channel estimation in fast fading mobile coded ofdm

8/14/2019 Channel Estimation in Fast Fading Mobile Coded OFDM

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Channel Estimation in Fast Fading Mobile CodedOFDM

Daniel N. Liu and Michael P. FitzDepartment of Electrical Engineering

University of California Los Angeles, Los Angeles, CA, 90095Email: {daniell and fitz}@ee.ucla.edu

Abstract-Orthogonal frequency division multiplexing(OFDM) systems suffer performance degradation in fast fadingchannels due to intercarrier interference (ICI). Combiningfrequency domain equalization and bit-interleaved codedmodulation (BICM), the iterative receiver is able to harvest bothtemporal and frequency diversity. In order to perform coherentdetection and estimation, channel state information (CSI) isindispensable. Being able to accurately and effectively acquireCSI in the fast time-varying channel is critical. Conventionalfrequency domain channel estimation (CE) methods have anirreducible error floor at high normalized Doppler frequency!dTc;, since ICI corrupts the orthogonality among subcarriers.Considering tha t the fast time-varying channel is also asource of temporal diversity, CE ought to take place in thepre-FFT time domain. Realizing channel variations in timeare often smooth, this paper proposes a time domain channelestimator using pilot symbol assisted modulation (PSAM) withcomplexity 0 where is the FFT size. Simulation resultsdemonstrate tha t a PSAM system with channel estimationprovides excellent trade-off between performance, complexityand spectral efficiency.

data and pilot symbols in the frequency domain. Stamoulis etal. [12] argue that in OFDM systems with non-negligible ICI,pilot tones should be grouped together into clumps equispacedwithin the the FFT grid. Despite accounting for ICI in the CEprocess, the method suggested in [12] remains a frequencydomain algorithm and only works well in the high NEb regime.To take full advantage of time diversity provided by the time-varying channels and completely avoid ICI, CE ought to beperformed in the pre-FFT time domain [2]. The simplest andhighest performing way to process and estimate time domainCSI is via linear filtering of the known pilot symbols [2], [6],[9]. The optimal linear filter in the minimum mean square error(MMSE) sense is well known as the Wiener filter [6] whichis often pre-computed and results in an open loop estimationstructure. Let y AI x 1 be time domain observation ofpilot values at the receiver, the estimated time domain CSIcan be simply obtained by:

where are the time domain Wiener filter coefficients. Thesize of the conventional time domain Wiener filter W 11\' is

x for estimating one symbol using pilotsymbols [2], where is the FFT size and 1 is the numberof discrete channel taps. Assuming pilot symbols are insertedevery Pins (i.e. pilot symbol insertion period) symbols for thepurpose of estimating CSI at the receiver side, the overall sizeof W conv in (1) becomes:

A careful examination of (2) reveals that the size of convquickly becomes impracticable when either or growslarge. In this paper, a computationally feasible and moreefficient time domain MMSE Wiener filter with size (AI l )PinsSrn N ~ A 1 is derived, where rn is the number ofmultipaths in the underlying channel model and is thenumber of sub-sampling points on N.Realizing the rapid time-varying channel is indeed a smoothfunction in time, a reduced size Wiener filter with manageable

dimension can be realized efficiently. Despite the capabilityof tracking and estimating the rapid time-varying channel, theconventional time domain Wiener filter has huge overheadsin terms of both actual physical memory and computationalcomplexity [2]. A straightforward implementation of (2) would

INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) attracts tremendous attention for high date rate communicationsystems. While OFDM systems have strong immunity totime-invariant (TI) FS multipath fading channels with thehelp of a guard interval it suffers severely from timevarying channels mainly due to user's mobility. Rapid channelvariation over a symbol duration destroys the orthogonalityamong subcarriers and gives rise to intercarrier interference(ICI). To suppress ICI effectively for high performing coherentdetection (and the references therein), channel stateinformation (CSI) is indispensable. Channel estimation (CE)techniques can roughly divide into two camps: one employsa transmitted reference which is commonly known as pilotsymbol assisted modulation [6]-[9]; the other is blind CE [10],[11]. This work is concerned with PSAM CE in high mobilitytime-varying channels.Being able to accurately and effectively estimate the high

mobility time-varying channel remains a challenging topic.In a highly mobile environment with a time-varying channel,ICI can be characterized by the normalized Doppler frequencyfdT.s' where fd is the maximum Doppler frequency and T.s isone OFDM symbol duration. Rapid channel variation withinan OFDM symbol duration creates an impossible task forconventional OFDM CEmethods [7], [8], which multiplex

