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Self-interference handling in OFDM based wireless communication systems

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Self-interference handling in OFDM based wireless communication systems
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Yücek, Tevfik
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University of South Florida
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channel estimation
frequency offset
inter-symbol interference
inter-carrier interference
frequency selectivity
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government publication (state, provincial, terriorial, dependent)   ( marcgt )
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ABSTRACT: Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier modulation scheme that provides efficient bandwidth utilization and robustness against time dispersive channels. This thesis deals with self-interference, or the corruption of desired signal by itself, in OFDM systems. Inter-symbol Interference (ISI) and Inter-carrier Interference (ICI) are two types of self-interference in OFDM systems. Cyclic prefix is one method to prevent the ISI which is the interference of the echoes of a transmitted signal with the original transmitted signal. The length of cyclic prefix required to remove ISI depends on the channel conditions, and usually it is chosen according to the worst case channel scenario. Methods to find the required parameters to adapt the length of the cyclic prefix to the instantaneous channel conditions are investigated. Frequency selectivity of the channel is extracted from the instantaneous channel frequency estimates and methods to estimate related parameters, e.g. coherence bandwidth and Root-mean-squared (RMS) delay spread, are given. These parameters can also be used to better utilize the available resources in wireless systems through transmitter and receiver adaptation. Another common self-interference in OFDM systems is the ICI which is the power leakage among different sub-carriers that degrades the performance of both symbol detection and channel estimation. Two new methods are proposed to reduce the effect of ICI in symbol detection and in channel estimation. The first method uses the colored nature of ICI to cancel it in order to decrease the error rate in the detection of transmitted symbols, and the second method reduces the effect of ICI in channel estimation by jointly estimating the channel and frequency offset, a major source of ICI.
Thesis:
Thesis (M.S.E.E.)--University of South Florida, 2003.
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by Tevfik Yücek.
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Self-in terference Handling in OFDM Based Wireless Comm unication Systems b y T evk Y ucek A thesis submitted in partial fulllmen t of the requiremen ts for the degree of Master of Science in Electrical Engineering Departmen t of Electrical Engineering College of Engineering Univ ersit y of South Florida Ma jor Professor: H useyin Arslan, Ph.D. Vija y K. Jain, Ph.D. Mohamed K. Nezami, Ph.D. Arth ur D. Snider, Ph.D., P .E. Date of Appro v al: No v em b er 14, 2003 Keyw ords: F requency oset, In ter-sym b ol in terference, In ter-carrier in terference, Channel estimation, F requency selectivit y c r Cop yrigh t 2003, T evk Y ucek

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DEDICA TION T o m y family

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A CKNO WLEDGMENTS I w ould lik e to thank m y advisor, Dr. H useyin Arslan for his guidance and encouragemen t throughout the course of this thesis. Learning from him has b een a v ery fruitful and enjo y able exp erience. The discussions w e had concerting the researc h often led to new ideas, taking researc h to new directions with successful results. I wish to thank Dr. Vija y K. Jain, Dr. Mohamed K. Nezami and Dr. Arth ur D. Snider for serving on m y committee and for oering v aluable suggestions. I w ould lik e to thank m y friends Ismail G uv en c, Sharath B. Reddy M. Kemal Ozdemir, F abian E. Aranda, Omer Dedeo~ glu and Oscar V. Gonzales in our researc h group for their supp ort as friends and for sharing their kno wledge with me. Sp ecial thanks to M. Kemal Ozdemir for fueling me up with his T urkish tea ev eryda y Last but b y no means least, I w ould lik e to thank m y paren ts for their con tin ued supp ort, encouragemen t and sacrice throughout the y ears, and I will b e forev er indebted to them for all that they ha v e done.

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T ABLE OF CONTENTS LIST OF FIGURES iii LIST OF A CR ONYMS v ABSTRA CT vii CHAPTER 1 INTR ODUCTION 1 1.1 Organization of thesis 5 CHAPTER 2 OR THOGONAL FREQUENCY DIVISION MUL TIPLEXING: AN O VER VIEW 6 2.1 In tro duction 6 2.2 System mo del 6 2.2.1 Cyclic extension of OFDM sym b ol 8 2.2.1.1 Cyclic prex or p ostx ? 10 2.2.2 Raised cosine guard p erio d 10 2.2.3 Filtering 10 2.2.4 Wireless c hannel 11 2.2.5 A simple system 13 2.3 OFDM impairmen ts 14 2.3.1 F requency oset 15 2.3.2 Time-v arying c hannel 19 2.3.3 Phase noise 21 2.3.4 Receiv er timing errors 22 2.3.5 P eak-to-a v erage p o w er ratio 24 CHAPTER 3 CHANNEL FREQUENCY SELECTIVITY AND DELA Y SPREAD ESTIMA TION 26 3.1 In tro duction 26 3.2 System mo del 29 3.3 Channel frequency selectivit y and dela y spread estimation 30 3.3.1 Channel frequency correlation estimation 31 3.3.2 Dela y spread estimation 32 3.3.2.1 Estimation of RMS dela y spread and c hannel coherence bandwidth 34 3.3.3 Eect of impairmen ts 37 3.3.3.1 Additiv e noise 37 3.3.3.2 Carrier-dep enden t phase shift in c hannel 37 i

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3.4 Short term parameter estimation 38 3.4.1 Obtaining CIR eectiv ely 38 3.4.2 Eect of impairmen ts 41 3.4.2.1 Additiv e noise 41 3.4.2.2 Constan t phase shift in c hannel 41 3.4.2.3 Carrier-dep enden t phase shift in c hannel 42 3.5 P erformance results 43 3.6 Conclusion 46 CHAPTER 4 INTER-CARRIER INTERFERENCE IN OFDM 48 4.1 In tro duction 48 4.2 Causes of ICI 48 4.3 Curren t ICI reduction metho ds 49 4.3.1 F requency-domain equalization 49 4.3.2 Time-domain windo wing 50 4.3.3 P artial transmit sequences & selected mapping 53 4.3.3.1 P artial transmit sequences 53 4.3.3.2 Selected mapping 54 4.3.4 M-ZPSK mo dulation 55 4.3.5 Correlativ e co ding 55 4.3.6 Self-cancellation sc heme 56 4.3.6.1 Cancellation in mo dulation 57 4.3.6.2 Cancellation in demo dulation 58 4.3.6.3 A div erse self-cancellation metho d 60 4.3.7 T one reserv ation 61 4.4 ICI cancellation using auto-regressiv e mo deling 62 4.4.1 Algorithm description 62 4.4.1.1 Auto-regressiv e mo deling 62 4.4.1.2 Estimation of noise sp ectrum and whitening 63 4.4.2 P erformance results 64 4.5 Conclusion 65 CHAPTER 5 ICI CANCELLA TION BASED CHANNEL ESTIMA TION 66 5.1 In tro duction 66 5.2 System mo del 67 5.3 Algorithm description 67 5.3.1 Prop erties of in terference matrix 68 5.3.2 Channel frequency correlation for c ho osing the b est h yp othesis 69 5.3.3 The searc h algorithm 70 5.3.4 Reduced in terference matrix 71 5.4 Results 72 5.5 Conclusion 75 CHAPTER 6 CONCLUSION 76 REFERENCES 78 ii

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LIST OF FIGURES Figure 1. Basic m ulti-carrier transmitter. 2 Figure 2. P o w er sp ectrum densit y of transmitted time domain OFDM signal. 7 Figure 3. P o w er sp ectrum densit y of OFDM signal when the sub carriers at the sides of the sp ectrum and at DCis set to zero. 8 Figure 4. Illustration of cyclic prex extension. 9 Figure 5. Resp onses of dieren t lo w-pass lters. 11 Figure 6. Sp ectrum of an OFDM signal with three c hannels b efore and after bandpass ltering. 12 Figure 7. An example 2D c hannel resp onse. 14 Figure 8. Blo c k diagram of an OFDM transceiv er. 15 Figure 9. Mo ose's frequency oset estimation metho d. 16 Figure 10. Constellation of receiv ed sym b ols when 5% normalized frequency oset is presen t. 19 Figure 11. The probabilit y that the magnitude of the discrete-time OFDM signal exceeds a threshold x 0 for dieren t mo dulations. 25 Figure 12. Estimation of coherence bandwidth B c of lev el K 35 Figure 13. RMS dela y spread v ersus coherence bandwidth. 36 Figure 14. Sampling of c hannel frequency resp onse. 42 Figure 15. Normalized mean squared error v ersus c hannel SNR for dieren t sampling in terv als. 43 Figure 16. Comparison of the estimated frequency correlation with the ideal correlation for dieren t RMS dela y spread v alues. 44 Figure 17. Normalized mean-squared-error p erformance of RMS dela y spread estimation for dieren t a v eraging sizes. 45 Figure 18. Dieren t p o w er dela y proles that are used in the sim ulation. 46 iii

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Figure 19. Normalized mean-squared-error p erformance of RMS dela y spread estimation for dieren t p o w er dela y proles. 47 Figure 20. Disp ersed pattern of a pilot in an OFDM data sym b ol. 49 Figure 21. P osition of carriers in the DFT lter bank. 51 Figure 22. F requency resp onse of a raised cosine windo w with dieren t roll-o factors. 52 Figure 23. All p ossible dieren t signal constellation for 4-ZPSK. 56 Figure 24. Real and imaginary parts of ICI co ecien ts for N=16. 57 Figure 25. Comparison of K ( m; k ), K 0 ( m; k ) and K 00 ( m; k ). 59 Figure 26. P o w er sp ectral densit y of the original and whitened v ersions of the ICI signals for dieren t AR mo del orders. 64 Figure 27. P erformance of the prop osed metho d for dieren t mo del orders. = 0 : 3. 65 Figure 28. Magnitudes of full and reduced in terference matrices for dieren t frequency osets. 72 Figure 29. V ariance of the frequency oset estimator. 73 Figure 30. Estimated and correct (normalized) frequency oset v alues. 74 Figure 31. Mean-square error v ersus SNR for con v en tional LS and prop osed CFR estimators. 75 iv

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LIST OF A CR ONYMS A CI Adjacen t Channel In terference ADSL Asymmetric Digital Subscrib er Line AR Auto-regressiv e A W GN Additiv e White Gaussian Noise BER Bit Error Rate BPSK Binary Phase Shift Keying CCI Co-c hannel In terference CF C Channel F requency Correlation CFR Channel F requency Resp onse CIR Channel Impulse Resp onse D AB Digital Audio Broadcasting DC Direct Curren t DFE Decision F eedbac k Equalizer DFT Discrete F ourier T ransform D VB-T T errestrial Digital Video Broadcasting FFT F ast F ourier T ransform FPGA Field-Programmable Gate Arra y GSM Global System for Mobile Comm unications ICI In ter-carrier In terference IDFT In v erse Discrete F ourier T ransform IEEE Institute of Electrical and Electronics Engineers IFFT In v erse F ast F ourier T ransform IMD In ter-mo dulation Distortion v

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ISI In ter-sym b ol In terference LAN Lo cal Area Net w ork LCR Lev el Crossing Rate LMMSE Linear Minim um Mean-square Error LS Least Squares ML Maxim ul Lik eliho o d MMSE Minim um Mean-square Error MSE Mean-squared-error OFDM Orthogonal F requency Division Multiplexing P APR P eak-to-a v erage P o w er Ratio PCC P olynomial Cancellation Co ding PDP P o w er Dela y Prole PICR P eak In terference-to-Carrier Ratio PSD P o w er Sp ectral Densit y PSK Phase Shift Keying PTS P artial T ransmit Sequences QAM Quadrature Amplitude Mo dulation QPSK Quadrature Phase Shift Keying RMS Ro ot-mean-squared SM Selected Mapping SNR Signal-to-noise Ratio TDMA Time Division Multiple Access WLAN Wireless Lo cal Area Net w ork WP AN Wireless P ersonal Area Net w ork WSSUS Wide-sense Stationary Uncorrelated Scattering ZPSK Zero-padded Phase Shift Keying vi

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SELF-INTERFERENCE HANDLING IN OFDM BASED WIRELESS COMMUNICA TION SYSTEMS T evk Y ucek ABSTRA CT Orthogonal F requency Division Multiplexing (OFDM) is a m ulti-carrier mo dulation sc heme that pro vides ecien t bandwidth utilization and robustness against time disp ersiv e c hannels. This thesis deals with self-in terference, or the corruption of desired signal b y itself, in OFDM systems. In ter-sym b ol In terference (ISI) and In ter-carrier In terference (ICI) are t w o t yp es of self-in terference in OFDM systems. Cyclic prex is one metho d to prev en t the ISI whic h is the in terference of the ec ho es of a transmitted signal with the original transmitted signal. The length of cyclic prex required to remo v e ISI dep ends on the c hannel conditions, and usually it is c hosen according to the w orst case c hannel scenario. Metho ds to nd the required parameters to adapt the length of the cyclic prex to the instan taneous c hannel conditions are in v estigated. F requency selectivit y of the c hannel is extracted from the instan taneous c hannel frequency estimates and metho ds to estimate related parameters, e.g. coherence bandwidth and Ro ot-mean-squared (RMS) dela y spread, are giv en. These parameters can also b e used to b etter utilize the a v ailable resources in wireless systems through transmitter and receiv er adaptation. Another common self-in terference in OFDM systems is the ICI whic h is the p o w er leak age among dieren t sub-carriers that degrades the p erformance of b oth sym b ol detection and c hannel estimation. Tw o new metho ds are prop osed to reduce the eect of ICI in sym b ol detection and in c hannel estimation. The rst metho d uses the colored nature of ICI to cancel it in order to decrease the error rate in the detection of transmitted sym b ols, and vii

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the second metho d reduces the eect of ICI in c hannel estimation b y join tly estimating the c hannel and frequency oset, a ma jor source of ICI. viii

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CHAPTER 1 INTR ODUCTION Wireless comm unication is not a new concept. Smok e signals and ligh thouses are all forms of wireless comm unication that ha v e b een around for y ears. In our era, wireless comm unication refers to accessing information without the need of a xed cable connection. Wireless comm unication con tin ue to gro w rapidly as the need for reac hing data an ywhere at an ytime rises. The increasing demand for high-rate data services along with the requiremen t for reliable connectivit y requires no v el tec hnologies. As wireless comm unication systems are usually in terference limited, new tec hnologies should b e able to handle the in terference successfully In terference can b e from other users, e.g. Co-c hannel In terference (CCI) and Adjacen t Channel In terference (A CI), or it can b e due to users o wn signal (self-in terference), e.g. In ter-sym b ol In terference (ISI). ISI is one of the ma jor problems for high data rate comm unications whic h is treated with equalizers in con v en tional single-carrier systems. Ho w ev er, for high data rate transmission, complexit y of equalizers b ecomes v ery high due to the smaller sym b ol time and large n um b er of taps needed for equalization. This problem is esp ecially imp ortan t for c hannels with large dela y spreads. Multi-carrier mo dulation is one of the transmission sc hemes whic h is less sensitiv e to time disp ersion (frequency selectivit y) of the c hannel. A basic m ulti-carrier transmitter diagram is sho wn in Fig. 1. In m ulti-carrier systems, the transmission bandwidth is divided in to sev eral narro w sub-c hannels and data is transmitted parallel in these sub-c hannels. Data in eac h sub-c hannel is mo dulated at a relativ ely lo w rate so that the dela y spread of the c hannel do es not cause an y degradation as eac h of the sub-c hannels will exp erience a rat resp onse in frequency Although, the principles are kno wn since early sixties [1, 2], m ulti-carrier 1

