MIMO-OFDM Beamforming for Improved Channel Estimation

Transcription

1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 1 MIMO-OFDM Beamforming for Improved Channel Estimation Cong Shen, Student Member, IEEE, and Michael P. Fitz, Senior Member, IEEE Abstract The MIMO-OFDM beamforming design problem is addressed from a system level standpoint. A beamforming method is proposed which helps improve the receiver channel estimation performance without degrading any benefit of a conventional beamformer. To that end, the Smoothed Singular Value Decomposition (SSVD) algorithm is first developed to get close effective channels after beamforming for two adjacent subcarriers. Based on the SSVD algorithm, the Frequency Smoothed Beamformer (FSB) design is then derived, in which smooth effective channels across all subcarriers are generated and thus the receiver can apply interpolation and smoothing to improve the channel estimation performance. The close singular value problem is discussed. Statistical characteristics of the effective channel are analyzed, which is used to design channel estimation for the beamformed channel. Simulation results show that the FSB design is efficient in the IEEE n setting. Index Terms MIMO, OFDM, Transmit beamforming, Singular Value Decomposition (SVD), channel estimation. I. INTRODUCTION Pursuing high spectral efficiency and high reliability is a main theme of today s wireless communication system design. Faced with this challenge, several new techniques have been developed during the last several years. For example, multiple-input multiple-output (MIMO) and orthogonal frequency division multiplexing (OFDM) systems have been extensively studied. These techniques are shown to provide high throughput together with good error performance. Also in order to achieve high spectral efficiency, high-order Quadrature Amplitude Modulation (QAM) such as 64QAM and even 256QAM are frequently adopted. Thanks to the rapid development of both signal processing algorithms and hardware implementation, most of these advanced techniques have been successfully deployed (or proposed) in practical wireless communication systems such as IEEE wireless LAN standards, e WiMax, 3GPP mobile telephony, and DVB. The spectral efficiency and error performance of a wireless system can be further improved if accurate (to some level) channel state information (CSI) is available at the transmitter. The benefit is even more pronounced in a multiple-antenna system, and has triggered enormous research activities in this This work was supported by Conexant and STMicroelectronics. Part of the material in this paper was presented at IEEE Globecom 2006, San Francisco, USA, Nov Cong Shen with the Department of Electrical Engineering, University of California Los Angeles (UCLA), Los Angeles, CA 90095, USA ( congshen@ee.ucla.edu). Michael P. Fitz is with Northrup Grummond Space Technology, Redondo Beach, CA 90278, USA, and also with the University of California, Los Angeles, CA 90095, USA ( Michael.Fitz@ngc.com). area. The idea can be traced back to the pioneering work [1], in which the Singular Value Decomposition (SVD) based beamformer together with the water-filling power allocation is proved to achieve the capacity of a multiple-antenna channel with transmitter CSI (CSIT). Since that point, there has been significant work studying how to design the precoder and decoder for MIMO channels from different perspectives, using different design criteria (e.g., maximizing capacity [2], [3], minimizing error probability [4], [5], maximizing received signal-to-noise ratio (SNR) [3]), different channel models (e.g., spatially independent or spatially correlated [6], frequency-flat or frequency-selective [5], [7]), and different CSIT assumptions (e.g., perfect or imperfect/limited feedback [6], [8] [10]). It should be mentioned that these precoder/decoder designs also have been adopted in industry standards. For example, the transmit precoding has been integrated into the IEEE e standard, and beamforming adapted to the MIMO channel at each subcarrier has been proposed in the ongoing IEEE n standardization [11]. Another system module that significantly affects the overall performance is the channel estimation. This is particularly important for a MIMO-OFDM system [12], [13], where multiple frequency selective channels make channel estimation more difficult. An accurate and unbiased channel estimator is also indispensable for decoding high-order constellations that are sensitive to channel estimation errors. Consequently not considering channel estimation in a spectrally efficient MIMO transmission carefully can lead to performance degradations that mitigate the throughput advantages. The major problem studied in this paper is how the transmit beamforming affects the receiver channel estimation. This paper shows that incorporating beamforming into the system, along with the advantages it brings such as increased capacity and decreased receiver complexity, significantly affects how channel estimation can be accomplished. Generally a broadband wireless channel in the frequency domain is very smooth due to the fact that channel responses at adjacent frequency tones are highly correlated. This correlation facilitates spectrally efficient channel estimation techniques. One can estimate the MIMO channel on one subcarrier by interpolating and smoothing over adjacent subcarriers, e.g., using the Wiener Filter channel estimation [14]. When beamforming is incorporated, the effective channel for estimation at the receiver is a composition of the beamformer and the physical MIMO channel in the frequency domain. A similar question then arises: can one still get smooth effective channels after beamforming? Being able to answer this question affirmatively is crucial to the overall system performance, as accurate

2 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 2 OFDM channel estimation relies on the smoothness. This problem was motivated from the IEEE n standardization [11]. It is worth noting that up until this point the general consensus of the n community is that beamforming is incompatible with smooth channels. The major contribution of this work is an efficient beamformer design that results in smooth effective channels. The socalled Frequency Smoothed Beamformer (FSB) design manages to control the SVD operation of the channel matrix on each subcarrier to generate smooth effective channels across all subcarriers. The key ingredient of the FSB design is the Smoothed Singular Value Decomposition (SSVD) algorithm. This algorithm ensures that every two channel matrices on adjacent subcarriers can generate two effective channel matrices that are close in terms of the Euclidean distance. Analysis of the statistical properties of the effective channel is presented to help derive an efficient channel estimation method. Simulation results using an n compatible MIMO-OFDM system validate the design. About 2 db gain can be obtained using the FSB design in a preamble setting, compared to the conventional method in which the smoothness of the effective channel is not preserved. The rest of the paper is organized as follows. Section II defines the system model and formulates the problem. Section III presents the major contribution of the paper: the derivation of the SSVD algorithm and the FSB design. Analysis of the statistical properties of the effective channel and derivation of the channel estimation method are addressed in Section IV. Section V reports the simulation results. Finally, Section VI concludes the paper and proposes possible future research directions. The following notations will be used throughout the paper. Matrices are denoted by bold capital letters, and vectors are denoted by letters with overhead arrows. A ij denotes the (i, j)- th element of a matrix A. A H denotes the Hermitian of a complex-valued matrix A, and A denotes the conjugate of a complex scalar A. E[ ] is used to denote expectation, P { } for probability, and min{ }/ max{ } denotes the minimum/maximum of a set. II. SYSTEM MODEL AND PROBLEM FORMULATION A. Signal Model Consider a MIMO-OFDM system in the frequency domain with L t transmit antennas, L r receive antennas and D subcarriers. D p payload subcarriers among all the D subcarriers are used for pilot and/or data transmission. The length of an OFDM packet is denoted as N f symbols. The discretefrequency signal model at the k-th subcarrier (k = 1,, D p ) and l-th time