Robust Constrained-Optimization-Based Linear
Receiver for High-Rate MIMO-OFDM Against
Channel Estimation Errors
Chih-Yuan Lin, Jwo-Yuh Wu, Member, IEEE, and Ta-Sung Lee, Senior Member, IEEE
Abstract—We consider multi-input multi-output–orthogonal
frequency division multiplexing (MIMO-OFDM) transmission in a scenario that the adopted cyclic-prefix (CP) length is shorter than the channel delay spread for boosting data rate and, moreover, the channel parameters are not exactly known but are estimated using the least-squares (LS) training technique. By exploiting the receiver spatial resource, we propose a constrained-opti-mization-based linear equalizer which can mitigate inter-symbol interference and inter-carrier interference incurred by insuf-ficient CP interval, and is robust against the net detrimental effects caused by channel estimation errors. The optimization problem is formulated in an equivalent unconstrained general-ized-sidelobe-canceller (GSC) setup. The channel parameter error is explicitly incorporated into the constraint-free GSC system model through the perturbation technique; this allows us to exploit the presumed LS channel error property for deriving a closed-form solution, and can also facilitate an associated analytic performance analysis. A closed-form approximate signal-to-inter-ference-plus-noise ratio (SINR) expression for the proposed robust scheme is given, and an appealing formula of the achievable SINR improvement over the nonrobust counterpart is further specified. Our analytic results bring out several intrinsic features of the proposed solution. Simulation study confirms the effectiveness of the proposed method and corroborates the predicted SINR results.
Index Terms—Constrained optimization, generalized sidelobe
canceller, interference suppression, least-squares channel estima-tion, multi-input multi-output (MIMO), orthogonal frequency division multiplexing (OFDM), perturbation analysis.
I. INTRODUCTION
A. Motivation and Paper Contributions
O
RTHOGONAL frequency division multiplexing (OFDM) combined with multiple transmit and receive antennas, aka, multi-input multi-output (MIMO) OFDM, has become a key communication technique over frequency-selective channels [21], [29]. At the transmitter, the OFDM modulator appends at each symbol block head a cyclic prefix (CP), with Manuscript received August 23, 2006; revised August 24, 2006. This work was supported in part by the National Science Council of Taiwan under Grant NSC 96-2752-E-002-009, in part by the MediaTek Research Center at National Chiao Tung University, Taiwan, and in part by the Ministry of Education of Taiwan under the MoE ATU Program. Part of this paper was presented at the 49th IEEE Global Telecommunications Conference, San Francisco, CA, November 2006. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Erchin Serpedin.The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: cylin.cm90g@nctu.edu.tw; jywu@cc.nctu.edu.tw; tslee@mail.nctu.edu.tw).
Digital Object Identifier 10.1109/TSP.2007.893206
length no shorter than the channel delay spread, for removing inter-symbol interference (ISI). The insertion of CP, however, will lower the effective data rate. A commonly used approach to rate boost in MIMO-OFDM is thus to conserve the length of CP, and rely on sophisticated receiver design for combating the residual ISI as well as the induced inter-carrier interference (ICI). Typical such proposals include the time-domain channel shortening mechanism [1] and the frequency-domain per-tone equalization scheme [13]. The former resorts to shortening filter for “squeezing” the composite channel memory within the CP interval, thus limiting the ISI effect; the latter, on the other hand, aims for direct ISI and ICI suppression in the frequency domain. Both of the above methods assume perfect channel information is available at the receiver. However, channel parameter mismatch due to imperfect estimation is inevitable in practice, and can further impair the system performance. To the best of our knowledge, robust receiver design for high-rate MIMO-OFDM against channel estimation errors remains scarce in the literature.
It is well known that multiple receive antennas can provide an array gain for interference rejection [32]. This thus motivates us to leverage the spatial resource for combating ISI-ICI effect in MIMO-OFDM when a limited CP is used for trading high-rate performance. This paper proposes one such solution via a linear constrained-optimization approach [7], assuming that the channel parameters are not exactly known but are estimated via the commonly used least-squares (LS) training technique [2], [19]. The proposed method relies on the block system model for MIMO-OFDM and exploits the inherent joint space-frequency degrees-of-freedom. The adopted interference mitigation mechanism is a simple linear weighting matrix; ISI and ICI are suppressed by minimizing the average power of the filtered interference, while a linear constraint is imposed for extracting the desired signals through maximal ratio combining. To tackle the impact of channel estimation errors, one natural strategy in the considered scenario is to model the channel mismatch as a random variable and exploit the presumed LS channel error characteristic to derive a solution. Toward this end, we resort to the generalized sidelobe canceller (GSC) principle [8], [25], [32] to reformulate the constrained-optimization problem into an equivalent unconstrained framework. The constraint-free GSC setup allows us to explicitly track each corrupted signal component resulting from imperfect channel estimation, and in turn leads to a very natural cost function for joint interference and channel error mitigation. We further leverage the perturba-tion technique [15], [37] to incorporate the channel parameter 1053-587X/$25.00 © 2007 IEEE
deviation into the solution equation; the optimal weighting ma-trix is then obtained by invoking the error characteristics of the LS channel estimate. The proposed optimal solution can miti-gate the aggremiti-gate impacts due to channel estimation errors like signal leakage and other background parameter perturbation effects. Further analysis reveals that signal leakage turns out to be a dominant factor and a suboptimal solution, in the form of diagonal loading (DL), can attain a near-optimal performance. Also relying on the perturbation technique, we then derive a closed-form signal-to-interference-plus-noise ratio (SINR) expression associated with the suboptimal DL scheme; due to the near-optimal nature of the DL solution, the established result can well predict the actual SINR tendency attained by the optimal one. Our analytic formula can further quantify the achievable average SINR improvement over the nonrobust GSC weight (i.e., the one derived under exact channel knowledge). In particular, we provide a closed-form expression of the SINR increment, and based on which several key features regarding the proposed robust solution can be inferred. Simulation study confirms the effectiveness of the proposed robust solution and also corroborates the presented SINR results.
B. Connection to Previous Works
Robust constrained-optimization based linear receiver design against channel estimation errors has also been addressed in the context of multiuser communication [22], [23], [24], [38]. By modeling the channel mismatch as a random variable with known statistical characteristics, the probability-constrained op-timization approach [22], [23] exploits the Gaussianality as-sumption on the estimation error, and the solution is obtained through linear [22] or nonlinear [23] programming. Although the proposed method in this paper adopts a similar stochastic setting, as we will see it relies entirely on the first- and second-order error statistics but not on the underlying assumption on error distribution. In [24] and [38], the channel error is, on the other hand, treated as a “deterministic” perturbation. Based on a min-max type formulation, the optimization problem in [24] is solved by using the convex programming technique; the so-lution in [38] admits a DL form, with the optimal loading factor determined via an iterative procedure. We note that the deter-ministic formulation of model error is also used in robust beam-former design [14], [18], [26], [33], in which exact statistical characterization of model uncertainty due to, e.g., array calibra-tion error, unknown antenna coupling effect, etc., is difficult (or even impossible) to track.
