Linear MMSE estimation of time–frequency variant channels for MIMO-OFDM systems☆
Introduction
Design of next-generation wireless systems is focusing mainly on the use of multiple-input multiple-output (MIMO) channels [26] with orthogonal frequency-division multiplexing (OFDM) [8] providing MIMO-OFDM systems [24]. Iterative receivers [28] have shown to be very attractive from complexity-performance point of view, and have been designed [1], [17], [19], [32] such to perform also pilot-assisted channel estimation [2]. In this paper we focus on channel estimation for MIMO-OFDM systems to be performed within the loop of an iterative receiver.
Channel estimation for OFDM systems has been proposed via singular value decomposition [4] and via discrete Fourier transform [5] exploiting frequency correlation, and via two-dimensional Wiener filtering [9] exploiting time and frequency correlations. Robustness to channel-statistics mismatch is analyzed in [13], while complexity issues have been taken into account also via parametric channel modeling [29]. Recently, a low-complexity channel estimator for MIMO-OFDM systems has been proposed exploiting the angle-domain representation [10], in which the trade-off between performance and complexity crucially depends on the knowledge of the correlation among the various angle-frequency domain beams.
Basis expansion models [7] have shown to be very effective in dealing with time-variant channels. Exploiting the works on discrete prolate spheroidal (DPS) sequences [23], a robust low-complexity channel estimator has been proposed [31] and applied in both iterative receivers for multi-carrier code-division multiple-access systems [32] and iterative receivers for MIMO-OFDM systems [17], [19]. Time and frequency variations of realistic wireless channels have been taken into account via the multidimensional DPS sequences [22], [27] in the extensions proposed in [3], [33] for multi-carrier code-division multiple-access systems, in [11] for MIMO systems, and in [18] for MIMO-OFDM systems.
Finally, alternative approaches to channel estimation [6], [21], [30] are based on blind-identification techniques exploiting higher order statistics of received signals. Blind channel estimation saves spectral efficiency with respect to pilot-assisted channel estimation at the cost of less reliable estimates.
In this paper we recall the estimator proposed in [18] and propose two similar approximations, comparing their performance and complexity on the basis of channel characteristics. The benefits of the two-dimensional Slepian-based approach proposed in [18] are (i) robustness—no assumption on channel statistics is needed but knowledge of the maximum delay spread and maximum Doppler spread; (ii) low complexity—less coefficients to be estimated due to the concentration of the space; (iii) accuracy—two-dimensional processing exploits both time and frequency correlations. Such an estimator is named in the following joint channel estimator (JCE) as it performs joint time–frequency processing. In order to further reduce the computational complexity, which can still remain high for channels with large Doppler and/or delay spreads, we design two serial channel estimators (SCEs) approximating the two-dimensional time–frequency processing via serially concatenated one-dimensional processing: (i) a serial time–frequency channel estimator (STFCE) in which time processing is performed for each subcarrier and then frequency processing is applied; (ii) a serial frequency–time channel estimator (SFTCE) in which frequency processing is performed for each OFDM block and then time processing is applied.
As we here mainly focus on channel estimation, when designing and testing the channel estimators all the transmitted symbols are assumed to be known at the receiver. However, in a real iterative receiver only pilot symbols are available at the first iteration, while soft estimates from the decoder are available at successive iterations to replace (initially unknown) data symbols. Soft estimates will converge, in a well-designed receiver, to the correct values of data symbols, thus the performance of the channel estimator shown represents the maximum achievable performance. Before concluding the paper, a final test of the channel estimator will be presented by showing the performance an iterative receiver in which both transmitted symbols (excluding pilots) and channel coefficients need to be estimated. Although, we are not exploring the problem of optimal pilot placement [14], [25], affecting mainly the performance of the channel estimator at the first iteration, it is worth noticing that the proposed estimators allow flexible pilot patterns. Usually, block-type (for some given time slots all subcarriers contain pilots) and comb-type (for some given subcarriers all time slots contain pilots) patterns are considered, while the proposed estimators allow, due to the two-dimensional processing, various pilot patterns that sparsely sample the time–frequency domain as long as they obey the limits of the sampling theorem set by delay spread and Doppler spread.
