Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
Chapter 2
Introduction to Spread Spectrum
Communications
As discussed in Chapter 0, a spread spectrum modulation produces a transmitted spectrum much wider
than the minimum bandwidth required. There are many ways to generate spread spectrum signals. We
are going to introduce some of the most common spread spectrum techniques such as direct sequence
(DS), frequency hop (FH), time hop (TH), and multicarrier (MC). Of course, one can also mix these
spread spectrum techniques to form hybrids which have the advantages of different techniques.
Spread spectrum originates from military needs and finds most applications in hostile communication environments. We will start by briefly looking at the advantage of spreading the spectrum in
the presence of a Gaussian jammer as our motivation to study spread spectrum communications. Toward the end of the chapter, we will also state some common spread spectrum applications. Detailed
treatments of some of these applications are left for the coming chapters.
2.1 Motivation—a jamming analysis
Consider the transmission of a bit stream fbk g1
k= 1 (bk
= 1) through an AWGN channel. We employ
BSPK modulation at the carrier frequency !c . The channel is also corrupted by an intentional jammer.
()
The received signal r t , in complex envelope representation, is given by
( ) = s(t) + j (t) + n(t);
r t
2.1
(2.1)
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
@ t=(k+1)T+ ∆
r(t)=s(t)+j(t)+n(t)
p Tc (T-t)
r k = sk + jk + nk
^
bk
sgn(Re[ ])
Figure 2.1: Matched filter receiver for BPSK data with jammer
s
where
1
X
( ) = 2PT T
s t
k=
c
()
1
(
bk pTc t
kT
)
(2.2)
( ) = N0 , and j (t) is the jamming
is the transmitted signal, n t is the AWGN with power spectrum n !
signal. In (2.2), T is the symbol duration, Tc is the symbol pulse width, and P is the average transmitted
()
power. From Section 1.7, we know that the spectrum of the transmitted signal s t is given by
s(!) = P Tc sin(!T(!T=2)c=22)
2
(2.3)
c
if we model the bits as iid random variables and
as a uniform random variable on [0; T ). By exam-
ining the spectrum of the transmitted signal, a reasonable jamming strategy is to put all the jamming
power PJ into the band coincides with the main lobe of the signal spectrum, i.e., from
2=Tc to
2=Tc rad/sec. For simplicity, we assume that j (t) is a zero-mean WSS Gaussian random process with
power spectrum j (! ) = PJ Tc for j! j < 2=Tc , and j (! ) = 0 otherwise. Moreover, n(t) and j (t)
are independent.
Neglecting the jamming signal, the ML receiver is the matched filter receiver developed in Section 1.2. We redraw the matched filter receiver in Figure 2.1 here for convenience. Let us consider
the performance of the matched filter when the jamming signal is present. Conditioning on
1, the
sampled output of the matched filter corresponding to the k th symbol is
rk
= sk + jk + nk ;
(2.4)
where
sk
1
=
Z1
1
() (
s pTc kT
We will see that the conditional error probability does not depend on
(averaged over
) is the same as the conditional error probability.
2.2
)d
.
Hence the unconditional error probability
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
Z kT ++Tc
= kT + s( )d
q
= 2P T Tcbk ;
jk
=
and
nk
=
Z1
1
Z1
1
(2.5)
j pTc () (
kT
)d;
(2.6)
() (
kT
)d:
(2.7)
n pTc From the assumptions above, we know that jk and nk are independent zero-mean Gaussian random
variables. It remains to determine their variances. The variance of nk is
n2 k
= 21 E [nk nk ] = N0Tc:
(2.8)
For jk , we note that its variance is equal to the value of the autocorrelation function of the matched filter
() 0
output component due to j t at . Using the Fourier relationship between autocorrelation function
and power spectral density, we have
j2k
= 21 E [jk jk ]
Z1
2
= 21 1 Tc2 sin(!T(!T=2)c=22) j (!)d!
c
Z
2
2
= PJTc sin!2 ! d!
= 0:9028PJ Tc2:
(2.9)
Now, we can calculate the symbol (bit) error probability of the communication system described
above. By symmetry, we know that the average symbol error probability is equal to the conditional
= 1. Under the condition that bk = 1, the decision statistic
p
Re[rk ] is a Gaussian random variable with mean 2P T Tc and variance j2k + n2 k . Therefore, the
symbol error probability given that, say, bk
symbol error probability is
Ps
=
=
0v
1
u
u
2
P T Tc
A
Q @t 2
jk + n2 k
0v
u
u
2E =N0
@
Q t
1 + 0:9028PJ Tc=N0
1
A;
(2.10)
= P T is the symbol energy. Comparing (2.10) to (1.22), we suffer a loss in SNR by a factor
of 1 + 0:9028PJ Tc =N0 with respect to the case where the jammer is not present.
where E
2.3
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
There are two ways to reduce the loss in SNR. For a bandwidth limited channel, we can increase
the transmitted power P of the signal. If power is the main constraint, we can reduce the pulse width
Tc . This corresponds to spreading the spectrum of the transmitted signal. In military applications, one
consideration is that we do not want our enemies to intercept or detect our transmission. The higher the
transmission power the more susceptible is the transmission being intercepted. Therefore, we usually
resort to spreading the spectrum of the transmitted signal instead of raising the transmission power.