C. Paper Organization and Notation List
The rest of this paper is organized as follows. Section II re-views the MIMO-OFDM system model and the LS channel esti-mation technique. Section III presents the multichannel system representation and shows the GSC-based ISI-ICI suppression scheme under perfect channel knowledge assumption. The pro-posed robust GSC filter is derived in Section IV; the related SINR performance analysis is provided in Section V. Section VI discusses the algorithm complexity. Section VII introduces an alternative formulation of the proposed approach. Section VIII is the simulation results. Finally, Section IX is the conclusion.
Notation List: Let and be the sets of -dimen-sional complex vectors and complex matrices. Denote by , and , respectively, the transpose, the com-plex conjugate, and the comcom-plex conjugate transpose. and denote the identity and zero matrices; is the
zero matrix. For , let be the
di-agonal matrix with the elements of on the main diagonal. The notation stands for the expected value of the random
vari-able . For and , denote by the vector
two-norm and the matrix Frobenius norm. The symbols and stand for the matrix Hadamard product ([10], p-298) and the matrix Kronecker product ([10], p-242), respectively.
II. PRELIMINARY
A. System Model and Basic Assumptions
We consider the discrete-time baseband model of a MIMO-OFDM system with subcarriers, transmit antennas, and receive antennas. At time , the time-domain symbol to be sent from the th transmit antenna is expressed as [35]
(2.1)
where is the frequency-domain OFDM symbol,
is the FFT matrix, ,
with being the last rows of , accounts for
the insertion of a CP with length . Let be the im-pulse response of the channel between the th transmit antenna and the th receive antenna; we assume without loss of gener-ality that all the MN channels are of the same order . Then the
received time-domain data vector from the th
antenna branch is [35]
(2.2)
where is lower triangular Toeplitz
with as the first column,
is upper triangular Toeplitz
with as the first row, and
is the noise vector. To process the received data , the leading guard samples is first discarded; this corresponds to post-multiplying by the CP-removal
matrix to get
When the length of CP is no less than the channel order, i.e., , the received signal is free from ISI and ICI, so
that and , which is a circulant
matrix with
(2.4) as the first row. In this paper, we focus on the case ; as such direct manipulation shows becomes upper
trian-gular Toeplitz with as the
first row, and is instead obtained from by setting
for and .
With (2.3), the received signal in the frequency domain reads
(2.5)
We note that, on the right-hand side (RHS) of (2.5), the second term is the ISI, whereas the first term is a mixture of the de-sired tone-by-tone signals and ICI. To realize low-complexity per-tone based signal recovery, which is one main advantage of MIMO-OFDM [21], a natural approach is to treat ISI and ICI as an overall composite interference and devise efficient schemes for joint ISI-ICI suppression [6], [31]. For this, it requires to further split ICI from the signal-ICI mixture in (2.5); since ICI is characterized via the deviation of from the circulant matrix [6], the splitting can be done according to the following decomposition:
(2.6)
where
(2.7) with denoting the circulant permutation matrix with as the first row [see Fig. 1 for schematic de-scriptions of (2.6) and (2.7)]. From (2.6), we can rewrite in (2.5) as
(2.8)
where is diagonal with
. The first term on the RHS of (2.8), which is composed of parallel tone-by-tone symbol streams from all transmit antennas, serves as the signal of interest. Since , the symbol in each tone is contaminated
Fig. 1. Schematic descriptions of (a) the decomposition (2.6) and (b) the rela-tion (2.7).
by both ISI from the previously transmitted block [the second term in (2.8)] as well as ICI due to the loss of channel cyclicity [the third term in (2.8)]. Based on (2.8),1we propose a method
for jointly suppressing ISI and ICI in the presence of channel mismatch. The following assumptions are made in the sequel.
1) The number of receive antennas is greater than the number of transmit antennas, i.e., .
2) The source symbols of each transmit antenna is zero mean, unit-variance, and
, where is the Kronecker delta. 3) The elements of ’s are i.i.d. complex circular
Gaussian with zero mean and variance .
B. Least-Squares Channel Estimation
We assume that during the training phase
and the channels are estimated based on the LS training technique; see [2], [19] for detailed treatments. Let , and be the corresponding optimal LS estimate, with modeling the estimation error. With
, it is known that [2]
(2.9) where is the transmit power dedicated to channel estimation; also, since the noises between different receive antennas are in-dependent, we have
(2.10) The channel error properties (2.9) and (2.10) will be used in our robust equalizer design.
1The proposed method can also instead rely on the raw signal model (2.5)
for ISI suppression first, followed by a space-frequency detector for removing ICI. Such an approach, however, can yield rather limited performance gain at the cost of significant extra computational complexity; this is further discussed in Section VII.
III. ISI-ICI MITIGATION
This section shows how the receive diversity can be ex-ploited for ISI-ICI suppression in the considered scenario. We will first collect all the received signal branches to form a multichannel system model, with which the interference subspace can be characterized; it will be seen that under quite mild conditions there will be sufficient inherent spatial and frequency degrees-of-freedom for combating ISI and ICI. Then we will show a constrained optimization based ISI-ICI sup-pression scheme under perfect channel knowledge assumption. The problem is formulated and solved within an equivalent unconstrained GSC setting, which lays the foundation of the proposed robust equalizer design.
A. Multichannel System Representation
Let be the vector
con-taining all the transmitted OFDM symbol blocks. By stacking , in (2.8) into a vector, we can form the -dimensional multichannel model:
(3.1) where (3.2) with (3.3) with (3.4) for , and
. Since , we may
as-sume without loss of generality that the channel tone matrix is of full column rank ; this assumption is valid whenever the subchannels between all transmit and receive antennas are uncorrelated. Since is upper triangular with all the nonzero entries clustering in the last columns, the matrix in (3.3) will have the last columns nonzero; by def-inition, the “big” in (3.2) then has only nonzero columns, and is of rank at most . From (2.7) and (3.3),
it is easy to check , and hence we have from
(3.2) that
(3.5)
where the last equality follows by the definition of Kronecker product. This asserts that the rank of does not exceed either since it is obtained by permuting the columns of . The relation (3.5) also implies the interference subspace, spanned by the columns of and , is of dimension no larger than . Indeed, from (3.5) and (3.1) it follows
(3.6)
With (3.6) and since [see (3.1)], the column
spaces of both and coincide with that of ,
whose rank is at most (as is orthonormal).
