Strictly speaking, the AVGN channeldoes not exist since no channelcan havean
`___
`infinite bandwidth. However,
`;
`Whenthe signal bandwidth is smaller than the channel
`bandwidth, many practical channels are approximately an AWGN channel. For ex-
`ample, the line-of-sight (LOS)radio channels, including fixedterrestrial microwave
`links and fixedsatellite links, are approximately AWGNchannels when the weather
`is good. Wideband coaxialcables are also approximately AWGNchannelssincethere
`is no other interference except the Gaussian noise.
`In this book, all modulation Schemes are studied for the AWGN channel. The
`reason of doing this is two-fold. First, some channels are approximately an AWGN
`channel, the results can be used directly. Second, additive Gaussian noise is ever
`presentregardless ofwhether other channel impairments such as limited bandwidth,
`fading, multipath, and other interferences exist or not. Thus the AWGNchannelis the
`best channelthat one can get. The performanceofa modulation scheme evaluatedin
`this channel is an upper boundon the performance. Whenother channel impairments
`exist, the system performancewill degrade. The extent ofdegradation may vary for
`different modulation schemes. The performance in AWGNcan serveas a standard
`in evaluating the degradation and also in evaluating effectiveness of impairment-
`combatting techniques.
`
`1.2.2
`
`Bandlimited Channel
`
`Whenthe channel bandwidth is smaller than the signal bandwidth, the channelis
`bandlimited. Severe bandwidth limitation causes intersymbolinterference (ISI) (i.c.,
`digital pulses will extend beyondtheir transmission duration (symbolperiod T,)) and
`interfere with the next symbol or even more symbols. The ISI causes an increase
`in the bit error probability (P,) or bit error rate (BER), as it is commonlycalled.
`Whenincreasing the channel bandwidth is impossible or notcost-efficient, channel
`equalization techniques are used for combatting ISI. Throughoutthe years, numerous
`equalization techniques have been invented and used. New equalization techniques
`are appearing continuously. We will not cover them in this book. For introductory
`treatmentofequalization techniques, the readeris referredto (1, Chapter 6] or any other
`communication systems books.
`
`Qualcomm Incorporated
`Exhibit 1009
`Page 1 of 58
`
`Qualcomm Incorporated
`Exhibit 1009
`Page 1 of 58
`
`

`

`individual multipath signals.
`In mobile communication channels, such as terrestrial mobile channeland satel-
`lite mobile channel, fading and multipath interference are caused by reflections from
`surrounding buildings and terrains.
`In addition, the relative motion between the
`transmitter and receiver results in random frequency modulation in the signal due
`to different Doppler shifts on each of the multipath components. The motion of
`surrounding objects, such as vehicles, also induces a time-varying Dopplershift on
`multipath component. However,if the surrounding objects moveat a speed less than
`the mobile unit, their effect can be ignored [2].
`Fading and multipath interference also exist in fixed LOS microwavelinks [3].
`On clear, calm summer evenings, normalatmospheric turbulence is minimal. The
`troposphere stratifies with inhomogeneous temperature and moisture distributions.
`Layering of the lower atmosphere creates sharp refractive index gradients which in
`turn create multiple signal paths with different relative amplitudes and delays.
`Fading causes amplitude fluctuations and phase variations in received signals.
`Multipath causes intersymbolinterference. Doppler shift causes carrier frequency
`drift and signal bandwidth spread. All these lead to performances degradation of
`modulations. Analysis of modulation performancesin fading channels is given in
`Chapter 10 where characteristics of fading channels will be discussed in more detail.
`
`13.
`
`BASIC MODULATION METHODS
`
`Digital modulation is a process that impresses a digital symbol onto a signal suitable
`for transmission. For short distance transmissions, baseband modulation is usually
`used, Baseband modulation is often called line coding. A sequenceof digital sym-
`bols are used to create a square pulse waveform with certain features which represent
`each type of symbol without ambiguity so that they can be recovered upon reception.
`These features are variations of pulse amplitude, pulse width, and pulse position.
`Figure 1.3 showsseveral baseband modulation waveforms. The first one is the non-
`return to zero-level (NRZ-L) modulation which represents a symbol| by a positive
`
`Page 2 of 58
`
`Page 2 of 58
`
`

