A Spectrally Efficient Anti-Jamming Technique ... - Semantic Scholar
A Spectrally Efficient Anti-Jamming Technique Based on Message Driven Frequency Hopping Lei Zhang, Jian Ren, and Tongtong Li Department of Electrical & Computer Engineering Michigan State University, East Lansing, MI 48824 {zhangle3,renjian,tongli}@egr.msu.edu
Abstract. This paper considers spectrally efficient anti-jamming system design based on message-driven frequency hopping (MDFH). Unlike conventional FH where the hopping frequencies are determined by a preselected pseudonoise (PN) sequence, in MDFH, part of the message acts as the PN sequence for carrier frequency selection. It is observed that MDFH has high spectral efficiency and is particularly robust under strong jamming. However, disguised jamming from sources of similar power strength can cause performance losses. To overcome this drawback, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) system. The main idea is to transmit a secure ID sequence along with the information stream. The ID sequence is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. It is then exploited by the receiver for effective signal detection and extraction. It is shown that AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral efficiency and/or more robust jamming resistance. Simulation example is provided to demonstrate the performance of the proposed approaches. Keywords: jamming resistance, physical layer security, message-driven frequency hopping.
1
Introduction
As a widely used spread spectrum technique, frequency hopping (FH) was originally designed for secure communication under hostile environments [1,2]. In conventional FH, each user hops independently based on its own PN sequence, a collision occurs whenever there are two users transmitting over a same frequency band. Mainly limited by the collision effect, the spectral efficiency of the conventional FH is very low [3]. To improve the spectral efficiency, FH systems that exploit high-dimensional modulation scheme have been studied in the literature [4,5]. However, the performance of these systems are still limited by the collision or self-jamming effect.
This research is partially supported by NSF under awards CNS-0746811 and CNS0845812.
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To break the low capacity barrier and allow secure high speed communications, a message-driven frequency hopping (MDFH) scheme has been proposed in [3,6]. The main idea of MDFH is that a large portion of the message acts as the PN sequence for carrier frequency selection at the transmitter. That is, selection of carrier frequencies is directly controlled by the encrypted information stream rather than by a pre-selected pseudo-random sequence as in the conventional FH. At the MDFH receiver, the carrier frequencies are captured using a filter bank which selects the strongest signals from all the frequency bands. Note that transmission through hopping frequency control essentially adds another dimension to the signal space, and the resulting coding gain enables MDFH to increase the system spectral efficiency by multiple times. It has been observed that MDFH is very robust under strong jamming scenarios, and outperforms the conventional FH by big margins. The underlying argument is that: strong jamming can enhance the power of the jammed signal and hence increases the correct detection probability. When the system experiences disguised jamming, that is, when the jamming power is close to the signal power, it is difficult for the MDFH receiver to distinguish jamming from true signal, resulting in performance losses. To improve the performance of MDFH under disguised jamming, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) scheme. The basic idea is to insert some signal identification (ID) information during the transmission process. This ID information is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. Therefore, it can be used by the receiver to locate the true carrier frequency or the desired channel. At the same time, it is computationally infeasible to be recovered by malicious users. Comparing with MDFH, AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming and deliver significantly better results when the jamming power is close to that of the signal power. At the same time, it is robust under strong jamming just as MDFH. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral efficiency and/or more robust jamming resistance. Simulation example is provided to demonstrate the effectiveness of the proposed approaches.
2
Anti-Jamming MDFH (AJ-MDFH) System
To enhance the jamming resistance of MDFH under disguised jamming, in this section, we will introduce the anti-jamming MDFH system, named AJ-MDFH. 2.1
Transmitter Design
The transmitter structure of AJ-MDFH is illustrated in Figure 1. The encrypted information sequence is transmitted through carrier frequency selection, and each user is assigned a secure ID sequence. This ID information is shared between the transmitter and the receiver, therefore, it can be used by the receiver to locate the true carrier frequency. Our design goal is to reinforce jamming resistance without sacrificing too much on spectral efficiency.
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Encrypted Information
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f Xn
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Fig. 1. AJ-MDFH transmitter structure
Let Nc be the total number of available channels, with {f1 , f2 , · · · , fNc } being the set of all available carrier frequencies. Without loss of generality, here we assume that Nc = 2Bc for an integer Bc . Let Ω be the selected constellation that consists of M symbols, each symbol in the constellation represents Bs = log2 M bits. Let Nh be the number of hops in one ordinary symbol period. At each symbol period, MDFH transmits a block of length L Nh Bc + Bs bits. Each block contains Nh Bc carrier bits and Bs ordinary bits. The carrier bits are used to determine the hopping frequencies, and the ordinary bits are mapped to a symbol which is transmitted through the selected channels successively. Comparing with MDFH, in AJ-MDFH, the spectral efficiency is only reduced 1 . by a factor Nh BBcs+Bs . Take Nh = 5, Bc = 8, Bs = 4, for example, Nh BBcs+Bs = 11 It should be noted that, in order to prevent impersonation attack, each user’s ID sequence needs to be kept secret from the malicious jammer. Therefore we generate the ID sequence through a reliable cryptographic algorithm, such as the Advanced Encryption Standard (AES) [7], so that it is computationally infeasible for the malicious user to recover the ID sequence. That is, we first generate a pseudo-random sequence using a linear shift feedback register, encrypt it using AES, and then take the AES output as our ID sequence. 2.2
Receiver Design
The receiver structure for AJ-MDFH is shown in Figure 2. The receiver regenerates the secure ID through the shared secret (including the initial vector, the LFSR information and the key). For each hop, the received signal is first fed into the bandpass filter bank. The output of the filter bank is then demodulated, and used for carrier bits (i.e., the information bits) detection. Demodulation. Let s(t), J(t) and n(t) denote the ID signal, the jamming interference and the noise, respectively. For AWGN channels, the received signal can be represented as r(t) = s(t) + J(t) + n(t). We assume that s(t), J(t) and n(t) are independent of each other. If the spectrum of J(t) overlaps with the frequency band of s(t), then the signal
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Fig. 2. AJ-MDFH receiver structure
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is jammed ; otherwise, the signal is jamming-free. If J(t) spreads over multiple channels, we have multi-band jamming; otherwise, we have single band jamming. Note that the true information is embedded in the index of the active carrier over which the ID signal s(t) is transmitted. For i = 1, 2, · · · , Nc , the output of the ith ideal bandpass filter fi (t) is ri (t) = fi (t) ∗ r(t) = αi (t)s(t) + Ji (t) + ni (t).
