UFDC Home  myUFDC Home  Help 



Full Text  
JAMMING MITIGATION THROUGH COLLABORATION By JANGWOOK MOON A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2005 Copyright 2005 by JangWook Moon I dedicate my dissertation to all of my family members... father MokYeun, mother JungOk, sister JeomSook, brothers JeEon, HyungJoon, my lovely wife HyunJung, and daughter Raina ACKNOWLEDGMENTS Four years in Gainesville was not easy for me. Especially, it was hard to overcome the feeling that I was alone in this foreign country. Tuition and living costs were also big problems for a poor student like me. Anyway, I managed to survive and now I finally graduate. I want to thank Dr. John Shea, Dr. Tan Wong, and all of my committee members for their academic advice. Especially, I could not have produced the research described in this dissertation without Dr. Shea's mentoring and instruction. I also thank all my lab colleagues, and wish them a bright future. This dissertation is a fruit of not only my research but also my family's support. My parents supported me consistently both financially and mentally throughout my life. I can not find any way to thank them too much. I also want to say thank you to my sister Jeom Sook, brothers JeEon, HyunJoon. I could not have gotten my degree without them. My wife, Hyunjung, devoted herself to me. She always treats me warmly. Whenever I was tired and had difficulty, I could overcome due to her. My daughter, Raina, always gave me hope and an objective. They were, and will be, my compass throughout my life. TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ iv LIST OF TABLES ............................. ...... vii LIST OF FIGURES ........ ............. ......... .... viii ABSTRACT ................................. ..... x CHAPTER 1 INTRODUCTION .............................. 1 1.1 Previous Work on Collaborative Decoding ............ 3 1.2 Previous Work on Jamming Mitigation .............. 4 1.3 Objectives ... ......... ... ... .. .. .. ... 5 1.4 M ain Contributions ......................... 5 1.5 Outline . . ..... .... . . .... 6 2 JAMMING MITIGATION IN AWGN CHANNELS ........... 7 2.1 System M odel ........... ......... ........ 7 2.2 Collaborative Jamming Mitigation ................ 9 2.2.1 Joint Density For Jammed Symbols ............ 10 2.2.2 Jamming Signal Cancellation ............... 12 2.3 Jammer State Estimation ..................... 17 2.3.1 MAP Algorithm for Jammer State Estimation ...... 18 2.3.2 ML Estimation of Jamming Parameters .......... 18 2.3.3 Collaborative Jamming Detection . . . 19 2.4 Overhead and Complexity of Jamming Mitigation . .... 20 2.5 Results . . ........ ... ...... 23 2.5.1 Collaborative Jammer Mitigation with Perfect CSI . 24 2.5.2 Estimation of Jamming ... . . . 27 2.5.3 Collaborative Jamming Estimation and Mitigation . 31 3 JAMMING MITIGATION IN QUASISTATIC FADING CHANNELS 33 3.1 Introduction . . . . . . . 33 3.2 System M odel . . ........ ..... . 34 3.3 Derivation of the EM Algorithm ... . . . 36 3.4 Softdecision Decoding and Jammer State Estimation . .... 41 3.4.1 MAP Decoder for Message . . ... 41 3.4.2 MAP Algorithm for Jammer State Estimation . .... 42 3.5 Improved Initial Estimation ...... . . . 43 3.5.1 Derivation of the new initial estimator . . 43 3.5.2 CramerRao Bounds ...... . . . 46 3.6 Blind Estimation Algorithm ... . . . 48 3.7 Joint Density Scheme In QuasiStatic Fading Channels . 49 3.8 Results . . . . . .. .. . . 50 3.8.1 Jamming Detection and Parameter Estimation with EM Algorithm ... ........... .. ..... . 50 3.8.2 Jamming Mitigation with Joint density Scheme in Quasi Static Fading Channels .... . . 56 4 ISSUES ON THE COLLABORATIVE JAMMING MITIGATION .... 58 4.1 ReducedOverhead Information Exchange Schemes . .... 58 4.2 The Effects of Quantization ..... . . . 63 4.3 The Effects of Imperfect Collaboration Channels . . 64 4.4 Analytical Upper Bounds . . . . . 66 5 CONCLUSIONS . . . . . . . 71 REFERENCES . . . ...... . .. . 73 BIOGRAPHICAL SKETCH . . . . . . . 78 LIST OF TABLES Table page 2.1 Comparison of overhead (fourbit quantization). . . ... 22 4.1 Comparison of the performance and overhead of the joint density scheme with various exchange schemes (Eb/No 10 dB, Eb/Nj = 0 dB, p 0.4, E{T } 50) . . ......... ..... . 62 LIST OF FIGURES Figure page 2.1 One transmitter is communicating with multiple nodes. . . 7 2.2 The overall system model . . . . . ... 8 2.3 Flowchart for collaborative jamming mitigation techniques. . .... 21 2.4 Performance of the joint density scheme and MRC for perfect CSI (Eb/Nj 6dB,p 0.6,E{Tj} 50). .... . . 24 2.5 Performance of the variations of the cancellation schemes vs. Eb/No when perfect CSI are given (Eb/Nj = 6dB, p 0.6, E{Tj} =50). .. . 25 2.6 Performance of the jammer mitigation schemes vs. Eb/No when perfect CSI are given (Eb/Nj = 6dB, p 0.6, E{Tj} =50). . .... 26 2.7 Required Eb/NJ in dB vs. p (Eb/No = 6dB, E{Tj} = 50). . .... 27 2.8 Estimation performance for block size 1000 (N = 1, p = 0.4, E{T j} 50). 28 2.9 Performance when some jammed bits are missed or some unjammed bits are falsely alarmed (N = 1, block size=1000, EbN/N = 0 dB, p = 0.4, E{Tj} 50) . . . ........... ..... . 29 2.10 Probability of miss vs. number of nodes (block size 1000, Eb/No = 5 dB, p 0.4, E{Tj} 50) . . .......... ..... . 30 2.11 Probability of false alarm vs. number of nodes (block size 1000, Eb/No = 5 dB, p 0.4, E{Tj} 50) .................. ..... . 30 2.12 Performance of the joint density scheme using estimated jammer state. 31 2.13 Performance of the cancellation scheme using estimated jammer state 32 3.1 The overall system model . . . . . . 35 3.2 Iterative estimation and decoding. . . . . . 40 3.3 Comparison of MSE for simple and improved estimates of channel gain a. 51 3.4 Comparison of MSE for simple and improved estimates of . . 52 3.5 Performance versus number of iterations in the EM algorithm. .. . 53 3.6 Performance of the improved estimate with various numbers of pilot symbols. 53 3.7 Performance of overall estimation and decoding process with various initial estim ates . . . . .......... ..... . 54 3.8 The required Eb/Nj to obtain a FER of 102. . . . . 55 3.9 Performance of jamming detection with various initial estimates in quasi static fading channels (Eb/NJ = 8 dB, p = 0.4). . . . 56 3.10 Performance of jamming mitigation with joint pdf scheme. The EM algorithm is used with the improved initial estimate (Eb/Nj = 0 dB, p 0.4) . . . . .... . ... . 57 4.1 Illustration of the window exchange scheme. In this example, nodes exchange information about jamming window of size > 4 . ... 59 4.2 Required Eb/Nj in dB to achieve a FER of 102 (N = 2) . ... 63 4.3 Overhead of joint pdf scheme with the modified exchange schemes. .. 64 4.4 Quantization effects in AWGN channels. ...... . . 65 4.5 Quantization effects in quasistatic fading channels. . . . 65 4.6 The effects of noise and jamming on collaboration among nodes . ... 66 4.7 Comparison of simulation and bounds for joint pdf scheme with nonfading message signals (ai) 1 for all i), Eb/Nj 0 dB, p 0.6. .. . 69 4.8 Comparison of simulation and bounds for joint pdf scheme with nonfading jamming signals (b) 1 for all i), Eb/Nj 0 dB, p 0.6. .. . 70 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy JAMMING MITIGATION THROUGH COLLABORATION By JangWook Moon August 2005 Chair: John M. Shea Major Department: Electrical and Computer Engineering We consider communications in the presence of partialtime jamming. Partialtime jamming or interference can severely affect the performance of communication systems. This is especially true for military communication systems, which may experience hostile jamming. For systems in which the radios employ a single receive antenna, the options for jamming mitigation are typically limited. For instance, most previous work has focused on identifying which symbols are jammed and using softdecision or errorsanderasures decoding to improve performance. One of the most effective ways to combat hostile in terference is to use multiple antennas. However, the spacing between antennas typically depends on the carrier wavelength, and the space required increases in proportion to the number of antennas. For mobile radios, the required antenna spacing may often exceed the size of the radio, and thus other approaches are needed to achieve spatial diversity. Re cently, collaborative reception schemes have been proposed that allow a group of radios to act as a distributed antenna array. Those schemes improve performance while exchang ing smaller amounts of information than conventional combining techniques. In this work, we extend these ideas by proposing new collaborative reception techniques for use in the presence of a partialtime jammer. These techniques are first derived and investigated in AWGN channels, and they are extended to blockfading channels. Several estimators are employed, and suboptimal exchange schemes are proposed to reduce complexity. The re sults show that these techniques can mitigate jamming and allow communications in much more severe jamming conditions than when collaboration is not employed. CHAPTER 1 INTRODUCTION Diversity techniques provide effective means to improve communications reliability in the presence of unknown or hostile channel conditions. Two typical examples of diver sity techniques are coding and the use of multiple antennas. Coding provides time diversity by correcting errors. Multiple antennas can be used to obtain spatial diversity if the signal at each antenna experiences different channel fading. Currently, much research is being performed to both obtain spatial diversity and increase system capacity using spacetime codes. Multiple antennas can also be used to mitigate interference. Therefore, smart an tenna techniques that employ multiple antennas are being considered as one of the core technologies of nextgeneration cellular systems. However, the antenna separation required to achieve maximum performance gain is dependent on the wavelength of the carrier fre quency, and thus the communication system must be able to physically accommodate mul tiple antennas at the required separation. Typically, this size constraint is not a problem for base stations, ships, aircraft, etc. However, mobile radios are often too small to provide sufficient antenna separation. Therefore, multiple antennas are typically deployed at only the communication devices that can accommodate their space requirements. To overcome these constraints, cooperative communication schemes have been pro posed. These new classes of communication utilize the inherent diversity among multiple transmitters or receivers. Cooperative schemes can generally be classified as either cooper ative transmission or cooperative reception. The classic example of a cooperative diversity scheme is the relay channel studied by Cover and El Gamal [1]. In this work, the authors investigate the capacity for a system in which a relay node aids communication between a transmitter and receiver. The authors propose several cooperative transmission schemes that form the basis for some later work by Laneman et al. [2]. Laneman and his coau thors develop two efficient protocols to achieve cooperative diversity. One is decodeand forward and the other is amplifyandforward. Because of the relay node, a destination node can obtain a different copy of the same message, which produces spatial diversity. It is shown that full diversity can be obtained via the amplifyandforward scheme. In the cooperative diversity schemes that rely on relays, the relay can either be con sidered to aid the transmitter in transmitting the message or aid the receiver in receiving the message. Thus, these schemes can be considered to be either examples of cooperative transmission schemes or cooperative reception schemes. Other cooperative schemes can be more easily categorized into these two categories. Cooperative transmission schemes are considered by Sedonaris et al. and Willems [36]. In these schemes, each user collaborates with a partner to form a codeword that is a function of both his own and his neighbor's information. In Sedonaris et al. [3, 5, 6], the concept of usercooperation diversity is proposed and investigated in the context of a cel lular communication system. It is shown that this new form of diversity can be achieved via cooperation of users in the same cell, and implementation issues are also considered. Hunter and Nosratinia investigate cooperative transmission using ratecompatible punc tured convolutional codes [79]. In cooperative reception schemes, virtual antenna arrays are formed by allowing sev eral receivers to collaborate to exploit the inherent spatial diversity among the nodes. An example of cooperative reception is collaborative decoding [1012]. In collaborative de coding, the collaborating nodes use iterative and distributed decoding of an errorcontrol code to achieve diversity reception. Collaborative decoding is discussed in more detail in Section 1.1. Although the collaborative schemes have been shown to be effective to improve per formance, they have not been investigated as a method to improve communications in the presence of jamming. Traditional cooperative techniques may suffer in the presence of jamming because, many times, the spatial diversity techniques require perfect decisions at a relay node or random errors at the cooperating nodes in a cooperative reception scheme. Generally, jamming can severely disrupt communications by concentrating its power over just a few symbols of a packet. If the jammer power is strong compared to the signal power, then it is impossible to communicate successfully. Therefore it is desirable to devise meth ods to combat or even cancel the detrimental effects. Currently, the most common ways to combat hostile jamming are the use of time diversity via repetition and/or errorcontrol coding and the use of beam forming with an antenna array. The cooperative techniques that we propose to combat jamming in this work have some aspect of both of these previous techniques. We extend the concept of collaborative reception to mitigate jamming. We assume multiple nodes are present, and they receive independent copies of the same message and correlated jamming signal. The nodes use iterative, distributed decoding of the code to achieve diversity and coding gain. The nodes also effectively do noncoherent beamforming on the received signal through proper choice of decoding metrics or interference cancella tion. 1.1 Previous Work on Collaborative Decoding In the collaborative reception techniques proposed in Shea et al. [1012], a cluster of radios acts as a distributed antenna array. Note that if we assume each node receives in dependent noise samples, then maximalratio combining (MRC) is the optimal combining technique in terms of minimizing the probability of error. However, implementing MRC requires the nodes to broadcast all of their symbols to some node that will do the combin ing. This will result in a very high communication overhead. The collaborative techniques proposed in Shea et al. [1012] can achieve performance close to that of MRC, while requiring a smaller amount of information to be exchanged than MRC. These reception schemes utilize the simple fact: not all the transmitted sym bols are in error; to correct a packet, additional information for only the unreliable bits is required. By increasing the amount of reliable information at each node, the probability of correct decoding can be increased and diversity can be achieved. In the simplest example of collaborative decoding, radios in the cluster exchange information about their least reliable bits (LRBs). Assuming that a MAP decoder that generates soft outputs for each symbol is employed, the reliability of each symbol can be estimated according to the magnitude of soft outputs. If the magnitude is small, then the symbol is unreliable, and the decision of the symbol is likely to be in error. Note that due to the nature of broadcast, the radios re ceive independent copies of the same message. Therefore, each node requests information about its LRBs, and other nodes that have better information (e.g., because of a smaller noise sample or larger channel gain) can provide useful information for those LRBs. The information received from other nodes is used in additional iterations of the decoding and information exchange processes. 