An FH/QPSK signal is characterized by transmitting (Figure 1.1)
(1.1) in the i-th signalling interval (i⫺1)Tsⱕt iTs, where is the particular carrier radian frequency selected by the frequency hopper for this inteval.2 According to the designated SS code,u(i)is the information symbol which ranges over the set of possible values
(1.2) andSis the transmitted average power.
um⫽ mp
4 ; m⫽1, 3, 5, 7
h1i2
s1i21t2 ⫽ 22S sin1h1i2t⫹u1i2
2We assume here the case of slow frequency hopping (SFH), i.e., the hop rate is equal to or a submultiple of the information symbol rate and that the frequency hopper and symbol clock are synchronous. Thus, in a given symbol interval, the signal frequency is constant and the jam- mer, if transmitting a tone at that frequency, affects the entire symbol interval.
At the receiver (Figure 1.2), the sum of additive white Gaussian noise n(t), the jammer J(t), and a random phase-shifted version of the transmitted sig- nals(i)(t;u) are first frequency dehopped, then coherently demodulated by a conventional QPSK demodulator.
The band-pass noise n(t) has the usual narrowband representation (1.3) whereNc(t) and Ns(t) are statistically independent low-pass white Gaussian noise processes with single-sided noise spectral density N0w/Hz. The partial- band multitone jamming J(t) is assumed to have a total power Jwhich is evenly divided among Qjammer tones. Thus, each tone has power
(1.4) Furthermore, since the jammer is assumed to have knowledge of the exact location of the spreading bandwidth Wssand the number Nof hops in this bandwidth, then, as was done in our previous discussions, we shall assume that he will randomly locate each of his Qtones coincident with Qof the N hop frequencies. Thus,
(1.5) represents the fraction of the total band which is continuously jammed with tones, each having power J0. Once again, the jammer’s strategy is to distrib- ute his total power J(i.e., choose randJ0) in such a way as to cause the com- municator to have maximum probability of error.
r⫽^ Q N J0⫽ J
Q .
n1t2 ⫽ 223Nc1t2 cos 1h1i2t⫹u2 ⫺Ns1t2 sin 1h1i2t⫹u2 4 Figure 1.1. Block diagram of a coherent FH/QPSK modulator.
Figure 1.2. Block diagram of a coherent FH/QPSK demodulator.
In view of the foregoing, the total received signal in a signalling interval which contains a jamming tone at the hop frequency is given by
(1.6) where
(1.7) n(t) is given by (1.3) and
(1.8) withuJuniformly distributed on (0, 2p) and independent of the information symbol phase u(i). Over an integral number of hop bands, the fraction rof the total number of signalling intervals will have a received signal charac- terized by (1.6). In the remaining fraction (1 ⫺r) of the signalling intervals, the received signal is simply characterized by
(1.9) After ideal coherent demodulation by the frequency hopper, the in-phase and quadrature components of the received signal become3
(1.10) These signals are then passed through integrate-and-dump filters of dura- tion equal to the information symbol interval Tsto produce the in-phase and quadrature decision variables
(1.11) where
(1.12) NQ⫽^ 冮1i⫺iTs12T
s
Nc1t2dt NI⫽^ ⫺冮1i⫺iTs12T
s
Ns1t2dt
⫽bi
B S
2Ts⫹ 2J0Ts cos uJ⫹NQ
zQ⫽^ 冮1i⫺iTs12T
s
eQ1t2dt⫽ 2S Ts sin u1i2⫹ 2J0Ts cos uJ⫹NQ
⫽aiB S
2Ts⫺ 2J0Ts sin uJ⫹NI
zI⫽^冮1i⫺iTs12T
s
eI1t2dt⫽ 2S Ts cos u1i2⫺ 2J0Ts sin uJ⫹NI
eQ1t2⫽^ y1i21t2 322 cos1vh1i2t⫹u2 4 ⫽ 2S sin u1i2⫹ 2J0 cos uJ⫹Nc1t2. eI1t2⫽^ y1i21t2 322 sin1vh1i2t⫹u2 4 ⫽ 2S cos u1i2⫺ 2J0 sin uJ⫺Ns1t2
y1i21t2 ⫽s1i21t;u2 ⫹n1t2. J1t2 ⫽ 22J0 cos1h1i2t⫹uJ⫹u2 s1i21t;u2 ⫽ 22S sin1h1i2t⫹u1i2⫹u2,
y1i21t2⫽s1i21t;u2 ⫹n1t2 ⫹J1t2
3We ignore double-harmonic terms.
are zero mean Gaussian random variables with variance N0Ts/2 and, in view of the possible values for u(i)given in (1.2), {ai} and {bi} are the equivalent independent in-phase and quadrature binary information sequences which take on values ⫾1.
