Synthesis of Band-Limited Orthogonal Signals for Multichannel Data Transmission

Synthesis of Band-Limited OrthogonalSignals for Multichannel Data

TransmissionBy ROBERT W. CHANG

(Manuscript received August 4, 1966)

This paper presents a principle of orthogonal multiplexing for trans-mitting a number of data messages simultaneously through a linear band-limited transmission medium at a maximum. data rate without interchanneland intersymbol interferences. A general method is given for synthesizingan infinite number of classes of band-limited orthogonal time functions ina limited frequency band. Stated in practical terms, the method permits thesynthesis of a large class of practical transmitting filter characteristics foran arbitrm'ily given amplitude characteristic of the transmission medium.Rectangular-shaped ideal filters are not required. The synthesis procedureis convenient. Furthermore, the amplitude and the phase characteristicsof the transmitting filters can be synthesized independently. Adaptive correla-tion reception can be used for data processing, since the received signalsremain orthogonalno matter what the phase distortion is in the transmissionmedium. The system provides the same signal distance protection againstchannel noises as if the signals of each channel were transmitted throughan independent medium and intersymbol interference in each channel wereeliminat.ed by reducing data rare.

I. INTRODUCTION

In data transmission, it is common practice to operate a number ofAM data channels through a single band-limited transmission medium.The system designer is faced with the problem of maximizing the overalldata rate, and minimizing interchannel and intersymbol interferences.In certain applications, the channels may operate on equally spacedcenter frequencies and transmit at the same data rate, and the signalingintervals of different channels can be synchronized. For these applica-tions, orthogonal multiplexing techniques can be considered. Several

1775

1776 THE BELL SYSTEM TECHNICAL JOURNAL, DECEMBER 1966

orthogonal-multiplexed systems developed':" in the past use specialsets of time-limited orthogonal signals. These signals have widelyspread frequency spectra (e.g., a sin xix spectrum). Consequently,when these signals are transmitted through a band-limited transmissionmedium at a data rate equivalent to that proposed in this paper, certainportions of the signal spectrum will be cut off and interferences willtake place. For instance, the interferences because of band-limitationhave been computed" for a system using time-limited orthogonal sineand cosine functions.

This paper shows that by using a new class of band-limited orthogonalsignals, the AM channels can transmit through a linear band-limitedtransmission medium at a maximum possible data rate without inter-channel and intersymbol interferences. A general method is given forsynthesizing an infinite number of classes of band-limited orthogonaltime functions in a limited frequency band. This method permits one tosynthesize a large class of transmitting filter characteristics for arbi-trarily given amplitude and phase characteristics of the transmissionmedium. The synthesis procedure is convenient. Furthermore, theamplitude and the phase characteristics of the transmitting filters canbe synthesized independently, i.e., the amplitude characteristics neednot be altered when the phase characteristics are changed, and viceversa. The system can be used to transmit not only binary digits (asin Ref. 1) or m-ary digits (as in Ref. 2), but also real numbers, such astime samples of analog information sources. As will be shown, the systemsatisfies the following requirements.

(i) The transmitting filters have gradual cutoff amplitude charac-teristics. Perpendicular cutoffs and linear phases are not required.

(ii) The data rate per channel is 2f. bauds,* where f. is the centerfrequency difference between two adjacent channels. Overall data rateof the system is [NI(N + 1)] Rm ax , where N is the total number ofchannels and Rm ax , which equals two times the overall baseband band-width, is the Nyquist rate for which unrealizable rectangular filterswith perpendicular cutoffs and linear phases are required. Thus, as Nincreases, the overall data rate of the system approaches the theoreticalmaximum rate Rm ax , yet rectangular filtering is not required.

(iii) When transmitting filters are designed for an arbitrary givenamplitude characteristic of the transmission medium, the receivedsignals remain orthogonal for all phase characteristics of the transmissionmedium. Thus, the system (orthogonal transmission plus adaptive

The speed in bauds is equal to the number of signal digits transmitted in onesecond.

OR'rHOGONAL SIGNALS FOR DATA TRANSMISSION 1777

correlation reception) eliminates interchannel and intersymbol inter-ferences for all phase characteristics of the transmission medium.

(iv) The distance in signal space between any two sets of receivedsignals is the same as if the signals of each AM channel were transmittedthrough an independent medium and intersymbol interference in eachchannel were eliminated by reducing data rate. The same distanceprotection is therefore provided against channel noises (impulse andGaussian noise). For instance, for band-limited white Gaussian noise,the receiver receives each of the overlapping signals with the sameprobability of error as if only that signal were transmitted. The distancesin signal space are also independent of the phase characteristics of thetransmitting filters and the transmission medium.