This is the reason why spread spectrum is originally considered for military communications.
In terms of jamming immunity, the spreading method described above is far from desirable. Simply
reducing Tc is effective only for the continuous Gaussian jammer assumed above. Since the continuous
Gaussian jammer spreads its power across the whole symbol period, for a small Tc , we only need to
integrate a small fraction of the symbol duration and, hence, pick up a small jamming energy. However,
because of the regularity of the transmitted signal, it is easy for the enemies to determine when the
pulses are sent. Hence, they can switch to a pulse jammer which outputs high power jamming pulses
coincide with our transmission pulses. By doing so, the jammer can cause maximum degradation to
our transmission without increasing the average jamming power. In this case, reducing Tc will not
help to combat the pulse jammer. To tackle the pulse jammer, we can randomize the transmission
time of the pulse within the symbol duration to make the detection of the transmission times of the
pulses difficult. Without the knowledge of the transmission times of the pulses, the pulse jammer
becomes ineffective. We once again force our enemies to spread the jamming power both in time and
in frequency. As a result, we arrive back at the case of continuous Gaussian jammer, and we can spread
the spectrum to combat the jammer. The spreading technique just described is known as time hopping.
The discussion above brings out an important characteristic of spread spectrum communications.
In order for the receiver to perform properly, it has to know the transmission times of the pulses. This
means that the transmission times cannot be truely random. Instead, a sequence of pseudo-random
transmission times is pre-assigned to both the transmitter and the receiver. This sequence is generally
referred to as a code. We will see that all spread spectrum techniques contain some forms of pseudorandom codes. In fact, we usually do not classify a spectral spreading technique, which does not
employ any form of codes (like the one in (2.2)), as a spread spectrum modulation.
2.4
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
2.2 Direct sequence spread spectrum
One non-trivial way of spreading the spectrum of the transmitted signal is to modulate the data signal
by a high rate pseudo-random sequence of phase-modulated pulses before mixing the signal up to the
carrier frequency for transmission. This spreading method is called direct sequence spread spectrum
(DS-SS). More precisely, suppose the data signal is
1
p X
( ) = 2P
b t
k=
1
(
bk pT t
)
kT ;
(2.11)
where fbk g1
k= 1 is the symbol sequence and T is the symbol duration. Note that all the signals here
()
are complex envelopes unless otherwise indicated. We modulate the data signal b t by a spreading
()
signal a t which is given by
()=
a t
1
X
l=
1
al
(t
(2.12)
(t) is called the chip waveform, which is time-
where fal g1
l= 1 is called the signature sequence and
limited to
)
lTc ;
[0; Tc). We impose that condition that T = N Tc , where N , which is usually referred to as
the processing gain or the spreading gain, is the number of chips in a symbol and Tc is the separation
between consecutive chips. For convenience, we normalize the energy of the chip waveforms to Tc .
The spread spectrum signal is given by
1
p X
( ) = a(t )b(t ) = 2P
s t
l=
1
bbl=N c al
(t
lTc
);
(2.13)
where bxc is the largest integer which is smaller than or equal to x. This general model for DS-SS
contains many different modulation and spreading schemes. Some of the common examples are listed
in Table 2.1. For example, a pictorial description of the BPSK modulation with BPSK spreading
scheme is given in Figure 2.2.
To obtain the power spectrum of the spread spectrum signal, we model the spreading elements al
[
as iid zero-mean random variables with E jal j2
] = 1, and the propagation delay as a uniform random
variable as described in Section 1.7. Moreover, we also normalize the average symbol energy to P T ,
[
i.e., E jbk j2
] = 1. Then the power spectrum of the spread spectrum signal s(t) is
s(!) = TP j(!)j2;
c
2.5
(2.14)
Tan F. Wong: Spread Spectrum & CDMA
BPSK
spreading
QPSK
spreading
2. Intro. Spread Spectrum
BPSK modulation
QPSK modulation
(t) = pTc (t)
bk 2 f1g
al 2 f1g
(t) = pTc (t)
bk 2 f1g
al 2 f p12 j p12 g
(t) = pTc (t)
bk 2 f p12 j p12 g
al 2 f1g
(t) = pTc (t)
bk 2 f p12 j p12 g
al 2 f1; jg
Table 2.1: Common spreading schemes
b(t)
a(t)
s(t)
Figure 2.2: BPSK modulation and BPSK spreading scheme
2.6
Tan F. Wong: Spread Spectrum & CDMA
r(t)
2. Intro. Spread Spectrum
@ t=(k+1)T+∆
zk
h k(t)
decision
device
^b
k
Figure 2.3: Matched filter receiver for the k th symbol of the DS-SS signal
where
(!) is the Fourier transform of the chip waveform (t).
