Digital Modulation Techniques
`
`Then the probability density function (PD?) of n can be written as
`
`1
`
`2
`
`P(n) = WNW-KT}
`
`(1.9)
`
`present regardless of whether other channel impairments such as limited bandWidth-
`fading. multipath, and other interferences exist or not. Thus the AWGN channel is the
`best channel that one can 861- The performance ofa modulation scheme evaluated in
`this channel is an upper bound on the performance. When other channel impairmcms
`exist, the system performance will degrade. The extent ofdegradation may vary for
`emes. The performance in AWGN can serve as a standard
`dation and also in evaluating effectiveness of impairment-
`
`.
`combatttng techniques.
`
`l.2.2
`
`Bandlimited Channel
`
`When the channel bandwidth is smaller than the signal bandwidth. the channel is
`bandlimited. Severe bandwidth limitation causes intersymbol interference (lSl) (i.c..
`digital pulses will extend beyond their transmission duration (symbol period T. )) and
`interfere with the next symbol or even more symbols. The 18] causes an increase
`in the bit error probability (Pb) or bit error rate (BER). as it is commonly called.
`When increasing the channel bandwidth is impossible or not cost-efficient. channel
`equalization techniques are used for combatting ISl. Throughout the 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
`treatment of equalization techniques. the reader is referred to [LChaptcr 6| or any other
`communication systems books.
`
`Qualcomm Incorporated
`Exhibit 1009
`
`Page 1 of 58
`
`Qualcomm Incorporated
`Exhibit 1009
`Page 1 of 58
`
`

`

`Chapter I
`
`Introduction
`
`7
`
`1.2.3
`
`Fading Channel
`
`Fading is a phenomena occurring when the amplitude and phase of a radio signal
`change rapidly over a short period of time or travel distance. Fading is caused by in-
`terference between two or more versions of the transmitted signal which arrive at the
`receiver at slightly different times. These waves, called multipath waves. combine
`at the receiver antenna to give a resultant signal which can vary widely in amplitude
`and phase. If the delays of the multipath signals are longer than a symbol period,
`these multipath signals must be considered as different signals. In this case, we have
`individual multipath signals.
`In mobile communication channels. such as ten'estrial mobile channel and 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 Doppler shifl on
`multipath component. However, if the surrounding objects move at a speed less than
`the mobile unit. their effect can be ignored [2].
`Fading and multipath interference also exist in fixed LOS microwave links [3]-
`On clear. calm summer evenings. normal atmospheric 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 intersymbol interference. Doppler shift causes carrier frequency
`drift and signal bandwidth spread. All these lead to performances degradation of
`modulations. Analysis of modulation performances in fading channels is given in
`Chapter I0 where characteristics of fading channels will be discussed in more detail.
`
`I.3
`
`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 sequence of 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 shows several baseband modulation waveforms. The first one is the non-
`return to zero-level (N RZ-L) modulation which represents a symbol I by a positive
`
`Page 2 of 58
`
`Page 2 of 58
`
`

`

`8
`
`Digital Modulation Techniques
`
`(21) NRZ-L
`
`1
`
`0
`
`l
`
`1
`
`l
`
`o
`
`o
`
`l
`
`A
`
`-A
`
`(b) Unipolar RZ
`
`A
`
`(c) Bi-Q-L (Manchester)
`A
`
`-A
`
`Figure L3 Baseband digital modulation examples.
`
`square pulse with length T and a symbol 0 by a negative square pulse with length T.
`The second one is the unipolar return to zero modulation with a positive pulse of T0
`for symbol l and nothing for 0. The third is the biphase level or Manchester. alter
`its inventor, modulation which uses a waveform consisting of a positive first-half T
`pulse and a negative second-half T pulse for l and a reversed waveform for 0. These
`and other baseband schemes will 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 shows three basic binary carrier modulations. They are amplitude
`shifl keying (ASK). frequency shift keying (FSK), and phase shift keying (PSK). In
`ASK, the modulator puts out a burst of carrier for every symbol I. and no signal
`for every symbol 0. This scheme is also called on-ofi' keying (00K). In a general
`ASK scheme, the amplitude for symbol 0 is not necessarily 0.
`ln FSK. for symbol
`l a higher frequency burst is transmitted and for symbol 0 a lower frequency burst
`
`Page 3 of 58
`
`h’
`
`ll
`
`
`
`)4/.
`
`4g
`
`Page 3 of 58
`
`

