A Structured Overview of Digital Communications--A Tutorial Review--Part I

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IEEE COMMUNICATIONSMAGAZINE

A Structured Overview of Digital

Communications-a TutorialuReview-Part IBERNARDSKLAR

Part I of a two-part overview f digital communications.

AIMPRESSIVE assortment of communications

signal processing techniques has arisen during the

past two decades. This two-part paper presents anoverview of some of these techniques, particularly as theyrelate to digital satellite communications. The material isdeveloped in the context of a structure used to race theprocessing steps from the information source to the nforma-tion ink. Transformations are organized according ofunctional classes: formatting and source coding, modulation,channel coding, multiplexing and multiple access, frequencyspreading , ncryption , nd synchronization. Th e paperbegins by treating formatting, source coding, modulation,and potential trade-offs for power-limited systems andbandwidth-limited systems.

Communications via satellites have two unique charac-teristics: the ability to cover the globe with a flexibility thatcannot be duplicated with terrestrial links, and the vailabilityof bandwidth exceeding anything previously available forintercontinental communications [ 11. Most satellite commu-nications syste ms o date have been analog in nature.However, digital communications is becoming increasinglyattractive because of the ever-growing demand for datacommunication, and because digital transmission offers dataprocessing options and flexibilities not available with analogtransmission [2].

This paper presents n overview of digital communicationsin general; for the most part, however, the treatment is in thecontext of a satellite communications link. The key feature ofa digital communications system DCS) is that it sends only afinite set of messages, in contrast to an analog communica-tions system , which ca n send a n infinite set of messages. In aDCS, the objective at the receiver is not o reproduce awaveform with precision; it is inste ad to determine from anoise-perturbed signal which of the finite set of waveformshad been sen t by the transmitter. An important measure ofsystem performance is the average number of erroneousdecisions made, or the probability of error ( P E ) .

Figure 1 illustrates a typical DCS. Let there be M symbolsor messages ml, m2, . . , mM to be transmitted. Let each

symbol be represented by transmitting a correspondingwaveformsl(t),s2(t), . . . , sM(t).Thesymbol(ormessage)

m i s sent by transmitting the digital waveform s i t ) for Tseconds, the ymbol period. Th e next symbol is sent over thenext period. Since the M symbols can be represented y k =log2M binary digits (bits), the data rate can be xpressed as

R = ( l / T )log,M = k/T b/s.

Data rate s usually expressed in bits per second (b/s) whetheror not binary digits ar e actually involved. A binary symbol isthe special case characterized y M = 2 and k = 1. A digitalwaveform is taken to mean a voltage or current waveformrepresenting a digital symbol. Th e waveform is endowed withspecially chosen amplitude, frequency, or phase characteri s-tics that allow the selection of a distinct-waveform for each

symbol from a finite set of symbols. At various points alongthe signal route, noise corrupts the waveforms t ) so that itsreception must be termed n estimate i t). Such noise, and itsdeleterious effect on system performance, will be treated inPart I of this paper, which will appea r in the October 1983IEEE Communications Magazine.

Signal Processing Steps

. .

The functional block diagr am shown in Fig. 1 illustrates theda ta flow through the DCS. The upper blocks, which arelabeled format, source encode, encrypt, channel encode,multiplex, modulate, frequency spread , and multiple access,dictate he signal transformations from the source o thetransmitter. The lower blocks dictate the signal transforma-tions from the receiver back to the source; the ower blocksessentially reve rse the signal processing steps performed bythe upper blocks. Th e blocks within the dashed ines initiallyconsisted only of the modulator a nd demodulator functions,hence the name MODEM. During the past two decades,other signal processing functions were frequently incorporatedwithin the same ssembly as the modulator nd dem.odulator.Consequently, he term MODEM often encompasses heprocessing steps shown within the dashe d lines of Fig. 1.When this is the case, he MODEM can be thought of as the

0163-6804/83/0800-0004 01 OO 1983 EEE

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AUGUST 1983

“brains” of the system, nd the transmitter and receiver as the“muscles.” While the transmitter^,consists of a frequencyup-conversion stage, a high-power amplifier, an d an ntenna,the receiver portion is occupied by an antenna, a low-noisefront-end amplifier, and a down-converter st age , typically toan intermediate frequency (IF).

Of all the signal processing steps, only formatting,

modulation, an d demodulation a re essential for all DCS’s; theother processing steps within the MODEM are considereddesign options for various system needs. Source encoding, asdefined here, removes information redundancy and performsanalog-to-digital A/D) conversion. Encryption preventsunauthorized users from understanding messages and frominjecting false messages into the system. Channel coding can,for a given data rat e, improve the PE performance at theexpense of power or bandwidth, reduce the ystem bandwidthrequirement a t the expense of power o r PE performance, orreduce the ower requirement at the expense f bandwidth orPEperformance. Frequency spreading renders the signal lessvulnerable to interference (both natu ral and intentional) a ndcanbe used to afford privacy to he ommunicators.Multiplexing and multiple ac cess combine signals that mighthave different characteristics or originate from different,sources.

The flow of the signal processing steps shown in Fig. 1represents a typical arrangement; however, the blocks aresometimes implemented in a different order. For example,multiplexing can ake plac e prior to channel encoding,

prior to modulation, or-with a two-step modulation process(subcarrier and carrier)-it can be performed between the twosteps. Similarly, spreading can take place anywhere alongthe transmission chain; its precise location depends on theparticular technique used. Figure 1 illustrates the reciprocalaspect of the procedure; a ny signal processing step s whichtake place in the transmitting chain must be reversed in the

receiving chain. The figure also indicates that, rom thesource to the modulator, a message takes the orm of a bitstream, also called a baseband signal. After modulation, themessage takes the orm of a digitally encoded sinusoid (digitalwaveform). Similarly, in the reverse direction, a receivedmessage appears as a digital waveform until it is demodu-lated. Thereaf ter it takes the orm of a bit stream for all furthersignal processing steps.

Figure 2 shows the basic signal processing functions, whichmay be viewed as transformations from one signal space toanother. The ransformations a re classified into seven basicgroups:

formatting and source codingmodulationchannel codingmultiplexing an d multiple acce ssspreadingencryptionsynchronization

The organization has some nherent overlap, but neverthe-less provides a useful structure for this overview. The text by

INFORMATION I SOURCE ISINK BITS I

OPTIONAL L - - - - - - --- - - - - - - - - - - _SSENTIAL T 3 OTHERDESTINATIONS

I Fig. 1. Block diagram of a typical digital commu nication system.

5



FORMARING/ SOURCEOOINGOHERENT

CHARACTER CODINGSAM PLIN G DELTA MODUL ATIONDM)QUANTIZATIONONTINUOUSARIABLE SLOPE D M CVSD)

PARTIAL RESPONSE CODINGUFFMANODING

r

x , . ,. I

. .

