In a differentially encoded frequency-hopped M-ary differential phase- shift-keyed (FH/MDPSK) system, the information to be transmitted in the i-th signalling interval (i⫺1)TsⱕtⱕiTsis conveyed by appropriately select- ing one of Mphases
(2.1) and adding it to the total accumulated phase in the (i⫺1)-st signalling inter- val of a constant amplitude (A), fixed frequency (assumed known at the receiver) sinusoid. Typically M⫽2KwithKinteger, and these are the only cases we shall consider in detail. Furthermore, since the derivation of the per- formance of FH/MDPSK in the presence of a partial-band multitone jam- mer will rely largely on certain geometric relations, it is expedient to deal with both the signal and the jammer as phasors. Thus, the transmitted signal s(i)(t) in the i-th signalling interval is conveniently represented in complex form by
(2.2) where is the total accumulated phase in the (i ⫺ 1)-st signalling interval and u(i)ranges over the set {um} of (2.1).
In the presence of multitone jamming inteference as characterized in the previous chapter, a jamming tone J(t), constant in both phase and magni- tude (amplitude), is added to the transmitted signal. Since, when the jam-
uT1i⫺12
S1i2⫽Aej1u1i2⫹uT1i⫺122 um⫽ 12m⫺12p
M ; m⫽1, 2,p,M
mer “hits,” he is assumed1to be of the same frequency as the signal, then we may also represent the jammer in complex form, namely,
(2.3) whereuJis a random phase uniformly distributed in the interval (0, 2p).Thus, in any hop interval which is hit by the jammer (the probability of this occur- ring is the partial-band fraction r), the signals on which a decision for the i- th signalling interval is to be based are given (in complex form) by
(2.4) Assuming a receiver structure that is optimum in the absence of the jam- mer, i.e., it employs the optimum decision rule for MDPSK against wideband noise, then in the presence of the on-tune jammer this rule would result in the estimate
(2.5) wherekis such that
(2.6) Then if unis indeed the true value of u(i), a symbol (phase) error is made, i.e.,
whenever
(2.7) Without loss in generality, we shall, for convenience, rotate the actual trans- mitted signal vectors by p/Mradians so that the possible transmitted signal phases of (2.1) become
(2.8) Finally, letting Q2pn/M;n⫽0;⫾1,⫾2, . . . ⫾(M⫺2)/2,M/2 denote the prob- ability of the error event in (2.7), namely,
(2.9) and noting that since we have assumed the absence of an AWGN back- ground, the probability of error in hop intervals which are not hit by the jam- mer is zero, then the average symbol error probability for MDPSK in the
Q2pn>M⫽Pre 0arg1Y1i2⫺Y1i⫺122⫺un0 7 p Mf um⫽ 2pm
M ; m⫽0,;1,;2,p,;aM⫺2 2 b,M
2 . 0arg1Y1i2 ⫺Y1i⫺122 ⫺un0 7 p
M . uˆ1i2⫽u1i2
0arg1Y1i2⫺Y1i⫺122⫺uk0 ⱕ p M . uˆ1i2⫽uk
Y1i2⫽Aej1u1i2⫹uT1i⫺12⫹IejuJ.
Y1i⫺12⫽AejuT1i⫺12⫹IejuJ
J⫽IejuJ
1The assumption of on-tune jamming is made solely to simplify the analysis, as has been done in previous chapters. Both the analytical technique and the sensitivity of the results that follow from its application depend heavily on this assumption. Some evidence of this statement will be discussed at the end of this section.
presence of multitone jamming is given by
(2.10) where the summation on nranges over the set n⫽0,⫾1,⫾2, . . . ,⫾(M⫺ 2)/2,M/2. Since uJis uniformly distributed, we can recognize the symmetry (2.11) Also if un⫽0 is transmitted, then, from (2.4),Y(i⫺1)andY(i)are identical vec- tors. Equivalently,
(2.12) and, from (2.9),
(2.13) Thus, using (2.11) and (2.13),Ps(M) of (2.10) simplifies to
(2.14) Finally, using the relation between average symbol and bit error probabili- ties, namely,
(2.15) the average bit error probability for MDPSK in the presence of multitone jamming is given by
(2.16)
Actually, the relation in (2.15) holds as an equality only for orthogonal sig- nal sets [5]. However, for low signal-to-jammer ratios, the right-hand side of (2.15) becomes a tight upper bound for the average bit error probability per- formance of FH/MDPSK. For binary DPSK (M⫽2), the equality in (2.15) is exact.
