Channe Estimation for SC-FDE Systems Using Frequency Domain utipexed Piots Chan-Tong am, David D. Faconer and Forence Danio-emoine BCWS, Careton Univ., Canada K1S 5B6 Emai: {amc, ddf, fdanio@sce.careton.ca Rui Dinis ISR-IST, Tech. Univ. of isbon, Portuga, Emai: rdinis@ist.ut.pt Abstract We investigate channe estimation for singe-carrierfrequency domain equaization (SC-FDE) system using the techniques typicay used for an orthogona frequency domain mutipexing (OFD) system. Two techniques of frequency domain mutipexed (FD) piot insertion using intereaved frequency domain mutipe access (IFDA) signa with a Chu sequence are considered. One caed frequency domain superimposed piot technique (FDSPT) scaes data-carrying tones and then superimposes them with piot tones. This technique preserves spectra efficiency at the expense of performance oss. The other, caed frequency expanding technique (FET), shifts groups of data frequencies for mutipexing of piot tones at the expense of spectra efficiency. Our resuts show that both techniques increase peak to average power ratio (PAPR) athough it is sti ower than that of an OFD system. The appication of FDSPT is imited by the piot overhead ratio, resuting from the remova of data frequencies for piot frequencies. It is shown that channe estimation using conventiona time domain mutipexed piots and FET piot tones produce the same BER, whie the FDSPT requires about 1.5 db more power for the same performance. Using FD piots in SC system faciitates fexibe and efficient assignment of signas to avaiabe spectrum. I. INTRODUCTION Next generation wireess systems wi ikey use fexibe combinations of frequency domain bock transmission methods such as orthogona frequency domain mutipe access (OFDA) and singe carrier with frequency domain equaization (SC-FDE). For exampe SC may be preferred for the upink of ceuar systems because of its ow peak to average power ratio (PAPR) and the resuting power ampifier efficiency in the user termina. The piot symbos for SC systems are traditionay time mutipexed within or in between fast Fourier transform (FFT) bocks and paced at the beginning of the packet [1][2]. In this paper, we consider channe estimation for SC-FDE systems using frequency domain mutipexed (FD) piot techniques which have been typicay used for OFD systems so that at the base station one estimator is sufficient. Instead of using the whoe OFD symbo for channe estimation, this piot assisted channe estimation (PACE) technique periodicay inserts piot tones with equidistant spacing, reducing the piot overheads [3][4]. Frequency domain signa generation and piot mutipexing faciitates fexibe and efficient assignment of signas to avaiabe spectrum. Aiming for appication in time and frequency seective channes, we mutipex mutipe piot tones within the signa bandwidth using an intereaved FDA (IFDA) 1 signa [6] with a Chu sequence [7], which has constant enveope and uniform spectrum. utipexing piot tones into the signa bandwidth affects the PAPR of the SC signa. We consider the effects of inserting the piots in terms of PAPR and compare with that of an OFD signa. We aso compare with the performance for conventiona time domain mutipexed (TD) piots. With one additiona FFT operation and using generaized muticarrier (GC) transmission technique [8], the SC signa with piot tones can be generated. Two techniques of piot tone insertion are considered. One is to scae datacarrying tones for superimposing of the piot tones, caed frequency domain superimposed piot technique (FDSPT). The other is to shift groups of data frequencies for mutipexing of the piot tones, caed frequency expanding technique (FET). The rest of the paper is organized as foows: Sec. II provides system description, incuding backgrounds on the generation of SC signas with piot tones. Sec. III presents signa anaysis for SC signas with piot tones using FDSPT and FET in terms of PAPR. Sec. IV describes the frequency domain channe estimation using FFT and inear SE equaization for systems with FD piots, whie Sec. V presents the simuation resuts and discussions, foowed by the concusions in Sec. VI. II. SYSTE DESCRIPTION Fig. 1 shows the bock diagram of the transmitter and receiver using GC signa generation. The transmitted bock, containing data and piots, consists of sampes pus a cp cycic prefix, which is assumed to be arger than the known channe impuse response ength. The compex data symbo a m has zero mean and variance σa. 2 We assume a size- data pus piot symbo bock transmission ( < ). The data tones A (-point FFT of {a m 1 for FDSPT and ( N)-point FFT of {a m N 1 ) and piot tones P (Npoint FFT of Chu sequence {c k N 1 ) are mutipexed into a singe frequency domain sequence, denoted as X of ength. Note that N < <. The kth eement of a ength-n Chu sequence is given by c k = e jπrk(k+i)/n,i = 0 for N even and i = 1 for N odd, where r is reativey prime to N. For equidistant piot spacing, each group of data has the same 1 IFDA has had many names, such as FDOSS[5].
