FUTURE wireless communication systems are expected to

Transcription

1 2508 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 Channel Estiation for Adaptive Frequency-Doain Equalization Michele Morelli, Meber, IEEE, Luca Sanguinetti, Student Meber, IEEE, and Uberto Mengali, Life Fellow, IEEE Abstract Frequency-doain equalization (FDE) is an effective technique for high-rate wireless counications because of its reduced coplexity copared to conventional tie-doain equalization (TDE). In this paper, we consider adaptive FDE for single-carrier (SC) systes with explicit channel and noise-power estiation. The channel response is estiated in the frequency doain following two different approaches. The first operates independently on each frequency bin while the second exploits the fading correlation across the signal bandwidth. Leastean-square (LMS) and recursive-least-square (RLS) algoriths are eployed to update the channel estiates. The noise power is estiated using a low-coplexity algorith based on ad hoc reasoning. Copared to other existing receivers eploying adaptive FDE, the proposed schees have better error-rate perforance and can be used even in the presence of relatively fast fading. Index Ters Channel estiation, channel tracking, frequency-doain equalization (FDE). I. INTRODUCTION FUTURE wireless counication systes are expected to support high-speed and high-quality ultiedia services. In these applications, the received signal is typically affected by frequency-selective fading and channel equalization is required to itigate the resulting intersybol interference (ISI) [1]. The classical approach in single-carrier (SC) systes is tie-doain equalization (TDE) [2]. However, the nuber of operations per signaling interval grows linearly with the nuber of interfering sybols or, equivalently, with the data rate. As a result, conventional tie-doain equalizers are not suitable for high-speed transissions with channel delay spreads extending over tens of sybol intervals. A proising alternative to TDE is the frequency-doain equalization (FDE) [3] [6]. Using the fast Fourier transfor (FFT) in conjunction with FDE leads to substantial coputational saving with respect to conventional TDE. Also, adaptive algoriths generally converge faster and are ore stable in the frequency doain [4]. Copared to orthogonal frequency division ultiplexing (OFDM), SC systes with FDE (SC-FDE) have siilar perforance and coplexity [5] [7], but the latter are less sensitive to carrier-frequency uncertainties and nonlinear distortions, thereby allowing the use of low- Manuscript received Noveber 24, 2003; revised April 14, 2004 and July 28, 2004; accepted August 22, The editor coordinating the review of this paper and approving it for publication is H. Li. This work was supported by the Istituto di Elettronica e di Ingegneria dell Inforazione e delle Telecounicazioni (IEIIT) of the Italian National Research Council (CNR). The authors are with the Departent of Inforation Engineering, University of Pisa, I Pisa, Italy (e-ail: ichele.orelli@iet.unipi.it). Digital Object Identifier /TWC cost power aplifiers. SC-FDE systes equipped with ultiple receive antennas have been discussed in [3] and [8], while the possibility of achieving transit diversity using Alaouti s space tie block coding [9] has been explored in [10]. Finally, novel FDE structures for SC ultiple-input ultiple-output (MIMO) systes are investigated in [11]. Mobile counication systes operating over tie-varying fading channels require adaptive signal processing to track the channel variations at the receiver. Adaptive FDE schees based on the iniu ean square error (MMSE) criterion have been investigated by Clark in [3]. They operate according to least-ean-square (LMS) or recursive-least-square (RLS) adaptation rules and do not require explicit channel estiation. Copared with their tie-doain counterparts, they have better stability and shorter acquisition ties. However, their perforance over fast-fading channels is unsatisfactory when a single receive antenna is eployed (i.e., without space diversity). In the present paper, we return to the proble discussed by Clark, but we consider adaptive SC-FDE schees in which estiates of the channel response are exploited to copute the equalizer coefficients according to the MMSE criterion. The channel response is estiated in the frequency doain using two different approaches. The first assues independently faded frequency bins and is referred to as unstructured channel estiation (UCE). The second is called structured channel estiation (SCE) and effectively exploits the fading correlation between adjacent bins. Both schees exploit training sybols to get initial channel estiates, whereas LMS or RLS algoriths are eployed to track channel variations. It is worth noting that in SC-FDE systes, the energy of each sybol is distributed over the whole signal bandwidth and pilots cannot be placed