From the past few years, there has been an exponential rise in the communication sector, especially wireless continuously fueled by the demand of higher data rates by users. The concept of multiple input multiple output (MIMO) systems is used extensively in currently deployed technologies to provide necessary data rates. MIMO systems using spatial multiplexing achieves higher data rates without wasting frequency or time resources. This happens even when there is channel state information only at the receiver.It was observed that the channel knowledge at the transmitter further increases the capacity of a system. Perfect channel state information at the transmitter (CSIT) is generally not feasible in FDD systems, however, partial CSIT may be used to further increase the capacity of a system. This is commonly referred to as pre-coding. In pre-coding, codebooks which are designed off-line and stored, known to both the transmitter and receiver, are used.The main objective of this thesis pertains to the designing of codebooks and the effect of codebooks on the capacity of limited feedback MIMO systems. A codebook contains code words in the form of matrices and the design of these matrices reduces to a quantization problem on certain manifolds. The algorithm employed for quantizing these manifolds is the Lloyd algorithm, one of the most widely used. We make use of specific geometrical properties of these manifolds to design codebooks.Previously, much work has been done on designing codebooks by quantizing a Grassmann manifold. Here, an alternate quantization of a permutation invariant flag manifold is considered. We derive semi analytical distortion bounds to evaluate our codebook. Through this work, codebooks thus obtained are used and their effect on capacity is observed and analyzed.
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