Least squares symbol detection for multi antenna fh mfsk systems in the presence of follower jamming

LEAST SQUARES SYMBOL DETECTION FOR MULTI ANTENNA SLOW FHSS/MFSK SYSTEMS IN THE PRESENCE OF FOLLOWER JAMMING ALAGUNARAYANAN NARAYANAN NATIONAL UNIVERSITY OF SINGAPORE 2011 LEAST SQUARES SYMBOL DETECTION FOR MULTI ANTENNA SLOW FHSS/MFSK SYSTEMS IN THE PRESENCE OF FOLLOWER JAMMING ALAGUNARAYANAN NARAYANAN (B.E., ANNA University) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2011 ACKNOWLEDGEMENTS First of all, I am grateful to God for giving me the strength and wisdom to finish this thesis. My sincere thanks goes to my supervisor Professor Ko Chi Chung for his excellent guidance, encouragement and insightful comments throughout the period of my research work. I also wish to express my thanks to the staff and students in the communication laboratory for their assistance and friendship. Finally, I also wish to express my sincere gratitude to my parents and family, who have always given me unconditional love and great support. i CONTENTS ACKNOWLEDGEMENTS i CONTENTS ii SUMMARY v LIST OF FIGURES vi LIST OF ABBREVIATIONS vii LIST OF SYMBOLS ix CHAPTER 1 INTRODUCTION 1 1.1 FREQUENCY SHIFT KEYING 1 1.2 FADING 3 1.3 AWGN 6 1.4 JAMMING 6 1.5 INTRODUCTION TO SPREAD SPECTRUM 1.6 1.7 COMMUNICATIONS 7 FREQUENCY HOPPED SPREAD SPECTRUM SYSTEMS 9 1.5.1 SLOW FHSS SYSTEMS 11 1.5.2 FAST FHSS SYSTEMS 11 PERFORMANCE OF FHSS SYSTEMS IN A JAMMING ENVIRONMENT 12 1.7 RESEARCH OBJECTIVE and CONTRIBUTIONS 13 1.8 STRUCTURE OF THE THESIS 14 ii CHAPTER 2 FHSS/MFSK SYSTEMS IN THE PRESENCE OF JAMMING 2.1 15 SYSTEM MODEL 15 2.1.1 TRANSMITTED SIGNAL MODEL 15 2.1.2 PARTIAL BAND JAMMING MODEL 16 2.1.3 RECEIVED SIGNAL MODEL 17 2.2 VECTOR REPRESENTATION 19 2.3 SUMMARY 20 CHAPTER 3 LEAST SQUARES BASED SYMBOL DETECTION SCHEME 21 3.1 LS BASED SYMBOL DETECTION SCHEME 21 3.2 THEORITICAL ANALYSIS OF THE PROPOSED SCHEME 23 3.3 SUMMARY 28 CHAPTER 4 PERFORMANCE OF LS BASED SYMBOL DETECTION SCHEME 29 4.1 SIMULATIONS 29 4.2 SUMMARY 35 CHAPTER 5 CONCLUSIONS AND PROPOSALS FOR FUTURE RESEARCH 36 5.1 CONCLUSION 36 5.2 FUTURE WORK 36 BIBILOGRAPHY 38 iii APPENDIX-I 43 LIST OF PUBLICATIONS 47 iv SUMMARY The focus of this thesis is the performance of frequency hopped M-ary frequency shift keying (MFSK) systems in the presence of follower partial band jamming (PBJN) over flat fading channels. Thermal and other wideband Gaussian noises have been modeled as additive white Gaussian noise (AWGN) at the receiver. Follower partial band jamming is a strong threat to the symbol error rate (SER) performance of FHSS systems. In order to overcome the effects of follower PBJN and carry out symbol detection in slow FHSS/MFSK systems over quasi-static flat fading channels, a least squares (LS) based method is proposed in this thesis. Specifically, using the principle of Least squares, the complex gain factor between the two jamming components is estimated. This estimate is then used to remove the jamming signal during the symbol detection process. The effect of AWGN on the channel estimation and symbol detection are theoretically analyzed. The symbol error rate performances of the proposed algorithm are compared with that of traditional maximum likelihood (ML) algorithm and the scheme proposed in [13]. The proposed algorithm is found to outperform the other algorithms, when signal to noise ratio (SNR) is greater than about 20dB. v LIST OF FIGURES Fig. 1.1 An example of Binary FSK 3 Fig. 1.2 Block diagram of frequency hoped spread spectrum transmitter 9 Fig. 1.3 Block diagram of frequency hopped spread spectrum receiver Fig. 4.1 Performance of various schemes against SNR for 0dB SJR, BFSK, and four samples per symbol 30 Fig. 4.2 Performance of various schemes against SNR for 0 dB SJR, 4-FSK and four samples per symbol 31 Fig. 4.3 Performance of various schemes against SNR for 0 dB SJR, 8-FSK and eight samples per symbol 31 Fig. 4.4 Performance of various schemes against SJR for 30dB SNR, 8-FSK and twelve samples per symbol 32 Fig. 4.5 Performance of the proposed LS based scheme with various number of samples per symbol for 0 dB SJR and 8FSK 32 Fig. 4.6 Performance of the proposed LS based scheme with various number of samples per symbol at 0 dB SJR and 16 FSK 33 Fig. 4.7 Plot of Mean percentage of absolute error between theoretical and simulated values of x against SNR, with BFSK, -10 dB SJR and four samples per symbol 34 Fig. 4.8 Performance of the theoretical and simulated SER of the proposed scheme for BFSK,-10dB SJR and four samples per symbol 10 35 vi LIST OF ABBREVIATIONS AFSK Audio Frequency Shift Keying AR Auto Regressive ARMA Auto Regressive Moving Average AWGN Additive White Gaussian Noise BER Bit Error Rate CDMA Code Division Multiple Access DS Direct Sequence DSSS Direct Sequence Spread Spectrum FH Frequency Hopping FHSS Frequency Hopped Spread Spectrum FSK Frequency Shift Keying GMSK Gaussian Minimum Shift Keying GSM Global System for Mobile communication i.i.d independent and identically distributed ISI Inter Symbol Interference LS Least Squares MAI Multiple Access Interference MFSK M-ary Frequency Shift Keying ML Maximum Likelihood vii MSK Minimum Shift Keying MTJ Multi Tone Jammer OFDM Orthogonal Frequency Division Multiplexing PBJN Partial Band Jamming Noise PN Pseudo noise PSD Power Spectral Density SER Symbol Error Rate SJR Signal to Jamming Power Ratio SNR Signal to Noise Power Ratio SS Spread Spectrum VSM Vector Similarity Metric viii LIST OF SYMBOLS l phase shift and attenuation for the desired signal, which is received in the lth antenna l phase shift and attenuation for the jamming signal, which is received in the lth antenna d data symbol d estimated data symbol f frequency of hopping fd frequency spacing between two adjacent MFSK tones h positive integer n j t  baseband equivalent band limited signal N sampling rate Pe theoretical bit error rate T duration of one information bit Tc chip duration Ts sampling period  Time interval d  cost function u pilot symbol vl jamming components of the signal received in the lth antenna ix w n added white Gaussian noise W bandwidth x CHAPTER 1 INTRODUCTION 1.1 FREQUENCY SHIFT KEYING Frequency shift keying (FSK) is a frequency modulation scheme in which digital information