CHANNELCODING.

. .

M - A RYS I G N A L I N GORTHOGONALBIORTHOGONALTRANSORTHOGONAL

SEaUEMCES .MULTIPLEXING/ MULTIPLECCESS

BLOCKCONVOLUTIONAL

MODUU\TION

PHASESHIFTKEYING PSK)

FREQUENCY SHIFT

KEYING FSK)AMPLITUDESHIFT

KEYING ASK)H YBR DSOFFSET QPSK

OQPSKIM I N I M U M S HI FT

KEYING MSK)L

FREQUENCY DIVIS ION FDMIF DMA IT I M ED I V I S I O N T D M I T D M A )C OD E D I V I S I O N C D M I C D M A )S PA C E D I V I S I O NSDMs)P O L A R I Z AT I O ND I V I S I O N P D M A )

. . , .. i... . ENCRYPTION

. .

, . .

NONCOHERENT. .

SHIFTKEYING DPSK)FREQUENCY SHIFT

KEYING FSK)A M P L I T U D E S H I F I

KEYING ASK)H Y B R I D S

SPREAOING

DIRECTSEQUENCING DS)FREQUENCY HOP PING FHIT I M EH O P P I N GH Y B R I D S

CARRIERSYNCHRONIZATIONSYMBOLSYNCHRONIZATIONFRAME SYNCHRONIZATIONNETWORK SYNCHRONIZATION

I

IEEE COMMUNICATIONS M AGAZINE

. >

- Fig. 2. Basic digital communication traneformationo.

Lindsey and Simon [3] is an excellent reference for the

modulation, coding, and synchronization transformationstreated here. The comprehensive books by Spilker [4] andBhargava et a]. 151specifically address digital communica-tions by satellite. The seven basic transformations ill now betreated individually, in the general order of their importancerather than in the order of the blocks shown in Fig. 1.

Formatting and Source Coding

The first essential processing, step, formatting, rende rs thecommunicated data compatible for digital processing. For-matting is defined as any operation that ransforms d ata intodigital symbols. Source coding means data compression inaddition to formatting. Some authors consider formatting to

be a special case of source coding (for which the datacompression amounts to zero), nstead of making a distinctionbetween the two. The source of most communicated dat a(except for computer-to-computer transmissions already indigital form) is either textual or analog n nature. If the dataconsists of alphanumeric text, it s character-encoded withone of several standard formats, such s American StandardCode for Information Interchange (AS CII), Extended BinaryCoded Decimal Interchange Code (EBCDIC), or Baud ot,and is thereby rendered nto digital form. If the dat a is analog,the (band-limited) waveform must first be sampled t a ra te ofat least 21, Hz (th e Nyquist frequency), where 1, is thehighest frequency contained n the waveform. Suc h sampling

6

insures perfect reconstruction of the analog signal; under-

sampling results in a phenomenon called aliasing, whichintroduces errors. However, the inimum sampling rate canbe less than 21, if the lowest signal frequency contained in thewaveform is nonzero [6]. Quantization of the time samplesallows each sample to be expressed .as a level from a finitenumber of predetermined levels; each such level can berepresented by a digital symbol. After qu antization, heanalog waveform c an still be recovered , but not precisely;improved reconstruction fidelity of the analog waveform ca nbe achieved by increasing the number of quantization levels(requiring increased transmission bandwidth).

Pulse code modulation (PCM), he classical and mostwidely used digital format, converts the quantized samples

into code groups of two-level pulses using fixed amplitudes.Each pulse group represents a quantized amplitude valueexpressed in binary notation. The re re eve ral PCMsubformats (such as nonreturn to zero, Manchester, a ndMiller), eac h providing som e special feature , such as self-clocking or a compac t spectral signature [3]. Duobinary, orpartial response coding (also called correlative coding), is aformatting technique that improves bandwidth efficiency byintroducing controlled interference between symbols. Thetechnique a1s.o provides error-detecting capabilities withoutintroducing redundancy into the da ta str eam 7-91.

Both source encoding and formatting mean.encoding thesource data with a digital format (A/D conversion); in this

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AUGUST 1983

sense alone, he two are dentical.However, the term “sourceencoding” has taken on additional meaning in DCS usage.Besides digital formatting, “source encoding” has also’cometo denote da ta compression (or da ta rat e reduction). Withstandard A/D conversion using PCM , dat a ompression canonly be achieved by lowering the sampling rate or reducingthe number of quantization levels per sample, each of whichincreases the mean squared rror of the reconstructed signal.Source encoding techniques accomplish rate reduction byremoving the redundancy hat is indigenous to most messagetransmissions; without sacrificing reconstruction fidelity. Adigital data source is said to possess edundancy if thesymbols ar e not equally likely or if.they a re not statisticallyindependent. Source encoding can reduce the data rate ifeither of these condi tions exists. A few descriptions ofcommon source coding techniques follow.

Differential, PCM (DPCM) utilizes the differences betweensamples rather han their actual amplitude. For most data,‘theavera ge amplitude variation from sample to sample s much

less than the total ‘amplitude variation; therefore, fewer bitsare needed to describe the difference. DPCM systemsactually encode the difference between a current amplitudesample and a predicted amplitude value estimated from pastsamples. The decoder utilizes a similar algorithm fordecoding. Delta modulation (DM) is the name given to thespecial case of DPCM where the quantization level of theoutput is taken to be one bit. Although DM can be easilyimplemented, it suffers from slope overload,” a condition inwhich the incoming signal slope exceeds he system’scapability to follow the analog source closely at the givensampling rate. To improve performance whenever slopeoverload is detected, the gain of the system can be variedaccording to a predetermined algorithm known to hereceiver. If the system s designed to adaptively.vary the gainover a continuous range, the modulation is termed continuousvariable slope delta (CVSD) modulation, or adaptive deltamodulation (ADM). Speech coding of good quality h as beendemonstrated with CVSD at bit rates less than 25 kb/s, anotable data rate eduction when compared with the 56-kb/ sPCM used with commercial telephone systems [lo].

