On the Use of FFH-MFSK for Ultra-Low Power Communications in a Jamming Environment

We are interested in a communication system that operates in the presence of an intelligent jammer, under stringent power constraints, but with flexible bandwidth constraints. We optimize some of the key elements in the transceiver design for low power consumption, and thus high complexity components of the system, such as matched filters (MF), forward error correction (FEC) that employs iterative decoders, coherent demodulators, and bandwidth-efficient modulation formats, are not feasible for this research. Rather, our system is designed using M-ary frequency shift keying (MFSK) with non-coherent detection and fast frequency hopping (FFH), optimized two-pole bandpass filters (BPF), and Reed-Solomon (RS) codes with hard-decision decoding. Among other things, we show that by properly optimizing the key parameters of the BPFs and RS codes, we can design the system to be significantly less complex than the MF system with a performance loss of less than 1.4 dB for most scenarios that we considered. Further, the 2-pole BPF system can actually outperform the corresponding MF system by up to 2.4 dB in the presence of multi-tone jamming.


I. INTRODUCTION
Frequency-hopping spread-spectrum (FHSS) frequency shift keying (FSK) is widely used in military communication systems because of its anti-jamming capability. In particular, fast FH with M -ary FSK (FFH/MFSK) is a typical non-coherent communication scheme with the potential for applications in both military and civilian communication systems [1], [2], [3], [4]. Among the intelligent jamming strategies are partial-band noise jamming (PBJ) and multi-tone jamming (MTJ).
Attempts have been made to study and combat various intelligent jammers and interference in different channel conditions with appropriate signal selection and error-correction coding [4], [5], [6], [7], [8], [9], [10], [11], [12]. The combined effects of diversity and coding to combat MTJ in a Rayleigh fading channel are studied in [4]; the performance The associate editor coordinating the review of this manuscript and approving it for publication was Xiaofan He . of an optimal maximum likelihood (ML) receiver in PBJ and frequency-selective Rician fading channels is derived in [5]; the composite effect of MTJ and PBJ in a Rayleigh fading channel with time and frequency offsets is analyzed in [6]; the performances of an FFH/MFSK system with various receivers under MTJ are compared in [7] and [8]. The performance of an FFH/BFSK system with a suboptimal ML receiver under MTJ in frequency-selective Rayleigh fading channels is studied in [9]. The performance of an FFH/MFSK system with product combining under PBJ in Rayleigh fading channels is analyzed in [10]. The use of RS codes to combat PBJ and MTJ in a slow FH system is studied in [11] and [12]. The use of index modulation-based FHSS to combat reactive symbol-level jammer is proposed in [13]. The design of waveforms to mitigate the effect of single tone jamming signals in time hopping SS systems is studied in [14]. The effect of phase noise in the frequency synthesizer on an FFH/MFSK system is analyzed in [15]. VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ In [16], we addressed the design and performance analysis of an ultra-low power communications system. As a concrete example of constraints that ultra-low power consumption impose on system design, such as in [16], consider a scenario where it is required to communicate over a range of at least 1km, with a data rate of at least 100kb/sec, and an uncoded bit error rate (BER) of 10 −3 or less, in an AWGN channel, subject to a total power constraint of 1mW. This constraint includes both transmitter and receiver power consumption, as well as the power consumption required for transmission [17]. If an attempt is made to implement this system with a coherent receiver, a phase-lock loop (PLL) is needed to achieve phase and frequency synchronization. However, it is shown in [18] that the minimum power consumption of a well-designed PLL is on the order of 0.55mW, i.e., the PLL alone consumes more than half of the total power-consumption budget. Thus, in [16], power-hungry elements, such as matched filters, were replaced with twopole BPFs, soft-decision iterative decoders were replaced by hard-decision decoders, etc. What we found was that if these ad-hoc components were locally optimized, significant reductions in power consumption could be achieved with relatively small increase in BER.
However, what was missing from [16] was the presence of intelligent jammers. Therefore, in this paper, we add the presence of a partial-band jammer, which maximizes the performance degradation by optimizing the fraction of total spread spectrum bandwidth jammed. We also add the presence of a multi-tone jammer, which maximizes the performance degradation by optimizing the amount of power in each jammed slot. Further, in order to more effectively combat the intelligent jammer, we revise our system design to include FFH. After analyzing the performance of the MF/2pole BPF system in different channel and jamming conditions, we concluded that under full-band jamming, the performance of the 2-pole system was mostly 1.0-1.4 dB worse than the corresponding MF system, in both fading and non-fading channels. Also, under PBJ, the performance degradation was between 0.8 and 0.9 dB, in both fading and non-fading channels. Note that the results for both full-band jamming and PBJ were consistent with what we found in [16]. Under MTJ, the 2-pole system was no worse than the corresponding MF system, and could actually outperform the corresponding MF system by more than 2 dB in a non-fading channel, due to the constructive use of ISI, whereas in a Rician fading channel, the two system performances are equivalent.
The rest of this paper is organized as follows. In Section II, we analyze the performance of an FFH-MFSK receiver in full-band noise jamming, whereby we compare matched filter and 2-pole BPF detection, in both a non-fading channel, and a slow, flat, Rician fading channel. We also improve the performance in fading channels with RS coding and/or diversity combining. In Section III, we compare the performance of an FFH-MFSK receiver under PBJ. For the matched filter case, we find the worst-case performance, and for the 2-pole BPF case, we find the Nash Equilibrium (NE) [19]  at a given probability of error, i.e., no player can be better off by changing its strategy. In Section IV, we compare the performance of an FFH-MFSK receiver under MTJ. Similar to the PBJ case, we find the worst-case performance with matched filter detection, and we find the parameters and corresponding performance at the NE with 2-pole BPF detection. In Section V, we present the numerical results.