on preassigned frequency bins. Accordingly, channel estiation cannot be accoplished with the sae ethods eployed in OFDM applications, where pilots are typically inserted in both tie and frequency diensions, and channel estiates are obtained by interpolation (see [12] and [13] and references therein). As explained later, coputing the equalizer coefficients requires knowledge of the noise power. The latter is estiated in the frequency doain using a axiu likelihood (ML) approach. The resulting schee has good perforance but it is coputationally deanding. Therefore, we also consider a sipler solution based on heuristic arguents. Siulation results indicate that the proposed SC-FDE schees outperfor Clark s adaptive detectors (CADs) (especially in a fast-fading environent) without a significant increase in coplexity. Diversity cobining using ultiple /$ IEEE

2 MORELLI et al.: CHANNEL ESTIMATION FOR ADAPTIVE FREQUENCY-DOMAIN EQUALIZATION 2509 Fig. 1. (a) SC-FDE transitter. (b) SC-FDE receiver. receive antennas is also considered. It is shown that this guarantees draatic perforance iproveents. The rest of the paper is organized as follows. The next section describes the signal odel and introduces basic notations. In Section III, the concept of FDE with explicit channel and noise-power estiation is discussed. Several channelestiation schees operating in the frequency doain are proposed in Section IV, whereas the proble of the noisepower estiation is addressed in Section V. Siulation results are discussed in Section VI and soe conclusions are offered in Section VII. II. SYSTEM MODEL A. SC-FDE Transitter Fig. 1(a) shows the transitter of the SC-FDE syste under investigation. The input sybols, belonging to a phaseshift keying (PSK) or quadratic-aplitude odulation (QAM) constellation, are partitioned into adjacent blocks of length N and each block is preceded by a cyclic prefix longer than the channel ipulse response (CIR). The prefix serves to eliinate interblock interference and akes the linear convolution of the sybols with the channel look like a circular convolution, which is essential for FFT-based deodulation. We denote c =[c (0) c (1) c (N 1)] T as the th block of sybols [the superscript ( ) T eans transpose operation] and assue that {c (n)} are independent and identically distributed (i.i.d.) with zero ean and unit variance. After insertion of the N G -point cyclic prefix, c is fed to a linear odulator with ipulse response g(t) and signaling interval T. The coplex envelope of the transitted signal is s(t) = = N 1 n= N G c (n)g(t nt T B ) (1) where counts the transitted blocks, n counts the data sybols within a block, T B =(N + N G )T is the duration of the cyclically extended data block, and c (n) =c (n + N) for N G n 1. We assue that g(t) has a root-raisedcosine Fourier transfor with soe roll-off α. B. SC-FDE Receiver The receiver has P diversity branches and its block diagra is sketched in Fig. 1(b). The coplex envelope of the received wavefor at the pth antenna is denoted r (t) and is expressed by r (t) =s R (t)+η (t) (2) where s R (t) is the signal coponent and η (t) is theral noise. The latter is odeled as a circularly syetric Gaussian process with two-sided power spectral density 2N 0 (possibly different fro branch to branch). As in [3], we assue negligible channel variations over a block (slow fading). Then, denoting h (t) as the CIR at the pth antenna (encopassing the transit filter and the physical channel) during the th data block, we have s R (t) = = N 1 n= N G c (n)h (t nt T B ). (3) In order to produce a discrete-tie signal, the wavefor fro each antenna is fed to a low-pass filter (LPF) and is sapled at a rate of 2/T to avoid aliasing distortion. For the sake of siplicity, the LPF is taken with a brick-wall transfer function of bandwidth 1/T. Note that the rectangular shape is not strictly necessary and could easily be ade ore realistic [14]. For exaple, we ay eploy a root-raised-cosine function with a suitable roll-off such that the signal coponent is passed undistorted and the noise saples at the filter output are uncorrelated. After carrier-frequency and block-tiing synchronization [not shown in Fig. 1(b)], the cyclic prefix is discarded and the received saples are arranged in blocks of 2N eleents. We denote x =[x (0) x (1) x (2N 1)] T as the th block of saples at the pth antenna and assue that h (t) has support (0,LT), with L N G. Then, the entries of x are found to be x (k) = N 1 n=1 L c (n)h (k 2n)+w (k), k =0, 1,...,2N 1 (4)