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mo dulation tec hniques, esp ecially Orthogonal F requency Division Multiplexing (OFDM), gained more atten tion in the last ten y ears due to the increased p o w er of digital signal pro cessors. ModulatedSignal DataSymbols Serial To Parallel Converter f c,n-1 c,n f f D f f c,2c,1 f c,N Modulators = Figure 1. Basic m ulti-carrier transmitter. OFDM is a m ulti-carrier mo dulation tec hnique that can o v ercome man y problems that arise with high bit rate comm unication, the biggest of whic h is the time disp ersion. In OFDM, the carrier frequencies are c hosen in suc h a w a y that there is no inruence of other carriers in the detection of the information in a particular carrier when the orthogonalit y of the carriers are main tained. The data b earing sym b ol stream is split in to sev eral lo w er rate streams and these streams are transmitted on dieren t carriers. Since this increases the sym b ol p erio d b y the n um b er of non-o v erlapping carriers (sub-carriers), m ultipath ec ho es will aect only a small p ortion of the neigh b oring sym b ols. Remaining ISI can b e remo v ed b y cyclically extending the OFDM sym b ol. The length of the cyclic extension should b e at least as long as the maxim um excess dela y of the c hannel. By this w a y OFDM reduces the eect of m ultipath c hannels encoun tered with high data rates, and a v oids the usage of complex equalizers. OFDM is used as the mo dulation metho d for Digital Audio Broadcasting (D AB) [3] and T errestrial Digital Video Broadcasting (D VB-T) [4] in Europ e, and in Asymmetric Digital Subscrib er Line (ADSL) [5]. Wireless Lo cal Area Net w orks (WLANs) use OFDM as their 2

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ph ysical la y er transmission tec hnique. Dieren t WLAN standards are dev elop ed in Europ e, USA, and Japan. The Europ ean standard is ETSI Hip erLAN/2 [6], American standard is IEEE 802.11a/g [7], and Japanese standard is ARIB HiSW ANa [8]; all of whic h has similar ph ysical la y er sp ecications based on OFDM. OFDM is also a strong candidate for IEEE Wireless P ersonal Area Net w ork (WP AN) standard [9] and for forth generation (4G) cellular systems (see e.g. [10]). Although OFDM has pro v ed itself as a p o w erful mo dulation tec hnique, it has its o wn c hallenges. Sensitivit y to frequency osets caused when a receiv er's oscillator do es not run at exactly the same frequency of transmitter's oscillator is one of the ma jor problems. This oset p erturbs the orthogonalit y of the sub-carriers, reducing the p erformance. Another problem is the large P eak-to-a v erage P o w er Ratio (P APR) of the OFDM signal, whic h requires p o w er ampliers with large linear ranges. Hence, p o w er ampliers require more bac k-o whic h, in turn, reduces the p o w er eciency Some other problems include phase distortion, time-v arying c hannel and time sync hronization. In Chapter 2, these problems will b e discussed in more details. Most standards emplo ying OFDM do not utilize the a v ailable resources eectiv ely Most of the time, systems are designed for the w orst case scenarios. The length of the cyclic prex, for example, is c hosen in suc h a w a y that it is larger than the maxim um exp ected dela y of the c hannel, whic h in tro duces a considerable amoun t of o v erhead to the system. Ho w ev er, it can b e c hanged adaptiv ely dep ending on the c hannel conditions, instead of setting it according to the w orst case scenario, if the maxim um excess dela y of the c hannel is kno wn. The information ab out the frequency selectivit y of the c hannel can also b e v ery useful for impro ving the p erformance of the wireless radio receiv ers through transmitter and receiv er adaptation. OFDM sym b ol duration, sub carrier bandwith, n um b er of sub-carriers etc. can b e c hanged adaptiv ely if the frequency selectivit y is estimated. Bandwidth of the in terp olation lters for c hannel estimation can b e adapted dep ending on the distortion in the en vironmen t [11]. Metho ds to estimate the parameters required b y these adaptation tec hniques are studied in Chapter 3. Both long term and instan taneous frequency selectivit y of wireless c hannels 3

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are studied 1 Metho ds to nd the coherence bandwidth, Ro ot-mean-squared (RMS) dela y spread, and Channel Impulse Resp onse (CIR) using Channel F requency Resp onse (CFR) are dev elop ed, and robustness of these metho ds against v arious OFDM impairmen ts are in v estigated. While OFDM solv es the ISI problem b y using cyclic prex, it has another self-in terference problem: In ter-carrier In terference (ICI), or the crosstalk among dieren t sub-carriers, caused b y the loss of orthogonalit y due to frequency instabilities, timing oset or phase noise. ISI and ICI are dual of eac h other o ccurring at dieren t domains; one in timedomain and the other in frequency-domain. ICI is a ma jor problem in m ulti-carrier systems and needs to b e tak en in to accoun t when designing systems. ICI can b e mo deled as Gaussian noise and results in an error ro or if it is not comp ensated for [15]. Therefore, ecien t cancellation of ICI is v ery crucial, and dieren t metho ds are prop osed b y man y authors in the literature. An o v erview of recen t literature on ICI cancellation is giv en and a no v el metho d based on the Auto-regressiv e (AR) mo deling is prop osed for ICI cancellation in Chapter 4. The prop osed metho d explores the colored nature of ICI. An AR pro cess is t to the colored ICI to nd the lter co ecien ts, whic h are then used to whiten the ICI. ICI also aects the c hannel estimation whic h is one of the most imp ortan t elemen ts of wireless receiv ers that emplo y coheren t demo dulation [16]. Previous c hannel estimation algorithms treat ICI as part of the additiv e white Gaussian noise and these algorithms p erform p o orly when ICI is signican t. A new metho d that mitigates the eects of ICI in c hannel estimation b y join tly estimating the frequency oset and c hannel resp onse is prop osed in Chapter 5 2 Unlik e con v en tional c hannel estimation tec hniques, where ICI is treated as part of the noise, the prop osed approac h tak es the eect of frequency oset, and hence ICI, in to accoun t in c hannel estimation. 1 This w ork is partly published in [12, 13] and it is curren tly under review for another publication [14]. 2 This w ork is published in [17]. 4

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1.1 Organization of thesis This thesis consists of six c hapters. Chapter 2 describ es the basic elemen ts of an OFDM system with signal pro cessing asp ects. Imp ortan t problems asso ciated with OFDM are analyzed as w ell. In Chapter 3, a practical metho d for frequency correlation estimation from CFR is giv en rst. Then, the exact mathematical relation b et w een the Channel F requency Correlation (CF C) and RMS dela y spread is deriv ed, and metho ds for estimating frequency selectivit y and RMS dela y spread are giv en. Moreo v er, estimation of time domain c hannel parameters from sampled CFR is studied and p erformance results of the prop osed algorithms are presen ted. Chapter 4 in tro duces the ICI problem. Impairmen ts that causes ICI are listed and some imp ortan t ICI reduction sc hemes are explained. Moreo v er, a no v el ICI reduction metho d based on AR mo deling is giv en. In Chapter 5, the join t c hannel and frequency oset estimation algorithm is explained and sim ulation results are giv en. The thesis is concluded in Chapter 6 whic h summarizes the thesis and discusses op en researc h areas. 5

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CHAPTER 2 OR THOGONAL FREQUENCY DIVISION MUL TIPLEXING: AN O VER VIEW 2.1 In tro duction In this c hapter, the basic principles of Orthogonal F requency Division Multiplexing (OFDM) is in tro duced. A basic system mo del is giv en, common comp onen ts for OFDM based systems are explained, and a simple transceiv er based on OFDM mo dulation is presen ted. Imp ortan t impairmen ts in OFDM systems are mathematically analyzed. 2.2 System mo del The Discrete F ourier T ransform (DFT) of a discrete sequence f ( n ) of length N F ( k ), is dened as [18], F ( k ) = 1 N N 1 X n =0 f ( n ) e j 2 k n N (1) and In v erse Discrete F ourier T ransform (IDFT) as; f ( n ) = N 1 X k =0 F ( k ) e j 2 k n N : (2) OFDM con v erts serial data stream in to parallel blo c ks of size N and uses IDFT to obtain OFDM signal. Time domain samples, then, can b e calculated as x ( n ) = I D F T f X ( k ) g = N 1 X k =0 X ( k ) e j 2 nk = N 0 n N 1 ; (3) 6

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where X ( k ) is the sym b ol transmitted on the k th sub carrier and N is the n um b er of subcarriers. Sym b ols are obtained from the data bits using an M -ary mo dulation e.g. Binary Phase Shift Keying (BPSK), Quadrature Amplitude Mo dulation (QAM), etc Time domain signal is cyclically extended to a v oid In ter-sym b ol In terference (ISI) from previous sym b ol. The sym b ols X ( k ) are in terpreted as frequency domain signal and samples x ( n ) are in terpreted as time domain signal. Applying the cen tral limit theorem, while assuming that N is sucien tly large, the x ( n ) are zero-mean complex-v alued Gaussian distributed random v ariables. P o w er sp ectrum of OFDM signal with 64 sub-carriers is sho wn in Fig. 2. Sym b ols are mapp ed using Quadrature Phase Shift Keying (QPSK) mo dulation. -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 -90 -80 -70 -60 -50 -40 -30 -20 -10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) Figure 2. P o w er sp ectrum densit y of transmitted time domain OFDM signal. Sometimes, the sub-carriers at the end sides of the sp ectrum are set to zero in order to simplify the sp ectrum shaping requiremen ts at the transmitter, e.g. IEEE 802.11a. These sub carriers are used as frequency guard band and referred as virtual c arriers in literature. 7

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T o a v oid diculties in D/A and A/D con v erter osets, and to a v oid DC oset, the sub carrier falling at DC is not used as w ell. The p o w er sp ectrum for suc h a system is sho wn in Fig. 3. Num b er of sub-carriers that are set to zero at the sides of the sp ectrum w as 11. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 90 80 70 60 50 40 30 20 10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) Figure 3. P o w er sp ectrum densit y of OFDM signal when the sub carriers at the sides of the sp ectrum and at DC is set to zero. 2.2.1 Cyclic extension of OFDM sym b ol Time domain OFDM signal is cyclically extended to mitigate the eect of time disp ersion. The length of cyclic prex has to exceed the maxim um excess dela y of the c hannel in order to a v oid ISI [19, 20]. The basic idea here is to replicate part of the OFDM time-domain sym b ol from bac k to the fron t to create a guard p erio d. This is sho wn in the Fig. 4. This gure also sho ws ho w cyclic prex prev en ts the ISI. As can b e seen from the gure, as long as maxim um excess dela y ( max ) is smaller than the length of the cyclic extension ( T g ), the 8

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distorted part of the signal will sta y within the guard in terv al, whic h will b e remo v ed later at the transmitter. Therefore ISI will b e prev en ted. Cyclicly Extended OFDM Symbol Original OFDM Symbol T g Multipath Component A Multipath Component B Multipath Component C T t max Figure 4. Illustration of cyclic prex extension. The ratio of the guard in terv al to the useful sym b ol duration is application dep enden t. If this ratio is large, then the o v erhead will increase causing a decrease in the system throughput. A cyclic prex is used for the guard time for the follo wing reasons; 1. to main tain the receiv er time sync hronization; since a long silence can cause sync hronization to b e lost. 2. to con v ert the linear con v olution of the signal and c hannel to a circular con v olution and thereb y causing the DFT of the circularly con v olv ed signal and c hannel to simply b e the pro duct of their resp ectiv e DFTs. 3. it is easy to implemen t in FPGAs. 9

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2.2.1.1 Cyclic prex or p ostx ? P ostx is the dual of prex. In p ostx, the b eginning of OFDM sym b ol is copied and app ended at the end. If w e use prex only w e need to mak e sure that the length of cyclic prex is larger than the maxim um excess dela y of the c hannel; if w e use b oth cyclic prex and p ostx, then the sum of the lengths of cyclic prex and p ostx should b e larger than the maxim um excess dela y 2.2.2 Raised cosine guard p erio d The OFDM signal is made up of a series of IFFTs that are concatenated to eac h other. A t eac h sym b ol b oundary there is a signal discon tin uit y due to the dierence b et w een the end of one sym b ol and the start of another one. These v ery fast transitions at the b oundaries increase the side-lob e p o w er. In order to smo oth the transition b et w een dieren t transmitted OFDM sym b ols, windo wing (Hamming, Hanning, Blac kman, Raised Cosine etc.) is applied to eac h sym b ol. 2.2.3 Filtering Filtering is applied b oth at the receiv er and at the transmitter. A t the transmitter, it is used to reduce the eect of side lob es of the sinc shap e in the OFDM sym b ol. This eectiv ely band pass lters the signal, remo ving some of the OFDM side-lob es. The amoun t of side-lob e remo v al dep ends on the sharpness of the lters used. In general digital ltering pro vides a m uc h greater rexibilit y accuracy and cut o rate than analog lters making them esp ecially useful for band limiting of an OFDM signal [21]. Some commonly used lters are rectangular pulse ( sinc lter), ro ot raised cosine lter, Cheb yshev and Butterw orth lter. The frequency resp onses of these lters are sho wn in Fig. 5. In the receiv er side, a matc hed lter is used to reject the noise and Adjacen t Channel In terference (A CI). The p o w er sp ectrum of an OFDM signal with adjacen t c hannel b efore and after passing through is a band-pass Cheb yshev lter is sho wn in Fig. 6. As clearly 10

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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 70 60 50 40 30 20 10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) (a) Raised-Cosine Filter. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 60 50 40 30 20 10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) (b) Rectangular Filter. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 90 80 70 60 50 40 30 20 10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) (c) Cheb yshev Filter. 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 80 70 60 50 40 30 20 10 0 10 Normalized FrequencyPower Specturm Magnitude (dB) (d) Butterw orth Filter. Figure 5. Resp onses of dieren t lo w-pass lters. sho wn from this gure, lter help us remo v e adjacen t c hannels and pic k the desired c hannel whic h carry useful information. 2.2.4 Wireless c hannel Comm unication c hannels in tro duce noise, fading, in terference, and other distortions in to the signals. The wireless c hannel and the impairmen ts in the hardw are of the receiv er and transmitter in tro duces additiv e noise on the transmitted signal. The main sources of noise are thermal bac kground noise, electrical noise in the receiv er ampliers, and in terference. 11

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 80 70 60 50 40 30 20 10 0 Normalized FrequencyPower Specturm Magnitude (dB) (a) Sp ectrum Before Filtering. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 80 70 60 50 40 30 20 10 0 Normalized FrequencyPower Specturm Magnitude (dB) Figure 6. Sp ectrum of an OFDM signal with three c hannels b efore and after band-pass ltering.In addition to this, noise can also b e generated in ternally to the comm unications system as a result of ISI, In ter-carrier In terference (ICI), and In ter-mo dulation Distortion (IMD) [21]. The noise due to these reasons decrease Signal-to-noise Ratio (SNR) resulting in an increase in the Bit Error Rate (BER). 12

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Most of the noises from dieren t sources in OFDM system can b e mo deled as Additiv e White Gaussian Noise (A W GN). A W GN has a uniform sp ectral densit y (making it white), and a Gaussian probabilit y distribution. Signals arriving to the receiv er via dieren t paths will ha v e dieren t dela ys, whic h causes time disp ersion. The amoun t of disp ersion is en vironmen t dep enden t. F or oce buildings, a v erage Ro ot-mean-squared (RMS) dela y spread is around 30{50 n s and maxim um RMS dela y spread is around 40{85 n s. This disp ersion is tak en care of using equalizers traditionally Ho w ev er, as data rate increases, the complexit y of these equalizers increases also, b ecause of the short sym b ol duration. OFDM solv es this problem, b y dividing the wide frequency band in to narro w er bands whic h can b e accepted as rat. On the other hand, mobilit y of the users causes frequency disp ersion (time selectivit y). An example 2dimensional c hannel resp onse is sho wn in Fig. 7. An exp onen tial P o w er Dela y Prole (PDP) with RMS dela y spread of 16 s is used and mobile sp eed is assumed to b e 100km/h. Channel taps are obtained using mo died Jak es' mo del [22]. 2.2.5 A simple system A blo c k diagram of a basic OFDM system is giv en in Fig. 8. Usually ra w data is co ded and in terlea v ed b efore mo dulation. In a m ultipath fading c hannel, all sub carriers will ha v e dieren t atten uations. Some sub carriers ma y ev en b e completely lost b ecause of deep fades. Therefore, the o v erall BER ma y b e largely dominated b y a few sub carriers with the smallest amplitudes. T o a v oid this problem, c hannel co ding can b e used. By using co ding, errors can b e corrected up to a certain lev el dep ending on the co de rate and t yp e, and the c hannel. In terlea ving is applied to randomize the o ccurrence of bit errors. Co ded and in terlea v ed data is then b e mapp ed to the constellation p oin ts to obtain data sym b ols. These steps are represen ted b y the rst blo c k of Fig. 8. The serial data sym b ols are then con v erted to parallel and In v erse F ast F ourier T ransform (IFFT) is applied to these parallel blo c ks to obtain the time domain OFDM sym b ols. Later, these samples are cyclically extended as explained in Section 2.2.1, con v erted to analog signal and up13