size. By padding enough zeros in the frequency domain to make a tota ength of and taking the IFFT, it is equivaent to use a sinc type puse for puse shaping in the time domain with an oversamping factor of I if = I. After adding the { a m a~ { m FFT { P IFFT { A U X { X SE Equaizer ~ H { FFT CE Fig. 1. Pad zeros D e U X { Y IFFT FFT { x n { y n add CP Remove CP TD Window Cycic Shift GC Transmitter and Receiver Structure cycic prefix (CP) to prevent inter-bock interference, a time domain window can be added to reduce the side-obes of the transmitted spectrum. In the receiver side, before removing CP, a cycic shift of the received sampes is necessary due to the rooff of the raised cosine time window skirt. Taking the - point FFT of the received baseband sampe y n and then removing the ast frequencies, we obtain the received piot and data tones * + { h n { v n Y = X H + V, = 0,1,..., 1 (1) where H and V represent the channe response at frequency and frequency domain noise sampes, respectivey. For FDSPT, X is the th eement of X S = [βp 0 +αa 0,A 1,...,βP 1 +αa K,A K+1,...,A 1 ] (2) where α and β are the scaing factors for the data and piot tones at piot ocations, respectivey, P is the th piot tone and K is the piot spacing. For FET, X is the th eement of X E = [P 0,A 0,...,P 1,A K 1,...,P N 1,...,A N 1 ] (3) Note that N more data symbos can be sent using FDSPT. The SE equaization with FDSPT and FET and FFT channe estimation wi be addressed in Sec. IV. The estimated data symbo ã m is obtained by taking the IFFT of the inear SE equaizer output. III. ANAYSIS OF SC SIGNAS WITH PIOT TONES A. Frequency Domain Superimposed Piot Technique The idea of FDSPT is to periodicay scae frequencies for superimposing of the IFDA piot tones. The advantage is not to expand the signa bandwidth, thus maintaining the spectra efficiency. However, as it wi be shown ater, it suffers performance degradation due to the oss of part of the usefu data frequencies and induces sighty higher PAPR. We want to obtain an expression for x n in Fig. 1 and anayze the impication of periodicay frequency scaing and superimposing in terms of PAPR. Define a frequency-scaing window as { α = 0,K,2K,...,(N 1)K Q = (4) 1 otherwise where 0 α 1. The baseband transmitted signa (with IFDA training signa) x n can then be expressed as x S (n) = 1 1 =0 A Q e j 2πn {{ distorted data signa + β N 1 P k e j 2πnkK {{ IF DA training signa where 0 β 1 and the subscript S represents the signa for FDSPT. The second term in (5) is a deterministic and common term to FDSPT, FET and OFD signa using an IFDA signa as a training signa. Therefore, for the interest of comparing the PAPR performance among the signas using these techniques, we focus on the first term in (5). It can be shown that the enveope fuctuation is the argest when α = 0. The transmitted signa with α = 0 and without the piot tones can be found as x S(n) = 1 ( a m g S n m ) where g S (x) = sinc( x I ) sinc( K 1 x) jπx sinc( ) x sinc( K x) e I is the samped impuse response of a channe with periodic nus, ( = N) and n = 0,1,..., 1. The high sideobes of the g S n m puse increase the PAPR. B. Frequency Expanding Technique The FET eveny expands groups of frequencies for mutipexing of IFDA piot tones. The FET has sighty ower spectra efficiency due to the expansion of data frequencies to accommodate for the piot tones, which resuts in no performance oss but sighty higher PAPR than that of the conventiona SC signa. Note that the FET is the frequency domain piot technique commony used in OFD systems. When using FET, the mutipexed data and piot tones are as defined in (3), where piot tones P are periodicay mutipexed with the data tones A. The baseband transmitted sampes x n in Fig. 1 is given as x E (n) = 1 1 =0 ik N 1 X e j 2πn 1 + (5) (6) P k e j 2πnkK (7) where the subscript E represents the signa for FET. Simiar to the FDSPT case, we consider the first term in (7), which can be further expanded as x E(n) = πn ej I where g E (m,n) = sinc((k 1)(n/ m/ )) sinc(n/ m/ ) 1 ( a m g E (m,n)e jπ ( 1)m )(8) sinc(n(kn/ m(k 1)/ )) sinc(kn/ m(k 1)/ ), = N and n = 0,1,..., 1. It is obvious that the time-varying modified puse shaping fiter g E (m,n) resuts in higher PAPR than a SC system without mutipexing piot tones, as further shown in Sec. V.