Hence, if 1) the -dimensional range space of the channel tone matrix does not overlap with the ISI-ICI subspace, and
2) , it is plausible to exploit the extra
degrees-of-freedom provided by the multi-channel space-fre-quency model (3.1) for ISI-ICI suppression. We note that con-dition 1) can be verified to hold unless all the subchannels lapse into the same direction, viz., for some and ; condition 2) is typically true since the number of subcarriers is often substantially larger than the channel order .
B. GSC-Based Interference Suppression: Perfect Channel Knowledge Case
We assume for the moment that the channel is perfectly known at the receiver. To exploit the extra degrees-of-freedom in model (3.1) for interference suppression, a commonly used approach is via constrained optimization [7], [25], [30]. Specifically, we will seek for a linear weighting matrix
which satisfies
(3.7) and minimizes the mean power of the filtered ISI-ICI, i.e., . With con-straint (3.7), the optimal weight will linearly combine the desired signal in the maximal-ratio sense (channel matched filtering), and suppress ISI-ICI via minimizes the total output interference-plus-noise power; the equalized signal then ap-proximates
(3.8) which can facilitate low-complexity tone-by-tone signal sepa-ration. To solve for the optimal , an efficient approach is to transform the constrained optimization problem into an uncon-strained one via the GSC principle [8], [32]. This amounts to decomposing the weighting matrix into
(3.9) where the signal signature represents the nonadaptive com-ponent for verifying constraint (3.7),
is the signal blocking matrix with , and
to be determined.2 With (3.9) and (3.1), the equalized output becomes (3.10) where (3.11) and (3.12) Since the matched filtered branch is contaminated by
(3.13) to effectively suppress interference equation (3.10) suggests that the matrix should be chosen to render as close to as possible; more precisely, we can determine by mini-mizing the following cost function
(3.14) The optimal weight, denoted by , must satisfy the linear equation (3.15) where (3.16) With (3.15), we have (3.17) and the optimal GSC weight is thus
(3.18) We note that 1) the GSC scheme solves a constrained optimiza-tion problem via a simple and elegant unconstrained formula-tion, 2) the decomposition (3.9) alleviates algorithm complexity
since only the computation of the matrix ,
other than the weight , is required, and 3) the matrix in (3.18) is obtained based on the crucial perfect channel knowledge assumption; when channel parameter mis-match occurs due to imperfect estimation, the performance of solution (3.18) will degrade since it does not take into account channel error mitigation.
2The resultant GSC based solution will coincide with the optimal one derived
under the original constrained-optimization based formulation ifrank(B) = (M 0 N)Q [3]; this condition is fulfilled if the columns of B form an or-thonormal basis for the left null-space associated withD.
IV. PROPOSEDROBUSTSOLUTION
This section studies the problem of robust equalizer design against channel estimation error. We will first introduce the de-sign formulation, and point out the challenge toward a solution. Then we will characterize the estimated blocking matrix via a perturbation analysis; this is crucial for solution derivation and subsequent performance analysis. Based on the established re-sults and assuming the channel is estimated via the LS criterion, a closed-form optimal weighting matrix is obtained and related discussions are given.
A. Problem Formulation
Our exposure will directly rely on the GSC setup. We first ob-serve that, when only an estimated is available, exact maximal-ratio combining of the desired signal is impos-sible; the best we can do, however, is to linearly combine through to get the approximation . This suggests us to fix as the nonadaptive portion of the GSC solution, and decompose the weighting matrix into
(4.1) where is the blocking matrix associated with , that is,
, and is to be determined. With
(4.1), the equalized output is instead
(4.2) where
(4.3) and
(4.4) Due to inexact channel knowledge, the desired signal in the matched filtered branch is noncoherently combined, and the contaminating interference becomes
(4.5) The channel estimation error also modifies the blocked signal characteristics in (4.4). In particular, since the estimated is
otherwise determined via (and hence in
general), there is a signal leakage into the blocking branch . To mitigate the aggregate impacts due to channel errors, a natural approach is to treat as a composite inter-ference, and to design by minimizing
(4.6) where the expectation is taken with respect to the source signal, channel estimation error, and background noise (assuming all are mutually independent). Based on (4.4), (4.5), and (4.6), and
by averaging the cost function over the source signal and noise, we have
(4.7) Since, for a given we have3
, and
, the derivative of with respect to is thus
(4.8)
With (4.8) and by setting , the
first-order necessary condition reads
(4.9) Note that, with perfect channel assumption ( and
), equation (4.9) reduces to (3.15), and the solution is simply given by (3.17). To determine the optimal from (4.9), it is nec-essary to explicitly evaluate all the involved expectation terms. This can be done if we can establish an expression linking the estimated blocking matrix with the channel matrix perturba-tion . Since is constructed as a basis of the left null space associated with through SVD (recalling that ), an exact relation between and appears highly intractable. In the next subsection, we will resort to the perturbation technique for developing an analytic (but approximate) expression.
B. Estimated Blocking Matrix: A Perturbation Analysis
Let us express the estimated channel tone matrix as
(4.10)
where models the estimation error and is
defined similarly as in (3.4), except that the th submatrix is
(4.11)
where contains the first columns of the
FFT matrix. Write an SVD of the exact signal matrix as
(4.12)
3These equalities follow immediately from the definition of the derivative of
a real valued function with respect to a matrix, e.g., [34, App. A], together with some straightforward manipulations.
and likewise for
(4.13) We note that each component matrix in (4.13) is of the same di-mension as the corresponding noise-free counterpart in (4.12); the blocking matrices with, and without, channel parameter error are, respectively, and , both of
dimen-sion . Let us further express as
(4.14) with modeling the deviation. When is small, we have the following linear first-order approximation of .
Lemma 4.1 [28]: The perturbed blocking matrix can be approximated by
(4.15)
where is such that .
To completely specify in the form (4.15), it remains to determine . This can be done by further taking into account the equality , which together with (4.10) and (4.15) implies
(4.16)
Since and [see (4.12)], (4.16) can
be rearranged into
(4.17) To determine from (4.17), we further observe that
(4.18) where the equality follows from the orthonormality of .
Since , inequality (4.18) asserts that
is bounded from above by some quantity quadratic in , which is small with small . We may thus neglect in (4.17) so that
(4.19) This technique is used in [15] for determining the first-order per-turbation. With (4.19), the matrix can thus be approximated by
(4.20) This immediately implies
, and
(4.21)
Equation (4.21) provides a closed-form expression of linear
in the estimation error . The linearity nature can consid-erably simplify the derivation of the optimal solution and will also lead to tractable procedures of performance analysis. We note that, instead of (4.15), more accurate approximation of can be obtained by incorporating higher order components [37].
