Journal of Electrical and Computer Engineering
Volume 2015 (2015), Article ID 687874, 8 pages
http://dx.doi.org/10.1155/2015/687874

Research Article

Adaptive Jamming Suppression in Coherent FFH System Using Weighted Equal Gain Combining Receiver over Fading Channels with Imperfect CSI

Yishan He, Yufan Cheng, Gang Wu, Binhong Dong, and Shaoqian Li

National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 611731, China

Received 2 December 2014; Revised 4 February 2015; Accepted 9 March 2015

Academic Editor: Tho Le-Ngoc

Copyright © 2015 Yishan He et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Fast frequency hopping (FFH) is commonly used as an antijamming communication method. In this paper, we propose efficient adaptive jamming suppression schemes for binary phase shift keying (BPSK) based coherent FFH system, namely, weighted equal gain combining (W-EGC) with the optimum and suboptimum weighting coefficient. We analyze the bit error ratio (BER) of EGC and W-EGC receivers with partial band noise jamming (PBNJ), frequency selective Rayleigh fading, and channel estimation errors. Particularly, closed-form BER expressions are presented with diversity order two. Our analysis is verified by simulations. It is shown that W-EGC receivers significantly outperform EGC. As compared to the maximum likelihood (ML) receiver in conventional noncoherent frequency shift keying (FSK) based FFH, coherent FFH/BPSK W-EGC receivers also show significant advantages in terms of BER. Moreover, W-EGC receivers greatly reduce the hostile jammers’ jamming efficiency.

1. Introduction

As a powerful antijamming method, fast frequency hopping (FFH) is widely used in military applications. FFH employs a number of advantages including capability of antijamming, robustness against multipath fading, and low probability of interception [1, 2].

In the presence of jamming, various diversity combining schemes have been proposed for noncoherent frequency shift keying (FSK) based FFH, including maximum likelihood (ML) combining [1–7], FFT based combining schemes [8], linear combining (LC) [9], self-normalization combining [10], noise-normalization combining [11], product combining [12–17], and clipped combining [18, 19]. Among the noncoherent FFH/FSK combining schemes, ML combining yields the best BER performance in the presence of jamming.

In spite of the low complexity in implementation, noncoherent FFH systems have inevitable shortcomings, for example, performance loss due to noncoherent diversity combining. With the growing demand of better performance in antijamming communications, coherent phase shift keying (PSK) based FFH system draws much attention. As indicated in [20] and the references therein, coherent reception has been made feasible by maintaining a continuous phase at the transmitter from hop to hop. Kang and Teh [20] studied the bit error ratio (BER) of coherent FFH/BPSK with partial band noise jamming (PBNJ) and AWGN channel. The authors considered coherent ML combining, LC combining, and hard-decision majority-vote combining, which significantly outperform various noncoherent FFH/FSK diversity combining schemes in terms of BER. However, the fading channels were not considered in [20]. In the presence of fading channels, we have proposed a novel FFH scheme [21], which enables reliable channel estimation for FFH signals. And we extended the study of [20] to the Rayleigh fading channels with imperfect channel state information (CSI) [22], where we analyzed the BER of FFH/BPSK with maximum ratio combining (MRC) and equal gain combining (EGC). It is illustrated that the two combining schemes have a close BER performance in the presence of PBNJ. However, the jamming suppression was not addressed in [21, 22].

This paper addresses the jamming suppression problem with coherent FFH/BPSK. In analysis, we consider PBNJ and frequency selective Rayleigh fading channels with imperfect CSI. Based on the studies of the EGC receiver [22], we give a further simplification on the BER expression. Then we propose adaptive jamming suppression schemes, namely, weighted EGC (W-EGC) with the optimum and suboptimum weighting coefficient, where the analytical BER expressions are also derived. Particularly, with diversity order , we work out closed-form BER expressions for the EGC and W-EGC. The theoretical results are validated by simulations. It is shown that the W-EGC receivers significantly outperform the noncoherent FFH/FSK ML receiver in terms of BER. It is also shown that with the increase of signal to jamming ratio (SJR), as compared with EGC, the optimum W-EGC lowers the error floor which is determined by the signal to noise ratio (SNR). Besides, W-EGC receivers reduce the hostile jammer’s efficiency, by forcing the jammer to take full-band jamming to achieve the worst case jamming.