C. PAPR Comparison For an OFD system, the probabiity that a PAPR vaue exceeds a certain vaue depends on the number of subcarriers. The arger the number of the subcarriers, the higher the probabiity [9]. We want to compare the number of data symbos contributing to the higher PAPR for (6) and (8) with that of an OFD system. First consider the transmitted sampes for OFD, generated using Fig. 1 without the FFT prior to the UX operation, given as x O(n) = 1 1 m ik a m e j 2πmn (9) where the subscript O represents the signa for OFD. From (9), there are ( N) random data symbos contributing to the PAPR. For SC moduated with FDSPT with α = 0 and FET, the number of random data symbos contributing significanty to the PAPR is much ess than ( N) due to the modified sinc type puse shaping fiters in both cases. Therefore, we can concude that the PAPR for a SC system with FDSPT or FET is ower than that of an OFD system with the same amount of piot tones. This is further justified using simuations in Sec. V, where it is aso shown that SC moduated signa with FET has sighty higher PAPR than that of FDSPT. IV. CHANNE ESTIATION AND EQUAIZATION A. Channe Estimation The channe estimation is accompished by estimating the channe response at the piot frequencies and then interpoating among these estimated frequencies to obtain the channe estimates for the whoe bock, using FFT and IFFT [10]. From (1), the received signa with FDSPT at the piot ocations can be written as Y = βp /K H +αa H +V, = 0,K,...,(N 1)K (10) et Y = [Y 0,Y K,...,Y (N 1)K ] T,Φ = βdiag{p 0,P 1,...,P N 1, H = [H 0,H K,...,H (N 1)K ] T, A = diag{a 0,A K,...,A (N 1)K and V = [V 0,V K,...,V (N 1)K ] T. (10) can be written in matrix form as Y = ΦH + U, where we treat U = αah + V as the combined noise term. Assuming that A and H are uncorreated and zero mean and that the data symbos and the channe noise are aso uncorreated, the covariance matrix of U is given by E{UU H = (α 2 σ 2 h σ2 a + σ 2 v)i, where we assume E{ A 2 = σ 2 a and E{ V 2 = σ 2 v. The east square estimates of H at the piot ocations are given by Ĥ = H + Φ 1 U. The mean square error (SE) of the channe estimates at the piot ocations can be shown to be E{(Ĥ H)(Ĥ H) H = α2 σh 2σ2 a + σv 2 β 2 σp 2 I (11) where E{ΦΦ H = β 2 σp 2 I and σ2 P is the variance of the piots. The interpoated channe estimates, denoted as H, are obtained by first taking the N-point IFFT of Ĥ and then padding N zeros before taking the -point FFT [10]. For the channe estimation of FET, it is equivaent to that of FDSPT when α = 0 and β = 1. Note that channe estimator using Wiener fitering can aso be empoyed for better channe estimates at the expense of higher compexity [4]. B. inear SE Equaization First consider FDSPT. Given the channe is known, the piot tones are removed from the received signa tones before equaization, Y = Y βh P /K = αh A + V, = 0,K,...,(N 1)K (12) At the non-piot frequencies, from (1), the received data frequencies are given as Y = H A + V. The inear SE equaizer taps for a singe received sampe can then be cacuated as H W = H 2 + σv 2, = 0,1,..., 1 (13) where ( ) denotes the compex conjugate and { H αh = 0,K,2K,...,(N 1)K = H 0,K,2K,...,(N 1)K (14) The corresponding SE of the inear equaizer can be shown to be J S = σ2 1 v 1 H =0 2 + σv 2 (15) The estimated data symbos using FDSPT can be obtained by taking the -point inverse FFT (IFFT) of {W Y 1 =0, where Y = Y at the piot ocations and Y = Y at the data ocations. Using (13), the equaization of SC signa with FET is performed as à = Y W, where = 0,1,..., 1 and 0,K,2K,...,(N 1)K. The SE of the inear equaizer for FET can be shown to be J E = σ2 1 v N =0 ik 1 H 2 + σv 2, i = 0,1,...,N 1 (16) et à = [Ã1,...,ÃK 1,ÃK+1,...,à 1]. The estimated data symbos using FET can be obtained by taking the ( N)- point IFFT of Ã. The BER for both cases can be found as where J is either J S or J E. Q( (1 J)/J) (17) V. SIUATION RESUTS AND DISCUSSIONS A. CCDF vs PAPR and Power Spectrum Consider Fig. 2, where x n is as defined in Fig. 1. At compementary cumuative distribution function (CCDF) =, the FDSPT with α = 0 and FET have about 2 db advantage over OFD and 1.5 db disadvantage over SC moduated signa without piot tones. The higher the vaue of α is, the better the PAPR. However, from (11), the higher the vaue of α the worse the performance of the channe estimator.
