`
`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 ofmultiple-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 access is a signal-processing operation that makes this pos-
`sible. In particular, it permits the simultaneous transmission of information-
`bearing signals from a number of independent users over the channel and on to
`their respective destinations.
`
`In wireless communications, the carrier, denoted by C(t), is typically sinusoidal and is
`written as
`
`C(t) = Accos(27rfcr+ 6)
`
`(3.1)
`
`where AC is the amplitude, fc is the frequency, and dis 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 wh at follows.
`
`3.2.1 Linear and Nonlinear Modulation Processes
`
`Figure 3.3 shows the block diagram of a modulator supplied with a sinusoidal carrier
`C(t). The modulating signal, acting as input, is denoted by 1710:). The modulated signal,
`acting as output, is denoted by s(t). The inputeoutput relation of the modulator is gov-
`erned by the manner in which the output s(t) depends on the input m(t). On this basis,
`we may classify the modulation process as one of two basic types: linear and nonlinear.
`
`|
`
`I
`
`Input
`(modulating)
`signal
`MU)
`
`Output
`(modulated)
`signal
`m)
`
`Modulator
`
`Sinusoidal
`carrier
`c(r)
`
`FIGURE 3.3 Block diagram of modulator.
`
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`

`5%
`
`l1
`
`i i l
`
`[Ii
`
`
`
`Section 3.2 Modulation
`
`107
`
`The modulation process is said to be linear if the inputwoutput relation of the
`modulator satisfies 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 number of 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 same factor.
`
`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 shall see in the remainder
`of the chapter.
`
`3.2.2 Analog and Digital Modulation Techniques
`
`
`
`
`
`Another way of classifying the modulation process is on the basis of whether the mes-
`sage signal m(r) 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 t. Consequently, the
`modulated signai s(t‘) 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 30‘) may exhibit discontinuiw
`ties at the instants of time at which the message signal m(t) switches from symbol 1 to
`symbol 0 or vice versa. Note, however (as we will find out later on in the chapter), that
`under certain conditions it is possible Jfor the modulated signal to maintain continuity
`even at the instants of switching.
`In other words, we may distinguish between analog and digital modulation pro—
`cesses as follows:
`
`0 All analog modulated signals are continuous functions of time.
`
`- Digitai modulated signals can be continuous or discontinuous functions of time,
`depending on how the modulation process is performed.
`
`3.2.3 Amplitude and Angie Modulation Processes
`
`Yet another way of classifying 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(t).
`Accordingly, we speak of two kinds of modulation:
`
`1. Amplitude modulation, in which the amplitude of the carrier, AC, is varied
`linearly with the message signal ma).
`
`
`
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`

`
`
`108
`
`Chapter 3 Modulation and Frequency—Division Multiple Access
`
`2. Angie modulation, in which the angle of the carrier, namely,
`
`1,110) = 27.11;: + 9
`
`(3.2)
`
`is varied linearly with the message signal m(t).
`
`Angle modulation may itself be classified into two kinds:
`
`2.1. Frequency modulation, in which the frequency of the carrier, fc, is varied linearly
`with the message signal m(r).
`
`2.2. Phase modulation, in which the phase of the carrier, 9, is varied linearly with the
`message signal m(z).
`
`In historical terms, the design of communication systems was dominated by analog
`modulation techniques. Nowadays, however, we find that the use of digital modulation
`techniques is the method of 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 Modulation1
`
`By definition, amplitude modulation (AM), produced by an analog message signal
`m(i), is described by
`
`s(t) = AC(1 + kam(t))cos(27rfct)
`
`(3.3)
`
`where ka is the sensitivity of the amplitude modulator. For convenience of presenta-
`tion, we have set the carrier phase 9 equal to 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
`shows that 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(t), denoted by MU), and the retention of the carrier, exemplified
`by the impulses at ifc, amplitude modulation has no other effect on the spectrum
`S(f) of 50?).
`
`Show that amplitude modulation is a nonlinear process, as it violates the prin-
`Problem 3.1
`ciple of superposition.
`I
`
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`