(v) When signaling intervals of different channels are not syn-chronized, at least half of the channels can transmit simultaneouslywithout interchannel or intersymbol interference.

II. ORTHOGONAL MULTIPLEXING USING BAND-LIMITED SIGNALS

Consider N AIVI data channels sharing a single linear transmissionmedium which has an impulse response h(t) and a transfer functionH(J) exp [J71(f)] (see Fig. 1).* H(J) and 71(J) will be referred to, respec-tively, as the amplitude and the phase characteristic of the transmissionmedium.

Since this analysis treats only transmission media having lineal'properties, the question of performance on real channels subject to suchimpairments as nonlinear distortion and carrier frequency offset is notconsidered here. Such considerations are subjects of studies beyond thescope of the present paper.

In deriving the following results, it is not necessary to assume thatthe transmitting filters and data processors operate in baseband. How-ever, this assumption will be made since in practice signal shaping anddata processing are usually performed in baseband. Carrier modulationand demodulation (included in the transmission medium) can be per-formed by standard techniques and need not be specified here.

Consider a single channel first (say, the ith channel). Let b, bi ,b, .. , be a sequence of m-ary (m 6; 2) signal digits or a sequence ofreal numbers to be transmitted over the ith channel. As is well known,'b, bi , bz , ... can be assumed to be represented by impulses withproportional heights. These impulses are applied to the ith transmittingfilter at the rate of one impulse every T seconds (data rate per channel

J denotes the imaginary number ";-1, while j is used as an index.


\ST TRANSMITTINGFILTER

AI (f) EXp[Ja l (fB

at (t)

A2

(f) EXp[Ja2 (f)]

a2 (t )

AN (f) EXp[JaN(f)]aN (t)

NTH TRANSMITTINGFILTER

TRANSMISSIONMEDIUM

H (f) EXP[J'I (f)]htt)

NOISE

DATAPROCESSING(RECEIVERS)

Fig. 1- N data channels transmitting over one transmission medium.

equals liT bauds). Let a;(t) be the impulse response of the ith trans-mitting filter, then the ith transmitting filter transmits a sequence ofsignals as

boa;(t) + bla;(t - T) + b2a;(t - 2T) + ....The received signals at the output of the transmission medium are

bou;(t) + blu;(t - T) + b2,u;(t - 2T) + ... ,where

k = 1, 2, .. .

These received signals overlap in time, but they are orthogonal if

L: u;(t)u;(t - kT) dt = 0, (1)As is well known, orthogonal signals can be separated at the receiver

by correlation techniques;" hence, intersymbol interference in the ithchannel can be eliminated if (1) is satisfied.

Next consider interchannel interference. Let Co, CI, C2, be the Correlation reception and its adaptive feature will be briefly discussed in

Appendix C.

ORTHOGONAL SIGNALS FOR DATA TRANSMISSION 1779

m-ary signal digits or real numbers transmitted over the jth channelwhich has impulse response aj(t). It has been assumed in Section I thatthe channels transmit at the same data rate and that the signaling inter-vals of different channels are synchronized, hence the jth transmittingfilter transmits a sequence of signals,

coaj(t) + claj(t - T) + C2,aj(t - 2T) + ....The received signals at the output of the transInission medium are

cO'Uj(t) + Cl'Uj(t - T) + C2'Uj(t - 2T) + ....These received signals overlap with the received signals of the ithchannel, but they are mutually orthogonal (no interchannel interference)if

k = 0, 1, 2, .. , . (2)

Thus, intersymbol and interchannel interferences can be simultane-ously eliminated if the transInitting filters can be designed (i.e., if thetransmitted signals can be designed) such that (1) is satisfied for all iand (2) is satisfied for all i andj (i ~ j).

Denote U,(1) exp [J1L;(1)] as the Fourier transform of u,(t). One canrewrite (1) as

L: U,2(f) exp (-J21rjkT) dj = 0k = 1, 2, ... (3)

i = 1,2, ... ,N,

(4)k = 0, 1, 2, ...

i,j = 1, 2, ... , N

i ~ i.Let A;{f) exp [Jai{f)] be the Fourier transform of a;(t). The transfer

function of the transmission medium is H{f) exp [J,.,(f)]. Equation (3)becomes

and rewrite (2) as

L: U,(f) exp [Ju;(f)]Uj(j) exp [-JlLif)]-exp [-J21rjkT] dj = 0


L: A/(j)H2(j) exp (-J27rfkT) df = 0k = 1, 2, ...

i = 1,2, ... ,N,or

f' A/(j)H2(j) cos 27rfkT df = 0k 1,2,3, .. ,i = 1,2, ... ,N.

