Channel estimation in OFDM system

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Baseband model Input Data Channel Coding Guard CP s() and modulation Band H S/P P/S (Signal Mapper)Insertion Insertion D/A Fading Channel g(t) y Output Data ua Channel Decoding Band pis:F: S/ CP and demodulation Deletion AD Deletion T r(t) Figure 1. A Digital Implementation of a Baseband OFDM System The binary information is first grouped coded and mapped according to the modulation in a"signal mapper After the guard band is inserted, an N-point inverse discrete-time Fourier transform(IdFTN) block transforms the data sequence into time domain(note that N is typically 256 or larger). Following the IdFT block, a cyclic extension of time length TG, chosen to be larger than the expected delay spread, is inserted to avoid intersymbol and intercarrier interferences. The D/A converter contains low-pass filters with bandwidth 1/Ts, where Ts is the sampling interval. The channel is modeled as an impulse response g(t)followed by the complex additive white Gaussian noise(AWGN)n(t), where On is a complex values and<tmTs STC M g Equation 1 n At the receiver, after passing through the analog-to-digital converter(ADC) and removing the CP, the dFTN is used to transform the data back to frequency domain. Lastly, the binary information data is obtained back after the demodulation and channel decoding T tX= x Y=LY(k=0,.,N-1) denote the input data of IdFt block at the transmitter and the output data of dft block at the receiver, respectively. Let 8 =[8n] and n=[n,(n=0,.,N-1) denote the sampled channel impulse response and AWGN, respectively, Define the input matrix X=diag(X) and the dft-matrix W 0(N-1) F N-1)0 (N-1)(N-1) quation 2 where N)J2n(ik/ N) Also definc H= DFTMa)=E8, andN= En Channel Estimation in OFDM Systems, Rev. 0 Freescale semiconductor 3 Block-Type pilot channel Estimation Under the assumption that the interferences are completely eliminated [ 1-3], you can derive Y=DFTN(IDFTN(X)68+n)=XE8+N= XH+N Equation 3 This equation demonstrates that an OFDM system is equivalent to a transmission of data over a set of parallel channe eIs As a result, the fading channel of the OFdM system can be viewed as a 2D lattice in a time-frequency plane, which is sampled at pilot positions and the channel characteristics between pilots are estimated by interpolation. The art in designing channel estimators is to solve this problem with a good trade-off between complexity and performance The two basic ID channel estimations in OFDM systems are illustrated in Figure 2. The first one, block-ty pe pilot channel estimation, is developed under the assumption of slow fading channel, and it is performed by inserting pilot tones into all subcarriers of OFDM symbols within a specific period. The second one, comb-type pilot channel estimation, is introduced to satisfy the need for equalizing when the channel changes even from one OFDM block to the subsequent one. It is thus performed by inserting pilot tones into certain subcarriers of each OFDM Symbol, where the interpolation is needed to estimate the conditions of data subcarriers. The strategies of these two basic types are analyzed in the next sections Block oo O oooooo ●pilo oo0●o o data ●000 ●。o。0● Time Ti Block-type pilot channel estimation Comb-type pilot estimation Figure 2. Two Basic Types of Pilot arrangement for ofDm channel Estimations 3 Block-Type Pilot Channel Estimation In block-type pilot-based channel estimation, as shown in Figure 2, OFDM channel estimation symbols are transmitted periodically, and all subcarriers are used as pilots The task here is to estimate the channel conditions (specified by H or g) given the pilot signals(specified by matrix X or vector X)and received signals(specified by Y), with or without using certain knowledge of the channel statistics. The receiver uses the estimated channel conditions to decode the received data inside the block until the next pilot symbol arrives. The estimation can be based on least square (Ls), minimum mcan-square error(MMSE), and modified MMSE. 3.1 LS Estimator The LS estimator minimizes the parameter(r-Xh)(r-XH), where()means the conjugate transpose operation. It is shown that the LS estimator of H is given by [2] HLS =X y=[(xk wI (k=0,,N-1) Equation 4 Channel Estimation in OFDM Systems, Rev. 0 Freescale Semiconductor Block-Type Pilot Channel Estimation Without using any knowledge of the statistics of the channels, the ls estimators are calculated with very low complexity, but they suffer from a high mean-square error. 