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Channel estimation techniques for OFDM systems in dispersive time-varying channels Song, Xuegui 2009-12-31

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CHANNEL ESTIMATION TECHNIQUES FOROFDM SYSTEMS IN DISPERSIVETIME-VARYING CHANNELSbyXUEGUI SONGB.Eng., Northwestern Polytechnical University, P. R. China, 2003A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe College of Graduate Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(OKANAGAN)May 2009c© Xuegui Song, 2009AbstractCoherent modulation is more effective than differential modulation for orthogonal frequency di-vision multiplexing (OFDM) systems requiring high data rate and spectral efficiency. Channelestimation is therefore an integral part of the receiver design. In this thesis, two iterative maximumlikelihood based channel estimation algorithms are proposed for an OFDM system in dispersivetime-varying channels. A multipath channel model is proposed for OFDM uplink transmissionin macrocellular systems. The multipath fading channel is modeled such that the channel statecan be determined by estimating the unknown channel parameters. A second-order Taylor seriesexpansion is adopted to simplify the channel estimation problem. Based on the system model, aniterative maximum likelihood based algorithm is first proposed to estimate the discrete-time chan-nel parameters. The mean square error performance of the proposed algorithm is analyzed usinga small perturbation technique. Based on a convergence rate analysis, an improved iterative maxi-mum likelihood based channel estimation algorithm is presented using a successive overrelaxationmethod. Numerical experiments are performed to confirm the theoretical analyses and show theimprovement in convergence rate of the improved algorithm.iiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 42 Wireless Channels and OFDM Technique . . . . . . . . . . . . . . . . . . . . . . . . 62.1 Wireless Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Large-Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.2 Small-Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 OFDM Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Basic Concept . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 OFDM Implementation Using FFT . . . . . . . . . . . . . . . . . . . . . 112.2.3 Advantages and Drawbacks of OFDM . . . . . . . . . . . . . . . . . . . 13iiiTable of Contents3 ML-basedChannelEstimationforOFDMSystemsinDispersiveTime-VaryingChan-nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2 Formulation of Channel Estimation Problem . . . . . . . . . . . . . . . . . . . . 163.3 An Iterative Channel Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . 223.4 Performance Analysis of the Channel Estimation Algorithm . . . . . . . . . . . . 233.5 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 284 Convergence Study of ML-based Channel Estimation Algorithms . . . . . . . . . . 354.1 Convergence Performance Analysis of the Proposed Algorithm . . . . . . . . . . 354.2 An Improved Channel Estimation Algorithm . . . . . . . . . . . . . . . . . . . . 414.3 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . 415 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52AppendicesA Derivation of the Discrete CIR Model . . . . . . . . . . . . . . . . . . . . . . . . . . 56B Derivation of (3.20) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58C Derivation of (3.30c) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61D Derivation of the Absolute Value for the Largest Eigenvalue of M−1η Nη . . . . . . . 63E Matlab Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65ivList of Tables3.1 Algorithm 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1 Algorithm 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41vList of Figures2.1 Signal power fluctuation versus range in wireless channels. . . . . . . . . . . . . . 72.2 Basic principle of OFDM modulation. . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Basic principle of OFDM demodulation. . . . . . . . . . . . . . . . . . . . . . . . 122.4 OFDM modulation by means of IFFT processing. . . . . . . . . . . . . . . . . . . 132.5 OFDM demodulation by means of FFT processing. . . . . . . . . . . . . . . . . . 143.1 An MSE comparison for three different approximation orders. . . . . . . . . . . . 203.2 An MSE performance comparison between Algorithm 1 and Moose’s estimator forˆfl when fl = 0.05, δh = 1% and δf = 1%. . . . . . . . . . . . . . . . . . . . . . . 303.3 An average MSE performance comparison between the simulated results and thetheoretical values for ˆhl of Algorithm 1 when δh = 1% and δf = 1%. . . . . . . . . 313.4 An MSE performance comparison between the simulated results and the theoreti-cal values for ˆfl of Algorithm 1 when fl = 0.02, δh = 1% and δf = 1%. . . . . . . 323.5 An MSE performance comparison between the simulated results and the theoreti-cal values for ˆfl of Algorithm 1 when fl = 0.04, δh = 1% and δf = 1%. . . . . . . 333.6 An MSE performance comparison between the simulated results and the theoreti-cal values for ˆfl of Algorithm 1 when fl = 0.06, δh = 1% and δf = 1%. . . . . . . 344.1 An average MSE performance comparison between Algorithm 1 and Algorithm 2,along with the theoretical values for ˆhl, when δh = 1% and δf = 1%. . . . . . . . . 434.2 An MSE performance comparison between Algorithm 1 and Algorithm 2, alongwith the theoretical values for ˆfl, when fl = 0.02, δh = 1% and δf = 1%. . . . . . 44viList of Figures4.3 An MSE performance comparison between Algorithm 1 and Algorithm 2, alongwith the theoretical values for ˆfl, when fl = 0.04, δh = 1% and δf = 1%. . . . . . 454.4 An MSE performance comparison between Algorithm 1 and Algorithm 2, alongwith the theoretical values for ˆfl, when fl = 0.06, δh = 1% and δf = 1%. . . . . . 464.5 An average number of iterations comparison between Algorithm 1 and Algorithm2 when η = 4/3, δh = 1% and δf = 1%. . . . . . . . . . . . . . . . . . . . . . . . 474.6 The average MSE performance versus number of iterations for ˆhl of Algorithm 1and Algorithm 2 when η = 4/3 and SNR = 25. . . . . . . . . . . . . . . . . . . . 484.7 The average MSE performance versus number of iterations for ˆfl of Algorithm 1and Algorithm 2 when η = 4/3 and SNR = 25 dB. . . . . . . . . . . . . . . . . . 49viiList of AcronymsAcronyms Definitions3GPP Third Generation Partnership ProjectBER Bit-Error RateBPSK Binary Phase-Shift KeyingCIR Channel Impulse ResponseCRB Cramer-Rao BoundDAB Digital Audio BroadcastingDFT Discrete Fourier TransformDVB Digital Video BroadcastingFFT Fast Fourier TransformICI Inter-Carrier InterferenceIDFT Inverse Discrete Fourier TransformIFFT Inverse Fast Fourier TransformISI Intersymbol InterferenceLOS Line-of-SightLS Least SquareLTE Long Term EvolutionML Maximum LikelihoodMMSE Minimum Mean Square ErrorMSE Mean Square ErrorOFDM Orthogonal Frequency Division MultiplexingP/S Parallel-to-SerialPAPR Peak-to-Average Power RatioviiiList of AcronymsRF Radio FrequencyS/P Serial-to-ParallelSC-FDMA Single-Carrier Frequency Division Multiple AccessSNR Signal-to-Noise RatioSOR Successive OverrelaxationWLAN Wireless Local Area NetworkWMAN Wireless Metropolitan Access NetworkixAcknowledgmentsI am deeply grateful to my supervisor Dr. Julian Cheng for his enthusiasm, guidance, advice,encouragement, and support. I will continue to be influenced by his rigorous scholarship, clarityin thinking, and professional integrity.I would like to thank Dr. Zheng Du from Huawei Technology Co. and Dr. Norman C. Beaulieufrom University of Alberta, Edmonton for their feedbacks, constructive comments, and valuablesuggestions on my research work.IwouldliketoexpressmythankstoDr. ParamjitGillforhiswillingnesstoserveasmyexternalexaminer. I would also like to thank Dr. Richard Klukas and Dr. Jonathan Holzman for serving onthe committee. I really appreciate their valuable time and constructive comments on my thesis.I owe many people for their generosity and support during my two-year study at the Universityof British Columbia. I would like to thank my dear colleagues Chiun-Shen Liao, Junfeng Zhao,James Jianchen Sun, Mingbo Niu, Nick Kuan-Hsiang Huang, Ning Wang, Xian Jin, and YeyuanXiao for sharing their academic experiences and constructive viewpoints generously with me dur-ing our discussions. I would also like to thank my dear friends Eric Jinglong Ma, Junning Li,Pan Luo, Yi Zhang, and Yu Hui for sharing in my excitement and encouraging me when I was infrustration during this journey. I would also like to thank all my friends in Vancouver, Beijing,Chengdu, and Xi’an.Finally, I would like to thank my parents for their patience, understanding and support over allthese years. All my achievements would not have been possible without their constant encourage-ment and support.xChapter 1Introduction1.1 Background and MotivationThe earliest prototype of wireless communications can be traced back to the pre-industrial age. Inthose years, information was transmitted over line-of-sight (LOS) distances using smoke signals,torch signaling, flashing mirrors, signal flares, or semaphore flags. However, true wireless com-munications, where information is transmitted in terms of electromagnetic waves through complexphysical mediums, started from the first experiments with radio communication by M. G. Marconiin the 1890s. Since then, wireless communication systems have been developing and evolving witha rapid pace. In the intervening hundred years, many types of wireless communication systemshave flourished, and often later disappeared. By far the most successful wireless communicationsystem has been the cellular system. It is reported that the number of mobile phone subscribersis on the order of 4 billion currently and will rise to 5.6 billion in 2013 worldwide. Undoubtedly,cellular phones have radically changed the life of people and become a critical business tool inmost countries.In the last two decades, with the explosive growth of wireless communications, wireless ser-vices have migrated from the conventional voice-centric services to data-centric services. There-fore, one main target for modern wireless communications is to provide the possibility for highend-user data rates. One straightforward way to meet this requirement is to increase the trans-mission bandwidth because it is well known that the achievable end-user data rates are limited bytransmission bandwidth and transmitter power. However, the transmitted signal in a wireless com-munication system with wider transmission bandwidth can incur increased corruption due to timedispersion (or equivalently frequency selectivity) of the radio channel. To counteract such signal11.2. Literature Reviewcorruption, researchers have proposed techniques such as receiver-side equalization, multicarriertransmission and specific single-carrier transmission. Orthogonal frequency division multiplexing(OFDM) is a typical multicarrier transmission scheme which can increase the overall transmissionbandwidthwithoutsufferingfromincreasedsignalcorruptionduetoradiochannelfrequencyselec-tivity. In OFDM systems, data is transmitted in parallel by modulating a number of closely-spacedorthogonal subcarriers, thereby converting a frequency-selective channel into multiple flat fadingsubchannels. Moreover, intersymbol interference (ISI) can be eliminated by inserting a guard in-terval between two consecutive OFDM symbols. With these attractive properties OFDM has beenadopted by wireless standards such as digital audio broadcasting (DAB), digital video broadcasting(DVB), wireless local area network (WLAN), and wireless metropolitan access network (WMAN)[1–3].There are several key challenging problems associated with OFDM systems: performance op-timization, time and frequency synchronization, channel estimation, and peak-to-average powerratio (PAPR) reduction [4]. In this thesis, we will focus on the channel estimation problem andstudy channel estimation techniques for an OFDM system operating in dispersive time-varyingchannels.1.2 Literature ReviewChannelestimationisachallengingprobleminwirelesssystemsbecausemobileradiochannelsarehighly dynamic. To avoid channel estimation, one can adopt a differential modulation techniqueinstead of coherent modulation. However, such a system typically results in lower data rates andcan incur a 3-4 dB penalty in signal-to-noise ratio (SNR) [5]. For OFDM systems which aimto provide high data rate and spectral efficiency, coherent modulation is more effective; hence,channel estimation is often required as an integral part of the receiver design.Channel estimation for OFDM systems in slow fading channels has been widely studied [6–8]. However, those channel estimation techniques were developed under the assumption that thechannel is constant over one OFDM symbol, an assumption that may not hold in some mobileapplications. Channel estimation for OFDM systems in fast fading channels has been studied in21.2. Literature Reviewthe following work. In [9], Li et al. presented a minimum mean square error (MMSE) estimator byexploiting both time-domain and frequency-domain correlations of the frequency response of rapiddispersive fading channels. In a related work, Moon and Choi introduced a channel estimationalgorithm by adopting a Gaussian interpolation filter or a cubic spline interpolation filter [10].However, algorithms in both [9] and [10] require knowledge of channel statistics, which may notbe available. To make the estimation algorithm independent of the channel statistics, Li discussedin[11]arobustimplementationoftheMMSEpilot-symbol-aidedestimator,whichdoesnotdependon channel statistics. In [12], Zhao and Huang proposed a method employing low-pass filtering ina transform domain to estimate the channel transfer function for the pilot subcarriers. Then a high-resolution interpolation is adopted to