1.2 Previous Work on Jamming Mitigation The most common techniques proposed for jamming mitigation are the use coding or beam forming. In the absence of multiple receive antennas, most previous research [1326] has focused on using time diversity via repetition and/or errorcontrol coding to improve performance in the presence of jamming. The performance of a ReedSolomon coded fre quency hop signal in partialband interference using a hybrid decoder structure with errors and erasures decoding is investigated in Pursley and Stark [15]. Stark [16, 17] investigates the performance of coding for frequencyhopped spreadspectrum system with partialtime jamming. Other researchers [2324] have used turbo codes for channel coding, and the jamming probability for each code symbol is estimated using the iterative structure of de coder. These estimated jamming probabilities are used to update the loglikelihood ratios for the received symbols at the input to the decoder. The use of low density parity check codes in the presence of jamming is also investigated [24,25]. If multiple antennas are available, the jamming signal can be reduced through nulling [27 29]. In Besson et al. [28], the use of an antenna array is investigated to mitigate intermittent jamming. The jamming is encountered in a slow frequencyhopping system with a partial time or partialband jammer. The direction of arrival of the jamming signal is estimated, and jamming is mitigated using beam forming. 1.3 Objectives In this work, communications in the presence of jamming is considered. We develop an efficient way to mitigate jamming. As stated, the use of multiple antennas is one of the most effective ways to improve performance. However, the spacing between each an tenna should be at least half the carrier wave length. As the number of antennas increases, the space required to install the antenna array increases. Because of this size constraint, multiple antennas are not practical in many communication systems. Therefore, using the collaborative reception concept, we propose a new class of jamming mitigation algorithm. This class includes two collaborative jamming mitigation techniques. In each of the pro posed techniques, copies of the demodulator outputs for only the jammed symbols are exchanged among the collaborative nodes. Iterative detection, estimation, and decoding algorithms are applied to achieve jamming mitigation. In the first method that we propose, the branch metrics in the maximum a posteriori (MAP) decoding algorithm [30, 31] are modified to use the joint density function for the jammed symbols from every node. In the second method, the jamming signal is estimated and then canceled from the jammed symbols. We also investigate the necessary techniques for determining the set of jammed symbols and the jamming parameters. 1.4 Main Contributions In this work, we develop an efficient way to combat hostile jamming. We utilize the concept of collaborative reception, and extend it to mitigate jamming. The main contribu tions of this work can be summarized as follows: 1. We propose two jamming mitigation schemes, which are the joint density scheme, and jamming cancellation scheme. The joint density scheme utilizes the joint density of the jammed symbols from all of the nodes in the branch metrics of a BCJR MAP decoder. The jamming cancella tion scheme directly estimates the jamming signal and subtracts it from the received symbols. 2. We propose collaborative jamming detection algorithms to allow a group of nodes to reach consensus on the set of jammed symbols. 3. We develop iterative estimation and decoding scheme in which many unknown pa rameters of the jamming signal and channel fading are estimated. 4. We develop suboptimal estimators that provide lower complexity for important esti mates, such as the fading coefficient in the presence of jamming. 5. We evaluate the performance of our schemes by simulation and analysis. 1.5 Outline The rest of this work is organized as follows. In Chapter 2, we propose jamming mit igation schemes for AWGN channels. The joint density scheme and cancellation scheme are proposed in this chapter. Several necessary estimators are derived for jamming detec tion and estimation. The performance of the proposed schemes are presented. In Chapter 3, the jamming mitigation schemes are extended and modified for blockfading channels. For blockfading channels, we need additional estimators for the fading coefficients. We apply the expectationmaximization (EM) algorithm to iteratively approximate the ML estimator for all of the parameters. Performance results for fading channels are also presented. In Chapter 4, several issues on the collaborative jamming mitigation are considered. Finally, conclusions are discussed in Chapter 5. CHAPTER 2 JAMMING MITIGATION IN AWGN CHANNELS 2.1 System Model We consider a scenario in which a transmitter communicates with N receivers in the presence of a partialtime jammer, as illustrated in Fig. 2.1. The overall system model is illustrated in Fig. 2.2. Binary phaseshift keying (BPSK) is used for modulation. A turbo code is used for channel coding and logMAP decoding is performed using the BCJR MAP decoding algorithm [30, 31]. Throughout this work, we assume that the turbo code has rate 1/3 and employs identical constituent codes. However, our schemes easily generalize to other codes. A rectangular interleaver is used to break up jamming bursts at the input to the decoder. We model the jammer's signal as a hidden Markov model. The jammer turns on and off according to a twostate discretetime Markov model. If at time k, the jammer is in state 0, then the kth bit is not jammed. If it is in state 1, then the bit is jammed, and the jamming signal is modeled as white Gaussian noise with power spectral density Nj/2. Let p be the probability that a bit is jammed, and let E{Tj} be the expected amount of time (in terms of number of bits) spent in the jamming state before returning to the unjammed state. The transition probability from state i to state j is denoted by Pij and can be determined Node 0 Node 2 Transmitter Q 0 4 0" V i Jammer Node 1 ' .. 'Node 3 Figure 2.1: One transmitter is communicating with multiple nodes. Figure 2.2: The overall system model. from p and E{Tj} according to E{Tj} = 1/Pio, and Po01 Pol P + PIO The received symbol at node i can be modeled as Y E suk + rnk +ZkJk'" ', (2.1) where E, is the symbol energy and Uk is the message bit which can be assumed to be 1. The parameter zk represents the jamming state at time k. Here, nk and Jk are the contributions from thermal noise and jamming, respectively, and are zeromean, circular symmetric Gaussian random variables. The variances of ni) and Jk are given by No and Ni/p, respectively. The total variance of the noise and jamming for state 1 is T A No + Nj/p. 0(i) is the relative phase of the jamming signal at node i with respect to the jamming signal at node 0 and is uniformly distributed on [0, 27). Without loss of generality, we can let 0() = 0. This does not imply that the jamming signal at node 0 is inphase with the information signal, as Jk itself is a complex random variable with phase uniformly distributed on [0, 27). We assume that the value of each 0(i) is fixed for the duration of each packet. A group of receivers acts as distributed array by exchanging the set of symbols that are estimated to be jammed and then individually applying jamming mitigation techniques. We note that in the general case, it is possible that the communications among the group of col laborating nodes may also be jammed. However, in this work, we assume that the receiving nodes are clustered in a relatively small area and thus have sufficiently high signaltonoise ratio to communicate in the presence of the jammer. The receivers use iterative estimation, information exchange, and decoding algorithms. First the nodes individually estimate which symbols are jammed and then exchange mes sages to reach a consensus on the set of jammed symbols. Then the nodes collaborate by exchanging information about the jammed symbols. Finally, each node uses all of the received information to mitigate the jamming through one of two collaborative jamming mitigation techniques, which are described in the next section. 2.2 Collaborative Jamming Mitigation The collaborative techniques used to mitigate jamming should be designed to take ad vantage of the highly correlated nature of the interference signal. In this section, we present two techniques of differing complexity for jamming mitigation. We begin by assuming that every node has perfect channel state information (CSI), which includes knowledge of which bits are jammed as well as the statistics of the received jamming signal. In Section 2.3 we consider the problem of estimating the jammer state and parameters of the jamming signal. The jamming mitigation techniques that we consider fall in the class of collaborative decoding techniques [1012,32]. As in previous work, we base our jamming mitigation techniques on the BCJR MAP decoding algorithm [30,33]. Under the BCJR algorithm, the a posteriori likelihood ratios for the messages bits can be calculated as p(uk +1y) HIu+ ak1(S')'k(s', s)/k(s) p(uk ly) Hl akl(s')7k(s', s)/k(s)' where U+ and U denote the branches that correspond to message bits with value +1 and 1, respectively. Here, aki(S') P(s', yk1) is the forwardlooking state probability, /k() = P(yk +Is) is the backwardlooking state probability, and 7k(s', s) P(s, k s') is the branch probability. Here, y] denotes the vector of received symbols between time indices a and b. It can be shown that ak (s') and fk (s) can be computed recursively using 7k(s', s) [30,33]. The rate 1/3 turbo codes considered in this paper are constructed from two rate 1/2 constituent convolutional codes that are decoded iteratively. In the absence of collaboration, the branch metric for each of these convolutional codes is given by (S', s) p(uk)py s', s)pI y s', s), (2.3) where y) and yk are the matched filter outputs at the ith radio for the kth systematic and parity bit, respectively. Since every node receives copies of the same message in the presence of phaserotated versions of the same jamming signal and independent thermal noise, it is desirable to uti lize the received symbols from every radio to discriminate against the jamming signal and simultaneously achieve combining gain. The greatest performance can be achieved if all of the received symbols from every radio are employed in calculating the branch metrics. However, to do so would require exchanging the soft decisions for every received symbol. As the performance is often dominated by the jammed symbols, we assume that only the jammed symbols are exchanged. We propose two techniques to utilize the different copies of the jammed symbols in the computation of the branch metrics. The two techniques of fer a tradeoff between complexity and communication overhead, which we investigate in Section 2.4. 2.2.1 Joint Density For Jammed Symbols In this section, we explain the first jammer mitigation technique, which we call the joint density approach. Consider the branch metrics in the BCJR MAP decoding algorithm. In the absence of collaboration, then at node i the branch metric is [30,33] (2.4) 7 )(s',s)= p(uk)p(yis',s)p {) s',s), where yi) and y(~ are the matched filter outputs at the ith node for the kth systematic and parity bit, respectively. Since every node receives different copies of the same message and jamming signal, it is desirable to utilize the received symbols from every node to dis criminate against the jamming signal and simultaneously achieve combining gain. We can utilize all the received symbols from every node by employing the joint density function for them in the branch metrics. However, to do so would require exchanging the soft decisions for every received symbol. As the performance is often dominated by the jammed symbols, we assume that only the jammed symbols are exchanged. The joint density technique is then applied to these symbols. Consider the conditional joint density function for the set of received symbols rep resenting a jammed information or parity bit given the transmitted symbol Uk. Under the assumption that the jamming parameters Nj and p and phases 0(i) are known, the conditional joint density function is Gaussian with mean mk and covariance matrix Ek. The mean of yi) is /ik where /lk = EUk. The variance of yi) is No + Nj/p. Let Yk [yk y yk N1)] Then the mean of yk is mk = [k, /ik, ..., .lk]. The covariance matrix is Ek = E[(yk Mk)(yk k)1H], where ^k(1, M) (M,1) E[(n^) + Jkt" ()nm) + JkEO)* = E[ JkJ2i(0_mq())] NJ( (o)()), l= O,...,N1, mn l+1,...,N1, P where H denotes complex conjugate transpose and denotes complex conjugate. In practice, the radios do not know a priori whether a symbol is jammed. Neither do they know Ek, nor the parameters needed to compute Ek, which are Nj, p, and 0('). The set of jammed symbols, as well as Nj and p can be estimated using the BaumWelch algorithm, which we discuss in Section 2.3. Then YEk can be estimated directly as Ek A ZkEJ(yk mk)(yk mk)H, where J is the set of jammed symbols. The mean mk is unknown because the correct symbol value Uk is unknown, so we calculate the mean using the a posteriori estimate for Uk from the previous iteration of the BCJR algorithm. 