interval (l = 1,, N f ) can be written as Y (k, l) = H(k)V(k) X(k, l) + N(k, l) = H(k) X(k, l) + N(k, l) (1) where H(k) represents the frequency-domain MIMO channel, H(k) = H(k)V(k) denotes the effective MIMO channel after beamforming, and V(k) is the beamformer, all at the k-th subcarrier. This paper focuses on a quasi-static frequency-selective channel where H(k) is constant within one OFDM packet and changes independently to another value in a different packet. The elements in H(k) are assumed to be spatially independent and identically distributed (i.i.d.) complex circular symmetric Gaussian variables with zero mean and variance σ 2 H, though this assumption is not critical to the algorithms but is done for simplicity. X(k, l) is the L t 1 signal vector transmitted at the l-th time interval and k-th subcarrier. The total transmit power is P regardless of the number of transmit antennas. N(k, l) is the L r 1 additive white Gaussian noise (AWGN) vector. The noise samples are independent circularly symmetric zeromean complex Gaussian random variables with variance N 0 /2 per dimension. A system diagram is shown in Fig. 1. In the following discussion the focus is on L t = 2 and L r = 2, which is a basic configuration in most MIMO-based standards. Section III-C will discuss the extension to an arbitrary number of antennas. B. Transmit Beamforming In this paper, perfect CSIT of each subcarrier is assumed such that optimal beamforming can be employed at each tone. Mathematically, the channel matrix H C Lr Lt at any subcarrier can be written in SVD form [15] H = UΛV H (2) where U C Lr Lr, V C Lt Lt are unitary, and Λ = diag (λ i,, λ Lt ) R Lr Lt is a diagonal matrix with the non-negative singular values (SV) of H. Telatar [1] showed that if the transmit beamformer is V, by multiplying U H at the receiver parallel Gaussian channels are generated. The MIMO channel capacity with full CSIT is then achieved with the water-filling power allocation across these parallel subchannels. From the signal processing perspective, the parallel subchannels created by transmit beamforming also greatly simplify the receiver decoding complexity. The reason is that inter-symbol interference (ISI) generated by transmitting several symbols simultaneously is completely eliminated by the orthogonality among subchannels. Thus maximum-likelihood (ML) decoding can be performed at each subchannel separately without degrading the overall performance. In fact, it is easily shown that with transmit beamforming, linear detectors such as Zero Forcing (ZF) and Minimum Mean-Square Error (MMSE) [16] with per-layer slicing are equivalent to ML. Another important property of transmit beamforming is that it generally results in a well-conditioned MIMO channel even if channel estimation errors exist at the receiver. This especially improves the performance of linear detectors using the estimated channel. If CSI at the receiver (CSIR) is imperfect and the channel estimation error is not significant (e.g., in the medium to high SNR regime), HH H is not diagonal, but almost diagonal in the sense that the off-diagonal elements are very small compared to the diagonal elements. Linear equalization theory tells us that a deep spectral null generated by an ill-conditioned MIMO channel matrix can lead to significant performance degradation in ZF and MMSE equalizers [17]. However, for a well-conditioned channel matrix such as the almost diagonal one generated by beamforming, the deep spectral null is typically not present, and thus performance is improved.

3 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 3 Effective Channel H BICM Beamformer V MIMO OFDM Channel H Y Soft output Detection Bit Deinterleaver Soft input Viterbi decoder H ˆ H Y p Channel Estimation Fig. 1. A MIMO-OFDM BICM system with transmit beamforming. C. The Effect of Beamforming on Receiver Channel Estimation An isolated view of the beamformer design would suggest that any valid SVD suffices in providing the aforementioned beamforming properties. However, from a system-level point of view, the beamformer also should be designed with a careful consideration of its impact on other system modules, such as the channel estimation, to achieve good overall performance. For a MIMO-OFDM system, interpolation-based channel estimation [18] such as Wiener Filtering is shown to provide excellent performance [14], [19]. This approach relies on the frequency correlation of the physical channel, or roughly speaking, each subchannel H ij (k) is smooth across subcarriers. Mathematically, each subchannel H ij (k), k = 1,, D p has a correlation function R Hij (m n) = E [H ij (m)h ij (n) ] (3) which has a low-pass characteristic, i.e., H ij (n) is highly correlated with H ij (m) when m n is small, but poorly correlated with increased frequency separation. In order to maintain good channel estimation performance in a beamformed system, the transmitter design should ensure that the effective channel H(k) = H(k)V(k) is smooth across subcarriers such that interpolation-based channel estimation can still be applied. On the other hand, failure to maintain smoothness requires the channel estimation to be performed on a single-subcarrier basis, i.e., estimating the k-th effective channel H(k) using the pilots and received signals only from the k-th subcarrier. This will lead to a dramatic performance loss, which will be demonstrated in Section V. Actually there were some debates in the IEEE n standardization process on this problem. We hope this work can at least partially answer the question. Maintaining the frequency smoothness also has another implementation advantage. In most of today s wireless standards, the closed-loop techniques such as beamforming are only optional instead of mandatory. In order to have a low-cost low-complexity implementation for both open and closed loop, it is desirable to have the same system module reusable in both situations, or modify as few modules from the openloop system as possible. If smoothness in the effective channel H(k) is preserved, the same open-loop interpolation-based channel estimation technique can be used in the closed-loop situation as well. Our work is able to provide this engineering advantage. A pertinent question is that since the channel is already known at the transmitter, why is there still a need for the receiver to estimate the effective channel since typically CSIT is obtained by feedback from the receiver. There are two major issues involved in this problem. 1) In the current IEEE n standardization, implicit feedback is proposed in the time-division duplex (TDD) mode. In this mode, the receiver sends its pilots back to the transmitter such that the transmitter can estimate the reverse link channel. Because the reverse and forward link channels are reciprocal, the transmitter implicitly obtains an estimate of the forward link channel, while the receiver still has to estimate the channel before decoding. To facilitate accurate channel estimation at the receiver, the transmitter must control the beamforming to generate smooth channels. 2) Even if the system is operating in the frequency-division duplex (FDD) mode and CSIT is obtained from receiver feedback, several practical system imperfections during the transmission could motivate the receiver to track or re-estimate the channel. The effective channel seen by the receiver may be time-varying due to the errors introduced in the transmission such as RF phase noise and frequency offset. Thus, the receiver needs to re-estimate the channel for better performance when it receives a beamformed transmission. III. THE SMOOTHED SVD ALGORITHM AND THE FSB DESIGN A. General Idea The ultimate goal of the smoothed beamforming design is to create an effective channel H(k) with low-pass autocorrelation functions without sacrificing the advantages of traditional beamforming. In this work minimizing the Euclidean distance between adjacent channel matrices is chosen as the objective. Since in all physically realizable channels the frequency domain channel response will be a low-pass stochastic process, adjacent channel gains will be generally close in Euclidean distance. Algorithms based on minimizing the Euclidean distance are shown to generate well shaped auto-correlation functions, and prove effective in the smoothed beamforming problem. Meanwhile, low complexity is another objective when developing the beamforming algorithm since beamforming is a real-time operation.

