chapter 1 review of digital communication theory · tan f. wong: spread spectrum & cdma 1....

Tan F. Wong:Spread Spectrum & CDMA 1. Digital Comm. Theory

Chapter 1

Review of Digital Communication Theory

We start by reviewing several important concepts which will be needed in the following chapters.

1.1 Maximum likelihood receiver

In this section, we assume that the communication channel is corrupted by an additive white Gaussian

noise (AWGN) with two-sided power spectral densityN0=2 W/Hz. The transmitter sends a signal cho-

sen from the set ofM signalsfsm(t)gM�1m=0 . We further assume that all theM signals are time-limited

to [0; T ], whereT is called thesymbol duration. The corresponding model for this communication

system is shown in Figure 1.1. The received signalr(t) is given by

r(t) = sm(t) + n(t); (1.1)

n(t)

m

r(t)=s (t)+n(t)ML

Receiverdecisions (t)

m

Figure 1.1: AWGN channel communication model

1.1


for somem 2 f0; 1; : : : ;M�1g. In (1.1),n(t) denotes the AWGN process with power spectral density

�n(f) = N0=2 W/Hz.

Our goal is to develop a receiver which observes the received signalr(t) and determines which one

of theM signals is being sent based on maximizing thelikelihood function. In order to proceed, we

need to define what the likelihood function is.

First, we try to represent the signals in a more convenient form. By employing the Gram-Schmidt

procedure, we can construct a set ofN (N � M ) orthonormal functionsf�n(t)gNn=1 (all are time-

limited to [0; T ]) which spans the signal space formed byfsm(t)gM�1m=0 . We augment this set of func-

tions by another set of orthonormal functionsf�n(t)g1n=N+1 so that the augmented setf�n(t)g1n=1

forms anorthonormal basisfor the space of square-integrable functions. Employing this basis, any

square-integrable function can be represented by a vector whose elements are coordinates with respect

to the basis functionsf�n(t)g1n=1. Based on this representation, we can rewrite (1.1) as

r = sm + n; (1.2)

where

r = [r1; r2; : : : ; rN ; : : :]T ;

sm = [sm1; sm2; : : : ; smN ; : : :]T for m = 0; 1; : : : ;M � 1;

n = [n1; n2; : : : ; nN ; : : :]T ;

are the vectors representingr(t), sm(t), andn(t)1, respectively. The coordinates are, respectively,

given by, fork = 1; 2; : : :,

rk =Z 1

�1r(t)�k(t)dt; (1.3)

smk =Z 1

�1sm(t)�k(t)dt for m = 0; 1; : : : ;M � 1; (1.4)

nk =Z 1

�1n(t)�k(t)dt: (1.5)

1We can represent a zero-mean WSS process with finite variance in a way similar to the vector representation of a

square-integrable function. Also, strictly speaking, the AWGN does not have finite energy and hence does not have such

an expansion. However, since the AWGN model is an approximation to the bandpass additive Gaussian noise, we abuse

the mathematics a little bit and assume that the vector representation for the AWGN we consider here is valid.

1.2


It can be shown (see Homework 1) that

rN = [r1; r2; : : : ; rN ]T

is a sufficient statistic for determining which signal is being sent, i.e., determining the value ofm.

Hence, we only need to deal with vectors of finite dimension. By rewriting (1.2) with these finite

dimensional vectors, we have

rN = smN + nN ; (1.6)

wheresmN andnN are the finite truncations ofsm andn, respectively. We also note thatnN is a zero

mean Gaussian random vector whose covariance matrix isN0

2I.

The maximum likelihood (ML) receiver makes a decision (selectm 2 f0; 1; : : : ;M � 1g) which

maximizes the likelihood function defined as the following conditional probability density function:

p(rN jsmN) =NYk=1

1p�N0

exp

"�(rk � smk)

2

N0

#: (1.7)

Since logarithm is a monotone increasing function, by taking logarithm ofp(rN jsmN ), it is easy to see

that the ML receiver picksm 2 f0; 1; : : : ;M � 1g such that the squared Euclidean distance between

the signal vectorsmN and the receiver vectorrN ,

d2(smN ; rN) =NXk=1

(smk � rk)2;

= sTmNsmN � 2sTmNrN + r

TNrN (1.8)

is minimized. Moreover, sincerTNrN is constant for all values ofm, we have

arg minm2f1;2;:::;Mg

d2(smN ; rN) = arg maxm2f1;2;:::;Mg

c(smN ; rN); (1.9)

where

c(smN ; rN) = sTmNrN � 1

2sTmNsmN ;