`

`
`
`Section 3.3 Linear Modulation Techniques
`
`109
`
`MU)
`
`Delta function
`
`3w so — f.)
`
`A.
`
`('3)
`
`FIGURE 3.4
`
`(a) Message spectrum. (b) Spectrum of. corresponding AM signai.
`
`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 transmission is
`concerned, retention of the carrier in the composition of the AM signal represents a
`loss of transmitted signal power. To mitigate this shortcoming of ampiitude modula-
`tion, the carrier is suppressed, in which case the process is referred to as double
`sidebandmsuppressed carrier (DSBMSC) modulation. Correspondingly, the DSB—SC
`modulated signai is defined simply as the product of the message signal m(r) and
`the carrier C(t); that is,
`
`30‘)
`
`C(t)m(r)
`Ach) cos(27rfct)
`
`(3'4)
`
`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 functions at ifc is testimony to the
`suppression of the carrier in Eq. (3.4). Nevertheless, the AM signal of Eq. (3.3) and
`the DSBWSC modulated signal of Eq. (3.4) do share a common feature: They both
`require the use of a transmission bandwidth equai to twice the message bandwidth,
`namely, 2W.
`
`
`
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`

`
`
`110
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`MU)
`
`W
`
`f
`
`— W
`
`0
`(a)
`
`Sif)
`
`‘1”: , W
`
`if.-
`
`if; t W
`
`0
`(b)
`
`)‘L- * W
`
`fr
`
`f: t W
`
`f
`
`FIGURE 3.5
`signal.
`
`(a) Message spectrum. (1)} Spectrum of corresponding DSB—SC modulated
`
`Problem 3.2 Consider the sinusoidal modulating signal
`
`mm = Amcos(2nf,nt)
`
`Show that the use of DSB-SC modulation produces a pair of side frequencies, one at fC +fm and
`the other at fE —fm ,where fc is the carrier frequency. What is the condition that the modulator
`has to satisfy in order to make sure that the two side-frequencies do not overlap?
`3'."
`Ans. .fL >f,
`
`I
`
`3.3.2 Binary Phase-Shift Keying
`
`Consider next the case of digital modulation in which the modulating signal 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 sequence of 1’s and 0’s, is described by
`
`m(t‘) = EbkpU—kfl
`k
`
`where
`
`bk : { +1
`
`#1
`
`for binary symbol 1
`
`tor binary symbol 0
`
`(3‘5)
`
`(3.6)
`
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`

`

`Section 3.3 Linear Modulation Techniques
`
`111
`
`MI)
`
`1.0
`
`O
`
`T
`
`t
`
`FIGURE 3.6 Rectangular pulse.
`
`For example, in the case of a rectangular pulse we have
`
`p(t) ={ +1
`
`0
`
`for OStET
`
`otherwise
`
`(37)
`
`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 6(t) = 0 radians,
`and the binary symbol 0 is represented by setting 9(r) = Irradians. Correspondingly,
`
`5(1) = Accos(2nfct)
`Accos(27cfct+a)
`
`1
`for binary symbol
`for binary symbol 0
`
`(3.8)
`
`Recognizing that
`
`cos(9(t)+n) = —cos(6(t)) for all time t,
`
`we may rewrite Eq. (3.8) as
`
`SO) I
`
`Accosanfct)
`—Accos(211:fct)
`
`for symbol 1
`for symbol 0
`
`(3‘9)
`
`In light of Eqs. (3.6), (3.7), and (3.9), we may express the BPSK signal in the compact
`form
`
`s(t) = c(t)m(r)
`
`(3.10)
`
`
`
`
`
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`