Pr( x n >λ) no piots, SC α=0, OFD α=0, FDSPT α=0.5, FDSPT α=0.707, FDSPT α=0, FET 2 3 4 5 6 7 8 9 10 λ[db] Fig. 2. Comparison of CCDF of PAPR for SC oduated Signas with FDSPT and FET ( = 826, N = 118, I = 12, η = N/ = 14.2%) using Probabiity Density Function There exists a fundamenta trade-off between the PAPR and that of the channe estimator. Fig. 3 shows the spectrum generated using the SC moduated signa with FDSPT with various α vaues, FET and OFD with IFDA piot signa with a Chu sequence, where ρ is the power backoff of the ampifier in db. The spectrum for the system without FD piots is the same as that for the TD piots. The ETSI 3GPP spectra mask is scaed I shows the urban macro power deay profie [8], whie Tabe II shows the parameters [8] used in the simuation for both the uncoded and coded cases. We assume the channe is static for the uncoded case. The theoretica resuts are consistent with the simuation resuts. The arger the vaue of α or the smaer the vaue of η is, the better the BER. The performance degradation for FDSPT with α = 0 is significant due to arge vaue of η and the presence of piots in every bock. This impies that if we add piots ess frequenty, e.g. every B bocks, where B > 1, we woud expect better BER performance. It is obvious that the required vaue of B depends on the maximum Dopper frequency. BER α=0, η=14.2%, simuation α=0, η=14.2%, theoretica α=0.2, η=14.2%, simuation α=0.2, η=14.2%, theoretica α=0.5, η=14.2%, simuation α=0.5, η=14.2%, theoretica α=0, η=7.1%, simuation α=0, η=7.1%, theoretica no piots 5 10 15 20 Es/No[dB] Power spectrum (db) 0 10 20 30 40 ETSI 3GPP Spectra ask ρ=100000 db OFDA, ρ=12 db SC without piots, ρ=9.3 db FET, ρ=10.5 db FDSPT(α=0), ρ=10 db FDSPT(α=0.707), ρ=9.6 db Fig. 4. Performance of FDSPT in Urban acro Channe, Uncoded TABE I URBAN ACRO POWER DEAY PROFIE Power[dB] -3.0-5.22-6.98-5.22-7.44-9.2-4.72-6.94-8.7-8.19-10.41-12.17-12.05-14.27-16.03-15.50-17.72-19.48 Deays[µs] 0.0 0.01 0.03 0.36 0.37 0.385 0.25 0.26 0.28 1.04 1.045 1.065 2.73 2.74 2.76 4.6 4.61 4.625 50 0 0.5 1 1.5 frequency normaized to data symbo rate Fig. 3. Raised Cosine time domain windowing rooff factor = 5.3%, =826 QPSK symbos/bock, Rapp mode (p= 2) to fit the generated spectrum. The backoff vaues, used for iustrative purposes, are obtained by tria and error such that the power spectrum is just within the mask. As expected, OFDA signa requires the argest backoff, whie the SC without piots requires the east. The SC with FET requires 0.5 db more backoff than that of SC with FDSPT (α = 0). Note that the spectra mask coud have been shifted sighty to the right for FDSPT, since its net data symbo rate is sighty higher, athough this has not been done in the figure. B. BER Performance Fig. 4. depicts the BER for an uncoded system using FDSPT with different vaues of α, given the channe is known. Tabe TABE II SIUATION SYSTE PARAETERS Parameter Urban acro Carrier frequency [GHz] 5.0 System bandwidth [Hz] 20.0 oduated symbos per bock 826 Symbo rate [sps] 16.25 RC time domain windowing rooff factor[%] 5.3 Upsamping factor 12 Cycic prefix ength [µs] 5.00 We then evauate the BER performance for a coded SC system with FFT channe estimation using FDSPT and FET. A 64- state, rate-1/2 convoutiona code with generators (133,171) o and a random bock intereaver are used for a frame with 10 bocks. We assume independent fading channe reaizations every frame and each independent fading mutipaths has cassica Jakes Dopper spectrum with vehice speed of 70 km/hour. For comparison, BER with channe estimation using TD is aso incuded. The piots are added in every frame according to