Although this can improve the solution accuracy, the resultant analysis would however become intractable.
C. Optimal Solution
Based on (4.21) and assuming that the channel is estimated in the optimal LS sense, we can explicitly determine the expec-tation quantities involved in (4.9); these are summarized in the next lemma (see Appendix A for a proof), and will be used for deriving an optimal weighting matrix.
Lemma 4.2: The following results hold.
1)
2)
3) Define the matrix
(4.22) Then we have
(4.23)
in which is block diagonal, with the th
block diagonal submatrix given by
(4.24)
and denotes the th diagonal block of .
Based on (4.9) and Lemma 4.2, the optimal can be ob-tained as
(4.25) where and are, respectively, defined in (3.16) and (4.24). Note that solution (4.25) is on average the optimal choice for ISI-ICI suppression under (white) LS channel estimation error assumption. In practical implementation when only an esti-mated channel is available, the sampled-version of the robust GSC weight is thus
(4.26) in which and are respectively the estimates of
and .
Discussions:
1) The proposed approach explicitly incorporates the channel mismatch effect into the GSC formulation; it aims for joint
mitigation of ISI-ICI and the net parameter mismatch ef-fects induced by channel estimation errors. The optimal differs from in (3.17) in additional two terms,
namely, and ,
in-volved in matrix inversion; the former accounts for the signal leakage effect driven by the white-noise-like LS channel estimation error [cf. (2.9) and (2.10)], whereas the latter is due to the parameter perturbation in the ISI-ICI signature matrices.
2) We should note that our design formulation is not exclusive to the case with LS channel estimation (which produces a white channel estimation error); it does provide a unified framework for robust GSC filter design regardless of the adopted channel acquisition techniques. Indeed, as long as the channel error (with known covariance matrix) is inde-pendent of the source signal and noise, (4.9) remains true and the proposed approach will yield a solution of the form (4.25), except that all the involved matrices are accordingly modified based on the actual channel error characteristics. 3) Since is the coherently combined signal signature, the magnitude of its nonzero entries would in general be substantially larger than those of and . This implies the entries of the matrix in (4.22), and as well, could be relatively small as compared with (through simulations it is found that the entries of are two-order less in magnitude in the medium-to-high SNR region). As a result, the achiev-able performance of in (4.25) can remain largely intact if we ignore the term , which reflects the ISI-ICI signature perturbation and consider the sampled DL solution
(4.27) with
(4.28) This would indicate that the signal leakage, on the other hand, is the dominant effect incurred by channel estima-tion errors. An intuitive reason for this is that, the leaking signal component into the blocking branch (4.4) will cause undesirable signal cancellation via the two-branch GSC in-terference-rejection mechanism, leading to a loss of the ef-fective SINR.
4) Robust constrained-optimization-based beamformer de-sign with background parameter error modeled as a white noise is addressed in [5]. The solution approach reported therein is to impose certain quadratic constraint on the beamforming weight so as to potentially keep down the white-noise amplification gain [5, pp. 1366–1367]. In light of this point, another plausible approach to robust GSC filter design in our context is thus
The expectation in (4.29) is taken with respect to the source signal and measurement noise. The solution to (4.29) is known to be
for some (4.30)
The performance of solution (4.30) depends crucially on the selection of [5], [18], [30], [33]; however, there are in general no tractable rules for explicitly determining an optimal , even when the uncertainty level is known [18]. We note that the suboptimal alternative (4.28) is exactly the sampled-version of the DL solution (4.30), with set to be (4.31) It is thus conjectured that, under white channel error as-sumption, derived based on the presented perturbation analysis is the best choice with respect to the design crite-rion (4.29). Our simulation results (see Simulation D) tend to confirm this postulation.
V. PERFORMANCEANALYSIS
This section investigates the SINR performance of the pro-posed robust filter (4.26). We will first derive an approximate average SINR expression in closed-form. Then we will show the proposed solution can yield an improved SINR gain over in (3.18); in particular, the achievable SINR increment will be quantified, and based on which several key features regarding the optimal solution can be inferred.
A. SINR Evaluation
To evaluate the average SINR attained by (4.26), we will resort to the perturbation-based technique. Specifically, we will explicitly link the estimated solution with channel mismatch , and then invoke the LS channel error property for mean SINR evaluation. To facilitate the underlying analysis, we will neglect the term in (4.26) and consider the diagonal loading solution (4.27); through simulation tests, the derived results based on the simplified solution form are seen to well predict the actual SINR tendency attained by the optimal one (4.26).
To proceed, let
(5.1) where is the exact solution of [by substituting the true parameters and in (4.27)] and models the deviation. With (5.1), the output from the robust GSC filter is expressed in equation (5.2), as shown at the bottom of the page,
where is the desired signal component and is the overall interference and noise. With (5.2), the average SINR is thus [12]
(5.3) where the expectation is taken with respect to the source signal, noise, and channel estimation errors. The signal power
can be directly computed by
(5.4) where the last equality in (5.4) follows from the definition of in (5.1) and . The crucial step is to determine the composite interference power . Based on fur-ther perturbation analysis, we have (see Appendix B for detailed derivations)
(5.5) where
(5.6) The expectation involved on the RHS on (5.5) is with respect to
the channel errors. The quantity in is the
filtered interference power under perfect channel knowledge; the remaining two arise due to channel mismatch and can be further computed as4(keeping only dominant terms)
(5.7) where the matrices , are provided in Table I. The average SINR attained by can then be evaluated based on (5.4), (5.5), and (5.7).
4The derivations can be found at
http://mimo.cm.nctu.edu.tw/Professor/Pa-pers/derivations.pdf.
TABLE I
FORMULAS OFX X ,WITHX = M Y M ; 1 i 7; X = N Y M ,ANDX = N Y M
B. Achievable Performance Advantage
Based on (5.5), the proposed scheme can be shown to yield an SINR advantage over in (3.18). To see this, let us write as an estimate of . By splitting the fil-tered output in the form (5.2), it is straight-forward to check that the signal power is
(5.8) where the last equality follows from (3.18). Also, by going through essentially the same perturbation analysis as in Appendix B, the filtered interference power is verified to be
(5.9) From (5.4) and (5.8), we can see that the average signal levels sustained by and are identical; the SINR are thus com-pletely determined by the respective interference powers
and . When is small, it can be shown that .
More precisely, we have the following result (see Appendix C for a proof).
