Suppose a Gaussian noise jammer chooses to restrict its total power J(ref- erenced to the FH receiver input) to a fraction r(0,r1) of the full SS bandwidth Wss. As shown in Figure 2.6, the jamming noise power is spread
Worst Case Jamming 475
Figure 2.5. Performance of conventional MFSK (M2K) system over AWGN channel, with bit energy Eband noise power density NJ.
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uniformly over WJrWss, resulting in an increased power density
(2.21) and a correspondingly degraded SNR level
(2.22) in the jammed band, where NJis still defined by (1.4).
Recall that an FH system can in principle avoid certain frequency bands that it determines are particularly noisy. Consequently, we assume as in Figure 2.6 that the jammer hops the jammed band over Wss, slowly relative to the FH dwell time 1/Rh, but often enough to deny the FH system the opportunity to detect that it is being jammed in a specific portion of Wssand take remedial action.Also, to simplify the analysis, we will assume that shifts in the jammed band coincide with carrier hop transitions, so that the chan- nel is stationary over each hop. Furthermore, we will assume that on a given hop, each M-ary band lies entirely inside or outside WJ. This last restriction is common to most analytical treatments of partial-band FH jamming and the dual case of pulsed DS jamming. (Since the FH carrier hops in and out of WJ, partial-band jamming can be regarded as pulsed jamming with non- uniformly spaced pulses of duration 1/Rh, under the simplifying assumptions made above.) The performance computed based on these assumptions is actually somewhat pessimistic, as Viterbi has noted [6, p. 14]: on a given M-ary symbol transmission, if only part of the M-ary band is jammed, and/or if it is only jammed over part of the symbol band, less noise is inter- cepted by the energy detectors, thereby reducing the probability of error.
Eb NJœ rEb
NJ NJœ J
WJ
J rWss
NJ
r ,
476 Non-Coherent Frequency-Hopped Systems
Figure 2.6. Partial-band noise jamming of FH system: jammer concentrates power in fraction rH(0, 1] of SS bandwidth, and hops noise band to prevent FH band avoid- ance countermeasure.
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We should add that it does not matter whether the jammed band WJis a sin- gle contiguous region as suggested by Figure 2.6: the analysis below is trans- parent to partitions of WJso long as each of them satisfies the assumptions above.
Because of the pseudorandom hopping, it is reasonable to model the FH/MFSK system in partial-band noise as a two-state channel, independent from hop to hop. With probability r, an M-ary transmission is jammed and the conditional Pbis determined by the SNR ratio of (2.22); but, since we are neglecting thermal noise, with probability (1 r), the transmission is noiseless and an error-free decision is made. Then the average error rate is simply
(2.23) where the term on the right denotes the expression of (2.20b) with Eb/NJ replaced by rEb/NJ.
What (2.23) tells us is that if ris reduced, the probability that an M-ary transmission is jammed is decreased, but jammed signals suffer a higher con- ditional error rate: the net effect may degrade the average FH/MFSK per- formance, depending on the values of Mand Eb/NJ. The utility of jamming only part of the RF band is illustrated in Figure 2.7 for M2. Suppose S, J,Wss, and Rbcombine to make Eb/NJ10.9 dB. In broadband noise (r 1), the resulting Pb103.
If the jammer concentrates the same noise power over half the SS band (r 1/2), only half the transmissions are jammed, but these have a conditional
PbrPbarEb NJb,
Worst Case Jamming 477
Figure 2.7. Illustration of partial-band jamming advantage against FH/MFSK sys- tems. Referring to (2.20b) and (2.23), if M2 and Eb/NJ10.9 dB, when r1 then Pb103; however, if r1/2, then conditional SNR ratio in jammed band is 7.9 dB, so that conditional Pb2.2 102, resulting in average Pb1.1 102.
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Pb2.2 102, which results in an average Pb 1.1 102according to (2.23). So in this example, reducing rto 1/2 degrades the performance more than an order of magnitude because of the steepness of the Pbcurve in the selected region. (The results below indicate that the worst performance for these parameters occurs at r.16, for which Pb3.0 102.)
Figure 2.8 illustrates the performance of an FH/BFSK system in partial- band noise for several partial-band jamming factors r. For small enough Eb/NJ, it is evident that broadband noise jamming (r1) is the most effec- tive. In general, for any value of Eb/NJ, there is an optimum value of rH(0, 1] from the jammer’s viewpoint which maximizes Pb, and this is denoted by rwc (for worst case jamming). The performance in worst case partial-band noise is the upper envelope (or supremum) of the family of Pbcurves for fixed values of r: as shown in Figure 2.8, when Eb/NJexceeds a threshold level,rwc1 indicating a partial-band jamming advantage, and the perfor- mance curve is a straight line. Of course, in practice, it may be difficult for the jammer to match rto the actual Eb/NJ.
The worst case partial-band noise jammer chooses rto maximize the Pb for a given M and Eb/NJ. From (2.20b) and (2.23), the resulting average
478 Non-Coherent Frequency-Hopped Systems
Figure 2.8. Performance of FH/BFSK system in partial-band noise for several fixed values of r. The performance in worst case partial-band noise is realized when the jammer chooses rrwcto maximize Pbfor a given Eb/NJ. Note that rwcdecreases as Eb/NJgets larger.
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performance can be expressed as
(2.24) For M2, this maximation is a simple mathematical calculation; for larger values of M, it must be evaluated numerically. The results have the form [5, (15) and (16)]
(2.25)
where the parameters band gare tabulated for 1 KSbelow and rwc denotes the jammer’s optimum r.
(2.25) demonstrates that so long as Eb/NJis not unusually small, worst case partial-band jamming converts the exponential relationship between Pband Eb/NJin (2.20) into an inverse linear dependence. As shown in Figure 2.9, the resulting degradation can be severe for small Pb’s: for example, the loss is 14.7 dB at Pb103for K1, and increases with K. At Pb’s of 105and lower, this gap exceeds 30 dB for any K, illustrating the effectiveness of worst-case partial-band noise jamming against uncoded FH/MFSK signals at typical operating points.
(2.25) indicates that rwc becomes very small for large Eb/NJ’s; that is, a worst case noise jammer concentrates its power in a small portion of Wssat low Pb’s. The signals do not get jammed most of the time, but those that do are likely to result in errors. This is an indication that some form of coding redundancy that causes data decisions to depend on multiple symbol trans- missions can reduce the effectiveness of partial-band jamming; the degree to which this statement is true will become evident in Section 1.3.
Pbf 1 21M12a
M
i2112iaM
ibe1KEb>NJ2111>i2,
and rwc1; Eb NJg b
Eb>NJ , and rwc g Eb>NJ ; Eb
NJg Pb max
06r1c r
21M12 a
M
i2112iaM
i be1rKEb>NJ2111>i2d.
Worst Case Jamming 479
Table 2.1
Parameters associated with performance of uncoded FH/MFSK signals in worst case partial-band noise, as defined in (2.25).
K b g, dB
1 e1.3679 3.01
2 .2329 .76
3 .1954 .33
4 .1812 .59
5 .1759 1.41
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