2. System Model

To guarantee reliable channel estimation, the so-called subset-based coherent FFH scheme [21] is adopted, where we partition the original hopping frequency set into a number of smaller subsets and choose only one of the frequency subsets as the hopping frequency set within a frame. The frame length is designed to be shorter than the channel coherence time . By controlling the subset size, the hopped frequencies are revisited within , which makes channel estimation feasible.

In this paper, perfect synchronization and multipath fading channels are assumed. With a hopping rate sufficiently fast, the current hop received from the second path usually falls into a posterior hop. After dehopping and filtering, only the signal from the first path will be received. Note that each modulated symbol is -fold hopped and the th equivalent baseband-form received signal is given by

where denotes the received signal which is contaminated by PBNJ. The Rayleigh fading channel coefficient is a zero mean complex Gaussian random variable (RV) with variance . For the hops of a modulated symbol, s are independent and identically distributed (i.i.d.). The BPSK modulated symbol is denoted by , with equal probability, where is the instant power of . The AWGN signal is a zero mean complex Gaussian RV with variance . The PBNJ signal is also a zero mean complex Gaussian RV, with variance . The jamming factor is defined as the ratio of the jamming bandwidth to the entire hopping bandwidth, which is also the probability of a hop contaminated by PBNJ. Within a frame, if a hopped frequency is disturbed by PBNJ, we assume that any hop with frequency will be jammed.

Similar to [21, 22], the channel estimate is assumed to be disturbed by Gaussian errors, as

where the estimation errors and are zero mean complex Gaussian RVs with variances and , respectively, which both are independent of . We have the following decomposition between and [23]:

where and are the second order moment between the real and imaginary part of and , as

where is the expectation of , is the real part of , and is the imaginary part of . and are i.i.d. zero mean Gaussian RVs, which are both independent of . The variance of or is , where is the complex correlation coefficient between and [23]:

From (5), . Considering the similarity between PBNJ and AWGN, there is a similar decomposition between and , with and . For each single hop, we define the average SNR and the signal to jamming plus noise ratio (SJNR) as

Considering the influence of channels estimation error, we further define the effective SNR and the effective SJNR as

3. Performance Analysis of EGC Receiver

In this section, we first derive the BER of FFH/BPSK with EGC receiver, which further simplifies the results obtained in [22]. Then we calculate a closed-form BER expression for the case with .

3.1. BER for an Arbitrary

In the presence of PBNJ, the EGC output is

where is the number of jammed hops of a symbol.

With the BPSK constellation, the decision statistic is the real part of the combining output. Error occurs with when is transmitted. Therefore, given and the set , the conditional error probability is

Using (1)–(5), the decision statistic is expanded as

According to (10), given , , and , is conditional Gaussian distributed. Hence, the of (9) is calculated to be

where is the variance of and is the Gaussian function calculated by

Using (10) and (11), we simplify as

where

By defining , the characteristic function (CHF) of is given by

where is the imaginary error function.

After averaging over the distribution of , we obtain the as

where the internal integration of (16) can be calculated in a closed form, as

Then is simplified to be

Since involves the CHF of a Rayleigh sum, a closed form for with an arbitrary has not yet been available so far as we know [24]. Compared with the quad-slope integration given by [22], we simplify to be a 1-tuple integration, which reduces the complexity of numerical calculation.

Finally, the average error probability of EGC receiver is

3.2. Closed-Form BER Expression with

In the special case with , we work out a closed-form expression for . When , is given by [25]

Similarly, is calculated to be

For the case with , we have , , and . Then is calculated by

which can be further expanded with (12), as

We first solve the internal integral with , as

Then we solve the internal integral with , as

Finally, we calculate the internal integral with , as

By substituting (20), (21), and (26) into (19), we obtain the closed-form BER expression for .

4. Adaptive Jamming Suppression Schemes

4.1. W-EGC Receiver

In the W-EGC receiver, the jammed and unjammed received signal are, respectively, sent into an EGC receiver. Then the two EGC outputs are weighted and combined, with final output as

where is the weighting coefficient. In the following analysis, we will optimize to minimize the BER.