Theorem 5.1: For small
(5.10)
Equation (5.10) shows that, in the high SNR regime, the pro-posed robust solution can provide an SINR increment
(5.11) Our simulation results show that (5.11) does remain valid for a wide range of SNR. The analytic SINR increment (5.11) not only quantifies the performance advantage of the robust scheme (4.26), but can also reveal several associated intrinsic features. To see this, we note that
(5.12) With (5.12), we can infer from (5.11) the inequality relation
(5.13)
where denotes the maximal eigenvalue
asso-ciated with . The lower bound (5.13) leads to the fol-lowing observations.
1) Let be small and fixed. Through manipulation it can be shown that the incremental SINR lower bound (5.13) will
increase as decreases. Hence, when is small and in-curs severe channel mismatch, the proposed robust equal-izer (4.26) would yield significant performance gain over solution (3.18). As increases, and hence the estimation accuracy improves, the performance gain would however become negligible (this is also seen in our simulation). 2) We can also see from (5.13) that, for fixed and ,
the performance improvement would be limited when , which reflects the maximal power of ISI and ICI with perfect channel knowledge, is large. This is intuitively reasonable since, under severe ISI and ICI, the equalizer will largely aim for interference suppression, rather than combating the channel mismatch effects.
VI. COMPLEXITYCOMPARISON
This section compares the algorithm complexity of the GSC solution (3.18) with that of the time-domain channel shortening/ equalization (TEQ) approach [1] and the frequency-domain per-tone equalizer (PEQ) [13]; we note that the complexity of the proposed robust solution (4.26) is essentially the same with that of (3.18).
The computational cost of solution (3.18) is in solving for the blocking matrix via and inverting the
matrix . A low-complexity scheme for obtaining which exploits the block diagonal structure of can be found in [17]. As such, the total number of flop counts for computing the GSC solution (3.18) (in terms of the number of complex multiplications) is approximately
(6.1) It is noted that the computational burden of the matrix inver-sion involved in (3.18) can be further alleviated by resorting to the partial adaptivity (PA) implementation [9] (the details are referred to [16], [17]); this instead calls for inverting an matrix, and can limit the flop cost to
(6.2)
The numbers of flop counts for TEQ [1] and PEQ [13], respec-tively, are obtained as
(6.3) and
(6.4) where and are, respectively, the TEQ and PEQ filter or-ders. From (6.1), (6.3), and (6.4), we have the following ob-servations: 1) TEQ method calls for the least algorithm com-plexity among the three; 2) the complexities of the GSC and PEQ methods are comparable for moderate numbers of sub-carriers ; 3) the GSC solution entails the highest computa-tional cost when is large. As we will see in the simulation section, the proposed GSC approach, although incurring more algorithm complexity, does yield significant performance im-provement over the other two comparative methods, even when perfect channel parameters are used for equalizer design.
VII. ALTERNATIVEFORMULATION
The proposed approach aims for joint mitigation of ISI and ICI through the GSC equalizer, and can facilitate low-com-plexity per-tone signal recovery after ISI-ICI suppression. This figure of merit benefits mainly from the decomposition (2.6), which splits the frequency-domain signal model into a sum of three components (namely, the ISI-ICI-free per-tone decoupled signals, along with ISI and ICI) as in (2.8); we note that a similar technique is also adopted in [6] and [31] for tackling interference in a similar high-rate scenario but for the SISO case. Without resorting to (2.6), the GSC based design can also be done instead via the original system (2.5), thus the resultant multichannel model
(7.1)
where is the signal signature. By following
the same procedures as in Sections III-B and IV, the alternative GSC filter under perfect channel knowledge can be obtained as (7.2)
where and ; the
associ-ated sampled robust solution is instead
in which is defined in (3.4) and is a diagonal matrix whose th element is given by
for
where denotes modulo- operation. Since (7.2) and (7.3) are obtained by treating as the interference to be mitigated, only ISI will be suppressed through the GSC fil-tering; also, as is no longer block diagonal, a space-frequency signal detector is then necessary for separating the NQ coupled streams. The performance of such an approach could benefit from larger space-frequency diversity gains in the signal separa-tion stage: if the VBLAST detector is used and inter-layer error propagation is negligible, the diversity gain per stage increases from 1 to NQ (the maximal achievable diversity gain for the per-tone based detection, however, is merely ). Our simulation re-sults indicate that it outperforms (but only slightly) the proposed joint ISI-ICI suppression scheme. The main drawback of this al-ternative solution is the intensive algorithm complexity. Indeed, with the MMSE-VBLAST detector [11], [36], the tone-by-tone based separation in the proposed approach requires
(7.4) flops. However, in the alternative solution the number of flop counts for signal detection is approximately
(7.5) Even if the fast recursive implementation [4] is employed, the overall flop cost can only be reduced to be of , which is significantly higher than that of the proposed scheme (especially when the number of subcarriers is large).
VIII. SIMULATIONRESULS
This section uses several numerical examples to illustrate the performance of the proposed method. We consider a
MIMO-OFDM system with transmit antennas,
re-ceive antennas, and subcarriers; the source symbols are drawn from the QPSK constellation. The background channel characteristics follow the standard wireless exponential decay model [20]: the channel impulse response is normalized such
that . The input SNR at the
th receive antenna is defined as . We consider the quasi-static environment, in which the channels are assumed to remain constant during per coherent interval of 300 OFDM symbol periods, and can vary independently between different inter-vals. In each data burst, the training pilots are placed in the entire heading OFDM symbol and are designed according to [2]. The outputs of both TEQ and GSC filters are fed into an MMSE-VBLAST detector [11], [36] for further separating the multi-antenna transmitted signals on each tone. All the simula-tions results are averaged over 800 trials.
Fig. 2. BER performances of the three methods (perfect channel knowledge).
Fig. 3. BER performances of the three methods at various channel orders (per-fect channel knowledge).
A. Comparison With Previous Works
We first compare the bit-error-rate (BER) performance of the proposed GSC based receiver with that of the TEQ [1] and PEQ [13] approaches. For a given channel order , the performances of TEQ and PEQ depend crucially on the equalizer order and the allowable decision delay. The TEQ is implemented using an -tap filter to shorten the composite channel order to the prescribed CP length (through simulation it is found that further increase in the filter order does not seem to improve per-formance); also, the resultant delay choice yielding the lowest BER is determined through exhaustive search and is then used in simulation. The order of PEQ, and the associated decision delay, are both set to be , as suggested in [13]. In the first simulation, we consider the case when the channel is perfectly known at the receiver [the GSC equalizer (3.18) is adopted]. For channel order and CP length , Fig. 2 shows the BER results of the three methods at different SNR levels. We can see that, among the three ISI-ICI mitigation schemes, the GSC equalizer (3.18) yields the best performance: it incurs no
Fig. 4. BER performances of the three methods at various CP lengths (perfect channel knowledge).
