1.51 Low Probability of Intercept (LPI)
1.5.3 High-Resolution Time-of-Arrival (TOA) Measurements
Not all interference waveforms satisfy the independence and randomly asynchronous assumptions used in Section 1.5.2 to reaffirm the energy gain capability of SS systems. Here are some examples which are illustrated pic- torially in Figure 1.5.
1. Multipath:Additional propagation paths from transmitter to receiver may produce undesirable interference in a correlator synchronized to a signal arriving via a specified path. For example, a single additional path may produce an interfering signal of the form (1.64) with
(1.82) mI1t2mT1t2,
2£ 1 TsTc2 qq
0P1f2 04df§
1
3Nc. SIRout1f sync2
SIRin 2c 1
Ts qqSm21f2dfd1
The Advantages of Spectrum Spreadinghttp://jntu.blog.com 29
at a fixed delay tIcorresponding to the incremental propagation delay between the interference path and the communication signal path.
2. Repeater Jamming:This is a form of artificial multipath, in which the jam- mer attempts to receive the SS signal, somehow alter the data modula- tion, and then broadcast the result. Hence, if the modulation multiplicatively changed, then the signal retransmitted by the jammer may be of the form (1.64) with
(1.83) In this case,tIcorresponds to the additional propagation delay encoun- tered over the propagation path through the surveillance/jamming sys- tem.
In both of these cases, the time shift parameter tIand freqnecy offset of (1.64) are nearly constant or slowly varying over restricted ranges. The inter- ference’s incremental delay tIis a positive quantity when a direct natural propagation path is used for communication. On the other hand, communi- cation via a friendly repeater or an indirect path may result in a negative value for tI.
The average response of a correlation detector to the above types of signal-related interference can be determined by evaluating (1.66) or (1.71), e.g.,
(1.84)
E5 0vI1k2 026hEe ` 1k12TkT mI1ttI2mR*1t2ej2p¢t dt`2f i.
mI1t2d¿1t2mT1t2.
30 A Spread-Spectrum Overview
Figure 1.5. A scenario for SS link operation.
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Here it is assumed that the receiver’s correlator reference signal mR(t) is syn- chronized to the signal arriving via the communication path, and that tIand are the incremental time and frequency shifts incurred by the interference signal during the k-th correlation interval. At this point, we must depart from the analysis of the previous section, because tIis fixed within a limited range in this interference scenario and cannot be used as an averaging variable.
The shape of 8E{0yI(k)02}9 as a function of tI and will indicate the time and freqnecy resolutioncapability of the SS signal structure, i.e., the abil- ity of the receiver’s correlation detector to discriminate against versions of the transmitted signal which do not arrive in synchronism with the receiver’s clocks. The mean-squared value of the integral in (1.84) is the cross-ambi- guity functionof the waveforms mI(t) and mR(t) at offsets tIand , and hence, we are embarking on a study of the time- and ensemble-averaged ambigu- ity function provided by an SS waveform/modulation system. The theory of auto-ambiguity functions (i.e.,mI(t) mR(t)) states that the time-width of the function’s central and largest peak at tI0 and 0 is inversely pro- portional to the rms bandwidth of the signal upon which the correlator acts.
Hence, one might expect under certain conditions that SS receivers are espe- cially sensitive to synchronization errors, and possess high time-of-arrival (TOA) resolution capabilities.
The evaluation of (1.84) for repeater modulation of the form (1.83), under the assumption that the added modulation d(t) is an independent sta- tionary random process, yields
(1.85) where Rd() and Sd() are the autocorrelation function and PSD, respec- tively, of d(t);yRJ(k) and yM(k) are the values of yI(k) for repeater jamming (1.83) and multipath (1.84) respectively, and
(1.86) Henceforth, for simplicity we set
(1.87) Equation (1.85) indicates that the effect of the jammer’s added modula- tion d(t) is to average an equivalent multipath interference measure 8E{0yM(k)02}9at offsets tIand f over f(weighted by Sd(f)) to determine the repeater jamming measure 8E{0yM(k)02}9.
n ^ f ¢.
0vM1k2 02 ` 1k12TkT mT1ttI2mR*1t2ej2p1f¢2tdt`2.
Sd¿1f2E5 0vM1k2 026df
mT1ttI2mR*1t2mT*1t¿tI2mR21t¿2dt dt¿f
E5 0vRJ1k2 026E51k12TkT Rd¿1tt¿2ej2p¢1tt¿2
The Advantages of Spectrum Spreadinghttp://jntu.blog.com 31
We will now evaluate the multipath interference measure for two basic SS waveform designs.
1. DS/BPSK modulation:In this case mT(t) is given by (1.45) and mR(t) is simply the SS code c(t) in (1.22). Breaking up the integral over the data symbol interval of duration TTsin (1.86) into a sum of integrals over the chip intervals of mR(t), gives
(1.88) The integral in (1.88) can be simplified by using the fact that the pulse shape p(t) is non-zero only in (0,Tc) and by defining the quantities NI (and integer) and tIto satisfy
(1.89) Then
(1.90)
where Xp(t,n;T) is the ambiguity function of the pulse waveform p(t) gen- erally defined as
(1.91) for pulse waveforms p(t) non-zero only in the time interval (0,T). The lim- ited values of mwhich produce non-zero results in (1.90) further simplify (1.88) to
(1.92) Equation (1.92) is a convenient starting point for time and ensemble averaging operations.
d:1NIn12>Nc;cNIn1cnXp1tITc, n; Tc2 4 `2.
0vM1k2 02 ` a
kNc1
n1k12Nc
ej2pnnTc3d:1NIn2>Nc;cNIncnXp1tI, n; Tc2 Xp1t, n; T2 0Tp1tt2p*1t2ej2pnt dt
e
Xp1tI, n; Tc2ej2pnnTc, for mNIn Xp1tITc, n; Tc2ej2pnnTc, for mNIn1 0, otherwise
nT1n12Tc
c
p1ttImTc2p*1tnTc2ej2pnt dt tINITctI, 0tI 6 Tc.
nT1n12Tc
c
p1ttImTc2p*1tnTc2#ej2pnt dt`2. 0vM1k2 02 ` a
kNc1 n1k12Nc
a
q
mq
d:m>Nc;cmcn
32 A Spread-Spectrum Overview
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We will carry out the ensemble-averaging process under the assump- tion that the spectrum-spreading sequence {cn} is composed of inde- pendent, identically distributed random variables, equally likely to be 1 or 1, and that the independent data sequence {dk} likewise, is composed solely of 1’s and 1’s. The expansion of (1.92) then requires the moments
(1.93)
(1.94) and the fact that
(1.95)
Expanding the squared sum in (1.92) as a double sum and simplifying, gives
(1.96) Because of the manner in which we modelled {cn} and {dk}, the above mean- squared value of yM(k) is not a function of k, and hence, no further averag- ing is necessary.
When the chip pulse p(t) is unit amplitude and rectangular in shape, with duration T, then
(1.97) 0Xp1t, n; Tc2 02 • `sin3pn1T 0t0 2 4
pn `2, 0t0 6 T 0, otherwise.
E5 0vM1k2 026i
`sin1pnTs2 sin1pnTc2 `
20Xp1tI, n; Tc2 02Nc0Xp1tITc, n; Tc2 02, for NI0,
`sin1pnTs2 sin1pnTc2 `
20Xp1tITc, n; Tc2 02Nc0Xp1tI, n; Tc2 02, for NI 1,
Nc5 0Xp1tI, n; Tc2 02 0Xp1tITc, n; Tc2 024,
` a
kNc1 n1k12Nc
ej2pnnTc`2 `sin1pnTs2 sin1pnTc2 `
2
.
E1d:1Nm2>Nc;d:1Nn12>Nc;cNmcmcNn1cn6 0 for all N, m, n,
E1d:1Nm2>Nc;d:1Nn2>Nc;cNmcmcNncn6 c
1 if N0 1 if N0, nm 0 otherwise
The Advantages of Spectrum Spreading 33
otherwise.
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With the aid of (1.89) and (1.97), the average ambiguity surface (1.96) can be evaluated as a function of tIand n, and the results are displayed in Figure 1.6 for the case Nc100.
When there is no time or frequency mismatch, i.e., when tI0 and n 0, then the normalized detector output (1.96) is . Alternatively, when 0tI0Tc, then (1.96) is upper-bounded by . Hence, multipath and repeater jamming are reduced by a factor 1/Ncwhen the interfer- ence’s incremental delay exceeds a chip time. This is a firm basis for stat- ing that the TOA resolution of a high energy gain DS/BPSK system is approximately one chip time.
2. FH/FSK modulation:When FH signalling of the form (1.56) is employed, a correlator set to detect the data frequency dRcorrelates the received signal with the reference
(1.98) where p(t) is a hop pulse of duration Th which is non-zero only on (0,Th), and is a sequence of uniformly distributed phase variables.