2.2.2 Jamming Signal Cancellation Now we develop the second technique for jamming mitigation. We use an iterative estimation, cancellation, and decoding process. Let J be a set containing the indices of the jammed symbols. In each iteration, the jamming signals J = {Jk : k CE J} and the relative phases 0(i) are estimated and subtracted from the received symbols in J before decoding. The information from the a posteriori probability decoder is used to improve the estimates in the next iteration. Estimators for 0(i) and Jk. We derive estimators for 0(i) and J under the assumption that u = {ui) : k cE i = 0, 1,..., N 1} is known. As u is not known at the receivers, we use the preliminary estimates { ii)} of {u'i)} that are generated by the iterative turbo decoder. Given u, the ML estimator for 0( and J is [, 0(1),0(2), .. .] argmax p (y(O), (), ...,y(N)J, (1), (2),..., (N1), u) ,(2.5) [J,o(1),o(2),...] where y(i) is a vector of the received symbols in J at node i. The logarithm of the condi tional joint density function in (2.5) can be written as N1 f C2 C1i Y k\i) Pk k _Jkk'*' '2, (2.6) keJ i=0 where C1 and C2 are constants that do not have any effect on the maximization. Taking the gradient and setting to zero, it is easy to see that the joint ML estimates for Jk and 0) must satisfy N1 Jk N Y k)e "", and (2.7) i=0o Stan 1 JmEj(yk Pk)J (2.8) Re Elj(yk Pk)Jk Thus, the ML estimate for 0) must satisfy sin 0(i) Im E" ( k lk) N( k)C (2.9) cos 0(i) Re Yk J(y lik) n( ) k Pk) * We can similarly expand (2.7). The numerator and denominator of R.H.S of (2.9) become N1 N1 SRe (Yi) Pk) (Y) k)*1 sin + Im () k)(k)* cos n0 kEJ n=0 kJ and N1 N1 SRe (i) ) ( k) ( k)*j cos (T) m (Yi) Pk)(Y() Pk)* sin 0( n0 keJ n=0 kJ respectively. Therefore by expanding (2.9), we have an equivalent set of N equations given by Fj ^ Im Ai,.e(on ) 0, i 0, 1,..., N 1, n0 where Am,n = kEJ(Yk 'k)( Yk) k)*. Note that NI Fo Aio sin(0(') ZAj,o) 0. (2.10) (2.11) When N = 2, one solution to (2.11) is 0) = ZAi,o, Vi. For N = 2, 0() = ZAi,o is a sufficient condition for (2.10) to hold for both i = 0, 1. We also know that this is a necessary condition for (2.10) and (2.8) to hold if we require that the tan1 function in (2.8) be defined as a "smart" tan1 that can distinguish between the different signs of the numerator and denominator of the RHS of (2.8) that result in the same sign for the ratio of the two. To show 0() = ZAi,o, Vi is a sufficient solution, we substitute into (2.10). By our assumption, Fo = 0 holds, and 1 Fi Im{A,inej(ZAnoZA1o)} n=0 = Im{A,oejZAIo} + Im{Al,lej(ZAIoZAd,o)} S0 1) 1) Therefore 0( = ZAi,o, Vi is a sufficient solution for N = 2. For N = 2, ML solution can be written as M t = tan 1M kE(Y Pk)(Yk k) (2.12) Re kEJ(Yk k)*(yk k) and, combining (2.7) and (2.12), we get Jk Y( k Y21) k p itan Im E(y) k)1) Ik) (2.13) 2 2 Re yE () _ lk) 1) lk)* Similarly, for any number of nodes, a solution to Fo is given by 0) = ZAi,o, Vi. This may not correspond to the ML estimate for N > 2. Since the ML estimator of 0( does not admit a simple form for general N, in what follows, we use the estimate given by o ZAi,o tan1 (y/i) k)(Y) P k)*, Vi. (2.14) kEJ We note that as N  oc, F  0, Vi for this estimate, and thus the estimator will satisfy the likelihood equations asymptotically. The value of Jk is then estimated using (2.7) with 0( replaced by its estimate, .0( Note that the dependence of the estimators for each Jk on the values of ('), 1 < i < N 1 may make Jk sensitive to the quality of 0). The estimates for 0(i) are typically much more accurate than for Jk. This is because more samples are typically used for 0(i) than for Jk. Each 0(i) is constant over the entire block, and the average is over time. The Jks change from symbol to symbol, and the estimator is averaged over the received values at different nodes. These estimates are also dependent on the values of the transmitted symbols, Uk. As these values are not known at the receivers, the a posteriori estimates from the softdecision decoders, it ), are used instead. For the jamming cancellation scheme, we found that the performance is sensitive to the values of it). Thus to improve performance, when the nodes exchange the received symbols, they also exchange a posteriori probability (APP) information {u'i)}, as described below. Subtraction of the Estimated Jammer Value. Jamming excision is performed by sub tracting the phasecorrected form of Jk from the received symbol for each node. Because the jamming signal estimation depends on the decoded message, it is important to have as accurate an estimate of the decoded message as possible. To achieve this, we assume that each node decodes the message independently, and generates separate aposteriori log likelihood ratios (LLRs) for each bit. The LLRs at different nodes for the same bit will be correlated because of the shared jamming samples. However, we use a suboptimal com bining technique to avoid the complexity of optimally combining the correlated LLRs. The LLRs from each node are added together to generate a morereliable LLR, which improves the probability of correct decoding. To calculate (2.7) and (2.14), every radio needs to broadcast the received values for the jammed symbols. This is the same information required for the joint density approach. This information needs to be exchanged only one time after the collaborators agree on the set of jammed symbols. However, the estimate (2.7) is very sensitive to errors in Uk from the output of the BCJR decoder. We consider the following approach to improving the estimate of 1ik for the jamming cancellation technique. Each radio performs independent decoding and generates separate a posteriori loglikelihood ratios (LLRs) for each coded bit. For the jamming cancellation technique, these LLRs are exchanged along with the received symbols. The LLRs at different radios for the same bit will be correlated because of the shared jamming samples. However, we use a suboptimal combining technique to avoid the complexity of optimally combining the correlated LLRs. The LLRs from each radio are added together to generate a morereliable LLR, which improves the probability of correct decoding. As these LLRs may change during the decoding, we investigate the effects of different exchange strategies in Section 2.5.1. The estimated codeword may contain errors, but we can use the LLRs to estimate the reliability of the estimated code bits and adapt the jamming cancellation accordingly, thereby reducing problems from error propagation. The branch metric is computed by conditioning on whether the bit decision is correct and averaging over the two cases. To illustrate this, consider the case of N = 2. In what follows, we assume that 0(1) is known or accurately estimated. We then have the following two cases: When the decision is correct. The probability that this happens can be approximated as a maximum of a posteriori probabilities (APPs), i.e. [34], Prob(correct decision for kth bit at node i y(i) C L() (k) 1 ( e max1'L ) (2.15) where y(i) is the total received vector at node i and L(i) (k) is the LLR for the kth message bit at node i. The reliability of L(') (k) can be improved by adding all L() (k), 0 < i < N as previously explained. From (2.1) and (2.13), the estimated Jk is i n k + A + n ee + A A n k + n ee Jk )+Jk+21J0 =J Jk (0) + (1) j0() 2 2 Using these estimates, we get the new random variable r ), which is defined as (i) (i) ik Note that ri) is the value after subtracting the estimated jamming signal from the received value yi). In the decoding process, we approximate ri) as Gaussian distributed with mean E{) I} k and variance Var{ri)} No(N 1)/N. When the decision is incorrect. For this case, Jk becomes 2Pk + n) + Jk + (21k + + Jke(O1) _jO(l) 2 Similarly, the random variables, r ..., r, can be approximated as being Gaussian with N1 E{r )} i) k " and o Ni Varf{r ki N No, for i = 0, 1,..., N 1, respectively. After jamming cancellation, the r) can be combined to improve the decoder perfor mance. The rj) are not independent, but in the interest of constraining the complexity of the algorithm, we use rk z 1 i) as our new decision statistic. Then rk and its characteristics can be utilized when calculating the branch metric in the BCJR algorithm. Therefore, the term p(rk s', s) in 7k (s', s) is calculated as p(rk s', s) p(rk s', s,correct decoding)P(k) +p(rk s', s, incorrect decoding) [1 P(k)], (2.16) where P(k) is the probability that the kth bit is correctly decoded as defined in (2.15). 2.3 Jammer State Estimation The two jammer mitigation techniques introduced in Section 2.2 require accurate knowledge of the set of jammed bits, as well as the variance of the jamming signal. We follow the general approach of iterative estimation and decoding that is described in Kang and Stark [22,23,35]. In this approach, the probability of jamming is determined for each received symbol by using another BCJRtype forwardbackward algorithm. Information from the turbo decoder is used to improve the performance of the detection scheme, and vice versa. A significant difference between our work and previous work is that we require that the collaborating nodes reach consensus on the set of jammed symbols. In addition, after information is exchanged, the nodes have more information that can be used to refine the estimates for the probability that a symbol is jammed. We note that alternative structures are possible for combining jamming estimation and decoding. In Worthen and Stark [24], a unified approach is presented that can perform both procedures by passing messages on a graph. However, because of the channel interleaver, the structure of the graph is extremely complicated, so this technique is not practical except for very small block sizes. If the channel interleaver were not present, jamming estimation and decoding could be performed by expanding the trellis for the code into a hypertrellis. Each state in the original trellis can be split into individual states that incorporate informa tion about the state of the jammer. However, again, the size of the trellis grows rapidly, making this approach unattractive because of its complexity. 2.3.1 MAP Algorithm for Jammer State Estimation Consider first the case that the state transition probabilities of the hidden Markov model and the variance of the jammer are known. This may be a reasonable assumption if the jammer does not change its parameters rapidly because these parameters may then be accurately estimated over many packets. As the partialtime jammer is modeled using a twostate hidden Markov model, the APP that a symbol is jammed can be estimated using the BCJR algorithm. Let Fk(zk, Zk+1) be the branch metric for the transition from jammer state Zk to state Zk+1 at time k. Then k(Zk, Zk+l) P(Zk+l Zk) [P(ykZk,Zk+1,Ck = 0) P(ck = 0) +P (ykzk, k+l, Ck = 1) P (ck = 1)], (2.17) where yk and Ck represent the received symbol and coded bit corresponding to jk, respec tively. The probabilities P(zk+l Zk) are the statetransition probabilities, which we have assumed are known, and the probabilities p(yk Zk, Zk+1, Ck) are Gaussian densities that de pend on the transmitted symbol and the jamming state. Note that we assume that we do not have the a priori probabilities P(ck = 0) and P(ck = 1), and thus these probabilities are approximated by the APPs from the output of the turbo decoder, P(ck = 0y) and P(ck 1= y). This process results in an overall iterative approximation for the APP for the jamming state and message. 2.3.2 ML Estimation of Jamming Parameters If the parameters of the hidden Markov model for the jammer are not known accu rately, then they, too, must be estimated. The algorithm to do so is a generalization of the BCJR algorithm that is usually called the BaumWelch algorithm [36, 31]. The first term in equation (3.18) is a transition probability of the hidden Markov model, which can be estimated by expected number of transitions from zk to Zk+1 P( ) expected number of transitions from zk k/ jAk (1zk)Fk(zk, zk+i)Ak(Zk+l) (2.18) E 1 Ai(zk)Fk(zkS)Ak(zk)' where L is the block length and R is the code rate. Here, Ak(zk, Zk+1) and Ak(zk, Zk+1) are the forward and backwardlooking state probabilities. Similarly, the estimator for the variance of the noise plus jamming in state 1 is ML1 = Yk k (2.19) Ek1 Ak(l)Ak(l) Note that in the first iteration of the decoder, the parameters of the jammer may be unknown. Therefore, we initially set the transition probabilities for the jammer state to 0.5, and we set the jammer variance to twice that of the thermal noise. The results in Sec tion 2.5.2 indicate that it does not severely affect the performance because these estimates are updated after each iteration. 2.3.3 Collaborative Jamming Detection Each node initially estimates the jammer state using the BCJR algorithm independent of other nodes. Thus the set of symbols which are estimated to be jammed may vary from node to node. To perform the jamming mitigation techniques introduced in Section 2.2, we require that every node share information about the same set of bits. To achieve the objective, we propose a scheme that allows the collaborating nodes to come to consensus on the set of jammed symbols. Each node independently estimates the jammer state for each received symbol. A hard decision on whether a symbol is jammed is made by comparing the a posteriori probability that the symbol is jammed to a threshold (0.5 for the results in this work). Each node broadcasts the symbol indices for the set of jammed symbols. In practice, these indices can be highly compressed by taking advantage of the bursty nature of the jammer. Finally, consensus on the set of jammed symbols is reached by voting. A group of N collaborating receivers will decide that a symbol is jammed if the number of receivers that estimate that bit is jammed is greater than or equal to a threshold T. We consider three voting rules, which can be described in simple terms: 1) union of each node's set (T = 1), 2) intersection of each node's set (T = N), and 3) majority voting (T = N/2). The union and intersection rules represent the extreme cases that minimize the probability of miss and probability of false alarm, respectively. The majority voting rule is a compromise between the two extremes. The performance of these algorithms is investigated in Section 2.5. Note that the consensus set of jammed bits may vary greatly from the set of jammed bits at the various nodes. We use the notation Ji to denote the set of jammed bits at node i and J to denote the consensus set of jammed bits. The cardinalities of these sets may also be quite different depending on the voting rule that is used. As we are usually interested in the consensus set of jammed symbols, we will usually just refer to this as the set of jammed symbols. 