(1)
Here αi (t) ∈ {0, 1} is a binary indicator for the presence of signal in channel i at time instant t. At each hopping period, αi (t) is a constant: αi (t) = 1 if and only if s(t) is transmitted over the ith channel during the mth hopping period; otherwise, αi (t) = 0. Ji (t) = fi (t) ∗ J(t) and ni (t) = fi (t) ∗ n(t). When there is no jamming presented in the ith channel, Ji (t) = 0. For demodulation, ri (t) is first shifted back to the baseband, and then passed through a matched filter. At the mth hopping period, for i = 1, · · · , Nc , the sampled matched filter output corresponds to channel i can be expressed as ri,m = αi,m sm + βi,m Ji,m + ni,m ,
(2)
where sm , Ji,m and ni,m correspond to the ID symbol, the jamming interference and the noise, respectively; αi,m , βi,m ∈ {0, 1} are binary indicators for the presence of ID signal and jamming, respectively. Note that the true information is carried in αi,m . Signal Detection and Extraction. Signal detection and extraction is performed at each hopping period. For notation simplicity, without loss of generality, we omit the subscript m in (2). That is, for a particular hopping period, (2) is reduced to: for i = 1, · · · , Nc . (3) ri = αi s + βi Ji + ni , Define r = (r1 , · · · , rNc ), α = (α1 , · · · , αNc ), β = (β1 , · · · , βNc ), J = (J1 , · · · , JNc ) and n = (n1 , · · · , nNc ), then (3) can be rewritten in vector form as: r = sα + β · J + n. For single carrier AJ-MDFH, at each hopping period, one and only one item in α is nonzero. That is, there are Nc possible information vectors: α1 = (1, 0, · · · , 0),· · · , αNc = (0, 0, · · · , 1). If αk is selected, and the binary expression of k is b0 b1 · · · bBc −1 , where Bc = log2 Nc , then estimated information sequence is b0 b1 · · · bBc −1 . So at each hopping period, the information symbol α, or equivalently, the hopping frequency index k, needs to be estimated based on the received signal and the ID information which is shared between the transmitter and the receiver. Here we use the maximum likelihood (ML) detector. If the input information is equiprobable, that is, p(αi ) = N1c for i = 1, 2, · · · , Nc , then MAP detector is reduced to the ML detector. For the ML detector, the estimated hopping frequency index kˆ is given by kˆ = arg max p{r|αi }. 1≤i≤Nc
(4)
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Recall that the information signal (including α and s), the jamming interference and the noise are independent to each other. Assume both the noise and the jamming interference are totally random, that is, n1 , · · · , nNc , J1 , · · · , JNc are all statistically independent, then r1 , · · · , rNc are also independent. In this case, the joint ML detector in (4) can be decomposed as: kˆ = arg max
Nc
1≤i≤Nc
Nc
p{rj |αi } = arg max
1≤i≤Nc
j=1
p{rj |αj = 0} · p{ri |αi = 1}(5)
j=1,j=i
Nc Since j=1 p{rj |αj = 0} is independent of i, (5) can be further simplified as i |αi =1} kˆ = arg max1≤i≤Nc p{r βi p(ri |αi = 1, βi )p(βi ) p{ri |αi =0} , where p{ri |αi = 1} = and p{ri |αi = 0} = βi p(ri |αi = 0, βi )p(βi ), with βi ∈ {0, 1}. Define Λi p{ri |αi =1} p{ri |αi =0}
be the likelihood ratio for channel i, then (5) can be rewritten as: kˆ = arg max Λi .
(6)
1≤i≤Nc
If we further assume that n1 , · · · , nNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σn2 , and J1 , · · · , JNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σJ2i , then it follows from (3) and (6) that: kˆ = arg max
1≤i≤Nc
P {βi =0} 2 πσn
2
exp{− riσ−s }+ 2
P {βi =0} 2 πσn
n
2
exp{− rσi2 } + n
P {βi =1} 2 +σ 2 ) π(σn J i
P {βi =1} 2 +σ 2 ) π(σn J i
2
i −s exp{− r } σ2 +σ2 n
Ji
2
i exp{− σr 2 +σ 2 } n
.
(7)
Ji
In the ideal case when the jammed channel indices are known, or equivalently, βi is known for i = 1, · · · , Nc , then the ML detector above can be further ˜ i is circularly symmetric Gaussian with simplified. Define n ˜ i = βi Ji + ni , then n zero mean and variance σi2 = βi σJ2i + σn2 . It follows from (3) and (6) that kˆ = 2
2
i −s arg max1≤i≤Nc R1 (ri ), where R1 (ri ) = ri −r . σi2 However, in reality, jamming side information is generally unknown. Here we develop the following two suboptimal detectors. Note that σi2 is generally unknown. If we replace the overall interference power σi2 in R1 (ri ) with the instantaneous received signal power ri 2 , then we can obtain another detector 2 kˆ = arg min1≤i≤Nc R2 (ri ), where R2 (ri ) = rri −s 2 . i For more tractable theoretical analysis, we can replace the instantaneous received signal power ri 2 in R2 (ri ) with average signal power observed in channel i, Pi = E{ri 2 }, then we obtain an alternative detector
kˆ = arg min R3 (ri ), 1≤i≤Nc
where R3 (ri ) =
ri −s2 . Pi
(8)
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ID Constellation Design and Its Impact on System Performance
For AJ-MDFH, ID signals are introduced to distinguish the true information channel from disguised channels invoked by jamming interference. The general design criterion of the ID constellation is to minimize the carrier detection error probability under a given signal power. Under this criterion, there are two questions need to be answered: (1) How does the size of the constellation impact the system performance? (2) How does the type or shape of the constellation influence the detection error and which type should we use for optimal performance? In this section, we will try to address these questions under different jamming scenarios. Recall that for AJ-MDFH, the message signal is embedded in the index of the hopping frequency or channel. In the worst case if the ID is known to the jammers, or can be easily guessed by the jammers, then the jammers can disguise itself by sending the same ID symbol over a different or fake channel. In this case, it would be difficult for the receiver to distinguish the true channel from the disguised channel, leading to high detection error probability. We define this kind of jamming as ID jamming or ID attack. In literature, jamming has generally been modeled as Gaussian noise. We refer this kind of jamming as noise jamming. Here we consider the constellation design problem under these two jamming scenarios separately: – Constellation Design under Noise Jamming: Without loss of generality, we assume that the ID symbol is transmitted through channel 1. When detection metric in (8) is used, it can be shown that the carrier detection error probability Pe is upper bounded by ⎡ Nc −1 ⎤ s2 (s2 +σ2 ) 2 1 − 2 1 σ 2 2 1 ⎣1 − 1 − ⎦, e σm (s +2σ1 ) Pe ≤ PeU = (9) |Ω| s2 + 2σ12 s∈Ω
where m = arg max{σl2 } for 2 ≤ l ≤ Nc and |Ω| is the size of the ID constellation Ω. Further mathematical analysis shows that: when SNR is high 2 enough, i.e., s 1, the upper bound of the detection error probabilσ2 1
ity, PeU , is minimized when the constellation is constant modulus, i.e., when s2 = Ps for all s ∈ Ω. An intuitive explanation for this result is that the signal power in constant modulus constellations always equals to the maximal signal power available. – Constellation Design under ID Jamming: In this case, the entropy or uncertainty of the ID symbol needs to be maximized. Under the assumption that all the symbol in a constellation Ω of size M are all equally probable, 1 = log2 |Ω| = log M. In the ideal case when the entropy H(s) = − log2 |Ω| the channel is noise-free, the optimal constellation size would be M = ∞. However, when noise is present, larger M also implies that there is a larger probability for an ID symbol to be mistaken for its neighboring symbols.