=Z T

0r(t)sm(t)dt� Em=2: (1.10)

In (1.10),c(smN ; rN) is called thecorrelation metricbetween the received signalr(t) and the trans-

mitted signalsm(t), and

Em =Z T

0s2m(t)dt; (1.11)

1.3


0

T( ) dt

0

T( ) dt

0

T( ) dt

- /2

M-1 ε- /2M-1

s (t)1 ε- /21

s (t)

Sele

ct M

axim

um

r(t) decision

s (t)0 ε0

Figure 1.2: ML receiver (correlation receiver) for AWGN channel

is the energy of the transmitted signalsm(t). The second equality follows from the orthonormal repre-

sentation (see Homework 1). In summary, we can implement the ML receiver as in the block diagram

shown in Figure 1.2. This implementation of the ML receiver is called thecorrelation receiver.

1.2 Matched filter receiver

In this section, we give another implementation, thematched filter receiver, of the ML receiver de-

veloped in Section 1.1. For simplicity, we restrict ourselves to the antipodal binary signaling case,

i.e.,M = 2 ands0(t) = s(t) = �s1(t) . It is easy to show (see Homework 1) that the correlators in

1.4


0

T( ) dt

@ t=T

s(t)

s(T-t)

Figure 1.3: Matched filter

Figure 1.2 can be replaced by the linear filters and samplers as shown in Figure 1.3. As a result, we can

employ these linear (matched) filters to implement the ML receiver in Figure 1.2. For the antipodal

binary case, it is easy to see that the receiver in Figure 1.4 is equivalent to the correlation receiver in

Figure 1.2. This form of implementation of the ML receiver is known as thematched filter receiver.

The matched filter has the optimal property that it is the linear filter that maximizes the output

signal-to-noise ratio (SNR). To establish this property, we replace the matched filter in Figure 1.4 by a

general linear filter with impulse responseh(t). Our goal is to determine the form ofh(t) maximizing

nY=Y + Ys

1s(T-t)

@ t=T

r(t) > 0, decide s (t)< 0, decide s (t)

0

Figure 1.4: Matched filter receiver for AWGN channel (antipodal binary signaling)

1.5


the output SNR defined by

SNR 4

=Y 2s

E[Y 2n ]; (1.12)

where

Ys =Z T

0s(t)h(T � t)dt; (1.13)

Yn =Z T

0n(t)h(T � t)dt: (1.14)

First, let us evaluateE[Y 2n ],

E[Y 2n ] =

Z T

0

Z T

0E[n(�)n(t)]h(T � �)h(T � t)d�dt

=N0

2

Z T

0

Z T

0Æ(� � t)h(T � �)h(T � t)d�dt

=N0

2

Z T

0h2(T � t)dt: (1.15)

Substituting (1.15) back into (1.12), we get

SNR =

hR T0 s(t)h(T � t)dt

i2N0

2

R T0 h2(T � t)dt

=

hR T0 s(T � t)h(t)dt

i2N0

2

R T0 h2(t)dt

: (1.16)

Now by employing the Cauchy-Schwartz inequality2, we have

SNR� 2

N0

Z T

0s2(T � t)dt =

2E0N0

; (1.17)

with equality holds if and only ifh(t) = Cs(T � t) for some constantC. Therefore, the matched filter

s(T � t), among all linear filters, maximizes the output SNR. We note that the choice of the constant

C is immaterial since it does not affect the value of the SNR. We chooseC = 1 in this case. An

interesting observation from (1.17) is that the maximum SNR achieved by the matched filter depends

only on the energy of the signal waveform, but not on other details.2Supposeg1(t) andg2(t) are square-integrable, then

�Z1

�1

g1(t)g2(t)dt

�2�

Z1

�1

g21(t)dt

Z1

�1

g22(t)dt;

with equality holds if and only ifg1(t) = Cg2(t) for some constantC.

1.6


1.3 Signal space representation

In Section 1.1, we represent the transmitted signals by vectors of finite dimension. It turns out that this

geometric viewpoint greatly facilitates the understanding and analysis of many modulation schemes.

Because of this, we study the geometric representation more carefully in this section.