`

`
`
`112
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`where m(t) 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 i1, centered on the origin. Determine the spectrum of the BPSK signal obtained by multi-
`plying m(r) by a sinusoidal carrier whose frequency is ten times that of the fundamental
`frequency of the square wave.
`Ans.
`
`similar/2)
`r =
`_._._..
`s() klgfim (kn/2) cos
`1
`sin(k9r/2)[
`= _
`—.—.—
`2k:l;5
`(kn/2)
`
`(
`
`COS
`
`1‘75 j
`2 __z
`1:10
`(2 :[
`” f.
`
`2
`,r
`cos( Jrf()
`kit-D
`.+_
`10
`
`cos
`
`[ ( kch]
`2 t —_
`’r If 10
`
`.
`.
`.
`kfc
`.
`.
`.
`.
`.
`The spectrum of the BPSK stgmri COHSISLE‘ of stde-fi'equencies at j(, i m With decreasmg ampli-
`tude in accordance with ésiMkJr/ZVWE/Z) , where k 2 1, 3,5....
`l
`
`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
`question lies 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 1(t) and "12(1). Next, note that, as
`the name implies, the phase of the carrier in QPSK assumes one of four equally spaced
`values, depending on the composition of each dibit, or gIOUp of Mo adjacent bits, in
`the original binary data stream. For example, We may use 0, 75/2, 71:, and 3m’2 radians as
`the set of four values available for phase-shift keying the carrier. Specifically, the val-
`ues 0 and Jr radians are used to phase-shift key one of the two substreams, m1(t), and
`the remaining values, 75/2 and 3211’2 radians, are used to phase-shift key the other sub-
`stream, m2(r).
`Accordingly, we can formulate the block diagram for the QPSK modulator as
`shown in Fig. 3.7. On the basis of this structure, we can then describe the QPSK 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
`
`ml“)
`
`510)
`
`
`
`
`Accos{2wfit)
`Binary data
`QPSK signal
`stream
`
`Demultiplexer
`5(1‘)
`Into
`
` Acsin(217fct)
`
`FIGURE 3.7 Block diagram of a Q?SK generator, using a phase-quadrature pair of carriers
`Accos(2flfct} and Acsinanfct).
`
`As remarked previously, m;(t) and m2(t) denote the two binary substreams that
`result from demuitipiexing of the binary data stream m(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(t) and m2(t) as follows:
`
`
`
`fill-(f) 2 2%,:PU—kfl
`k
`
`fort: 1,2
`
`Fori=1,2 we have
`
`bin 2 { +1
`
`W1
`
`and for the case of a rectangular puise,
`
`for symbol
`
`for symbol 0
`
`1
`
`PU) = { +1
`
`0
`
`for OS fSZT
`
`otherwise
`
`Then the BPSK signal produced in the upper path of Fig. 3.7 is described by
`
`The BPSK signai produced in the lower path of Fig. 3.7 is described by
`
`510‘) = Acm1(r)cos(27rfct)
`
`520?) = Acm2(r)sin(27rfct)
`
`The QPSK signal is obtained by adding these two BPSK signals:
`If
`
`3(3)
`
`31(r) +320)
`
`I!
`
`Acm1(t)c0s(2xfct) + Acm2(t)sin(27zfct)
`
`(3.11)
`
`(312)
`
`(313)
`
`(3.14)
`
`(3.15)
`
`(3.16)
`
`
`
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`
`114
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`Since both BPSK signals 51(r) and 520‘) are linear, the QPSK signal s(t) is likewise
`linear.
`
`The transmission bandwidth requirement of the QPSK signal s(t) is the same as
`that of the original binary data stream m(r). We justify this important property of
`QPSK signals as follows:
`
`The original binary data stream m(t) is based on bits, whereas the substreams
`m](r) and m2(t) are based on dibits. The symbol duration of both m1(t) and m2(t)
`is therefore twice the symbol duration of m(t).
`
`The bandwidth of a rectangular pulse is inversely proportional to the duration
`of the pulse. Hence, the bandwidth of both m1(t) and m2(t) is one—half that of
`m(r).
`
`The BPSK signals 51(1) and 32(1) have a common transmission bandwidth equal
`to twice that of m1(t) or m2(t).
`
`The QPSK signal 50:) has the same transmission bandwidth as s10) or 520).
`Hence, the transmission bandwidth of the QPSK signal is the same as that of the
`original binary data stream m(t).
`
`EXAMPLE 3.1 QPSK Waveform
`
`Figure 3.8(a) depicts the waveform of a QPSK signal 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 t S 107':
`0
`
`The input dibit (i.e., the pair of adjacent bits in m(t)) changes in going from the interval
`0 S r S 2T to the next interval 2T S t S 4T.
`