obtain the channel transfer function for non-pilot subcarriers.In [13], Chang and Su discussed a pilot-aided channel estimation method for OFDM systemsoperating in Rayleigh fading channels. The channel responses at pilot subcarriers are estimatedusing a least square (LS) estimator and then interpolated to non-pilot subcarriers using a 2-Dregression surface function. This estimator also does not require channel statistics. Recently, MerliandVitettaproposedamaximumlikelihood(ML)basedalgorithmforjointcarrierfrequencyoffsetand channel impulse response (CIR) estimation for OFDM systems in [14]. The authors adopted asecond-order Taylor series to simplify the estimation problem and derived an approximate closed-form solution to the estimation problem. However, it was derived under the assumption that thechannel does not change over one OFDM symbol. In [14], the frequency offset is assumed to beconstant for all multipaths implying that the Doppler frequency shift is neglected for individualmultipath.In this thesis, we study the channel estimation problem for OFDM systems in dispersive time-varying fading channels. Two ML-based channel estimation algorithms which do not require chan-nel statistics are proposed. Unlike the existing solutions in the literature, our channel estimationalgorithms are based on a unique channel model which is particularly appropriate for OFDM up-links in a macrocellular system. In this channel model, the channel state can be determined byestimating a number of unknown channel parameters which do not change over several OFDMsymbols. This property of our developed channel model allows us to estimate the channel pa-31.3. Thesis Outline and Contributionsrameters once and then employ them to determine the channel state for the next several OFDMsymbols.1.3 Thesis Outline and ContributionsThis thesis consists of five chapters. Chapter 1 presents background knowledge of wireless com-munication developments and technologies. In modern wireless communication, a high end-userdata rate is one main objective of system design and therefore wider transmission bandwidth isdesired. However, an increased transmission bandwidth will increase the frequency selectivity ofa radio channel and thus cause severe corruption to the transmitted signal. This problem can besolved by using multicarrier transmission techniques such as OFDM.Chapter 2 provides detailed technical background for the entire thesis. Firstly, fading character-istics of wireless propagation environments are presented and classified into large-scale fading andsmall-scale fading from the view of propagation terrain. Secondly, the basic concept of OFDMand its implementation are explained. Finally, we list the advantages and drawbacks of OFDMcompared with a conventional single-carrier transmission.In Chapter 3, a channel estimation problem is formulated based on a discrete-time CIR model.This channel model is particularly appropriate for OFDM uplinks in a macrocellular system. Thechannel model developed here allows the CIR to vary within one OFDM symbol. To facilitate thechannel estimation problem we adopt a truncated Taylor series expansion to approximate the inter-carrier interference (ICI) caused by Doppler frequency shifts. We find that a second-order Taylorseries expansion is sufficient for our estimation problem. Then an iterative ML-based algorithm isproposed to estimate the discrete-time channel parameters. Our proposed channel estimation al-gorithm is particularly useful for an OFDM system which has already compensated the frequencyoffset due to local oscillator mismatch. The mean square error (MSE) performance of the pro-posed algorithm in large SNR regions is analyzed using a small perturbation technique, and thendemonstrated by simulated results.In Chapter 4, the convergence performance of the proposed algorithm is analyzed. Basedon the analysis, an improved fast converging iterative channel estimation algorithm is presented41.3. Thesis Outline and Contributionsusing a successive overrelaxation (SOR) method to accelerate the proposed algorithm in Chapter3. Simulated results are also presented to demonstrate the MSE and convergence performance forthe improved algorithm.Chapter 5 summarizes the whole thesis and lists our contributions in this work. In addition,some future work related to our current research is suggested.5Chapter 2Wireless Channels and OFDM TechniqueIn this chapter, we will present some background knowledge concerning wireless channels andthe OFDM technique. We first address the fading characteristics of wireless propagation envi-ronments, then introduce the basic concept and fast Fourier transform (FFT) implementation ofOFDM. Finally, advantages and drawbacks of the OFDM technique are compared with conven-tional single-carrier modulation.2.1 Wireless ChannelsIn a wireless communication system, the transmitted signal typically undergoes attenuation anddistortion over the transmission path. The overall effect on the transmitted signal caused by thetransmission path is one main source of system performance degradation in any wireless system.To address and compensate for the attenuation and distortion caused by the transmission path,researchers have studied wireless channels extensively and proposed different channel models [5],[15–18]. These effects, which are mainly due to path loss, shadowing, scattering and reflectingeffects caused by unpredictable objects between the transmitter and receiver, can primarily becategorized into large-scale fading and small-scale fading (see Fig. 2.1). In this section, we willbriefly describe both types of fading.2.1.1 Large-Scale FadingLarge-scale fading, which is caused by path loss of the signal, can be characterized as a functionof transmitted distance and shadowing effects of large obstacles such as buildings and hills. Thisphenomenon happens when the mobile moves over a large distance (of the order of the cell size),62.1. Wireless ChannelsSignal Level (dB)Antenna DisplacementSmall-scale fadingPass lossLarge-scale fadingFigure 2.1: Signal power fluctuation versus range in wireless channels.and is typically frequency independent [18]. In an ideal LOS channel with no obstructions betweenthe transmitter and the receiver, the received signal power Pr is given byPr = Ptparenleftbigg λ4pidparenrightbigg2GtGr (2.1)where Pt is the transmitted signal power, λ is the wavelength, d is the distance from the transmitterto the receiver, and Gt, Gr are respectively the power gains of the transmit and receive antennas.The power attenuation PL = Pr/Pt is also referred to as free space path loss and it is obviousfrom (2.1) that the received signal power Pr is inversely proportional to the square of the distanced between the transmit and receive antennas. However, in real transmission environments, thereceived signal power Pr does not obey this free space path loss model and it varies randomly dueto the terrain. Usually a ray tracing method can be used to trace the signal propagation througha wireless channel. Unfortunately, the free space model and ray tracing method cannot modelcomplex propagation environments accurately. Based on empirical measurements, some empiricalmodels for path loss in typical wireless environments have been developed to predict the averagereceived signal power as the transmitted distance d varies, i.e., the Okumura model, the Hata72.1. Wireless Channelsmodel, the European Cooperative for Science and Technology (COST) model, and the piecewiselinear (multi-slope) model [17], [19]. In addition to path loss, the transmitted signal is also subjectto shadowing, which is caused by changes in reflecting surfaces and scattering objects along thetransmission path. The shadowing causes random attenuation to the transmitted signal. Typically,the log-normal shadowing model, which has been confirmed empirically to accurately model thevariation in received power, is used to characterize this random attenuation.2.1.2 Small-Scale FadingSmall-scale fading is due to the constructive and destructive addition of different multipath compo-nents introduced by the channel between the transmitter and receiver. Therefore, it is also referredto as multipath fading. This phenomenon usually occurs over a distance of several signal wave-lengths and is frequency dependent. Since the transmitted signal over a multipath fading channelexperiences randomness we must characterize multipath fading channels statistically. Frequencyselectivity and the time-varying nature, which depend on the relative relation between parametersof the transmitted signal (i.e., signal bandwidth and symbol duration) and parameters of multi-path fading channels (i.e., delay spread and Doppler spread), are two important characteristics ofmultipath fading channels. In the following, we will briefly describe these two characteristics ofmultipath fading channels.Depending on the relative relation between transmitted signal bandwidth and delay spread(or equivalently coherence bandwidth BC), multipath fading channels can be categorized intofrequency-nonselective (flat) fading channels and frequency-selective fading channels. The pa-rameter coherence bandwidth is the reciprocal of the delay spread which is defined as the span ofthe delays of duplicates of the transmitted signal arriving at the receiver via different paths. Whenthe transmitted signal bandwidth is small compared with the coherence bandwidth BC, the channelis called frequency-nonselective or flat fading channel. For a flat fading channel, the spectral com-ponents of the transmitted signal are affected in a similar manner so that the multipath componentsare not resolvable. Otherwise, if the transmitted signal bandwidth is large compared with the co-herence bandwidth BC, the channel is said to be frequency-selective. For a frequency-selective fad-82.2. OFDM Techniqueing channel, the spectral components of the transmitted signal are affected by different amplitudegains and phase shifts. In a frequency-selective fading channel, different multipath componentswith delay differences significantly exceeding the inverse of the transmitted signal bandwidth areresolvable. Typically, such a frequency-selective fading channel can be modeled as a tapped delayline filter with time-variant tap coefficients.In a similar way, the multipath fading channel can be categorized as slow fading or fast fadingbased on the relative relation between symbol duration and Doppler spread (or equivalently coher-ence time TC). The parameter coherence time, which measures the period of time over which thechannel effect on the transmitted signal does not change, is defined as the reciprocal of the Dopplerspread. The fading channel is said to be slow fading if the symbol duration is small compared withthe channel coherence time TC; otherwise, it is considered to be fast fading. In a slow fading chan-nel, the transmitted signal is affected by the same amplitude gain and phase shift over at least onesymbol duration, while the amplitude gain and phase shift vary within one symbol duration in afast fading channel.According to the discussion above, we have four types of multipath channel, i.e., slow flat fad-ing channel, slow frequency-selective fading channel, fast flat fading channel, and fast frequency-selectivefadingchannel. Inthisthesis, wefocusonthechannelstateestimationofafastfrequency-selective fading channel for an OFDM system. The channel is characterized using the CIR as [17]h(τ,t) =L(t)∑lγl(t)δ(τ−τl(t)) (2.2)where L(t) is the number of resolvable multipaths, τl(t) is the delay of the lth multipath, and γl(t)is the corresponding complex amplitude. We will look into the details of this channel model in thenext chapter.2.2 OFDM TechniqueOFDM has become a promising multicarrier modulation technique and has received considerableinterest in the past decade. In OFDM systems, data is transmitted in parallel by modulating a92.2. OFDM Techniquenumberof closely-spaced orthogonalsubcarriers, thereby convertingafrequency-selectivechannelintomultipleflatfadingsubchannels[4], [20]. Moreover, ISIcanbeeliminatedbyinsertingaguardinterval between two consecutive OFDM symbols. With these attractive properties OFDM hasbeenadoptedbywirelessstandardssuchasDAB,DVB,WLAN,andWMAN[1–3]. Inthissection,we will present the basic concept and FFT implementation of OFDM and list the advantages anddrawbacks of the OFDM technique compared with conventional single-carrier modulation.2.2.1 Basic ConceptOFDM is a special case of multicarrier transmission which uses parallel data transmission andfrequency division multiplexing. The earliest development of this concept can be traced back tothe 1950s [21] and the idea was published in the mid-1960s [22], [23]. In an OFDM system, theavailable bandwidth is divided into a collection of narrow sub-bands, and the data is transmitted inparallel by modulating these closely-spaced orthogonal subcarriers. Let X[k] denote the complexsymbols to be transmitted by an OFDM system, such that the OFDM (modulated) signal can beexpressed ass(t) =N−1∑k=0X[k]ej2pi fkt, 0≤t ≤Ts (2.3)where fk = f0+k∆f, j2 =−1, and N denotes the number of subcarriers in the system. ParametersTs and ∆f are called symbol duration and subcarrier spacing of the OFDM system, respectively.These two parameters must satisfy the orthogonality condition (Ts∆f = 1) to guarantee that theOFDM signal can be demodulated properly by the receiver.To demonstrate this orthogonal property, we first letϕk(t) =ej2pi fkt, 0≤t ≤Ts0, otherwise,k = 0,1,···,N−1 (2.4)and rewrite (2.3) ass(t) =N−1∑k=0X[k]ϕk(t). (2.5)102.2. OFDM TechniqueUsing the definition of ϕk(t), one can show1Tsintegraldisplay Ts0ϕk(t)ϕ∗l (t)dt = 1Tsintegraldisplay Ts0ej2pi(fk−fl)tdt= 1Tsintegraldisplay Ts0ej2pi(k−l)∆ftdt= δ[k−l] (2.6)where δ[k−l] is the delta function defined asδ[n] =1, n = 00, otherwise.