is transmitted through changing the frequency of a carrier wave. M-ary frequency shift keying (MFSK) is a variation of FSK that uses more than two frequencies. MFSK is a form of M-ary orthogonal modulation, where each symbol consists of one element from an alphabet of orthogonal waveforms. M , the size of the alphabet is usually a power of two, so that each symbol has log 2 M bits. An example of Binary FSK is shown in fig. 1.1. Minimum Shift keying (MSK) and audio frequency shift keying (AFSK) are two other forms of FSK. MSK is a particular form of coherent FSK, and it has better spectrum usage when compared to FSK. In MSK, the waveforms that are used to represent the bits 0 and 1 will differ from each other by exactly half a carrier period. This is the smallest FSK modulation index that can be chosen such that the waveforms for 0 and 1 are orthogonal. Another form of MSK called Gaussian minimum shift keying (GMSK), is used in the global system for mobile communication (GSM) phone standard [30]. 1 In Audio frequency-shift keying (AFSK) modulation technique, digital data is represented by changes in the frequency (pitch) of an audio tone, yielding a signal that has been encoded suitably for transmission via radio or telephone. Normally, the transmitted audio shuffles between two tones: "mark" and “space", representing a binary one and a binary zero respectively. In AFSK, modulation is done at baseband frequencies. This is the difference between regular frequency-shift keying methods and AFSK. Even though Phase Shift Keying (PSK) modulation gives better performance than FSK in an additive white Gaussian noise (AWGN) channel, it is difficult to maintain phase coherence in the synthesis of the frequencies used in the hopping pattern. Therefore, FSK modulation with non coherent detection is used in frequency hopped spread spectrum (FHSS) systems [2]. 2 1 0 1 0 1 Data Carrier Modulated signal Figure 1.1 An example of Binary FSK 1.2 FADING It is the deviation that a carrier modulated communication signal experiences when it travels through certain propagation media. In general, fading tends to vary with time, geographical position and radio frequency, and it can be modelled as a random process. A channel that experiences fading is called as a fading channel. The two main reasons for fading in wireless systems are multipath propagation (referred to as 3 multipath fading) and shadowing from obstacles affecting the wave propagation (referred to as shadow fading). Multiple paths, in which a signal can traverse, are created by reflectors present in the environment surrounding the transmitter and receiver. Multipath propagation results in the superposition of multiple copies of the transmitted signal at the receiver. While travelling from the source to the receiver, each copy of the signal will be experiencing differences in terms of attenuation, delay and phase shift, which will lead to constructive and destructive interferences at the receiver end. This can cause amplification or attenuation of the signal power at the receiver. Strong destructive interference (also known as deep fades) can cause temporary failure of communication due to a severe drop in the channel signal to noise ratio. The effects of fading can be overcome by using transmit diversity where the signal travels over different channels that experience independent fading and then coherently combining them at the receiver. Now, the probability of experiencing a fade in this channel is proportional to the probability that all the component channels simultaneously experience a fade. Different types of fading are discussed below.  Slow fading – It arises when the coherence time of the channel is large when compared to the delay constraint of the channel. In this type of fading, the amplitude and phase variations imposed by the channel can be considered as constant with respect to the symbol period. Slow fading can be caused when there is a large obstruction such as a hill or large building, obscuring the main 4 signal path between the transmitter and the receiver. Log normal distribution is often used to model the amplitude change that is caused by shadowing [30].  Fast Fading – This occurs when the coherence time of the channel is small compared to the delay constraint of the channel. In this type of fading, the amplitude and phase variations imposed by the channel vary considerably with respect to the symbol period. In a fast-fading channel, the transmitter may use time diversity to take advantage of the variations in the channel conditions, and thereby increase the robustness of the communication to a temporary deep fade. A deep fade may temporarily erase some of the information that was transmitted. By using an error-correcting code coupled with successfully transmitted bits during other time instances (interleaving), the erased bits can be recovered [30].  Flat fading – In this type of fading, the coherence bandwidth of the channel is larger than the bandwidth of the signal. With flat fading all frequency components will be affected in the same way.  Frequency selective fading – When the coherence bandwidth of the channel is smaller than the bandwidth of the signal, frequency selective fading occurs. It is highly improbable that all parts of the signal will be simultaneously affected by a deep fade because different frequency components of the signal will be affected independently. Frequency selective fading channels are dispersive resulting delay spreads in the received signal. As a result, the transmitted symbols that are adjacent in time interfere with each other. In such 5 channels equalizers can be used to compensate for the effects of the inter symbol interference (ISI). Modulation schemes such as orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) use frequency diversity to provide robustness to frequency selective fading. In OFDM, the wideband signal is divided into many narrowband modulated subcarriers with each of them being exposed to flat fading rather than frequency selective fading [32]. CDMA uses the Rake receiver to deal with each echo separately [30]. 