`In going from the interval 2T S tS 4T to the next interval 47" S rS 6T, there is no change in
`the input dibit.
`The input dibit changes again in going from the interval 4TS t S 67‘ to 6T S t S ST.
`The input dibit is unchanged in going from the interval 6T S t S 87‘ to the next interval
`ST S t S 107".
`I
`
`Examining the waveform of Fig. 3.8(a), we see that QPSK signals exhibit two unique
`properties:
`
`1. The carrier amplitude is maintained constant.
`2. The carrier phase undergoes jumps of 0°, :90" or i180° every 2T seconds, where
`T is the bit duration of the incoming binary data stream.
`
`3.3.4 Offset Quadriphase-Shif‘t Keying
`
`Property 2 of the conventional QPSK signal, namely, the fact that the carrier phase
`may jump by 190" or :1800 every two bit durations can be of particular concern when
`the QPSK signal is filtered during the course of transmission over a wireless channel.
`
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`
`
`Section 3.3 Linear Modulation Techniques
`
`115
`
`Amplitude
`
`Amplitude
`
`Amplitude
`
`o
`
`0
`
`2T
`
`4T
`
`6T
`Time
`
`(c)
`
`8T
`
`101“
`
`FiGURE 3.8 Waveforms of (a) conventional QPSK {b) offset QPSK, and (c) 75/4-shifted
`QPSK.
`
`The filtering action can, in turn, cause the carrier amplitude (i.e., the envelope of the
`QPSK signal) to fluctuate, thereby making the receiver produce additional symbol
`errors over and ab0ve those due to channei noise.
`
`The extent of amplitude fluctuations exhibited by conventional QPSK signals
`may be reduced by using offset quadriphase-Shifz keying (OQPSK), which is also
`referred to as staggered QPSK. In this variant of QPSK, the second substrearn m2(t),
`multiplied by the 90° phase-shifted carrier Acsin(2rg’cr), is delayed (i.e., offset) by a bit
`duration T with respect
`to the first substream 1721(1), multiplied by the carrier
`Accos(2afcr). Accordingly, unlike the phase transitions in conventional QPSK, the
`phase transitions likely to occur in offset QPSK are confined to 0°, $900, as illustrated
`in the next example. However, the $900 phase jumps in OQPSK occur twice as fre-
`quently, but with a reduced range of amplitude fluctuations, compared with those of
`conventional QPSK. Since, in addition to :900 phase jumps, there are $1800 phase
`jumps in conventional QPSK, we usually find that ampiitude fluctuations in OQPSK
`due to filtering have a smaller amplitude than in conventional QPSK.
`
`
`
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`116
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`EXAMPLE 3.2 OQPSK Waveform
`
`Part (b) of Fig. 3.8 depicts the waveform of the OQPSK for the same binary data stream respon—
`sible for generating the conventional QPSK waveform depicted in part (a) of the figure. Here
`again, we see that the carrier amplitude of OQPSK is maintained constant. However, unlike the
`carrier phase in the conventional QPSK of Fig. 3.8(a). the carrier phase of the OQPSK showri in
`Fig. 3.8(b) has jumps of only i90°.
`l
`
`3.3.5 n/4-Shifted Quadriphase-Shift Keying
`
`As mentioned previously, ordinarily the carrier phase of a conventional QPSK signal
`may reside in one of two possible discrete settings:
`
`1. 0, 77.12, 7:, or 3M2 radians.
`
`2. m'4, 3rd4,51r/4, or 77:!4 radians.
`
`These two phase settings are shifted by M4 radians relative to each other. The QPSK
`waveform depicted in Fig. 3.8(a) follows setting 1. In another variant of QPSK known
`as JI/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 trill-shifted QPSK signals is that amplitude fluctuations
`due to filtering are significantly reduced, compared with their frequency of occurrence
`in conventional QPSK signals. Thus, the use of JIM—shifted QPSK provides the band—
`width efficiency of conventional QPSK, but with a reduced range of amplitude fluctua-
`tions. The reduced amplitude fluctuations become important when the transmitter
`includes a slightly nonlinear amplifier, as we shall see in Section 3.9. Indeed, it is for this
`reason that Mai—shifted QPSK has been adopted in the North American digital cellular
`time-division multiple access (TDMA) standard, 15-54 as well as the Japanese digital
`cellular standard.2
`
`EXAMPLE 3.3 arm-Shifted QPSK Waveform
`
