Numerical simulations have been conducted to validate the performance of the proposed interference suppression scheme for a slow FH system. In this system, each hop has 4 MFSK symbols, the symbol rate is 200000 symbols per second, and the hop rate is 50000 hops per second. The frequency spacing is 100 kHz. The ratio of the unjammed interval to the hop duration, RU, is given by 0.025 for all except the last result (Figure 2.5). Channel gains of jamming and desired signals are complex
Gaussian random variables with variance values of 1. The jammer’s bandwidth is equal to the bandwidth occupied by the all M data tones in each hop.
Figure 2.1 shows the SER of the proposed scheme versus the signal-to-noise ratio (SNR) when the signal-to-jamming ratio (SJR) is -25dB and -40dB. BFSK modulati- on is used and the number of samples per symbol is N = 4. For comparison, the results of using the conventional beamformer [28] and the SMI-based beamformer are also plotted. As can be seen, the performance of the proposed scheme differs only slightly for the various SJRs used, which is highly desirable in military communications.
Also, unlike the conventional beamformer, no error floor exists for the proposed scheme. This is because the latter regards the jamming components as deterministic quantities to be estimated while the conventional beamformer simply treats the jamm- ing components as receiver noise. Furthermore, the proposed scheme is able to offer a better performance than the other methods since it is a ML-based approach.
However, in the unlikely event that αp =βp, as when both signal and jammer are from the same direction or there is no distinction between the signal and the jammer in terms of channel gains, all the algorithms will fail. In fact, since there is no distinc- tion between the signal and the jammer in terms of transmission characteristics and the jamming signal is unknown, it will not be possible for any statistical signal proce- ssing algorithm to reject the jamming signal. Similarly, when two jammers are present and both are unknown, it will not be possible for the proposed scheme, the SMI meth- od and other similar techniques to work properly. This is because the array is a two- element one and the presence of two jammers will give rise to an under-determined system where the number of unknown parameters is more than number of the degrees of freedom that the system has.
Figure 2.2 illustrates the performance of the proposed detection scheme under vari-
ous modulation levels. The SJR is -10 dB and the number of samples per symbol is
=4
N . As observed, the performance of the proposed scheme degrades as the modulation level increases.
Figure 2.3 investigates the performance of the proposed scheme as the number of samples per symbol is changed. BFSK modulation is used and SJR is -10 dB. It can be seen that the proposed scheme has a better performance as the number of samples per symbol is increased. The average conditional error probabilities of the proposed scheme are also plotted in Figure 2.3. The validity of the performance analysis for the proposed scheme is also demonstrated in Figure 2.3 from noting that the SER values from simulation are remarkably close to the corresponding analytical curve.
0 5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
SER
Conventional beamformer
SMI method Proposed approach
+ : SJR = -25 dB : SJR = -40 dB
Figure 2.1: Performance of the proposed approach under various SJRs with BFSK modulation and N = 4.
0 5 10 15 20 25 30 10-4
10-3 10-2 10-1 100
SNR(dB)
SER
Simulated SER
+ Theoretical SER SJR = -10dB BFSK N = 2
N = 4
N = 6 N = 8
Figure 2.3: Performance of the proposed scheme under various numbers of samples per symbol and the tightness of the theoretical and simulated SER values for BFSK signaling.
0 5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
SER
32-FSK 16-FSK 8-FSK 4-FSK BFSK SJR = -10dB
N = 4 samples/symbol
Figure 2.2: Performance of the proposed scheme under various modulation levels and N=4 samples/symbol.
The results from Figures 2.1, 2.2 and 2.3 have been obtained by assuming perfect channel estimation. To investigate the effect of imperfect channel estimation, Figure 2.4 shows the performance of the proposed scheme with imperfect knowledge of the desired signal’s channel gains, blindly estimated by using the ML technique (as desc- ribed in Appendix A) within the unjammed interval of a hop. Obviously, at SJR=- 10dB and using just 4 received samples in a very short unjammed interval of a hop to estimate the channel gains, the resulting SER performance in the case of imperfect channel estimation is very close to that in the case of perfect channel estimation.