The power spectra of the spread
signals for the four schemes shown in Table 2.1 are all given by
s(!) = P Tc sin(!T(!T=2)c=22) :
2
(2.15)
c
Comparing this to the power spectrum of the original data signal
=2)
b (!) = P T sin(!T(!T
;
=2)2
2
(2.16)
we see that the spectrum is spread N times wider by the direct sequence technique in (2.13). In
practice, the spreading sequences are pseudo-random. We will discuss, in Chapter 3, different ways to
generate sequences which have properties close to those of random sequences.
In an AWGN channel, the ML receiver for the spread spectrum signal is the matched filter receiver
shown in Figure 2.3. We note that the matched filter is time-varying unless the spreading sequence
fal g1l=
()
1 is periodic with period N . For the kth symbol, the impulse response of the matched filter
hk t is given by
()=
hk t
X1 al+kN (T
N
l=0
lTc
)
t :
(2.17)
^ = sgn(Re[zk ]). For QPSK modulation,
the decision device gives decision ^bk = p12 sgn(Re[zk ]) + j p12 sgn(Im[zk ]). The general case is left as
For BPSK modulation, the decision device gives decision bk
an exercise. Alternatively, we can implement (see Homework 2) the matched filter receiver as shown
in Figure 2.4.
The spreading method described in (2.13) is by no mean the only possible DS-SS technique. For
example, one can spread the in-phase and quadrature channels independently instead of spreading the
complex channel as in (2.13). Suppose the complex data signal is
( ) = bx (t) + jby (t);
b t
2.7
(2.18)
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
@ t=(l+1)Tc +∆
Σ
(k+1)N-1
ψ (Tc -t)
*
r(t)
zk
decision
device
l=kN
^
bk
a*l
Figure 2.4: Equivalent implementation of the matched filter receiver in Figure 2.3
where
() =
bx t
() =
by t
1
p X
P
k=
1
k=
1
1
p X
P
(
kT ;
(
kT
bxk pT t
byk pT t
)
(2.19)
)
(2.20)
are the (real-valued) in-phase and quadrature data signals. The in-phase and quadrature data signals
()
are spread respectively by the (real-valued) in-phase spreading signal ax t and quadrature spreading
()
signature ay t , where
1
X
() =
ax t
l=
1
1
X
() =
ay t
()
l=
1
(
lTc ;
(
lTc ;
axl x t
ayl y t
)
(2.21)
)
(2.22)
()
[0 )
and x t and y t are the in-phase and quadrature chip waveforms (both are time-limited to ; Tc ),
respectively. The resulting spread spectrum signal is given by
( ) = ax(t )bx (t ) + jay (t
s t
T0
)by (t
T0
):
(2.23)
With the same model for the data symbols and spreading sequences we have before2 , the power spectral
density of the spread spectrum signal in (2.23) is given by
n
s (!) = 2PT jx(!)j2 + jy (!)j2
c
where
o
;
x(!) and y (!) are the Fourier transforms of the chip waveforms x(t) and y (t), respec-
tively.
2
(2.24)
In this case,
is still modeled as uniformly distributed on [0; Tc) and T0 is assumed to be a constant.
2.8
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
@ t=(l+1)Tc + ∆
rx (t)
Σ
decision
device
^b
xk
Σ
decision
device
^b
yk
(k+1)N-1
ψx(Tc -t)
l=kN
a xl
@ t=(l+1)Tc +T0 + ∆
ry (t)
ψy(Tc -t)
r(t) = rx (t) + j ry (t)
(k+1)N-1
l=kN
a yl
Figure 2.5: Matched filter receiver for independent-channel-spreading DS-SS
This DS-SS model includes offset QPSK spreading and serial-MSK spreading, which we are not
going to discuss in detail. The matched filter receiver for this form of DS-SS is shown in Figure 2.5.
We note that the two different forms of DS-SS described in (2.13) and (2.23) overlap, but they are not
subsets of each other.
