GENERAL ANALYSIS OF ANTI-JAM COMMUNICATION SYSTEMS
4.8 FH/MFSK AND PARTIAL-BAND JAMMING
The previous section also applies to FH/MFSK with broadband noise jam- ming where N0⫽NJandEb/NJis given by (4.3). Here the case of partial- band noise jamming is examined to illustrate our general coded error bound evaluation.
Assume a jammer with power Jthat transmits Gaussian noise with con- stant power spectral density over a total bandwidth WJ. This jammed band- width is some subset of the total FH/MFSK signal bandwidth Wssand it is assumed that the transmitter and receiver do not know this jammed band- width before transmission. Indeed, the jammed subset of total bandwidth WJ may be changed randomly by the jammer to prevent the transmitter and receiver from knowing what frequencies will be jammed. The fraction of the jammed band is denoted
(4.82) Thus, defining NJ⫽J/Wss, the true noise power spectral density is
(4.83) forWJHz and zero for the rest of the spread bandwidth.
Assuming that an MFSK signal is transmitted during the time interval of Tsseconds, the hop time interval is denoted Thseconds, then
(4.84) wheredis a fraction or a positive integer. Thus, there are dMFSK symbols transmitted during each hop interval.3Assume that each hop is independent of other hops and equally likely to be in any part of the total spread-spec- trum signal frequency band of WssHz. Thus, the probability of transmitting an FH/MFSK symbol in a jammed frequency band is given by r.
Throughout this analysis, also assume that
(4.85) so that during any hop interval the whole set of Mpossible tones is either totally in the jammed band or not. This ignores the unlikely cases where the Mpossible tones straddle the edge of the jammed band leaving only some of the Mtone positions with jamming noise present. When (4.85) holds, this edge effect can be ignored.
M
Ts V WJ
Th⫽dTs
NJœ ⫽NJ
r ⫽ J WJ
r⫽ WJ
Wss
.
3Later in the text, we refer to the case where d⬎1 as slow frequency hopping (SFH) and dⱕ 1 as fast frequency hopping (FFH).
As always, ideal interleaving is assumed so that the coding channel is mem- oryless (see Part 1, Chapter 3, Section 3.8). In this FH/MFSK example, it means that each transmitted M-ary symbol is jammed with probability rand not jammed with probability 1 ⫺rindependent of other transmitted sym- bols. As before, define the jammer state random variable Zwhere
and
(4.86) This is the jammer state information that may be available at the receiver whereZ⫽1 indicates that the transmitted symbol hopped into the jammed band while Z⫽0 indicates that it hopped outside the jammed band.
With zavailable at the receiver, the metric we consider is
(4.87) which is a weighted version of the energy detector output corresponding to inputx.
For the above metric the parameter (4.18) has the form
(4.88) From (4.67), (4.68), and (4.69) we have
(4.89) for 0 ⬍2lc(1)⬍1 and
(4.90) Thus,
(4.91) When side information is available, the metric can be chosen with c(0) large enough so that the second term in (4.91) is negligible and c(1)⫽1/2 chosen for normalization. Then, the parameer Dbecomes
(4.92) D⫽ min
0ⱕlⱕ1e r
1⫺l2e⫺lr1Es>NJ2>11⫹l2f.
⫹ 11⫺r2e⫺2lc102r1Es>NJ2.
D1l2 ⫽ r
1⫺12lc112 22e⫺2lc112r1Es>NJ2>31⫹2lc1124 E5elc1023exˆ⫺ex40x,Z⫽06 0ˆx⫽x⫽e⫺2lc102r1Es>NJ2. E5elc1123exˆ⫺ex40x,Z⫽16 0x⫽xˆ ⫽ 1
1⫺ 12lc11222e⫺2lc112r1Es>NJ2>31⫹2lc1124
⫹ 11⫺r2E5elc1023exˆ⫺ex40Z⫽06 0xˆ⫽x.
⫽rE5elc1123exˆ⫺ex40x,Z⫽16 0ˆx⫽x
D1l2⫽E5elc1Z23exˆ⫺ex40x6 0xˆ⫽x
m1y,x;z2 ⫽c1z2ex
P5Z⫽06⫽1⫺r. P5Z⫽16⫽r
With no side information,c(1)⫽c(0) which can be normalized to 1/2 to get (4.93) Figures 4.15a and 4.15b show Dgiven by (4.92) and (4.93), respectively, for various values of r. Next, in Figures 4.16 through 4.18, the correspond- ing cutoff rates given by (4.27) for M⫽2, 4, and 8 are shown. The special case of r⫽1.0 is the broadband noise jamming case discussed in the previ- ous section.4There Dis given by (4.70) hich is the same as (4.92) and (4.93)
D⫽ min
0ⱕlⱕ1e r
1⫺l2e⫺lr1Es>NJ2>11⫹l2⫹ 11⫺r2e⫺lr1Es>NJ2f.
4The broadband noise jammer case is the same as having an additive white Gaussian noise chan- nel with N0⫽NJ.
Figure 4.15a. Parameter D for soft decision with jammer state knowledge—
FH/MFSK.
forr⫽1.0. Thus, in Figures 4.15 through 4.18, we can compare the broad- band noise jammer (r⫽1.0) with various partial-band noise jammers. For any code, the bit error bounds for all these cases can be directly compared using these figures.
As an example, suppose we assume the receiver has no jammer state knowledge and the jammer is a partial-band noise jammer which jams a frac- tion
(4.94) of the total spread-spectrum band. Assume the conventional non-coherent MFSK metric of (4.62).
With M ⫽ 8 and using the error bound on the Trumpis code given by (4.78), what is the Eb/NJgiven by (4.3) required to achieve 10⫺5bit error
r⫽.05
Figure 4.15b. Parameter Dfor soft decision with no jammer state knowledge—
FH/MFSK.
Figure 4.16a.Cutoff rate of FH/BFSK for soft decision with jammer state knowledge.
Figure 4.16b.Cutoff rate of FH/BFSK for soft decision with no jammer state knowledge.
Figure 4.17a.Cutoff rate of FH/4FSK for soft decision with jammer state knowledge.
Figure 4.17b.Cutoff rate of FH/4FSK for soft decision with no jammer state knowledge.
Figure 4.18a.Cutoff rate of FH/8FSK for soft decision with jammer state knowledge.
Figure 4.18b.Cutoff rate of FH/8FSK for soft decision with no jammer state knowledge.
probability? From the curve labelled Din Figure 4.12, the required Eb/N0 for the additive white Gaussian noise channel is
(4.95) Here, since,R⫽1,Es⫽Ebso this is also the value of Es/N0, the coded sym- bol energy-to-noise ratio. Figure 4.14 shows that the cutoff rate is
(4.96) This is also shown in the r⫽1.0 curve in Figure 4.18b since broadband noise jamming and the additive white Gaussian noise channels are the same. To achieve the same value of R0and, thus, the same 10⫺5bit error probability forr⫽0.05 we require
(4.97) This is determined from Figure 4.18b. By repeating this procedure for sev- eral values of the bit error probability, the bit error probability curve for the Trumpis code with an additive white Gaussian noise channel can be trans- lated into the corresponding bit error probability curve for a partial-band noise jammer with r⫽.05.