3.2 MMSE EStimator The MMSE estimator employs the second-order statistics of the channel conditions to minimize the mean-square error Denote by 88:HH, and ryy the autocovariance matrix of 8,H and y, respectively, and by .y the cross covariance matrix bctwcen g and Y. Also denote by of the noise variance EIN J. Assumc the channel vector g and the noise N are uncorrelated, it is derived that HH=E出H}=EKF2)(E3)}=h的 H R Equation 5 E。y=E园y}=E区(E+N)}=RE2y Equation 6 Ryy= erY =XFr FX+oIN Equation 7 Assume gg(thus HH) and n are known at the receiver in advance, the mmse estimator of g is given by mmse RoyRyy r [2-5]. Note that if 8 is not Gaussian, mmse is not necessarily a minimum mean square error estimator, but it is still the best linear estimator in the mean-square error sense. At last, it is calculated that H-H HMMSE F mMSE FI( R +XFI 88 =EE(ⅩXE)∝+ RUUF HLS Equation 8 1-1 HHLAHH +O(XX) The MMSE estimator yields much better performance than LS estimators, especially under the lOw SNR scenarios A major drawback of the Mmse estimator is its high computational complexity, especially if matrix inversions are needed each time the data in x changes 3. 3 Modified MMSE Estimator Modified MMSE estimators are studied widely to reduce complexity [2-4]. Among them, an optimal low-rank MMSE(OLR-MMSE) estimator is proposed in this paper, which combines the following three simplification techniques 1.The first simplification pf MMSE estimator is to replace the term(de in equation 8 with its expectation E(XX )) Assuming the same signal constellation on all tones and equal probability on all constellation points we have ELXX ))=Eq1/ XujI Equation 9 Defining the average Snr as Snr=EqX y dxy, and the termp EIXY Eq1/ X) The term ON(XX) is then approximate by (B/ SNR)I, where B is a constant depending only on the signal constellation. For example, for a 16-QAM transmission, B= 17/9 Channel Estimation in OFDM Systems, Rev. 0 Freescale semiconductor 5 Block-Type pilot channel Estimation 2. The second simplification is based on the low-rank approximation. As indicated in Section 2 that Equation I has Os TG, most of the energy in8 is contained in, or near, the first (L+I taps, where=TG/ TSN and N is the DFT size. Therefore, we can only consider the taps with size of matrix is reduced dramatically after the low-rank approximation is used.l 802.11 and IEEE Std 802.16[131,TG/ Tsl is chosen among( 1/32, 1716, 178,174 so the effective 3. The third simplification uses the singular value decomposition(SVD). The SVd of R II ls UAU, where U is a unitary matrix containing the singular vectors and A is a diagonal matrix containing the singular values 202n,.22N- on its diagonal. The SVD also dramatically reduces the calculation complexity of matrices Combining all simplification techniques, the OLR-MMSe estimator is explained as follows. The system first determines the number of ranks required by the estimator, denoted by p, which should be no smaller than(L+ 1) Then, given the signal constellation, the noise variance and the channel autocovariance matrix R hh, the receiver pre-calculates B, SNR, the unitary matrix U, and the singular values n S. It thus obtains the(Nx N)diagonal matrix△ p With entries k=0,1,…,p-1 SNR Equation 10 P N During the transmission, using the transmitted pilots X and received signals Y, the His is calculated according to Equation 4, and the olr-MMsE estimator with rank p is given by HoLR-MMSE=y△UHLs uation 11 P The OLr-MMSE estimator can be interpreted as first projecting the ls estimates onto a subspace and then performing the estimation. Because the subspace has a small dimension(as small as(L+1)and still describes the channel well, the complexity of OLR-MMsSE estimator is much lower than MMsE estimator with a good performance. However, the low-rank estimators introduce an irreducible error floor due to the part of the channel that does not belong to the subspace. a legitimate question is what if (L 1) is too large to deal with--for example if the dft size n is 2048, and TG/ Ts is 1/4, (L+ 1)is still as large as 513. One solution to this problem is to partition the tones into reasonably-sized blocks and perform the estimation independently in these blocks. For example, the 2048-tone system can be approximately described by 32 parallel 64-tone systems, and each channel attenuation can be estimated independently by Olr-MMsE estimator with rank P=(6474+ 1)=17. In the scenarios when(L + 1)is large, this strategy reduces the complexity significantly at the expense of certain performance loss because it neglects the correlation between tones in different subsystems 3. 