UNCODED FREQUENCY-HOPPED BINARY

Taking a basic modulation technique and changing the carrier frequency in some pseudorandom manner is the frequency-hopping approach to gener- ating a spread-spectrum signal. The most common modulations used with frequency hoping are the M-ary frequency-shift-keying (MFSK) modula- tions together with non-coherent reception. This section illustrates some

Figure 3.20. Repeat code m⫽3 with known jammer state/hard decision.

additional basic concepts with the frequency-hopped binary frequency- shift-keying (FH/BFSK) spread-spectrum signals.

Ordinary BFSK signals have the form

(3.71) HereTbis the data bit time and {dn} are the independent data bits where

(3.72) Typically we choose

(3.73)

¢␻Tb⫽p

dn⫽ e 1, with probability 12

⫺1, with probability 12. nTbⱕt 6 1n⫹12Tb, n⫽integer.

s1t2 ⫽ 22S sin3␻0t⫹dn¢␻t4;

Figure 3.21. Repeat code m⫽5 with known jammer state/hard decision.

so that the two possible transmitter tones are orthogonal for all relative phase shifts over the Tbsecond interval.

Frequency hopping of this BFSK signal is done with a pseudorandom binary sequence that is used to select a set of carrier frequency shifts result- ing in the frequency-hopped signal

(3.74) wherevnis the particular hop frequency chosen for the n-th transmission interval. Generally, if Lpseudorandom binary symbols are used to select a frequency shift each Tbseconds, then there are at most 2Ldistinct frequency shift values possible. The range of values taken by these frequency shifts defines the total spread-spectrum signal bandwidth Wss. Although this total

nTbⱕt 6 1n⫹12Tb, n⫽integer x1t2⫽ 22S sin3␻0t⫹ ␻nt⫹dn¢␻t4;

Figure 3.22. Repeat code m⫽9 with known jammer state/hard decision.

spread bandwidth need not be contiguous, it is assumed here that this band is a contiguous frequency range.

Figure 3.23 illustrates the basic uncoded FH/BFSK system. For simplic- ity, assume the receiver’s PN sequence generator is synchronized with that of the transmitter and, thus, frequency dehop at the receiver removes the effects of the pseudorandom frequency shifts. A conventional non-coherent BFSK receiver follows the frequency dehop. Essentially, the transmitted sig- nal is a conventional BFSK signal that has a shifting carrier frequency and the receiver has a conventional BFSK receiver that merely shifts its center frequency together with that of the transmitter.

The outputs of the energy detectors in Figure 3.23 are denoted e⫹ande⫺. If there were no jamming signal present and if d⫽1 were transmitted, these outputs would be e⫺⫽0 and e⫹⫽STb, the BFSK pulse energy. In general, the non-coherent decision rule based on the additive white Gaussian noise channel [8] is

(3.75) During any Tbsecond interval, the transmitted signal is a tone of dura- tionTbseconds and has a (sin2x)/x2spectrum of bandwidth roughly 2/Tbcen- tered at frequency v0⫹vn. The transmitted signal would then be one of two possible tones separated in frequency by 2⌬v. This “instantaneous band- width” is generally a small fraction of the total spread-spectrum signal bandwidthWss, which is primarily determined by the range of frequency shift values generated by the frequency hopping.

For each Tbsecond interval, the particular bit error probability is deter- mined by the amount of jammer power in the “instantaneous bandwidth”

of the signal that contributes to the energy terms e⫹ande⫺. The overall bit error probability is then the average of these particular bit error probabili- ties where the average is taken over all frequency-hopped shifts.

3.6.1 Constant Power Broadband Noise Jammer

Assume that the jammer transmits broadband noise over the total spread- spectrum band with constant power J. Thus, dring any Tbsecond interval, regardless of the carrier frequency shift, there will be an equivalent white Gaussian noise process in the “instantaneous bandwidth” of the transmit- ted signal. The one-sided noise spectral density is NJ⫽J/Wss.

Since an equivalent white Gaussian noise process is encountered in all parts of the total spread-spectrum band, the bit error probability for the uncoded FH/BFSK system of Figure 3.23 is the same as that for conventional BFSK in white Gaussian noise, namely

(3.76) whereEb/NJis still given by (3.39). This is the baseline performance of the FH/BFSK system.

Pb⫽12e⫺1Eb>2NJ2 dˆ ⫹ ⫽ e 1, e⫹ 7 e⫺

⫺1, e⫹ⱕe⫺.

Figure 3.23.Uncoded FH/BFSK system.

3.6.2 Partial-Band Noise Jammer

Next, consider the impact of partial-band noise jamming where the jammer transmits noise over a fraction rof the total spread-spectrum signal band.

Denoting the jammed frequency band by WJ, then ris given by (3.10) and in the jammed part of the band, the equivalent single-sided noise spectral density is given by (3.11).