`

`A
`
`(c) Bi-®-L (Manchester)
`A
`
`-A
`
`Figure 1.3. Baseband digital modulation examples.
`
`square pulse with length T and a symbol 0 by a negative square pulse with length T.
`The second oneis the unipolar return to zero modulation with a positive pulse of T/2
`for symbol | and nothing for 0. The third is the biphase level or Manchester, after
`its inventor, modulation which uses a waveform consisting of a positive first-half T
`pulse and a negative second-half T pulse for | and a reversed waveform for 0. These
`and other baseband schemeswill be discussed in detail in Chapter 2.
`For long distance and wireless transmissions, bandpass modulation is usually
`used. Bandpass modulation is also called carrier modulation. A sequence of dig-
`ital symbols are used to alter the parameters of a high-frequency sinusoidal signal
`called carrier.
`It is well known that a sinusoidal signal has three parameters: am-
`plitude, frequency, and phase. Thus amplitude modulation, frequency modulation,
`and phase modulation are the three basic modulation methods in passband modula-
`tion. Figure 1.4 showsthree basic binary carrier modulations. They are amplitude
`shift keying (ASK), frequency shift keying (FSK), and phase shift keying (PSK). In
`ASK,the modulator puts out a burst of carrier for every symbol 1, and nosignal
`for every symbol 0. This schemeis also called on-off keying (OOK). In a general
`ASKscheme, the amplitude for symbol0 is not necessarily 0.
`In FSK, for symbol
`| a higher frequency burst is transmitted and for symbol 0 a lower frequency burst
`
`Page 3 of 58
`
`q .
`
`—_—
`
`==a
`
`Page 3 of 58
`
`

`

`FSK
`
`PSK
`
`Figure 1.4 Three basic bandpass modulation schemes.
`
`In PSK, a symbol1 is transmitted as a burst of carrier
`is transmitted, or vice versa.
`with 0 initial phase while a symbol0 is transmitted as a burst of carrier with 180°
`initial phase.
`Based on these three basic schemes, a variety of modulation schemescan be de-
`rived from their combinations. For example, by combining two binary PSK (BPSK)
`signals with orthogonal carriers a new schemecalled quadrature phaseshift keying
`(QPSK)can be generated. By modulating both amplitude and phaseofthe carrier,
`we can obtain a schemecalled quadrature amplitude modulation (QAM),etc.
`
`1.4
`
`CRITERIA OF CHOOSING MODULATION SCHEMES
`
`The essenceofdigital modem designisto efficiently transmitdigital bits and recover
`them from corruptions from the noise and other channel impairments. There are
`three primary criteria of choosing modulation schemes: powerefficiency, bandwidth
`
`Page 4 of 58
`
`Page 4 of 58
`
`

`

`n= |
`
`
`/2E,
`
`(1.10)
`
`where E,is the averagebit energy, N,is the noise powerspectral density (PSD), and
`(Q(z) is the Gaussian integral, sometimes referred to as the Q-function.It is defined
`as
`
`(1.1)
`Q(x) = Le edu
`whichis a monotonically decreasing function of x. Therefore the power efficiency
`of a modulation scheme is defined straightforwardly as the required E,/No for 4
`certain bit error probability (P,) over an AWGN channel. P, = 10~°is usually used
`as the referencebit error probability.
`
`1.4.2
`Bandwidth Efficiency
`The determination of bandwidth efficiency is a bit more complex. The bandwidth
`efficiency is defined as the numberof bits per second that can be transmitted in
`one Hertz ofsystem bandwidth. Obviouslyit depends on the requirementofsystem
`bandwidthfor a certain modulated Signal. For example, the one-sided power spectral
`density ofan ASKsignal modulated by an equiprobable independentrandom binary
`sequenceis given by
`
`2
`
`
`2
`
`WA) = TPsinc? iri — 4.914 acy — 4.
`andis shown in Figure 1.5, where T is the bit duration, A is the carrier amplitude,
`and fc is the carrier frequency. From the figure we can see that the signal spectrum
`stretches from —oo to oo. Thus to perfectly transmit the signal an infinite syste™
`bandwidth is required, whichis impractical. Thepractical system bandwidth require-
`mentis finite, which varies depending on different criteria, For example, in Figure
`1.5, most of the signal energy concentrates in the band between twonulls, thus 4
`null-to-null bandwidth requirement seems adequate. Three bandwidth efficiencies
`
`Page 5 of 58
`
`
`
`er
`
`| i
`
`e4)
`
`Page 5 of 58
`
`

`

`
`
`Figure 1.5
`
`Powerspectral density ofASK.
`
`used in the literature are as follows:
`Nyquist Bandwidth Efficiency—Assumingthe system uses Nyquist (ideal rec-
`tangular) filtering at baseband, which has the minimum bandwidth required for in-
`tersymbolinterference-free transmission ofdigital signals, then the sedate
`baseband is 0.5.R,, R, is the symbolrate, and the bandwidth at carrier frequency
`is W = R,. Since R, = R,/logg M, Ry = bitrate, for M-ary modulation, the
`bandwidth efficiency is
`
`(1.12)
`Ry/W = logs M
`Null-to-Null Bandwidth Efficieney—For modulation schemesthat have power
`density spectral nulls such as the one of ASKin Figure 1.5, defining the bandwidth
`as the width of the main spectral lobe is a convenient way of bandwidth definition.
`Percentage Bandwidth Efficiency—If the spectrum of the modulated signal
`does not have nulls, as in general continuous phase modulation (CPM), null-to-null
`bandwidth no longer exists. In this case, energy percentage bandwidth may beused.
`Usually 99% is used, even thoughotherpercentages(¢..,ila
`
`System Complexity
`1.4.3
`System complexity refers to the amountofcircuits involved and the technical dif-
`ficulty of the system. Associated with the system complexity is the cost of manu-
`
`Page 6 of 58
`
`Page 6 of 58
`
`