1.3 AWGN In additive white Gaussian noise model, the only impairment to proper communication is the linear addition of wideband noise with a constant spectral density and a Gaussian distribution of amplitude. It produces simple and tractable mathematical models which are useful for gaining insight into the underlying behavior of the system. Thermal vibrations of atoms in conductors, shot noise, black body radiation from the earth and other warm objects are the main sources for wideband Gaussian noise. Background noise of the channel under study is normally represented using AWGN. 6 1.4 JAMMING Jamming refers to the deliberate transmission of signals that disrupt communications, by decreasing the signal to noise ratio at the receiver. Generally, the jammer sends out a signal at the same frequency as the transmitter and causes interference to the received signal. The purpose of jamming is to block out the reception of transmitted signals. Jamming models considered in frequency hopped spread spectrum (FHSS) systems include partial band multi tone jamming and partial band Gaussian noise jamming [31]. A follower jammer has the capability to determine which portion of the spread spectrum bandwidth is being used during some time interval  , and transmits its jamming signal in that portion of the spectrum. 1.5 INTRODUCTION TO SPREAD SPECTRUM COMMUNICATIONS In this technique, a communication signal is transmitted in a bandwidth which is significantly larger than the original frequency content of the signal. The main feature of this technique is that it decreases the probability of interference to other receivers while maintaining the privacy. Spread Spectrum generally uses a sequential noise like signal structure to spread the narrowband information signal over a relatively wideband of frequencies. At the receiver, the received signal is given to a correlator to retrieve the original information signal. 7 Major features of spread spectrum communication are  Resistance to jamming (interference) - The transmitted signal will have an element of pseudo-randomness (unpredictability) associated with it. This randomness will be known only to the intended receiver and not to the jammer. As a result, the jammer will transmit an interfering signal without the knowledge of the pseudo random pattern. This reduces the vulnerability of the transmitted signal to jamming.  Resistance to fading- Since spread spectrum signals occupy high bandwidth it is unlikely that the signal will encounter multipath fading over its whole bandwidth.  Multiple access capability- Multiple users can transmit simultaneously on the same frequency (range) as long as they use different spreading codes. The different types of spread spectrum communications are 1. Direct Sequence (DS) – A sine wave is pseudo randomly phase modulated with a string of pseudo noise code symbols called chips. The duration of the chip is shorter than that of the information bit. 2. Frequency hopping (FH) - The carrier frequency is pseudo randomly changed over a wide range of frequency for transmitting radio signals. A detailed description of frequency hopping system is given in the next section. 3. Time hopping – In this technique, the carrier is turned on and off by a pseudo random sequence. 8 4. Chirp Spread – Here, wideband frequency modulated chirp pulses are used to encode information. A chirp is a sinusoidal signal whose frequency increases or decreased over a certain amount of time. 1.6 FREQUENCY HOPPED SPREAD SPECTRUM SYSTEMS In this system, each carrier frequency is chosen from a set of 2h (where h is a positive integer) frequencies that are placed over the width of the available data modulation spectrum. The pseudo-random code is used to control the sequence of carrier frequencies. A block diagram of a frequency hopped spread spectrum system transmitter and receiver are given in fig. 1.2 and 1.3. Figure 1.2 Block diagram of frequency hopped spread spectrum transmitter 9 Figure 1.3 Block diagram of frequency hopped spread spectrum receiver Normally binary or M-ary FSK mosulation schemes are used in FHSS. Based on the symbol transmitted , any one of the M frequencies will be used. The output signal from the modulator will be translated in frequency by an amount that is determined by the pseudo noise (PN) sequence, which in turn , is used to selsct a frequency that is synthesized by the frequency synthesizer. The frequency translated signal is mixed with the output from the FSK modulator and transmitted. If the PN generator output has m bits then 2  1 frequency translations are possible. m In the receiver , an identical PN generator, that is synchronised with the received signal, is used to control the output of the frequency synthesizer [19]. By mixing the synthesizer output with the received signal, the frequency translation introduced at the transmitter can be removed. The resultant signal is demodulated by means of an FSK 10 demodulator. A signal for maintaining synchronism of the PN generator with the frequency translated received signal is usually extracted from the received signal. FHSS systems are mainly used in miltary communication [3], wireless personal communications [20] and satellite communications [21-23]. Two different types of frequency hopped spread spectrum systems are discussed below. 1.6.1 SLOW FHSS SYSTEMS When MFSK data modulation is used with FHSS systems, the data modulator output is one of the 2h tones, each lasting hT seconds, where T is the duration of the information bit. Each of these tones will be orthogonal with respect to the other tones. Hence, the frequency spacing between two tones should be at least 1 . Assume that, hT in each Tc (chip duration) seconds the modulated data output is transmitted in a new frequency by the frequency hop modulator. When Tc  hT , the FHSS system is called a slow frequency hopping system. 1.6.2 FAST FHSS SYSTEMS In fast FHSS systems, the hopping frequency band changes many times per symbol. That is Tc  hT . A major advantage of fast FHSS systems is that frequency 11 diversity gain can be achieved in each transmitted symbol, which is particularly beneficial in a partial jamming environment. 