Secure OFDM System Design and Capacity Analysis under Disguised Jamming

1 Secure OFDM System Design and Capacity Analysis under Disguised Jamming Y uan Liang Jian Ren T ongtong Li Department of Electrical & Computer Engineering, Michigan State Uni versity Email: { liangy11, renjian, tongli } @egr .msu.edu Abstract —In this paper , we propose a secur ely pr ecoded OFDM (SP-OFDM) system for efficient and r eliable transmission under disguised jamming, where the jammer intentionally mis- leads the recei ver by mimicking the characteristics of the autho- rized signal, and causes complete communication failure. More specifically , we bring off a dynamic constellation by introducing secure shared randomness between the legitimate transmitter and recei ver , and hence break the symmetricity between the authorized signal and the disguised jamming. W e analyze the channel capacities of both the traditional OFDM and SP-OFDM under hostile jamming using the arbitrarily varying channel (A VC) model. It is sho wn that the deterministic coding capacity of the traditional OFDM is zero under the worst disguised jamming. On the other hand, due to the secure randomness shar ed between the authorized transmitter and rec eiver , SP-OFDM can achieve a positive capacity under disguised jamming since the A VC channel corresponding to SP-OFDM is not symmetrizable. A remarkable feature of the proposed SP-OFDM scheme is that while achieving strong jamming resistance, it has roughly the same high spectral efficiency as the traditional OFDM system. The robustness of the proposed SP-OFDM scheme under disguised jamming is demonstrated through both theoretic and numerical analyses. Index T erms —OFDM, disguised jamming, arbitrarily varying channel, channel capacity . I . I N T RO D U C T I O N In wireless systems, one of the most commonly used techniques for limiting the effecti v eness of an opponent’ s communication is referred to as jamming, in which the autho- rized user’ s signal is deliberately interfered by the adversary . Along with the wide spread of various wireless devices, especially with the advent of user configurable intelligent devices, jamming attack is no longer limited to battlefield or military related ev ents, but has become an urgent and serious threat to civilian communications as well. In literature [1]–[4], jamming has widely been modeled as Gaussian noise. Based on the noise jamming model and the Shannon capacity formula, C = B log(1 + S N R ) , an intuitiv e impression is that jamming is really harmful only when the jamming po wer is much higher than the signal po wer . Howe v er , this is only partially true. More recently , it has been found that disguised jamming [5]–[8], where the jamming is highly correlated with the signal, and has a power lev el close or equal to the signal power , can be much more destructive than the noise jamming; it can reduce the system capacity to zero even when the jamming po wer equals the signal po wer . Consider the following example: R = S + J + N where S is the authorized signal, J the jamming interference, N the noise independent of J and S , and R the recei ved signal. If the jammer is capable of eav esdropping on the symbol constellation and the codebook of the transmitter, it can simply replicate one of the sequences in the codebook of the legitimate transmitter , the receiver , then, would not be able to distinguish between the authorized sequence and the jamming sequence, resulting in a complete communication failure [9, ch 7.3]. Orthogonal frequency di vision multiplexing (OFDM), due to its high spectral efficiency and robustness under fading channels, has been widely used in modern high speed multi- media communication systems [10], such as L TE and WiMax. Howe v er , unlike the spread spectrum techniques [11], OFDM mainly relies on channel coding for communication reliability under hostile jamming, and has very limited built-in resilience against jamming attacks [12]–[18]. For example, in [12], the bit error rate (BER) performance of the traditional OFDM was explored under full-band and partial band Gaussian jamming, as well as multitone jamming. It was shown that OFDM is quite fragile under jamming, as BER can go abov e 10 − 1 when the jamming power is the same as the signal power . In [15]–[17], the jamming attacks aiming at the pilots in OFDM systems were studied. It was shown that when the system standard is public and no encryption is applied to the transmitted symbol sequence, pilot attacks can completely nullify the channel estimation and synchronization of OFDM, and hence result in complete communication failure. Most existing w ork [12], [13], [17] has been focused on the jamming attacks which damage OFDM by minimizing the signal-to- interference power ratio (SIR). In this paper, we identify the threat to OFDM from the disguised jamming: when the jamming interference is also OFDM modulated, the receiv er can easily be decei ved into synchronizing with the jamming interference instead of the legitimate signal, hence paralyzing the legitimate transmission. In [14], the anti-jamming performance of Frequency Hopped (FH) OFDM system was e xplored. Like the traditional FH system, this approach achiev es jamming resistance through large frequency diversity and sacrifices the spectral efficiency of OFDM. In [18], a collision-free frequency hopping (CFFH) scheme was proposed, where the basic idea was to randomize the jamming interference through frequency domain interleav- ing based on secure, collision-free frequency hopping. The most significant feature of CFFH based OFDM is that it is very effecti ve under partial band jamming, and at the same 2 time, has the same spectral efficiency as the original OFDM. Howe v er , CFFH based OFDM is still fragile under disguised jamming [6]–[8], [19]. T o combat disguised jamming in OFDM systems, a pre- coding scheme was proposed in [8], where extra redundancy is introduced to achiev e jamming resistance. Howe ver , lack of plasticity in the precoding scheme results in inadequate reliability under cognitiv e disguised jamming. As OFDM being identified as a major modulation technique for the 5G systems, there is an ev er increasing need on the de velopment of secure and efficient OFDM systems that are reliable under hostile jamming, especially the destructiv e disguised jamming. If we examine disguised jamming carefully , we can see that the main issue there is the symmetricity between the authorized signal and the jamming interference. Intuitiv ely , to design the corresponding anti-jamming system, the main task is to break the symmetricity between the authorized signal and the jamming interference, or make it impossible for the jammer to achiev e this symmetricity . For this purpose, encryption or channel coding at the bit le vel will not really help, since the symmetricity appears at the symbol level. That is, instead of using a fixed symbol constellation, we hav e to introduce secure randomness to the constellation, and utilize a dynamic constellation scheme, such that the jammer can no longer mimic the authorized user’ s signal. At the same time, the authorized user does not have to sacrifice too much on the performance, efficiency and system complexity . Motiv ated by the observ ations abov e and our pre vious research on anti-jamming system design [6]–[8], [18], [20], in this paper , we propose a securely precoded OFDM (SP- OFDM) system for efficient and reliable transmission under disguised jamming. By integrating adv anced cryptographic techniques into OFDM transceiv er design, we design a dy- namic constellation by introducing shared randomness be- tween the legitimate transmitter and receiver , which breaks the symmetricity between the authorized signal and the jamming interference, and hence ensures reliable performance under disguised jamming. A remarkable feature of the proposed SP- OFDM scheme is that it achie ves strong jamming resistance, but has the same high spectral efficienc y as the