`

`tations and
`
`de
`
`ve-
`
`them in the analysis ofModulation techniquesin therest
`ddescriptive
`OVERVIEWoFDIGITALMODULATIONS
`Toprovidethereaderwith anoverview, welist2iantable 1.1 andsueioes
`namesofvariousdigitalmodulationsthat wewill c oftheschemescanberae
`theminarelationshiptreediagraminFigure 1.6. Some
`differentialencoding
`frommorethanone“parent” scheme. TheschemesSeamanSeal
`beusedarelabeledbyletterDandthosethatcanaedemodulated.
`2
`into
`two
`labeledWithaletterN.Allschemescanbecoheren hetreeareclassifiedinto
`:
`ThemodulationSchemeslistedinthetableandhiatal pehsenechsaasiaa
`large categories: Constant envelope and Seaens and CPM. Under nenconstan
`velope class, there are three subclasses: FSK, PSK,
`envelope Class, t
`here are three subclasses: ASK, QAM,and oth
`lope modulations.
`Amongthelisted schemes, ASK, PSK, and FSK are ba
`d
`;
`sic modulations, an
`MSK, GMS
`K, CPM, MHPM d QAM, etc. are advanced schemes
`. The advanced
`schemesareVariationsandcombinationsoeRoadfecommunicationgens
`‘
`.
`,an
`»
`ete.
`ic
`schemes.
`ie
`onDowieanneaeoperate in thepeehaa Anexampleisthe
`class is generally
`:
`ion
`ofthe inpu'
`|
`A
`characteristicinordertoachieveeeSecereitiine gia
`eeemeeeaginareinappropriateaeee BinaryFSK
`i
`lifier)
`in sa’
`:
`licationsi
`eeverylowbandwidthefficiencyincomparisonoecellularsystems,AMPS
`isusedinthelow-ratecontrolchannelsoffirstCS(Europeantotalaccesseral
`(advancemobilephoneserviceofUS.)andoe AMPSand 8 Kbps for ETACS.
`nication System). The data rates are 10 Kbps Ga and MSKhave been used in
`ThePSKschemes, includingBPSK,QPSK,OQPSK,
`waren
`Satellitecommunicationsystems.
`ion
`duetoitsabilitytoavoi As a
`thatdemodulation, Ithasbeen us
`-
`i
`special attention a
`ed in digitalmobi
`P
`cellular systems, such as the United States digital cellular (U
`ee?
`
`er nonconstant en
`
`1.5
`
`SDC) system.
`
`Page 7 of 58
`
`Page 7 of 58
`
`

`

`
`
`2/2/H3\5)83)3|7|4/8|* .
`
`,
`
`-
`
`
` =é2
`|S‘ 77/4 Quadrature Phase Shift Keying
`
`ORKSsfo—|M-ary Phase Shift Keying
`
`
`Continuous Phase Modulations (CPM)
`SPM]Sileh(modulationindex)PhaseModulation
`[MHPM|_| _Multi-h Phase Modulation
`PLREC|~——S~S=S-sRectanguilar Pullse of Length L
`ceESK[|ominuosPhaseFrequeneySRKEying
`PMsk[FESR]MinimoShitKeying,FastFSK
`rSMSK|—~——__| Serial Minimum Shift Keying
`
`
`
`PERC|__|RaisedCosinePulseofLength
`
`
`
`PESRC|__|SpectrallyRaisedCosinePulseofLength|
`PoMsk||asstnMinimumSuiKeying
`
`
`
`
`
`PTFM|__|TamedFrequencyModulation
`
`Amplitude and Amplitude/Phase modulations
`
`
`ASK[AmplitudeshitKeyinggenericname)|
`
`
`Pook| ASK
`BeyOnOiKeying
`
`
`/MASK|MAM_| Mary ASK, M-ary Amplitude Modulation
`Fam|__|QuadratureAmplitudeModulation|
`
`
`Nonconstant Envelope Modulations
`
`
`
`Sea Quadrature Overlapped Raised Cosine Modulation
`egEaa 8zBQO
`
`
`adrature Overlapped Squared Raised Cosine Modulation
`2
`ture Quadrature Phase Shift Keying
`
`
`Intersymbol-Interference/Jitter-Free OQPSK
`
`|__| Two-Symbol-Interval OQPSK
`
` Superposed-QAM
`
`
`
`
`
`
`Crosscorrelated QPSK
`
`Table 1.1 Digital modulation schemes (Abbr=Abbreviation).
`
`Page 8 of 58
`
`Page 8 of 58
`
`