2.4 Overhead and Complexity of Jamming Mitigation The two proposed schemes offer a tradeoff between overhead and complexity, which we briefly investigate in this section. We define the average overhead as the average amount of information, in bits, that must be transmitted by each radio to share informa tion with its neighbors. In Fig. 2.3 we provide a flowchart that illustrates the operation of the two jamming mitigation algorithms along with the information exchanges required. As the flowchart illustrates, the overhead comes from several sources. First, every radio broadcasts the indices for the set of symbols that are estimated to be jammed in order to reach consensus on the set of jammed bits. If L is the number of information bits in the block and R is the rate of the code, then 1[og2(L/R)] is the number of bits required to index an arbitrary jammed symbol within the block. Let Ji denote the set symbols that are estimated to be jammed at radio i, and the expected number of such symbols is E [Ji\]. The set of jammed symbols is highly compressible because of the Markovian nature of the jammer. In what follows we present analytical values for optimum compression and for Figure 2.3: Flowchart for collaborative jamming mitigation techniques. Table 2.1: Comparison of overhead (fourbit quantization). Joint Density Jamming Cancellation MRC Opt. Analysis Simul. Opt. Analysis Simul. Analysis N = 2 29835 30617 31900 44752 45046 47139 49056 N 3 44264 45925 47804 66396 67569 70432 72144 compression based on sending the indices of the starting and stopping times of jamming bursts, both of which are considered in detail in Roongta et al. [37]. The analytical results are based on exact knowledge of which symbols are jammed. For the simulation results, the radios do not know the set of jammed symbols, but select the set using the majority voting scheme described in Section 2.3.3. The second set of information that is exchanged consists of the complex received val ues for the consensus set of jammed symbols, J. Let Q denote the number of bits needed to represent a real number. Then the number of bits required to exchange the received val ues for the jammed symbols is N (2Q) E [5J\]. For the jamming cancellation technique, the a posteriori LLRs are also exchanged for the jammed symbols. In Fig. 2.3, we show only one exchange of these LLRs (which has the best performance), but we also considered schemes in which the LLRs are updated in several iterations. These are real values, and thus each time the LLRs are exchanged, a total of N (Q) E [J,71] bits are required. The results in Table 2.1 show the overhead required for the two schemes with two or three collaborating radios and fourbit quantization. For comparison, the overhead re quired for MRC is also shown. For the analytical results, p = 0.6. For the simulation results, p = 0.6, Eb/Nj = 6 dB, and Eb/No = 6 dB. The joint density scheme offers a significant overhead savings over MRC, especially for N = 3. The jamming signal can cellation technique requires more overhead, approximately equal to that of MRC. These comparisons will depend on the parameters of the jamming. For smaller p, the proposed schemes will require much less overhead than MRC, while for larger p, it may be more efficient to exchange all of the received symbols (as in MRC). The complexity of the two approaches can be estimated by considering the number of multiplications required for each jammed symbol in one iteration of the BCJR decoder. Except where noted, all operations are assumed to be on complex numbers. For the joint density approach, the complexity is dominated by the matrix multiplications in the Gaus sian density. The total number of complex multiplications is easily seen to be N(N + 1). For the jamming signal cancellation approach, estimating 0( and Jk each requires 1 and N multiplications per jammed symbol, respectively. Computing (2.16) requires 2 complex and 2 real multiplications per symbol. Thus, the equivalent number of complex multipli cations required is approximately N + 4. Thus, although the jamming signal cancellation approach requires higher overhead than the joint density approach, it is equally compu tationally efficient for N = 2 and more computationally efficient than the joint density approach for N > 2. Since the jamming signal estimates can be updated over multiple iterations, the jam ming signal cancellation technique will require the number of computations estimated above times the number of iterations in which the jamming signal estimates are updated. By comparison, the joint density approach does not require the density to be updated in later iterations. Thus, the actual tradeoff in complexity depends not only on N but also on the average number of iterations in which the jamming estimates are updated for the jamming signal cancellation technique. However, for large N, the O(N2) complexity of the joint density scheme will generally be much larger than the O(N) complexity of the jamming cancellation scheme. 2.5 Results For the simulation results presented in this work, the turbo codes are constructed from identical, memorytwo, recursive, systematic convolutional codes with feedforward poly nomial 1 + D2 and feedback polynomial 1 + D + D2. The code rate is 1/3, the information block size is 1000, and a random interleaver is used. A total of 15 decoder iterations are performed. Except where otherwise stated, Eb/Nj = 6 dB, p = 0.6, and E{Tj} = 50. W' 10.  U LLJ 103 MRC (N=2) ,, MRC (N=3)  Joint Pdf (N=2) Joint Pdf (N=3) 10" i 0 3 6 9 12 15 18 Eb/N0 (dB) Figure 2.4: Performance of the joint density scheme and MRC for perfect CSI (Eb/Nj  6dB, p 0.6, E{Tj} 50). For the following graphs, we use the labels "Joint PDF" and "Cancel" to refer to the joint density and jamming signal cancellation schemes, respectively. 2.5.1 Collaborative Jammer Mitigation with Perfect CSI We first consider the performance of the two proposed jamming mitigation algorithms for the case of perfect CSI in the sense that the radios know exactly which symbols are jammed and the statistics of the jamming signal. The results in Figure 2.4 show the frame error rates (FERs) for the joint density scheme and for MRC with two or three collaborating radios. The results show that MRC is not an effective approach for mitigating the jamming signal because it does not take into account the correlated nature of the jamming signal among the radios. Even for N = 3, the FER has an error floor above 102. The joint density scheme is better able to mitigate the jamming signal, achieving error rates close to 103 and 104 for N = 2 and N = 3, respectively. We next consider the cancellation scheme because we wish to show the performance effects of exchanging the a posteriori LLR information from other radios. We consider three different approaches. In the first approach, no LLRs are exchanged; that is, each W 10.' LU N=3 10.  No Jamming (N=1) + One LLR exchange Three LLR exchange 10, i i , 0 3 6 9 12 15 Eb/NO (dB) Figure 2.5: Performance of the variations of the cancellation schemes vs. Eb/No when perfect CSI are given (Eb/Nj = 6dB, p = 0.6, E{Tj} = 50). radio uses its own a posteriori estimate of the jammed symbols to do the cancellation. In the second approach, LLRs are exchanged after only the tenth iteration. In the third approach, updated LLRs are exchanged after 4, 8, and 12 iterations. The performance for these variations of the jamming cancellation scheme is illustrated in Fig. 2.5. The results show that the performance degrades significantly if no a posteriori LLR information is exchanged. However, the results are relatively insensitive to whether the LLRs are updated in future iterations. A performance comparison between the two jamming mitigation schemes proposed in this paper is illustrated in Fig. 2.6. Consider the performance at Eb/No = 9 dB. Without collaboration (N = 1), the FER is 101. With the joint density scheme, the FER is less than 5 x 103 and 2 x 104 for N = 2 and N = 3, respectively. While the performance of the jamming cancellation scheme is significantly better than no collaboration, it is worse than the joint density scheme. For example, the performance of the jamming cancellation scheme for N = 3 is only slightly better than the joint density scheme with N = 2. W 10 101 : Cancel (N=2) I Joint Pdf (N=2) Cancel (N=3) Joint Pdf (N=3) 10"4 0 3 6 9 12 15 18 Eb/NO (dB) Figure 2.6: Performance of the jammer mitigation schemes vs. Eb/No when perfect CSI are given (Eb/Nj = 6dB, p = 0.6, E{Tj} = 50). As previously mentioned, this can be attributed to error propagation from incorrect bit decisions used in jamming cancellation. As suggested in Pursley and Stark [15], the antijam capability of the jamming mit igation schemes can be measured by determining p*, which is the value of p required to prevent the receivers) from achieving an acceptable error probability. The higher the value of p*, the more symbols that must be jammed in order to significantly degrade communi cations. In Fig. 2.7, the value of Eb/Nj that is required to achieve a frame error rate of 102 is shown as a function of p, the probability that a symbol is jammed. For these re sults, Eb/N = 6 dB. The curve labeled "Perfect CSI" corresponds to the performance of a single receiver that is able to do perfect detection of which bits are jammed and that knows Eb/Nj and p exactly. The value of p* for this case is approximately 0.42. For two collaborating receivers, the value of p* is approximately 0.48 and 0.56 for the cancellation and joint density function approaches, respectively. If three receivers collaborate, then p* increases to 0.64 and 0.75 for the cancellation and joint density approaches to jamming "sb 2. m Cr 4. DC' L2 a ) i?  Figure 2.7: Required Eb/Nj in dB vs. p (Eb/No = 6dB, E{Tj} = 50). mitigation. Thus, these schemes do offer a significant advantage in partialtime jamming, with the joint density approach providing the best performance. 2.5.2 Estimation of Jamming For the results presented up to this point, we have assumed that the receivers know not only exactly which symbols are jammed but also the signaltonoise plus interference ratio in the jamming state, Eb/(No + p1Nj). In this subsection, we first investigate the perfor mance of algorithms for estimating the probability that a bit is jammed and the parameters of the jamming signal in the absence of collaboration/mitigation. The frame error rate (FER) is illustrated in Fig. 2.8 for jamming estimation under several different conditions. For the curve labeled "No Estimation", the receiver treats all symbols as if they are received in additive white Gaussian noise with bit energytonoise density ratio given by Eb/No. For the curves labeled, "ML Est.", and "Param. Known", iterative MAP jamming detection and decoding is applied. The distinction between these curves is how the parameters of the Markov model are determined for use in the iterative MAP algorithm. For the "Param. Known" curve, it is assumed that the receiver knows the 100  * No Estimation 4 ML Est.  Param. Known 101 A Perfect CSI  No Jamming W 10  JLL \u U A^ 10 0 1 2 3 4 5 6 7 EbNO Figure 2.8: Estimation performance for block size 1000 (N 1, p = 0.4, E{Tj} = 50). values of Eb /N, p, and E[Tj]. For the "ML Est." curve, these parameters are estimated using the BaumWelch algorithm, as described in Section 2.3.2. Finally, for the curve la beled "Perfect CSI", the receiver knows exactly which symbols are jammed as well as the signaltonoise ratios for the jammed and unjammed symbols. From the results in Fig. 2.8, we can see that jamming detection and estimation provides performance close to that of perfect CSI, even when the jamming parameters are unknown. As previously mentioned in Section 2.3.2, using the naive initial values for the jamming parameters does not signifi cantly degrade the performance when BaumWelch estimation is employed. Now consider the performance of the voting schemes that allow the collaborating nodes to determine a consensus set of jammed bits. A miss is defined to be the event in which jammed bits are incorrectly identified as being unjammed. Similarly, a false alarm is defined to be the event in which unjammed bits are identified as jammed. The effects of false alarm and miss can be most easily investigated by evaluating the performance of these two phenom ena separately. To do so, we degrade the performance of the system with perfect CSI by providing incorrect information about a random set of 10 or 20 bits. For example, to in vestigate the effects of missed symbols, 10 or 20 bits that were jammed were incorrectly 10 I I 0 Miss 20 bits * Miss 10 bits v False alarm 20 bits 101 v False alarm 10 bits S: Perfect CSI 01 102 10"0.4,,,... 10  2 3 4 5 6 7 8 Eb/NO Figure 2.9: Performance when some jammed bits are missed or some unjammed bits are falsely alarmed (N 1, block size=1000, Eb/Nj = 0 dB, p = 0.4, E{Tj} = 50). indicated as unjammed at the input to the decoder. The results of this investigation are illustrated in Fig. 2.9. As seen, the effects of false alarm in terms of FER are negligible, but miss events cause a significant degradation in performance. However, the false alarm prob ability should be small because it results in additional overhead when information about the set of jammed symbols are exchanged among multiple nodes. Thus, when error rate performance is most important, the probability of miss should be small, and if overhead is most important, the false alarm probability should be small. Typically, there is a tradeoff between these two probabilities such that reducing one increases the other. As the proba bility of miss has the greatest effect on performance, this performance measure is plotted in Fig. 2.10 for the three collaborative detection schemes described in Section 2.3.3. The union scheme provides the lowest probability of miss, while the intersection scheme pro vides the highest probability of miss. The majority scheme provides a compromise that can achieve lower overhead than the union scheme while achieving performance close to that of the union. The probability of false alarm is plotted in Fig. 2.11. The union scheme provides the highest probability of false alarm, while the intersection scheme provides the 10' N=1 10  N=2 N=3 103 ML Est.  Perfect CSI 10 0 3 6 9 12 15 18 Eb/NO (dB) Figure 2.12: Performance of the joint density scheme using estimated jammer state. lowest probability of false alarm. The majority scheme also provides a good compromise between the union and intersection scheme. Thus, the majority technique is used for the rest of the results in this work. 2.5.3 Collaborative Jamming Estimation and Mitigation In this section, we consider the scenario in which the collaborating receivers must estimate the set of jammed symbols and the jamming parameters in conjunction with the collaborative jamming mitigation techniques. The system model for this scenario is illus trated in Fig. 2.2. The operation and information exchange for this scenario are illustrated in Fig. 2.3. After five iterations of jammer state and parameter estimation, the nodes trans mit the indices of their jammed symbols and use the majority voting technique to determine the consensus set of jammed symbols. Following this, the received values and (for jam ming cancellation only) the a posteriori LLRs for these symbols are exchanged. Then each node iterates between two algorithms: 1) jamming mitigation and decoding, and 2) jammer state and parameter estimation. S10 ,T...,, N=3 10' ML Est.  