(2.7)From (2.6) one can see that {ϕk(t)}N−1k=0 is a set of orthogonal functions. Using this orthogonalproperty, the received OFDM signal r(t) (in an ideal case r(t) = s(t)) can be demodulated at thereceiver by1Tsintegraldisplay Ts0r(t)e−j2pi fktdt = 1Tsintegraldisplay Ts0s(t)e−j2pi fktdt= 1Tsintegraldisplay Ts0parenleftBiggN−1∑l=0X[l]ϕl(t)parenrightBiggϕ∗k(t)dt=N−1∑l=0X[l]δ[l−k]= X[k]. (2.8)According to the discussion above, we provide an illustrative description of a basic OFDMmodulator in Fig. 2.2 and the basic principle of OFDM demodulation in Fig. 2.3.2.2.2 OFDM Implementation Using FFTThe relationship between OFDM and the discrete Fourier transform (DFT) was first addressed in[24]. It has been shown that the DFT can be applied to an OFDM system as part of the modulationand demodulation processes. In practice, the DFT can be implemented with the computationallyefficient FFT. Here we will briefly discuss the FFT implementation of OFDM.112.2. OFDM Techniqueg86g11g87g12g54g18g51g87g73g77g72 g19g21pig87g73g77g72 g20g21pig87g73g77 g49g72 g20g21 −pig166g59g62g19g64g15g3g59g62g20g64g15g3 g59g62g49g16g20g64g59g62g19g64g59g62g20g64g59g62g49g16g20g64Figure 2.2: Basic principle of OFDM modulation.g85g11g87g12g179 g86g55 g71g87g19 g22g179 g86g55 g71g87g19 g22g87g73g77g72 g19g21pi−g87g73g77g72 g20g21pi−g179 g86g55 g71g87g19 g22g87g73g77 g49g72 g20g21 −− pig59g62g19g64g59g62g20g64g59g62g49g16g20g64Figure 2.3: Basic principle of OFDM demodulation.122.2. OFDM Techniqueg54g18g51 g49g16g83g82g76g81g87g44g41g41g55 g51g18g54 g39g18g36g3g38g82g81g89g72g85g87g72g85g59g62g19g64g15g3g59g62g20g64g15g3 g59g62g49g16g20g64g59g62g19g64g59g62g20g64g59g62g49g16g20g64g86g62g19g64g86g62g20g64g86g62g49g16g20g64g86g11g87g12g171g171g17g17g17g171g171g17g17g17Figure 2.4: OFDM modulation by means of IFFT processing.From (2.3), we can sample s(t) at an interval of Tsa = Ts/N. Then the sampled OFDM signalbecomess[n] triangle= s(t)|t=nTsa=N−1∑k=0X[k]e j2pi fknTsN , n = 0,1,...,N−1. (2.9)Without loss of generality, we can set f0 = 0 so that fkTs = k ands[n] =N−1∑k=0X[k]e j2piknN= IDFT{X[k]}, n = 0,1,...,N−1 (2.10)where IDFT denotes the inverse discrete Fourier transform. Therefore, as illustrated in Fig. 2.4,OFDMmodulationcanbeimplementedbymeansofIDFT(orthecomputationallyefficientinversefast Fourier transform (IFFT)) processing followed by digital-to-analog conversion. Similarly, thedemodulation at the receiver can be carried out using efficient FFT processing, as illustrated inFig. 2.5.2.2.3 Advantages and Drawbacks of OFDMAs a special case of multicarrier transmission scheme, OFDM has the following key advantagesover conventional single-carrier transmission [20]:132.2. OFDM Techniqueg54g18g51 g49g16g83g82g76g81g87g41g41g55 g51g18g54g85g62g81g64g85g62g19g64g85g62g20g64g85g62g49g16g20g64g59g62g19g64g59g62g20g64g59g62g49g16g20g64g171g171g17g17g17g171g171g17g17g17g59g62g19g64g15g3g59g62g20g64g15g3 g59g62g49g16g20g64g85g11g87g12g86g68g81g55Figure 2.5: OFDM demodulation by means of FFT processing.• OFDMisanefficientwaytodealwithmultipathfading; foragivenchanneldelayspread, theimplementation complexity is much lower than that of a conventional single-carrier systemwith an equalizer.• In relatively slow time-varying channels, it is possible to significantly enhance the capacityby adapting the data rate per subcarrier according to the SNR of that particular subcarrier.• OFDM is robust against narrowband interference, because such interference only affects asmall percentage of the subcarriers.• OFDM makes single-frequency networks possible, which is especially attractive for broad-casting applications.However, OFDM also has some drawbacks compared with conventional single-carrier modulation[20]:• OFDM is more sensitive to frequency offset and phase noise. The frequency offset and phasenoise will destroy the orthogonality among subcarriers and hence introduces ICI.• OFDM signals have relatively large PAPR, which tends to reduce the power efficiency of theRF amplifier.• OFDM needs an adaptive or coded scheme to overcome spectral nulls in the channel.14Chapter 3ML-based Channel Estimation for OFDMSystems in Dispersive Time-VaryingChannelsIn this chapter, by taking into account the ICI component, we propose an ML-based channel es-timation algorithm (which does not require channel statistics) for OFDM systems in dispersivetime-varying fading channels. A channel estimation problem is formulated based on a discrete-time CIR model. This channel model is particularly appropriate for OFDM uplinks in a macro-cellular system. The channel model developed here allows the CIR to vary within one OFDMsymbol. To facilitate the channel estimation problem we adopt a truncated Taylor series expansionto approximate the ICI caused by Doppler frequency shifts. We find that a second-order Taylorseries expansion is sufficient for our estimation problem. Then an iterative ML-based algorithm isproposed to estimate the discrete-time channel parameters. Using a small perturbation technique,we analyze the performance of the proposed algorithm in large SNR regions. Our computer sim-ulations demonstrate that the proposed algorithm can estimate the desired parameters accurately,and the simulated performances agree with our theoretical analysis.3.1 System ModelWe consider an OFDM system with N subcarriers. For each OFDM symbol, we denote the trans-mitted symbols as X[0],X[1],...,X[N−1]. After the IFFT, the time-domain OFDM signal can be153.2. Formulation of Channel Estimation Problemexpressed as [25]s[n] = 1NN−1∑k=0X[k]e j2piknN , n = 0,1,...,N−1 (3.1)where j2 = −1. Each OFDM symbol is extended with a cyclic prefix and transmitted after appro-priate pulse shaping.Recalling (2.2), we can describe a multipath CIR byh(τ,t) =L∑l=1γl (t)δ (τ−τl) (3.2)where γl(t) is the complex amplitude of the lth multipath. The number of resolvable multipathsL(t) and the delay of the lth multipath τl(t) are assumed to be L and τl, respectively. According to[6], [9], the CIR can be well approximated by a sampled discrete-time CIR h[m,n] triangle=h(mTsa,nTsa),where Tsa is the sampling interval defined as Tsa = Ts/N and Ts is the OFDM symbol duration. Inthis thesis, we assume a time-varying multipath channel and express the discrete-time CIR as [12],[26]h[m,n] =L∑l=1hle j2pi flnN δ[m−nl] (3.3)where hl is the complex amplitude for the lth multipath, fl is the corresponding Doppler frequencyshift normalized by 1/Ts, and nl is the delay in samples for the lth multipath. The detailed deriva-tion of this discrete-time CIR model is given in Appendix A. It should be emphasized that thediscrete-time CIR in (3.3) is only valid for OFDM uplink transmission in macrocellular systems.Without loss of generality, we assume n1 ≤ n2 ≤···≤ nL. It is seen from (3.3) that the CIR canvary within one OFDM symbol duration.3.2 Formulation of Channel Estimation ProblemOur goal is to estimate the channel parameters hl, fl, for l = 1,2,...,L, in the discrete-time CIR in(3.3). We assume that the length of the cyclic prefix is longer than the delay spread of the multipathchannel and that the receiver accurately removes the cyclic prefix before implementing the FFT.163.2. Formulation of Channel Estimation ProblemFor the channel model given in (3.3), one can express the output of the FFT asY[k] = 1NL∑l=1N−1∑kprime=0N−1∑n=0X[kprime]e j2pikprime(n−nl)N hlej2pi flnN e−j2piknN +w[k]=L∑l=1N−1∑kprime=0e− j2pikprimenlN X[kprime]hl 1−ej2pi(kprime−k+fl)Nparenleftbigg1−e j2pi(kprime−k+fl)Nparenrightbigg +w[k]=L∑l=1N−1∑kprime=0e− j2pikprimenlN X[kprime]hlR(kprime,k, fl)+w[k]= αkX[k]+βk +w[k], k = 0,1,...,N−1 (3.4a)whereαk =L∑l=1e− j2piknlN hlR(k,k, fl) (3.4b)andβk =L∑l=1N−1∑kprimenegationslash=kkprime=0e− j2pikprimenlN X[kprime]hlR(kprime,k, fl). (3.4c)In (3.4a) w[k] is an additive complex Gaussian noise component with zero mean and variance σ2,i.e.,CNparenleftbig0,σ2parenrightbig, andR(kprime,k, fl) = 1NN−1∑n=0e j2pin(kprime−k+fl)N= 1−ej2pi(kprime−k+fl)Nparenleftbigg1−e j2pi(kprime−k+fl)Nparenrightbigg (3.5)represents the interference of X[kprime] on the kth subcarrier caused by a normalized Doppler fre-quency shift fl. In (3.4b) and (3.4c), αk and βk denote the multiplicative distortion at the desiredsubchannel and the additive ICI term, respectively. In the absence of Doppler frequency shifts(fl = 0,l = 1,2,...,L), we have R(kprime,k,0) = δ[kprime−k] resulting in the conventional model withoutICI among subcarriers. In the special case when fl = ε, for l = 1,2,...,L, our estimation problem173.2. Formulation of Channel Estimation Problemis mathematically equivalent to the joint channel and frequency offset estimation problem that hasbeen considered in [14], [27] and [28].In practical mobile communication scenarios, fl is typically less than 0.1. Consider an OFDMsystem, for example, with N = 256 subcarriers and a subcarrier spacing of 7.81 kHz. A mobileterminal moving at a speed of 84.4 km/h will result in a normalized Doppler frequency fl = 0.05with a carrier frequency fc = 5 GHz. Exploiting the fact that fl has small values, one can considera Taylor series expansion for R(kprime,k, fl) asR(kprime,k, fl) =1+ jpiN−1N fl −pi22N2−3N+13N2 f2l − jpi3(N−1)23N2 f3l +···, kprime = k− j2pi(1−ω)N fl +2pi2(2−N)ω+N(1−ω)2N2 f2l + j4pi3N2(1−ω)2+3ω(1+ω−Nω+N)3N3(1−ω)3 f3l +···, kprime negationslash= k(3.6a)whereω = expparenleftbigg j2pi(kprime−k)Nparenrightbigg. (3.6b)Applying (3.6a) to (3.4a) and collecting the terms with the same power of fl, one obtainsY[k] =L∑l=1hlbraceleftbigAl[k]+Bl[k]fl +Cl[k]f2l +Dl[k]f3l +···bracerightbig+w[k] (3.7)for k = 0,1,...,N−1, or, in vector form,vectorY = L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +vectorDl f3l +···parenrightBig+vectorw. (3.8)The first three leading coefficients Al[k], Bl[k],Cl[k] in (3.7) can be shown to beAl[k] = X[k]e−j2piknlN (3.9a)andBl[k] = jpiN−1N X[k]e−j2piknlN − j2pi(1−ω)NN−1∑kprimenegationslash=kkprime=0X[kprime]e−j2pikprimenlN (3.9b)183.2. Formulation of Channel Estimation ProblemandCl[k] =−pi22N2−3N +13N2 X[k]e−j2piknlN +2pi2(2−N)ω +N(1−ω)2N2N−1∑kprimenegationslash=kkprime=0X[kprime]e−j2pikprimenlN . (3.9c)The channel estimation technique in this thesis will be based on (3.8). In (3.8), the coefficient vec-tors vectorAl, vectorBl, vectorCl, ...can be derived given the transmitted training sequence X[k], k = 0,1,...,N−1.We will assume the receiver has knowledge of the training sequence X[k] and performs parameterestimation of the hl’s and fl’s based on (3.8).We note that the Taylor expansion in (3.6a) is an approximation for R(kprime,k, fl) and the accuracyof the approximation depends on the order of the Taylor expansion. On the other hand, the orderof the Taylor expansion also determines the computational complexity of the channel estimationalgorithm. In this work we employ a second-order Taylor expansion for |fl|< 0.1.In order to assess the accuracy of the second-order Taylor series expansion, we compare theMSE of the received OFDM symbol using several approximation orders. Here the MSE is definedas the mean squared difference between the exact value of the signal at the receiver (Y[k]) and itsapproximation (hatwideY[k])MSE triangle= EbraceleftbiggvextendsinglevextendsinglevextendsingleY[k]−hatwideY[k]vextendsinglevextendsinglevextendsingle2bracerightbigg (3.10)whereE{x}istheensembleaverageofarandomsequencex. Y[k]andhatwideY[k]arecalculatedaccordingto (3.4a) and (3.7) assuming w[k] = 0, respectively. From Fig. 3.1 one observes that the MSEincreases with the normalized Doppler frequency fl as expected. Fig. 3.1 also shows that thefirst-order approximation may be inadequate. However, the difference between the second-orderapproximation and third-order approximation is negligible. Furthermore, with a typical value offl, e.g. fl = 0.05, the MSE of the second-order approximation is about −50 dB and, thus, can beignored in our estimation problem. Therefore, we can conclude that a second-order approximation193.2. Formulation of Channel Estimation Problem0.02 0.04 0.06 0.08 0.1 0.12 0.14−100−90−80−70−60−50−40−30−20−100fl (Normalized Doppler Frequency)MSE (dB)  p=1p=2p=3Figure 3.1: An MSE comparison for three different approximation orders.is sufficient for our problem and modify (3.8) to bevectorY ≈ L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBig+vectorw. (3.11)The likelihood function of the received symbol is then given byfparenleftBigvectorYvextendsinglevextendsinglevextendsingle{hl, fl}Ll=1parenrightBig= 1piNσN exp−bracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBiggH×bracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBigg/σ2bracerightBigg (3.12)203.2. Formulation of Channel Estimation Problemwhere (·)H denotes the Hermitian transpose of the argument matrix. One can therefore express theML estimation for the parameters hl’s and fl’s asbraceleftbigˆhl, ˆflbracerightbigLl=1 = arg min{hl,fl}Ll=1vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevectorY −L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle2(3.13)where ˆhl’s and ˆfl’s are the solutions to∂∂hlprime ln fparenleftBigvectorYvextendsinglevextendsinglevextendsingle{hl, fl}Ll=1parenrightBig= 0 (3.14)and∂∂ flprime ln fparenleftBigvectorYvextendsinglevextendsinglevextendsingle{hl, fl}Ll=1parenrightBig= 0, (3.15)respectively. From (3.12), we can getln fparenleftBigvectorYvextendsinglevextendsinglevextendsingle{hl, fl}Ll=1parenrightBig=ln 1piNσN − 1σ2bracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBiggH×bracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBigg (3.16)and show that (3.14) and (3.15) are equivalent to the following equationsbracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBiggHparenleftBigvectorAlprime +vectorBlprime flprime +vectorClprime f2lprimeparenrightBig= 0 (3.17)ℜbracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2lparenrightBigbracketrightBiggHhlprimeparenleftBigvectorBlprime +2vectorClprime flprimeparenrightBig= 0, (3.18)respectively. Hereℜ{·} denotes the real part of the complex argument. It is worth mentioning thatwe need to pay attention to the properties of complex derivatives [29] when obtaining (3.17) and(3.18). Furthermore, we also need to note that (3.17) and (3.18) are non-linear in the hl’s and fl’s.A technique to solve these two equations will be presented in the next section.213.3. An Iterative Channel Estimation AlgorithmTable 3.1: Algorithm 1Initialization: Initialize h(0)l and f(0)l , for l = 1,2,...,L. Set the iteration counter i = 0.Iterations:Step 1: For each l = 1,2,...,L, calculate˜h(i+1)l = argminhlLparenleftBighl, ˜f(i)lparenrightBig˜f(i+1)l = argminfl LparenleftBig˜h(i+1)l , flparenrightBigStep 2: If maxlparenleftBiggvextendsinglevextendsinglevextendsingle˜h(i+1)l −˜h(i)lvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingle˜h(i)lvextendsinglevextendsinglevextendsingleparenrightBigg×100% > δh or maxlparenleftBiggvextendsinglevextendsinglevextendsingle ˜f(i+1)l − ˜f(i)lvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingle ˜f(i)lvextendsinglevextendsinglevextendsingleparenrightBigg×100% > δf, leti = i+1 and go to Step 1, otherwise output ˜h(i+1)l ’s and ˜f(i+1)l ’s.3.3 An Iterative Channel Estimation AlgorithmIn this section, we propose an iterative approach to estimate the ˆhl’s and ˆfl’s based on (3.13).The most straightforward approach is the coordinate-ascent algorithm [30] as shown in Table 3.1,where the cost function L(hl, fl) is defined asL(hl, fl) =vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevectorY −l−1∑j=1h(i+1)jparenleftbiggvectorAj +vectorBj f(i+1)j +vectorCjparenleftBigf(i+1)j parenrightBig2parenrightbigg−L∑j=l+1h(i)jparenleftbiggvectorAj +vectorBj f(i)j +vectorCjparenleftBigf(i)j parenrightBig2parenrightbigg−hlparenleftBigvectorAl +vectorBl fl +Cl fl2parenrightBigvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle2.