Equation (4) becomes

L: A;(j)A j(j)H2(f) exp {J[a;(j) - aj(f) - 27rfkT]} df = 0

(5)

(6)

(7)k = 0, 1, 2, ...i,j = 1,2, ... ,N

i j.

Writing (7) in real and imaginary parts and comparing parts for k =1,2, ... and k = -1, -2, ... , it is seen that (7) holds if and only if

and

where

k = 0, 1,2, .

i,j = 1, 2, ,N

i j.

It will be recalled that the transmitting filters and the data processorsoperate in baseband. Let!;, i = 1,2, .... ,N, denote the equally spacedbaseband center frequencies of the N independent channels. One canchoose

fl = (h + !)f., (10)


where h is any positive integer (including zero), and f. is the differencebetween center frequencies of two adjacent channels. Thus,

fi = f1 + (i - l)f. = (h + i - !)f.. (11)Carrier modulation will translate the baseband signals to a given fre-quency band for transmission.

Each AM data channel transmits at the data rate 2f. bauds. Hence,

1T = 2f. seconds. (12)

For a given amplitude characteristic H(f) of the transmission medium,band-limited transmitting filters can be designed (i.e., band-limitedtransmitted signals can be designed) such that (6), (8), (9), and (12)are simultaneously satisfied (no intersymbol and interchannel inter-ference for a data rate of 2f. bauds per channel). In addition, the fiverequirements in Section I are also satisfied. A general method of design-ing these transmitting filters is given in the following theorem.

Theorem: For a given H(!), let Alf), i = 1,2, ... ,N, be shaped such that

c. + os > 0,0, f < fi - i.,

fi - f. < f < fi + f.f > Ii + t.,

(13)

where C. is an arbitrary constant and Qi(J) is a shaping function havingodd symmetries oboui ], + (f./2) and/; - (f./2) , i.e.,

o. [Vi +~) +1] = -o. [Vi +~) -I] 'a. [Vi - ~) + 1] = -Q. [Vi -~) -IJ.

o


Let the phase characteristic ali), i = 1, 2, ... ,N, be shaped such that

fi < f < Ii +f.(17)

i = 1, 2, .... ,N - 1,

where 'Yi(f) is an arbitrary phase function with odd symmetry about [ +(j./2).

If Ai(f) and ai(f) are shaped as in (13) through (17) and !I is set ac-cording to (10), then (6), (8), (9), and (12) are simultaneously satisfied(no intersymbol 01' interchannel interference for a data rate of 2f. bauds perchannel). Furthermore, the five requirements in Section I are also satisfied.

The proof of this theorem will be broken down into two parts. Thefirst part [showing that (6), (8), (9), and (12) are simultaneously satis-fied] will be given in Appendix A. The second part (showing that thefive requirements in Section I are satisfied) will be given in Section IIIfollowing a discussion of the various choices of the shaping functionsand transmitting filter characteristics.

III. TRANSMITTING FILTER CHARACTERISTICS

Consider first the shaping of the amplitude characteristics Ai(j) ofthe transmitting filters. Equations (13), (14), and (15) in the theoremcan be easily satisfied. Equation (16) can be satisfied in many ways.For instance, a simple, practical way to satisfy (16) is stated in the fol-lowing corollary.

Corollary 1: Under the simplifying condition that(i) C, should be the same for all i

(ii) Qi(f), i = 1, 2, .... ,N, should be identically shaped, i.e.,

i = 1, 2, ... ,N - 1, (18)

(16) holds when Qi(f) is an even function about Ii , i.e.,

Qi(fi + f') = Qi(fi - f'), o < f' < f . (19)The proof of this corollary is straightforward and need not be given here.Two examples are given for illustration purpose. The first example isillustrated in Fig. 2 where Qi(f) is chosen to be

Qi(f) 1 i-Ii= -cos '/1'--2 f.' Ii - t. < f < fi + f. i = 1,2, .... ,N.

A~ (f) H (f)


Fig. 2 - First example of shaping the amplitude characteristic A I (j) of thetransmitting filters.

