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
2
noise (AWGN) with two-sided power spectral density N0 = W/Hz. The transmitter sends a signal cho-
( )gMm
sen from the set of M signals fsm t
to
1
=0
. We further assume that all the M signals are time-limited
[0; T ], where T is called the symbol duration.
The corresponding model for this communication
()
system is shown in Figure 1.1. The received signal r t is given by
( ) = sm (t) + n(t);
r t
r(t)=sm (t)+n(t)
s m(t)
ML
Receiver
n(t)
Figure 1.1: AWGN channel communication model
1.1
(1.1)
decision
Tan F. Wong: Spread Spectrum & CDMA
01
for some m 2 f ; ; : : : ; M
n(f ) = N =2 W/Hz.
1. Digital Comm. Theory
1g. In (1.1), n(t) denotes the AWGN process with power spectral density
0
()
Our goal is to develop a receiver which observes the received signal r t and determines which one
of the M signals is being sent based on maximizing the likelihood 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
M ) orthonormal functions fn(t)gNn (all are timelimited to [0; T ]) which spans the signal space formed by fsm (t)gM
. We augment this set of funcm
tions by another set of orthonormal functions fn (t)g1
so that the augmented set fn (t)g1
n N
n
procedure, we can construct a set of N (N
=1
1
=0
=
+1
=1
forms an orthonormal basis for 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 functions fn t g1
n=1 . Based on this representation, we can rewrite (1.1) as
r
= sm + n;
(1.2)
where
r
sm
n
= [r ; r ; : : : ; rN ; : : :]T ;
= [sm ; sm ; : : : ; smN ; : : :]T
= [n ; n ; : : : ; nN ; : : :]T ;
1
2
1
1
2
for m
= 0; 1; : : : ; M 1;
2
()
()
()
are the vectors representing r t , sm t , and n t 1 , respectively. The coordinates are, respectively,
given by, for k
= 1; 2; : : :,
rk
smk
nk
1
=
=
=
Z
Z
Z
1
r (t)k (t)dt;
1
1
sm (t)k (t)dt for m = 0; 1; : : : ; M
1
1
n(t)k (t)dt:
1
(1.3)
1;
(1.4)
(1.5)
We 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
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
It can be shown (see Homework 1) that
rN
= [r ; r ; : : : ; rN ]T
1
2
is a sufficient statistic for determining which signal is being sent, i.e., determining the value of m.
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)
where smN and nN are the finite truncations of sm and n, respectively. We also note that nN is a zero
mean Gaussian random vector whose covariance matrix is
N0
2
I.
The maximum likelihood (ML) receiver makes a decision (select m
2 f0; 1; : : : ; M 1g) which
maximizes the likelihood function defined as the following conditional probability density function:
( j
p rN smN
)=
N
Y
k =1
1 exp (rk
pN
"
0
smk
N0
)
2
#
:
(1.7)
(
)
Since logarithm is a monotone increasing function, by taking logarithm of p rN jsmN , it is easy to see
that the ML receiver picks m
2 f0; 1; : : : ; M 1g such that the squared Euclidean distance between
the signal vector smN and the receiver vector rN ,
(
d2 smN ; rN
) =
=
N
X
(smk
)
rk 2 ;
k =1
sTmN smN
2sTmN rN + rTN rN
(1.8)
is minimized. Moreover, since rTN rN is constant for all values of m, we have
arg
min d (smN ; rN ) = arg
max c(smN ; rN );
m2f ; ;:::;M g
m2f ; ;:::;M g
2
12
(1.9)
12
where
(
c smN ; rN
(
) =
=
1 sT s ;
mN mN
2
T
r (t)sm (t)dt Em =2:
T
smN rN
Z
0
)
(1.10)
()
In (1.10), c smN ; rN is called the correlation metric between the received signal r t and the trans-
()
mitted signal sm t , and
Em =
Z T
0
()
s2m t dt;
1.3
(1.11)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
T
( ) dt
s 0 (t)
-
ε0 /2
-
ε1 /2
-
εM-1 /2
T
( ) dt
0
r(t)
s 1 (t)
Select Maximum
0
decision
T
( ) dt
0
s M-1 (t)
Figure 1.2: ML receiver (correlation receiver) for AWGN channel
()
is the energy of the transmitted signal sm t . The second equality follows from the orthonormal representation (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 the correlation receiver.