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Figure 7. An example 2D c hannel resp onse. con v erted to the RF frequencies using mixers. The signal is then amplied b y using a p o w er amplier (P A) and transmitted through an tennas. In the receiv er side, the receiv ed signal is passed through a band-pass noise rejection lter and do wncon v erted to baseband. After frequency and time sync hronization, cyclic prex is remo v ed and the signal is transformed to the frequency domain using F ast F ourier T ransform (FFT) op eration. And nally the sym b ols are demo dulated, dein terlea v ed and deco ded to obtain the transmitted information bits. 2.3 OFDM impairmen ts This section giv es the main impairmen ts that exist in OFDM systems with underlying mathematical details. 14

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MOD Data-IN ADD Cyclic Ext Lowpass D/AA/D Sampling Remove Cyclic Ext P S S P S P P S Data-Out frequency domain Processing in the Processing in thetime domain DEMODFFT IFFT CHANNELBaseband Signal HF Signal Up-Conv. Down-Conv RFRF Transmitter Receiver Figure 8. Blo c k diagram of an OFDM transceiv er. 2.3.1 F requency oset F requency oset is a critical factor in OFDM system design. It results in in ter-carrier in terference (ICI) and degrades the orthogonalit y of sub-carriers. F requency errors will tend to o ccur from t w o main sources. These are lo cal oscillator errors and common Doppler spread. An y dierence b et w een transmitter and receiv er lo cal oscillators will result in a frequency oset. This oset is usually comp ensated for b y using adaptiv e frequency correction (AF C), ho w ev er an y residual (uncomp ensated) errors result in a degraded system p erformance. The c haracteristics of ICI are similar to Gaussian noise, hence it leads to degradation of the SNR. The amoun t of degradation is prop ortional to the fractional frequency oset whic h is equal to the ratio of frequency oset to the carrier spacing. F requency oset can b e estimated b y dieren t metho ds e.g. using pilot sym b ols, the statistical redundancy in the receiv ed signal, or transmitted training sequences. In [23], a frequency oset estimator whic h uses the rep eated structure of training signal is giv en. 15

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This metho d is illustrated in Fig. 9. The a v erage phase dierence b et w een the rst and second part of the long training sequences is calculated and then normalized to obtain the frequency oset. Offset Part I Part II Repeated Training Sequence Conjugate Angle Normalize Frequency Figure 9. Mo ose's frequency oset estimation metho d. Assume that w e ha v e the sym b ols X ( k ) to b e transmitted using an OFDM system. These sym b ols are transformed to the time domain using IDFT as sho wn earlier in (3). This baseband signal (OFDM sym b ol) is then up-con v erted to RF frequencies and transmitted o v er the wireless c hannel. In the receiv er, the receiv ed signal is do wn-con v erted to baseband. But, due to the frequency mismatc h b et w een the transmitter and receiv er, the receiv ed signal has a frequency oset. This signal is denoted as y ( n ). The frequency oset is added to the OFDM sym b ol in the receiv er. Finally to reco v er the data sym b ols, DFT is applied to the OFDM sym b ol taking the signal bac k to frequency domain. Let Y ( k ) denote the reco v ered data sym b ols. This pro cess is sho wn b elo w. X ( k ) I D F T x ( n ) f r eq uency of f set y ( n ) D F T Y ( k ) 16

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Let us apply the ab o v e op erations to X ( k ) in order to get Y ( k ). First nd x ( n ) using (2) x ( n ) = I D F T f X ( k ) g (4) = N 1 X k =0 X ( k ) e j 2 k n N (5) The eect of frequency oset on x ( n ) will b e a phase shift of 2 n= N where is the normalized frequency oset. Therefore; y ( n ) = x ( n ) e j 2 n N (6) = N 1 X k =0 X ( k ) e j 2 k n N e j 2 n N (7) = N 1 X k =0 X ( k ) e j 2 n N ( k + ) (8) Finally w e need to apply DFT to y ( n ) with a view to w ard reco v ering the sym b ols. Y ( k ) = D F T ( y ( n ) ) (9) = 1 N N 1 X n =0 ( N 1 X m =0 X ( m ) e j 2 n N ( m + ) ) e j 2 k n N (10) = 1 N N 1 X n =0 N 1 X m =0 X ( m ) e j 2 n N ( m k + ) (11) = 1 N N 1 X m =0 X ( m ) ( N 1 X n =0 e j 2 n N ( m k + ) ) (12) 17

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The term within the curly braces can b e calculated using geometric series expansion, S n P nk =0 r k = 1 r n +1 1 r Using this expansion w e ha v e Y ( k ) = 1 N N 1 X m =0 X ( m ) ( N 1 X n =0 e j 2 n N ( m k + ) ) (13) = 1 N N 1 X m =0 X ( m ) 1 e j 2 ( m k + ) 1 e j 2 ( m k + ) N (14) = 1 N N 1 X m =0 X ( m ) e j ( m k + ) ( e j ( m k + ) e j ( m k + ) ) e j ( m k + ) N ( e j ( m k + ) N e j ( m k + ) N ) (15) = 1 N N 1 X m =0 X ( m ) e j ( m k + ) e j ( m k + ) N 2 j sin( ( m k + )) 2 j sin( ( m k + ) N ) (16) N 1 X m =0 X ( m ) e j ( m k + ) N 1 N sin ( ( m k + )) ( m k + ) (17) N 1 X m =0 X ( m ) sin ( ( m k + )) ( m k + ) e j ( m k + ) (18) In the ab o v e deriv ation w e used the fact that sin( x ) x for small x v alues, and N 1 N 1 for large v alues of N These appro ximations are reasonable since usually N is a large in teger. W e can no w relate the receiv ed sym b ols to the transmitted sym b ols using (18). But rst dene S ( m; k ) = sin ( ( m k + )) ( m k + ) e j ( m k + ) Therefore; Y ( k ) = N 1 X m =0 X ( m ) S ( m; k ) (19) = X ( k ) S ( k ; k ) + N 1 X m =0 ;m 6 = k X ( m ) S ( m; k ) (20) The rst term in (20) is equal to the originally transmitted sym b ol shifted b y a term that corresp onds to This term, S ( k ; k ), in tro duces a phase shift of and an atten uation of sin ( ) = in magnitude. Actually this term only dep ends on the v alue of oset, but not carrier index k so the eect of frequency oset on eac h sub-carrier will b e the same. 18

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The second term in (20) represen ts the in terference from other sub-carriers whic h is a dual to ISI in time domain due to timing oset. Constellation diagram of the receiv ed sym b ols in the case of 5% normalized frequency oset ( = 0 : 05) is plotted in Fig. 10 where a noiseless transmission is assumed. Constellation of original sym b ols ( = 0) are also sho wn in this gure. F rom this gure, the rotation of the constellation and the noise-lik e distortion (ICI) can b e seen easily 1.5 1 0.5 0 0.5 1 1.5 1.5 1 0.5 0 0.5 1 1.5 Real PartImaginary Part e = 0.05 e = 0.00 Figure 10. Constellation of receiv ed sym b ols when 5% normalized frequency oset is presen t. 2.3.2 Time-v arying c hannel OFDM systems are kno wn to lo ose their orthogonalit y when the v ariation of the c hannel o v er an OFDM sym b ol duration is not negligible. Assuming an L -tap sym b ol-spaced timev arying c hannel, the time domain receiv ed OFDM signal can b e obtained b y con v olving the 19

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transmitted signal with Channel Impulse Resp onse (CIR) h ( l ; n ) as y ( n ) = L 1 X l =0 h ( l ; n ) x ( n l ) 0 n N 1 : (21) By taking FFT of (21) and ignoring the c hannel noise for the momen t, frequency domain receiv ed sym b ols can b e obtained as Y ( k ) = 1 N N 1 X n =0 y ( n ) e j 2 k n N (22) = 1 N N 1 X n =0 L 1 X l =0 h ( l ; n ) x ( n l ) # e j 2 k n N (23) = 1 N N 1 X n =0 L 1 X l =0 h ( l ; n ) N 1 X m =0 X ( m ) e j 2 m ( n l ) N !# e j 2 k n N (24) = N 1 X m =0 X ( m ) 8>>>><>>>>: L 1 X l =0 1 N N 1 X n =0 h ( l ; n ) e j 2 n ( m k ) N | {z } H l ( m k ) e j 2 ml N 9>>>>=>>>>; : (25) Note that when h ( l ; n ) = h ( l ), i.e. when the c hannel is constan t o v er the OFDM sym b ol, H l ( m k ) = h ( l ) and there is no ICI. Let us dene the in terference matrix for this case as a matrix with elemen ts giv en b y D ( m; k ) = 1 N L 1 X l =0 N 1 X n =0 h ( l ; n ) e j 2 n ( m k ) N e j 2 ml N : (26) Note that (26) also includes the eect of frequency selectiv e c hannel. Therefore, Y ( k ) = N 1 X m =0 X ( m ) ( 1 N L 1 X l =0 N 1 X n =0 h ( l ; n ) e j 2 n ( m k ) N e j 2 ml N ) (27) = N 1 X m =0 X ( m ) D ( m; k ) (28) = X ( k ) D ( k ; k ) + N 1 X m =0 ;m 6 = k X ( m ) D ( m; k ) (29) 20

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As in (20), the second term in (29) represen ts the in terference b et w een the sub carriers. while the rst term is equal to the originally transmitted sym b ol m ultiplied b y D ( k ; k ) whic h, in this case, dep ends on the carrier index. This term can b e re-written as D ( k ; k ) = 1 N L 1 X l =0 N 1 X n =0 h ( l ; n ) e j 2 n ( k k ) N e j 2 k l N (30) = L 1 X l =0 1 N N 1 X n =0 h ( l ; n ) e j 2 k l N : (31) The term in the paren theses is just an arithmetic a v erage of the v arying CIR taps within an OFDM sym b ol. Hence, the whole expression is F ourier transform of the a v erage CIR, whic h giv es the frequency domain c hannel i.e the transmitted sym b ol is m ultiplied with the v alue of frequency domain c hannel at that sub-carrier. 2.3.3 Phase noise Phase noise is in tro duced b y lo cal oscillator in an y receiv er and can b e in terpreted as a parasitic phase mo dulation in the oscillator's signal. Phase noise can b e mo deled as a zero mean random v ariable. If w e assume the c hannel is rat and the signal is only eected b y phase noise ( n ) at the receiv er, the receiv ed time domain signal can b e written as r ( n ) = x ( n ) e j ( n ) (32) If w e assume phase oset is small ( e j ( n ) 1 + j ( n )), the reco v ered sym b ols will ha v e the form ^ X k X ( k ) + j N N 1 X r =0 X ( r ) N 1 X n =0 ( n ) e j (2 = N )( r k ) n X ( k ) + j X ( k ) 1 N N 1 X n =0 ( n ) | {z } j X ( k ) + j N N 1 X r =0 ;r 6 = k X ( r ) N 1 X n =0 ( n ) e j (2 = N )( r k ) n | {z } I C I ter m (33) 21

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In (33), the second term represen ts a common error added to ev ery sub carrier that is prop ortional to its v alue m ultiplied b y a complex n um b er j that is a rotation of the constellation. This rotation is the same for all sub carriers, so it can b e corrected b y using a phase rotation equal to the a v erage of the phase noise, = 1 N N 1 X n =0 ( n ) : (34) The last term in (33) represen t the leak age from neigh b oring sub carriers to the useful signal of eac h sub carrier, i.e. ICI. This term can not b e corrected, since b oth phase oset ( n ) and input data sequence X ( k ) are random. Therefore it will cause SNR degradation of the o v erall system. The only w a y to reduce in terference due to the phase noise is to impro v e the p erformance of the oscillator, with asso ciated cost increase [24]. A more detailed study of the eects of phase noise on OFDM system can b e found in [15, 24]. 2.3.4 Receiv er timing errors The eect of sampling time oset in OFDM is the rotation of the sym b ols whic h can b e folded in to the c hannel estimate and corrected easily The next section will deriv e the amoun t of rotation assuming a small timing oset (timing oset is smaller than the un used part of the cyclic prex). As in the frequency oset case, w e are going to relate the transmitted and reco v ered data sym b ols with resp ect to time oset. Let us use the same notation used in frequency oset case, represen t transmitted sym b ols b y X ( k ), and represen t the baseband equiv alen t of the time domain signal b y x b ( n ). T o prev en t confusion, w e will sho w the impaired signals with a tilde this time, so ~ x b ( n ) will represen t the baseband signal with time oset and ~ a [ k ] will represen t the reco v ered sym b ols. Therefore, new OFDM pro cess c hain will b e as follo ws; X ( k ) I D F T x b ( n ) timing er r or ~ x b ( n ) D F T ~ X ( k ) 22

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The form ula for x b ( n ) is already deriv ed, and is giv en in (5). Timing error is caused b y sampling the receiv ed signal at a wrong time. So ~ x b ( n ) will b e nothing but the shifted v ersion of x b ( n ) in time domain, i.e. if w e ha v e timing error of ; ~ x b ( n ) = x b ( n ) = N 1 X k =0 X ( k ) e j 2 k N ( n ) Here the sign of dep ends on whether w e are sampling b efore or after the correct time instan t. Assuming to b e p ositiv e, w e will use a min us sign hereafter. No w w e can calculate ~ X ( k ) from ~ x b ( n ) using DFT. ~ X ( k ) = 1 N N 1 X n =0 ( N 1 X m =0 X ( m ) e j 2 m N ( n ) ) e j 2 k n N = 1 N N 1 X n =0 ( N 1 X m =0 X ( m ) e j 2 n N ( m k ) e j 2 m N ) = 1 N N 1 X m =0 ( N 1 X n =0 X ( m ) e j 2 n N ( m k ) ) | {z } N ( m k ) e j 2 m N = N 1 X m =0 X ( m )( m k ) e j 2 m N = X ( k ) e j 2 k N (35) Equation 35 sho ws that a timing oset of causes only a rotation on the reco v ered data sym b ols. The v alue of the reco v ered sym b ol dep ends only on the transmitted data, but not the neigh b oring carriers, this means timing error do es not destro y the orthogonalit y of carriers and the eect of timing error is a phase rotation whic h linearly c hanges with carriers' order. Therefore, timing sync hronization is not a v ery serious problem in OFDM based systems. In the rest of this thesis, p erfect timing sync hronization is assumed. 23

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2.3.5 P eak-to-a v erage p o w er ratio One of the ma jor dra wbac ks of OFDM is its high P eak-to-a v erage P o w er Ratio (P APR). Sup erp osition of a large n um b er of sub carrier signals results in a p o w er densit y with Ra yleigh distribution whic h has large ructuations. OFDM transmitters therefore require p o w er ampliers with large linear range of op eration whic h are exp ensiv e and inecien t. An y amplier non-linearit y causes signal distortion and in ter-mo dulation pro ducts resulting in un w an ted out-of-band p o w er and higher BER [25]. The Analog to Digital con v erters and Digital to Analog con v erters are also required to ha v e a wide dynamic range whic h increases complexit y Discrete-time P APR of m th OFDM sym b ol x m is dened as [26] P AP R m = max 0 n N 1 j x m ( n ) j 2 =E fj x m ( n ) j 2 g : (36) Although the P APR is mo derately high for OFDM, high magnitude p eaks o ccur relativ ely rarely and most of the transmitted p o w er is concen trated in signals of lo w amplitude, e.g. maxim um P AP R for an OFDM system with 32 carriers and QPSK mo dulation will b e observ ed statistically only once in 3.7 million y ears if the duration of an OFDM sym b ol is 100 s [27]. Therefore, the statistical distribution of the P APR should b e tak en in to accoun t. Fig. 11 sho ws the probabilit y that the magnitude of the discrete-time signal exceeds a threshold x 0 for dieren t mo dulations. The n um b er of sub carriers w as 128. Applying the cen tral limit theorem, while assuming that N is sucien tly large, x ( n ) is zero-mean complex-v alued near Gaussian distributed random v ariables for all mo dulation options. Therefore, P APR is indep enden t of mo dulation used. This can b e seen from Fig. 11. One w a y to a v oid non-linear distortion is to op erate the amplier in its linear region. Unfortunately suc h solution is not p o w er ecien t and th us not suitable for battery op erated wireless comm unication applications. Minimizing the P APR b efore p o w er amplier allo ws a higher a v erage p o w er to b e transmitted for a xed p eak p o w er, impro ving the o v erall signal to noise ratio at the receiv er. It is therefore imp ortan t to minimize the P APR. In 24

