Abstract-This paper investigates quantization methods for feeding back the channel information through a low-rate feedback channel in the context of multiple-input single-output (MISO) systems. We propose a new quantizer design criterion for capacity maximization and develop the corresponding iterative vector quantization (VQ) design algorithm. The criterion is based on maximizing the mean-squared weighted inner product (MSwIP) between the optimum and the quantized beamforming vector. The performance of systems with quantized beamforming is analyzed for the independent fading case. This requires finding the density of the squared inner product between the optimum and the quantized beamforming vector, which is obtained by considering a simple approximation of the quantization cell. The approximate density function is used to lower-bound the capacity loss due to quantization, the outage probability, and the bit error probability. The resulting expressions provide insight into the dependence of the performance of transmit beamforming MISO systems on the number of transmit antennas and feedback rate. Computer simulations support the analytical results and indicate that the lower bounds are quite tight.Index Terms-Bit error probability, channel capacity, channel state information, multiple antennas, transmit beamforming, outage probability, vector quantization (VQ).

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In this paper, we propose two efficient lowcomplexity quantization methods for multiple-input multipleoutput (MIMO) systems with finite-rate feedback based on proper parameterization of the information to be fed back followed by quantization in the new parameter domain. For a MIMO channel which has multiple orthonormal vectors as channel spatial information, we exploit the geometrical structure of orthonormality while quantizing the spatial information matrix. The parameterization is of two types: one is in terms of a set of unit-norm vectors with different lengths, and the other is in terms of a minimal number of scalar parameters. These parameters are shown to be independent for the i.i.d. flatfading Rayleigh channel, facilitating efficient quantization. In the first scheme, each of the unit-norm vectors is independently quantized with a finite number of bits using an optimal vector quantization (VQ) technique. Bit allocation is needed between the vectors, and the optimum bit allocation depends on the operating SNR of the system. In the second scheme, the scalar parameters are quantized. In slowly time-varying channels, the scalar parameters are also found to be smoothly changing over time, leading to the development of a simple quantization and feedback method using adaptive delta modulation. The results show that the proposed feedback scheme has a channel tracking feature and achieves a capacity very close to perfect feedback with a reasonable feedback rate.