1.2 Matched filter receiver
In this section, we give another implementation, the matched filter receiver, of the ML receiver developed in Section 1.1. For simplicity, we restrict ourselves to the antipodal binary signaling case,
i.e., M
= 2 and s (t) = s(t) =
0
()
s1 t . It is easy to show (see Homework 1) that the correlators in
1.4
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
T
( ) dt
0
s(t)
@ t=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 the matched 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 response h t . Our goal is to determine the form of h t maximizing
@ t=T
r(t)
Y=Ys + Yn
s(T-t)
> 0, decide s0 (t)
< 0, decide s1 (t)
Figure 1.4: Matched filter receiver for AWGN channel (antipodal binary signaling)
1.5
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
the output SNR defined by
=4 EY[Ys ] ;
n
2
SNR
(1.12)
2
where
Z T
=
=
Ys
Yn
Z
0
0
( ) ( t)dt;
T
n(t)h(T
t)dt:
s t h T
(1.13)
(1.14)
[ ]
First, let us evaluate E Yn2 ,
Z T Z T
[ ] =
= N2
= N2
E Yn2
0
0
0
Z
0
0
Z
[ ( ) ( )] (
T
T
Æ (
t)h(T
T
h (T
t)dt:
)(
)h(T
En n t h T
h T
Z
0
)
t)d dt
t d dt
2
0
(1.15)
Substituting (1.15) back into (1.12), we get
( ) ( t)dt
= N T h (T t)dt
T
s(T
t)h(t)dt
=
T
N
h (t)dt
hR
T
SNR
0
R
0
2
0
2
hR
i2
s t h T
i2
0
R
0
2
0
2
:
(1.16)
Now by employing the Cauchy-Schwartz inequality2, we have
2
SNR N
Z T
0
0
) = 2NE
(
s2 T
0
t dt
;
(1.17)
0
( ) = Cs(T t) for some constant C . Therefore, the matched filter
with equality holds if and only if h t
(
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 choose C
= 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.
2
Suppose g1 (t) and g2 (t) are square-integrable, then
Z
1
1
2
g1 (t)g2 (t)dt
Z
1
1
2
g1 (t)dt
with equality holds if and only if g1 (t) = C g2 (t) for some constant C .
1.6
Z
1
1
2
g2 (t)dt;
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
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.
( )gNn
Suppose fn t
=1
is an orthonormal basis for the signal space spanned by a set of square-
( )gMm
integrable signal waveforms fsm t
1
=0
. We represent the signal waveforms by a set of M N -
( )gNn . More precisely, for m = 0; 1; : : : ; M 1,
sm (t) is represented by the N -dimensional vector sm = [sm ; sm ; : : : ; smN ]T whose n-th coordinate
smn , for n = 1; 2; : : : ; N; is given by the inner product of sm (t) and n (t), i.e.,
dimensional vectors with respect to the basis fn t
=1
1
smn
= (sm; n) =4
Z
2
1
sm (t)n (t)dt:
1
(1.18)
()
Given the basis, we can uniquely3 determine the signal sm t from the vector sm 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
4
d (sm ; sk ) =
2
for m; k
Z
Z
1
sm (t)sk (t)dt = sTm sk ;
1
1
[sm(t) sk (t)]2dt = ksm
1
(1.19)
sk
k;
2
(1.20)
= 0; 1; : : : ; M 1. The notation kk 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
3
p
2P cos(t=T + =4)pT (t);
There can be many bases for the signal space. Different bases give rise to different vector representations.
1.7
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
φ2
s1
s0
PT/2
φ1
PT/2
s2
s3
Figure 1.5: QPSK constellation
p
( ) = 2P cos(t=T + 3=4)pT (t);
p
s (t) =
2P cos(t=T + 5=4)pT (t);
p
2P cos(t=T + 7=4)pT (t);
s (t) =
s1 t
2
3
( ) = 1 for 0 t < T , and pT (t) = 0 otherwise. By inspection, a simple basis for this signal
where pT t
set is
s
() =
1 t
s
() =
2 t
2 cos(t=T )p (t);
T
T
2 sin(t=T )p (t):
T
T
Using this basis, the corresponding signal vectors are
s0
s1
s2
s3
=
=
=
=
q
[
[
[
[
2
q
2]T ;
P T =2; P T =2]T ;
P T =2;
P T =2]T ;
P T =2;
P T =2]T :
PT= ;
PT=
q
q
q
q
q
q
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
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
1.4 ML receiver error analysis
In this section, we analyze the performance of the ML receiver by evaluating the symbol error probability. 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. For m
()
= 0; 1; : : : ; M 1,
let Psjm denotes the conditional symbol error probability given that sm t is being transmitted, and Pm
()
denotes that probability that the transmitter sends sm t . Then the average symbol error probability,
Ps , is given by
Ps
=
M
X1
m=0
Psjm Pm :
(1.21)
= 1=M , for
1. Then the problem reduces to evaluating Psjm, for m = 0; 1; : : : ; M 1. We start
For simplicity, we assume all the signals are equally likely to be transmitted, i.e., Pm
m
= 0; 1; : : : ; M
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 output Y of the matched filter to the threshold zero. If
0
( ) = s(t) is sent. Otherwise, it decides that s (t) = s(t) is sent.