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0.1 0.15 0.2 0.25 0.3 10 5 10 4 10 3 10 2 10 1 x oP(|xm (n)| > xo) BPSKQPSK16QAM64QAM Figure 11. The probabilit y that the magnitude of the discrete-time OFDM signal exceeds a threshold x 0 for dieren t mo dulations. the rest of this thesis, the distortion of the signal due to non-linear eects is ignored. In the literature, dieren t approac hes w ere used to reduce P APR of OFDM signals. Some of these includes clipping, scram bling, co ding, phase optimization [26], tone reserv ation [28] and tone injection. 25

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CHAPTER 3 CHANNEL FREQUENCY SELECTIVITY AND DELA Y SPREAD ESTIMA TION 3.1 In tro duction In digital wireless comm unication systems, transmitted information reac hes the receiv er after passing through a radio c hannel, whic h can b e represen ted as an unkno wn, time-v arying lter. T ransmitted signals are t ypically rerected, diracted and scattered, arriving at the receiv er through m ultiple paths. When the relativ e path dela ys are on the order of a sym b ol p erio d or more, images of dier ent symb ols arriv e at the same time, causing In ter-sym b ol In terference (ISI). T raditionally ISI due to time disp ersion is handled with equalization tec hniques. As the wireless comm unication systems making transition from v oice cen tric comm unication to in teractiv e In ternet data and m ulti-media t yp e of applications, the desire for higher data rate transmission is increasing tremendously Ho w ev er, higher data rates, with narro w er sym b ol durations exp erience signican t disp ersion, requiring highly complex equalizers. New generations of wireless mobile radio systems aim to pro vide higher data rates to the mobile users while serving as man y users as p ossible. Adaptation metho ds are b ecoming p opular for optimizing mobile radio system transmission and reception at the ph ysical la y er as w ell as at the higher la y ers of the proto col stac k. These adaptiv e algorithms allo w impro v ed p erformance, b etter radio co v erage, and higher data rates with lo w battery p o w er consumption. Man y adaptation sc hemes require a form of measuremen t (or estimation) of one or more v ariable(s) that migh t c hange o v er time. The information ab out the frequency selectivit y of the c hannel, and the corresp onding time domain Ro ot-mean-squared (RMS) dela y spread can b e v ery useful for impro ving the p erformance of the wireless radio receiv ers 26

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through transmitter and receiv er adaptation. F or example, in a Time Division Multiple Access (TDMA) based GSM system, the n um b er of c hannel taps needed for equalization migh t v ary dep ending on c hannel disp ersion. Instead of xing the n um b er of c hannel taps for the w orst case c hannel condition, it can b e c hanged adaptiv ely [29], allo wing simpler receiv ers with reduced battery consumption and impro v ed p erformance. Similarly in [30] a TDMA receiv er with adaptiv e demo dulator is prop osed using the measuremen t ab out the disp ersiv eness of the c hannel. Disp ersion estimation can also b e used for other parts of transmitters and receiv ers. F or example, in c hannel estimation with c hannel in terp olators, instead of xing the in terp olation parameters for the w orst exp ected c hannel disp ersion as commonly done in practice, the parameters can b e c hanged adaptiv ely dep ending on the disp ersion information [31]. In some applications, maxim um excess dela y of the c hannel ma y b e needed. In suc h cases, maxim um excess dela y can b e obtained b y m ultiplying RMS dela y spread b y four as a rule of th um b [32]. Although disp ersion estimation can b e v ery useful for man y wireless comm unication systems, w e b eliev e that it is particularly crucial for Orthogonal F requency Division Multiplexing (OFDM) based wireless comm unication systems. Cyclic prex extension of the OFDM sym b ol a v oids ISI from the previous OFDM sym b ols if the cyclic prex length is greater than the maxim um excess dela y of the c hannel. Since the maxim um excess dela y dep ends on the radio en vironmen t, the cyclic prex length needs to b e designed for the w orst case c hannel condition whic h mak es cyclic prex a signican t p ortion of the transmitted data, reducing sp ectral eciency One w a y to increase sp ectral eciency is to adapt the length of the cyclic prex dep ending on the radio en vironmen t. In [33], b oth the n um b er of carriers and the length of cyclic prex is c hanged adaptiv ely Ob viously this adaptation requires estimation of maxim um excess dela y of the radio c hannel, whic h is also related to the frequency selectivit y Other OFDM parameters that could b e c hanged adaptiv ely using the kno wledge of the disp ersion are OFDM sym b ol duration and OFDM sub-carrier bandwidth. Disp ersion estimation can also b e used for receiv er adaptation. A t w o-dimensional Wiener lter, whic h is implemen ted as a cascade of t w o one-dimensional lters, is used for c hannel estimation in [11]. The bandwidth of the second lter, whic h is in frequency 27

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direction, is c hanged dep ending on the estimated dela y spread of the c hannel to k eep the noise lo w and th us to impro v e the c hannel estimation. In summary adaptation through disp ersion estimation pro vides b etter o v erall system p erformance, impro v ed radio co v erage, and higher data rates. Characterization of the frequency selectivit y of the radio c hannel is studied in [34{36] using Lev el Crossing Rate (LCR) of the c hannel in frequency domain. F requency domain LCR giv es the a v erage n um b er of crossings p er Hz at whic h the measured amplitude crosses a threshold lev el. Analytical expression b et w een LCR and the time domain parameters corresp onding to a sp ecic m ultipath P o w er Dela y Prole (PDP) is giv en. LCR is v ery sensitiv e to noise, whic h increases the n um b er of lev el crossing and sev erely deteriorates the p erformance of the LCR measuremen t [36]. Filtering the Channel F requency Resp onse (CFR) reduces the noise eect, but nding the appropriate lter parameters is an issue. If the lter is not designed prop erly one migh t end up smo othing the actual v ariation of frequency domain c hannel resp onse. In [37], instan taneous RMS dela y spread, whic h pro vides information ab out lo cal c hannel disp ersion, is obtained b y estimating the Channel Impulse Resp onse (CIR) in time domain. The detected sym b ols in frequency domain are used to re-generate the time domain signal through In v erse Discrete F ourier T ransform (IDFT). This signal, then, is used to correlate the actual receiv ed signal to obtain CIR, whic h is then used for dela y spread estimation. Since, the detected sym b ols are random, they migh t not ha v e go o d auto correlation prop erties, whic h can b e a problem esp ecially when the n um b er of carriers is small. In addition, the use of detected sym b ols for correlating the receiv ed samples to obtain CIR pro vides p o or results for lo w Signal-to-noise Ratio (SNR) v alues. In [11], RMS dela y spread is also calculated from the instan taneous time domain CIR, where in this case the CIR is obtained b y taking IDFT of the frequency domain c hannel estimate. In a recen t pap er [38], a tec hnique based on the cyclic-prex for dela y spread estimation is prop osed. This tec hnique uses c hange of gradien t of the correlation b et w een cyclic prex and the last part of the OFDM sym b ol as a strategy to detect the disp ersion parameters. Ho w ev er, computationally complex optimization is required. Also, 28

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the accuracy of the tec hnique can b e exp ected to degrade for closely spaced and w eak (in magnitude) m ultipath comp onen ts. In this c hapter, b oth global (long term) and lo cal (instan taneous) frequency selectivit y of the wireless comm unication c hannel will b e discussed. First, a v eraged Channel F requency Correlation (CF C) estimate is used to describ e the global frequency selectivit y of the c hannel. A no v el and practical algorithm for CF C estimation will b e describ ed. CF C estimates will then b e used to get a v erage RMS dela y spread of the c hannel. The p erformance of the estimates will b e sho wn in noise limited scenarios. The eect of noise v ariance and robustness of the estimates against dieren t c hannel PDPs will b e discussed. F or measuring lo cal frequency selectivit y CFR estimate will b e exploited. Unlik e [11], CIR is obtained b y taking IDFT of the sample d c hannel frequency resp onse. Sampling the CFR b efore taking IDFT reduces the computational complexit y The sampling rate of the CFR is c hosen based on ecien t IDFT implemen tation, and according to the Nyquist criterion assuming the kno wledge of the w orst case maxim um excess dela y v alue. The eect of c hannel estimation error will then b e discussed. The c hapter is organized as follo ws. First, a generic OFDM system description will b e giv en with a brief discussion of time and frequency domain c hannel mo dels in Section 3.2. Then, estimation of frequency correlation from the CFR will b e discussed in Section 3.3. Deriv ation of the mathematical relation b et w een the CF C and RMS dela y spread will also b e presen ted in this section. Later, estimation of time domain c hannel parameters from sampled CFR will b e studied in Section 3.4. Finally p erformance results of the prop osed algorithms will b e presen ted in Section 3.5, follo w ed b y the concluding remarks. 3.2 System mo del After the addition of cyclic prex and D/A con v ersion, the OFDM signal is passed through the mobile radio c hannel. Assuming a Wide-sense Stationary Uncorrelated Scattering (WSSUS) c hannel, the c hannel H ( f ; t ) can b e c haracterized for all time and all frequencies b y 29

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the t w o-dimensional frequency and time correlation function as ( f ; t ) = E f H ( f ; t ) H ( f + f ; t + t ) g : (37) In this c hapter, the c hannel is assumed to b e constan t o v er an OFDM sym b ol, but timev arying across OFDM sym b ols, whic h is a reasonable assumption for lo w and medium mobilit y A t the receiv er, the signal is receiv ed along with noise. After sync hronization, do wnsampling, and remo v al of the cyclic prex, the simplied baseband mo del of the receiv ed samples can b e form ulated as y m ( n ) = L 1 X l =0 x m ( n l ) h m ( l ) + z m ( n ) ; (38) where L is the n um b er of c hannel taps, z m ( n ) is the Additiv e White Gaussian Noise (A W GN) sample with zero mean and v ariance of 2 z and the time domain CIR for m th OFDM sym b ol, h m ( l ), is giv en as time-in v arian t linear lter. After taking Discrete F ourier T ransform (DFT) of the OFDM sym b ol, the receiv ed samples in frequency domain can b e written as Y m ( k ) = D F T f y m ( n ) g = X m ( k ) H m ( k ) + Z m ( k ) ; (39) where H m and Z m are DFT of h m and z m resp ectiv ely 3.3 Channel frequency selectivit y and dela y spread estimation In this section, rst a practical metho d for frequency correlation estimation from CFR is in tro duced. Then, an analytical expression for the correlation as a function of RMS dela y spread is deriv ed. Calculation of coherence bandwidth and the corresp onding RMS dela y spread from CF C is explained next. Finally eects of OFDM impairmen ts on the prop osed metho d are in v estigated. 30

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3.3.1 Channel frequency correlation estimation CF C pro vides information ab out the v ariation of the CFR across frequency carriers. Channel estimates in frequency domain can b e obtained using OFDM training sym b ols, or b y transmitting regularly spaced pilot sym b ols em b edded within the data sym b ols and b y emplo ying frequency domain in terp olation. In this c hapter, transmission of training OFDM sym b ols is used. Using the kno wledge of the training sym b ols, CFR can b e estimated using (39) ^ H m ( k ) = Y m ( k ) X m ( k ) = H m ( k ) + W m ( k ) ; (40) where W m ( k ) is the c hannel estimation error whic h is mo deled as A W GN with zero mean and v ariance of 2 w The ratio b et w een the p o w ers of H m ( k ) and W m ( k ) is dened as the c hannel SNR. F rom the c hannel estimates, the instan taneous c hannel frequency correlation v alues can b e calculated as ^ H () = E k f ^ H m ( k ) ^ H m ( k + ) g ; (41) where E k is the mean with resp ect to k (a v eraging within an OFDM sym b ol). These instan taneous correlation estimates are noisy and needs to b e a v eraged o v er m ultiple OFDM training sym b ols. The n um b er of sym b ols to b e a v eraged dep ends on the c hannel conditions (Doppler spread). If the c hannel is c hanging v ery fast in time domain, less n um b er of sym b ols are needed for a v eraging and vice v ersa. F or the rest of this c hapter, enough a v eraging is assumed to b e done. In Section 3.5, the eect of a v eraging in terv al will b e in v estigated. 31

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By a v eraging (41) o v er OFDM training sym b ols and using (40), the a v eraged correlation estimates can b e deriv ed as ~ H () = 8><>: H () if 6 = 0 H (0) + 2 w if = 0 : (42) 3.3.2 Dela y spread estimation The PDP estimate, ~ h ( ), can b e obtained b y taking IDFT of the a v eraged frequency correlation estimates ~ h ( ) = I D F T f ~ H () g = h ( ) + 2 w : (43) The statistics lik e RMS dela y spread and maxim um excess dela y can b e calculated from PDP. Ho w ev er, this requires computationally complex IDFT op eration. Instead, the desired parameters can b e calculated directly from the a v eraged CF C estimates. In the rest of this section, the direct relation b et w een RMS dela y spread and CF C is deriv ed. Equation 41 can b e re-written as ^ H m () = E k f H m ( k ) H m ( k + ) g = E k f R m ( k ) R m ( k + ) g + E k f Q m ( k ) Q m ( k + ) g + j E k f R m ( k ) Q m ( k + ) g j E k f Q m ( k ) R m ( k + ) g ; (44) where R m ( k ) and Q m ( k ) are the r e al and imaginary parts of H m ( k ). Going through mathematical details, the terms in (44) can b e expressed as E k f ( R m ( k ) R m ( k + ) g = E k f Q m ( k ) Q m ( k + ) g = 1 2 N 1 X l =0 cos 2 l N r 2 m ( l ) + q 2 m ( l ) (45) 32

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and E k f ( R m ( k ) Q m ( k + ) g = E k f R m ( k + ) Q m ( k ) g = 1 2 N 1 X l =0 sin 2 l N r 2 m ( l ) + q 2 m ( l ) ; (46) where r m ( l ) and q m ( l ) are the real and imaginary parts of the l th tap of CIR, h m ( l ). In order to deriv e an analytical expression b et w een frequency correlation and RMS dela y spread, an appropriate and generic mo del for PDP needs to b e assumed. Exp onen tially deca ying PDP is the most commonly accepted mo del for indo or c hannels. It has b een sho wn theoretically and exp erimen tally as the most accurate mo del [39]. In this section, a form ula for RMS dela y spread is deriv ed assuming an exp onen tially deca ying PDP. Later, in Section 3.5 the robustness of this assumption against other PDPs is tested using computer sim ulation. Exp onen tial dela y prole can b e expressed as h ( l ) = P e 0 R M S l ; (47) where 0 is time duration b et w een t w o consecutiv e discrete taps, and R M S is the RMS dela y spread v alue, and P is a constan t for normalizing the p o w er. Instan taneous c hannel correlation function estimates can b e obtained using (44), (45) and (46) as ^ H m () = 2 E k f ( R m ( k ) R m ( k + ) g + j 2 E k f ( R m ( k ) Q m ( k + ) g = N 1 X l =0 cos 2 l N r 2 m ( l ) + q 2 m ( l ) + j N 1 X l =0 sin 2 l N r 2 m ( l ) + q 2 m ( l ) : (48) The real and imaginary parts of the taps in CIR, r m ( l ) and q m ( l ), can b e written as r m ( l ) = a m ( l ) e 0 2 R M S l and q m ( l ) = b m ( l ) e 0 2 R M S l where a m ( l ) and b m ( l ) are Gaussian distributed indep enden t random v ariables with zero mean and iden tical v ariances. 33