This simple choice satisfies (14), (15), (18), and (19). Let C, be ~ for alli, then (14), (15), and (16) are all satisfied. From (13)

ANI)H2(f) = Ci + Qi(j)III - Ii

= 2 + 2 cos 'II" ----:r.- 'and

I - i.Ai(j)H(j) = cos 'II"~ , t, - I. < I < Ii + I.i = 1,2, .... , N.

This Ai(f)H(f) is similar to the amplitude characteristic of a standardduobinary filter (except shift in center frequency). The second exampleis illustrated in Fig. 3 where Q;(f) is chosen such that Ai(j)H(j) has ashape similar to that of a multiple tuned circuit. It can be seen fromthese two examples that there is a great deal of freedom in choosing

f2

o

r-~

) \

Fig. 3 - Second example of shaping the amplitude characteristic Al(j) of thetransmi tting filters. -


the shaping function Qi(f). Consequently, Ai(J)H(J) can be easily shapedinto various standard forms. Ai(f) would have the same shape asAi(f)H(f), if H(f) is flat in the frequency band fi - f. to fi + f. of theith channel. If H(J) is not flat in this individual band, Ai(f) can be ob-tained from Ai(f)H(f)/H(J), provided that H(f) 0 for any f in theband.

It is also noted from the preceding that if C, is chosen to be the samefor all i and if Qi(f) , i = 1,2, ... ,N, are chosen to be identically shaped(i.e., identical in shape except shifts in center frequencies), thenAi(f)H(f), i = 1,2, ,N, will also be identically shaped. Consequently,A;(f), i = 1, 2, , N, will be identically shaped if H(f) is flat or ismade flat. An advantage of having identically shaped filter characteristicsis that each filter can be realized by using an identical shaping filter plusfrequency translation.

H(f) can be made flat by using a single compensating network whichcompensates the variation of H(f) over the entire band. As an alternative,note that Ai(f) exists only from 1; - f. to 1; + f . Hence, for the ithreceiver, the integration limits of (6), (8), and (9) can be changed to1; - f. and fi + f . Therefore, the signal at the ith receiver only has tosatisfy the theorem in the limited frequency band 1; - f. to 1; + f .This permits one to design the transmitting filters for flat H (f) and thencompensate the variation of H(f) individually at the receivers, i.e., usean individual network at the ith receiver to compensate only for thevariation of H (f) in the limited frequency band 1; - f. to fi + f .

Finally, note that if the channels are narrow, each channel will usuallybe approximately flat. In these cases, one may design the transmittingfilters for flat H(f) without using compensating networks. This designshould lead to only small distortion.

Consider next the shaping of the phase characteristics OIi(j) of thetransmitting filters. It is only required in the theorem that (17) besatisfied. However, if it is desired to have identically shaped transmittingfilter characteristics, one may consider a simple method such as that inthe following corollary.

Corollary 2: Under the simplifying condition that OIi(f) , i = 1,2, ... , N,be identically shaped, i.e.,

(17) holds when

i 1,2, ... ,N - 1 (20)

ORTHOGONAl, SIGNALS FOR DATA TRA.NSMISSION 178.5

m 1, 2, 3, 4, 5, '"

n = 2,4,6, ...

t, - f. < .f < .f; + f. ,where h is an arbitrary odd integer and the other coefficients ('Po; 'Pm , m =1, 2, 3, 4, 5, ... ; t/;n , n = 2, 4, 6, ... ) can all be chosen arbitrarily.

This corollary is proven in Appendix B. Note that if the index n in thecorollary were not required to be even, ai(f) would be completely arbi-trary (a Fourier series with arbitrary coefficients). This shows that thereis a great deal of freedom in shaping ai(f) even if the additional constraintof identical shaping is introduced (three-fourths of the Fourier coefficientscan be chosen arbitrarily). The linear term hll-[(f - .f;)/2f.] is introducednot only to give the term 71"/2 in (17), but also because a linear com-ponent is usually present in filter phase characteristics. A simple exampleis given in Fig. 4 to illustrate (21). For clarity, the arbitrary Fouriercoefficients are all set to zero, except Vt2 , and It is set to -1.

As can be seen in the theorem, the requirement on ai(f) is independentof the requirements on Ai(f). Hence, the amplitude and the phase char-acteristics of the transmitting filters can be synthesized independently.This gives even more freedom in designing the transmitting filters.