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More specifically, it can be shown that: for a given SNR and assuming PSK constellation is used, the carrier detection error probability Pe will converges to a limit P¯e as constellation size M increases. That is, for any > 0, there always exists an Mt such that for all M > Mt , |Pe − P¯e | < .
4
Multi-carrier AJ-MDFH
For more efficient spectrum usage and more robust jamming resistance, in this section, we extend the concept of AJ-MDFH to multi-carrier AJ-MDFH (MCAJ-MDFH). The transmitter and receiver structure of the MC-AJ-MDFH system are illustrated in Figure 3. The idea is to split all the Nc channels into Ng non-overlapping groups, and each subcarrier hops within the assigned group based on the AJ-MDFH scheme. To ensure hopping randomness of all the subcarriers, the groups need to be reorganized or regenerated securely after a prespecified period, named group period. In the following, we will first describe the secure group generation algorithm, and then discuss the design of MC-AJ-MDFH with and without additional frequency diversity. X 1,n
S/P
Encrypted Information Sequence
Channel Coding and Interleaving
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Channel Coding and Interleaving Initial Vector, Key Initial Vector, Key Secure ID
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(a) Transmitter structure
Recovered Information Message Reconstruction
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BPF, fNc
Initial Vector, Key
Secure ID Generation
Y1,n , , YNg ,n
s1,n , , sNg ,n Symbol Mapper
(b) Receiver structure
Fig. 3. Transmitter and receiver structure of MC-AJ-MDFH
4.1
Secure Group Generation
In this section, we propose a secure group generation algorithm to ensure that: (i) Each subcarrier hops over a new group of channels during each group period, so that it eventually hops over all the available channels in a pseudo-random manner; (ii) Only the legitimate receiver can recover the transmitted information correctly. Secure group generation is synchronized at the transmitter and the receiver. At the receiver, the received signal is fed to a bank of single-carrier AJ-MDFH receivers for signal extraction and recovery. Recall that we assume there are a total of Nc available channels and there are Ng subcarriers in the system. For l = 0, · · · , Ng − 1, the number of channels assigned to subcarrier i is denoted as Ngi . As different subcarriers transmit over Ng non-overlapping set of channels, we have Ngi = Nc . i=0
For secure group generation, first, generate a pseudo-random binary sequence using a 32-bit linear feedback shift register (LFSR) as in Section 2, which is initialized by a secret sequence shared between the transmitter and receiver.
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Encrypt the generated sequence into a ciphertext using the AES algorithm and a secure key. Pick an integer L ∈ [ N2c , Nc ] and let q = L log2 Nc = LBc . Take q bits from the ciphertext and put them as a q-bit vector e = [e1 , e2 , · · · , eq ]. Second, partition the ciphertext sequence e into L groups, such that each group contains Bc bits. For k = 1, 2, · · · , L, the partition of the ciphertext is represented as pk = [e(k−1)∗Bc +1 , e(k−1)∗Bc +2 , · · · , e(k−1)∗Bc +Bc ], where pk corresponds to the kth Bc -bit vector. For k = 1, 2, · · · , L, denote Pk as the decimal number corresponding to pk . And denote P = [P1 , P2 , · · · , PL ] as the permutation index vector. For k = 0, 1, 2, · · · , L, denote Ik = [Ik (0), Ik (1), · · · , Ik (Nc − 1)] as the index vector at the kth step. The secure permutation scheme of the index vector is achieved through the following steps: 1. Initially, the index vector is I0 = [I0 (0), I0 (1), · · · , I0 (Nc − 1)] and the permutation index is P = [P1 , P2 , · · · , PL ]. We start with I0 = [0, 1, · · · , Nc −1]. 2. For k = 1, switch I0 (0) and I0 (P1 ) in index vector I0 to obtain I1 . In other words, I1 = [I1 (0), I1 (1), · · · , I1 (Nc − 1)], where I1 (0) = I0 (P1 ), I1 (P1 ) = I0 (0), and I1 (m) = I0 (m) for m = 0, P1 . 3. Repeat the previous step for k = 2, 3, · · · , L. In general, if we already have Ik−1 = [Ik−1 (0), Ik−1 (1), · · · , Ik−1 (Nc − 1)], then we can obtain Ik = [Ik (0), Ik (1),· · · , Ik (Nc − 1)] through the permutation defined as Ik (k − 1) = Ik−1 (Pk ), Ik (Pk ) = Ik−1 (k − 1), and Ik (m) = Ik−1 (m) for m = k − 1, Pk . 4. After L steps, we obtain the channel center frequency vector as FL = [fIL (0) , fIL (1) , · · · , fIL (Nc −1) ]. 5. Vector FL is used to assign the channels to Ng groups. We assign channels {fIL (0) , fIL (1) , · · · , fIL (Ng0 −1) } to the first group; Assign {fIL (Ng0 ) , fIL (Ng0 +1) , · · · , fIL (Ng0 +Ng1 −1) } to the second group, and so on. Because each frequency index appears in FL once and only once, the proposed algorithm ensures that all the subcarriers are transmitting on non-overlapping sets of channels. 4.2
Multi-Carrier AJ-MDFH without Diversity
In this case, each subcarrier transmits independent bit stream. The spectral efficiency of the AJ-MDFH system can be increased significantly. Let Bc = log2 Nc and Bg = log2 Ng , then the number of bits transmitted by the MC-AJMDFH within each hopping period is BMC = (Bc − Bg )Ng = (Bc − log2 Ng )Ng . BMC is maximized when Bg = Bc − 1 or Bg = Bc − 2, which results in BMC = 2Bc −1 . Note that the number of bits transmitted by the AJ-MDFH within each hopping period is Bc , it can be seen that BMC > Bc as long as Bc > 2. Take Nc = 256 for example, then the transmission efficiency of AJ-MDFH can be Bc −1 increased by BBMcC = 2 Bc = 16 times. We assume that jamming is random and equally distributed among all the groups. Then the overall carrier detection error probability Pe of MC-AJ-MDFH is equal to that corresponding to each subcarrier fk for k = 1, · · · , Ng . Let Pe,k
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denote the carrier detection error probability corresponding to the kth subcarrier or the kth group, then we have Pe = Pe,k , and Pe,k = P0,k · P {incorrect carrier detection|signal not jammed} + P1,k · P {incorrect carrier detection|signal jammed},
