Research Article Estimation of FBMC/OQAM Fading ...downloads.hindawi.com › journals › tswj › 2014 › 586403.pdflter banks with o set quadrature amplitude modulation (FBMC/OQAM),

Research ArticleEstimation of FBMCOQAM Fading Channels Using DualKalman Filters

Mahmoud Aldababseh and Ali Jamoos

Department of Electronic Engineering Al-Quds University PO Box 20002 Jerusalem Palestine

Correspondence should be addressed to Ali Jamoos aliengalqudsedu

Received 3 August 2013 Accepted 2 January 2014 Published 18 February 2014

Academic Editors G Jovanovic Dolecek and A Materka

Copyright copy 2014 M Aldababseh and A Jamoos This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We address the problem of estimating time-varying fading channels in filter bank multicarrier (FBMCOQAM) wireless systemsbased on pilot symbols The standard solution to this problem is the least square (LS) estimator or the minimum mean squareerror (MMSE) estimator with possible adaptive implementation using recursive least square (RLS) algorithm or least meansquare (LMS) algorithm However these adaptive filters cannot well-exploit fading channel statistics To take advantage of fadingchannel statistics the time evolution of the fading channel is modeled by an autoregressive process and tracked by Kalmanfilter Nevertheless this requires the autoregressive parameters which are usually unknown Thus we propose to jointly estimatethe FBMCOQAM fading channels and their autoregressive parameters based on dual optimal Kalman filters Once the fadingchannel coefficients at pilot symbol positions are estimated by the proposed method the fading channel coefficients at data symbolpositions are then estimated by using some interpolationmethods such as linear spline or low-pass interpolationThe comparativesimulation study we carried out with existing techniques confirms the effectiveness of the proposed method

1 Introduction

Wireless multicarrier (MC) communication systems areparallel data transmission techniques in which high datarates can be achieved by the simultaneous transmission overmany orthogonal subcarriers [1] UsingMCcommunicationsa wide-band frequency-selective fading channel is dividedinto many narrow-band frequency nonselective flat fadingsubchannels facilitating channel estimation and equalizationaswell as time synchronization andnarrow-band interferencemitigation In addition the division of the whole bandwidthinto many subchannels provides scalability and flexibilitywhen configuring the communication link

Themost widely used multicarrier modulation techniqueis orthogonal frequency division multiplexing (OFDM) withcyclic prefix (CP) [2] Due to the various advantages ofOFDM it has become the key physical layer transmissiontechnology adopted in current broadband communicationsystems such as wireless local area networks (WLAN) world-wide interoperability for microwave access (WiMAX) andlong term evolution (LTE) aswell as in digital video and audio

broadcasting (DVB and DAB) [2 3] The OFDM systemoffers simple implementation using the inverse fast Fouriertransforms (IFFT) and the fast Fourier transform (FFT) pairsin the modulator and demodulator respectively Howeverthe large side lobes of the frequency response of the filtersthat characterize the subcarrier channels result in significantinterference among subcarriers As an alternative to OFDMfilter bank multicarrier (FBMC) modulation which is alsoimplemented based on IFFTFFT pairs has several advan-tages over OFDM [4 5] Firstly it does not require CP result-ing in better use of the allocated spectrum Secondly insteadof using a rectangular window a prototype filter designedwith the Nyquist pulse shaping principle is adopted whichcan reduce greatly the spectral leakage problem of OFDMThis results in negligible intersymbol interference (ISI) andintercarrier interference (ICI) Thirdly the combination offilter banks with offset quadrature amplitude modulation(FBMCOQAM) known also as OFDMOQAM leads to amaximum data transmission rate [6]

Fading channel estimation and equalization techniquesare crucial for coherent symbol detection at the receiver

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 586403 9 pageshttpdxdoiorg1011552014586403

2 The Scientific World Journal

In the framework of multicarrier systems many channelestimation techniques were proposed in the literature par-ticularly for OFDM systems [7ndash11] They can be classifiedas training sequencepilot symbols based techniques andblind methods In this paper as blind techniques requirea large amount of data and have higher complexity thantrainingpilot based techniques we will focus our attentionon the latter techniques

Compared to OFDM few FBMCOQAM studies haveaddressed the problem of channel estimation and equal-ization [12ndash18] Indeed when classical channel estimationmethods used for OFDM systems are applied directly toFBMCOQAM system an intrinsic intersymbol-interferenceis observed This results in system performance degradationTo reduce this intrinsic interference several methods arerecently proposed including both preamble-based [12ndash14]and scattered pilots-based [15 16] channel estimators In [17]a two-stage minimum mean square error (MMSE) equal-izer is proposed to cope more efficiently with intercarrierinterference (ICI) A fractionally spaced adaptive decisionfeedback equalizer (DFE) based on the least mean square(LMS) algorithmwas proposed in [18]where the input of eachequalizer comprises only the output of each subcarrier

When dealing with Kalman filter based channel estima-tors several approaches were developed for multicarrier sys-tems such as multicarrier code division multiple access (MC-CDMA) and OFDM [10 11 19] Indeed the Kalman filtercombined with an autoregressive (AR) model to describe thetime evolution of the fading channel is shown to track channelvariations and to provide superior bit error rate (BER) per-formance over the standard LMS and recursive least square(RLS) estimators [19] Nevertheless the AR parameters areunknown and should be estimated To jointly estimate theautoregressive process and its parameters from noisy obser-vations one of the authors of this paper has recently proposeddual Kalman filters based structure [11 20] This structureconsists of two cross-coupled Kalman filters where the firstfilter uses the latest estimated AR parameters to estimatethe fading process while the second filter uses the estimatedfading process to update the AR parameters In this paper wepropose to investigate the relevance of the dual Kalman filtersfor the estimation of FBMCOQAM time-varying fadingchannels based on pilot symbols Once the fading channelcoefficients at pilot symbol positions are estimated by the pro-posed method the fading channel coefficients at data symbolpositions are then to be estimated by using some interpola-tionmethods such as linear spline or low-pass interpolation

The remainder of the paper is organized as follows InSection 2 we explain the FBMCOQAM system model Theestimation of the fading channels based on dual Kalman fil-ters is introduced in Section 3 Simulation results are reportedin Section 4 Conclusion remarks are drawn in Section 5

2 FBMCOQAM System Model

There are different filter bank multicarrier structures thatare proposed in the literature Among them are discretewavelet multitone [21] filtered multitone (FMT) [22] cosinemodulated multitone (CMT) [23] modified discrete Fourier

transform (MDFT) [24] exponentially modulated filter bank(EMFB) [25] and FBMCOQAM [6] The last is the mostpopular filter bank scheme that results in high bandwidthefficiency

The fundamental parts of the FBMCOQAM system arethe synthesis filter bank (SFB) at the transmitter and theanalysis filter bank (AFB) at the receiver arranged in theso-called transmultiplexer (TMUX) configuration as shownin Figures 1 and 2 It should be noted that the SFB and AFBpairs can be efficiently implemented using FFT and IFFTof size 119872 aided by polyphase filtering structures Whenusing uniformly modulated filter banks [26] a prototypefilter 119901[119898] of length 119871119901 is shifted in frequency to generatesubchannels which cover the whole system bandwidthThusthe discrete-time baseband signal at the output of the FBMCtransmitter with OQAMmodulation can be expressed as