Y > , the receiver decides that s0 t
1
From (1.14) and (1.15), we know that the noise sample Yn is a zero mean Gaussian random variable
with variance
2
Yn
()
= 2
N0
( ) = E N2
Z T
s2 t dt
0
0
:
Suppose s0 t is being sent, then Y is a Gaussian random variable with mean
From the decision rule stated above, the receiver makes an error when Y
Psj0
=
=
=
=
Pr(Y 0js (t)sent)
(x E )
p1
exp
2Yn 1 p 2Yn
Q(E =Yn ) = Q( SNR)
Q( 2E =N );
0
Z 0
"
2
2
E and variance Yn .
2
0. Hence,
#
dx
q
0
1.9
(1.22)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
R1
s1
R0
R2
s0
s2
s5
R3
R5
s3
s4
R4
Figure 1.6: Voronoi diagram
where
( ) = p12
Q x
With the same argument, we can show that Psj1
Z
1
x
exp(
2)du:
u2 =
(1.23)
= Psj = Q( 2E =N ). Therefore, Ps = Q( 2E =N ).
q
0
q
0
0
1.4.2 General case (a geometric approach)
Now assume that we employ M -ary signaling, i.e., the transmitter sends a signal out from the set
fsm(t)gMm
1
=0
. Using the vector representation in Section 1.1, we know that the ML receiver decides that
(
)
the m-th signal is sent when the Euclidean distance d rN ; smN is the smallest among all the M 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 signal smN closest to the receiver vector rN 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 vector rN falls into the corresponding decision region. A diagram showing the signal points
and their corresponding decision regions is known as the Voronoi diagram of a modulation scheme.
1.10
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
Now we have all the tools needed to calculate the symbol error probability of a general M -ary
()
01
1g, is being sent, and
modulation with the ML receiver. Suppose sm t , for some m 2 f ; ; : : : ; M
()
let Rm denotes the decision region for sm t . We make an error if the received vector rN falls outside
Rm . Therefore, the conditional symbol error probability given that sm(t) is sent,
= Pr(rN 2 <N n Rm jsm(t) sent)
= <N nRm p(rN jsmN )drN
= 1 Rm p(rN jsmN )drN ;
Psjm
Z
Z
(
(1.24)
)
where p 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 points s0 and s1 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). Suppose s0 t is being sent, we know that
Psj0
=
Z
( j )
= 1 p21 exp (x
n
= Q(d(s N ; s N )=2n);
R1
p rN s0N drN
"
Z 0
0
where n2
(
) 2)
d s0N ; s1N =
2n
2
2
#
(1.25)
1
= N =2 is the variance of an element of the noise vector nN . We note that the second equality
0
in (1.25) above is obtained by a change of variable which corresponds to a suitable rotation and translation of the axis. Clearly, Psj1 can be calculated in the same way. Thus Ps
= Q(d(s N ; s N )=2n).
0
1
Moreover, it can be seen that the result in (1.25) reduces to (1.22) for BPSK (antipodal binary signaling).
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. Suppose s0 t is being sent, then
Psj0
= 1
Z
R0
( j )
p r2 s02 dr2
1.11
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
2) + (r
P T =2)
1 exp
= 1
2n
2n
= 1 Q ( P T =N ) = 2Q( P T =N ) Q ( P T =N ):
0
4
2
0
q
PT=
1
q
0
0 and Ps
3
2
2
2
0
= Psj = Psj = Psj
2
q
2
2
q
2
Similarly, we have Psj1
(r
2
1Z 1
Z
=1
Q2
(
3
5
dr1 dr2
q
0
q
(1.26)
)
P T =N0 .
1.4.3 Union bound
When the exact symbol error probability is too difficult to evaluate, we resort to bounds and approximations. One of such methods is the union bound.