The correlator resets every hop interval, i.e.,TTh, and the squared 5fnœ6
mR1t2 a
n
ej32p1fndR2tfnœ4p1tnTh2. Nc2Tc2
Nc2Tc2
34 A Spread-Spectrum Overview
Figure 1.6. Normalized RMS detector output as a function of normalized time mis- match tI/Tcfor several values of normalized frequency mismatch nTc. Random DS modulation is assumed, with an energy gain of Nc100 and square chip pulses.
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k-th correlator output (1.86) is
(1.99) Only two terms in the sum over mcontribute to the value of 0yM(k)02for a given k. Defining the integer NIand tIto satisfy
(1.100) and using techniques similar to those seen earlier, result in
(1.101) where uand uare independent, uniformly distributed phase variables, and Xp(,;) is defined by (1.91).
We continue on the assumption that {fn} is a sequence of random vari- ables, each chosen from a set of possible frequencies according to the probability distribution PF(). Averaging over u, u, and {fn} in (1.101), produces a variety of expressions depending on the values of Nand the data-induced frequency shifts. For simplicity, we assume that all the involved data variables are equal. Ensemble-averaging (1.101) produces
(1.102) for where,
(1.103) We will use (1.102) as a basis for evaluating the TOA resolution capa- bility of FH signals.
Before proceeding, we will develop a useful upper bound on (1.103).
For this purpose we assume that f is composed of MF frequencies, 0Xp1t, n; Th2 02¢
a
f¿f a
f–f
PF1f¿2PF1f–2 0Xp1t, f¿f–n; Th2 02.
dRd:1NIk12>MD;d:1NIk2>MD;,
E5 0vM1k2 026 •0Xp1tI, n; Th2 02 0Xp1tITh, n; Th2 02, NI0 0Xp1tITh, n; Th2 02 0Xp1tI, n; Th2 02, NI 1 0Xp1tI, n; Th2 02 0Xp1tITh, n; Th2 02, otherwise
F fk1dRn; Th2 02,
eju¿Xp1tITh, fNIkd:1NIk2>MD;
0vM1k2 02 0ejuXp1tI, fNIk1d:1NIk12>MD;fk1dRn; Th2
tINIThtI, 0tI 6 Th,
ej32p1fk1dR2tfkœ14p*1t 1k12Th2ej2pnt dt`2. 0vM1k2 02 ` 1kkTh12T
h
a
m
ej32p1fmd:m>MD;21ttI2fm4p1ttImTh2
The Advantages of Spectrum Spreadinghttp://jntu.blog.com 35
equally likely and uniformly spaced 1/ThHz apart (the minimum spac- ing required for orthogonality over a hop time). For this calculation’s purposes, this assumption is conservative in the sense that the hop fre- quencies in an operational system may be further apart to support FSK data and maintain the orthogonality of all possible waveforms.Assuming that the pulse shape is rectangular, the corresponding ambiguity func- tion (1.97) can be overbounded by
(1.104)
Now define the integer k0such that
(1.105) or equivalently
(1.106) Noting that there are MFk0frequency pairs (f,f, each from , for which f f k0/Th, it can be shown that
(1.107)
The last inequality in (1.107) follows from (1.105) and (1.106), and worst case choice of d. The weakest bound (1.107) occurs when nis close to zero and k00, and hence, considering that case and letting mk1
a
MF
k11 a
MF
k21
Th2
p21 0k1k2k00 1>222
k1k2k0
T 1
MF2 D1MFk02Th2 0Xp1t, n; Th2 02 1
MF2 £1MFk02Th2 a
f¿f a
f–f
1
p21f¿f–n22
f'f–k0>Th § F n k0
Th d
Th , 0d0 1 2 .
`k0
Thn` min
k ` k Thn 0Xp1t, n; Th2 02 à
Th2, n 6 1 pTh ,
a 1
pnb2, n 1 pTh .
36 A Spread-Spectrum Overview
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k2, gives
(1.108)
The resolution capability of FH/SS communication system can now be esti- mated by noting from (1.102) and (1.108) that
(1.109) This is the basis for stating that a high energy gain FH/FSK communication system has TOA resolution on the order of Th. That is, (1.109) indicates that frequency-synchronous correlation detectors should be able to reject the desired FH/FSK signal, if it arrives out of time synchronism by more than Thseconds.