119910 [119898] =

119872minus1

sum

119896=0

infin

sum

119899=minusinfin

119909119896 [119899] 120573119896 [119899] 119901 [119898 minus119899119872

2] 119890119895(2120587119872)119896119898

(1)

where119898 is the sample index at SFB outputAFB input 119899 is thesample index atOQAMpreprocessing outputpostprocessinginput119872 is the number of subcarriers in the filter bank and

120573119896 [119899] = (minus1)119896119899 exp(minus119895

2120587119896

119872(

119871119901 minus 1

2)) (2)

119909119896 [119899] = 119889119896 [119899] 120579119896 [119899] (3)

120579119896 [119899] = 119895119896+119899

(4)

In (3) 119889119896[119899] are real-valued data symbols at subcarrier119896 transmitted at a rate of 2119879 where the signaling period isdefined as 119879 = 1Δ119891 with Δ119891 being the subcarrier spacingThepair of symbols119889119896[119899] and119889119896[119899+1] represent respectivelythe in-phase and quadrature parts of the complex-valuedinput symbol 119888119896[119897] This complex-to-real conversion opera-tion is considered to introduce an up-sampling by two Theinverse operation of real-to-complex conversion is performedat the receiver and results in down-sampling the signal by twoThus the complex-to-real operation performs the followingmapping

119889119896 [119899] = Re (119888119896 [119897]) 119896 evenIm (119888119896 [119897]) 119896 odd

119889119896 [119899 + 1] = Im (119888119896 [119897]) 119896 evenRe (119888119896 [119897]) 119896 odd

(5)

where 119897 is the sample index at OQAM preprocessinginputpostprocessing output Note that the complex-valuedinput symbols 119888119896[119897] belong to QAM alphabet with the realand imaginary parts being interleaved by a time offset of 1198792resulting in OQAM modulation To maintain orthogonality119889119896[119899] is multiplied by 120579119896[119899]

In order to achieve high spectral efficiency complexmodulated filter banks are usually used which means thatall subchannel filters are frequency shifted versions of the

The Scientific World Journal 3

OQAM preprocessing

Inverse

fast

Fourier

transform

Transform block Synthesis polyphase filtering

Parallelserial conversion

Complexto real

Complexto real

Complexto real

120579Mminus1[n]

dMminus1[n] xMminus1[n]cMminus1[l]

1205790[n]

1205791[n]

c1[l]

d0[n] x0[n]

d1[n] x1[n]

120573Mminus1[n]

1205730[n]

1205731[n]

AMminus1[Z2]

A1[Z2]

A0[Z2] uarr

M

2

uarrM

2

uarrM

2

Zminus1

Zminus1

y[m]c0[l]

Figure 1 Block diagram of the FBMCOQAM transmitter

Real

Kalman

filtering

based

channel

estimation

and

equalization

Fast

Fourier

transform

Real

Real

OQAM postprocessingTransform blockAnalysis polyphasefiltering

Serialparallelconversion

r[m]

darrM

2

darrM

2

darrM

2

Zminus1

Zminus1

B0[Z2]

B1[Z2]

BMminus1[Z2]

s1[n]

s0[n]

sMminus1[n]

dMminus1[n]cMminus1[l]

Realto complex

Realto complex

Realto complex

d0[n]

d1[n]

c0[l]

c1[l]x1[n]

x0[n]

xMminus1[n]

120573lowast1 [n]

120573lowast0 [n] 120579lowast0 [n]

120579lowast1 [n]

120573lowastMminus1[n] 120579lowastMminus1[n]

Figure 2 Block diagram of the FBMCOQAM receiver

prototype filter 119901[119898] So the 119896th synthesis filter 119892119896[119898] canbe expressed as

119892119896 [119898] = 119901 [119898] exp(1198952120587119896

119872(119898 minus

119871119901 minus 1

2)) (6)

where 119898 = 0 1 119871119901 minus 1 The length 119871119901 of the prototypefilter 119901[119898] depends on the size of the filter bank and thenumber of FBMCOQAM symbol waveforms119870 that overlapin time as 119871119901 = 119870119872

The 119896th analysis filter119891119896[119898] is simply a time-reversed andcomplex-conjugated version of the corresponding synthesisfilter

119891119896 [119898] = 119892lowast

119896[119871119901 minus 1 minus 119898] (7)

It should be noted that the design of the prototype filtermust satisfy perfect reconstruction (PR) conditions or at leastprovide nearly perfect reconstruction (NPR) characteristics


However the PR property is only achieved under the con-dition of ideal transmission channel As interferences in thewireless channel are unavoidable there is no way to have PRconditionsThus prototype filters are designed to satisfyNPRcharacteristics

In this paper we used frequency sampling technique [27]or windowing based approach [28] to design NPR prototypefilter In these methods the prototype filter coefficientscan be obtained using a closed-form representation thatincludes only a few adjustable design parametersThe impulseresponse coefficients of the filter are obtained according tothe desired frequency response which is sampled on a 119870119872

uniformly spaced frequency point 120596119896 = 2120587119896119870119872 So thefinite impulse response (FIR) of the low-pass prototype filtercan be expressed as [29]

119901 [119898]

= 119875 [0]

+ 2

119880

sum

119896=1

(minus1)119896119875 [119896] cos( 2120587119896

119870119872(119898 +

119870119872 minus (119871119901 minus 1)

2))

(8)

where119898 = 0 1 119871119901minus1119875[0] = 1 and119875[119888]2+119875[119870 minus 119888]

2= 1

for 119888 = 1 2 1198702 119875[0] = 0 for 119896 = 119870 119870 + 1 119880 =

(119870119872 minus 2)2When polyphase filtering structures are used with the

FFT and IFFT pairs to implement the FBMCOQAM systemthe synthesis polyphase filter at the transmitter is defined as

119886119896 [119898] = 119901 [119896 + 119898119872] (9)

At the receiver the analysis filter polyphase is given by

119887119896 [119898] = 119886119872minus1minus119896 [119898] = 119901 [119872 minus 1 minus 119896 + 119898119872] (10)

The transmitted FBMCOQAM signal 119910[119898] in (1) isassumed to go through a time-varying frequency-selectiveRayleigh fading channel with background additive whiteGaussian noise (AWGN) Assuming that the allocated band-width for each subcarrier is less than the coherence band-width of the wireless channel each subcarrier undergoesfrequency nonselective flat-fading Thus the received signalsample over the 119896th subcarrier for the 119898th FBMCOQAMsymbol can be expressed as

119903119896 [119898] = 119910119896 [119898] ℎ119896 [119898] + 119908119896 [119898] (11)

where ℎ119896[119898] is a complex valued fading process over the119896th subcarrier for the 119898th FBMCOQAM symbol and119908119896[119898] is an additive white Gaussian noise (AWGN) processThe noise processes 119908119896[119898]