II. DEMODULATOR PERFORMANCE IN FULL-BAND NOISE JAMMING
In this section, we analyze the performance of an FFH-MFSK non-coherent demodulator in full-band noise jamming. The analysis in this section serves as a basis so that we can easily generalize it to partial-band jamming in later sections. The transmitted signal is given by where A 0 is the amplitude of the transmitted signal, T is the symbol duration, so that the symbol energy is given by E s = A 2 T /2, bit energy is given by E b = E s / log 2 M , and P a (x) ≜ 1 for |x| ⩽ a and 0 elsewhere. Also, the random phase of the m th symbol, θ m ∼ U [0, 2π], and f (m) ∈ {f 1 , f 2 , . . . , f M } is the frequency of the m th symbol. The block diagram of an FFH-MFSK non-coherent demodulator is shown in Fig. 1 [20], [21], [22], [23], [24], where the received dehopped waveform, r(t), is given by In (2), n J (t) and n w (t) are additive Gaussian noise of the jammer and thermal noise, with single-sided power spectral density N J and N 0 , respectively, T c is the hop duration, L ⩾ 1 is the number of hops/symbol, and (·) 2 denotes square law detection. The dehopped FSK signal s(t) is given by where A is the amplitude of the received signal, f (l) is the frequency of the l th hop, and the random phase of the l th hop, 2π]. Note that here the L hops of a symbol have the same frequency prior to hopping and after dehopping, i.e., f (1) = · · · = f (L) , f (L+1) = · · · = f (2L) and so on, but they do not need to have the same frequency, as we will see later, when we use chip-interleaving. As in the previous section, the filters are either matched filters or 2-pole BPFs. The transfer function and impulse response of the i th 2-pole BPF are given by respectively, for i = 1, 2, . . . , M , where W is the filter bandwidth and f ≜ f 2 − f 1 = 1/T c is the tone spacing, i.e., we use the minimum tone spacing that satisfies orthogonal signaling, since the optimal performance we found in [16] was almost always achieved with this tone spacing. In (4), we assumed f 1 ≫ W when we derived the impulse response from the transfer function. We optimize the time-bandwidth parameter z ≜ WT c to minimize the E b /N J required to achieve a predetermined BER for the 2-pole BPF system, and we compare the results to the corresponding matched filter system. Lastly, we ignore thermal noise in most of the following analysis, which is consistent with the typically dominant jamming power, as is done in references such as [20], [25], and [26]. However, we will come back to show how insignificant the thermal noise is when we consider PBJ in Section III.

