Alias-Component Matrices of Multirate Systems

We consider a multirate system, which is a generalization of linear time-invariant systems. Such a system is invariant to a certain shift in the input sequence. In particular, assume that
p
and
q
are coprime. A multirate system with the property that a delay of
mq
samples in its input sequence results in a delay of
mp
samples in its output sequence is called an (
mp
,
mq
)-periodic system. This multirate system can be obtained by cascading an upsampler, followed by a linear periodically time-varying (LPTV) kernel system, then followed by a downsampler. Here, we study the alias-component matrices of multirate systems. We show that they can be obtained from the alias-component matrices of their LPTV kernels by some row and column additions. An example shows the use of the method to design rate changers for a specified frequency band swap.