`
`106
`
`Chapter 3. Modulation and Frequency-Division Multiple Access
`
`off increased transmission bandwidth for improved system performance in the
`presence of noise.
`3. Modulation permits the use of multiple-access techniques.
`A cellular radio channel represents a major capital investment and must there-
`fore be deployed in a cost-effective manner, permitting mobile users access to
`the channel. Multiple accessis a signal-processing operation that makesthis pos-
`sible. In particular, it permits the simultaneous transmission of information-
`bearingsignals from a numberof independentusers over the channel and on to
`their respective destinations.
`
`In wireless communications, the carrier, denoted by c(t), is typically sinusoidal andis
`written as
`
`c(t) = A,cos(27f,t+ @)
`
`(3.1)
`
`whereA,is the amplitude, f. is the frequency, and @ is the phase. With these three car-
`rier parameters individually available for modulation, we have three basic methods of
`modulation whose specific descriptions depend on whether the information-bearing
`signal is of an analog or a digital nature. These two families of modulation are dis-
`cussed in whatfollows.
`
`3.2.1 Linear and Nonlinear Modulation Processes
`Figure 3.3 shows the block diagram of a modulator supplied with a sinusoidalcarrier
`c(t). The modulating signal, acting as input, is denoted by m(t). The modulatedsignal,
`acting as output, is denoted bys(¢). The input-outputrelation of the modulatoris gov-
`erned by the mannerin which the output s(¢) depends on the input m(t). On this basis,
`we mayclassify the modulation process as one of twobasic types:linear and nonlinear.
`
`i
`
`|
`
`Input
`(modulating)
`signal
`m(t)
`
`Output
`(modulated)
`signal
`s()
`
`Modulator
`
`Sinusoidal
`carrier
`e(t)
`
`FIGURE 3.3 Block diagram of modulator.
`
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`3
`
`4
`
`|j
`
`
`
`Section 3.2. Modulation
`
`107
`
`The modulation processis said to be linear if the input-output relation of the
`modulatorsatisfies the principle ofsuperposition. According to this principle, the mod-
`ulation process, or equivalently, the modulator,satisfies two conditions:
`
`1. The output of the modulator produced by a numberof inputs applied simulta-
`neously is equal to the sum of the outputs that result when the inputs are applied
`one at a time.
`
`2. If the input is scaled by a certain factor, the output of the modulator is scaled by
`exactly the samefactor.
`
`The modulation process, or equivalently, the modulator, is said to be nonlinear if the
`principle of superposition is violated in part or in full.
`The linearity or nonlinearity of a modulation process has important conse-
`quences, in both theoretical as well as practical terms, as we shail see in the remainder
`of the chapter.
`
`3.2.2 Analog and Digital Modulation Techniques
`
`
`
`
`
`Another wayofclassifying the modulation process is on the basis of whether the mes-
`sage signal! m(¢) is derived from an analog or a digital source of information. In the ana-
`log case, the message signal m(r) is a continuous function of time ¢. Consequently, the
`modulated signal s(¢) is, likewise, a continuous function of time t. It is for this reason
`that a modulation process of the analog kind is commonly referred to as continuous-
`wave (CW) modulation.
`In the digital case, by contrast, the modulated signal s(¢) may exhibit discontinui-
`ties at the instants of time at which the message signal m(f) switches from symbol1 to
`symbol 0 or vice versa. Note, however(as wewill find out later on in the chapter), that
`under certain conditionsit is possible for the modulated signal to maintain continuity
`even at the instants of switching,
`In other words, we maydistinguish between analog and digital modulation pro-
`cesses as follows:
`
`* All analog modulated signals are continuous functions of time.
`* Digital modulated signals can be continuous or discontinuous functionsof time,
`depending on how the modulation processis performed.
`
`3.2.3 Amplitude and Angle Modulation Processes
`
`Yet another wayofclassifying modulation processes is on the basis of which parame-
`ter of the sinusoidal carrier c(t) is varied in accordance with the message signal m(/).
`Accordingly, we speak of two kinds of modulation:
`
`1. Amplitude modulation, in which the amplitude of the carrier, A,, is varied
`linearly with the message signal m(2).
`
`
`
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`
`
`108
`
`Chapter 3. Modulation and Frequency-Division Multiple Access
`
`2. Angle modulation, in which the angle of the carrier, namely,
`
`w(t) = 2af,t+ 0
`
`(3.2)
`
`is varied linearly with the message signal z(t).
`
`Angle modulation may itself be classified into two kinds:
`
`2.1. Frequency modulation, in which the frequency of the carrier, f,, is varied linearly
`with the messagesignal m(t).