`

`Chapter I
`
`Inlrvducnon
`
`9
`
`ASK
`
`FSK
`
`PSK
`
`Figure l 4 Three basic bandpass modulation schemes.
`
`In PSK, a symbol 1 is transmitted as a burst of carrier
`is transmitted. or vice versa.
`with 0 initial phase while a symbol 0 is transmitted as a burst of carrier with 180°
`initial phases
`Based on these three basic schemes. a variety of modulation schemes can be de-
`rived from their combinations. For example. by combining two binary PSK (BPSK)
`signals with orthogonal carriers a new scheme called quadrature phase shift keying
`(QPSK) can be generated. By modulating both amplitude and phase of the carrier.
`we can obtain a scheme called quadrature amplitude modulation (QAM), etc.
`
`1.4
`
`CRITERIA OF CHOOSINC MODULATION SCHEMES
`
`The essence of digital modern design is to efficiently transmit digital bits and recover
`them from corruptions from the noise and other channel impairments. There are
`three primary criteria of choosing modulation schemes: power efficiency, bandwidth
`
`Page 4 of 58
`
`Page 4 of 58
`
`

`

`IO
`
`Digital Modulation Techniques
`
`efficiency, and system complexity.
`
`1.4.1
`
`Power Efficiency
`
`The bit error rate, or bit error probability of a modulation scheme is inversely related
`to Eb/No. the bit energy to noise spectral density ratio. For example. Pb 0f ASK m
`the AWGN channel is given by
`
`
`2E
`
`Pb=Q( Nob)
`
`(1‘10)
`
`where E, is the average bit energy, No is the noise power spectral density (PSDX and
`(2(1) is the Gaussian integral, sometimes referred to as the Q-function. It is defined
`as
`
`0-“)
`Q(x) = /°" LIE—“2dr;
`which is a monotonically decreasing function of 1. Therefore the power efficiency
`of a modulation scheme is defined straightforwardly as the required Eli/Na for a
`certain bit error probability (Pb) over an AWGN channel. Pi, = 10—5 is usually used
`as the reference bit error probability.
`
`1.4.2
`
`Bandwidth Efficiency
`
`The determination of bandwidth efficiency is a bit more complex. The bandwidth
`efficiency is defined as the number of bits per second that can be transmitted In
`one Hertz of system bandwidth. Obviously it depends on the requirement of SYS‘cm
`bandwidth for a certain modulated signal. For example. the one-sided power Specml
`density ofan ASK signal modulated by an equiprobable independent random binary
`sequence is given by
`
`2
`
`mm = A T
`
`A2
`
`varies depending on different criteria. For example" in Figul’:
`al energy concentrates in the band between two nulls,_thuf
`th requrrement seems adequate. Three bandwidth emc'cnc'es
`
`Page 5 of 58
`
` ,
`
`‘0'
`
`'——'l
`
`Page 5 of 58
`
`

`

`
`
`Chapter I
`
`Introduction
`
`ll
`
`1
`T
`
`tc -
`
`I
`~
`tc’T
`
`fc
`
`f°++
`
`2
`fo’ T
`
`Figure 1.5 Power spectral density of ASK.
`
`.
`used in the literature are as follows:
`Nyquist Bandwidth Efficiency—Assuming the system uses Nyquis‘ “deal rec'
`tangular) filtering at baseband. which has the minimum bandwidth required for in-
`tersymbol interference-free transmission of digital signals, then the bandWidth at
`baseband is 0_ 512,, R3 is the symbol rate, and the bandwidth at carrier frequency
`is W = R,. Since R, : lib/1082 M, Rb = bit rate, for M-al')’ modulation, the
`bandwidth efficiency is
`
`Rb/W = logz M
`
`(”2)
`
`Null-to—Null Bandwidth Efficiency—For modulation schemes that have power
`density spectral nulls such as the one of ASK in 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 be used.
`Usually 99% is used, even though other percentages (e.g., 90%, 95%) are also Used'
`
`1.4.3
`
`System Complexity
`
`System complexity refers to the amount of circuits involved and the technical dif-
`ficu't)’ 0f the system. Associated with the system complexity is the COSt 0f manu—
`
`Page 6 of 58
`
`
`
`Page 6 of 58
`
`