Fig. 5. BER performances of the three methods (LS channel estimate).
more than 1-dB penalty as compared with the ISI-free bench-mark result, and even the associated low-complexity PA imple-mentation can outperform TEQ and PEQ. The performance ad-vantage of the GSC approach would lie in the exploitation of the joint space-frequency degrees-of-freedom for interference sup-pression. With SNR dB, Fig. 3 compares the respective BER as the CP length fixed at and channel order in-creases from 4 to 21; on the other hand, Fig. 4 shows the BER results with fixed at 13 and CP length varying from 2 to 10. As we can see, the GSC filter in both cases yields the lowest BER. In the second simulation, we repeat the above three ex-periments but instead use the LS channel estimate for equalizer design; the results are shown in Figs. 5–7 (the power dedicated for training is ). Compared with Figs. 2–4, we can see that TEQ, PEQ, and the nonrobust GSC filter (3.18) all suffer performance degradation due to imperfect channel estimation. The proposed robust solution (4.26) is seen to improve the per-formance over the one (3.18); still, it maintains less than 1-dB SNR gap with respect to the ISI-free case, and can also relieve the BER penalty against long delay spread channels and short CP lengths. Finally, we demonstrate the BER performances of the three comparative methods under distinct subchannel orders:
Fig. 6. BER performances of the three methods at various channel orders (LS channel estimate).
Fig. 7. BER performances of the three methods at various CP lengths (LS channel estimate).
, and ,
where denotes the order of the subchannel between the
th transmit and th receive antennas, .
In implementing the three methods we thus set (for the subchannels with orders smaller than 13, zeros are padded in the respective impulse response tails). For , Figs. 8 and 9 show the BER performances at different SNR under perfect and imperfect channel estimates, respectively; as we can see, the BER tendencies are essentially the same with those in the common subchannel order case (cf. Figs. 2 and 5).
B. Performance of the Suboptimal Diagonal Loading Scheme (4.27)
This simulation illustrates the achievable performance of the suboptimal diagonal loading solution (4.27). For , and two distinct transmit powers during the training phase and , Fig. 10 compares solution (4.27) with the optimal weighting matrix (4.26) in terms of SINR. The results show that the respective performances are almost identical. Since the diagonal loading weight aims exclusively for signal leakage reduction, this simulated results would imply
Fig. 8. BER performances of the three methods with distinct subchannel orders (perfect channel knowledge).
Fig. 9. BER performances of the three methods with distinct subchannel orders (LS channel estimate).
Fig. 10. SINR performances of the optimal and suboptimal DL solutions.
that the leakage effect is the prime negative factor induced by channel mismatch.
Fig. 11. Output SINR at different transmit powers in the training phase.
Fig. 12. SINR gain at different transmit powers in the training phase.
C. Corroboration of the Analytic SINR Result
This simulation validates the predicted SINR results in Section V. We consider two different transmit powers in the
training phase and . For and ,
Fig. 11 shows the theoretical SINR, computed using the for-mulas in Table I, and the corresponding simulated outcomes. It can be seen that our analytic formula based on perturbation analysis well predicts the actual SINR tendency, even if the channel would be poorly estimated using a small transmit power. Fig. 12 shows the SINR gain, in terms of difference in dB, attained by the proposed robust solution (4.26) over the one (3.18); the theoretical solution is computed based on (5.11). As we can see, although the theoretical solution is derived based on the high SNR assumption, it appears very close to the experimental results over the medium-to-high SNR region ( dB). It is also observed that, for a fixed SNR, the SINR gain is larger for smaller (hence, a less accurate channel estimate). This confirms the effectiveness of proposed robust GSC filter against severe parameter uncertainty; such a tendency has been deduced based on the lower bound relation (5.13) [see the first discussion following (5.13)].
Fig. 13. Output SINR at different regularization factors (P = 2).
Fig. 14. Output SINR at different regularization factors (P = 14).
D. On Selection of Regularization Factor
This simulation investigates the performance of the regular-ization based design (4.30) at different factors; the channel order, CP length and noise variance are, respectively, set to be
and (this corresponds to
SNR dB). We consider two different transmit powers and in the training phase; the respective con-jectured optimal computed using (4.31) are 0.044268 and 0.006324. Figs. 13 and 14 show the respective SINR perfor-mances of the regularized solution
(8.1) at different values of . It can be seen that the SINR peaks attain
at for the case, and at when
: this tends to indicate that in (4.31) is optimal with respect to the regularization-based design under a (stochastic) white estimation error assumption (however, will no longer be
Fig. 15. BER performances of the proposed and the alternative schemes (per-fect channel knowledge).
Fig. 16. BER performances of the proposed and the alternative schemes (LS channel estimate).
optimal when different channel error models and design criteria are considered).
E. Performance of Alternative Formulation
This simulation compares the proposed joint ISI-ICI sup-pression scheme with the alternative GSC equalizer discussed in Section VII in terms of BER performance; for the latter an MMSE-VBLAST detector is used for separating the NQ coupled streams (the channel order and CP length are set to be and ). Figs. 15 and 16 show the respective BER results under perfect and imperfect channel knowledge assumptions (in Fig. 16, the training power for channel estima-tion is ). Although the alternative solutions (7.2) and (7.3) could benefit from a potentially larger space-frequency diversity gain in the signal detection stage, they are seen to only slightly outperform the proposed schemes (this would be due to inter-layer error propagation). We should note that the limited performance advantage attained by (7.2) and (7.3), however, comes at the price of increased computational cost (as discussed in Section VII).
IX. CONCLUSION
We propose a robust constrained-optimization based ISI-ICI mitigation scheme for supporting high-rate MIMO-OFDM transmission when the channels are not exactly known but are estimated via the LS training technique. The proposed constraint-free GSC design formulation yields a very natural cost function, and can facilitate the exploitation of the pre-sumed LS channel error property toward a solution through simple first-order perturbation analysis. The proposed robust GSC filter can jointly mitigate ISI-ICI and the net detrimental factors caused by channel estimation errors. Numerical study reveals that the signal leakage is the dominant impairment and a suboptimal diagonal loading solution can attain almost all the performance gain. Based on perturbation techniques, we further derive a closed-form mean approximate SINR expres-sion for the proposed robust scheme, and also an informative formula for quantifying the achievable SINR increment over the nonrobust solution. The analytic SINR gain reveals that prominent performance advantage can be attained by the robust solution under severe channel mismatch. Simulation results confirm the effectiveness of the proposed GSC-based equal-izer: it outperforms existing methods under either perfect or imperfect channel assumption (at the cost of complexity) and, in both cases, yields a performance very close to the ISI-ICI free benchmark.