3 2510 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 Fig. 2. Frequency-doain equalizer/cobiner with explicit channel and noise-power estiation. where h (l) is the saple of h (t) at t = lt/2 and w (k) is white Gaussian noise with variance σ 2 =4N 0 /T. Each block x (p =1, 2,...,P) is transfored in the frequency doain using a 2N-point discrete Fourier transfor (DFT) unit and the DFT outputs are passed to the channel equalizer/cobiner to for the N-diensional vector Y = [Y (0) Y (1) Y (N 1)] T (details on the channel equalizer/cobiner are given in the next section). After an N-point inverse DFT (IDFT), the tie-doain saples y = [y (0) y (1) y (N 1)] T are finally fed to a threshold device that delivers the data decisions ĉ =[ĉ (0) ĉ (1) ĉ (N 1)] T corresponding to the th block. C. Channel Model We consider a ultipath channel with N p distinct paths and we assue that the P receive antennas are arranged in a unifor linear array (ULA) with intereleent spacing d. Then, the baseband ipulse response of the syste at the pth antenna takes the for N p ξ (t, τ) = a l (t)e j(p 1)ωl(t) δ(τ τ l (t)) (5) l=1 where δ(t) is the Dirac delta function, τ l (t) is the delay of the lth path, a l (t) is the corresponding coplex aplitude, and ω l (t) is defined as ω l (t) = 2π λ d sin [ϕ l(t)]. (6) In the above equation, λ is the free-space wavelength and ϕ l (t) is the direction of arrival (DOA) of the lth path. In the following, we assue that the path delays and the DOAs do not change significantly with tie, i.e., we set τ l (t) τ l and ϕ l (t) ϕ l. Conversely, the path gains are odeled as narrowband, independent, and coplex-valued Gaussian processes with zero ean and average power σ 2 l = E{ a l(t) 2 }. The CIR at the pth antenna is the convolution of ξ (t, τ) with the transit pulse g(t). Recalling that the gains {a l (t)} are practically constant over a block, we have h (t) = N p a l (T B )e j(p 1)ω l g(t τ l ). (7) l=1 Note that the length of h (t) (expressed in sybol intervals) is L =int{(τ ax + T g )/T }, where T g is the duration of g(t), τ ax = ax l {τ l } is the axiu path delay, and int(x) denotes the axiu integer not exceeding x. Since τ ax is usually unknown, in practice, L is estiated as taking the axiu expected value of τ ax. III. FREQUENCY-DOMAIN EQUALIZATION (FDE) Fig. 2 illustrates an FDE schee with explicit channel and noise-power estiation. The DFT output at the pth branch is denoted =[X (0) X (1) X (2N 1)] T, with (n) = 1 2N 1 2N k=0 x (k)e j2πnk (2N). (8) Substituting (4) into (8) and bearing in ind that h (t) has duration LT produces (n) =C (n)h (n)+w (n) (9) where H (n) is the DFT of the channel response H (n) = 1 2L 1 2N l=0 h (l)e j2πnl (2N), 0 n 2N 1 (10)

4 MORELLI et al.: CHANNEL ESTIMATION FOR ADAPTIVE FREQUENCY-DOMAIN EQUALIZATION 2511 while C (n) is defined as TABLE I COMPUTATIONAL COMPLEXITY PER DETECTED SYMBOL C (n) = N 1 k=0 Finally, the quantity W (n) = 1 c (k)e j2πnk N, 0 n 2N 1. (11) 2N 1 2N k=0 w (k)e j2πnk (2N), 0 n 2N 1 (12) is additive white Gaussian noise (AWGN) with variance σ 2. Bearinginindthatg(t) [and hence, h (t)] is bandliited to f (1 + α)/2t, it is seen that H (n) is 0 for N α n 2N N α, with N α =1+int[N(1 + α)/2]. Vector isfedtothepth channel equalizer (a bank of 2N coplex-valued ultipliers {F (n); 0 n 2N 1}) and is then cobined with the other branch outputs to for Z (n) = P p=1 (n)f (n), 0 n 2N 1. (13) Coputing the 2N-point IDFT of Z (n) produces the equalized sequence in the tie doain z (k) = 2N 1 n=0 Z (n)e j2πnk (2N), 0 k 2N 1 (14) fro which the decision statistics y =[y (0) y (1) y (N 1)] T are obtained by deciation, i.e., taking y (k) = z (2k) for k =0, 1,...,N 1. As shown in Fig. 2, this is tantaount to passing the saples {Z (n)} to an aliasing operator that produces the quantities [3] Y (n) =Z (n)+z (n + N), 0 n N 1 (15) and coputing y as the N-point IDFT of {Y (n)}. This approach can be explained observing that deciating in the tie doain corresponds to an aliasing operation in the frequency doain. The MSE at the input of the decision device is E{ y (k) c (k) 2 }, where the expectation is taken over the transitted data sequence and additive noise (i.e., the MSE is defined for a static channel). Assuing i.i.d. data sybols with zero ean and unit variance, the optiu equalizer coefficients iniizing the MSE are coputed with ordinary anipulations and read [2] F (n) = 1+ N N ] [H (n) σ 2 P l=1 i (l) H (n+in) 2 σ 2(l), 0 n 2N 1; p =1, 2,...,P. (16) Note that the denoinator in (16) is independent of p so that the equalizer coefficients are proportional to [H (n)] /σ 2. Therefore, fro (13), it is seen that Z (n) is a axiu ratio cobination of {X (n); p =1, 2,...,P}. Fro (16), we see that coputing F (n) requires knowledge of the channel response and the noise power at each branch. In practice, these quantities are unknown. A way out is discussed in [3], where the equalizer coefficients are updated in the frequency doain using LMS or RLS algoriths without explicit channel and noise-power estiation. As indicated in Fig. 2, here, we propose an alternative approach in which the estiates of σ 2 and H (n),sayˆσ 2 and Ĥ (n),areeployed in (16) to approxiate the equalizer coefficients. Soe ethods to copute ˆσ 2 and Ĥ (n) arediscussedinthe next section. The coputational load of the FDE is assessed as follows. The DFT operator in (8) needs N