`

`l2
`
`Digital Modulation "Techniques
`
`c0"
`
`. r conce
`rse a map
`
`"
`
`.
`in choosmg a mo
`
`facturing, which is of
`dulation tec
`hnique'
`-
`Usually the demodul
`oherent dem°d“'?'
`tor is much more co
`ator is more complex than the modulalgch cmicr recovery is
`mplex than noncoherem demodulatorl orimms like the Viterbi
`required. For some demodulation methOdS' 5°phisnca-Ed :rgnpafison‘
`algorithm isrequired. Allthesearebasis forComplex”:vstem complexi‘)’ are the
`Since powar efiiciency, bandwidth efficiency, andwi'll always pay attention to
`main criteria ofchoosing a modulation teChnigue’ we tofthe book-
`them in the analysis ofmodulation techniques m the res
`d a scriptive
`t.s
`ovaavuaworDIGITALM0DULATIONs
`To provide the reader with an overview, we "St-film:fzreiraT;ble "l and miss:
`names ofvarious digital modulations that we w“ c oftheSchemescanbeSiencan
`them inarelationshiptreediagram inFigure l'6‘ some
`e differential encoding
`from morethan one “parent“ scheme. The “nameswsgoherenilydemOdula‘ed are
`beusedarelabeledbyletterDandthosethatcanbeta,demodula‘ed'
`.
`-
`t
`two
`labeledwithaletterN.Allschemescanbewhere“
`hetree are classified in o
`.
`The modulation schemes listed in ‘he “ableand‘envelope- Underconstflm el"t
`large categories: constant envelope and nonconswlgt and CPM- Under nonconslfln
`velope class, there are three subclasses: FSK' PS '
`envelope class.
`there are three subclasses: ASK. QAM. and 0th
`lope modulations.
`Among the listed schemes, ASK. P5K° 3:323:22: s:hemes. The advanced
`MSK. GMSK, CPM, MHPM, andQAM.etc.
`.
`hemeS-
`.
`schemesarevariationsandcombinationsoi:hes:?t:l:|:cf°" communiC3‘_'°“ 3:51:11:
`whoihxvgte‘fmpii‘gtSrgpneiust operate in the nonlincaEEfiEy. AnexampleISthe
`1
`class is genera y
`.
`-
`0f the mpu
`.
`characteristic inordertoachievemilxi')“.uma:£::§f:lotettmuni°a‘i°ns- Bowen:?.m1:;
`TX;(#:1133313:Eltzzltzfareinappropriatefor'Sztggrgfimes. BinaryFSK
`-
`Iifier m 58
`-
`licatlon st
`lgiaveverylowbandwidthefi‘tciency incomparisonWl‘tion cellularsystems. AMPS
`isused inthe low-ratecontrol channels offirst gegfigauropean total accesscammt;
`(advance mobile phone service ofUS.) and ETAfC AMPS and 8 Kbps for E‘mC. .
`nication system). The data rates are l0 KbPS 01:8K and MSK have been “53“ m
`The PSKschemes, including BPSK, QPSK. 0Q
`'
`-d 180° abrupt
`satellitecommugglzu‘gvsg’rsthesnpsecialattentionduetoitsabiltlyge‘ajviz'digimlmobile
`h 35:52::chenalbledifferentialdemodulation. “11ml;bfsgDC)system.
`scalar systems such as the United States digital CC ‘1
`
`cHEMEs
`
`. fonS an
`
`e
`
`er noncoltswnt en
`
`ve-
`
`.
`b sic modulations
`
`and
`
`Page 7 0f58
`
`Page 7 of 58
`
`

`

`Chapter I
`
`Introduction
`
`13
`
`Abbreviation
`
`AltcmaIeAbbr
`
`-
`
`;
`
`
`Frequency Shifl Keying (FSk)
`E- Binary Frequency Shift Keying
`— M-ary Frequency Shifl Keying
`
`
`
`
`“——
`manna-—
`
`———
`m— M-ary Pm: Shift Keying
`
`Continuous Phase Modulnions (CPM)
`
`
`
`
`
`
`Single-h (modulation index) Phase Modulation
`ulii-h Phase Modulation
`
`
`
`iU
`
`!83'5
`
`2°82'U7:” (n3
`
`Table l.l Digital modulation schemes (Abbt.=Abbrevietion).
`
`Page 8 of 58
`
`Page 8 of 58
`
`