978-1-4244-2644-7/08/$25.00 2008 IEEE

conv

(1)

(2)

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Iterative Receiver

t) ,.....------,(f)---.AWGN t ~ - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Fig. 1. A baseband equivalent model for coded OFDM system

(4)

require an enormous storage space even for OFDM parameters with nominal values. For example, using the OFDMparameters suggested in [5] and a COST typical urban (TU)channel model [13], a communication system using 16QAMmodulation with rate-1/2 Binary Convolution Code (BCC),which achieves an overall spectral efficiency of 1.25 BPCU(i.e. l'vl == 6 and Pins == 3), would need to store 2.0 10complex constants as filter coefficients. Since the FFT sizeis usually very large, N == 2048 or N == 8192 in applicationssuch as [14], the conventional time domain Wiener filterproposed in [2] may not even be feasible in a practicalsystem. Though the channel varies rapidly in time accordingto the normalized Doppler frequency fdT.." the variations areusually smooth. [12]. This crucial observation suggests thatonly a finite number of points of the CSI need be estimatedand the rest of the CSI is recovered by interpolation. Let

C be the result of uniform sampling of true CSIH C(N+N,q)xrrt on each symbol duration in time for eachmultipath. That is:

H==H ( s ( l ) ~ 1) H(s(1),2) H(s(l),rnp )H(s(2),1) H(s(2),2) H(s(2),rnp ) (3)H(s(5),1) H(s(5),2) H(s(5), p )

where = vec are the channel estimates for a n W CU'Y:! - 1 ) P i n . ~ Sm x N Recovering the estimated CSI Hfrom r e l i e s ~ o n an interpolation filter R E q suchthat == RH. In this paper, both an efficient filter Wandthe interpolation filter are derived.

II. SYSTEM MODEL AND NOTATIONSA. System ModelConsider a single-input single-output coded OFDM system

illustrated in Fig. 1. A set of -coded QAM "frequencydomain" symbols d == forms the input tothe IFFT. This paper further assumes that the average symbolenergy E s 1

2== 1 and thta symbols are equallylikely chosen from a complex constellation with cardinality== 2f1.1c , where Ale denotes number of bits per constellation

symbol. Assuming is the smallest power of 2 greater thanthe sampling time e is defined as: c == Equivalently,the discrete time domain signal can be described as:

1 '27Tkn== - L..-t 7V "IN k=Owhere N denotes the number of samples for the guard intervaland is d ~ f i n e d as: JV:q == i ~ This paper assumes a time-varying wireless multipath chan-nel with an impulse response:

where 70 :::; 71 :::; .. . :::; -1 with 7 i being the tap-delayon ith tap, and == are the randomly timevarying tap gains. Moreover, t) is modeled as a wide sensestationary uncorrelated-scattering (WSSUS) channel. The tap

where s == [s(l) s(2) .. . is a set of 5 equispaced points on N JV:q is the number of samplesfor guard interval (GI) and N Srnp Let vec denotesa column vector formed by stacking the columns of A. Withthe assumption of pilot symbols inserted every Pins symbolsand using 1\J pilot symbols for CE, (3) allows efficientimplementation of the time domain Wiener filter: h ==

rrl,p -1t, 7 ) == L t )6(7 - 7i),

i=O(5)



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N-1Y L diag ( f ~ ) [P(l)Hg1 P(2)Hg2

k=O== [ONX(N g- i+1) IN ONx(i-1) J,

(12)(13)

where R E C(N+Ng)xS . The use of channel interpolation filterR not only resolves the channel observabili ty problem completely, it also provides exceptional performance. Appendix

and k E eNx1 is the kth column of By substituting(4) into (10) and collecting the pre-FFT time-domain samplesinto y E C

NX 1 , (10) can be re-expressed as in (12)where diag denotes diagonal matr ix with entr ies read fromthe column a, E JRNx(N+Ng) V iE L is apermutation matrix which is defined in (13), gi E JRrnpx lV iE is the i th column of and E ce(L+1)X1is the first entries of f To recover the informationbearing symbols FFT is performed on the time-domainsignal y, that is:

(15)== RH,

where 11 E CNxN is the effective frequency-domain channeland Vi e N x is still AWGN. If t remains constantover Ts, then 11 is diagonal . On the other hand, 11 becomesa banded matrix as the normalized Doppler frequencyincreases. As shown in [5], the upper and lower band width of1t varies depending on the level of ICI. This paper focuses onthe development of the channel estimation and leaves detailsof equalization to the literature [5].