`

`
`
`x] se] [om|BooneGx] few]
`
`
`
`
`
`BPSK
`(D)
`
`MPSK.
`(D)
`
`OOK
`(N)
`
`(N)
`
`QPSK
`
`anre)
`
`[Sar
`
`Logs]
`
`71/4-QPSK
`
`LRC
`
`pee
`
`|GMsK|
`
`1REC
`(CPFSK
`
`} h-0.5
`' Sinusoidal
`sneeeoesce
`
`| pulse-shaping
`
`h=0.5 —>
`
`OORC
`
`8avo)4
`
`38 é
`
`z
`
`53)|
`
`Solid lines indicate “can be derived from”
`
`Dashedlines indicate “alternatively can be derived from”
`
`Can be differentially encoded and decoded
`
`Can be noncoherently detected
`
`Figure 1.6 Digital Modulation Tree. After [4].
`
`Page 9 of 58
`
`BFSK
`(N)
`
`''''''''''' +
`
`:
`'
`
`:
`
`''''''!''''
`
`Page 9 of 58
`
`

`

`bandwidth efficiency. Its modulator and demodulator are also not too complex. MSK
`has been used in NASA's Advanced Communication Technology Satellite (ACTS).
`GMSK has a Gaussian frequency pulse. Thus it can achieve even better bandwidth
`efficiency than MSK. GMSKis used in the UScellular digital packet data (CDPD)
`system and European GSM (global system for mobile communication) system.
`MHPMis worth special attention since it has better error performance than
`single-h CPM by cyclically varying the modulation index h.
`The generic nonconstant envelope schemes, such as ASK and QAM,are gen-
`erally not suitable for systems with nonlinear power amplifiers. However QAM,
`with a large signal constellation, can achieve extremely high bandwidth efficiency.
`QAM has been widely used in modemsused in telephone networks, such as computer
`modems. QAM can even be considered for satellite systems. In this case, however,
`back-off in TWTA’s input and output power must be provided to ensurethelinearity
`of the power amplifier.
`The third class under nonconstant envelope modulation includes quite a few
`schemes. These are primarily designed for satellite applications since they have very
`good bandwidth efficiency and the amplitude variation is minimal. All of them ex-
`cept Q?PSKare based on 27, amplitude pulse shaping and their modulatorstructures
`are similar to that of OQPSK. The scheme Q*PSKis based on four orthogonal car-
`riers.
`
`References
`
`[1]
`
`[2]
`
`[3]
`
`[4]
`
`Proakis, J., Digital Communication, New York: McGraw-Hill, 1983.
`
`Rappaport, T., Wireless Communications: Principles and Practice, Upper Saddle River, New
`Jersey; Prentice Hall, 1996.
`
` Siller, C., “Multipath propagation,” JEEE Communications Magazine, vol. 22, no.2, Feb. 1984,
`pp. 6-15.
`
`Xiong, F,“Modem techniques in satellite communications,” JEEE Communications Magazine,
`vol. 32, no.8, August 1994, pp. 84-98.
`
`Page 10 of 58
`
`Page 10 of 58
`
`

`

`
`It is expected that the error performance ofthe noncoherentreceiversis
`to that ofthe coherent ones. However, the degradation is only a fractic
`
`COae eee ane cae error probebilicies arealin Salenaaaa
`Finally we explored other possible demodulations. The d
`lator is simple and efficient. It is oven better then the noncobllrallad
`
`i yi
`
`References
`
`Anderson, R. R., and J. Salz, “Spectra ofdigital FM," Bell System TechnicalJo
`July-August, 1965, pp.1165-1189.
`
`Tjhung, T. T., and PH. Wittke, “Carrier transmission of binary data in a rest
`Trans. Comm. Tech., vol. 18, no. 4, August 1970, pp. 295-304.
`
`Pawula, R. F, “On the theory oferror rates for narrow-band digital FM,” EEE 7hans.
`
`[2]
`
`B)
`
`[4]
`
`vol. 29, no. 11, Nov. 1981, pp. 1634-1643.
`
`7
`
`Selected Bibliography
`
`ulator for BFSK.
`
`Mazo, J. E., “Theory oferror rates foe digital FM,” BellSystem Technical
`1966, pp. 1511-1535.
`
`
`
`Couch II, L. W, DigitalandAnalog Communication Systems, 3rd Ed., NewYork: Macmti
`Haykin, S., DigitalCommunications, New York: John Wiley & Sons, Inc., 1988.
`Salz, J., “Performance of multilevel narrow-band FM digital communication systems,” |
`Trans. Comm. Tech., vol. 13, no.4, Dec. 1975, pp. 420-424.
`“—
`
`Sklar, B., DigitalCommunications, FundamentalsandApplications, EnglewoodCli
`:
`Prentice Hall, 1988.
`
`Smith, D. R., Digital Transmission Systems, Second Edition, New York: Van Nostrand Reinh
`1993.
`
`Sunde, E. D., “Ideal binary pulse transmission by AM and FM,” Bell System
`vol. 38, Nov, 1959,pp. 1357-1426.
`Van Trees, H, L., Detection, Estimation, and Modulation Theory, Part I, New York:
`Sons, Inc., 1968.
`.
`
`Jo
`
`
`
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`
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`