`

`14
`
`Digital Modulation Techniques
`
`Digital Modulations
`
`Constant Envelope
`
`Nonconstam Envelope
`
`m m
`BFSKIIIW
`
`(D)
`
`IREC
`(CPFSK
`.
`_
`. smusondal
`‘h=0.5
`: pulse-shaping
`.
`—>
`.............. MSK
`- ........................ mm
`
`L-l
`
`$
`
`LRC
`
`LSRC
`
`m
`
`TSI-
`
`OQPSK
`
`SQAM
`
`m
`
`m
`
`W ""0?“
`.
`(D)
`I
`'
`:
`u
`
`I :
`
`h=0 s —>
`
`(D)
`
`(N)
`
`Can be difl’erentially encoded and decoded
`
`Can be noncohctcntly detected
`
`Figure L6 Digital Modulation Trcc~ Aflcr [4].
`
`Page 9 of 58
`
`Page 9 of 58
`
`

`

`Chapter I
`
`Introduction
`
`15
`
`The PSK schemes have constant envelope but discontinuous phase transitions
`from symbol to symbol. The CPM schemes have not only constant envelope, but also
`continuous phase transitions. Thus they have less side lobe energy in their spectra
`in comparison with the PSK schemes. The CPM class includes LREC. LRC. LSRC.
`GMSK, and TFM. Their difi’erences lie in their different frequency pulses which are
`reflected in their names. For example. LREC means the frequency pulse is a rectan-
`gular pulse with a length of L symbol periods. MSK and GMSK are two important
`schemes in C PM class. MSK is a special case of C PFSK, but it also can be derived
`from OQPSK with extra sinusoidal pulse-shaping. MSK has excellent power and
`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. GMSK is used in the US cellular digital packet data (CDPD)
`system and European GSM (global system for mobile communication) system.
`MHPM is 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 efi'iciency.
`QAM has been widely used in modems used in telephone networks, such as computer
`modems. QAM can even be considered for satellite systems. In this case, however.
`back-off in TW'llA's input and output power must be provided to ensure the linearity
`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 QQPSK are based on 2T, amplitude pulse shaping and their modulator structures
`are similar to that of OQPSK. The scheme Q2PSK is based on four orthogonal car-
`riers.
`
`References
`
`[I]
`
`[2]
`
`[3]
`
`[4]
`
`Proakis. 1., Digital Communication. New Ybrk: McGraw-Hill. 1983.
`
`Rappaport. T.. "ire/es: Communications: Principles and Practice. Upper Saddle River. New
`Jersey: Prentice Hall. l996.
`
`Siller. C .. “Multipath propagation.“ IEEE Communications Magazine. vol. 22. no.2. Feb. I984.
`pp. 6-l5.
`
`Xiong. F.."Modem techniques in satellite communications.” IEEE Communications Magazine.
`vol. 32. no.8. August I994, pp. 84-98.
`
`Page 10 of 58
`
`Page 10 of 58
`
`

`

`l22
`
`Digiul Modulation 1mm
`
`
`
`
`costly, most of FSK receivus use noncoherem demodulliou.
`.,
`
`I'ls“New"“mu”efl’Ofpetformanoeofthcnoneohetutrocdxvdl
`tomatofthecohmiona. However thedegmhuontsonlyl
`4‘
`
`exprasiomandcurvcs fordremprobabilitiesueahoptw
`Finally we explored other possible demodulation. The
`‘
`Intorissimpleandefi'lcient. ltisevenbenermmew- '
`
`ulator fat BFSK.
`
`[I] MILK. M1. Sill. 'smormm'uuww‘}
`
`Judy-August. 1965. pp. "65-1189.
`
`
`
`References
`
`’ .'_>
`
`.
`3!
`
`[21 MIL-unnm. mmama—m f
`memmvol. l8.no.....4Augul970pp29$-300
`
`[3] MILE. “mumormmmmwmm'm
`vol29..nollNou|98l...pp1634-l6‘3
`'
`
`[4] M1.E.‘11wyofammfoedWFM.'hRS)uu-W
`I966. pp. fill-1535.
`
`Selocud Blbllognply
`
`
`
`
`, f";
`.
`
`o ma.uw.wmwmcmw.mu.mfik '~ "l
`o mms..mmcmmmmunyammlsfi;"
`o sag]..'mdmmkvdmwmmdwm -.
`MCMMWmIJMJMIWSm4M
`Skin. 3. mnemm. FMadWau.W
`Prentice Hfll. I988.
`
`0
`
`
`
`
`o smmn.x.pmunmmmm.wwmmmu a
`1993.
`
`'
`
`0
`
`I
`
`Suite. E. 0.. ‘Idcdbhl'ypullemiuhnbyAMmdFM'MM
`v01. 38. Nov. I939. pp ”574426.
`\h'n'euJ‘LL,Wmmmmwmmmamw
`Smlmlm
`9
`
`
`
`Page 11 ofSé
`
`Page 11 of 58
`
`