III. CHANNEL ESTIMATIONTo effectively and accurately estimate the time-varying high

mobility channel H in (12), this paper relies on sub-samplingand linear interpolation. The estimation of N,q) xperfect CSI per OFDM symbol poses a seemingly impossible task even when all are pilot subcarriers, since only

observation (i.e. values are available for N,q )rnpunknowns. Conventional techniques [2] approach the observabil ity problem by accumulating pilot symbols in time untilthe observations outnumber the channel unknowns. That is,using the nearest AI pilot symbols in time to estimate Hfor the desired symbol such that A1N (N Ng)rnThis is certainly impractical as the size of the estimationproblem grows exponentially with N. To reduce the numberof parameters needed for estimation from N,q )rnp to lessthan per OFDM symbol, it is important to realize that thechannel variation in time (i.e. each column of is generallysmooth [12]. This crucial observation implies that only finitenumbers of points of in time need to be est imated and ther ~ s t of can be recovered by interpolation. As shown in (3),H is the undersampled version of perfect CSI H with thenumber of sampling points S N. Thus, the original CSIcan be expressed as:

(7)

gains (t)} are complex Gaussian with zero mean andvariance rr;, where L ~ O - l a"T == 1. The autocorrelation ofthe WSSUS channel is

~ l 72)t] == R h ( ~ t ) c P T ( 7 1 ) 6 ( 7 1 72),with the assumption that the angle of arrival of the receivedsignal waveform is a uniformly distributed random variableand C)t means Hermitian transpose. ~ T2)t]is separable in time and delay. In (6), ( ~ t ) is the normalized time-correlation function and is the power delayprofile with a; == dJ Since the signal is band limited,the time-varying wireless multipath channel in (5) can berepresented as a tapped delay line (TDL) with time-varyingcoefficients and fixed tap spacing [2], [4], [15]. In this TDLchannel model, the length of the TDL is determined by themaximum delay spread (i.e. 7 and the tap spacing can beequal to or less than the reciprocal of the passband bandwidth[16]. The discrete channel coefficients at sample time andlth tap are:

where == m; 1 1. Collecting into a matrixHd E e (N+Ng )X(L+1) and using C)T to denote matrixtranspose, (7) can be rewritten as:Hd ==

where H E C(N+Ng)xm is the perfect CSI and G EJR(L+1)xrnp is defined as:

= sine ( (9)

A careful examination of (7) and (8) reveals the fact thatE L+ 1) 1 (i.e. the row of is in general

correlated through the sinc interpolator G, because the channeltap delay 7i is generally not a multiple of c Veach i.With the assumption of N,q 2 L, the discrete received signal

linearly depends on via,== L

1=0where 11)( is an i. i.d. complex Gaussian random variablewith zero mean and variance N /2 per dimension. This paperfurther denotes F N E CNxN as the discrete fast Fouriertransform (FFT) matrix which can be defined as:

(m= 1r



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A k = diagL ~ : / fk(i)G(i.1)B(i)(1)L ~ ~ l f (i)G(i, 1)B(i) (2) L ~ : / fk(i)G(i, 2)B(i)(1)1 fk(i)G(i, 2)B(i)(2) L ~ : / f i)G( i, rnp)B(i)L f ~ l fk(i)G(i, rnp)B(i)(2) (17)

fk(i)G(i, l)B(i)(N) L f ~ l fk(i)G(i, 2)B(i)(N)B(i) == P(i)R (18)

==L ~ = - O l

ONxSm pONXS111,p

L ~ = - O l

Dk=O k (Al-1)P1n.s

(21 )

p ==L ~ : O l

ON X S111,pONxSrn p

L ~ : O lO N x S n ~ pONXS111,p (22)

A gives a detai led derivat ion of R. With the help of (15),the received samples corresponding to the jt h informationbearing symbol in (12) can then be expressed in a matrix form:

(16)

where CSmp is defined as = vee and A kcNxS rnp , defined in (17) with 1X S , denotes thekth row of B(i). The received signal Y I - 1)PinsNx is acollection of each jt h individual information bearing symbolwithin the CE processing window and can be defined as:

[ T T T ]T== 1 2 (AI -1 )Pi n s (where each CE processing widow contains l)Pinsinformation bearing symbols and j [1,2, .. ,(A1 - l)Pins ].is evident that Y can be also expressed in matrix form as:

(20)where AD CN(AI-1)PinsxSrnp(M-1)Pins and h

S 111,p(Al-1)Pins x1 are defined as in (21) and[hi hf ... h ~ 1 - 1 ) P ; n 8 ] T.Aside from the information bearing symbols, a set of

known sequences (i.e. d are inserted every Pins symbolsfor the purpose of channel state estimation at the receiver. Asillustrated in Fig. 2, pilot symbols are inserted every Pins == 3symbols and the channel est imator uses == 6 nearestpilot symbol observations to acquire CSI for a particularinformation bearing symbol d In a similar manner to (21),the pilot symbol sequences are collected into a N x SrnpAJ

matrix as in (22) where Pi are the known complex pi lo tsymbols transmitted on the ith OFDM symbol time and the kthfrequency subcarrier. By collecting the received pilot signals

Alxl'Into == ~ I \0 , Itis obvious that Y linearly depends on in (22) via:

== (23)where h c Srnp Al 1 is the undersampled version of the CSIfor the pilot symbols. Notice the overall spectral efficiencyachieved by this particular PSAM communication system is:

where denotes number of information bearing symbolswithin one particular packet and R c is code rate for the outerchannel code.After extraction of the nearest pilot symbols, the initial

phase of the turbo channel estimator t r ~ s to interpolate thesesamples to construct an est imate of (i.e. as in (20 atevery sample t ime instance for every channel tap. Given theaggregate data and pilot models in (20) and (21), t ~ linearminimum mean square error (LMMSE) estimation of in (20)takes the form:

(25)The solut ion to (25) is the LMMSE estimator W which is thewell known Wiener filter [6], [9]:

E [hyt] E [ypytJ -1 yp,= R h h } - ~ ( A p R h p h ) \ ~ N INA1 ) -1

W



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Frequencysubcarrier

N -3

53

Fig. 2. Signaling scheme for the high mobility OFDM system

where R hh" = E [hht] is the cross-correlation matrix between data channel and pilot channel and R hphp =E [hphtis the auto-correlation matrix of the pilot channel. T h covan-

~ ~ ance of the channel state estimation error he == h - h can becomputed as:

E[heh!] = E [ ( h - h ) ( h ~ h ) t ]=R hii - E [ h Y ~ ] E [ Y p Y ~ ] E [ypht], (27)

where E [heh!] in ", in A IV. NUMERICAL RESULTS

This section presents computer simulation results of the proposed front-end equalizers with application to IEEE 802.16emobile WiMAX standard. The number of subcarriers is assumed to be K == 256 and the length of the guard intervalis T.q == T.s /4. The outer channel code is the de factostandard 64-state rate-I/2 binary convolutional code (BCC)with polynomials (133,171)8' With different BCC code ratesand constellation mappings (i.e. QPSK, 16QAM and 64QAM),a variety of spectral efficiencies can be achieved. This paperconsiders the COST typical urban channel model [13]with independent Rayleigh faded rays. The ray's relative powerand delay are:

dB ==[-4, -3.0, -2.6, - 3, - 5, - 7, - 5, -6 .5 , -8 .6 , -11, -10] ,Tits == [0, 0.1, 0.3, 0.5, 0.8, 1.1, 1.3, 1.7, 2.3, 3.1, 3.2, 5].It is further assumed that perfect timing synchronization isused for the iterative receiver. At the receiver side, up to

1 0 - ' O ~ ~ ~ ~ ~ - - - : l = - - - - + - 1 5 - - ~ - - - - : ! : - - - - - - - - - J

Fig. 3. BER Performance for Reduced-Sta te MAP Equalizer (RM) withPSAM, = fdT.


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Fig. 4. BER Performance for Reduced-State MAP (RM) Equalizer withPSAM, = 256, fdT.s = 20.46%, 16QAM. 64-State Rate-1/2 BCC withCOST-TU Channel

PSAM system with up to Pins == 6 (i.e. assuming 1\:1 == 6),which achieved 1.61 BPCU, still outperforms the conventionfrequency domain CE by r-v 2.5 dB at 10 -4 with 27.8% betterspectral efficiency.

Fig. 4 shows the performance of CE with various spectralefficiencies of the coded OFDM system with normalizedDoppler frequency fdT.5 == 20.46%. At normalized Dopplerfrequency as high as 20.46%, conventional frequency domainCE shows an error floor at a BER of 10-2 due to theunaccounted ICI. Be ing able to harvest both temporal andfrequency diversity, the i terative receiver [5] with PCSI atthe 4th iterat ion provides a gain of r-v 2.2 dB at a BER of10 -4 compared to no ICI. The PSAM system with Pins == 3and Pins == 4, which achieved 1.25 BPCU and 1.43 BPCUrespectively, are shown.

