COHERENT DIRECT-SEQUENCE SYSTEMS
1.3 UNCODED BIT ERROR PROBABILITY FOR SPECIFIC
Consider the set of orthonormal basis
(1.39) The transmitted direct-sequence spread BPSK signal during the data bit time interval [0,Tb] has the form
(1.40) The jammer would like to place all of its signal energy into the same sig- nal space as the transmitted signal, otherwise its power would be wasted. It, however, can only know the signal bandwidth and not the signal phase. Thus, in general it has the form
(1.41) where recall J⫽(J0,J1, . . . ,JN⫺1) consists of the cosine components. Note that only these cosine components enter into the bit error probability bound. The total jammer signal energy during the data bit time is
(1.42)
(1.43) NJ⫽ 1
N a
N⫺1 k⫽0
Jk2⫹ 1 N a
N⫺1 k⫽0
J苲k2.
⫽ a
N⫺1 k⫽0
Jk2⫹ a
N⫺1 k⫽0
J苲k2
. JTb⫽ 冮0TbJ21t2dt
J1t2⫽ a
N⫺1
k⫽0Jkfk1t2 ⫹ a
N⫺1 k⫽0J苲kf苲k1t2 0ⱕtⱕTb.
⫽d2STc a
N⫺1 k⫽0
ckfk1t2 x1t2 ⫽c1t2d1t222S cos 0t k⫽0, 1,p,N⫺1.
f苲k1t2 ⫽ •B 2 Tc
sin 0t, kTcⱕt 6 1k⫹12Tc
0, elsewhere
fk1t2 ⫽ •B 2
Tc cos 0t, kTcⱕt 6 1k⫹12Tc
0, elsewhere
The worst case jammer is one that places all its energy in the cosine coor- dinates so that
(1.44) Normally, however, the jammer can only place equal energy in the cosine and sine coordinates with the result that
(1.45) Figure 1.3 shows the bit error probability for the Gaussian assumption and
a
N⫺1
k⫽0Jk2⫽ JTb
2 . a
N⫺1 k⫽0
Jk2 ⫽JTb.
Figure 1.3. Gaussian assumption and Chernoff bound.
the general Chernoff bound for these two cases. Note that there is only about a one dB difference for fixed bit error rates. This suggests that the Gaussian assumption is reasonable for all N.
The impact of jamming signals by characterizing two types of jammers will now be illustrated. The first model assumes a deterministic jammer wave- form that is characterized by a set of parameters u. The second model assumes the jammer is a stationary random process with statistical charac- terizations. In both cases the jammer is assumed to be transmitting contin- uously with constant power J. In the next section pulse jamming will be examined along with its generalization where the jammer power can be var- ied in time while maintaining a time-averaged power J.
1.3.1 CW Jammer
The most harmful jammer waveform is one that maximizes Jkgiven by (1.22) for each value of k. Since the jammer does not know the PN sequence {ck} this means the jammer should place as much energy as possible in the cosine coordinate which is achieved with a CW signal. Generally, the jammer may not know the transmitted signal carrier phase so consider the deterministic jammer waveform model
(1.46) which is characterized by the phase parameter u. Thus
(1.47) which is maximized when u⫽0.
For this CW jammer the conditional variance of ngiven in (1.35) is (1.48) For the Gaussian approximation (Central Limit Theorem applied for large N⫽Tb/Tc) the conditional bit error probability is
(1.49) UsingJTc⫽NJthis becomes
(1.50) for the CW jammer. The choice of u⫽0 maximizes Pb(u) yielding the bound (1.51) Pb1u2 ⱕQa
B Eb
NJ
b. Pb1u2 ⫽Qa
B Eb
NJ cos2ub
⫽Qa B
Eb
JTc cos2ub. Pb1u2⫽Pb1J2
Var5n0J6⫽JTc cos2u. Jk⫽ 2JTc cos u, all k J1t2 ⫽ 22J cos30t⫹u4
This upper bound is the same as that for binary orthogonal signals in an addi- tive white Gaussian noise channel for single-sided spectral density NJ. For a jammer with constant power Jthis is the worst performance of the direct- sequence spread BPSK system.
Against a CW jammer an effective technique is to use BPSK data mod- ulation with QPSK PN spreading. This is a DS/BPSK signal of the form
(1.52) wherecc(t) and cs(t) are PN waveforms. This is the special case of dc⫽ds⫽ din the QPSK modulation given in (1.13) where the QPSK symbol energy is also the bit energy (one bit per QPSK signal). For this case (1.14) and (1.17) have the form
(1.53) and
(1.54) wherencandnsare zero mean independent with conditional variances
(1.55) and
(1.56) Next use
(1.57) as the statistic for the decision rule given in (1.19). Since
(1.58) the bit error probability for the Gaussian approximation is
(1.59) This is independent of uand is a 3 dB improvement over the worst case (u⫽ 0) BPSK PN spreading system.Thus, to minimize the maximum possible degradation due to a CW jammer, one should use QPSK modulation with the same data in both inphase and quadrature coordinates. (For further dis- cussion on this see Kullstam [20].)