traditional OFDM system. Moreover , the change to the physical layer transceiv ers is minimal, feasible and affordable. The robust- ness of the proposed SP-OFDM scheme under disguised jamming is demonstrated through both theoretic and numerical analyses. More specifically , the main contributions of this paper can be summarized as follows: • W e design a highly secure and ef ficient OFDM sys- tem under disguised jamming, named securely precoded OFDM (SP-OFDM), by exploiting secure symbol-lev el precoding basing on phase randomization. The basic idea is to randomize the phase of transmitted symbols using the secure PN sequences generated from the Advanced Encryption Standard (AES) algorithm. The security is guaranteed by the secret key shared only between the legitimate transmitter and receiv er . While SP-OFDM achiev es strong jamming resistance, it does not introduce too much extra coding redundancy into the system and can achiev e roughly the same spectral efficiency as the traditional OFDM system. • W e identify the vulnerability of the synchronization al- gorithm in the original OFDM system under disguised jamming, and propose a secure synchronization scheme for SP-OFDM which is robust ag ainst disguised jamming. In the proposed synchronization scheme, we design an encrypted c yclic prefix (CP) for SP-OFDM, and the synchronization algorithm utilizes the encrypted CP as well as the precoded pilot symbols to estimate time and frequency offsets in the presence of jamming. • W e analyze the channel capacity of the traditional OFDM and the proposed SP-OFDM under hostile jamming using the arbitrarily varying channel (A VC) model. It is shown that the deterministic coding capacity of the traditional OFDM is zero under the worst disguised jamming. At the same time, we prov e that with the secure randomness shared between the authorized transmitter and receiv er , the A VC channel corresponding to SP-OFDM is not symmetrizable, and hence SP-OFDM can achiev e a pos- itiv e capacity under disguised jamming. Note that the authorized user aims to maximize the capacity while the jammer aims to minimize the capacity , we show that the maximin capacity for SP-OFDM under hostile jamming is given by C = log  1 + P S P J + P N  bits/symbol, where P s denotes the signal power , P J the jamming power and P N the noise power . Numerical examples are provided to demonstrate the ef- fectiv eness of the proposed system under disguised jamming and channel fading. Potentially , SP-OFDM is a promising modulation scheme for high speed transmission under hostile en vironments. Moreover , it should be pointed out that the secure precoding scheme proposed in this paper can also be applied to modulation techniques other than OFDM. The rest of this paper is organized as follows. The design of the proposed SP-OFDM system is described in Section II. The synchronization procedure of SP-OFDM is presented in Section III. The symmetricity analysis and capacity ev aluation of SP-OFDM are presented in Section IV. Numerical examples are provided in Section V and we conclude in Section VI. I I . S E C U R E O F D M S Y S T E M D E S I G N U N D E R D I S G U I S E D J A M M I N G In this section, we introduce the proposed anti-jamming OFDM system with secure precoding and decoding, named as securely procoded OFDM (SP-OFDM). A. T ransmitter Design with Secur e Precoding The block diagram of the proposed system is shown in Fig. 1. Let N c be the number of subcarriers in the OFDM system and Φ the alphabet of transmitted symbols. For i = 0 , 1 , · · · , N c − 1 and k ∈ Z , let S k,i ∈ Φ denote the symbol transmitted on the i -th carrier of the k -th OFDM block 1 . 1 In literature, the term OFDM symbol is often used to denote the symbol block transmitted in one OFDM symbol period. In this paper , to avoid the ambiguity with the data symbols transmitted at each subcarrier, we choose to use the term OFDM block instead. 3 C hannel C od i ng & M a ppi ng S e c ur e P r e c odi ng I F F T I ns e r t S e c ur e C P C ha nne l D e c odi ng & D e m a ppi ng S e c ur e D e c odi ng FFT R e m ove S e c ur e C P + S k S k ~ P r e - FFT S ync P os t - FFT S ync j a m m i ng i nt e r f e r e nc e x ( t ) + no i s e n ( t ) s ( t ) r ( t ) Fig. 1: Anti-jamming OFDM design through secure precoding and decoding. P N S eq u en c e G en er a t o r A E S E n c r y p t i o n S e c r e t K e y M - P S K M a p p i n g exp ( - j Θ 0 , 0 ) , exp ( - j Θ 0 , 1 ) , ... T i m e C l o c k Fig. 2: Secure phase shift generator W e denote the symbol vector of the k -th OFDM block by S k = [ S k, 0 , S k, 1 , · · · , S k,N c − 1 ] T . The input data stream is first fed to the channel encoder, mapped to the symbol vector S k , and then fed to the proposed symbol-le vel secure precoder . As pointed out in [7], [20]–[22], a key enabling factor for reliable communication under disguised jamming is to intro- duce shared randomness between the transmitter and receiv er , such that the symmetry between the authorized signal and the jamming interference is broken. T o maintain full spectral efficienc y of the traditional OFDM system, the precoding is performed by multiplying an in vertible N c × N c precoding matrix P k to the symbol vector S k , i.e., ˜ S k = P k S k . (1) In this paper, we design the precoding matrix P k to be a diagonal matrix as P k = diag ( e − j Θ k, 0 , e − j Θ k, 1 , · · · , e − j Θ k,N c − 1 ) . (2) That is, a random phase shift is applied to each transmitted symbol; more specifically , for i = 0 , 1 , · · · , N c − 1 and k ∈ Z , a random phase shift − Θ k,i is applied to the symbol transmitted on the i -th carrier of the k -th OFDM block. The phase shift changes randomly and independently across sub- carriers and OFDM blocks, and is encrypted so that the jammer has no access to it. More specifically , { Θ k,i } is generated through a secure phase shift generator as shown in Fig. 2. The secure phase shift generator consists of three parts: (i) a pseudo-noise (PN) sequence generator; (ii) an Advanced Encryption Standard (AES) [23] encryption module; and (iii) an M -PSK mapper . The PN sequence generator generates a pseudo-random sequence, which is then encrypted with AES. The encrypted sequence is further con verted to PSK symbols using an M - PSK mapper, where M is a power of 2 , and e very log 2 M bits are conv erted to a PSK symbol. T o facilitate the syn- chronization process, the PN sequence generator is initialized in the follo wing way: each party is equipped with a global time clock, and the PN sequence generators are reinitialized at fixed intervals. The new state for reinitialization, for example, T s T CP , 2 T CP , 1 180 ° pha s e s hi f t O F D M B l oc k B ody CP 2 CP 1 Fig. 3: An OFDM wav eform example with secure cyclic prefix, illustrated with a 180 ◦ phase shift on CP1. can be the elapsed time after a specific reference epoch in seconds for the time being, which is public. As the initial state changes with each reinitialization, no repeated PN sequence will be generated. The security , as well as the randomness of the generated phase shift sequence, are guaranteed by the AES encryption algorithm [23], for which the secret encryption key is only shared between the authorized transmitter and receiv er . Hence, the phase shift sequence is random and unaccessible for the jammer . The resulted symbol vector from the secure precoding, ˜ S k , is then used to generate the body of OFDM block through IFFT , whose duration is T s . In OFDM transceiver design, the synchronization module plays a crucial role: OFDM requires both accurate time and frequency synchronization to av oid inter-symbol interference (ISI) and inter-carrier interference (ICI). In SP-OFDM, we propose a cyclic prefix (CP) based synchronization algorithm, as in traditional OFDM. Ho wev er , SP-OFDM differs in that its CP is encrypted to ensure the security under disguised jamming. B. Cyclic Prefix Design with Secure Pr ecoding In traditional OFDM, CP has three major functions: (i) eliminating the ISI between neighboring blocks; (ii) con verting the linear con volution of OFDM block body with the channel impulse response into circular con volution under multi-path channel fading; and (iii) eliminating the ICI introduced by multipath propagation. As CP is a copy of the tail of OFDM block body , we can calculate the correlation between CP and the tail of OFDM block to estimate the starting point of each OFDM block [24] when disguised jamming is absent. Howe ver , as to be shown in Section III, the traditional CP based synchronization is fragile under disguised jamming. As shown in Fig. 3, to ensure the robustness of synchronization, in SP-OFDM, we apply a secure phase shift to part of the CP for each OFDM block. More specifically , the CP of each OFDM block is divided into two parts: for the first part, with a duration of T C P, 1 , a secure phase shift is applied to the signal. W e name this part of CP as CP1; while for the second part, which is of length T C P, 2 , no special processing is applied. W e name the second part as CP2. CP1 is used for ef fectiv e synchronization under disguised jamming; CP2 maintains the functions of the original CP . T o avoid ISI and ICI, both T C P, 1 and T C P, 2 are chosen to be longer than the maximum delay spread of the channel. 4 T s - T CP , 2 - T CP t u k ( t ) T CP , 1 T b 1 - 1 0 Fig. 4: The wav eform of u k ( t ) with C k = − 1 . T o ensure the security , the phase shift applied to CP1 is en- crypted and varies for each OFDM block. The corresponding secure phase shift sequence can be generated using the same phase shift generator proposed in Fig. 2, with a much lower generation rate, since only one phase shift symbol is needed per OFDM block. Let s k ( t ) denote signal of the k -th OFDM block in the time domain by aligning the beginning of the OFDM block body at t = 0 , and C k denote the phase shift symbol applied to its CP1; let u ( t ) be the unit step function, T C P = T C P, 1 + T C P, 2 and T s denote the duration of OFDM block body . Define function u k ( t ) as u k ( t ) 4 = C k [ u ( t + T C P ) − u ( t + T C P, 2 )] + u ( t + T C P, 2 ) − u ( t − T s ) . (3) An example of u k ( t ) with C k = − 1 is plotted in Fig. 4. For SP-OFDM with secure CP , s k ( t ) can be expressed as s k ( t ) = 1 N c N c − 1 X i =0 ˜ S k,i e j 2 πi T s t u k ( t ) , (4) where ˜ S k,i = S k,i e − j Θ k,i . Let T b = T s + T C P denote the duration of an OFDM block. Then the entire OFDM signal in the time domain can be expressed as s ( t ) = ∞ X k = −∞ s k ( t − k T b ) . (5) Even though the receiver can generate identical phase shift sequences used in CP1 generation from the design of Fig. 2, there will still be an offset between the two generated sequences considering the delays in communication and the mismatch between the time clocks. Let C k and ˜ C k denote the phase shift symbols generated at the transmitter and receiv er respectiv ely , and we hav e C k = ˜ C k + k 0 , ∀ k . (6) Since the phase shift sequences are generated from the global time clock, the of fset k 0 is bounded. The of fset k 0 can be estimated by the synchronization module at the recei ver . Note that synchronization is needed for the precoding matrix sequence P k as well; for the ease of synchronization, we pair the CP phase shift symbol C k with the precoding matrix P k for each OFDM block k ; that is, for each CP phase shift symbol generated, we generate N c phase shift symbols in parallel as the sub-carrier phase shifts. In this way , the two phase shift sequences are synchronized, in the sense that once the synchronization on the CP phase shift sequence is obtained, the synchronization on the precoding matrices is achiev ed automatically . C. Receiver Design with Secur e Decoding W e consider an additive white Gaussian noise (A WGN) channel under hostile jamming. The transmitted OFDM signal is subject to an A WGN term, denoted by n ( t ) , and an additive jamming interference x ( t ) . The receiv ed OFDM signal can be expressed as r ( t ) = s ( t − t 0 ) e j ( ω 0 t + φ 0 ) + x ( t ) + n ( t ) , (7) where t 0 , ω 0 and φ 0 denote the time, frequency and phase off- sets between the transmitter and receiv er , respecti vely . Without loss of generality , we can assume that t 0 ∈ [0 , T b ) . As in the traditional OFDM system, the synchronization module of SP-OFDM consists of tw o stages: a pr e-FFT synchr onization , which makes use of the correlation between the secure CP and the OFDM body tail to roughly estimate the offsets, and a post-FFT synchr onization , which makes use of the pilot symbols inserted to certain sub-carriers to obtain a fine estimation. The phase shift of fset k 0 is also estimated in the pre-FFT stage. The detailed algorithm and analysis on the synchronization of SP-OFDM will be presented in Section III. The demodulation module at the recei ver will crop the CP to obtain the body of each OFDM block, and apply FFT to obtain the frequency component at each sub-carrier . Under perfect synchronization, the receiv ed signal of the k -th OFDM block body can be expressed as r k ( t ) = s k ( t ) + x k ( t ) + n k ( t ) , t ∈ [0 , T s ) , (8) where x k ( t ) and n k ( t ) are the jamming interference and noise ov erlaid on the k -th OFDM block, respectively . The frequency components of jamming and noise can be calculated as J k,i = N c − 1 X m =0 x k ( mT s N c ) e − j 2 πi N c m , i = 0 , 1 , · · · , N c − 1 , (9) ¯ N k,i = N c − 1 X m =0 n k ( mT s N c ) e − j 2 πi N c m , i = 0 , 1 , · · · , N c − 1 , (10) where T s N c is the sampling interval. For an A WGN chan- nel, ¯ N k,i ’ s are i.i.d. circularly symmetric comple x Gaus- sian random variables with variance σ 2 . After applying FFT to the received signal, a symbol vector ˜ R k = [ ˜ R k, 0 , ˜ R k, 1 , · · · , ˜ R k,N c − 1 ] T is obtained for the k -th transmit- ted OFDM block. That is, ˜ R k = P k S k + J k + ¯ N k . (11) where J k = [ J k, 0 , J k, 1 , · · · , J k,N c − 1 ] T , (12) and ¯ N k = [ ¯ N k, 0 , ¯ N k, 1 , · · · , ¯ N k,N c − 1 ] T . (13) The secure decoding module multiplies the inv erse matrix of P k to ˜ R k , which results in the symbol vector R k = S k + P − 1 k J k + P − 1 k ¯ N k , (14) where R k = [ R k, 0 , R k, 1 , · · · , R k,N c − 1 ] T , with R k,i = S k,i + e j Θ k,i J k,i + N k,i , (15) 5 Normalized time offset ( τ / T b ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Normalized correlation coefficient 0 0.2 0.4 0.6 0.8 1 1.2 Disguised jamming time offset Legitimate signal time offset Fig. 5: Correlation coefficients of the original OFDM under disguised jamming. where N k,i = e j Θ k,i ¯ N k,i , and Θ k,i is uniformly distributed ov er { 2 π i M | i = 0 , 1 , · · · , M − 1 } . Note that for any circularly symmetric Gaussian random variable N , e j θ N and N have the same distrib ution for any angle θ [25, p66]; that is, N k,i is still a circular symmetric complex Gaussian random variable of zero-mean and variance σ 2 . T aking the delay in the communication system into consideration, in this paper, we assume that the authorized user and the jammer do not hav e pre-knowledge on the sequence of each other . I I I . S Y N C H RO N I Z AT I O N I N S P - O F D M U N D E R D I S G U I S E D J A M M I N G In this section, first, we show the vulnerability of the synchronization process in tradition OFDM under disguised jamming attacks; then we propose the synchronization algo- rithm of SP-OFDM and prov e its effecti veness under hostile jamming. In modern OFDM systems, there are generally two kinds of approaches to achiev e signal synchronization: (i) making use of the correlation between the CP and the tail of each OFDM block [24]; or (ii) inserting certain training symbols in ev ery OFDM frame [26]. Howe ver , neither of these two approaches is rob ust under malicious jamming, especially disguised jamming, where the jammer modulates the inference with OFDM and decei ve the receiv er into synchronizing with the disguised jamming instead of the legitimate