Supposef�n(t)gNn=1 is an orthonormal basisfor the signal space spanned by a set of square-

integrable signal waveformsfsm(t)gM�1m=0 . We represent the signal waveforms by a set ofM N -

dimensional vectors with respect to the basisf�n(t)gNn=1. More precisely, form = 0; 1; : : : ;M � 1,

sm(t) is represented by theN -dimensional vectorsm = [sm1; sm2; : : : ; smN ]T whosen-th coordinate

smn, for n = 1; 2; : : : ; N; is given by the inner product ofsm(t) and�n(t), i.e.,

smn = (sm; �n)4

=Z 1

�1sm(t)�n(t)dt: (1.18)

Given the basis, we can uniquely3 determine the signalsm(t) from the vectorsm or vice versa. As

a result, the vector representation provides a geometric viewpoint of the signal space. Since we are

much more familiar with Euclidean geometry than the square-integrable function space, this geometric

viewpoint allows us to visualize the underlying structure of the signal space easily. There are two

important identities which greatly simply the analyses in the following sections:

(sm; sk)4

=Z 1

�1sm(t)sk(t)dt = s

Tmsk; (1.19)

d2(sm; sk)4

=Z 1

�1[sm(t)� sk(t)]

2dt = ksm � skk2; (1.20)

form; k = 0; 1; : : : ;M�1. The notationk�k denotes the Euclidean norm of a vector. The first identity

states that the inner products in the function space and the vector space are equivalent. The second

identity states that the squared distance in the function space is the same as the squared Euclidean

distance in the vector space.

We see from Section 1.1 that the Gram-Schmidt procedure can be employed to find an orthonormal

basis from the signal sets. However, we do not always need to employ the Gram-Schmidt procedure to

obtain a convenient basis for a signal set. For example, consider the following signal set of the QPSK

scheme:

s0(t) =p2P cos(�t=T + �=4)pT (t);

3There can be many bases for the signal space. Different bases give rise to different vector representations.

1.7


PT/2

PT/2

1

s3

s0s1

s2

φ

φ2

Figure 1.5: QPSK constellation

s1(t) =p2P cos(�t=T + 3�=4)pT (t);

s2(t) =p2P cos(�t=T + 5�=4)pT (t);

s3(t) =p2P cos(�t=T + 7�=4)pT (t);

wherepT (t) = 1 for 0 � t < T , andpT (t) = 0 otherwise. By inspection, a simple basis for this signal

set is

�1(t) =

s2

Tcos(�t=T )pT (t);

�2(t) =

s2

Tsin(�t=T )pT (t):

Using this basis, the corresponding signal vectors are

s0 = [qPT=2;

qPT=2]T ;

s1 = [�qPT=2;

qPT=2]T ;

s2 = [�qPT=2;�

qPT=2]T ;

s3 = [qPT=2;�

qPT=2]T :

The corresponding constellation diagram is drawn in Figure 1.5. We will use this simple orthonormal

basis to represent the signal spaces of all quadrature modulation schemes.

1.8


1.4 ML receiver error analysis

In this section, we analyze the performance of the ML receiver by evaluating the symbol error prob-

ability. We begin by defining what a symbol error is. We say that a symbol error event occurs when

the decision made by the receiver is different from the transmitted symbol. Form = 0; 1; : : : ;M � 1,

letPsjm denotes the conditional symbol error probability given thatsm(t) is being transmitted, andPm

denotes that probability that the transmitter sendssm(t). Then the average symbol error probability,

Ps, is given by

Ps =M�1Xm=0

PsjmPm: (1.21)

For simplicity, we assume all the signals are equally likely to be transmitted, i.e.,Pm = 1=M , for

m = 0; 1; : : : ;M�1. Then the problem reduces to evaluatingPsjm, form = 0; 1; : : : ;M�1. We start

by working through the simple cases of BPSK and QPSK, for which exact symbol error probabilities

can be found. In general, it is often very hard to obtain the exact symbol error probabilities. Therefore,

we introduce the method of union bound to upper-bound the symbol error probability.

1.4.1 BPSK

For the case of BPSK (binary antipodal signaling), the matched filter receiver in Section 1.2 is the ML

receiver. The receiver compares the sampled outputY of the matched filter to the threshold zero. If

Y > 0, the receiver decides thats0(t) = s(t) is sent. Otherwise, it decides thats1(t) = �s(t) is sent.

From (1.14) and (1.15), we know that the noise sampleYn is a zero mean Gaussian random variable

with variance

�2Yn =N0

2

Z T

0s2(t)dt =

EN0

2:

Supposes0(t) is being sent, thenY is a Gaussian random variable with meanE and variance�2Yn .