119896=01119872minus1are assumed to be

mutually independent and identically distributed zero-meancomplex Gaussian processes with equal variances 120590

2

119908 The

fading process ℎ119896[119898] = |ℎ119896[119898]|119890minus119895120593119896(119898) is assumed to be

a zero-mean complex Gaussian process with uniformly dis-tributed phase120593119896(119898) on [0 2120587] andwith Rayleigh distributedenvelope |ℎ119896[119898]| The variances of the fading processesℎ119896[119898]

119896=01119872minus1are all assumed to be equal to 120590

2

ℎ

After processing the received signal 119903119896[119898] with theanalysis filter bank block the resulting signal at the input ofchannel estimation and equalization is given by

119904119896 [119899] = [119903119896 [119898] lowast 119891119896 [119898]]darr1198722

= 119909119896 [119899] sdot 119902119896 [119899] + 120578119896 [119899]

(12)

where 119902119896[119899] = [ℎ119896[119898]119892119896[119898] lowast 119891119896[119898]]darr1198722

with lowast denotingthe convolution operator and 120578119896[119899] being a Gaussian noiseprocess with variance 120590

2

120578 Under the assumption of nearly

perfect reconstruction (NPR) conditions of the prototypefilter it follows that 119902119896[119899] = [ℎ119896[119898]119892119896[119898] lowast 119891119896[119898]]

darr1198722cong

ℎ119896[119899] Thus (12) is reduced to

119904119896 [119899] = 119909119896 [119899] ℎ119896 [119899] + 120578119896 [119899] 119896 = 0 1 119872 minus 1 (13)

The statistical properties of the fading process ℎ119896[119899] aregiven by its power spectrum density (PSD) and autocorrela-tion function (ACF) Indeed the PSD of ℎ119896[119899] is defined bythe well-known U-shaped band-limited Jakes spectrum withmaximum Doppler frequency 119891119889 [30]

119878 [119891] =

1

120587119891119889radic1 minus (119891119891119889)

2

10038161003816100381610038161198911003816100381610038161003816 le 119891119889

0 elsewhere(14)

where 119891119889 = V120582 with V is the mobile speed and 120582 is the wavelength of the carrier wave The corresponding normalizeddiscrete-time autocorrelation function (ACF) hence satisfies

119877119896 = 1198690 (2120587119891119889119879119904 |119896|) (15)

where 1198690(sdot) is the zero-order Bessel function of the first kind119879119904 is the symbol period and 119891119889119879119904 denotes the Doppler rate

To exploit the statistical properties of the fading channelgiven by its PSD and ACF the fading process ℎ119896[119899] can bemodeled by a 119901th order AR process denoted by AR(119901) anddefined as follows [31]

ℎ119896 [119899] = minus

119901

sum

119894=1

119886119894ℎ119896minus119894 [119899] + 120592119896 [119899] (16)

where 119886119894119894=12119901 are the AR model parameters and 120592119896[119899]

denotes the zero-mean complex white Gaussian drivingprocess with equal variance1205902

120592over all subcarriers Increasing

the AR mode order 119901 will provide better fitting betweenthe resulting model and the theoretical PSD and ACF [1131] However the computational complexity of the resultingchannel estimation algorithm will also increase Thereforea compromise between the accuracy of the model and thecomputational complexity of the estimation algorithm hasto be found When the Doppler rate 119891119889119879119904 is available at thereceiver the AR parameters 119886119894119894=12119901 can be computed bysolving the so-called Yule-Walker (YW) equations Howeveras the Doppler rate 119891119889119879119904 is usually unknown we propose tocomplete the joint estimation of the fading process ℎ119896[119899] andits AR parameters 119886119894119894=12119901 based on dual Kalman filters


Time

Freq

uenc

y

Pilot

Data

Figure 3 Comb-type pilot arrangement

3 Dual Kalman Filters BasedChannel Estimation

The estimation of the fading process ℎ119896[119899] along the 119899thFBMCOQAMsymbolwill be performed in two steps Firstlythe fading process ℎ119896[119899] at the pilot symbol position isestimated using dual Kalman filters Secondly the fadingprocess at data symbol position will then be estimated byusing some interpolation methods such as linear spline orlow-pass interpolation

When using comb-type pilot arrangement [8] as shownin Figure 3 119873119901 pilots are uniformly inserted into 119909119896[119899] asfollows

119909119896 [119899] = 119909119906119871+119894 [119899]

=

119909119906119871pilot [119899] 119894 = 0

119909119906119871+119894data [119899] 119894 = 1 2 119871 minus 1

(17)

where 119909119906119871pilot[119899] is the 119906th pilot symbol with 119906 = 0 1

119873119901 minus 1 and 119871 = 119872119873119901 is the pilot interval spacing with 119872

being the number of subcarriersThus using known comb-type pilot symbols 119909119896[119899] =

119909119896pilot[119899] we propose to jointly estimate the fading processℎ119896[119899] and itsARparameters 119886119894119894=12119901 based ondualKalmanfilters as shown in Figure 4 Indeed the first Kalman filter inFigure 4 uses the pilot symbol 119909119896pilot[119899] the output of theanalysis filter bank 119904119896[119899] and the latest estimated AR para-meters 119886119894119894=12119901 to estimate the fading process ℎ119896[119899] whilethe second Kalman filter uses the estimated fading processℎ119896[119899] to update the AR parameters

31 Fading Process Estimation at Pilot Symbol Positions Inthis subsection our purpose is to estimate the fading processℎ119896[119899] over the 119899th FBMCOQAM symbol based on Kalmanfiltering with known pilot symbols To this end let us definethe state vector as follows

h119896 = [ℎ119896 ℎ119896minus1 sdot sdot sdot ℎ119896minus119901+1]119879 (18)

Kalman filter (1)Fading process estimation

Kalman filter (2)AR parameter estimation

hk[n]sk[n]

xkpilot[n]

aii=1 2 p

Figure 4 Dual Kalman filters based structure for the joint esti-mation of the fading process and its AR parameters over the 119899thFBMCOQAM symbol

Note that for the sake of simplicity and clarity of present-ation the time index subscript is dropped Then (16) can bewritten in the following state-space form

h119896 = 120601h119896minus1 + g120592119896 (19)

where

120601 =[[[[

[

minus1198861 minus1198862 sdot sdot sdot minus119886119901

1 0 sdot sdot sdot 0

d

0 sdot sdot sdot 1 0

]]]]

]

g = [1 0 sdot sdot sdot 0]119879

(20)

In addition given (13) and (18) it follows that

119904119896 = x119879119896h119896 + 120578119896 (21)

where x119896 = [119909119896pilot 0 sdot sdot sdot 0]119879

Hence (19) and (21) represent the state-space modeldedicated to the 119899th FBMCOQAM symbol fading channelsystem (13) and (16) A standard Kalman filtering algorithmcan then be carried out to provide the estimation h119896119896 ofthe state vector h119896 given the set of observations 119904119894119894=1119896 Tothis end let us introduce the so-called innovation process 120572119896which can be obtained as follows

120572119896 = 119904119896 minus x119879119896120601h119896minus1119896minus1 (22)