4 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 4 Since the SVD of a complex-valued matrix is not unique, selecting the best from all possible singular vectors in each subcarrier will be crucial to the overall smoothness. First of all, the following lemma characterizes the constraints on the non-uniqueness of SVD. Lemma 1: If H = UΛV H = U ΛV H are both valid SVDs of H, then there exist unitary matrices W U, W V such that 1) U = UW U, V = VW V ; 2) W U ΛW V H = Λ. Proof: Since UΛV H = U ΛV H are both valid SVDs, U, V, U, V are all unitary. Thus we can always let W U = U H U and W V = V H V. Both W U and W V are unitary because they are multiplications of unitary matrices. Finally, W U ΛW V H = U H U ΛV H V H = U H HV H = Λ (4) This lemma suggests that for the same matrix, any two valid SVDs can be related by unitary rotations. This observation is further generalized to two matrices H(1) and H(2) that are very close in terms of the Euclidean distance. Assume H(1) has a SVD H(1) = U(1)Λ(1)V(1) H. Then it is desirable to have a SVD for H(2): H(2) = U(2)Λ(2)V(2) H such that the singular vectors are close to those of H(1). The observation from Lemma 1 suggests to take carefully-controlled unitary rotations of U(1) and V(1) to formulate U(2) and V(2), respectively. That is, U(2) = U(1)W U (5) V(2) = V(1)W V (6) This idea is pursued in details in the following. B. The Smoothed SVD Algorithm for L t = L r = 2 The problem can be formulated as following: Problem 1: Given matrices H(1) and H(2) where H ij (2) H ij (1) ɛ ij, i, j = 1, 2 and ɛ ij are small. Can the SVDs of the two matrices H(1), H(2) be designed such that H(1) = U(1)Λ(1)V(1) H H(2) = U(2)Λ(2)V(2) H 1) H ij (2) H ij (1) are made as small as possible, where H(k) = H(k)V(k), k = 1, 2; and 2) the SVDs have reasonable complexity for implementation? The Smoothed SVD (SSVD) algorithm is a solution to this problem. Consider two adjacent channel matrices H(1) and H(2) in the frequency domain. Assume H(1) has any valid SVD: H(1) = U(1)Λ(1)V(1) H. Rewrite the SVD of H(1) as ( ) λ1 (1) 0 = U(1) H H(1)V(1) (7) 0 λ 2 (1) where λ 1 (1), λ 2 (1) 0. Note that U(1) and V(1) are in general not the singular vectors of H(2). Thus ( ) A = U(1) H A11 A H(2)V(1) = 12 (8) A 21 A 22 has A 12 0 and A Recall from Section III-A that the idea is to obtain U(2) and V(2) via unitary rotations of U(1) and V(1), respectively. However, since real-time beamforming is desired, directly searching over all unitary matrices to find out the best W U and W V is computationally infeasible. Thus some constraints are posed on the unitary rotations to reduce the number of unknowns and simplify the computation. It should be noted that putting extra constraints certainly does not favor the performance, but a careful selection of constraints shows very little degradation in the MIMO-OFDM beamforming application. The first constraint is that the unitary rotation belongs to the special unitary group SU(2) [20]. Any matrix B from SU(2) is a 2 2 unitary matrix with determinant 1. It has a general representation ( ) α β B = β α (9) where α 2 + β 2 = 1. The second constraint is α being real. Notice that both constraints are aimed at reducing the number of unknowns in the unitary rotation, which is important in the real-time computation. With these constraints, U(2) and V(2) can be written in the forms of and U(2) = U(1)W ( U ) 1 Q = U(1) (1 + Q Q) 1 2 (10) Q 1 V(2) = V(1)W ( V ) 1 P = V(1) (1 + P P ) 1 2 (11) P 1 where P and Q are complex unknown scalars. Now denote ( ) A = U(2) H A 11 A 12 H(2)V(2) = (12) A 21 A 22 Then in order to get a valid SVD, P and Q should be chosen such that A 12 = 0, A 21 = 0. This leads to the following equations: { A22P + QA 11 = QA 12P A 21 P A 11 + A 22Q = P A 21Q + A (13) 12 In case of a 2 2 system, these equations can be easily solved and closed-form solutions are given in Equations (14) and (15). Obtaining (P, Q) does not immediately give a valid SVD of H(2), because U(2) H H(2)V(2) is only diagonal, not necessarily non-negative real. However this is easy to handle: one can separate the amplitude of the diagonal elements and combine the remaining phase items into V(2). This will not affect the unitarity of V(2), but will formulate a valid SVD of H(2): H(2) = U(2)Λ(2)V(2) H where U(2) and V(2) are