1.7 PERFORMANCE OF FHSS SYSTEMS IN A JAMMING ENVIRONMENT FHSS systems are known to be robust against interference. However, their performance will be severely affected by multi tone jamming (MTJ) and partial band jamming. Among the two, MTJ can cause more damage to the FHSS signal. In partial band jamming, the frequency that is currently assigned to the receiver is measured by the jammer and then a jamming signal is transmitted in the frequency slot used. The jamming signal will be sent as soon as possible, once the current frequency slot is determined [4]. Fast frequency hopping may be seen as a viable solution to overcome the detrimental effects of partial band jamming, because of the fact that hopping frequency changes at a very high rate, making it difficult for the jammer to find out the current frequency slot used. But when fast frequency hopping is used the synchronization requirements will become more stringent as hopping rate is increased and it may be impossible to decrease the dwell interval of the hop. Due to such practical limitations fast hopping is difficult to be implemented in some applications and scenarios. The effect of the jammer causes the interference component in the received signal to be very high. So, symbol detection at the receiver end gets complex. Many anti-jamming algorithms have been proposed for slow FHSS system to reduce the effect of jamming. But the focus of most of these algorithms is the 12 elimination of partial band jamming [5-11], with the problem of follower jamming addressed to a smaller extent in [12-13] and [24]. In [12], an antenna array using the sample matrix inversion algorithm is exploited to separate the desired signal and the jamming signal. But, in this case the antennas have been assumed to be having equal gains. These assumptions will not hold good in a quasi static flat fading channel. The technique proposed in [24] performs better in a jamming dominant scenario. But in this technique, the received jamming signals are treated as deterministic quantities to be estimated. So this algorithm will produce less accurate jamming estimates at lower jamming power regions. This causes deterioration in the performance of the algorithm. Even though vector similarity based symbol detection scheme proposed in [1] gives good symbol detection performance in the presence of follower jamming in a quasi static flat fading channel, it assumes that the receiver has complete knowledge about the channel parameters. This places a restraint on the system. 1.8 RESEARCH OBJECTIVE and CONTRIBUTION Least squares (LS) method is a standard way of estimating the unknown parameters from the received data set. In this thesis, we investigate how a least squares based approach can be formulated for carrying out symbol detection in the presence of jamming and AWGN in FHSS communication systems. Specifically, the proposed approach uses a two element array to reject single follower jamming signal interference and carry out symbol detection in slow FHSS/MFSK systems over quasi static flat fading channels. Using the principle of Least squares, the complex gain 13 factor between the two jamming components is estimated. This estimate is used to remove the jamming signal during the symbol detection process. The effect of AWGN on the channel estimation and symbol detection are theoretically analyzed. The SER performances of the proposed algorithm are compared with that of traditional maximum likelihood (ML) algorithm and the scheme proposed in [13]. The proposed algorithm has been found to outperform the other algorithms, when signal to noise ratio (SNR) is greater than about 20dB. 1.9 STRUCTURE OF THE THESIS In CHAPTER 2, the transmitted signal model and the received signal model are discussed. The proposed LS based algorithm and the associated theoretical calculations are given in CHAPTER 3. Performance of the proposed scheme is discussed in CHAPTER 4 CHAPTER 5 concludes this thesis and suggests some future work. 14 CHAPTER 2 FHSS/MFSK SYSTEM IN THE PRESENCE OF JAMMING 2.1 SYSTEM MODEL In this thesis an MFSK modulated slow FHSS system is considered. In order to reduce the harmful effects of follower partial band jamming in a flat fading environment, a simple two-element receiver array, where the signal from each element is down converted and sampled at N times the symbol rate is used. Using the samples collected from the two elements the relative gain between the jamming components in the two elements can be determined, and this in turn can be used to for symbol detection. In CHAPTER 3, the process of symbol detection will be discussed in detail. 2.1.1 TRANSMITTED SIGNAL MODEL Without loss of generality symbols are taken to be transmitted in hops each consisting of k symbols. The first symbol in each hop is a pilot symbol, while the remaining ones are data symbols. Consider the detection of the symbol in a hop over 15 the interval 0  t  Ts , where Ts is the symbol duration. The complex envelop of the transmitted pilot signal is given by s p  t   exp  j 2  f  uf d  t  , (1) where f is the hopping frequency, u  0,1,...., M  1 represents the pilot symbol that is known at the receiver and f d denotes the frequency spacing between two MFSK tones. Similarly the complex envelope of the transmitted data signal is s  t   exp  j 2  f  df d  t  , (2) where d  0,1,....M  1 represents the data symbol. 2.1.2 PARTIAL BAND JAMMING MODEL As shown in [14], the follower jammer first measures the hopping frequency and the spectrum of the desired hop and then directs the available transmitting power to the currently used frequency slot. With just the knowledge of the hopping frequency of the desired signal ,this jammer will transmit a noise like signal which will cover the entire band of the desired signal. The complex envelope of the follower partial band jamming signal is therefore c  t   n j  t  exp  j 2 ft  , (3) where n j  t  is the baseband equivalent band-limited signal which can be modeled as a zero mean band-limited Gaussian random process. In equation (3) n j  t  is 16 multiplied by an exponential term, so as to up convert the baseband signal to the frequency slot occupied by the desired signal. 