CODING & MODULATION Prof. Ing. Anton Čižmár, PhD.

also from Digital Communications 4th Ed., J. G. Proakis, McGraw-Hill Int. Ed. 2001

CONTENT

1. REPRESENTATION OF BAND-PASS SIGNALS AND SYSTEMS 2. SIGNAL SPACE REPRESENTATION 3. REPRESENTATION OF DIGITALLY MODULATED SIGNALS 4. SPECTRAL CHARACTERISTIC OF DIGITALLY MODULATED SIGNALS

4Characterization of Communication Signals and Systems

Signals can be categorized in a number of different ways, such as

• random versus deterministic, • discrete time versus continuous time, • discrete amplitude versus continuous amplitude, • low-pass versus band-pass, • finite energy versus infinite energy, • finite average power versus infinite average power.

4.1REPRESENTATION OF BAND-PASS SIGNALS AND SYSTEMS

Many digital information-bearing signals are transmitted by some type of carrier modulation. Signals and channels (systems) that satisfy the condition that their bandwidth is much smaller than the carrier frequency are termed narrowband band-pass signals and channels (systems). With no loss of generality and for mathematical convenience, it is desirable to reduce all band-pass signals and channels to equivalent low-pass signals and channels. As a consequence, the results of the performance of the various modulation and demodulation techniques presented in the subsequent chapters are independent of carrier frequencies and channel frequency bands. 4.1.1Representation of Band-Pass Signals

Suppose that real valued signal s(t) has a frequency content concentrated in a narrow band of frequencies in the vicinity of a frequency fc as show in Figure 4.1-1

Our objective is to develop a mathematical representation of such signals. First, we consider a signal that contains only positive frequencies in s(t). Such a signal may be expressed as

)()(2)( fSfufS =+ (4.1-1)

0f

fc-fc

|S(f )|

Figure 4.1.-1 Spectrum of a bandpass signal

where S(f) is the Fourrier transform of s(t) and u(f) is the unit step function. The equivalent time domain expression is

)]([)](2[

)()(

11

2

fSFfuF

dfefSts ftj

−−

∞

∞−++

∗=

= ∫ π

(4.1-2)

The signal s+(t) is called the analytic signal or pre-envelope of s(t). We note that

tjtfuF

tsfSF

πδ +=

=

−

−

)()](2[

)()]([

1

1

(4.1-3)

Hence,

)(1)(

)(])([)(

tst

jts

tstjtts

∗+=

∗+=+

π

πδ

(4.1-4)

We define

τ

ττ

π

π

∫∞

∞− −=

∗=

dts

tst

ts

)(1

)(1)()

(4.1-5)

The signal )(ts) may by viewed as the output of the filter with impulse response

t

thπ1)( = ∞<<∞− t (4.1-6)

when excited by the input signal s(t). Such a filter is called a Hilbert transformer. The frequency response of this filter is simply

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

<=>−

==

=

∫

∫

∞

∞−

−

∞

∞−

−

)0.......()0.......(0)0....(

11

)()(

2

2

fjffj

dtet

dtethfH

ftj

ftj

π

π

π

(4.1-7)

We observe that |H(f)| = 1 and that the phase response π21)( −=Θ f for f > 0 and

π21)( =Θ f for f < 0. Therefore, this filter is basically a 90o phase shifter for all

frequencies in the input signal. The analytical signal s+(t) is a band-pass signal. We may obtain an equivalent low-pass representation by performing a frequency translation of S+(f). Thus we define Sl(f) (l=low) as

)()( cl ffSfS += + (4.1-8) The equivalent time-domain is

tfjtfjl

cc etsjtsetsts ππ 22 )]()([)()( −−+ +== ) (4.1-9)

or

tfjl

cetstsjts π2)()()( =+ ) (4.1-10) In general, the signal is complex-valued and may be expressed as

)()()( tjytxtsl += (4.1-11) After substitution and equation real and imaginary part

tftytftxts cc ππ 2sin)(2cos)()( −= (4.1-12) tftytftxts cc ππ 2cos)(2sin)()( +=) (4.1-13)

The expression (4.1-12) is the desired form for the representation of a band-pass signal. The low-frequency signal components x(t) and y(t) may be viewed as amplitude modulation impressed on the carrier components tfcπ2cos and tf cπ2sin . Since these carrier components are in phase quadrature, x(t) and y(t) are called the quadrature components of the band-pass signal s(t). Another representation of the signal is

])(Re[})]()(Re{[)( 22 tfjl

tfj cc etsetjytxts ππ =+= (4.1-14)

A third possible representation is expressing

)()()( tjl etats θ= (4.1-15)

where

)()()( 22 tytxta += (4.1-16

)()(tan)( 1

txtyt −=θ (4.1-17)

Then

)](2cos[)()( ttftats c θπ += (4.1-18)

Therefore, 4.1-12, 4.1-14 and 4.1-18 are quivalent representation of band-pass signals. The Fourier transform of s(t) is

dteetsdtetsfS ftjtfjl

ftj c πππ 222 ]})({Re[)()( −∞

∞−

−∞

∞−∫∫ == (4.1-19)

Use of the identity

)(21)Re( ∗+= ξξξ (4.1-20)

in 4.1-19 yields the results

)]()([21

])()([21)( 222

clcl

ftjtfjl

tfjl

ffSffS

dteetsetsfS cc

−−+−=

+=

∗

−−∗∞

∞−∫ πππ

(4.1-21)

This is the basic relationship between the spectrum of the real band-pass signal S(f) and the spectrumof the equivalent low-pass signal Sl(f). The energy in the signal s(t) is defined as

dtetsdtts tfjl

c 222 ]})({Re[)( π∫∫∞

∞−

∞

∞−

==Ε (4.1-22)

When the identity in Eq 4.1.-20 is used in Eq. 4.1-22, we obtain the following results:

∫∫∞

∞−

∞

∞−

++=Ε dtttftsdtts cll )](24cos[)(21)(

21 22 θπ (4.1-23)

Consider the second integral in Eq.4.1-23. Since the signal s(t) narrow-band, the real envelope )()( tsta l≡ or, equivalently, a2(t) varies slowly relative to the rapid variations

exhibited by the cosine function. A graphical illustration of the integrand in the second integral of Eq.4.1-23 is shown in Fig.4.1-2. The value of the integral is just the net area under the cosine function modulated by a2(t). Since the modulating waveform a2(t) varies slowly relative to the cosine function, the net area contributed by the second integral is very small relative to the value of the first integral in Eq.4.1-23 and, hence, it can be neglected. Thus, for all practical purposes, the energy in the band-pass signal s(t), expressed in terms of the equivalent low-pass signal sl(t) is

dttsl∫∞

∞−

=Ε 2|)(|21 (4.1-24)

where is just the envelope a(t) of s(t). |)(| tsl

Example:

4.1.4 Representation of Band-Pass Stationary Stochastic Processes Suppose that n(t) is a sample function of a wide-sense stationary stochastic process with zero mean and power spectral density )( fnnΦ .The power spectral density is assumed to be zero outside of an interval of frequencies centered aroud fc. The stochastic process n(t) is said to be a narrowband band-pass process if the width of the spectral density is much smaller than fc. Under this condition, a sample function of n(t) can be represented by any of the three equivalent forms

)](2cos[)()( ttftatn c θπ += (4.1-37) tftytftxtn cc ππ 2sin)(2cos)()( −= (4.1-38)