The variance of the innovation process can be expressed as

119862119896 = 119864 [120572119896120572lowast

119896] = x119879119896P119896119896minus1x119896 + 120590

2

120578119896 (23)

where the so-called a priori error covariance matrix P119896119896minus1can be recursively obtained as follows

P119896119896minus1 = 120601P119896119896minus1120601119867

+ g1205902120592g119879 (24)

The Kalman gain is calculated in the following manner

K119896 = P119896119896minus1x119896119862minus1

119896 (25)


The a posteriori estimate of the state vector and the fadingprocess are respectively given by

h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)

The error covariance matrix is updated as follows

P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)

It should be noted that the state vector and the errorcovariance matrix are initially assigned to zero vector andidentity matrix respectively that is h00 = 0 and P00 = I119901

Equations (22)ndash(27) can be evaluated providing the avail-ability of theARparameters that are involved in the transitionmatrix 120601 and the driving process variance 120590

2

120592 They will be

estimated in the next subsections

32 Estimation of the AR Parameters To estimate the ARparameters from the estimated fading process ℎ119896 (26) arefirstly combined such as the estimated fading process is afunction of the AR parameters

h119896 = g119879120601h119896minus1 + g119879K119896120572119896 = h119879119896minus1

a119896 + 120589119896 (28)

where h119896minus1 = [ℎ119896minus1 ℎ119896minus2 sdot sdot sdot ℎ119896minus119901] and a119896 =

[minus1198861 minus1198862 sdot sdot sdot minus119886119901]119879 In addition the variance of the pro-

cess 120589119896 = g119879K119896120572119896 is given by

1205902

120589119896= g119879K119896119862119896K

H119896g (29)

When the channel is assumed stationary the AR param-eters are invariant and satisfy the following relationship

a119896 = a119896minus1 (30)

As (28) and (30) define a state-space representation forthe estimation of the AR parameters a second Kalman filtercan be used to recursively estimate a119896 as follows

a119896 = a119896minus1 + Ka119896 (ℎ119896 minus ℎ119879

119896minus1a119896minus1) (31)

where the Kalman gain Ka119896 and the update of the errorcovariance matrix Pa are respectively given by

Ka119896 = Pa119896minus1 hlowast

119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1

Pa119896 = Pa119896minus1 minus Ka119896 h119879

119896minus1Pa119896minus1

(32)

with initial conditions a0 = 0 and Pa0 = I119901

33 Estimation of the Driving Process Variance To estimatethe driving process variance 120590

2

120592 the Riccati equation is first

obtained by inserting (24) in (27) as follows

P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)

Taking into account that P119896119896minus1 is a symmetric Hermitianmatrix one can rewrite (25) in the following manner

b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)

By combining (33) and (34) 1205902120592can be expressed as follows

1205902

120592= f [P119896119896 minus 120601P119896minus1119896minus1120601

119867+ K119896119862119896K

119867

119896] f119879 (35)

where f = [g119879g]minus1g119879 = g119879 is the pseudoinverse of gThus we propose to estimate 120590

2

120592recursively as follows

2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]

times f [P119896119896 minus 120601P119896minus1119896minus1120601119867

+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)

where the variance of the innovation process119862119896 is replaced byits instantaneous value |120572119896|

2 and 120582 is the forgetting factor Itshould be noted that 120582 can be either constant or time-varying(eg 120582119896 = (119896 minus 1)119896)

34 Fading Process Estimation at Data Symbol PositionsOnce the fading process at pilot symbol positions is estimatedby the dual Kalman filters the fading process at data symbolpositions will then be estimated by using some interpolationmethods such as linear spline or low-pass interpolation [8]

(1) Linear Interpolation Using linear interpolation methodthe channel estimates at data positions 119906119871 lt 119896 lt (119906 + 1)119871are given by

ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)

where ℎ119901119896is the estimated fading process at pilot symbol

position

(2) Low-Pass InterpolationThe low-pass interpolation is car-ried out by inserting zeros into the data symbol positions ofthe original sequence and then applying a special low-passfilter In this paper we use the Matlab function interp

(3) Spline Interpolation Spline interpolation produces asmooth and continuous polynomial fitted to the estimatedfading process at pilot symbol positions In this paper we usethe Matlab function spline

35 Fading Channel Equalization Once the fading processat pilot and data symbols is estimated using the proposedapproach channel equalization can be performed by mul-tiplying (13) with a normalized version of the complexconjugate of the channel estimate as follows

119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)

Known channel envelopeDual Kalman filters estimationRLS estimationLMS estimation

minus25

minus15

minus5

minus10

minus20

Figure 5 Envelope of estimated fading process using the variousestimators SNR = 15 dB119872 = 2048119873119901 = 1024 and 119891119889119879119904 = 00167

4 Simulation Results

In this section we carry out a comparative simulation studyon the estimation of FBMCOQAM fading channels betweenthe proposed dual Kalman filters based channel estimatorand the conventional LMS and RLS channel estimators Inaddition we test the performance of three interpolationtechniques namely low-pass spline and linear interpolationFurthermore we investigate the effect of the number of pilotsymbols119873119901 and Doppler rate 119891119889119879119904 on the BER performance

In all of our simulations the fading channels are gener-ated according to the autoregressive based method presentedin [31]The autoregressivemodel order119901 in the proposed dualKalman filters based estimator is set to 119901 = 2 The step sizefor LMS algorithm is set to 120583 = 05 and the forgetting factorfor RLS algorithm is set to 120582 = 01

Figure 5 shows the envelope of estimated fading processusing the various channel estimators with low-pass interpola-tion SNR = 15 dB119872 = 2048119873119901 = 1024 and119891119889119879119904 = 00167One can notice that the dual Kalman filters based estimatorprovidesmuch better estimation than the LMS andRLS basedones

Figure 6 shows the BER performance versus SNR forthe FBMCOQAM system when using the various channelestimators with 119872 = 2048 119873119901 = 1024 pilots 119891119889119879119904 =

00167 and using low-pass interpolationAccording to Figure6 the proposed dual Kalman filters based estimator out-performs the conventional LMS and RLS estimators withthe performance difference increasing as the SNR increasesFigure 7 displays the BER performance of the FBMCOQAMsystem versus Doppler rate when using the various channelestimators with low-pass interpolation 119872 = 2048 and119873119901 = 1024 pilots One can notice that the dual Kalman filters

0 2 4 6 8 10 12 14 16 18 20Signal-to-noise ratio (SNR) (dB)

Bit e

rror

rate

(BER

)

Known channelLMS estimationRLS estimationDual Kalman filters estimation

10minus3

10minus2

10minus1

100

Figure 6 BER performance versus SNR for the FBMCOQAMsystem with the various channel estimators 119872 = 2048 119873119901 = 1024and 119891119889119879119904 = 00167

001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100

Figure 7 BER performance versus Doppler rate for theFBMCOQAM system with the various channel estimatorsSNR = 15 dB119872 = 2048 and 119873119901 = 1024

based estimator yields lower BER performance than the LMSand RLS estimators with the BER performance differenceincreasing as the Doppler rate increases This confirms thatthe dual Kalman filters based estimator can exploit the fadingchannel statistics and can track fast variations of rapidly vary-ing fading channels