(3.19)Note that in Algorithm 1, the first equation in Step 1 is a linear least-square problem and thesecond equation in Step 1 requires finding the roots for a third-order polynomial. Both equationscanbesolvedusingwell-knownnumericalmethods. Asdiscussedin[30], thecoordinate-ascental-gorithm is a special case of the space-alternating generalized expectation-maximization algorithmand has the attractive property of increasing the likelihood value at each iteration.223.4. Performance Analysis of the Channel Estimation Algorithm3.4 Performance Analysis of the Channel EstimationAlgorithmIn this section, we analyze the MSE performance of the proposed algorithm in large SNR regionsassuming the fl’s are small.From the discussion in Section 3.2 we can see that the ML-based estimation algorithm aimsto minimize the cost function in (3.13). Using a small perturbation technique [31], one can obtainfrom (3.13) a linear equation for the perturbation of the estimates asR11 R12 ··· R1LR21 R22 ··· R2L... ... ... ...RL1 RL2 ··· RLLbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightR∆vectorθ1∆vectorθ2...∆vectorθLbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright∆vectorθ=vectorn1vectorn2...vectornLbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprightvectorn(3.20a)whereRij =ℜbraceleftBigvectorAHj vectorAibracerightBigℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigh∗jvectorBHj vectorAibracerightBig−ℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigvectorAHj vectorAibracerightBig−ℑbraceleftBigh∗jvectorBHj vectorAibracerightBigℜbraceleftBighivectorAHj vectorBibracerightBigℑbraceleftBighivectorAHj vectorBibracerightBigℜ{hih∗jvectorBHj vectorBi}(3.20b)and∆vectorθl =ℜ{∆hl}ℑ{∆hl}∆flvectornl =ℜbraceleftBigvectorwHvectorAlbracerightBig−ℑbraceleftBigvectorwHvectorAlbracerightBigℜbraceleftBighlvectorwHvectorBlbracerightBig(3.20c)where ℜ{·} and ℑ{·} respectively denote the real part and imaginary part of the argument, {·}∗denotes the complex conjugate of the argument, ∆hl = ˆhl −hl and ∆fl1 = ˆfl − fl are the estimateperturbations caused by Gaussian noisevectornl. A detailed derivation of (3.20) is given in Appendix B.When the Gaussian noise is absent, we have ˆhl = hl and ˆfl = fl.Equation (3.20) indicates that the perturbations ∆vectorθl subject to the Gaussian noise components1Here ∆fl does not denote the subcarrier spacing.233.4. Performance Analysis of the Channel Estimation Algorithmvectornl, l = 1,2,...,L, are also Gaussian, and one can express ∆vectorθl as∆vectorθ1∆vectorθ2...∆vectorθL=R11 R12 ··· R1LR21 R22 ··· R2L... ... ... ...RL1 RL2 ··· RLL−1vectorn1vectorn2...vectornL(3.21)or in a more compact form∆vectorθ = R−1vectorn. (3.22)To determine the variance of ∆vectorθl subject to Gaussian noise componentsvectornl, we havevectornivectornHj =ℜbraceleftBigvectorwHvectorAibracerightBig−ℑbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBighivectorwHvectorBibracerightBigℜbraceleftBigvectorwHvectorAjbracerightBig−ℑbraceleftBigvectorwHvectorAjbracerightBigℜbraceleftBighjvectorwHvectorBjbracerightBigH=ℜbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBigvectorwHvectorAjbracerightBig−ℜbraceleftBigvectorwHvectorAibracerightBigℑbraceleftBigvectorwHvectorAjbracerightBigℜbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBighjvectorwHvectorBjbracerightBig−ℑbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBigvectorwHvectorAjbracerightBigℑbraceleftBigvectorwHvectorAibracerightBigℑbraceleftBigvectorwHvectorAjbracerightBig−ℑbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBighjvectorwHvectorBjbracerightBigℜbraceleftBighivectorwHvectorBibracerightBigℜbraceleftBigvectorwHvectorAjbracerightBig−ℜbraceleftBighivectorwHvectorBibracerightBigℑbraceleftBigvectorwHvectorAjbracerightBigℜbraceleftBighivectorwHvectorBibracerightBigℜbraceleftBighjvectorwHvectorBjbracerightBig(3.23)andEbraceleftbigvectornivectornHj bracerightbig = σ22 EℜbraceleftBigvectorAHj vectorAibracerightBigℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigh∗jvectorBHj vectorAibracerightBig−ℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigvectorAHj vectorAibracerightBig−ℑbraceleftBigh∗jvectorBHj vectorAibracerightBigℜbraceleftBighivectorAHj vectorBibracerightBigℑbraceleftBighivectorAHj vectorBibracerightBigℜbraceleftBighih∗jvectorBHj vectorBibracerightBig= σ22 EbraceleftbigRijbracerightbig= σ22 Rij. (3.24)In obtaining (3.24), we have used the fact that vectorw is a complex Gaussian noise vector whose el-ements have independent real and imaginary parts with zero mean and variance equal to σ2/2.243.4. Performance Analysis of the Channel Estimation AlgorithmTherefore, using (3.20a), (3.22) and (3.24), one can show straightforwardly thatEvectorn1...vectornLparenleftbigvectornH1 ···vectornHLparenrightbig= σ22 R (3.25)andE∆vectorθ1...∆vectorθLparenleftBig∆vectorθH1 ···∆vectorθHLparenrightBig= σ22 R−1. (3.26)Generally speaking, the elements in Rij, i negationslash= j, are non-zero and, thus, the small perturbationsfor different taps, ∆vectorθl, are correlated. When L << N and a pseudo-random sequence X[k] is usedas the training symbols, the values of Rij, i negationslash= j, are generally much smaller than those in Rii and,thus, negligible [25]. For such cases, one can rewrite (3.26) asEbraceleftBig∆vectorθHl ∆vectorθlbracerightBig= σ22 R−1ll , l = 1,2,...,L (3.27)whereRll =ℜbraceleftBigvectorAHl vectorAlbracerightBigℑbraceleftBigvectorAHl vectorAlbracerightBigℜbraceleftBigh∗l vectorBHl vectorAlbracerightBig−ℑbraceleftBigvectorAHl vectorAlbracerightBigℜbraceleftBigvectorAHl vectorAlbracerightBig−ℑbraceleftBigh∗l vectorBHl vectorAlbracerightBigℜbraceleftBighlvectorAHl vectorBlbracerightBigℑbraceleftBighlvectorAHl vectorBlbracerightBigℜ{h∗l hlvectorBHl vectorBl}=vectorAHl vectorAl 0 ℜbraceleftBigh∗l vectorBHl vectorAlbracerightBig0 vectorAHl vectorAl −ℑbraceleftBigh∗l vectorBHl vectorAlbracerightBigℜbraceleftBighlvectorAHl vectorBlbracerightBigℑbraceleftBighlvectorAHl vectorBlbracerightBig|hl|2vectorBHl vectorBl}(3.28)253.4. Performance Analysis of the Channel Estimation Algorithmand R−1ll is shown to beR−1ll = 1|hl|2vectorAHl vectorAlparenleftbiggζ −vextendsinglevextendsinglevextendsinglevectorBHl vectorAlvextendsinglevextendsinglevextendsingle2parenrightbigg×ζ |hl|2−parenleftBigℑbraceleftBighlvectorAHl vectorBlbracerightBigparenrightBig2ℜbraceleftBighlvectorAHl vectorBlbracerightBigℑbraceleftBighlvectorAHl vectorBlbracerightBig−vectorAHl vectorAlℜbraceleftBighlvectorAHl vectorBlbracerightBigℜbraceleftBighlvectorAHl vectorBlbracerightBigℑbraceleftBighlvectorAHl vectorBlbracerightBigζ |hl|2−parenleftBigℜbraceleftBighlvectorAHl vectorBlbracerightBigparenrightBig2−vectorAHl vectorAlℑbraceleftBighlvectorAHl vectorBlbracerightBig−vectorAHl vectorAlℜbraceleftBighlvectorAHl vectorBlbracerightBig−vectorAHl vectorAlℑbraceleftBighlvectorAHl vectorBlbracerightBig parenleftBigvectorAHl vectorAlparenrightBig2(3.29)where ζ is defined as ζ =vectorAHl vectorAlvectorBHl vectorBl for the sake of convenience. Equation (3.27) indicates thatthe perturbations ∆vectorθl for different taps can be treated independently. However, for the same l,the perturbations ∆hl = ℜ{∆hl}+ jℑ{∆hl} and ∆fl are still correlated because the non-diagonalelements in Rll are not negligible.It is of interest to obtain some explicit results for the estimation performance. Motivated by[32], we consider the asymptotic behavior with a white training sequence. Under this condition,we havevectorAHl vectorAl = 1 (3.30a)vectorBHl vectorAl =− jpi(N−1)N (3.30b)vectorBHl vectorBl = 2pi2(N−1)(2N−1)3N2 (3.30c)where the derivations of (3.30a)-(3.30b) are trivial, and a detailed derivation of (3.30c) can befound in Appendix C. According to the definition of ∆vectorθl, we can obtain∆vectorθl∆vectorθHl =ℜ{∆hl}ℑ{∆hl}∆flℜ{∆hl}ℑ{∆hl}∆flH=(ℜ{∆hl})2 ℜ{∆hl}ℑ{∆hl} ℜ{∆hl}∆flℜ{∆hl}ℑ{∆hl} (ℑ{∆hl})2 ℑ{∆hl}∆flℜ{∆hl}∆fl ℑ{∆hl}∆fl ∆f2l(3.31)263.4. Performance Analysis of the Channel Estimation Algorithmand∆h∗l∆hl = |∆hl|2= (ℜ{∆hl})2 +(ℑ{∆hl})2. (3.32)Therefore, we can obtain the expression for the estimation performance asE{∆h∗l∆hl} = EbraceleftBig(ℜ{∆hl})2 +(ℑ{∆hl})2bracerightBig= EbraceleftBig(ℜ{∆hl})2bracerightBig+EbraceleftBig(ℑ{∆hl})2bracerightBig. (3.33)Applying (3.27), (3.29) and (3.30) to (3.33), one can obtainE{∆h∗l∆hl} =σ2parenleftbigg2vectorAHl vectorAlvectorBHl vectorBl −vextendsinglevextendsinglevextendsinglevectorBHl vectorAlvextendsinglevextendsinglevextendsingle2parenrightbigg2vectorAHl vectorAlparenleftbiggvectorAHl vectorAlvectorBHl vectorBl −vextendsinglevextendsinglevextendsinglevectorBHl vectorAlvextendsinglevextendsinglevextendsingle2parenrightbigg= (5N2−6N−1)σ22(N2−1) . (3.34)Similarly, we can obtainEbraceleftbig∆f2l bracerightbig = σ2vectorAHl vectorAl2|hl|2parenleftbiggvectorAHl vectorAlvectorBHl vectorBl −vextendsinglevextendsinglevextendsinglevectorBHl vectorAlvextendsinglevextendsinglevextendsingle2parenrightbigg= 3N2σ22pi2(N2−1)|hl|2. (3.35)When N →∞, we havelimN→∞E{∆h∗l∆hl}= limN→∞ (5N2−6N−1)σ22(N2−1) =5σ22 (3.36)andlimN→∞Ebraceleftbig∆f2l bracerightbig= limN→∞ 3N2σ22pi2(N2−1)|hl|2 =3σ22pi2|hl|2. (3.37)273.5. Numerical Results and DiscussionsIn comparison, the Cramer-Rao bound (CRB) at large SNR for the classical frequency offset esti-mator of Moose is [33], [34]CRBMoose(fl) =parenleftbigg 22piparenrightbigg2 2σ2|hl|2vectorAHl vectorAl =2σ2pi2|hl|2. (3.38)Thus, from (3.37) and (3.38), we can see that the estimation algorithm considered in this paper canachieve 10log10 23/2 = 1.25 dB gain over Moose’s estimator. In Section 3.5, it will be seen thatsimulations performed to test the analytical results confirm the theoretical analysis.3.5 Numerical Results and DiscussionsIn this section, we present some numerical results by considering a discrete baseband OFDMuplink in our simulation. The complex amplitude hl is randomly chosen, and the normalizedDoppler frequency fl is chosen to be less than 0.1. The total number of subcarriers is fixed atN = 256. A binary phase-shift keying (BPSK) signaling scheme is adopted in the simulation.We set the stopping threshold values for the proposed algorithm as δh = 1% and δf = 1%. Asecond-order Taylor series expansion is adopted in the simulation. Here the MSE is adopted as theperformance measure.Fig. 3.2 plots the MSE performance comparison between Algorithm 1 and Moose’s estimatorfor fl in a fading channel with L = 1. Here hl is randomly chosen and fl is set to be fl = 0.05.The MSEs are obtained by averaging over 2000 trials, and the theoretical performance predictionsof ˆfl are obtained using (3.37) and (3.38). From Fig. 3.2, we observe that the simulated MSEperformance of both algorithms and their theoretical predictions are in excellent agreement overa wide range of SNR values. The simulated performance curve for Algorithm 1 exhibits an errorfloor, which is caused by the approximation error when the average SNR value is greater than 25dB. This is the cause of discrepancy between the simulated curve and the theoretical curve for SNRgreater than 25 dB. However, the values of MSE in these regions are small and the error floor isnegligible. Fig. 3.2 also shows that Algorithm 1 outperforms Moose’s estimator by about 1.2 dBover a wide range of SNR values.283.5. Numerical Results and DiscussionsFig. 