`

`Phase shift keying (PSK)is a large class of digital modulation schemes. PSK is
`widely used in the communication industry. In this chapter we study each PSK mod-
`ulation schemein a single section where signal description, power spectral density,
`modulator/demodulator block diagrams, and receiver error performanceareall in-
`cluded. First we present coherent binary PSK(BPSK) and its noncoherent coun-
`terpart, differential BPSK (DBPSK), in Sections 4.1 and 4.2. Then we discuss in
`Section 4.3 M-ary PSK (MPSK)andits PSD in Section 4.4. The noncoherentver-
`sion, differential MPSK (DMPSK)istreated in Section 4.5. We discussin great detail
`quadrature PSK (QPSK) and differential QPSK (DQPSK)in Sections4.6 and 4.7,re-
`spectively. Section 4.8 is a brief discussion of offset QPSK (OQPSK). An important
`variation of QPSK, the 7/4—DQPSK which has been designated as the American
`standard of the second-generation cellular mobile communications, is given in Sec-
`tion 4.9. Section 4.10 is devoted to carrier and clock recovery. Finally, we summarize
`the chapter with Section 4.11.
`
`4.1
`
`BINARY PSK
`
`Binary data are represented by twosignals with different phases in BPSK. Typically
`these two phasesare (0) and 7, the signals are
`
`s(t) = Acos2zf.t, O<t<T,
`8o(t) = —Acos2rf.t, O0<t<T,
`
`for!
`for0
`
`(4.1)
`
`These signals are called antipodal. The reason that they are chosenis that they have
`a correlation coefficient of —1, which leads to the minimumerrorprobability for the
`same E/N, as we will see shortly. These two signals have the same frequency and
`energy.
`As wewill see in later sections, all PSK signals can be graphically represented
`
`123
`
`Page 12 of 58
`
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`
`

`

`(0)
`
`Figure 4.1 BPSK signal constellation.
`
`by a signal constellation in a two-dimensional coordinate system with
`¢,(t) = 1/2cos2mft, O<t<T
`
`and
`
`¢(t) = ~\/2sin @nf.t; OS¢tST
`
`(4.2)
`
`(4.3)
`
`as its horizontal andvertical axis, respectively. Note that we deliberately add a minus
`Sign in @9(t) so that PSKsignalexpressionswill be a sum insteadofa difference (see
`(4.14)), Many othersignals, especially QAMsignals, can also be represented in the
`same way. Therefore we introduce the signal constellation of BPSK here as shown
`in Figure 4.1 where s;(t) and s2(t) are represented by two points on the horizontal
`axis, respectively, where
`
`A°T
`2
`
`The waveform of a BPSKsignal generated by the modulator in Figure 4.3 for a
`data stream {10110} is shownin Figure 4.2. The waveform hasa constantenvelope
`like FSK.Its frequency is constant too.
`In general the phase is not continuousat
`
`Page 13 of 58
`
`Page 13 of 58
`
`

`

`(a) f, = 2/T
`
`(b) f. = 1.8/T
`
`Figure 4.2 BPSK waveforms.
`
`If the f; = m Ry = m/T, where m is an integer and Ryis the
`bit boundaries.
`data bit rate, and the bit timing is synchronouswith thecarrier, then the initial phase
`at a bit boundary is either 0 or 7 (Figure 4.2(a)), corresponding to data bit 1 or 0.
`However,if the f. is not an integer multiple of R,, the initial phase at a bit boundary
`is neither 0 nor x (Figure 4.2(b)). In other words, the modulated signals are not the
`onesgiven in (4.1). We will show next in discussion of demodulationthat condition
`f. = m Rg is necessary to ensure minimumbit error probability. However, if f, >>
`R,, this condition can be relaxed and the resultant BER performance degradationis
`negligible.'
`The modulator which generates the BPSKsignalis quite simple (Figure 4.3 (a)).
`First a bipolar data stream a(t) is formed from the binary data stream
`
`Cc
`a(t)= S~ agp(t — kT)
`k=—co
`
`(4.4)
`
`Thisis true for all PSK schemes and PSK-derived schemes, including QPSK, MSK, and MPSK. We
`will not mention this again when wediscuss other PSK schemes.
`
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`
`