2.1.3 RECEIVED SIGNAL MODEL It is assumed that the desired signal and the follower jamming signal experience a quasi-static flat fading channel. The received pilot signal at the lth antenna element is therefore given by pl  t   l s p  t   l c  t   wl  t  , (4) where l  1, 2 , wl  t  represents the complex additive white gaussian noise (AWGN) in the receiver, and the complex coefficients  l , l account for the overall effects of phase shifts , fading and antenna response on the desired signal and the jamming signal at the lth antenna element respectively. As discussed in [15] and [16], the fading gains can be taken to be non-selective and remain unchanged within hop duration in slow FH systems. As mentioned earlier, the first symbol in each hop is a pilot symbol, while the remaining ones are data symbols. Following (4), the received data signal at the lth antenna element is given by rl  t   l s  t   l c  t   wl  t  (5) At the lth antenna element the received signal is sampled at N times the symbol. Using (1), (3) and (4), the nth sample of the received pilot signal is pl ,n  l exp  jn  u    l cn  wl ,n , (6) where 17  n   pl ,n  pl    Ts   N   (7)  n    Ts   N   n  u   2  f  uf d    (8)  n   cn  c    Ts   N   (9) and  n   wl ,n  wl    Ts   N   for n  0,1,....N  1 (10) Using (2), (3) and (5) the nth sample of the received data signal is similarly given by rl ,n  l exp  jn  d    l cn  wl ,n , (11) where  n    Ts   N   n  d   2  f  df d    (12) From (6) and (11), the signal to jamming ratio (SJR) and signal to noise power (SNR) ratio are given by  E c  ,   J  E l  2 and  N  E wp ,n SJR= 2 n 2 S S J and S N SNR= , respectively with  , 2 2  E  l exp  jn  u    E  l exp  jn  d   = E  l     2 . 18 2.2 VECTOR REPRESENTATION For ease of analysis, equations (6) and (11) can be written in vector form for the N samples from the two sensors as follows p1  1s p  u   v1  w1 , (13) p2   2s p  u    v1  w 2 , (14) r1  1s  d   v1  w3 , (15) r2   2s  d    v1  w 4 , (16) where T pl   pl ,0 , pl ,1 ,.... pl , N 1  , T rl  rl ,0 , rl ,1 ,....rl , N 1  , l  1, 2, (17) (18) s p  u   exp  j0  u   , exp  j1  u   ,....exp  jN 1  u   (19) s  d   exp  j0  d   ,exp  j1  d   ,....exp  jN 1  d   , (20) v1  1 c0 , c1 ,.....cN 1   (21) 2 , 1 (22) and T w k   wk ,0 , wk ,1 ,....wk , N 1  , k  1, 2,3, 4 (23) Note that the receiver does not have any knowledge about the channel parameters affecting the transmitted signal. 19 2.3 SUMMARY In each hop, the first symbol that is transmitted is the pilot symbol and the remaining symbols are data symbols. These symbols are FSK modulated and transmitted through the channel. The channel parameters will be same throughout one hop. The jamming signal is generated at the hopping frequency of the transmitted symbol. This signal is then multiplied by a zero mean band limited Gaussian random process. It is assumed that the channel is a quasi static flat fading channel and AWGN is added at the receiver. So the received signal is the sum of the transmitted signal, jamming signal and AWGN. Finally, all the signals are represented as vectors for ease of analysis. 20 CHAPTER 3 LEAST SQUARES BASED SYMBOL DETECTION ALGORITHM 3.1 LS BASED SYMBOL DETECTION SCHEME We will now consider the detection of the symbol in one hop. Since MFSK signaling is employed, the desired symbol d can only take value in the range 0,1,.....M  1 . Specifically, we will derive a scheme that first makes use of the pilot symbol to estimate the relative gain  between the jamming components in the two elements before eliminating the jamming components for data detection. Since p2   2s p  u     p1  1s p  u   (24) when SNR tends to infinity, consider v  p2   2s p  u    p1  1s p  u   . (25) As equation (25) will be zero when noise is not present, one approach to find  is to 2 adopt a least squares approach to minimize v . To do this, (25) can be simplified to become v  p2 p1   s p  u  , (26) 21 where   1   2 . (27) To carry out the least squares formulation, this can be written in terms of an unknown parameter vector x as follows: v  Ax  p2 , (28) A   p1 s p  u  (29)   x   .   (30) where and Using the least squares approach described in [17], the unknown parameter x that 2 minimizes v is given by   x      A †p 2 ,   (31) A†   AT A  AT (32) where 1 is the pseudo inverse of A . An example derivation for (31) has been given in Appendix-I. On substituting (31) into (28), we get v  A  A†p 2   p 2 . (33) The estimated values of  and  , obtained from the transmitted pilot symbol, can now be used to do symbol detection. Specifically, the value of channel parameters will not change in a flat fading scenario for all the symbols in the hop, where the 22 hopping frequency remains constant. For this purpose, (15) and (16) can be used to represent the received data signal. Similar to (24), the received data signals satisfies r2   2s  d     r1  1s  d   in the absence of noise, with (34) s  d  now equal to the estimated value of the transmitted signal. Again, (34) can be written in matrix form as, Γ  d   Ax  r2 (35) where A   r1 s  d  , (36) and the left hand side of (35) would be zero in the absence of noise. Using the least squares approach with the objective function given by (35), the transmitted symbol can then be detected by using   d  arg min Γ  d  ; d  0,1,...M  1 . d 2 (37) 3.2 THEORETICAL ANALYSIS OF THE PROPOSED SCHEME In this section, we will analyze the effect of noise on the channel parameters estimated using the above method, when noise power is low. Let p1 and p 2 be the noise free components of p1 and p 2 , such that p1  p1  w1 (38) 23 and p2  p2  w 2 . (39) Using (38), (29) can be modified and written as A  A  w1D , (40) D  1 0 (41) where and A  p1 s p  u  . (42) Therefore, we can rewrite x as  x   A  w1D    A  w1D  p † 2  w2    A  w D  A  w D p 1 H H 1 1    A H A  A H w1D  D H w1H A  D H w1H w1D 2  w2   A 1 H p 2  D H w1H p 2  D H w1H w 2  A H w 2 .  (43) According to Woodbury matrix identity [18],  B  YZ  is given by 1  B  YZ  1  B1  B1Y  I  ZB 1Y  ZB 1 , 1 (44) where B is a n  n matrix, Y is a n  k matrix, I is a k  k identity matrix and Z is a k  n matrix. To use (44) on (43), we can first rewrite (43) as 24   x   E  DH w1H A  w1D   A 1 H  p 2  DH w1H p 2  DH w1H w 2  A H w 2 , (45) where E  A H A  A H w1D    A H A   A H w1D . (46) Then, (45) can be simplified using Woodbury matrix identity to yield     x   E1  E1  DH w1H  I  A  w1D E1  D H w1H    1  H H H   A p 2  D w1 p 2   A  w1D E 1  H H  H  D w1 w 2  A w 2   , (47)  where I is an N  N identity matrix. Using Woodbury matrix identity for E1 then, gives   1 1 1 1    H 1 H  A H I   w1D  A H A A H  w1D  A H A   A A  A A     1 1 1 1 1  H H H H    AH A  AH A A I   w1D  A A A  w1D  A A     x    1  1  1  1   H   H H H  A H I   w1D  A H A A H  w1D  A H A     D w1   I   A  w1D  A A  A A     1   1 1 1 1    A H I   w1D  A H A A H    A  w1 D  A H A  A H A  w1D  A H A                                                            H H H    A p 2  D w1 p 2  1     H H 1    D w1 w 2  A H w 2  H H   D w1             (48) Performing some simplifications and ignoring all the second order noise terms in (48), we get     x  A H A 1 A H w  A H A 1 D H w H p 2 1 2  opt x 1 1  A H A A H  w1D  A H A A H p 2  A H A           , 1 1 H H H H D w A A A A p  1 2      (49) 25 where  xopt   A H A  A 1 H  p2 . (50) In (49) xopt represents the value of x in the absence of noise. The remaining terms in (49) gives the contribution of noise terms in determining the value of x .Now we will derive an approximate SER expression for the proposed LS scheme. We will be considering BFSK signaling only for the sake of simplicity and noting that the case of MFSK signaling can be similarly analyzed. The two BFSK symbols will be taken to be equally probable with the transmitted symbol being d  0 . Using (37), the SER of the proposed scheme can be shown to be Pe  Pr Γ 2  0   Γ 2 1 | d  0  Pr Γ 2  0   Γ 2 1  0 | d  0 , (51) where Γ  0  and Γ 1 are the objective functions for the correct and incorrect decisions, with Γ  0   r1 s  0  x  r2 (52) Γ 1  r1 s 1 x  r2 . (53) and Using (15) and (16), we have 26   Γ 2  0   Γ2 1   A0  w 3D  x   r2  w 4    A1  w 3D  x   r2  w 4  2 2 , (54) where A0   r1 s  0  , (55) A1   r1 s 1 , (56) r1  r1  w1 (57) r2  r2  w 2 . (58) and On performing some simplifications on (54), we get x H A 0H A 0 x  x H A 0H w 3Dx  x H D H w 3H A 0 x   H H  H H H  x A 0 r2  x A 0 w 4  r2 A 0 x   H  H H H H    w 4 A 0 x  x A1 A1x  x A1 w 3Dx .  H H H  H H H H  x D w 3 A1x  x A1 r2  x A1 w 4  r H A x  w H A x  4 1  2 1  (59) Ignoring second order terms, we have 27 H H xopt A 0H A 0  A1H A1  xopt  r2H  A 0  A1  xopt  xopt A 0H  A1H  r2      H   xopt   A 0H A 0 H H       A A x  A A x 0 0 opt 1 1 opt   r2H  A 0  A1   E  E H     AH r  AH r    H H  , 0 2 1 2    xopt A1 A1       H  H H H H H  xopt   A 0  A1  w 3Dxopt  xopt D w 3   A 0  A1  xopt   H  H H H  xopt  A 0  A1  w 4  w 4  A 0  A1  xopt  (60) where 1 1  H H H H H   A A A w 2  A A D w1 p 2  1 1 E    A H A A H  w1D  A H A A H p 2  1 1  H H H H H   A A  D w1  A A A A p 2                   .     (61) From (51) and (60) we can conclude that, symbol error probability is given by Pe  Pr   0 | d  0 . (62) 3.3 SUMMARY In this chapter, a least squares based algorithm is proposed to do symbol detection in slow FH/MFSK systems in the presence of partial band jamming. The proposed scheme estimates the complex gain factor between the two jamming components using the pilot symbol that was transmitted. Since this factor will be constant throughout the hop, it can then be used to do detect the successive data symbols in the hop. Finally, analytical expressions for the estimated parameters and the SER were derived. 28 CHAPTER 4 PERFORMANCE OF LS BASED SYMBOL DETECTION SCHEME 4.1 SIMULATIONS Numerical simulations have been performed to analyze the performance of the proposed LS based detection scheme for the case of slow FHSS systems with various MFSK modulations. In the simulation study, each hop has ten MFSK symbols, the symbol rate is 200,000 symbols per second and the hop rate is 25,000 hops per second. A total of 20 million symbols were used for each simulation run, and the fading coefficients are modelled as independent identically distributed (i.i.d) complex random variables with zero mean and variance of 0.5 per dimension. Noise samples were generated using random number generator. In figure 4.1, the SER performance of the proposed LS based scheme is compared with that of traditional ML algorithm and a scheme proposed in [13]. The SER performances have been plotted against SNR, with the SJR fixed at 0 dB and the number of samples per symbol given by N  4 . As illustrated from the figure, the proposed LS based scheme outperforms other algorithms when SNR is greater than about 20 dB. When noise power is high, the traditional ML algorithm performs slightly better when compared to the performance of the LS based scheme. This is 29 because of the fact that the traditional ML algorithm uses a simpler signal model, which requires fewer parameters to be estimated. As a result, better estimation accuracy can be obtained, which in turn can lead to better performance. 0 10 Proposed Scheme Traditional ML Scheme in [13] -1 SER 10 -2 10 -3 10 -4 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.1: Performance of various schemes against SNR for 0dB SJR, BFSK, and four samples per symbol Figures 4.2 and 4.3 show the SER performances of the proposed LS based scheme, traditional ML algorithm and the scheme in [13] under 4-FSK (with four samples per symbol) and 8-FSK (with eight samples per symbol) conditions. It is clear from the figures that the proposed LS based scheme performs better than other algorithms when SNR is greater than about 20 dB. 30 0 10 Proposed Scheme Traditional ML Scheme in [13] -1 SER 10 -2 10 -3 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.2: Performance of various schemes against SNR for 0 dB SJR, 4FSK and four samples per symbol 0 10 Proposed Scheme Traditional ML Scheme in [13] -1 SER 10 -2 10 -3 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.3: Performance of various schemes against SNR for 0 dB SJR, 8FSK and eight samples per symbol SER performances of the various schemes against SJR, with the SNR fixed at 30dB and 8-FSK has been shown in figure 4.4. It is seen form the figure that the proposed scheme performs well even in high interference regions. 