Table Of ContentSource: SPREAD SPECTRUM COMMUNICATIONS HANDBOOK
Part 1
INTRODUCTION TO
SPREAD-SPECTRUM
COMMUNICATION
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INTRODUCTION TO SPREAD-SPECTRUM COMMUNICATION
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Source: SPREAD SPECTRUM COMMUNICATIONS HANDBOOK
Chapter 1
A SPREAD-SPECTRUM OVERVIEW
Over thirty years have passed since the terms spread-spectrum (SS) and
noise modulation and correlation (NOMAC) were first used to describe a
class of signaling techniques possessing several desirable attributes for com-
munication and navigation applications,especially in an interference envi-
ronment.What are these techniques? How are they classified? What are
those useful properties? How well do they work? Preliminary answers are
forthcoming in this introductory chapter.
We will motivate the study of spread-spectrum systems by analyzing a sim-
ple game,played on a finite-dimensional signal space by a communications
system and a jammer,in which the signal-to-interference energy ratio in the
communication receiver’s data detection circuitry serves as a payoff func-
tion.The reader is hereby forewarned that signal-to-interference ratio cal-
culations alone cannot illustrate many effects which,in subtle ways,degrade
more realistic performance ratios,e.g.,bit-error-rate in coded digital SS sys-
tems.However,the tutorial value of the following simple energy calculations
soon will be evident.
1.1 A BASIS FOR A JAMMING GAME
The following abstract scenario will be used to illustrate the need for spec-
trum spreading in a jamming environment, to determine fundamental
design characteristics, and to quantify one measure of SS system perfor-
mance.Consider a synchronous digital communication complex in which
the communicator has Ktransmitters available with which to convey infor-
mation to a cooperating communicator who possesses Kmatching receivers
(see Figure 1.1).Assume for simplicity that the communication signal space
has been “divided equally”among the Ktransmitters.Hence,with a band-
widthW available for communicating an information symbol in a T sec-
ss s
ond interval (0, T), the resultant transmitted-signal function space of
s
dimension approximately 2TW is divided so that each transmitter has a
s ss
3
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A SPREAD-SPECTRUM OVERVIEW
4 A Spread-Spectrum Overview
Figure 1.1. The scenario for a game between a jammer and a communication sys-
tem complex.
D-dimensional subspace,D (cid:1) 2TW /K,in which to synthesize its output
s ss
signal.Denote an orthonormal basis for the total signal space by c (t),k(cid:1)
k
1,2,...,2TW ,i.e.,
s ss
(cid:1)Ts 1, j(cid:1)k
cj1t2c*k1t2dt(cid:1) e0, j(cid:5)k (1.1)
0
where the basis functions may be complex valued,and ( )* denotes conju-
gation.Then the signal emitted by the k-th transmitter is of the form
mk1t2 (cid:1) a ajcj1t2, (1.2)
jHN
k
where
N (cid:1) 5j:1k(cid:3)12D 6 j(cid:4)kD6 (1.3)
k
and {a} is a data-dependent set of coefficients.We will refer to the above as
j
anorthogonal communication system complex of multiplicity K.
Of course,real systems generally radiate real signals.The reader may wish
to view m (t) as the modulation on the radiated signal Re{m (t) exp (jvt(cid:2)
k k c
u)}.Without loss of generality,we can dispense with the shift to RF during
this initial discussion.
In a simplified jamming situation, the signal z(t) observed at the i-th
i
receiver in the receiving complex might be
K
zi1t2 (cid:1) a mk1t2 (cid:2)J1t2 (cid:2)ni1t2. (1.4)
k(cid:1)1
wheren(t) represents internally generated noise in the i-th receiver,J(t) is
i
an externally generated jamming signal,and the K-term sum represents the
total output signal of the transmitter complex. One signal processing
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A SPREAD-SPECTRUM OVERVIEW
A Basis for a Jamming Game 5
strategy for the i-th receiver is to project the received signal onto the set of
basis functions for the i-th transmitter’s signal space,thereby calculating
(cid:1)Ts
zj(cid:1) zi1t2c*j 1t2dt for all jH Ni. (1.5)
0
In the absence of jamming and receiver noise,the properties of the ortho-
normal basis insure that z (cid:1)a,and thus,the i-th receiver correctly discov-
j j
ers the data dependent set of coefficients {a}, jHN, used by the i-th
j i
transmitter.
