary 'signal digits
`(mÈZ) or a sequence of analog samples to be trans
`mitted over an arbitrary ith channel. Each of bo, 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 l/T bauds). Let aí(t) be the impulse
`response of the associated ith transmitting filter. Then
`this filter transmits a sequence of signals as
`
`The received signals at the output of transmission medium
`18 are
`
`4.o) :fl hu-nmrmzf
`(T is a dummy variable of integration.)
`These received signals overlap in time, but they are
`orthogonal (noninterfering) if
`
`Intersymbol interference in the ith channel is eliminated
`if Equation 1 is satisfied.
`Now let co, c1, c2 .
`,
`
`. be the m-ary signal digits or
`
`40
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`50
`
`60
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`65
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`70
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`75
`
`Facebook's Exhibit No. 1013
`Page 5
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`

`

`3,488,445
`5
`6
`analog samples transmitted over an adjacent jth channel
`dium, a channel amplitude characteristic .41(7’) (ir-’1, 2
`which has a transmitting filter impulse response of aj(t).
`.
`.
`.
`, N) exists such that
`Since all signaling channels are assumed to be synchro
`nized, the jth transmitting filter transmits a sequence of
`signals
`
`5
`
`CVI-Q1( f) is greater than zero for :all f in the range
`fìztfs and zero outside this range. C, is an arbitrary con
`stant and Qi(f) is a `shaping function having odd symme
`tries about fiifS/Z. Furthermore, the products of the
`shaping functions for adjacent channels [C14-Q1 (f)]
`[C1+1+Q1+1(f)] are even functions about the frequency
`(fi-j-fs/Z) midway between adjacent channel center fre
`quencies fi and fi+1.
`As a further part of my theorem the channel phase
`characteristic «1(1‘) (z'=l, 2 .
`.
`.
`, N) can be shaped
`such that
`
`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,e-1¢T)fzt=o,k=o, i1, i2 . _.
`Intersymbol and interchannel interference can be
`simultaneously eliminated if Equation l is satisfied for
`all i and Equation 2, for all ì and j (í-fej).
`By well known principles of Fourier transform analysis
`Equations 1 and 2 can be transformed into the frequency
`domain, such that Equation l becomes
`
`
`
`fork=1,2,3 .. .,í=1,2,3 becomes
`
`(3)
`N; and Equation 2
`
`Los Aa(f)Aj(f)H2(f) COS [aiffl-al-(ßl COS 21rfkTdfr0
`(real part)
`(4)
`
`and
`
`Los AÁÜAJ'UÜHQU) Sill loliff) _wiwi Sill 21ffkTdf=0
`(imaginary part)
`(5)
`
`for
`
`k-_«0, 1, 2 .
`.
`.
`i, f=1, 2 . ..N, ig;
`
`20
`
`30
`
`35
`
`in the frequency range between channel center frequen
`cies. 71(1‘) is an arbitrary phase function having odd sym
`metry about the frequency (fj-j-fs/Z) midway between
`adjacent channel center frequencies fi and fr_H.
`If A1( f) and '11(1‘) are -shaped in accordance with my
`theorem and f1 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 2fs bauds per channel.
`Furthermore, the transmitting filters have gradual rolloffs,
`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
`err-or 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 (C0) for all i and all Qi(f) (i=1„ 2 .
`.
`.
`, N) are
`identically shaped, the product function [C1-|(cid:173)Qi(f)]
`[Ci+1-j-Qi+1(f)] is an even function about the frequency
`midway between adjacent channel center frequencies
`(fi-l-fs/Z) provided Qi(f) is an odd function about
`(fi-i-fs/Z) 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
`
`In Equations 3, 4, and 5 Ai(f) is the `amplitude char
`acteristic and «1(1‘) is the phase characteristic of the ith
`transmitting filter. ASU) and atl-(f) for the jth transmitting
`filter are similarly defined. HU) is the amplitude char
`acteristic of medium 18.
`Let fì(i=l, 2, 3 .
`.
`. N) denote the equally spaced
`center frequencies of the N independent signaling chan~
`nels. Let the lowest channel center frequency »be
`
`40
`
`45
`
`where h is zero or any positive integer and fs is the dif
`ference between the center frequencies of adjacent
`channels. Thus, the center frequency of the ith channel is
`
`fl=f1+o(cid:173)1>fr<h+t 2));
`
`_
`
`._„l
`
`.
`
`55
`
`(7)
`
`Each amplitude-modulated data channel is assumed to
`transmit at 2fS bauds (symbols per second). Hence
`1
`T523.;
`(8)
`Since the 'bandwidth of each channel is 21‘5, there is
`no inherent difficulty in transmitting at 2fs bauds for an
`arbitrary channel shaping.
`For a given amplitude characteristic H(f) of transmis
`sion medium 18, band-limited transmitting filters (1S, 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 H( f) of a transmission rne
`
`60
`
`65
`
`1
`
`i
`
`(cid:173)” i
`
`Qitf) =§ cos 21rf2ff
`where j lies between f1» fs and fi-j-fs and z' is any positive
`integer.
`Choose
`
`1
`
`70
`
`and FIG. 2(B) is identical to FIG 2(A) with the zero
`intercept changed to coincide with the minimum value of
`the Ifunction. Then waveform 22 is the raised cosine
`function
`
`Facebook's Exhibit No. 1013
`Page 6
`
`