Assume that WJ is large compared to the bandwidth of the unhopped BFSK signal and the effects of the signal hopping onto the edge of this band are negligible. That is, ignore the possibility that when a signal is sent it is frequency-hopped to the edge where only part of the instantaneous band of the signal is jammed. This assumes either a signal is hopped into the jammed band or not. In addition, the jammer is allowed to change the band it is jam- ming and so the transmitter and receiver never known a prioriwhich fre- quency range is being jammed.

We again introduce the jammer state parameter Zfor each Tbinterval where now

(3.77) with probability distribution as in (3.41). The bit error probability is then given by

(3.78) where there are no errors when the signal hops out of the jammed band.

Figure 3.24 illustrates the bit error probability for various values of r. The value of rthat maximizes Pbis easily obtained by differentiation and found to be

(3.79) This yields the maximum value of Pbgiven by

(3.80) Figure 3.25 shows this worst case value of the bit error probability. Here at 10⫺6bit error probability there is a 40 dB difference between broadband

Pb⫽ • e⫺1

Eb>NJ , Eb>NJ 7 2

1

2e⫺1Eb>2NJ2, Eb>NJⱕ2.

r*⫽ • 2 Eb>NJ

, Eb>NJ 7 2

1, Eb>NJⱕ2.

⫽ r

2e⫺r1Eb>2NJ2

⫹Pr5e⫹ 7 e⫺0d⫽ ⫺1,Z⫽06Pr5Z⫽06

⫽Pr5e⫹ 7 e⫺0d⫽ ⫺1,Z⫽16Pr5Z⫽16 Pb⫽Pr5e⫹ 7 e⫺0d⫽ ⫺16

Z⫽ e1, signal in jammed band 0, signal not in jammed band

noise jamming and the worst case partial-band noise jamming for the same jammer power.

The partial-band noise jammer effect on the uncoded FH/BFSK system is analogous to the pulse noise jammer effect on the uncoded DS/BPSK sys- tem of Section 3.4. In both systems, these jammers cause considerable degra- dation by concentrating more jammer power on a fraction of the transmitted uncoded symbols. This potentially large degradation is explained by the fact that the uncoded bit error probability varies dramatically with small changes in the effective bit energy-to-jammer noise ratio,Eb/NJ. Thus, the jammer can cause high error probabilities for a fraction of the transmitted bits resulting in a high average bit error probability.

For the uncoded FH/BFSK system, pulse noise jamming and partial-band noise jamming have the same effect on performance. These are essentially equivalent ways of concentrating more jammer power on some fraction of

Figure 3.24. FH/BFSK—Partial-band noise jammer.

the uncoded transmitted symbols. Using pulse noise jamming or a combi- nation of pulse noise and partial-band noise jamming would give the same results as we found for partial-band noise alone.

3.6.3 Multitone Jammer

Recall that each signal tone of Tbsecond duration has one-sided first null bandwidth 1/Tb. For the total spread-spectrum signal bandwidth Wss, there are

possible orthogonal tone positions. Each FH/BFSK signal would then use an adjacent pair of these tone positions to transmit one data bit. The pair of

N⫽WssTb

Figure 3.25. FH/BFSK—Against jammers.

tone positions selected at any data bit time is determined by the PN sequence generator.

Consider a jammer that transmits many tones each of energy SJTb. With total power Jthere are at most

(3.81) jammer tones randomly scattered across the band. The probability that any given signal tone position is jammed with a jammer tone is, thus,

(3.82) Hereris also a fraction of the signal tone positions that are jammed.

Assume that the jammer has exact knowledge of the Npossible signal tone positions and places the Ntjamming tones in some subset of these Nposi- tions, where Nt,Nis always assumed.

During the transmission of a data bit, one of two possible adjacent tone positions is used by the transmitter. An error occurs if the detected energy is the alternate tone position not containing the transmitted signal tone is larger than the detected energy in the transmitted tone position. This can occur only if a jammer tone occurs in this alternative tone position. Here, ignore the smaller probability of a jammer tone in both positions and assume an error occurs if and only if a jammer tone with power SJⱖSoccurs in the alternative tone position. Thus, the probability of a bit error is

(3.83) providedSJ⬎S. From the communicator’s standpoint, the worst choice of SJisSJ⫽S resulting in the maximum bit error probability

(3.84) This bit error probability is slightly larger than the worst partial-band noise jammer performance; the results are essentially the same. Figure 3.25 shows the bit error probabilities for broadband noise jamming, worst partial-band noise jamming, and worst multitone jamming. Part 2, Chapter 2, will exam- ine these cases in greater detail.

⫽ 1 Eb>NJ . Pb*⫽ J

SWssTb

Pb⫽r⫽ J SJWssTb

⫽ J SJWssTb . r⫽ Nt

N Nt⫽ J

SJ