`2.2. Phase modulation, in which the phase of the carrier, 0, is varied linearly with the
`message signal m(t).
`
`In historical terms, the design of communication systems was dominated by analog
`modulation techniques. Nowadays, however, wefind that the use of digital modulation
`techniquesis the methodof choice, due to the pervasive use of silicon chips and digital
`signal-processing techniques. For this reason, the focus in what follows is on digital
`modulation techniques.
`
`3.3.
`
`LINEAR MODULATION TECHNIQUES
`
`3.3.1 Amplitude Modulation!
`
`By definition, amplitude modulation (AM), produced by an analog message signal
`m(t),is described by
`
`s(t) = A.(1+k,m(t))cos(27f,t)
`
`(3.3)
`
`where k, is the sensitivity of the amplitude modulator. For convenience of presenta-
`tion, we have set the carrier phase @ equalto zero,as it has no bearing whatsoever on
`the transmission of information.
`Appendix A briefly reviews Fourier theory, which is basic to the spectral analy-
`sis of signals. In light of the theory presented therein, we may portray the spectral
`characteristics of amplitude modulation as illustrated in Fig. 3.4. The figure clearly
`showsthat the bandwidth of the AM signal s(t) is 2W, where W is the bandwidth of
`m(t) itself. Most important, except for the frequency shift in the spectrum of the
`message signal m(r), denoted by M(f), and the retention of the carrier, exemplified
`by the impulses at +f., amplitude modulation has no other effect on the spectrum
`S(f) of s(t).
`
`Show that amplitude modulationis a nonlinear process, as it violates the prin-
`Problem 3.1
`ciple of superposition.
`a
`
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`

`
`
`Section 3.3. Linear Modulation Techniques
`
`109
`
`M(f)
`
`
`A.
`
`5 BF + f)
`
`
`
`Delta function
`A,
`v3” OCF fe)
`
`“fe W “fe —ft+W 0G
`(b)
`
`f,-W te
`
`i+w f
`
`FIGURE 3.4
`
`(a) Message spectrum. (b) Spectrum of corresponding AM signal.
`
`Note, however, that the violation of the principle of superposition described in
`Problem 3.1 is of a mild sort which permits the application of the Fourier transform to
`an AM signal, as described in Fig. 3.4.
`Another important point to note is that, insofar as information transmissionis
`concerned, retention ofthe carrier in the composition of the AM signal represents a
`loss of transmitted signal power. To mitigate this shortcoming of amplitude modula-
`tion, the carrier is suppressed, in which case the process is referred to as double
`sideband-suppressed carrier (DSB-SC) modulation. Correspondingly, the DSB-SC
`modulated signal is defined simply as the product of the message signal m(t) and
`the carrier c(t); that is,
`
`s(t)=c(t)m(t)
`Am(t) cos(2.Af,t)
`G4)
`
`Figure 3.5 depicts the spectrum of the new s(t). Comparing this spectrum with that of
`Fig. 3.4, we see clearly that the absence of the delta functionsat +f, is testimonyto the
`suppression of the carrier in Eq. (3.4). Nevertheless, the AM signal of Eq. (3.3) and
`the DSB-SC modulated signal of Eq. (3.4) do share a common feature: They both
`require the use of a transmission bandwidth equal to twice the message bandwidth,
`namely, 2W.
`
`
`
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`

`
`
`110
`
`Chapter 3. Modulation and Frequency-Division Multiple Access
`
`M(f)
`
`W
`
`f
`
`-W
`
`0
`(a)
`
`S(f)
`
`|
`
`“.-W fh
`
`k+W 0
`
`(b)
`
`f-W ff.
`
`ftW ff
`
`FIGURE 3.5
`signal.
`
`(a) Message spectrum. (b) Spectrum of corresponding DSB-SC modulated
`
`Problem 3.2 Consider the sinusoidal modulating signal
`
`m(t) = A,,cos(2Tf,,t)
`
`Showthat the use of DSB-SC modulation producesa pairof side frequencies, one at f,+/,, and
`the other at f.—/,,, where f, is the carrier frequency. Whatis the condition that the modulator
`hasto satisfy in order to make sure that the two side-frequencies do not overlap?
`n°
`Ans. f.>f,
`
`a
`
`3.3.2 Binary Phase-Shift Keying
`
`Consider next the case of digital modulation in which the modulatingsignal is in the
`form of a binary data stream. Let p(t) denote the basic pulse used in the construction
`of this stream. Let T denote the bit duration (i.e., the duration of binary symbol 1 or 0).