log 2 (2N) coplex products and 2N log 2 (2N) coplex additions for each diversity branch. Also, a total of 2NP coplex ultiplications and 2NP N coplex additions are involved in the coputation of Z (n) and Y (n) in (13) and (15), respectively. The IDFT of {Y (n)} needs (N/2) log 2 (N) coplex products and N log 2 (N) coplex additions. Finally, coputing the equalizer coefficients in (16) requires 5NP real products and 4NP real additions. The overall operations per detected sybol are suarized in the first line of Table I. In writing these figures, we have borne in ind that a coplex product aounts to four real products plus two real additions, while a coplex addition is equivalent to two real additions. The results of Table I indicate that the coputational load involved in the FDE is proportional to P log 2 N. For coparison, we recall that the coplexity of a tie-doain equalizer with P diversity branches is on the order of PL [3]. Since in typical applications the block length N is about 5L (corresponding to an overhead of 20%), we see that in highly dispersive channels (where large values of L are expected), FDE ay achieve significant coputational savings with respect to TDE. IV. CHANNEL ESTIMATION We begin by rewriting (9) in atrix for = C H + W (17)

5 2512 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 where H =[H (0) H (1) H (2N 1)] T, C is a diagonal atrix C = diag { C (0) C (1) C (2N 1) } (18) and W =[W (0) W (1) W (2N 1)] T is a Gaussian vector with zero ean and covariance atrix C W = σ 2 I 2N (I 2N is the identity atrix of order 2N). Fro (10), we see that H can also be written as where h H = Fh (19) =[h (0) h (1) h (2L 1)] T is the CIR vector at the pth antenna and F is a 2N 2L atrix with entries [F ] n,l = 1 e j2πnl (2N), 2N 0 n 2N 1; 0 l 2L 1. (20) In the following, we discuss two iterative schees for estiating the channel frequency response at each diversity branch. The first, tered UCE, is based on odel (17) and considers the entries of H as unknown independent paraeters. The second takes advantage of the correlation between adjacent frequency bins and effectively exploits the structure of H shown in (19). For this reason, it is called the SCE. As we shall see, both UCE and SCE require knowledge of the transitted data sybols. To this end, we assue that the data blocks are organized in fraes, and each frae is preceded by soe training blocks. During the data section of the frae, the channel estiators are switched to a decision-directed ode and the transitted sybols are replaced by data decisions. A. LMS Unstructured Channel Estiation (LMS-UCE) LMS-UCE eploys the LMS algorith to iniize the cost function ( ) { X J H LMS-UCE = E C H 2}, with respect to produces where Ĥ is given by H p =1, 2,...,P (21) ( denotes Euclidean nor). This Ĥ +1 = Ĥ + µe, p =1, 2,...,P (22) is the estiate of H, µ is the step size, and e e = C H [ ] C Ĥ (23) with ( ) H denoting Heritian transpose. The perforance of LMS-UCE over a static channel (i.e., H = H ) is assessed in Appendix A, assuing i.i.d. data sybols with zero ean and unit variance. It turns out that is unbiased and has the following ean square estiation error (MSEE) { } E Ĥ H 2 =4σ 2 B L T B (24) Ĥ where B L T B = µn/[2(2 µn)] is the noise equivalent bandwidth [15, p. 126] of the recursion (22), noralized to 1/T B. The coplexity of LMS-UCE is assessed as follows. In the decision-directed ode, the entries of C are coputed fro ĉ through an N-point FFT involving (N/2) log 2 N coplex products and N log 2 N coplex additions. Also, evaluating C H C Ĥ in the right-hand side (RHS) of (23) needs 6N real products plus N real additions. Copleting the coputation of e requires 2N coplex products and additions. Finally, updating the channel estiates in the RHS of (22) needs 2N coplex additions. The overall operations per detected sybol are suarized in the second line of Table I. B. RLS Unstructured Channel Estiation (RLS-UCE) The RLS-UCE ais at iniizing the exponentially weighted su J RLS-UCE ( H ) = λ i i C i H 2, p =1, 2,...,P (25) where 0 <λ<1 is the forgetting factor. The iniu is achieved for H = Ĥ, with Ĥ satisfying the recursive equation Ĥ +1 = Ĥ + K e, p =1, 2,...,P (26) where e is defined in (23) and K = diag{k (0) K (1) K (2N 1)}. TheterK (n) is the Kalan gain over the nth frequency bin and it is expressed by S (n) K (n) = λ + C (n) 2, 0 n 2N 1 (27) S (n) with S (n) satisfying the recursion S +1 (n) = 1 [ λ S (n) 1 K (n) C (n) 2], 0 n 2N 1. (28) Note that K (n) and S (n) do not depend on the index p, which eans that they are the sae at each antenna. The overall operations required by RLS-UCE per detected sybol are shown in the third line of Table I. In writing this figures, we have taken into account that K (n) and S (n) are real quantities that need to be coputed only for 0 n N 1. This is a consequence of the identities K (n + N) = K (n) and S (n + N) =S (n), which are easily derived fro (11), (27), and (28).