A. MATCHED FILTER DETECTION
The symbol error rate (SER) of an FFH-MFSK-MF system in full-band noise jamming, P s,MF , is given in [20]. The numerical results for M = 16 and L = 2 are presented in [27]. In a slow, flat, Rician fading channel, the symbol amplitude R is Rician distributed with the well known pdf If we let γ denote the instantaneous E s /N J , andγ denote the averageĒ s /N J with Rician fading, then the pdf of γ is given by [28] where K is the Rician K-factor defined as K ≜ A 2 2σ 2 . The FFH-MFSK-MF system performance in full-band noise jamming and slow, flat, Rician fading is given by Equation (5) of [27]. Numerical results for M = 16 and L = 2 with both no fading and Rician (K = 10) fading, are presented Section IV of [27].

B. TWO-POLE BPF DETECTION
One problem with 2-pole BPF detection is that there exists both inter-carrier interference (ICI) and inter-symbol interference (ISI), where ISI primarily comes from the previous pulse [16]. Let Ps m in denote the probability that the current symbol is at frequency f i but is incorrectly detected as f n , n ̸ = i, and where the previous symbol is at frequency f m . With square-law combining of the output signals from the corresponding 2-pole BPFs for the L hops, the output of either branch i or branch n is the sum of 2L correlated non-central chi-square random variables (rv). We employ the union bound and thus do pairwise comparison. The test statistic is a linear combination of 4L correlated non-central chi-square rv's, i.e., we are interested in the quadratic form where X, defined in Appendix I, has a multi-normal distribution with mean vector µ = E[X] and covariance matrix = Cov(X). The weighting matrix A is given by where I 2L and 0 2L denote the 2L × 2L identity matrix and zero matrix, respectively. Detailed analysis on computing µ and can be found in Appendix I. It turns out that, as shown in [29], Q(X) can be represented as a linear combination of 4L independent non-central chi-square rv's: where λ 1 , . . . , λ 4L are eigenvalues of Then the characteristic function of the random variable Q(X) can be found as the product of the individual characteristic functions, and Ps m in = Prob(Q(X) < 0) can be evaluated numerically.
The union bound on the average symbol error rate is, by symmetry, In a Rician fading channel, the union bound on performance is given byP where f (γ ) was defined in (6).
The optimal time-bandwidth parameter, z, and the corresponding performance of our ad-hoc receiver in comparison with the matched filter system performance for various 2-pole BPF systems, are listed in Table 1 of [27], and the numerical results for M = 16 and L = 2 with both no fading and Rician (K = 10) fading are presented in Section VI of [27].

C. DEMODULATOR PERFORMANCE WITH DIVERSITY AND/OR RS CODING
Diversity can be achieved by interleaving the hops and thus improving the performance in a slow, flat, Rician fading channel. Detailed discussion including the optimal parameters and performance can be found in Section II of [27] and detailed analysis is shown in Appendix II of this paper. Alternatively, we can use FEC to improve performance. Detailed analysis including the optimal parameters (time-bandwidth product and code dimension) and performance can be found in Section II of [27].