A simple set of Ai(f) and ai(f) is sketched in Fig. 5 for three adjacentchannels. This illustrates that the frequency spectrum of each channelis limited and overlaps only with that of the adjacent channel. H(f) isassumed flat and the transmitting filters are identically shaped. Asmentioned previously, these filters can be realized either by differentnetworks or simply by using identical shaping filters plus frequencytranslations.

Now consider the five requirements in Section 1. The first requirementis satisfied since the transmitting filters designed are of standard forms(see the examples in Figs. 2 and 3). Perpendicular cutoffs and linearphase characteristics are not required.

As for the second requirement, it can be seen from Fig. 5(a) that theoverall baseband bandwidth of N channels is (N + l)f. Since data rate


....--- .....,,' I-J11.::;-c

tUJIII

~Q.

Fig. 5 - Example of transmitting filter characteristics for orthogonal multi-plexing data transmission.


per channel is 2!, bauds, the overall data rate of N channels is 2!,Nbauds. Hence,

overall data rate = N2~ 1 X overall baseband bandwidth

N= N + 1 Rmax ,

where Rmax , which equals two times overall baseband bandwidth, is theNyquist rate for which unrealizable filters with perpendicular cutoffsand linear phases are required. Thus, for moderate values of N, theoverall data rate of the orthogonal multiplexing data transmissionsystem is close to the Nyquist rate, yet rectangular filtering is not re-quired. This satisfies requirement (ii).

Now consider the third requirement. As has been shown, the receivedpulses are orthogonal if (6), (8), and (9) are simultaneously satisfied.Note that the phase characteristic 11(1) of the transmission medium doesnot enter into these equations. Hence, the received signals will remainorthogonal for all 11(!), and adaptive correlation reception (see AppendixC) can be used no matter what the phase distortion is in the transmissionmedium. Also note that so far as each receiver is concerned, the phasecharacteristics of the networks in each receiver (including the bandpassfilter at the input of each receiver) can be considered as part of 11(!),and hence has no effect on the orthogonality of the received signals.

In the case of the fourth requirement, let

and

b,,', k = 0, 1,2, ... ;

Cki, k = 0, 1, 2, ... ;

i = 1,2, ... , N,

i = 1,2, ... ,N

be two arbitrary distinct sets of m-ary signal digits or real numbers to betransmitted by the N AM channels. The distance in signal space betweenthe two sets of received signals

L: L: b,,'u,(t - kT)i k

and

is


In an ideal case where interchannel and intersymbol interferences areeliminated by transmitting the signals of each channel through an inde-pendent medium and slowing down data rate such that the receivedsignals in each channel do not overlap, the distance d can be written as

didoal = [~ L 1"" (b/ - Cki)2U i2(t - kT)dtJl., k -co

In this study, the N channels transmit over the same transmissionmedium at the maximum data rate T = 1/2f . If the transmitting filterswere not properly designed, the distance d could be much less thandidoal and the system would be much more vulnerable to channel noises(impulse and Gaussian noises). However, if the transmitting filters aredesigned in accordance with the theorem in Section II, the receivedsignals will be orthogonal and d = didoal. Thus, the distance betweenany two sets of received signals is preserved and the same distance pro-tection is provided against channel noises. For instance, since d = d id oa ! ,it follows from maximum likelihood detection principle that for band-limited white Gaussian noise and m-ary transmission the receiver willreceive each of the overlapping signals with the same probability of erroras if only that signal is transmitted.

Note further that didoal can be written as

didoal = [~ L (bk i - Cki )21 "" A/(f)H2(f)dfJl., k -~

Thus, didoal is independent of the phase characteristics (Xi(f) of the trans-mitting filters and the phase characteristic 'TI(f) of the transmissionmedium. Since d = didoal, it follows that d is also independent of (Xi(f)and 'TI(f) and the same distance protection is provided against channelnoises for all (Xi(f) and'TI(j).

Finally, consider the fifth requirement. It is assumed in this paper thatsignaling intervals of different channels are synchronized. However, it isinteresting to point out that the frequency spectra of alternate chan-nels (for instance, i = 1,3,5, ... ) do not overlap (see Fig. 5). Hence, ifone uses only the odd- or the even-numbered channels, one can transmitwithout interchannel and intersymbol interferences and without syn-chronization among signaling intervals of different channels.* Theoverall data rate becomes ~ Rm ax for all N. A very attractive feature isobtained in that the transmitting filters may now have arbitrary phase

*For instance, signal digits are applied to the ith transmitting filter at 0, T,~T, ST,"', while signal digits are applied to the (i + 2)th transmitting filter at'T, T + 'T, 2T + 'T, ST + 'T, , where 'T is an unknown constant.

bauds


characteristics OI.i(f). [This is because OI.i(f) is not involved in (6) andintersymbol interference is eliminated for all OI.i(f).] Thus, only the ampli-tude characteristics of the transmitting filters need to be designed as inthe theorem and the transmitting filters can be implemented very easily.