6 MORELLI et al.: CHANNEL ESTIMATION FOR ADAPTIVE FREQUENCY-DOMAIN EQUALIZATION 2513 C. LMS Structured Channel Estiation (LMS-SCE) LMS-SCE ais at estiating h by looking for the iniu of ( h ) { J LMS-SCE = E C F h 2}, with respect to h. This leads to the recursion p =1, 2,...,P (29) ĥ +1 = ĥ + µf H e, p =1, 2,...,P (30) where e is defined in (23) and ĥ is the CIR estiate at the th step. Preultiplying both sides of (30) by F and bearing in ind (19) produces Ĥ +1 = Ĥ + µff H e, p =1, 2,...,P. (31) Following the sae arguents as in Appendix A, it can be shown that LMS-SCE is unbiased and its MSEE is given by { E Ĥ } H 2 4Lσ2 B L T B =. (32) N Note that the only difference between LMS-SCE and LMS- UCE is the presence of the atrix FF H in (31), which perfors a better noise filtering by taking into account that h has the duration L<N. This leads to a reduction of the MSEE by a factor N/L, as seen by coparing (32) with (24). For L = N, LMS-SCE boils down to LMS-UCE since, in this case, we have FF H = I 2N. The third line in Table I shows the overall operations involved in LMS-SCE. In writing this line, we have borne in ind that FF H e is efficiently coputed by feeding e to a 2N-point IDFT unit, setting to 0 the last 2N 2L outputs, and finally passing the resulting vector to a 2N-point DFT. D. RLS Structured Channel Estiation (RLS-SCE) In this case, we look for the iniu of J RLS-SCE ( h ) = λ i i C i F h 2, p =1, 2,...,P (33) with respect to h. As shown in Appendix B, this leads to the recursion ĥ +1 = ĥ + R 1 F H e, p =1, 2,...,P (34) where e is still given in (23) and R C 2L 2L is defined as R = λ i F H C H i C i F. (35) Bearing in ind that Ĥ = F ĥ, fro (34), we get Ĥ +1 = Ĥ + FR 1 F H e, p =1, 2,...,P. (36) Note that, although R can be coputed recursively as R = λr 1 + F H C H C F (37) there sees to be no recursive way to copute R 1 in (36). This akes RLS-SCE prohibitively coplex and, for this reason, it is not considered in the sequel. V. N OISE-POWER ESTIMATION We assue that the noise power is constant over a frae and, therefore, the noise variance σ 2 at each antenna can be estiated frae by frae, exploiting the available training blocks. In the sequel, we discuss two ethods for estiating σ 2. The first is based on ML reasoning while the second is derived fro an ad hoc arguent. A. ML-Based Estiation (MLBE) Collecting (17) and (19) yields = C Fh + W (38) where W is Gaussian distributed with zero ean and covariance atrix σ 2 I 2N. Recalling that C is known (training block), the joint ML estiates of σ 2 and h, based on the observation of, are found by axiizing the loglikelihood function ( Λ σ 2, ) ( h = 2N ln π σ 2) 1 σ 2 C F with respect to the trial values σ 2 fixed and axiizing Λ( σ 2, produces ĥ h 2 (39) h. Keeping σ 2 and h ) with respect to h = ( D H D ) 1 D H (40) with D = C F. Next, substituting (40) into (39) and axiizing with respect to σ 2 gives the ML estiate of σ 2 ˆσ ML 2 = 1 D 2N 2 (41) where D = I 2N D (D H D ) 1 D H is the orthogonal copleent of D. Note that D depends on the training sybols (through C ) and can be precoputed and stored in the receiver. It can be shown that { E } ˆσ ML 2 = N L σ2 N (42) which eans that ˆσ ML 2 is a biased estiator. On the other hand, averaging (41) over the available training blocks

7 2514 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 TABLE II COMPUTATIONAL COMPLEXITY OF THE NOISE-POWER ESTIMATOR PER FRAME (to iprove the estiation accuracy) produces the following unbiased estiate ˆσ 2 MLBE = 1 2N T (N L) N T 1 =0 D 2 (43) with N T being the nuber of training blocks. In the sequel, (43) is referred to as the MLBE. Its variance is coputed using ordinary anipulations and reads { } var ˆσ MLBE 2 = [σ 2] 2 2N T (N L). (44) The overall operations required by MLBE for each frae are shown in the first line of Table II. The ajor coplexity arises fro the coputation of vectors {D } in (43), which is cubersoe for large values of N. For this reason, it is worth looking for a sipler suboptial solution. B. Ad Hoc Estiation Returning to (9) and recalling that H (n) =0for n I = {N α,n α +1,...,2N N α } yields (n) =W (n), n I; p =1, 2,...,P (45) where W (n) are statistically independent Gaussian rando variables with zero ean and variance σ 2. Inspection of (45) suggests the following estiate of σ 2 ˆσ 2 AHE = 1 N T N I N T 1 =0 n I (n) 2 (46) with N I being the cardinality of I. Since (46) is not based on an optiality criterion, it is referred to as ad hoc estiator (AHE) in the sequel. With standard calculations, it is found that AHE is unbiased, with variance { } [σ 2] 2 var ˆσ AHE 2 =. (47) N T N I To copare the accuracy of AHE and MLBE, we introduce the ratio Γ=var{ˆσ AHE 2 }/var{ˆσ2 MLBE }. Recalling that N I = 2(N N α )+1 and N α N(1 + α)/2, fro (44) and (47), we have Γ 2 ( ) 1 L N 1 α. (48) In practical applications, L is saller than N (say L N/5) and MLBE outperfors the AHE since Γ is greater than unity. However, AHE is uch sipler to ipleent (the overall operations are shown in the second line of Table II). Note that Γ grows to infinity as α approaches unity, eaning that AHE fails with a large excess bandwidth. In these circustances, the MLBE is indispensable. VI. SIMULATION RESULTS Coputer siulations have been run to assess the perforance of an SC-FDE receiver eploying the proposed channel and noise-power-estiation schees. The syste paraeters are as follows. A. Syste Paraeters The transitted sybols belong to a quaternary PSK (QPSK) constellation and are related to the inforation bits through a Gray ap. The odulation pulse g(t) is a root-raised-cosine function with roll-off = 0.35 and duration T g =6T. Assuing a axiu expected path delay of 10 µs, this corresponds to a CIR length of L = int(10r +6), where R =1/T is the signaling rate in egabaud. The length N G of the cyclic prefix is set equal to L and results in an effective bit rate of R b = 2RN/(L + N) (ignoring the overhead due to training blocks). The carrier frequency is f 0 =2 GHz (corresponding to a wavelength λ =15 c) and the intereleent spacing in the antenna array is d =2λ. Each frae is ade of 100 blocks [3] and is preceded by a preable of five blocks for channel and noise-power estiation. As indicated in (7), the CIR at the pth antenna is generated with six paths (N p =6). At