signal. For the training sequence based synchronization approach, ev en if the training sequence is not public, there is still a chance for the jammer to eavesdrop on the training sequence, and then generate the OFDM modulated disguised jamming with the true training sequence. Synchronization of traditional OFDM under disguised jamming: T o demonstrate the damage of disguised jamming, we calculate the CP based correlation coef ficients of the traditional OFDM signal at dif ferent time of fsets in the A WGN channel under an OFDM modulated disguised jamming. W e av erage the correlation coef ficients over multiple OFDM blocks, and the result is shown in Fig. 5. W ithout pr oper encryption applied to the signal, the legitimate signal and the jamming interfer ence are completely symmetric; we can observe peaks of the correlation coefficients at two differ ent time offsets, one corresponding to that of the legitimate signal and the other corr esponding to that of the disguised jamming. If the jamming power is the same as the signal power , then the probability that the recei ver chooses to synchronize with jamming is 50% . Ob viously , a complete communication failure occurs when the receiver chooses to synchronize with the disguised jamming instead of the legitimate signal. T o address this problem, in the synchronization algorithm of SP-OFDM, we apply encrypted phase shifts to the sub- carriers and CP . For the ease of analysis, in the following, we consider an A WGN channel model; the effecti veness of the proposed algorithm in multi-path fading channels will be verified through numerical analysis in Section V. Even though our goal is to guarantee the robustness of SP-OFDM under disguised jamming, in the following analysis, we do not assume any specific form on the jamming interference x ( t ) , that is, we prove the robustness of our algorithm under any form of jamming attacks. W ithout loss of generality , we denote the combined term of jamming and noise as z ( t ) = x ( t )+ n ( t ) , and the received signal can be expressed as r ( t ) = s ( t − t 0 ) e j ( ω 0 t + φ 0 ) + z ( t ) . (16) A. Pre-FFT Synchr onization In the pre-FFT stage, we estimate the encrypted phase shift sequence of fset k 0 , time of fset t 0 and the fractional part of w 0 T s / 2 π for frequency of fset w 0 . Since the phase shift sequence C k is generated from the global time clock, the receiv er has rough bounds on k 0 relativ e to the arriv al time of the signal. W e denote the finite candidate set of offset k 0 by K . In the traditional OFDM system, the CP correlation based synchronization algorithm is deri ved from the maximum- likelihood (ML) rule [24], [27]. Howe ver , since the jamming distribution is unspecified in our case, the ML rule is not appli- cable. Instead, we prove the robustness of the synchronization algorithm of SP-OFDM using the Chebychev inequality [28, Theorem 5.11]. In the pre-FFT stage, the recei ver calculates the following correlation coefficient Y k ( τ , d ) 4 = Z τ − T C P, 2 + kT b τ − T C P + kT b r ( t ) r ∗ ( t + T s ) ˜ C ∗ k + d d t, k ∈ Z ∗ , (17) for τ ∈ [0 , T b ) , d ∈ K . W e have the following proposition on Y k ( τ , d ) , whose proof is given in the appendix. Proposition 1. If the fourth moment of z ( t ) is bounded for any time instant t , i.e., E {| z ( t ) | 4 } < ∞ , ∀ t ∈ R , then as K → + ∞ , we have 1 K K − 1 X k =0 Y k ( τ , d ) =        P S N c v ( τ + T b − t 0 ) e − j ω 0 T s , d = k 0 − 1 , P S N c v ( τ − t 0 ) e − j ω 0 T s , d = k 0 , P S N c v ( τ − T b − t 0 ) e − j ω 0 T s , d = k 0 + 1 , 0 , other wise, (18) almost surely (a.s.), wher e v ( τ ) 4 =    τ + T C P, 1 , − T C P, 1 ≤ τ < 0 , T C P, 1 − τ , 0 ≤ τ < T C P, 1 , 0 , other wise, (19) 6 and P S is the average symbol power of constellation Φ . Basing on Proposition 1, to estimate t 0 and k 0 , we search for τ and d which can maximize | 1 K P K − 1 k =0 Y k ( τ , d ) | for some K . Meanwhile, after we obtain t 0 and k 0 , the phase of the av erage correlation coefficient 1 K P K − 1 k =0 Y k ( t 0 , k 0 ) is − w 0 T s mo d 2 π , (20) where we can estimate the fractional part of w 0 T s / 2 π as well. In practice, the jamming interference should be peak power bounded considering the constraints in RF , so we can ensure that the fourth moment of z ( t ) is bounded. The selection of K depends on the power and the form of the jamming interference. In Section V, we will show that under a disguised jamming, SP-OFDM is able to obtain relati vely accurate estimation results with 25 to 30 OFDM blocks. As in the traditional OFDM, the CP based synchronization is only able to provide a coarse estimation of time offset t 0 , especially under multi-path fading, and it requires a fine estimation on the time of fset at the post-FFT stage. In addition, from (21), it can be seen that even for a very minor estimation error on the carrier frequency , there still may be an essential phase offset. As long as the range of the time estimation error is smaller than the duration of CP2, without loss of generality , we can model the signal after pre-FFT synchronization as r 0 ( t ) = s ( t − t 0 0 ) e j ( 2 π ( n 0 + ζ 0 ) T s t + φ 0 ) + z 0 ( t ) , (21) where z 0 ( t ) is the jamming interference after pre-FFT syn- chronization, t 0 0 ∈ [0 , T C P, 2 ) is the remaining time offset, 2 π ( n 0 + ζ 0 ) /T s is the remaining frequency offset, n 0 is an integer and | ζ 0 |  1 . B. P ost-FFT Synchr onization In this stage, we first estimate n 0 + ζ 0 after demodulating the synchronized signal r 0 ( t ) in (21) using FFT . Suppose n 0 satisfies N l ≤ n 0 ≤ N u , (22) where integers N l and N u are determined by the maximal frequency offset between the transmitter and receiv er . Basing on (21), to demodulate the k -th OFDM block, the receiv er applies FFT to signal r 0 ( t ) within interval [ k T b , k T b + T s ) . The received signal of k -th OFDM block after alignment can be expressed as r 0 k ( t ) = s k ( t − t 0 0 ) e j ( 2 π ( n 0 + ζ 0 ) T s t + φ k ) + z 0 k ( t ) , t ∈ [0 , T s ) , (23) where φ k = φ 0 + 2 π ( n 0 + ζ 0 ) T b T s k , (24) and z 0 k ( t ) = z 0 ( t + k T b ) . (25) Considering the frequency offset n 0 , the receiv er samples the receiv ed signal with a sampling frequency N c + N u − N l T s . Let N 0 c 4 = N c + N u − N l . For 0 ≤ i < N 0 c , the FFT applied to r 0 k ( t ) can be expressed as R k ( i ) = N 0 c − 1 X m =0 r 0 k ( mT s N 0 c ) e − j 2 πi N 0 c m = e j φ k N c N 0 c − 1 X i 0 =0 ˜ S k,i 0 e − j 2 πt 0 0 T s i 0 (1 − e j 2 πζ 0 ) 1 − e j 2 π ( n 0 + ζ 0 + i 0 − i ) N 0 c + Z 0 k ( i ) , (26) where Z 0 k ( i ) = N 0 c − 1 X m =0 z 0 k ( mT s N 0 c ) e − j 2 πi N 0 c m . (27) Since we assume | ζ 0 |  1 , for 0 ≤ i < N 0 c , we can neglect the ICI in (26) and approximate R k ( i ) as R k ( i ) = N 0 c N c e j φ k e − j 2 πt 0 0 T s [( i − n 0 ) mo d N 0 c ] ˜ S 0 k,i − n 0 + Z 0 k ( i ) , (28) where ˜ S 0 k,i =  ˜ S k, ( i mo d N 0 c ) , 0 ≤ i mo d N 0 c < N c , 0 , otherwise . (29) The post-FFT synchronization generally utilizes the pilot symbols inserted at certain sub-carriers. For the ease of analysis, we assume a pilot symbol p is placed at sub-carrier i p of each OFDM block. Note that, as the precoding matrix sequence is synchronized with the CP phase shift sequence, the precoding matrix sequence is synchronized at the receiv er after pre-FFT synchronization. W e calculate the follo wing correlation coefficients for each OFDM block k : Γ k ( i ) 4 = R k ( i ) R ∗ k +1 ( i ) e j (Θ k,i p − Θ k +1 ,i p ) . (30) W e hav e the following proposition on Γ k ( i ) . Proposition 2. If the fourth moment of z ( t ) is bounded for any time t , then as K → + ∞ , we have 1 K K − 1 X k =0 Γ k ( i ) = (  N 0 c N c  2 e j 2 π ( n 0 + ζ 0 ) T b T s | p | 2 , i = n 0 + i p mo d N 0 c , 0 , otherwise , a.s.. (31) Pr oof. Note that Γ k ( i ) can be deriv ed as Γ k ( i ) = [( N 0 c /N c ) 2 e j 2 π ( n 0 + ζ 0 ) T b T s ˜ S 0 k,i − n 0 ˜ S 0∗ k +1 ,i − n 0 + N 0 c N c e j φ k ˜ S 0 k,i − n 0 Z 0∗ k +1 ( i ) + N 