3,438,445
`R. W. CANG
`Jara. 6, 1970
`ORTHOGONAL FREQUENCY MULTPLEX DATA TRANSMISSION SYSTEM
`Filed Nov. 4, 1966
`3. Sheets-Sheet 3
`
`AAAA/5// AG
`AF/A-7AAPS
`?t- a y(t)
`
`O
`
`- 5
`
`- /48
`
`Afts. A
`
`f
`
`244.
`SOLACAS
`
`7AAASM/SS/OA.
`AAAAAAAf
`-A, (e)
`
`2O
`
`: - AACE//A
`
`
`
`g
`
`WOfSE
`
`Af. 2
`
`v-u--mum-
`
`v----
`
`---
`
`--
`
`-
`
`---
`
`(a)
`A 2(f) A 2 (f)
`
`(C)
`A (f) # (f)
`
`A.
`(r
`2
`
`it; |
`
`2
`
`O
`
`\
`
`?
`
`- a - an Oland-
`fi-fi
`fi
`firfs fifs
`FREouency
`
`-
`fi
`
`
`
`:
`6
`Q
`t
`
`i
`
`FREQUENCY
`
`-y
`
`(c)
`
`133
`
`f
`
`an.
`f +fs
`fAW/AA/7CA
`A. B. CAA Af
`(ZA. 21
`Az
`
`A 77"OAR/WE'
`
`IPR2018-01477
`Apple Inc. EX1010 Page 1
`
`

`

`3,488,445
`R. W. CHANG
`Jan. 6, 1970
`ORTHOGONAL FREQUENCY MULTIPLEX DATA TRANSMISSION SYSTEM
`Filed Nov. 4, 1966
`3 Sheets-Sheet 3
`
`44
`A/47AA
`H, (f)
`
`AfG. 4
`
`46
`
`O
`7AAAWSAM/SS/OW
`MAA/C//f
`
`
`
`
`
`
`
`C
`
`A/A7AAR
`A, (7)
`
`
`
`(C)
`
`Y /56
`X
`
`l
`
`a
`
`is
`
`ai.
`
`is
`
`45.
`
`(k - O.5) is
`
`AAPEQUAA/CY
`
`IPR2018-01477
`Apple Inc. EX1010 Page 2
`
`

`

`Jan. 6, 1970
`R. W. CHANG
`3,438,445
`ORTHOGONAL FREQUENCY MULTPLEX DATA TRANSMISSION SYSTEM
`Filed Nov. 14, 1966
`
`S
`ry s
`
`- ------
`
`24
`
`É Z --
`
`9 979
`
`99
`
`OZ
`
`979
`
`)
`
`IPR2018-01477
`Apple Inc. EX1010 Page 3
`
`

`

`United States Patent Office
`
`3,488,445
`Patented Jan. 6, 1970
`
`5
`
`O
`
`1.
`
`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. C. H04 1/00, H04b 1/66
`U.S. C. 79-15
`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. 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 overal data
`rate approaches the theoretirical 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.
`m
`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-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 S(t) and deemed
`to be orthogonal if
`T
`2T J. Sa(t) S(t) di = 0
`for mizn 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 (t=0), is said to be even if
`f(-i)=f(t)
`for all values of t in the assigned interval, and odd if
`f(-t)=-f(t)
`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, or combinations thereof, are well .
`3 5
`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
`limitation 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
`
`60
`
`70
`
`55
`
`IPR2018-01477
`Apple Inc. EX1010 Page 4
`
`