In order to implement the matched filter receivers shown above, we need to achieve timing and
phase synchronization which will be discussed in Chapter 5.
2.3 Frequency hop spread spectrum
Another common method to spread the transmission spectrum of a data signal is to (pseudo) randomly
hop the data signal over different carrier frequencies. This spreading method is called frequency hop
spread spectrum (FH-SS). Usually, the available band is divided into non-overlapping frequency bins.
The data signal occupies one and only one bin for a duration Tc and hops to another bin afterward.
When the hopping rate is faster than the symbol rate (i.e., T > Tc ), the FH scheme is referred to
as fast hopping. Otherwise, it is referred to as slow hopping. A typical FH-SS transmitter and the
corresponding receiver are shown in Figures 2.6 and 2.7, respectively.
Because it is practically difficult to build coherent frequency synthesizers, modulation schemes,
2.9
Tan F. Wong: Spread Spectrum & CDMA
data
2. Intro. Spread Spectrum
b(t)
Data
Modulator
Highpass
Filter
s(t)
a(t)
Frequency
Synthesizer
FH code
clock
Code
Generator
Figure 2.6: Transmitter for FH-SS
Image
Rejection
Filter
Bandpass
Filter
a(t)
Frequency
Synthesizer
FH code
clock
Code
Generator
Figure 2.7: Receiver for FH-SS
2.10
Data
Demodulator
estimated
data
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
such as M -ary FSK, which allow noncoherent detection are usually employed for the data signal. For
M -ary FSK, the data signal3 can be expressed as
1
p X
( ) = 2P
b t
where !k
k=
(
pT t
kT
) cos(!k t + k );
(2.25)
2 f!s0; !s1; : : : ; !sM 1g. The frequency synthesizer outputs a hopping signal
()=2
a t
where !k0
1
1
X
l=
1
(
pTc t
lTc
) cos(!l0t + 0l);
(2.26)
2 f!c0; !c1; : : : ; !cL 1g. This means that there are L frequency bins in the FH-SS system.
We impose the constraint that T
= N Tc for fast hopping, or Tc = N T for slow hopping. The FH-SS
signal is given by
p
( ) = 2P
s t
1
X
l=
1
(
pTc t
lTc
h
) cos (!bl=N c + !l0)(t ) + bl=N c + 0l
i
(2.27)
for the case of fast hopping (T > Tc ), or
1
p X
( ) = 2P
s t
k=
1
(
pT t
kT
h
) cos (!k + !b0 k=N c )(t ) + k + 0bk=N c
i
(2.28)
for the case of slow hopping (T < Tc ). The orthogonality requirement for the FSK signals forces the
2
2
separation between adjacent FSK symbol frequencies be at least =Tc for fast hopping, or =T for
2
slow hopping. Hence, the minimum separation between adjacent hopping frequencies is M =Tc for
2
fast hopping, or M =T for slow hopping. For example, Figure 2.8 depicts the operation of a fast
2
FH-SS system with -FSK modulation (M
(T
= 2Tc, N = 2).
= 2), 8 hopping bins (L = 8), and 2 hops per symbol
To obtain the power spectrum of the M -ary FH-SS signal, we model the phases k and 0l as iid
[0; 2). The hopping frequencies !l0 are modelled as iid
random variables taking values from the set f!c0 ; !c1 ; : : : ; !cL 1 g with equal probabilities. The FSK
symbol frequencies !k are iid random variables taking values from the set f!s0 ; !s1 ; : : : ; !sM 1 g with
equal probabilities. The delay is assumed to be uniformly distributed on [0; Tc ) for fast hopping, or
[0; T ) for slow hopping. We also assume that all the random variables mentioned above are indepenrandom variables uniformly distributed on
dent. With these assumptions, we can show (see Homework 2) that the power spectrum of the FH-SS
3
All the signals in this section are real bandpass signals
2.11
Tan F. Wong: Spread Spectrum & CDMA
data sequence
code sequence
hop bin
2. Intro. Spread Spectrum
0
1
1
0
+1+1-1 -1+1-1 +1-1-1 -1-1-1 +1+1+1 -1-1+1 -1+1+1 +1-1+1
1
5
3
7
0
6
4
2
7
Frequency bin
6
5
4
3
2
1
0
Transmitted frequency pattern
T
Tc
1/Tc
Dehopped frequency pattern
Figure 2.8: Fast FH-SS system with 2-FSK modulation, 8 hopping bins, and 2 hops per symbol
2.12
Tan F. Wong: Spread Spectrum & CDMA
()
signal s t is given by
8
X1 <"
P Tc LX1 M
sin(!
s(!) = 2M L
:
(!