`Figure 3.8(c) depicts the arm-shifted QPSK waveform produced by the same binary data stream
`used to generate the conventional QPSK waveform of Fig. 3.8(a). Comparing these two wave-
`forms, we see that (1) the phase jumps in the JIM-shifted QPSK are restricted to inc/4 and flit/4
`radians and (2) the in phase jumps of QPSK are eliminatedfihence the advantage of 15’4—
`shifted QPSK over conventional QPSK. However. this advantage is attained at the expense of
`increased complexity.
`I
`
`3.4
`
`PULSE SHAPING
`
`The pulse defined in Eq. (3.7) for representing binary symbol 1 or 0 is rectangular in
`shape. From a practical perspective, the use of a rectangular pulse shape is undesirable
`for two fundamental reasons:
`
`1. The spectrum (i.e.. Fourier transform) of a rectangular pulse is 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 intersymbol interference (181), which refers 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 overcome the signal distortion and 131 problems
`cited under points 1 and 2.
`The design criteria for pulse shaping are covered by the fundamental theoretical
`work of. Nyquist.3 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 sinusoidal rolloff
`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=Rf2 to an adjustable
`value from W to 2W by defining P(f) as follows:
`
`
`
`
`
`PU") m
`
`_...
`
`1
`W
`i
`.
`( 7t
`r
`4W[1 + cos ZWpcrl Wu pa]
`O
`
`_.__
`
`_
`
`W
`
`<
`0 — ifi sfl
`.
`<
`f1 m < W 11
`|ff 2 2 Wmf1
`
`2
`
`k
`
`__
`
`The frequency parameter f1 and bandwidth W are related by the parameter
`
`g
`
`fl
`= . - _
`1 W
`
`p
`
`(3.17)
`
`3.18
`
`(
`
`)
`
`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(t) (i.e., the overall impulse response of
`the transmit filter, the channel, and the receive filter) has the value of unity at the cur-
`rent signaling instant (i.e., 13(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(t) ensure that the ISI problem is reduced to zero.
`The frequency response of Fig. 3.9(a) is called the raised-cosine (RC) spectrum, so
`called because of its 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 ,0 over the range (0,1) allows the designer to
`tradeoff transmitted signal bandwidth for robustness of the pulse shape.
`
`2WP(f)
`
`.00)
`
`
`
`
`
`(3) Frequency response of the raised cosine spectrum for varying roll—off rates
`FIGURE 3.9
`(b) Impulse response of the Nyquist shaping filter (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(f) of Eq. (3.17), evaluate the inverse Fourier transform
`of P(f) and thus show that.
`
`11(3) : [cosmrrthgSmCQH/I}
`1— i6p2W2z‘
`
`(319)
`
`(b) Determine P(f) and p(t) for the special case of ,0 =1, which is known as the full-cosine
`roll—offpluse.
`
`Ans. (b) pm : sinc(4Wt)/(l—16W2t2)
`
`a
`
`3.4.1 Root Raised-Cosine Pulse Shaping;4
`
`A more sophisticated form of pulse shaping uses the root raised-cosine (RC) spec-
`trum rather than the reguiar 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
`
`C0529 : %(i + c0526)
`
`where, for the problem at hand,
`
`gmi
`ZWp
`
`(lfl - W(1-p))
`
`and retaining P(f ) as the symbol for the root RC spectrum, we may Write
`
`
`
`
`«El—MI»?
`
`Pm =
`
`1
`
`fiwdmm— th—pn)
`
`n
`
`O
`
`OSlfl 5.13
`
`flslfl <2W f]
`
`_
`
`lflzZWmfl
`
`(3.20)
`
`where, as before, the roll-off factor p is defined in terms of the frequency parameter
`f] and the bandwidth W as in 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 wilt experience the spectrum P2 (f) ,which is the regular raised cosine
`spectrum. In effect, by adopting the root RC spectrum P0“) of Eq. (3.20) for pulse
`shaping, we would be working with P (f) in an overali transmitter-receiver sense. On
`this basis, we find that in the context of wireless communications, if the channei is
`affected by both flat fading and additive white Gaussian noise, and the pulse-shape fil-
`tering is partitioned equally between the transmitter and the receiver in the manner
`described herein, then effectiveiy 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
`
`
`
`.
`
`2/ T
`
`(b)
`
`FIGURE 3.10
`spectrum.
`
`(a) P(f) for root raised — cosine spectrum. (b) p(t) for root raised - cosine
`
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`

