Reduction of recursive filters to representations by sparse matrices

A recursive filter as a part of a recursive convolutional code is of practical importance in composite interleaved code circuits. We consider a matrix description of recursive filters in the time domain over the finite field $\mathbb F_2$. We analyze and formalize the reduction of matrices describing recursive filters (with puncturing) to sparse matrices of a special form. We mainly address the analysis of binary sequences of recursive filters with puncturing every second bit. We describe the application of the obtained sparse matrices to finding punctured transfer functions for such filters. We propose an approach to the minimal circuit realization of the punctured transfer functions. We give examples of circuit realizations of punctured turbo codes as duo-binary turbo codes.