`Then the binary data stream, consisting of a sequenceof 1’s and 0’s, is described by
`
`where
`
`m(t) = ¥byp(t—kT)
`k
`
`b, = | +1
`
`-1
`
`for leary symbol 1
`
`for binary symbol 0
`
`(3.5)
`
`(3.6)
`
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`

`Section 3.3 Linear Modulation Techniques=111
`
`AD
`
`6
`
`T
`
`t
`
`FIGURE 3.6 Rectangularpulse.
`
`For example,in the case of a rectangular pulse we have
`
`ptt) _
`
`+]
`0
`
`for O<¢<T
`otherwise
`
`(3.7)
`
`
`
`which is depicted in Fig.3.6.
`In binary phase-shift keying (BPSK), the simplest form of digital phase modula-
`tion, the binary symbol 1 is represented by setting the carrier phase @(¢) =0 radians,
`and the binary symbol0 is represented by setting 6(f) = radians. Correspondingly,
`
`s(t) =
`
`A,cos(27f,t)
`A,cos(2nf,t+#)
`
`1
`for binary symbol
`for binary symbol 0
`
`(3.8)
`
`Recognizing that
`
`cos(O(4) +7) = —cos(@(1)) for all time z,
`
`we may rewrite Eq. (3.8) as
`
`s(t) =
`
`A,cos(2nf.é)
`-A ,cos(2mf.t)
`
`for symbol 1
`for symbol 0
`
`(3.9)
`
`In light of Egs. (3.6), (3.7), and (3.9), we may express the BPSK signal in the compact
`form
`
`s(t) = e(Hm(t)
`
`(3.10)
`
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`

`
`
`112
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`where m/(f) is itself defined by Eq. (3.5). Most important, Eq. (3.10) shows that BPSK
`is another example of linear modulation.
`
`Problem 3.3 Consider a binary data stream m(r) in the form of a square wave with ampli-
`tudes +1, centered on the origin. Determine the spectrum of the BPSK signal obtained by multi-
`plying m/(t) by a sinusoidal carrier whose frequency is ten times that of the fundamental
`frequency of the square wave.
`Ans.
`
`t=
`
`Se 2n—t
`
`2af.t
`
`
`
`==—>= + 2nt| f_-—
`
`
`
`( Kf, )
`sin(km/2)
`cos|
`207 cos(27/1)
`s(t) _ Tka7D)
`( ( 2)
`ane (2 ( =)
`1
`3 EATLOSE * Ty)PM Ve“TH
`
`;
`; he,
`.
`;
`.
`The spectrum of the BPSK signal consists of side-frequencies at f,, = To with decreasing ampli-
`tude in accordance with sin (k/2)/(kn/2) , Where k =1,3,5....
`a
`
`3.3.3 Quadriphase-Shift Keying
`
`As with DSB-SC modulation, BPSK requires a transmission bandwidth twice the
`message bandwidth. Now, channel bandwidth is a primary resource that should be
`conserved, particularly in wireless communications. How then can we retain the
`property of linearity that characterizes BPSK, yet accommodate the transmission of
`a digital phase-modulated signal over a channel whose bandwidth is equal to the
`bandwidth of the incoming binary data stream? The answer to this fundamental
`questionlies in the use of a digital modulation technique known as quadriphase-shift
`keying (QPSK).
`To proceed with a description of QPSK, suppose the incoming binary data
`stream is first demultiplexed into two substreams, m,(f) and (ft). Next, note that, as
`the name implies, the phase of the carrier in QPSK assumesone of four equally spaced
`values, depending on the composition of each dibit, or group of two adjacentbits, in
`the original binary data stream. For example, we may use 0, 7/2, 2, and 37/2 radians as
`the set of four values available for phase-shift keying the carrier. Specifically, the val-
`ues 0 and radians are used to phase-shift key one of the two substreams, m,(f), and
`the remaining values, 7/2 and 32/2 radians, are used to phase-shift key the other sub-
`stream, m>(f).
`Accordingly, we can formulate the block diagram for the QPSK modulator as
`shownin Fig. 3.7. On the basis ofthis structure, we can then describe the OPSK modu-
`lator as the parallel combination of two BPSK modulators that operate in phase
`quadrature with respect to each other. By phase quadrature, we mean the arrangement
`of phases such that the phase of the carrier in the lower path of the modulator is 90°
`out of phase with respect to the carrier in the upper path.
`
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`

`Section 3.3. Linear Modulation Techniques
`
`113
`
`m4
`
`
`Acos{2af.t)
`Binary data
`
`stream
`mir)
`
`5)
`Demultiplexer A sin{27ft)
`
`OPSKsignal
`s(Z)
`
`FIGURE 3.7 Block diagram of a OPSK generator, using a phase-quadrature pair of carriers
`A,cos(2af.2} and A,sin(2af.0).