III. DEMODULATOR PERFORMANCE IN PARTIAL-BAND NOISE JAMMING
The previous section focused on full-band noise jamming, where the jammer jams the entire spectrum. In reality, the jammer may jam a fraction of the spectrum to degrade system performance. In this section, we consider an FFH-MFSK system using either matched filter or 2-pole BPF detection under PBJ. To be specific, for the matched filter system, we find the worst-case performance, and for the 2-pole BPF system, we find the NE at a given BER.
We assume the partial-band interference is a zero-mean Gaussian random process with a flat power spectral density over a fraction ρ of the total spread spectrum bandwidth, W ss , and zero elsewhere. Furthermore, as is common in the literature, we assume that on a given hop, each M -ary band lies either entirely inside or entirely outside W J [25]. In the region or regions where the power spectral density is nonzero, its value is N J /ρ.
As in previous sections, we ignore thermal noise. To justify this, we compare the performance with and without thermal noise when M = 2 and L = 1 with matched filter detection both in a non-fading channel and in a Rician fading channel. Let a ≜ N J /N 0 , and since jamming is usually dominant over thermal noise, we assume a > 1 (typically a ≫ 1). We denote Let [30], so that the approximate BER, obtained by ignoring thermal noise, is given by P b1 = ρ ·f (ργ ) for the non-fading channel, and P b1 = ρ · g(ργ ) for the Rician fading channel. Since Thus, the BER with jamming and noise is given by for a non-fading channel, and for a Rician fading channel, The exact BER and approximate BER, P b and P b1 , respectively, with different ρ ′ s in non-fading and Rician (K = 10) fading channels, are shown in Figs. 2 and 3, respectively, where a = 5 is used. Note that the performance with and without thermal noise are almost identical in both figures and as a consequence not all the curves are distinguishable. The noise-free approximation is valid as long as E b /N J is reasonably large, and the approximation is better for smaller ρ. As we will see later, both conditions are satisfied in this paper. As a result, we will continue ignoring thermal noise, the consequence of which will be error-free detection with a probability of 1 − ρ.
Detailed discussion of the FFH-MFSK system with MF/2pole BPF detection under PBJ can be found in [27] and detailed analysis on finding the worst-case performance of the FFH-MFSK-MF system both in a non-fading channel and in a Rician fading channel is shown in Appendix III of this paper.

IV. DEMODULATOR PERFORMANCE IN MULTI-TONE JAMMING
A second, sometimes more effective, class of intelligent FH jamming than PBJ is MTJ. In this category, the jammer divides its total received power into a number of distinct, equal power, random-phase continuous-wave (CW) tones, and distributes them over the spread spectrum bandwidth according to strategies as will be discussed shortly. Because CW tones are the most efficient way for a jammer to inject energy into the non-coherent detectors, MTJ is particularly effective against a FH/MFSK system. Unlike PBJ, where the performance improves with the alphabet size M , the performance in MTJ degrades with M . In this section, we analyze the performance of an FFH-MFSK non-coherent system under MTJ, both with MF detection and with 2-pole BPF detection, in a non-fading or slow, flat, Rician fading channel. To be specific, for the matched filter system, we find the worst-case partial-band jamming performance, and for the 2pole BPF system, we find the NE at a given BER.
Some simplifying assumptions that allow us to focus on the issues of interest are [25] 1) Thermal noise is dominated by jamming interference and thus negligible. 2) Each jamming tone coincides exactly in frequency with one of the N t available FH slots (no frequency offset). 3) Changes of location of slots that are jammed coincides with hop transitions. While these assumptions can never be achieved in practice, they simplify the analysis and yield somewhat pessimistic performance results.
There are two MTJ strategies: independent MTJ, where the jammer distributes the tones pseudorandomly over all slots, and band MTJ, where the jammer places n ∈ [1, M ] tones in each jammed M -ary band. Independent MTJ and band MTJ with n = 1 are shown to result in the same performance when E b /η J is large, and band MTJ with n = 1 is shown to result in more performance degradation than MTJ with n > 1. Therefore, we will focus on MTJ with n = 1 in this paper.

A. MATCHED FILTER DETECTION
The asymptotic BER of an FFH-MFSK system with MF detection as E b /η J → ∞ can be shown to be where detailed analysis is provided in Appendix IV. In (15), α is the SJR to be optimized by the jammer and the worst-case performance is found as follows: with the worst-case parameter α In a Rician fading channel, we assume that the system and the jammer experience independent fades, with a Rician Kfactor K s for the signal, and K j for the jammer. Then the performance is given as a function of c ≜ α r L (α r is the average SJR), to be optimized by the jammer: where Detailed analysis is shown in Appendix V. The worst-case performance is found by differentiating P b with c and setting to zero: The optimum value of c, denoted by c * , along with the worst-case performance, characterized by G(c * ), for K s , K j ∈ {0, 10, 10 4 } is shown in Table 1 below. Note that we have a sanity check that when K s , K j → ∞, c * = G(c * ) = 1 − ϵ (the numbers are slightly off because K s and K j are not large enough). The K s = 0 scenario is slightly different, as shown in Appendix VI.