5 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 5 { 1 ( A11 Q = 2 2 (A 12 A 22 + A 11A 21 ) + A 12 2 A 21 2 A 2) 22 ± } ( A A 12 2 A 21 2 A 22 2 ) (A 12 A 22 + A 11A 21 ) (A 21A 11 + A 22A 12 ), P = A 21 A 11 Q A 22 + A 12 Q. (14) (15) unitary, and Λ(2) is diagonal with the non-negative real SVs of H(2). Another issue is that since the equations are quadratic, there are two sets of solutions (P 1, Q 1 ) and (P 2, Q 2 ). One needs to determine which set to choose. Notice that the ultimate goal is to make the effective channel H(2) close to H(1). Thus one method would be to calculate U(2) and V(2) from both sets of solutions, formulate the effective channels H(2)V(2), and then choose the one in which the effective channel is closer to the previous one by minimizing the Frobenius norm H(2)V(2) H(1)V(1) F. However, choosing (P, Q) to minimize H(2)V(2) H(1)V(1) F requires matrix multiplications and computation of the Frobenius norms, which leads to high complexity. A low-complexity approach that works very well in practice is to choose (P, Q) such that P (or Q ) is smaller in order to make V(2) close to V(1) (or U(2) close to U(1)). Simulations suggest that this simple approach has a performance that is almost the same as the minimizing Frobenius norm approach. The Smoothed SVD algorithm is compactly described in Algorithm 1. A majority of SSVD s complexity comes from calculating (P, Q) in (14) and (15), and several other matrix multiplications. In a 2 2 system the complexity of SSVD is comparable with the conventional SVD algorithms [21]. Algorithm 1 The 2 2 Smoothed SVD Algorithm Input U(1), V(1), H(2) A = U(1) H H(2)V(1) Calculate (P 1, Q 1 ), (P 2, Q 2 ) in (14) and (15) Q = arg min Qi Q i 2, find corresponding P Calculate U(2), V(2) in (10) and (11), respectively B = U(2) ( H H(2)V(2) ) B11 0 Λ(2) = V(2) = V(2) 0 B 22 ( B 11 B 11 0 B 0 22 B 22 Output U(2), Λ(2), V(2) It should be noted that the SSVD algorithm is not claimed as the optimal solution to Problem 1. This algorithm is offered as a practical solution with very good performance and reasonable complexity. In fact, even if the complexity condition of Problem 1 is not considered, the optimum performance is still not clear, i.e., what is the upper bound of H ij (2) H ij (1) given H ij (2) H ij (1) ɛ ij. There are numerical examples in matrix perturbation theory where the smoothness is difficult to achieve. This will be discussed in Section III-E. ) C. SSVD for Arbitrary L t and L r Extension of this method to more than 2 antennas is accomplished by utilizing the representation theory of SU(n), which is a compact, simply-connected Lie group of dimension n 2 1 [22]. For example, for L t = L r = 3 the parametrization of SU(3) [23] can be utilized to represent any B SU(3) as B = [ Φ ] a 0 1 a a 2 0 a [ ] (16) 0 21 Ψ where 0 mn denotes the m n all-zero matrix, Φ, Ψ SU(2), and a is a complex scalar with a < 1. It is easily verified that such B has 4 complex unknowns. Another possible approach uses the same argument as in Section III-B, but with W U and W V having the general form ( ) ( ) I Q H (I + Q W U = H Q) Q I 0 (I + QQ H ) 1 2 (17) and ( ) ( ) I P H (I + P W V = H P) P I 0 (I + PP H ) 1 2 (18) where I is the identity matrix, 0 is the zero matrix, and P,Q are also sub-matrices of compatible dimensions. Extra constraints that some unknowns are real can be posed in a similar fashion to reduce the number of unknowns. There is a tradeoff between the number of unknowns and the corresponding SSVD/FSB performance. More unknowns enhances the SSVD/FSB performance, but requires higher complexity to solve the equations. Such a complexity-performance tradeoff should be considered when determining how many unknowns to use in the SSVD/FSB design. Nevertheless, the methods still lead to unacceptable complexity as the number of antennas becomes very large. Fortunately this does not affect the practical value of the algorithm. In a typical wireless system with many antennas, the system designer will not use all the available degrees of freedom (spatial channels), since generally there are only a small number of SVs that are large enough for data transmission. In typical wireless systems only the largest 2 4 spatial channels are used. Within this range, the proposed algorithm is still computationally feasible.