(10)
where P0,k , P1,k denote the probability that the kth subcarrier is jamming-free or jammed, respectively. 4.3
Multi-carrier AJ-MDFH with Diversity
Under the multi-band jamming, diversity needs to be introduced to the AJMDFH system for robust jamming resistance especially. A natural solution to achieve frequency diversity is to transmit the same or correlated information through multiple subcarriers. The number of subcarriers needed to convey the same information differs in different jamming scenarios. Ideally, the number of correlated signal subcarriers should not be less than the number of jammed bands. At the receiver, the received signals from different diversity branches are often combined for joint signal detection. To achieve better performance, appropriate diversity combination schemes need to be selected for different metrics used. In this paper, we propose to use the equal gain combination scheme [8] while choosing R3 (ri ) be the detection metric, since it can also be regarded as a normalized square-law metric. Assume that the same infor10 MDFH mation is transmitted through conventional FH AJ−MDFH the hopping frequency index of MC−AJ−MDFH: without diversity MC−AJ−MDFH: with diversity 10 Nd subcarriers over Nd groups {Gn1 , Gn2 , · · · , GnNd } simultaneously, each group has the 10 same number of channels, denoted as Ngc . Note that the se10 cure group generation algorithm ensures that the channel index 10 −20 −15 −10 −5 0 5 10 15 20 in each group is random and JSR(dB) does not necessarily come in ascending or descending order. Let Fig. 4. The performance comparison under 2band ID jamming. R3 (rinl ) denote detection metric value for ith channel in group Gnl , then the active hopping frequency index can be estimated as kˆ = d nl arg min1≤i≤Ngc N l=1 R3 (ri ). The diversity order Nd can be dynamic in different jamming scenarios to achieve tradeoff between performance and efficiency. 2
BER
0
−2
−4
−6
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Simulation Results
In this section, we illustrate the performance of the proposed AJ-MDFH and MCAJ-MDFH through a Monte Carlo simulation example using Matlab. We assume
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that the signal is transmitted through AWGN channels and experiences 2-band ID jamming. For conventional FH, BFSK modulation is used; For MDFH, 8-PSK is adopted for ordinary bits modulation; For AJ-MDFH and MC-AJ-MDFH, 32PSK is adopted for ID constellation to combat possible ID jamming. The SNR is taken as Eb /N0 = 10dB. JSR is defined as the ratio of the jamming power to signal power per channel. The jamming bands of the multi-band ID jamming are independently and randomly selected, and each ID jamming randomly transmits a symbol from the constellation of corresponding modulation scheme. The number of available channels is Nc = 64 (Bc = 6). Choose Ng = Nd = 4 and Ng = 32 for MC-AJ-MDFH with and without diversity, respectively. From Figure 4, it can be seen that AJ-MDFH and MC-AJ-MDFH deliver much better performance than MDFH and conventional FH.
6
Conclusion
In this paper, we proposed a highly efficient anti-jamming scheme AJ-MDFH based on message-driven frequency hopping. It was shown that AJ-MDFH is robust under strong jamming and can effectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to multi-carrier AJ-MDFH for higher spectral efficiency and more robust jamming resistance. The proposed approaches can be applied to both civilian and military applications for reliable communication under jamming interference.
References 1. Viterbi, A.: A processing satellite transponder for multlple access by low rate mobile users. In: Proc. Digital Satellite Commun. Conf., Montreal, Canada (October 1978) 2. Pickholtz, R., Schilling, D., Milstein, L.: Theory of spread-spectrum communications: A tutorial. IEEE Trans. Commun. 30(5), 855–884 (1982) 3. Ling, Q., Ren, J., Li, T.: Spectrally efficient spread spectrum system design: Message-driven frequency hopping. In: Proc. IEEE Intl. Conf. Commun., May 2008, pp. 4775–4779 (2008) 4. Cho, J., Kim, Y., Cheun, K.: A novel frequency-hopping spread-spectrum multipleaccess network using M-ary orthogonal Walsh sequence keying. IEEE Trans. Commun. 51(11), 1885–1896 (2003) 5. Choi, K., Cheun, K.: Maximum throughput of FHSS multiple-access networks using MFSK modulation. IEEE Trans. Commun. 52, 426–434 (2004) 6. Ling, Q., Li, T.: Message-driven frequency hopping: Design and analysis. IEEE Trans. Wireless Commun. 8(4), 1773–1782 (2009) 7. Advanced Encryption Standard, FIPS-197, National Institute of Standards and Technology Std. (November 2001) 8. Miller, L., Lee, J., Kadrichu, A.: Probability of error analyses of a BFSK frequencyhopping system with diversity under partial-band jamming interference–part III: Performance of a square-law self-normalizing soft decision receiver. IEEE Trans. Commun. 34(7), 669–675 (1986)
Abstract. This paper considers spectrally efficient anti-jamming system design based on message-driven frequency hopping (MDFH). Unlike conventional FH where the hopping frequencies are determined by a preselected pseudonoise (PN) sequence, in MDFH, part of the message acts as the PN sequence for carrier frequency selection. It is observed that MDFH has high spectral efficiency and is particularly robust under strong jamming. However, disguised jamming from sources of similar power strength can cause performance losses. To overcome this drawback, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) system. The main idea is to transmit a secure ID sequence along with the information stream. The ID sequence is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. It is then exploited by the receiver for effective signal detection and extraction. It is shown that AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral efficiency and/or more robust jamming resistance. Simulation example is provided to demonstrate the performance of the proposed approaches. Keywords: jamming resistance, physical layer security, message-driven frequency hopping.