Similar to (13), the conditional error probability of the W-EGC receiver can be written as

where

Note that the Gaussian function is a monotone decreasing function; that is, minimizing is equivalent to maximizing . By solving , the optimum can be obtained as

After substituting (30) into (28), we simplify as

where and . Once again, we use the CHF method and calculate the as

where

Note that is eliminated in (28) with or , which indicates that is used only for . Due to the Rayleigh sum involved, a closed-form expression for is not available for an arbitrary . Similar to (19), the BER for W-EGC is

We would like to compare the BER between EGC and W-EGC. Due to the monotonicity of the Gaussian Q function, we only need to compare the internal fraction of (13) and (31), which is calculated to be

Take the expectation of with regard to and , as

From (36), it is shown that, as compared with EGC, W-EGC has a lower conditional error probability. Besides, with the increase of SJR, we have

From (37), it is indicated that, in a high SJR region, W-EGC will lower the error floor, which is determined by the SNR.

In the special case with , has a closed-form expression. For and , we have . For , is calculated to be

4.2. Suboptimum W-EGC Receiver

Note that the optimum weighting coefficient of (30) contains the instantaneous channel estimates. To achieve a simpler result, we replace the and with the corresponding mathematical expectations and , respectively. The suboptimum weighting coefficient is then calculated to be

The conditional error probability of the suboptimum W-EGC (SW-EGC) receiver is calculated to be

where

Since and are in similar form, can be simply obtained by substituting (41) into (18). Similarly, the closed-form expressions for with are given by and

Similar to (35), the comparison between the SW-EGC and EGC can be denoted as . In the high SJR region, we have

For a moderate large SNR, that is, , we can see

From (44), it is indicated that, with a moderate large SNR, the SW-EGC and EGC receivers approach exactly the same error floor.

5. Numerical Results and Discussions

In this section, we present some numerical results and corresponding discussions. In simulation, the channel coefficient for each frequency and the location of jammed bandwidth are assumed to be unchanged within a frame. The simulation parameters are given in Table 1.

Table 1: Simulation parameters.

Type
Value

Frequency subset size
15

Frame length
0.9 ms

Hopping rate
 hops/s

Number of pilot groups
2

Pilot power

Fading type
Frequency selective Rayleigh fading

1

Jamming type
PBNJ

Channel estimation method
Pilot-based LS estimation

In Figure 1, the analytical results match the simulation very well. Both W-EGC and SW-EGC outperform EGC in terms of BER, especially in the low region. When , for example, W-EGC and SW-EGC show, respectively, 3.5 dB and 2 dB gain over EGC. In a high region, W-EGC shows a lower error floor than that of EGC, while the SW-EGC receiver approaches the same error floor, which have been explained in Section 4. As compared to the noncoherent FFH/BFSK with ML receiver, which is the optimum receiver for the noncoherent FFH in the presence of jamming, coherent FFH/BPSK with W-EGC and SW-EGC receivers shows performance gain even with imperfect CSI. As seen in Figure 1, the performance gain is 2.5 dB and 1 dB, respectively, when . And this performance improvement increases with the increase of .

Figure 1: BER comparison, with dB, , and .

Figure 2 shows the influence of the jamming factor on the performance of EGC, W-EGC, and SW-EGC, respectively. For the EGC receiver, there is a worst case jamming factor, which is less than 1. With the worst case jamming factor, the hostile jammer achieves the worst case jamming effect by jamming only a small fraction of the bandwidth. In contrast, as seen in Figure 2, the worst case jamming factor for the W-EGC and SW-EGC receiver is 1. Then the jammer has to take full-band jamming to achieve the worst case jamming, whose jamming effectiveness is greatly reduced.

Figure 2: BER of FFH/BPSK with W-EGC and EGC versus for and 4, dB, and dB.

6. Conclusion

We have proposed jamming suppression schemes for coherent FFH/BPSK system, which is based on the weighted equal gain combining. And we analyzed the BER performance in the presence of PBNJ, frequency selective Rayleigh fading channel, and imperfect CSI. From theoretical analysis and simulation validation, it is shown that the proposed schemes significantly outperform EGC and noncoherent ML receiver in terms of BER. Besides, the proposed schemes greatly reduce the jammer’s efficiency, where the jammer has to implement full-band jamming to achieve the worst case jamming.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported in part by Natural Science Foundation of China (NSFC) Grant no. 61201126, Ministry Sponsored Project 4010103020201-2, Program for New Century Excellent Talents in University (no. NCET-11-0058), and Program for Sichuan Youth Science and Technology Fund (no. 2012JQ0020).

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