Another example of source coding is linear predictivecoding (LPC). Th is technique is useful where the waveformresults from a process that an be modeled as a linear system.Rather than encode samples of the waveform, significant

features of the process ar e encoded. For speech, these includegain, pitch, a nd voiced or unvoiced information. Wh ere as inPCM each sample is processed independently, a predictivesystem such as DPCM uses a weighted sum of the n-pastsamples to predict ea ch present sample; it then transmits the“error” signal. The weights are calculated to minimize theaverage energy in the error signal that represents thedifference between the predicted and ctua l amplitude.For speech, the weights are calculated over short waveformsegments of 10 to 30 ms, and thus change as the speechstatistics vary. The LPC technique ha s been used to produceacce ptab le speech quality at a data rate f 2.4 kb/s, and ighquality at 7.2 kb/s [ 1 1 131. For \current perspectives in

7

digital formatting of speech, see Crochiere and Flanagan

Some source coding techniques employ code sequences ofunequal length so as to minimize the average number of bitsrequired per da ta sample. A useful coding procedure, calledHuffm an coding [15,161, can be used for effecting da tacompression upon any symbol set, provided the Q prioriprobability of symbol occurrence is known and not equallylikely. Huffman coding genera tes a binary sequence for eachsymbol so as to achieve the smallest avera ge number of bitsper sample, for the iven Q priori probabilities. The techniqueinvolves assigning shorter code sequences to the symbols ofhigher probability, and longer code sequences to those oflower probability. Th e price paid for achieving data ratereduction in this’way s a commensurate increase n decodercomplexity. In addition, there is a tendency for symbol errors,once made , to propagate for several symbol periods.

~ 4 1 .

Digital Modulation FormatsModulation, in general, is the process by which some

characteristic of a waveform is varied in accordance withanother waveform. A sinusoid has just three features whichcan be used to distinguish it from other sinusoids-phase,frequency, and amplitude. For the purpose of radio trans-mission, modulation is defined as the process whereby thephase, frequency, or amplitude of a radio frequency (RF)carrier wave is varied in accordance with the information tobe transmitted. Figure 3 illustrates examples of digitalmodulation formats: phase shift keying PSK), frequencyshift keying (FSK), amplitude shift keying (ASK), and ahybrid combination of ASK and PSK sometimes calledquadrature amplitude modulation (QAM). The first columnlists the analyti c expression, the second s a pictorial of the ’

waveform, and the hird is a vectorial picture. In the generalM-ary signaling case , the rocessor accepts k source bits at atime, and instructs the modulator to produce one of an

A N A L n l C WAVEFORM VECTOR

M - 8 *2lII

Fig. 3. Digital modulation ormats.





For the signal detection problem, the noise can be partitionedinto two components

n(t) = ;I(t) + ii(t)

whereN

j = lis taken to be the noise within the signal spac e, or heprojection of the noise components on the ignal axes 4 1( t) ,42(t), . , @N(t), and

Y t ) = nj 4i(t) (5)

ii(t) = n t) - i t)

is defined as the noise outside the signal space. In other words,h(t) may be thought of a s the noise that is effectively tunedout by the detector. he symbol i( t) represents the noise thatwill interfere with he detection process, and it will henceforthbe referred to simply as n(t). Once a convenient set of Northonormal functions has been adopted (note that t) is notconstra ined to any specific form), each of the transmittedsignal waveforms si(t) is completely determined by the vectorof its coefficients

s i = a i l , ai2, . . , i N ) i = I 2, . . . , M

Similarly, the noise n(t) can be expressed by the vector of itscoefficients

n = n l , n2, . . . , nN)

where n is a random vector with zero mean and Gaussian

distribution.Since any arbitrary waveform set, as well as noise, can be

represented as a linear combination of orthonormal wave-forms (see 1)- 5)), we are justified in using (Euclidean-like)distance in such an orthonormal space, as a decisioncriterion for the detection of any signal set in the presence ofAWGN.

Detection in the Presence of AWCNFigure 4 illustrates a two-dimensional signal sp ace, the

locus of two noise-perturbed prototype binary signals (s +n )and ( s2 +n ) , and a received signal r. Th e received signal invector notation s: r=si +n, where i = 1 or 2. This geometric

or vector view of signals an d noise facilitates the discussion ofdigital signal detection. The vectors s and 8 2 are fixed, sincethe waveforms sl t) and s2(t) are nonrandom. Th e vector orpoint n is a random vector; hence, is also a random vector.

The detector’s task aft er receiving r is to decide whethersignal s l or s2 was actually ransmitted. Th e method isusually to decide upon the signal classification that yields theminimum PE, although other strategies re possible [20] orthe case where M equals two signal cla sses, with S I and s2

being equally likely and he noise being AWGN, theminimum-error decision rule turns out to be: Whenever thereceived signal r lands in region 1 choose signal s ; when itlands in region 2, choose signal s2 see Fig. 4). An equivalentstatement is: Choose the signal class such that the distanced r, s i ) = I r - i is minimal, where is called the‘‘norm” of.vecto r x and generalizes the concept of length.

Detection of Coherent PSKThe receiver structure implied by the above rule s

illustrated in Fig. 5 . There is one product integrator(correlator) for each prototype waveform M inall); thecorrelators are followed by a decision stage. The receivedsignal is correlated with each prototype waveform knowna priori to the eceiver. The decision sta ge chooses the ignalbelonging to the orrelator with the largest output largest z i ) .For example, let:

s I(t) = sin w ts p(t) = - in w tn(t)= a random process with zero mean and aussian

distribution

Assume s l(t) was transmitted, so that:

r(t) = sl(t) +n(t) and z i = r(t) si(t) i t i = 1 2’

9



Fig. 7. Signal space and decision regions for 3-ary coherentFSK detection.

The expected alues of the roduct ntegrators, asillustrated in Fig. 5, are found as follows:

E {zz(t = T }= E u t +n(t) sin wt dt

where E is the statistical average.The decision stage must ecide which signal was

transmitted by measuring its location within the signal space.Th e decision rule is to choose the ignal with the largest value

of zi . Unless the noise is large and of a nature liable to causean error, the eceived signal is judged to be s t). Note tha t inthe presence of noise this process is statistical; the optimaldetector is one that makes the fewest errors on the average.The only strat egy that he detector can employ is to “guess”using some;optimized decision rule.

Figure 6 shows the detection process with the signal spacein mind. It represents a coherent four-level (4-ary) PSK orquadraphase shift keying (QPSK) system. In the terms weused earlier for M-ary signaling; k = 2 and M = 22 = 4.Binary source digits are collected two at a time, and for eachsymbol interval the two sequential digits instruct themodulator as to which of the four waveforms to produce. In

general, for coherent M-ary PSK (MPSK) systems, si(t) canbe expressed as

si(t)= d m c o s ( w o t - 7ri/M) (for 0 < < Ti 1,2, ... , M

Here, E is the energy content of s , t ) , and w o s an integralmultiple of Z.rr/T. We can choose a convenient set oforthogonal axes scaled to fulfill 3) as follows

4,(t) = m c o s ot 6), . +’ t) = m s i n ot

Now si(t) can be writ& n terms of these orthogonalcoordinates, giving:

.. ,

si t) = ,/i?cos(2mi/M) + , t ) + fi i n ( 2 d M ) + d t ) (7)

The decision rule for the detector (see Fig. 6) is to decidethat s t ) was transmitted if the received signal point falls nregion 1, hat sp(t) was transmitted if the received signal pointfalls n region 2 and so forth. In other words, the decision rule

is to choose the ith waveform with the largest value ofcorrelator output z i (see Fig. 5 ) .

