Blind MIMO communication based on Subspace Estimation T. Dahl, S. Silva, N. Christophersen, D. Gesbert T. Dahl, S. Silva, and N. Christophersen are at the Department of Informatics, University of Oslo, P.O. Box 080, N-036 Blindern, Norway, while D. Gesbert is at the Mobile Communication Department, Eurecom, BP 93, F-06904, France. Their e-mails are: tobias@ifi.uio.no, silvanam@ifi.uio.no, nilsch@ifi.uio.no, gesbert@eurecom.fr, respectively.
2 Abstract A new method is proposed for blindly estimating the top singular modes in a reciprocal MIMO (Multiple Input Multiple Output) channel, while at the same time using these modes for multi-stream communication without the need for training data. The uplink and downlink parties obtain the relevant singular modes from the received data blocks as the eigenvectors/eigenvalues of the spatial empirical correlation matrices. The only requirement is that the separate data streams are statistically uncorrelated. The approach relies on a key and simple need to know observation about MIMO transmission : In order both to transmit and receive, one party needs only the singular values and one set of singular vectors, say the left set, while the other party needs only the corresponding right set of singular vectors in addition to the singular values; no party needs both sets of vectors, and other aspects of H are irrelevant to both parties. Advantages of this blind approach include no need for higher order statistic based estimation and convergence in one iteration. EDICS: -MAPP MULTICHANNEL SIGNAL PROCESSING APPLICATIONS -ACOM Antenna arrays and multichannel processing for communications Keywords MIMO systems, channel identification, singular value decomposition (SVD), singular modes, eigen-modes. I. Introduction Wireless MIMO (Multiple Input Multiple Output) systems are capable of delivering large increases in capacity through utilization of parallel communication channels [5], [6], [3]. Most wireless communication systems assume knowledge of the channel at the receiver, in the MIMO case a channel matrix. Channel estimation techniques can be divided into training-based techniques (e.g. V-BLAST [5], [7]) and blind or semi-blind methods (e.g. [4],[0],[],[2],[4], [5]). Training-based methods have the advantage of relative computational simplicity at the cost of a reduced data rate. Blind methods are typically more complex and may not be accurate enough, but avoid the use of training data and thus have the potential for increased payload rates. If the channel matrix is known at both the transmitter and the receiver, the singular modes of the matrix channel can be used to transport independent data streams (to increase data rate), while maximizing the SNR on each stream. For example in one extreme, one may choose to exploit the top mode only (associated with the largest singular value) in order to maximize the spatial diversity advantage. Also, when combined with an optimization of the number of streams, one can realize the optimum trade-off between rate and diversity maximization. Traditionally, the problems of (blind) estimation and that of transmission are addressed in a decoupled manner, but here the two problems are solved simultaneously. In [2] and [3], we proposed a technique for direct blind identification of the main singular modes, without estimating the channel matrix itself. The technique is related to the iterative numerical Power method for finding eigenvectors of a matrix and basically required only a QR decomposition ([8]). In estimating the eigenstructure directly, it overcomes many of the problems associated with other classes of algorithms
3 ([4],[0],[],[4],[5]), such as slow convergence, use of higher order statistics and requirements about statistical independence. J.B. Andersen ([]) had previously noted the that vectors transmitted iteratively forwards and back converge towards the singular vectors of the channel matrix. In this paper, we present a new and conceptually simpler blind method achieving the same goal. The new approach is based on standard ideas from signal subspace estimation in array processing which, to our knowledge, have so far not been utilized in the context of blind MIMO transmission of the spatial multiplexing type. The present method relies on a Singular Value Decomposition (SVD) of the received data block iterated between the transmitter and the receiver. The paper is laid out as follows: In Sec. 2.A we present the channel model, and in Sec. 2.B we recapitulate MIMO communication using singular modes of H and H. In Sec. 2.C we show how array signal subspace estimation in a simple way is brought to bear on the problem of singular mode transmission without prior knowledge of H. In Sec. 3 some initial results are given and discussed. II. Mathematical Background A. Channel Model We assume two-way communication through a N(receive) M(transmit) flat-fading MIMO channel matrix H C N M : Y Rec = HX Send (uplink) () X Rec = H T Y Send (downlink) (2) where X Send C M n and Y Send C N n are the transmitted data blocks of length n uplink (UL) and downlink (DL), respectively. Such a model describes a TDD (Time Division Duplex) system provided the ping-pong time - the time between the beginning of a DL frame and the beginning of the next UL frame - is small compared to the channel coherence period. For algorithm derivation purposes, we assume no channel noise, but return to in Sec. 2.C. It is also convenient to work with a channel where H T in (2) is effectively replaced by the complex conjugate transpose H. This is achieved by letting the transmit data Y Send be complex conjugated prior to transmission, and the received data block X Rec be complex conjugated prior to any further processing. In this way, (2) may be replaced by X Rec = H Y Send (downlink) (3) B. Communication using singular vectors The rank of H is denoted by K 0 min(n,m) and its SVD is H = USV. S is the diagonal matrix of singular values σ σ 2 σ K0 >0,and U=[u,...,u K0 ] C N K0 (4) V =[v,...,v K0 ] C M K0 (5)