`
`As remarked previously, m,(f} and s(t) denote the two binary substreams that
`result from demultiplexing of the binary data stream w(t). Extending the mathemati-
`cal description of Eq. (3.5) to the situation at hand, we may express the corresponding
`descriptions of binary substreams m1(/) and m7>(¢) as follows:
`
`
`
`mi) = Sb, :p(t- kT)
`k
`
`forf=1,2
`
`For i= 1,2 we have
`
`be, =
`
`+1
`~1l
`
`1
`for symbol
`for symbol 0
`
`and for the case of a rectangular pulse,
`
`p(t} =
`
`+1
`0
`
`for O<¢<2T
`otherwise
`
`Then the BPSKsignal produced in the upper path of Fig. 3.7 is described by
`
`The BPSKsignal produced in the lower path of Fig. 3.7 is described by
`
`8,(1) = Amy (t}cos(2 77,1)
`
`So(t) = A,my(t)sin(2Af,0)
`
`The OPSKsignal is obtained by adding these two BPSKsignals:
`li
`
`s(t)
`
`31(t) + s2(4)
`= Am,(t)cos(2 aft) + A,m,(fsin (2772)
`
`BAL)
`
`(3.12)
`
`(3.13)
`
`(3.14)
`
`(3.15)
`
`(3.16)
`
`
`
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`114
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`Since both BPSKsignals s;(t) and s(t) are linear, the QPSK signal s(t) is likewise
`linear.
`The transmission bandwidth requirement of the QPSK signals(t) is the same as
`that of the original binary data stream m(f). We justify this important property of
`OPSKsignals as follows:
`
`The original binary data stream m/(t) is based on bits, whereas the substreams
`m(t) and m(t) are based on dibits. The symbol duration of both (tf) and m»(t)
`is therefore twice the symbol duration of m(f).
`The bandwidth of a rectangular pulse is inversely proportional to the duration
`of the pulse. Hence, the bandwidth of both m (rt) and 77(f) is one-half that of
`m(t).
`The BPSKsignals s;(t) and s(t) have a common transmission bandwidth equal
`to twice that of m4(t) or m(t).
`The OPSKsignals(t) has the same transmission bandwidth as s;(¢) or s(t).
`Hence, the transmission bandwidth of the OPSKsignalis the sameas that of the
`original binary data stream m(f).
`
`EXAMPLE 3.1 QPSK Waveform
`
`Figure 3.8(a) depicts the waveform of a OPSKsignal for which the carrier phase assumes one of
`the four possible values 0°, 90°, 180°, and 270°. Moreover, the waveform is the result of transmit-
`ting a binary data stream with the following composition over the interval 0 <¢s 107:
`e
`
`The input dibit (i.e., the pair of adjacent bits in m(r)) changes in going from the interval
`0<1<2T to the next interval 27 <¢<4T.
`
`In going from the interval 27 <1<4T to the next interval 47 <¢< 67, there is no change in
`the input dibit.
`The input dibit changes again in going from the interval 47 <¢< 67 to 67 <t<8T.
`The input dibit is unchanged in going from the interval 6T < ¢ < 8T to the next interval
`8T <t<10T.
`a
`
`Examining the waveform of Fig. 3.8(a), we see that OPSKsignals exhibit two unique
`properties:
`
`1.
`
`2.
`
`The carrier amplitude is maintained constant.
`The carrier phase undergoes jumps of 0°, +90° or +180° every 2T seconds, where
`T is the bit duration of the incoming binary data stream.
`
`3.3.4 Offset Quadriphase-Shift Keying
`
`Property 2 of the conventional QPSKsignal, namely, the fact that the carrier phase
`may jump by 90° or +180° every two bit durations can be of particular concern when
`the OPSKsignalis filtered during the course of transmission over a wireless channel.
`
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`
`
`Section 3.3. Linear Modulation Techniques
`
`115
`
`Amplitude
`
`Amplitude
`
`Amplitude
`
`>
`
`0
`
`2T
`
`4F
`
`6T
`Time
`
`(c)
`
`8T
`
`10F
`
`FIGURE 3.8 Waveforms of (a} conventional OPSK (b) offset OPSK, and (c) 2/4-shifted
`OPSK.
`
`Thefiltering action can, in turn, cause the carrier amplitude (Le., the envelope of the
`OPSKsignal) to fluctuate, thereby making the receiver produce additional symbol
`errors over and above those due to channel noise.