1024 pilots512 pilots256 pilots

128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100

Figure 8 BER performance versus SNR for the FBMCOQAMsystem with different number of pilot symbols 119872 = 2048 and119891119889119879119904 = 00167

Low-pass interpolationSpline interpolationLinear interpolation


Bit e

rror

rate

(BER

)

10minus2

10minus1

100

Figure 9 BER performance versus SNR for the FBMCOQAMsystem with the various interpolation methods 119872 = 2048 119873119901 =

256 and 119891119889119879119904 = 00167

The effect of changing the number of pilot symbols 119873119901

on the BER performance of the FBMCOQAM system whenusing dual Kalman filters based channel estimator with low-pass interpolation is shown in Figure 8 Indeed increasingthe number of pilot symbols 119873119901 will improve the BERperformance of the system

Figure 9 shows the BER performance of the FBMCOQAM system when using dual Kalman filters with thevarious interpolation methods 119872 = 2048 119873119901 = 256 pilotsand 119891119889119879119904 = 00167 It is confirmed that the low-pass inter-polation yields the best BER performance

5 Conclusions

This paper addresses the estimation and equalization ofFBMCOQAM fading channels based on pilot symbols Adual Kalman filters based structure is proposed for the jointestimation of the fading process and its AR parameters overeach subcarrier at pilot symbol positions The fading processat data symbol positions is then obtained by using some inter-polation methods such as linear spline or low-pass interpo-lation The simulation results showed that the proposed dualKalman filters based estimator outperforms the LMS andRLSbased ones In addition the low-pass interpolation is con-firmed to outperform both spline and linear interpolation

Conflict of Interests

The authors declare that they have no conflict of interestsregarding the publication of this paper

References

[1] ZWang andG B Giannakis ldquoWireless multicarrier communi-cations where Fourier meets Shannonrdquo IEEE Signal ProcessingMagazine vol 17 no 3 pp 29ndash48 2000

[2] G Li andG StuberOrthogonal FrequencyDivisionMultiplexingfor Wireless Communications Springer New York NY USA2006

[3] K Baum B Classon and P Sartori Principles of BroadbandOFDMCellular System Design Wiley-Blackwell New York NYUSA 2009

[4] B Farhang-Boroujeny ldquoOFDM versus filter bank multicarrierrdquoIEEE Signal Processing Magazine vol 28 no 3 pp 92ndash112 2011

[5] H Zhang D Le Ruyet D Roviras Y Medjahdi and H SunldquoSpectral efficiency comparison of OFDMFBMC for uplinkcognitive radio networksrdquoEurasip Journal onAdvances in SignalProcessing vol 2010 Article ID 621808 2010

[6] P Siohan C Siclet and N Lacaille ldquoAnalysis and designof OFDMOQAM systems based on filterbank theoryrdquo IEEETransactions on Signal Processing vol 50 no 5 pp 1170ndash11832002

[7] M K Ozdemir and H Arslan ldquoChannel estimation for wirelessOFDM systemsrdquo IEEE Communications Surveys amp Tutorialsvol 9 no 2 pp 18ndash48 2007

[8] S Coleri M Ergen A Puri and A Bahai ldquoChannel estimationtechniques based on pilot arrangement in OFDM systemsrdquoIEEE Transactions on Broadcasting vol 48 no 3 pp 223ndash2292002

[9] J H Manton ldquoOptimal training sequences and pilot tones forOFDM systemsrdquo IEEE Communications Letters vol 5 no 4 pp151ndash153 2001

[10] W Chen and R Zhang ldquoEstimation of time and frequencyselective channels in OFDM systems a Kalman filter structurerdquoin Proceedings of the IEEE Global Telecommunications Confer-ence (GLOBECOM rsquo04) pp 800ndash803 December 2004


[11] A Jamoos A Abdo and H Abdel Nour ldquoEstimation of OFDMtime-varying fading channels based on two-cross-coupledKalman filtersrdquo in Proceedings of the International Joint Con-ferences on Computer Information and Systems Sciences andEngineering (CISSE rsquo07) University of Bridgeport December2007

[12] C Lele J-P Javaudin R Legouable A Skrzypczak and P Sio-han ldquoChannel estimationmethods for preamble-basedOFDMOQAMmodulationsrdquo European Transactions on Telecommuni-cations vol 19 no 7 pp 741ndash750 2008

[13] E Kofidis D Katselis A Rontogiannis and S TheodoridisldquoPreamble-based channel estimation in OFDMOQAM sys-tems a reviewrdquo Signal Processing vol 93 no 7 pp 2038ndash20542013

[14] E Kofdisa and D Katselisb ldquoImproved interference approx-imation method for preamble based channel estimation inFBMCOQAMrdquo in Proceedings of the 19th European SignalProcessing Conference (EUSIPCO rsquo11) Barcelona Spain 2011

[15] T H Stitz T Ihalainen A Viholainen and M Renfors ldquoPilot-based synchronization and equalization in filter bank multi-carrier communicationsrdquo Eurasip Journal on Advances in SignalProcessing vol 2010 Article ID 741429 14 pages 2010

[16] C Lele ldquoIterative scattered-based channel estimation methodfor OFDMOQAMrdquo EURASIP Journal on Advances in SignalProcessing vol 2012 p 42 2012

[17] A Ikhlef and J Louveaux ldquoAn enhanced MMSE per subchan-nel equalizer for highly frequency selective channels for FBMCOQAM systemsrdquo in Proceedings of the IEEE 10th Workshopon Signal Processing Advances in Wireless Communications(SPAWC rsquo09) pp 186ndash190 Perugia Italy June 2009

[18] D S Waldhauser L G Baltar and J A Nossek ldquoAdaptive deci-sion feedback equalization for filter bank based multicarriersystemsrdquo in Proceedings of the IEEE International Symposium onCircuits and Systems (ISCAS rsquo09) pp 2794ndash2797 May 2009

[19] D N KalofonosM Stojanovic and J G Proakis ldquoPerformanceof adaptive MC-CDMA detectors in rapidly fading rayleighchannelsrdquo IEEE Transactions on Wireless Communications vol2 no 2 pp 229ndash239 2003

[20] A Jamoos E Grivel N Shakarneh and H Abdel-Nour ldquoDualoptimal filters for parameter estimation of amultivariate autore-gressive process from noisy observationsrdquo IET Signal Proc-essing vol 5 no 5 pp 471ndash479 2011

[21] S D Sandberg and M A Tzannes ldquoOverlapped discrete multi-tone modulation for high speed copper wire communicationsrdquoIEEE Journal on Selected Areas in Communications vol 13 no9 pp 1571ndash1585 1995

[22] G Cherubini E Eleftheriou S Olcer and J M Cioffi ldquoFilterbank modulation techniques for very high-speed digital sub-scriber linesrdquo IEEE Communications Magazine vol 38 no 5pp 98ndash104 2000

[23] L Lin and B Farhang-Boroujeny ldquoCosine-modulated mul-titone for very-high-speed digital subscriber linesrdquo EurasipJournal on Applied Signal Processing vol 2006 Article ID 1932916 pages 2006