1
Introduction
As a widely used spread spectrum technique, frequency hopping (FH) was originally designed for secure communication under hostile environments [1,2]. In conventional FH, each user hops independently based on its own PN sequence, a collision occurs whenever there are two users transmitting over a same frequency band. Mainly limited by the collision effect, the spectral efficiency of the conventional FH is very low [3]. To improve the spectral efficiency, FH systems that exploit high-dimensional modulation scheme have been studied in the literature [4,5]. However, the performance of these systems are still limited by the collision or self-jamming effect.
This research is partially supported by NSF under awards CNS-0746811 and CNS0845812.
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To break the low capacity barrier and allow secure high speed communications, a message-driven frequency hopping (MDFH) scheme has been proposed in [3,6]. The main idea of MDFH is that a large portion of the message acts as the PN sequence for carrier frequency selection at the transmitter. That is, selection of carrier frequencies is directly controlled by the encrypted information stream rather than by a pre-selected pseudo-random sequence as in the conventional FH. At the MDFH receiver, the carrier frequencies are captured using a filter bank which selects the strongest signals from all the frequency bands. Note that transmission through hopping frequency control essentially adds another dimension to the signal space, and the resulting coding gain enables MDFH to increase the system spectral efficiency by multiple times. It has been observed that MDFH is very robust under strong jamming scenarios, and outperforms the conventional FH by big margins. The underlying argument is that: strong jamming can enhance the power of the jammed signal and hence increases the correct detection probability. When the system experiences disguised jamming, that is, when the jamming power is close to the signal power, it is difficult for the MDFH receiver to distinguish jamming from true signal, resulting in performance losses. To improve the performance of MDFH under disguised jamming, in this paper, we propose an anti-jamming MDFH (AJ-MDFH) scheme. The basic idea is to insert some signal identification (ID) information during the transmission process. This ID information is generated through a cryptographic algorithm using the shared secret between the transmitter and the receiver. Therefore, it can be used by the receiver to locate the true carrier frequency or the desired channel. At the same time, it is computationally infeasible to be recovered by malicious users. Comparing with MDFH, AJ-MDFH can effectively reduce the performance degradation caused by disguised jamming and deliver significantly better results when the jamming power is close to that of the signal power. At the same time, it is robust under strong jamming just as MDFH. Moreover, AJ-MDFH can be extended to a multi-carrier scheme for higher spectral efficiency and/or more robust jamming resistance. Simulation example is provided to demonstrate the effectiveness of the proposed approaches.
2
Anti-Jamming MDFH (AJ-MDFH) System
To enhance the jamming resistance of MDFH under disguised jamming, in this section, we will introduce the anti-jamming MDFH system, named AJ-MDFH. 2.1
Transmitter Design
The transmitter structure of AJ-MDFH is illustrated in Figure 1. The encrypted information sequence is transmitted through carrier frequency selection, and each user is assigned a secure ID sequence. This ID information is shared between the transmitter and the receiver, therefore, it can be used by the receiver to locate the true carrier frequency. Our design goal is to reinforce jamming resistance without sacrificing too much on spectral efficiency.
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Encrypted Information
Initial Vector, Key
Channel Coding
PN Sequence Generation
Interleaving
Encryption
Yn
Xn
Carrier Frequency Selection
Symbol Mapper
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f Xn
sn Baseband Signal
Modulation
s (t )
Generation
Secure ID Generation
Fig. 1. AJ-MDFH transmitter structure
Let Nc be the total number of available channels, with {f1 , f2 , · · · , fNc } being the set of all available carrier frequencies. Without loss of generality, here we assume that Nc = 2Bc for an integer Bc . Let Ω be the selected constellation that consists of M symbols, each symbol in the constellation represents Bs = log2 M bits. Let Nh be the number of hops in one ordinary symbol period. At each symbol period, MDFH transmits a block of length L Nh Bc + Bs bits. Each block contains Nh Bc carrier bits and Bs ordinary bits. The carrier bits are used to determine the hopping frequencies, and the ordinary bits are mapped to a symbol which is transmitted through the selected channels successively. Comparing with MDFH, in AJ-MDFH, the spectral efficiency is only reduced 1 . by a factor Nh BBcs+Bs . Take Nh = 5, Bc = 8, Bs = 4, for example, Nh BBcs+Bs = 11 It should be noted that, in order to prevent impersonation attack, each user’s ID sequence needs to be kept secret from the malicious jammer. Therefore we generate the ID sequence through a reliable cryptographic algorithm, such as the Advanced Encryption Standard (AES) [7], so that it is computationally infeasible for the malicious user to recover the ID sequence. That is, we first generate a pseudo-random sequence using a linear shift feedback register, encrypt it using AES, and then take the AES output as our ID sequence. 2.2
Receiver Design
The receiver structure for AJ-MDFH is shown in Figure 2. The receiver regenerates the secure ID through the shared secret (including the initial vector, the LFSR information and the key). For each hop, the received signal is first fed into the bandpass filter bank. The output of the filter bank is then demodulated, and used for carrier bits (i.e., the information bits) detection. Demodulation. Let s(t), J(t) and n(t) denote the ID signal, the jamming interference and the noise, respectively. For AWGN channels, the received signal can be represented as r(t) = s(t) + J(t) + n(t). We assume that s(t), J(t) and n(t) are independent of each other. If the spectrum of J(t) overlaps with the frequency band of s(t), then the signal
BPF, f1(t) BPF, f2(t) r (t )
Signal Detection & Extraction
Demodulation
Recovered Information
BPF, fNc(t)
sn Initial Vector, Key
Secure ID Generation
Yn
Symbol Mapper
Fig. 2. AJ-MDFH receiver structure
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is jammed ; otherwise, the signal is jamming-free. If J(t) spreads over multiple channels, we have multi-band jamming; otherwise, we have single band jamming. Note that the true information is embedded in the index of the active carrier over which the ID signal s(t) is transmitted. For i = 1, 2, · · · , Nc , the output of the ith ideal bandpass filter fi (t) is ri (t) = fi (t) ∗ r(t) = αi (t)s(t) + Ji (t) + ni (t).