`

`Chapter 4
`
`Phase Shift Keying
`
`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 scheme in a single section where signal description, power spectral density.
`modulator/demodulator block diagrams, and receiver error performance are all in-
`cluded. First we present coherent binary PSK(BPSK) and its noncoherent coun-
`terpart, differential BPSK (DBPSK), in Sections 4.l and 4.2. Then we discuss in
`Section 4.3 M-ary PSK (MPSK) and its PSD in Section 4.4. The noncoherent ver—
`sion, differential MPSK (DMPSK) is treated in Section 4.5. Vie discuss in great detail
`quadrature PSK (QPSK) and differential QPSK (DQPSK) in Sections 4.6 and 4.7, re-
`spectively. Section 4.8 is a brief discussion of offset QPSK (OQPSK). An important
`variation of QPSK, the 7r/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. l l.
`
`4.1
`
`BINARY PSK
`
`Binary data are represented by two signals with different phases in BPSK. Typically
`these two phases are 0 and 7r. the signals are
`
`31(t) = Acos21rfct. OStST.
`
`for!
`
`32(t) = —A00321rfct. ogtsT.
`
`for0
`
`(4.1)
`
`These signals are called ann'padal. The reason that they are chosen is that they have
`a correlation coefficient of — l, which leads to the minimum error probability for the
`same Eb/No, as we will see shortly. These two signals have the same frequency and
`energy.
`
`As we will see in later sections. all PSK signals can be graphically represented
`
`123
`
`Page 12 of 58
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`Page 12 of 58
`
`

`

`124
`
`Digital Modulation Techniques
`
`‘92“)
`
`
`
`Figure 4.1 BPSK signal constellation.
`
`by a signal constellation in a two-dimensional coordinate system With
`
`aw): \fgcosfi'fct,
`
`0_<_t§T
`
`and
`
`L1
`
`¢,(t) = —\/——,12:sin 2an,
`
`o g t s T
`
`(41)
`
`(4.3)
`
`as its horizontal and vertical axis. respectively. Note that we deliberately add a minus
`sign in c520) so that PSK signal expressions will be a sum instead of a difference (sec
`(4.14)). Many other signals, especially QAM signals, can also be represented in the
`same way. Therefore we introduce the signal constellation of BPSK here as shown
`
`in Figure 4.1 where 31(t) and 32(t) are represented by two points on the horizon!!!
`axis, respectively, where
`
`A’T
`E:—
`2
`
`1
`
`I I
`
`l
`
`.
`:}
`i
`
`The waveform of a BPSK signal generated by the modulator in Figure 4.3 fora
`data stream {10110} is shown in Figure 4.2. The waveform has a constant envelope
`like FSK. lts frequency is constant too.
`In general the phase is not continuous I
`
`h
`
`Page 13 of58
`
`‘
`
`Page 13 of 58
`
`

`

`('hapler 4
`
`Phase Shift Keying
`
`I25
`
`Date
`
`I
`
`U
`
`l
`
`l
`
`(l
`
`(a) rc = 2n
`
`(b)fc= l.8ff
`
`Figure 4.2 BPSK waveforms.
`
`If the fC = m Rb = m/T, where m is an integer and Rb is the
`bit boundaries.
`data bit rate, and the bit timing is synchronous with the carrier, then the initial phase
`at a bit boundary is either 0 or 71' (Figure 4.2(a)), corresponding to data bit 1 or 0.
`However. if the fc is not an integer multiple of Rb , the initial phase at a bit boundary
`is neither 0 nor 7r (Figure 4.2(b)). In other words, the modulated signals are not the
`ones given in (4.1). We will show next in discussion of demodulation that condition
`fC = m Rb is necessary to ensure minimum bit error probability. However. if fa >>
`Rb, this condition can be relaxed and the resultant BER performance degradation is
`negligible.‘
`The modulator which generates the BPSK signal is quite simple (Figure 4.3 (a)).
`First a bipolar data stream a(t) is formed from the binary data stream
`
`00
`
`a(t)= 2 akp(t—-kT)
`k=—oo
`
`(4.4)
`
`‘ This is true for all PSK schemes and PSK-derived schemes. including QPSK. MSK. and MPSK. We
`will not mention this again when we discuss other PSK schemes.
`
`Page 14 of 58
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`