Perfect CSI 10" 0 3 6 9 12 15 18 Eb/NO (dB) Figure 2.13: Performance of the cancellation scheme using estimated jammer state. The performance of the joint density scheme is shown in Fig. 2.12 for perfect CSI and for majority detection of jammed symbols. For perfect CSI, the radios know which symbols are jammed and the parameters of the jamming signal. For majority detection, the jamming parameters are estimated using the BaumWelch algorithm. The performance for both two and three collaborators degrades from imperfect knowledge of which symbols were jammed and the jamming parameters. The degradation results in an increase in the FER by a factor of approximately 2. However, the collaborative system still achieves most of the performance gain over reception without collaboration. The performance of the cancellation scheme is illustrated in Fig. 2.13 for perfect CSI and for majority detection of jammed symbols and ML parameter estimation. Again, the performance degrades from imperfect knowledge of which symbols are jammed and the jamming parameters. CHAPTER 3 JAMMING MITIGATION IN QUASISTATIC FADING CHANNELS 3.1 Introduction In the previous chapter, the proposed collaborative jamming mitigation schemes are shown to be effective in AWGN channels. In this chapter, we extend the jamming mitiga tion techniques to quasistatic fading channels. Communication in quasistatic fading channels requires high average signaltonoise ratios (SNRs). This is especially true for military communications in which it can be even more difficult when a jammer is present. So there has been much research aimed at overcoming the jammer's effect (see for example, [38, 39, 23]). For best performance, it is important to have a good estimate of the fading coefficient, so many algorithms have been presented (see for example, [3942]). In this chapter, we consider the communication scenario illustrated in Fig. 3.1 where the channels experience quasistatic fading. We focus on the joint density scheme because it provides better performance with smaller overhead than the cancellation scheme. The overhead is very important when wireless communication is used between nodes. More over, the joint density scheme is less susceptible to errors in the preliminary bit decision. Note that the number of errors in a packet can be very large when a quasistatic fading coefficient is small. Therefore, we extend only the joint density scheme to quasistatic fad ing channels and address the problems of phase acquisition and channel estimation in the presence of jamming. In quasistatic fading channels, the detection and estimation problems dealt with in the previous chapter becomes much more difficult because they depend on the fading coef ficients. The receiver must estimate fading coefficient for best performance. To deal with this difficult detection and estimation problem, we employ the expectationmaximization (EM) algorithm [4347] to approximately obtain the joint maximumlikelihood (ML) es timates for the message or jamming parameters. The EM algorithm has previously been applied to channel estimation [4850]. We show that under the EM approach, the problem of detecting the message in the presence of unknown channel and jamming parameters results in an iterative detection and estimation procedure in which two separate BCJR [30] or BaumWelch [47] algorithms are used for the message and jamming states. Thus, the overall detector structure is same as the one in the previous chapter. The EM update process requires an initial estimate of the parameters in order to avoid converging to a local minima rather than the ML estimate. In our derivation of the EM estimators, we propose a simple initial estimate. However, the performance of this naive initial estimate is not good enough for the scenarios that we consider, so we propose a new estimator that provides better result. We also present a blind estimation algorithm that does not require the use of pilot symbols. This chapter is organized as follows. In Section 3.2, the system model for fading channels is introduced. In Section 3.3, we show how the EM algorithm can be applied to estimate the unknown parameters of the message and jamming signals. In Section 3.4, we explain the overall detection and estimation process. In Section 3.5, to improve con vergence of the EM algorithm, we develop an improved initial estimator for the channel coefficient. The CramerRao bound (CRB) for this estimator is also derived in this section. In Section 3.6 we extend to our previous estimation algorithm to a blind algorithm that does not require pilot symbols. In Section 3.7, we extend the joint density scheme and the necessary detection and estimation schemes for the fading and jamming parameters. In Section 3.8, simulation results are presented. 3.2 System Model We consider the same scenario in which a single transmitter is communicating with a distant cluster of N nodes in the presence of jamming. Each node receives independent copies of the same message and correlated jamming signal. Both the message and jamming Figure 3.1: The overall system model. signals undergo quasistatic fading that is independent from each other and independent between nodes. In this chapter, we assume that the nodes in the cluster are assumed to be close enough to reliably exchange messages in the presence of the jamming signal. We consider more realistic channels for collaboration in Chapter 4. As in Chapter 2, we consider a partialtime jammer that is modeled by a twostate hidden Markov model. The jamming signal is only present in state 1 (ON state), and is modeled as white Gaussian noise with power spectral density (PSD) Nj/p. Here, p is the steadystate probability of being in the ON state. The expected time (in the number of channel symbols) in the ON state is E{Tj}. Gaussian thermal noise with power spectral density No is present in both states 0 and 1. To provide diversity against jamming, the message is encoded by a turbo code, and the coded symbols are passed through a rectangular interleaver before transmission to break up jamming bursts at the input to the decoder. The overall system model is illustrated in Fig. 3.1. The received symbol at node i can be modeled as y^) a^ ESUk + n W + zkbJk, (3.1) where a(i) and b) are complex quasistatic fading coefficients for message and jamming signal, respectively. E, is the symbol energy, and Uk is the message bit which is 1. The parameter Zk represents the jamming state at time k. Here, ni) is complex Gaussian noise with variance No, and Jk is the jamming signal, which is a circularly symmetric complex Gaussian random variable with zeromean and variance Nj/p. Therefore in the view of the ith node, the variance of state 1 is No + b0) 2Nj/p. 3.3 Derivation of the EM Algorithm In this section, we use the EM algorithm to estimate the unknown jamming and chan nel parameters at a single node. Let there be L BPSK symbols in a packet. Some of the UkS maybe pilot symbols and are hence known. Let S and D be the set of the indices of the pilot and data symbols, respectively. Without loss of generality, we assume P(uk = 1) =1 for all the pilot symbols. We further assume that No is known to the receiver. Let Oo No/2, r = Nj/(2p), and / = o 0/o. Note that none of the parameters related to the jamming or signal amplitude and phases, i.e. a, b, Nj and p, are known to receiver. Only the power spectral density of the thermal noise, No, is known to the receiver. Because of the channel interleaving, we treat the Uk as independent and let P(uk = 1) = Pk and P(uk +1) 1 Pk. Let Zk denote the jammer state at time k. The four transition probabilities for the jammer state are given by P(zk 1 IZk1 = 0) = 0o, P(zk lZk 1) 1, P(zk = 0zk1 0) 1 To, and P(zk 01zk 1) 1 71. We also define the probabilities for the initial jamming state as P(zo = 1) = T* and P(zo = 0) =1 T*. In the following analysis, we use bold letters to denote vectors of parameters or symbols, such as the vector of message probabilities P = [PI, P2,... PL] and similarly the vectors of jamming states z, received symbols y, and transmitted symbols U. Consider estimating 0 = (a, i, T7o, 71, T*, P) from y, the vector of the received sym bols. The ML estimator can be written as OML = argmax p(y 0). We use the EM algo 0 rithm with the transmitted information u and jamming states z treated as missing data to iteratively obtain OML. Note that p(y, u, z 0) = p(y 0, u, z)p(u O)p(z 0), where LT1  \yk aVEsUk 2 p(y0, u, z) = 2 (1 + ) exp 2 Y2(t+ k) , k= 1 L p(ZI) [zoT* + (1 zo)(1 7*)] [Zk(1 Zk1)O + ZkZk171 k=1 +(1 k)(1  k_)(1 70) + (1 Zk)Zk(1 71)]. (3.2) Let 0(') denote the parameter estimates at the nth iteration of the EM algorithm. Then the EM algorithm amounts to updating 0') = argmax Q(0, 0(' 1)) starting from an initial 0 estimate 0(0). Here, Q(0, 0') is Baum's auxiliary function which is given by Q(0,0') = E[logp(y,u, zO)y,0'] =E[logp(yju,z,O) y,0'] +E[logp(u0) y,0'] +E[logp(z\0) y,0']. (3.3) For convenience, consider the nth iteration and let 0' 0( ). Then the first term in (3.3) can be written as E[logp(ylu, z, 0) y, 0'] L L lYk a/ETuk2 1 E[log 2ro(1 + zk) y, 0'] E (1 ) y2 0' k2 1 kZk 1 L [p(zk = 1y, 0') log 27r (1 + + p(zk 01y, 0') log 27e7o2] k 1 L aI E 1 2 S[P(uk tY, O')P(zk ay,0) 2( +) k l1 2 0 +p(uk ly, 0')p(zk ly, 0')Yk + a ' 2 2o(21 + ) +p(uk l y, 0')p(zk Oly, 0') k aE2 2oo 0oo where in obtaining the second equality, we have approximated p(uk, Zk 0(n 1)) = p(uk y, O)p(zk y, 0). This approximation is justified by the channel interleaving illus trated in Fig. 2.2. The other two terms of (3.2) can be written as E[logp(uO)y, 0'] = E log (1Uk Pk) + ukPk) Y,0 k= L = [p(uk =ly,0')0og(1 Pk) k=1 E[logp(zlO) ly, 0 +p(uk 1 ty, 0') og Pk], L '] [pzk Zk1 0y, 0')log o k=1 +p(zk = 0, k 0Oy, 0') log(1 7o) +p(zk , Zk1 ly, 0') log1 +p(zk 0, Zk = 1 y, 0') log(1 71) +p(zo 1= ly,0')logr* +p(zo = 0y,0')log(1 r*). (3.5) (3.6) (3.4) Because the three terms of E[logp(y, u, z0) y, 0'], i.e. (3.4), (3.5), and (3.6), are additive and depend on different components of 0, we can maximize them separately with respect to the corresponding components in 0. First, consider (3.5). It is easy to see that the choice of Pkn) p(uk y, 0(n)) (3.7) maximizes (3.5). Similarly, to maximize (3.6), we should choose (n) YZk1 p(zk ,Zk1 0 oy,0 O)) S E p(zk 0,Zk1 0 y, )) + p(zk lk1 0 y, 1)) (n) 1 k p(zk lZk1 1 iy, 1) (39) Sk lP(zk 0,Z 1 y, )+p(zk 1,Zk1 y,0(1))' (3.) By differentiating (3.4) with respect to a and ( and setting the derivative to zero, we also obtain k 1 P PU(1) p( 1)] [pzI(1)/(1 + P) + (0)]yk VE; k 1 [Pz (1)/(1 + 0 +p" (0)] ELk1 {lYk l2 ~s +2Re(i,,V/ )[puk(1) p.(1)] pz (1) 2 2 y ((3.12) where p (i) p(zk = iy, 0) and p, (i) = p(uk = i y, 0). Thus, we need to solve for a and ( simultaneously from (3.11) and (3.12). Simultaneous maximization requires a numerical search, so instead, we employ the following approximate maximization: L pn(1) p l(1)p (1)/(1 / 1) + pl) (0) yk EYk 1 p[ )(1)/(1 + (nl))+ p (0) ( ) L [jYk 1 + l() 2E + 2Re(i, ,(n) ){p ( ) p 1)(1) }] P (1) S2 o2 L p (nl)/ 2o k lZk M1 Figure 3.2: Iterative estimation and decoding. where p (i) p(zk i yO(n)) and pY (i) p(uk y,O(n1)). To update these parameters, we need to calculate the three kinds of probabilities, which are p(uk 1Y, O()),p(zk y, 0(1)), and p(zk Zk1 y, (1)), (3.13) where i, j 0 or 1. Since we use convolutional codes and the jamming signal is modeled using a twostate Markov chain, the codeword and jamming state can be directly estimated using two separate BCJR algorithms given the previousiteration parameter estimates. The values of (3.13) are generated in the two BCJR algorithms. The consequent overall EM algorithm is illustrated in Fig. 3.2. Note that the iterative decoding process is a byproduct of the EM algorithm. An initial estimate 0(0) is needed to run the BCJR algorithms for the jamming and message and update the estimates. Let a() be the average of the pilot symbols, a() 1 lk, (3.14) where S is the cardinality of 5, and then use a() in (3.12) with p(uk 1) = p(uk 1) 0.5 for k D andp(zk 1) p(zk 1) 0.5, k = 1, 2,... ,L, which yields (o) 2 k a(O) 2 + (I ly2 + (a(O Es) l. (3.15) 0 kES kED For later use, we call this initial estimate 0(0) the simple estimate. 3.4 Softdecision Decoding and Jammer State Estimation In this section, the overall iterative decoding procedure is explained. The receiver employs two BCJR algorithms to provide updates for (3.13) given the current 0 (), as described in the previous section. One BCJR algorithm provides the a posteriori proba bilities for the message and coded bits, and the other estimates the a posteriori probability that each symbol is jammed. They are connected in serial as illustrated in Fig. 3.2, and use each other's information to refine the estimates for the message and jamming. 3.4.1 MAP Decoder for Message We consider first the BCJR decoder for the message [30, 33], which gives as output the a posteriori probability for a bit in terms of three types of probabilities, ak (s'), 3k (), and 7k (s', s). Each of these probabilities is computed for different states of the convolu tional code or transitions between these states. Consider the branch connecting state s' to state s corresponding to the kth bit. Then ak(s') is the forwardlooking state probabil ity, 3k (s) is the backwardlooking state probability, and 7k (s', s) is the branch probability. It can be shown that the ak(s') and 3k(s) can be determined in a recursive manner us ing 7yk(s', s) [30]. Assuming the use of a rate 1/2 nonsystematic convolutional code, the branch metric for iteration n + 1 conditioned on the previous parameter estimate 0( ) is given by 7yk(S, ) P(s s) p(yl s', s, 0()) p(y s, ()), (3.16) where 1,1 and /:, are the received symbol values for the two parity bits corresponding to the kth message bit. Note that p(yl s', s, 0(T)) and p(,u Is', s, O)) are Gaussian densities if we further condition on whether the bit is jammed. Let zJ and z be the states of the jammer for the two parity symbols of kth message bit. Then for i = 1, 2, p(y s', s, ) = p(y s', s, O z 0)P(z 00 () +p(y s',s,0 ,z, = 1)P(z 1 ). (3.17) The probabilities P(4z 010 ()) and P(z 1 0(')) are replaced by the aposteriori prob abilities generated by the MAP algorithm for the jammer state. As illustrated in Fig. 3.2, in each iteration the estimator for the jamming state runs before the BCJR algorithm for the message, so P(z 01 0')) and P(z 11 0')) are replaced by P(z4 0y, 0(')) and P(z = 1 y, 0(')), respectively. The MAP algorithm for the jammer state is explained in the next subsection. Decoder performance depends on the accuracy of 7k(S', s 0). Thus it is important to have accurate knowledge of the jammer state and the channel coefficient. 3.4.2 MAP Algorithm for Jammer State Estimation The BCJR algorithm directly applies for jammer state estimation because the partial time jammer is modeled using a Markov chain. The a posteriori probability that a symbol is jammed can be estimated in a similar way as in the previous subsection. Then in iteration n+1, the estimated parameters from the previous iteration will be used in the MAP estimate of the current jamming state probabilities. As for the BCJR for the message, all of the probabilities in the MAP algorithm for the jammer state can be determined from Fk (z', z), which is the branch metric for the transition from jammer state z' to state z at time k. Then Fk(Z',ZI(T)) ( PtZZ') pYktZ', Z,o()) Pz z') [p yk I ZTUk 1)P(Uk I() +p(yk ',0(Uk )P(uk 1 )], (3.18) where yk and Uk represent the received symbol and code bit corresponding to the time instance k, respectively. For the parameters of the form P(zlz'), the estimates from the (n 1)th iteration of the EM algorithm given by (3.8) and (3.9) are used. In place of the probabilities P(uk 1 I0()) and P(uk 1 l(T)), we use the a posteriori probabilities from the BCJR algorithm for the message from the previous iteration, which are P(uk = 1 y, 00(T1)) and P(uk l y, 0(T1)), respectively. Thus, we see that the EM algorithm is an iterative algorithm for estimating 0 and detecting the jamming state, and message bits. 3.5 Improved Initial Estimation For some cases, the simple initial estimate 0(0) in (3.14) and (3.15) may not be good enough because some of the ykks can have a very large variance due to the jamming signal. As a result, the EM algorithm may be stuck at a local minimum, giving poor results [51]. In Section 2.5, we will present simulation results that illustrate the performance problems of the simple estimate. To increase the accuracy of the initial guess, we propose a new estimator. 