[24] S Mirabbasi and K Martin ldquoOverlapped complex-modulatedtransmultiplexer filters with simplified design and superiorstopbandsrdquo IEEE Transactions on Circuits and Systems II vol50 no 8 pp 456ndash469 2003

[25] J Alhava and M Renfors ldquoExponentially-modulated filterbank-based transmultiplexerrdquo in Proceedings of the 2003 IEEEInternational Symposium on Circuits and Systems vol 4 pp233ndash236 Bangkok Thailand May 2003

[26] P P VaidyanathanMultirate Systems and Filter Banks Prentice-Hall Englewood Cliffs NJ USA 1993

[27] M G Bellanger ldquoSpecification and design of a prototype filterfor filter bank based multicarrier transmissionrdquo in Proceedingsof the IEEE Interntional Conference on Acoustics Speech andSignal Processing pp 2417ndash2420 Salt Lake City Utah USAMay2001

[28] P Martin-Martin R Bregovic A Martin-Marcos F Cruz-Roldan and T Saramaki ldquoA generalized window approach fordesigning transmultiplexersrdquo IEEE Transactions on Circuits andSystems I vol 55 no 9 pp 2696ndash2706 2008

[29] A Viholainen T Ihalainen T Stitz M Renfors and MBellanger ldquoPrototype filter design for filter bank based multi-carrier transmissionrdquo in Proceedings of the 17th European SignalProcessing Conference (EUSIPCO rsquo09) 2009

[30] W C Jakes Microwave Mobile Communications Wiley NewYork NY USA 1974

[31] K E Baddour and N C Beaulieu ldquoAutoregressive modelingfor fading channel simulationrdquo IEEE Transactions on WirelessCommunications vol 4 no 4 pp 1650ndash1662 2005

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



In the framework of multicarrier systems many channelestimation techniques were proposed in the literature par-ticularly for OFDM systems [7ndash11] They can be classifiedas training sequencepilot symbols based techniques andblind methods In this paper as blind techniques requirea large amount of data and have higher complexity thantrainingpilot based techniques we will focus our attentionon the latter techniques

Compared to OFDM few FBMCOQAM studies haveaddressed the problem of channel estimation and equal-ization [12ndash18] Indeed when classical channel estimationmethods used for OFDM systems are applied directly toFBMCOQAM system an intrinsic intersymbol-interferenceis observed This results in system performance degradationTo reduce this intrinsic interference several methods arerecently proposed including both preamble-based [12ndash14]and scattered pilots-based [15 16] channel estimators In [17]a two-stage minimum mean square error (MMSE) equal-izer is proposed to cope more efficiently with intercarrierinterference (ICI) A fractionally spaced adaptive decisionfeedback equalizer (DFE) based on the least mean square(LMS) algorithmwas proposed in [18]where the input of eachequalizer comprises only the output of each subcarrier

When dealing with Kalman filter based channel estima-tors several approaches were developed for multicarrier sys-tems such as multicarrier code division multiple access (MC-CDMA) and OFDM [10 11 19] Indeed the Kalman filtercombined with an autoregressive (AR) model to describe thetime evolution of the fading channel is shown to track channelvariations and to provide superior bit error rate (BER) per-formance over the standard LMS and recursive least square(RLS) estimators [19] Nevertheless the AR parameters areunknown and should be estimated To jointly estimate theautoregressive process and its parameters from noisy obser-vations one of the authors of this paper has recently proposeddual Kalman filters based structure [11 20] This structureconsists of two cross-coupled Kalman filters where the firstfilter uses the latest estimated AR parameters to estimatethe fading process while the second filter uses the estimatedfading process to update the AR parameters In this paper wepropose to investigate the relevance of the dual Kalman filtersfor the estimation of FBMCOQAM time-varying fadingchannels based on pilot symbols Once the fading channelcoefficients at pilot symbol positions are estimated by the pro-posed method the fading channel coefficients at data symbolpositions are then to be estimated by using some interpola-tionmethods such as linear spline or low-pass interpolation

The remainder of the paper is organized as follows InSection 2 we explain the FBMCOQAM system model Theestimation of the fading channels based on dual Kalman fil-ters is introduced in Section 3 Simulation results are reportedin Section 4 Conclusion remarks are drawn in Section 5

2 FBMCOQAM System Model

There are different filter bank multicarrier structures thatare proposed in the literature Among them are discretewavelet multitone [21] filtered multitone (FMT) [22] cosinemodulated multitone (CMT) [23] modified discrete Fourier

transform (MDFT) [24] exponentially modulated filter bank(EMFB) [25] and FBMCOQAM [6] The last is the mostpopular filter bank scheme that results in high bandwidthefficiency

The fundamental parts of the FBMCOQAM system arethe synthesis filter bank (SFB) at the transmitter and theanalysis filter bank (AFB) at the receiver arranged in theso-called transmultiplexer (TMUX) configuration as shownin Figures 1 and 2 It should be noted that the SFB and AFBpairs can be efficiently implemented using FFT and IFFTof size 119872 aided by polyphase filtering structures Whenusing uniformly modulated filter banks [26] a prototypefilter 119901[119898] of length 119871119901 is shifted in frequency to generatesubchannels which cover the whole system bandwidthThusthe discrete-time baseband signal at the output of the FBMCtransmitter with OQAMmodulation can be expressed as

119910 [119898] =

119872minus1

sum

119896=0

infin

sum

119899=minusinfin

119909119896 [119899] 120573119896 [119899] 119901 [119898 minus119899119872

2] 119890119895(2120587119872)119896119898

(1)

where119898 is the sample index at SFB outputAFB input 119899 is thesample index atOQAMpreprocessing outputpostprocessinginput119872 is the number of subcarriers in the filter bank and

120573119896 [119899] = (minus1)119896119899 exp(minus119895

2120587119896

119872(

119871119901 minus 1

2)) (2)

119909119896 [119899] = 119889119896 [119899] 120579119896 [119899] (3)

120579119896 [119899] = 119895119896+119899

(4)

In (3) 119889119896[119899] are real-valued data symbols at subcarrier119896 transmitted at a rate of 2119879 where the signaling period isdefined as 119879 = 1Δ119891 with Δ119891 being the subcarrier spacingThepair of symbols119889119896[119899] and119889119896[119899+1] represent respectivelythe in-phase and quadrature parts of the complex-valuedinput symbol 119888119896[119897] This complex-to-real conversion opera-tion is considered to introduce an up-sampling by two Theinverse operation of real-to-complex conversion is performedat the receiver and results in down-sampling the signal by twoThus the complex-to-real operation performs the followingmapping

119889119896 [119899] = Re (119888119896 [119897]) 119896 evenIm (119888119896 [119897]) 119896 odd

119889119896 [119899 + 1] = Im (119888119896 [119897]) 119896 evenRe (119888119896 [119897]) 119896 odd

(5)

where 119897 is the sample index at OQAM preprocessinginputpostprocessing output Note that the complex-valuedinput symbols 119888119896[119897] belong to QAM alphabet with the realand imaginary parts being interleaved by a time offset of 1198792resulting in OQAM modulation To maintain orthogonality119889119896[119899] is multiplied by 120579119896[119899]