the row vector [1100110011], where an 1 represents piots are added and a 0 represents no piots are added in the corresponding bocks. east square inear ine fitting is used to obtain the channe estimates for the bocks without piots. BER Fig. 5. 10 0 10 5 TD FET, K=7 FET, K=8 FDSPT, α=0 FDSPT, α=0.2 FDSPT, α=0.35 FDSPT, α=0.5 FDSPT, α=0.707 Idea CSI 0 2 4 6 8 10 12 14 16 E /N b o Performance of FD Piots with Coding in Urban acro Channe Fig. 5 shows the BER for the system with channe estimation using TD and FD piots, where N = 118 piots are used for bocks with piots. We observe that with the same number of piots, TD and FD piot arrangements have the same BER performance. Note that using the TD piots, the spectra efficiency is sighty ower than that of using the FD piots. In time domain, an extra cp data ocations are fied with piot symbos for the cycic prefix of the bock. On the other hand, the two dimensiona FD piot arrangement requires that a maximum piot spacing in frequency and time axis is satisfied, which depends on the maximum deay of the channe and the maximum Dopper frequency, respectivey. Exceeding this maximum spacing resuts in overapping of the orgina spectrum with its aias [11]. For this particuar case, the maximum piot spacing is K = 7. For K = 8, an error foor occurs at around BER = 5 as a resut of spectrum aiasing. Athough the higher vaue of α gives better PAPR, it however produces worse BER performance for arge vaue of α. Tabe III summarizes the advantages and disadvantages among different piot arrangement schemes, where higher PAPR means more power backoff. TABE III COPARISON OF DIFFERENT PIOT ARRANGEENT SCHEES Advantages Disadvantages No BER degradation, Higher overhead, TD ow PAPR ess spectrum fexibiity No BER degradation, Higher PAPR, FET good spectrum fexibiity sighty higher overhead east overhead, sighty higher PAPR, FDSPT good spectrum fexibiity BER degradation VI. CONCUSIONS Two techniques of FD piot insertion have been presented: FDSPT, where piots are superimposed on scaed data-carrying tones and FET, where groups of data carrying tone are shifted for mutipexing of piot tones. It was shown that both techniques yied arger enveope variations of the SC signa. However, the SC signas with FDSPT and FET have sti ower PAPR than that of an OFD signa due to the smaer number of random data symbos contributing to the peak enveope variations. Tabe III summarizes the advantages and disadvantages of using FDSPT and FET. The appication of FDSPT depends on the piot overhead ratio and the time variation of the channe. Channe estimation with FET and that using TD piot symbos have the same BER performance, given that the piot tone spacing does not exceed the maximum vaue and the number of piots are arger than the maximum deay spread. Using FD piots in SC system provides the option of channe estimation in frequency domain at the expense of sighty increasing the PAPR. Aso, frequency domain signa generation and piot mutipexing faciitates fexibe and efficient assignment of signas to avaiabe spectrum. ACKNOWEDGENT Part of this work has been performed in the framework of the IST project IST-2003-507581 WINNER, which is party funded by the European Union and party by the Natura Sciences and Engineering Research Counci (NSERC) of Canada. The authors CT, DF and FD woud ike to acknowedge the contributions of their coeagues in WINNER, athough the views expressed are those of the authors and do not necessariy represent the project. REFERENCES [1] D. Faconer, S.. Ariyavisitaku, A. Benyamin-Seeyar, and B. 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