The receiver estimates of aiand bi are obtained by passing zIand zQ through hard limiters, giving
(1.13) Hence, given ai,bianduJ, the probability that the i-th symbol is in error is the probability that either or is in error, i.e.,
(1.14) Since the signal set is symmetric, we can compute (1.14) for any of the four possible signal points and obtain the average probability of symbol error conditioned on the jammer phase Ps(uJ). Thus, assuming for simplicity that ai⫽1,bi⫽1, we compute Ps(uJ) from (1.14), combined with (1.11) and (1.13), as
(1.15) where
(1.16) withQ(x) the Gaussian probability integral as used in previous chapters.
Finally, the unconditional average probability of symbol error for symbol intervals which are jammed is obtained by averaging Ps(uJ) of (1.15) over the uniform distribution of uJ. Thus,
(1.17) PsJ⫽ 1
2p冮02p3PI1uJ2 ⫹PQ1uJ2⫺PI1uJ2PQ1uJ2 4duJ.
PsJ
⫽Qc B
STs
N0
a1⫹
B 2J0
S cos uJb d PQ1uJ2⫽PreNQ 6 ⫺B
S
2Ts⫺ 2J0Ts cos uJf
⫽Qc B
STs
N0
a1⫺
B 2J0
S sin uJb d PI1uJ2 ⫽PreNI 6 ⫺B
S
2Ts⫹ 2J0Ts sin uJf
⫽PI1uJ2⫹PQ1uJ2⫺PI1uJ2PQ1uJ2
⫺Pr5zI 6 00ai⫽16Pr5zQ 6 00bi⫽16 Ps1uJ2⫽Pr5zI 6 00ai⫽16⫹Pr5zQ 6 00bi⫽16
⫺Pr5aˆi⫽ai6Pr5bˆi⫽bi6.
⫽Pr5aˆi⫽ai6 ⫹Pr5bˆi⫽bi6 Psi1uJ2⫽Pr5aˆi⫽ai or bˆi⫽bi6
bˆ aˆi i
aˆi⫽sgnzI; bˆ
i⫽sgnzQ.
Recognizing that, for a QPSK signal, the symbol time Tsis twice the bit timeTb, letting Eb⫽STbdenote the bit energy, we then have
(1.18) Furthermore, from (1.4) and (1.5),
(1.19) Now, if the hop frequency slots are 1/Tswide, in terms of the total hop fre- quency band Wssand the number of hop slots Nin that band, we then have (1.20) Substituting (1.20) into (1.19) gives
(1.21) As in previous chapters, the quantity J/Wssrepresents the effective jammer power spectral densityin the hop band; thus, we have again introduced the notationNJto represent this quantity.
Finally, rewriting (1.16) using (1.18) and (1.21) gives
(1.22) For the fraction (1 ⫺r) of symbol (hop) intervals where the jammer is absent, the average symbol error probability is given by the well-known result [1]
(1.23) Thus, the average error probability over all symbols (jammed and unjammed) is simply
(1.24) where is given by (1.17), together with (1.22), and is given in (1.23).
Before presenting numerical results illustrating the evaluation of (1.24), it is of interest to examine its limiting behavior as N0S0. Clearly, from (1.23) we have
(1.25)
Eb>Nlim0SqPs0⫽0
Ps0
PsJ
Ps⫽rPsJ⫹ 11⫺r2Ps0, Ps0⫽2Qa
B 2Eb
N0
b ⫺Q2a
B 2Eb
N0
b. PQ1uJ2⫽Qc
B 2Eb
N0
a1⫹
B NJ
rEb
cos uJb d. PI1uJ2⫽Qc
B 2Eb
N0
a1⫺B
NJ
rEb
sin uJb d 2J0
S ⫽J>Wss
rSTb
⫽ J>Wss
rEb
⫽^ NJ
rEb
. N⫽ Wss
1>Ts
⫽WssTs⫽2WssTb. 2J0
S ⫽ 2J rNS . STs
N0
⫽ 2STb
N0
⫽ 2Eb
N0
.
Also,
(1.26)
This result can be obtained directly from the graphical interpretation given in Figure 1.3. Finally, substituting (1.25) and (1.26) in (1.24) gives the desired
Eb>Nlim0SqPsJ⫽f
0; rEb
NJ
7 1 2
p cos⫺1 B
rEb
NJ
; 1
2 6 rEb
Nj
ⱕ1 1
p cos⫺1 B
rEb
NJ
⫹ 1
4; 0 6 rEb
NJ
ⱕ 1 2 .