Detection o f Coherent F S K

FSK modulation is characterized by the information beingcontained in the frequency of the carrier wave. A typical set ofsignal waveforms is described by

s i ( t ) = d m c o s w i t ( fo rO <t<T ) =1,2, ..., M= O (otherwise)

where E is the energy content of si(t), and w ~ + ~a i s anintegral multiple of Z.rr/T. The most useful form for theorthonormal coordinates +l(t), &(t) , . . , +N(t) is

+j(t) = @cos wjt j = I , 2, . . ’ . Nand, from 2)

a , ., = j COS wit J2/ Tco s wjt dt.

Therefore

a i j = fl (for i = )

= O (otherwise)

In other words, the ith signal point is located on the ithcoordinate ax is at displacement *from the origin of thesignal space. Figure 7 illustrates the signal vectors (points)and the decision regions for a 3-ar y coherent FSK modulation( M = ). In this scheme, the distance between a ny two signalpoints s i and si is constant

d(s,sj) = s i - j = (for i )

As in the coherent SK case, the signal space is partitionedinto M distinct regions, each containing one prototype signalpoint. The optimum decision rule is to decide that thetransmitted signal elongs to the class whose index number isthe same as the region where the received signal was found. InFig. 7, a received signal point r is shown in region 2. Using thedecision rule, the detector classifies it as signal s2. Since the

‘

Fig. 8. Signal space or DPSK detection.. .

10



noise is a random vector, there is a probability greater thanzero that the location f r is due to some signal ther than s2

For example, if the transmitter sent 82, then r.is the sum ofs2 +n,, and the decision o choose s2 s correct; however, fthe transmitter actually sent 3, hen r must be the sum of s3 i

nb (see Fig. 7), nd the decision to select 2 is an error.

Detection of DPSKWith noncoherent systems, no rovision is made to phase-

synchronize the receiver ith the transmitter. Therefore, f thetransmitted wavefohn is

Si(t) = J r n c o s ( w ( + +i i = 1 , 2 , . . . , M

r t) = J m c o s wet + chi+ a + n(t)

where CY is unknown and is assumed to be randomlydistributed between zero and 27~.

For coherent etection, product integrators or theirequivalents) are used; for noncoherent detection, his practice

is generally inadequate because he output of a productintegra tor is a function of the unknown angle CY. owever, ifwe assume that CY varies slowly enough to be consideredconstant over two, period times ZT), the elative phasedifference between two successive waveforms is independentof a, hat is,

the received signal can be cha racterized by

This is the basis or DPSK modulation. The carrier phase fthe previous signaling interval s used s a phase reference ordemodulation. Its use requires differential encoding of themessage sequence at the transmitter since he information s

carried by the difference in phase between two successivewaveforms. To send the ith message (i 1,2, . . , M), thecurrent signal waveform must have its phase advanced byZrri/M radians over the previous waveform. The detectorcan then calculate the coordinates f the incoming signal byproduct-integrating it with the locally generated waveformsm c o s o t and m s i n gt. In this way it measures theangle between the current and the reviously received signalpoints (see Fig., 8) 191.

One way of viewing the difference between cohe rent PSKand DPSK is that the former compares the received signalwith a clean reference; n the latter however, wo noisy ignalsare compared with each other. Thus, we might say there is

twice as much noise in DPSK as in PSK. Consequently,DPSKmanifests a degradation f approximately 3 dB whencompared with PSK; this number decrea ses rapidly withincreasing signal-to-noise atio. In gene ral, the errors tend topropagate (to adjacent period times) due to the correlationbetween signaling waveforms. Th e trade-off or his per-formance loss is reduced system complexity.

Detection of Noncoherent F S K

A noncoherent FSK dete ctor can be implemented withcorrelators such as those shown in Fig. However, hehardware must be configured s an nergy detector, without

exploiting phase measurements. For his reason, it isimplemented with twice as ma ny channel branches a s thecoherent detector. Figure 9 illustrates the in-phase (1)channels and quadrature ( Q )channels used o detect thesignal se t noncoherently. Another possible implementation

uses filters followed by envelope detec tors; the detectors arematched o he signal envelopes and not to the signalsthemselves. The phase of the ca rrier is of no importance indefining +e envelope; hence, o phase information is used. Inthe case of binary FSK, the decision as to whether a “1” or a“0” as transmitted s made on the basis f which of the twoenvelope detectors has the largest amplitude t the moment ofmeasure ment. Similarly , for a multifrequency shift keying(M-ary FSK, or MFSK) system, the decision as to which ofthe M signals was transmitted s made on the basis f which ofthe M envelope detectors has maximum output.

Probability of Error

The calculations for probability of error (PE), which can beviewed geometrically (see Fig. 4), involve finding heprobability that given a particular signal, say s l , the noisevector n will give rise to a received signal falling outside region1 ; all PE calcu lations have this goal. For the general M-arysignaling case , the probability of making an incorrect decisionis termed the probability of symbol error , or simply (PE). It isoften convenient to pecify system performance by theprobability of bit erro r (PB), ven when decisions are madeon the basis f symbols for which > 1. PE nd PB are relatedas follows: For orthogonal signals [21],

pB pE = (2 k- ‘)/(2 - ) .

For nonorthogonal schemes, such as MPSK signaling, oneoften uses a binary-to-M-ary code such tha t binary sequencescorresponding to adjacent symbols (phase shifts) differ n onlyone bit position; one such code s the G ray code. When anM-ary symbol error occurs, t is more likely that only one ofthe k input bits will be in error. For such signals [3],

Ps P,/log,M = PE/k for PE<< I)

For convenience, this discussion is restricted to BPSK ( k =1 , M = 2) modulation. For the binary case, the symbol errorprobability equal s he bit error probability. Assume thatsignal s ( t ) has been transmitted and that (t) = s 1 t ) +n(t).

11



IEEE COMMUNICATIONS MAGAZINE.