Pb1u2⫽Qa B
2Eb
NJb.
⫽ JTc
4 , Varenc⫹ns
2 `uf ⫽ 1
45JTc cos2u⫹JTcsin2u6 r⫽ rc⫹rs
2 ⫽d2Eb>2⫹ nc⫹ns
2 Var5ns0u6 ⫽JTc sin2u Var5nc0u6 ⫽JTc cos2u rs⫽d2Eb>2⫹ns
rc⫽d2Eb>2⫹nc
x1t2⫽d1t22S3cc1t2cos0t⫹cs1t2sin0t4
1.3.2 Random Jammer
Consider next characterizing the jammer as a stationary random process with autocorrelation
(1.60) and power spectral density
(1.61) where
(1.62) the constant power of the jammer.
The PN waveform c(t) is also stationary (introduce a uniformly distrib- uted time shift) with autocorrelation
(1.63) and power spectral density
(1.64) Naturally,J(t) and c(t) are independent of each other.
Sc1f2 ⫽ 冮⫺qqRc1t2e⫺j2pftdt.
Rc1t2 ⫽E5c1t⫹t2c1t2 6⫽ •1⫺ 0t0 Tc
, 0t0 ⱕTc
0, 0t0 7 Tc
冮⫺qqSJ1f2df⫽J,
SJ1f2 ⫽ 冮⫺qqRJ1t2e⫺j2pftdt
RJ1t2 ⫽E5J1t⫹t2J1t2 6
Figure 1.4. Power spectral densities.
Recall from (1.9) that when received waveform x(t)⫹J(t) is multiplied by the PN waveform c(t), the resulting noise term is
(1.65) SinceJ(t) and c(t) are independent, the autocorrelation of n(t) is
(1.66) which has the power spectral density
(1.67) where * indicates the convolution operation.
Figure 1.4 illustrates the power spectra of c(t),J(t), and the product n(t)
⫽c(t)J(t). Here SJ(f) is arbitrary and Sc(f) is a broad sin2x/x2spectrum of bandwidth roughly 1/Tc. The resulting noise spectrum Sn(f) has value at f⫽ 0 given by
(1.68) since
(1.69) andJis the total jammer power given by (1.62). There is equality in (1.68) whenJ(t) has a narrow power density spectrum compared to the PN wave- formc(t). In addition
(1.70) Thus
(1.71) whereWssis given by (1.6) and NJis our usual definition given as J/Wss.
Note that the equivalent noise of power spectral density bounded by NJ represents the total interference power. If we were to divide this power equally between the sine and cosine coordinates of the narrowband
⫽NJ Sn102 ⱕJTc
⫽Tc.
⫽ 冮⫺TTc
c
a1⫺ 0t0 Tc bdt Sc102 ⫽ 冮⫺qqRc1t2dt
Sc1f2 ⱕSc102
⫽Sc102J
ⱕSc102冮⫺qqSJ1f2df
Sn102⫽ 冮⫺qqSc1f2SJ1f2df
Sn1f2⫽Sc1f2*SJ1f2
⫽Rc1t2RJ1t2,
⫽E5c1t⫹t2c1t2 6E5J1t⫹t2J1t2 6 Rn1t2 ⫽E5n1t⫹t2n1t2 6
n1t2 ⫽c1t2J1t2.
(unspread) signal, then each coordinate would have a noise component of variance less than or equal to NJ/2. With BPSK data modulation and QPSK PN spreading, equal distribution of the jammer power is guaranteed. In this case the jammer appears as white noise of power spectral density bounded byNJ/2.
Multiplying the channel output signal by the PN waveform results in the narrowband signal s(t) in broadband noise n(t). Here s(t) has bandwidth 1/Tb whilen(t) has bandwidth greater than Wss⫽1/Tc⫽N/Tbwhich is Ntimes wider than the signal bandwidth. Since Sn(f) is essentially flat over the nar- rowband signal bandwidth, the detection problem reduces to demodulation of a BPSK signal in white noise of double-sided power spectral density less than or equal to NJ/2. Thus, making the Gaussian approximation, the uncoded bit error probability is bounded by
(1.72) Here equality is achieved for narrowband jamming signals. This gives the same result as with a CW jammer and QPSK modulation where the same data is entered in both the inphase and quadrature components (BPSK data modulation with QPSK PN spreading). It also is the baseline performance where the jammer simply transmits broadband Gaussian noise.
Based on these results for CW and random jammers, we would expect the direct-sequence BPSK anti-jam systems to be robust and insensitive to all jammers that produce waveforms which are independent of the transmis- sion. They appear to always give as good a performance as the basline jam- mer case and that is all one can expect from a good anti-jam communication system. This, however, is true only for constant power jammers.