4 are unitary matrices whose columns can be used as receive and transmit vectors {u i } and {v i }, respectively. Clearly, one can select a number K (K K 0 ) of vectors for communication through orthogonal singular modes. Assume, initially, that the singular vectors are known: One party (e.g. the base station) knows the top K left singular vectors {u i }, i =,,K of H and the other side (subscriber unit) knows the corresponding subset of right singular vectors {v i }, i =,,K. The top singular values {σ,σ 2,...,σ K }are known to both parties. Let U K =[u,u 2,...,u K ], V K =[v,v 2,...,v K ] and S K = diag{σ,σ 2,...,σ K },denote these subsets arranged into matrices, and let C x, C y C K n (n K) bethesymbol matrices comprising the UL and DL symbol blocks, respectively. Each row of a symbol matrix represents an individual data stream. The elements of these matrices are symbols from a modulation constellation (e.g. BPSK, QPSK, 6PSK,6QAM). Using these (known) singular vectors and values and neglecting noise, the UL transmit data block X Send = V K C x would be received as Y Rec = HX Send = USV V K C x = U K S K C x. The last equality is obtained by noting that V V K = I K0 K which has s inthekmain diagonal elements and zeros elsewhere. From this it follows that USI K = U K S K. Decoding is then simply performed through Ĉx = S K U K Y Rec. The DL transmission block is set up as Y Send = U K C y and decoding is carried out at the subscriber unit correspondingly. Note that both parties require S K, but the key remark is that no party requires knowledge of H. In fact, the base station both receives and transmits using only U K, while the subscriber unit only needs V K. In addition, if only phase modulation is used (e.g. QPSK), the singular values are not even necessary for decoding. Next we show how each party, based on what they receive during normal operation in a noisy environment, can estimate the singular values and the required singular vectors in only one iteration. C. Obtaining the singular modes through subspace estimation using noisy data Starting from, say, the subscriber unit, one transmits X Send = V Ini C x where V Ini C M M is an arbitrary unitary matrix. This is received as Y Rec = HV Ini C x + N, where each element of the complex AWGN term N has zero mean and variance σn 2. The spatial correlation matrix at the base station is then Y Rec YRec = HV IniC x C x V Ini H + NN + HV Ini C x N + NC x V Ini H Assuming a large enough block length n this simplifies to n Y RecYRec HH + σn 2 I = USV VSU + σn 2 I = US2 U + σn 2 I (6) where we have assumed independent data streams leading to C x C x ni (for an average squared symbol modulo of ), used the fact that V Ini VIni = I since V Ini is unitary, and finally assuming that the noise is statistically independent of the data. The first data block C x goes to waste. Estimates of the K top left singular vectors, Û K, and singular values, ˆσ i of H are obtained as the corresponding eigenvectors and square root of the eigenvalues of the normalized spatial correlation matrix n Y RecYRec. Note that the estimates of the singular vectors will be unbiased whereas the singular values
5 will be incremented by the noise variance. If only phase modulation is used, the singular values are not of concern. Once this is completed, DL information is transmitted as Y Send = Û K C y. Forming the normalized spatial correlation matrix at the subscriber unit, we have: n X RecX Rec H U K U KH + σni 2 = VSU U K U KUSV + σni 2 (7) Here U K is in general not a unitary matrix and U K U K I. However, U U K = I K0 K implying n X RecX Rec V KS 2 K V K + σ2 N I From this correlation matrix, the subscriber unit estimates the K required singular values and vectors, decodes the received information, and transmits uplink X Send = V K C x. Note that the method gives consistent estimates of the eigenvectors, even in the presence of noise. The stability of this approach deserves a separate comment. Once started, the procedure transmits approximate singular vectors of the channel matrix through the channel. As noted in [3], this is essentially identical to the method of orthogonal iteration ([8]) used to find a set of dominant eigenvectors of a matrix. The procedure is therefore numerically stable and also robust in the presence of noise as demonstrated in the simulations below. Note that the singular vectors can only be determined up to multiplication by a complex number of unit norm. This is a standard ambiguity in blind estimation methods, and is overcome by using differential coding. III. Results and discussion Figure shows the effect of block length on BER vs. SNR for a 4TX x 4Rx channel using QPSK modulation. The results are for the average of the two top singular modes and also includes the case for known singular vectors. For each block length, a plateau in BER is reached for increasing SNR and the plateau decreases with block length. This is because the approximations in equations 6 and 7 improve with increasing block length and for any finite length there remains an effective noise component. Differential coding leads to a 3 db loss in SNR and this explains the difference between the case for true singular vectors and the two largest block cases at low SNR. Figure 2 shows the BER against SNR for each of the two top singular modes in 4TX x 4Rx system for both true and estimated singular vectors, using a block length of 000. The difference in BER between the two modes is small in the plateau phase. A. Discussion We have demonstrated blind communication via subspace estimation for a reciprocal MIMO channel. We exploit the fact that the data streams are uncorrelated in a spatial multiplexing scenario. Singular vectors and values are estimated on a "need-to-know" basis. No party knows the full matrix H, only one set of singular vectors and the singular values. The algorithm is also suitable for tracking, since it
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