3.5.1 Derivation of the new initial estimator Let us revisit p(y 0), which is given by p(y0) = p(yyu,'z', )p(uO)p(z0) Z {o* + (1 zo)(1 *)] U Z UHZ t ( Yk aVEsUkl2 k L 27o(1 + (Zk) exp 2jo(1 +2) ) 2 ( t2 Pk) 1 2 Pk) (Zk(l Zk1)Qo + (1 Zk)(l 1)(l 7o) + ZkZk1w1 +(l Zk)Zl(l 71)) 1 }. The direct ML estimation of 0 is difficult. For the initial estimate 0(0), the decoder has no knowledge of rT*, 7ro, 7r1, or P (except for the pilot symbols), and we have observed from simulation that the initial estimates for these values do not contribute significantly to the performance of the EM algorithm. Thus we set the initial values of these probabilities to 0.5. Then under these assumptions, p(y 0) reduces to Sexp k Vk 4] L exp(1'+ 1I1 p(yla, ) S 22(1 + ~Zk) 4 2ea ( + ( k) 2 u,z k6D kES Hence we have logp(y a, ) Sexp{ Yf } exp{ a log 8j2 + 2 exp l 2ex(1p) + I 12 I] + 2a (1+ 2) 1 Y ,+ ) 1 exp I 2aT } exp{ 2y a (1+) + log 4o + 4o 2(10 (3 kES 0 A + log !exp Yk 2 +a2Es coshRe(/,I Es) + log exp +  2 cosh +/o kED t yk la 2E1o a E(I 1 Elog t + exp{2 ;(1 }osh '' kED 0ac2 Js 2 + 812 + log 1 + 7exp o }/ kES 2 1 + 2 L log 4 o . For high SNRs, where VE//o4 > 1, cosh[Re(y^a / )/o] 4 0.5exp{ Re(y*a V )/o}, cosh [Re(a )/{o(1 +)}] a 0.5exp [ Re(ya l/{oa(1 + )}]. Therefore logp(ya,) Ak + k D log 2 Nlog 47o2, kED kES where D is the cardinality of the set of data symbols, and S d(yk,a) log1 exp d(Yk, a)} Ak = 2 + log 1 + /exp2.g /o Bk Yk + log + T exp ( + /Yka) where d(yk, a) A 1aEs 2 and d(yk, a) A min{Yk a /8 2, Yk + a /Es 2}. For fixed (a, ) and k such that d(yk, a) > T A 2o0(1 + 1/i) log(1 + ), we can approximate .19) Ak by Ak  [ 22 log(1 + ) (k, a) (3.20) 2Qo 1 + S On the other hand, for k such that d(yk, a) < ', Ak d(yk, a)/(20o). (3.21) Similarly Bk can be approximated by using (3.20) and (3.21) with d replaced by d. Here, we can further approximate logp(yla, ) by partitioning D into subsets {k E D : d(yk, a) < T} and {k E D : d(yk, a) > T}. Similar approximations are also applied for the set S. Eventually, we have logp(ya,0) 12 d(yk, a) 2 d(yk, a) 0 kcAoo 0 kcAlo 2 d(yk, a) d(yk, a) 2o (1+ () kA2(t + () (Aoil + A) log(1 + ) D log2 Llog 47re, (3.22) where Aoo = {k E D d(yk, a) < '}, Ao = {k E D d(yk, a) > '}, Ao = {k S : d(yk, a) < ''}, and All {k E 5 : d(yk, a) > T'}. Then the approximate ML estimator for (a, i) can be obtained by minimizing c(a0 A 5 d(yk,a)+ d(yk,a)+ 1 d(yka) 1 1 kcAoo kcAlo kcAol + Y d(yk, a) + (AoI + A1 )2o2 log(1 + t ). (3.23) The overall steps to find and a from this approximate ML estimator are as follows: 1. Start with small initial guess for and find the corresponding 4. 2. Partition the complex plane into squares of size 24'. 3. Assume that the center of each bin is the temporary a 8E. Calculate (3.23) for each bin. Determine the four regions Ai, i, j = 0, 1 for each a. 4. Choose the bin that gives minimum value of (3.23) and store the corresponding a, , and c(a, (). 5. Repeat steps 2) ~ 5) by increasing ( and updating the optimal a, ( and c(a, (). 6. After finishing the iterations, the stored a is the improved initial estimate. We call this estimate the improved estimate. Note that for the improved estimate, it is important to have enough pilot symbols to ensure that the estimate is not 7r radians out of phase. To better understand this, we revisit (3.23). Note that the first and third summation generate the same value for the bins at a 7r radians offset. In the absence of pilot symbols, two possible candidate bins will have the same value of (3.23). The second and fourth summation, which depend on the pilot symbols, decide between the two bins. Because the pilot symbols are also susceptible to jamming, we need enough pilots to ensure an accurate initial estimate. The performance with various numbers of pilot symbols is presented in Section 3.8. 3.5.2 CramerRao Bounds In this section, we derive the CramerRao bound (CRB) for the variance of the pro posed initial estimators for a and Assume that we estimate the vector parameter [ar ai ], where a, and a, are the real and imaginary part of a, respectively. The CRB for each com ponent is the diagonal element of the inverse of the Fisher Information Matrix (FIM). We show how to calculate the element of the FIM that is in the first row and first column. From (3.19), let MD and Ms be the argument of the log for the data and pilot symbols respec tively, and M'D, MD, M', and /' be the first and second partial derivatives of MD and Ms with respect to ar, respectively. For both MD and Ms, it can be shown that E(y+ aE4 ) exp [ y  a~ E2/(2o2)] VE (y, + aV) exp ) y + aVr 2 /(2 7) 87_o o Es(yr a E) exp y a, 2/ (2o (1 + Vi 8v (1+ 2 exp [ly + a 2/(2,o(1t + and Sy / aR VEs 2 Es(yr ar rEs) 2 ) / 4 S=vo exp( 2 ) 2 t)/(8 ) vlexp( y+ar /2 E s(yr + ar E)2 vi Iexp ( 2 ) 1)_ )]/(87 ) y arVE 12 E (yr ar E)2 1 74t + +vo exp y r t)] /(87 0(1+ ) +Vo l(1 ) o(1 j2( 2 (t vi /yI + ar 2 E (yr + ar 2E)2 4 vi exp 2(1 2(t+1)/8 (1+ 0 2 where vo = 1 for AMD and 2 for Ms, v1 1 for MD and 0 for A /s. Then it can be shown that a21 ogp(y a, D M D1 S MM M' M2 (3.24) ThereforeD1 MD D2] + ISI Ms 3.24) Therefore L2 logp(yja,0i D J J D6MD M MDdy dy, + / J ; IK'Ms MMsM2] Msdyrdy,. 00l J ...002  .25) Note that the other required elements of the FIM can be obtained in a similar manner. Then, the CRB for a is obtained by adding the first two diagonal elements of the inverse of the Fisher Information Matrix, and the CRB for is the last diagonal element. 3.6 Blind Estimation Algorithm The initial estimators proposed in the previous sections require pilot symbols to re solve a w radian ambiguity in the phase of the message signal. This ambiguity results from the use of BPSK modulation, which causes some symbols to have mean a and others to have mean a. Pilot symbols allow this ambiguity to be resolved because the mean values of those symbols are known a priori. However, the use of pilot symbols reduces the overall code rate. Therefore, in this section, we propose a blind decoding algorithm to avoid the use of pilot symbols. The approach we use to deal with this ambiguity is to decode for both of the cases, a and a. In each iteration, the two decodings will result in two possible decoded message sequences. We use the path metrics for the message sequence to select a candidate sequence for use in further iterations or as the final decoder output. The use of path metrics to choose the correct codeword is reasonable because of the following reason. Note that the flipped sequence (0 to 1, 1 to 0) of a valid codeword is not a valid codeword, because the all 1 sequence is not a valid codeword for the rate 1/2 convolutional code that we use. So with high probability, the path metric obtained from the a with the correct sign will be greater than the other path metric with the a with the incorrect sign. Note that the path metric is easily obtained using the Viterbi algorithm and thus from the maxlogMAP algorithm [52], which provides the same hard decision as the Viterbi algorithm. The ML path metric in the Viterbi algorithm is the metric of the terminating state, and this value is the summation of all branch metrics along the ML path. For the maxlogMAP algorithm, the summation of all the branch metrics on the ML path is 7ck(Skl,Sk\1O) ) P(, 0) p(YkSk_1, Sk, 6), (3.26) k k where P(uk\ 0) is independent of 0 and is the a priori probability for the message bit Uk, which we assume is 0.5. Here the sequence of states {sk} corresponds to the states on the ML path. Therefore, the ML path metric from the Viterbi algorithm is a linear function of the summation of all k (s', s) along the ML path. Exploiting these facts, we perform decoding as follows. 1. Calculate improved initial estimate as described in Section 3.5 without using pilot symbols. 2. Decode for two cases, a and a. 3. Select one of the two a that gives the greater path metric. Stop and output the decoded sequence if this is the last iteration. 4. Update estimates of a and the jamming parameters and probabilities. Return to step 2). We call this decoding procedure the blind scheme. Performance results for this scheme are presented in Section 2.5. 3.7 Joint Density Scheme In QuasiStatic Fading Channels Collaborative jamming mitigation using joint pdf scheme among the nodes is done in basically the same manner as in Chapter 2. It is straightforward to derive the covariance matrix between each node's received symbols. Let Yk [yO), y, , k ] Then the conditional mean of yk given a() a(N1) is mk [ , .. (N, p1)], where p(i) = a(i) EUk. The covariance matrix Ek conditioned on b, is defined as SE[(yk mk)(yk mk)H], (3.27) where the diagonal elements are given by E No + b) 2NJ/p, I =0, ,N 1. (3.28) Note that No is assumed to be known to each node, and 1b) '2NJ/p is estimated using ML estimator during decoding process and updated after every iteration. The upper off diagonal elements are given by Im= b(l)b(m)N, l=0,.. ,N1, m= l+l,... ,N1. (3.29) P The lower offdiagonal elements are conjugates of the upper offdiagonal elements. Then the joint density function of the received symbols becomes 1 f(yk uk, a, b) N Yk exp [(yk mk) HYk mk)] (3.30) The joint density can also be utilized when calculating the received LLR for ex changed symbols which is defined as log f(yYkUk +1, a, b) f(yk Uk = 1,a,b) 3.8 Results 3.8.1 Jamming Detection and Parameter Estimation with EM Algorithm In this subsection, assuming N = 1, we first present performance results for the proposed iterative estimation and decoding algorithms using EM algorithm. For the results presented in this subsection, the rate 1/2 convolutional code with memory 6 and generator polynomials 133 and 171 (in octal) is used. The information block size (without tail bits) is 1000. Therefore L = 2012 +  S where I S is the number of pilot symbols. The channel interleaver is a rectangular interleaver of size 50x43. We consider a quasistatic fading AWGN channel. We set a = 1 and b 1. The phase of a varies from packet to packet. First, we compare performance of the proposed initial estimates for a and with the CRB. We compare the performance with independent jamming and 7ro = 71 = 7* 0.5, which is the assumption under which the initial estimators and thus the CRB are derived. For 50 and 100 pilot symbols and 10 dB, the mean squared error (MSE) of the initial estimates (before decoding) for a are compared with the CRB in Fig. 3.3. The MSE of the 100 10 =10dB simple #of pilot 10 .50 . 10 U0 152'02 Figure 3.3: Comparison of MSE for simple and improved estimates of channel gain a. improved estimate converges to the CRB at high SNR. The MSE of the simple estimate and the CRB is shown in Fig. 3.4. Again, the improved estimate shows better performance 10 U incgure 3ases. In comparison, theof MSE for simple and improved estimate is relatively flat.nnel gain a. impFor the resutimate converges to the CRB at high R. The Monsider the performance for estimate 0 dB, is much largernd than that of the impresults in Fstimate. 3.5 illustrate the performance of the simple estalgorithm with different numbers of iterations. Here we show the results for the a local minim if thproved initial estimate is not accurate. The comparison between the MSd of the initial estimates for and the CRB is shown in Fig. 3.4. Again, the improved estimate shows better performance than the simple estimate. It can be shown that the simple estimate of is biased, and the bias term increases as SNR increases. This is the reason that the MSE of the simple estimate increases. In comparison, the MSE of the improved estimate is relatively flat. For the results in Figs. 3.53.7, we consider the performance for ELb/NJ 0 dB, p 0.2, and E{Tj} 35. The results in Fig. 3.5 illustrate the performance of the EM algorithm with different numbers of iterations. Here we show the results for the improved estimate only. The performance gradually converges to that of perfect side information (PSI) as the number of iterations is increased, regardless of Eb/No. As Eb/No is increased, the performance is eventually dominated by the fixed jamming power, which results in the error floor evident in the figure. The results show that 5 iterations of the EM algorithm 105 1 0l. II. i . 4=10dB # of pilots u150 00 100 103" 7 102 Simple Estimate 0tU? 10" : i^^ Improved Estimate 10, 101 . . 102" : . ;l      10 0 5 10 15 20 25 Eb/NO (dB) Figure 3.4: Comparison of MSE for simple and improved estimates of (. provides performance close to PSI. To ensure performance close to PSI as other parameters are varied, we use 10 EM iterations for the remainder of the results presented in this paper. We next consider the performance of the improved estimate with various numbers of pilot symbols. These results are illustrated in Fig. 3.6. As previously explained, the improved estimate will not work properly if we do not use enough pilot symbols. We can see that the performance with 10 to 20 pilot symbols is very close to the performance when PSI is available. Note that further increasing the number of pilot symbols will decrease the overall code rate. This decreases the effective Es/No, which will increase the error probabilities. The bit error rates for the pilotassisted and blind detection and decoding processes are illustrated in Fig. 3.7. The curve labeled 'PSI' illustrates the results with PSI for the channel and jamming parameters. For the curves labeled 'Simple' and 'Improved', the receiver uses the pilotassisted estimation algorithm with the simple estimate or the im proved estimate, respectively. For the curve labeled 'Blind', the receiver uses the improved estimator with blind detection. For these results, the total number of iterations is 10, and 0 3 6 9 12 15 18 21 Eb/NO (dB) Figure 3.5: Performance versus number of iterations in the EM algorithm. p. A I A A PSIA A PSI # of pilots  2 A 5 V 10 9 20 0 3 6 9 12 15 18 21 Eb/N0 (dB) Figure 3.6: Performance of the improved estimate with various numbers of pilot symbols. WD 10 \,,, 10 106 0 3 6 9 12 15 18 21 Eb/N0 (dB) Figure 3.7: Performance of overall estimation and decoding process with various initial estimates. the number of pilot symbols is 20. We observe that the performance with the simple esti mate is very poor at high Eb/No. From Figs. 3.3 and 3.4, we can see that the variance of the the simple estimate is too large, and hence the EM algorithm often does not converge. Jamming detection is difficult without an accurate estimate of a, and inaccurate jamming information leads to inaccurate decoding and inaccurate channel estimates. This vicious cycle continues, causing the poor performance. However, using our improved estimator, we see that the decoding algorithm offers performance close to PSI. The performance of the blind scheme is nearly as good as the performance with pilots. Thus, the use of the path metric is an effective way to select the candidate codeword to be retained at the end of each iteration. The disadvantage of this scheme is the decoder complexity and decoding time. The label 'No estimation' represents the case that only the channel coefficient is estimated using pilot symbols and the receiver operates as if there is no jamming signal. The poor performance of this approach illustrates the need to accurately identify jammed symbols and the jamming parameters. 