Another case of interest is where part of the channels are synchronized.As a simple example, assume that there are five channels and that channel1 is synchronized with channel 2; channel 4 is synchronized with channel5; while channel 3 cannot be synchronized with other channels. If theamplitude characteristics of the five channels plus the phase character-istics of channels 1, 2, 4, and 5 are designed as in the theorem, one cantransmit simultaneously through channels, 1, 2, 4, and 5 or simultane-ously through channels 1, 3, and 5 without interchannel or intersymbolinterferences. The overall data rate is then between ~Rmllx and(N/N + I)Rmllx.IV. CONCLUSION

This paper presents a principle of orthogonal multiplexing for trans-mitting N(N ~ 2) AM data channels simultaneously through a linearband-limited transmission medium. The channels operate on equallyspaced center frequencies and transmit at the same data rate withsignaling intervals synchronized. Each channel can transmit binarydigits, m-ary digits, or real numbers. By limiting and stacking the fre-quency spectrums of the channels in a proper manner, an overall datarate of

N2~ 1 X overall baseband bandwidth

is obtained which approaches the Nyquist rate when N is large. Inter-channel and intersymbol interferences are eliminated by a new methodof synthesizing the transmitting filter characteristics (i.e., designingband-limited orthogonal signals). The method permits one to synthesizea large class of transmitting filter characteristics in a very convenientmanner. The amplitude and the phase characteristics can be synthesizedindependently. The transmitting filter characteristics obtained arepractical in that

(i) The amplitude characteristics may have gradual rolloffs, and thephase characteristics need not be linear.

(ii) The transmitting filters may be identically shaped and can berealized simply by identical shaping filters plus frequency transla-tions.


It is noted that the principle presented in this paper uses band-limitedorthogonal signals as opposed to other orthogonal multiplexing schemesusing nonband-limited orthogonal signals. The chief advantage of usingband-limited signals is that (as mentioned in Section I) these signals canbe transmitted through a band-limited transmission medium at a maxi-mum data rate without interchannel and intersymbol interferences.Other advantages of using band-limited signals over methods usingnonband-limited signals are

(i) Permitting the use of a narrowband bandpass filter at the inputof each receiver (see Appendix C) to reject noises and signalsoutside the band of interest. This is particularly important insuppressing impulse noises and in preventing overloading thefront ends of the receivers.

(~'i) Permitting unsynchronized operations at data rates betweeniRm ax and (NIN + l)Rm ax

It has been shown that the received signals remain orthogonal for allphase characteristics of the transmission medium; hence, adaptive cor-relation reception can be used to separate the received signals no matterwhat the phase distortion is in the transmission medium. These correla-tors adapt not only to the phase distortions in the system (includingtransmission medium, bandpass receiving filters, etc.), but also (seeAppendix C) to the phase difference between modulation and demodula-tion carriers (easing synchronization requirements).

V. ACKNOWLEDGMENTS

I wish to thank A. B. Brown, Jr., R. A. Gibby, and J. W. Smith forstimulating discussions and helpful suggestions.

APPENDIX A

In this appendix, it will be proven that if the transmitting filtersA.(J) exp [Ja.(f)], i = 1, 2, ... , N, are shaped as in the theorem inSection II and j; is set according to (10), then equations (6), (8), (9),and (12) are simultaneously satisfied.

First consider (6). From (13)

1'" A/(j)H2(j) cos 211-jkT df =i: [0. + Q.(j)] cos 27rfkTdf. (22)Since T = 1/2f., one has


f,H CC, cos 27rfkT df = 2 kIT [sin 27r(fi + f.)kT

I.-I. 7r

- sin 27r(ji - f.)kT]

= 7r~~ sin 7rk cos 27rfikT

= 0, k = 1, 2, 3, '"i = 1, 2, 3, ... ,N.

(23)

(24)

Since f; = (h + i - ~)f., one has

27rVi -1) kT = 27r(h + i - l)f.k 2~.= (h + i - l)k7r.

