I. Introduction
To Fully exploit the advantages of multiple-input multiple-output (MIMO) communications, the accurate knowledge of channel state information at transmitter (CSIT) is crucial. In time-division duplex (TDD) systems, CSIT is obtained by uplink channel estimation based on the uplink and downlink channel reciprocity. On the other hand, in frequency division duplexing (FDD) systems, uplink and downlink channels are usually independent of each other [1]–[4] and thus feedback on CSI from receivers is required. A popular feedback technique is codebook based approach where an index of a quantized CSI in a pre-designed codebook is fed back to a transmitter. In the Long Term Evolution (LTE) of the Third Generation Partnership Project (3GPP) standard, each receiver reports rank indicator (RI) which corresponds to the number of independent data streams, precoding matrix indicator (PMI) from the pre-designed codebook, and channel quality indicator (CQI) which represents the channel quality correponding to the modulation and coding scheme [5], [6]. On the other hand, explicit feedback schemes which directly reports a quantized CSI allow more flexibility of transmission and the reception methods and achieve a higher scheduling gain but they require larger overhead than implicit feedback [2], [6]. With the codebook based feedback, the accuracy of CSI depends on the codebook structure and the amount of allowed feedback bits. On this account, one of the main research thrusts of MIMO communications has been on efficient codebook design with limited feedback for various channels.