Time-frequency transfer function and realization algorithm for discrete periodic linear systems

A realization of a linear periodically time-varying system of order n with m inputs, p outputs and period N is given by x(k+1)=A_[/k]x(k)+Bsub [/k]u(k), y(k)=C_[/k]x(k)+Dsub [/k]u(k), Where [k] = k mod N. The parameters of the realization are denoted by ((A_k, B_k , C_k, D_k))N. The construction of a state variable representation, from a given pulse response, is outlined. Time-frequency analysis/synthesis transfer function are defined by means of proper z-transforms of the pulse response h(l,r), and their N-phase decomposition is introduced. The realisation problem is reduced to solving the matrix equation H_e(z)=C_e(zI-A_e )^-1B_e+D_e where H_e(z) is the extended matrix transfer function constructed from h(l,r) through the N-phase decomposition F(l,z) or G(z,r)