()
Suppose s0 t is being transmitted, we know from Section 1.4.2 that
Psj0
= Pr
"
M[ 1
fd(rN ; smN ) < d(rN ; s N )g
s0N
0
m=1
M
X1
Pr [fd(rN ; smN ) < d(rN ; s N )gjs N ] :
0
m=1
#
(1.27)
0
(
) (
)g in (1.27) is exactly the same as the error event
as if there were only two signals, s (t) and sm (t) (m 1), in the signal set. The probability of this
We notice that the event fd rN ; smN < d rN ; s0N
0
event has been calculated in (1.25). Hence, we obtain the union bound of the conditional symbol error
probability as
Psj0
where n2
M
X1
((
)2 )
Q d s0N ; smN = n ;
m=1
(1.28)
= N =2. Similarly, we can find union bounds for the conditional error probabilities given
0
that other signals are sent. By averaging over all the signals, we obtain the union bound for the average
symbol error probability as
Ps
M1
M
X1 M
X1
m=0 n=0
n=m
((
)2 )
Q d smN ; snN = n :
(1.29)
6
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
((
) 2 ) + Q(d(s N ; s N )=2n) + Q(d(s N ; s N )=2n)
= Q( P T =N ) + Q( 2P T =N ) + Q( P T =N ):
Q d s0N ; s1N = n
0
q
q
0
2
0
3
q
0
1.12
0
(1.30)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
By symmetry, we have
Ps
2Q(
q
q
P T =N0
) + Q( 2P T =N );
(1.31)
0
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. Instead, we mix the baseband signal with a carrier up to a certain frequency, which matches the electromagnetic 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
()
Suppose s 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
()
( ) = 0 for j!j 2!c. We refer to this condition as the narrowband assumption.
For a bandpass signal s(t) satisfying the narrowband assumption stated above, it can be shown [1]
that s(t) can be represented by an in-phase component x(t) and a quadrature component y (t),
of s t satisfies S !
( ) = x(t) cos(!ct) y(t) sin(!ct)
e !c t + e !c t
e !c t e !c t
= x(t)
y (t)
2
2j
= x(t) +2 jy(t) e !ct + x(t) 2 jy(t) e !ct:
Now we define the complex envelope s~(t) of the signal s(t) as
s t
j
!
j
j
!
!
j
!
j
j
~( ) =4 x(t) + jy(t):
s t
(1.32)
(1.33)
Then from (1.32), we have
( ) = s~(t)e !ct =2 + s~(t)e
s t
j
1.13
j!
2 = Re[~s(t)e !ct]:
ct=
j
(1.34)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
Taking Fourier transform on both sides of (1.34), we get
( ) = S~(!
S !
~( )
) 2 + S~(
!c =
)2
!
!c = ;
~( )
(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 signal s t back from its complex envelope s t .
~( )
() ()
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 transform S (! ) of s(t) is given (or s(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 that S (! ) 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),
The remaining question is how to obtain the complex envelope s t from the signal s t . If s t is
( + !c) = S~(!)=2 + S~ (
S !
~(
We notice S !
!
2!c) is nonzero only on the interval ( 3!c;
2!c)=2:
(1.36)
)
!c . Therefore,
~( ) = 2L!c [S (! + !c)];
S !
(1.37)
[] 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 of S (! ) and shift it
down to baseband to obtain S~(! ). This is the reason why the complex envelope s~(t) is sometimes
called the lowpass equivalent signal of s(t).
where L!c
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,
~( ) = 2L!c [s(t)e !ct ]
= 2L!c [s(t) cos(!ct) js(t) sin(!ct)]
= L!c [2s(t) cos(!ct)] jL!c [2s(t) sin(!ct)]:
s t
j
1.14
(1.38)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
Lw
c
s(t)
2cos(wc t)
~s(t)
Lw
c
-j
2sin(w c t)
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 operator L!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
()
~ (t) is the
bandpass filter given that h(t) satisfies the narrowband assumption stated before. Hence, if h
complex envelope of h(t), then
~ (t)e !ct ]:
h(t) = Re[h
(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 filter so (t) also satisfies the narrowband assumption and
We can use the complex envelope in the previous section to represent the impulse response h t of a
j
( ) = h(t) si(t);
so t
(1.40)
where denotes the convolution operator. In the frequency domain,
( ) = H (!)Si(!);
So !