Figure 2.5 investigates the timing of the jamming signal on the system performance.
The values of RU used for the three sets of results are 0.025, 0.25 and 0.5, and the results are obtained as follows. The dotted curves are obtained from using 10 samples of the received signals at the beginning of each hop in the ML approach (as described
0 5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
SER
Imperfect channel information Perfect channel information
SJR = -10dB
8-FSK, N = 4
BFSK, N = 8
BFSK, N = 4
Figure 2.4: Performance of the proposed scheme when the desired signal’s channel gains are blindly estimated by using the ML technique in Appendix A within the unjammed
interval of a hop.
in Appendix A) to estimate the desired signal’s channel response. Then, a simple bea- mforming structure is employed to place a null toward the desired signal (as described in Appendix B). Using the technique in [21], the onset of jamming can then be detect- ed by determining the time when a significant increase in the signal power at the beamformer’s output has occurred.
Based on the detected jammed or unjammed status of the system, detection of the jammed symbols are carried out by the proposed approach, while that for the unjamm- ed symbols are performed by using the conventional ML technique. The curves in Figure 2.5 denote the overall SER results, including the SER performance in both the jammed and unjammed portions of each hop.
As described, the dotted curves in Figure 2.5 are obtained with imperfect channel estimates. On the other hand, the solid curves are based on using the exact channel response of the desired signal. The minor performance degradation between the two
0 5 10 15 20 25 30
10-4 10-3 10-2 10-1 100
SNR(dB)
SER
SJR = -25 dB BFSK
N = 4 samples/symbol : Imperfect channel estimation : Perfect channel estimation
RU = 0.5
RU = 0.25 RU = 0.025
Figure 2.5: Performance of the proposed scheme with various unjammed intervals in a hop.
sets of curves again indicates that the new algorithm does not require very accurate channel information.
The effect of the timing of jamming signal can be studied in more detail by comparing the three sets of results in Figure 2.5, each for a different value of RU. Note that the lower the value of RU, the more jammed the hop will be. As can be seen, while an increase in the jamming duration will worsen the SER performance, the use of the new algorithm has the effect that such deterioration becomes rather insignificant.
Finally, Figure 2.6 examines the issue of jamming timing estimation. Specifically, the result is obtained from using the blind ML channel estimation algorithm given in Appendix A to estimate the channel gains of the desired signal, followed by impleme- nting the beamformer in Appendix B to reject the desired signal based on these estimated gains, and then using the algorithm in [21] to detect the onset of jamming.
0 5 10 15 20 25 30
10-6 10-5 10-4 10-3 10-2
SNR(dB)
Mean error of jamming timing estimate (hop duration)
4 samples used in blind channel estimation 10 samples used in blind channel estimation
SJR = -25dB Ru = 0.025 BFSK
Figure 2.6: Estimation of jamming timing.
The two curves in the figure show how the mean jamming timing estimate error, normalized with respect to the hop duration, changes as a function of SNR when 4 and 10 samples are used in the blind ML channel estimation procedure. As can be seen, using 10 received samples will give a more accurate timing estimation. Howev- er, this difference is rather insignificant, especially when the SNR is large. The reason is that we can obtain highly accurate timing estimation with a small number of used samples under high SNR regimes. Also, even with a small number of samples, accurate timing estimate can be quite readily performed under low SNR regimes.
It should also be noted that other mitigation techniques, such as channel coding and interleaving, could also be used for the anti-jamming purpose. In fact, channel coding and interleaving are effective to intermittent jamming, such as a pulsed noise or a partial band jammer. However, even with channel coding and interleaving, the performance of FHSS systems will still deteriorate significantly in the presence of a follower jammer which is on most of the time. On the other hand, the proposed algorithm is able to suppress such a jammer. On the issue of complexity, the proposed algorithm operates only at the receiver and, as discussed in Section 2.3, the implementation complexity is low. Comparatively, channel coding and interleaving techniques need to be used at both the transmitter and receiver, while interleaving will increase delay. Nevertheless, to further enhance performance, an appropriate channel coding and interleaving scheme may be used on top of the proposed algorithm.