We shall see shortly that, for the evaluation of Q2pn/M, it is convenient to renormalize the problem in terms of the ratio of jamming power per tone J0⫽J/Qto signal power S. Let b2denote this ratio, i.e.,
(2.17) b2⫽J>Q
S . Pb1M2⫽ r
21M⫺12 £Qp⫹2 a
M⫺2 2 n⫽1
Q2pn>M§. Pb1M2⫽ c M
21M⫺12dPs1M2, Ps1M2⫽ r
M £Qp⫹2 a
M⫺2 2 n⫽1
Q2pn>M§. Q0⫽0.
0arg1Y1i2⫺Y1i⫺12 ⫺u00 ⫽0
Q2pn>M⫽Q⫺2pn>M; n⫽1, 2,p,M⫺2 2 . Ps1M2⫽ r
M a
n
Q2pn>M
Then, recalling from the previous chapter that the number of hop slots Nin the total hop frequency band Wssis
(2.18) then, the partial-band fraction rcan be expressed in terms of b2and the bit energy-to-jammer noise spectral density ratio Eb/NJby
(2.19) Using (2.19), we can rewrite (2.16) in the form
(2.20) Before proceeding to the evaluation of Q2pn/M;n⫽1, 2, . . . ,M/2 we make the final observation that the per tone jamming-to-signal power ratio b2can also be expressed in terms of the vector definitions of the signal and tone jamming interference. Since from (2.2) and (2.3) the signal power and jam- mer power per tone are given by
(2.21) then equivalently from (2.17) we have that
(2.22)
2.1.1 Evaluation of Q2pn/m
In view of (2.22), Q2pn/Mof (2.9) may be restated in the normalized form (2.23) where
(2.24) and
(2.25) Z1i2⫽ejnu⫹bejuJ⫽^ R2ej1⫺nu⫹c22.
Z1i⫺12⫽e⫺jnu⫹bejuJ⫽^ R1ej1⫺nu⫹c12
u⫽^ p M
Q2pn>M⫽Pr5 0arg1Z1i2⫺Z1i⫺122 ⫺2nu0 7 u6 b2⫽ I2
A2 . S⫽ A2
2 ; J Q⫽I2
2 , Pb1M2⫽ 1
21M⫺121log2M2b2Eb>NJ
£Qp⫹2 a
M⫺2 2
n⫽1
Q2pn>M§. r⫽ Q
N ⫽^ J
b2SWssTblog2M⫽ 1 1log2M2b2Eb>NJ
N⫽ Wss
1>Ts
⫽WssTblog2M
Note that, in obtaining (2.25) from (2.4), we have substituted for u(i) its assumed true value, namely,un⫽2pn/M⫺2nu, and, since uJis uniformly distributed, we have arbitrarily established the symmetry . Figure 2.1 is a graphical representation of (2.25) where we have further introduced the notation
(2.26) Thus, using (2.25) and (2.26),
(2.27) and, hence,
(2.28) Consider the product (asterisk denotes complex conjugate)
(2.29)
⫽R1R2ej1c2⫺c12.
1Z1i⫺122*Z1i2e⫺j2nu⫽R1R2ej1nu⫺c12ej1nu⫹c22e⫺j2nu Q2pn>M⫽Pr5 0c2⫺c10 7 u6 ⫽1⫺Pr5 0c2⫺c10 ⱕu6.
⫽c2⫺c1
⫽nuc2⫺ 1nuc12⫺2nu arg1Z1i2⫺Z1i⫺122⫺2nu⫽c⫺2nu
c⫽^ arg1Z1i2⫺Z1i⫺122.
uT1i⫺12⫽ ⫺pn>M⫽ ⫺nu
Figure 2.1. A graphical representation of (2.25).