Both the jamming and receiver noise signals can be expanded in terms of
the orthonormal basis as
2TsWss
J1t2 (cid:1) a Jjcj1t2 (cid:2)J01t2, (1.6)
j(cid:1)1
2TsWss
ni1t2 (cid:1) a nijcj1t2 (cid:2)n0i1t2, (1.7)
j(cid:1)1
whereJ(t) represents that portion of the jamming signal orthogonal to all
0
of the 2TW basis functions used in producing the composite signal.The
s ss
receiver noise component n (t) likewise is orthogonal to all possible trans-
0i
mitted signals.These representations indicate that,in general,the projection
(1.5) of z(t) onto c(t) in the i-th receiver produces
i j
z (cid:1)a (cid:2)J (cid:2)n for all jHN . (1.8)
j j j ij i
The everpresent thermal noise random variable n , assumed complex
ij
Gaussian,independent,and identically distributed for different values of i
and/or j, represents the relatively benign receiver perturbations in the
absence of jamming.The jamming signal coefficients J are less easily clas-
j
sified,and from the jammer’s point of view,hopefully are unpredictable by
the receiver.
The total energy E in the jamminig signal J(t) over the time interval
J
(0,T) is given by
s
(cid:1)Ts 2TsWss (cid:1)Ts
EJ(cid:1) ƒJ1t2ƒ2dt(cid:1) a ƒJjƒ2(cid:2) ƒJ01t2ƒ2dt. (1.9)
0 j(cid:1)1 0
Obviously,the energy term involviong J (t) serves no useful jamming pur-
9
pose,and henceforth,will be assumed zero.(In keeping with this conserva-
tive aspect of communication system design,we also assume that the jammer
has full knowledge of timing and of the set {c(t)} of basis functions.) The
j
sum in (1.9) can be partitioned into K parts,the i-th part representing the
energyE used to jam the i-th receiver.Thus,
Ji
K
EJ(cid:1) aEJi, EJi(cid:1) a ƒJjƒ2. (1.10)
i(cid:1)1 jHNi
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A SPREAD-SPECTRUM OVERVIEW
6 A Spread-Spectrum Overview
A similar partition holds for the total transmitted signal energy E,namely
s
K
ES (cid:1) aESi, ESi(cid:1) a ƒajƒ2, (1.11)
i(cid:1)1 jHNi
E being the energy used by the i-th transmitter.The additive partitions
Si
(1.10),(1.11) are a direct result of the orthogonality requirement placed on
the signals produced by the transmitter complex.
The above signal representations and calculations have been made under
the assumption that the channel is ideal,causing no attenuation,delay,or
distortion in conveying the composite transmitted signal to the receiver com-
plex, and that synchronous clocks are available at the transmitter and
receiver for determining the time interval (0,T) of operation.Hence,impor-
s
tant considerations have been suppressed in this initial discussion,so that
we may focus on one major issue facing both the communication system
designer and the jammer designer,namely their allocations of transmitter
energy and jammer energy over the Korthogonal communication links.
1.2 ENERGY ALLOCATION STRATEGIES
Within the framework of an orthogonal communication system complex of
multiplicityK,let’s consider the communicator and jammer to use the fol-
lowing strategies for allocating their available energies,E and E respec-
S J
tively,to the Klinks.
Communicators’ strategy: Randomly select K links, K (cid:4) K, for equal
S S
energy allocations,each receiving E /K units.The remaining links are not
S S
utilized.
Jammer’s strategy:Randomly select K receivers for equal doses of jam-
J
ming energy,each receiving E/K units.The remaining channels are not
J J
jammed.