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Replacing r m ( l ) and q m ( l ) with the ab o v e dened v alues, (48) can b e re-written as ^ H m () = N 1 X l =0 cos 2 l N e 0 R M S l a 2m ( l ) + b 2m ( l ) + j N 1 X l =0 sin 2 l N e 0 R M S l a 2m ( l ) + b 2m ( l ) : (49) CF C is obtained from the instan taneous CF C function with a v eraging o v er OFDM training sym b ols, whic h can also b e form ulated as ~ H () = E m f ~ H m () g = A N 1 X l =0 cos 2 l N e 0 R M S l + j A N 1 X l =0 sin 2 l N e 0 R M S l ; (50) where the constan t A is equal to E m f a 2m + b 2m g Absolute v alue of frequency correlation is obtained from (50) as j ~ H () j = vuut A 2 N 1 X l =0 N 1 X u =0 cos 2 ( l u ) N e 0 R M S ( l + u ) : (51) W e will assume N 1 to simplify the results. This is a reasonable assumption since maxim um excess dela y of PDP is m uc h smaller then OFDM training sequence duration L N After expanding the cosine term in to exp onen tials, Geometric series is used to simplify the equation. Note that the frequency correlation v alues should b e normalized to 1 to get j ~ H (0) j = 1. After going through these steps, the absolute v alue of the CF C is obtained as j ~ H () j = s 1 2 e 0 = R M S + e 2 0 = R M S 1 2 e 0 = R M S cos 2 N + e 2 0 = R M S : (52) 3.3.2.1 Estimation of RMS dela y spread and c hannel coherence bandwidth Coherence bandwidth ( B c ), whic h is a statistical measure of the range of frequencies o v er whic h the t w o sub carriers ha v e a strong correlation, can b e calculated from the a v eraged CF C function estimates. Coherence bandwidth at a sp ecied lev el K 2 (0 ; 1] is dened to 34

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b e the minim um frequency separation f suc h that the normalized frequency correlation drops b elo w K i.e. B c := inf f f 0 : j H ( f ) j = j H (0) j < K g [40]. P opularly used v alues for K are 0.9 and 0.5 [39]. F or sim ulations w e ha v e used K = 0 : 9, since the estimated correlation for small v alues are more reliable, as more data p oin ts are used to obtain these v alues. Fig. 12 sho ws ho w to calculate B c for a giv en coherence lev el, K 5 10 15 20 25 30 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Discrete frequency ( D )Channel frequency correlation K =0.9 D 1 D 2 D 3 t rms =0.6 t 0 t rms = t 0 t rms =2 t 0 Figure 12. Estimation of coherence bandwidth B c of lev el K from absolute correlation estimates corresp onding to dieren t RMS dela y spread v alues. F or giv en K and the corresp onding B c v alue, RMS dela y spread can b e deriv ed from (52) as R M S = 0 ln 2 2 K 2 cos 2 B c N + q (2 K 2 cos 2 B c N 2) 2 4(1 K 2 ) 2 2(1 K 2 ) : (53) Equation 53 is v ery complex to calculate since it requires cosine, square ro ot, and logarithm op erations. A simple expression can b e obtained b y appro ximating this equation 35

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with a simpler function. A go o d function to appro ximate is R M S = C B c ; (54) as B c and R M S are kno wn to b e in v ersely prop ortional [39]. Fig. 13 sho ws the v alue of RMS dela y spread as a function of coherence bandwidth. Results obtained b y using the exact relation (53) and the appro ximation (54) are sho wn for the coherence lev els of K = 0 : 5 and K = 0 : 9. The constan t C is obtained b y minimizing the Mean-squared-error (MSE) b et w een exact relation and the appro ximation. As can b e seen from this gure, appro ximation is nearly p erfect and giv es accurate results. If B c is kno wn, the coherence bandwidth B c can b e calculated b y m ultiplying B c with sub carrier spacing. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 25 30 35 40 Rms delay spread ( trms) x t0Coherence bandwidth (B c ) x 1/ t 0 Hz. K=0.5 TrueK=0.5 ApproximationK=0.9 TrueK=0.9 Approximation K=0.5 K=0.9 Figure 13. RMS dela y spread v ersus coherence bandwidth. The appro ximation and actual results are sho wn for t w o dieren t coherence lev els, K=0.5 and K=0.9. 36

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Some relations b et w een coherence bandwidth and RMS dela y spread is also dened in [39, 40]. The true relationship b et w een B c and R M S is an uncertain t y relationship and is giv en in [40] as B c cos 1 K 2 R M S (55) whic h can also b e written as B c R M S cos 1 K 2 ; (56) putting a lo w er b ound on the pro duct of coherence bandwidth and RMS dela y spread. The v alues of the constan t C whic h is obtained b y sim ulation, is found to b e alw a ys ab o v e this lo w er limit. 3.3.3 Eect of impairmen ts 3.3.3.1 Additiv e noise Additiv e noise is one of common limiting factors for most algorithms in wireless comm unications and it is often assumed to b e white and Gaussian distributed. In our system mo del w e ha v e also made the same assumptions. The eect of noise on the CF C is giv en in (42), where it app ears as a DC term whose magnitude dep ends on noise v ariance. Extrap olation is used to calculate the actual v alue of DC term using the correlation v alues around DC v alue. This w a y some inheren t information ab out the c hannel SNR can also b e obtained. In ter-carrier In terference (ICI) is biggest impairmen t in OFDM systems whic h can b e caused b y carrier frequency oset, phase noise, Doppler shift, m ultipath, sym b ol timing errors and pulse shaping. It is commonly mo deled as white Gaussian noise [41, 42], and considered as part of A W GN. 3.3.3.2 Carrier-dep enden t phase shift in c hannel Timing oset is another impairmen t in OFDM whic h is also folded in to the c hannel. It in tro duces a sub-carrier dep enden t phase oset on the c hannel [43, 44]. Channel frequency 37

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resp onse that includes the eect of timing oset can b e written as H m ( k ) = H m ( k ) e j 2 k N ; (57) where is time oset v alue. Using (57) CF C in the presence of timing error can b e calculated as H () = E m;k f H m ( k ) H m ( k + ) g = H () e j 2 N : (58) This equation sho ws that timing error causes a constan t phase shift in the CF C. Ho w ev er, this do es not aect the prop osed algorithm since the magnitude of CF C, whic h is not aected from timing oset, is used. 3.4 Short term parameter estimation In the previous sections, an algorithm to nd the global parameters of wireless c hannel w ere describ ed. Ho w ev er, some applications ma y require instan taneous parameters for adaptation. Esp ecially in lo w mobilit y scenarios, where wireless c hannel do es not c hange frequen tly instan taneous c hannel parameters should b e used. In this section, a metho d for obtaining the instan taneous c hannel parameters in a computationally eectiv e w a y b y using the CFR is explained and the eects of OFDM impairmen ts on this metho d are discussed. Time domain parameters, e.g. RMS dela y spread, can b e calculated if CIR is kno wn. Therefore, w e will concen trate on the calculation of CIR eectiv ely in the next section. 3.4.1 Obtaining CIR eectiv ely Channel frequency resp onse for an OFDM system can b e calculated using DFT of time domain CIR. Assuming that w e ha v e an L tap c hannel, and the v alue of l th tap for the m th 38

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OFDM sym b ol is represen ted b y h m ( l ). Then CFR can b e found as H m ( k ) = 1 N N 1 X l =0 h m ( l ) e j 2 k l = N 0 k N 1 : (59) The rev erse op eration can b e done as w ell, i.e. CIR can b e calculated from CFR with IDFT op eration. Channel estimation in frequency domain is studied extensiv ely for OFDM systems [45, 46]. W e can use estimated CFR of receiv ed samples, (40), to calculate time domain CIR. This metho d is used in [11] to obtain the co ecien ts of c hannel estimation lter adaptiv ely Ho w ev er, it requires IDFT op eration with a size equal to the n um b er of sub carriers. CFR can b e sampled to reduce the computational complexit y In this case, w e need to sample CFR according to Nyquist theorem in order to prev en t aliasing in time domain. W e can write this as max f S f 1 ; (60) where max is maxim um excess dela y of the c hannel, f is sub carrier spacing in frequency domain, and S f is the sampling in terv al. Note that the righ t hand side of the ab o v e equation is 1 and not 1 = 2. This is b ecause PDP is nonzero b et w een 0 and max W e can represen t frequency spacing in terms of OFDM sym b ol duration ( f = 1 =T u ), then w e can re-write (60) as max T u S f : (61) F rom the ab o v e equation b y assuming w orst case maxim um excess dela y sampling rate can easily b e calculated. Alternativ ely sampling rate can also b e adaptiv ely calculated b y using maxim um excess dela y calculated in the previous steps instead of using the w orst case maxim um excess dela y of the c hannel. 39

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Using (40) and (59), estimate of CFR can b e written as ^ H m ( k ) = H m ( k ) + W m ( k ) = 1 N L 1 X l =0 h m ( l ) e j 2 k l = N + W m ( k ) ; (62) where W m ( k ) are indep enden t iden tically distributed complex Gaussian noise v ariables. Note that w e ha v e replaced the upp er b ound of summation with L 1 since h m ( l ) is zero for l L The CFR estimate is sampled with a spacing of S f The sampled v ersion of the estimate can, then, b e written as ^ H 0 m ( k ) = 1 N L 1 X l =0 h m ( l ) e j 2 ( S f k ) l = N + W m ( S f k ) 1 k N S f : (63) Without loss of generalit y w e can assume N S f = L No w, CIR can b e obtained b y taking IDFT of the sampled estimate CFR. IDFT size is reduced from N to N =S f b y using sampling. As a result of this reduction, the complexit y of the IDFT op eration will decrease at least S f times. F or wireless LAN (IEEE 802.11a), for example, the w orst case scenario S f w ould b e 4 (assuming a maxim um excess dela y equal to guard in terv al, 0.8 s ), whic h decreases original complexit y b y at least 75 p ercen t. An IDFT of size N =S f = L is applied to (63) in order to obtain the estimate of CIR as ^ h m ( l ) = I D F T ( 1 N L 1 X n =0 h m ( n ) e j 2 k n=L + W m ( S f k ) ) = h m ( l ) + w 0 m ( l ) ; (64) where w 0 m ( l ) is the IDFT of the noise samples. Equation 64 giv es the instan taneous CIR. Ha ving this information, PDP can b e calculated b y a v eraging the magnitudes of instan taneous CIR o v er OFDM sym b ols. 40

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Channel estimation error will result in additiv e noise on the estimated CIR. The signalto-estimation error ratio for CIR will b e equal to signal-to-estimation error ratio for CFR since IDFT is a linear op eration. 3.4.2 Eect of impairmen ts 3.4.2.1 Additiv e noise The errors on the frequency domain c hannel estimation, whic h can b e mo deled as white noise, will eect the calculated CIR whic h in turn will eect the estimated parameters. Only the taps where energy is concen trated will b e used for CIR after IDFT is tak en. Therefore, for small sampling p erio ds, i.e. small S f noise p o w er will b e spread o v er more taps while CIR p o w er is concen trated in the same n um b er of taps alw a ys, increasing SNR. This can b e understo o d more clearly b y analyzing Fig. 14. Sampled CFRs and corresp onding CIRs obtained b y taking IDFT are sho wn in this gure. When no sampling is p erformed, noise p o w er is spread o v er 64 taps while signal p o w er (CIR) is concen trated in the rst 16 taps. As more sampling is p erformed, the same noise p o w er is no w spread o v er less taps, increasing MSE. This observ ation matc hes with the results sho wn in Fig. 15. 3.4.2.2 Constan t phase shift in c hannel A constan t phase shift in the CFR will not c hange when IDFT op eration is applied. Hence, if there is a phase oset, in the CFR, the CIR calculated using the sampled v ersion of this c hannel will b e h m ( l ) = h m ( l ) e j : (65) This phase shift in the CIR has no signicance since the statistics lik e RMS dela y spread and maxim um excess dela y dep ends only on the magnitude of CIR. 41

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10 20 30 40 50 60 0 1 2 3 Subcarrier indexMagnitudeChannel Frequency Responce (CFR) 0 20 40 60 0 2 4 6 Channel Impulse Responce (CIR) TapsMagnitude 10 20 30 40 50 60 0 1 2 3 Subcarrier indexMagnitude 0 20 40 60 0 2 4 6 TapsMagnitude 10 20 30 40 50 60 0 1 2 3 Subcarrier indexMagnitude 0 20 40 60 0 2 4 6 TapsMagnitude Figure 14. Sampling of c hannel frequency resp onse. Sampled frequency resp onse and corresp onding c hannel impulse resp onse is sho wn for dieren t sampling p erio ds. 3.4.2.3 Carrier-dep enden t phase shift in c hannel In the presence of timing oset, CFR can b e written as (57). If w e tak e IDFT of this CFR, w e obtain the follo wing relation h m ( l ) = I D F T ( H m ( k )) = L l X n =0 h m ( n ) sin ( n l ) ( n l ) e j ( n l ) : (66) Equation 66 implies an in terference b et w een the taps of CIR. This the time domain dual of ICI whic h happ ens in frequency domain and the only w a y to prev en t this in terference is to estimate the timing oset precisely 42

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0 5 10 15 20 25 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Channel SNR (dB)Mean squared error S f =1, Sim S f =1, Theo S f =2, Sim S f =2, Theo S f =4, Sim S f =4, Theo S f =5, Sim Figure 15. Normalized mean squared error v ersus c hannel SNR for dieren t sampling in terv als. Sim ulation results and theoretical results are sho wn. 3.5 P erformance results P erformance results of the prop osed algorithms are obtained b y sim ulating an OFDM system with 64 sub carriers. Wireless c hannel is mo deled with a 16-tap sym b ol-spaced CIR with an exp onen tially deca ying PDP. The c hannel taps are obtained b y using a mo died Jak es' mo del [22]. Sp eed of the mobile is assumed to b e 30 km/h. Fig. 16 sho ws the dierence b et w een the frequency correlation estimates and ideal correlation v alues for dieren t RMS dela y spreads. Ideal c hannel frequency correlation is obtained b y taking the F ourier transform of PDP. As can b e seen, the correlation estimates are v ery close to the ideal correlation v alues. As describ ed in previous sections, correlation estimate is used to nd the coherence bandwidth for a giv en correlation v alue of K This is illustrated in Fig. 12. Notice that 43

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2 4 6 8 10 12 14 16 18 20 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency ( x 1/T OFDM Hz)Channel frequency correlation IdealEstimated t rms = t 0 t rms =2 t 0 t rms =3 t 0 Figure 16. Comparison of the estimated frequency correlation with the ideal correlation for dieren t RMS dela y spread v alues. 4000 OFDM sym b ols are a v eraged and mobile sp eed w as 30km/h. as RMS dela y spread increases, coherence bandwidth decreases. Three dieren t coherence bandwidth estimates that corresp onds to three dieren t RMS dela y spread v alues are sho wn in this gure for K = 0 : 9. Fig. 17 sho ws the p erformance of the prop osed RMS dela y spread estimator as a function of c hannel SNR. Normalized MSE p erformances are giv en for dieren t n um b er of OFDM sym b ols that are used to obtain the CF C. As exp ected, the estimation error decreases as the n um b er of a v erages increases since calculated CF C is closer to the actual one. Figures 18 and 19 sho w PDPs used in the sim ulations and corresp onding MSE p erformances of the dela y spread estimator resp ectiv ely Dieren t PDPs are used in order to test the robustness of the prop osed metho d in dieren t en vironmen ts. Sm ulders' PDP is included as it has b een considered b y man y authors as an alternativ e to exp onen tially 44