4 Estimation with Decision Feedback block-type pilot-based channels, the estimators are usually calculated once per block and are used until the next pilot symbol arrives. The channel estimation with decision feedback is proposed to improve the performance where the estimators inside the block are updated using the decision feedback equalizer at each subcarrier. The receiver first estimates the channel conditions using the pilots and obtains H= Hk(k=0,., N-1), which is based on Ls, MMSE, or modified MMSE. Inside the block, for each coming symbol and for its each subcarrier the estimated transmitted signal is found by the previous Hk according to the formula Xk =Y, Hk. kis mapped to the binary data through the demodulation according to the "signal demapper, and then obtained back though "signal mapper"as Xk j. The estimated channel Hk is updated by Hk =Y,/ Xk and is used in the next Channel Estimation in OFDM Systems, Rev. 0 Freescale Semiconductor Comb-Type pilot channel Estimation Note: The block-type channel estimation is suitable for slow fading channels; the fast fading channel causes the complete loss of estimated channel parameters 4 Comb-Type Pilot Channel Estimation In comb-type pilot based channel estimation, as shown in Figure 2, for each transmitted symbol, N, pilot signals are uniformly inserted into X with S with subcarriers apart from each other, whereS=ni Ne v The receiver knows the pilots locations P=[Pi](k=0,, N-1), the pilot values X=d (k=0, ., N-1), and the received signal Y. The LS estimates to the channel conditions at the pilot subcarriers are calculated by f=[Y(P)/X0,Y(、%”m-+, P Equation 12 The task here is to estimate the channel conditions at the data subcarriers(specified by h with length M), given the LS estimates at pilot subcarriers HLs, received signals Y, and maybe certain additional knowledge of the channel statistics. The solutions include ls estimator with ID interpolation, the maximum likelihood (ML) estimator, and the parametric channel modeling-based(PCMB)estimator. [5-71 4.1 LS Estimator with 1D Interpolation I D interpolation is used to estimate the channel at data subcarriers, where the vector Fls With length N he ID interpolated to the vector H with length N, without using additional know ledge of the channel statistics interpolation methods are summarized in the remainder of this section 4.1.1 Linear Interpolation (LI) The LI method performs better than the piecewise-constant interpolation, where the channel estimation at the data subcarrier between two pilot His(k)and HIS(k 1)is given by H(KS+t)=Hls(k)+(Hls(k+1)-HiS((t/ s) (0<t<S) Equation 13 4.1.2 Second-Order Interpolation(SOl) The soi method performs better than the li method, where the channel estimation at the data subcarrier is obtained by weighted linear combination of the three adjacent pilot estimates 4.1.3 Low-Pass Interpolation(LPI) The LPI method is performed by inserting zeros into the original His sequence and then applying a low-pass finite-length impulse response(FIR) filter(the interp function in MATLAB), which allows the original data to pass through unchanged. This method also interpolates such that the mean-square error between the interpolated points and their ideal values is minimized 4.1.4 Spline Cubic Interpolation( scl) The SCI method produces a smooth and continuous polynomial fitted to given data points(the spline function in MATLAB Channel Estimation in OFDM Systems, Rev. 0 Freescale semiconductor 7 Comb-Type Pilot Channel Estimation 4.1.5 Time Domain Interpolation (TDI) The TDi method is a high-resolution interpolation based on zero-padding and DFT/IDFT. It first converts his to time domain by IDFt and then interpolate the time domain sequence to N points with simple piecewise-constant method [5]. Finally, the dFT converts the interpolated time domain sequence back to the frequency domain In [5], the performance among these estimation techniques usually ranges from the best to the worst, as follows LPI, SCl, TDI, SOl, and LI Also, LPI and sci yield almost the same best performance in the low and middle snr scenarios, while lPI outperforms sCI at the high SNR scenario In terms of the complexity, TDI, LPI and SCi have roughly the same computational burden, while Soi and Li have less complexity. As a result, LPI and SCi are usually recommended because they yield the best trade-off between performance and complexity. 