APPENDIXA PROOF OFLEMMA4.2
Proof of (1): With and defined in (4.21), we have
(A.1)
Since the elements of the noise is circular Gaussian, the entries of the LS channel estimation error vector are i.i.d. circular Gaussian [see (2.9)]. By definitions of [see (4.11)] and [see (4.21)] and with the circularity condition of , the last three terms on the RHS of (A.1) is identically
zero. Also, since is zero-mean, so are and .
Equation (A.1) thus reduces to
(A.2)
where the last equality follows since .
Proof of (2): With given in (4.21), we have , where the equality follows by definition of in (4.21) and since
. Denote by the th
block submatrix of , for and
; then we have
where
(A.3)
For , let be the diagonal matrix obtained
by keeping only the diagonal entries of . With (A.3), it follows
(A.4)
Equations (2.9) and (2.10) together imply
, and (A.4) thus becomes
(A.5) where the last equality follows due to
. Based on (A.5), the th block
submatrix of is given by
(A.6) The result follows directly from (A.6) and since
.
Proof of (3): With given in (4.21) and since , direct manipulation shows
(A.7)
where the matrix is defined in (4.22). We note that the th block submatrix of equals
. Since the channel estimation errors between different receive antennas are independent [From (2.9) and
(2.10)], we have
, which implies equation (A.8), as shown at the bottom of the page. Equation (A.8) shows is a block diagonal matrix, and (4.24) follows by expanding the block diagonal terms in (A.8) and using
. APPENDIXB DERIVATION OF(5.5) To derive (5.5), we need the following lemma.5
Lemma B.1: Let be defined in (3.16) and be the exact solution of defined in (4.28). Also, let an SVD of the channel tone matrix be given in (4.12). Then the first-order approximation to is (B.1) with (B.2) and (B.3)
Derivation of (5.5): By definitions of [see (4.11)], it is easy to verify from (B.1) that ; (5.5) can be obtained via substituting into in (5.2) followed by some direct manipulations.
APPENDIXC PROOF OFTHEOREM5.1
We shall note that the matrix will be sparse for small [cf. (5.6) and (3.16)]. This implies that, as
and are small, both and
are close to zero in the high SNR region, and hence
(C.1)
5The detailed proof can be found at http://mimo.cm.nctu.edu.tw/Professor/
Papers/derivations.pdf.
With in (B.1) and since , we have
, and similarly . This implies
(C.2) and
(C.3)
The circularity condition of implies the last two terms on the RHS of both (C.2) and (C.3) are identically zero. From
(C.2), (C.3), and since [see (A.3) and
(A.6)], we have
(C.4) With (C.1) and (C.4), it follows
(C.5) According to the matrix inversion lemma [27], we have
(C.6) Using (C.6) and by definitions of and [see Lemma B.1 and (3.17)], it can be verified that
(C.7)
and hence
(C.8) Substituting (C.7) and (C.8) into (C.5), we have
(C.9) We observe that the second and the third terms on the RHS of (C.9) can be further combined into
(C.10) With (C.9) and (C.10), it follows that
REFERENCES
[1] N. Al-Dhahir, “FIR channel-shortening equalizer for MIMO ISI channels,” IEEE Trans. Commun., vol. 49, no. 2, pp. 213–218, Feb. 2001.
[2] I. Barhumi, G. Leus, and M. Moonen, “Optimal training design for MIMO-OFDM systems in mobile wireless channels,” IEEE Trans.
Signal Process., vol. 51, no. 6, pp. 1615–1624, Jun. 2003.
[3] B. R. Breed and J. Strauss, “A short proof of the equivalence of LCMV and GSC beamforming,” IEEE Signal Process. Lett., vol. 9, no. 6, pp. 168–169, Jun. 2002.
[4] J. Benesty, Y. Huang, and J. Chen, “A fast recursive algorithm for op-timum sequential signal detection in a BLAST system,” IEEE Trans.
Signal Process., vol. 51, no. 7, pp. 1722–1730, Jul. 2003.
[5] H. Cox, R. Zeskind, and M. Owen, “Robust adaptive beamforming,”
IEEE Trans. Acoust., Speech, Signal Process., vol. 35, no. 10, pp.
1365–1376, Oct. 1987.
[6] S. Chen and C. Zhu, “ICI and ISI analysis and mitigation for OFDM systems with insufficient cyclic prefix in time-varying channels,”
IEEE Trans. Consum. Electron., vol. 50, no. 2, pp. 78–83, Feb.
2004.
[7] O. L. Frost, “An algorithm for linearly constrained adaptive array pro-cessing,” Proc. IEEE, vol. 60, no. 8, pp. 926–935, Aug. 1972. [8] L. J. Griffiths and C. W. Jim, “An alternative approach to linearly
con-strained adaptive beamforming,” IEEE Trans. Antenna Propag., vol. AP-30, no. 1, pp. 27–34, Jan. 1982.
[9] S. Haykin and A. Steinhardt, Adaptive Radar Detection and
Estima-tion. New York: Wiley, 1992.
[10] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1995.
[11] H. Jafarkhani, Space-Time Coding: Theory and Practice. Cambridge, U.K.: Cambridge Univ. Press, 2005.
[12] A. Leke and J. M. Cioffi, “Impact of imperfect channel knowledge on the performance of multicarrier systems,” Proc. IEEE GLOBECOM, vol. 2, pp. 951–955, Nov. 1998.
[13] G. Leus and M. Moonen, “Per-tone equalization for MIMO OFDM systems,” IEEE Trans. Signal Process., vol. 51, no. 11, pp. 2965–2975, Nov. 2003.
[14] J. Li, P. Stoica, and Z. Wang, “On robust Capon beamforming and diagonal loading,” IEEE Trans. Signal Process., vol. 51, no. 7, pp. 1702–1715, Jul. 2003.
[15] F. Li, H. Liu, and R. J. Vaccaro, “Performance analysis for DOA estimation algorithms: Unification, simplification, and observations,”
IEEE Trans. Aerosp. Electron. Syst., vol. 29, no. 4, pp. 1170–1184,
Oct. 1993.
[16] C. Y. Lin, J. Y. Wu, and T. S. Lee, “GSC-based frequency-domain equalizer for CP-free OFDM systems,” in Proc. IEEE ICC, Seoul, Korea, May 2005, vol. 2, pp. 1132–1136.
[17] C. Y. Lin and T. S. Lee, “An efficient interference cancellation scheme for CP-free MIMO-OFDM systems,” in Proc. IEEE VTC-Fall 2005, Dallas, TX, Sep. 2005, vol. 1, pp. 439–443.
[18] R. G. Lorenz and S. P. Boyd, “Robust minimum variance beam-forming,” IEEE Trans. Signal Process., vol. 53, no. 5, pp. 1684–1696, May 2005.