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10 5 0 5 10 15 20 25 30 10 2 10 1 10 0 10 1 Channel SNR (dB)Normalized mean squared error 1,0002,0004,00010,00060,000 Figure 17. Normalized mean-squared-error p erformance of RMS dela y spread estimation for dieren t a v eraging sizes. deca ying PDP in indo or c hannels [35]. Although rectangular PDP is not a commonly used mo del for wireless c hannels, it pro vides a w orst case scenario for measuring the robustness of the prop osed algorithm. Fig. 19 sho ws that the prop osed metho d p erforms w ell not only for exp onen tially deca ying PDP but also for other PDPs. As exp ected, rectangular PDP giv es the w orst results and exp onen tially deca ying PDP giv es the b est results. The p erformance of instan taneous CIR estimation dep ends on the sampling rate. As sampling rate increases MSE of the estimates will decrease b ecause of the noise rejection eect. This w as already presen ted in Fig. 15, whic h giv es the MSE of the instan taneous CIR as a function of c hannel SNR. Since w e ha v e a 16-tap c hannel, Nyquist sampling p erio d will b e S f = 4. This gure sho ws the normalized mean squared error for dieren t sampling p erio ds. Results are giv en for unsampled ( S f = 1), o v ersampled ( S f = 2), Nyquist rate ( S f = 4) and under-sampled ( S f = 5) cases. Note that sampling the CFR b elo w Nyquist 45

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5 10 15 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Normalized powerTaps a) Exponential PDP 5 10 15 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Normalized powerTaps a) Rectangular PDP 5 10 15 0 0.05 0.1 0.15 0.2 Normalized powerTaps a) Triangular PDP 5 10 15 0 0.05 0.1 0.15 0.2 Normalized powerTaps a) Smulders` PDP Figure 18. Dieren t p o w er dela y proles that are used in the sim ulation. rate causes an irreducible error ro or. This is b ecause of the aliasing in time domain due to under-sampling.3.6 Conclusion In this c hapter, dieren t metho ds to estimate the instan taneous and global time domain parameters of wireless comm unication c hannels w ere in v estigated. A practical algorithm for a v eraged CF C estimation has b een presen ted. Coherence bandwidth and RMS dela y spread, whic h are commonly used measures for frequency selectivit y are obtained from the correlation estimates. Exact relation b et w een coherence bandwidth and RMS dela y spread is analytically deriv ed for exp onen tially deca ying PDP, then the prop osed algorithm is tested for a v ariet y of c hannel PDPs. 46

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10 5 0 5 10 15 10 2 10 1 10 0 10 1 Channel SNR (dB)normalized mean squared error ExponentialRectangularTriangularSmulders` Figure 19. Normalized mean-squared-error p erformance of RMS dela y spread estimation for dieren t p o w er dela y proles. F or measuring lo cal frequency selectivit y CFR estimate is exploited. Time domain CIR is obtained b y taking IDFT of the sample d CFR. The optimal sampling rate for sampling the c hannel resp onse is in v estigated and sim ulation results for dieren t sampling rates are giv en. The p erformance of the estimates are obtained in noise limited situations using Mon te Carlo sim ulations. It is observ ed that prop osed CF C and RMS dela y spread estimation algorithms w ork v ery w ell in v arious en vironmen ts with dieren t PDPs. Lo cal parameter estimation is also sho wn to b e p erforming w ell as long as CF C is sampled o v er Nyquist rate. 47

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CHAPTER 4 INTER-CARRIER INTERFERENCE IN OFDM 4.1 In tro duction In Orthogonal F requency Division Multiplexing (OFDM) based systems, the loss of orthogonalit y among sub carriers causes In ter-carrier In terference (ICI). ICI is often mo deled as Gaussian noise and aects b oth c hannel estimation [16] and detection of the OFDM sym b ols [15]. If not comp ensated for, ICI will result in an error ro or. In this c hapter, the impairmen ts causing ICI will b e analyzed and commonly used ICI reduction metho ds will b e giv en. 4.2 Causes of ICI Some OFDM impairmen ts are giv en in Chapter 2. As explained in that c hapter; carrier frequency sync hronization errors [47], time v arying c hannel [42] and phase noise [24] causes ICI in OFDM systems. ICI term for frequency oset is giv en b y (20) Similarly ICI terms for Doppler spread and phase noise are giv en b y (29) and (33) resp ectiv ely Usually in OFDM systems a rectangular pulse shaping is applied. The imp ortan t adv an tage of this c hoice is that the sub carrier are orthogonal to eac h other, and therefore, ICI do es not o ccur. Ho w ev er, there are also disadv an tages of this rectangular pulse shaping related to the sinc shap e of the corresp onding sub carrier sp ectra. The disadv an tages can b e a v oided if pulse shaping lters are applied. Pulse shaping lters ma y not b e orthogonal and can cause in terference b et w een sub carrier, i.e. ICI. 48

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4.3 Curren t ICI reduction metho ds Curren tly a few dieren t approac hes for reducing ICI ha v e b een dev elop ed. These approac hes includes, frequency-domain equalization, time-domain windo wing, and the ICI self-cancellation sc heme. In the follo wing sections these metho ds along with some others will b e discussed. 4.3.1 F requency-domain equalization F requency domain equalization can b e used to remo v e the eect of distortions causing ICI. In [48], frequency domain equalization is used to remo v e the fading distortion in an OFDM signal where a frequency non-selectiv e, time v arying c hannel is considered. Once the co ecien ts of the equalizer is found, linear or decision feedbac k equalizers are used in frequency domain. One in teresting p oin t here is ho w the co ecien ts are calculated. Since ICI is dieren t for eac h OFDM sym b ol, the pattern of ICI for eac h OFDM sym b ol needs to b e calculated. ICI is estimated through the insertion of frequency domain pilot sym b ols in eac h sym b ol. A pilot sym b ol is inserted to adjacen t a silence among t w o sub-blo c ks. This metho d is demonstrated in Fig. 20. Data symbols Guard symbols Data symbols Pilot symbol Dispersed pattern of pilot symbol Figure 20. Disp ersed pattern of a pilot in an OFDM data sym b ol. In [49], a nonlinear adaptiv e lter in frequency domain is also used to reduce ICI. This ltering is applied to reduce ICI due to the frequency oset. A nonline ar lter is used since it uses higher order statistics. Ho w ev er it con v erges slo wly 49

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4.3.2 Time-domain windo wing Time domain windo wing is used to reduce the sensitivit y to linear distortions and to reduce the sensitivit y to frequency errors (ICI). Windo w ma y b e realized with a raised cosine or other kind of function that fullls the Nyquist criterion. Raised cosine windo w is used in order to reduce the ICI eects in [50]. Ho w ev er, this in tuitiv e windo w is sho wn to b e sub-optim um and a closed solution for optim um windo w co ecien ts is deriv ed in [51]. A condition for orthogonalit y of windo wing sc hemes in terms of the DFT of the windo wing function is deriv ed in [47]. The FFT can b e considered as a lter bank with N lters where N is the FFT size. The frequency resp onse of the n th lter H n ( F ) is j H n ( F ) j = sin [ ( F n )] sin [ ( F n ) = N ] (67) where F := N f =f s and f s is the sampling rate at the receiv er. This lter has the shap e of a p erio dic sinc function. The DFT op eration in the receiv er p erforms transform in blo c ks of only N samples. This is equiv alen t to using a square windo w of length T s in time domain corresp onding to a sinc function in frequency domain. The lter bank consisting of N lters ha ving sinc shap e is plotted in Fig. 21(a). Carriers are represen ted b y ideal Dirac distributions placed on the lter maxima. The maxim um of one lter coincides with the zero crossing of all others; this fact allo ws to separate the carriers without suering an y ICI. As explained un windo w ed OFDM system has rectangular sym b ol shap es and hence, in the frequency domain the individual sub-c hannels will ha v e the shap e of sinc functions. The use of a windo w on N samples (in time domain) b efore the FFT reduces the side lob e amplitude of this sinc function but also leads to an orthogonalit y-loss b et w een carriers. A windo w whic h reduces the side lob es and preserv es the orthogonalit y is called Nyquist windo w. This windo w will reduce the amplitude of the lter side lob es dep ending on the roll-o factor. The side lob e magnitudes of the frequency resp onse of a raised cosine windo w for dieren t roll-o factors are giv en in Fig. 22. 50

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(a) Filter bank for rectangular windo wing. (b) Filter bank for a 2N rectangular windo w. Figure 21. P osition of carriers in the DFT lter bank. In [50], an adaptiv e Nyquist windo w is used. The windo wing uses the part of the guard in terv al that is not disturb ed b y m ultipath reception. If w e can estimate the maxim um ec ho dela y w e can calculate the length of undisturb ed guard p erio d and hence w e can c ho ose the roll-o factor of the windo w accordingly Therefore, the length of the windo w adapts to the transmission conditions. T o reduce the sensitivit y to frequency errors, useful part of the signal and un used part of the guard p erio d is shap ed with the Nyquist windo w function. After Nyquist windo wing the sub-carriers has lost their orthogonalit y Th us a symmetrical zero padding is p erformed in order to complete a total of 2 N samples. Therefore, 2 N lters will b e used in the FFT pro cess. The adv an tage of ha ving t wice as man y lters (2 N ) on the lter bank (Fig. 21(b)) is that the area under the lter curv e is one half of that of the N -lter case for the same maxima v alue. Th us the o dd or ev en lters in tegrate the same carrier p o w er but only one half of the white noise p o w er, leading to an impro v emen t in carrier to noise ratio. In the receiv er the outputs of the DFT with ev en-n um b ered subscripts are then used as estimates of the transmitted data and the o dd-n um b ered ones are discarded. Since not all 51

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40 30 20 10 0 10 20 30 40 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Frequency response of a raised cosine window with different rolloffs a =0 a =0.5 a =1 Figure 22. F requency resp onse of a raised cosine windo w with dieren t roll-o factors. of the receiv ed p o w er is b eing used in generating data estimates, windo wing reduces o v erall Signal-to-noise Ratio (SNR) compared with OFDM without windo wing. With Nyquist windo wing, the whole lter bank is less sensitiv e to frequency deviations, disturbances, etc The reason for the impro v emen t can also b e explained through a decrease of the DFT-leak age. Since the leak age is resp onsible in sev eral cases for an OFDM signal degradation, an o v erall impro v emen t in demo dulation is exp ected. A n um b er of dieren t windo ws (Hanning, Nyquist, Kaiser etc. ) ha v e b een describ ed in the literature. All of the windo ws giv e some reduction in the sensitivit y to frequency oset. But only Nyquist windo ws (of whic h the Hanning windo w is one particular example) ha v e no ICI for the case of no frequency oset [47]. 52

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In the receiv er 2 N -p oin t FFT m ust b e tak en in order to reco v er the sym b ols. But since only o dd or ev en sub carriers will carry data, w e can calculate only those p oin ts. The complexit y will b e appro ximately the same with N -p oin t FFT. 4.3.3 P artial transmit sequences & selected mapping These t w o approac hes are in tro duced in [52]. Both metho ds are adapted from P eak-toa v erage P o w er Ratio (P APR) reduction tec hniques. Since the denition of P APR and P eak In terference-to-Carrier Ratio (PICR) are analogous to eac h other [52], w e can adapt P APR reduction sc hemes to PICR reduction problem. In these t w o metho ds the goal is to reduce ICI b y minimizing PICR. Assume that the mo dulated data sym b ol sen t at sub carrier k is X ( k ) and X = f X (0) ; X (1) ; X (2) : : : X ( N 1) g The second term in (20) sho ws the ICI on the k th sub carrier due to carrier frequency oset. Let us call this term as I ( k ), whic h will ha v e the form I ( k ) = N 1 X m =0 ; m 6 = k X ( m ) K ( m; k ) (68) This term dep ends only on the transmit data sequence, X and complex co ecien ts, K ( m; k ), whic h dep ends on the normalized frequency oset and the v alue of m k P eak In terference-to-Carrier Ratio is dened as P I C R ( X ) = max 0 k N 1 j I ( k ) j 2 j K ( m; m ) a ( m ) j 2 (69) No w the goal is to minimize this ratio. 4.3.3.1 P artial transmit sequences In P artial T ransmit Sequences (PTS), the input data blo c k is partitioned in to disjoin t sub-blo c ks or clusters whic h are com bined to minimize the p eaks. Then eac h sub-blo c k is m ultiplied b y a constan t phase factor and these phase factors (w eigh ts) are optimized to reduce PICR. 53

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Let us partition X in to D disjoin t sub-blo c ks, represen ted b y the v ectors f X d ; d = 1 ; 2 ; : : : ; D g suc h that X = [ X 1 ; X 2 ; : : : ; X D ]. The ob jectiv e of the PTS approac h is to form a w eigh ted com bination of the D blo c ks, X new = D X d =1 b d X d (70) where b d ; d = 1 ; 2 ; : : : ; D are w eigh ting factors and are assumed to b e pure rotations. No w the resulting ICI can b e written as, I P T S ( k ) = D X d =1 b d I d ( k ) (71) where I d ( k ) is the in terference on k-th sub carrier due to blo c k d Th us, the total ICI is the w eigh ted sum of ICI from eac h sub-blo c k. Therefore, ICIcan b e reduced b y optimizing the phase sequence b = [ b 1 ; b 2 ; : : : ; b D ], and nally the optimal PICR can b e found as P I C R optimal = min b 1 ;b 2 ;::: ;b D max 0 k N 1 j I P T S ( k ) j 2 j S 0 X ( k ) j 2 (72) The receiv er m ust kno w the generation pro cess of the generated OFDM signal, and therefore the phase factors m ust b e transmitted to the receiv er as side information. 4.3.3.2 Selected mapping In Selected Mapping (SM) approac h, sev eral indep enden t OFDM sym b ols represen ting the same information are generated (b y m ultiplying the information sequence b y a set of xed v ectors, as explained b elo w) and the OFDM sym b ol with lo w est PICR is selected for transmission. Assume U statistically indep enden t alternativ e transmit sequences X ( u ) represen t the same information. The sequence with lo w est PICR, (69), is selected for transmission. The data sym b ol X = f X (0) ; X (1) ; X (2) : : : X ( N 1) g is m ultiplied sym b ol b y sym b ol b y a xed v ector P ( u ) = [ P ( u ) 0 ; : : : ; P ( u ) N 1 ]. Since with an ordinary P ( u ) it w ould b e v ery 54

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complex, eac h elemen t in P ( u ) is selected as P ( u ) v 2 [ 1 ; j ] for 0 v N 1, 1 u U No w resulting ICI can b e expressed as I S L ( k ) = N 1 X m =0 ;m 6 = k P ( u ) m X ( m ) K ( m; k ) (73) whic h is a function of the w eigh ting sequence P ( u ) Finally the optimal PICR can b e found as P I C R optimal = min P 1 ;::: ; P ( U ) max 0 k N 1 j I S M ( k ) j 2 j S 0 X ( k ) j 2 (74) 4.3.4 M-ZPSK mo dulation This metho d w as in tro duced in [53] and it can b e used to reduce b oth P APR and ICI. MZPSK means M-p oin t zero-padded PSK, whic h includes a signal p oin t of zero amplitude in the constellation as mo dulation sc heme. Th us, some terms in the summation in (20) v anish. Therefore, the M-ZPSK sc heme is less sensitiv e to frequency oset errors than con v en tional sc hemes. The frequency of the bit pattern of l og 2 M bits in an input sym b ol can b e coun ted. And the most lik ely bit pattern is mapp ed to a signal constellation of zero amplitude. This increases the n um b er of v anishing terms in the summation in (20) and th us reduces the ICI eects more. The p ossible signal constellations are giv en in Fig. 23 for Quadrature Phase Shift Keying (QPSK) mo dulation. Therefore, w e need only one mapping and one IFFT calculation, as in the con v en tional system. Ho w ev er, transmission of side information is necessary to let the receiv er whic h mapping is used. 4.3.5 Correlativ e co ding Correlativ e co ding is another metho d used to compress the in ter-carrier in terference caused b y c hannel frequency errors [54]. It do es not reduce the bandwidth eciency In this co ding new sym b ols are determined from old sym b ols using the correlation p olynomial F ( D ) = 1 D 55