B. TWO-POLE BPF DETECTION
In a non-fading channel with 2-pole BPF detection, the average SER is given by where P 1 denotes the conditional SER when the jammed hop is the last one, P 2 denotes the conditional SER when the jammed hop is not the last one, and α 2 = where r s and r j are the amplitudes of the signal and jamming waveforms, respectively. Detailed analysis can be found in Appendix VII. Note that the SER is a function of the system's parameter, z (time-bandwidth parameter), and the jammer's parameter α 2 . From (21), in order to improve the performance, the system optimizes z to minimize h(z, α 2 ) (but subject to z ⩾ 0.24 for M = 2 or z ⩾ 0.2 for M = 16 to keep the eye open), and in order to degrade the performance the jammer optimizes α 2 to maximize h(z, α 2 ). The two players iteratively VOLUME 11, 2023   optimize their strategies until a NE is reached. The numerical results for L = 2 hops/symbol are summarized in Table 2 below.
We see that the 2-pole BPF detection results in better performance compared to MF detection, for both M = 2 and M = 16. This is because ISI is helping the system. With MF detection, there is no ISI, so the jammer power only needs to be ϵ greater than the signal power for each tone it jams, while with 2-pole BPF detection, since there is ISI, the jammer needs to put more power in each tone it jams in order to make sure jamming is effective. Therefore, with fixed jamming power, the jammer can jam fewer tones and the performance is better. Moreover, M = 16 results in a performance that is roughly 3 dB worse than that for M = 2, and this is consistent between MF and 2-pole BPF detection.
With Rician fading, the amplitudes of the signal and jamming waveforms, r s and r j , are now random variables. The pdf of r s is given by  and for the pdf of r j , we simply replace K s by K j in (22). The SJR is now defined as the ratio between the total power of the signal and jamming waveform: α r2 = s j . The performance with 2-pole BPF detection is found by integrating out the    where for i = 1, 2,

Rician amplitudes as
It can be shown that P r s is independent of s and j , using the same method described in Appendix V. Then, similar to the non-fading case, the system optimizes z to minimize h r (z, α r2 ) and the jammer optimizes α r2 to maximize it. The two players iteratively optimize their strategies until an NE is reached. Different from the non-fading case, now the NE is achieved when z → ∞, and α r2 and the corresponding performance are the same as the MF case, since ISI is diminishing for large z. However, we notice that when z ⩾ 0.6, h r (z, α r2 ) is almost independent of z since ISI is small, and thus we use z = 0.6 to generate the numerical results. The numerical results for L = 2 hops/symbol and K s = K j = 10 are summarized in the table below.
We can use FEC to improve performance under MTJ. The idea is the same -we want to find the NE, but now the system has two parameters, the time-bandwidth parameter z and the code dimension k. The uncoded and (15,5) RS coded performances under MTJ in a non-fading or Rician (K s = K j = 10) fading channel are presented in Fig. 8 in Section V.

V. NUMERICAL RESULTS
In this section, we present the numerical results of previous sections. In all the figures of this section, M = 16, L = 2 and (15,5) RS code is used. Figure 4 shows the performance analysis and the simulation results for both matched filter and 2-pole BPF detection for both non-fading and Rayleigh fading channels, with and without diversity, under full-band noise jamming. Figs. 5 and 6 show the performance where RS coding is used in a Rayleigh fading channel, with and without diversity, under full-band noise jamming, respectively. Fig. 7 shows the performance comparison for K = 0, 10, ∞, with and without coding, with and without diversity, under full-band noise jamming. Fig. 8 shows the results under MTJ in either a non-fading channel or a Rician (K s = K j = 10) fading channel, with and without coding. Fig. 9 shows the performance comparison of a FFH-16FSK system with MF or 2-pole BPF detection under PBJ or MTJ, in a non-fading or Rician (K = 10) fading channel. The 2-pole BPF system performs worse than the MF system by about 1 dB under PBJ, but is no worse than the MF system under MTJ.