`The extent of amplitude fluctuations exhibited by conventional QPSK signals
`may be reduced by using offset quadriphase-shift keying (OQ@PSK), which is also
`referred to as staggered QPSK.In this variant of OPSK, the second substream miy(/),
`multiplied by the 90° phase-shifted carrier A,sin(2af.0), is delayed (i.e., offset) by a bit
`duration 7 with respect
`to the first substream om,(f), multiplied by the cartier
`A,cos(2xf,i). Accordingly, unlike the phase transitions in conventional QPSK, the
`phase transitionslikely to eccur in offset QPSK are confined to 0°, 190°, as illustrated
`in the next example. However, the +90° phase jumps in OOPSK occur twice as fre-
`quently, but with a reduced range of amplitude fluctuations, compared with those of
`conventional QPSK. Since, i addition to 290° phase jumps, there are +180° phase
`jumps in conventional OPSK,we usually find that amplitude fluctuations in OOPSK
`due to filtering have a smaller amplitude than in conventional OPSK.
`
`
`
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`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`EXAMPLE 3.2 OQPSK Waveform
`
`Part (b) of Fig. 3.8 depicts the waveform of the OOPSKfor the same binary data stream respon-
`sible for generating the conventional OPSK waveform depicted in part (a) of the figure. Here
`again, we see that the carrier amplitude of OQPSKis maintained constant. However, unlike the
`carrier phase in the conventional OPSKofFig. 3.8(a), the carrier phase of the OQPSK shownin
`Fig. 3.8(b) has jumpsof only +90°.
`=
`
`3.3.5 a/4-Shifted Quadriphase-Shift Keying
`
`As mentioned previously, ordinarily the carrier phase of a conventional QPSKsignal
`mayreside in one of two possible discrete settings:
`
`1. 0, 2/2, x, or 37/2 radians.
`2. 1/4, 37/4, 57/4, or 77/4 radians.
`
`These two phase settings are shifted by 7/4 radians relative to each other. The OPSK
`waveform depicted in Fig. 3.8(a) follows setting 1. In another variant of OPSK known
`as m/4-shifted QPSK,the carrier phase used for the transmission of successive dibits is
`alternatively picked from settings 1 and 2.
`An attractive feature of 7/4-shifted OPSKsignals is that amplitude fluctuations
`duetofiltering are significantly reduced, compared with their frequency of occurrence
`in conventional QPSKsignals. Thus, the use of 2/4-shifted QPSK provides the band-
`width efficiency of conventional OPSK, but with a reduced range of amplitudefluctua-
`tions. The reduced amplitude fluctuations become important when the transmitter
`includesa slightly nonlinear amplifier, as we shall see in Section 3.9. Indeed,it is for this
`reason that 7/4-shifted QPSK has been adopted in the North American digital cellular
`time-division multiple access (TDMA) standard, IS-54 as well as the Japanese digital
`cellular standard.”
`
`EXAMPLE 3.3 2/4-Shifted QPSK Waveform
`
`Figure 3.8(c) depicts the z/4-shifted OPSK waveform produced by the same binary data stream
`used to generate the conventional OPSK waveform of Fig. 3.8(a). Comparing these two wave-
`forms, we see that (1) the phase jumpsin the z/4-shifted QPSKarerestricted to +m/4 and +372/4
`radians and (2) the +m phase jumps of OPSKare eliminated—hence the advantage of z/4-
`shifted OPSK over conventional OPSK. However,this advantageis attained at the expense of
`increased complexity.
`=
`
`3.4
`
`PULSE SHAPING
`
`The pulse defined in Eq. (3.7) for representing binary symbol 1 or 0 is rectangular in
`shape. Fromapractical perspective, the use of a rectangular pulse shape is undesirable
`for two fundamental reasons:
`
`1. The spectrum(i.e., Fourier transform) of a rectangular pulseis infinite in extent.
`Correspondingly, the spectrum of a digitally modulated signal based on the use
`
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`

`

`Section 3.4 Pulse Shaping
`
`117
`
`of a rectangular pulse is shifted by an amount equal to the carrier frequency but,
`most important, its frequency content is also infinite in extent. However, a wire-
`less channel is bandlimited, which means that the transmission of such a digi-
`tally modulated signal over the channel will introduce signal distortion at the
`receiving end.
`2. A wireless channel has memory due to the presence of multipath. Consequently,
`the transmission of a digitally modulated signal over the channel results in a spe-
`cial form of interference called intersymbolinterference (ISI), whichrefers to inter-
`ference between consecutive signaling symbols of the transmitted data sequence.