the start of each frae, a new set of path delays, coplex gains, and DOAs are randoly generated. The path delays and DOAs are uniforly distributed within [0,10 µs] and [ 60, 60 ], respectively, and are kept constant over a frae. The path gains with power σl 2 =exp( l/2) (0 l 5) vary independently of each other within a frae. They are generated by passing coplex-valued and statistically independent white Gaussian processes through a third-order lowpass Butterworth filter. The 3-dB bandwidth of the filter is taken as a easure of the Doppler rate f D = f 0 v/c, where v is the obile speed and c denotes the speed of light. As in a welldesigned syste the channel coherence tie is uch larger than the block duration, we assue that the path gains are static within a single block [3]. Siulations results are given for R =2Mbaud, L =26, and N = 128. The obile velocity and the nuber of diversity branches are given different values to assess their ipact on the syste perforance. The optial selection of the step size µ and the forgetting factor λ for the adaptive channel estiators depends on the fading rate. Siulations indicate that for obile speeds between 25 and 140 k/h, a good choice of the adaptation paraeters is µ = for LMS- UCE, µ = for LMS-SCE, and λ =0.5for RLS-UCE. As entioned earlier, RLS-SCE is not considered due to its coplexity. The noise power spectral density is the sae at each diversity branch (i.e., we set N 0 = N 0 for p =1, 2,...,P).

8 MORELLI et al.: CHANNEL ESTIMATION FOR ADAPTIVE FREQUENCY-DOMAIN EQUALIZATION 2515 Fig. 3. Perforance of the noise-power estiators. Fig. 4. BER versus E b /N 0 with a single-antenna receiver and v =25k/h. Finally, bearing in ind that h (t) is bandliited to f (1 + α)/2t, Ĥ (n) is forced to 0 for n I. B. Perforance Assessent We begin by coparing the perforance of the noise-power estiators. Fig. 3 shows the accuracy of AHE and MLBE versus 1/σ 2 in the case of a single-antenna receiver. Marks indicate siulations while solid lines represent analytical results as given by (44) and (47). Good agreeent is observed between siulations and theory. As expected, MLBE gives the best results. However, extensive siulations (not shown for space liitations) indicate that using MLBE instead of AHE does not produce significant iproveents in the error-rate perforance. For this reason, MLBE is not considered further due to its coplexity. The syste perforance has been assessed in ters of bit error rate (BER) versus E b /N 0, where E b is the energy per bit. Fig. 4 illustrates the BER of a receiver eploying the proposed channel estiators. The obile velocity is 25 k/h (corresponding to f D =47Hz) and the receiver is equipped with a single antenna (P =1). The curve labeled ICI (ideal channel inforation) corresponds to a perfect knowledge of the channel response and noise power and serves as a benchark. The perforance of CADs [3], using either the LMS (LMS-CAD) or RLS (RLS-CAD) adaptation rules, is also shown for coparison. Finally, the curve labeled LMS-TDE indicates the perforance of a conventional T/2-spaced tiedoain equalizer eploying the LMS algorith. For an error probability of 10 3, LMS-UCE and RLS-UCE have siilar perforance and are approxiately 3.5 db fro ICI. LMS-SCE gives the best results while LMS-TDE and LMS-CAD have poor perforance due to their liited tracking capabilities. It is likely that the convergence rate of LMS-CAD can be iproved by using a different step size for each frequency bin, as suggested in [4]. Figs. 5 and 6 show analogous results with obile speeds of 70 and 140 k/h. Note that the siulation results with ICI do not depend on the fading rate, as the channel is assued Fig. 5. BER versus E b /N 0 with a single-antenna receiver and v =70k/h. constant within each block. We see that the BER deteriorates as the obile speed increases and all detectors exhibit an error floor. LMS-SCE is always superior but it becoes unsatisfactory as the obile speed increases. The perforance of LMS- TDE (not shown in the figures) is siilar to that of LMS-CAD. Figs. 7 and 8 illustrate siulations obtained in the sae operating conditions of Figs. 4 and 5, except that four antennas are now eployed. We see that the ultiple antennas draatically iprove the syste perforance. For an error probability of 10 3 and a obile speed of 70 k/h, the loss of LMS-SCE with respect to ICI is 1.5 db while it is 5 db with either LMS-UCE or RLS-UCE. When the obile speed increases to 140 k/h, LMS-SCE is 2.5 db fro ICI while both LMS-UCE and RLS-UCE exhibit a floor. Clark s detectors and LMS-TDE cannot track fast fading and have the worst perforance. Fig. 9 shows the learning curves of a receiver eploying the proposed channel estiators. The MSE at the detector input is coputed by averaging over 1000 siulation runs. The obile speed is 70 k/h and E b /N 0 is set to 15 db. Four antennas are eployed at the receiver. We see that RLS-UCE achieves an

9 2516 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 Fig. 6. BER versus E b /N 0 with a single-antenna receiver and v = 140 k/h. Fig. 8. BER versus E b /N 0 with four receiving antennas and v = 140 k/h. Fig. 7. BER versus E b /N 0 with four receiving antennas and v =70k/h. MSE of approxiately after only two training blocks while LMS-UCE takes ore than ten blocks to converge. LMS-SCE has the lowest MSE in the steady state, but its acquisition tie is longer than that of RLS-UCE. Fig. 10 shows the coputational coplexity of the various detection and channel-estiation schees expressed in illions of floating operations per second (FLOPS) versus the bit rate R b (expressed in egabits per second). The curves are coputed fro Tables I and II with P =1 (single-antenna receiver), assuing that AHE is eployed for noise-power estiation. We see that FDE affords substantial coputational savings with respect to a conventional TDE, especially at high bit rates. Also, the frequency-doain equalizer with LMS-SCE is only slightly ore coplex than the other schees. Fig. 9. Learning curves with four receiving antennas, v =70 k/h, and E b /N 0 =15dB. VII. CONCLUSION We have discussed three channel-estiation schees for adaptive FDE in SC systes. They exploit a sequence of training blocks placed at the beginning of each data frae and Fig. 10. Coplexity of the proposed schees.