0 c N c e j φ k +1 ˜ S 0∗ k +1 ,i − n 0 Z 0 k ( i ) + Z 0 k ( i ) Z 0∗ k +1 ( i )] e j (Θ k,i p − Θ k +1 ,i p ) . (32) Since the phase shifts Θ k,i ’ s are independent across the sub- carriers, following the approach in the pre-FFT analysis, we hav e E { Γ k ( i ) } = (  N 0 c N c  2 e j 2 π ( n 0 + ζ 0 ) T b T s | p | 2 , i = n 0 + i p mo d N 0 c , 0 , otherwise . (33) 7 while the variance of 1 K P K − 1 k =0 Γ k ( i ) con ver ges to 0 as K → + ∞ . Therefore (31) is obtained accordingly . W e skip the details here for brevity . Follo wing Proposition 2, n 0 can be estimated by finding the i which maximizes 1 K P K − 1 k =0 Γ k ( i ) . With the n 0 obtained, we can further estimate the frequenecy estimation error ζ 0 in the pre-FFT stage by e valuating the phase of 1 K P K − 1 k =0 Γ k (( n 0 + i p ) mo d N 0 c ) . After n 0 is estimated, without loss of generality , we can assume n 0 = 0 in the following deri vation. In terms of the time offset t 0 0 , given two pilot symbols p 1 and p 2 located at sub-carriers i p 1 and i p 2 , respecti vely , we ev aluate the following correlation coefficient for each OFDM block k : Υ k ( i p 1 , i p 2 ) = R k ( i p 1 ) R ∗ k ( i p 2 ) p ∗ 1 p 2 e j (Θ k,i p 1 − Θ k,i p 2 ) , (34) and we have the following proposition. Proposition 3. If the fourth moment of z ( t ) is bounded for any time t , then as K → + ∞ , we have 1 K K − 1 X k =0 Υ k ( i p 1 , i p 2 ) =  N 0 c N c  2 e − j 2 πt 0 0 T s ( i p 1 − i p 2 ) | p 1 | 2 | p 2 | 2 , (35) a.s. The proof of Proposition 3 follo ws a similar approach as Proposition 1, and we skip it for brevity . Note that t 0 0 ∈ [0 , T C P, 2 ) , so t 0 0 can be estimated from the phase of 1 K P K − 1 k =0 Υ k ( i p 1 , i p 2 ) . Like wise, the phase offset φ 0 can be estimated by av eraging R k ( i p ) e j Θ k,i p after compensating for the frequency offset. Discussions: Note that under disguised jamming, the estimator av erages multiple OFDM blocks to make use of the encrypted signal for an accurate synchronization. In prac- tice, estimation errors always exist in synchronization, so the receiv er has to keep track of all the offsets, which can be implemented by the moving average approach. The pre-FFT synchronization exploits the correlation be- tween secure CP and the OFDM body tail. The data-aided synchronization approach, i.e., inserting independent training sequence in each OFDM frame, is still an option under disguised jamming if encryption is applied to the training sequence. Ho wev er , the CP based approach experiences less delay in synchronization. By inserting secure CP for each OFDM block, it is easier to keep track of the time offset continuously . In the post-FFT stage, inserting more pilots can accelerate the synchronization process; meanwhile, under fading chan- nels, the channel estimation process necessitates pilot symbols ov er different sub-carrier locations. Channel estimation can be implemented by averaging the received pilot symbols at each sub-carrier location follo wing the approach in synchronization. Howe ver , an important point here is that for time v arying channels, the duration of the OFDM blocks used for averaging should be smaller than the coherence time so that the channel does not change significantly during each estimation. This is guaranteed in practical systems where the whole OFDM frame duration is shorter than the channel coherence time [26]. I V . S Y M M E T R I C I T Y A N D C A PAC I T Y A NA LY S I S U S I N G T H E A V C M O D E L In this section, we analyze the symmetricity and capacity of the proposed SP-OFDM system using the arbitrarily varying channel (A VC) model. Recall that from Section II, under perfect synchronization, the equiv alent channel model of SP- OFDM can be expressed as R = S + e j Θ J + N , (36) where S ∈ Φ , J ∈ C , N ∼ C N (0 , σ 2 I ) , Θ is uniformly distributed over { 2 π i M | i = 0 , 1 , ..., M − 1 } , and C N ( µ , Σ ) denotes a circularly symmetric complex Gaussian distribution with mean µ and variance Σ . For generality , in this section, we do not assume any a priori information on the jamming J , except a finite average power constraint of P J , i.e., E {| J | 2 } ≤ P J . W e will show that the A VC corresponding to SP-OFDM is nonsymmetrizable, and hence the A VC capacity of SP-OFDM is positive under disguised jamming. A. A VC Symmetricity Analysis The arbitrarily varying channel (A VC) model, first intro- duced in [22], characterizes the communication channels with unknown states which may v ary in arbitrary manners across time. For the jamming channel (36) of interest, the jamming symbol J can be viewed as the state of the channel under consideration. The channel capacity of A VC ev aluates the data rate of the channel under the most adverse jamming interference among all the possibilities [29]. Note that unlike the jamming free model where the channel noise sequence is independent of the authorized signal and is independent and identically distributed (i.i.d.), the A VC model considers the possible correlation between the authorized signal and the jamming, as well as the possible temporal correlation among the jamming symbols, which may cause much worse damages to the communication. T o prove the effecti veness of the proposed SP-OFDM under disguised jamming, we need to introduce some basic concepts and properties of the A VC model. First we re visit the definition of symmetrizable A VC channel. Definition 1. [29] [30] Let W ( r | s , x ) denote the con- ditional PDF of the received signal R given the transmitted symbol s ∈ Φ and the jamming symbol x ∈ C . The A VC channel (36) is symmetrizable iff for some auxiliary channel π : Φ → C , ∀ s , s 0 ∈ Φ , r ∈ C , we have Z C W ( r | s , x ) d F π ( x | s 0 ) = Z C W ( r | s 0 , x ) d F π ( x | s ) , (37) wher e F π ( ·|· ) is the pr obability measur e of the output of channel π given the input, i.e., the conditional CDF F π ( x | s ) = Pr { Re ( π ( s )) ≤ Re ( x ) , I m ( π ( s )) ≤ I m ( x ) } , (38) for x ∈ C , s ∈ Φ , wher e π ( s ) denotes the output of channel π given input symbol s . 8 W e denote the set of all the auxiliary channels, π ’ s, that can symmetrize channel (36) by Π , that is, Π =  π | Eq. (37) is satisfied w .r .t. π ∀ s , s 0 ∈ Φ , r ∈ C  . (39) W ith the a verage jamming po wer constraint considered in this paper , we further introduce the definition of l - symmetrizable channel. Definition 2. [30] The A VC channel (36) is called l - symmetrizable under average jamming power constraint iff ther e exists a π ∈ Π such that Z C | x | 2 d F π ( x | s ) < ∞ , ∀ s ∈ Φ . (40) In [30], it was shown that reliable communication can be achiev ed as long as the A VC channel is not l -symmetrizable. Lemma 1. [30, Cor ollary 2] The deterministic coding ca- pacity 2 of A VC channel (36) is positive under any hostile jamming with finite averag e power constraint iff the A VC is not l -symmetrizable. Furthermor e, given a specific average jamming power constraint P J , the channel capacity C in this case equals C = max P S min F J I ( S, R ) , s.t. R C | x | 2 d F J ( x ) ≤ P J , (41) wher e I ( S, R ) denotes the mutual information (MI) between the R and S in (36), P S denotes the pr obability distribution of S over Φ and F J ( · ) the CDF of J . First, we sho w that the traditional OFDM system is l - symmetrizable under disguised jamming. Theorem 1. The traditional OFDM system is l -symmetrizable . Ther efor e, the deterministic coding capacity is zer o under the worst disguised jamming with finite average jamming power . Pr oof. The A VC model of the traditional OFDM system is R = S + J + N . (42) W e will show that when S and J have the same constellation Φ , hence the same finite average power , the A VC channel is l -symmetrizable. It follows from (42) that W ( r | s , s 0 ) = W ( r | s 0 , s ) , ∀ s , s 0 ∈ Φ , r ∈ C . (43) Since Φ has finite average po wer , the av erage power constraint (40) is satisfied by disguised jamming. Hence, channel (42) is l -symmetrizable. From Lemma 1 , a necessary condition for a positiv e A VC deterministic coding capacity is that the channel is not l -symmetrizable. So the traditional OFDM system has zero deterministic coding capacity under disguised jamming with finite average jamming power . Next, we show that with the proposed secure precoding, it is impossible to l -symmetrize the A VC channel (36) corre- sponding to the SP-OFDM system. 