In the last decade, there is an increasing use of widely linear processing (WLP) in various signal processing fields. WLP is applied in orthogonal complex space time block code (OCSTBC), binary phase shift keying (BPSK), code division multiple access (CDMA) with offset quadrature phase shift keying (OQPSK) modulations to name a few. It is noticed that in these systems, its structure is a time varying linear system which can be represented by a filter bank (FB).

The goal of this thesis is to bridge the theory for oversampling, WLP, precoding in the framework of FB. It is well known that oversampling, WLP, precoding can improve the performance of equalization. Through representing these combined systems as a FB, the design of an equalizer is similar to the design of the synthesis polyphase matrix.

Moreover, the condition of the equalization can also be analyzed using the Smith Normal Form Decomposition for the resulting analysis polyphase matrix. In terms of FB theory, the condition of equalization is similar to perfect reconstruction (PR) FB. We analyze various communications systems which can be represented as a FB to see whether its PR can be achieved using finite impulse response (FIR) synthesis polyphase matrix. With the existence of FIR solution, it is expected that the equalization performance can be improved through the use of MMSE WLE. We also characterized the performance gap between the WLE and the linear equalizer (LE) and studied some channel conditions when the performance gain is vanished. Computer simulations are provided to demonstrate the superior performance of the proposed oversampled MMSE WLE in various channel conditions.