])(Re[)( 2 tfj cetztn π= (4.1-39)

where a(t) is the envelope and )(tθ is the phase of the real-valued signal, x(t) and y(t) are the quadrature components of n(t), and z(t) is called the complex envelope of n(t). Let us consider 4.1-38 in more detail. First, we observe that if n(t) is zero mean, then x(t) and y(t) must also have zero mean values. In addition, the stationarity of n(t) implies that the autocorrelation and cross-correlation functions of x(t) and y(t) satisfy the following properties:

)()( τφτφ yyxx = (4.1-40) )()( τφτφ yxxy −= (4.1-41)

The autocorrelation function of the band-pass stochastic process is

])(Re[)( 2 τπτφτφ cfjzznn e= (4.1-50)

The power density spectrum of the stochastic process n(t) is the Fourier transform of

)(τφnn

))]()([21

]})({Re[)( 22

czzczz

fjfjzznn

ffff

deef c

−−Φ+−Φ=

=Φ ∫∞

∞−

− ττφ τπτπ

(4.1-51)

Representation of white noise White noise is a stochastic process that is defined to have a flat (constant) power spectral density over entire frequency range. This type of noise cannot be expressed in terms of quadrature components, as result of wide-band character. In problems concerned with the demodulation of narrowband signals in noise, it is mathematically convenient to model the additive noise process as white and to represent the noise in terms of quadrature components. This can be accomplished by postulating that the signals and noise at the receiving terminal have passed through an ideal band-pass filter, having a pass-band that includes the spectrum of the signals but is much wider. Such a filter will introduce negligible, if any, distortion on the signal, but it does not eliminate the noise frequency components outside of the passband. The noise resulting from passing the white noise process through a spectrally flat (ideal) band-pass filter is termed band-pass white noise and has power spectral density depicted in Figure 4.1-3

„White light contains all frequencies“

„After filtering the only remaining noise power is that contained within the filter

bandwidth (B)“

Fig4.1-3 The equivalent low-pass noise z(t) has a power spectral density

⎪⎩

⎪⎨

⎧

>

≤=Φ

)21(

)21(

0)( 0

Bf

BfNfzz (4.1-56)

and its autocorrelation function is

πττπτφ BNzz

sin)( 0= (4.1-57)

The limiting form of )(τφzz as B approaches infinity is

)()( 0 τδτφ Nzz = (4.1-58)

4.2 SIGNAL SPACE REPRESENTATION In this section, we demonstrate that signals have characteristics that are similar to vectors and develop a vector representation for signal waveforms. (For graphical presentation go to file: vector space.pdf) 4.2.1 Vector Space Concepts A vector v in n-dimensional space is characterized by its n components [v1,v2, … , vn]. It may also be represented as a linear combination of unit vectors or basis vectors ei ,

∑=

=n

iiievv

1

(4.2-1)

The inner product of two n-dimensional vectors v1=[v11 v12 …v1n] and v2=[v21 v22 …v2n] is

∑=

=⋅n

iiivvvv

12121 (4.2-2)

Orthogonal vectors are if

021 =⋅ vv (4.2-3) The norm of a vector (simply its length)

∑=

=n

iivv

1

2 (4.2-4)

A set of m vectors is said to be orthonormal if the vectors are orthogonal and each vector has a unit norm. A set of m vectors is said to be linearly independent if no one vector can be represented as a linear combination of the remaining vectors. The norm square of the sum of two vectors may be expressed as

212

22

12

21 2 vvvvvv ⋅++=+ (4.2-7)

if v1 and v1 are orthogonal, then

22

21

221 vvvv +=+ (4.2-8)

This is the Pythagorean relation for two orthogonal n-dimensional vectors. Finaly, let us review the Gram-Schmidt procedure for constructing a set of orthonormal vectors from a set of n-dimensional vectors vi ; mi ≤≤1 . We begin by arbitrarily selecting a vector from the set, say v1 . By normalizing its length, we obtain the first vector, say

1

11 v

vu = (4.2-11)

Next, we may select v2 and, first, substract the projection of v2 onto u1. Thus, we obtain

1122'2 )( uuvvu ⋅−= (4.2-12)

Then, we normalize the vector u2’ to unit length. This yields

'2

'2

2 uuu = (4.2-13)

The procedure continues by selecting v3 and subtracting the projections of v3 into u1 and u2. Thus, we have

2231133'3 )()( uuvuuvvu ⋅−⋅−= (4.2-14)

Then, the orthonormal vector u3 is

'3

'3

3 uuu = (4.2-15)

By continuing this procedure, we construct a set of n1 orthonormal vectors, where , in general. If , then , and if , then

nn ≤1

nm < mn ≤1 nm > nn ≤1 .

4.2.2 Signal Space Concepts The inner product of two generally complex-valued signals is

∫ ∗=b

a

dttxtxtxtx )()(),(),( 2121 (4.2-16)

The norm of the signal is

21

2 )|)(|()( ∫=b

a

dttxtx (4.2-17)

The triangle inequality is

)()()()( 2121 txtxtxtx +≤+ (4.2-18)

Cauchy-Schwarz inequality is

21

22

21

2121 |)(||)(|)()( ∫∫∫ ≤∗

b

a

b

a

b

a

dttxdttxdttxtx (4.2-19)

4.2.3 Orthogonal Expansions of Signals In this section, we develop a vector representation for signal waveforms, and, thus, we demonstrate an equivalence between a signal waveform and its vector representation. Suppose that s(t) is a deterministic real-valued signal with finite energy

∫∞

∞−

= dttsEs2)]([ (4.2-20)

There exists a set of function {fn(t), n=1,2,…,K} that are orthonormal. We may approximate the signal s(t) by a weighted linear combination of these functions, i.e.,

∑=

=K

kkk tfsts

1)()() (4.2-22)

where sk are the coefficients in the approximation of s(t). The approximation error is

)()()( tstste )−= (4.2-23)

Let us select the coefficients {sk} so as to minimize the energy Ee of the approximation error.

dttfstsdttstsEK

kkke ∑∫∫

=

∞

∞−

∞

∞−

−=−=1

22 )]()([)]()([ ) (4.2-24)

The optimum coefficients in the series expansion of s(t) may be found by differentiating Eq.4.2-24 with respect to each of the coefficients {sk} and setting the first derivatives to zero. Under the condition that Emin= 0 we may expres s(t) as

∑=

=K

kkk tfsts

1)()( (4.2-29)

When every finite energy signal can be represented by a series expansion of the form(4.2-29) for which Emin= 0, the set of orthonormal functions {fn(t)} is said to be complete.

Example 4.2-1. Trigonometric Fourier series Consider a finite energy signal s(t) that is zero everywhere except in the range 0≤ t≤T and has a finite number of discontinuities in this interval. Its periodic extension can be represented in a Fourier series as

∑∞

=

++=1

0 )2sin2cos(2

)(k

kk Tktb

Tktaats ππ (4.2-30)

where the coefficients {ak,bk} that minimize the mean square error are given by

∫=T

k dtTktts

Ta

0

;2cos)(2 π 0≥k

(4.2-31)

∫=T

k dtTktts

Tb

0

;2sin)(2 π 1>k

The set of trigonometric functions { )/2sin(,/2cos( TktTkt ππ } is complete, and, hence, the series expansion results in zero mean square error.