2 0 0g 2 v No estimation V Simple 4 Blind Improved 6 PSI 8 , 0.0 0.2 0.4 0.6 0.8 1.0 P Figure 3.8: The required Eb/Nj to obtain a FER of 102. As suggested in Pursley and Stark [15], the antijam capability of the jamming mit igation schemes can be measured by determining p*, which is the value of p required to prevent the receivers) from achieving an acceptable error probability. The higher the value of p*, the more symbols that must be jammed in order to significantly degrade communi cations. In Fig. 4.2, the value of Eb/Nj that is required to achieve a frame error rate (FER) of 102 is shown as a function of p, the probability that a symbol is jammed. For these results, Eb/No = 15 dB. The curve labeled 'PSI' corresponds to the performance with perfect detection of which bits are jammed and perfect knowledge of Eb/Nj and p. The value of p* for all of the estimates except the simple estimate is approximately 0.4. The blind scheme again shows approximately the same performance as the improved estimate. The EM algorithm with the simple estimate offers poor performance under moderate jam ming (p < 0.4). For the 'No estimation' scheme, the required Eb/Nj is a monotonically decreasing function of p. This is because the decoder treats all symbols as unjammed; as p increases, the jamming signal is spread more evenly over all the symbols, so the perfor mance improves. The proposed improved estimate and jamming detection scheme can be No estimation v Simple 4 Improved 10" Blind 0 PSI No jamming 105 0 10 20 30 40 50 Eb/NO (dB) Figure 3.9: Performance of jamming detection with various initial estimates in quasistatic fading channels (Eb/Nj = 8 dB, p = 0.4). applied to quasistatic fading fading channels (where the fading coefficient is constant over each codeword) without any modification. We consider the case of independent fading for the message and jamming signal. So, a and b are independent, complex Gaussian random variables with mean 0 and variance 0.5 per each dimension. The bit error rates for the var ious iterative detection and decoding processes are illustrated in Fig. 3.9 for an average bit energytojamming power spectral density of EbN/N = 8 dB and p = 0.4. Again, the EM algorithm with the improved estimate and either pilotassisted estimator or blind estimator provides performance close to PSI. The EM algorithm with the simple estimate offers poor performance. 3.8.2 Jamming Mitigation with Joint density Scheme in QuasiStatic Fading Channels In this subsection, we present the performance results for the collaborative jamming mitigation using the joint density scheme. For the results presented in this subsection, the rate 1/3 turbo codes are used. The generator polynomials is 5 and 7 (in octal). The information block size is 1000. For the initial estimate in the EM algorithm, the proposed 10 " I I S*EM w/ improved : Perfect CSI 1 0 N = 2 of 10 2 S\N=3 10  10, , ^ 0 5 10 15 20 25 Eb/NO (dB) Figure 3.10: Performance of jamming mitigation with joint pdf scheme. The EM algorithm is used with the improved initial estimate (Eb/Nj = 0 dB, p = 0.4). improved estimate is used. The jamming mitigation performance with joint density scheme is illustrated in Fig. 3.10. We see that the performance gain significantly increases as the number of collaborating nodes increases. The gain is approximately 15 dB for N = 2 and 20 dB for N = 3 at the FER of 102. The performance with EM algorithm achieves good performance, which is almost the same as the case with perfect CSI. CHAPTER 4 ISSUES ON THE COLLABORATIVE JAMMING MITIGATION In this chapter, several issues that are related to the collaborative jamming mitigation schemes are investigated. The issues include Reducing the overhead of the information exchange scheme The effects of quantization for the information to be exchanged The effects of imperfect collaboration channels Analytical performance bounds. Each issue is dealt with onebyone in the following sections. 4.1 ReducedOverhead Information Exchange Schemes To perform collaborative jamming mitigation, the nodes need to exchange informa tion over wireless channels. Therefore the amount of information that is exchanged in the collaborative process should be kept as small as possible. In this section, we propose two modified exchange schemes to reduce the overhead. The scheme that is proposed in Chapter 2, 3, and Moon et al. [53] is to exchange all jammed symbols. For notational con venience, we call it the all jammed symbols exchange scheme (AES). It does not consider whether a jammed symbol is likely to be in error or not. Notice that all jammed symbols are not likely to be in error, and we don't have to apply jamming mitigation scheme to those symbols. We exploit this fact by only exchanging a subset of the jammed symbols. As illustrated in Fig. 2.2, the effect of deinterleaving is to break up the bursts of jam ming at the input to the decoder. From the decoder's perspectives, these are new "broken" bursts of jammed symbols that have different characteristics from the bursts before dein terleaving. If p is small and the interleaver size is well matched to the jamming charac teristics, most of the jamming bursts after deinterleaving should consist of single isolated jammed symbols. However, longer broken bursts may significantly degrade the decoder y Order for the Input to decoder Window of length 2 Window of length 4 0 U O E Window of length 1 unjammed symbol jammed symbol jammed symbols that are exchanged Figure 4.1: Illustration of the window exchange scheme. In this example, nodes exchange information about jamming window of size > 4 performance. As p increases, it becomes more difficult to break up the jamming bursts. In addition, the interleaver cannot typically be adapted to the jamming characteristics, as these characteristics are often unknown at the time of transmission. As the first modified scheme, we propose to exchange information for only these broken jamming bursts that consist of i or more consecutive symbols, where i > 1. Note, these are bursts that are generated after deinterleaving. We call this scheme the window exchange scheme of size i (WESJ). If i = 1, this modified scheme is identical to AES. If i > 1, we don't exchange information for short broken jamming bursts (i.e. windows of size 1 to i 1). This scheme is illustrated in Fig. 4.1 for i = 4. For the second variation, we consider the a posteriori reliabilities of the received sym bols. When selecting symbols to be exchanged, we choose the least reliable bits (LRB) [1012] among the jammed symbols. The LRBs are those that have a small magnitude of their LLR, which indicates that the bit is likely to be in error. To reduce the overhead, we exchange p'. of jammed symbols, where 0 < p < 100. We call this the LRB exchange scheme p (LESp). So, if p = 100, this scheme is identical to AES. Note that the LRBs at each node may be different. So after broadcasting the indices for the LRBs, the received symbol values for the union of the indices are exchanged. Next, we investigate the amount of overhead that has to be exchanged between the nodes. To reduce the overhead, we can compress it by utilizing the statistics of the jammed symbols. By sending only the starting bit index and ending index of each burst, we can significantly reduce the overhead. It can be calculated that the average number of windows of jammed symbols is Wavg (= M/R 1)PoPoi + Pi [37], where M is the number of information bits, and R is the code rate. Therefore, the overhead of the joint pdf scheme with AES becomes OAES = 2NW,,g log2 L] + 2NQpL, (4.1) where L = M/R is the length of a packet and Q is the number of bits to represent a floating number. The first term of (4.1) is for the indices for the jammed symbols, and the second term is for the floating values. The WES scheme operates on the deinterleaved symbols, so we need the statistics of the deinterleaved symbols to calculate the overhead. Define pk [pko Pkl] to be jammer's state probability vector, which means that at time k, the jammer is in state 0 with probability Pko, and in state 1 with probability Pkl. The state probability vector at time k + K can be expressed as r K Poo00 P Pk+K Pk (4.2) P10 P11 where Pi is the transition probability of the jammer from state i to j. We approximate the jammer's state sequence after deinterleaving as the output of a Markov chain with transition probabilities governed by (4.2). Then, in a similar manner as in Roongta et al. [37], we can calculate the average number of jamming windows of size i after deinterleaving as W1.,(i) (L 1)PoPo1PI 1Pio + PiPi P1io + PoPoiP 1, where P, is the steadystate probability of the jammer at state i. Therefore it is possible to estimate the overhead of the joint pdf scheme employing WESi as i1 () IO lW IW (1)N2Q (4.3) l=1 where the second term is for the windows we don't exchange. We can also consider another scheme to exchange the indices of the jammed symbols. If each node represents a jammed symbol using '1' and unjammed symbol using '0', a sequence composed of 'O's and 'l's can be obtained and this sequence can be exchanged. The overhead of WES using this scheme can be approximated by i1 w2(i) NL + N2Q pL lW.,g(1) (4.4) l=0 Then, we can obtain OWES min {l (i), 2(i)}. The overhead of the mitigation schemes employing LESp is upper bounded by OLESp < min NL, NpPL \log2 L] } + min N22QpPL, 2NQpL (4.5) where P = p/100. When p = 50, we exchange 25% of the total jammed symbols at the first broadcast, and the decoder iterates with the exchanged information. After some number of iterations, the remaining 25% are broadcast. We exchange in this way because the first set of broadcast symbols may be enough for decoder to converge to a correct codeword. Note that the LRBs at each node may be different and the union of the LRBs are exchanged. The second term in (4.5) explains the worst case that the estimated jammed symbols are different at each node in which the overhead is N22QpIL. However it may be greater than the overhead of the case where all jammed symbols are broadcast. Therefore the minimum 62 Table 4.1: Comparison of the performance and overhead of the joint density scheme with various exchange schemes (Eb/No 10 dB, Eb/Nj = 0 dB, p = 0.4, E{Tj} = 50). N 2 N= 3 FER Overhead FER Overhead AES 1.40 x 102 37832 1.69 x 103 56748 WES i = 2 1.48 x 102 32721 2.58 x 103 49082 i = 4 3.35 x 102 20386 5.22 x 103 30578 i =6 4.66 x 102 11564 1.13 x 102 17346 LES p = 50 1.58 x 102 42084 3.03 x 103 63126 p = 25 4.18 x 102 24048 5.88 x 103 49599 of the two cases is selected. Note that the distribution of the LRBs is difficult to derive, so the compression technique used in AES and WES can not be utilized. This fact results in inefficient exchange for bit indices. Note that this is an upper bound, so it may be greater than the overhead of AES. The performance and overhead of the joint pdf scheme at Eb/No 10 dB with WESi and LESp are tabulated in Table 4.1. The jamming parameters are specified in the caption of the table. The relative performances at other values of Eb/No are similar, so they are not included here. As expected, the performance degrades according to i. There is a tradeoff between the amount of overhead and performance. So a system designer is required to carefully choose the parameters i or p. The value p*, which is the value of p required to prevent receiver to get acceptable performance is a standard measure to compare antijamming capability. Assuming a target FER of 102, the required Eb/Nj to achieve the target error rate is illustrated in Fig. 4.2. For this result, Eb/No 15 dB, and E{Tj} = 50. For simplicity, we assume that perfect CSI is available at receiver. When N = 2, the value of p* using AES is 0.7. When joint density scheme is employed with WESi or LESp, it results in some performance degradation, but the value of p* does not change drastically. For WES,, the required Eb/Nj merges to the original scheme's from p >= 0.9. This is because the number of windows of size less than i becomes less and less as p gets large. So every jammed symbols are exchanged at the region of high p. Note that for N = 3 with Eb/No 15 dB, the joint 2 "0 0 o 4 i=4 6 8 10 0.5 0.6 0.7 0.8 0.9 1.0 Figure 4.2: Required Eb/Nj in dB to achieve a FER of 102 (N = 2). density scheme with any of the exchange schemes achieve much lower error rate than the target error rate. Fig. 4.3 illustrates the overhead of the joint density scheme employing the proposed exchange schemes. For this result, N = 2 and Q = 5, where Q is the number of bits required to represent a real number. The overhead with WESi gets significantly smaller as i becomes large at most of the range of p. The upper bounds of the overhead with LESp is slightly loose when i = 50. The average overhead of LESp scheme is also much smaller than AES scheme. An appropriate exchange scheme can be chosen by considering these overheads. 4.2 The Effects of Quantization The exchange of information among the collaborating nodes requires quantization of the information, which is a floating number. In this subsection, the effects of quantization are investigated assuming errorfree channels for collaboration. Quantization is done using the LloydMax quantizer [54]. The effects of quantization using various number of bits are illustrated in Fig. 4.4 and 4.5 for AWGN and quasistatic fading channels, respectively. We can see that 5bit quantization gives approximately the same performance with the 60k 1 I I 1 I  SLES (bounds) AES 50k v LES (avg.) S WES i=2 40k i4 0) p=50 p=25 0 p=25 20k  10k 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 P Figure 4.3: Overhead of joint pdf scheme with the modified exchange schemes. quantization with an infinite number of bits, and 3bit quantization only performs slightly worse for very high Eb/No. 4.3 The Effects of Imperfect Collaboration Channels Until now, we assume that the collaborating nodes are clustered in a relatively small area so that their communication (exchange of information) is errorfree. However, this assumption is not necessarily true in practical situations. The signaltonoise ratio (SNR) among the collaborating nodes can not be infinite, so the effects of noise may not be ig nored. Sometimes the collaboration may be even hindered by jamming. Therefore in this subsection, the effects of imperfect collaboration channels are investigated. For the re sults in this subsection, 5bit quantization used. The effects of collaborative information exchange imperfect channels are illustrated in Fig. 4.6. For this result, Eb/Nj = 0 dB, p = 0.4, and E{Tj} = 50. In the figure, we consider three cases: When channel coding is not used, and there is no jamming (C = 0, J = 0) When channel coding is not used, and there is jamming (C = 0, J = 1) When channel coding is used, and there is jamming (C = 1, J = 1), x 10  10 10 .0 0 1 2 3 4 5 Eb/N0 (dB) Figure 4.4: Quantization effects in AWGN channels. 