From the decision rule stated above, the receiver makes an error whenY � 0. Hence,

Psj0 = Pr(Y � 0js0(t)sent)

=1p

2��Yn

Z 0

�1exp

"�(x� E)2

2�2Yn

#dx

= Q(E=�Yn) = Q(p

SNR)

= Q(q2E=N0); (1.22)

1.9


s1

s4

s0

s3

s2

s5

R0

R1

R5

R2

R3

R4

Figure 1.6: Voronoi diagram

where

Q(x) =1p2�

Z 1

xexp(�u2=2)du: (1.23)

With the same argument, we can show thatPsj1 = Psj0 = Q(q2E=N0). Therefore,Ps = Q(

q2E=N0).

1.4.2 General case (a geometric approach)

Now assume that we employM -ary signaling, i.e., the transmitter sends a signal out from the set

fsm(t)gM�1m=0 . Using the vector representation in Section 1.1, we know that the ML receiver decides that

them-th signal is sent when the Euclidean distanced(rN ; smN ) is the smallest among all theM signal

vectors. If we draw the signal vectors as points in the constellation diagram as shown in Figure 1.6, the

geometric meaning of the ML decision rule is that the signalsmN closest to the receiver vectorrN is

selected. Equivalently, we can construct a decision region (based on the minimum distance principle)

for each of the signal point in the constellation diagram, and decide a specific signal point is sent if the

received vectorrN falls into the corresponding decision region. A diagram showing the signal points

and their corresponding decision regions is known as theVoronoi diagramof a modulation scheme.

1.10


Now we have all the tools needed to calculate the symbol error probability of a generalM -ary

modulation with the ML receiver. Supposesm(t), for somem 2 f0; 1; : : : ;M � 1g, is being sent, and

letRm denotes the decision region forsm(t). We make an error if the received vectorrN falls outside

Rm. Therefore, the conditional symbol error probability given thatsm(t) is sent,

Psjm = Pr(rN 2 <N n Rmjsm(t) sent)

=Z<NnRm

p(rN jsmN )drN

= 1�ZRm

p(rN jsmN )drN ; (1.24)

wherep(rN jsmN ) is given in (1.7). Although the expression in (1.24) looks simple, it is generally

difficult to construct the Voronoi diagram and evaluate the integral in (1.24). However, for some

special cases closed form solutions can be found.

The first special case we consider is the binary signaling case (M = 2, N � 2). It is intuitive that

the decision regions for the signal pointss0 ands1 are separated by the hyperplane half-way between

the signal points and perpendicular to the line joining the two signal points. The next step is to evaluate

the integral in (1.24). Supposes0(t) is being sent, we know that

Psj0 =ZR1

p(rN js0N)drN

=Z 0

�1

1p2��n

exp

"�(x� d(s0N ; s1N )=2)

2

2�2n

#

= Q(d(s0N ; s1N)=2�n); (1.25)

where�2n = N0=2 is the variance of an element of the noise vectornN . We note that the second equality

in (1.25) above is obtained by a change of variable which corresponds to a suitable rotation and trans-

lation of the axis. Clearly,Psj1 can be calculated in the same way. ThusPs = Q(d(s0N ; s1N)=2�n).

Moreover, it can be seen that the result in (1.25) reduces to (1.22) for BPSK (antipodal binary signal-

ing).

The next special case we consider is the QPSK example given in Section 1.3 (M = 4, N = 2). It

is again obvious that the decision region for a signal point is the quadrant in which the signal point is

located. Supposes0(t) is being sent, then

Psj0 = 1�ZR0

p(r2js02)dr2

1.11


= 1�Z 1

0

Z 1

0

1

2��2nexp

24�(r1 �

qPT=2)2 + (r2 �

qPT=2)2

2�2n

35 dr1dr2

= 1�Q2(�qPT=N0) = 2Q(

qPT=N0)�Q2(

qPT=N0): (1.26)

Similarly, we havePsj1 = Psj2 = Psj3 = Psj0 andPs = 1�Q2(�qPT=N0).

1.4.3 Union bound

When the exact symbol error probability is too difficult to evaluate, we resort to bounds and approxi-

mations. One of such methods is the union bound.