3.3 plots the average MSE performance of ˆhl in a multipath fading channel with L =3 for the proposed algorithm and Figs. 3.4-3.6 plot MSE performance of ˆfl for the same sys-tem with three different fl values. Here the hl’s are arbitrarily chosen to be vectorhl = [0.2944 +1.6236j,−1.3362−0.6918j,0.7143+0.858j], and the fl’s are set to be vectorfl = [0.04,0.02,0.06], re-spectively. The MSEs are obtained by averaging over 1500 trials, and the theoretical performancepredictions of ˆhl and ˆfl are obtained using (3.36) and (3.37), respectively. From Figs. 3.3-3.6, weobserve that the simulated MSE performance of ˆhl and ˆfl for Algorithm 1 and their theoreticalpredictions have excellent agreement over a wide range of SNR values. Figs. 3.4-3.6 also showthat the MSE performance degrades with an increasing value of fl, owing to increased (determin-istic) approximation errors. From Fig. 3.4, it is seen for fl = 0.02 that the simulated result and thetheoretical result agree over the entire SNR range. Figs. 3.3, 3.5 and 3.6 suggest that the simulatedresults overestimate the theoretical results when the average SNR value is large. This is becausewe adopt a second-order Taylor series expansion to simplify the estimation problem, and the errorfloor is caused by the approximation error. However, the values of MSE in these regions are smalland the error floor is negligible.293.5. Numerical Results and Discussions−10 0 10 20 30 40−80−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Simulation result for Algorithm 1Simulation result for Moose estimatorTheoretical value for Algorithm 1Theoretical value for Moose estimatorFigure 3.2: An MSE performance comparison between Algorithm 1 and Moose’s estimator for ˆflwhen fl = 0.05, δh = 1% and δf = 1%.303.5. Numerical Results and Discussions−10 0 10 20 30 40−60−50−40−30−20−100Average SNR (dB)AverageMSEofˆ hl(dB)  Simulated resultTheoretical valueFigure 3.3: An average MSE performance comparison between the simulated results and the the-oretical values for ˆhl of Algorithm 1 when δh = 1% and δf = 1%.313.5. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Simulated resultTheoretical valueFigure 3.4: An MSE performance comparison between the simulated results and the theoreticalvalues for ˆfl of Algorithm 1 when fl = 0.02, δh = 1% and δf = 1%.323.5. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Simulated resultTheoretical valueFigure 3.5: An MSE performance comparison between the simulated results and the theoreticalvalues for ˆfl of Algorithm 1 when fl = 0.04, δh = 1% and δf = 1%.333.5. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Simulated resultTheoretical valueFigure 3.6: An MSE performance comparison between the simulated results and the theoreticalvalues for ˆfl of Algorithm 1 when fl = 0.06, δh = 1% and δf = 1%.34Chapter 4Convergence Study of ML-based ChannelEstimation AlgorithmsIn Chapter 3, by taking into account the ICI component, we proposed an ML-based channel esti-mation algorithm (which does not require channel statistics) for OFDM systems in dispersive time-varying fading channels. Our proposed channel estimation algorithm is iterative. In this chapter,we will study the convergence performance of the iterative algorithm presented in Chapter 3 ana-lytically. Furthermore, we propose an improved fast converging channel estimation algorithm forOFDM systems in dispersive time-varying fading channels. An SOR method [35] is adopted toaccelerate our proposed channel estimation algorithm. Our computer simulations demonstrate thatthe improved algorithm, which achieves the same MSE performance as the algorithm proposed inChapter 3, has better convergence performance.4.1 Convergence Performance Analysis of the ProposedAlgorithmIn this section we first analyze the convergence rate of Algorithm 1 proposed in Chapter 3. Basedon the convergence rate analysis we propose an improved fast converging algorithm using an SORmethod [35] in the next section. We first rewrite the linear expression in (3.20a) for small pertur-354.1. Convergence Performance Analysis of the Proposed Algorithmbation as a linear Gauss-Seidel iteration [35, eq. (3.15), page 72]D∆vectorθ(i+1)1∆vectorθ(i+1)2...∆vectorθ(i+1)L=−L∆vectorθ(i+1)1∆vectorθ(i+1)2...∆vectorθ(i+1)L−U∆vectorθ(i)1∆vectorθ(i)2...∆vectorθ(i)L+vectorn1vectorn2...vectornL(4.1)or, equivalently,D∆vectorθ(i+1) =−L∆vectorθ(i+1)−U∆vectorθ(i) +vectorn (4.2)where D is a diagonal matrix whose entries are the diagonal elements of R; L and U are the lowerand upper triangular matrices of R, i.e., L+D+U = R. Applying (3.20a) to (4.1), we can obtainD∆vectorθ(i+1)1∆vectorθ(i+1)2...∆vectorθ(i+1)L=−L∆vectorθ(i+1)1∆vectorθ(i+1)2...∆vectorθ(i+1)L−U∆vectorθ(i)1∆vectorθ(i)2...∆vectorθ(i)L+R∆vectorθ1∆vectorθ2...∆vectorθL, (4.3)i.e.,(D+L)∆vectorθ(i+1)1 −∆vectorθ1∆vectorθ(i+1)2 −∆vectorθ2...∆vectorθ(i+1)L −∆vectorθL=−U∆vectorθ(i)1 −∆vectorθ1∆vectorθ(i)2 −∆vectorθ2...∆vectorθ(i)L −∆vectorθL(4.4)or, in a more compact form as(D+L)parenleftBig∆vectorθ(i+1)−∆vectorθparenrightBig=−UparenleftBig∆vectorθ(i)−∆vectorθparenrightBig. (4.5)The convergence rate2 of (4.5) is determined by the largest (in an absolute value sense) eigenvalueof (D+L)−1U [35].The Gauss-Seidel iteration is a simple and efficient algorithm. However, by exploiting the2With this convention, the smaller the largest eigenvalue is, the faster the iterative algorithm will converge.364.1. Convergence Performance Analysis of the Proposed Algorithmstructure of the Fisher information matrix R, one may adopt an overrelaxation philosophy to accel-erate the convergence rate. We will next introduce an SOR method [35] to achieve faster conver-gence.Corresponding to the linear iteration (4.2), an overrelaxed linear iteration can be expressed as[35, eq. (3.22), page 73]D∆vectorθ(i+1) = ηparenleftBig−L∆vectorθ(i+1)−U∆vectorθ(i) +vectornparenrightBig+(1−η)D∆vectorθ(i) (4.6)where η is the relaxation factor. Applying (3.20a) to (4.6), we can obtain(D+ηL)parenleftBig∆vectorθ(i+1)−∆vectorθparenrightBig= [(1−η)D−ηU]parenleftBig∆vectorθ(i)−∆vectorθparenrightBig. (4.7)Similarly to the Gauss-Seidel iteration in (4.5), the convergence rate of the successive overrelaxediteration in (4.7) depends on the largest absolute value of the eigenvalues of (D+ηL)−1[(1−η)D−ηU] [35]. To achieve the fastest convergence rate for (4.7), one is required to minimizethe largest eigenvalue of (D+ηL)−1[(1−η)D−ηU] by choosing the optimal value for η. Tosimplify the analysis, we again assume that Rij = 0 for i negationslash= j. Under this assumption, D, L andU are block diagonal matrices composed of 3×3 matrices Dl, Ll and Ul, l = 1,...,L. Thus, forthe first step, we consider the optimization on the eigenvalues of (Dl +ηLl)−1[(1−η)Dl −ηUl]separately. For the sake of convenience, we denoteal =vectorAHl vectorAl,bl =|hl|2vectorBHl vectorBl,cl =ℜbraceleftBigh∗l vectorBHl vectorAlbracerightBig,dl =ℑbraceleftBigh∗l vectorBHl vectorAlbracerightBig,374.1. Convergence Performance Analysis of the Proposed Algorithmand obtainMη triangle= Dl +ηLl=al 0 00 al 0ηcl −ηdl bl=1 0 00 1 0ηcl/al −ηdl/al 1al 0 00 al 00 0 bl(4.8)Nη triangle= (1−η)Dl −ηUl=(1−η)al 0 −ηcl0 (1−η)al ηdl0 0 (1−η)bl. (4.9)It is straightforward to show thatM−1η Nη =1−η 0 −ηcl/al0 1−η ηdl/al−η(1−η)cl/bl η(1−η)dl/bl (1−η)+η2(c2l +d2l )/albl. (4.10)The eigenvalues of (4.10) are the roots of the characteristic equationλ3−bracketleftbigg3(1−η)+ η2(c2l +d2l )alblbracketrightbiggλ2+(1−η)bracketleftbigg3(1−η)+ η2(c2l +d2l )alblbracketrightbiggλ −(1−η)3 = 0. (4.11)For the sake of simplicity, we denotekl triangle= c2l +d2lalbl =vextendsinglevextendsinglevextendsinglevectorBHl vectorAlvextendsinglevextendsinglevextendsingle2vectorAHl vectorAlvectorBHl vectorBl (4.12)384.1. Convergence Performance Analysis of the Proposed Algorithmand simplify (4.11) toλ3−bracketleftbig3(1−η)+η2klbracketrightbigλ2 +(1−η)bracketleftbig3(1−η)+η2klbracketrightbigλ −(1−η)3 = 0. (4.13)It is shown in Appendix D that the optimal value of η which minimizes the largest eigenvalue (inabsolute value) of M−1η Nη, isη = 2(1−√1−kl)kl , (4.14)and the corresponding largest absolute value for the eigenvalues of M−1η Nη is|λ|=|1−η|=vextendsinglevextendsinglevextendsinglevextendsingle1− 2(1−√1−kl)klvextendsinglevextendsinglevextendsinglevextendsingle. (4.15)Generally, an SOR method requires knowledge about the structure of the Fisher informationmatrix. It is observed in (3.20b) that this matrix (the Fisher information matrix Rll) depends on thechannel realization hl. However, (4.12) and (4.14) indicate that for a given 3×3 Fisher informationmatrix Rll, the optimal η value does not depend on the channel realization hl. Instead, it only relieson the training sequence. In other words, given the training sequence, the receiver can employ ηin (4.14) to achieve a fast convergence rate for the overrelaxed iteration.TypicallyconvergenceacceleratingalgorithmsrequirefullknowledgeoftheFisherinformationmatrix Rll to achieve better convergence rates. This implies that one needs to estimate hl andit renders such algorithms less useful. In contrast, the overrelaxation method proposed in thiswork only requires information about the training sequence and, therefore, is more robust. Thisproperty makes the overrelaxation method an attractive approach to achieve a fast convergence ratein practical receivers.From (4.12) and (4.14), it is observed that the η value depends on kl and may vary for differentl. Substitution of (3.30) into (4.12) yieldskl = pi2(N−1)2/N22pi2(N−1)(2N−1)/3N2= 3(N−1)2(2N−1) (4.16)394.1. Convergence Performance Analysis of the Proposed Algorithmwhich is, in fact, independent of l assuming a white training sequence is employed. By lettingN →∞, we have the limiting value η, from (4.14) and (4.16), asη = limN→∞ 2parenleftbig1−√1−klparenrightbigkl= 43 (4.17)which leads to a convergence rate of |1−η|= 1/3.In comparison, considering the Gauss-Seidel iteration by letting η = 1, one can express (4.10)asM−1η Nη =0 0 00 0 00 0 kl(4.18)and conclude that the largest eigenvalue for the iteration is kl. Thus, the largest eigenvalue for theGauss-Seidel iteration is kl = 3/4, which is larger than that of 1/3 for the SOR method, implying afaster convergence for the later3.In each Gauss-Seidel iteration, a value that maximizes the likelihood function is produced asan output. In contrast, an SOR method adopts a less aggressive strategy by combining the Gauss-Seidel solution and the previous value linearly ash(i+1)l = η˜h(i+1)l +(1−η)h(i)l (4.19a)f(i+1)l = η ˜f(i+1)l +(1−η)f(i)l (4.19b)where η is an overrelaxation parameter, and ˜h(i+1)l and ˜f(i+1)l are the solutions to the Gauss-Seideliteration. When η = 1, the SOR method degenerates to a Gauss-Seidel iteration.3We should comment that the η value in (4.17) is only asymptotically optimal. In Section 4.3, we will demonstratethat this suboptimal solution already shows significant improvement over the Gauss-Seidel iteration.404.2. An Improved Channel Estimation Algorithm4.2 An Improved Channel Estimation AlgorithmThe above discussion is based on the linear approximation (3.20a) near the ML estimation result.Algorithm 1 is, however, a nonlinear estimator. Using the overrelaxation method, we can proposean improved new iterative ML algorithm as shown in Table 4.1. Note that the function L(hl, fl)in Step 1 is the cost function defined in (3.19) and η = 4/3, which is the asymptotically optimalvalue given by (4.17). Our numerical results will show that the improved algorithm can achievegood performance, both in terms of MSE and convergence rate.Table 4.1: Algorithm 2Initialization: Initialize h(0)l and f(0)l , for l = 1,2,...,L. Set the iteration counter i = 0.Iterations:Step 1: For each l = 1,2,...,L, calculate˜h(i+1)l = argminhlLparenleftBighl, f(i)lparenrightBig˜f(i+1)l = argminfl LparenleftBigh(i+1)l , flparenrightBigLet h(i+1)l = η˜h(i+1)l +(1−η)h(i)l and f(i+1)l = η ˜f(i+1)l +(1−η)f(i)l .Step 2: If maxlparenleftBiggvextendsinglevextendsinglevextendsingleh(i+1)l −h(i)lvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsingleh(i)lvextendsinglevextendsinglevextendsingleparenrightBigg×100% > δh or maxlparenleftBiggvextendsinglevextendsinglevextendsinglef(i+1)l −f(i)lvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevextendsinglef(i)lvextendsinglevextendsinglevextendsingleparenrightBigg×100% > δf, leti = i+1 and go to Step 1, otherwise output h(i+1)l ’s and f(i+1)l ’s.4.3 Numerical Results and DiscussionsIn this section, we present some numerical results by considering the same discrete basebandOFDM uplink transmission discussed in Chapter 3 in our simulation. All the simulation param-eters adopted here are the same as those considered in Section 3.5. Here we compare the MSEperformance and convergence performance of Algorithm 1 and Algorithm 2 to demonstrate theperformance of the improved algorithm.Fig.4.1plotstheaverageMSEperformanceof ˆhl forthetwoproposedalgorithmsandFigs.4.2-4.4 plot MSE performance of ˆfl for the same system with three different fl values. The MSEs areobtained by averaging over 2000 trials, and the theoretical performance predictions of ˆhl and ˆfl are414.3. Numerical Results and Discussionsobtained using (3.36) and (3.37), respectively. From Figs. 4.1-4.4, we observe that the simulatedMSE performance of ˆhl and ˆfl for Algorithm 1 and Algorithm 2, along with their theoreticalpredictions, have excellent agreement over a wide range of SNR values. We can notice a slightdifference between the MSE performance of Algorithm 1 and Algorithm 2 when the SNR valuesare greater than 30 dB. However, the MSEs of both algorithms in this SNR region are less than−50 dB and the differences are therefore negligible. We can conclude that the two algorithms havealmost the same performance in terms of MSE.Fig. 4.5 plots the average number of iterations required by the two proposed algorithms toachieve the MSE performance shown in Figs. 4.1-4.4. As expected, Fig. 4.5 shows that the averagenumber of iterations of both algorithms decreases with increasing value of SNR up to 20 dB. Theaverage number of iterations of both algorithms is unchanged when the average SNR value isgreater than 20 dB. It is shown in Fig. 4.5 that Algorithm 2 converges faster than Algorithm 1, andit achieves 63% improvement in the number of iterations in the large SNR region. From Fig. 4.5, itis seen that the asymptotically optimal η value obtained under a high SNR assumption also workswhen the SNR value is low.Figs. 4.6 and 4.7 respectively plot the average MSE performance of ˆhl and ˆfl versus the numberof iterations. For both figures, the average SNR value is fixed at 25 dB, and we observe that theMSE for Algorithm 2 decreases faster than that of Algorithm 1. Our numerical results clearlydemonstrate that Algorithm 2, which uses an SOR technique, can achieve faster convergence thanAlgorithm 1.424.3. Numerical Results and Discussions−10 0 10 20 30 40−60−50−40−30−20−100Average SNR (dB)AverageMSEofˆ hl(dB)  Algorithm 1Algorithm 2Theoretical valueFigure 4.1: An average MSE performance comparison between Algorithm 1 and Algorithm 2,along with the theoretical values for ˆhl, when δh = 1% and δf = 1%.434.3. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Algorithm 1Algorithm 2Theoretical valueFigure 4.2: An MSE performance comparison between Algorithm 1 and Algorithm 2, along withthe theoretical values for ˆfl, when fl = 0.02, δh = 1% and δf = 1%.444.3. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Algorithm 1Algorithm 2Theoretical valueFigure 4.3: An MSE performance comparison between Algorithm 1 and Algorithm 2, along withthe theoretical values for ˆfl, when fl = 0.04, δh = 1% and δf = 1%.454.3. Numerical Results and Discussions−10 0 10 20 30 40−70−60−50−40−30−20−10Average SNR (dB)MSEofˆ fl(dB)  Algorithm 1Algorithm 2Theoretical valueFigure 4.4: An MSE performance comparison between Algorithm 1 and Algorithm 2, along withthe theoretical values for ˆfl, when fl = 0.06, δh = 1% and δf = 1%.464.3. Numerical Results and Discussions0 5 10 15 20 25 30 35 408910111213141516Average SNR (dB)AverageNumberofIterations  Algorithm 1Algorithm 2Figure 4.5: An average number of iterations comparison between Algorithm 1 and Algorithm 2when η = 4/3, δh = 1% and δf = 1%.474.3. Numerical Results and Discussions0 5 10 15 20−40−35−30−25−20−15−10−505Number of IterationsAverageMSEofˆ hl(dB)  Algorithm 1Algorithm 2Figure 4.6: The average MSE performance versus number of iterations for ˆhl of Algorithm 1 andAlgorithm 2 when η = 4/3 and SNR = 25.484.3. Numerical Results and Discussions0 5 10 15 20−55−50−45−40−35−30−25−20Number of IterationsAverageMSEofˆ f l(dB)  Algorithm 1Algorithm 2Figure 4.7: The average MSE performance versus number of iterations for ˆfl of Algorithm 1 andAlgorithm 2 when η = 4/3 and SNR = 25 dB.49Chapter 5ConclusionsIn this chapter, we first summarize the contributions of this work, and then suggest some futurework.5.1 Summary of ContributionsThrough the numerical simulations and mathematical analyses, the performance of our proposedalgorithms have been verified. In the simulation, we do not compare the performance of ourproposed algorithms with any other solutions because of the uniqueness of our channel model.However, we do compare the simulated estimate’s performance with the CRB, which is the bestone can achieve. The contributions of this thesis can be summarized as follows.1. A discrete CIR model, which allows the CIR to vary within one OFDM symbol, has been de-veloped for OFDM uplink transmission in macrocellular systems. The CIR is characterizedby the channel parameters hl and fl, where hl is the complex amplitude for the lth multipathand fl is the corresponding Doppler frequency shift normalized to the duration of an OFDMsymbol. In this case, the channel state can be determined by estimating the unknown channelparameters so that the channel estimation problem can be simplified.2. Taking into account the ICI component, we formulate the channel estimation problem basedon our proposed discrete CIR model. Due to the fact that fl is typically less than 0.1 in prac-tical mobile communication scenarios, the ICI factor R(kprime,k, fl) is represented by a truncatedTaylor series expansion to facilitate the channel estimation problem. We also demonstratethat a second-order Taylor series approximation is sufficient for a typical channel estimationproblem.505.2. Future work3. We propose two iterative ML-based channel estimation algorithms, which do not requirechannel statistics, for OFDM systems in dispersive time-varying fading channels. These twoalgorithms can estimate the desired parameters accurately in order to provide channel stateinformation for the receiver. Our proposed channel estimation algorithms are particularlyuseful for an OFDM system which has already compensated for the frequency offset dueto local oscillator mismatch. The MSE performance of the proposed algorithms in largeSNR regions has been analyzed using a small perturbation technique. Since our algorithmsare iterative, the convergence performance is also analyzed. Based on the analysis, an SORmethod is adopted to accelerate the proposed algorithm. The performance analyses are veri-fied by our simulated results.5.2 Future workIn this work, we have proposed two iterative ML-based channel estimation algorithms and demon-strated their performance in terms of MSE. 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M. Young, Iterative Solution of Large Linear Systems. New York: Academic Press, 1971.[36] G.T.RAN, “SpatialchannelmodelforMultipleInputMultipleOutput(MIMO)simulations,”Technical Report, Dec. 2008.[37] G. Cardano and T. R. Witmer, The Great Art, or: The Rules of Algebra, 1st ed. Cambridge:MIT Press, 1968.55Appendix ADerivation of the Discrete CIR ModelFor a multipath CIR described in (3.2), each resolvable path consists of a number of non-resolvablecomponents so that γl(t) can be expressed asγl(t) =∑iαl,ie−jφl,i(t) (A.1)where αl,i and φl,i(t) are respectively the complex amplitude and the phase shift incurred by theith non-resolvable component of the lth resolvable path. The phase shift φl,i(t) is due to the timedelay and Doppler frequency shift and is given byφl,i(t) = 2pi fcτl −2pi fDl,it (A.2)where fc is the carrier frequency and fDl,i is the Doppler frequency shift at the ith non-resolvablecomponent of the lth resolvable path. In this work, we consider an uplink transmission in a macro-cellular system where the angular spread is small for those non-resolvable components [36]. Con-sequently, we can assume fDl,i = fDl andφl,i(t) = 2pi fcτl −2pi fDlt. (A.3)Applying (A.3) to (A.1), one can obtainγl(t) = ej2pi fDlt∑iαl,ie−j2pi fcτlbracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipuprighthl(A.4)56Appendix A. Derivation of the Discrete CIR Modeland rewrite (3.2) ash(τ,t) =L∑l=1hlej2pi fDltδ(τ−τl). (A.5)Therefore, the discrete-time CIR can be expressed ash[m,n] triangle= h(mTsa,nTsa)=L∑l=1hlej2pi fDlTsnTsaNTsa δ(mTsa−Tsaτl/Tsa)=L∑l=1hle j2pi flnN δ[m−nl] (A.6)where Tsa is the sampling interval defined as Tsa = Ts/N and Ts is the OFDM symbol duration, flis the Doppler frequency shift at the lth multipath normalized by 1/Ts, i.e., fl = fDlTs, and nl is thecorresponding delay in samples, i.e., nl =floorleftτl/Tsafloorright.57Appendix BDerivation of (3.20)Recalling (3.8) and (3.13), the cost function we try to minimize isJ(hl, fl) =vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevectorY −L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBigvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle2=vextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevectorY −L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBigvextendsinglevextendsinglevextendsinglevextendsinglevextendsingleHvextendsinglevextendsinglevextendsinglevextendsinglevextendsinglevectorY −L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBigvextendsinglevextendsinglevextendsinglevextendsinglevextendsingle(B.1)where (·)H denotes the Hermitian transpose of the argument matrix. According to (30), (31), and(32) in [31], we can obtain∆vectorθi =ℜ{∆hi}ℑ{∆hi}∆fi(B.2)vectorni =−∂J(hl,fl)∂ℜ{hi}∂J(hl,fl)∂ℑ{hi}∂J(hl,fl)∂ fi(B.3)andRij =∂2J(hl,fl)∂ℜ{hi}∂ℜ{hj}∂2J(hl,fl)∂ℜ{hi}∂ℑ{hj}∂2J(hl,fl)∂ℜ{hi}∂ fj∂2J(hl,fl)∂ℑ{hi}∂ℜ{hj}∂2J(hl,fl)∂ℑ{hi}∂ℑ{hj}∂2J(hl,fl)∂ℑ{hi}∂ fj∂2J(hl,fl)∂ fi∂ℜ{hj}∂2J(hl,fl)∂ fi∂ℑ{hj}∂2J(hl,fl)∂ fi∂ fj(B.4)where ℜ{·} and ℑ{·} respectively denote the real part and imaginary part of the argument, and∆hi = ˆhi−hi and ∆fi = ˆfi− fi are the perturbation of tne estimates caused by Gaussian noisevectorni.58Appendix B. Derivation of (3.20)In the following, we will calculate the elements ofvectorni.Using (B.1), one can obtain∂J(hl, fl)∂ℜ{hi} =bracketleftBig−parenleftBigvectorAi +vectorBi fi +vectorCi f2i +···parenrightBigbracketrightBigHbracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBigbracketrightBigg+bracketleftBiggvectorY − L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBigbracketrightBiggHbracketleftBig−parenleftBigvectorAi +vectorBi fi +vectorCi f2i +···parenrightBigbracketrightBig.(B.5)Recalling from (3.8), we havevectorw =vectorY −L∑l=1hlparenleftBigvectorAl +vectorBl fl +vectorCl f2l +···parenrightBig(B.6)and can rewrite (B.5) as∂J(hl, fl)∂ℜ{hi} =bracketleftBig−parenleftBigvectorAi +vectorBi fi +vectorCi f2i +···parenrightBigbracketrightBigHvectorw+vectorwHbracketleftBig−parenleftBigvectorAi +vectorBi fi +vectorCi f2i +···parenrightBigbracketrightBig. (B.7)Using the assumption that fl’s are small, we can neglect the terms containing fi in (B.7) and furthersimplify (B.7) as∂J(hl, fl)∂ℜ{hi} ≈parenleftBig−vectorAiparenrightBigHvectorw+vectorwHparenleftBig−vectorAiparenrightBig=parenleftBig−vectorwHvectorAiparenrightBigH+parenleftBig−vectorwHvectorAiparenrightBig= −2ℜbraceleftBigvectorwHvectorAibracerightBig. (B.8)In a similar way, we can obtain∂J(hl, fl)∂ℑ{hi} = 2ℑbraceleftBigvectorwHvectorAibracerightBig(B.9)and∂J(hl, fl)∂ fi =−2ℜbraceleftBighivectorwHvectorBibracerightBig(B.10)59Appendix B. Derivation of (3.20)so that we obtainvectorni =−∂J(hl,fl)∂ℜ{hi}∂J(hl,fl)∂ℑ{hi}∂J(hl,fl)∂ fi= 2ℜbraceleftBigvectorwHvectorAibracerightBig−ℑbraceleftBigvectorwHvectorAibracerightBigℜbraceleftBigvectorhivectorwHvectorBibracerightBig. (B.11)Based on the derivation above, it is straightforward to calculate the elements of Rij in a similar wayand obtainRij = 2ℜbraceleftBigvectorAHj vectorAibracerightBigℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigh∗jvectorBHj vectorAibracerightBig−ℑbraceleftBigvectorAHj vectorAibracerightBigℜbraceleftBigvectorAHj vectorAibracerightBig−ℑbraceleftBigh∗jvectorBHj vectorAibracerightBigℜbraceleftBighivectorAHj vectorBibracerightBigℑbraceleftBighivectorAHj vectorBibracerightBigℜ{hih∗jvectorBHj vectorBi}(B.12)where {·}∗ denotes the complex conjugate of the argument. Applying (B.11) and (B.12) to (32) in[31], we can obtain (3.20).60Appendix CDerivation of (3.30c)From (3.7) and (3.8), we havevectorBHl vectorBl = N−1∑k=0BHl [k]Bl[k]. (C.1)Applying (3.9b) to (C.1) and using the uncorrelation property of white training sequence , oneobtainsvectorBHl vectorBl = pi2(N−1)2N2N−1∑k=0|X[k]|2 + 4pi2N2N−1∑k=0N−1∑kprimenegationslash=kkprime=01|1−ω|2vextendsinglevextendsingleX[kprime]vextendsinglevextendsingle2 (C.2)where ω = expparenleftBigj2pi(kprime−k)NparenrightBig. Assuming |X[k]| = 1√N, for k = 0,1,...,N−1, we can simplify (C.2)to bevectorBHl vectorBl = pi2(N−1)2N2 +4pi2N3N−1∑k=0N−1∑kprimenegationslash=kkprime=01|1−ω|2. (C.3)For the sake of simplicity, we denotes[k] triangle=N−1∑kprimenegationslash=kkprime=01|1−ω|2=N−1∑kprimenegationslash=kkprime=01vextendsinglevextendsinglevextendsingle1−e j2pi(kprime−k)N vextendsinglevextendsinglevextendsingle2, k = 0,1,...,N−1 (C.4)and rewrite (C.3) asvectorBHl vectorBl = pi2(N−1)2N2 +4pi2N3N−1∑k=0s[k]. (C.5)When N is fixed, exploiting the fact that s[k] is constant for k = 0,1,...,N−1, one can obtains[k] triangle= s =N−1∑n=11vextendsinglevextendsinglevextendsingle1−e j2pinN vextendsinglevextendsinglevextendsingle2, k = 0,1,...,N−1 (C.6)61Appendix C. Derivation of (3.30c)and further simplify (C.5) to bevectorBHl vectorBl = pi2(N−1)2N2 +4pi2N2 s. (C.7)Using Parseval’s relationship, one can shows =N−1∑n=11vextendsinglevextendsinglevextendsingle1−e j2pinN vextendsinglevextendsinglevextendsingle2= 14N−1∑n=11sin2parenleftbignpiN parenrightbig= (N−1)(N +1)12 . (C.8)Finally substituting of (C.8) into (C.7), one obtains (3.30c) as desired, i.e.,vectorBHl vectorBl = 2pi2(N−1)(2N−1)3N2 . (C.9)62Appendix DDerivation of the Absolute Value for theLargest Eigenvalue of M−1η NηAccording to algebraic theory, the roots of a cubic equationx3 +αx2 +βx+γ = 0 (D.1)are given by [37]x = v−u− α3 (D.2)whereu = 3radicalBiggq2 ±radicalbiggq24 +p327 (D.3a)v = p3u (D.3b)p = β − α23 (D.3c)q = γ + 2α3−9αβ27 . (D.3d)After some manipulations, one can calculate that the roots for the characteristic equation in (4.12)are given byp =−3(1−η)+η2kl3 η2kl (D.4a)63Appendix D. Derivation of the Absolute Value for the Largest Eigenvalue of M−1η Nηandq =− 127bracketleftbig9(1−η)+2η2klbracketrightbigη4k2l . (D.4b)To minimize the largest absolute value of the roots of (4.12), we letδ = q24 +p327 = 0 (D.5)and have three equal roots. After applying (D.4) into (D.5), one can show that η is required tosatisfyη2kl −4η +4 = 0 (D.6)orη = 2(1−√1−kl)kl . (D.7)After substitution of η into u and v, it can be shown thatu = 3radicalbiggq2 =−23(1−η) (D.8a)v = 23(1−η) (D.8b)x = v−u− α3 = α = 1−η. (D.8c)Thus, the absolute value for the largest eigenvalue of M−1η Nη is|λ|=|1−η|=vextendsinglevextendsinglevextendsinglevextendsingle1− 2(1−√1−kl)klvextendsinglevextendsinglevextendsinglevextendsingle. (D.9)64Appendix EMatlab Code%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% Main Function %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%clc ;clear ;%% Parameter Initialization %%REPEAT = 2000; % Number of repeat times , for purpose of averageNfft = 256; % Size of FFTL = 3; % L−path channel% f l = ones(1 , L ); % Normalized Doppler frequency (1 by L vector )% hl = zeros (1 , L ); % Amplitude for the path (1 by L vector )nl = 0 : (L − 1); % Delay of the path (1 by L vector )n modu type = 1; % The type of modulation scheme% 1=BPSK, 2=QPSK, 4=16QAM, 6=64QAMTx = 1; % Tx=1 => Transmitor ; Tx=0 => ReceiverSNR = −10: 5: 40;hl = [(0.2944+1.6236 j ) (−1.3362−0.6918 j ) (0.7143+0.8580 j ) ] ;f l = [0.02 0.04 0.06];delta h 2 = 0.01; % Stoping thresholddelta f 2 = 0.01; % Stoping thresholdh l i n i t i a l = ones(2 , L );f l i n i t i a l = ones(2 , L );h l i n i t i a l = (1 + 1j ) ∗ h l i n i t i a l ; % I n i t i a l valuef l i n i t i a l = 0.05 ∗ f l i n i t i a l ; % I n i t i a l value%% For Algorithm 1 %%hl hat ave p2 A1 = zeros ( length (SNR) , L );fl hat ave p2 A1 = zeros ( length (SNR) , L );hl MSE p2 A1 = zeros ( length (SNR) , L );fl MSE p2 A1 = zeros ( length (SNR) , L );hl err ave p2 A1 = zeros ( length (SNR) , L );65Appendix E. Matlab Codefl err ave p2 A1 = zeros ( length (SNR) , L );iteration p2 A1 = zeros ( length (SNR) , 1);%% For Algorithm 2 %%hl hat ave p2 A2 = zeros ( length (SNR) , L );fl hat ave p2 A2 = zeros ( length (SNR) , L );hl MSE p2 A2 = zeros ( length (SNR) , L );fl MSE p2 A2 = zeros ( length (SNR) , L );hl err ave p2 A2 = zeros ( length (SNR) , L );fl err ave p2 A2 = zeros ( length (SNR) , L );iteration p2 A2 = zeros ( length (SNR) , 1);%% For theoretical value %%nvar = zeros ( length (SNR) , 1); % Noise variancetheoretic fl = zeros ( length (SNR) , (L + 1));for snr index = 1 : length (SNR)%% For Algorithm 1 %%hl hat temp p2 A1 = zeros (1 , L );fl hat temp p2 A1 = zeros (1 , L );hl MSE temp p2 A1 = zeros (1 , L );fl MSE temp p2 A1 = zeros (1 , L );hl err temp p2 A1 = zeros (1 , L );fl err temp p2 A1 = zeros (1 , L );Iteration temp p2 A1 = 0;%% For Algorithm 2 %%hl hat temp p2 A2 = zeros (1 , L );fl hat temp p2 A2 = zeros (1 , L );hl MSE temp p2 A2 = zeros (1 , L );fl MSE temp p2 A2 = zeros (1 , L );hl err temp p2 A2 = zeros (1 , L );fl err temp p2 A2 = zeros (1 , L );Iteration temp p2 A2 = 0;%% For theoretical value %%nvar temp = 0; % Noise variancefor repeat index = 1 : REPEAT%% Generate Data %%Xbit = round(rand(1 , ( Nfft ∗ n modu type ) ) ) ;%Xbit = randint (1 , Nfft ∗ n modu type );66Appendix E. Matlab Code%% Modulation (Gray Mapping) %%X = mapper( Xbit , n modu type , Tx );X = X / sqrt ((X ∗ X’ ) ) ; %%% Normalize X%% Calculate Coefficiences %%[ Al , Bl , Cl , Dl ] = calcu coeff ( Nfft , L, nl , X);%% Calculate the output of FFT (Y[k ]) using Eq. 4 %%[Y, nvar i ] = calcu corrupted ( Nfft ,L, fl , hl , nl ,X,SNR( snr index ));nvar temp = nvar temp + nvar i ;%% Channel Estimation using Algorithm 1 %%temp iter p2 = 0;p = 2; % The order of the Taylor expansion .