`

`3,488,445
`4.
`3
`FIG. 1 is a block diagram of the basic orthogonal fre
`functions is even whenever both functions are even or
`quency-multiplex transmission system of this invention;
`both are odd, and is odd whenever one of the functions
`FIG. 2 is a waveform diagram showing the develop
`is even and the other is odd. Summarizing,
`ment of a shaping filter characteristic satisfying the con
`dition of orthogonality according to this invention;
`(Even) (Even) = (Odd) (Odd)=Even
`FIG. 3 is another waveform diagram showing the
`(Even) (Odd) = (Odd) (Even) =Odd
`development of a shaping filter characteristic satisfying
`It can therefore be further stated from the orthogonal
`the condition of orthogonality according to this invention;
`ity integral above that whenever the function S(t) is
`FIG. 4 is a block diagram of a representative three
`channel orthogonal frequency multiplex transmitter
`of opposite parity to the function S(t) and both are
`centered in a common interval, they are mutually orthog
`according to this invention using identical shaping filters
`onal. Since the interval is common to both functions and
`for all channels;
`both functions are periodic with respect to this interval,
`FIG. 5 is a series of waveform diagrams useful in
`the implication is that the two functions are Synchron
`explaining the operation of the system of FIG. 4; and
`FIG. 6 is a block diagram of a representative correla
`ized.
`The orthogonality concept is not limited to two func
`tion detection system capable of recovering the data
`tions. Any number of functions can be mutually orthog
`signals generated in the transmitting system of FIG. 4.
`onal and mutually synchronized in a common interval.
`FIG. 1 is a generalized block diagram of an orthogonal
`Orthogonality with respect to time within each chan
`multiplex data transmission system according to this
`nel and with respect to frequency between channels is
`invention. From data sources on the left (not shown)
`preserved by shaping the signals applied to each channel
`impulse samples are applied in synchronism on a plurality
`such that the integral of the mathematically transformed
`of lines such as those designated 10, 11 and 12. Each
`product of the squares of the shaping function applied
`impulse is shaped in associated transmitting filters 15,
`to the individual channel and the channel transfer func
`16, and 17 and others not shown for additional sources.
`tion and the integral of the transformed products of the
`Line 13 symbolically indicates such other signaling
`shaping functions applied to adjacent channels and the
`channels. The passbands of the several transmitting filters
`square of the channel transfer function are each Zero.
`are centered at equally spaced frequencies with the spac
`These conditions are met in practical cases by shaping
`ing equal to half the data rate per channel. Their outputs
`functions whose squares have even symmetry about the
`are combined on line 14 and applied to common trans
`channel center frequencies and odd symmetry about fre
`mission medium 18, having an impulse response h(t)
`quencies located halfway between the channel center
`and a transfer function
`frequency and the channel band-edge frequencies. At the
`same time the phase characteristics of adjacent chan
`H (f)e.J. (f)
`nels may be arbitrary, provided only that their phase
`where H(f) and n(f) are respectively the amplitude and
`characteristics differ by ninety electrical degrees plus an
`phase characteristics of medium 18, e is the base of
`arbitrary phase function with odd symmetry about the
`frequency midway between the channel center frequen
`natural logarithms and J is the imaginary number V-1.
`Noise is also added at various points in the system as
`CICS. The required symmetries are achievable in a half-cycle
`indicated symbolically by adder 19. The several signal
`of the cosine wave whose square is the raised cosine
`ing channels are separately detected in receiver 20. It
`40
`shaping function as one readily definable illustrative ex
`is assumed for the present that the channel with the
`ample.
`lowest frequency is operating at baseband. Carrier modul
`Preservation of orthogonality within each channel per
`lation and demodulation at passband can be accomplished
`mits establishing individual channel data transmission
`by standard techniques.
`rates equal to the channel bandwidth. This is half the
`Channel shaping is the critical element here. Let
`bo, b1, b2 . . . be a sequence of m-ary signal digits
`ideal Nyqquist rate. However, due to the fact that ad
`jacent channels are synchronized, they can be overlapped
`(m22) or a Sequence of analog samples to be trans
`by 50 percent. The overall data transmission rate for the
`mitted over an arbitrary ith channel. Each of bo, b1,
`full channel bandwidth then becomes the ideal Nyquist
`b2 . . . can be represented by an impulse with height
`proportional to that of the corresponding sample. These
`rate times the ratio of the number of channels to the
`impulses are applied to the ith transmitting filter at the
`number of channels plus one.
`Inasmuch as the amplitudes of the shaping functions
`rate of one impulse every T seconds (data rate per
`are proportional to the amplitudes of the samples by
`channel equals 1/T bauds). Let ai(t) be the impulse
`which they are multiplied, transmission is in no way re
`response of the associated ith transmitting filter. Then
`stricted to binary digits. Multilevel symbols and symbols
`this filter transmits a sequence of signals as
`of arbitrary height derived from analog samples are equal
`boa;(t), b1ai(t-T), bai(t-2T) . . .
`ly transmissible.
`Orthogonal signals are readily detectable by correlation
`The received signals at the output of transmission medium
`procedures using matched filter techniques.
`A feature of this invention is that the band-limited
`18 are
`shaping filters for each channel can be identical.
`bou;(t), bui(t-T), bui(t-2T) o
`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:
`
`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
`
`0.
`
`20
`
`25
`
`30
`
`50
`
`60
`
`65
`
`70
`
`75
`
`where
`
`(T is a dummy variable of integration.)
`These received signals overlap in time, but they are
`orthogonal (noninterfering) if
`
`?ui () ui(t-kT)dl=0, k= + 1, +2 ...
`
`IPR2018-01477
`Apple Inc. EX1010 Page 5
`
`