1.15
(1.41)
Tan F. Wong: Spread Spectrum & CDMA
( ) ( )
1. Digital Comm. Theory
( )
() ()
()
(1.37), the Fourier transform of the complex envelope, s~o (t), of so (t) is given by
where So ! , H ! , and Si ! are the Fourier transforms of so t , h t , and si t , respectively. From
~( ) =
=
=
=
So !
2L!c [So (! + !c)]
2L!c [H (! + !c)Si(! + !c)]
2L!c [H (! + !c)]L!c [Si(! + !c)]
1 H~ (!)S~ (!);
i
2
(1.42)
~( ) ~ ( ) ( )
~( ) ~ ( )
and si (t), respectively. The third equality in (1.42) is due to the fact that both h(t) and si (t) share the
where H ! and Si ! are the Fourier transforms of the complex envelopes, h t and si t , of h t
same passband. By taking inverse Fourier transform on both sides of (1.42), we obtain
~ ( ) = 12 h~ (t) s~i(t):
so t
()
(1.43)
()
Hence, we can convolute the complex envelopes of h t and so t and then convert the result back to
obtain the output bandpass signal.
1.5.3 Narrowband process
()
n(!). If n (!) satisfies the narrowband assumption, then n(t) is called a narrowband process.
turns out [2] that n(t) can also be written as
Suppose n t is a wide-sense stationary (WSS) process with zero mean and power spectral density
( ) = nx(t) cos(!ct)
n t
()
( ) sin(!ct);
ny t
()
It
(1.44)
()
()
where nx t and ny t are zero-mean jointly WSS processes. Moreover, if n 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 ny ( ) =
Rn ( ) =
Rnx ()
Rny ;
(1.45)
()
Rnx ( ) cos(!c )
Rny nx ;
1.16
(1.46)
( ) sin(!c );
Rnx ny (1.47)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
( ) =4 E [n(t)n(t + )] is the autocorrelation function of the random process n(t), Rnx ( ) =4
4
E [nx (t)nx (t + )] and Rny ( ) = E [ny (t)ny (t + )] are, respectively, the autocorrelation functions of
4
4
the processes nx (t) and ny (t), and Rnx ny ( ) = E [nx (t)ny (t + )] and Rny nx ( ) = 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
4 1 E [~n (t)~n(t + )]
Rn ( ) =
2
= 12 E [(nx(t) jny (t))(nx (t + ) + jny (t + ))]
= 12 Rnx ( ) + 12 Rny ( ) + 2j Rnxny ( ) 2j Rny nx ( )
= Rnx ( ) + jRnxny ( ):
(1.49)
Now compare (1.47) with (1.32) and (1.49) with (1.33). If we treat the autocorrelation function Rn ( )
as a bandpass signal (by definition, it satisfies the narrowband assumption since n(t) is a narrowband
process), then Rn ( ) is its complex envelope. Hence, we can use the results in Section 1.5.1 to convert
between Rn ( ) and Rn ( ).
A common example of narrowband process is the bandpass additive Gaussian noise n(t) with zero
mean and power spectral density n (! ) = N =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(!c t) ny (t) sin(!c t);
(1.50)
where nx (t) and ny (t) are zero-mean jointly WSS Gaussian processes. The complex envelope of n(t)
where Rn ~
~
~
0
is given by
~( ) = 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
n t
~
given by
n (!) = 2L!c [n (! + !c)]
= N if j!j < !c;
0 otherwise:
~
8
>
<
0
>
:
1.17
(1.52)
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
Taking inverse Fourier transform, we get
( ) = N 2fc sin(!!c ) ;
Rn~ (1.53)
c
( ) = 0 and hence the processes nx(t) and ny (t)
are uncorrelated. Moreover, we have Rnx ( ) = Rny ( ) = Rn ( ). For the case where bandpass
transmitted signals are sent through a channel corrupted by n(t) and the bandwidths of the transmitted
signals are much smaller than the carrier frequency !c , we approximate Rn ( ) in (1.53) by N Æ ( ).
where !c
= 2fc.
0
()
Since Rn~ is real, Rnx ny ~
~
0
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 a non-dispersive
channel, for which we can model the received signal as
( ) = Ae sm(t) + n(t); 4
r t
j
0
(1.54)
where A > represents the channel gain (attenuation), represents the carrier phase shift due to prop-
()
agation delay, local oscillator mismatch, and etc., and n t is the complex AWGN with autocorrelation
( ) = N Æ( ). Suppose the receiver knows the value of 5, the problem reduces to the
function Rn 0
one in Section 1.1. Hence we can use the correlation receiver in Figure 1.2 (or its complex equivalent)
4
5
From now on, we drop the ˜ symbol for complex envelopes.