Supposes0(t) is being transmitted, we know from Section 1.4.2 that

Psj0 = Pr

"M�1[m=1

fd(rN ; smN ) < d(rN ; s0N)g��s0N

#

�M�1Xm=1

Pr [fd(rN ; smN) < d(rN ; s0N )gjs0N ] : (1.27)

We notice that the eventfd(rN ; smN ) < d(rN ; s0N)g in (1.27) is exactly the same as the error event

as if there were only two signals,s0(t) andsm(t) (m � 1), in the signal set. The probability of this

event has been calculated in (1.25). Hence, we obtain theunion boundof the conditional symbol error

probability as

Psj0 �M�1Xm=1

Q(d(s0N ; smN)=2�n); (1.28)

where�2n = N0=2. Similarly, we can find union bounds for the conditional error probabilities given

that other signals are sent. By averaging over all the signals, we obtain the union bound for the average

symbol error probability as

Ps � 1

M

M�1Xm=0

M�1Xn=0n6=m

Q(d(smN ; snN)=2�n): (1.29)

As an illustration, we work out the union bound for the symbol error probability for the QPSK example

in Section 1.3. From (1.28), we have

Psj0 � Q(d(s0N ; s1N )=2�n) +Q(d(s0N ; s2N )=2�n) +Q(d(s0N ; s3N )=2�n)

= Q(qPT=N0) +Q(

q2PT=N0) +Q(

qPT=N0): (1.30)

1.12


By symmetry, we have

Ps � 2Q(qPT=N0) +Q(

q2PT=N0); (1.31)

which is slightly larger than the exact symbol error probability given in (1.26).

1.5 Complex envelope

Very often in a communication system, we do not transmit the lowpass baseband signal directly. In-

stead, we mix the baseband signal with a carrier up to a certain frequency, which matches the elec-

tromagnetic propagation characteristic of the channel. As a result, the actual transmitted signal is

a bandpass signal. In this section, we introduce the concept of complex envelope which provides a

convenient way to represent bandpass signals.

1.5.1 Narrowband signal

Supposes(t) is a (real-valued) bandpass signal with most of its frequency content concentrated in a

narrow band in the vicinity of a center frequency!c. A sufficient condition is that the Fourier transform

of s(t) satisfiesS(!) = 0 for j!j � 2!c. We refer to this condition as thenarrowband assumption.

For a bandpass signals(t) satisfying the narrowband assumption stated above, it can be shown [1]

thats(t) can be represented by an in-phase componentx(t) and a quadrature componenty(t),

s(t) = x(t) cos(!ct)� y(t) sin(!ct)

= x(t)

ej!ct + e�j!ct

2

!� y(t)

ej!ct � e�j!ct

2j

!

=

x(t) + jy(t)

2

!ej!ct +

x(t)� jy(t)

2

!e�j!ct: (1.32)

Now we define thecomplex envelope~s(t) of the signals(t) as

~s(t)4

= x(t) + jy(t): (1.33)

Then from (1.32), we have

s(t) = ~s(t)ej!ct=2 + ~s�(t)e�j!ct=2 = Re[~s(t)ej!ct]: (1.34)

1.13


Taking Fourier transform on both sides of (1.34), we get

S(!) = ~S(! � !c)=2 + ~S�(�! � !c)=2; (1.35)

where ~S(!) is the Fourier transform of the complex envelope~s(t). Using (1.34) and (1.35), we can

reconstruct the real-valued bandpass signals(t) back from its complex envelope~s(t).

The remaining question is how to obtain the complex envelope~s(t) from the signals(t). If s(t) is

given in the form of (1.32), then we can simply set the complex envelope as~s(t) = x(t)+jy(t). If only

the Fourier transformS(!) of s(t) is given (ors(t) is not in the convenient form of (1.32)), more work

is needed. First, let us notice that~S(!) = 0 for j!j � !c. We can conclude this fact easily from (1.35).

Based on our narrowband assumption,S(!) = 0 for j!j � 2!c. From (1.35), we know thatS(!) is

the sum of two shifted (and scaled) versions of~S(!). If ~S(!) does not vanish outside(�!c; !c), S(!)cannot vanish outside(�2!c; 2!c). This, of course, contradicts the narrowband assumption. Next, we

shift ~S(!) to the left by!c rad./s in (1.35),

S(! + !c) = ~S(!)=2 + ~S�(�! � 2!c)=2: (1.36)

We notice~S�(�! � 2!c) is nonzero only on the interval(�3!c;�!c). Therefore,

~S(!) = 2L!c[S(! + !c)]; (1.37)

whereL!c [�] is the ideal lowpass filter (with bandwidth!c rad./s) operator, which removes all the

frequency components outside the band(�!c; !c). Finally, we can take the inverse Fourier transform

of 2L!c[S(! + !c)] to get~s(t). Pictorially, we take the positive frequency part ofS(!) and shift it

down to baseband to obtain~S(!). This is the reason why the complex envelope~s(t) is sometimes

called thelowpass equivalent signalof s(t).