6 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 6 D. Frequency smoothed beamformer (FSB) design The SSVD algorithm only works for two adjacent subcarriers. The beamformers across all D p subcarriers can be easily designed by using the SSVD algorithm serially through subcarriers. At the first subcarrier any valid SVD can be performed. All the following subcarriers are input into the SSVD algorithm in a pair-wise and serial fashion to get the corresponding beamformers. This design is described in Algorithm 2, where svd ( ) is any valid SVD function, and ssvd ( ) is the SSVD algorithm described in Algorithm 1. The complexity of Algorithm 2 is proportional to that of SSVD, and thus is reasonable in a 2 2 MIMO-OFDM system. Algorithm 2 The FSB Design Input H(k), k = 1,, D p [U(1), Λ(1), V(1)] = svd (H(1)) for i = 2 to D p do [U(i), Λ(i), V(i)] = ssvd (U(i 1), V(i 1), H(i)) end for Output V(k), k = 1,, D p The key observation in developing the FSB design is that high correlation between adjacent subcarriers typically leads to small Euclidean distances between adjacent channel matrices. This observation helps shift our attention from the frequency correlation function to minimizing the Euclidean distances, which is a much more tractable target. This simplification has resulted in the proposed SSVD and FSB design. However, since the ultimate target is still the low-passed shaped frequency correlation for H ij, the effectiveness of FSB needs to be verified by calculating R Hij. This is addressed in Section IV. E. The Close Singular Value Problem The SSVD and FSB algorithms discussed in previous sections work very well when the SVs of H(k) are not close to each other. Some disturbing and unwanted phenomena might happen when SVs are close. This section discusses these phenomena, and their impact on the SSVD/FSB design. For the sake of simplicity, L t = L r = 2 is used as an exemplary configuration. The argument can be directly extended to general (L t, L r ) with high-dimensional geometry. Two major problems are present when SVs come close to each other. 1) Switch of SV ordering: Let us revisit the SVD of H from the singular space perspective: H = UΛV H = 2 λ i u i v i H (19) i=1 where u i / v i are left/right singular vectors of λ i, and ( u 1, u 2 )/( v 1, v 2 ) expands the left/right singular space of H. One can view ( u 1, u 2 ) as the two orthonormal bases of the left singular space, and H has length (λ 1, λ 2 ) on each direction, respectively. The similar geometric illustration can also be extended to ( v 1, v 2 ), but the discussion is restricted to the left singular space to make the argument simple. Consider the smoothness problem of SVDs on adjacent subcarriers indexed by (k, k + 1). A natural question is which pair of SVs λ i (k) λ j (k + 1) and singular vectors u i (k) u j (k + 1) should be matched together to make the SVD smooth. Typically, if the two SVs are not close to each other, this problem can be solved by always matching the larger/smaller SVs and the corresponding singular vectors together. The intuition is that the larger SV cannot suddenly become the smaller one in adjacent subcarriers due to the inherent frequency correlation. However, this argument is no longer valid when the two SVs are close to each other. In this case, the larger/smaller SVs can switch from one subcarrier to the next. If the same SV ordering is used to determine which pair to be made smooth, there will be a problem when such a switch happens. Fig. 2(a) shows one such example, which is a snapshot taken from the IEEE n channel model D [24]. Obviously if one insists on the SV ordering, the smoothness will be lost at the switch. Fortunately, this problem will be automatically taken care of by the SSVD/FSB design. The reason is that there is essentially no SV ordering required in the SSVD algorithm 1. The only objective in SSVD is to make the adjacent U or V smooth, regardless of whether the ordering of SVs switches or not. Fig. 2(b) shows the SVs created by the FSB design across subcarriers, using the same channel realizations as in Fig. 2(a). Clearly this shows the advantage of the SSVD/FSB design in dealing with such problem. 2) There is no way to maintain the smoothness in some extreme situations: When the SVs come close to each other, matrix perturbation theory points out that the smoothness cannot always be guaranteed. For example, the following is a famous construction in matrix perturbation theory [25]: ( ) 1 0 H(1) = ɛ ( ) 1 ɛ H(2) = ɛ 1 V(1) = ( ) V(2) = 1 2 ( (20) ). (21) Since ɛ can be made arbitrarily small, this gives an example for which arbitrarily small perturbation in H can completely change the direction of the singular space V. A geometric method is developed in the following to understand and analyze this phenomenon. The discussion is focused on the left singular space and SVs. Based on the illustration in Section III-E.1, H = UΛ can be represented by an ellipse with 1) major axis: direction u 1, length λ 1 ; and 2) minor axis: direction u 2, length λ 2. An illustration of the matrix perturbation (20)(21) is given in Fig. 3(a) with this geometric representation. The target is to find out why the directions of major/minor axes can change dramatically when H(1) smoothly changes to H(2). To facilitate the analysis, we artificially add an intermediate state, H(1.5), for which min{λi (k), i, k = 1, 2} λ 1 (1.5) = λ 2 (1.5) max{λ i (k), i, k = 1, 2}. Note that the geometric representation of H(1.5) is a circle instead of an ellipse. 1 Notice that the input of Algorithm 1 does not include the singular values Λ(1).

7 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 7 10 Larger SV Smaller SV 10 Spatial channel 1 by FSB Spatial channel 2 by FSB Singular Value [db] 4 2 Singular Value [db] Smoothness is lost 0 Smoothness is preserved subcarrier index (a) SV ordering subcarrier index (b) SSVD/FSB Fig. 2. An example showing how SSVD/FSB can deal with the singular value switch problem. This is an instance taken from realizations of the IEEE TGn n 2 2 channel model D. Although the intermediate state H(1.5) is artificial, there is no problem since it is a smooth intermediate state in between H(1) and H(2), due to the fact that the SVs of H(1) or H(2) are close to each other and the ellipses of H(1) and H(2) are already close to a circle. With the help of this intermediate state, the problem with (20)(21) is obvious: when going into the circular intermediate state, the distinction between the major and minor axes disappears; when leaving the circular intermediate state, the major and minor axes can be any orthogonal directions. In other words, the circular intermediate state totally eliminates directions of the singular vectors. In fact, the circular intermediate state is a degraded ellipse where the dimension reduces from 2 to 1. This is the fundamental reason why smoothness cannot be guaranteed in a close SV situation. Fortunately, this close SV problem is not the bottleneck of the overall performance in practice. The reason is that in a typical wireless LAN environment, the probability that two SVs come close to each other is very small. Meanwhile, even if the close SV event happens, it does not necessarily lead to completely changed directions. For example, simulations based on the IEEE n TGn channel models suggest that the probability of close SV is almost on the same order of magnitude as the deep fading probability. Since the deep fading event is the dominant factor of decoding errors in the high SNR regime, this implies that the close SV events do not happen more frequently in comparison to other error-causing physical events, especially in the SNR range of practical interests. The simulation results reported in Section V will also prove that the FSB design is effective in practical MIMO- OFDM systems. IV. STATISTICAL PROPERTIES OF H AND CHANNEL ESTIMATION DESIGN Assuming that the frequency correlation of H(k) is known, a natural question to ask is how to analyze the secondorder statistics of the effective channels H(k). There are two motivations for considering this problem. First of all, the proposed FSB design is based on minimizing the Euclidean distance between adjacent SVDs. Recall that the ultimate goal is to generate a highly correlated beamformed channel H(k) for a given physical channel H(k) with a low-pass frequency correlation. Thus it is necessary to compute the frequency correlation functions for H(k) to verify whether the FSB design is effective in achieving this ultimate goal. The second purpose is to perform channel estimation. In order to utilize the smoothness in channel estimation, the receiver has to obtain the correlation functions for the beamformed channel H(k). The calculation in this section will be used in Section V. Strictly speaking, H(k) loses almost all the statistical properties of H(k). To name a few: 1) Hij (k) is no longer Gaussian. 2) H(k) is spatially correlated. 3) Even if R H (n) is known, it is difficult to derive the analytical auto-correlation functions for H ij (k). The main reason behind these facts is that the beamformer V(k) at the k-th subcarrier is not only determined by the current channel matrix H(k), but also related to all the previous channel matrices H(i), i = 1,, k 1 through the serially adopted SSVD algorithm. A few assumptions are made to help discuss the secondorder statistics of H(k) and obtain the empirical frequency correlation functions for each random process H ij (k). Simulation results shown in Section V back up the validity of these assumptions. The first assumption is to ignore the crosssubchannel correlations. Each subchannel is further assumed to be a stationary Gaussian process. However due to the effect of beamforming, different subchannels may have different autocorrelation functions. Rewrite H = UΛ ( ) λ1 u = 11 λ 2 u 12. (22) λ 1 u 21 λ 2 u 22 Then all the u ij and λ i involved are random variables. If

8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 8 u 2 (2) u 2 (1) u 1 (2) H(1) H(2) u 2 (2) u 2 (1) u 1 (2) H(1) H(1.5) H(2) u 1 (1) u 1 (1) (a) H(1) H(2) (b) H(1) H(1.5) H(2) Fig. 3. Geometric illustration of why close singular values can lead to dramatic change of directions of the singular vectors. H(1.5) is artificially added as an intermediate state to understand the problem. one further assumes that u ij is independent of λ j, the autocorrelation function of H ij can be written as R Hij (n) = E [λ j (m + n)u ij (m + n)λ j (m) u ij (m) ] = E [λ j (m + n)λ j (m) ] E [u ij (m + n)u ij (m) ] = R λj (n) R uij (n). (23) In order to fully characterize the second-order statistics of H ij, one needs the auto-correlation functions R λj (n) and R uij (n). Unfortunately, to the authors best knowledge, these two sets of auto-correlation functions have not been solved in the field of stochastic matrix processes. Nevertheless, analytical solution is not the only way to solve this problem in the engineering practice. The proposed solution is to obtain the frequency auto-correlation functions for H ij (k) by Monte-Carlo simulations. For a given MIMO- OFDM channel model with a specific frequency correlation function R H (n), the procedure is as following: many channel realizations of {H(k), k = 1,, D p } are first generated according to the distribution, and then H(k) is obtained by applying the FSB design, and then the empirical autocorrelation functions for each subchannel are calculated, and finally all the realizations are averaged over to get the empirical frequency correlation functions for each H ij (k). The correlation functions R Hij (n), i = 1, 2; j = 1, 2 are saved for future channel estimation use. In this paper, the ideal assumption that the frequency correlation of H is perfectly known a priori is made, and thus the proposed method also gives the accurate frequency correlation of H. This assumption is not true in practice, and there are existing solutions to deal with the problem. For example, [26] suggests a worst-case design method where the correlation is estimated from the largest possible delay spread. It is shown that such method is robust to the model mismatch. As another example, [27] proposes an adaptive Wiener filter channel estimator that adaptively chooses the best Wiener coefficients from a set of pre-computed coefficients based on the received signal. The idea is to parameterize channels based on the delay spread and use channel observation to select a channel model in an off-line algorithm. It has been shown [27] that such method works well in a wireless LAN system when many consecutive transmissions have roughly the same channel geometry. All these solutions, although original proposed for the physical channel H, can be directly applied to the FSB effective channel H. The reason is that the effective channel has similar low-pass correlation functions thanks to the FSB design. Meanwhile, wireless LAN typically has a stationary indoor environment, where the mobility is extremely limited and the surroundings could stay unchanged for a long time. Since the frequency domain characteristics is largely dependent on the propagation geometrical characteristics, the problem of estimating the frequency correlation of H can be addressed from the existing solutions. Fig. 4 gives an example of the empirical correlation functions for the two spatial channels 2 from H. The analytical correlation function for H is also plotted for comparison. The IEEE n channel model D [24] with D = 64 total subcarriers and D p = 52 payload subcarriers is used. An interesting observation from Fig. 4 is that although the physical channel H(k) has independent subchannels H ij (k) with an identical auto-correlation R H (n), the subchannels of H(k) associated with λ 1 and λ 2 have different auto-correlations: R λ1 (n) R λ2 (n). This is caused by two issues. 1) Initial SV ordering effect. Any valid SVD can be performed at the first subcarrier in Algorithm 2. However, in practice some ordering of the singular values of H(1) has to be determined such that the SVD algorithm has 2 Two spatial channels here means the spatial channels corresponding to singular values λ 1 and λ 2. With some approximations on (23), one can show that R H11 (k) R H21 (k) which is defined as R λ1 (k), and R H12 (k) R H22 (k) which is defined as R λ2 (k). This is why only two correlation functions are plotted with the terminology two spatial channels.