Gram-Schmidt procedure We have a set of finite energy signal waveforms {si(t), i=1,2,…,M} and we wish to construct a set of orthonormal waveforms. The first orthonormal waveform is simply constructed as

1

11

)()(Etstf = (4.2-32)

Thus, f1(t) is simply s1(t) normalized to unit energy. The second waveform is constructed from s2(t) by first computing the projection of f1(t) onto s2(t), which is

dttftsc ∫∞

∞−

= )()( 1212 (4.2-33)

Then

)()()( 11222 tfctstf −=′ (4.2-34)

and again

2

22

)()(E

tftf′

= (4.2-35)

in general

k

kk E

tftf )()(′

= (4.2-36)

where

∑−

=

−=′ 1

1)()()(

k

iiikkk tfctstf (4.2-37)

and

dttftsc ikik ∫∞

∞−

= )()( (4.2-38)

Thus, the orthogonalization process is continued until all the M signal waveforms {si(t)} have been exhausted and orthonormal waveforms have been constructed. The dimensionality N of the signal space will be equal to M if all the signal waveforms are linearly independent, i.e., none of the signal waveform is a linear combination of the other signal waveforms.

MN ≤

Once we have constructed the set of orthonormal waveforms {fn(t)}, we can express the M signals {sn(t)} as linear combinations of the {fn(t)}. Thus we may write

∑=

=N

nnknk tfsts

1)()( (4.2-39)

and

∑∫=

∞

∞−

===N

nkknkk ssdttsE

1

222)]([ (4.2-40)

Based on the expression in Equation 4.2-39, each signal may be represented by the vector

}...{ 21 kNkkk ssss = (4.2-41)

or, equivalently as a point in N-dimensional signal space with coordinates {ski, i=1,2,,…N}. The energy in the kth signal is simply the square of the length of the vector or, equivalently, the square of the Euclidean distance from the origin to the point in the N-dimensional space. Thus, any signal can be represented geometrically as a point in the signal space spanned by the orthonormal functions {fn(t)}.

Example 4.2-2 Let us apply the Gram-Schmidt procedure to the set of four waveforms illustrated in Fig.4.2-1a.

The waveform has energy )(1 ts 21 =E , so that 2)()( 1

1tstf = . Next, we observe that

; hence and are orthogonal. Therefore 012 =c )(2 ts )(1 tf2

)()( 22

tstf = . To obtain ,

we compute and , which are

)(3 tf

13c 23c 213 =c and 023 =c . Thus,

)32(;1)(;013

'3 {)(2)()( ≤≤−=−= t

otherwisetftstf Since has unit energy, it follows that . In determining , we find

that )('3 tf )()( '

33 tftf = )(4 tf

214 −=c , , and 024 =c 134 =c . Hence

0)()(2)()( 314'

4 =−+= tftftstf Consequently, is a linear combination of and and, hence, . The three orthonormal functions are illustrated in Fig4.2-1b.

)(4 ts )(1 tf )(3 tf 0)(4 =tf

Example 4.2-3 Let us obtain the vector representation of the four signals shown in Fig.4.2-1a by using the orthonormal set of functions in Fig4.2-1b. Since the dimensionality of the signal space is N=3, each signal is described by three components. The signal is characterized by the vector s

)(1 ts

1 = ( 0,0,2 ). Similarly, the signals , and are characterized by the vectors s

)(2 ts )(3 ts )(4 ts

2 = ( 0,2,0 ), s3 = ( 1,0,2 ), and s4 = ( 1,0,2− ),

respectively. These vectors are shown in Fig4.2-2. Their lengths are 21 =s , 22 =s ,

33 =s , and 34 =s , and the corresponding signal energies are 2kk sE = , k=1,2,3,4.

(note mistake in localization s3) We have demonstrated that a set of M finite energy waveforms {sn(t)} can be represented by a weighted linear combination of orthonormal functions {fn(t)} of dimensionality

. The functions {fMN ≤ n(t)} are obtained by applying the Gram-Schmidt orthogonalization procedure on {sn(t)}. It should be emphasized, however, that the functions {fn(t)} obtained from Gram-Schmidt procedure are not unique. If we alter the order in which the orthogonalization of the signals {sn(t)} is performed, the orthonormal waveforms will be different and the corresponding vector representation of the signals {sn(t)} will depend on the choice of the orthonormal functions {fn(t)}. Nevertheless, the vectors {sn} will retain their geometrical configuration and their lengths will be invariant to the choice of orthonormal functions {fn(t)}. The orthogonal expansions described above were developed for real-valued signal waveforms. Finally, let us consider the case in which the signal waveforms are band-pass and represented as

])(Re[)( 2 tfjlmm

cetsts π= (4.2-42)

where {slm(t)} denote the equivalent low-pass signals. Signal energy may be expressed either in terms of sm(t) or slm(t), as

dttsdttsE lmmm ∫∫∞

∞−

∞

∞−

== 22 |)(|21)( (4.2-43)

The similarity between any pair of signal waveforms, say sm(t) and sk(t), is measured by the normalized cross correlation

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

= ∫∫∞

∞−

∞

∞−

dttstsEE

dttstsEE lklm

kmkm

km

)()(2

1Re)()(1 * (4.2-44)

We define the complex-valued cross-correlation coefficient kmρ as

∫∞

∞−

= dttstsEE lklm

kmkm )()(

21 *ρ ( 4.2-45)

Then,

∫∞

∞−

= dttstsEE km

kmkm )()(1)Re(ρ (4.2-46)

or, equivalently

km

km

km

kmkm EE

ssssss ⋅

=⋅

=)Re(ρ (4.2-47)

The cross-correlation coefficients between pairs of signal waveforms or signal vectors comprise one set of parameters that characterize the similarity of a set of signals. Another related parameter is the Euclidean distance between a pair of signals

21

21

2)(

)}Re(2{

})]()([{

kmkmkm

kmkme

km

EEEE

dttstsssd

ρ−+=

−=−= ∫∞

∞− (4.2-48)

When Em=Ek=E for all m and k, this expression simplifies to

21

)( )]}Re(1[2{ kme

km Ed ρ−= (4.2-49)

Thus, the Euclidean distance is an alternative measure of the similarity (or dis-similarity) of the set of signal waveforms or the corresponding signal vectors. In the following section, we describe digitally modulated signals and make use of the signal space representation for such signals. We shall observe that digitally modulated signals, which are classified as linear, are conveniently expanded in terms of two orthonormal basis functions of the form

tfT

tf

tfT

tf

c

c

π

π

2sin2)(

2cos2)(

2

1

−=

= (4.2-50)

Hence, if slm(t) is expressed as slm(t) = xl(t) +jyl(t), it follows that sm(t) in Eq. 4.2-42 may be expressed as

)()()()()( 21 tftytftxts llm += (4.2-51)

where xl(t) and yl(t) represent the signal modulations.

SELECTED PROBLEMS: 4.4 Determine the autocorrelation function of the stochastic process

)2sin()( θπ += tfAtx c where fc is a constant and θ is a uniformly distributed phase, i.e.,

,21)(π

θ =p πθ 20 ≤≤

Sol:

4.8 Determine the correlation coefficient kmρ among the four signal waveforms {si(t)} shown in Fig.4.2-1, and the corresponding Euclidean distances. Sol:

4.17 Carry out the Gram-Schmidt orthogonalization of the signals in Fig.4.2-1a in the order s4(t), s3(t), s2(t), s1(t), and, thus, obtain a set of orthonormal functions. Then, determine the vector representations of the signals and determine the signal energies. Sol:

4.3 REPRESENTATION OF DIGITALLY MODULATED SIGNALS In the transmission of digital information over a communication channel, the modulator is the interface device that maps the digital information into analog waveforms that match the characteristics of the channel. The mapping is generally performed by taking blocks of binary digits at a time from the information sequence Mk 2log= { }na and selecting one of kM 2= deterministic, finite energy waveforms { }Mmtsm ,...,2,1),( = for transmission over the channel.