`
`Now, in a wireless communication system, the goal is to accommodate the largest pos-
`sible number of users in a prescribed channel bandwidth. To satisfy this important
`requirement we must use a premodulation filter, whose objective is that of pulse
`shaping. Specifically, the shape of the basic pulse used to generate the digitally modu-
`lated signal must be designed so as to overcomethe signal distortion and ISI problems
`cited under points 1 and 2.
`The design criteria for pulse shaping are covered by the fundamental theoretical
`work of Nyquist.> Let P(f) denote the overall frequency response made up of three
`components: the transmit filter, the channel, and the receive filter. According to
`Nyquist, the effect of intersymbol interference can be reduced to zero by shaping the
`overall frequency response P(f) so as to consist of a flat portion and sinusoidalrodloff
`portions, as illustrated in Fig. 3.9(a). Specifically, for a data rate of R bits/second, the
`channel bandwidth may extend from the minimum value W=A/2 to an adjustable
`value from W to 2W by defining P(/) as follows:
`
`
`
`
`
`Pf} sy
`
`—_
`
`<
`i
`Os If Shi
`Ti
`}
`ng
`( Th
`c
`1
`gpttsylfl-WA-py)) Asifl<2MA
`[f, 22V-f,
`0
`
`—_
`
`_
`
`~
`
`=
`
`2h
`
`(3.17)
`
`The frequency parameter f, and bandwidth W are related by the parameter
`
`fi
`(3.18)
`p=1l-_
`;
`3,18
`=l-Z
`Called the roll-offfactor, p indicates the excess bandwidth over the ideal solution cor-
`responding to p = 0. An important characteristic of the frequency response P(f) is
`that its inverse Fourier transform, denoted by p(s) (Le., the overall impulse response of
`the transmit filter, the channel, and the receivefilter) has the value of unity at the cur-
`rent signaling instant (ic, p(0) = 1) and zero crossings at all other consecutive signal-
`ing instants (i.e., p(nT)} = 0 for nonzero integer n), as shown in Fig. 3.9(b). The zero
`crossings of the impulse response p(r) ensure that the [SI problem is reduced to zero,
`The frequency response of Fig. 3.9(a) is called the raised-cosine (RC) spectrum, so
`called becauseofits trigonometric form as defined in Eq. (3.17).
`
`
`
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`118
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`Chapter 3. Modulation and Frequency-Division Multiple Access
`
`The variability of the rolloff factor p over the range (0,1) allows the designer to
`tradeoff transmitted signal bandwidth for robustness of the pulse shape.
`
`
`
`
`
`2WP(f)
`
`P(t)
`
`(a) Frequency responseof the raised cosine spectrum for varying roll-off rates.
`FIGURE 3.9
`(b) Impulse response of the Nyquist shapingfilter (i.e., inverse Fourier transform of the
`spectrum plotted in part (a) for varying roll-off rates).
`
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`
`
`Section 3.4 Pulse Shaping
`
`119
`
`Problem 3.4
`
`(a) Starting with the RC spectrum P(/) of Eq. (3.17), evaluate the inverse Fourier transform
`of P(f) and thus show that.
`
`p(t) = (<eeCxpihe|sinc (20
`1-t6p° Hs
`(b) Determine P(f) and p(t) for the special case of p = 1, which is known as the fili-cosine
`roll-offpluse.
`Ans, (b) p(t) = sinc(4W/(1 - 169°)
`
`(3.19)
`
`J.
`
`3.4.1 Root Raised-Cosine Pulse Shaping*
`
`A more sophisticated form of pulse shaping uses the root raised-cosine (RC) spec-
`frum vather than the regular RC spectrum of Eq. (3.17). Specifically, the spectrum of
`the basic pulse is now defined by the square root of the right-hand side of this equa-
`tion. Thus, using the trigonometric identity
`cos’6 = stl + cos?@)
`
`where, for the problem at hand,
`
`g= " (f|-wa-
`a9Blt |-W1-p))
`and retaining P(f) as the symbol for the root RC spectrum, we may write
`
`Oslfish
`aR
`PO = eglawgll/l- ae] A Slfi<2W-s
`
`0
`
`If|22W¥-f,
`
`(3.20)
`
`where, as before, the roll-off factor p is defined in terms of the frequency parameter
`f, and the bandwidth W asin Eq. (3.18).