VI. CONCLUSION
The essence of this work is the use of a spectrally-inefficient, but power-efficient, waveform, in conjunction with low complexity implementation. We consider intentional jamming, in both fading and non-fading environments. The goal is to minimize overall power consumption, which includes both transmitter and receiver power consumption, as well as the power consumption required for transmission, subject to a predetermined average BER. To achieve this, we chose noncoherent MFSK, RS encoding with hard-decision decoding, and 2-pole BPFs.
We compared the performance of an FFH-MFSK noncoherent receiver when the matched filters are replaced by 2-pole BPFs, for AWGN, Rayleigh, and Rician (K =10) channels. We further considered a coded system with RS encoding and hard-decision decoding. What we found was that by carefully optimizing the system parameters, namely filter bandwidth and code dimension, we could achieve the desired performance with a penalty in E b /N J of 0.8-1.8 dB compared to the more conventional design using non-coherent FSK with matched filter detection. To be specific, with full-band noise jamming, the 1.8 dB, performance degradation only occurs when we use diversity combining in a Rayleigh fading channel. In all other cases, the performance degradation is between 1.0 dB and 1.4 dB, which is consistent with what we found in [16]. With PBJ, the performance degradation is between 0.8 dB and 0.9 dB. With MTJ, we found that the worst-case performance of an FFH-MFSK system with 2pole BPF detection is no worse than the corresponding MF system: in a non-fading channel, the 2-pole system outper-forms the corresponding MF system by 2.4 dB for M = 2 and 2.2 dB for M = 16, and in a Rician fading channel, the performances of the two systems are identical at the NE, because an NE is reached when filter bandwidth is large, and thus ISI is negligible.

APPENDIX I. FFH-MFSK SYSTEM WITH 2-POLE BPF DETECTION
As stated in Section II, we use the union bound and, thus do pairwise comparison. The block diagram is shown in Fig. 1, where, as discussed in more detail in Appendix I of [16], the output of the i th filter can be expressed in terms of the in-phase and quadrature components as where a i (t) and b i (t) are the in-phase and quadrature parts of the output signal, respectively, as will be defined in (27) and (29), n ci (t) and n si (t) are the in-phase and quadrature parts of the output noise, respectively, and we sample at T c , . . . , LT c . Without loss of generality, we assume ''i ′′ is transmitted and ''n ′′ is the ICI branch. Thus, after the square law detector and summation, both the i th and the n th branch are the sum of 2L correlated non-central chi-square random variables, with different non-centrality parameters. We take the difference to form a new random variable, which is a linear combination of 4L correlated non-central chi-square random variables. Let X be a multi-normal random vector defined as We compute the mean and covariance matrix of X, i.e., µ = E[X] and = Cov(X). It was shown in [16] that only the adjacent hop gives significant ISI, so that only the first hop of each symbol experiences ISI from the previous symbol, while all other hops experience ISI from the previous hop of the current symbol. From Appendix II of [16], given that the current symbol is ''i'', the ICI branch is ''n'' and the previous symbol is ''m'', we have, for the first hop, Note that the terms in (28) are very similar to the corresponding parameters in Appendix II of [16], and θ 1 is the difference between the phase of the first hop of the current symbol and the last hop of the previous symbol. Let θ ′ 1 , θ ′ 2 , . . . , θ ′ L denote the random phases associated with each hop of the current symbol, and let θ ′ 0 denote the random phase associated with the last hop of the previous symbol. We define Then for all other hops, we have and thus which can be easily reshaped to µ in the form of a column vector.
Representing the in-phase and quadrature components of the filtered noise of the i th branch, using Hilbert transforms gives [31] n ci (t) = n i (t) cos(ω i t) +n i (t) sin(ω i t), n si (t) =n i (t) cos(ω i t) − n i (t) sin(ω i t).
Since we use orthogonal signaling and sample at the end of each hop, (31) can be simplified to The covariance matrix can be represented as where A, B, C are L × L matrices with the (x, y) th element, respectively, and 0 is the L × L all zero matrix. The diagonal terms in A, B and C can be found using the same technique described in Appendix I of [16] as For the non-diagonal terms, we have the following relationship: where x is a non-negative integer. The impulse response of the i th filter is Letting x ≜ t/T c , then From (32), Similarly, for a non-negative integer x, Now (40) can be rewritten as With µ and , we can numerically find the pdf of Q(X), defined in (7), using characteristic functions. The probability of error, where i is sent but n is detected, and where m is the previous symbol, conditioned on the set of random phases, θ 1 , . . . , θ L is given by Then finally, the conditional SER, Ps m in , is found by averaging over the L random phases.