Based on (1.37), we can easily construct a circuit to convert a real-valued signal to its complex

envelope. To do so, we start by rewriting (1.37) in the time domain,

~s(t) = 2L!c [s(t)e�j!ct]

= 2L!c [s(t) cos(!ct)� js(t) sin(!ct)]

= L!c [2s(t) cos(!ct)]� jL!c [2s(t) sin(!ct)]: (1.38)

1.14


wL

cwL

c

2sin(w t) -j

s(t) s(t)~c2cos(w t)

c

Figure 1.7: Complex envelope conversion circuit

We note that the third equality in (1.38) is due to the linearity of the lowpass filter operatorL!c . Now

it is obvious that we can use the circuit in Figure 1.7 to convert a real-valued signal to its complex

envelope.

1.5.2 Bandpass filter

We can use the complex envelope in the previous section to represent the impulse responseh(t) of a

bandpass filter given thath(t) satisfies the narrowband assumption stated before. Hence, if~h(t) is the

complex envelope ofh(t), then

h(t) = Re[~h(t)ej!ct]: (1.39)

Now, if a bandpass signal (satisfying the narrowband assumption)si(t) is the input to the bandpass

filter h(t), then the output from the filterso(t) also satisfies the narrowband assumption and

so(t) = h(t) � si(t); (1.40)

where� denotes the convolution operator. In the frequency domain,

So(!) = H(!)Si(!); (1.41)

1.15


whereSo(!),H(!), andSi(!) are the Fourier transforms ofso(t), h(t), andsi(t), respectively. From

(1.37), the Fourier transform of the complex envelope,~so(t), of so(t) is given by

~So(!) = 2L!c[So(! + !c)]

= 2L!c[H(! + !c)Si(! + !c)]

= 2L!c[H(! + !c)]L!c[Si(! + !c)]

=1

2~H(!) ~Si(!); (1.42)

where ~H(!) and ~Si(!) are the Fourier transforms of the complex envelopes,~h(t) and~si(t), of h(t)

andsi(t), respectively. The third equality in (1.42) is due to the fact that bothh(t) andsi(t) share the

same passband. By taking inverse Fourier transform on both sides of (1.42), we obtain

~so(t) =1

2~h(t) � ~si(t): (1.43)

Hence, we can convolute the complex envelopes ofh(t) andso(t) and then convert the result back to

obtain the output bandpass signal.

1.5.3 Narrowband process

Supposen(t) is a wide-sense stationary (WSS) process with zero mean and power spectral density

�n(!). If �n(!) satisfies the narrowband assumption, thenn(t) is called anarrowband process. It

turns out [2] thatn(t) can also be written as

n(t) = nx(t) cos(!ct)� ny(t) sin(!ct); (1.44)

wherenx(t) andny(t) are zero-mean jointly WSS processes. Moreover, ifn(t) is Gaussian,nx(t) and

ny(t) are jointly Gaussian. By employing the stationarity of the random processes involved, we can

show (see Homework 1) that

Rnx(�) = Rny(�); (1.45)

Rnxny(�) = �Rnynx(�); (1.46)

Rn(�) = Rnx(�) cos(!c�)�Rnxny(�) sin(!c�); (1.47)

1.16


whereRn(�)4

= E[n(t)n(t+ �)] is the autocorrelation function of the random processn(t), Rnx(�)4

=

E[nx(t)nx(t + �)] andRny(�)4

= E[ny(t)ny(t + �)] are, respectively, the autocorrelation functions of

the processesnx(t) andny(t), andRnxny(�)4

= E[nx(t)ny(t + �)] andRnynx(�)4

= E[ny(t)nx(t+ �)]

are the cross-correlation functions. Now, let us define the complex envelope~n(t) of the random process

n(t),

~n(t)4

= nx(t) + jny(t): (1.48)

Obviously,~n(t) is a zero-mean WSS complex random process with autocorrelation function

R~n(�)4

=1

2E[~n�(t)~n(t+ �)]

=1

2E[(nx(t)� jny(t))(nx(t+ �) + jny(t+ �))]

=1

2Rnx(�) +

1

2Rny(�) +

j

2Rnxny(�)�

j

2Rnynx(�)

= Rnx(�) + jRnxny(�): (1.49)

Now compare (1.47) with (1.32) and (1.49) with (1.33). If we treat the autocorrelation functionRn(�)

as a bandpass signal (by definition, it satisfies the narrowband assumption sincen(t) is a narrowband

process), thenR~n(�) is its complex envelope. Hence, we can use the results in Section 1.5.1 to convert

betweenRn(�) andR~n(�).