`

`3,488,445
`5
`6
`analog samples transmitted over an adjacent jth channel
`dium, a channel amplitude characteristic A(f) (iF1, 2
`which has a transmitting filter impulse response of a(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
`
`10
`
`Ci-Q, (f) is greater than zero for all f in the range
`fi-Efs and Zero outside this range, C is an arbitrary con
`stant and Q(f) is a shaping function having odd symme
`tries about fifs/2. Furthermore, the products of the
`shaping functions for adjacent channels C--O (f)]
`IC 1--O, (f)] are even functions about the frequency
`(f--fs/2) midway between adjacent channel center fre
`quencies fi and f1.
`As a further part of my theorem the channel phase
`characteristic ai(f) (i=1, 2 . . . , N) can be shaped
`Such that
`
`Coa;(t), c1a (t-T), c2a (t-2T) O
`The received signals at the output of medium 18 are now
`coui(t), clu(t-T), cauj (t-2T) . . .
`Although these signals overlap those of the ith channel
`in both time and frequency, they are nevertheless
`mutually orthogonal if
`(2)
`2 . . .
`J. u;(t) ui(t-kT) dt = 0, k=0, d. 1,
`Intersymbol and interchannel interference can be
`simultaneously eliminated if Equation 1 is satisfied for
`all i and Equation 2, for all i and j (izi).
`By well known principles of Fourier transform analysis
`Equations 1 and 2 can be transformed into the frequency
`domain, such that Equation 1 becomes
`J.A.(1)H-(1) cos 2nfkT df = 0
`(3)
`N; and Equation 2
`for k=1, 2, 3 . . ., i=1, 2, 3 . . .
`becomes
`
`20
`
`Ai(f) A(f) H2(f) cos Iovi (f) -cy (f)) cos 2arfkTclf=0
`(real part)
`(4)
`
`and
`A (f) A (f) H2(f) sin avi (f)-cy, (f) sin 2afkT df = 0
`(imaginary part)
`(5)
`
`for
`
`k=0, 1, 2 . . .
`i, i=1, 2 . . . N, izi
`In Equations 3, 4, and 5 A(f) is the amplitude char
`acteristic and oxi(f) is the phase characteristic of the ith
`transmitting filter. A(f) and a (f) for the jth transmitting
`filter are similarly defined. H(f) is the amplitude char
`acteristic of medium 18.
`Let f(i=1, 2, 3 . . . N) denote the equally spaced
`center frequencies of the N independent signaling chan
`nels. Let the lowest channel center frequency be
`
`30
`
`35
`
`40
`
`45
`
`in the frequency range between channel center frequen
`cies. Yi (f) is an arbitrary phase function having odd sym
`metry about the frequency (f--f/2) midway between
`adjacent channel center frequencies fi and f1.
`If Ai(f) and ai(f) are shaped in accordance with my
`theorem and f 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 2f 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
`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 Oi (f) (i=1, 2 . . . , N) are
`identically shaped, the product function C--O, (f)]
`EC-1--O1(f) is an even function about the frequency
`midway between adjacent channel center frequencies
`(f--fs/2) provided Q, (f) is an odd function about
`(f--fs/2) and is an even function about the channel
`center frequency f. 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
`Qi (f) = cos 2nd
`" 2f,
`where flies between f-f and f--fs and i is any positive
`integer.
`Choose
`
`50
`
`55
`
`()
`
`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
`Ji-fit (i-1)?.-(+i-).f.
`-?h-Li
`
`Each amplitude-modulated data channel is assumed to
`transmit at 2f bauds (symbols per second). Hence
`---
`(8)
`T-3.
`Since the bandwidth of each channel is 2fs, 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 (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 2fs 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 me
`
`60
`
`65
`
`70
`
`Ci=;
`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
`
`IPR2018-01477
`Apple Inc. EX1010 Page 6
`
`