Assuming qually likely signals, and recalling that thedecision of region 1 versus region 2 depends on the productintegrators and the decision stage (see ig. 5 ) , we can write

EJ binary= PB = Pr1 (t)sz(t) dt >

0

lo7 t)sI * ) dt r(t) = s I t) + n t)

for 0 < < T. The solution for he PBexpression can e shownto be

P B = I Jz;;j0

exp(-u2/2) du

E b / N o ( l-cos 6 )where Eb is the signal energy er bit in oules, N o is the noisedensity at the receiver in watts per Hz, and 0 is the anglebetween s and s z (see Fig. 4). When 8 = T he signals aretermed antipodal, and the PB becomes

t

/ - w

The same kind of analysis is pursued in finding the PBexpressions for the other types f modulation. The parameterE b / N o in ( 8 ) an be expressed as the ratio of average signalpower to average noise power, S / N (or SNR). By arbitrarily'introducing the baseband signal bandwidth , we can writethe following identities, showing the elationship between

/ N o and SNR

TABLE 1PROBABILITY OF BITERRORORSELECTEDINARYODULATIONCHEMES

.Modulation

Coherent PSK

Coherent FSK

~

whereE b Lergy/Bit

N o Noiseensity

S Signal Power

NOR Noise Density X Bit Rate_ - =

and Q x) = J-x:xp(. 2/2)d

where S = averagemodulating ignal powerT = bit time durationR = 1 / T = bit rateN = N O W

The dimensionless ratio E b / N o (required o achieve aspecified P a , isuniformly used for character izing digitalcommunicat ions system performance. Note hat optimumdigital signal detection mplies a correlator (or matched ilter)implementation, in which case the signal bandwidth is equalto the noise bandwidth. Often we are faced with a systemmodel for which this is not the case ( less than optimum); npractice, we ust reflect a factor into the required E b / N oparameter hat' ccounts for the uboptimal etection

performance. Therefore, required b / N ocan be considered ametric hat characterizes he performance of one systemversus another; the smaller the required / N O , the moreefficient the system modulation and detection process.

The PB expressions for the binary modulation schemesdiscussed above are listed n Table I and are graphicallycompared in Fig. 10. At large N Rs , it can be seen that thereis approximately a 4-d B difference between the best (cohe rentPSK) and the orst (noncoherent FSK). n some cases, dBis a small price o pay for the mplementat ion simplicitygained in going from a coherent PSK to a noncoherent FSK;however, for some applications, even a 1-dB saving isworthwhile. There a re other considerations besides PB and

system complexity; for example, in some cases (such asrandomly ading propagation conditions), a noncoherentsystem is more robust and desirable because there may bedifficulty in establishing a coherent reference.

An exception o Table I and Fig. 10 s worth mentioning, inlight of today's bandwidth efficient modulation schemes.MSK modulation, which can be regarded as coherent FSK,manifests rror-rate performance qual o BPSK whendetected with the appropriate receiver 181.

Digital Transmission Trade-offs

System trade-offs are fundamental to ll digital communi-cations designs. The goals of the designer are: (1) to

12



AUGUST 1983

maximize transmission bit rate R, 2) o minimize probabilityof bit error PB, 3) o minimize required power, or relatedly, ominimize required bit energy per noise density E b / N o ,(4) tominimize required system bandwidth W, 5 ) to maximizesystem utilization, that is, to provide reliable service for amaximum umber of users, with minimum delay andmaximum resistance to interference, and 6) tominimizesystem complexity, computational load , and system cost.The designer usually se eks o achieve all these goals.However, goals (1) and 2) are clearly in conflict with goals3) and 4); they call for simultaneously maximizing R, while

minimizing f , E b / N o ,and w. The re are everal constraintsand theoretical limitations that necessitate the trading-off ofany one requirement with each of the others. Some of theconstraints are: the Nyquist theoretical minimum bandwidthrequirement, the Shannon-Hartley capacity theorem, theShannon limit, government regulations (for example, fre-quency allocations) , technological limitations (for example,state-of -the-ar t omponents), and othe r ystem requirements

(for example, satellite orbits).

M-ary Signaling an d the Error-Rate Plane

Figure 11 a) illustrates the family of waterfall-like curvescharacterizing f~ versus , / N o for orthogonal signaling.Figure 11 b) illustrates similar urves for multiphase signaling[3]. A Sdescribed in the earlier section on “Digital ModulationFormats”, the signaling is called Mary for modulation orcoding schemes that process k bits at a time. The systemdirects the modulator to choose one of its M = z kwaveformsfor each k bit sequence, where Mis the symbol-set size, and kis the number of binary digits that each symbol represents.Figure 1 (a) illustrates potential PB improvement as k (orM) increases. For orthogonal signal sets, uch as FSKmodulation, M-ary signaling, compared o binary, canprovide an improved Ps performance or a reduced E b / N orequirement, at the cost of an increased bandwidth require-ment. Figure 1 (b ) illustrates potential PB degradation as k(or M) ncreases. For nonorthogonal signal sets, such asmultiphase shift keying MPSK) modulation, M-ary signaling,compared o binary, can provide a reduced bandwidthrequirement, a t the cost of a degraded-PB performance or anincreased , / N orequirement. Theappropriate Fig. 11 curve,from the amily of curves depicting system performance, is afunction of the system designer’s choice of the parameter k =

log2M. We shall refer to either of these curve families (Fig.11 a) or Fig. 11 b)) as error-rate performance curves, nd tothe plane upon which they are plotted as an error-rate plane.Such a plane describes the locus of operating points availab lefor a particular type of modulation and coding. For a givensystem, each curve in the plane can be associated with adifferent fixed bandwidth; therefore, the set,of curves can betermed equi-bandwidth curves. As the curves move in thedirection of the ordinate, he required bandwidth grows, untilit goes to infinity n the limit. As the curves move in theobposite direction, the required bandwidth decreases. Oncemodulation, coding scheme, nd available E b / N oare chosen,system operat ion is characte rized by a particular point in he

error-rate plane. Possible trade-offs can be iewed as changesin the operating point on one of the curves, or s changes inthe operating point from one curve to another curve of thefamily. Such potential trad e-of fs are een in Figs. 1 (a) and1 1(b) as chan ges in operating point in the direction shown bythe arrows. Movement of the operating point along line 1,between points a and b, can be viewed as trading PB versus

/ N o performan ce (with w fixed). Similarly, movementalong line 2, between points c and d, is seen as trading PBversus Wper formance with E b / N ofixed). Finally, movementalong line 3, between points e and f, illustrates trading Wversus E b / N operformance (with PB fixed). Movement alongline 1 is effected simply by increasing or decreasing theavailable E b / N o .Movement along ine 2 or line 3 is effectedthrough an appropri ate change to the ystem modulation or

, coding scheme.

The Nyquist and Shannon Constraints

Symbol detection in a realizable system, even in theabsence of noise, suffers from intersymbol interference, ISI;the tail of one pulse spills over into adjacent symbol intervalsso as to interfere with correct detection: Nyquist [22,23]showed that the theoretical minimum bandwidth needed totransmit x symbols per second (Symbols/s) without IS1 isx / 2 Hz; this s a basic theoretical constraint, limiting thedesigner’s goal to expend as little bandwidth as possible. Inpractice, it ypically equires x Hz bandwidth for thetransmission of x symbols/s. In other words, typical digitalcommunication throughput without IS1 is limited to 1symbol/s/Hz. For a fixed bandwidth, as k (and M) Increases,the bandwidth efficiency R/W, measured in b/s/Hz, in-

creases . For example, movement along ine 3, from point e opoint f, in Fig. 11 b) represents rading , / N ofor a reducedbandwidth requirement; in other words, with the same systembandwidth one can ransmit at an ncreased data rate, henceat an ncreased R/W.