(1)
Here αi (t) ∈ {0, 1} is a binary indicator for the presence of signal in channel i at time instant t. At each hopping period, αi (t) is a constant: αi (t) = 1 if and only if s(t) is transmitted over the ith channel during the mth hopping period; otherwise, αi (t) = 0. Ji (t) = fi (t) ∗ J(t) and ni (t) = fi (t) ∗ n(t). When there is no jamming presented in the ith channel, Ji (t) = 0. For demodulation, ri (t) is first shifted back to the baseband, and then passed through a matched filter. At the mth hopping period, for i = 1, · · · , Nc , the sampled matched filter output corresponds to channel i can be expressed as ri,m = αi,m sm + βi,m Ji,m + ni,m ,
(2)
where sm , Ji,m and ni,m correspond to the ID symbol, the jamming interference and the noise, respectively; αi,m , βi,m ∈ {0, 1} are binary indicators for the presence of ID signal and jamming, respectively. Note that the true information is carried in αi,m . Signal Detection and Extraction. Signal detection and extraction is performed at each hopping period. For notation simplicity, without loss of generality, we omit the subscript m in (2). That is, for a particular hopping period, (2) is reduced to: for i = 1, · · · , Nc . (3) ri = αi s + βi Ji + ni , Define r = (r1 , · · · , rNc ), α = (α1 , · · · , αNc ), β = (β1 , · · · , βNc ), J = (J1 , · · · , JNc ) and n = (n1 , · · · , nNc ), then (3) can be rewritten in vector form as: r = sα + β · J + n. For single carrier AJ-MDFH, at each hopping period, one and only one item in α is nonzero. That is, there are Nc possible information vectors: α1 = (1, 0, · · · , 0),· · · , αNc = (0, 0, · · · , 1). If αk is selected, and the binary expression of k is b0 b1 · · · bBc −1 , where Bc = log2 Nc , then estimated information sequence is b0 b1 · · · bBc −1 . So at each hopping period, the information symbol α, or equivalently, the hopping frequency index k, needs to be estimated based on the received signal and the ID information which is shared between the transmitter and the receiver. Here we use the maximum likelihood (ML) detector. If the input information is equiprobable, that is, p(αi ) = N1c for i = 1, 2, · · · , Nc , then MAP detector is reduced to the ML detector. For the ML detector, the estimated hopping frequency index kˆ is given by kˆ = arg max p{r|αi }. 1≤i≤Nc
(4)
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Recall that the information signal (including α and s), the jamming interference and the noise are independent to each other. Assume both the noise and the jamming interference are totally random, that is, n1 , · · · , nNc , J1 , · · · , JNc are all statistically independent, then r1 , · · · , rNc are also independent. In this case, the joint ML detector in (4) can be decomposed as: kˆ = arg max
Nc
1≤i≤Nc
Nc
p{rj |αi } = arg max
1≤i≤Nc
j=1
p{rj |αj = 0} · p{ri |αi = 1}(5)
j=1,j=i
Nc Since j=1 p{rj |αj = 0} is independent of i, (5) can be further simplified as i |αi =1} kˆ = arg max1≤i≤Nc p{r βi p(ri |αi = 1, βi )p(βi ) p{ri |αi =0} , where p{ri |αi = 1} = and p{ri |αi = 0} = βi p(ri |αi = 0, βi )p(βi ), with βi ∈ {0, 1}. Define Λi p{ri |αi =1} p{ri |αi =0}
be the likelihood ratio for channel i, then (5) can be rewritten as: kˆ = arg max Λi .
(6)
1≤i≤Nc
If we further assume that n1 , · · · , nNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σn2 , and J1 , · · · , JNc are i.i.d. circularly symmetric Gaussian random variables of zero mean and variance σJ2i , then it follows from (3) and (6) that: kˆ = arg max
1≤i≤Nc
P {βi =0} 2 πσn
2
exp{− riσ−s }+ 2
P {βi =0} 2 πσn
n
2
exp{− rσi2 } + n
P {βi =1} 2 +σ 2 ) π(σn J i
P {βi =1} 2 +σ 2 ) π(σn J i
2
i −s exp{− r } σ2 +σ2 n
Ji
2
i exp{− σr 2 +σ 2 } n
.