A common example of narrowband process is the bandpass additive Gaussian noisen(t) with zero

mean and power spectral density�n(!) = N0=2 for j!j < 2!c and�n(!) = 0 otherwise. Since

�n(!) satisfies the narrowband assumption,n(t) can be written as

n(t) = nx(t) cos(!ct)� ny(t) sin(!ct); (1.50)

wherenx(t) andny(t) are zero-mean jointly WSS Gaussian processes. The complex envelope ofn(t)

is given by

~n(t) = nx(t) + jny(t): (1.51)

Using the result above and (1.37), the power spectral density�~n(!) of the complex envelope~n(t) is

given by

�~n(!) = 2L!c[�n(! + !c)]

=

8><>:N0 if j!j < !c;

0 otherwise:(1.52)

1.17


Taking inverse Fourier transform, we get

R~n(�) = N02fcsin(!c�)

!c�; (1.53)

where!c = 2�fc. SinceR~n(�) is real,Rnxny(�) = 0 and hence the processesnx(t) andny(t)

are uncorrelated. Moreover, we haveRnx(�) = Rny(�) = R~n(�). For the case where bandpass

transmitted signals are sent through a channel corrupted byn(t) and the bandwidths of the transmitted

signals are much smaller than the carrier frequency!c, we approximateR~n(�) in (1.53) byN0Æ(�).

This means that the lowpass equivalent of the additive bandpass Gaussian noise looks white to the

lowpass equivalents of the transmitted signals. In this way, we are back to the communication model

of transmitting baseband signals over an AWGN channel as in Section 1.1. Of course, all the signal are

complex instead of real now. By using the same method in Section1.1, we can develop (see Homework

1) the ML receiver for the complex baseband communication system.

1.6 Noncoherent receiver

The ML receiver is developed in Section 1.1 based on the assumption that the channel does nothing

to the transmitted signal except adding the AWGN to it. This model is obviously too simple to model

any real life communication channel. As we mentioned before, since most communication systems

transmit bandpass signals instead of baseband ones, we focus on this kind of signals and use the

complex envelopes to represent them here. Again, we consider the simple case of anon-dispersive

channel, for which we can model the received signal as

r(t) = Aej�sm(t) + n(t); 4 (1.54)

whereA > 0 represents the channel gain (attenuation),� represents the carrier phase shift due to prop-

agation delay, local oscillator mismatch, andetc., andn(t) is the complex AWGN with autocorrelation

functionRn(�) = N0Æ(�). Suppose the receiver knows the value of� 5, the problem reduces to the

one in Section 1.1. Hence we can use the correlation receiver in Figure 1.2 (or its complex equivalent)4From now on, we drop the ˜ symbol for complex envelopes.5The value ofA is not needed when the signals have equal energies

1.18


0

T( ) dt

0

T( ) dt

s (t)0*

0

T( ) dt | | or | |

r(t)s (t)1*

s (t)*M-1

Sele

ct M

axim

um

decision

2

2

2

| | or | |

| | or | |

Figure 1.8: Envelope / square-law receiver forM -ary orthogonal signals

to detect the received signalr(t). Generally, receivers that make use of the phase information are re-

ferred to ascoherentreceivers. Therefore, the correlation receiver in Figure 1.2 and the matched filter

receiver in Figure 1.4 are coherent receivers.

For coherent reception, we need to estimate the carrier phase�. This estimation can sometimes

be hard to perform, and inaccurate estimation of the carrier phase will significantly degrade the per-

formance of the coherent ML receiver. One alternative to coherent reception is to avoid using the

phase information. To do so, we model the carrier phase� as a random variable uniformly distributed

on [0; 2�). Following steps similar to those in Section 1.2, we can develop the ML receiver for this

case. The resulting receiver is known as thenoncoherentML receiver. For the case where the trans-

mitted signalsfsm(t)gM�1m=0 have equal energies, the ML receiver assumes the simple form [1] shown

in Figure 1.8. This receiver is usually referred to as theenvelope receiveror thesquare-law receiver

1.19


depending on whether the envelope or the square-law detecting device is employed. It is difficult to

evaluate the symbol error probability for a generalM -ary signal set received by the noncoherent ML

receiver. For the special case of equal-energy binary orthogonal signals, we state that the average

symbol error probability (assuming equala priori probabilities) is given by [1]

Ps =1

2e�E=2N0 ; (1.55)

whereE is the signal energy.