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1011 01 00 a) 1011 01 00 b) 11 01 00 10 c) 10 01 00 11 d) Figure 23. All p ossible dieren t signal constellation for 4-ZPSK. The expression for carrier to in terference ratio (CIR) with correlativ e co ding is deriv ed and compared with the con v en tional OFDM in [54]. Without an y loss in the bandwidth 3.5dB impro v emen t in CIR lev el is gained with this metho d (for Binary Phase Shift Keying (BPSK)).4.3.6 Self-cancellation sc heme Self-Cancellation metho d is studied most among other ICI reduction metho ds. The metho d is in v estigated b y dieren t authors in [55{57]. It is also called as P olynomial Cancellation Co ding (PCC) or (half-rate) rep etition co ding. 56

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The main idea in self-cancellation is to mo dulate one data sym b ol on to a group of subcarriers with predened w eigh ting co ecien ts to minimize the a v erage carrier to in terference ratio (CIR). 4.3.6.1 Cancellation in mo dulation Fig. 24 giv es the real and imaginary parts of the ICI co ecien ts. W e can observ e that for a ma jorit y of m k v alues, the dierence b et w een K ( m k ) and K ( m k 1) is v ery small (this is more realizable as N increases). Therefore, if a data pair ( a a ) is mo dulated on to t w o adjacen t sub carriers ( m m + 1), where a is a complex data, then the ICI signals generated b y sub carrier m will b e canceled out signican tly b y the ICI generated b y sub carrier m + 1. 0 5 10 15 0.2 0 0.2 0.4 0.6 0.8 1 1.2 Real Part of K(mk)K(mk) (Real)mk e =0.1 e =0.3 0 5 10 15 0.2 0 0.2 0.4 0.6 Imaginary Part of K(mk)K(mk) (Imaginary)mk e =0.1 e =0.3 Figure 24. Real and imaginary parts of ICI co ecien ts for N=16. 57

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Assume the transmitted sym b ols are constrained so that X (1) = X (0), X (3) = X (2), . X ( N 1) = X ( N 2), then the receiv ed signal on sub carrier k b ecomes ^ X 0 ( k ) = N 2 X m =0 ; m = ev en X ( m ) f K ( m; k ) K ( m + 1 ; k ) g + Z ( k ) (75) and on sub carrier k + 1 is ^ X 0 ( k + 1) = N 2 X m =0 ; m = ev en X ( m ) f K ( m 1 ; k ) K ( m; k ) g + Z k +1 (76) In suc h a case, the new ICI co ecien t will turn out to b e K 0 ( m; k ) = K ( m; k ) K ( m + 1 ; k ) : (77) Fig. 25 sho ws a comparison b et w een K and K 0 W e can easily see that new ICI co ecien ts, K 0 are m uc h smaller. In addition, the summation in (75) only tak es ev en m v alues, the total n um b er of in terference signals is reduced to half compared with that in (20). Consequen tly the ICI signals in (75) are m uc h smaller than those in (20) since b oth the n um b er of ICI signals and the amplitudes of the ICI co ecien ts ha v e b een reduced. But this will yield a decrease in bandwidth usage b y half. 4.3.6.2 Cancellation in demo dulation By using the ICI cancellation mo dulation, eac h pair of sub carriers, in fact, transmit only one data sym b ol. The signal redundancy mak es it p ossible to impro v e the system p erformance at the receiv er side. The demo dulation for self-cancellation is suggested to w ork in suc h a w a y that eac h signal at the ( k + 1)th sub carrier ( k is ev en) is m ultiplied b y 1 and then summed with the one at the k th sub carrier. Then the resultan t data sequence is used for making sym b ol 58

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20 40 60 80 100 120 10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 0 mk KK ¢ K ¢ ¢ Figure 25. Comparison of K ( m; k ), K 0 ( m; k ) and K 00 ( m; k ). decision. It can b e represen ted as ^ X 00 ( k ) = ^ X 0 ( k ) ^ X 0 ( k + 1) = N 2 X m =0 ; m = ev en X ( m ) f K ( m 1 ; k ) + 2 K ( m; k ) K ( m + 1 ; k ) g + Z ( k ) Z ( k + 1) : (78) The corresp onding ICI co ecien ts then b ecomes K 00 ( m; k ) = K ( m 1 ; k ) + 2 K ( m; k ) K ( m + 1 ; k ) : (79) An amplitude comparison of this new co ecien ts with the other ones for N = 128 and = 0 : 2 is giv en in Fig. 25. Th us, the ICI signal b ecome smaller when applying ICI cancellation 59

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mo dulation. On the other hand, the ICI canceling demo dulation can further reduce the residual ICI in the receiv ed signals. ICI canceling demo dulation also impro v es the system signal-to-noise ratio. The signal lev el increases b y a factor of 2, due to coheren t addition, whereas the noise lev el is prop ortional to p 2 b ecause of non-coheren t addition of the noise on dieren t sub carriers. W e can obtain more ICI reduction b y mapping one sym b ol on to more than t w o sub carrier (three, four . ). Although it will yield a b etter ICI reduction, it will cause a larger bandwidth loss. W e can also map t w o mo dulated sym b ols on to three adjacen t sub-c hannels or three mo dulated sym b ols on to four adjacen t sub-c hannels, etc [57]. But in this case, ICI reduction is not uniform and w e exp ect t w o lev el of ICI reduction among these sub carriers, one for rep eated sym b ols and the other one for non-rep eated sym b ols. Due to the rep etition co ding, the bandwidth eciency of the ICI self-cancellation sc heme is reduced b y half. T o fulll the demanded bandwidth eciency it is suggested to use a larger signal alphab et size. W e can use a larger the signal alphab et size since the in terference is decreased with the use of self-cancellation sc heme. 4.3.6.3 A div erse self-cancellation metho d This metho d is v ery similar to the self-cancellation sc hemes, the only dierence is that, in this metho d the o dd symmetry of in terference term K ( m; k ) = K ( m; k ) is used b y mapping data to the sub carriers at the p oin ts k and ( N 1 k ) [58]. Since it is highly unlik ely that b oth sub carrier k and ( N 1 k ) exp ose to same fade together, this metho d oers a frequency div ersit y eect in a m ultipath fading c hannel. Ho w ev er, the ICI term do es not v anish with the appro ximation but getting reduced. This is b ecause of the dieren t fading on the sub carrier k and ( N 1 k ). If the normalized frequency oset is smaller than 0.35, this metho d giv es a b etter CIR then ordinary self-cancellation. 60

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4.3.7 T one reserv ation T one reserv ation is another metho d whic h is also adopted from P APR reduction [57]. It is based on adding a sym b ol dep enden t time domain signal to the original OFDM sym b ol to reduce ICI. The transmitter do es not send data on a small subset of carriers, whic h are used to insert the optimized tones. The complex baseband signal (with inserted pilots) ma y no w b e represen ted as x b ( n ) = X k 2 I inf o X ( k ) e j 2 k n N + X k 2 I tones X ( k ) e j 2 k n N (80) where I inf o and I tones are t w o disjoin t sets suc h that I inf o [ I tones = 0 ; 1 ; : : : ; N 1 (81) The ICI term can b e denoted in v ector form as I = K X (82) where K is N N dimensional in terference matrix with K ij = K j i and K ii = 0. T o eliminate the ICI completely I should b e zero. T o ac hiev e that, lets set X P = [ X (0) ; X ( m ) ; X (2 m ) ; : : : ; X (( P 1) m )]. Therefore, I ( k ) should b e zero for only ( N P ) subset of k that b elongs to data. Th us for ICI free c hannel, X P should satisfy the follo wing condition: K 0 a = 0 (83) where K 0 results from eliminating P ro ws, corresp onding to pilot tone p ositions, from K and a = a d [ a p is the IFFT input consisting data ( a d ) and pilots ( a p ). No w the pilots, i.e. a p can b e calculated from (83) using optimization tec hniques. In [57], least squares error optimization and standard linear programming approac hes are used for optimization. 61

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This sc heme oers 3 to 5 dB gain in PICR o v er normal OFDM with 75% of data throughput. Ho w ev er, the complexit y of transmitter is gradually increased b ecause of the complex optimization pro cess. 4.4 ICI cancellation using auto-regressiv e mo deling Most of the ICI reduction metho ds mo dels ICI as additiv e white Gaussian noise. Although ICI has the statistics of Gaussian distribution due to cen tral limit theorem for sucien tly large sub carriers, it is not white. In this researc h w e exploit the colored nature of ICI b y mo deling ICI as an Auto-regressiv e (AR) pro cess and whitening it. Although the prop osed algorithm is in tended to reduce the ICI, it will also reduce an y kind of in terference whic h is colored. Adjacen t Channel In terference (A CI) and Co-c hannel In terference (CCI) are t w o examples that are colored in nature with high-pass and lo w-pass c haracteristics resp ectiv ely Reduction of A CI will relax the F CC requiremen ts for sideband p o w er of OFDM signal. On the other hand, if w e reduce CCI, the OFDM transmitters can b e placed closer increasing the capacit y of the o v erall system. 4.4.1 Algorithm description 4.4.1.1 Auto-regressiv e mo deling W e sa y that the time series u ( n ), u ( n 1), . u ( n M ) represen ts the realization of an AR pro cess of order M if it satises the dierence equation u ( n ) + a 1 u ( n 1) + + a M u ( n M ) = v ( n ) (84) where a 1 a 2 . a M are constan ts called the AR parameters, and v ( n ) is a white-noise pro cess. Equation 84 implies that if w e kno w the parameters, a 1 a 2 . a M then w e can whiten the signal u ( n ) b y con v olving it with the sequence of parameters a m 62

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The relationship b et w een the parameters of the mo del and the auto correlation function of u ( n ), r xx ( l ), is giv en b y the Y ule-W alk er equations r xx ( l ) = 8><>: P Mk =1 a k r xx ( l k ) f or n 1 P Mk =1 a k r xx ( k ) + 2 v f or n = 0 (85) where 2 v = E fj v ( n ) j 2 g Therefore if w e are kno w the input sequence u ( n ), w e can obtain the auto correlation and w e can solv e for the mo del parameters, a k b y using Levinson-Durbin algorithm. 4.4.1.2 Estimation of noise sp ectrum and whitening ICI samples of dieren t carriers are correlated since the summation in (68) dep ends on the same transmitted sym b ols, whic h mak es ICI colored. Fig. 26 sho ws the P o w er Sp ectral Densit y (PSD) of ICI sequence, whic h has lo w-pass c haracteristics. In Fig. 26, the sp ectral p o w er densities of ICI sequences whitened with AR lters of dieren t mo del orders is also giv en. As mo del order increases the sp ectrum b ecomes less colored, ho w ev er this increases the computational complexit y W e whiten the ICI signal since the receiv ers will p erform m uc h b etter in the presence of white noise. Since ICI for eac h OFDM sym b ol dep ends on the instan taneous carrier frequency oset or Doppler shift, w e need to estimate the ICI samples for eac h OFDM sym b ol indep enden tly A t w o stage detection tec hnique will b e emplo y ed. In the rst stage, ten tativ e sym b ol decisions will b e p erformed using initially receiv ed signal. Then, these initial estimates will b e used to estimate the ICI presen t on the curren t OFDM sym b ol. These estimates, then, will b e used to nd the AR mo del parameters and to whiten the in terference. After this pro cess, the receiv ed signal with white noise will b e used in a second stage to pro vide sym b ol decisions. Since w e can not distinguish ICI from other impairmen ts ( e.g. additiv e noise, CCI, A CI, etc. ), w e calculated ICI + other in terferences and whitened this sum. Assuming w e made correct sym b ol decisions in the rst stage and assuming p erfect c hannel kno wledge, w e can 63

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1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 x 10 7 40 35 30 25 20 15 10 Frequency (Hz)PSD (dB) ICI signalAR 1AR 2AR 15 Figure 26. P o w er sp ectral densit y of the original and whitened v ersions of the ICI signals for dieren t AR mo del orders. nd the total impairmen ts b y subtracting the re-mo dulated sym b ols from the impaired receiv ed sym b ols. W e t the sp ectrum of colored noise b y an AR sto c hastic pro cess of order M and calculated the AR parameters. Ha ving the AR lter co ecien ts, w e can whiten the colored noise b y passing it through the AR lter. Although, ltering will whiten the colored signal, it will eect the desired signal also. T o reco v er the desired signal bac k, w e can use M tap Decision F eedbac k Equalizer (DFE), with M is equal to the order of the AR lter. 4.4.2 P erformance results The gain obtained using the prop osed algorithm is prop ortional to the AR mo del order. Ho w ev er, as the mo del order increases the computational complexit y is also increasing. Fig. 27 sho ws the bit error rate for dieren t AR mo del orders. This gure is obtained with 64

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10 20 30 40 50 60 10 2 10 1 BER For Different AR ModelsBit Error Rate (BER) Eb/No ConventionalAR 1AR 2AR 5AR 10AR 15 Figure 27. P erformance of the prop osed metho d for dieren t mo del orders. = 0 : 3. a normalized frequency oset of 0.3. W e obtain more gain with higher mo del orders with the increase in computational complexit y 4.5 Conclusion In this c hapter, impairmen ts that causes ICI is describ ed briery and some recen t ICI cancellation tec hniques are describ ed. Later, an ICI cancellation algorithm based on AR mo deling is giv en. This algorithm explores the colored nature of ICI in OFDM systems. ICI is mo deled as the output of a lter for whic h the input is the transmitted sym b ols (assumed to b e white). The co ecien ts of this lter is calculated and receiv ed signal is whitened b y passing through an in v erse lter. Filter co ecien ts are found b y tting an AR pro cess to the ICI. 65

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CHAPTER 5 ICI CANCELLA TION BASED CHANNEL ESTIMA TION 5.1 In tro duction Channel estimation is one of the most imp ortan t elemen ts of wireless receiv ers that emplo ys coheren t demo dulation. F or Orthogonal F requency Division Multiplexing (OFDM) based systems, c hannel estimation has b een studied extensiv ely Approac hes based on Least Squares (LS), Minim um Mean-square Error (MMSE) [45], and Maxim ul Lik eliho o d (ML) [59] estimation are studied b y exploiting the training sequences that are transmitted along with the data. The previous c hannel estimation algorithms treat In ter-carrier In terference (ICI) as part of the additiv e white Gaussian noise and these algorithms p erform p o orly when ICI is signican t. Linear Minim um Mean-square Error (LMMSE) estimator is analyzed in [46] to suppress the ICI due to mobilit y (Doppler spread). Ho w ev er, it is sho wn that non-adaptiv e LMMSE estimator giv en in [46] is not capable of reducing ICI and the design of an adaptiv e LMMSE is relativ ely dicult since b oth Doppler prole and noise lev el need to b e kno wn. A c hannel estimation sc heme whic h uses time-domain ltering to mitigate the ICI eect of time-v arying c hannel is prop osed in [60]. This c hapter presen ts a no v el c hannel estimation metho d that eliminates ICI b y join tly nding the frequency oset and Channel F requency Resp onse (CFR). The prop osed metho d nds c hannel estimates b y h yp othesizing dieren t frequency osets and c ho oses the b est c hannel estimate using correlation prop erties of CFR. In the rest of this c hapter, the prop osed algorithm will b e describ ed briery and sim ulation results will b e giv en. 66

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5.2 System mo del Time domain represen tation of OFDM signal is giv en is (3). This signal is cyclically extended to a v oid In ter-sym b ol In terference (ISI) from previous sym b ol and transmitted. A t the receiv er, the signal is receiv ed along with noise. After sync hronization, do wn sampling, and remo v al of cyclic prex, the baseband mo del of the receiv ed frequency domain samples can b e written in matrix form as y = S p Xh + z ; (86) where y is the v ector of receiv ed sym b ols, X is a diagonal matrix with the transmitted (training) sym b ols on its diagonal, h = [ H (1) H (2) H ( N )] T is the v ector represen ting the CFR to b e estimated, and z is the additiv e white Gaussian noise v ector with mean zero and v ariance of 2 z The N N matrix, S p is the in terference (crosstalk) matrix that represen ts the leak age b et w een sub carriers, i.e. ICI. If there is no frequency oset, i.e. p = 0, S p b ecomes S 0 = I whic h implies no in terference from neigh b oring sub carriers. If ICI is assumed to b e caused only b y frequency oset, en tries of S h can b e found using the follo wing form ula [47] S p ( m; n ) = sin ( m n + p ) N sin N ( m n + p ) e j ( m n + p ) ; (87) where p is the pr esent normalized carrier frequency oset (the ratio of the actual frequency oset to the in ter-sub carrier spacing). 5.3 Algorithm description The in terference matrix S p is not kno wn to the receiv er as it dep ends on the unkno wn carrier frequency oset, p In this section, w e will try to matc h to S p b y S h where h is the h yp othesis for the true frequency oset. 67