(7)
Ji
In the ideal case when the jammed channel indices are known, or equivalently, βi is known for i = 1, · · · , Nc , then the ML detector above can be further ˜ i is circularly symmetric Gaussian with simplified. Define n ˜ i = βi Ji + ni , then n zero mean and variance σi2 = βi σJ2i + σn2 . It follows from (3) and (6) that kˆ = 2
2
i −s arg max1≤i≤Nc R1 (ri ), where R1 (ri ) = ri −r . σi2 However, in reality, jamming side information is generally unknown. Here we develop the following two suboptimal detectors. Note that σi2 is generally unknown. If we replace the overall interference power σi2 in R1 (ri ) with the instantaneous received signal power ri 2 , then we can obtain another detector 2 kˆ = arg min1≤i≤Nc R2 (ri ), where R2 (ri ) = rri −s 2 . i For more tractable theoretical analysis, we can replace the instantaneous received signal power ri 2 in R2 (ri ) with average signal power observed in channel i, Pi = E{ri 2 }, then we obtain an alternative detector
kˆ = arg min R3 (ri ), 1≤i≤Nc
where R3 (ri ) =
ri −s2 . Pi
(8)
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ID Constellation Design and Its Impact on System Performance
For AJ-MDFH, ID signals are introduced to distinguish the true information channel from disguised channels invoked by jamming interference. The general design criterion of the ID constellation is to minimize the carrier detection error probability under a given signal power. Under this criterion, there are two questions need to be answered: (1) How does the size of the constellation impact the system performance? (2) How does the type or shape of the constellation influence the detection error and which type should we use for optimal performance? In this section, we will try to address these questions under different jamming scenarios. Recall that for AJ-MDFH, the message signal is embedded in the index of the hopping frequency or channel. In the worst case if the ID is known to the jammers, or can be easily guessed by the jammers, then the jammers can disguise itself by sending the same ID symbol over a different or fake channel. In this case, it would be difficult for the receiver to distinguish the true channel from the disguised channel, leading to high detection error probability. We define this kind of jamming as ID jamming or ID attack. In literature, jamming has generally been modeled as Gaussian noise. We refer this kind of jamming as noise jamming. Here we consider the constellation design problem under these two jamming scenarios separately: – Constellation Design under Noise Jamming: Without loss of generality, we assume that the ID symbol is transmitted through channel 1. When detection metric in (8) is used, it can be shown that the carrier detection error probability Pe is upper bounded by ⎡ Nc −1 ⎤ s2 (s2 +σ2 ) 2 1 − 2 1 σ 2 2 1 ⎣1 − 1 − ⎦, e σm (s +2σ1 ) Pe ≤ PeU = (9) |Ω| s2 + 2σ12 s∈Ω
where m = arg max{σl2 } for 2 ≤ l ≤ Nc and |Ω| is the size of the ID constellation Ω. Further mathematical analysis shows that: when SNR is high 2 enough, i.e., s 1, the upper bound of the detection error probabilσ2 1
ity, PeU , is minimized when the constellation is constant modulus, i.e., when s2 = Ps for all s ∈ Ω. An intuitive explanation for this result is that the signal power in constant modulus constellations always equals to the maximal signal power available. – Constellation Design under ID Jamming: In this case, the entropy or uncertainty of the ID symbol needs to be maximized. Under the assumption that all the symbol in a constellation Ω of size M are all equally probable, 1 = log2 |Ω| = log M. In the ideal case when the entropy H(s) = − log2 |Ω| the channel is noise-free, the optimal constellation size would be M = ∞. However, when noise is present, larger M also implies that there is a larger probability for an ID symbol to be mistaken for its neighboring symbols.
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More specifically, it can be shown that: for a given SNR and assuming PSK constellation is used, the carrier detection error probability Pe will converges to a limit P¯e as constellation size M increases. That is, for any > 0, there always exists an Mt such that for all M > Mt , |Pe − P¯e | < .
4
Multi-carrier AJ-MDFH
For more efficient spectrum usage and more robust jamming resistance, in this section, we extend the concept of AJ-MDFH to multi-carrier AJ-MDFH (MCAJ-MDFH). The transmitter and receiver structure of the MC-AJ-MDFH system are illustrated in Figure 3. The idea is to split all the Nc channels into Ng non-overlapping groups, and each subcarrier hops within the assigned group based on the AJ-MDFH scheme. To ensure hopping randomness of all the subcarriers, the groups need to be reorganized or regenerated securely after a prespecified period, named group period. In the following, we will first describe the secure group generation algorithm, and then discuss the design of MC-AJ-MDFH with and without additional frequency diversity. X 1,n
S/P
Encrypted Information Sequence
Channel Coding and Interleaving
X Ng ,n
Channel Coding and Interleaving Initial Vector, Key Initial Vector, Key Secure ID
Generation
Y1,n , , YNg ,n
Subcarrier Selection Over Group Gn1
Subcarrier Selection Over Group GnNg
f1,n
Group Gn1 Subcarrier Detection
BPF, f1
f Ng ,n
s (t )
r (t )
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Modulation
Gn1 , , GnNg Secure Group Generation
s1 (t ), , sNg (t ) Symbol Mapping and Baseband Signal Generation
Secure Group De-assignment
Initial Vector, Key
(a) Transmitter structure
Recovered Information Message Reconstruction
Group GnNg Subcarrier Detection
BPF, fNc
Initial Vector, Key
Secure ID Generation
Y1,n , , YNg ,n
s1,n , , sNg ,n Symbol Mapper
(b) Receiver structure
Fig. 3. Transmitter and receiver structure of MC-AJ-MDFH
4.1
Secure Group Generation
In this section, we propose a secure group generation algorithm to ensure that: (i) Each subcarrier hops over a new group of channels during each group period, so that it eventually hops over all the available channels in a pseudo-random manner; (ii) Only the legitimate receiver can recover the transmitted information correctly. Secure group generation is synchronized at the transmitter and the receiver. At the receiver, the received signal is fed to a bank of single-carrier AJ-MDFH receivers for signal extraction and recovery. Recall that we assume there are a total of Nc available channels and there are Ng subcarriers in the system. For l = 0, · · · , Ng − 1, the number of channels assigned to subcarrier i is denoted as Ngi . As different subcarriers transmit over Ng non-overlapping set of channels, we have Ngi = Nc . i=0
For secure group generation, first, generate a pseudo-random binary sequence using a 32-bit linear feedback shift register (LFSR) as in Section 2, which is initialized by a secret sequence shared between the transmitter and receiver.