`

`Polar NRZ source a(t) Aa(t)cos2nft
`
`
`
` Acos2nf.t
`
`
`(b)
`
`Figure 4.3 BPSK modulator (a), and coherent BPSK demodulator (b).
`
`where a,x € {+1,~1}, p(t) is the rectangular pulse with unit amplitude defined 0”
`(0,7). Then a(t) is multiplied with a sinusoidal carrier A cos 27fet. The result is
`the BPSKsignal
`
`s(t) = Aa(t)cos27f.t, —co<t< oo
`
`(4.5)
`
`?
`Notethatthe bit timing is not necessarily synchronous with the carrier.
`The coherent demodulator of BPSKfalls in the class ofcoherentdetectors for bi-
`nary signals as described in Appendix B. The coherentdetector could be in the form
`ofa correlator or matchedfilter. The correlator’s reference signalis the difference sig-
`nal (s4(t) = 2A cos 27f,t). Figure 4.3(b) is the coherentreceiver using a correlator
`where the reference signal is the scaled-down version of the difference signal. The
`reference signal must be synchronousto the received signal in frequency and phase.
`
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`

`

`kT
`
`= i
`Qx cos” 2af.tdt
`3
`a,(1 + cos 4zf,t)dt
`= ae ca na [sin4af.(k + 1)T — sin4xf.kT]
`
`kT
`
`If f. = m Rp, the second term is zero, thus the original signal a(t) is perfectly
`recovered (in the absence of noise). If f, + m Ry, the second term will not be zero.
`However, as long as f. >> Ry, the second term is much smaller than thefirst term
`so that its effect is negligible.
`The bit error probability can be derived from the formula for general binary
`signals (Appendix B):
`
`P=Q
`
`EB, + Eo — 2pyoVE2Ey
`QN.o
`
`For BPSK p,. = —Land E; = E2 = E>», thus
`
`P,=Q ( | , (coherent BPSK)
`
`o
`
`(4.6)
`
`A typical exampleis that, at E/N, = 9.6 dB, P, = 10~°. Figure 4.4 showsthe P,
`curve of BPSK. The curves of coherent and noncoherent BFSK are also shownin the
`figure. Recall the P, expression for coherent BFSK is P, = Q (
`ey) whichis 3
`dB inferior to coherent BPSK. However, coherent BPSK requiresthat the reference
`signalat the receiver to be synchronized in phase and frequency with the received
`signal. This will be discussed in Section 4.10. Noncoherent detection of BPSK is
`also possible. It is realized in the form ofdifferential BPSK which will be studied in
`the next section.
`Next we proceed to find the powerspectral density of the BPSK signal.
`
`It suf-
`
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`

`

`Coherent BPSK 0
`
`-..
`
`Noncoherent BFSK
`~,
`*.
`.
`
`Coherent BFSK
`
`5
`
`10
`
`15
`
`E,/ N, (dB)
`
`Figure 4.4
`
`P,, of BPSK in comparison with BFSK.
`
`fices to find the PSD ofthe baseband shaping pulse. As shown in Appendix A, the
`PSD ofa binary, bipolar, equiprobable, stationary, and uncorrelated digital waveform
`isjust equal to the energy spectral density ofthe pulse divided by the symbolduration
`(see (A.19)). The basic pulse of BPSKis just a rectangular pulse?
`= { 0,
`otherwise
`—_ Pity|CSie a
`
`
`or
`
`Its Fourier transform is
`
`i
`-onsr/2
`Apia tsT
`=
`Gf)
`(f) AT——rrfT e
`Thusthe PSD of the baseband BPSK signalis
`) , (BPSK)
`W3(f) = er = A?T (
`
`sinafT
`nfT
`2 The bipolarity ofthe baseband waveform of BPSKis controlled by the bipolar data a, = +1.
`
`x
`
`2
`
`(4.8)
`
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`