l=0 m=0
for fast hopping, or
2. Intro. Spread Spectrum
!cl
!cl
8
X1 <"
P T LX1 M
9
) 2 #2 + " sin(! + !cl + !sm)Tc=2 #2=
) 2
(! + !cl + !sm)Tc=2 ;
!sm Tc =
!sm Tc =
(2.29)
9
) 2 #2 + " sin(! + !cl + !sm)T =2 #2= (2.30)
) 2
(! + !cl + !sm)T =2 ;
for slow hopping. Therefore, the spectrum of the original data signal b(t) is approximately spread by
sin(!
s(!) = 2M L
: (!
l=0 m=0
!cl !sm T =
!cl !sm T =
a factor of LN for fast hopping, or by a factor of L for slow hopping.
2.4 Time hop spread spectrum
In time hop spread spectrum (TH-SS), we spread the spectrum by modulating the data signal by a
pseudo-random pulse-position-modulated spreading signal. We have described the general idea in
()
Section 2.1. Here, we give the precise definition. Suppose the data signal b t is
1
p X
( ) = 2P
b t
k=
1
(
bk pT t
)
kT :
(2.31)
We modulate the data signal by the spreading signal
()=
a t
where ak
s
T
1
X
Tc k=
1
(
pTc t
kT
)
ak Tc ;
(2.32)
2 f0; 1; : : : ; N 1g and T = N Tc. The resulting spread spectrum signal s(t) is given by
s
1
X
( ) = a(t )b(t ) = 2PT T
s t
c k=
1
(
bk pTc t
kT
ak Tc
):
(2.33)
()
To obtain the power spectrum of the TH-SS signal s t , we model the data symbols bk as iid zero-
[ ] = 1. The pulse location indices ak are assumed to be iid random
variables taking values from f0; 1; : : : ; N 1g with equal probabilities. The propagation delay is
modelled as a uniform random variable on [0; T ) as usual. It can be shown (see Homework 2) that the
power spectral density of the spread spectrum signal s(t) is the same as the one obtained in Section 2.1,
mean random variables with E jbk j2
i.e.,
s(!) = P Tc sin(!T(!T=2)c=22) :
2
c
2.13
(2.34)
Tan F. Wong: Spread Spectrum & CDMA
r(t)
2. Intro. Spread Spectrum
decision
device
pTc(T-t)
^b
k
@ t=(k+1)T+akTc + ∆
Timing
Circuit
TH code
clock
Code
Generator
Figure 2.9: Matched filter receiver for TH-SS
The matched filter receiver for this spreading method is shown in Figure 2.9. It is the same as the
receiver in Figure 2.1 except that the sampler is controlled by a timing circuit which is in turn driven
by the pseudo-random pulse-location code.
We note that there are other types of TH-SS techniques. For example, one can use pulse-position
modulation for the data signal. As a result, the spread spectrum signal will be purely pulse-position
modulated. The hopping scheme is similar to the M -ary FSK FH-SS system described in Figure 2.8
with frequency bins replaced by time bins.
2.5 Multicarrier spread spectrum
In FH-SS, only one of many possible frequencies is transmitted at a time. The other extreme is that
we transmit all the possible frequencies simultaneously. The resulting spreading method is called
()
multicarrier spread spectrum (MC-SS). More precisely, suppose the data signal b t is given by
1
p X
( ) = 2P
b t
k=
1
2.14
(
bk pT t
)
kT :
(2.35)
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
We modulate the data signal by the spreading signal
1 NX1
X
( ) = p1N
an;k ej!n t pT (t
a t
k=
The resulting spread spectrum signal is
)
kT :
1 n=0
s
1 NX1
X
( ) = a(t )b(t ) = 2NP
bk an;k ej!n (t
s t
) pT (t
1 n=0
k=
(2.36)
kT
):
(2.37)
The data and spreading sequences are phase-modulated as in the case of DS-SS. The carrier frequencies
!n should be chosen so that signals at different frequencies do not interfere each other. The minimum
2
frequency separation is =T . To obtain the power spectrum of the spread spectrum signal, we model
[
the spreading elements an;k as iid zero-mean random variables with E jan;k j2
[0 )
] = 1, and the propagation
delay as a uniform random variable on ; T . Moreover, we also normalize the average symbol energy
[
to P T by setting E jbk j2
] = 1.
()
Then the power spectrum of the spread spectrum signal s t is (see
"
P T NX1
Homework 2)
s (!) =
N n=0
sin(!
(!