31 0 10 -1 10 SER Proposed Scheme Traditional Ml Scheme in [13] -2 10 -3 10 -4 10 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 SJR(dB) Figure 4.4: Performance of various schemes against SJR for 30 dB SNR, 8FSK and twelve samples per symbol In Fig. 4.5 and 4.6, the effect of varying the number of samples per symbol for the LS based algorithm with 0 dB SJR and 8FSK/16-FSK is shown. It is evident from the figures, that increasing the number of samples per symbol leads to better parameter estimation, thereby improving the symbol error rate. 0 10 6 samples 12 samples -1 SER 10 -2 10 -3 10 -4 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.5: Performance of the proposed LS based scheme with various number of samples per symbol for 0 dB SJR and 8-FSK 32 -1 10 8 samples 16 samples -2 SER 10 -3 10 -4 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.6: Performance of the proposed LS based scheme with various number of samples per symbol at 0 dB SJR and 16-FSK The validity of the analytical expression of x obtained in Section 4 is shown in figure 4.7. The mean % of error between the theoretical and simulated values of x has been plotted against SNR, for the case of BFSK signaling with four samples per symbol and when SJR is fixed at 0 dB. As illustrated from the figure, even when SNR is 10 dB, the percentage of error between the theoretical and analytical values of x is small. When noise power is decreased, the percentage of error becomes negligible. 33 Mean percentage of absolute error between theoritical and simulated values of x -1 10 -2 10 -3 10 -4 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.7: Plot of Mean percentage of error between theoretical and simulated values of x against SNR, with BFSK, -10 dB SJR and four samples per symbol In Fig.4.8, the theoretical and simulated SER performance of the proposed LS based symbol detection scheme has been compared for BFSK (with four samples per symbol) condition, when the SJR has been fixed at -10dB. As seen from the figure, there is a very small difference between the theoretical and the simulated SER performances. This is a clear indication of the validity of the theoretical SER expression in (62). 34 0 10 Simulated Theoritical -1 SER 10 -2 10 -3 10 10 12 14 16 18 20 22 24 26 28 30 SNR(dB) Figure 4.8: Performance of the theoretical and simulated SER of the proposed scheme for BFSK,-10dB SJR and four samples per symbol. 4.2 SUMMARY Numerical simulations analyzing the SER performance of the proposed LS based symbol detection scheme is discussed in this chapter. It is clear from the simulation results that the SER performance of the proposed scheme outperforms that of the traditional ML method and the scheme in [13] when SNR is greater than about 20dB. Also it is seen that when the number of samples is increased the SER performance of the scheme gets better. The narrow margin of difference between the theoretical and simulated values of SER indicates the validity of the proposed scheme. 35 CHAPTER 5 CONCLUSIONS AND PROPOSALS FOR FUTURE RESEARCH 5.1 CONCLUSION In this thesis, a LS based method has been proposed for symbol detection in slow FHSS/MFSK systems, without the knowledge of channel parameters, in the presence of a follower partial band jammer, over quasi-static flat fading channels. By finding out the complex gain factor between the received jamming components, the proposed scheme is able to overcome the effects of jamming. The performance of the proposed scheme has been compared with that of traditional ML algorithm and the scheme in [13]. Theoretical analysis for the estimated channel parameters and the SER were also done and they were shown to agree well with the simulated results. 5.2 FUTURE WORK In our scheme, a pilot symbol will be transmitted at the start of each hop. This was done to get information about the parameters affecting the signal. Research work can be carried out to find whether parameter knowledge can be obtained without the usage of pilot symbol. 36 Least squares method could also be used in digital filter design, adaptive noise cancellation, auto regressive (AR) parameter estimation for auto regressive moving average (ARMA) model and phase locked loop. Multi user communications have become common in modern communication systems [25]. Significant effort has been spent on analyzing the capacity of such channels for AWGN and Rayleigh fading environments [26-29].Over coming MAI (Multiple Access Interference) in OFDM systems is an interesting research area. For future work, using the proposed system to eliminate multi access interference (MAI) in OFDM systems can be considered. In this work, symbol by symbol detection method has been used. 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[32] Hui Liu and Guoqing Li, OFDM – Based Broadband Wireless Networks, John Wiley and Sons, Inc., Hoboken, New Jersey, 2005. 42 APPENDIX –I v   p2   2s p  u     p1  1s p  u   (I-1) 1    j 90  e  s p u    1     e j 90  (I-2) 0.60e j 23     0.76e  j 73  p1    j166 0.92e     j 91 0.76e  (I-3) 0.73e j 80     0.67e  j 39  p2    j134 0.53e     j117 0.33e  (I-4) Let   Equation (I-1) can be rewritten as   0.60e j 23  0.73e j 80  1  1         j 90    j 90     j 73  0.67e j 39    e 0.76 e   e  v           2 1  1   1   , j134 j166 0.92 e  0.53e           j 90       j117 e  e j 90    0.76e j 91 0.33e     (I-5) 43 which is equivalent to 0.60e  j157    0.76e j107 v  j14  0.92e   j 89  0.76e  0.73e j 80  1         j 39   e j 90   0.67 e  .  1      j134 1     0.53e   2   j 90  j117  e  0.33e   1  e  j 90 1  e j 90 (I-6) Let   1   2 . (I-7) v is the error vector. Equation (I-6) can be written as 0.60e  j157    0.76e j107 v  j14  0.92e   j 89  0.76e  0.73e j 80  1       e  j 90    0.67e  j 39  .     1    0.53e j134     j117  e j 90  0.33e   (I-8) Let   A   .   (I-9) So equation (I-8) can be written as 0.60e  j157    0.76e j107 v  j14  0.92e   j 89  0.76e  0.73e j100 1      0.67e j141 e  j 90  A  j 46 1  0.53e     j 63 e j 90  0.33e     .    (I-10) 44 2 Therefore, v is given by 0.60e  j157    0.76e j107 2 v    j14  0.92e   j 89  0.76e    0.60e  j157      0.76e j107 H A   j14   0.92e    j 89   0.76e  H   0.73e  j100       0.67e j141      j 46   0.53e    j 63    0.33e 0.73e  j100 1       0.67e j141 e  j 90  A  j 46 1  0.53e     j 63 e j 90  0.33e  1    e  j 90   1    e j 90  H 0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e  0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e        1    e  j 90  H AA 1    e j 90  1    e  j 90  A 1    e j 90   2  0.73e  j100    0.67e j141   j 46  0.53e   j 63  0.33e         0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e  1    e  j 90   1    e j 90  H 0.73e  j100    0.67e j141   j 46  0.53e   j 63  0.33e  2                 (I-11) 2 Now we will differentiate v with respect to A and equate the result to zero.   0.60e  j157      0.76e j107 H A   j14   0.92e    j 89  0.76e   H   0.73e  j100       0.67e j141      j 46   0.53e   2  j 63  v   0.33e   A 0.60e    0.76e j107  2   j14  0.92e   j 89  0.76e  j157  2.35   2.93e  j175  1    j 90  e  1    e j 90  H 1    j 90  e  1    e j 90  H 0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e  1    j 90  e H AA 1    e j 90  0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e 1    j 90  e A 1    e j 90  0.73e   0.67e j141   j 46 0.53e   j 63 0.33e  0.60e      0.76e j107   2  j14   0.92e    j 89   0.76e  0.73e  j100   0.67e j141   j 46 0.53e   j 63 0.33e         0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e  1    j 90  e  1    e j 90  H 0.73e  j100   0.67e j141   j 46 0.53e   j 63 0.33e 2 A  j100  j157 1    j 90  e  1    e j 90  H  0.60e  j157    0.76e j107   j14  0.92e   j 89  0.76e                   0 1    e A 0 1    e j 90   j 90 1.388e j 21  2.93e j175  A     1.57e  j140  4      45 (I-12) From equation (I-12) we can write     2.35  2.93e j175   1.388e j 21 (I-13)  2.93e (I-14)  j175   4  1.57e  j140 On simultaneously solving equation (I-13) and (I-14), we get    1.96e j 25 (I-15)   0.995 (I-16) 46 LIST OF PUBLICATIONS [1] N.Alagunarayanan, F. Liu, C. C. Ko, “Least squares symbol detection for multi antenna FH/MFSK systems in the presence of follower jamming” ,under review for possible publication in IET on Signal Processing. 47 [...]... Specifically, the proposed approach uses a two element array to reject single follower jamming signal interference and carry out symbol detection in slow FHSS /MFSK systems over quasi static flat fading channels Using the principle of Least squares, the complex gain 13 factor between the two jamming components is estimated This estimate is used to remove the jamming signal during the symbol detection process The. .. this in turn can be used to for symbol detection In CHAPTER 3, the process of symbol detection will be discussed in detail 2.1.1 TRANSMITTED SIGNAL MODEL Without loss of generality symbols are taken to be transmitted in hops each consisting of k symbols The first symbol in each hop is a pilot symbol, while the remaining ones are data symbols Consider the detection of the symbol in a hop over 15 the interval... component in the received signal to be very high So, symbol detection at the receiver end gets complex Many anti -jamming algorithms have been proposed for slow FHSS system to reduce the effect of jamming But the focus of most of these algorithms is the 12 elimination of partial band jamming [5-11], with the problem of follower jamming addressed to a smaller extent in [12-13] and [24] In [12], an antenna. .. beneficial in a partial jamming environment 1.7 PERFORMANCE OF FHSS SYSTEMS IN A JAMMING ENVIRONMENT FHSS systems are known to be robust against interference However, their performance will be severely affected by multi tone jamming (MTJ) and partial band jamming Among the two, MTJ can cause more damage to the FHSS signal In partial band jamming, the frequency that is currently assigned to the receiver... BASED SYMBOL DETECTION ALGORITHM 3.1 LS BASED SYMBOL DETECTION SCHEME We will now consider the detection of the symbol in one hop Since MFSK signaling is employed, the desired symbol d can only take value in the range 0,1, M  1 Specifically, we will derive a scheme that first makes use of the pilot symbol to estimate the relative gain  between the jamming components in the two elements before eliminating... for the overall effects of phase shifts , fading and antenna response on the desired signal and the jamming signal at the lth antenna element respectively As discussed in [15] and [16], the fading gains can be taken to be non-selective and remain unchanged within hop duration in slow FH systems As mentioned earlier, the first symbol in each hop is a pilot symbol, while the remaining ones are data symbols... knowledge about the channel parameters This places a restraint on the system 1.8 RESEARCH OBJECTIVE and CONTRIBUTION Least squares (LS) method is a standard way of estimating the unknown parameters from the received data set In this thesis, we investigate how a least squares based approach can be formulated for carrying out symbol detection in the presence of jamming and AWGN in FHSS communication systems. .. mathematical models which are useful for gaining insight into the underlying behavior of the system Thermal vibrations of atoms in conductors, shot noise, black body radiation from the earth and other warm objects are the main sources for wideband Gaussian noise Background noise of the channel under study is normally represented using AWGN 6 1.4 JAMMING Jamming refers to the deliberate transmission of. .. deterministic quantities to be estimated So this algorithm will produce less accurate jamming estimates at lower jamming power regions This causes deterioration in the performance of the algorithm Even though vector similarity based symbol detection scheme proposed in [1] gives good symbol detection performance in the presence of follower jamming in a quasi static flat fading channel, it assumes that the. .. antenna array using the sample matrix inversion algorithm is exploited to separate the desired signal and the jamming signal But, in this case the antennas have been assumed to be having equal gains These assumptions will not hold good in a quasi static flat fading channel The technique proposed in [24] performs better in a jamming dominant scenario But in this technique, the received jamming signals are .. .LEAST SQUARES SYMBOL DETECTION FOR MULTI ANTENNA SLOW FHSS /MFSK SYSTEMS IN THE PRESENCE OF FOLLOWER JAMMING ALAGUNARAYANAN NARAYANAN (B.E., ANNA University) A THESIS SUBMITTED FOR THE DEGREE... proposed for slow FHSS system to reduce the effect of jamming But the focus of most of these algorithms is the 12 elimination of partial band jamming [5-11], with the problem of follower jamming addressed... deterioration in the performance of the algorithm Even though vector similarity based symbol detection scheme proposed in [1] gives good symbol detection performance in the presence of follower jamming in