`
`
`126
`
`Digital Modulation Techniques
`
`Polar NRZ some a(t)
`Aa(t)cosZ1rQ
`
`
` Acosanct
`
`
`
`(b)
`
`Figure 43 BPSK modulator (a). and coherent BPSK demodullor (b)-
`
`whm 0k 6 {+1. —-1},p(t) is the rectangular pulse with unit amplitude defined m
`[0.7"]. Then a(t) is multiplied with a sinusoidal carrier Acos21rfct. The rest!!! 3
`the BPSK signal
`
`30) = Aalt) €0821rfct,
`
`——oo < t < oo
`
`(45)
`
`.
`Note that the bit timing is not necessarily synchronous with the carrier.
`The coherent demodulator of BPSK falls in the class of coherent detectors for '1'
`nary signals as described in Appendix B. The coherent detector could be in the m
`ofa correlator or matched filter. The correlator’s reference signal is the difference SIE-
`nal (34(t) = 2A cos 21rfct). Figure 4.3(b) is the coherent receiver using a cone“
`Where the reference signal is the scaled-down version of the difference signal. The
`reference signal must be synchronous to the received signal in frequency and phe-
`
`Page 15 of58
`
`Page 15 of 58
`
`

`

`Chapter 4
`
`Phase Shift Keying
`
`l27
`
`It is generated by the carrier recovery (CR) circuit. Using a matched filter instead of a
`correlator is not recommended at passband since a filter with h.(t) = cos 271fc(T - t)
`is dificult to implement.
`In the absence ofnoise. setting A = 1. the output ofthe correlator at t = (k+l)T
`
`IS
`
`(k+l)T
`
`/
`
`kT
`(k+l)T
`
`r(t)cos 27rfctdt
`
`= /
`
`kT
`
`ak cos2 27rfctdt
`
`kT
`
`-
`2
`
`0;.
`T
`Eak + 81rfC
`
`ak(1 + cos 47rfct)dt
`
`[sin 41rfc(k + 1)T - sin 47rfckT]
`
`If IC = m Rb. the second term is zero, thus the original signal a(t) is perfectly
`recovered (in the absence of noise). If fC 9e m Rb, the second term will not be zero.
`However, as long as fc > > Rb, the second term is much smaller than the first term
`so that its effect is negligible.
`The bit error probability can be derived from the formula for general binary
`signals (Appendix B):
`
`Pb
`
`Q (
`
`l
`
`2
`
`P12V
`
`2N0
`
`2
`
`1 )
`
`For BPSK p12 = —1 and E, = E2 = Eb, thus
`
`Pb = Q <
`
`
`
`215”) , (coherent BPSK)
`
`(4.6)
`
`A typical example is that, at Eb/No = 9.6 dB, Pb = 10‘5. Figure 4.4 shows the Pb
`curve of BPSK. The curves of coherent and noncoherent BPSK are also shown in the
`figure. Recall the Pb expression for coherent BPSK is P5 = Q (‘ / %) which is 3
`dB inferior to coherent BPSK. However, coherent BPSK requires that the reference
`signal at the receiver to be synchronized in phase and frequency with the received
`signal. This will be discussed in Section 4.!0. Noncoherent detection of BPSK is
`also possible. It is realized in the form of differential BPSK which will be studied in
`the next section.
`
`Next we proceed to find the power spectral density of the BPSK signal.
`
`It suf-
`
`Page 16 of 58
`
`Page 16 of 58
`
`