V. CONCLUSIONChannel estimation of the wireless mobile coded OFDM

system is considered. A robust and efficient channel estimator using PSAM is developed which provide excellentperformance with linear complexity. Time domain CE alongwith the iterative receiver is able to exploit the time-varyingchannel as a source of temporal diversity. The performanceof CE has been evaluated over the COST-TU channel modelwith application to WiMAX. The results suggest that iterativeprocessing at the receiver end allows full exploitation of bothtemporal and frequency diversity avai lable in a spectrallyefficient manner.

10'"

1 0

-- .....\\ .....~ "-

....", '\..\\ ,

\--

interpolation filter is:== -1 , (29)

where R = E [Hilt] and RHH = E [ililt].REFERENCES

11] Y. H. Kim, Song, H. G. Kim, Chang, and H. M. Kim, "Performanceanalysis of a coded OFDM system in time-varying multipath Rayleighfading channels," IEEE Trans. Veh. Tech., vol. 48, pp. 1610-1615, Sep.1999.

[2] Y .-S . Choi, P. J. Voltz. and A. Cassara, "On channel estimation anddetection for multicarrier signals in fast and selective Rayleigh fadingchannels," IEEE Trans. Commun., vol. 49, pp. 1375-1387, Aug. 2001.[3] S. Kim and G. Pottie. "Robust OFDM in fast fading channels." in

Proc. IEEE Global Telecommunications ConI, 2003, pp. 1074-1078.[4J X. Cai and G. B. Giannakis. "Bounding performance and suppressingintercarrier interference in wireless mobile OFDM," IEEE Trans. Com-

mun .. vol. 51, pp. 2047-2056, Dec. 2003.[5] D. N. Liu and M. P. Fitz. "Iterative MAP equalization and decoding inwireless mobile coded OFDM:' submitted to IEEE Trans. Commun.[6] J. K. Cavers, "An analysis of pilot symbol assisted modulation for

Rayleigh faded channels." IEEE Trans. Veh. Tech .. vol. vol. 40. pp. 686693, Nov. 1991.[7J R. Negi and Cioffi, "Pi lo t tone selec tion for channel estimation in amobile OFDM system." IEEE Trans. Consumer Electronics, vol. 44, pp.1122-1128, Aug. 1998.[8] Y. Li, L. J. Cimini. and N. R. Sollenberger, "Robust channel estimationfor OFDM systems with rapid dispersive fading channels," IEEE Trans.

Commun., vol. 7. pp. 902-915. July 1998.[9] J. C. Guey, M. P. Fitz, M. R. Bell. and W. Y. Kuo, "Signal designfor transmitter diversity wireless communication systems over Rayleighfading channels." IEEE Trans. Commun., vol. 47. pp. 527-537, Apr.1999.[10] M. C. Necker and G. L. Stiiber, "Totally blind channel estimation forOFDM on fast varying mobile radio channels," IEEE Trans. Wireless

Commun., vol. 3. pp. 1514-1525, Sep. 2004.[11] Cui and C. Tellambura, ""Joint data detection and channel estimationfor OFDM systems," IEEE Trans. Commun., vol. 54, pp. 670-679, Apr.

2006.[12] A. Stamoulis. S. N. Diggavi, and N. AI-Dhahir. "Estimation of fastfading channels in OFDM," in Proc. IEEE Wireless Communicationsand Networking ConI. vol. 1, Mar. 2002. pp. 465-470.

[13] Commission of the European Communities, "Digital Land Mobile RadioCommunica tions - COST 207, Final Report ." Office for OfficialPublications of the European Communities, Luxembourg, Tech. Rep..1989.

[14J ETSI EN 300 744 v1.5.1 (2004-2006), "Digital Video Broadcasting(DVB); Framing structure, channel coding and modulation for digitalterrestrial television ," European Broadcasting Union, Tech. Rep.. 2006.[15J P. A. Bello. "Characterization of randomly time-variant linear channels,"

IEEE Trans. Commun.. vol. 11. pp. 360-393, Dec. 1963.[16J J. G. Proakis. Digital Communications. 4th ed. Boston: McGraw-Hill ,2001.

ApPENDIXDERIVATION OF CHANNEL INTERPOLATION FILTER RWith the assumption of a priori information about the

underlying channel model is available, the optimal linearinterpolator in the minimum mean square error sense can befound by solving the following optimization problem:

minimizesubject to H == (28)

where H C(N+Ng)xm p is PCSI, if cSxrnp is the subsampled version of H. It can be shown that the optimal

channel estimation in fast fading mobile coded ofdm

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