The above product can also be written in the form
(2.30) Thus, using (2.29) and (2.30) in (2.28) results in the equivalent relation
(2.31) Equation (2.31) can be given a geometric interpretation as in Figure 2.2.
Here the vector OQ(a line drawn from point Oto either point Q) represents
⫽1⫺Pr5⫺uⱕarg31⫹b2e⫺j2nu⫹2b cos uJe⫺jnu4 ⱕu6. Q2nM⫽1⫺Pr5⫺uⱕarg3Zi⫺1*Zie⫺j2nu4 ⱕu6
⫽1⫹b2e⫺j2nu⫹2b cos uJe⫺jnu.
⫽ej2nu⫹b2⫹bejnu1ejuJ⫹e⫺juJ2e⫺j2nu 1Z1i⫺122*Z1i2e⫺j2nu⫽ 1ejnu⫹be⫺juJ21ejnu⫹bejuJ2e⫺j2nu
Figure 2.2. A geometric interpretation of (2.31).
the complex number whose argument is required in (2.31), i.e.,.
(2.32) Thus, in terms of the geometry in Figure 2.2, (2.31) may be written in the alternate form
(2.33) Considering separately the cases where point Pfalls outside and inside the 2uwedge, which expressed mathematically corresponds to the inequal- ities (see Figure 2.3)
(2.34) then after much routine trigonometry, it can be shown that
(2.35) whereu(b) is the unit step function and
(2.36) bn⫺1⫽^ ⫺sin3 1n⫺12u4 ⫹sinnu
sin3 12n⫺12u4
⫽f 1
p cos⫺1cb2 sin3 12n⫹12u4 ⫹sinu
2b sin3 1n⫹12u4 du1b⫺bn2
⫹ 1
p cos⫺1csinu⫺b2 sin3 12n⫺12u4
2b sin3 1n⫺12u4 d u1b⫺bn⫺12; 0 6 b 6 1 1; bⱖ1 n⫽2, 3,p,M
2 ⫺1, Q2pn>M⫽Q2nu
b2 sinu sin3 12n⫺12u4 ,
⫽1⫺pr5pointQ lies along the line AB6. Q2pn>M⫽1⫺Pr5pointQ is within the 2u wedge6
OQ⫽1⫹b2e⫺j2nu⫹2b cos uJejnu.
Figure 2.3. The geometry needed to establish (2.34).
Note that
(2.37) For n⫽1, the appropriate result analogous to (2.35) is
(2.38)
Here
(2.39) Finally, for n⫽M/2, we have the result
(2.40)
Also,
(2.41) As an example, Figure 2.4 is a plot of Q2nu;n⫽1, 2, 3, 4, 8 versus bforM
⫽16. These probabilities are computed from (2.35), (2.38), and (2.40). Using these results in (2.20), Figure 2.5 illustrates the product (Eb/NJ)⫻Pb(16) versusb. This curve has a maximum value of 1.457 at b⫽0.1614, which, from
bS1limQp⫽1.
bM>2⫺1⫽
1⫺cos p M sin p
M QMu⫽Qp⫽f
0; 0 6 b 6 bM>2⫺1
2
p cos⫺1≥ asin p
Mb 11⫺b22 2b cos p
M
¥; bM>2⫺1ⱕb 6 1
1; bⱖ1
bS1limQ2u⫽ 1
p cos⫺1csin 3u⫹sinu 2 sin 2u d 6 1.
b1⫽ sin 2u⫺sinu sin 3u ⫽
sin2p
M ⫺sin p M sin3p
M . Q2u⫽ à
0; 0 6 b 6 b1
1
p cos⫺1cb2 sin 3u⫹sinu
2b sin 2u d; b1ⱕb 6 1
1; bⱖ1
⫽1
⫹cos⫺1csinu⫺sin3 12n⫺12u4 2sin3 1n⫺12u4 d f
BSlim1Q2nu⫽ 1
p ecos⫺1csin3 12n⫹12u4 ⫹sinu 2 sin3 1n⫹12u4 d
(2.19), corresponds to the optimal (worst case) jamming strategy.