`

`f_fi
`
`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 fil-fs and maximum
`transmission at the center frequency fi.
`Waveforms 21 and 22 meet the symmetry properties
`about
`
`fi, mi; and fi-â
`
`25
`
`30
`
`50
`
`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.
`It may be observed from these waveforms that the
`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 Qi(f). Consequently A1(]‘)H(f) can also assume
`various forms. If H(f) is flat over the narrow frequency
`band of the individual channel, A1(f) may have the
`same shape as Ai(f)H(f). If H(f) is not flat in the indi
`vidual channel band, Ai(f) can be obtained from a
`division of the product A1(f)H(f) by I(cid:173)I(f).
`My theorem also places constraints on the phase char
`acteristic ai(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
`
`3,488,445
`8
`7
`It is readily appreciated that the phase characteristic
`From trigonometric identities the square root of this
`@,(f) is independent of the amplitude characteristic AM).
`equation is
`Further, the phase function of the channel is absent from
`both corollaries l 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(f) 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 Q1(f) can be
`chosen in an identical manner for all channels, according
`to my corollaries l and 2. Then A1(f)I-I(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(f) and a
`phase characteristic 0:1(1’) as shown in FIG. 5. Char
`acteristics H1(f) and œ1(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-|(cid:173)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)fs. 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-f-0.5)fs. The phase characteristics «1(1‘) in broken
`line form are superimposed on the amplitude character
`istics as identical waveforms `56, 58, and 60. The average
`slope is linear and the difference in slope between chan
`nels is equal to i-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 f1 is chosen equal to (k-1) fs to form
`a lower sideband centered on a frequency of 1.5y‘s as
`shown by waveform 51 on line (A) of FIG. 5. This
`waveform has a bandwidth extending from 0.5y‘s to 2.5fs.
`Similarly, frequencies f2 and f3 are chosen respectively
`to be (k--2)fs and (k-3)]‘s to form lower sidebands
`52 and 53 on lines (B) and (C) of FIG. 5. The new cen
`ter frequencies are 2.5,7‘S 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 4.51‘S 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.
`
`-l-Epm cos m21r
`
`. . in the range
`. . and n=2, 4, 6` .
`for mf=l, 2, 3 .
`)Qi-fs, where h is an arbitrary odd integer and rpo, om, i//n
`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 fiifs/Z. The only real
`constraint is that n be even. If n were allowed to assume
`odd as well as even values, the form of «1(1‘) would be
`completely arbitrary.
`In FIG. 4 to be discussed more fully later the phase
`function oq(f) is sketched as identical curves S6, 58, and
`60. For this particular choice lz is set to `--1, <po, and om
`equal zero, m equals l, n equals 2 and \[/2=l. The linear
`term is thus --1r/2 and a sine function with odd sym
`metry about hijs/2 is superimposed thereon.
`
`55
`
`60
`
`65
`
`70
`
`75
`
`Facebook's Exhibit No. 1013
`Page 7
`
`