1.7 Power spectrum

In all the previous sections, we assume that a single time-limited signal (pulse) is sent. In this section,

we consider a more realistic model in which a train of pulses are transmitted. For simplicity, we ignore

the white noise and assume that the (complex envelope of the) received signal is given by

s(t) = x(t) + jy(t); (1.56)

x(t) =1X

k=�1

ak x(t� kTs ��); (1.57)

y(t) =1X

k=�1

bk y(t� kTs ��); (1.58)

whereak’s are independent identically distributed (iid) random variables with mean zero and variance

A2, andbk’s are also iid random variables with mean zero and varianceB2. Moreover, we assume

that the two data streamsfakg1k=�1 andfbkg1k=�1 are independent. In above,� can be interpreted

as the propagation delay, and x(t) and y(t) are the pulses for the in-phase and quadrature channels,

respectively. We notice thats(t) is a zero-mean random process. This model almost covers all practical

quadrature modulation schemes.

Our objective is to evaluate the autocorrelation function ofs(t). First, let us model� as a random

variable which is uniformly distributed on[0; Ts), and is independent to bothfakg1k=�1 andfbkg1k=�1.

Then the autocorrelation function ofs(t) is given by

Rs(t; t+ �)4

=1

2E[s�(t)s(t+ �)]

=1

2fE[x(t)x(t + �)]� jE[x(t)y(t+ �)]� jE[y(t)x(t+ �)] + E[y(t)y(t+ �)]g

=1

2fE[x(t)x(t + �)] + E[y(t)y(t+ �)]g : (1.59)

1.20


The last equality in (1.59) follows from the fact that the two data streams consist of zero-mean inde-

pendent random variables. Now, it suffices to evaluateE[x(t)x(t + �)],

E[x(t)x(t + �)] =1X

k=�1

1Xl=�1

E[akal]E[ x(t� kTs ��) x(t+ � � lTs ��)]

=1X

k=�1

A2 1

Ts

Z Ts

0 x(t� kTs ��) x(t+ � � kTs ��)d�

=A2

Ts

1Xk=�1

Z (k+1)Ts

kTs x(t��) x(t+ � ��)d�

=A2

Ts

Z 1

�1 x(t��) x(t+ � ��)d�

=A2

Ts

Z 1

�1 x(��) x(� ��)d�

=A2

Ts x(��) � x(�): (1.60)

Similarly, we have

E[y(t)y(t+ �)] =B2

Ts y(��) � y(�): (1.61)

Therefore, the processs(t) is WSS and

Rs(�) = Rs(t; t+ �) =1

2Ts

hA2 x(��) � x(�) +B2 y(��) � y(�)

i: (1.62)

The power spectral density (power spectrum) ofs(t) is given by

�s(!) =1

2Ts

hA2jx(!)j2 +B2jy(!)j2

i; (1.63)

wherex(!) andy(!) are the Fourier transforms of x(t) and y(t), respectively.

For example, we consider the BPSK scheme where x(t) = pTs(t) and y(t) = 0. We consider

two cases:Ts = T andTs = T=10. In both cases, we letA2 = 2. For the first case, the power spectrum

is

�s(!) = Tsin2(!T=2)

(!T=2)2: (1.64)

For the second case, the power spectrum is

�s(!) =T

10

sin2(!T=20)

(!T=20)2: (1.65)

The two spectra are plotted in Figure 1.9 for comparison. From Figure 1.9, we observe that if we

decrease the pulse duration, we will obtain a wider and lower power spectrum. This observation forms

the basis for spread spectrum communications.

1.21


−40 −30 −20 −10 0 10 20 30 4010

−5

10−4

10−3

10−2

10−1

100

ω (π/T Hz)

pow

er s

pect

ral d

ensi

ty

Ts=T/10

Ts=T

Figure 1.9: Power spectra of BPSK schemes:Ts = T andTs = T=10

1.22


1.8 References

[1] J. G. Proakis,Digital Communications, 3rd Ed., McGraw-Hill, Inc., 1995.

[2] W. B. Davenport and W. L. Root,An Introduction to the Theory of Random Signals and Noise,

McGraw-Hill, Inc., 1958.

1.23

chapter 1 review of digital communication theory · tan f. wong: spread spectrum & cdma 1....

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