Figure 1.3. Graphical interpretation of (1.26).
limiting behavior for the average symbol error probability, namely,
(1.27)
The partial-band fraction r corresponding to the worst case jammer (maximumPs) can be obtained by differentiating (1.27) with respect to rand equating to zero. Assuming that, for a fixed Eb/NJ, this worst case roccurs when 1/2 ⬍rEb/NJⱕ1, then
(1.28) implies
(1.29) where
(1.30)
The solution to (1.29) may be numerically found to be
(1.31) or
(1.32)
Note that, since QandNare integers, then ras defined in (1.5) is not a con- tinuous variable. Thus, for a given N, the true worst case rwould be the rational number nearest to (1.32) which yields an integer value of Q. Also, the second part of (1.32) comes about from the fact that Q is constrained to be less than or equal to N. Thus, when Eb/NJis such that the solution of (1.30) and (1.31) gives a value of r⬎1, we take r⫽1 (full-band jamming) as the worst case jammer. Substituting (1.32) into (1.27) gives the limiting average symbol error probability performance corresponding to the worst
rwc⫽ • 0.6306
Eb>NJ
; Eb>NJ 7 0.6306
1; Eb>NJⱕ0.6306.
Z⫽0.7654 Z⫽^
B
1⫺rEb>NJ
rEb>NJ
. tan⫺1Z⫽ 1
2Z d
dr c2r p cos⫺1
B rEb
NJ
d ⫽0
Eb>Nlim0SqPs⫽f
0; rEb
NJ
7 1 2r
p cos⫺1 B
rEb
NJ
; 1
2 6 rEb
NJ
ⱕ1 r
p cos⫺1 B
rEb
NJ
⫹r
4; 0 6 rEb
NJ
ⱕ1 2 .
case jammer, namely,
(1.33) The final step in the characterization of the performance of FH/QPSK in the presence of multitone jamming is the conversion of average symbol error probability to average bit error probability. If one encodes the infor- mation symbols using a Gray code, the average bit error probability,Pb, for a multiple phase-shift-keyed (MPSK) signal is then approximated for large
Eb>Nlim0SqPsmax⫽f
0.2623 Eb>NJ
; Eb>NJ 7 0.6306 2
p cos⫺1 B
Eb
NJ
; 0.5 6 Eb>NJⱕ0.6306
1 p cos⫺1
B Eb
NJ⫹1
4; 0 6 Eb>NJⱕ0.5.
Figure 1.4. Pbversusr—FH/QPSK (tone jamming).
Eb/N0by
(1.34) where log2Mis the number of bits/symbol.The approximation in (1.34) refers to the fact that only errors in symbols whose corresponding signal phases are adjacent to that of the transmitted signal are accounted for. Since a Gray code has the property that adjacent symbols differ in only a single bit, then an error in an adjacent symbol is accompanied by one, and only one, bit error.
Since QPSK is the particular case of MPSK corresponding to M⫽4, then from (1.34),
(1.35) wherePsis given by (1.24) or its limiting form in (1.27).
Fortunately, for the case of QPSK, it is straightforward to account for the diagonal symbol errors which result in two bit errors and, thus, arrive at an exactexpression for Pb. In fact,Pbfor QPSK is identical to Pbfor binary PSK (BPSK) and is given by
(1.36) wherePI(uJ) are given in (1.22). Thus, comparing the approximate result of (1.35) (using (1.17), (1.22), and (1.24)) with the exact result of (1.36), we observe that the difference between the two resides in the terms resulting from the productof error probabilities, namely,
and . Also, by analogy with (1.27), the exact limiting form of Pbbecomes
(1.37)
with a worst case ras in (1.32) and corresponding maximum error proba- bility
Eb>Nlim0SqPb⫽ à
0; rEb
NJ 7 1 r
p cos⫺1 B
rEb
NJ
; 0 6 rEb
NJ ⱕ1 Q2122Eb>N02
1
2p冮02pPI1uJ2PQ1uJ2duJ
⫽rc 1
2p冮02pPQ1uJ2duJd ⫹ 11⫺r2QaB 2Eb
N0
b Pb⫽rc 1
2p冮02pPI1uJ2duJd ⫹11⫺r2QaB 2Eb
N0
b Pb⬵12Ps
Pb⬵ Ps
log2M
Figure 1.5. Worst case rversusEb/NJ—FH/QPSK (tone jamming).
Figure 1.6. Worst case PbversusEb/NJ—FH/QPSK (tone jamming).
(1.38)
Figure 1.4 is a typical plot of Pbversusr, with Eb/NJas a parameter for the case Eb/N0⫽20 dB. It is seen that, for fixed Eb/N0andEb/NJ, there exists a value of rwhich maximizes Pband, thus, represents the worst case multi- tone jammer situation. In the limit, as Eb/N0approaches infinity, this value ofrbecomes equal to that given by (1.32). Figure 1.5 is a plot of worst case rversusEb/NJ, with Eb/N0as a parameter. Figure 1.6 illustrates the corre- sponding plot of versus Eb/NJ, with Eb/N0fixed.