Hence, cos 27rfkT is an even function about fi - (f./2). This, togetherwith the fact that Qi(f) is an odd function about f; - (f./2) [see (15)],gives

fli Qi(f) cos 27rfkT elf = 0,I.-I.Silnilarly, one can show

i: Qi(f) cos 27rfkT elf = 0,Ii

k = 1,2,3, ...

i = 1,2, ... ,N.

k = 1,2,3, ...

i = 1,2, ... ,N.

(25)

(26)

Substituting (23), (25), and (26) into (22) gives

{O A/(f)H2(f) cos 27rfkT elf = 0, k = 1, 2, 3, ...i = 1, 2, 3, ... ,N.

Thus, (6) is satisfied and intersymbol interference is eliminated.Next consider interchannel interference and (8) and (9). From (13),

Ai(f)H(f) = 0, f < f; - f. , f > I, + f. ,so

A i(f)Aj(f)H2(f) = for j = i 2, i 3, i 4, ... ,


or

100 A i(f)A j(f)H2(f) cos [ai(f) - aj(f)] cos 27rlkT dl = 0100 A i(f)A j(f)H2(f) sin [ai(f) - aj(f)] sin 27rlkT dl = 0

(27)k = 0,1,2, .

i = 1,2,3, ,Nj = i 2, i 3, i 4, ....

Equation (27) shows that (8) and (9) are satisfied for j = i 2, i 3,i 4, .... It remains to show that (8) and (9) hold for j = i 1. Con-sider j = i + 1. It is seen from (13) that

(29)

Ii < I Ii + I.

One can write from (17) and (28)

100 A i(f)A Hi(f)H2(f) cos [ai(f) - aHi(f)] cos 27rlkT dlJ';+'. 1 1 [7r ]= I. [Ci + Qi(f)] [CHi + Qi+l(f)] cos '2 + "tiCf)

.cos 27rlkT dl

k = 0,1,2, .i = 1,2, ,N.

It is required in the theorem that

[Ci + Qi(f)]l[Ci+i + Qi+l{f)]lbe an even function about Ii + ([./2). Furthermore, cos [(7r/2) +'Yi(f)] and cos 27rfkT are, respectively, odd and even functions aboutIi + (f./2). Hence, from (29)100 A i(f)AHi(f)H2(f) cos [ai(f) - ai+l(f)] cos 27rfkT df = 0

k = 0, 1, 2, ... (30)

i = 1,2, ... ,N.


Equation (30) shows that (8) is satisfied for j = i + 1. In a similarmanner, one can show that (8) holds for j = i-I and that (9) holdsfor j = i 1. These, together with (27), prove that (8) and (9) hold forall k, i, and j.

APPENDIX B

Proof of Corollary 2

From (20) and (21)

(XHl(f) = (Xi(f - f.)

_ h f - f. -Ii +- 1r 2f. 1,00

" f - f. - fi+ LJ tp,n cos m1r j.m

m = 1, 2, 3, 4, 5, ...

n = 2,4,6, '"

fi < f < fi + 2f.For f.; < f < fi + f. , one has from (21) and (31)

(Xi(f) - (XHl(f) = ~;. [f - t, - (f - f. - f;)]

+~ tpm [cos m1r f 1. fi- cos m1r f - f. - f i]

f.

,,[ f - f+ L;: Y;n sin n1r T

(31)

(32)


h7r + " [2 1( 2f - 2fi - f.) . m7rJ= - .." 'Pm - SIn - m7r SIn -2 m 2 f. 2

+ ~ 1/In [2 sin n; cos ~ ( n7r 2f - Xi - fs)Jh7r 2 " . l7r . l7r(2f - 2fi - f.)= 2" - "7' 'PI sm "2 sm -.....;...-2"'"'1.,;:-.-----'--

l = 1,3,5, .h = 1, 3, .

Since sin [l7r(2f - 2fi - f.) /2f.] is an odd function about f = Ii + (f./2) ,(32) is equivalent to (17) and corollary 2 is proven.

APPENDIX C

This appendix briefly describes a possible receiver structure for receiv-ing the multichannel orthogonal signals.

The receiver of a single channel (say, the fifth channel) is shown inFig. 6(a). When viewed at point B toward the transmitter, the channelshave amplitude characteristics as shown in Fig. 6(b). The bandpassfilter at the input of the fifth receiver has a passband from f6 - f. tof6 + f. [Fig. 6(c)]. This filter serves the important purpose of rejectingnoises and signals outside the band of interest. Sharp impulse noises withbroad frequency spectra are greatly attenuated by this filter. Signals inother channels are rejected to prevent overloading and cross modulation.