) 2 #2 :
!n )T =2
!n T =
The matched filter receiver for MC-SS is shown in Figure 2.10. When !n
(2.38)
= 2n=T , the operation
of the correlator branches in Figure 2.10 can be approximately performed by a single FFT. Hence the
matched filter receiver can be implemented very efficiently. To see this, consider the output of the nth
correlator for the k th symbol in Figure 2.10 and denote it by zn;k . Then,
zn;k
=
=
Z (k+1)T +
()
r t e
j 2nt
T + dt
kT +
N 1 kT ++(l+1)Tc
XZ
kT ++lTc
l=0
()
r t e
j 2nt
T + dt:
(2.39)
[ + ; (k + 1)T + ) into N sub-intervals of length Tc. In the lth
sub-interval, for l = 0; 1; : : : ; N 1, we approximate4 the integral
In above, we divide the interval kT
Z kT ++(l+1)Tc
kT ++lTc
()
j 2nt
T + dt Tc r (kT
Tc
N
r t e
Hence,
zn;k
X1
+ + lTc)e
( + + lTc)e
r kT
j 2nl
N :
c
j 2nlT
T :
(2.40)
(2.41)
l=0
4
Better approximations can be made, say, by using the trapezoidal and Simpson’s methods instead. We still get similar
FFT implementation of the correlator branches by using these approximations.
2.15
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
(k+1)T+ ∆
( ) dt
kT+ ∆
e-jw0 t+∆
a*0,k
(k+1)T+ ∆
( ) dt
r(t)
kT+ ∆
e-jw 1t+ ∆
a*1,k
Σ
(k+1)T+ ∆
( ) dt
kT+ ∆
e-jwN-1 t+
∆
a*N-1,k
Figure 2.10: Matched filter receiver for MC-SS
2.16
decision
device
^b
k
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
1
We note that (2.41) says that for each k , the sequence fzn;k gN
n=0 is approximately the DFT (scaled)
( + + lTc)gNl=0 1. Therefore, we can implement the correlator branches approximately by sampling r (t) at the chip rate and passing the samples through an FFT circuit. Similar
of the sequence fr kT
approximations can also be applied on the transmitter side.
2.6 Applications
As discussed in Section 2.1, anti-jamming is an important application for spread spectrum modulations. In addition to anti-jamming, we will briefly introduce several other spread spectrum applications in this section. In describing these applications, we focus on DS-SS systems. One should note
that other spread spectrum techniques also have similar applications since the main idea behind these
applications is the spreading of the spectrum.
2.6.1 Anti-jamming
We know that we can combat a wide-band Gaussian jammer by spreading the spectrum of the data
signal. Here we consider another kind of jammers—the continuous wave (CW) jammers. Suppose
the spread spectrum signal is given by (2.13) and it is jammed by a sinusoidal signal with frequency
+ !0 and power PJ . The received signal is given by
q
r (t) = s(t) + 2PJ e ! t + n(t);
(2.42)
where n(t) represents the AWGN. When one of the four DS-SS schemes in Table 2.1 is used, we can
easily see that the power spectrum of the received signal r (t) is given by
2
r (!) = P Tc sin(!T(!T=2)c=22) + PJ Æ(! !0 ) + N0:
(2.43)
!c
j 0
c
We consider the matched filter receiver in Figure 2.3. For convenience, we redraw the receiver in
()
the equivalent correlator form in Figure 2.11. At the output of the despreader, the signal z t can be
expressed as
( ) = r(t)a(t )
q
= s(t)a (t ) + 2PJ e ! t a(t ) + n(t)a (t )
q
= b(t ) + 2PJ e ! t a(t ) + n(t)a (t ):
z t
j 0
j 0
2.17
(2.44)
Tan F. Wong: Spread Spectrum & CDMA
r(t)
z(t)
2. Intro. Spread Spectrum
(k+1)T+ ∆
( ) dt
kT+ ∆
decision
device
^b
k
a* (t- ∆)
Figure 2.11: Matched filter receiver (correlator form) for DS-SS
()
It can be shown (see Homework 2) that the power spectrum of the despread signal z t is
"
2 (!T =2)
sin
!
z (!) = P T (!T =2)2 + PJ Tc sin(
(!
) 2 #2 + N :
0
) 2
!0 Tc =
!0 Tc =
(2.45)
Now the anti-jamming property of the spread spectrum modulation can be explained by comparing
the spectra of the signals before and after despreading in Figure 2.12. Before despreading, the jammer power is concentrated at frequency !0 and the signal power is spread across a wide frequency
[ 2=Tc; 2=Tc]). The despreader spreads the jammer power into a wide frequency band
([ 2=Tc ; 2=Tc ]) while concentrates the signal power into a much narrower band ([ 2=T; 2=T ]).
band (
The integrator acts like a low-pass filter to collect power of the despread signal over the frequency
band
[ 2=T; 2=T ]. As a result, almost all of the signal power is collected, but only 1=N th of the
jammer power is collected. The effective power of the jammer is reduced by a factor of N . This is the
reason why N is called the spreading gain.