The value of A is not needed when the signals have equal energies
1.18
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
T
( ) dt
| | 2 or | |
s*0 (t)
T
( ) dt
| | 2 or | |
0
r(t)
s*1 (t)
Select Maximum
0
decision
T
( ) dt
| | 2 or | |
0
s*M-1 (t)
Figure 1.8: Envelope / square-law receiver for M -ary orthogonal signals
()
to detect the received signal r t . Generally, receivers that make use of the phase information are referred to as coherent receivers. 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 performance 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 the noncoherent ML receiver. For the case where the trans-
( )gMm
mitted signals fsm t
1
=0
have equal energies, the ML receiver assumes the simple form [1] shown
in Figure 1.8. This receiver is usually referred to as the envelope receiver or the square-law receiver
1.19
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
depending on whether the envelope or the square-law detecting device is employed. It is difficult to
evaluate the symbol error probability for a general M -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 equal a priori probabilities) is given by [1]
Ps
where E is the signal energy.
= 21 e
E =2N0 ;
(1.55)
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
( ) = x(t) + jy(t);
1
x(t) =
ak x (t
s t
X
k=
() =
y t
1
1
X
k=
1
bk
y
(t
(1.56)
kTs
);
(1.57)
kTs
);
(1.58)
where ak ’s are independent identically distributed (iid) random variables with mean zero and variance
A2 , and bk ’s are also iid random variables with mean zero and variance B 2 . Moreover, we assume
1
that the two data streams fak g1
k = 1 and fbk gk = 1 are independent. In above,
as the propagation delay, and
x
()
can be interpreted
(t) and y (t) are the pulses for the in-phase and quadrature channels,
respectively. We notice that s t is a zero-mean random process. This model almost covers all practical
quadrature modulation schemes.
()
as a random
1
variable which is uniformly distributed on [0; Ts ), and is independent to both fak g1
k
1 and fbk gk 1.
Then the autocorrelation function of s(t) is given by
4 1 Rs (t; t + ) =
2 E [s (t)s(t + )]
= 12 fE [x(t)x(t + )] jE [x(t)y(t + )] jE [y(t)x(t + )] + E [y(t)y(t + )]g
= 12 fE [x(t)x(t + )] + E [y(t)y(t + )]g :
(1.59)
Our objective is to evaluate the autocorrelation function of s t . First, let us model
=
1.20
=
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
The last equality in (1.59) follows from the fact that the two data streams consist of zero-mean inde-
[ ( ) ( + )],
E [ak al ]E [ x (t kTs ) x (t + pendent random variables. Now, it suffices to evaluate E x t x t
[ ( ) ( + )] =
Ex t x t
=
1
k=
1
k=
A2
=
A2
=
X
1 l= 1
X
=
=
1
X
Ts
Ts
A2
Ts
A2
Ts
1
A
1
2
1
Ts
X
1
1
1
1
1
Z
x
s
x
0
(t
s
x
kTs
) x(t + kTs
Z (k +1)T
k=
Z
Z T
kTs
lTs
)]
)d
(t ) x (t + )d
x
(t ) x(t + )d
x
( ) x ( )d
( ) x( ):
(1.60)
Similarly, we have
2
[ ( ) ( + )] = BT y ( ) y ( ):
Ey t y t
()
1
Rs ( ) = Rs (t; t + ) =
2T
(1.61)
s
Therefore, the process s t is WSS and
(1.62)
x ( ) x ( ) + B
y ( ) y ( ) :
s
The power spectral density (power spectrum) of s(t) is given by
s(!) = 21T A jx(!)j + B jy (!)j ;
(1.63)
s
where x (! ) and y (! ) 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 and Ts = T =10. In both cases, we let A = 2. For the first case, the power spectrum
h
A2
h
i
2
2
2
2
2
i
2
is
=2)
s (!) = T sin(!T(!T
:
=2)
2
2
(1.64)
For the second case, the power spectrum is
T sin (!T =20)
s(!) = 10
(!T =20) :
2
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
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
0
10
T =T/10
s
Ts=T
−1
power spectral density
10
−2
10
−3
10
−4
10
−5
10
−40
−30
−20
−10
0
ω (π/T Hz)
10
Figure 1.9: Power spectra of BPSK schemes: Ts
1.22
20
= T and Ts = T =10
30
40
Tan F. Wong: Spread Spectrum & CDMA
1. Digital Comm. Theory
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