`

`
`
`‘28
`
`Digital Modulation Techniques
`
`Coherent BFSK
`
`.
`
`. _
`
`Noncoherent BFSK
`‘.
`‘u .
`
`‘-
`
`Coherent BPSK
`
`0
`
`5
`
`lo
`
`15
`
`Eb/ N. (dB)
`
`Figure 4.4
`
`P5 of BFSK in comparison with BFSK.
`
`fices to find the PSD of the 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 symbol dilution
`(see (A. 19)). The basic pulse of BPSK is just a rectangular pulse2
`P“) “ { 0,
`otherwise
`
`_
`
`A,
`
`0 < t < T
`
`(*7)
`
`Its Fourier transform is
`
`
`C(f) = ATsin WfTe_2'/T/2
`7rfT
`
`Thus the PSD of the baseband BPSK signal is
`
`
`|G(f)|2
`2
`sin 1rfT 2
`K
`=
`-
`=
`4.8
`
`
`
`
`
`
` A T 7rfT . (BPS ) (T‘I'.(f) l\
`2 The bipolarity of the baseband waveform of BPSK is controlled by the bipolar data a. = :H.
`
`Page 17 of 58
`
`Page 17 of 58
`
`

`

`(‘Imprer 4
`
`Phase Shift Keying
`
`l29
`
`which is plotted in Figure 4.5. From the figure we can see that the null-to—null band-
`width 8,...” = 2/ T = 2R5. (Keep in mind that the PSD at the carrier frequency is
`two-sided about fa.) Figure 4.5(c) is the out-of-band power curve which is defined
`by (2.21). From this curve we can estimate that 890% x 1.712;, (corresponding to
`—l0 dB point on the curve). We also calculated that 899% 2: mm.
`
`4.2
`
`DIFFERENTIAL BPSK
`
`In Chapter 2 we first introduced differential encoding and decoding of binary data.
`This technique can be used in PSK modulation. we denote differentially encoded
`BPSK as DEBPSK. Figure 4.6 (a) is the DEBPSK modulator. DEBPSK signal can
`be coherently demodulated or differentially demodulated. We denote the modula-
`tion scheme that uses differential encoding and differential demodulation as DBPSK,
`which is sometimes simply called DPSK.
`DBPSK does not require a coherent reference signal. Figure 4.6(b) is a simple,
`but suboptimum, differential demodulator which uses the previous symbol as the ref-
`erence for demodulating the current symbol.3 The front-end bandpass filter reduces
`noise power but preserves the phase of the signal. The integrator can be replaced by
`an LPF. The output of the integrator is
`
`(k+l)T
`
`1: /
`
`kT
`
`mm: - T)dt
`
`in the absence of noise and other channel impairment,
`
`_
`(k+l)T
`l- f"
`
`E, ifs (t) =3k— (t)
`5k(t)3k—l(t)dt ={ -5: ifs:(t) = —3kl—l(t)
`
`where sk(t) and sk_1(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 encoded as the
`difference between adjacent signals. which is exactly what the differential encod-
`ing can accomplish. Table 4.l shows an example of differential encoding, where an
`arbitrary reference bit 1 is chosen. The encoding rule is
`
`dk = 0-1: (D dk—r
`
`3 This is the commonly referred DPSK demodulator Another DPSK demodulator is the optimum
`differentially coherent demodulator. Differentially encoded PSK can also be coherently detected. These
`will be discussed shortly.
`
`Page 18 of 58
`
`Page 18 of 58
`
`

`

`l36
`
`Digital Modulation Techniques
`
`4.10). This is not usually meant by the name DBPSK. DBPSK refers to the schemeof
`difl‘erential encoding and differentially coherent demodulation as we have discuwd
`above.
`
`In the case of DEBPSK. the bit error rate of the final decoded sequence {3.}.
`Pa is related to the bit error rate of the demodulated encoded sequence {(7.}, PM,
`by
`
`Pb = 23.4(1 - Pm)
`
`(4.")
`
`as we have shown in Section 2.4. I of Chapter 2. Substituting PM as in (4.6) intothe
`above expression we have
`
`Pa = 2Q (
`
`
`
`215°) [1 — Q (
`
`
`
`2155)] . (DEBPSK)
`
`(4J2)
`
`for coherently detected differentially encoded PSK. For large SNR. this is just about
`two times that of coherent BPSK without differential encoding.
`Finally we need to say a few words of power spectral density of difi'etentially
`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 of the differentially encoded BPSK is the sameas
`BPSK which we assume is 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 BPSK is not the one in Figure 4.5. but the PSD of the differentially encoded
`PSK is still the one in Figure 4.5.
`
`4.3
`
`M-ARY PSK
`
`The motivation behind MPSK is to increase the bandwidth efficiency of the PSK
`modulation schemes.
`In BPSK, a data bit is represented by a symbol.
`In MPSK,
`n = log2 M data bits are represented by a symbol. thus the bandwidth efficiency
`is increased to n times. Among all MPSK schemes. QPSK is the most-often-used
`scheme since it does not suffer from BER degradation while the bandwidth efficiency
`is increased. we will see this in Section 4.6. Other MPSK schemes increase band-
`width efi'iciency at the expenses of BER performance.
`M-ary PSK signal set is defined as
`
`5,-(t) = Acos(21rfct + 9;),
`
`0 S f S T.
`
`i=1.2.....1\!
`
`(4J3)
`
`Page 19 of 58
`
`
`
`
`
`——3—“”I‘-
`
`-—...—
`it
`
`Page 19 of 58
`
`