(2.42) Thus, the average bit error probability performance of FH/MDPSK (M⫽ 16) in the presence of the worst case tone jammer is given by
(2.43) Pbmax⫽ à
1
30 cQp⫹2a
7 n⫽1
Qnp>8d `
b⫽1>22Eb>NJ
; 0.25 6 Eb>NJ 6 9.597
1.457 Eb>NJ
; Eb>NJⱖ9.597
0.5; Eb>NJ 6 0.25.
rwc⫽ • 9.597 Eb>NJ
; Eb>NJⱖ9.597
1; Eb>NJ 6 9.597
Figure 2.4. Individual signal point error probability components as a function of square root of jamming (per tone)-to-signal power ratio.
Figure 2.5.Bit error probability performance of FH/MDPSK (M⫽16) as a function of square root of jamming (per tone)-to-signal power ratio.
Similar results can be obtained2for FH/MDPSK with M⫽2, 4, and 8. The asymptotic behavior of these results (i.e.,randPbinversely related to Eb/NJ) is given in Table 2.1. Using the results in Table 2.1 and the fact that, for any M,
(2.44) withPb(M) given by (2.20), Figure 2.6 is an illustration of the average bit error probability performance of FH/MDPSK for worst case partial-band multitone jamming.
Before concluding this section, we wish to alert the reader to a point of pathological behavior that is directly attributable to the assumption of an on-tune tone jammer and is perhaps not obvious from the analytical or graphical results given. In particular, we observe from Figure 2.4 that Qp/8 has a jump discontinuity at b ⫽ 1 and thus of (2.43) will have a similar jump discontinuity at Eb/NJ⫽0.25 (⫺6 dB). In fact, since from (2.39), Qp/8⫽0.625 as bapproaches one from below, then at Eb/NJ⫽ ⫺6 dB, Pbmax
Pbmax
Pbmax⫽Pb1M2 0b⫽1>21log2M2Eb>NJ ; Eb>NJⱕKr
Table 2.1.
Asymptotic performance of FH/MDPSK for worst case partial-band multitone jamming.
Worst case partial-band fraction
Maximum average bit error probability
M b Kr KP
2 1 1 0.50
4 0.5220 1.835 0.2593
8 0.2760 4.376 0.5280
16 0.1614 9.597 1.457
Pbmax⫽ KP
Eb>NJ rwc⫽ Kr
Eb>NJ
2It should be noted here that the results in [1] for the performance of FH/MDPSK (M⫽4) in the presence of the worst case tone jammer are partially incorrect. In particular, Houston finds b⫽0.52 as the maximizing value. However, since the fraction of the band jammed, which is given by r⫽1/(2b2Eb/NJ), cannot exceed one, the value b⫽0.52 can only be achieved if Eb/NJ
⬎1.85. For smaller values of Eb/NJ, the relation must be used.Thus, we arrive at the following corrected results for M⫽4.
Pbmax⫽e 0.2592 Eb>NJ
; Eb>NJ 71.85
1
3p ccos⫺1a2Eb>NJ⫺1
222Eb>NJ
b ⫹cos⫺1a2Eb>NJ⫹1
42Eb>NJ
b d; 0.56 Eb>NJ 6 1.85
0.5 Eb >NJ 60.5
b⫽1>22Eb>NJ
jumps from .4375 to .5. For other values of Mⱖ4, a similar jump disconti- nuity in the worst case tone jammer bit error probability will occur at b⫽ 1 and r⫽1, or equivalently, from (2.19),Eb/NJ⫽1/log2M. Since the range ofEb/NJin Figure 2.6 extends only down to ⫺2 dB, these jump discontinu- ities are not visible on this plot, i.e., the largest value of Eb/NJat which a dis- continuity occurs would correspond to M⫽4 (Eb/NJ⫽1/2 ⫽ ⫺3 dB).
Figure 2.6. Worst case bit error probability performance of FH/MDPSK for partial- band multitone jamming.