• Modulator with memory – when the mapping from the digital sequence { to waveforms is performed under constraint that a waveform transmitted in any time interval depends on one or more previously transmitted waveforms.

}na

• Memoryless modulator – when the mapping from the sequence { to the waveforms

}na{ })(tsm is performed without any constraints on previously transmitted

waveforms. • Linear modulator – principle of superposition • Non-linear modulator - principle of superposition does not exist

4.3.1 Memoryless Modulation Methods

• ASK, FSK, PSK Pulse-amplitude-modulated (PAM) signals (ASK) In digital PAM, the signal waveforms may be represented as

tftgAetgAts cmtfj

mmc ππ 2cos)(])(Re[)( 2 == ; m = 1,2,…,M (4.3-1)

where { denote the set of M possible amplitudes corresponding to }MmAm ≤≤1, kM 2= possible k-bit block of symbols. The signal amplitudes take the discrete values (levels)

dMmAm )12( −−= ; m = 1,2,…,M (4.3-2)

where 2d is the distance between adjacent signal amplitudes. The waveform g(t) is a real-valued signal pulse whose shape influences the spectrum of the transmitted signal. The symbol rate for the PAM is R/k. This is the rate at which changes occur in the amplitude of the carrier to reflect the transmission of new information. The time interval Tb=1/R is called the bit interval and the time interval T=k/R is called the symbol interval.

The M PAM signals have energies

gm

T

m

T

mm EAdttgAdttsE 2

0

22

0

2

21)(

21)( === ∫∫ (4.2-3)

where Eg denotes the energy in the pulse g(t). These signals are one-dimensional (N=1) and are represented by the general form

)()( tfsts mm = (4.3-4) where f(t) is defined as the unit-energy signal waveform given as

tftgE

tf cg

π2cos)(2)( = (4.3-5)

and

gmm EAs21

= ; m = 1,2,…,M (4.3-6)

The corresponding signal space diagram for M=2, M=4, and M=8 are shown in Fig.4.3-1. Digital PAM is also called amplitude-shift keying (ASK).

Gray encoding mapping of the bits.

Euclidean distance between any pair of signal points is

||2||21

)( 2)(

nmEdAAE

ssd

gnmg

nme

mn

−=−=

−= (4.3-7)

and the minimum Euclidean distance is

ge Edd 2)(

min = (4.3-8)

The carrier-modulated PAM signal represented by Eq. 4.3-1 is a double-sideband (DSB) signal and requires twice the channel bandwidth of the equivalent low-pass signal for transmission. The digital PAM signal is also appropriate for transmission over a channel that does not require carrier modulation. In this case the signal is called baseband signal represented as

)()( tgAts mm = ; m = 1,2,…,M (4.3-10)

If M=2 than such signals are called antipodal signal with property

)()( 21 tsts −= and cross-correlation coefficient of –1.

Phase-modulated signal (PSK) In digital phase modulation, the M signal waveforms are represented as

tfmM

tgtfmM

tg

mM

tftg

eetgts

cc

c

tfjMmjm

c

ππππ

ππ

π

2sin)1(2sin)(2cos)1(2cos)(

)]1(22cos[)(

])(Re[)( 2/)1(2

−−−=

−+=

= −

; m = 1,2,…,M (4.3-11)

where g(t) is the signal pulse shape and Mmm /)1(2 −= πθ ; m = 1,2,…,M , are the M possible phases of the carrier. These signal waveforms have equal energy

g

TT

m EdttgdttsE21)(

21)(

0

2

0

2 === ∫∫ (4.3-12)

Furtheremore, the signal waveforms may be represented as linear combination of two orthonormal signal waveforms f1(t) and f2(t)

)()()( 2211 tfstfsts mmm += (4.3-13)

where

tftgE

tf cg

π2cos)(2)(1 = (4.3-14)

tftgE

tf cg

π2sin)(2)(2 −= (4.3-15)

and the two-dimensional vectors [ ]21 mmm sss = are given by

⎥⎥⎦

⎤

⎢⎢⎣

⎡−−= )1(2sin

2);1(2cos

2m

ME

mM

Es gg

mππ

(4.3-16)

If M=2

⎥⎥⎦

⎤

⎢⎢⎣

⎡= 0;

21gE

s ; ⎥⎥⎦

⎤

⎢⎢⎣

⎡−= 0;

22gE

s

Signal space diagrams for M=2,4,8 are shown in Fig.4.3-3. We note that M=2 corresponds to one-dimensional signals, which are identical to binary PAM signals.

The Euclidean distance (ED) between any pair of signal points

21

)(

)]}(2cos1[{ nmM

E

ssd

g

nme

mn

−−=

−=

π (4.3-17)

The minimum ED corresponds to the case in which 1=− nm , i.e., adjacent signal phases. In this case,

)2cos1()(min M

Ed ge π

−= (4.3-18)

A variant of 4-phase PSK (QPSK), called QPSK−4/π is obtained by introducing an additional

4/π phase shift in the carrier phase in each symbol interval. This phase shift facilitates symbol synchronization.

Quadrature Amplitude Modulation (QAM) The bandwidth efficiency of PAM/SSB can also be obtained by simultaneously impressing two separate k-bit symbols from the information sequence {an) on two quadrature carriers cos and sin. The resulting modulation technique is called QAM, and the corresponding signal waveforms is expressed as

tftgAtftgAetgjAAts

cmscmc

tfjmsmcm

c

ππ

π

2sin)(2cos)(])()Re[()( 2

−=+=

(4.3-19)

where Amc and Ams are the information-bearing signal amplitudes of the quadrature carriers and g(t) is the signal pulse. Alternatively, the QAM signal waveforms may be expressed as

)2cos()(])(Re[)( 2

θπ

πθ

+==

tftgVetgeVts

cm

tfjmjmm

c

(4.3-20)

where

)/(tan 1

22

mcmsm

msmcm

AA

AAV−=

+=

θ

QAM signal waveforms may be viewed as combined amplitude and phase modulation.

As in the case of PSK signals, the QAM signal waveforms may be represented as a linear combination of two orthonormal signal waveforms

)()()( 2211 tfstfsts mmm += (4.3-21)

where

tftgE

tf

tftgE

tf

cg

cg

π

π

2sin)(2)(

2cos)(2)(

2

1

−=

=

(4.3-22)

and

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡==

2;

221g

msg

mcmmm

EA

EAsss (4.3-23)

Euclidean distance between any pair of signal points is

( ) ([ ) ]22

)(

21

nsmsncmcg

nme

mn

AAAAE

ssd

−+−=

−= (4.3-24)

In the special case where the signal amplitudes take the set of discrete values

, the signal diagram is rectangular, as shown in Fig4.3.-5. In this case, the ED between adjacent points, i.e, the minimum distance is ( ){ MmdMm ,...2,1,12 =−− }

ge Edd 2)(

min = (4.3-25)

which is the same result as for PAM.

