We present new results on decomposing the transfer function t(z) of a linear, asymptotically stable, discrete-time SISO system as a difference t(z) = t(1)(z) - t(2)(z) of two positive linear systems. We extend the results of  to a class of transfer functions t(z) with multiple poles. One of the appearing positive systems is always 1-dimensional, while the other has dimension corresponding to the location and order of the poles of t(z). Recently, in , a universal approach was found, providing a decomposition for any asymptotically stable t(z). Our approach here gives lower dimensions than  in certain cases but, unfortunately, at present it can only be applied to a relatively small class of transfer functions, and it does not yield a general algorithm.