2 The deterministic coding capacity is defined by the capacity that can be achiev ed by a communication system, when it applies only one code pattern during the information transmission. In other words, the coding scheme is deterministic and can be readily repeated by other users [31]. Theorem 2. The A VC channel corr esponding to the pr oposed SP-OFDM is not l -symmetrizable . Pr oof. W e prove this result by contradiction. Suppose that there exists a channel π ∈ Π such that the A VC channel is l -symmetrizable. Denote the output of channel π giv en input x by π ( x ) , and define the corresponding A VC channel output for inputs s and s 0 as ˆ R ( s , s 0 ) = s + π ( s 0 ) e j Θ + N , (44) where ˆ R ( s , s 0 ) denotes the channel output. Follo wing (37), ˆ R ( s , s 0 ) and ˆ R ( s 0 , s ) ha ve the same distrib ution. Let ϕ X ( ω 1 , ω 2 ) denote the characteristic function (CF) of a com- plex random variable X . So we have ϕ ˆ R ( s , s 0 ) ( ω 1 , ω 2 ) ≡ ϕ ˆ R ( s 0 , s ) ( ω 1 , ω 2 ) , (45) and ϕ ˆ R ( s , s 0 ) ( ω 1 , ω 2 ) = ϕ [ s + π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) ϕ N ( ω 1 , ω 2 ) , (46) where, for the complex Gaussian noise N , we have ϕ N ( ω 1 , ω 2 ) = e − σ 2 4 ( w 2 1 + w 2 2 ) , ω 1 , ω 2 ∈ ( −∞ , + ∞ ) , (47) which is non-zero ov er R 2 . Thus by eliminating the character- istic functions of the Gaussian noises on both sides of equation (45), we have ϕ [ s + π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) = ϕ [ s 0 + π ( s ) e j Θ ] ( ω 1 , ω 2 ) . (48) for ω 1 , ω 2 ∈ ( −∞ , + ∞ ) . Let s = s 1 + j s 2 , we can then express ϕ [ s + π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) as ϕ [ s + π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) = e j s 1 ω 1 + j s 2 ω 2 ϕ [ π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) , (49) and ϕ [ π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) = E { e j ω 1 Re ( π ( s 0 ) e j Θ )+ j ω 2 I m ( π ( s 0 ) e j Θ ) } = Z C E { e j ω 1 Re ( x e j Θ )+ j ω 2 I m ( x e j Θ ) } d F π ( x | s 0 ) . (50) Recall that under the proposed secure precoding scheme, Θ is uniformly distributed ov er { 2 π i M | i = 0 , 1 , ..., M − 1 } , where M is a po wer of 2 . W e hav e E { e j ω 1 Re ( x e j Θ )+ j ω 2 I m ( x e j Θ ) } = 1 M M − 1 X i =0 e j ω 1 | x | cos( 2 πi M +arg( x ))+ j ω 2 | x | sin( 2 πi M +arg( x )) = 2 M M / 2 − 1 X i =0 cos { ω 1 | x | cos[2 π i/ M + arg( x )] + ω 2 | x | sin[2 π i/ M + arg( x )] } , (51) which is of real value for ω 1 , ω 2 ∈ ( −∞ , + ∞ ) . So ϕ [ π ( s 0 ) e j Θ ] ( ω 1 , ω 2 ) and ϕ [ π ( s ) e j Θ ] ( ω 1 , ω 2 ) are also real-valued ov er R 2 . For s 6 = s 0 and s 0 = s 0 1 + j s 0 2 , e j [( s 1 − s 0 1 ) ω 1 +( s 2 − s 0 2 ) ω 2 ] has non-zero imaginary part for ( s 1 − s 0 1 ) ω 1 + ( s 2 − s 0 2 ) ω 2 6 = nπ , n ∈ Z . Without loss of generality , we assume s 1 6 = s 0 1 . 9 From (48), (49) and (51), for ω 1 + s 2 − s 0 2 s 1 − s 0 1 ω 2 6 = nπ s 1 − s 0 1 , ∀ n ∈ Z , we have ϕ [ π ( s ) e j Θ ] ( ω 1 , ω 2 ) = 0 . (52) On the other hand, the characteristic function of an R V should be uniformly continuous in the real domain [28, Theorem 15.21]. So for any fixed ω 2 ∈ ( −∞ , ∞ ) , we should hav e ϕ [ π ( s ) e j Θ ] ( nπ − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1 , ω 2 ) = lim ω 1 → nπ − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1 ϕ [ π ( s ) e j Θ ] ( ω 1 , ω 2 ) , ∀ n ∈ Z . (53) For ω 1 ∈  ( n − 1) π − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1 , nπ − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1  ∪  nπ − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1 , ( n +1) π − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1  , ϕ [ π ( s ) e j Θ ] ( ω 1 , ω 2 ) ≡ 0 , so ϕ [ π ( s ) e j Θ ] ( nπ − ( s 2 − s 0 2 ) ω 2 s 1 − s 0 1 , ω 2 ) = 0 , ∀ n ∈ Z . (54) Combining (52) and (54), we hav e ϕ [ π ( s ) e j Θ ] ( ω 1 , ω 2 ) = 0 , ∀ ω 1 , ω 2 ∈ ( −∞ , ∞ ) . (55) Howe ver , (55) cannot be a valid characteristic function for any R V . Therefore, the auxiliary channel π does not exist, and Π is empty . Hence, the A VC channel is not l -symmerizable. Follo wing Lemma 1, the result in Theorem 2 implies that the proposed SP-OFDM will always have positiv e capacity under any hostile jamming with finite av erage power constraint. The next subsection is focused on how to calculate the channel capacity of SP-OFDM under hostile jamming. B. Capacity Analysis From Lemma 1, the capacity of channel R = S + e j Θ J + N is given by C = max P S min F J I ( S, R ) , s.t. R C | x | 2 d F J ( x ) ≤ P J . It is hard to obtain a closed form solution of the channel capac- ity for a general discrete transmission alphabet Φ . Howe ver , if we relax the distribution of the transmitted symbol S from the discrete set Φ to the entire complex plane C under an average power constraint, we are able to obtain the follo wing result on channel capacity . Theorem 3. The deterministic coding capacity of SP-OFDM is positive under any hostile jamming. More specifically , let the alphabet Φ = C and the average power of S being upper bounded by P S , then the maximin channel capacity in (41) under average jamming power constraint P J and noise power P N = σ 2 is given by C = log  1 + P S P J + P N  . (56) The capacity is achieved at input distribution C N (0 , P S ) and jamming distribution C N (0 , P J ) . T o prove Theorem 3, we need the following lemma [30, Lemma 4]. Lemma 2. Mutual information I ( S, R ) is concave with r e- spect to the input distribution F S ( · ) and con vex with respect to the jamming distribution F J ( · ) . Pr oof of Theor em 3. First, following Lemma 1 and Theorem 2, we can get that the deterministic coding capacity of SP- OFDM is positive under any hostile jamming. Second, we will ev aluate the channel capacity of SP-OFDM under hostile jamming. When the support of S is Φ = C , the whole complex plane, following Lemma 1, the channel capacity in (41) equals C = max F S min F J I ( S, R ) , (57) s.t. R C | x | 2 d F S ( x ) ≤ P S , (58) R C | x | 2 d F J ( x ) ≤ P J , (59) where F S ( · ) denotes the CDF function of S defined on C , and (58) and (59) denote the av erage power constraints on the input and the jamming, respectiv ely . W e denote the I ( S, R ) w .r .t the input distribution F S ( · ) and the jamming distribution F J ( · ) by φ ( F S , F J ) . Follo wing Lemma 2, φ ( F S , F J ) is concave w .r .t. F S ( · ) and conv ex w .r .t. F J ( · ) . As shown in [32], if we can find the input distribution F ∗ S and the jamming distribution F ∗ J such that φ ( F S , F ∗ J ) ≤ φ ( F ∗ S , F ∗ J ) ≤ φ ( F ∗ S , F J ) , (60) for any F S and F J satisfying the av erage po wer constraints (58) and (59), respectiv ely , then φ ( F ∗ S , F ∗ J ) = C. (61) That is, the pair ( F ∗ S , F ∗ J ) is the saddle point of the max-min problem in equation (57) [33]. Assume the jamming interference is circularly symmetric complex Gaussian with average po wer P J , that is, F ∗ J = C N (0 , P J ) . Note that the phase shift would not change the distribution of a complex Gaussian R V , and the fact that the jamming J and the noise N are independent, hence the jammed channel in this case is equiv alent to a complex A WGN channel with noise power P J + P N , where the capacity achieving input distribution is also a complex Gaussian with power P S , that is, F ∗ S = C N (0 , P S ) . It follows that for any input distribution F S satisfying the power constraint P S , φ ( F S , C N (0 , P J )) ≤ φ ( C N (0 , P S ) , C N (0 , P J )) . (62) On the other hand, when the input distribution is F ∗ S = C N (0 , P S ) , the worst noise in terms of capacity for Gaussian input is Gaussian [9]. Since e j Θ J + N is complex Gaussian with power P J + P N if F ∗ J = C N (0 , P J ) , then for any jamming distribution F J satisfying the power constraint P J , φ ( C N (0 , P S ) , C N (0 , P J )) ≤ φ ( C N (0 , P S ) , F J ) . (63) So the saddle point ( F ∗ S , F ∗ J ) is achie ved at ( C N (0 , P S ) , C N (0 , P J )) , where the corresponding channel capacity is C = log  1 + P S P J + P N  , (64) which completes the proof. 