APPENDIX II. FFH-MFSK SYSTEM WITH DIVERSITY
We again use the block diagram shown in Fig. 1. The only difference now is that the input hops have i.i.d. amplitudes, that is, where R l are i.i.d. Rician distributed. As in Appendix I, we compute in this appendix the mean and covariance matrix of the test statistic X, i.e., µ = E[X] and = Cov(X).

A. MATCHED FILTER DETECTION
Let R 1 , . . . , R L denote the amplitude of the 1 st , . . . , L th hop, respectively, of the current symbol. For matched filter detection, since there is no ISI or ICI, the parameters are where 0 3L is the 3L × 1 zero column vector and I 4L is the 4L × 4L identity matrix. After a simple normalization, the parameters are equivalent to 2N 0 and the pdf of γ i , f (γ i ), was defined in (6).

B. 2-POLE BPF DETECTION
Interleaving has no effect on the noise components, so we retain the same as in Appendix I. Let R 0 denote the amplitude of the last hop of the previous symbol, and R 1 , . . . , R L denote the amplitude of the 1 st , . . . , L th hop of the current symbol, respectively. The construction of the mean vector µ is similar to that in Appendix I -just replace the constant amplitude A is by Rician distributed amplitudes R 0 , R 1 , . . . , R L .
Similar to Appendix I, with µ and , we can numerically find the probability of error, where i is sent but n is detected and m is the previous symbol, conditioned on the set of random phases θ 1 , . . . , θ L , and the set of instantaneous E b /N J , γ 0 , . . . , γ L , denoted as Ps(i, n, m, θ 1 , . . . , θ L , γ 0 , . . . , γ L ). Then finally, the conditional probability of error can be represented by the following integral: To find the union bound on average SER, we average over i and m, and sum over n.

APPENDIX III. FFH-MFSK-MF SYSTEM UNDER PBJ A. NON-FADING CHANNELS
Taking the derivative of P s (ρ) defined in Equation (10) of [27] with respect to ρ yields where y ≜ ργ = ρ(log 2 M ) E b N J , and the derivative of the modified Bessel function of the first kind, n th order, is given by . Let y 0 be the value of y resulting in P ′ s (ρ) = 0. It is known that when E b /N J is sufficiently small, full-band jamming is optimal. Let γ 0 be the threshold so that when E b /N J < γ 0 , full-band jamming is optimal. Then γ 0 is found by setting ρ = 1, and thus γ 0 = y 0 log 2 M . Finally, plugging y = y 0 in (48) yields P s,worst The worst-case bit error rate can be expressed as

B. SLOW, FLAT, RICIAN FADING CHANNEL
Taking the derivative of P s (ρ) defined in equation (13) of [27] with respect to ρ yields where y ≜ 1+K ργ . Let y 0 be the value of y resulting in P ′ s (ρ) = 0. Then γ 0 is found by setting ρ = 1, and thus γ 0 = 1+K y 0 log 2 M is the threshold for partial-band jamming as the optimal strategy, i.e., when E b /N J < γ 0 , full-band jamming is optimal. The worst case SER is given by The worst-case BER can be expressed as