Multidimensional signals. It is apparent from the discussion above that the digital modulation of the carrier amplitude and phase allows us to construct signal waveforms that correspond to two-dimensional vectors and signal space diagrams. If we wish to construct signal waveforms corresponding to higher-dimensional vectors, we may use either the time domain or the frequency domain or both in order to increase the number f dimensions. Suppose we have N-dimensional signal vectors. For any N, we may subdivide a time interval of length T1=NT into N subintervals of length T=T1/N. In each subinterval of length T, we may use binary PAM (a one-dimensional signal) to transmit an element of the N-dimensional signal vector. Thus, the N time slots are used to transmit the N-dimensional signal vector. If N is even, a time slot of length T may be used to simultaneously transmit two components of the N-dimensional vetor by modulating the amplitude of quadrature carriers independently by the corresponding components. In this manner, the N-dimensional signal vector is transmitted in (1/2)NT seconds (1/2 - N times slots). Alternatively, a frequency band of width fN∆ may be subdivided into N frequency slots each of width . An N-dimensional signal vector can be transmitted over the channel by simultaneously modulating the amplitude of N carriers, one in each of the N frequency slots. Care must be taken to provide sufficient frequency separation

f∆

f∆ between successive carriers so that there is no cross-talk interference among signals on the N carriers. If quadrature carriers are used in each frequency slot, the N-dimensional vector (even N) may be transmitted in (1/2)N frequency slots, thus reducing the channel bandwidth utilization by a factor of 2. More generally, we may used both the time and frequency domains jointly to transmit an N-dimensional signal vector. For example, Fig.4.3-6 illustrates a subdivision of the time and frequency axes into 12 slots. Thus, an N=12 – dimensional signal vector may be transmitted by PAM or an N=24 – dimensional signal vector may be transmitted by use of two quadrature carriers (QAM) in each slot.

Orthogonal multidimensional signals (FSK) As a special case of the construction of multidimensional signals, let us consider the construction of M equal-energy orthogonal signal waveforms that differ in frequency and are represented as

]22cos[2

])(Re[)( 2

tfmtfTE

etsts

c

tfjlmm

c

∆+=

=

ππ

π

(4.3-26)

where the equivalent low-pass signal waveforms are defined as

ftmjlm e

TEts ∆= π22)( (4.3-27)

This type of frequency modulation is called frequency-shift keying (FSK). These waveforms are characterized as having equal energy and cross-correlation coefficients

∫ ∆−∆−

∆−∆−

==T

fkmTjftkmjkm e

fkmTfkmTdte

ETE

0

)()(2

)()(sin

2/2 ππ

ππρ (4.3-28)

The real part of kmρ is

fkmTfkmT

fkmTfkmT

fkmTkmr

∆−∆−

=

∆−∆−∆−

=≡

)(2])(2sin[

])(cos[)(

])(sin[)Re(

ππ

πππρρ

(4.3-29)

First, we observe that 0)Re( =kmρ when Tf 2/1=∆ and km ≠ . Since |m-k|=1 corresponds to adjacent frequency slots, represents the minimum frequency separation between adjacent signals for orthogonality of the M signals.

Tf 2/1=∆

Note that 0=kmρ for multiples of 1/T whereas 0)Re( =kmρ for multiples of 1/2T. For the case in which , the M FSK signals are equivalent to the N-dimensional vectors

Tf 2/1=∆

]...0000[

]0...000[

]0...000[

]0...000[

3

2

1

Es

Es

Es

Es

N =

=

=

=

(4.3-30)

where N=M. The distance between pairs of signals is

Ed ekm 2)( = for all m,k (4.3-31)

which is also the minimum distance.

Biorthogonal signals A set of M biorthogonal signals can be constructed from (1/2)M orthogonal signals by simply including the negatives of the orthogonal signals. Thus, we require N = (1/2)M dimensions for the construction of a set of M biorthogonal signals.

We note, that the correlation between any pair of waveforms is either 0,,1 orr −=ρ . The

corresponding distances are Ed 2= or Ed 2= , with the latter being the minimum distance.

Simplex signals Suppose we have a set of M orthogonal waveforms{sm(t)} or, equivalently, their vector representation{sm}. Their mean is

∑=

=M

mms

Ms

1

1 (4.3-32)

Now, let us construct another set of M signals by substracting the mean from each of the M orthogonal signals. Thus

sss mm −=' (4.3-33) The effect of the subtraction is to translate the origin of the m orthogonal signals to the point s . The resulting signal waveforms are called simplex signals and have the following properties. First, the energy per waveform is

)11(

)/1()/2(

22'

ME

EMEME

sss mm

−=

+−=

−=

(4.3-34)

Second, the cross correlation of any pair of signals is

11

/11/1)Re(

−−=

−−

=′′′⋅′

=MM

Mssss

nm

nmkmρ (4.3-35)

for all m, n. Hence, the set of simplex waveforms is equally correlated and requires less energy, by the factor 1-1/M, than the set of orthogonal waveforms. Since only the origin was translated, the distance between any pair of signal points is maintained at Ed 2= , which is the same as the distance between any pair of orthogonal signals.

Signal waveforms from binary codes A set of M signaling waveforms can be generated from a set of M binary code words of the form

[ mNmmm cccC ...21= ] m=1,2,…M (4.3-36) where cmj=0 or 1 for all m and j. Each component of a code word is mapped into an elementary binary PSK waveform as follows:

tfTEtsc cc

cmjmj π2cos2)(1 =⇒= , cTt ≤≤0

(4.3-37)

tfTEtsc cc

cmjmj π2cos2)(0 −=⇒= , cTt ≤≤0

where Tc=T/N and Ec=E/N. Thus, the M code words { }mC are mapped into a set of M waveforms { }. )(tsm

The waveforms can be represented in vector form as

[ mNmmm ssss ...21= ], m=1,2,…M (4.3-38)

where NEsmj ±= for all m and j. N is called the block length of the code, and it is also

the dimension of the M waveforms.

We note that there are 2N possible waveforms that can be constructed from the 2N possible binary code words. We may select a subset of NM 2< signal waveforms for transmission of information. We also observe that 2N possible signal points correspond to the vertices of an N – dimensional hypercube with its center at the origin. Figure 4.3-11 illustrates the signal points in N=2 and 3 dimensions. Each of the M waveforms has energy E. The cross correlation between any pair of waveforms depends on how we select the M waveforms from the 2N possible waveforms. Clearly, any adjacent signal points have a cross-correlation coefficient

NN

ENE

r2)/21( −

=−

=ρ (4.3-39)

and a corresponding distance of

NEEd re /4)1(2)( =−= ρ (4.3-40)

This concludes our discussion of memoryless modulation signals.

4.3.2 Linear Modulation with Memory In this section, we present some modulation signals in which there is dependance between the signals transmitted in successive symbol intervals. This signal dependence is usually introduced for the purpose of shaping the spectrum of the transmitted signal so that it matches the spectral characteristics of the channel. We shall present examples of modulation signals with memory and characterize their memory in terms of Markov chains. We shall confine our treatment to baseband signals. The generalization to band-pass signals is relatively straightforward.

NRZ modulation is memoryless and is equivalent to a binary PAM or a binary PSK signal in a carrier-modulated system. The NRZI signal is different from NRZ signal in that transitions from one amplitude level to another occur only when a 1 is transmitted. The amplitude level remains unchanged when zero is transmitted. NRZI is called differential encoding. The encoding operation is described mathematically by the relation

1−⊕= kkk bab (4.3-41) where ak is the input into the encoder and bk is the output of the encoder. When bk=1, the transmitted waveform is a rectangular pulse of amplitude A, when bk=0, the transmitted waveform is a rectangular pulse of amplitude –A. Hence, the output of the decoder is mapped into one of two waveforms in exactly the same manner as for NRZ signal. The differential encoding operation introduces memory in the signal. The combination of the encoder and the modulator operations may be represented by a state diagram (a Markov chain) as shown in Fig.4.3-13.