4.2 ML Estimator As mentioned in Section 3.3, most of the energy in g is contained in, or near, the first(L 1)taps, where L=TG/ TSN. Define gL+1=[80,8L+1] is the first(L+ 1)taps of g. Similarly to the definition of the square DFT matrix F, we define the non-square DFT matrix b AB NA×B (0≤a<A,0≤b<B) Equation 14 Also, we define the uniform-spaced-DFT matrix with space S as follows F(S S b A B W NA×B (0≤a<A,0≤b<B) Equation 15 It is obvious that hP=ESNpX(L+181+1, where S is the space between pilot subcarriers.Thus,the maximum likelihood estimator (MLE)of 8L+/ given the estimate to Hp(we use HLS) is obtained by 2+1=((S Np, (L+1) F (F(S P Vp, (L+1))nLS Equation 16 Finally, the complete channel estimate H of all the subcarriers is computed from gL+l by MLE N,(L+1)SL+1 Equation 17 4.3 PCMB Estimator As shown in Equation 1, the channel is modeled by a multipath fading channel with M resolvable paths with different path complex gain amj and time delays itmTs. we assume different path gains are uncorrelated with respect to each other and denoted by ro(M) the channel auto-covariance matrix, and Ro(M)= diag/ool M In [7 a channel estimation scheme based on the parametric channel modeling is proposed. In this estimator, knowledge of the channel is required; that is, M, and it mTsf are required. The estimate of M, denoted by M, is obtained by the criterion of minimum description length(MDL). The estimation of signal parameters by rotational invariance(ESPRIT)[ 8 method is used to acquire the initial multipath time delays, and an inter-path interference cancellation(iPic)delay locked loop Dll) tracks the channel multipath time delays. We define two nonuniform spaced-DFT matrices as follows Np. M Np×M (0≤k<Nn,1≤m≤M) Equation 18 N, M =W,t(m) B N×M (0≤i<N,0≤m≤M) Equation 19 Channel Estimation in OFDM Systems, Rev. 0 Freescale Semiconductor Other pilot- Aided channel estimations where(P(k))are the pilot locations, and t(m) are the estimated multipath time delays The MMSE estimator is given by [5] PCMB B R(M)+B,B Np. M Np. M-Np. M B SNR Np. M X HLS Equation 20 where snr is the average SNR, and y is the ratio of average signal power to the pilot power When these three channel estimation schemes are compared, the ls estimator with ID interpolation shows the lowest complexity Without counting the complexity of the mDl scheme and IPIC-dll to track the channel parameters, the PCMB estimator is usually simpler than the ML estimator, if 2M<N, [6]. Also, the LS estimator with 1D interpolation scheme is worse than the other two in terms of both Mse and ser. the performance of the PCMB estimator and ML estimator i almost the same, though the former performs slightly better in mse at small SNRS 5 Other pilot-Aided channel estimations Other channel estimation schemes include the simplified 2D channel estimators, the iterative channel estimators and the channel estimators for the ofdm systems with multiple transmit-and-receive antennas 5.1 Simplified 2D Estimators In 2D channel estimation, the pilots are inserted in both the time and frequency domains, and the estimators are based on 2D filters. In general, 2D channel estimation yields better performance than the 1D scheme, at the expense of higher computational complexity and processing delay. The optimal solution in terms of mean-square error is based on 2D Wiener filter interpolation, which employs the second-order statistics of the channel conditions. However, such a 2D estimator structure suffers from a huge computational complexity, especially when the dft size n is several hundred or larger. a proposed algorithm with two concatenated ld linear interpolations on frequency and time sequentially minimizes the system complexity. In [9], channel estimators based on 2D least square (Ls) and 2D normalized