The contributions of this paper are mainly:
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the design of two low-complexity channel estimators that fit the same characteristics of the estimator in [18];
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the analysis of their performance with respect to the channel characteristics (such as delay and Doppler spreads) and system parameters (such as number of subcarriers and blocks in a frame);
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the verification of their performance in an iterative receiver for MIMO-OFDM systems.
The rest of the paper is organized as follows: Section 2 introduces the system model; the Slepian-basis expansion models are described in Section 3; the channel estimators are presented in Section 4; Section 5 shows the performance obtained via computer simulation in terms of normalized mean square error (NMSE) vs. signal-to-noise ratio (SNR), as well as the performance in terms of bit error rate (BER) vs. SNR of an iterative receiver for MIMO-OFDM systems employing such estimators within the loop; some concluding remarks are given in Section 6.
Notation: Column vectors (resp. matrices) are denoted with lower-case (resp. upper-case) bold letters; an (resp. An,m denotes the nth (resp. (n,m)th) element of vector (resp. matrix ); denotes a diagonal matrix whose main diagonal is ; denotes the N×N identity matrix; denotes the N×N null matrix; denotes the column of ; denotes a vector of length N whose components are 1; denotes a vector of length N whose components are 0; , (.)⁎, (.)T and (.)H denote expectation, conjugate, transpose, and conjugate transpose operators; denotes an estimate of a; denotes the expected value of a; is the Kronecker delta; denotes the Kronecker product; denotes the smallest integer value greater than or equal to a; j denotes the imaginary unit; denotes a circular symmetric complex normal distribution with mean vector and covariance matrix ; the symbol means “distributed as”.
Section snippets
Analytical model
We assume a wireless MIMO-OFDM system with K transmit antennas and N receive antennas. For data transmission, each transmit antenna uses OFDM with M subcarriers. Data are assumed to be encoded within a frame composed of S OFDM blocks, and each OFDM block is composed of M symbols. In the following, for the generic frame, xk[m,s] denotes the (frequency domain) symbol1 transmitted by the kth transmit antenna on the mth subcarrier during the transmission of the sth
Slepian-basis expansion
We consider a wireless channel with maximum normalized delay spread and maximum normalized Doppler spread , i.e. for each transmit/receive antennas pair, is the rectangular support of the scattering functionIt is worth noticing that and represent delay and Doppler as they correspond via a Fourier transformation to frequency index m and time index s, respectively.
Let and denote
Channel estimation
JCE proposed in [18] was based on the following signal model for channel estimationwherewherewhereand whereJCE assumes the following expression (see [18] for details)where
Simulation results
Performance of the various channel estimators is evaluated and compared by means of NMSE, computed via numerical simulations as follows: Two typologies of channels are considered, namely: (i) “square channel”, presenting a rectangular support in the delay-Doppler domain; (ii) “V-stripe channel”, presenting two parallel vertical segments as support in the delay-Doppler domain,3 i.e.
Conclusion
Two low-complexity two-dimensional channel estimators for MIMO-OFDM systems have been designed in order to exploit in a serial way both time and frequency correlations of the wireless channel via use of a Slepian expansion. Their complexity and their performance have been compared to an analogous two-dimensional channel estimator performing joint processing of time and frequency correlations. Performance in terms of NMSE vs. SNR has been analyzed for the case in which both pilots and data are
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This work has been supported by the Research Council of Norway and by the Swedish Governmental Agency for Innovation Systems under the projects WILATI and WILATI+ within the NORDITE framework. The material in this paper was presented in part at the IEEE Global Telecommunications Conference (GLOBECOM), Honolulu, HI, US, December 2009 (see [20]).