`If, now, the transmitter includes a pre-modulation filter with the transfer func-
`tion defined in Eq. (3.20) and the receiver includes an identical filter, then the overall
`pulse waveform will experience the spectrumP(f) , which is the regular raised cosine
`spectrum. In effect, by adopting the root RC spectrum P(f) of Eq. (3.20) for pulse
`shaping, we would be working with P’(/) in an overall transmitter-receiver sense. On
`this basis, we find that in the context of wireless communications, if the channel is
`affected by both flat fading and additive white Gaussian noise, and the pulse-shapefil-
`tering is partitioned equally between the transmitter and the receiver in the manner
`described herein, then effectively the receiver would maximize the output signal-to-
`noise ratio at the sampling instants,
`
`
`
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`
`
`120
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`oT
`
`(b)
`
`FIGURE 3.10
`spectrum.
`
`(a) P(f) for rootraised - cosine spectrum.(b) p(t) for root raised - cosine
`
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`

`

`The inverse Fourier transform of Eq. (3.20) defines the root RC shaping pulse
`
`Section 3.4 Pulse Shaping
`
`121
`
`pti) = OW(sinap)0+ 4Pcos(2K + py) (3.21)
`
`
`(1- (8p) )
`Jat
`t
`
`
`
`The important point to note here is the fact that the root RC shaping pulse p(t) of
`Eq.(3.21) is radically different from the standard RC shaping pulse of Eq. (3.19). In
`particular, the new shaping pulse satisfies an orthogonality constraint under T-shifts,
`as shown by
`
`Fp@ptt~nTat =0forn=+41, #2...
`
`©
`
`(3.22)
`
`where Tis the symbol duration. Yet, p(‘) has exactly the same excess bandwidth as the
`standard RC pulse.
`It is important to note, however, that despite the added property of orthogonal-
`ity, the root RC shaping pulse of Eq.(3.21) lacks the zero-crossing property of the reg-
`ular RC shaping pulse defined in Eq. (3.19).
`Figure 3.10(a) plots the root RC spectrum P(f) for roll-off factor p = 0, 0.5, 1; the
`corresponding time-domain plots are shown in Fig. 3.10(b). These plots are different
`from those of Fig. 3.9 for nonzero p. The following example contrasts the waveform of
`a specific binary sequence using the root RC shaping pulse with the corresponding
`waveform using the regular RC shaping pulse.
`
`EXAMPLE 3.4 Pulse Shaping Comparison
`
`Using the root RC shaping pulse p(#) of Eq.(3.21) with roll-off factor p= 0.5, plot the waveform
`for the binary sequence 01100, and compare it with the corresponding waveform obtained by
`using the regular RC shaping pulse p(4) of Eq. (3.19) with the same roll-off factor.
`Using the root RC pulse p(Z) of Eq. (3.21) with a multiplying plus sign for binary symbol1
`and multiplying minussign for binary symbol 0, we get the dashed and dotted pulse train shown
`in Fig. 3.11 for the sequence 01100. The solid pulse train shownin the figure corresponds to the
`use of the regular RC pulse p(¢) of Eq. (3.15). The figure shows the root RC waveform occupies
`a larger dynamic range than the regular RC waveform.
`a
`
`Problem 3.5
`
`(a) Starting with Eq. (3.20), derive the root RC pulse shape p(t) of Eq. (3.21).
`{b) Evaluate p(i) at (i}r=0, and (ii) ¢ = 41/(8pF).
`(c) Show that the p(s) derived in part (a) satisfies the orthogonality constraint described in
`Eq. (3.22)
`Ans.(b)
`(i) p(0) = im ~p+ 4e)
`(i) = PAEa(t Zh)
`
`
`
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`122
`
`Chapter 3. Modulation and Frequency-Division Multiple Access
`
`
`
`Raised Cosine
`—-— Sqrt Raised Cosine
`
`
`
`
`
`
`
`
`
` Pulsep(é)
`
`
`
`Binary sequence
`
`el:
`
`=i
`
`+]
`Timet/T;,
`
`—
`
`ol
`
`FIGURE 3.11 Two pulse trains for sequence 01100, one using regular RC pulse and the other
`using root RC pulse.
`
`3.5 COMPLEX REPRESENTATION OF LINEAR MODULATED SIGNALS
`AND BAND-PASS SYSTEMS?
`
`The linear modulation schemes considered in Section 3.3 may be viewed as special
`cases of the canonical representation of a band-pass signal:
`
`s(t) = s/(t)cos(22f,1) —so()sin (2af,t)
`
`(3.23)
`
`It is customaryto refer to sj/(¢) as the in-phase componentof s(t) and to s¢(t) as the
`quadrature component. This terminology follows from the definition of the sinusoi-
`dal carrier c(t) in Eq. (3.1). Table 3.1 summarizes descriptions of AM, DSB-SC, BPSK,
`and QPSKin terms of the componentss(t) and s¢(v).