2.6.2 Low probability of detection
Another military-oriented application for spread spectrum is low probability of detection (LPD), which
means that it is hard for an unintentional receiver to detect the presence of the signal. The idea behind
this can be readily seen from Figure 2.12. When the processing gain is large enough, the spread
spectrum signal hides below the white noise level. Without knowledge of the signature sequences, an
unintentional receiver cannot despread the received signal. Therefore, it is hard for the unintentional
receiver to detect the presence of the spread spectrum. We are not going to treat the subject of LPD
any further than the intuition just given. A more detailed treatment can be found in [1, Ch. 10].
2.18
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
Before despreading
Φj (ω)
PJ
Φn (ω)
No
PTc
-2 π /Tc
Φs (ω)
2 π/Tc
ω0
After despreading
PT
No
P J Tc
-2 π /Tc
-2 π /T
2 π/T
Figure 2.12: Spectra of signals before and after despreading
2.19
2 π/Tc
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
2.6.3 Multipath combining
Another advantage of spreading the spectrum is frequency diversity, which is a desirable property
when the channel is fading. Fading is caused by destructive interference between time-delayed replica
of the transmitted signal arise from different transmission paths (multipaths). The wider the transmitted
spectrum, the finer are we able to resolve multipaths at the receiver. Loosely speaking, we can resolve
1
multipaths with path-delay differences larger than =W seconds when the transmission bandwidth is
W Hz. Therefore, spreading the spectrum helps to resolve multipaths and, hence, combats fading.
The best way to explain multipath fading is to go through the following simple example. Suppose
+1” in the BPSK format, i.e., the transmitted signal envelope
the transmitter sends a bit with the value “
()
is pT t , where T is the symbol duration. Assume that there are two transmission paths leading from
the transmitter to the receiver. The first path is the direct line-of-sight path which arrives at a delay
0 seconds and has a unity gain. The second path is a reflected path which arrives at a delay of 2Tc
seconds and has a gain of 0:8, where Tc = T =10 is the chip duration of the DS-SS system we are
of
going to introduce in a moment. The overall received signal can be written as
( ) = pT (t) 0:8pT (t 2Tc) + n(t);
r t
()
(2.46)
where n t is AWGN. To demodulate the received signal, we employ the matched filter receiver,
( ) = pT (T
which is matched to the direct line-of-sight signal, i.e., h t
)
t . The output of the matched
filter is plotted in Figure 2.13. We can see from the figure that the contribution from the second path
partially cancels that from the first path. We sample the matched filter output at time t
0 36
= T.
The
signal contribution in the sample is : T and the noise contribution is a zero-mean Gaussian random
variable with variance N0 T . Compared to the case where only the direct line-of-sight path is present,
the signal energy is reduced by
87%, while the noise energy is the same.
Therefore, the bit error
probability is greatly increased.
Now, let us spread the spectrum by the spreading signal
()=
a t
where fa0 ; a1 ; : : : ; a9 g
9
X
l=0
(
ak pTc t
)
lTc ;
= f+1; 1; +1; +1; +1; 1; 1; 1; +1; 1g.
(2.47)
The received signal for this
DS-SS system is
( ) = a(t) 0:8a(t 2Tc) + n(t):
r t
2.20
(2.48)
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
1
1st path
0.8
0.6
matched filter output (T)
0.4
0.2
overall
0
−0.2
−0.4
2nd path
−0.6
−0.8
−1
0
0.5
1
1.5
2
time (T sec)
Figure 2.13: Matched filter output for the two-path channel without spreading
2.21
2.5
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
1
1st path
2nd path
overall
0.8
0.6
matched filter output (T)
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.5
1
1.5
2
2.5
time (T sec)
Figure 2.14: Matched filter output for the two-path channel with spreading
()
Again, we consider using the matched filter receiver, which is matched to a t . The output of the
matched filter is shown in Figure 2.14. We can clearly see from the figure that the contributions from
the two paths are separated since the resolution of the spread system is ten times finer than that of
the unspread system. If we sample at t
= T , we get a signal contribution of T , which is the same as
what we would get if there was only a single path. Hence, unlike what we saw in the unspread system,
multipath fading does not have a detrimental effect on the error probability. In fact, we will show in
Chapter 4 that we can do better by taking one more sample at t
= T + 2Tc to collect the energy of the
second path. If we know the channel gain of the second path, we can combine the paths coherently.
Otherwise we can perform equal-gain noncoherent combining. This ability of the spread spectrum
2.22
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
modulation to collect energies from different paths is called multipath combining.