The state diagram may be described by two transition matrices corresponding to the two possible input bits {0,1}. We note that when ak=0, the encoder stays in the same state. Hence, the state transition matrix for a zero is simply

⎥⎦

⎤⎢⎣

⎡=

1001

1T (4.3.-42)

where tij=1 if ak results in a transition from state i to state j, i=1,2 and j=1,2; otherwise, tij=0. Similarly, the state transition matrix for ak=1 is

⎥⎦

⎤⎢⎣

⎡=

0110

2T (4.3-43)

Thus, these two state transition matrices characterize the NRZI signal. Another way to display the memory is by a trellis diagram. The trellis provides the same information concerning the signal dependence as the state diagram, but also depicts a time evolution of the state transitions.

The signal generated by delay modulation also has memory. Delay modulation is equivalent to encoding the data sequence by a run-length-limited code called a Miller code and using NRZI to transmit the encoded data this type of modulation has been used extensively in binary PSK. The signal may be described by a state diagram that has four states as shown in Fig.4.3.-15a. There are two elementary waveforms s1(t) and s2(t) and their negatives -s1(t) and -s2(t), which are used for transmitting the binary information. These waveforms are illustrated in Fig.4.3-15b. The mapping from bits to corresponding waveforms is illustrated in the state diagram. The state transition matrices that characterize the memory of this encoding and modulation method are easily obtained from the state diagram in Fig.4.3.-15. When ak=0, we have

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0001000110001000

1T (4.3-44)

and when ak=1, the transition matrix is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0100001001000010

2T (4.3-45)

Modulation techniques with memory such as NRZI coding are generally characterized by a K-state Markov chain with stationary state probabilities {pi=1,2,…,K} and transition probabilities {pij, i,j = 1,2,…,K}. Thus, the transition probability pij denotes the probability that signal waveform sj(t) is transmitted in a given signaling interval after the

transmission of the signal waveform si(t) in the previous signaling interval. The transition probabilities may be arranged in matrix form as

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

KKKK

K

K

ppp

pppppp

P

.....

.

.

21

22221

11211

(4.3-46)

where P is called the transition probability matrix. The transition probability matrix is easily obtained from the transition matrices {Ti} and the corresponding probabilities of occurrence of the input bits (or, equivalently, the stationary state transition probabilities {pi}. The general relationship may be expressed as

∑=

=2

1iiiTqP (4.3-47)

where q1=P(ak=0) and q2=P(ak=1). For the NRZI signal with equal state probabilities p1=p2=1/2 and transition matrices given by Equations 4.3-42 and 4.3-43, the transition probability matrix is

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

21

21

21

21

P (4.3-48)

Similarly, the transition probability matrix for the Miller-coded signal with equally likely symbols (q1=q2=1/2 or, equivalently, p1=p2=p3=p4=1/4) is

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

02/102/1002/12/12/12/1002/102/10

P (4.3-49)

The transition probability matrix is useful in the determination of the spectral characteristics of digital modulation techniques with memory, as we shall observe in Section 4.4.

4.3.3 Non-Linear Modulation Methods with Memory Continuous-phase FSK (CPFSK) To avoid the use of signále having large spectral side lobes, the information-bearing signal frequency modulates a single carrier whose frequency is changed continuously (CPFSK). This type of FSK signal has memory because the phase of the carrier ic constrained to be continuous. In order to represent a CPFSK signal, we begin with a PAM signal

∑ −=n

n nTtgItd )()( (4.3-50)

where {In} denotes the sequence of amplitudes obtained by mapping k-bit blocks of bojary digits from the information sequence {an} into the amplitude levels

and g(t) is a rectangular pulse of amplitude 1/2T and duration T seconds. The signal d(t) is used to frequency-modulate the carrier. Consequently, the equivalent low-pass waveform v(t) is expressed as

)1(,...,3,1 −±±± M

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎥⎦

⎤⎢⎣

⎡+= ∫

∞−

t

d ddTfjTEtv 0)(4exp2)( φττπ (4.3-51)

where fd is the peak frequency deviation and 0φ is the initial phase of the carrier. The carrier-modulated signal corresponding to Eq. 4.3.-51 may be expressed as

]);(2cos[2)( 0φφπ ++= IttfTEts c (4.3-52)

where );( Itφ represents the time-varying phase of the carrier, which is defined as

ττπττπφ dnTgITfddTfItt

nnd

t

d ∫ ∑∫∞−∞−

⎥⎦

⎤⎢⎣

⎡−== )(4)(4);( (4.3-53)

Note that, although d(t) contains discontinuities, the integral of d(t) is continuous. Hence, we have a continuouse-phase signal. The phase of the carrier in the interval

is determined by integrating Eq. 4.3-53. Thus, TntnT )1( +≤≤

)(2)(22);(1

nTtqhIInTtfITfIt nnnd

n

kkd −+=−+= ∑

−

−∞=

πθππφ (4.3-54)

where h, nθ and q(t) are defined as

dTfh 2= (4.3-55)

∑−

−∞=

=1n

kkn Ihπθ (4.3-56)

(4.3-57)

))0(

)0(

2/12/0

)(Tt

Ttt

Tttq>≤≤

<

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=

We observe that nθ represents the accumulation (memory) of all symbols up to time (n-1)T. The parameter h is called the modulation index. Continuous-phase modulation (CPM) When expressed in the formo f Eq.4.3-54, CPFSK becomes a speciál case of a general vlase of continuous-phase modulated (CPM) signále in which the carrier phase is

)(2);(1

kTtqhIIt k

n

kk −= ∑

−

−∞=

πφ , TntnT )1( +≤≤ (4.3-58)

where {Ik} is the sequence of M-ary information symbols selected from the alphabet

, {h)1(,...,3,1 −±±± M k} is a sequence of modulation indices, and q(t) is some normalized waveform shape. When hk=h for all k, the modulation index is fixed for all symbols. When the modulation index varies from one symbol to another, the CPM signa lis called multi-h. In such a case, the {hk} are made to vary in a vycliv manner through a set of indices. The waveform q(t) may be represented in general as the integral of some pulse g(t), i.e.,

∫=t

dgtq0

)()( ττ (4.3-59)

If g(t) = 0 for t > T, the CPM signal is called full response CPM. If for t > T, the modulated signa lis called partial response CPM. Fig 4.3-16 illustrates several pulses shapes for g(t), and corresponding q(t). I tis apparent that an infinite variety of CPM signále can be generated by choosing different pulse shapes g(t) and by varying the modulation index h and the alphabet size M.