The quantity K is referred to as the diversity factorof the communication
S
system complex.When K exceeds unity,the receiver must employ a diver-
S
sity combining algorithm to convert the outputs of the K chosen links into
S
a single output for the system user. The performance measure to be
employed here,in determining the effectiveness of these strategies,will not
depend on specifying a particular diversity combining algorithm.
The randomness required of these strategies should be interpreted as
meaning that the corresponding adversary has no logical method for pre-
dicting the choice of strategy,and must consider all strategies equally likely.
Furthermore,random selection of communication links by the transmitter
should not affect communication quality since all available links are assumed
to have equal attributes. (Examples of link collections with non-uniform
attributes will be considered in Part 2,Chapter 2.)
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A SPREAD-SPECTRUM OVERVIEW
Energy Allocation Strategies 7
The receiving complex, having knowledge of the strategy selected for
communication, will collect all E units of transmitted energy in the K
S S
receivers remaining in operation.However,the amount of jamming energy
collected by those same KS receivers is a random variable whose value# is
K
determined by which of the a b jamming strategies is selected (a#b
K
J
denotes a binomial coefficient).Under the equally likely strategy assump-
tion,the probability that the jammer strategy will include exactly Nof the
K receivers in use,is given by
S
K K(cid:3)K
a Sb a Sb
N K (cid:3)N
Pr1N2 (cid:1)e KJ , Nmin(cid:4)N(cid:4)Nmax (1.12)
a b
K
J
0, otherwise
where
N (cid:1)max10,K (cid:2)K (cid:3)K2 (1.13)
min J S
N (cid:1)min1K ,K 2. (1.14)
max S J
Using (1.12)—(1.14),it is possible to compute the expected total effective jam-
ming energy E sensed by the K receivers,namely
Jeff S
E
E (cid:1) J E5N6, (1.15)
Jeff K
J
E being the expected value operator. Despite the complicated form of
Pr(N),it can be verified that
K K
E5N6 (cid:1) J S, (1.16)
K
and hence,that
E K
E (cid:1) J S . (1.17)
Jeff K
More generally,it can be verified that when the communicators use the strat-
egy described above,(1.17) is the average total effective jamming energy for
any arbitrary distribution of jamming energy.
This idealized situation leads one to conclude,based on (1.17),that the
receiver can minimize the jammer’s effectiveness energy-wise by not using
diversity, i.e., by using K (cid:1) 1. Furthermore, the multiplicity K of the
S
orthogonal communication system complex should be made as large as pos-
sible to reduce E ,i.e.,the complex should be designed to use all of the
Jeff
available bandwidth.The energy-optimal communication strategy (K (cid:1)1)
S
using a single one of the Kavailable communication links,is called a pure
spread-spectrum strategy.This strategy,with its accompanying threat to use
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A SPREAD-SPECTRUM OVERVIEW
8 A Spread-Spectrum Overview
any of Korthogonal links,increases the total signal-to-jamming ratio from
E /E at each receiving antenna’s terminals to KE /E at the output of the
S J S J
designated receiver,and therefore qualifies as an anti-jam(AJ) modulation
technique.
The improvement E/E in signal-to-jamming ratio will be called the
J Jeff
energy gain EG of the signalling strategy played on the orthogonal com-
munication system complex.
E K 2TW
EG(cid:1) J (cid:1) (cid:1) s ss. (1.18)
E K K D
Jeff S S
Hence,the energy gain for a pure SS strategy is the multiplicity factor of the
complex.In this fundamental form (1.18),the energy gain is the ratio of the
signal space dimension 2TW perceived by the jammer for potential com-
s ss
munication use to the total dimension K Dof the K links’D-dimensional
S S
signal spaces which the receiver must observe.For a fixed TW product,this
s ss
definition of energy gain makes no distinction between diversity and SS
strategies using the same signal space dimension K D.The reader may rec-
S
ognize the fact that the quantity called the multiplicity factor,or energy gain
in this chapter,is sometimes referred to as the processing gain of the SS sys-
tem.This nomenclature is by no means universally accepted, and we will
instead identify the term processing gain PGwith the ratio W /R ,where R
ss b b
is the data rate in bits/second.It is easily verified from (1.18) that process-
ing gain and energy gain are identical when R (cid:1) K D/2T,e.g.,for binary
b S s
orthogonal signalling (D(cid:1)2) with no diversity (K (cid:1)1).
S
Two key assumptions were made in showing that the pure SS strategy is
best:(1) The channel is ideal and propagates all signals equally well,and (2)
the proper performance measure is the total effective jamming energy.If
either of the above assumptions is not acceptable,then the jammer’s strat-
egy may influence the performance measure,and the optimum diversity fac-
torK may be greater than one.Indeed,in later chapters it is shown that the
S
use of bit-error rate (BER) as a performance measure implies that the opti-
mum diversity factor can exceed unity.
Let’s summarize the requirements characteristic of a digital spread-spec-
trum communication system in a jamming environment:
1. The bandwidth (or equivalently the link’s signal-space dimension D)
required to transmit the data in the absence of jamming is much less than
the bandwidth W (or equivalently the system’s signal space dimension
ss
2TW ) available for use.
s ss
2. The receiver uses inner product operations (or their equivalent) to con-
fine its operation to the link’sD-dimensional signal space,to demodu-
late the signal, and thereby to reject orthogonal jamming waveform
components.
3. The waveforms used for communication are randomly or pseudoran-
domly selected,and equally likely to be anywhere in the available band-
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A SPREAD-SPECTRUM OVERVIEW
Spread-Spectrum System Configurations and Components 9
width (or equivalently,anywhere in the system’s 2TW dimensional sig-
s ss
nal space).
The term pseudorandomis used specifically to mean random in appearance
but reproducible by deterministic means.
We will now review a sampling of the wide variety of communication sys-
tem designs which possess SS characteristics.
1.3 SPREAD-SPECTRUM SYSTEM CONFIGURATIONS
AND COMPONENTS
A pure spread-spectrum strategy,employing only a single link at any time,
can be mechanized more efficiently than the system with potential diversity
factor K,shown in Figure 1.1.In an SS system,the K transmitter-receiver
pairs of Figure 1.1 are replaced by a single wideband communication link
having the capability to synthetize and detect all of the waveforms poten-
tially generated by the orthogonal communication system complex.The pure
SS strategy of randomly selecting a link for communication is replaced with
an equivalent approach, namely, selecting a D-dimensional subspace for
waveform synthesis out of the system’s 2TW -dimensional signal space.This
s ss
random selection process must be independently repeated each time a sym-
bol is transmitted.Independent selections are necessary to avoid exposing
the communication link to the threat that the jammer will predict the sig-
nal set to be used,will confine his jamming energy to that set,and hence,
will reduce the apparent multiplicity and energy gain to unity.
Three system configurations are shown in Figure 1.2,which illustrate
basic techniques that the designer may use to insure that transmitter and
receiver operate synchronously with the same apparently random set of
signals.The portions of the SS system which are charged with the respon-
sibility of maintaining the unpredictable nature of the transmission are
double-boxed in Figure 1.2.The modus operandi of these systems is as
follows:
1. Transmitted reference(TR) systems accomplish SS operation by trans-
mitting two versions of a wideband, unpredictable carrier, one (x(t))
modulated by data and the other (r(t)) unmodulated (Figure 1.2(a)).
These signals,being separately recovered by the receiver (e.g.,one may
be displaced in frequency from the other),are the inputs to a correla-
tion detector which recovers the data modulation.The wideband carrier
in a TR-SS system may be a truly random, wideband noise source,
unknown by transmitter and receiver until the instant it is generated for
use in communication.
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A SPREAD-SPECTRUM OVERVIEW
10 A Spread-Spectrum Overview
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