10 MORELLI et al.: CHANNEL ESTIMATION FOR ADAPTIVE FREQUENCY-DOMAIN EQUALIZATION 2517 operate in an iterative fashion. Two of the, LMS-UCE and RLS-UCE, assue independently faded frequency bins while the third schee, LMS-SCE, uses a structured approach that iproves the quality of the channel estiates. In addition to channel-state inforation, frequency-doain MMSE equalization requires knowledge of the noise power. To this purpose, a siple algorith based on ad hoc reasoning has been proposed. The perforance of all these schees has been investigated analytically and by siulation. It has been found that LMS- SCE outperfors the other ethods. The price to pay is a slight increase in coplexity, which, however, is still uch saller than that of a conventional TDE. Copared with other existing schees based on adaptive FDE, the proposed ethods have better perforance due to their enhanced tracking capabilities. In particular, a four-branch receiver eploying LMS-SCE can handle a obile speed of 140 k/h with only a 3-dB loss with respect to an ideal syste with perfect channel knowledge. APPENDIX A In this appendix, we highlight the ajor steps leading to the perforance of LMS-UCE. For siplicity, we assue that the channel is static and we drop the superscript ( ) designating the diversity branch. We begin by coputing the conditional expectation E{e Ĥ}. To this purpose, we substitute (17) into (23) to obtain e = C H C Ĥ + C H W (A1) where Ĥ = H Ĥ is the estiation error at the th step and {W } are statistically independent Gaussian vectors with zero ean and covariance atrix σ 2 I 2N. Then, using the identity E{C H C } = N (which is valid for i.i.d. data sybols with zero ean and unit variance) produces E{e Ĥ} = N Ĥ. (A2) Fro above, we see that e ay be thought of as the su of N Ĥ plus soe zero-ean disturbance ter η. Accordingly, recursion (22) ay be rewritten as Ĥ+1 =(1 µn) Ĥ µη (A3) with η =(C H C N I) Ĥ + C H W. Since in the steady state Ĥ H (i.e., Ĥ 0), it is reasonable to approxiate η as η C H W. (A4) Inspection of (A3) reveals that Ĥ aybeviewedasthe response to η of a digital filter with ipulse response { µ(1 µn) p k = k 1, k 1 (A5) 0, otherwise. Thus, (A3) becoes Recalling that η has zero ean, fro (A6), we see that E{ Ĥ} = 0, eaning that Ĥ is an unbiased estiate of H. Returning to (A4), we observe that vectors {η } are independent for different values of and have covariance atrix C η = σ 2 N I 2N. Putting these facts together, fro (A6), we have [ { } E Ĥ ĤH = σ 2 N ] p 2 i I 2N. (A7) i Next, substituting (A5) into (A7) and using the identity Ĥ 2 =tr{ Ĥ ĤH } produces { E Ĥ 2} = 2µN 2 µn σ2. (A8) At this stage, we introduce the noise equivalent bandwidth of the filter p k [15, p. 126] B L = µn 2(2 µn)t B. (A9) Then, collecting (A8) and (A9) yields (24) in the text. APPENDIX B In this appendix, we derive an iterative procedure to iniize J RLS-SCE( h) = λ i X i C i F h 2 (B1) with respect to h. We begin by setting the gradient of J RLS-SCE( h) to zero and solving for h = ĥ+1. This produces where R = d = R ĥ +1 = d λ i F H C H i C i F λ i F H C H i X i. (B2) (B3) (B4) Next, we observe that R and d ay be coputed iteratively as R = λr 1 + F H C H C F d = λd 1 + F H C H X. (B5) (B6) Then, replacing ĥ+1 with [ĥ+1 ĥ]+ĥ in (B2) and using (B5) and (B6) yields Ĥ = i p i η i. (A6) R [ĥ+1 ĥ]+ [ λr 1 + F H C H C F ] ĥ = λd 1 + F H C H X. (B7)

11 2518 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 4, NO. 5, SEPTEMBER 2005 Finally, bearing in ind that R 1 ĥ = d 1, (B7) reduces to R [ĥ+1 ĥ] =F H C H [X C F ĥ] fro which (34) in the text follows easily. (B8) REFERENCES [1] J. G. Proakis, Digital Counications, 2nd ed. New York: McGraw- Hill, [2] S. U. H. Qureshi, Adaptive equalization, Proc. IEEE, vol. 73, no. 9, pp , Sep [3] M. V. Clark, Adaptive frequency-doain equalization and diversity cobining for broadband wireless counications, IEEE J. Sel. Areas Coun., vol. 16, no. 8, pp , Oct [4] J. J. Shynk, Frequency-doain and ultirate adaptive filtering, IEEE Signal Process. Mag., vol. 9, no. 1, pp , Jan [5] D. Falconer, S. L. Ariyavisitakul, A. Benyain-Seeyar, and B. Eidson, Frequency doain equalization for single-carrier broadband wireless systes, IEEE Coun. Mag., vol. 40, no. 4, pp , Apr [6] H. Sari, G. Kara, and I. Jeanclaude, Frequency doain equalization of obile radio and terrestrial broadcast channels, in