`

`Ali (f) H(f)
`
`COS Ji
`2fs
`
`5
`
`10
`
`25
`
`30
`
`3 5
`
`40
`
`50
`
`2f --2, sin n2r. -fi
`--2pm cosm2ar
`
`55
`
`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 fifs and maximum
`transmission at the center frequency f.
`Waveforms 21 and 22 meet the symmetry properties
`about
`f, f-- and f-,
`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 A(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 H(f) is not flat in the indi
`vidual channel band, A(f) can be obtained from a
`division of the product A(f)H(f) by H(f).
`My theorem also places constraints on the phase char
`acteristic oxi(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 o (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
`oi (f) is independent of the amplitude characteristic A(f).
`equation is
`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(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 Q, (f) can be
`chosen in an identical manner for all channels, according
`to my corollaries 1 and 2. Then A(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 H(f) and a
`phase characteristic ox (f) as shown in FIG. 5. Char
`acteristics H(f) and ox (f) have the properties described
`in corollaries 1 and 2. Filters 44 are bandpass filters of
`2fs 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,
`R 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-1-0.5)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--0.5) fs. The phase characteristics ori (f) 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 -r/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 f, fa, and f in modulators
`45. The frequency f1 is chosen equal to (k-1)f to form
`a lower sideband centered on a frequency of 1.5f as
`shown by waveform 51 on line (A) of FIG. 5. This
`waveform has a bandwidth extending from 0.5f to 2.5f.
`Similarly, frequencies f, and f are chosen respectively
`to be (k-2)fs and (k-3)fs 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 f. 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 4.5f 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 f. and then appears on line
`49 for application to a connected transmission medium.
`
`for m=1, 2, 3 . . . and n=2, 4, 6 . . . in the range
`fifs, where h is an arbitrary odd integer and po, p.m., bn
`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 fitf/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 oxi(f) would be
`completely arbitrary.
`In FIG. 4 to be discussed more fully later the phase
`function a (f) is sketched as identical curves 56, 58, and
`60. For this particular choice h is set to -1, p and sp
`equal Zero, in equals 1, n equals 2 and t2=1. The linear
`term is thus -r/2 and a sine function with odd sym
`metry about fi-Efs/2 is superimposed thereon.
`
`80
`
`35
`
`70
`
`75
`
`IPR2018-01477
`Apple Inc. EX1010 Page 7
`
`

`

`5
`
`O
`
`20
`
`25
`
`30
`
`40
`
`3,488,445
`10
`The passband of the transmission medium is assumed to
`Weighting resistors are adjusted before data transmission
`be centered on the frequency f.
`by taking samples of the desired waveform after it has
`Since the transmission rate in each channel is 2fs
`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 T when the peak
`6fs 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, 50 percent greater
`response of the matched filter will be a maximum for
`than that possible by use of conventional nonoverlap
`the waveform for which the filter 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 T 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 IRE
`N
`Transactions on Information Theory of June 1960 for
`N-1
`more details on the use of matched filters as correlators.
`times 2 bauds per cycle of bandwidth is obtained, where
`At the output of each matched filter 78 is a sampler 79
`controlled by sampling pulses at the data rate 2fs derived
`N is the number of channels used.
`The data in individual channels of a composite signal
`from pick-off 70. The sampling pulses are delayed by the
`as shown on line (D) of FIG. 5 can be demodulated
`time t to coincide with the arrival of the peak response
`and detected by the use of adaptive correlation techniques
`on the reference tap of the matched filter. On the basis of
`as shown in the block diagram of FIG. 6. The composite
`the summed sample the decision is made as to the nature
`signal arriving on line 65 after having traversed the trans
`of the data bit transmitted. The data output is made
`mission medium has the reference frequency f. removed
`available on leads 80, 81, and 82 for the respective chan
`in pickoff device 70. Pickoff 70 may comprise a narrow
`nels. Thus, the system of this invention operates in real
`band filter and frequency multipliers by means of which
`time and no signal storage, other than a fixed delay, is
`the sampling frequency 2fs and the several demodulating
`needed.
`carriers are derived for application to conductor 66.
`Where larger numbers of channels than three are used,
`Pickoff 70 can alternatively be placed to the right of
`the three-channel method above may be extended in a
`upper modulator 74, if desired. The received signal is next
`straightforward manner. For example, the channels may
`applied to modulators 74, having demodulating fre
`be separated in groups of three using bandpass filters and
`quencies chosen to translate the respective channel band
`each group of three may then be translated down to base
`widths to a common frequency range. This frequency
`band in the manner previously described. The group of
`range is defined by the characteristic H(f) of low-pass
`three passing through the bandpass filter is initially trans
`lated to baseband by using a demodulating signal of the
`filters 76. The characteristics of filters 76 are identical
`as shown in waveform 75. The characteristic is flat to 2.5f
`form cos 2T (i-1)ft(E) where i is the center frequency
`cycles and falls off to zero beyond that frequency.
`of the center channel of the group of three and (E) is an
`35
`arbitrary phase angle accounting for carrier phase shift
`On the top line channel 1 is translated back to its base
`band position centered on the frequency 1.5fs by de
`in the transmission medium.
`modulation with a frequency f, the same carrier fre
`The orthogonal multiplex system of this invention can
`also be operated without synchronization among channels
`quency used at the transmitter. In passing through filter
`76 channel 3 is