Closed-Form Expressions for Channel Shortening Receivers Using $A\ Priori$ Information

Channel shortening has been studied in the context of intersymbol interference and multi-in multi-out channels as a means to compute a posteriori probabilities with a BCJR algorithm at a reduced computational complexity. This is done by considering an approximate channel response of reduced length. In a turbo receiver, soft a priori information can be linearly combined with the received sequence to form a new input to the BCJR trellis-based processing. In this letter, we provide closed-form expressions for the channel shortening filters using a generalized mutual information objective function. The proposed receiver allows a complexity reduction with respect to numerical optimization approaches which may also present stability, precision, and convergence issues.