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The estimate of CFR is obtained b y m ultiplying b oth sides of (86) with ( S h X ) 1 as ( S h X ) 1 y = ( S h X ) 1 S p Xh + ( S h X ) 1 z h h = X 1 S h 1 S p Xh + z h : (88) The in v ersion of the matrix S h X is simple since the in terference matrix S h is unitary and the data matrix X is diagonal. In this c hapter, w e assume that all of the sub-carriers are used in training sequence i.e. no virtual carriers. This assumption ensures the in v ertibilit y of training data matrix X Equation 88 will yield sev eral c hannel estimates for dieren t frequency oset h yp otheses. F or the oset h yp othesis, h whic h is closest to the actual frequency oset, p (88) will yield the b est estimate of the CFR. F or c ho osing the b est h yp othesis, c hannel frequency correlation is used as a decision criteria. In the rest of this section, prop erties of the in terference matrix will b e describ ed rst. Then, the metho d for c ho osing the b est h yp othesis will b e explained follo w ed b y the description of the searc h algorithm to nd the b est h yp othesis. 5.3.1 Prop erties of in terference matrix The follo wing prop erties related to the in terference matrix can b e deriv ed using (87). 1. S H S = I : In terference matrix is a unitary matrix. Therefore, the in v erse of the in terference matrix can b e calculated easily b y taking the conjugate transp ose since S 1 = S H Note that the sup erscript H represen ts conjugate transp ose. 2. S 1 S 2 = S 1 + 2 : If t w o in terference matrices corresp onding to t w o dieren t frequency osets are m ultiplied, another in terference matrix corresp onding to the sum can b e obtained. This prop ert y is exploited in the searc h algorithm. 3. S = S H : The in terference matrix for a negativ e frequency oset can b e obtained from the in terference matrix corresp onding to a p ositiv e frequency oset with the same magnitude b y nding the complex transp ose. 68

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5.3.2 Channel frequency correlation for c ho osing the b est h yp othesis The m ultiplication of t w o in terference matrices in (88) can b e written using the prop erties of in terference matrix as S 1 h S p = S h S p = S p h = S r ; (89) where r is the dierence b et w een the actual frequency oset and frequency oset h yp othesis, i.e. r esidual frequency error. Using (88) and (89), the estimate of the c hannel frequency resp onse can b e written as H h ( k ) = 1 X k N X l =1 X ( l ) H ( l ) S r ( k ; l ) + 1 X k N X l =1 z ( l ) S h ( k ; l ) 1 k N : (90) Using (90), the frequency correlation of the estimated c hannel for eac h OFDM sym b ol can b e calculated as h h () = 1 N 2 N X k =+1 n H h ( k ) H h ( k ) o = 1 N 2 N X k =+1 ( 1 X ( k ) N X l =1 X ( l ) H ( l ) S r ( k ; l ) 1 X ( k ) N X u =1 X ( u ) H ( u ) S r ( k ; u ) + 1 X ( k ) N X l =1 z ( l ) S h ( k ; l ) 1 X ( k ) N X u =1 z ( u ) S h ( k ; u ) ) : (91) 69

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If w e assume that the n um b er of sub carriers, N is large, (91) can b e simplied as h h () = 8><>: h (0) + 2 z 2 s = 0 h () j S r (0) j 2 6 = 0 (92) where j S r (0) j = sin ( r ) N sin ( r = N ) is the magnitude of the diagonal elemen t of in terference matrix of residual frequency oset, S r and 2 s is the v ariance of the receiv ed signal. Note that as the residual frequency oset increases, the v alue of j S r (0) j decreases, causing the correlation to decrease. As (92) implies, the correlation magnitude of the CFR dep ends on the residual frequency oset. F or a giv en CFR, c hannel frequency correlation b ecomes maxim um when the frequency oset h yp othesis, h matc hes to the actual frequency oset. Therefore, the correlation v alues can b e used as a decision criteria for c ho osing the b est h yp othesis. F or c ho osing the b est h yp othesis among sev eral h yp otheses, this criteria is used in the searc h algorithm According to (92), all the lags of c hannel correlation can b e used for obtaining the b est h yp othesis. Ho w ev er, as increases c hannel correlation decreases, this degrades the p erformance of the estimation since the ratio of useful signal p o w er to the noise p o w er b ecomes smaller. Also, for large v alues, correlations are more noisy since less samples are used to obtain these correlations. Moreo v er, increasing the n um b er of lags increases the computational complexit y as more correlations need to b e estimated. Therefore, selection of the n um b er of lags to b e used is a design criteria and needs to b e further in v estigated. In our sim ulation, only the rst correlation v alue, h h (1), is used. Ho w ev er, b etter results can b e obtained b y eectiv ely com bining the information from other correlation lags.5.3.3 The searc h algorithm Finding the frequency domain c hannel for all of the h yp otheses and c ho osing the b est h yp othesis require enormous computation. The in terference matrices for eac h frequency oset 70

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h yp othesis should also b e precomputed and stored in memory Ho w ev er, these requiremen ts can b e relaxed b y emplo ying an optim um searc h algorithm. Instead of trying all p ossible frequency osets, the correct frequency oset is calculated b y using a binary searc h algorithm. The magnitude of the correlation is estimated at the maxim um and minim um exp ected frequency oset v alues rst. If the v alue at the minim um p oin t is smaller, correct frequency oset is exp ected to b e at the b ottom half of the initial in terv al. Therefore, maxim um p oin t is mo v ed to the p oin t b et w een the previous t w o p oin ts and minim um is not c hanged. If maxim um p oin t is smaller, opp osite op eration is p erformed. In the second step the same op eration is rep eated for the new in terv al. Then, this pro cess is rep eated for a predened n um b er of iterations. Note that CFR needs to b e obtained only for one more h yp othesis in eac h iteration after the rst iteration. Therefore the total n um b er of CFRs estimated is total n um b er of iterations plus one. T o calculate the CFR for a h yp othesis, w e do not need to ha v e all the in terference matrices. If the in terference matrices for max max = 2, max /4, max /8, . are calculated, where max is the maxim um exp ected frequency oset, the required in terference matrices can b e found b y using the second prop ert y of in terference matrix. Moreo v er, CFR estimates can b e calculated without ha ving all of the in terference matrices. In (88), receiv ed sym b ols are m ultiplied b y S 1 h and then m ultiplied with the diagonal matrix X 1 The result of m ultiplication with S 1 h can b e stored and m ultiplied with S 1 2 in the next step to obtain the same result whic h w ould b e obtained b y m ultiplying S 1 h + 2 5.3.4 Reduced in terference matrix The in terference matrix S is an N N matrix. Ho w ev er, most of the energy is concen trated around the diagonal, i.e. in terference is mostly due to neigh b oring sub carriers. The en tries a w a y from the diagonal are set to zero in order to decrease the n um b er of m ultiplications and additions p erformed during the searc h algorithm. This will also decrease the memory requiremen t. The amplitudes of the full and reduced in terference matrices are sho wn in Fig. 28 for normalized frequency osets of 0.1 and 0.3. 71

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As seen in Fig. 28, the eect of round-o b ecomes more noticeable as frequency oset increases, since the energy will b e spread a w a y form the diagonal at high frequency osets. The gain in computational complexit y is more noticeable as the n um b er of sub carriers increases. 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Carrier indexAmplitude of coefficientsNorm. Freq. Offset = 0.3 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Carrier indexAmplitude of coefficients 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Carrier indexAmplitude of coefficientsNorm. Freq. Offset = 0.1 10 20 30 40 50 60 0 0.2 0.4 0.6 0.8 1 Carrier indexAmplitude of coefficients Figure 28. Magnitudes of b oth full and reduced in terference matrices for dieren t frequency osets. Second ro w sho ws the reduced matrix. Only one ro w is sho wn. 5.4 Results Sim ulation results are obtained in an OFDM based wireless comm unication system with 64 sub carriers whic h emplo ys Quadrature Phase Shift Keying (QPSK) mo dulation. A 6-tap sym b ol-spaced time domain c hannel impulse resp onse with exp onen tially deca ying p o w er dela y prole is used. 72

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Num b er of iterations for the searc h algorithm w as 8, whic h means that CFR is estimated for 8 + 1 = 9 dieren t frequency oset h yp otheses to nd the b est CFR. Fig. 29 sho ws the v ariance of the frequency oset estimator as a function of Signal-tonoise Ratio (SNR). Results for full and reduced in terference matrices are sho wn. Reduced matrix is obtained using the 32 en tries of full in terference matrix, reducing computational complexit y b y 50%. The Cr amer-Rao b ound [61] C R B ( ) = 1 2 2 3( S N R ) 1 N (1 1 = N 2 ) (93) is also pro vided for comparison. As can b e seen from this gure, truncating the in terference matrix has little eect on the p erformance. 5 10 15 20 25 30 35 10 7 10 6 10 5 10 4 10 3 SNR (dB)Mean squared error Full MatrixReduced MatrixCR Bound Figure 29. V ariance of the frequency oset estimator. Results obtained b y using full and reduced in terference matrices and Cr amer-Rao lo w er b ound is sho wn. 73

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The frequency range in whic h the frequency oset is b eing searc hed is c hosen adaptiv ely dep ending on the history of the estimated frequency osets. If the v ariance of previous frequency oset estimates is small, the range is decreased to increase the p erformance with the same n um b er of iterations; and if it is large the range is increased in order to b e able to trac k the v ariations of the frequency oset. Fig. 30 sho ws the correct and estimated frequency oset v alues that are obtained b y xing the frequency oset range and c hanging it adaptiv ely It can b e seen from this gure that the algorithm con v erges to the correct frequency oset and c hanging the range adaptiv ely helps tracing the frequency oset. 0 20 40 60 80 100 120 0.2 0 0.2 0.4 0.6 0.8 1 1.2 OFDM framesNormalized frequency offset Corect frequency offsetAdaptive offset rangeFixed offset range Figure 30. Estimated and correct (normalized) frequency oset v alues at 10 dB. Results for adaptiv e and xed initial frequency oset ranges are sho wn. Mean-square error p erformances of the prop osed and con v en tional LS estimators are sho wn in Fig. 31 as a function of SNR, where a normalized frequency oset of 0.05 is used. Obtained c hannel estimates can b e further pro cessed to decrease the mean-square error, ho w ev er this is out of the scop e of this this study 74

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5 10 15 20 25 30 35 10 3 10 2 10 1 10 0 SNR (dB)Mean squared error Least Squares MethodProposed Method Figure 31. Mean-square error v ersus SNR for con v en tional LS and prop osed CFR estimators. Normalized carrier frequency is 0.05. 5.5 Conclusion A no v el frequency-domain c hannel estimator whic h mitigates the eects of ICI b y join tly nding the frequency oset and CFR is describ ed in this c hapter. Unlik e con v en tional c hannel estimation tec hniques, where ICI is treated as part of the noise, the prop osed approac h considers the eect of frequency oset in estimation of CFR. Metho ds to nd the b est CFR eectiv ely with lo w complexit y is discussed. It is sho wn via computer sim ulations that the prop osed metho d is capable of reducing the eect of ICI on the frequency domain c hannel estimation. 75

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CHAPTER 6 CONCLUSION The demand for high data rate wireless comm unication has b een increasing dramatically o v er the last decade. One w a y to transmit this high data rate information is to emplo y w ellkno wn con v en tional single-carrier systems. Since the transmission bandwidth is m uc h larger than the coherence bandwidth of the c hannel, highly complex equalizers are needed at the receiv er for accurately reco v ering the transmitted information. Multi-carrier tec hniques can solv e this problem signican tly if designed prop erly Optimal and ecien t design leads to adaptiv e implemen tation of m ulti-carrier systems. Examples to adaptiv e implemen tation metho ds in m ulti-carrier systems include adaptation of cyclic prex length, sub-carrier spacing etc These tec hniques are often based on the c hannel statistics whic h need to b e estimated. In this thesis, metho ds to estimate parameters for one of the most imp ortan t statistics of the c hannel whic h pro vide information ab out the frequency selectivit y has b een studied. These parameters can b e used to c hange the length of cyclic prex adaptiv ely dep ending on the c hannel conditions. They can also b e v ery useful for other transceiv er adaptation tec hniques. Although m ulti-carrier systems handle the disp ersion in time, they bring ab out other problems lik e In ter-carrier In terference (ICI). In this thesis, ICI problem is studied for impro ving the p erformance of b oth data detection and c hannel estimation at the receiv er. ICI problem is created to solv e the problem with time disp ersion, i.e. In ter-sym b ol In terference (ISI). Dep ending on the application and the c hannel statistics, one problem will b e more signican t than the other. F or example for high data rate applications, ISI app ears to b e more signican t problem. On the other hand, for high mobilit y applications, 76

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ICI is a more dominan t impairmen t. F or high data rate and high mobilit y applications, the systems should b e able to handle these in terference sources ecien tly as they will app ear one w a y or another. Adaptiv e system design and adaptiv e in terference cancellation tec hniques, therefore, are v ery imp ortan t to ac hiev e this goal. Curren t applications of Orthogonal F requency Division Multiplexing (OFDM) do not require high mobilit y F or next generation applications, ho w ev er, it is crucial to ha v e systems that can tolerate high Doppler shifts caused b y high mobile sp eeds. Curren t OFDM systems assume that the c hannel is time-in v arian t o v er OFDM sym b ol. As mobilit y increases, this assumption will not b e v alid an ymore, and v ariations of the c hannel during the OFDM sym b ol p erio d will cause ICI as explained in Chapter 2. In the prop osed c hannel estimation metho d giv en in Chapter 5, only ICI due to frequency oset is considered. ICI due to timev arying c hannel should b e in v estigated further and eectiv e c hannel estimation metho ds that are imm une to ICI due to mobilit y should b e dev elop ed to ha v e OFDM ready for high mobilit y applications. 77

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Self-interference handling in OFDM based wireless communication systems
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ABSTRACT: Orthogonal Frequency Division Multiplexing (OFDM) is a multi-carrier modulation scheme that provides efficient bandwidth utilization and robustness against time dispersive channels. This thesis deals with self-interference, or the corruption of desired signal by itself, in OFDM systems. Inter-symbol Interference (ISI) and Inter-carrier Interference (ICI) are two types of self-interference in OFDM systems. Cyclic prefix is one method to prevent the ISI which is the interference of the echoes of a transmitted signal with the original transmitted signal. The length of cyclic prefix required to remove ISI depends on the channel conditions, and usually it is chosen according to the worst case channel scenario. Methods to find the required parameters to adapt the length of the cyclic prefix to the instantaneous channel conditions are investigated. Frequency selectivity of the channel is extracted from the instantaneous channel frequency estimates and methods to estimate related parameters, e.g. coherence bandwidth and Root-mean-squared (RMS) delay spread, are given. These parameters can also be used to better utilize the available resources in wireless systems through transmitter and receiver adaptation. Another common self-interference in OFDM systems is the ICI which is the power leakage among different sub-carriers that degrades the performance of both symbol detection and channel estimation. Two new methods are proposed to reduce the effect of ICI in symbol detection and in channel estimation. The first method uses the colored nature of ICI to cancel it in order to decrease the error rate in the detection of transmitted symbols, and the second method reduces the effect of ICI in channel estimation by jointly estimating the channel and frequency offset, a major source of ICI.
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channel estimation.
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frequency selectivity.
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