Cyclic LTI systems in digital signal processing

Cyclic signal processing refers to situations where all the time indices ar e interpreted module some integer L, In such cases, the frequency domain is defined as a uniform discrete grid (as in L-point DFT), This offers more f reedom in theoretical as well as design aspects. While circular convolution has been the centerpiece of many algorithms in signal processing for decad es, such freedom, especially from the viewpoint of linear system theory, ha s not been studied in the past. In this paper, we introduce the fundamental s of cyclic multirate systems and filter banks, presenting several importan t differences between the cyclic and noncyclic cases. Cyclic systems with a llpass and paraunitary properties are studied. The paraunitary interpolatio n problem is introduced, and it is shown that the interpolation does not al ways succeed, State-space descriptions of cyclic LTI systems are introduced , and the notions of reachability and observability of state equations are revisited. It is shown that unlike in traditional Linear systems, these two notions are not related to the system minimality in a simple way. Througho ut the paper, a number of open problems are pointed out from the perspectiv e of the signal processor as well as the system theorist.