least square ( nls) are proposed and a parallel 2D (nls channel estimation scheme solves the realization problem due to the high computational complexity of 2D adaptive channel estimation 5.2 terative channel estimators Two efficient iterative channel estimators are proposed in [10]. To reduce complexity, the 2D transmission lattice is divided by 2D blocks, and the pilots are uniformly inserted inside each block Channel estimation proceeds on a block-by-block basis. The first estimator is based on iterative filtering and decoding, which consists of two cascaded lD Wiener filters to interpolate the unknown time-varying 2D frequency response between the known pilot symbols. The second estimator uses an a posteriori probability(APP)algorithm, in which the two APP estimators, one for the frequency and the other for the time direction are embedded in an iterative loop similar to the turbo decoding principle. These iterative estimators yield robust performance even at low SNR scenarios, but with high computation complexity and certain iteration time delay. Channel Estimation in OFDM Systems, Rev. 0 Freescale semiconductor 9 Performance evaluation 5. 3 Channel Estimators for OFDM with Multiple antennas Multiple transmit-and-receive antennas in OFDM systems can improve communication quality and capacity For the ofdm systems with multiple transmit antennas, each tone at each receiver antenna is associated with multiple channel parameters, which makes channel estimation difficult. Fortunately, channel parameters for different tones of each channel are correlated and the channel estimators are based on this correlation Several channel estimation schemes have been proposed for the ofdm systems with multiple transmit-and-receive antennas for space diversity, or multiple input multiple output(MIMO) systems for high-rate wireless data access. For example, in [ll], channel estimation is based on a Id block-type pilot arrangement, and optimal training sequences are constructed not only to optimize, but also to simplify channel estimation during the training period In [12], channel estimation in 2D for OFDM systems with multiple transmit antennas is discussed. The approach estimates and separates Nr superimposed received signals, corresponding to Nr transmit antennas, by exploiting the correlation in 2D of the received signal. More specifically, it uses two ID estimators instead of a true 2D estimator, by dividing the estimation and the separation task into two stages. The first stage separates a subset of the superimposed signals and estimates the channel response in the first dimension. The second stage further separates the signals of each subset in the second dimension, yielding an estimate for all transmit antennas. For space-time block-coded OFDM systems, this proposed estimator can track the channel variations even at high Doppler frequencies 6 Performance Evaluation This section summarizes the computational complexity of the proposed channel estimation schemes and provides simulation results to demonstrate performance 6.1 Complexity Analysis In general, ID channel estimation schemes have a much lower computational complexity than 2D schemes because they avoid computing 2D matrices. Also, block-type pilot-channel estimation schemes are usually simpler than comb-type pilot schemes because they calculate the estimators once per block. In the block-type pilot schemes with decision feedback, the estimators are updated for each symbol by simple vector division. Com-type pilot schemes calculate the estimators for every OFDM symbol. Algorithm complexity, ranking from low to high, is summarized in Table I and Table 2 for the block-type pilot arrangement and comb-type pilot arrangement, respectively Table 1. Computational Complexity Analysis: Channel Estimation Schemes with Block-Type Pilot Arrangement Estimation scheme Complexity Comments LS EStimator LOW Simple vector division OLR-MMSE EStimator Moderate Avoid matrix inversion and also simplify the matrix operations to the calculations between a low-rank diagonal matrix and a unitary matrix. MMSE EStimator High Matrix inversion and other operations with size N, where N is the dft size ( typically256,512,1024,or2048 Channel Estimation in OFDM Systems, Rev. 0 10 Freescale Semiconductor


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