9 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 9 Amplitude of frequency auto correlation R(n) n R H (n) R λ1 (n) R λ2 (n) Fig. 4. Amplitude of the frequency auto-correlations of the two spatial channels of H and of the original channel H, 2 2 IEEE TGn n channel model D, D = 64 total subcarriers, and D p = 52 payload subcarriers. a unique output. For example the results in Fig. 4 are obtained with λ 1 (1) λ 2 (1). 2) Finite-length effect. The initial effect of SV ordering will disappear if the number of subcarriers is large enough. The SV switch event (c.f., Section III-E.1) will happen with probability one as D goes to infinity, and this will eliminate the initial difference posed by SV ordering. However in Fig. 4 and in the simulation reported in Section V, only D = 64 and D p = 52 subcarriers are used. Thus together with the initial effect of SV ordering λ 1 (1) λ 2 (1), the event {λ 1 (k) λ 2 (k), k = 1,, D} will happen with high probability. Fig. 4 suggests that the effective channel H still looks similar to H: they are both low-pass channels, and the correlation functions are much alike. In fact, this should not be a surprising result. The goal of the FSB/SSVD design is to make adjacent effective channel as close (in terms of Euclidean distance) as possible, while high correlation means two channel matrices are statistically close to each other. These observations imply that Linear MMSE (LMMSE) channel estimation can be directly applied to estimating H(k) [28]. Details on the LMMSE channel estimation method can be found in [14]. Due to the non-gaussianess and spatial correlations of Hij (k), LMMSE is no longer the optimal channel estimator in this problem setting. However simulation results demonstrate that its performance is still very good. Meanwhile, keeping the same channel estimator for both open and closed loop has implementation advantages, as has been discussed in Section II-C. V. SIMULATION RESULTS In this section, simulation results for two system configurations are reported to validate our algorithm. An IEEE n compatible MIMO-OFDM simulation system is built. We set D = 64, D p = 52, and skip the DC subcarrier. The OFDM structure is identical to that in a/g. At the transmitter, a Bit-Interleaved Coded Modulation (BICM) spatial-multiplexing transmission is implemented. BICM is chosen because it is the main scheme considered for next generation wireless standards. The standard 64-state rate-1/2 binary convolutional code with octal generators (133, 171) is used, which is followed by a block interleaver. High-order constellations such as 64QAM and 256QAM are adopted for 2 transmit antennas to achieve high spectral efficiency. The receiver implements both ML and linear MMSE soft-output detectors with max-log bit log-likelihood ratio (LLR) computation concatenated with a bit deinterleaver and a soft-input Viterbi algorithm (VA) convolutional decoder. The ML softoutput detector is implemented with the state-of-the-art LORD algorithm [29] when channel estimation error exists. 3 Standard IEEE n channel models [24] are used throughout this section. As has been mentioned before, we make the ideal assumption that the frequency correlation of H is perfectly known. Thus the receiver has perfect knowledge of the channel statistics. Notice that aside from beamforming, the transmitter can also deploy adaptive power allocation and/or adaptive coded modulation based on CSIT. However, since the target of this work is to examine how channel estimation is affected by beamforming, and adaptive power/rate control do not affect the effective channel, they are not considered in the following simulations. A. Simulation of Short Packet PSAM Structure The first simulation configuration is a short packet Pilot Symbol Assisted Modulation (PSAM) structure where the OFDM packet length is N f = 2 time intervals. Orthogonal pilots are scattered into different subcarriers so that the channels on data subcarriers can be estimated by interpolating and smoothing over adjacent pilot subcarriers. The antenna configuration is (L t = 2, L r = 2). The frame structure uses 28 out of the total 52 subcarriers for pilots, and the remaining for data transmission. In particular, all the odd-indexed subcarriers are employed for pilots together with one even-indexed subcarrier at each edge to give better performance at both edges of the frequency band. Simulations that compare the bit error rate (BER) performance of both perfect CSIR and LMMSE channel estimation are reported in Fig. 5. An important note is that a standard SVD (without considering the smoothing issue) can not operate in this environment. The reason is that no interpolation is possible if the channel smoothness is destroyed. Thus CSIR of the data subcarriers cannot be obtained by channel estimation from adjacent pilot subcarriers. When perfect CSIR is used, one can perform ML decoding by simply multiplying H(k) H to the received signal to fully decouple the two spatial streams, and then perform exhaustive search for each spatial stream to generate bit LLRs. However, when the estimated channel is used, such a simple ML decoding is not available due to channel estimation errors. In this case the LORD algorithm 3 As has been discussed in Section II-B, ML is equivalent to ZF and MMSE in case of perfect CSIR and beamforming. LORD is unnecessary in this case.

10 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR 10 Bit Error Rate (BER) SNR [db] perfect CSIR LMMSE channel estimation Fig. 5. BER vs. SNR plot of the short packet PSAM simulation, 64QAM n channel model D with ML soft output detection. [29] is implemented to generate ML soft output. LORD has been proved to be a low complexity detector that can efficiently generate soft outputs, and when L t = 2 it is equivalent to ML. The conclusion from this simulation result is that in the short packet PSAM structure with MIMO-OFDM beamforming, not only is the FSB design the only known high performance solution (standard SVD does not work), but its performance is about 2 db away from the perfect CSIR case, which is a reasonable result for a short packet (N f = 2) transmission in an n environment. B. Simulation of Long Packet Preamble Structure The second simulation setting is a long packet preamble structure, which is a typical wireless LAN implementation. Each OFDM packet carries 1000 information bytes. Three different antenna configurations are simulated: (L t = 2, L r = 2), (L t = 2, L r = 4), and (L t = 4, L r = 4). The first L t symbol times are devoted entirely for channel estimation, and the remaining are used for data transmission. Two different frequency-selective channel models [24] are considered: channel B, characterized by a 9-tap tapped delay line profile with 15 ns root mean square (rms) delay spread; and channel D with 18-tap and 50 ns rms delay spread. Channel B is a useful model for scenarios with limited frequency selectivity, like home residential environment with line of sight (LOS) and small delay spread. Channel D has a large amount of frequency diversity as typical of indoor office (e.g., cubicles) with nonline of sight (NLOS). The channel is assumed to be constant within one OFDM packet, and changes independently from one packet to another. Three different receiver configurations are considered to show the performance advantage of FSB: perfect CSIR. This serves as the benchmark, which is the most ideal situation; FSB design. In this case H is smooth across subcarriers thanks to the FSB design. When estimating H(k), one can use the pilots and received signals from all subcarriers i = 1,, D p, together with the pre-calculated frequency correlation functions, to perform the