10 T ABLE I: SP-OFDM parameters in numerical results ( T s : duration of OFDM body) Carrier number N c 128 CP1 duration T C P, 1 T s / 8 CP2 duration T C P, 2 T s / 16 Number of candidate phase shift offset |K| 50 Signal-to-noise ratio (dB) 15 Phase shift constellation size M 16 50 40 Phase shift offset 30 20 10 0 0 0.2 Normalized time offset ( τ / T b ) 0.4 0.6 0.8 0 1.2 1 0.8 0.6 0.4 0.2 1 Normalized correlation coefficient (Legitimate signal time offset, Phase shift offset) Fig. 6: Correlation coefficients of SP-OFDM at different time and phase shift sequence offsets under disguised jamming. Normalized time estimation error 10 -4 10 -3 10 -2 10 -1 CDF 0 0.2 0.4 0.6 0.8 1 K = 1 K = 5 K = 10 K = 15 K = 20 K = 25 Normalized frequency estimation error 10 -4 10 -3 10 -2 10 -1 10 0 CDF 0 0.2 0.4 0.6 0.8 1 K = 1 K = 5 K = 10 K = 15 K = 20 K = 25 K = 30 Fig. 7: The synchronization error distribution under A WGN channels with disguised jamming attack. V . N U M E R I C A L R E S U LT S In this section, we ev aluate the synchronization and bit error rate (BER) performances of the proposed SP-OFDM system under disguised jamming attacks through numerical examples. Throughout this section, we consider the case where the malicious user generates disguised jamming using OFDM, with the same format and power le vel as that of the legitimate signal. Example 1: Synchronization performance under dis- guised jamming in A WGN channels: In this example, we verify the robustness of SP-OFDM under disguised jamming in terms of synchronization for A WGN channels. The system parameters are listed in T able I. W e first compute the average correlation coefficients at different time offsets and phase shift sequence offsets for the receiv ed signal as in (17), and the result is plotted in Fig. 6 for K = 40 3 . Here, K denotes the number of OFDM blocks used for estimation. It shows that with the secure precoding scheme, ev en under disguised jam- 3 In the 802.11a WLAN [26], 40 OFDM blocks correspond to 1440 data bytes with 64QAM mapping, while the OFDM frame length can be as large as 2312 bytes. Normalized time estimation error 10 -4 10 -3 10 -2 10 -1 CDF 0 0.2 0.4 0.6 0.8 1 K = 5 K = 10 K = 15 K = 20 K = 25 Normalized frequency estimation error 10 -4 10 -3 10 -2 10 -1 10 0 CDF 0 0.2 0.4 0.6 0.8 1 K = 5 K = 10 K = 15 K = 20 K = 25 K = 30 Fig. 8: The synchronization error distribution under static multi-path fading channels with disguised jamming attack. Normalized time estimation error 10 -4 10 -3 10 -2 10 -1 CDF 0 0.2 0.4 0.6 0.8 1 K = 5 K = 10 K = 15 K = 20 K = 25 K = 30 Normalized frequency estimation error 10 -4 10 -3 10 -2 10 -1 10 0 CDF 0 0.2 0.4 0.6 0.8 1 K = 5 K = 10 K = 15 K = 20 K = 25 K = 30 Fig. 9: The synchronization error distribution under time varying multi-path fading channels with disguised jamming attack. ming, the receiver is able to correctly estimate the time offset as well as the phase shift sequence of fset of the legitimate signal. Then we simulate the synchronization accuracy of SP- OFDM by calculating the cumulati ve distribution functions (CDFs) of the estimation errors with different numbers of OFDM blocks K to average the correlation coefficients. W e normalize the time offset by the duration of one OFDM block T b and the frequency offset by the sub-carrier spacing 1 /T s , and the results are shown in Fig. 7. It can be observed that under the giv en setup, with 25 OFDM blocks to compute the correlation coef ficients, the synchronization algorithm is robust under disguised jamming, where 99% cases hav e less than 0 . 01 normalized time offset estimation errors and 98% cases hav e less than 0 . 04 normalized frequency offset estimation errors. Example 2: Synchronization performance under dis- guised jamming in multi-path fading channels: In this example, we simulate the synchronization accuracy of SP- OFDM under disguised jamming in static and time v arying multi-path fading channels, which are modeled as 4 paths fading channels with a maximum delay spread of 3 T s / 256 . Fig. 8 sho ws the estimation error distribution in the static chan- nel. A slight performance loss is observed compared with the A WGN case, where 98% cases have less than 0 . 02 normalized time of fset estimation errors and 96 . 5% cases have less than 0 . 04 normalized frequency offset estimation errors using 25 OFDM blocks in estimation. T o demonstrate the effecti veness of the synchronization algorithm under slow time varying 11 Code Rate 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 BER 10 -4 10 -3 10 -2 10 -1 10 0 SP-OFDM, SNR = 4 dB OFDM, SNR = 4 dB SP-OFDM, SNR = 10 dB OFDM, SNR = 10 dB Fig. 10: BER performance comparison under disguised jam- ming in A WGN channels: SP-OFDM versus the traditional OFDM system, signal to jamming po wer ratio (SJR) = 0 dB. channels, we introduce a Doppler shift to each path with a maximum value of 2% sub-carrier spacing ( 0 . 02 /T s ) in the multi-path fading channel. Fig. 9 shows the estimation error distribution under the time-varying multi-path fading channel, where around 98% cases ha ve less than 0 . 02 normalized time of fset estimation errors and 96 . 5% cases have less than 0 . 04 normalized frequency of fset estimation errors using 30 OFDM blocks in estimation. The simulation results illustrate the rob ustness of SP-OFDM against disguised jamming attacks under various channel conditions. Example 3: BER performance under disguised jamming in A WGN channels: In this example, we analyze the bit error rate (BER) of the proposed system under disguised jamming in A WGN channels. Perfect synchronization is assumed. W e use the lo w density parity check (LDPC) codes for channel coding, and adopt the parity check matrices from the D VB-S.2 stan- dard [34]. The coded bits are mapped into QPSK symbols. The random phase shifts in the proposed secure precoding are ap- proximated as i.i.d. continuous R Vs uniformly distributed over [0 , 2 π ) . W e observe that such an approximation has negligible difference on BER performance compared with a sufficiently large M . The jammer randomly selects one of the code words in the LDPC codebook and sends it to the receiver after the mapping and modulation. On the receiv er side, we use a soft decoder for the LDPC codes, where the belief propagation (BP) algorithm [35] is employed. The likelihood information in the BP algorithm is calculated using the likelihood function of a general Gaussian channel, where the noise po wer is set to 1 + σ 2 considering the existence of the disguised jamming, and σ 2 is the noise power . That is, the signal to jamming power ratio (SJR) is set to be 0 dB. It should be noted that for more complicated jamming distributions or mapping schemes, customized likelihood functions basing on the jamming dis- tribution will be needed for the optimal performance. Fig. 10 compares the BERs of the communication system studied with and without the proposed secure precoding under dif ferent code rates and SNRs. It can be observed that: (i) under the disguised jamming, in the traditional OFDM system, the BER cannot really be reduced by decreasing the code rate or the noise power , which indicates that without appropriate anti- K 0 parameter 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 BER 10 -4 10 -3 10 -2 10 -1 10 0 SP-OFDM, SNR = 7 dB OFDM, SNR = 7 dB SP-OFDM, SNR = 10 dB OFDM, SNR = 10 dB Fig. 11: BER performance comparison under disguised jam- ming in Rician channels: code rate = 1 / 3 , SJR = 0 dB. Here the K 0 parameter refers to the power ratio between