13

HOGONAL SIGNALING

NORMALIZEDENERGYPER INFORMATION BIT$ I N ~ I ~ B I

a) b)

Fig. 11. The error-rate plane.



Fig. 12. Various bandwidth crfterfa.

Shannon [24] showed that he system capacity C , for

channels perturbed by AWCN, is a function of the averagereceived signal power S ; the aver age noise power N ; and thebandwidth W .The capacit y relationship (Shannon-Hartleytheorem) can be stated as:

It is possible to transmit information over such a channel at arate R, where R < C, with an arbitrarily small error rate byusing a sufficiently complicated coding scheme. For a rateR > C , it is not possible o find a code which can achie ve anarbitrarily small error rate. Shannon's work showed that the

values of S, N, nd W set a limit on transmission rate, not onaccuracy. It can also be shown, rom lo), hat the requiredE b / N o approaches the Shannon limit of -1.6 dB as Wincreases without bound. At the Shannon imit, shown in Fig.1 1 a), and PB curve is discontinuous, going rom a value ofPB = 1/2 to P B = 0. It is not possible to reach the Shannonlimit, because, as k increases without bound, the bandwidthrequirement and delay become infinite and the implementa-tion complexity increases without bound. Shannon's workpredicted the existence of codes that could improve the PBperformance or reduce the E b / N orequired from the levels ofthe uncoded binary modulation schemes up to the limitingcurve. For P B = BPSK modulationequires an

/No of 9.6 dB the optimum uncoded binary case).Shannon 's work therefore promised a theoretical performanceimprovement of 1 1.2 dB over the performance of optimumuncoded binary modulation, hrough he use of codingtechniques. Today, most of that promised improvement(approximately 7 dB) is realizable [25]. Optimum systemdesign can best be described as a search for rationalcompromises or trade-offs amongst the various constraintsand conflicting goals.

Bandwidth of Digital DataThe theor ems of Nyquist and Shannon, though concise

and fundamental , are based on the assumption of strictly

band-limited channels, which means that no signal powerwhatever is allowed outside the defined band. We are facedwith the dilemma that strictly band-limited signals are notrealizable since they imply infinite transmission-time delay;non-band-limited signals, having energy at arbitrarily highfrequencies, appear just as unreasonable [26]. t is no wonder

that there is no single universal, definition f bandwidth.All criteria of bandwidth have in common the at tempt to

specify a measure of the width W of a non-negative real-valued spectral density efined for all frequencies I f < m .

Figure 12 illustrates some of the most common efinitions ofbandwidth; in general, the various criteria are not inter-changable [27]. The power spectral density S f) for a singlepulse takes the analytical form

. ~~

IEEE COMMUNICATIONS MAGAZlNE

14

where f is the carrier frequency and is the symbol duration.This same spectral density, whose general appearance issketched in Fig. 12 , characterizes a sequence of randomdigital data, assuming the averaging time is long, relative tothe symbol duration 28]. The spect ral ensity plot consists ofa main lobe and smaller symmetrical ide lobes. The generalshape of the plot is valid for most digital modulat ion formats ;some formats, however, do ot have well defined lobes [28].The bandwidth criteria depicted in Fig. 12 are:

1) Half-Power Bandwidth: This is the nterval betweenfrequencies at which S f) has dropped to half ower, or3 dB below the peak value.

2) Equivalent Rectangular or Noise Equivalent Band-width: Th e noise equivalent bandwidth was originallyconceived to permit rapid computation of output noise-power from an amplifier with a wide-band noise input;the oncept an similarly e applied to a signalbandwidth. The noise equivalent bandwidth f a signalis defined as the value of bandwidth which satisfies therelationship P = WN S f,), where P is the total signalpower over all frequencies, WN is the noise equivalentbandwidth, and S fJ is the value of S f) at the bandcenter (assumed to be the maximum value over allfrequencies).

3) Null-to-Null Bandwidth: The most popular measure of

bandwidth is the width of the main spectral lobe, heremost of the signal power is contained. This criterionlacks omplete enerality ince some modulationformats lack well-defined lobes.

4) Fractional Power Containment andwidth: This band-width criterion has been dopted by the FederalCommunications Commission FCCRules and Regula-tions Section 2.202) and states that the occupiedbandwidth is the band which leaves exactly 0.5%of the signal power above the upper band limit andexactly 0.5% of the signal ower below the lower bandlimit. Thus, 99% of the signal power is nside theoccupied band.



5) Bounded Power Spectral Density: A popular methodof specifying bandwidth is to state that everywhereoutside the specified band S(f) must have fallen atleast o a certain stated level below hat ound atthe band center . Typical attenuation levels might be35 or 50 B.

The Bandwidth-Efficiency PlaneEquation 1 0) an be written as

E b / N o= w/c 2'Iw - . 1 1)

Equation 1 1) as been plotted o n the R/W versus E b / N oplane in Fig. 13. W e shall term this plane the bandwidth-efficiency plane. The ordinate R/W is a measure of howmuch data can'be ransmitted in a specified bandwidth withina given time; t therefore reflects how efficiently he bandwidthresource is utilized. The abscissa is / N o in decibels. ForC = R in 1 l ) , the plotted curve in the plane represents aboundary that sepa rate s arameter combinations supportingpotential error-free communication from regions where suchcommunication is not possible. Upon he bandwidth-efficiencyplane of Fig. 13 are plotted th e operat ing points for MPSKand MFSK modulation, e ach at PB = Notice that forMPSK modulation, R/W increases with increasing M ;however, for MFSK modulation, R / W.decreases withincreasing M. Notice also that the location of the MPSKpoints indicate that BPSK ( M 2) and QPSK M = 4)require the same b / N o .That is, for he same alue of , / N O ,QPSK has a bandwidth efficiency of 2 b/s/Hz, compared to1 b/s/Hz for BPSK. This unique feature stems from thefact hat QPSK is effectively a composite of two BPSK

REGION FOR

16 -CAPACITY. BOUNDARY

1p ~6 MPSK pB =

m MFSK noncoherent) pB= lo->

LIMITEDt REGION I

Fig. 13. The bandwidth-ettlciency plane.

signals, transmitte d on waveforms orthogonal to one anotherand having the same spectral ccupancy. This same featureis illustrated in Fig. 1 I b), where it can be seen that QPSK( k = 2) signaling has the same PB (not the same symbolerror rate) as does BPSK ( k = 1) signaling. Each of thetwo orthogonal BPSK signals comprising QPSK yieldshalf the bit rate and alf the signal power of the QPsK signal;hence the required / N o for a given PB s identical or BPSKand QPSK.Also plotted on the andwidth-efficiency plane ofFig. 13 are the operating points for noncoherent MFSKmodulation a t a BER of Notice tha t the position of theMFSK points indicates that binary FSK, BFSK ( M = ) andquarternary FSK (QFSK M = 4)) have the same band-width efficiency, even though the former requires greaterE b / N o for the same error rate. The bandwidth efficiencyvaries with the modulation index; if we assume that an equalincrement of bandwidth is required for each MFSK tone thesystem must support, t can be een that for M = 2, R/W1 b/s/2 Hz = 1/2; and or M = 4, similarly, R / W = 2 b/s/4Hz = 1/2.