[19] J. H. Manton, “Optimal training sequences and pilot tones for OFDM systems,” IEEE Commun. Lett., vol. 5, no. 4, pp. 151–153, Apr. 2001.
[20] B. O’Hara and A. Petrick, The IEEE 802.11 Handbook: A Designer’s
Companion. New York: IEEE Press, 1999.
[21] A. J. Paulraj, D. Gore, R. U. Nabar, and H. Boelcskei, “An overview of MIMO communications-A key to Gigabit wireless,” Proc. IEEE, vol. 92, no. 2, pp. 198–218, Feb. 2004.
[22] Y. Rong, S. A. Vorobyov, and A. B. Gershman, “Robust linear receivers for multi-access space-time block coded MIMO systems: A probabilis-tically constrained approach,” IEEE J. Sel. Areas Commun., vol. 24, no. 8, pp. 1560–1570, Aug. 2006.
[23] Y. Rong, S. A. Vorobyov, and A. B. Gershman, “A robust linear re-ceiver for multi-access space-time block-coded MIMO systems based on probability-constrained optimization,” Proc. IEEE Vehic. Technol.
Conf., vol. 1, pp. 118–122, May 2004.
[24] Y. Rong, S. Shahbazpanahi, and A. B. Gershman, “Robust linear re-ceivers for space-time block coded multi-access MIMO systems with imperfect channel state information,” IEEE Trans. Signal Process., vol. 53, no. 8, pt. 2, pp. 3081–3090, Aug. 2005.
[25] J. B. Schodorf and D. B. Williams, “A constrained optimization ap-proach to multiuser detection,” IEEE Trans. Signal Process., vol. 45, no. 1, pp. 258–262, Jan. 1997.
[26] S. Shahbazpanahi, A. B. Gershman, Z. Luo, and K. M. Wong, “Robust adaptive beamforming for general-rank signal models,” IEEE Trans.
Signal Process., vol. 51, no. 9, pp. 2257–2269, Nov. 2004.
[27] J. R. Schott, Matrix Analysis for Statistics, 2nd ed. New York: Wiley, 2005.
[28] G. W. Stewart, “Error and perturbation bounds for subspaces asso-ciated with certain eigenvalue problems,” SIAM Rev., no. 15–33, pp. 1413–1416, 1973.
[29] G. L. Stuber, J. R. Barry, S. W. Mclaughlin, Y. Li, M. A. Ingram, and T. G. Pratt, “Broadband MIMO-OFDM wireless communications,” Proc.
IEEE, vol. 92, no. 2, pp. 271–294, Feb. 2004.
[30] Z. Tian, K. L. Bell, and H. L. Van Trees, “Robust constrained linear receivers for CDMA wireless systems,” IEEE Trans. Signal Process., vol. 49, no. 7, pp. 1510–1522, Jul. 2001.
[31] S. Trautmann and N. J. Fliege, “A new equalizer for multitone systems without guard time,” IEEE Comm. Lett., vol. 6, no. 1, pp. 34–36, Jan. 2002.
[32] B. D. Van Veen and K. M. Buckley, “Beamforming: A versatile ap-proach to spatial filtering,” IEEE ASSP Mag., vol. 5, pp. 4–24, Apr. 1988.
[33] S. A. Vorobyov, A. Gershman, and Z. Luo, “Robust adaptive beam-forming using worst-case performance optimization: A solution to the signal mismatch problem,” IEEE Trans. Signal Process., vol. 51, no. 2, pp. 313–324, Feb. 2003.
[34] H. L. Van Trees, Optimum Array Processing. New York: Wiley , 2002.
[35] Z. Wang and G. B. Giannakis, “Wireless multicarrier communications: Where Fourier meets Shannon,” IEEE Signal Process. Mag., vol. 17, no. 3, pp. 29–48, May 2000.
[36] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over rich scattering wireless channels,” in Proc. IEEE ISSSE, Sep. 1998, pp. 295–300.
[37] Z. Xu, “Perturbation analysis for subspace decomposition with appli-cations in subspace-based algorithms,” IEEE Trans. Signal Process., vol. 50, no. 11, pp. 2820–2830, Nov. 2002.
[38] K. Zarifi, S. Shahbazpanahi, A. B. Gershman, and Z. Luo, “Robust blind multiuser detection based on the worse-case performance opti-mization o the MMSE receiver,” IEEE Trans. Signal Process., vol. 53, no. 1, pp. 295–305, Jan. 2005.
Chih-Yuan Lin was born in Yilan, Taiwan, R.O.C.,
in 1979. He received the B.S. degree from National Chung Hsing University, Taichung, Taiwan, in 2000. He is currently working toward the Ph.D. degree in the Department of Communication Engineering at National Chiao Tung University, Hsinchu, Taiwan.
His current research interests include space-time signal processing for wireless communications and statistical signal processing.
Jwo-Yuh Wu (M’04) received the B.S. degree in
1996, the M.S. degree in 1998, and the Ph.D. degree in 2002 from National Chiao Tung University, Hsinchu, Taiwan, R.O.C, all in electrical and control engineering.
He is currently a Postdoctoral Research Fellow in the Department of Communication Engineering, National Chiao Tung University. His research interests are in signal processing and information theory, with current emphasis on communications and networking.
Ta-Sung Lee (M’94–SM’05) was born in Taipei,
Taiwan, R.O.C., in 1960. He received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1983, the M.S. degree from the University of Wisconsin, Madison, in 1987, and the Ph.D. degree from Purdue University, West Lafayette, IN, in 1989, all in electrical engineering.
In 1990, he joined the Faculty of National Chiao Tung University (NCTU), Hsinchu, Taiwan, where he holds a position as Professor and Chairman of De-partment of Communication Engineering. His other positions include Technical Advisor at Information and Communications Re-search Laboratories (CCL), Industrial Technology ReRe-search Institute (ITRI), Taiwan, Managing Director of MINDS Research Center, College of Electrical and Computer Engineering, NCTU, and Managing Director of Communications and Computer Training Program, NCTU. He is active in research and develop-ment in advanced techniques for wireless communications, such as smart an-tenna and MIMO technologies, cross-layer design, and SDR prototyping of ad-vanced communication systems. He has co-led several National Research Pro-grams, such as the Program for Promoting Academic Excellence of Univer-sities—Phases I & II and the 4G Mobile Communications Research Program Sponsored by Taiwan Government.
Dr. Lee has won several awards for his research, engineering, and teaching contributions; these include two times the National Science Council (NSC) Su-perior Research Award, the 1999 Young Electrical Engineer Award of the Chi-nese Institute of Electrical Engineers, and the 2001 NCTU Teaching Award.