0)( ≠tg

We note that the CPM signal has memory that is introduced through the phase kontinuity. For L > 1, additional memory is introduced in the CPM signal by the pulse g(t). Three popular pulse shapes are given:

LREC ⎪⎩

⎪⎨⎧ ≤≤

=)()10(

,0

,2

1)(

otherwiseLT

LTtg

LRC ⎪⎩

⎪⎨⎧ ≤≤−=

)()10(

,0

),2cos1(2

1)(

otherwiseLT

LTt

LTtgπ

GMSK ]})2/(ln)2

(2[])2/(ln)2

(2[{)( 2/12/1 TtBQTtBQtg +−−= ππ

∫∞

−=t

x dtetQ 2/2

21)(π

LREC denotes a rectangular pulse duration LT, where L is a positive integer. In this case, L = 1 results in a CPSK signal, with the pulses as shown in Fig.4.3-16a. The LREC pulse for L = 2 is shown in Fig.4.3-16c. LRC denotes a raised cosine pulse of duration LT. The LRC pulses corresponding to L = 1 and L = 2 are shown in Fig.4.3-16b and d, respectively. The third pulse is called a Gaussian minimum-shift keying (GMSK) pulse with bandwidth parameter B, which represents the -3dB bandwidth of the Gaussian pulse. Fig.4.3-16e illustrates a set of GMSK pulses with time-bandwidth products BT ranging from 0.1 to 1. We observe that the pulse duration increases as the bandwidth of the pulse decreases, as expected. In practical applications, the pulse is ussually truncated to some

specific fixed duration. GMSK with BT = 0.3 is used in the European digital cellular communication systém, called GSM. From Fig.4.3.-16e we observe that when BT = 0.3, the GMSK pulse may be truncated at |t| = 1.5T with a relatively small error incurred for T > 1.5T. It is instructive to sketech the set of phase trajectories );( Itφ generated by all possible values of the information sequence . For example, in the case of CPFSK with bojary symbols , the set of phase trajectories beginning at time t = 0 is shown in Fig.4.3-17. For comparison, the phase trajectories for quaternary CPFSK are illustrated in Fig.4.3-18. These phase diagrams are called phase trees. We observe that the phase trees for CPFSK are picewise linear as a consequence of the fact that the pulse g(t) is rectangular. Smoother phase trajectories and phase trees are obtained by usány pulses that do not contain discontinuities, such as the vlase of raised cosine pulses. For example, a phase trajectory generated by the sequence (1,-1,-1,-1,1,1,-1,1) for a partial response, raised cosine pulse of length 3T is illustrated in Fig.4.3-19. For comparison, the corresponding phase trajectory generated by CPFSK is also shown.

}{ nI1±=nI

Minimum-shift keying (MSK) MSK is a special formo f bojary CPFSK (and, therefore, CPM) in which the modulation index h = ½. The phase of the carrier in the interval TntnT )1( +≤≤ is

⎟⎠⎞

⎜⎝⎛ −

+=−+= ∑−

−∞= TnTtInTtqIIIt n

n

knnk πθππφ

21)(

21);(

1

, TntnT )1( +≤≤ (4.3-63)

and the modulated carrier signal is

⎥⎦

⎤⎢⎣

⎡+−⎟

⎠⎞

⎜⎝⎛ +=⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −

++= nnncnnc IntIT

fATnTtItfAts θπππθπ

21

412cos

212cos)( (4.3-64)

The expression 4.3-64 indicates that the binary CPFSK signal can be expressed as a sinusoid having one of two possible frequencies in the interval TntnT )1( +≤≤ . If we devone these frequencies as

Tff c 4

11 −= ,

Tff c 4

12 += (4.3-65)

Then the binary CPFSK signal given by Eq.4.3-64 may be written in the form

⎥⎦⎤

⎢⎣⎡ −++= −1)1(

212cos)( i

nii ntfAts πθπ , i = 1,2 (4.3-66)

The frequency separation Tfff 2/112 =−=∆ . Recall that Tf 2/1=∆ is the minimum frequency separation that is necessary to ensure the orthogonality of the signále s1(t) and s2(t) over a signalling interval of length T. This explains why bojary CPFSK with h = ½ is called minimum-shift keying (MSK). Signal space diagrams for CPM. In general, continuous-phase signále cannot be represented by discrete points in signal space as in the case of PAM, PSK, and QAM, because the phase of the carrier is time-variant. Instead, a continuous-phase signa lis described by the various paths or trajectories from one phase state to another. For a konstant-amplitude CPM signal, the various trajectories form a circle. For example, Fig4.3-25 illustrates the signal space (phase trajectory) diagram for CPFSK signále with h = ¼, h = 1/3 , h = ½, and h = 2/3 . The beginning and ending point sof these phase trajectories are marked in the figure by dots. Note that the length of the phase trajectory increases with an increase in h. An increase in h also results in a increase of the signal bandwidth, as demonstrated in the following section. A linear representation of CPM. As desribed above, CPM is a non-linear modulation technique with memory. However, CPM may also be represented as a linear superposition of signal waveforms (see Proakis pp.195).

Multiamplitude CPM. Multiamplitude CPM is a generalization of ordináty CPM in which the signal amplitud eis allowed to vary over a set of amplitude values while the phase of the signa lis constrained to be continuous (see Proakis pp. 199).

4.4 SPECTRAL CHARACTERISTIC OF DIGITALLY MODULATED SIGNALS In most digital communication systems, the available channel bandwidth is limited. Consequently, the system designer must consider the constraints imposed by the channel bandwidth limitation in the selection of the modulation technique used to transmit the information. For this reason, it is important for us to determine the spectral content of the digitally modulated signals. Since the information sequence is random, a digitally modulated signal is a stochastic process. We are interested in determining the power density spectrum of such a process. From the power density spectrum we can determine the channel bandwidth required to transmit the information-bearing signal. 4.4.1 Power Spectra of Linearly Modulated Signals Beginning with the form

])(Re[)( 2 tfj cetvts π=

which relates the band-pass signal s(t) to the equivalent low-pass signal v(t), we may express the autocorrelation function of s(t) as

])(Re[)( 2 τπτφτφ cfjvvss e= (4.4-1)

where )(τφvv is the autocorrelation function of the equivalent low-pass signal v(t). The Fourier transform of Eq. 4.4-1 yields the desired expression for the power density spectrum in the form

)]()([21)( cvvcvvss fffff −−Φ+−Φ=Φ (4.4-2)

where is the power density spectrum of v(t). )( fvvΦ First we consider the linear digital modulation method for which v(t) is represented in the general form

∑∞

−∞=

−=n

n nTtgItv )()( (4.4-3)

where the transmission rate is 1/T=R/k symbols/s and {In} represents the sequence of symbols that results from mapping k-bit blocks into corresponding signal points selected from the appropriate signal space diagram. Observe that in PAM, the sequence {In} is real and corresponds to the amplitude values of the transmitted signal, but in PSK, QAM, and combined PAM-PSK, the sequence {In} is complex-valued, since the signal points have a two-dimensional representation. The average power density spectrum of v(t) is

)()(1)( 2 ffGT

f iivv Φ=Φ (4.4-12)

where G(f) is the Fourier transform of g(t), and )( fiiΦ denotes the power density spectrum of the information sequence, defined as

∑∞

−∞=

−=Φm

fmTjiiii emf πφ 2)()( (4.4-13)

The result 4.4-12 illustrates the dependence of the power density spectrum of v(t) on the spectral characteristics of the pulse g(t) and the information sequence {In}. That is, the spectral characteristics of v(t) can be controlled by design of the pulse shape g(t) and by design of the correlation characteristics of the information sequence. The desired power density spectrum of v(t) when the sequence of information symbol sis uncorrelated is in the form

∑∞

−∞=

−+=Φm

iivv T

mfTmG

TfG

Tf )()()()(

2

2

22

2

δµσ (4.4-18)

where denotes the variance of an information symbol and denotes the mean. 2

iσ iµ The expression 4.4-18 for the power density spectrum is purposely separated into two terms to emphasize the two different types of spectral components. The first term is the continuous spectrum, and its shape depends only on the spectral characteristic of the signal pulse g(t). The second term consists of discrete frequency components spaced 1/T apart in frequency. Each spectral line has a power that is proportional to |G(f)|2 evaluated at f =m/T. Note that the discrete frequency components vanish when the information symbols have zero mean, i.e., 0=iµ . This condition is usually desirable for the digital modulation techniques under consideration, and it is satisfied when the information symbols are equally likely and symmetrically positioned in the complex plane. Thus, the system designer can control proper selection of the characteristics of the information sequence to be transmitted.