The bandwidth-efficiency plane in Fig. 13 s analogous tothe error-rate plane shown in Fig. 11. The Shannon limit ofthe Fig. 1 1 plane is.analogous to the capacity boundary f theFig. 13 plane. The curves in Fig. 1 1 were referred to asequi-bandwidth curves. In Fig. 13, we can analogouslydescribe equi-error-probability curves for various modulationand coding schemes. The curves labeled F B I , PBn, and P B ~are hypothetical constructions for some arbitrary modulationand coding scheme; the PSI curve represents he largest errorprobability of the three curves, and the B 3 curv e representsthe smallest. The general direction in which the curves movefor improved P B is indicated on the figure.

Just as potential trade-offs amongst PB, , / N o , and wwere considered for the error-rate plane, o too we can viewthe same trade-offs on the bandwidth-efficiency plane. Suchpotential trade-offs ar e seen in Fig. 13 s changes in operatingpoint in the direction shown by the arrows. Movement of theoperating point along line 1 can be viewed as trading Psversus , / N o performance, with R/W fixed.Similarly,

' movement along line 2 is seen as trading P B versus W (orR / W )performance, with , / N ofixed. Finally, movementalong line 3 llustrates trading w or R / W )versus E b / N operformanc e, with PB fixed. n Fig. 13, as 'inFig. 11 ,

movement along line 1 is effected simply by increasing ordecreasing the available E b / N o .Movement along line 2 orline 3 s effected through appropriate changes o the systemmodulation or coding scheme.

Power-Limited Systems and Bandwidth-Limited Systems

For the case of power-limited systems, in which power isscarce but system bandwidth is available (for example, aspac e communication link), the following tradeoff s might bemade:

Improved P B performance can.. be achieved by ex-pending bandwidth (for a given E b / N o ) .

15



Required E b / N o can be reduced by expending band-width (for a given PB).

The error-rat e lane of Fig. 1 1a) s most useful for examiningsuch potential trade-offs. t is on this plane that we can verifywhether or not a candidate code offers improvement inrequired E b / N o (coding gain) for a specified PB (or whetherthe code offers improvement in PB for a given E i / N o ) .

Any digital scheme that transmits R = logzM bits in Tseconds, using a bandwidth of W Hz, always operates at abandwidth efficiency of R/W = (logZM)/ WTb/s/Hz. Fromthis expression, it can be seen that signals with small WTproducts are most bandwidth-efficient. Such ignals aregenerally associated ith bandwidth-limited systems in whichchannel bandwidth s constrained but ower is abundant. Forthis cas e, the usual objective’ is to design the link SO as tomaximize he ransmitted data rate over the band-limitedchannel, at the expense of E b / N o (while maintaining aspecified Ps performance level). For band-limited operation,bandwidth efficiency s a useful criterion of system per-formance, and the bandwidth-efficiency plane of Fig. ‘13 suseful for examining potential trade-offs, such as E b / N oforimproved R / W,or degraded PB for improved R / W.

The bandwidth-limited and power-limited regions areshown on the bandwidth-efficiency plane of Fig. 13. Noticethat the desirable trade-offs associated with each of theseregions are not equitable. For the bandwidth-limited region,large R/W is desired; however as E b / N o is continually

increased, the capacity boundary curve flattens out, andever-increasing amounts of , / N o are required to achieveimprovement in R / W.A similar law of nature seems o be atwork in the power-limited egion. Here, a savings in / N o isdesired, but the capacity boundary curve s steep; to achieve asmall relief in required E b / N o requires a large reduction in

R/W (increase in bandwidth for a given data rate).

Digital Communication Tradeoffs

Figure 14 has been configured for pointing out analogiesbetween the two performance planes, the error-rate plane ofFig. 11 , and he bandwidth-efficiency plane of Fig. 13.Figures 1 4( a) and 14(b) represent ‘the same planes as Figs.1 1nd 13, espectively. They have een redrawn, purposelysymmetrical, by choosing appropriat e scales. The arrows andtheir labels, in each case, describe he general effect ofmoving an operating point in the direction of the arrow bymeans of appropriat e modulation and coding techniques . Thenotations C, C, and F stand for the trade-off considerat ions“Gained or achieved,” “Cost or expended,” and “Fixed orunchanged,’’ respectively. The parameters being traded arePB,W, R / W,nd P(power or S/N). Just as the movement fan operating point toward the Shannon limit in Fig. 14 (a )gains improved P B or lower transmit ter power a t the cost ofbandwidth, so toodoes movement oward he capacityboundary in Fig. 14 (b) gain improved bandwidth efficiency atthe cost of increased power or degraded P B .

16



~ ~ ~ ~

AUGUST 1983

Most often, such trade-offs are examined with a fixed P B

(constrained by the system requirement) in mind. Therefore ,the most interesting arrows are those having fixed bit-errorprobability (marked F: P B ) .There ar e four such arrows n Fig.,l4, wo on the error-rate plane and two on the bandwidth-efficiency plane. System operation can be characterized byeither of these two planes. The planes represent two ways oflooking at some of the key system par ameter s; each planehighlights lightly different aspects of the overall designproblem. The error-rate plane tends to find most use withpower-limited systems; here, as we move from curve to curve,the bandwidth requirements are only inferred, but the PB isclearly displayed. The bandwidth-efficiency plane is generallymore useful for examining bandwidth-limited systems; here,as we move from curve to curve, P B is only nferred, butthe bandwidth requirements are explicit, since the ordinateis RIW.

Additional Constraints

We are not a s free to make trade-offs as we might like;Government regulations dictate choice of frequencies,bandwidths, transmission power levels, and- in the case o f

satellites, orbit selection. The satellite orbit and geometry ofcoverage fixes the satellite antenna gain. Technologicalstate-of-the-art onstrains such items as satellite power rans-mission and earth station antenna gain. There may e othersystem requirements (for example, the need to operate underscintillation or interference conditions) that can nfluence thechoice of modulation and coding. The effect of theseadditional constraints3 to limit the regions of realizable

.operation within the error-rate plane and the bandwidth-efficiency plane.