`

`Section 3.4 Pulse Shaping
`
`121
`
`The inverse Fourier transform of Eq. (3.20) defines the root RC shaping pulse
`
`pa) = fi£Wfl+ 47:9cos(2aW(1+p)t))
`
`(3.21)
`
`The important point to note here is the fact that the root RC shaping pulse p(r) of
`Eq. (3.21) is radically different from the standard RC shaping pulse of Eq. (3.1.9). in
`particular, the new shaping pulse satisfies an orthogonality constraint under T—shifts,
`as shown by
`
`meprziprnnnd: = 0 for n = :1, 12,
`
`'
`
`(3.22)
`
`where T is the symbol duration. Yet, p(r) 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 z 0, 0.5, 1; the
`corresponding time-domain plots are shown in Fig. 3.10(b). These piots 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 PuEse Shaping Comparison
`
`Using the root RC shaping pulse p(t) of Eq. (3.21) with roll-off factor ,0 = 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(r) of Eq. (3.19) with the same roll-off factor.
`Using the root RC pulse p(t) of Eq. (3.21) with a multiplying plus sign for binary symbol 1
`and multiplying minus sign 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 shown in the figure corresponds to the
`use of the regular RC pulse 19(3) of Eq. (3.15). The figure shows the root RC waveform occupies
`a Earger dynamic range than the regular RC waveform.
`l
`
`i’roblem 3.5
`
`(3) Starting with Eq. (3.20), derive the root RC pulse shape p(z) of Eq. (3.21).
`
`(h) Evaluate p(t) at (i) r: 0, and (ii) t = il/(Sp W) .
`
`(c) Show that the p(t) derived in part (3) satisfies the orthogonality constraint described in
`Eq.(3.22)
`Ans. (b)
`(i)p(0) z JzTr/(i—p+ie)
`
`(11)!)[8Wl=mal+2alfin£42l [1 :lcoslzi—pll
`
`'
`
`
`
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`