Example 4.4-1 To illustrate the spectral shaping resulting from g(t), consider the rectangular pulse shown in Fig.4.4-1a. The Fourier transform of g(t) is

fTjefT

fTATfG π

ππ −=

sin)()( 2

Hence

222 sin)()( ⎟⎟

⎠

⎞⎜⎜⎝

⎛=

fTfTATfG

ππ (4.4-19)

This spektrum is illustrated in Fig4.4-1b.

Note that it contains zeros at multiples of 1/T in frequency and that it decays inversely as the square of the frequency variable. As a consequence of the spectral zeros in G(f), all but one of the discrete spectral components in Eq.4.4-18 vanish. Thus upon substitution for |G(f)|2 from Eq.4.4-19, Eq.4.4-18 reduces to

)(sin)( 222

22 fAfT

fTTAf iivv δµππσ +⎟⎟

⎠

⎞⎜⎜⎝

⎛=Φ (4.4-20)

Example 4.4-2 As a second illustration of the spectral shaping resulting from g(t), we consider the raised cosine pulse

⎥⎦⎤

⎢⎣⎡ −+= )

2(2cos1

2)( Tt

TAtg π , Tt ≤≤0 (4.4-21)

This pulse is graphically illustrated in Fig.4.4-2a. Its Fourier transform is easily derived, and it may be expressed in the form

fTje

TffTfTATfG π

ππ −

−=

)1(sin

2)( 22 (4.4-22)

Consequently, all the discrete spectral components in Eq.4.4-18, excerpt the one at f=0 and , vanish. When compared with the spektrum of the rectangular pulse, the spektrum of the raised cosine pulse has a broader main lobe but tha tails decay inversely as f

Tf /1±=

6. Example 4.4-3 To illustrate that spectral shaping can also be accomplished by operations performed on the input information sequence, we consider a binary sequence {bn} from which we form the symbols

1−+= nnn bbI (4.4-23) The {bn} are assumed to be uncorrelated random variables, each having zero mean and unit variance. Then the autocorrelation function of the sequence {In} is

⎪⎩

⎪⎨

⎧±==

== +

)()1()0(

012

)()(otherwise

mm

IIEm mnniiφ (4.4-24)

Hence, the power density spectrum of the input sequence is

fTfTfii ππ 2cos4)2cos1(2)( =+=Φ (4.4-25) and the corresponding power density spectrum for the (low-pass) modulated signal is

fTfGT

fvv π22 cos)(4)( =Φ (4.4-26)

4.4.2 Power Spectra of CPFSK and CPM Signals

4.4.3 Power Spectra of Modulated Signals with Memory

SELECTED PROBLEMS: 4.14 Consider an equivalent low-pass digitally modulated signal of the form

[ ]∑ −−−−=n

nn TnTtgjbnTtgatu )2()2()(

where {an} and {bn}are two sequences of statistically independent binary digits and g(t) is a sinusoidal pulse defined as

⎩⎨⎧ <<

=)(0)20()2/sin(

)(otherwise

TtTttg

π

This type of signal is viewed as a four-phase PSK signal in which the pulse shape is one-half cycle of a sinusoid. Each of the information sequences {an} and {bn} is transmitted at rate of 1/2T bits/s and, hence, the combined transmission rate is 1/T bits/s. The two sequences are staggered in time by T seconds in transmission. Consequently, the signal u(t) is called staggered four-phase PSK.

a) Show that the envelope )(tu is a constant, independent of the information an on the in-phase component and information bn on the quadrature component. In other words, the amplitude of the carries used in transmitting the signal is constant.

b) Determine the power density spectrum of u(t). c) Compare the power density spectrum obtained from (b) with the power density

spectrum of the MSK signal. What conclusion can you draw from this comparison?

Sol:

4.15 Consider a four-phase PSK signal represented by the equivalent low-pass signal

∑ −=n

n nTtgItu )()(

where In takes on one of the four possible values )1(21 j±± with equal probability. The

sequence of information symbols {In} is statistically independent. a) Determine and sketch the power density spectrum of u(t) when

⎩⎨⎧ <<

=)(0)20(

)(otherwise

TtAtg

b) Repeat (a) when

⎩⎨⎧ ≤≤

=)(0)0()/sin(

)(otherwise

TtTtAtg

π

c) Compare the spectra obtained in (a) and (b) in terms of the 3-dB bandwidth and

the bandwidth to the first spectral zero.

Sol:

4.19 π- QPSK may be considered as two QPSK systems offset by π/4 radians.

a) Sketch the signal space diagram for a π/4-QPSK signal b) Using Gray encoding, label the signal points with the corresponding data bits.

Sol:

4.20 The power density spectrum of the cyclostationary process

)()( ∑∞

−∞=

−=n

n nTtgItv

was derived in Section 4.4.1 by averaging the autocorrelation function Φvv(t+τ,t) over the period T of the process and the evaluating the Fourier transform of the average autocorrelation function. An alternative approach is to change the cyclostationary process into a stationary process ν∆(t) by adding a random variable ∆, uniformly distributed over 0≤∆<T, so that

)()( ∑∞

−∞=

∆ ∆−−=n

n nTtgItv

And defining the spectral density of ν∆(t) as the Fourier transform of the autocorrelation function of the stationary process ν∆(t).Derive the result in Equation 4.4 -11 by evaluating the autocorrelation function of ν∆(t) and its Fourier transform.

Sol:

4.22 The low-pass equivalent representation of PAM signal is

∑ −=n

n nTtgItu )()(

Suppose g(t) is a rectangular pulse and

2−−= nnn aaI where {an}is a sequence of uncorrelated binary-valued (1,-1) random variables that occur with equal probability.

a) Determine the autocorrelation function of the sequence {In}. b) Determine the power density spectrum of u(t). c) Repeat (b) if the possible values of the an are (0,1).

Sol:

4.30 Show that 16 QAM can be represented as a superposition of two four-phase constant envelope signals where each component is amplified separately before summing, i.e.,

)2sin2cos()2sin2cos()( tfDtfCtfBtfAGts cncncncn ππππ +++= where {An}, {Bn}, {Cn}, and {Dn} are statistically independent binary sequences with element from the set {+1,-1} and G is the amplifier gain. Thus, show that the resulting signal is equivalent to

tfQtfIts cncn ππ 2sin2cos)( += and determine In and Qn in terms of An, Bn, Cn, and Dn. Sol:

4.36 The two signal waveforms for binary FSK signal transmission with discontinuous phase are

⎥⎦⎤

⎢⎣⎡ +

∆−= 00 )

2(2cos2)( θπε tff

Tts c

b

b 0≤t≤T

⎥⎦⎤

⎢⎣⎡ +

∆−= 11 )

2(2cos

2)( θπ

εtff

Tts c

b

b 0≤t≤T

where ∆f=1/T<<fc, and θ0 and θ1 are uniformly distributed random variables on the interval (0,2π). The signals s0(t) and s0(t) are equally probable.

a) Determine the power spectral density of the FSK signal. b) Show that the power spectral density decays as 1/f2for f>>fc.

Sol: