Communications over the Best Singular Mode of a Reciprocal MIMO Channel

Transcription

1 Communications over the Best Singular Mode of a Reciprocal MIMO Channel Saeed Gazor and Khalid AlSuhaili Abstract We consider two nodes equipped with multiple antennas that intend to communicate i.e. both of which transmit and receive data. We model the responses of the communication channels between these nodes as linear and reciprocal (time invariant or with very slow time variations). In practice, we exploit the closed loop conversation between these nodes and present an efficient algorithm allowing to adaptively identify the Best Singular Mode (BSM) of the channel. We consider two scenarios. In the first scenario, the initial communication link is established over the BSM assuming that the exchanged data is partially known at both nodes. This scenario is suitable for channel training. In the second scenario, the BSM is adaptively updated while the real unknown data is exchanged between the nodes i.e. no capacity is wasted for channel identification. The proposed adaptive algorithm is robust to noise as the involved step-size allows a trade-off to reduce the impact of the additive noise at the expense of some estimation delay. Our computer simulations show that the proposed algorithm works efficiently in both modes of operations (training mode and simultaneous training/data transmission mode) for both static and slow fading MIMO channels and for both white and colored noises. I. INTRODUCTION The performance of the MIMO systems depends on the statistical properties of the channel matrix response H, especially on its rank. For example, we know that correlation among the sub-channels has important impact on the communication performance. If H is known at both transmitter and receiver, the channel is equivalent to r parallel SISO AWGN channel where r is the rank of H [1]. The gain of each of these r channels is the corresponding singular value of H. The water filling theorem indicates that more power should be allocated for transmission of signals over stronger singular modes of the channel in order to maximize the capacity. However,

2 this optimal mechanism requires the full knowledge of H at both transmitter and receiver in order to design optimal pre-coder and optimal decoder [2]. One sub-optimal approach is to transmit only over the best singular mode (BSM). This approach uses only one optimal beamformer at the transmitter and one at the receiver. These beamformers represent the singular vectors of H associated with the largest singular valus. For highly correlated channels with a rank r = 1, this approach is optimal. Also when the ratio of the transmit power to receive noise variance is smaller than 1 1, the optimal water-filling σ2 2 σ1 2 algorithm suggests using only the BSM, and in such a case, this approach is capacity optimal, where σ 1 and σ 2 are the two most dominant singular values of the channel response. This is because the water filling allocates power to less number of the best modes as the transmit power is reduced, which is equivalent to reducing the SNR. The drawback of these methods is that the channel could be only estimated at the receivers. Therefore, optimal or suboptimal methods could be implemented only if some information is fed-back from the receiver to the transmitter. This requires the establishment and use of expensive control channels. In [3] [5], a scenario is considered where two nodes communicate iteratively back and forth. Such an iterative communication allows alleviating the need for feedback. The BIMA (Blind Iterative MIMO Algorithm) in [3] [5] exploits the reciprocity of the communication channels and identifies the required optimal beamformers at both sides of a communication link. The stability analysis of the BIMA is addressed in [6]. The performance of the algorithm is poor in low SNRs and there is no way to increase the accuracy of the estimates. In this paper, we extend this idea and propose an adaptive algorithm for estimating the BSM, allowing to make reasonable trade-off between the speed of convergence and the accuracy of the estimate like in other adaptive algorithms. In addition, in this paper we estimate/track the BSM while the data stream is also transmitted at the same time. Interestingly, our results reveal that only one training symbol is required to synchronize/align the two sides of the link. The procedure in [4], [5] is that each side normalizes, conjugate and retransmit the received vector. Instead, in this paper, we choose to combine the previous estimates with the received information. This allows us to derive a weighted sum of the previous estimate and the current received signal. The proposed approach has two main advantages. First, we could incorporate data stream while the BSM is being identified. Second, we could control the trade-off between the 2

3 convergence speed and the accuracy of the result by controlling the parameter of the algorithm. The algorithm in [4], [5] represents a special case of our proposed algorithm. This article comprises the following sections. We introduce our assumed MIMO communication system model and formulate the problem we are working on in Section II. In Sec.III we introduce our method and establish its algorithm. The performance of the proposed method is studied in Section IV via computer simulations. Finally, we conclude our discussion in Section V. We have used small bolded letters to show vectors and capital bolded letters to show matrices. The set of m-dimensional complex vectors are shown by C m and C m n shows the set of (m n)- dimensional complex matrices. The expected value of a random variable is denoted by E{ }. The i th column of matrix A is shown with [A] i and [A] ij shows the (i, j) th entry of A. The k dimensional identity matrix is denoted by I k. Notations, ( ) T, ( ) and ( ) H are used for the Frobenius norm, transpose, conjugate and hermitian of a vector/matrix, respectively. II. SYSTEM MODEL AND PROBLEM FORMULATION We consider an N M MIMO system (N and M are the number of antennas at node Y and X respectively) over slow block fading channels. Both Nodes are capable of transmitting and receiving data. We assume that the baseband channel matrix representing the path from node Y to node X is the reciprocal of the channel matrix representing the path from node X to node Y. That is H yx = H T xy = H. (1) The received signal matrices, R y C N L at node Y and R x C M L at node X (L is the length of the sent data frame), are given by R y = HT x + N y, R x = H T T y + N x, (2) where T x C M L and T y C N L are the transmitted signal matrices from node X and Y respectively (T x and T y are mappings of the frames d x C L 1 and d y C L 1 intended to be sent by node X and Y, respectively). N y C N L and N x C M L are the complex white Gaussian noise matrices with i.i.d., zero mean and ν 2 variance elements. For slow block fading channels, the channel matrix H C N M is constant during multiple frames d x or d y. The elements of H are assumed to be i.i.d. circularly symmetrical complex Gaussian random 3

4 variables, with zero mean and unit variance. The channels are assumed to be unknown to both, the transmitter and the receiver. form The Singular Value Decomposition theorem states that there exists a factorization of H of the H = UΣV H = Rank(H) i=1 σ i u i v H i (3) where U = [u 1,u 2,,u Rank(H) ] C N Rank(H) and V = [v 1,v 2,,v Rank(H) ] C M Rank(H) are two matrices with orthonormal columns, the matrix Σ = diag(σ 1, σ 2,, σ Rank(H) ) is a Rank(H) Rank(H) diagonal matrix with nonnegative real numbers on the diagonal such that σ 1 σ 2 σ Rank(H). We make the following remarks on the decomposed matrix, H: The matrix V contains a set of ortho-normal input or analyzing basis vector directions for H, and its columns are often called the right singular vectors. The matrix U contains a set of ortho-normal output basis vector directions for H, and its columns are often called the left singular victors. The matrix Σ contains the singular values, which can be thought of as scalar gain controls by which corresponding inputs are multiplied to give corresponding outputs. Definition 1: We call the triplet (σ i,v i,u i ) as the i th singular-mode of the channel H. We note that the singular vectors (SVs) are not unique as {e jθ u i } Rank(H) i=1 and {e jθ v i } Rank(H) i=1 are also sets of left and right SVs for H for any arbitrary {θ i [0, 2π)}. Figure 1 depicts the decomposition of the channel matrix H into its singular-modes, the set of triplets {(σ n,v n,u n ) : n = 1, 2, Rank(H)}. Pre-multiplying H by v j and post-multiplying it by u H j result in nulling out all singular modes but the j th one, for the fact that U and V comprise ortho-normal columns. That is u H j UΣVH v j = u H j Rank(H) i=1 σ i u i v H i v j = σ j. (4) Thus a MIMO channel is equivalent to multiple Rank(H) SISO parallel chancels provided that singular modes are known at both nodes. This portrays that if we want to select one singularmode of H to transmit data between nodes, we should pick the best singular-mode, (σ 1,v 1,u 1 ), as σ 1 is the largest singular value (gain). The optimal water-filling solution for power allocation 4

5 over multiple parallel channels suggests to only use the BSM if the transmit power is limited (such that the transmit power to receive noise variance is less than 1 1 ). One main motivation σ2 2 σ1 2 to only communicate through the BSM is that the BSM can be estimated much more accurately than other modes with very low computational cost [6]. In addition, in a multiuser environment, the spectrum can be better exploited by avoiding the usage of weaker modes. In this case, provided that (σ 1,v 1,u 1 ) is known at both nodes, the following signals are transmitted at nodes X and Y, respectively T x = v 1 d T x, T y = u 1 dt y. (5) Under our assumptions, the unbiased minimum variance estimate of data vectors at Y and X are as follows respectively ˆd T x = 1 σ 1 u H 1 R y, ˆdT y = 1 σ 1 v T 1 R x. (6) The transmitters in (5) and receivers in (6) are known as optimal beamformers. Using (2), (5) and (6), the covariance of the elements of ˆd T x dt x and ˆd T y dt ν2 y can be easily calculated as. σ1 2 Thus the receive SNRs at both nodes are given by σ2 1 σ2 d. It is evident that the BSM beamformers ν 2 in (5) and (6) optimize the receive SNRs, as they align the transmitted signals with the direction that maximizes the channel gain. Our goal in this paper is to accurately estimate the BSM of H, (σ 1,v 1,u 1 ), in a distributed manner and simultaneously use the BSM for data transmission. Interestingly, the knowledge of all components of the BSM is not required at both nodes. In particular, node X is required to only know (σ 1,v 1 ) while the knowledge of (σ 1,u 1 ) is required at node Y. Furthermore, if we choose to use an M-PSK signal constellation (for example), then the decision of the demodulator/detector at the receivers will not be affected by the signal amplitude, which results in exempting the receivers from knowing σ 1. III. ESTIMATING THE BEST SINGULAR-MODE: THE PROPOSED ALGORITHM To achieve the capacity promised by MIMO communication systems, one has to know the channel singular-modes both at the transmitter and at the receiver. In fact, many beam-forming and decoding schemes in MIMO systems use the channel singular-modes for their purpose (for 5

6 example see [7], [8] and references therein). As an application, to achieve the MIMO channel capacity when the channel is known both at the transmitter and the receiver, and also to facilitate the transmit signal detection, designing of optimal pre-coder and decoder is addressed in several works (e.g., see [9], [10]). For this purpose, the singular value decomposition (SVD) of the channel matrix, H, is required to be known at both transmitter and receiver. One possible scenario is to obtain the channel matrix through channel training at the receiver side [11] and then apply a matrix decomposition. Obviously, some information can be sent back to the transmitter through a low bandwidth feedback channel, to inform the transmitter about channel state matrix or the channel singular-modes, e.g., see [12], [13]. In this case, one proposed method for reducing the bandwidth of feedback channel is using quantized beam-forming [7]. Blind channel estimation is also another proposed method for estimating the channel between transmitter and receiver [14], [15]. However, this method is time consuming and long data record has to be used for acceptable accuracy of these methods. In addition, this kind of channel estimation is effective for situations in which the channel is stationary for a sufficiently long period of time. In [3] [5], a simple, yet powerful algorithm was proposed to identify the triplet (σ 1,v 1,u 1 ) of a channel matrix, H, in an environment where the reciprocity condition (H yx = H T xy = H) is valid. The convergence of this algorithm is proven in a noise free environment to the BSM of H. However, the performance of the algorithm degrades dramatically in a noisy environment as the signal to noise ratio (SNR) decreases. Therefore, we propose an algorithm that allows to combat the impact of the noise and make a tradeoff between the accuracy and learning time. In this paper, we also explore the possibility of data transmission while BSM is being identified. We assume that a previous estimate of the triplet (σ 1,v 1,u 1 ) or a guess is available from previous iteration. To update the estimates, the algorithm must take into considerations both, the previous estimate of the triplet and the current received signal. In other words, the current estimate should be a function of the previous estimate and the current received signal. Interestingly, we will obtain a weighted sum of the two. In particular, we propose to update the estimate of the current u 1 at node Y, denoted by û 1,k (where k is the time or iteration index), by minimization 6

7 of the following criterion f(u) = µ u û 1,k Ry,k ˆσ 1,y,k 1 ud T 2 x,k, û 1,k = arg min(f(u)), u s.t. û 1,k 2 = 1. where µ is a positive number larger than zero and ˆσ 1,y,k 1 is a previous estimate of the largest singular value σ 1 at node Y. The criterion f has two components. The first component ensures that the new estimate is close enough to the previous estimate. The second component makes use of the received signal. The significance of µ stems from the fact that the larger it is, the more weight we put on previous estimate, and that is a sound choice, especially in a noisy environment. Notice that the solution of (7) depends on whether d x is known or otherwise. We consider the solution for the two cases in subsequent subsections. In the first case, the knowledge of d x is used for the initial establishment of the communication link. In the second case, the data stream d x is also estimated along with the tracking of the BSM. (7) A. Estimation of the Best Singular-Mode Using a Training Sequence Here, we assume that the receiver knows a priori the transmitted data d x,k in the optimization problem in (7). To minimize f in (7), we use the lagrange multiplier method and set the gradient of the lagrangian function with respect to u to zero ( µ λ + ˆσ 2 1,y,k 1 d x,k 2) u µû 1,k 1 ˆσ 1,y,k 1 R y,k d x,k = 0, (8) where λ is the lagrange multiplier. Using (8) and the constraint in (7) the optimal solution of (7) is given by û 1,k = µû 1,k 1 + ˆσ 1,y,k 1 R y,k d x,k µ + ˆσ1,y,k 1 R y,k d. (9) x,k Note that the denominator in (9) normalizes the length of the numerator to one. This suggests that (9) can be written as ũ 1,k = µû 1,k 1 + ˆσ 1,y,k 1 R y,k d x,k, (10a) û 1,k = ũ1,k ũ 1,k. (10b) 7

8 The vector ũ 1,k is a linear combination of two terms, û 1,k 1 and R y,k d x,k. The first term is the previous estimate and the second term is calculated from the newly observed signal. Obviously, the larger the µ, the more memory the system has. The above update equation can be written in terms of the the previous estimates û 1,k 1 and ˆσ 1,y,k 1 as follows E y,k = R y,k ˆσ 1,y,k 1 û 1,k 1 d T x,k, (11a) ũ 1,k = û 1,k 1 + û 1,k = ũ1,k ũ 1,k. ˆσ 1,y,k 1 µ + ˆσ 2 1,y,k 1 d x,k 2E y,kd x,k, (11b) (11c) where E y,k is the a priori prediction error of R y using û 1,k 1, ˆσ 1,y,k 1 and d x,k. The equations in (11) represent a Normalized Least Mean Square (NLMS) algorithm where µ controls the step-size and (11c) imposes the unitary constraint on û 1,k. The same procedure could be applied at node X: E x,k = R x,k ˆσ 1,x,k 1ˆv 1,k 1 d T y,k, (12a) ṽ 1,k = ˆv 1,k 1 + ˆσ 1,x,k 1 µ + ˆσ 2 1,x,k 1 d y,k 2E x,kd y,k, (12b) ˆv 1,k = ṽ1,k ṽ 1,k. In the following, at node Y, we propose to update ˆσ 1,y,k as follows f(σ) = µ (σ ˆσ 1,y,k 1 ) 2 + Ry,k σû 1,k d T 2 x,k, (12c) (13a) Setting ˆσ 1,y,k = arg min σ R + (f(σ)). f ˆσ 1,y,k = 0, the optimization problem yields ), 0 ) (13b) ˆσ 1,y,k = max( µˆσ 1,y,k 1 + Re ( û H 1,k R y,kd x,k µ + d x,k 2, (14) where, the max( ) operator in (14) is necessary to satisfy the constraint of σ R +. Similarly, the above update equation can be rewritten in terms of the a posteriori error as follows E y,k = R y,k ˆσ 1,y,k 1 û 1,k d T x,k (15a) ( ) 1 ( ) = max ˆσ 1,y,k 1 + µ + d x,k 2Re û H 1,kE y,kd x,k, 0, (15b) ˆσ 1,y,k 8

9 The same procedure could be applied at node X, which yields E x,k = R x,k ˆσ 1,x,k 1ˆv 1,k d T y,k (16a) ( ) 1 ( ) = max ˆσ 1,x,k 1 + µ + d y,k 2Re ˆv 1,k H E x,k d y,k, 0. (16b) ˆσ 1,x,k The closed loop NLMS update equations in (11), (12), (15), and (16) has a computational complectly of order of 3(M + N) per symbol. Close examination of these update equations shows that the larger the µ is, the less fluctuation we encounter in our estimation, and hence smaller variance. However, the larger it is, the slower the convergence of the triplet (ˆσ 1, ˆv 1,û 1 ). Thus, µ regulates the trade off between the estimation error fluctuations and the learning time lag. Intuitively, µ should be inversely proportional to the SNR. In the special case for µ = 0 and where the frame has only one symbol d = 1, this algorithm becomes identical to the one proposed in [3]. The proposed extended algorithm allows to make use of previous estimates of the BSM at both nodes which in return significantly reduces the impact of the noise. This is intuitively verified as follows by substituting R y,k = UΣV Hˆv 1 d T x,k + N y,k in (10a) and obtaining Rank(H) ũ 1,k = µû 1,k 1 + ˆσ 1,y,k 1 d x,k 2 σ i γ i,k u i + (ˆσ 1,y,k 1 N y,k d x ), (17) where γ i,k = vi Hˆv 1,k. Note that the estimator in (17) is a waited sum of three components, namely Rank(H) the previous estimate û 1,k 1, the inter-mode-interference σ i γ i,k u i and the noise component N y,k d x. By increasing µ, more wait is given to the first component compared to the other two, which in turn reduces the impact of the noise and the inter-mode-interference components and results in a more accurate estimation. However, a slower convergence is experienced, as the effect of the desired component, γ 1,k, is also reduced. The same argument applies in the case of the estimator of σ 1 in (14). i=1 i=2 B. Blind Joint Estimation of the BSM and Data Stream Now, we assume that the data frame is unknown to the receiver, and hence, we propose to optimize f in (7) not only in terms of u 1 but also in terms of d x,k. This scenario is applicable after having established the communication link (i.e., after obtaining some reasonable estimate of 9

10 the BSM using a method such as the one proposed in the previous section), where we must not only detect the transmitted data stream but also track the triplet (σ 1,v 1,u 1 ). In this subsection, we assume no a priori knowledge about the set of constellation points at the transmitter. Thus the procedure in this subsection allows the transmitters to change their modulation schemes adaptively. In the next subsection, we explain how the knowledge of the set of constellation points can be used to estimate σ 1. At node Y, we first estimate d x,k assuming that û 1,k 1 and ˆσ 1,y,k 1 are some available previous estimates of u 1 and σ 1, respectively. Then, we use this estimation in (7) to update û 1,k. To this end, we propose to set f d x,k = 0 and solve it for d x,k using previous estimates of u 1 and σ 1. Then replace this solution back into (7) and optimize the resulting f in order to update u 1. Thus, we have f d x,k = σ 2 1d T x,k σ 1 u H 1,k 1R y,k = 0. (18) Using the previous available estimates, û 1,k 1 and ˆσ 1,y,k 1, the above yields ˆd T x,k = 1 ˆσ 1,y,k 1 û H 1,k 1R y,k, (19) To proceed and update the estimate of u 1 without use of hard detector, we substitute the term ˆσ 1,y,k 1 d T x,k in (7) by ˆσ 1,y,k 1ˆd T x,k, and minimize f(u) = µ u û 1,k Ry,k uû H 1,k 1 R 2 y,k. (20) under the unitary constraint u H u = 1. We set the gradient of Lagrangian to zero with respect to u and obtain the following updating estimate for u 1 at node Y ũ 1,k = µû 1,k 1 + R y,k (û H 1,k 1 R y,k) H, (21a) û 1,k = ũ1,k ũ 1,k, (21b) Note that (21b) imposes the unitary constraint. The update equation in (21a) represents a linear combination of the previous estimate, û 1,k 1, and R y,k (û H 1,k 1 R y,k) H. Similar to previous case, 10

11 we can rewrite the update equations as follows E y,k = (I û 1,k 1 û H 1,k 1 )R y,k, (22a) ũ 1,k = û 1,k 1 + E y,k (ûh 1,k 1 R y,k) H µ + û H 1,k 1 R 2, y,k (22b) û 1,k = ũ1,k ũ 1,k. (22c) The same procedure could be applied at node X, which yields similar update equations for ˆv 1,k. Note that this blind adaptive estimator for u 1 does not need any estimate of σ 1 nor provides an updating estimator for σ 1. C. Semi-Blind Joint Estimation of the BSM and Data Stream In this subsection, we propose a semi-blind algorithm (using decision feedback approach) assuming that the modulation scheme (i.e., partial information about data stream) used at the transmitter is known to the receiver. It is obvious that we feed the estimate of d T x,k in (19) to a hard detector (which maps the elements of ˆd x,k to the closest constellation point) to detect the transmitted symbols and also to reduce the noise component of this estimate. The output of the hard detector denoted by d T x,k can be then substituted for dt x,k in (11) and (15) to update the estimates of u 1 and σ 1 respectively. Similar procedure can be used, at node X to update the estimates of v 1 and σ 1. Table I summarizes the proposed algorithms which after initialization can be switched to operate in either training, blind or semi-blind. IV. SIMULATION RESULTS In this section, we evaluate the performance of the proposed algorithms. In all computer simulations, the elements of the channel matrix H C N M are generated randomly for each run as i.i.d. circularly symmetrical complex Gaussian random variables with zero mean and unit variance. In addition, we generate the noise samples randomly as i.i.d. circularly symmetrical complex Gaussian random variables with zero mean and variance ν 2. In this paper, we define the Signal to Noise Ratio (SNR) as the transmitted power to the received noise variance, (i.e., 1 since the transmitted power is 1). ν 2 11

12 We first evaluate the average of the estimated principal (best/largest) singular value ˆσ 1,x and compare it to the true value σ 1. For the average of ˆσ 1,x, we consider one random realization of H and run the simulation for independent noise realizations and average the estimated results ˆσ 1,x,k. Figure 2 illustrates the average of the estimated values versus iteration index in the training mode. For this realization of H, the true value of σ 1 is This results reveal that the proposed estimator for σ 1 converges in the mean sense but provides a biased estimator in the steady state. The bias decreases as the SNR or the regulating parameter, µ, is increased. In fact, our extensive simulations also show that increasing the frame length has similar effect on the bias. The initial transition increases as we increase the value of µ. The remaining of this section has three subsections. Subsection IV-A evaluates the behavior of the training mode algorithm when the transmitted data is known at the receiver. In Subsection IV-B, we demonstrate the performance of the decision-feed-back algorithm when the sent data is unknown to the receiver (tracking mode). And in Section IV-C, we illustrate the performance of the proposed algorithm for a slowly varying MIMO channel and for colored or white noises. In order to better evaluate the convergence of ˆσ 1,x to σ 1, we also consider ( ˆσ 1,x,k σ 1 σ 1 ) 2 as a criterion which is the normalized squared estimation error. Similarly, to evaluate the estimation error between the principal singular vector u 1 (or for v 1 ) and its estimate û 1,k, we define φ k = (u 1,û 1,k ) = cos 1 ( u H 1 û 1,k ) as the angle φ k in radian between u 1 and û 1,k at node Y. In particular, we use the average of the squared angle φ k except in Section IV-C wherein we use the Root Mean Square (RMS) of this angle. To this end, in each run the channel matrix H is generated independently and the collected data for ( ˆσ 1,x,k σ 1 σ 1 ) 2 and φ k 2 are averaged over runs. (23) A. The training mode performance in the covariance sense In this subsection, we assume that the transmitted data is known at the receiver and evaluate the training mode performance of the algorithm. For a 4 4 MIMO system, we study the performance for different values of the regulating parameter µ {0, 5, 10, 20} and for SNRs of zero and 20 db, versus the iteration index. Both nodes sends only one symbol at a time to 12

13 the other node, that is the frame length of the training data is one. Figures 3 and 4 depict the performance of our algorithm versus the iteration index for different µ at SNR values of zero and 20 db, respectively. To compare the proposed method with open loop training we also simulated an open loop channel estimator, where only node X sends known training symbols to node Y. At node Y, these training symbols are used to first estimate the channel matrix H using optimal least-squares algorithm, referred to herein as Least Square Open Loop Training (LS-OLT) [16], which then is decomposed into its SVD form. The normalized error of estimating the largest singular value, σ 1, as well as the square of the angle, φ are plotted for LS-OLT on the same figures. From these figures, we observe the convergence of the proposed algorithm. The average of ( ˆσ 1,x,k σ 1 σ 1 ) 2 in the steady state is inversely proportional to µ and the SNR. The proposed algorithm illustrates considerable improvement compared with the results of the algorithm in [4] (which corresponds to µ = 0). The price paid for this improvement is the increases of initial transition lag which is also proportional with µ. We observe also similar trend in the average of the squared of (u 1,û 1,k ). In other words, the larger the regulating parameter is, the smaller the steady state error between the true BSM and the estimated one, yet the slower the initial convergence. The higher the SNR is, the lower the steady state errors. Depending on how slow the channel response can vary with time, we must choose the value of µ, i.e., the slower the variation of H with time, the larger the value we can choose for µ to improve the performance. In a high SNR scenario, the bias tend to be small, and therefore, we could chose smaller values for µ to accelerate the convergence rate. The regulating parameter µ could be chosen as a function of the convergence time and the SNR. For example, for convergence in 50 iterations at SNR of zero db, we choose µ = 20 and at SNR of 20 db, we choose µ = 5. Alternatively, to accelerate the convergence and improve the performance, one could use variable parameter µ k, starting with a small value to expedite the initial transition and increase the value of µ k with time. To demonstrate this, we have added the performance curves for µ k = 1.5k for an SNR of 0 db and µ k = 0.2k for an SNR of 20 db. Notice that the optimization of µ k is an interesting problem that is not the scope of this paper. The impact of the frame length is similar to that of the SNR on the steady state. However, the initial learning time is proportional with the frame length. We observe that the proposed closed loop algorithm significantly out performs the open loop optimal training. This is intuitively justified by arguing that the closed loop method 13

14 at each iteration attempts to concentrates the transmit power in the desired direction and avoids extracting information about unnecessary modes. B. The performance of the algorithm in a tracking mode (the sent data is unknown): In the previous subsection we studied the performance of the training mode algorithm. Here, we evaluate the performance of the proposed tracking mode algorithm used for the case in which the transmitted data are unknown to the receivers. The tracking mode algorithm needs to be initialized. In our simulations, only one of the nodes initially transmits only one known symbol to the other node in order to initialize the estimated values of the triplet (σ 1, v 1, u 1 ). After this first symbol they start exchanging unknown data frames. The receiver role is to detect the received frame while simultaneously update the BSM. We fix the regulating parameter to µ = 100 and plot the performance results versus the iteration index. The iteration index here represents a whole frame. Figures 5(a) and 5(b) compare the results for a 2 2 MIMO system with a 4 4 system at zero db SNR and frame lengths of 100 and 1000 symbols. Our simulations show that, interestingly, only one symbol is enough for initial training of the two nodes prior to exchanging data and that the distributed algorithms synchronize themselves. The estimation errors in the tracking mode exhibit similar trends to those of the training mode. For example, the bias level is reduced when either or both of SNR and frame length are increased. From Figures 5(a) and 5(b), we see that the bias in estimating σ 1 decreases as the number of antennas increase. In fact, the impact of increasing the number of antennas on the normalized squared estimation error is similar to the impact of the increase in the SNR. This is because, the total energy collected by the receiver is proportional with the number of antennas which is analogous to increasing the SNR. Note that the increase of the number of antennas has more significant impact than the SNR does on the performance of communication systems. This is because the system will choose the best mode among all possible singular modes where the number of modes which is Rank(H) determines the diversity gain [17]. However, Figure 5(b) shows that the effect is not the same when estimating and updating the vectors û 1 and ˆv 1, for the fact that increasing the number of receiving antennas increases the number of coordinates of the singular victors, which results in more degree of freedom and produces a slightly higher squared error. From Figures 5(a) and 5(b), we see that for a fixed values of SNR, µ and MIMO system size, the 14

15 steady state error of estimating the triplet decreases as the frame size increases because of the fact that the bigger the frame length, the higher the average energy per frame. The bias curves for the tracking mode algorithm are similar to that of the training mode. C. The training mode performance in slow fading channels Figures 6(a) and 6(b) illustrates the performance of the proposed algorithm for a time varying channel and for colored noise. We generate the entries of the MIMO channel matrix as independent circularly symmetrical complex Gaussian random variables with zero means and unit variance, however, they vary slowly as a rayleigh channel having a Clarke s Doppler spectrum, with a normalized maximum Doppler shift of f d = We consider a random realization of the slowly varying channel matrix response, H k, over a period of 1000 samples (iteration). In the first 500 samples we generated the noise processes N x,k and N y,k as spatially and temporally white with variance of νw 2 = To demonstrate the behavior of our algorithm in a colored noise environment in the last 500 samples, we add an additional noise components 1 N n x,k [1,, 1] T and 1 M n y,k [1,, 1] T to receivers where n x,k and n y,k are generated as white Gaussian processes with variance of ν 2 c = 0.1 independent of all other processes. It can easily be shown that the noise covariance matrix has an eigenvalue of ν 2 c + ν 2 w and all other eigenvalues are equal to ν 2 w. Thus in the last 500 iterations, the maximum ratio of the eigenvalues of the noise covariance matrix is 1 + ν2 c = 11. νw 2 For time varying channels, the second largest singular value occasionally approaches the dominant one. Therefore in order to better understand the performance of the algorithm, we have added the curve of σ 1 σ 2 on these figures. In addition to (23), in Figure 6(b), we have added two more curves of φ 2,k = cos 1 ( u H 2 û 1,k 2 ), and φ 12,k = cos 1 ( u H 1 û1,k 2 + u H 2 û1,k 2 ), (24) where φ 2,k is the angle between û 1,k and u 2 for µ = 10; and φ 12,k is the angle between û 1,k and the subspace spanned by the two most dominant singular vectors {u 1,u 2 } for µ = 10. The effect of adding the extra colored noise component appears as a jump around iteration 500 in both figures. We notice that the algorithm is able to track a time varying channel and remains robust to colored noise. We observe an interesting phenomenon; that is, the errors are inversely 15

16 proportional to the separation of two dominant singular values σ 1,k σ 2,k. When the second singular value approaches the dominant one, the errors of both estimators increase. However in Figure 6(b), we notice more significant impact on the estimation error of the singular vector (compared with singular values) at two instances around iteration indices 220 and 870. We notice also as σ 1,k σ 2,k approaches zero, the error increases more significantly for larger regulating parameter. This is justified since the memory (time constant) of the algorithm is proportional to µ σ 1,k σ 2,k. For µ = 10, the curves of φ 2,k and φ 12,k show that as σ 1,k σ 2,k approaches zero, the estimated vector, û 1 starts to slowly diverge from the direction of u 1 towards the direction of u 2 while maintaining the angel with their subspace in a rate proportional to the regulating parameter. This suggests to use a smaller value of µ as the normalized Doppler shift increases. There is a trade off between the tracking ability and the reduction of the impact of noise. V. CONCLUSION We formulated the problem of estimating the strongest/best singular mode and presented an efficient distributed algorithm allowing to adaptively identify the Best Singular Mode of the channel. The proposed algorithm allows to self tune and identify the best transmit and receive beamformers at both nodes in parallel. The proposed algorithm has two modes. In the first mode, the initial communication link is established assuming that the exchanged data is known at both nodes. This mode is useful when we require to train the channel to identify the BSM prior to starting exchanging data frames. In the second mode, the BSM is adaptively updated while the real unknown data is exchanged between the nodes, which means that no capacity is wasted for channel identification. In fact, our computer simulations reveals that exchanging only one known symbol is adequate to initialize the algorithms in both nodes and the algorithms are self synchronizing after initialization. We have shown the convergence of the proposed algorithm for static and slow fading MIMO channels and for white and color noise throw computer simulations and observed that the proposed algorithm is robust to noise as the involved regulating parameter allows a trade-off to combat the effect of the additive noise at the expense of some estimation delay. REFERENCES [1] D. Tse and P. Viswanath, Fundamentals of wireless communication. Cambridge University Press,

17 TABLE I SUMMARY OF THE PROPOSED ALGORITHMS WHICH CAN OPERATE EITHER IN INITIALIZATION, TRAINING, BLIND OR Node X Choose a random unitary vector v 0 and send it through the channel H. SEMI-BLIND MODE. Initialization Node Y Receive R y,0 = Hv 0 + N y,0. û 1,0 R y,0 R y,0 ; ˆσ1,y,0 Ry,0. Receive R x,0 = H H û 1,0 + N x,0. Send û 1,0 through the channel H T. ˆv 1,0 R x,0 R x,0 ; ˆσ1,x,0 Rx,0. BSM Estimation Send T x,k = ˆv 1,k 1 d T x,k through H. Receive R y,k = HT x,k + N y,k. Receive R x,k = H T T y,k + N x,k. Send T y,k = û 1,k 1d T y,k through H T. Training (for known data d y,k ): use similar versions of(9) and (14) to update ˆv 1,k and ˆσ 1,x,k. Blind (for unknown data d y,k ): use similar version of (21) Training (for known data d x,k ): use (9) and (14) to update û 1,k and ˆσ 1,y,k. Blind (for unknown data d x,k): use(21) to update û 1,k. to update ˆv 1,k Semi-Blind (for known constellation of d y,k ): feed ˆd T y,k = ˆv 1,k 1 T R x,k ˆσ 1,x,k 1 to a hard detector to extract the unknown data d y,k and substitute it for d y,k in node X s version of (9) and (14) to update ˆv 1,k and ˆσ 1,x,k. Semi-Blind (for known constellation of d x,k ): feed ˆd T x,k = û H 1,k 1 R y,k ˆσ 1,y,k 1 to a hard detector to extract the unknown data d x,k and substitute it for d x,k in (9) and (14) to update û 1,k and ˆσ 1,y,k. [2] C. Chuah, D. Tse, J. Kahn, and R. Valenzuela, Capacity scaling in MIMO wireless systems under correlated fading, IEEE Transactions on Information Theory, vol. 48, no. 3, pp , [3] T. Dahl, N. Christophersen, and D. Gesbert, BIMA: Blind iterative MIMO algorithm, IEEE International Conference on Acoustics Speech and Signal Processing, vol. 3, [4], Blind MIMO eigenmode transmission based on the algebraic power method, IEEE Transactions on Signal Processing, vol. 52, no. 9, pp , [5] Y. Tang, B. Vucetic, and Y. Li, An iterative singular vectors estimation scheme for beamforming transmission and detection in MIMO systems, IEEE Communications Letters, vol. 9, no. 6, pp , [6] S. Gazor and M. Biguesh, A Cooperative Closed-Loop Channel SVD Training of Reciprocal MIMO Systems, submitted to IEEE Transactions on Wierless Communications, [7] D. Love, R. Heath Jr, and T. Strohmer, Grassmannian beamforming for multiple-input multiple-output wireless systems, IEEE Transactions on Information Theory, vol. 49, no. 10, pp , [8] D. Palomar, J. Cioffi, and M. Lagunas, Joint Tx-Rx beamforming design for multicarrier MIMO channels: A unified framework for convex optimization, IEEE Transactions on Signal Processing, vol. 51, no. 9, pp , [9] H. Sampath, P. Stoica, and A. Paulraj, Generalized linear precoder and decoder design for MIMO channelsusing the 17

18 input T x H + N y output R y σ 1 u 1 v H 1 input σ 2 u 2 v T 2 H + + x N y output R y σ r u r v H r Fig. 1. Block diagram of the equivalent SVD representation of the input-output model in (2) where r = Rank(H). weighted MMSE criterion, IEEE Transactions on Communications, vol. 49, no. 12, pp , [10] A. Scaglione, P. Stoica, S. Barbarossa, G. Giannakis, and H. Sampath, Optimal designs for space-time linear precoders and decoders, IEEE Transactions on Signal Processing, vol. 50, no. 5, pp , [11] M. Biguesh and A. Gershman, Training-based MIMO channel estimation: a study of estimator tradeoffs and optimal training signals, IEEE Transactions on Signal Processing, vol. 54, no. 3, pp , [12] K. Mukkavilli, A. Sabharwal, E. Erkip, B. Aazhang, Q. Inc, and C. San Diego, On beamforming with finite rate feedback in multiple-antenna systems, IEEE Transactions on Information Theory, vol. 49, no. 10, pp , [13] S. Zhou, Z. Wang, and G. Giannakis, Performance analysis for transmit-beamforming with finite-rate feedback, in Proc. Conf. on Inform. Sciences and Systems, [14] Z. Xu and M. Tsatsanis, Blind channel estimation for long code multiuser CDMA systems, IEEE Transactions on Signal processing, vol. 48, no. 4, pp , [15] M. Torlak and G. Xu, Blind multiuser channel estimation in asynchronous CDMA systems, IEEE Transactions on Signal Processing, vol. 45, no. 1, pp , [16] M. Biguesh, S. Gazor, and M. Shariat, Optimal Training Sequence for MIMO Wireless Systems in Colored Environments, [17] A. C. Tracy and H. Widom, Distribution functions for largest eigenvalues and their applications, Proc. International Congress of Mathematicians (Beijing, 2002), p ,

19 Average of estimates ˆσ1,k SNR = 10dB µ = 0 SNR = 0dB µ = 2 µ = 5 average of ˆσ 1,k using µ = 10 True σ 1 = Bias Iteration index Fig. 2. Average of the estimate of σ 1 for one random realization of H over runs (noises are generated independently in each run) plotted for two SNR values, 0dB (upper curves) and 10dB (lower curves). ( ) 2 ˆσ1,k σ1 σ1 Average of µ = 0 µ = 5 µ = 10 µ = 20 µ = 1.5k LS-OLT Average of φk µ = 0 µ = 5 µ = 10 µ = 20 µ = 1.5k LS-OLT Iteration index Iteration index (a) (b) Fig. 3. Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ between u 1 and its estimate û 1,k as defined in (23) versus iteration index for different values of µ in a low SNR regime of 0dB for a 4 4 MIMO system. 19

20 ( ) 2 ˆσ1,k σ1 σ1 Average of µ = 0 µ = 5 µ = 10 µ = 20 µ = 0.2k LS-OLT Average of φk µ = 0 µ = 5 µ = 10 µ = 20 µ = 0.2k LS-OLT Iteration index Iteration index (a) (b) Fig. 4. Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ between u 1 and its estimate û 1,k as defined in (23) versus iteration index for different values of µ in a high SNR regime of 20dB for a 4 4 MIMO system L = 100, M = N = 2 L = 100, M = N = 4 L = 1000, M = N = 2 L = 1000, M = N = 4 L = 100, M = N = 2 L = 100, M = N = 4 L = 1000, M = N = 2 L = 1000, M = N = ˆσ1 σ1 2 σ1 Average of Average of φ Iteration index Iteration index (a) (b) Fig. 5. Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ as defined in (23) versus the iteration index for µ = 100, SNR = 0dB and different frame lengths L (100 or 1000 symbols) and two different antenna sizes (2 2 and 4 4). 20

21 ( ) 2 ˆσ1,k σ1 σ1 Average of σ 1 σ 2 3 x µ = 0 µ = 1 µ = 5 µ = 10 RMS of φ1,k µ = 0 µ = 1 µ = 5 µ = σ 1 σ 2 φ 2,k φ 12,k Iteration index Iteration index (a) (b) Fig. 6. (a) Average of the squared normalized error in estimating σ 1 (b) the root mean square of the difference angle φ as defined in (24), versus the iteration index for a 4 4 MIMO system. The channels are slowly varying with maximum normalized Doppler shift of

22 LIST OF TABLES I Summary of the proposed algorithms which can operate either in initialization, training, blind or semi-blind mode LIST OF FIGURES 1 Block diagram of the equivalent SVD representation of the input-output model in (2) where r = Rank(H) Average of the estimate of σ 1 for one random realization of H over runs (noises are generated independently in each run) plotted for two SNR values, 0dB (upper curves) and 10dB (lower curves) Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ between u 1 and its estimate û 1,k as defined in (23) versus iteration index for different values of µ in a low SNR regime of 0dB for a 4 4 MIMO system Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ between u 1 and its estimate û 1,k as defined in (23) versus iteration index for different values of µ in a high SNR regime of 20dB for a 4 4 MIMO system Average of (a) the normalized square error in estimating σ 1 (b) the square of the difference angle φ as defined in (23) versus the iteration index for µ = 100, SNR = 0dB and different frame lengths L (100 or 1000 symbols) and two different antenna sizes (2 2 and 4 4) (a) Average of the squared normalized error in estimating σ 1 (b) the root mean square of the difference angle φ as defined in (24), versus the iteration index for a 4 4 MIMO system. The channels are slowly varying with maximum normalized Doppler shift of