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- width Wssavailable for communicating an information symbol in a Tssec- ond interval (0, Ts), the resultant transmitted-signal function space of dimension approximately 2TsWssis divided so that each transmitter has a
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D-dimensional subspace,D 2TsWss/K, in which to synthesize its output signal. Denote an orthonormal basis for the total signal space by ck(t),k 1, 2, . . . , 2TsWss, i.e.,
(1.1) 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
(1.2) where
(1.3) and {aj} is a data-dependent set of coefficients. We will refer to the above as an orthogonal communication system complex of multiplicity K.
Of course, real systems generally radiate real signals. The reader may wish to view mk(t) as the modulation on the radiated signal Re{mk(t) exp (jvct u)}. Without loss of generality, we can dispense with the shift to RF during this initial discussion.
In a simplified jamming situation, the signal zi(t) observed at the i-th receiver in the receiving complex might be
(1.4) where ni(t) represents internally generated noise in the i-th receiver,J(t) is an externally generated jamming signal, and the K-term sum represents the total output signal of the transmitter complex. One signal processing
zi1t2 a
K k1
mk1t2J1t2ni1t2. Nk5j: 1k12D 6 jkD6
mk1t2 a
jHNk
aj cj 1t2,
0Tscj1t2c*k1t2 dt e1,0, jjkk
4 A Spread-Spectrum Overview
Figure 1.1. The scenario for a game between a jammer and a communication sys- tem complex.
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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
(1.5) In the absence of jamming and receiver noise, the properties of the ortho- normal basis insure that zjaj, and thus, the i-th receiver correctly discov- ers the data dependent set of coefficients {aj}, used by the i-th transmitter.
Both the jamming and receiver noise signals can be expanded in terms of the orthonormal basis as
(1.6)
(1.7) where J0(t) represents that portion of the jamming signal orthogonal to all of the 2TsWssbasis functions used in producing the composite signal. The receiver noise component n0i(t) likewise is orthogonal to all possible trans- mitted signals. These representations indicate that, in general, the projection (1.5) of zi(t) onto cj(t) in the i-th receiver produces
(1.8) The everpresent thermal noise random variable nij, assumed complex 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 Jjare less easily clas- sified, and from the jammer’s point of view, hopefully are unpredictable by the receiver.
The total energy EJ in the jamminig signal J(t) over the time interval (0,Ts) is given by
(1.9) Obviously, the energy term involviong J9(t) serves no useful jamming pur- 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 {cj(t)} of basis functions.) The sum in (1.9) can be partitioned into Kparts, the i-th part representing the energy EJiused to jam the i-th receiver. Thus,
(1.10) EJ a
K i1
EJi , EJi a
jHNi
ƒJjƒ2.
EJ 0TsƒJ1t2ƒ2 dt2Tj1asWss ƒJjƒ2 0TsƒJ01t2ƒ2 dt.
zjajJjnij for all jHNi . ni1t2 a
2TsWss
j1
nij cj1t2n0i1t2, J1t2 a
2TsWss
j1
Jjcj1t2J01t2, jHNi, zj 0Tszi1t2 c*j1t2 dt for all jH Ni .
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A similar partition holds for the total transmitted signal energy Es, namely (1.11) ESi being the energy used by the i-th transmitter. The additive partitions (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,Ts) of operation. Hence, impor- 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.