`
`
`122
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`1.5
`
`
`
`Raised Cosine
`— - — Sqrt Raised Cosine
`
`
`
`0.5
`
`o
`
` Pulsep(t)
`
`
`
`—-0.5
`
`“1.5
`
`
`
`
`
`
`Binary sequence
`
`*1
`
`e .1
`
`+1
`Time t/Tb
`
`—I
`
`—1
`
`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 SYSTEMS5
`
`The linear modulation schemes considered in Section 3.3 may be viewed as special
`cases of the canonical representation of a band-pass signal:
`
`5(2‘) = 51(t)cos(27rfct) —SQ(I)Sln(2KfCI)
`
`(3.23)
`
`It is customary to refer to 51(1) as the in-phase component of 5(t) and to SQU) 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 QPSK in terms of the components S10) and SQU).
`We may simplify matters further by introducing the complex signal
`
`E0) = 51(1) +.st(t)
`
`(3.24)
`
`Page 142 of 474
`
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`
`

`

`Section 3.5 Complex Representation of Linear Modulated SignaEs and Band-Pass Systems
`
`123
`
`
`
`Special Cases of the Canonical Equation (3.23).
`
`Type of modulation
`
`In—phase component
`$10)
`
`Quadrature component
`SQU)
`
`Defining
`equation
`
`Amplitude modulation
`
`AC(1 + kam(:))
`
`O
`
`TABLE 3.1
`
`Analog
`
`
`
`3.3
`
`3.4
`
`'
`
`3.5
`
`3.1.1
`
`Q,
`i
`
`
`
`é
`l
`
`Double sidebandn-
`suppressed carrier modulation
`
`ACmU)
`
`Di ital
`g
`
`Binar
`
`base-shift ke in
`Y g
`
`yp
`
`AczkaUWle
`k
`
`0
`
`D
`
`Quadri base-shift ke in
`P
`y
`
`g
`
`£1,2ka ,pn—zkr)
`k
`
`7ACZbk59’pU—2kT)
`
`k
`
`where j is the square root of -—1. For obvious reasons, the new signal 30) is referred to
`as the complex envelope of the modulated signal 3(t). Next, we invoke Euler’s for-
`mula
`
`exp(j2rtfct) = cos(27tfcr)+jsin(21tfct)
`
`(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
`
`30) = Re{§(t)exp(j27tfcr)}
`
`(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, be it in the context of analog or digital modula-
`tion techniques. Specifically, we may make the following four statements:
`
`1. The iii-phase component 51(t) and the quadrature component SQ(I) are both real—
`valued functions of time that are uniquely defined in terms of. the baseband (i.e.,
`message) signal m(t). Given the two components 51(t) and sQ(t), we may thus use
`the scheme shown in Fig. 3.12(a) to synthesize the modulated signal s(t).
`2. Given the modulated signal s(t), we may use the scheme shown in Fig. 3.1203) to
`analyze the modulated signal s(t) and thereby construct the in-phase component
`.910) and the quadrature component SQU).
`3. The in-phase and quadrature components are orthogonal to each other, occupy-
`ing exactly the same bandwidth as the message signal m(t).
`4. The complex envelope 3*(1‘) given in Eq. (3.24) completely preserves the infor—
`mation content of the modulated signal 5(3‘) except for the carrier frequency fc.
`
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`

`
`
`124
`
`Chapter 3 Modulation and Frequency-Division Multiple Access
`
`In-phase component
`51(3)
`
`§
`
`Modulated signal
`e m
`
`
`
`
`
`90" phase
`shifter
`
`Quadrature component
`we
`
`>3
`
`(a)
`
`
`
`Low—pass
`filter
`
`In-phase component
`510‘)
`
`
`Zoos (21rfct)
`Modulated signal
`S“)
`
`90° phase
`
`shifter
`
`423in(2qrfct)
`
`
`Quadrature component
`Low- ass
`
`filter
`p
`Self)
`
`(13)
`
`
`
`
`(a) Synthesizer for constructing a modulated signal from its in-phase and
`FIGURE 3.12
`quadrature components. (b) Analyzer for deriving the in-phase and quadrature components
`of the modulated band—pass signal.
`
`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
`other linear 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