the direct path and the scattered path. jamming procedures, the traditional OFDM cannot achieve reliable communications under disguised jamming; (ii) with the proposed SP-OFDM scheme, when the code rates are below certain thresholds, the BER can be significantly reduced with the decrease of code rates using the proposed secure precoding. This demonstrates that the proposed SP-OFDM system can achiev e a positiv e deterministic channel coding capacity under disguised jamming. Example 4: BER performance under disguised jamming in Rician channels: In this example, we verify the effecti ve- ness of the proposed system in fading channels. W e consider a Rician channel, where the multipath interference is introduced and a strong line of sight (LOS) signal exists [36]. The fading effect is slow enough so that the channel remains unchanged for one OFDM symbol duration. In the simulation, we set the power of the direct path of Rician channel to be 1 and vary the K 0 parameter , which is the ratio between the power of the direct path and that of the scattered path. Fig. 11 shows the BERs for LDPC code rate 1 / 3 under disguised jamming. It can be observed that the proposed system is still effecti ve under the fading channel with a sufficient large K 0 parameter . For a small K 0 parameter , i.e., when the fading is se vere, channel estimation and equalization will be needed to guarantee a reliable communication. V I . C O N C L U S I O N S In this paper , we designed a highly secure and efficient OFDM system under disguised jamming, named securely precoded OFDM (SP-OFDM), by exploiting secure symbol- lev el precoding basing on phase randomization. W e demon- strated the destructive effect of disguised jamming on the traditional OFDM system, and prov ed the robustness of SP- OFDM against disguised jamming in terms of synchronization and channel capacity . First, we showed that the traditional OFDM cannot distinguish between the legitimate signal and disguised jamming in the synchronization process, while SP- OFDM, with the secure CP , can achieve accurate synchro- nization under disguised jamming. Second, we analyzed the channel capacity of the traditional OFDM and the proposed 12 SP-OFDM under hostile jamming using the arbitrarily varying channel (A VC) model. It was shown that the deterministic coding capacity of the traditional OFDM is zero under the worst disguised jamming; on the other hand, with the secure randomness shared between the authorized transmitter and receiv er, the A VC channel corresponding to SP-OFDM is not symmetrizable, and hence SP-OFDM can achiev e a positive capacity under disguised jamming. Both our theoretical and numerical results demonstrated that SP-OFDM is robust under disguised jamming and frequency selectiv e fading. Potentially , SP-OFDM is a promising modulation scheme for high speed transmission under hostile en vironments, and the secure pre- coding scheme proposed in this paper can also be applied to modulation techniques other than OFDM. A P P E N D I X A. Pr oof of Pr oposition 1 Pr oof. Note that r ( t ) r ∗ ( t + T s ) can be calculated as r ( t ) r ∗ ( t + T s ) = s ( t − t 0 ) s ∗ ( t + T s − t 0 ) e − j ω 0 T s + z ( t ) s ∗ ( t + T s − t 0 ) e − j ( ω 0 t + ω 0 T s + φ 0 ) + s ( t − t 0 ) e j ( ω 0 t + φ 0 ) z ∗ ( t + T s ) + z ( t ) z ∗ ( t + T s ) . (65) In the following we analyze the four terms on the right-hand- side (RHS) of (65) respectiv ely . First, define Y k, 1 ( τ ) 4 = Z τ − T C P, 2 + kT b τ − T C P + kT b s ( t − t 0 ) s ∗ ( t + T s − t 0 ) d t, (66) for k ∈ Z ∗ , τ ∈ [0 , T b ) . W e ev aluate the expectation of Y k, 1 ( τ ) ˜ C ∗ k + d for d ∈ K . Note that for t ∈ [ τ − T C P + k T b , τ − T C P, 2 + k T b ] , where τ ∈ [0 , T b ) , we have s ( t − t 0 ) = k +1 X l = k − 1 s l ( t − t 0 − l T b ) , (67) s ( t + T s − t 0 ) = k +1 X l = k − 1 s l ( t + T s − t 0 − l T b ) . (68) Note that since the OFDM blocks are zero-mean and indepen- dent, for k 1 6 = k 2 , we have E  s k 1 ( t 1 ) s ∗ k 2 ( t 2 )  = 0 , ∀ t 1 , t 2 ∈ R . (69) So we focus on Z τ − T C P, 2 + kT b τ − T C P + kT b k +1 X l = k − 1 s l ( t − t 0 − lT b ) s ∗ l ( t + T s − t 0 − lT b ) d t = 1 N 2 c 1 X l = − 1 Z τ − lT b − t 0 − T C P, 2 τ − lT b − t 0 − T C P N c − 1 X i 1 =0 ˜ S l + k,i 1 e j 2 πi 1 T s t u l + k ( t ) N c − 1 X i 2 =0 ˜ S ∗ l + k,i 2 e − j 2 πi 2 T s t u ∗ l + k ( t + T s ) d t. (70) Since for i 1 6 = i 2 , E { ˜ S k,i 1 ˜ S ∗ k,i 2 } = 0 , we further focus on 1 N 2 c 1 X l = − 1 N c − 1 X i =0 | ˜ S l + k,i | 2 Z τ − lT b − t 0 − T C P, 2 τ − lT b − t 0 − T C P u l + k ( t ) u ∗ l + k ( t + T s ) d t. (71) Define function v k ( τ ) as v k ( τ ) 4 = Z τ − T C P, 2 τ − T C P u k ( t ) u ∗ k ( t + T s ) d t =            ( τ + T C P, 1 ) C k , − T C P, 1 ≤ τ < 0 , τ + ( T C P, 1 − τ ) C k , 0 ≤ τ < T C P, 2 , T C P, 2 + ( T C P, 1 − τ ) C k , T C P, 2 ≤ τ < T C P, 1 , T C P − τ , T C P, 1 ≤ τ < T C P , 0 , other wise. (72) So (71) can be expressed as 1 N 2 c 1 X l = − 1 N c − 1 X i =0 | ˜ S l + k,i | 2 v l + k ( τ − l T b − t 0 ) . (73) In addition, since the phase shift symbols are zero-mean and independent, for τ ∈ R , we have E { v k 1 ( τ ) C ∗ k 2 } =  v ( τ ) , k 1 = k 2 , 0 , k 1 6 = k 2 . (74) So the expectation of Y k, 1 ( τ ) ˜ C ∗ k + d is E { Y k, 1 ( τ ) ˜ C ∗ k + d } =        P S N c v ( τ + T b − t 0 ) , d = k 0 − 1 , P S N c v ( τ − t 0 ) , d = k 0 , P S N c v ( τ − T b − t 0 ) , d = k 0 + 1 , 0 , other wise. (75) whose maximum is achie ved at τ = t 0 and d = k 0 . In addition, since constellation Φ is a finite set, the variance of Y k, 1 ( τ ) ˜ C ∗ k + d is bounded for any possible k , τ and d , while giv en τ and d , E { Y k 1 , 1 ( τ ) ˜ C ∗ k 1 + d Y k 2 , 1 ( τ ) ˜ C ∗ k 2 + d } = 0 , for | k 1 − k 2 | > 1 . (76) So as K → ∞ , the variance of 1 K P K − 1 k =0 Y k, 1 ( t 0 ) ˜ C ∗ k + k 0 con verges to 0 , and using the Chebychev inequality , we hav e 1 K K − 1 X k =0 Y k, 1 ( t 0 ) ˜ C ∗ k + k 0 = P S T C P, 1 N c , a.s.. (77) Second, define Y k, 2 ( τ ) 4 = Z τ − T C P, 2 + kT b τ − T C P + kT b z ( t ) s ∗ ( t + T s − t 0 ) e − j ( ω 0 t + ω 0 T s + φ 0 ) d t, (78) and Z k,l ( ω , τ , t 0 ) 4 = Z τ − T C P, 2 τ − T C P z ( t + k T b ) e j ωt u l ( t − lT b + T s − t 0 ) d t. (79) It can be deriv ed that Y k, 2 ( τ ) = 1 X l = − 1 N c − 1 X i =0 e j 2 πi T s [ − lT b − t 0 ] ˜ S k + l,i Z k,l ( 2 πi T s − ω 0 , τ , t 0 ) N c e j ( kω 0 T b + ω 0 T s + φ 0 ) . (80) Considering the delay in signal processing, we assume the jamming term Z k,l ( 2 π i T s − ω 0 , τ , t 0 ) is independent of the transmitted symbol ˜ S k + l,i in (80). Therefore, we hav e E { Y k, 2 ( τ ) ˜ C ∗ k + d } = 0 , ∀ k ∈ Z ∗ , τ ∈ [0 , T b ) , d ∈ K . (81) 13 Note that the fourth moment of jamming interference z ( t ) is bounded, so are the variances of z ( t ) of Y k, 2 ( τ ) ˜ C ∗ k + d . In addition, for τ ∈ [0 , T b ) , d ∈ K , we have E { Y k 1 , 2 ( τ ) ˜ C ∗ k 1 + d Y ∗ k 2 , 2 ( τ ) ˜ C k 2 + d } = 0 , ∀| k 1 − k 2 | > 1 . (82) Therefore, 1 K K − 1 X k =0 Y k, 2 ( τ ) ˜ C ∗ k + d = 0 , ∀ τ ∈ [0 , T b ) , d ∈ K , a.s.. (83) Third, define Y k, 3 ( τ ) 4 = Z τ − T C P, 2 + kT b τ − T C P + kT b s ( t − t 0 ) e j ( ω 0 t + φ 0 ) z ∗ ( t + T s ) d t. Follo wing the same argument as in the deri vation of (83) on Y k, 2 ( τ ) , we have 1 K K − 1 X k =0 Y k, 3 ( τ ) ˜ C ∗ k + d = 0 , ∀ τ ∈ [0 , T b ) , d ∈ K , a.s.. (84) At last, we define Y k, 4 ( τ ) 4 = Z τ − T C P, 2 + kT b τ − T C P + kT b z ( t ) z ∗ ( t + T s ) d t. Considering the security of phase shift sequence C k and the delay in signal processing, we assume that for t ≤ ( k + 1) T b + T s − T C P, 2 , the jammer is unable to recov er ˜ C k + d , ∀ d ∈ K . Since the fourth moment of z ( t ) is bounded, we can hav e 1 K K − 1 X k =0 Y k, 4 ( τ ) ˜ C ∗ k + d = 0 , ∀ τ ∈ [0 , T b ) , d ∈ K , a.s.. (85) In conclusion, by av eraging the correlation coefficients Y k ( τ , d ) over multiple OFDM blocks, (18) can be ob- tained. R E F E R E N C E S [1] T . 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