2.6.4 Code division multiple access
We end our survey of spread spectrum applications by introducing the most investigated application
of spread spectrum techniques—code division multiple access (CDMA). For simplicity, let us focus on
DS-SS. We can allow different users to use the channel simultaneously by assigning different spreading code sequences to them. Thus there is no physical separation in time or in frequency between
signals from different users. The physical channel is divided into many logical channels by the spreading codes. Different from TDMA and FDMA, spread signals from different users do interfere each
other unless the transmissions from all users are perfectly synchronized and orthogonal codes are used.
The interference from other users is known as mulitple access intereference (MAI). In general, synchronization across users is hard to achieve in the uplinks of most practical wireless systems. In some
situations, we may not want to restrict ourselves to orthogonal codes. Therefore, we are interested in
investigating how MAI affect the performance of the system and how we can eliminate the effect of
MAI. Detailed discussions on these two issues will be provided in Chapters 6 and 7. Here we present
a crude analysis on how MAI affect the bit error probability performance of the system.
Suppose we employ DS-SS with BPSK modulation and BPSK spreading. There are K simultaneous users, each having a distinct pseudo-random sequence, in the system. We select one of the
users, call it the desired user, and try to determine the bit error probability for this user. We assume
that the desired user employs the correlator receiver, which is matched to the spreading code of the
desired user. To simplify the discussion, we assume that the received powers of signals from all the
users are identical. Recall from the discussion in Section 2.6.1 that the despreader removes the effect
of spreading from the desired signal. Since the despreader is not matched to the signals from other
users, it cannot remove the effect of spreading for those signals. As a result, the signals from other
users remain wideband after despreading, while the despread desired signal is the same as the original
unspread narrowband data signal. Since the integrator following the despreader acts like a lowpass
filter collecting power from the passband of the data signal, a crude but reasonable approximation is to
assume the signals from other users are independent and identical white Gaussian random processes
2.23
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
with power spectral density P Tc 5 , where P is the received power and Tc is the chip duration. Thus
the combined effect of the MAI and AWGN is the same as that of the AWGN with power spectral
(
density raised from N0 to K
1)P Tc + N0.
error probability is given by
Substituting this back into (1.22), we see that the bit
s
!
2
PT
Ps = Q
(K 1)P Tc + N0 1
0v
u
u
0
A
= Q @t 1 + (K2E =N
(2.49)
1)E =N N0 ;
where T = N Tc is the symbol duration and E = P T is the symbol energy. We suffer a loss in SNR
by a factor of 1 + (K 1)E =N N0 with respect to the single-user system. If the number of users in the
system is fixed, we can reduce the loss in SNR by increasing the processing gain, i.e., the bandwidth
of the system. Unlike the case of anti-jamming, we cannot reduce the loss by merely increasing the
signal power.
Using (2.49), we can have a preliminary estimate on the capacity of a DS-SS CDMA system. First,
we note that the signal-energy-to-white-noise ratio E =N0 is large in most practical situations. Hence,
we can further approximate the error probability by
0s
Ps = Q @
For example, a bit error probability of
10
1
2N A :
K
1
(2.50)
3 is considered sufficient in voice communications. Using
the result in (2.50), we see that the CDMA system with processing gain N can accommodate about
5
N= users without using any error-correcting code. With a powerful error-correcting code, we expect
this number to increase.
2.7 References
[1] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to Spread Spectrum Communications,
Prentice Hall, Inc., 1995.
5
The reason we set the power spectral density to P Tc is that the power spectra of the DS-SS signals are squared “sinc”
functions which peak at !
= 0 with the value P Tc.
2.24
Tan F. Wong: Spread Spectrum & CDMA
2. Intro. Spread Spectrum
[2] M. B. Pursley, “Performance evaluation for phase-coded spread-spectrum multiple-access communication — Part I: System analysis,” IEEE Trans. Commun., vol. 25, no. 8, pp. 795–799,
Aug. 1977.
[3] R. A. Scholtz, “Multiple access with time-hopping impulse modulation,” Proc. MILCOM ’93,
pp. 11-14, Boston, MA, Oct. 1993.
[4] N. Yee, J. M. G. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor wireless radio
networks,” IEICE Trans. Commun., vol. E77-B, no. 7, pp. 900–904, Jul. 1994.
[5] S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans.
Commun., vol. 44, no. 2, pp. 238–246, Feb. 1996.
[6] R. L. Pickholtz, L. B. Milstein, and D. L. Schilling, “Spread spectrum for mobile communications,” IEEE Trans. Veh. Technol., vol. 40, no. 2, pp. 313–321, May 1991.
2.25