`We maysimplify matters further by introducing the complex signal
`
`S(t) = s,(t) +jso()
`
`(3.24)
`
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`

`
`
`Special Cases of the Canonical Equation (3.23),
`
`Type of modulation
`
`In-phase component
`st)
`
`Quadrature component
`sg(t}
`
`Defining
`equation
`
`Amplitude modulation
`
`AU + k(t)
`
`0
`
`3.3
`
`
`
`Section 3.5 Complex Representation of Linear Modulated Signals and Band-Pass Systems
`
`123
`
`TABLE 3.1
`
`Analog
`
`|
`|
`
`
`
`Double sideband-
`suppressed carrier modulation
`
`Am(t)
`
`Digital
`
`Binary phase-shift keying
`
`A> byp(t~ kT)
`k
`
`a
`
`G
`
`3.4
`
`|
`
`35
`
`Quadriphase-shitt keying Ayby wp(t—2kT)
`
`k
`
`ASby op(t—2kT)
`
`3.11
`
`& w
`
`here j is the square root of -1. For obvious reasons, the new signal $(f) is referred to
`as the complex envelope of the modulated signal s(r). Next, we invoke Euler's for-
`mula
`
`exp(j2af_t) = cos(2xf,t) +/sin(2af.p
`
`(3.25)
`
`Hence,in light of Eqs. (3.24) and (3.25), we may considerably simplify the formulation
`of the modulated signal s(t) of Eq. (3.23) as
`
`s(f) = Re{sespti2ar|
`
`(3.26)
`
`Equation (3.26)is referred to as a single-carrier transmission.
`‘The material presented in this section is of profound theoretical importance in
`the study of linear modulation theory, beit in the context of analog ordigital modula-
`tion techniques. Specifically, we may make the following four statements:
`
`1. The in-phase components/(t) and the quadrature component So(t) are both real-
`valued functionsof time that are uniquely defined in terms of the baseband(i.c.,
`message) signal m(t). Given the two components st) and Sg(4), we may thus use
`the scheme shownin Fig, 3.12(a) to synthesize the modulated signals(t).
`2. Given the modulated signal s(t), we may use the scheme shownin Fig. 3.12(b) to
`analyze the modulated signals(f) and thereby construct the in-phase component
`s,(t) and the quadrature component sald).
`3. The in-phase and quadrature components are orthogonal to each other, occupy-
`ing exactly the same bandwidth as the message signal m/().
`4. The complex envelope 5(t) given in Eq. (3.24) completely preserves the infor-
`mation contentof the modulatedsignals(#) exceptfor the carrier frequencyf..
`
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`124
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`sit)
`
`x)
`
`Modulatedsignal
`One
`
`In-phase component
`
`90° phase
`shifter
`
`
`
`QQ)
`
`5)
`
`Quadrature component
`
`(a)
`
`
`In-phase component
`s(t)
`Low-pass
`filter
`
`
`
`
`
`Modulated signal
`s(t)
`
`
`Quadrature component
`Low-pass
`
`y
`so(t)
`filter
`
`(b)
`
`(a) Synthesizer for constructing a modulatedsignalfrom its in-phase and
`FIGURE 3.12
`quadrature components. (b) Analyzer for deriving the in-phase and quadrature components
`of the modulated band-passsignal.
`
`3.5.1 Complex Representation of Linear Band-Pass Systems
`
`In a communication system, modulators exist alongside linear band-pass systems rep-
`resented by band-pass filters and narrowband communication channels. As with any
`otherlinear system, these band-pass systems are uniquely characterized by an impulse
`response in the time domain and the corresponding transfer function in the frequency
`domain. From an analytic viewpoint, we find it highly instructive to develop complex
`representations of linear band-pass systems in a manner analogous to that followed
`for linear modulated(i.e., band-pass) signals. This form of representation not only sim-
`plifies the mathematical analysis of communication systems, but also provides the
`basis for their simulation on a digital computer.
`Consider, then, a linear band-pass system with impulse response A(t) and fed by
`an input signal x(f) to produce an output signal y(s), as depicted in Fig. 3.13. Two
`assumptions are made:
`
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`

`Section 3.5 Complex Representation of Linear Modulated Signals and Band-Pass Systems
`
`125
`
`0
`
`a
`
`A(t)
`
`(a)
`
`hit)
`(b)
`
`wo)
`
`2¥()
`
`|
`
`(a) Block diagram oflinear band-pass system driven by a modulatedsignal x(1)
`FIGURE 3.13
`to preduce