`

`("hapler 4
`
`Phase Shift Keving
`
`137
`
`where
`
`9i
`
`_ (2i — l)7r
`—
`M
`
`The carrier frequency is chosen as integer multiple of the symbol rate. therefore in
`any symbol interval. the signal initial phase is also one of the M phases. Usually M
`is chosen as a power of 2 (i.e.. M = 2", n = log2 M). Therefore binary data stream
`is divided into n-tuples. Each of them is represented by a symbol with a particular
`initial phase.
`The above expression can be written as
`
`s.(t.) = Acos 0.- cos21rfct — Asin 91 sin 277fct
`
`= Sii¢1(t)+ 5.202(1)
`
`(414)
`
`where 01 (t) and (92(8) are orthonormal basis functions (see (4.2) and (4.3)). and
`T
`
`Sn =/ 3.(t)¢1(t)dt = VEcosgi
`
`0
`
`T
`
`3.2 = /0 5,-(t)d‘)2(t)dt = s/Esinei
`
`_l2
`E—2AT
`
`where
`
`is the symbol energy of the signal. The phase is related with s” and 3.2 as
`
`The MPSK signal constellation is therefore two-dimensional. Each signal si(t)
`is represented by a point (8.1.8.2) in the coordinates spanned by ¢1(t) and (:5 (t).
`The polar coordinates of the signal are (x/E 9.). That is. its magnitude is E and
`its angle with respect to the horizontal axis is 6;. The signal points are equally spaced
`on a circle of radius x/E and centered at the 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 symbol error occurs, it is more likely that the signal is detected as the adjacent
`signal on the constellation, thus only one of the 11 input bits is in error. Figure 4.9 is
`the constellation of 8-PSK, where Gray coding is used for bit assignment. Note that
`
`Page 20 of 58
`
`Page 20 of 58
`
`

`

`‘33
`
`Digital Modulation Techniques
`
`
`
`Figure 4.9
`
`8-PSK constellation with Gray coded hit assignment
`
`BPSK and QPSK are special cases of MPSK with M = 2 and 4, respectively. On
`the entire time axis, we can write MPSK signal as
`
`8(1) = 51(t) C0821l’fct — 52(t)sin 21rfct, —00 < t < oo
`
`(4.15)
`
`where
`
`51m = A Z cos(9k)p(t — kT)
`k=—oo
`
`32m = A Z sin(9k)p(t — kT)
`k=-oo
`
`(4.16)
`
`(4.17)
`
`where 9,, is one of the M phases determined by the input binary n-tuple, p(t) isthe
`rectangular pulse with unit amplitude defined on [0.1"]. Expression (4.15) implies
`that the carrier frequency is an integer multiple of the symbol timing so that the initial
`phase of the signal in any symbol period is 0*.
`Since MPSK signals are two-dimensional, for M 2 4, the modulator can be
`implemented by a quadrature modulator. The MPSK modulator is shown in Figure
`4.l0. The only difference for different values of M is the level generator. Bad!
`n-tuple of the input bits is used to control the level generator.
`It provides the I-
`
`Page 21 of 58
`
`Page 21 of 58
`
`

`

`Chapter 4
`
`Phase Shift Keying
`
`I39
`
`5m. T= nTh
`
`n bits of {at}
`
`
`
`
`$20). T= n1},
`
`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 generator is 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, MPSK signals are digitally synthesized and fed to a D/A converter whose
`output is the desired phase modulated signal.
`The coherent demodulation of MPSK could be implemented by one of the coher-
`ent detectors for M-ary signals as described in Appendix 8. Since the MPSK signal
`set has only two basis functions, the simplest receiver is the one that uses two cor-
`relators (Figure 8.8 with N = 2). D