APPENDIX IV. FFH-MFSK-MF SYSTEM UNDER MTJ IN A NON-FADING CHANNEL
Let R c be the hop rate, R b be the bit rate, R s = R b / log 2 M be the symbol rate and N t be the total number of slots. Then the total spread spectrum bandwidth is given by W ss = N t R s . Letting J be the total jamming power, Q be the number of jamming tones, and S be the signal power, we have E b = S/R b as the energy per bit, η J = J /W ss as the power spectral density of the jammer, and finally we define α ≜ S J /Q as the SJR.
For each hop, the probability that the M -ary band containing the keyed tone is jammed is When E b /η J is large, µ ′ is small and the probability that multiple hops of the M -ary band with the keyed tone are jammed is orders of magnitude smaller, so we can assume that, at most, one hop of the M -ary band with the keyed tone is jammed at large E b /η J . Therefore, the probability that any hop of the M -ary band with the keyed tone is jammed is A symbol error occurs when a hop of the M -ary band with the keyed tone is jammed in an unkeyed slot, and 1 α > L ⇒ α < 1 L . Thus, the bit error rate is Obviously, the worst case performance is achieved by maximizing α under the constraints α < 1/L and µ ⩽ 1: Therefore, and letting α = α wc in (54), we find the worst-case BER as This happens to be the same as the SFH case analyzed in [25]. Note that P b,wc increases with M since the key parameter, µ, increases with M .

APPENDIX V. FFH-MFSK-MF SYSTEM UNDER MTJ IN A SLOW, FLAT, RICIAN FADING CHANNEL
In a Rician fading channel, let s and j be the total power of the signal and jamming power per jammed slot, respectively, and denote α r ≜ s / j as the SJR. Then, at large E b /η J , the probability that any hop of the M -ary band with the keyed tone is jammed is roughly The BER as a function of α r is given by VOLUME 11, 2023 where f (r s ) was defined in (22) and f (r j ) follows in a similar manner. If we let where h(r, K ) was defined in (19).

APPENDIX VI. FFH-MFSK-MF SYSTEM UNDER MTJ WHEN SIGNAL EXPERIENCES RAYLEIGH FADING
For K s = 0, G(c) in (61) is monotonically increasing, for all K j . Therefore, the jammer would use the largest permissible c. The constraint on c is and thus the worst-case BER is given by letting c = On the other hand, the worst-case BER is also Since it can be shown that, for any K j , lim c→∞ G(c) = 1, we have the asymptotic performance

APPENDIX VII. FFH-MFSK-2POLE SYSTEM UNDER MTJ IN A NON-FADING CHANNEL
We consider two conditional SER's: when the jammed hop is the last one (with probability 1 L ), denoted by P 1 , and when the jammed hop is not the last one (with probability L−1 L ), denoted by P 2 , and thus the average SER is given by (21).
First consider the case that the last (L th ) hop is jammed. The filter output of the jamming waveform in terms of the in-phase and quadrature components is, for the L th hop, and zero for all other hops, where r j is the amplitude of the jamming waveform. Thus the filter output of the signal plus jamming waveform is where [µ a µ b ] was found in (30) (replace A by r s ) and we reshape µ ′ 1 to the form of a column vector, denoted by µ 1 . A symbol error is made if the square-law detector output of the jammed branch exceeds that of the signal branch, i.e., if µ 1 T Aµ 1 < 0, where A was defined in (8). Finally, the conditional SER is found by averaging over the transmitted symbol i, the jammed symbol n, and the previous symbol m. We then integrate over the L + 1 random phases, and multiply by µ, which from (53) is the probability that the M -ary band with the keyed tone is jammed, as can be seen in (68) The other case is that the jammed hop is not the last one. We assume the l th hop is jammed where 1 ⩽ l ⩽ L − 1.
It is easy to see that the performance is not a function of l.
The filter output of the jamming waveform in terms of the in-phase and quadrature components is µ aJ for the l th hop, and for the l +1 st hop. The filter output of the signal plus jamming waveform is 1 ⩽ l ⩽ L − 1, and we reshape µ ′ 2 to the form of a column vector, denoted by µ 2 . The conditional SER, P 2 , can be found using (68) -just replace µ 1 by µ 2 .