The product device translates the frequency spectra further towardthe origin so that the signal can be represented by a minimum number ofaccurate time samples and the adaptive correlator can operate in digitalfashion. The transmitter can transmit a reference frequency f. or a knownmultiple of f. to the receivers for deriving the signals cos [27r(i - l)f.t +8i] for the product devices. It is important to note that the transmittercan lock this frequency f. to the data rate 2f. so that the arbitrary phaseangle 8i is time invariant and can be taken into account by adaptive cor-relation. Furthermore, the receiver can also derive the sampling rate2f. from this reference frequency.

When observed at point D, the channels have amplitude characteristicsas shown in Fig. 6(d). Note that the fifth channel now has a centerfrequency at 1.5!. [satisfying (11)] and an undistorted amplitude char-acteristic; hence, the signals in channel 5 remain orthogonal. The over-lapping frequency spectra between channel 5 and channels 4 and 6remain undistorted, and the phase differences Ci4(f) - Ci6(f) and Ci6(f) -Ci6(f) are unchanged; therefore, the signals in channels 4 and 6 remain


(a)

(b)

f

0 f 1 fs l+fs-.l(15fs ) (s.sfs)

~ rt\ (e) f0 fs

_-------Ofo t.!)fs

Fig. 6 - Reception of the signals in channel 5.

(d)

orthogonal to those in channel 5. Other channels produce no interferencesince their spectra do not overlap with that of channel 5.

Let bou(t), bluet - T), b2u(t - 2T), ... be the signals in channel 5 atpoint D, where bo , b1 , b2 , are the information digits. These signalscan be represented by vectors of time samples as

Since u(t), u(t - T), ... differ only in time origin, it is only necessary tolearn Yo for correlation purposes. The received signal at point D can bewritten as

where !l represents the sum of the signals in other channels. From discus-sions in the preceding paragraph


~k':Hi = A k =j= k ~jUk'!! 0.

Thus, the adaptive correlator can learn the vector Yo prior to data trans-mission and then correlate the received signal with ~k , k = 0, 1, 2, ...to obtain the information digits b , k = 0, 1, 2, ... .

In order to describe the operation more clearly we assume that thesignal at point D is fed to a delay line tapped at T13-second intervals(signal at D is band-limited between and 3i.). Assuming that u(t)is essentially time-limited to mT seconds for all possible phase character-istics of the transmission system, then 3m taps are sufficient. The ithtap is connected to a gain control G; . In the training period prior to datatransmission, the ith tap is also connected to a sampler 8; In the trainingperiod, the transmitter transmits a series of identical test pulses at t = 0,IT,2lT, ... The integer l is chosen large enough such that the receivedtest pulses u(t), u(t - IT), u(t - 2lT), ... do not overlap. The sampler8; samples at t = T, IT + T, 2lT + T, '. The only requirement on T isthat u(t) should be approximately centered on the tapped delay line att = T. The output of 8; (without noise) is a series of samples each repre-senting the ith time sample Ui of u(t). Since noise is always present, thesesamples are passed through a network (probably a simple RC circuit)such that the output -a i of this network is an estimate of u;-a; is in theform of a voltage or current and hence can be used to set the gain controlG; of the ith tap. Thus, at the end of the training period, the gain con-trols of the successive taps are set according to the magnitudes of thesuccessive time samples of u(t).

During data transmission, the transmitter transmits the informationdigits b , b1 , bs , ... sequentially at t = 0, T, 2T, .... A sampler atthe receiver samples the sum of the outputs of all the tap gain controls att = T, T + T, 2T + T, to recover b, bi , bs, ... The time delay Tremains the same as in the training period. The data transmission oper-ates in real time.

REFERENCES

1. Mosier, R. R., A Data Transmission System Using Pulse Phase Modulation,IRE Convention Record of First National Convention on Military Elec-tronics, June 17-19, 1957, Washington, D. C.

2. Dynamic Error-Free Transmission, General Dynamics/Electronics-Rochester,N.Y.

3. Harmuth, H. F. On the Transmission of Information by Orthogonal TimeFunctions, AlEE Trans., pt. I (Communication and Electronics), July, 1960,pp. 248-255.

4. Bennett, W. R. and Davey, J. R., Data Transmission, McGraw-Hill BookCompany, Inc., New York, 1965, p. 65.

Engineering

Synthesis of Band-Limited Orthogonal Signals for Multichannel Data Transmission