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Encrypt the generated sequence into a ciphertext using the AES algorithm and a secure key. Pick an integer L ∈ [ N2c , Nc ] and let q = L log2 Nc = LBc . Take q bits from the ciphertext and put them as a q-bit vector e = [e1 , e2 , · · · , eq ]. Second, partition the ciphertext sequence e into L groups, such that each group contains Bc bits. For k = 1, 2, · · · , L, the partition of the ciphertext is represented as pk = [e(k−1)∗Bc +1 , e(k−1)∗Bc +2 , · · · , e(k−1)∗Bc +Bc ], where pk corresponds to the kth Bc -bit vector. For k = 1, 2, · · · , L, denote Pk as the decimal number corresponding to pk . And denote P = [P1 , P2 , · · · , PL ] as the permutation index vector. For k = 0, 1, 2, · · · , L, denote Ik = [Ik (0), Ik (1), · · · , Ik (Nc − 1)] as the index vector at the kth step. The secure permutation scheme of the index vector is achieved through the following steps: 1. Initially, the index vector is I0 = [I0 (0), I0 (1), · · · , I0 (Nc − 1)] and the permutation index is P = [P1 , P2 , · · · , PL ]. We start with I0 = [0, 1, · · · , Nc −1]. 2. For k = 1, switch I0 (0) and I0 (P1 ) in index vector I0 to obtain I1 . In other words, I1 = [I1 (0), I1 (1), · · · , I1 (Nc − 1)], where I1 (0) = I0 (P1 ), I1 (P1 ) = I0 (0), and I1 (m) = I0 (m) for m = 0, P1 . 3. Repeat the previous step for k = 2, 3, · · · , L. In general, if we already have Ik−1 = [Ik−1 (0), Ik−1 (1), · · · , Ik−1 (Nc − 1)], then we can obtain Ik = [Ik (0), Ik (1),· · · , Ik (Nc − 1)] through the permutation defined as Ik (k − 1) = Ik−1 (Pk ), Ik (Pk ) = Ik−1 (k − 1), and Ik (m) = Ik−1 (m) for m = k − 1, Pk . 4. After L steps, we obtain the channel center frequency vector as FL = [fIL (0) , fIL (1) , · · · , fIL (Nc −1) ]. 5. Vector FL is used to assign the channels to Ng groups. We assign channels {fIL (0) , fIL (1) , · · · , fIL (Ng0 −1) } to the first group; Assign {fIL (Ng0 ) , fIL (Ng0 +1) , · · · , fIL (Ng0 +Ng1 −1) } to the second group, and so on. Because each frequency index appears in FL once and only once, the proposed algorithm ensures that all the subcarriers are transmitting on non-overlapping sets of channels. 4.2
Multi-Carrier AJ-MDFH without Diversity
In this case, each subcarrier transmits independent bit stream. The spectral efficiency of the AJ-MDFH system can be increased significantly. Let Bc = log2 Nc and Bg = log2 Ng , then the number of bits transmitted by the MC-AJMDFH within each hopping period is BMC = (Bc − Bg )Ng = (Bc − log2 Ng )Ng . BMC is maximized when Bg = Bc − 1 or Bg = Bc − 2, which results in BMC = 2Bc −1 . Note that the number of bits transmitted by the AJ-MDFH within each hopping period is Bc , it can be seen that BMC > Bc as long as Bc > 2. Take Nc = 256 for example, then the transmission efficiency of AJ-MDFH can be Bc −1 increased by BBMcC = 2 Bc = 16 times. We assume that jamming is random and equally distributed among all the groups. Then the overall carrier detection error probability Pe of MC-AJ-MDFH is equal to that corresponding to each subcarrier fk for k = 1, · · · , Ng . Let Pe,k
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denote the carrier detection error probability corresponding to the kth subcarrier or the kth group, then we have Pe = Pe,k , and Pe,k = P0,k · P {incorrect carrier detection|signal not jammed} + P1,k · P {incorrect carrier detection|signal jammed},
(10)
where P0,k , P1,k denote the probability that the kth subcarrier is jamming-free or jammed, respectively. 4.3
Multi-carrier AJ-MDFH with Diversity
Under the multi-band jamming, diversity needs to be introduced to the AJMDFH system for robust jamming resistance especially. A natural solution to achieve frequency diversity is to transmit the same or correlated information through multiple subcarriers. The number of subcarriers needed to convey the same information differs in different jamming scenarios. Ideally, the number of correlated signal subcarriers should not be less than the number of jammed bands. At the receiver, the received signals from different diversity branches are often combined for joint signal detection. To achieve better performance, appropriate diversity combination schemes need to be selected for different metrics used. In this paper, we propose to use the equal gain combination scheme [8] while choosing R3 (ri ) be the detection metric, since it can also be regarded as a normalized square-law metric. Assume that the same infor10 MDFH mation is transmitted through conventional FH AJ−MDFH the hopping frequency index of MC−AJ−MDFH: without diversity MC−AJ−MDFH: with diversity 10 Nd subcarriers over Nd groups {Gn1 , Gn2 , · · · , GnNd } simultaneously, each group has the 10 same number of channels, denoted as Ngc . Note that the se10 cure group generation algorithm ensures that the channel index 10 −20 −15 −10 −5 0 5 10 15 20 in each group is random and JSR(dB) does not necessarily come in ascending or descending order. Let Fig. 4. The performance comparison under 2band ID jamming. R3 (rinl ) denote detection metric value for ith channel in group Gnl , then the active hopping frequency index can be estimated as kˆ = d nl arg min1≤i≤Ngc N l=1 R3 (ri ). The diversity order Nd can be dynamic in different jamming scenarios to achieve tradeoff between performance and efficiency. 2
BER
0
−2
−4
−6
5
Simulation Results
In this section, we illustrate the performance of the proposed AJ-MDFH and MCAJ-MDFH through a Monte Carlo simulation example using Matlab. We assume
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that the signal is transmitted through AWGN channels and experiences 2-band ID jamming. For conventional FH, BFSK modulation is used; For MDFH, 8-PSK is adopted for ordinary bits modulation; For AJ-MDFH and MC-AJ-MDFH, 32PSK is adopted for ID constellation to combat possible ID jamming. The SNR is taken as Eb /N0 = 10dB. JSR is defined as the ratio of the jamming power to signal power per channel. The jamming bands of the multi-band ID jamming are independently and randomly selected, and each ID jamming randomly transmits a symbol from the constellation of corresponding modulation scheme. The number of available channels is Nc = 64 (Bc = 6). Choose Ng = Nd = 4 and Ng = 32 for MC-AJ-MDFH with and without diversity, respectively. From Figure 4, it can be seen that AJ-MDFH and MC-AJ-MDFH deliver much better performance than MDFH and conventional FH.
6
Conclusion
In this paper, we proposed a highly efficient anti-jamming scheme AJ-MDFH based on message-driven frequency hopping. It was shown that AJ-MDFH is robust under strong jamming and can effectively reduce the performance degradation caused by disguised jamming. Moreover, AJ-MDFH can be extended to multi-carrier AJ-MDFH for higher spectral efficiency and more robust jamming resistance. The proposed approaches can be applied to both civilian and military applications for reliable communication under jamming interference.
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