`

`In Chapter 2 wefirst introduced differential encoding and decoding of binary data.
`This technique can be used in PSK modulation. We denote differentially encoded
`BPSKas DEBPSK. Figure 4.6 (a) is the DEBPSK modulator. DEBPSKsignal can
`be coherently demodulated or differentially demodulated. We denote the modula-
`tion schemethatuses differential encoding and differential demodulation as DBPSK,
`which is sometimes simply called DPSK.
`DBPSKdoesnot require a coherentreference signal. Figure 4.6(b)is a simple,
`but suboptimum,differential demodulator which uses the previous symbolas theref-
`erence for demodulating the current symbol.’ The front-end bandpassfilter reduces
`noise powerbutpreservesthe phase ofthe signal. Theintegrator can be replaced by
`an LPF. The outputof the integratoris
`
`(k+1)T
`
`t= i)
`
`kT
`
`r(t)r(t —T)dt
`
`In the absence of noise and other channel impairment,
`
`2
`(k41)T
`l= Ne
`
`Ey, if sx (t) = sx—1(t)
`Sx(t)8x—1(t)dt = { ae if54(t) = aye
`
`where x(t) and s(t) are the current and the previous symbols. The integrator
`output is positive if the current signal is the same as the previous one, otherwise the
`output is negative. This is to say that the demodulator makes decisions based on the
`difference between the two signals, Thus information data must be encodedas the
`difference between adjacent signals, which is exactly what the differential encod-
`ing can accomplish. Table 4.1 shows an example of differential encoding, where an
`arbitrary reference bit 1 is chosen. The encodingruleis
`
`dy = ay GB dp_1
`
`3 This is the commonly referred DPSK demodulator. Another DPSK demodulatoris the optimum
`differentially coherent demodulator. Differentially encoded PSK can also be coherently detected. These
`will be discussed shortly.
`
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`

`

`above.
`In the case of DEBPSK,thebit error rate of the final decoded sequence {Gx},
`P, is related to the bit error rate of the demodulated encoded sequence {d,}, Pi,
`by
`
`P, = 2Ps,a(1 — Pb,a)
`
`(4.11)
`
`as we have shownin Section 2.4.1 of Chapter 2. Substituting Py4 as in (4.6)into the
`above expression we have
`
`P, = 2Q ( ) f = ( =) , (DEBPSK)
`
`
`
`(4.12)
`
`for coherently detected differentially encoded PSK. For large SNR, this is just about
`two timesthat of coherent BPSK without differential encoding.
`Finally we need to say a few words of powerspectral density ofdifferentially
`encoded BPSK.Since the difference of differentially encoded BPSK from BPSK is
`differential encoding, which always produces an asymptotically equally likely data
`sequence(see Section 2.1), the PSD ofthe differentially encoded BPSKis the same as
`BPSKwhich we assumeis equally likely. The PSD is shown in Figure 4.5. However,
`it is worthwhile to point out that if the data sequence is not equally likely the PSD
`of the BPSKis notthe one in Figure 4.5, but the PSD ofthe differentially encoded
`PSKisstill the one in Figure 4.5.
`
`The motivation behind MPSKis to increase the bandwidth efficiency of the PSK
`modulation schemes.
`In BPSK,a data bit is represented by a symbol.
`In MPSK,
`n = log, M data bits are represented by a symbol, thus the bandwidth efficiency
`is increased to n times. Among all MPSK schemes, QPSKis the most-often-used
`schemesinceit does not suffer from BER degradation while the bandwidth efficiency
`is increased. We will see this in Section 4.6. Other MPSK schemesincrease band-
`width efficiency at the expenses of BER performance.
`M-ary PSKsignalset is defined as
`
`8;(t) = Acos(27f.t+6;), O<t<T,
`
`i=1,2,...,M
`
`(4.13)
`
`Page 19 of 58
`
`
`,—"—oor
`
`4.3>M-ARY PSK
`
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`

`The above expression can be written as
`
`s;(t)
`
`Acos 6; cos2rf,t — Asin 6; sin 27f,t
`81101 (t) + Si2@0(t)
`
`(4.14)
`
`where @,(t) and @,(t) are orthonormalbasis functions (see (4.2) and (4.3)), and
`T
`
`$i] -| s;(t)@,(t)dt = VEcos8;
`
`0
`
`7
`
`Sio = [ 8;(t)d(t)dt = VEsin6;
`
`where
`
`1
`42
`E= 54 -
`is the symbol energy of the signal. The phase is related with s;; and s;2 as
`
`@; = tan oe
`Sil
`
`The MPSKsignalconstellation is therefore two-dimensional. Each signal s;(t)
`is represented by a point (s;1, 8:2) in the coordinates spanned by ,(t) and ¢(t).
`The polar coordinates of the signalare ( VE, 6;). Thatis, its magnitude is
`FE and
`its angle with respectto the horizontalaxis is @;. The signal points are equally spaced
`on acircle ofradius VE andcenteredatthe origin. The bits-signal mapping could be
`arbitrary provided that the mapping is one-to-one. However, a method called Gray
`coding is usually used in signal assignment in MPSK. Gray coding assigns n-tuples
`with only one-bit difference to two adjacent signals in the constellation. When an
`M-ary symbolerroroccurs,it is more likely that the signal is detected as the adjacent
`signal on the constellation, thus only one of the n inputbits is in error. Figure 4.9 is
`the constellation of 8-PSK, where Gray coding is used for bit assignment. Note that
`
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`
`