ConclusionIn the first part of this paper , we have generated a structure

and hierarchy of key signal processing transformations. Wehave used this structure as a guide for overviewing theformatting, source coding, an d modulation steps. We havealso examined potential trade-offs for power-limited systemsand bandwidth-limited systems. In Pa rt I1we will continue toexamine the remainder f the signal processing steps outlinedin Figs. 1 and 2. Also in Part 11, we will review fundamentallink analysis relationships in the context f a satellite repeaterchannel.

References

[ l ] W.L. Pritchard , “Satellite comm unication- an overview of theproblems and program s,” Proc.I E E E ,vol. 65 , pp. 294-3 07, March1977.

[2] M.P. Ristenbatt, “Alternatives in digital communications,” Proc.IEEE, June 1973.

[3] W.C. Lindsey, and M.K. Simon, Telecommunication SystemsEngineering, Englewood Cliffs, NJ: Prentice-Hall, 19 73 .

[4] J.J. Sp ilker, Jr., Digital Com munic ations by Satellite, EnglewoodCliffs, NJ: Prentice-Hall, 197 7.

[5] V.K. Bharg ava, D. Haccoun , R. Matya s, and P.P. Nuspl, DigitalCommunications bg Sate llite, New York, NY: John Wiley and Sons,1981.

[6] C.B. Feldm an’a nd W.R. Bennett, “Band width and transmissionperformance,”Bell Syst. Tech.J . , vol. 28, appendix V, pp. 594-5 95,1949.

[7] A. Lender, “The duobinary technique for high speed data transmis-sion,” IEEE Trans. Commun. Electron., vol. 82 , pp. 21 4-2 18, May1963.

[8] E.R. Kretzm er, “Generalizatio n of a techniq ue for binary dat acommunication,” I E E E Trans. Commun. Tech., pp. 67 , 68 , Feb.1966.

[9] S. Pasupa thy, “Correlative coding: a bandwidth-efficient signalingscheme,” IEEE Communications Magazine, pp. 4-11, July 1977.

[ 101 N.S. Jaya nt, “Digital coding of speech waveforms: PCM , DPCM, and

DM quantizers,” Proc.IEEE,vol. 6 2, no.5, pp. 61 1-632 , May 1974.[ 111 B.S. Atal and S.L. H anover , “Speech analysis and synthesis by linear

prediction of the speech wave,”J. Acoust. SOC. m., vol. 50, pp.637-655 , 1971 .

[121 J.W . Bayless, S.J. Campanella, andA.J. Coldberg, “Voice signals: bitby bit,” I E E E Spectrum, vol. 10, pp. 28-39, October 1973.

[13] J.L. F lanagan , Speech Analysis, Synthesis, and Perception, 2nd ed.,New York, NY: Springer-Verlag, 1972.

[14] R.E. Crochiere and J.L. Flanagan, “Current perspectives in digitalspeech,”I E E E Communications Magazine, vol. 21 , no. 1, pp. 32- 40 ,January 1983.

[15] D. Huffman,“A method for constructing minimum redundancycodes,” Proc. IRE, vol. 40, pp. 1098-11 01, May 1952.

[IS] R.C. Callager, Information Theory and Reliable Communication,New York, NY: J ohn Wiley and Sons Inc.,1968.

[17] S.A. Groae meye r nd A.L. McBride, “MSK and offset QPS Kmodulation,”IEEE Trans. Commun., August 1976.

[18] S. Pasupa thy, “Minimum Shift Keying: a spectrally efficient modula-tion,” IEEE Communications Magazine, pp. 14-22, July 1979.[ 191 E. Arthurs and H. ym, “On the optimum detection of digital signals in

the presence of white gaussian noise-a geom etric interpretatio nofthree basic data transmission systems,”IRE Trans. Commun. Syst.,December 1962.

1201 H.L. Van Trees. Detection. Estimation, and M odulation Theory-. - - -Part I , New York, NY: John Wiley and Sons,1968.

1211 A .J. Viterbi, Principlesof Coherent Communications, New York, NY:McCraw-Hill Book Co., 1966.

[22] H. yquist, “Certain factors affecting telegraph speed,” Bell Syst.Tech. J . , vol. 3, pp. 324- 326, April 19 24.

[23]H. Nyquist, “Certain topics on telegraph transmission theory,”Trans. AIEE, vol. 47,pp. 617- 644, April i928.

[24] C.E. Shannon, “Amathematical theoryof communication,” Bell Syst.Tech. J . , vol. 27, pp. 379-423 and 623-65 6,1948. ’

1251 J.P. Odenwalder, Error Control Coding Handbook, San Diego, CA:Linkabit Corporation, July 15, 1976.

[26] D. Slepian, “On bandwidth,”Proc.EEE,vol. 64 , no.3, pp. 292-300,March 1976.

[27] R.A. Scholtz, “How do you define bandwidth,” Proc. Int. TelemeteringConf., Los Angeles, C A, vol.8 pp. 281-288, October 1972.

[28] F. A moroso,The bandwidth of digital data signals,” I E E ECommunications Magazine, vol. 1 8, no. 6 , pp. 1 3-24,Nove mber1980.

Bernard Sklar was born in’New York, NY on September 11, 192 7. Hereceived theB.S. degree in mathematics and.sciencerom the UniversityofMichigan, AnnArbo r, MI, in 194 9; the M.S.E.E. degree from thePolytechnic Instituteof New York, Brooklyn, NY, in 19 58 ; and the Ph.D.degree in engineering from the University of California, Los Apgeles, CA , in1971.

Dr. Sklar has30 yea rs of experienc e with the aerospac e/defense industryin a variety of technical design and manag emen t positions: from1953 to1 9 5 8he was a research enginee r with RepublicAviation Corp., Farmingdale,NY; from 19 58 o 1959 he was a mem ber of the technical staff at Hug hesAircraft Co., Culver City, CA; and rom 19 59 to 19 68e was a sen ior staffengineer at Litton System s, Inc., Canog a Park, CA. n 1 96 8 he joined Th eAero space Corp., El Segund o, CA , where he s currently employed.AsMan ager of System Analysis, heis involved in the develop ment of satellitecommunication systems.

H ehas augh t engineering courses during the past 2 5 yyars at theUniver sity of C alifornia,Los Angeles and lrvine: the Universityof SouthernCalifornia, Los Angeles; and We stCoast U niversity,L o s Angeles. Dr. Sklaris a past chairmanof the Los Angeles Council IEEE Education Committee,andas been a Senior Member of the IEEE since 195 8. rn

17.

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A Structured Overview of Digital Communications--A Tutorial Review--Part I

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