% It can not work properly when p=1.[ hl hat ave p2 A1 ( snr index , :) , fl hat ave p2 A1 ( snr index , :) , iteration p2 A1 ( snr index ) ] . . .= channel est A1 ( Nfft , L, p, Al , Bl , Cl , Dl , Y, delta h 2 , delta f 2 , h l init ial , f l i n i t i a l );iteration p2 A1 ( snr index ) = 1000 − iteration p2 A1 ( snr index );%% Channel Estimation using Algorithm 2 %%[ hl hat ave p2 A2 ( snr index , :) , fl hat ave p2 A2 ( snr index , :) , iteration p2 A2 ( snr index ) ] . . .= channel est A2 ( Nfft , L, p, Al , Bl , Cl , Dl , Y, delta h 2 , delta f 2 , h l init ial , f l i n i t i a l );iteration p2 A2 ( snr index ) = 1000 − iteration p2 A2 ( snr index );%% Calculate the MSE %%%% For Algorithm 1 %%hl hat temp p2 A1=hl hat temp p2 A1+hl hat ave p2 A1 ( snr index , : ) ;fl hat temp p2 A1=fl hat temp p2 A1+fl hat ave p2 A1 ( snr index , : ) ;hl temp p2 A1 = abs (( hl hat ave p2 A1 ( snr index , :) − hl )) .ˆ 2;fl temp p2 A1 = abs (( fl hat ave p2 A1 ( snr index , :) − f l )) .ˆ 2;hl MSE temp p2 A1 = hl MSE temp p2 A1 + hl temp p2 A1 ;fl MSE temp p2 A1 = fl MSE temp p2 A1 + fl temp p2 A1 ;Iteration temp p2 A1 = Iteration temp p2 A1 + iteration p2 A1 ( snr index );%% For Algorithm 2 %%hl hat temp p2 A2=hl hat temp p2 A2+hl hat ave p2 A2 ( snr index , : ) ;fl hat temp p2 A2=fl hat temp p2 A2+fl hat ave p2 A2 ( snr index , : ) ;hl temp p2 A2 = abs (( hl hat ave p2 A2 ( snr index , :) − hl )) .ˆ 2;fl temp p2 A2 = abs (( fl hat ave p2 A2 ( snr index , :) − f l )) .ˆ 2;hl MSE temp p2 A2 = hl MSE temp p2 A2 + hl temp p2 A2 ;fl MSE temp p2 A2 = fl MSE temp p2 A2 + fl temp p2 A2 ;Iteration temp p2 A2 = Iteration temp p2 A2 + iteration p2 A2 ( snr index );end67Appendix E. Matlab Code%% For Algorithm 1 %%hl hat ave p2 A1 ( snr index , :) = hl hat temp p2 A1 / REPEAT;fl hat ave p2 A1 ( snr index , :) = fl hat temp p2 A1 / REPEAT;hl MSE p2 A1( snr index , :) = hl MSE temp p2 A1 / REPEAT;fl MSE p2 A1 ( snr index , :) = fl MSE temp p2 A1 / REPEAT;hl err ave p2 A1 ( snr index , :) = hl hat ave p2 A1 ( snr index , :) − hl ;fl err ave p2 A1 ( snr index , :) = fl hat ave p2 A1 ( snr index , :) − f l ;iteration p2 A1 ( snr index ) = Iteration temp p2 A1 / REPEAT;%% For Algorithm 2 %%hl hat ave p2 A2 ( snr index , :) = hl hat temp p2 A2 / REPEAT;fl hat ave p2 A2 ( snr index , :) = fl hat temp p2 A2 / REPEAT;hl MSE p2 A2( snr index , :) = hl MSE temp p2 A2 / REPEAT;fl MSE p2 A2 ( snr index , :) = fl MSE temp p2 A2 / REPEAT;hl err ave p2 A2 ( snr index , :) = hl hat ave p2 A2 ( snr index , :) − hl ;fl err ave p2 A2 ( snr index , :) = fl hat ave p2 A2 ( snr index , :) − f l ;iteration p2 A2 ( snr index ) = Iteration temp p2 A2 / REPEAT;%% For theoretical value %%nvar ( snr index ) = nvar temp / REPEAT;end%% Plot Figure %%hl temp p2 A1 = zeros ( length (SNR) , 1);fl temp p2 A1 = zeros ( length (SNR) , 1);theoretic hl = nvar ∗ 5 /2;hl normal = abs( hl ) .ˆ 2;for i = 1 : Ltheoretic fl (: , i ) = nvar / hl normal ( i );theoretic fl (: , (L + 1)) = theoretic fl (: , (L + 1))+ theoretic fl (: , i );endtheoretic fl = theoretic fl / 2 / pi ˆ2 ∗ 3;theoretic fl (: , (L + 1)) = theoretic fl (: , (L + 1)) / L;hl temp p2 A2 = zeros ( length (SNR) , 1);fl temp p2 A2 = zeros ( length (SNR) , 1);for i = 1 : Lhl temp p2 A1 = hl MSE p2 A1 (: , i ) + hl temp p2 A1 ;hl temp p2 A2 = hl MSE p2 A2 (: , i ) + hl temp p2 A2 ;endhl ave MSE p2 A1 = hl temp p2 A1 ’ / L;hl ave MSE p2 A2 = hl temp p2 A2 ’ / L;hl ave MSE p2 A1 = 10 ∗ log10 (hl ave MSE p2 A1 );68Appendix E. Matlab Codefl ave MSE p2 A1 = 10 ∗ log10 ( fl MSE p2 A1 );hl ave MSE p2 A2 = 10 ∗ log10 (hl ave MSE p2 A2 );fl ave MSE p2 A2 = 10 ∗ log10 ( fl MSE p2 A2 );theoretic hl final = 10 ∗ log10 ( theoretic hl );theoretic fl final = 10 ∗ log10 ( theoretic fl );for p = 2 : 3switch pcase 2plot (SNR, hl ave MSE p2 A1 , ’−ko ’ ) ;hold onplot (SNR, hl ave MSE p2 A2 , ’−mp’ ) ;hold onplot (SNR, theoretic hl final , ’−bd ’ ) ;grid on;legend ( ’ Algorithm 1 ’ , ’ Algorithm 2 ’ , ’ Theoretical value ’ );xlabel ( ’Average SNR (dB) ’ , ’ interpreter ’ , ’ latex ’ );ylabel ( ’Average MSE of $\hat h l$ (dB) ’ , ’ interpreter ’ , ’ latex ’ );endendfor i = 1 : Lfigureplot (SNR, fl ave MSE p2 A1 (: , i ) , ’−ko ’ ) ;hold onplot (SNR, fl ave MSE p2 A2 (: , i ) , ’−mp’ ) ;hold onplot (SNR, theoretic fl final (: , i ) , ’−bd ’ ) ;grid on;legend ( ’ Algorithm 1 ’ , ’ Algorithm 2 ’ , ’ Theoretical value ’ );xlabel ( ’Average SNR (dB) ’ , ’ interpreter ’ , ’ latex ’ );ylabel ( ’MSE of $\hat f l$ (dB) ’ , ’ interpreter ’ , ’ latex ’ );endx d = SNR(3:end );y d A1 = iteration p2 A1 (3:end );y d A2 = iteration p2 A2 (3:end );figureplot (x d , y d A1 , ’−ko ’ ) ;hold onplot (x d , y d A2 , ’−mp’ ) ;hold ongrid onlegend ( ’ Algorithm 1 ’ , ’ Algorithm 2’ );xlabel ( ’Average SNR (dB) ’ , ’ interpreter ’ , ’ latex ’ );69Appendix E. Matlab Codeylabel ( ’Average Number of Iterations ’ , ’ interpreter ’ , ’ latex ’ );% The End %70Appendix E. Matlab Codefunction data mapped = mapper( data input , n modu type , Tx )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : mapear.m %%%% %%%% Function : Map the input bits into the modulated symbols %%%% according to the modulation type decided by %%%% the parameter ’n modu type ’ at the transmitor . %%%% Or demap the received symbols into bits at the %%%% receiver . %%%% %%%% Parameters : data input => input bits %%%% n modu type => modulation type %%%% (1=BPSK, 2=QPSK, 4=16QAM, 6=64QAM) %%%% Tx => flag indicates the transmitor or receiver %%%% (1=transmitor , 0=receiver ) %%%% %%%% Outputs : A matrix of the mapped symbols according to the %%%% modulation scheme used at the transmitor side . %%%% Or a vector of the demapped bits according to the %%%% modulation scheme. %%%% %%%% Notice : The length of ’ data input ’ must be a integer multiple %%%% of ’n modu type ’ at the transmitor side . %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Decide the constellation f i r s t %%[M, M1, M2, type mapping , c] = parameters constellation ( n modu type );alphabet = bit symbol (M, type mapping , M1, M2);i f n modu type ˜= 1constellation gray = alphabet (: ,3) + 1j ∗ alphabet (: ,2);elseconstellation gray = [−1 1] ’;endl = length ( data input );%% Realize the mapping or demapping %%i f Tx==1matrix data = reshape ( data input , n modu type , l / n modu type );m data decimal = bi2de ( matrix data ’ , ’ left−msb’ ) ;for i = 1 : ( l / n modu type)71Appendix E. Matlab Codev data decimal = m data decimal ( i );v encoded = genqammod( v data decimal , constellation gray );output (: , i ) = v encoded ;enddata mapped = c .∗ output ;elseif Tx==0data normalized = data input ./ c;for i = 1 : lv data mapped = data normalized ( i );v decoded = genqamdemod(v data mapped , constellation gray );data decimal (: , i ) = v decoded ;data mapped = de2bi ( data decimal , n modu type , ’ left−msb’ ) ’ ;data mapped = data mapped ( : ) ’ ;endend72Appendix E. Matlab Codefunction [M, M1, M2, type mapping , c] = parameters constellation ( n modu type )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : parameters constellation .m %%%% %%%% Function : Decide the constellation parameters according %%%% to modulation scheme to be used. %%%% %%%% Parameters : n modu type => type of modulation scheme %%%% (1=BPSK, 2=QPSK, 4=16QAM, 6=64QAM) %%%% %%%% Outputs : The parameters which are used to call another %%%% function to calculate the constellation using %%%% Gray labeling . %%%% %%%% Notice : Here we should pay attention to the parameter ’c ’ %%%% We use i t to normalize the constellation . %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%switch n modu typecase 1 % For BPSK (As a PAM)type mapping = ’MPSK’ ; % Only take into account AicM = 2;M1 = 0;M2 = 0;c = 1;case 2 % For QPSK ( Like 4−QAM)type mapping = ’QAM’ ;c = 1/ sqrt (2);case 4 % For 16−QAMtype mapping = ’QAM’ ;c = 1/ sqrt (10);case 6 % For 64−QAMtype mapping = ’QAM’ ;c = 1/ sqrt (42);endi f n modu type ˜= 1M = 0;M1 = sqrt (2ˆ n modu type );M2 = sqrt (2ˆ n modu type );end73Appendix E. Matlab Codefunction [ alphabet ] = bit symbol ( M, type , M1, M2)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : bit symbol .m %%%% %%%% Function : Return the alphabet of the constellation according %%%% to the modulation scheme used. Here the encoding %%%% is done by Gray labeling %%%% %%%% Parameters : M => total number of points in the constellation %%%% It should equal to 2ˆk , where k is the number of %%%% bits that are grouped together to form a symbol , %%%% except QAM, which can come as the product of the %%%% arguments M1 and M2. %%%% type => type of modulation scheme %%%% PAM, MPSK, QAM( rectangular ). %%%% M1 & M2 => number of points on each axle %%%% %%%% Outputs : the alphabet of the constellation according %%%% to the modulation scheme used. %%%% %%%% Notice : When using QAM, M is the product of M1 by M2. %%%% Here the encoding is done by Gray labelin %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Block i ni ti al iz at io n %%i f strcmp ( type , ’QAM’ )k1 = ceil ( log2 (M1) );k2 = ceil ( log2 (M2) );M1 = 2 ˆ k1;M2 = 2 ˆ k2;M = M1 ∗ M2;Aicd = zeros (1 , k1 );Aisd = zeros (1 , k2 );table1 = zeros (M1, 2);table2 = zeros (M2, 2);alphabet = zeros (M, 3);d1 = 0 : 1 : M1−1;d1 = d1 ’ ;d2 = 0 : 1 : M2−1;d2 = d2 ’ ;74Appendix E. Matlab Codeind1 = bi2de ( f l i p l r ( gray2bi ( f l i p l r ( de2bi (d1 ) ) ) ) ) ;table1 = [d1, ind1 +1];ind2 = bi2de ( f l i p l r ( gray2bi ( f l i p l r ( de2bi (d2 ) ) ) ) ) ;table2 = [d2, ind2 +1];elsek = ceil ( log2 (M));M = 2 ˆ k;Aicd = zeros (1 , k );Aisd = zeros (1 , k );table = zeros (M, 2);alphabet = zeros (M, 3);d = 0 : 1 : M−1;d = d ’ ;ind = bi2de ( f l i p l r ( gray2bi ( f l i p l r ( de2bi (d ) ) ) ) ) ;table = [d, ind +1];end%% Block computation %%i f strcmp ( type , ’PAM’ )Aicd = −(M−1) : 2 : M−1;Aisd = [ ] ;% Use the alphabetfor i=1 : Mindex = find index ( i −1,table );alphabet ( i , :) = [ i −1, Aicd ( index ) , 0];endelseif strcmp ( type , ’MPSK’ )angle = 0:2∗ pi /M:2∗ pi ∗(M−1)/M;Aicd = cos( angle );Aisd = sin ( angle );% Use the alphabetfor i=1 : Mindex = find index ( i −1,table );alphabet ( i , :) = [ i −1, Aicd ( index ) , Aisd ( index ) ] ;endelseif strcmp ( type , ’QAM’ )Aicd = −(M1−1) : 2 : M1−1;Aisd = (M2−1) : −2 : −(M2−1);% Use the alphabetfor i=1 : M1for j=1 : M2index1 = find index ( i −1,table1 );index2 = find index ( j −1,table2 );75Appendix E. Matlab Codel = i + M1 ∗ ( j −1);alphabet ( l , :) = [ l −1, Aicd ( index1 ) , Aisd ( index2 ) ] ;endendend76Appendix E. Matlab Codefunction b = gray2bi ( g )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% GRAY2BI converts Gray encoded sequence into the binary %%%% sequence. It is assumed that the most significant bit is %%%% the left hand side bit . %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% copy the msb:b(: ,1) = g(: ,1);for i = 2: size (g,2) ,b(: , i ) = xor ( b(: , i −1), g(: , i ) );endreturn ;77Appendix E. Matlab Codefunction indice = find index (num, table )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : find index .m %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%dim = size ( table );num col = dim (1);for i =1:num coli f num==table ( i ,1);indice = table ( i ,2);returnendend78Appendix E. Matlab Codefunction [ Al , Bl , Cl , Dl ] = calcu coeff ( Nfft , L, nl , X)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : calcu coeff .m %%%% %%%% Function : Calculate the coefficients of Equation 8. %%%% Then output Al [k] , Bl [k] , Cl [k] and Dl [k ]. %%%% %%%% Parameters : Nfft => Size of FFT %%%% L => L−path channel %%%% nl => Delay of the path %%%% X => Training sequence %%%% %%%% Outputs : Al , Bl , Cl and Dl %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%Al = zeros (L, Nfft );for l = 1 : Lfor k = 1 : NfftAl ( l , k) = X(k) ∗ exp(−1j ∗ 2 ∗ pi ∗ (k − 1) ∗ nl ( l ) / Nfft );w = exp(1 j ∗ 2 ∗ pi ∗ ((1 : Nfft ) − k + 1e−13) / Nfft );temp = X .∗ exp(−1j ∗ 2 ∗ pi ∗ (0 : ( Nfft − 1)) ∗ nl ( l ) / Nfft ) . . ..