In order to achieve high spectral efficiency complexmodulated filter banks are usually used which means thatall subchannel filters are frequency shifted versions of the


OQAM preprocessing

Inverse

fast

Fourier

transform



Complexto real

Complexto real

Complexto real

120579Mminus1[n]


1205790[n]

1205791[n]

c1[l]

d0[n] x0[n]

d1[n] x1[n]

120573Mminus1[n]

1205730[n]

1205731[n]

AMminus1[Z2]

A1[Z2]

A0[Z2] uarr

M

2

uarrM

2

uarrM

2

Zminus1

Zminus1

y[m]c0[l]


Real

Kalman

filtering

based

channel

estimation

and

equalization

Fast

Fourier

transform

Real

Real



r[m]

darrM

2

darrM

2

darrM

2

Zminus1

Zminus1

B0[Z2]

B1[Z2]

BMminus1[Z2]

s1[n]

s0[n]

sMminus1[n]


Realto complex

Realto complex

Realto complex

d0[n]

d1[n]

c0[l]

c1[l]x1[n]

x0[n]

xMminus1[n]

120573lowast1 [n]


120579lowast1 [n]




119892119896 [119898] = 119901 [119898] exp(1198952120587119896

119872(119898 minus

119871119901 minus 1

2)) (6)



119891119896 [119898] = 119892lowast

119896[119871119901 minus 1 minus 119898] (7)






119901 [119898]

= 119875 [0]

+ 2

119880

sum

119896=1

(minus1)119896119875 [119896] cos( 2120587119896

119870119872(119898 +

119870119872 minus (119871119901 minus 1)

2))

(8)


2= 1

for 119888 = 1 2 1198702 119875[0] = 0 for 119896 = 119870 119870 + 1 119880 =



119886119896 [119898] = 119901 [119896 + 119898119872] (9)




119903119896 [119898] = 119910119896 [119898] ℎ119896 [119898] + 119908119896 [119898] (11)




2

119908 The




2

ℎ


119904119896 [119899] = [119903119896 [119898] lowast 119891119896 [119898]]darr1198722

= 119909119896 [119899] sdot 119902119896 [119899] + 120578119896 [119899]

(12)



2



darr1198722cong


119904119896 [119899] = 119909119896 [119899] ℎ119896 [119899] + 120578119896 [119899] 119896 = 0 1 119872 minus 1 (13)


119878 [119891] =

1

120587119891119889radic1 minus (119891119891119889)

2

10038161003816100381610038161198911003816100381610038161003816 le 119891119889

0 elsewhere(14)


119877119896 = 1198690 (2120587119891119889119879119904 |119896|) (15)



ℎ119896 [119899] = minus

119901

sum

119894=1

119886119894ℎ119896minus119894 [119899] + 120592119896 [119899] (16)






Time

Freq

uenc

y

Pilot

Data





119909119896 [119899] = 119909119906119871+119894 [119899]

=

119909119906119871pilot [119899] 119894 = 0

119909119906119871+119894data [119899] 119894 = 1 2 119871 minus 1

(17)









hk[n]sk[n]

xkpilot[n]

aii=1 2 p



h119896 = 120601h119896minus1 + g120592119896 (19)

where

120601 =[[[[

[



d


]]]]

]


(20)


119904119896 = x119879119896h119896 + 120578119896 (21)





119862119896 = 119864 [120572119896120572lowast

119896] = x119879119896P119896119896minus1x119896 + 120590

2

120578119896 (23)


P119896119896minus1 = 120601P119896119896minus1120601119867

+ g1205902120592g119879 (24)


K119896 = P119896119896minus1x119896119862minus1

119896 (25)



h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)


P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)



2

120592 They will be




a119896 + 120589119896 (28)




1205902

120589119896= g119879K119896119862119896K

H119896g (29)


a119896 = a119896minus1 (30)



119896minus1a119896minus1) (31)



119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1



(32)



2



P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)


b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)


1205902


119867+ K119896119862119896K

119867

119896] f119879 (35)


2


2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]


+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)





ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)


position




119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










OQAM preprocessing

Inverse

fast

Fourier

transform



Complexto real

Complexto real

Complexto real

120579Mminus1[n]


1205790[n]

1205791[n]

c1[l]

d0[n] x0[n]

d1[n] x1[n]

120573Mminus1[n]

1205730[n]

1205731[n]

AMminus1[Z2]

A1[Z2]

A0[Z2] uarr

M

2

uarrM

2

uarrM

2

Zminus1

Zminus1

y[m]c0[l]


Real

Kalman

filtering

based

channel

estimation

and

equalization

Fast

Fourier

transform

Real

Real



r[m]

darrM

2

darrM

2

darrM

2

Zminus1

Zminus1

B0[Z2]

B1[Z2]

BMminus1[Z2]

s1[n]

s0[n]

sMminus1[n]


Realto complex

Realto complex

Realto complex

d0[n]

d1[n]

c0[l]

c1[l]x1[n]

x0[n]

xMminus1[n]

120573lowast1 [n]


120579lowast1 [n]




119892119896 [119898] = 119901 [119898] exp(1198952120587119896

119872(119898 minus

119871119901 minus 1

2)) (6)



119891119896 [119898] = 119892lowast

119896[119871119901 minus 1 minus 119898] (7)






119901 [119898]

= 119875 [0]

+ 2

119880

sum

119896=1

(minus1)119896119875 [119896] cos( 2120587119896

119870119872(119898 +

119870119872 minus (119871119901 minus 1)

2))

(8)


2= 1

for 119888 = 1 2 1198702 119875[0] = 0 for 119896 = 119870 119870 + 1 119880 =



119886119896 [119898] = 119901 [119896 + 119898119872] (9)




119903119896 [119898] = 119910119896 [119898] ℎ119896 [119898] + 119908119896 [119898] (11)




2

119908 The




2

ℎ


119904119896 [119899] = [119903119896 [119898] lowast 119891119896 [119898]]darr1198722

= 119909119896 [119899] sdot 119902119896 [119899] + 120578119896 [119899]

(12)



2



darr1198722cong


119904119896 [119899] = 119909119896 [119899] ℎ119896 [119899] + 120578119896 [119899] 119896 = 0 1 119872 minus 1 (13)


119878 [119891] =

1

120587119891119889radic1 minus (119891119891119889)

2

10038161003816100381610038161198911003816100381610038161003816 le 119891119889

0 elsewhere(14)


119877119896 = 1198690 (2120587119891119889119879119904 |119896|) (15)



ℎ119896 [119899] = minus

119901

sum

119894=1

119886119894ℎ119896minus119894 [119899] + 120592119896 [119899] (16)






Time

Freq

uenc

y

Pilot

Data





119909119896 [119899] = 119909119906119871+119894 [119899]

=

119909119906119871pilot [119899] 119894 = 0

119909119906119871+119894data [119899] 119894 = 1 2 119871 minus 1

(17)









hk[n]sk[n]

xkpilot[n]

aii=1 2 p



h119896 = 120601h119896minus1 + g120592119896 (19)

where

120601 =[[[[

[



d


]]]]

