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

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 andW-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.


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 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 harddecision 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].

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  f is designed to be shorter than the channel coherence time  c .By controlling the subset size, the hopped frequencies are revisited within  c , 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 2 2  .For the  hops of a modulated symbol,   s are independent and identically distributed (i.i.d.).The BPSK modulated symbol is denoted by ,  = ±√ d with equal probability, where  d is the instant power of .The AWGN signal   is a zero mean complex Gaussian RV with variance 2 2  .The PBNJ signal   is also a zero mean complex Gaussian RV, with variance 2 2  .The jamming factor  PBNJ 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 2 2  and 2 2   , respectively, which both are independent of   .We have the following decomposition between ĝ and   [23]: where  c and  q are the second order moment between the real and imaginary part of ĝ and   , as where E{} is the expectation of , R() is the real part of , and I() is the imaginary part of .  and V  are i.i.d.zero mean Gaussian RVs, which are both independent of ĝ .The variance of , where  is the complex correlation coefficient between ĝ and   [23]: From ( 5), || =  =   / ĝ.Considering the similarity between PBNJ and AWGN, there is a similar decomposition between   and ĝ  , with   =   / ĝ and  2   =  2  (1 −  2 ).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

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  = 2.

BER for an Arbitrary 𝐿.
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 R( EGC ) < 0 when  = √ d is transmitted.Therefore, given  and the set G = { 1 , . . .,   , ĝ 1 , . . ., ĝ  , ĝ+1 , . . ., ĝ }, the conditional error probability is Using ( 1)-( 5), the decision statistic R( EGC ) is expanded as According to (10), given , , and G, R( EGC ) is conditional Gaussian distributed.Hence, the  EGC (, G) of ( 9) is calculated to be where var() is the variance of  and Q() is the Gaussian Q function calculated by Using ( 10) and ( 11), we simplify  EGC (, G) as where By defining  =  +   , the characteristic function (CHF) of  is given by where erfi() is the imaginary error function.
After averaging  EGC (, G) over the distribution of , we obtain the  EGC () as where the internal integration of ( 16) can be calculated in a closed form, as Then  EGC () is simplified to be Since   () involves the CHF of a Rayleigh sum, a closed form for  EGC () 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  EGC () to be a 1-tuple integration, which reduces the complexity of numerical calculation.

Adaptive Jamming Suppression Schemes
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 Q function is a monotone decreasing function; that is, minimizing  W-EGC is equivalent to maximizing  +   .By solving ( +   )/ = 0, the optimum  can be obtained as After substituting (30) into (28), we simplify  W-EGC as where  1 =   /( − ) and  2 =    /.Once again, we use the CHF method and calculate the  W-EGC as where Note that  is eliminated in (28) with  = 0 or  = , which indicates that  opt is used only for 1 ≤  ≤  − 1. Due to the Rayleigh sum involved, a closed-form expression for  W-EGC 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 lim From (37), it is indicated that, in a high SJR region, W-EGC will lower the error floor, which is determined by the SNR.

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.
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  b / 0 region.When  BER = 1 × 10 −3 , for example, W-EGC and SW-EGC show, respectively, 3.5 dB and 2 dB gain over EGC.In a high  b / 0 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 BER = 1 × 10 −3 .And this performance improvement increases with the increase of  b / 0 .Figure 2 shows the influence of the jamming factor  PBNJ 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
Note that the optimum weighting coefficient of (30) contains the instantaneous channel estimates.To achieve a simpler result, we replace the | ĝ | and | ĝ  | with the corresponding mathematical expectations √/2 ĝ and √/2 ĝ , respectively.The suboptimum weighting coefficient  Sopt is then calculated to be  Sopt =        .