∗ (−2 ∗ 1j ∗ pi / Nfft ./ (1 − w));temp(k) = Al ( l , k) ∗ 1j ∗ pi ∗ ( Nfft − 1) / Nfft ;Bl ( l , k) = sum(temp );temp = X .∗ exp(−1j ∗ 2 ∗ pi ∗ (0 : ( Nfft − 1)) ∗ nl ( l ) / Nfft ) . . ..∗ (2 ∗ pi ˆ2 .∗ (1 − 2 ∗ w ./ (w−1) / Nfft ) ./ (1−w) / Nfft );temp(k) = −Al ( l , k) ∗ pi ˆ2 ∗ (2 / 3 − 2 / 3 / Nfft / Nfft − ( Nfft −1) / Nfft / Nfft );Cl ( l ,k) = sum(temp );temp = X .∗ exp(−1j ∗ 2 ∗ pi ∗ (0 : ( Nfft − 1)) ∗ nl ( l ) / Nfft ) . . ..∗ (1 j ∗ 4 ∗ pi ˆ3 ∗ ( Nfft ˆ2 ∗ (1−w).ˆ2 + 3 ∗ w .∗ ( Nfft + 1 . . .+ w + Nfft ∗ w)) / 3 / Nfft ˆ3 / (1−w).ˆ3);temp(k) = −Al ( l , k) ∗ 1j ∗ pi ˆ3 ∗ ( Nfft −1)ˆ2 / 3 / Nfft ˆ2;Dl ( l , k) = sum(temp );endend79Appendix E. Matlab Codefunction [Y, nvar ] = calcu corrupted ( Nfft , L, fl , hl , nl , X, SNR)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : calcu corrupted .m %%%% %%%% Function : Calculate the corrupted signal , undergoing fading , %%%% adding AWGN according to Equation (4). %%%% %%%% Parameters : Nfft => Size of FFT %%%% L => L−path channel %%%% f l => Normalized Doppler frequency %%%% hl => Complex valued channel gain %%%% nl => Delay of the path %%%% X => Training sequence %%%% SNR => SNR value %%%% %%%% Outputs : The corrupted signal at the receiver side . %%%% %%%% Notice : We do not add AWGN i f we just want the corrupted %%%% signal for calculating the equivalent SNR to %%%% quantify the approximation . Then we only have %%%% 6 input arguments without SNR. %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%switch nargincase 6Y = zeros (1 , Nfft );R = zeros (L, 2∗Nfft −1);kk = (1 − Nfft ) : ( Nfft − 1);% Calculate the received signal without AWGNfor l = 1 : LR( l , :) = (1 − exp( j ∗ 2 ∗ pi ∗ f l ( l ))) ./ (1 − exp( j ∗ 2 . . .∗ pi .∗ (kk + f l ( l ) + 1e−13) / Nfft )) / Nfft ;endfor k = 1 : Nfftfor l = 1 : Lfor k = 1 : NfftY(k) = Y(k) + exp(−j ∗ 2 ∗ pi ∗ ( k −1) ∗ nl ( l ) / . . .Nfft ) ∗ X( k ) ∗ hl ( l ) ∗ R( l , ( Nfft−k+k ));end80Appendix E. Matlab Codeendendnvar = 0;case 7SNR l = 10 ˆ (SNR / 10);Y = zeros (1 , Nfft );R = zeros (L, 2∗Nfft −1);kk = (1 − Nfft ) : ( Nfft − 1);% Calculate the received signal without AWGNfor l = 1 : LR( l , :) = (1 − exp( j ∗ 2 ∗ pi ∗ f l ( l ))) ./ (1 − exp( j ∗ 2 . . .∗ pi .∗ (kk + f l ( l ) + 1e−13) / Nfft )) / Nfft ;endfor k = 1 : Nfftfor l = 1 : Lfor k = 1 : NfftY(k) = Y(k) + exp(−j ∗ 2 ∗ pi ∗ ( k −1) ∗ nl ( l ) / . . .Nfft ) ∗ X( k ) ∗ hl ( l ) ∗ R( l , ( Nfft−k+k ));endendend% Add AWGN to the signalEs = mean(abs(Y) .ˆ 2);nvar = Es / SNR l;AWGNoise = sqrt ( nvar / 2) .∗ (randn(1 , Nfft ) + 1j .∗randn(1 , Nfft ));Y = Y + AWGNoise;end81Appendix E. Matlab Codefunction [ hl hat , fl hat , count ] = channel est A1 ( Nfft , L, p, Al , Bl , . . .Cl , Dl , Y, delta h , delta f , hl ini tia l , f l i n i t i a l )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : channel est A1 .m %%%% %%%% Function : Estimate the channel parameters ( hl & f l ) using Algorithm 1.%%%% %%%% Parameters : %%%% Nfft => Size of FFT %%%% L => L−path channel %%%% p => the order of Taylor Series Expansion %%%% Al , Bl , Cl , Dl => the coefficients of the approximation %%%% Y => the received signal %%%% delta h => the threshold of hl ( stops the iteration ) %%%% delta f => the threshold of f l %%%% h l i n i t i a l => the i n i t i a l value of estimate ’ hl hat ’ %%%% f l i n i t i a l => the i n i t i a l value of estimate ’ fl hat ’ %%%% %%%% Outputs : The esimates ’ hl hat ’ , ’ fl hat ’ and the iteration ’count ’ %%%% %%%% Notice : The order of accuracy depends on Nfft and p. We have %%%% different accuracy for different path . The iteration time %%%% depends on the i n i t i a l value and the threshold . %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I n i t i a l i z e hl and f l %%%i f nargin == 10YYY = Y. ’;AAA = Al . ’;BBB = Bl . ’;temp H = inv ( Al ∗ AAA) ∗ Al ∗ YYY;temp F = inv ( Bl ∗ BBB) ∗ Bl ∗ (YYY − AAA ∗ temp H) ./ temp H;h l i n i t i a l = [temp H temp H ];f l i n i t i a l = [ temp F temp F ];h l i n i t i a l = h l i n i t i a l . ’;f l i n i t i a l = f l i n i t i a l . ’;endcount = 0;hl hat = zeros (1 , L );fl hat = zeros (1 , L );82Appendix E. Matlab Codeswitch pcase 2 % second order approximationiteration = 1000;while ( iteration )for l = 1 : LAlpha h = zeros (1 , Nfft );Alpha f = zeros (1 , Nfft );i f l == 1for k = 2 : LAlpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endelseif l == Lfor k = 1 : L − 1Alpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endelsefor k = 1 : l − 1Alpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endfor k = l + 1 : LAlpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endend% update hlAlpha h = Y − Alpha h ;Beta h = Al ( l , :) + f l i n i t i a l (2 , l ) .∗ Bl ( l , :) + f l i n i t i a l (2 , l )ˆ2 .∗ Cl ( l , : ) ;temp = h l i n i t i a l (2 , l );h l i n i t i a l (2 , l ) = Beta h ∗ Alpha h ’ / ( Beta h ∗ Beta h ’ ) ;h l i n i t i a l (2 , l ) = conj ( h l i n i t i a l (2 , l ));h l i n i t i a l (1 , l ) = temp;% update f lAlpha f = Alpha h − h l i n i t i a l (2 , l ) .∗ Al ( l , : ) ;Beta f = h l i n i t i a l (2 , l ) .∗ Bl ( l , : ) ;Eta f = h l i n i t i a l (2 , l ) .∗ Cl ( l , : ) ;a = Eta f ∗ Eta f ’ ∗ 4;b = ( Beta f ∗ Eta f ’ + Eta f ∗ Beta f ’ ) ∗ 3;c = ( Beta f ∗ Beta f ’ − Alpha f ∗ Eta f ’ − Eta f ∗ Alpha f ’ ) ∗ 2;d = −(Alpha f ∗ Beta f ’ + Beta f ∗ Alpha f ’ ) ;ppp = [a b c d ];83Appendix E. Matlab Coderrr = roots (ppp );temp = f l i n i t i a l (2 , l );f l i n i t i a l (2 , l ) = rrr ( length ( rrr ));f l i n i t i a l (1 , l ) = temp;endi f (max(abs( h l i n i t i a l (2 , :) − h l i n i t i a l (1 , :)) ./ abs( h l i n i t i a l (1 , : ) ) ) < delta h ) . . .&& (max(abs( f l i n i t i a l (2 , :) − f l i n i t i a l (1 , :)) ./ abs( f l i n i t i a l (1 , : ) ) ) < delta f )count = iteration ;iteration = 0;elseiteration = iteration − 1;endendhl hat = h l i n i t i a l (2 , : ) ;fl hat = f l i n i t i a l (2 , : ) ;case 3 % third order approximationiteration = 1000;while ( iteration )for l = 1 : LAlpha h = zeros (1 , Nfft );Alpha f = zeros (1 , Nfft );i f l == 1for k = 2 : LAlpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endelseif l == Lfor k = 1 : L − 1Alpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endelsefor k = 1 : l − 1Alpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endfor k = l + 1 : LAlpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endend% update hl84Appendix E. Matlab CodeAlpha h = Y − Alpha h ;Beta h = Al ( l , :) + f l i n i t i a l (2 , l ) .∗ Bl ( l , :) + f l i n i t i a l (2 , l )ˆ2 .∗ Cl ( l , : ) . . .+ f l i n i t i a l (2 , l )ˆ3 .∗ Dl ( l , : ) ;temp = h l i n i t i a l (2 , l );h l i n i t i a l (2 , l ) = Beta h ∗ Alpha h ’ / ( Beta h ∗ Beta h ’ ) ;h l i n i t i a l (2 , l ) = conj ( h l i n i t i a l (2 , l ));h l i n i t i a l (1 , l ) = temp;% update f lAlpha f = Alpha h − h l i n i t i a l (2 , l ) .∗ Al ( l , : ) ;Beta f = h l i n i t i a l (2 , l ) .∗ Bl ( l , : ) ;Eta f = h l i n i t i a l (2 , l ) .∗ Cl ( l , : ) ;Theta f = h l i n i t i a l (2 , l ) .∗ Dl ( l , : ) ;a = Theta f ∗ Theta f ’ ∗ 6;b = ( Eta f ∗ Theta f ’ + Theta f ∗ Eta f ’ ) ∗ 5;c = ( Beta f ∗ Theta f ’ + Eta f ∗ Eta f ’ + Theta f ∗ Beta f ’ ) ∗ 4;d = ( Beta f ∗ Eta f ’ + Eta f ∗ Beta f ’ − Alpha f ∗ Theta f ’ − Theta f ∗ Alpha f ’ ) ∗ 3;e = ( Beta f ∗ Beta f ’ − Alpha f ∗ Eta f ’ − Eta f ∗ Alpha f ’ ) ∗ 2;f = −(Alpha f ∗ Beta f ’ + Beta f ∗ Alpha f ’ ) ;ppp = [a b c d e f ];rrr = roots (ppp );temp = f l i n i t i a l (2 , l );f l i n i t i a l (2 , l ) = rrr ( length ( rrr ));f l i n i t i a l (1 , l ) = temp;endi f (max(abs( h l i n i t i a l (2 , :) − h l i n i t i a l (1 , :)) ./ abs( h l i n i t i a l (1 , : ) ) ) < delta h ) . . .&& (max(abs( f l i n i t i a l (2 , :) − f l i n i t i a l (1 , :)) ./ abs( f l i n i t i a l (1 , : ) ) ) < delta f )count = iteration ;iteration = 0;elseiteration = iteration − 1;endendhl hat = h l i n i t i a l (2 , : ) ;fl hat = f l i n i t i a l (2 , : ) ;end85Appendix E. Matlab Codefunction [ hl hat , fl hat , count ] = channel est A2 ( Nfft , L, p, Al , Bl , . . .Cl , Dl , Y, delta h , delta f , hl ini tia l , f l i n i t i a l )%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%% File : channel est A2 .m %%%% %%%% Function : Estimate the channel parameters ( hl & f l ) using Algorithm 2.%%%% %%%% Parameters : %%%% Nfft => Size of FFT %%%% L => L−path channel %%%% p => the order of Taylor Series Expansion %%%% Al , Bl , Cl , Dl => the coefficients of the approximation %%%% Y => the received signal %%%% delta h => the threshold of hl ( stops the iteration ) %%%% delta f => the threshold of f l %%%% h l i n i t i a l => the i n i t i a l value of estimate ’ hl hat ’ %%%% f l i n i t i a l => the i n i t i a l value of estimate ’ fl hat ’ %%%% %%%% Outputs : The esimates ’ hl hat ’ , ’ fl hat ’ and the iteration ’count ’ %%%% %%%% Notice : The order of accuracy depends on Nfft and p. We have %%%% different accuracy for different path . The iteration time %%%% depends on the i n i t i a l value and the threshold . %%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% I n i t i a l i z e hl and f l %%%i f nargin == 10YYY = Y. ’;AAA = Al . ’;BBB = Bl . ’;temp H = inv ( Al ∗ AAA) ∗ Al ∗ YYY;temp F = inv ( Bl ∗ BBB) ∗ Bl ∗ (YYY − AAA ∗ temp H) ./ temp H;h l i n i t i a l = [temp H temp H ];f l i n i t i a l = [ temp F temp F ];h l i n i t i a l = h l i n i t i a l . ’f l i n i t i a l = f l i n i t i a l . ’endeta = 4 / 3;count = 0;hl hat = zeros (1 , L );86Appendix E. Matlab Codefl hat = zeros (1 , L );switch pcase 2 % second order approximationiteration = 1000;while ( iteration )for l = 1 : LAlpha h = zeros (1 , Nfft );Alpha f = zeros (1 , Nfft );i f l == 1for k = 2 : LAlpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endelseif l == Lfor k = 1 : L − 1Alpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endelsefor k = 1 : l − 1Alpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endfor k = l + 1 : LAlpha h = Alpha h + h l i n i t i a l (2 , k) .∗ ( Al (k , : ) . . .+ f l i n i t i a l (2 , k) .∗ Bl (k , :) + f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , : ) ) ;endend% update hlAlpha h = Y − Alpha h ;Beta h = Al ( l , :) + f l i n i t i a l (2 , l ) .∗ Bl ( l , :) + f l i n i t i a l (2 , l )ˆ2 .∗ Cl ( l , : ) ;temp1 = Beta h ∗ Alpha h ’ / ( Beta h ∗ Beta h ’ ) ; % h( i +1)temp1 = conj (temp1 );temp2 = eta ∗ temp1 + (1 − eta ) ∗ h l i n i t i a l (2 , l );temp = h l i n i t i a l (2 , l );h l i n i t i a l (1 , l ) = temp;h l i n i t i a l (2 , l ) = temp2;% update f lAlpha f = Alpha h − h l i n i t i a l (2 , l ) .∗ Al ( l , : ) ;Beta f = h l i n i t i a l (2 , l ) .∗ Bl ( l , : ) ;Eta f = h l i n i t i a l (2 , l ) .∗ Cl ( l , : ) ;87Appendix E. Matlab Codea = Eta f ∗ Eta f ’ ∗ 4;b = ( Beta f ∗ Eta f ’ + Eta f ∗ Beta f ’ ) ∗ 3;c = ( Beta f ∗ Beta f ’ − Alpha f ∗ Eta f ’ − Eta f ∗ Alpha f ’ ) ∗ 2;d = −(Alpha f ∗ Beta f ’ + Beta f ∗ Alpha f ’ ) ;ppp = [a b c d ];rrr = roots (ppp );temp1 = rrr ( length ( rrr )); % f ( i +1)temp2 = eta ∗ temp1 + (1 − eta ) ∗ h l i n i t i a l (2 , l );temp = f l i n i t i a l (2 , l );f l i n i t i a l (1 , l ) = temp;f l i n i t i a l (2 , l ) = temp1;endi f (max(abs( h l i n i t i a l (2 , :) − h l i n i t i a l (1 , :)) ./ abs( h l i n i t i a l (1 , : ) ) ) < delta h ) . . .&& (max(abs( f l i n i t i a l (2 , :) − f l i n i t i a l (1 , :)) ./ abs( f l i n i t i a l (1 , : ) ) ) < delta f )count = iteration ;iteration = 0;elseiteration = iteration − 1;endendhl hat = h l i n i t i a l (2 , : ) ;fl hat = f l i n i t i a l (2 , : ) ;case 3 % third order approximationiteration = 1000;while ( iteration )for l = 1 : LAlpha h = zeros (1 , Nfft );Alpha f = zeros (1 , Nfft );i f l == 1for k = 2 : LAlpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endelseif l == Lfor k = 1 : L − 1Alpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endelsefor k = 1 : l − 1Alpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;88Appendix E. Matlab Codeendfor k = l + 1 : LAlpha h = Alpha h + h l i n i t i a l (2 ,k) .∗ ( Al (k , : ) + f l i n i t i a l (2 ,k) .∗ Bl (k , : ) . . .+ f l i n i t i a l (2 , k)ˆ2 .∗ Cl (k , :) + f l i n i t i a l (2 , k)ˆ3 .∗ Dl (k , : ) ) ;endend% update hlAlpha h = Y − Alpha h ;Beta h = Al ( l , :) + f l i n i t i a l (2 , l ) .∗ Bl ( l , :) + f l i n i t i a l (2 , l )ˆ2 .∗ Cl ( l , : ) . . .+ f l i n i t i a l (2 , l )ˆ3 .∗ Dl ( l , : ) ;temp1 = Beta h ∗ Alpha h ’ / ( Beta h ∗ Beta h ’ ) ; % h( i +1)temp1 = conj (temp1 );temp2 = eta ∗ temp1 + (1 − eta ) ∗ h l i n i t i a l (2 , l );temp = h l i n i t i a l (2 , l );h l i n i t i a l (1 , l ) = temp;h l i n i t i a l (2 , l ) = temp2;% update f lAlpha f = Alpha h − h l i n i t i a l (2 , l ) .∗ Al ( l , : ) ;Beta f = h l i n i t i a l (2 , l ) .∗ Bl ( l , : ) ;Eta f = h l i n i t i a l (2 , l ) .∗ Cl ( l , : ) ;Theta f = h l i n i t i a l (2 , l ) .∗ Dl ( l , : ) ;a = Theta f ∗ Theta f ’ ∗ 6;b = ( Eta f ∗ Theta f ’ + Theta f ∗ Eta f ’ ) ∗ 5;c = ( Beta f ∗ Theta f ’ + Eta f ∗ Eta f ’ + Theta f ∗ Beta f ’ ) ∗ 4;d = ( Beta f ∗ Eta f ’ + Eta f ∗ Beta f ’ − Alpha f ∗ Theta f ’ − Theta f ∗ Alpha f ’ ) ∗ 3;e = ( Beta f ∗ Beta f ’ − Alpha f ∗ Eta f ’ − Eta f ∗ Alpha f ’ ) ∗ 2;f = −(Alpha f ∗ Beta f ’ + Beta f ∗ Alpha f ’ ) ;ppp = [a b c d e f ];rrr = roots (ppp );temp1 = rrr ( length ( rrr )); % f ( i +1)temp2 = eta ∗ temp1 + (1 − eta ) ∗ h l i n i t i a l (2 , l );temp = f l i n i t i a l (2 , l );f l i n i t i a l (1 , l ) = temp;f l i n i t i a l (2 , l ) = temp1;endi f (max(abs( h l i n i t i a l (2 , :) − h l i n i t i a l (1 , :)) ./ abs( h l i n i t i a l (1 , : ) ) ) < delta h ) . . .&& (max(abs( f l i n i t i a l (2 , :) − f l i n i t i a l (1 , :)) ./ abs( f l i n i t i a l (1 , : ) ) ) < delta f )count = iteration ;iteration = 0;elseiteration = iteration − 1;end89Appendix E. Matlab Codeendhl hat = h l i n i t i a l (2 , : ) ;fl hat = f l i n i t i a l (2 , : ) ;end90

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