]


(20)


119904119896 = x119879119896h119896 + 120578119896 (21)





119862119896 = 119864 [120572119896120572lowast

119896] = x119879119896P119896119896minus1x119896 + 120590

2

120578119896 (23)


P119896119896minus1 = 120601P119896119896minus1120601119867

+ g1205902120592g119879 (24)


K119896 = P119896119896minus1x119896119862minus1

119896 (25)



h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)


P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)



2

120592 They will be




a119896 + 120589119896 (28)




1205902

120589119896= g119879K119896119862119896K

H119896g (29)


a119896 = a119896minus1 (30)



119896minus1a119896minus1) (31)



119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1



(32)



2



P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)


b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)


1205902


119867+ K119896119862119896K

119867

119896] f119879 (35)


2


2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]


+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)





ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)


position




119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119901 [119898]

= 119875 [0]

+ 2

119880

sum

119896=1

(minus1)119896119875 [119896] cos( 2120587119896

119870119872(119898 +

119870119872 minus (119871119901 minus 1)

2))

(8)


2= 1

for 119888 = 1 2 1198702 119875[0] = 0 for 119896 = 119870 119870 + 1 119880 =



119886119896 [119898] = 119901 [119896 + 119898119872] (9)




119903119896 [119898] = 119910119896 [119898] ℎ119896 [119898] + 119908119896 [119898] (11)




2

119908 The




2

ℎ


119904119896 [119899] = [119903119896 [119898] lowast 119891119896 [119898]]darr1198722

= 119909119896 [119899] sdot 119902119896 [119899] + 120578119896 [119899]

(12)



2



darr1198722cong


119904119896 [119899] = 119909119896 [119899] ℎ119896 [119899] + 120578119896 [119899] 119896 = 0 1 119872 minus 1 (13)


119878 [119891] =

1

120587119891119889radic1 minus (119891119891119889)

2

10038161003816100381610038161198911003816100381610038161003816 le 119891119889

0 elsewhere(14)


119877119896 = 1198690 (2120587119891119889119879119904 |119896|) (15)



ℎ119896 [119899] = minus

119901

sum

119894=1

119886119894ℎ119896minus119894 [119899] + 120592119896 [119899] (16)






Time

Freq

uenc

y

Pilot

Data





119909119896 [119899] = 119909119906119871+119894 [119899]

=

119909119906119871pilot [119899] 119894 = 0

119909119906119871+119894data [119899] 119894 = 1 2 119871 minus 1

(17)









hk[n]sk[n]

xkpilot[n]

aii=1 2 p



h119896 = 120601h119896minus1 + g120592119896 (19)

where

120601 =[[[[

[



d


]]]]

]


(20)


119904119896 = x119879119896h119896 + 120578119896 (21)





119862119896 = 119864 [120572119896120572lowast

119896] = x119879119896P119896119896minus1x119896 + 120590

2

120578119896 (23)


P119896119896minus1 = 120601P119896119896minus1120601119867

+ g1205902120592g119879 (24)


K119896 = P119896119896minus1x119896119862minus1

119896 (25)



h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)


P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)



2

120592 They will be




a119896 + 120589119896 (28)




1205902

120589119896= g119879K119896119862119896K

H119896g (29)


a119896 = a119896minus1 (30)



119896minus1a119896minus1) (31)



119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1



(32)



2



P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)


b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)


1205902


119867+ K119896119862119896K

119867

119896] f119879 (35)


2


2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]


+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)





ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)


position




119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time

Freq

uenc

y

Pilot

Data





119909119896 [119899] = 119909119906119871+119894 [119899]

=

119909119906119871pilot [119899] 119894 = 0

119909119906119871+119894data [119899] 119894 = 1 2 119871 minus 1

(17)









hk[n]sk[n]

xkpilot[n]

aii=1 2 p



h119896 = 120601h119896minus1 + g120592119896 (19)

where

120601 =[[[[

[



d


]]]]

]


(20)


119904119896 = x119879119896h119896 + 120578119896 (21)





119862119896 = 119864 [120572119896120572lowast

119896] = x119879119896P119896119896minus1x119896 + 120590

2

120578119896 (23)


P119896119896minus1 = 120601P119896119896minus1120601119867

+ g1205902120592g119879 (24)


K119896 = P119896119896minus1x119896119862minus1

119896 (25)



h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)


P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)



2

120592 They will be




a119896 + 120589119896 (28)




1205902

120589119896= g119879K119896119862119896K

H119896g (29)


a119896 = a119896minus1 (30)



119896minus1a119896minus1) (31)



119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1



(32)



2



P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)


b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)


1205902


119867+ K119896119862119896K

119867

119896] f119879 (35)


2


2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]


+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)





ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)


position




119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











h119896119896 = 120601h119896minus1119896minus1 + K119896120572119896

ℎ119896 = ℎ119896119896 = g119879h119896119896(26)


P119896119896 = P119896119896minus1 minus K119896x119879

119896P119896119896minus1 (27)



2

120592 They will be




a119896 + 120589119896 (28)




1205902

120589119896= g119879K119896119862119896K

H119896g (29)


a119896 = a119896minus1 (30)



119896minus1a119896minus1) (31)



119896minus1(h119867119896minus1

Pa119896minus1 h119896minus1 + 1205902

120589119896)minus1



(32)



2



P119896119896 = 120601P119896minus1119896minus1120601119867

+ g1205902120592119896g119879 minus K119896x

119879

119896P119896119896minus1 (33)


b119879119896P119896119896minus1 = 119862119896K

119867

119896 (34)


1205902


119867+ K119896119862119896K

119867

119896] f119879 (35)


2


2

120592119896= 1205821205902

120592119896minus1+ [1 minus 120582]


+ K1198961003816100381610038161003816120572119896

1003816100381610038161003816

2K119867119896] f119879

(36)





ℎ119896 = (ℎ119901119896+1minus ℎ119901119896

)119897

119871+ ℎ119901119896

0 le 119897 lt 119871 (37)


position




119909119896 [119899] = 119904119896 [119899](ℎlowast

119896 [119899]

10038161003816100381610038161003816ℎ119896[119899]

10038161003816100381610038161003816

2) (38)


100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










100 110 120 130 140 150 160 170 180 190 200

0

5

10

Symbols

Enve

lope

(dB)


minus25

minus15

minus5

minus10

minus20









Bit e

rror

rate

(BER

)


10minus3

10minus2

10minus1

100


001 002 003 004 005 006 007 008 009 01


Doppler rate fd Ts

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100






128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












128 pilots64 pilots

Bit e

rror

rate

(BER

)

10minus3

10minus2

10minus1

100




Bit e

rror

rate

(BER

)

10minus2

10minus1

100


256 and 119891119889119879119904 = 00167




5 Conclusions




References



































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









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Research Article Estimation of FBMC/OQAM Fading ...downloads.hindawi.com › journals › tswj › 2014 › 586403.pdflter banks with o set quadrature amplitude modulation (FBMC/OQAM),