LMMSE channel estimation; standard SVD. In this case it is not possible to benefit from the smoothness across subcarriers, since it is destroyed by a standard beamformer design which has no requirement on the smoothness. Instead one has to perform channel estimation of each subcarrier separately. Only the signal at the k-th subcarrier can be utilized when estimating H(k). Simulation results for (L t = 2, L r = 2) are reported in Figs. 6 and 7, in which the above three receiver settings for channel model B and D are compared for both ML and MMSE detectors with both 64QAM and 256 QAM modulations. Several interesting observations can be made from these simulation results. First, as the number of pilot subcarriers increases compared to the short packet structure, the channel estimation performance of interpolation and smoothing is further improved: the gap between channel estimation and perfect CSI is only about 2/3 db for channel B in Fig. 6, and about 1 db for channel D as can been seen in Fig. 7, compared to the previous 2 db gap reported in Fig. 5. Second, the performance is dramatically improved if the beamformer generates a smooth H(k) and the receiver channel estimation benefits from interpolation and smoothing across frequency, compared to the case where smoothing is impossible if the beamformer is not properly designed. From Figs. 6 and 7, there are approximately 2.5 db and 2 db gain by using the proposed algorithm compared to the single-subcarrier channel estimation for both channel B and D, respectively. This shows the efficiency of the FSB design in an n setting. Third, an interesting observation is that when the receiver uses the estimated beamformed channel ˆ H(k) as if it was H(k), it seems that MMSE soft-output detector outperforms ML detector in a MIMO-OFDM BICM configuration. Notice that in the channel estimation setting, both ML and MMSE soft-output detectors are using imperfect effective channel matrices without any compensation for the estimation errors, and are both strictly suboptimal. MMSE detector in this setting might enjoy the benefit of a well-conditioned effective MIMO channel, as has been discussed in Section II-B. Finally, Figs. 8 and 9 report the simulation results for (L t = 2, L r = 4) with 64QAM and (L t = 4, L r = 4) with 16QAM, respectively. It is clear from the figures that benefits of deploying FSB are still remarkable in these antenna configurations. VI. CONCLUSIONS AND FUTURE WORK The FSB design has been developed to generate smooth effective channels across all frequencies in a MIMO-OFDM system. The receiver can utilize this smoothness to improve the channel estimation performance. Algorithms are presented and some related problems are discussed. Statistical characteristics of the effective channel have been studied. Efficient channel estimation is proposed using these results. Simulations are shown to support our claims. The work in this paper also has some limitations, and they can be the subject of future work. One key assumption in

11 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, TO APPEAR Bit Error Rate (BER) 10 2 Bit Error Rate (BER) QAM perfect CSIR, ML=MMSE 64QAM FSB, ML QAM standard SVD, ML 64QAM FSB, MMSE 64QAM standard SVD, MMSE 256QAM perfect CSIR, ML=MMSE 256QAM FSB, ML 256QAM standard SVD, ML 256QAM FSB, MMSE QAM standard SVD, MMSE SNR [db] 10 3 perfect CSIR, ML=MMSE FSB, ML standard SVD, ML FSB, MMSE 10 4 standard SVD, MMSE SNR [db] Fig. 6. BER vs. SNR plot of the long packet preamble simulation, 64QAM and 256QAM, n channel model B. Fig. 8. BER vs. SNR plot of the long packet preamble simulation, 64QAM n channel model D Bit Error Rate (BER) 10 2 Bit Error Rate (BER) QAM perfect CSIR, ML=MMSE 64QAM FSB, ML QAM standard SVD, ML 64QAM FSB, MMSE 64QAM standard SVD, MMSE 256QAM perfect CSIR, ML=MMSE 256QAM FSB, ML 256QAM standard SVD, ML 256QAM FSB, MMSE QAM standard SVD, MMSE SNR [db] Fig. 7. BER vs. SNR plot of the long packet preamble simulation, 64QAM and 256QAM, n channel model D perfect CSIR, ML=MMSE FSB, ML standard SVD, ML FSB, MMSE 10 4 standard SVD, MMSE SNR [db] Fig. 9. BER vs. SNR plot of the long packet preamble simulation, 16QAM n channel model D. the present work is perfect CSIT, which is not valid in a practical system. We plan to examine whether the proposed FSB is still effective when CSIT is imperfect, especially in a FDD system, and how we can further improve the design. In the current n proposal [11], there are both implicit and explicit feedbacks, and three feedback methods for explicit feedback have been proposed: CSI quantized matrix feedback, non-compressed steering matrix feedback, and compressed steering matrix feedback. We plan to investigate how the proposed algorithm will be affected by these different feedback techniques. Another ideal assumption is the perfect knowledge of channel statistics at the receiver. Although we have argued that this should not be a problem in the wireless LAN environment, it would be interesting to see how well existing solutions perform on the FSB effective channel H, and how to further improve the performance. ACKNOWLEDGMENT The authors would like to thank the editor, Dr. Howard Huang, and the anonymous reviewers for their comments that improve the quality of this paper. REFERENCES [1] I. E. Telatar, Capacity of multi-antenna Gaussian channels, AT&T Bell Labs, Tech. Rep., June [2] M. Skoglund and G. Jöngren, On the capacity of a multiple-antenna communication link with channel side information, IEEE J. Select. Areas Commun., vol. 21, no. 3, pp , Apr [3] A. Narula, M. Lopez, M. Trott, and G. Wornell, Efficient use of side information in multiple-antenna data transmission over fading channels, IEEE J. Select. Areas Commun., vol. JSAC-16, pp , Oct [4] H. Sampath, P. Stoica, and A. Paulraj, Generalized linear precoder and decoder design for MIMO channels using the weighted MMSE criterion, IEEE Trans. Commun., vol. 49, no. 12, pp , Dec

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Fitz, Layered orthogonal lattice detector for two transmit antenna communications, in Forty-Third Annual Allerton Conference on Communication, Control, and Computing, Sept Cong Shen (S 01) received the B.S. and M.S. degrees, in 2002 and 2004 respectively, from the Department of Electronic Engineering, Tsinghua University, Beijing, China. He is currently working towards the Ph.D. degree in the Electrical Engineering Department, University of California, Los Angeles (UCLA). His research interest is on general communication theory with emphasis on wireless communications. Michael P. Fitz (S 82-M 83-SM 02) received the B.E.E. degree (summa cum laude) from the University of Dayton, Dayton, OH, in 1983 and the M.S. and Ph.D. degrees in electrical engineering from the University of Southern California, Los Angeles, in 1984 and 1989, respectively. From 1983 to 1989, he was a Communication Systems Engineer with Hughes Aircraft and TRW Inc. Since 1989, he has been with the Faculty of Purdue University, West Lafayette, IN, The Ohio State University (OSU), Columbus, and the University of California, Los Angeles. He is currently with Northrop Grumman Corp. as a Senior Systems Engineer working on satellite communications. His research is in the broad area of statistical communication theory and experimentation. He is the author of Fundamentals of Communications Systems (New York: McGraw Hill, 2007). His research group at UCLA currently is interested in the theory of space-time modems and operates an experimental wireless widearea network and a space-time coding testbed. Prof. Fitz received the 2001 IEEE Communications Society Leonard G. Abraham Prize Paper Award in the field of communications systems.