100 . ^v Q=3 4 Q=5 101 N=2 Q=infinite X, 102 N =3  10 10= 3 Eb/N0 (dB) Figure 4.5: Quantization effects in quasistatic fading channels. FER can be written as OO FER < APd, (4.6) d=dfree where Ad is the total number of codewords with weight d. The pairwise error probability in the block fading channels, Pd, can be derived assuming ML decoding. The derivation is done assuming perfect side information for jamming state and fading coefficient are available. We also assume that a() and b) are known for conciseness, where a( and b) are the fading coefficients for message and jamming signal at node i, respectively. For simplicity, assume N = 2 and the allones codeword (denoted by 1) is transmitted. If c is a competing codeword at distance d from the true codeword, the pairwise error probability is Pd =pE(c) = Pr logp(y1l,z) < log p(i z)), where z is jammer's state sequence. We condition on di symbols being jammed out of the d symbols that differ between c and the correct word. Thus, do = d di symbols are unjammed. Let Do and D1 be the set that contains the bit indices of the do unjammed and di jammed symbols, respectively. Then after some manipulations, we can write PE(C) = Pr( Ti + 2C'o U, + 2C', V, iEDO iED1 iED1 +2 W, < 0), (4.7) iED1 where C' is the ith row, jth column element of Y1, Ti Re(y)*a(o)), U, = Re(y o)*a(o)), V, = Re(yl)*a(1)), and W, = Re ({y(o)*a(1) + y )*}C~i Note that T, and U, have the same mean but different variance. It can be easily calculated that E[Ti] E[Ui] E[Vi] E[Wj] Var[Wi]  la)2 E, Var[T1] a(0) 2N0/2 = a(0)2 2 E, Var[U1] a(0) 21 00/2  l()2 E Var[V] a()1 2 11/2 = 2 ERe (a(o)*a (1)COI S 21a(0) 2 0C 2 1+ a(l) 2 02 0 + Re a()b(o)*b* N P Note that Ui, Vi, and Wi are not independent of each other. So we calculate the covariance between them as Cov(U,, ) E(UVV) E(U,)E(V,) = Re(a(o)*b(O)a(1)b(l)*) NJ 2p Cov(Ui, W) = Re(a(o)*a(1)C0O1)oo/2 +la(O)12 Re(b(o)*b(l)Ci,1) N 2p Cov(i, Wi) Re(a(o)*a(1)CO1)E71/2 +a(l) 2 Re(b(o)* b(1)COl )N The terms in (4.7) except the first summation can be represented as iED1 where A 2CoU, + 2CllV + 2Wi. 100 S^ *e Bound 101 ^I Simulation 10"2 10 ' A=27E/5 A=7t/5 10'5 i i , 0 2 4 6 8 10 Eb/NO Figure 4.7: Comparison of simulation and bounds for joint pdf scheme with nonfading message signals (a(i) 1 for all i), Eb/Nj 0 dB, p 0.6. Define T be the argument in (4.7), then we can obtain 2(d d) a(0)12 V + E[Tj = + dlE[ ] No V2(d dl) la()0 2 Var[T] + d Var[Y]. No So, the pairwise error probability Pd conditioned on a and b is P d [T] d d ,(1 p)dd Note that the offdiagonal elements C, can be written as b()b()*K, where K is a con stant that does not depend on the phases of the b) s, so every expectation and (co)variance can be represented as the function of the magnitudes of a('), b), and the relative phase difference AA Za() Za(o) + Zb() Zb(o). Note that this traditional transfer function upper bound becomes too loose as indicated in Malkamaki and Leib [55]. Therefore we use the modified techniques proposed in Malka maki and Leib [55]. However, it requires us to do 4N dimensional numerical integration 100II Average A performance 10 / ' Bound using SMonte Carlo JU .1...r A=4m5 S iBoundl  10"  0 5 10 15 20 25 Eb/NO Figure 4.8: Comparison of simulation and bounds for joint pdf scheme with nonfading jamming signals (b) 1 for all i), Eb/Nj 0 dB, p 0.6. (due to complex number) over all a(, b0), i 0, N 1. This integration becomes prohibitive as N grows. To avoid this complexity, we assume N = 2, and present bounds for two special cases. For the first case, we set all a(') to one, and average over the mag nitude of b) by fixing the A to a constant. For the second case, we set all b) to one, and average over the magnitude of a(') by fixing the A to a constant. Fig. 4.7 and 4.8 illustrate the upper bounds of the joint pdf scheme with the first and second case, respectively. For these result, Eb/Nj = 0 dB, p = 0.6, and AES is employed. The bounds and simulation results shows good matches. In Fig. 4.8, the average performance (over all a(i) and b(0)) by simulation and the average bound using MonteCarlo method are also plotted. We can see that the average bound is good estimate for the average performance. CHAPTER 5 CONCLUSIONS In this work, two techniques to mitigate jamming are investigated. In the first tech nique, the joint density for the jammed symbols at different receivers is applied directly to the calculation of the branch metrics in the BCJR algorithm. In the second technique, the jamming signals are estimated and subtracted from the received symbols. The results show that both techniques are effective at improving performance in the presence of a partialtime Gaussian jammer. It is shown that the joint density technique provides better performance with smaller overhead than the jamming cancellation technique. The two schemes also offer a tradeoff in terms of complexity that will depend on the number of collaborating receivers and the number of times that the jamming estimates are updated for the jamming cancellation technique. Overall, the results show that collaborative jamming mitigation techniques can significantly improve performance in the presence of a partialtime jammer. We also presented approaches to estimate the probability of jamming for each symbol as well as the statistics of the jamming signal. We showed that such techniques can provide performance close to that of perfect CSI. Moreover, we presented techniques that allow a group of collaborating nodes to achieve consensus on the set of jammed symbols. The symbols in this set are those that are exchanged and operated on by the jamming mitigation algorithms. We extend our mitigation schemes to quasistatic fading channels. Due to the ad ditional estimator for fading coefficient, the overall ML estimate for every parameter be comes prohibitive. Therefore, we use EM algorithm to overcome the difficult estimation problem. For the initial estimate in the EM algorithm, we show that sometimes, the sim ple estimate is not good enough for our scenario. So we propose an improved estimate that requires pilot symbols. We also propose a blind scheme that does not require pilot symbols. 72 We also investigate several issues related to our mitigation schemes. We propose two schemes to reduce the overhead of the information exchange. We investigate the effects of quantization for the information to be exchanged and the effects of imperfect collaboration channels. We also derive the performance bounds for the joint density scheme. REFERENCES [1] T. M. Cover and A. A. El Gamal, "Capacity theorems for the relay channel," IEEE Trans. Info. Theory, vol. IT25, pp. 572584, Sept. 1979. [2] N. Laneman, D. Tse, and G. Wornell, "Cooperative diversity in wireless networks: Efficient protocols and outage behavior," IEEE Trans. Inform. Theory, vol. 50, pp. 30623080, Dec. 2004. [3] A. Sendonaris, E. Erkip, and B. Aazhang, "Increasing uplink capacity via user coop eration diversity," in Proc. 1998 IEEE Int. Symp. Inform. Theory, Boston, Aug. 1998, p. 156. [4] F. M. J. Willems and E. C. Van Der Meulen, "The discrete memoryless channels with cribbing encoders," IEEE Trans. Inform. Theory, vol. 31, pp. 313327, May 1985. [5] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversitypart I: System description," IEEE Trans. Commun, vol. 51, pp. 19271938, Nov. 2003. [6] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversitypart II: Im plementation aspects and performance analysis," IEEE Trans. Commun, vol. 51, pp. 19391948, Nov. 2003. [7] T. E. Hunter and A. Nosratinia, "Coded cooperation under slow fading, fast fading and power control," in Proc. Asilomar C m f, rt, c on Signals, Systems and Computers, Pacific Grove, CA, Nov. 2002, pp. 118122. [8] T. E. Hunter and A. Nosratinia, "Cooperative diversity through coding," in Proc. 2002 IEEE Int. Symp. Infor Theory, Palais de Beaulieu, Lausanne, Switzerland, Jul. 2002, p. 220. [9] T. E. Hunter and A. Nosratinia, "Performance analysis of coded cooperation diver sity," in Proc. 2003 Int. Conf Commun., Anchorage, AK, May 2003, pp. 26882692. [10] T. F. Wong, X. Li, and J. M. Shea, "Distributed decoding of rectangular paritycheck code," IEE Electronics Letters, vol. 38, no. 22, pp. 13641365, Oct. 2002. I 11] A. Avudainayagam, J. M. Shea, T. F. Wong, and X. Li, "Reliability exchange schemes for iterative packet combining in distributed arrays," in Proc. 2003 Wireless Com munications and Networking Ci f, r, c',. New Orleans, LA, Mar. 2003, vol. 1, pp. 832837. [12] A. Avudainayagam, J. M. Shea, and T. F. Wong, "Cooperative diversity through reliability filling," in Proc 2003 Allerton Conf on Commun., Control, and Computing, Monticello, IL, Oct. 2003, pp. 15861587. 1131 A. J. Viterbi and I. M. Jacobs, Advances in Communication Systems, vol. 4, chapter Advances in coding and modulation for noncoherent channels affected by fading, partial band, and multiple access interference, pp. 279308, Academic, New York, 1975. 1141 J. K. Omura and B. K. Levitt, "Coded error probability evaluation for antijam com munication systems," IEEE Trans. Commun., vol. COM30, pp. 896903, May 1982. 1151 M. B. Pursley and W. E. Stark, "Performance of ReedSolomon coded frequencyhop spreadspectrum communications in partialband interference," IEEE Trans. Com mun., vol. 33, pp. 767774, Aug. 1985. [16] W. E. Stark, "Coding for frequencyhopped spreadspectrum communication with partialband interferencepart I: Capacity and cutoff rate," IEEE Trans. Commun., vol. 33, pp. 10361044, Oct. 1985. 1171 W. E. Stark, "Coding for frequencyhopped spreadspectrum communication with partialband interferencepart II: Coded performance," IEEE Trans. Commun., vol. 33, pp. 10451057, Oct. 1985. [18] C. W. Baum and M. B. Pursley, "Bayesian methods for erasure insertion in frequency hop communication systems with partialband interference," IEEE Trans. Commun., vol. 40, pp. 12311238, July 1992. 1191 M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Com munications Handbook, McGrawHill, New York, 2001. 1201 K. Cheun and W. E. Stark, "Performance of robust metrics with convolutional cod ing and diversity in fhss systems under partialband noise jamming," IEEE Trans. Commun., vol. 41, pp. 200209, Jan. 1993. [21] C. D. Frank and M. B. Pursley, "Concatenated coding for frequencyhop spread spectrum with partialband interference," IEEE Trans. Commun., vol. 44, pp. 377 387, Mar. 1996. 1221 J. H. Kang and W. E. Stark, "Turbo codes for coherent FHSS with partial band interference," in Proc. 1997 IEEE Military Commun. Conf., Monterey, CA, 1997, pp. 59. [23] J. H. Kang and W. E. Stark, "Turbo codes for noncoherent FHSS with partial band interference," IEEE Trans. Commun., vol. 46, pp. 1451 1458, Nov. 1998. [24] A. P. Worthen and W. E. Stark, "Unified design of iterative receivers using factor graphs," IEEE Trans. Inf, vol. 47, pp. 843849, Feb. 2001. [25] S. H. Schremmer, "Low density parity check codes in a frequency hopped communi cation system with partialband interference," in Proc. 2002 IEEE Military Commun. Conf., Anaheim, CA, Oct. 2002, vol. 1, pp. 894898. 1261 T. G. Macdonald and M. B. Pursley, "Staggered interleaving and iterative errorsand erasures decoding for frequencyhop packet radio," IEEE Trans. Wireless Commun., vol. 1, pp. 9298, Jan. 2003. [27] D. Torrieri and K. Bakhru, "An anticipative adaptive array for frequencyhopping communications," IEEE Trans. Aerosp. Electron. Syst., vol. 24, pp. 449456, July 1988. 1281 0. Besson, P. Stoica, and Y. Kamiya, "Direction finding in the presence of an inter mittent interference," IEEE Trans. Sig. Proc., vol. 50, pp. 15541564, Jul. 2002. [29 Y. Kamiya and 0. Besson, "Interference rejection for frequencyhopping communi cation systems using a constant power algorithm," IEEE Trans. Commun., vol. 51, pp. 627633, Apr. 2003. [30] L. R. Bahl, J. Cocke, E Jelinek, and J. Raviv, "Optimal decoding of linear codes for minimizing symbol error rates," IEEE Trans. Inform. Theory, vol. IT20, pp. 284287, Mar. 1974. [311 P. L. McAdam, L. R. Welch, and C. L. Weber, "M.A.P. bit decoding of convolutional codes," in Proc. 1972 Int. Symp. Inform. Theory, Asilomar, CA, 1972, p. 91. [32] T. E Wong, X. Li, and J. M. Shea, "Iterative decoding in a twonode distributed array," in Proc. 2002 IEEE Military Commun. Conf., Anahiem, CA, Oct. 2002, vol. 2, pp. 13201324. [33] W. E. Ryan, "Concatenated convolutional codes and iterative decoding," in Wiley Encyclopedia of Telecommunications. (J. G. Proakis, ed.) New York:Wiley and Sons, 2003. [34] S. Benedetto and G. Montorsi, "Unveiling turbo codes: Some results on parallel concatenated coding schemes," IEEE Trans. Inform. Theory, vol. 42, pp. 409428, Mar. 1996. [35] J. H. Kang, W. E. Stark, and A. 0. Hero, "Turbo codes for fading and burst channels," in IEEE Commun. Theory Mini Conf. at Globecom, Sydney, Australia, Nov. 1998, pp. 4045. [36] L. E. Baum and T. Petrie, "Statistical inference for probabilistic functions of finite state markov chains," Ann. Math. Stat., vol. 37, pp. 15541563, 1966. [37] A. Roongta, J. Moon, and J. M. Shea, "Reliabilitybased hybrid ARQ as an adaptive response to jamming," IEEE Journ. Select. Area. Commun., vol. 23, pp. 10451055, May 2005. [38] M. B. Pursley and J. S. Skinner, "Turbo product coding in frequencyhop wireless communications with partialband interference," in Proc. 2002 IEEE Military Com mun. Conf., Anaheim, CA, 2002, pp. 774779. [391 H. E. Gamal and E. Geraniotis, "Iterative channel estimation and decoding for con volutionally coded antijamming FH signals," IEEE Trans. Commun., vol. 50, pp. 321331,Feb.2002. [ 1] M. C. Valenti and B. D. Woemer, "Iterative channel estimation and decoding of pilot assisted turbo codes over flatfading channels," IEEE Journ. Select. Area. Commun., vol. 19, no. 9, pp. 16971705, Sep. 2001. 1411 J. H. Kang and W. E. Stark, "Iterative estimation and decoding for FHSS with slow Rayleigh fading," IEEE Trans. Commun., vol. 48, pp. 20142023, Dec. 2000. [421 Q. Li, C. N. Georghiades, and X. Wang, "An iterative receiver for turbocoded pilot assisted modulation in fading channels," IEEE Commun. Letters, vol. 5, pp. 145147, Apr. 2001. [43] A. P. Dempster, N. M. Laird, and D. B. Rubin, "Maximum likelihood from incomplete data via the EM algorithm," Journ. of the Royal Statistical Society, vol. 39, pp. 138, 1977. [441 L. R. Rabiner, "A tutorial on hidden Markov models and selected applications in speech recognition," Proc. IEEE, vol. 77, pp. 257 286, 1989. [45] J. A. Bilmes, "A gentle tutorial on the EM algorithm and its application to parameter estimation for Gaussian mixture and hidden Markov models," 1998, Available at http://citeseer.ist.psu.edu/bilmes98gentle.html, Accessed on Jun. 20, 2005. [46] G. K. Kaleh and R. Vallet, "Joint parameter estimation and symbol detection for linear and nonlinear unknown channels," IEEE Trans. Commun., vol. 42, pp. 24062413, 1994. [47] L. E. Baum, T. Petrie, G. Soules, and N. Weiss, "A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains," Ann. math. Stat., vol. 41, pp. 164171, 1970. [i~.] H. ZamiriJafarian and S. Pasupathy, "EMbased recursive estimation of channel parameters," IEEE Trans. Commun., vol. 47, pp. 12971302, 1999. [j A. Kocian and B.H. Fleury, "EMbased joint data detection and channel estimation of DSCDMA signals," IEEE Trans. Commun., vol. 51, pp. 17091720, 2003. [50] Y. Xie and C.N.Georghiades, "Two EMtype channel estimation algorithms for OFDM with transmitter diversity," IEEE Trans. Commun., vol. 51, pp. 106115, 2003. [51 ] C. F. J. Wu, "On the convergence properties of the EM algorithm," The Annals of Statistics, vol. 11, pp. 95103, 1983. [52] M. P. C. Fossorier, F Burkert, S. Lin, and J. Hagenauer, "On the equivalence between SOVA and maxlogMAP decodings," IEEE Commun. Letters, vol. 2, no. 5, pp. 137 139, May 1998. 1531 J. Moon, J. Shea, and Tan F. Wong, "Collaborative mitigation of partialtime jamming on nonfading channels," IEEE Trans. Wireless Commun., Acepted for publication, Available at http://plaza.ufl.edu/moonjw/paper/Twireless2004.pdf, Accessed on Jun. 01 2005. 1541 J. G. Proakis, Digital Communications, McGrawHill, 4th edition, 2000. [55] E. Malkamaki and H. Leib, "Evaluating the performance of convolutional codes over block fading channels," IEEE Trans. Inform. Theory, vol. 45, pp. 1643 1646, Jul. 1999. BIOGRAPHICAL SKETCH Jangwook Moon(S'03) received the B.S in electrical engineering from SungKyun Kwan University in Korea in 1999 and the M.S. degrees in electrical and computer en gineering from Yonsei University in Korea in 2001. He is currently in Ph.D program in electrical and computer engineering at University of Florida. His research interest is wire less communications/CDMA, with emphasis on turbo coding and iterative decoding, anti jamming communication protocol, collaborative communications, and ad hoc networks. 