Proc. Global Telecounications (GLOBECOM), San Francisco, CA, Nov. Dec. 1994, pp [7] A. Czylwik, Coparison between adaptive OFDM and single carrier odulation with frequency doain equalization, in Proc. IEEE Vehicular Technology Conf. (VTC), New York, Spring 1998, vol. 2, pp [8] G. Kadel, Diversity and equalization in frequency doain A robust and flexible receiver technology for broadband obile counication systes, in Proc. Vehicular Technology Conf. (VTC), Phoenix, AZ, May 1997, vol. 2, pp [9] S. Alaouti, A siple transit diversity technique for wireless counications, IEEE J. Sel. Areas Coun., vol. 16, no. 8, pp , Oct [10] N. Al-Dhahir, Single-carrier frequency-doain equalization for spacetie-coded transissions over broadband wireless channels, in Proc. Personal Indoor and Mobile Radio Counications (PIMRC), San Diego, CA, Sep./Oct. 2001, pp. B143 B146. [11] X. Zhu and R. D. Murch, Novel frequency-doain equalization architectures for a single-carrier wireless MIMO syste, in Proc. Vehicular Technology Conf. (VTC), Vancouver, BC, Canada, Sep. 2002, pp [12] Y. Li, L. J. Ciini, Jr., and N. R. Sollenberger, Robust channel estiation for OFDM systes with rapid dispersive fading channels, IEEE Trans. Coun., vol. 46, no. 7, pp , Jul [13] Y. Le, Pilot-sybol-aided channel estiation for OFDM in wireless systes, IEEE Trans. Veh. Technol., vol. 49, no. 4, pp , Jul [14] H. Meyr, M. Oerder, and A. Polydoros, On sapling rate, analog prefiltering and sufficient statistics for digital receivers, IEEE Trans. Coun., vol. 42, no. 12, pp , Dec [15] U. Mengali and A. N. D Andrea, Synchronization Techniques for Digital Receivers. New York: Plenu, Michele Morelli (M 04) received the Laurea degree (cu laude) in electrical engineering and the Preio di Laurea SIP degree fro the University of Pisa, Pisa, Italy, in 1991 and 1992, respectively, and the Ph.D. degree in electrical engineering fro the Departent of Inforation Engineering, University of Pisa, in In Septeber 1996, he was a Research Assistant at the Centro Studi Metodi e Dispositivi per Radiotrasissioni (CSMDR), Italian National Research Council (CNR), Pisa, Italy. Since 2001, he has been with the Departent of Inforation Engineering, University of Pisa, where he is currently an Associate Professor of Telecounications. His research interests are in wireless counication theory, with ephasis on equalization, synchronization, and channel estiation in ultiple-access counication systes. Luca Sanguinetti (S 04) received the Laurea degree (cu laude) in inforation engineering fro the University of Pisa, Pisa, Italy, in 2002, and is currently working toward the Ph.D. degree in inforation engineering in the Departent of Inforation Engineering, University of Pisa. In 2004, he was a Visiting Ph.D. Student at the Geran Aerospace Center (DLR), Oberpfaffenhofen, Gerany. His research interests span the areas of counications and signal processing, estiation, and detection theory. Current research topics focus on transitter and receiver diversity techniques for single- and ultiuser fading counication channels, antenna array processing, channel estiation and equalization, ultiple-input ultiple-output (MIMO) systes, ulticarrier systes, and linear and nonlinear prefiltering for interference itigation in ultiuser environents. Uberto Mengali (M 69 SM 85 F 90 LF 03) received the degree in electrical engineering fro the University of Pisa, Pisa, Italy, and the Libera Docenza degree in telecounications fro the Italian Education Ministry, Italy, in Since 1963, he has been with the Departent of Inforation Engineering, University of Pisa, where he is a Professor of Telecounications. In 1994, he was a Visiting Professor at the University of Canterbury, New Zealand, as an Erskine Fellow. His research interests are in digital counications and counication theory, with ephasis on synchronization ethods and odulation techniques. He coauthored the book Synchronization Techniques for Digital Receivers (Plenu Press, 1997). Prof. Mengali is a eber of the Counication Theory Coittee and was the Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS fro 1985 to 1991 and of the European Transactions on Telecounications fro 1997 to He has served on the technical progra coittees of several international conferences and was Co-Chair of the 2004 International Syposiu on Inforation Theory and Applications (ISITA).