`

`0,0
`
`Figure 4.9
`
`8-PSK constellation with Gray coded bit assignment.
`
`BPSK and QPSKare special cases of MPSK with M = 2 and 4,respectively. On
`the entire time axis, we can write MPSKsignal as
`s(t) = s;(t) cos 2mf.t — so(t)sin2xf,t, —oo<t< oo
`
`(4.15)
`
`where
`
`si(t)= A }> cos(0x)p(t — kT)
`k=-—0o
`
`so(t)= A }~ sin(Ox)p(t — kT)
`k=—oo
`
`(4.16)
`
`(4.17)
`
`where6; is one of the M phases determined by the input binary n-tuple, p(t) is the
`rectangular pulse with unit amplitude defined on [0,7]. Expression (4.15) implies
`that the carrier frequencyis an integer multiple ofthe symboltimingso that theinitial
`phaseofthe signal in any symbol period is 4,.
`Since MPSKsignals are two-dimensional, for M > 4, the modulator can be
`implemented by a quadrature modulator. The MPSK modulator is shown in Figure
`4.10. The only difference for different values of \Mis the level generator. Each
`n-tuple of the input bits is used to control the level generator.
`It provides the I-
`
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`

`

` n bits of {a,}
`
`St), T= oT,
`
`Figure 4.10 MPSK modulator.
`
`and Q-channels the particular sign and level for a signal’s horizontal and vertical
`coordinates, respectively. For QPSK,the level generatoris particularly simple,it is
`simply a serial-to-parallel converter (see Section 4.6).
`Modern technology intends to use completely digital devices. In such an envi-
`ronment, MPSKsignals are digitally synthesized and fed to a D/A converter whose
`outputis the desired phase modulated signal.
`The coherent demodulation of MPSKcould be implementedby oneofthe coher-
`ent detectors for M-ary signals as described in Appendix B. Since the MPSKsignal
`set has only twobasis functions, the simplest receiver is the one that uses two cor-
`relators (Figure B.8 with N = 2). Dueto the special characteristic of the MPSK
`signal, the general demodulator of Figure B.8 can be further simplified. For MPSK
`the sufficientstatistic is
`
`T
`
`0
`
`T
`
`0
`
`a
`
`| r(t)e(tae = [ r(t)[sir;(t) + si2ea(t)]dt
`if r(t)[VEcos6,4,(t) + VEsin4;¢9(t)lat
`0
`= VE[r,cos6; + r2sin§;]
`
`(4.18)
`
`li
`
`where
`
`i
`
`>
`
`r af rite(that = [s(t) + n(t)]o,(t)dt = sy +4
`
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`

`

`
`distribution of AG; is equally likely too. In turn the absolute phases of
`signals are also equally likely. This satisfies the condition for derivin;
`the PSD of DEMPSKis the sameas that of MPSK given in (4.26)
`likely original data sequence.
`‘a
`ha sratarve ptovel a Chee and mentioned in Section4.2,t eC
`encoding in DEBPSK always produces an equally likely data
`sec
`cally regardless ofthe distribution ofthe original data. This leadsto
`(4.8) for DEBPSKeven ifthe original data is not evenly distributed.
`
`4.6
`
`QUADRATURE PSK
`
`
`
`
`
`Amongall MPSK schemes, QPSKis the most often used schemes! er
`suffer from BER degradation while the bandwidth efficiency is
`increased
`MPSKschemes increase bandwidth efficiency at the expensesofBER|perfor
`In this section we will study QPSKin great detail.
`ae |
`Since QPSKis a special case of MPSK, its signals are definedas
`si(t) = Acos(2nfet +0), OStST, #=1,2,84
`S
`
`where
`
`
`
`_ (2i-—1)n
`6,=
`The initial signal phases are 7, 3%, 9, 2". The carrierfrequencyis ch
`multiple ofthe symbolrate, thereforein anysymbolinterval [kT, (k+
`initial phase is also one of the four phases.
`The above expression can be written as
`
`s(t) = Acos6;cos2rf,t — Asin, sin 2xfet
`= 8:1,(t) + siao(t)
`
`where ,(t) and ¢(t) are defined in (4.2) and (4.3),
`
`and
`
`si) = VEcos 6;
`
`82 = VEsin 6;
`
`6; = tan™ae
`$1
`
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`
`
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`

`

`combination of two orthonormalbasis functions: @,(t) and @.(t). On a coordinate
`system of @,(t) and #2(t) we can represent these four signals by four points or vec-
`tors: s; = (5) ,i = 1,2,3,4. The angle of vector s; with respect to the horizontal
`axis is the signalinitial phase 6;. The length of the vectorsis VE.
`The signal constellation is shown in Figure 4.18.
`In a QPSK system, databits
`are divided into groupsof twobits, called dibits. There are four possible dibits, 00,
`01, 10, and 11. Each of the four QPSKsignals is used to represent one of them. The
`mappingofthe dibits to the signals could b