`

`3,488,445
`The passband of the transmission medium is assumed t0
`weighting resistors are adjusted before data transmission
`be centered on the frequency fc.
`by taking samples of the desired waveform after it has
`Since the transmission rate in each channel is 2j‘s
`been transmitted through the channel. The output of the
`bauds, the total transmission rate for three channels is
`summing circuit is observed at a time »r when the peak
`67‘s bauds in a transmission band of 4fs. This is a rate
`response is noted at the reference tap. Thereafter, the
`of 1.5 bauds per cycle of bandwidth, 50i percent greater
`response of the matched ñlter will be a maximum for
`than that possible by use of conventional nonoverlap
`the waveform for which the ñlter has been adjusted.
`ping frequency spectra. By extension of the principle of
`The signal in the adjacent channel is orthogonal to that
`this invention it is apparent that the more channels used,
`in the channel of interest and its contribution to the
`the closer is the approach to the theoretical maximum
`output of the summing circuit at time 1- will be zero.
`of 2 bauds per cycle of bandwidth. In general
`Reference is made to the paper of G. L. Turin entitled
`“An Introduction to Matched Filters” published in IIRE
`Transactions on Information Theory of June 1960 for
`more details on the use of matched filters as correlators.
`At the output of each matched filter 78 is a sampler 79
`controlled by sampling pulses at the data rate 27’s derived
`from pick-off 70. The sampling pulses are delayed by the
`time -r to coincide with the arrival of the peak response
`on the reference tap of the matched filter. On the basis of
`the summed sample the decision is made as to the nature
`of the data bit transmitted. The data output is made
`available on leads 80, 81, and 82 for the respective chan
`nels. Thus, the system of this invention operates in real
`time and no signal storage, other than a fixed delay, is
`needed.
`Where larger numbers of channels than three are used,
`the three-channel method above may be extended in a
`straightforward manner. For example, the channels may
`be separated in groups of three using bandpass filters and
`each group of three may then be translated down to base
`band in the manner previously described. The group of
`three passing through the bandpass filterl is initially trans
`lated to baseband by using a demodulating signal of the
`form cos [21r(ì-1) fstCrDí] 'Where i is the center frequency
`of the center channel of the group of three and @i is an
`arbitrary phase angle accounting for carrier phase shift
`in the transmission medium.
`The orthogonal multiplex system of this invention can
`also be operated without synchronization among channels
`if transmission is restricted to odd or even numbered
`channels. The overall data rate is then half the theoreti
`cal maximum or one baud per cycle of bandwidth. The
`system may also be operated with some adjacent channels
`synchronized and others unsynchronized, provided that
`an unused channel is left between synchronized groups.
`While this invention has been described generally in
`theoretical terms and specifically by way of a three
`channel example, its principles are susceptible of wide
`application by those skilled in the art within the spirit
`and scope of the appended claims.
`What is claimed is:
`1. In a frequency-multiplex communications system:
`a transmitting station for forming a plurality of equally
`spaced band-limited mutually orthogonal communi
`cation channels in said system;
`a transmission medium having a substantially fiat am
`plitude-frequency characteristic and connected to said
`transmitting station;
`a plurality of input circuits for data signals;
`means synchronizing said input circuits at a common
`transmission rate equal to one data lsymbol per cycle
`of bandwidth; and
`a transmitting bandpass filter for each of said input cir
`cuits, each of said filters having a -center frequency
`equally spaced by half the synchronizing rate from
`that of an adjacent filter, a bandwidth equal to the
`synchronizing rate, even symmetry about the center
`frequencies for its amplitude response, odd sym
`rnetry about frequencies midway between the cen
`ter frequencies and band-edge frequencies for the
`square of its yamplitude response characteristic, and
`phase-frequency characteristics differing between ad
`jacent communication channels by 90 electrical de
`grees plus an arbitrary phase function, with odd sym
`metry for said phase-frequency characteristic about
`
`N + 1
`times 2 bauds per cycle of bandwidth is obtained, where
`N is the number of channels use