Optimal Recursive Spatial Multiplexing Strategies for Slowly Time-Varying MIMO Channels T. Edlich, T. Hunziker, I. A. Shah, D. Dahlhaus Communications Laboratory, University of Kassel Wilhelmsh¨ oher Allee 73, Kassel, 34121 Germany E-mail: edlich@uni-kassel.de Abstract—We consider a simple closed-loop spatial multiplex- ing architecture, which can be incorporated into an outer system with off-the-shelf single-input single-output encoders/decoders. Instead of relying on transmitter-side channel state informa- tion for matching the layers of the multidimensional signal to the eigenmodes of the multiple-input multiple-output channel through a precoding, the eigenmodes with high attenuation are identified in the receiver and a retransmission of the affected signal parts requested through a feedback channel. The requested backup facilitates a linear array signal reconstruction subject to limited noise amplification. Since the backup is always embedded in a subsequent signal frame, the multiplexing becomes a recur- sive procedure. We propose appropriate transmit antenna selec- tion policies for limiting the number of recursions and thereby the memory requirement and incurred delay. Optimal policies are obtained using the theory of Markov decision processes. Moreover, the use of a simple unsupervised learning method lets the system adapt to unknown channel conditions while always guaranteeing a target signal-to-noise ratio at the decoder input. I. I NTRODUCTION Multiple-input multiple-output (MIMO) channels offer great capacities in theory, but their realization by adequate MIMO channel codes, such as space-time trellis codes, is often impractical in view of the high complexity. In order to pro- vide high data rates, simple architectures are needed which rely on off-the-shelf single-input single-output (SISO) en- coders/decoders and linear pre/post-processors. In the presence of channel state information (CSI) at the transmitter end, proper precoding decomposes the MIMO channel into inde- pendent SISO channels [1]. However, providing the transmitter with accurate CSI may be difficult, especially in environments with time-varying channels. Without precoding, a linear recon- struction of the array signal at the receiver end, based on the zero forcing (ZF) or minimum mean-squared error (MMSE) criteria, boosts additive noise over the parallel signals in cases with badly conditioned channel matrices. Interference cancellation as part of the Vertical Bell Lab Layered Space- Time (V-BLAST) proposal [2] limits noise amplification, but the successive layer-wise decoding increases the system complexity and brings the danger of error propagation. An alternative spatial multiplexing architecture, where ex- cessive noise amplification arising from ”weak” eigenmodes of a channel is evaded by means of a retransmission involving the critical signal subspace, has been proposed in [3], [4]. To show the key idea, suppose the transmitted signal was distorted in a narrow-band MIMO channel represented by the (N×M )- matrix H ℓ (with N ≥ M ). A ZF in the receiver would amplify the average noise power by a factor tr((H † ℓ H ℓ ) -1 )/M [5], where (·) † denotes conjugate transposition and tr(·) the trace of a square matrix. The presence of a backup in the form of orthogonal projections of the transmitted vector signals onto an m ℓ -dimensional subspace with m ℓ ≤ M , spanned by the orthonormal columns of an (M×m ℓ )-matrix Q ℓ , results in the augmented channel matrix  H ℓ =  H ℓ √ αQ † ℓ  (1) with α a positive factor as discussed below. Proper choice of Q ℓ limits the eigenvalue spread of (  H † ℓ  H ℓ ), thereby mitigating noise amplification by the ZF. The matrix Q ℓ can be computed at the receiver end and sent through a feedback channel to the transmitter, which embeds the backup in the array signal transmitted in a subsequent time slot. As the reception of the latter may again require a backup, this approach results in a recursive spatial multiplexing (RSM) procedure. The ergodic performance of such a closed-loop architecture with unconstrained recursion is investigated in [4]. In this paper we employ RSM to ensure the compliance with a given target signal-to-noise ratio (SNR) at the SISO decoder input as explained in Sect.II. We derive optimal transmit antenna selection policies that meet constraints imposed on the recursion, with the purpose of limiting latency and memory re- quirements, using the framework of Markov decision processes in Sect. III. Moreover, we discuss a simple learning method to make the architecture self-adaptable in Sect. IV. Finally, we present numerical results in Sect. V and draw conclusions in Sect. VI. II. RECURSIVE SPATIAL MULTIPLEXING At the transmitter side of our architecture in Fig. 1, a demultiplexer transforms the serial signal stream from a SISO encoder into array signal frames to be sent successively over a MIMO channel. Every frame comprises a certain number of vector signals. The transmitter may use only t ℓ out of the M transmit antennas for the ℓth frame transmission, and allocate the t ℓ dimensions of the vector signals as follows: t ℓ -m ℓ-1 dimensions (i.e., layers) are provided for serial/parallel (S/P) converted signals from the SISO encoder, while m ℓ-1 dimen- sions are dedicated to the retransmission of critical signal parts of the previous frame, serving as a backup in the 978-1-61284-231-8/11/$26.00 ©2011 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings