I understand that interpolation is a linear transformation of a signal vector that combines interleaving the elements of the input vector with zeros followed by a filtering operation to remove any newly introduced signal components.
Call the matrix that does this interpolation $U$. It is $M\times{N}$ matrix that maps a $N$-element input vector onto a longer $M$-element output vector with $M>N$.
A similar matrix for decimation, $V$, can be formed. A $P\times{M}$ matrix with $P<M$ that maps an $M$-element input vector onto a shorter $P$-element output vector.
The composition of these interpolation and decimation matrices $VU$ has dimension $P\times{N}$.
How do I construct this $P\times{N}$ matrix explicitly, without needing to multiply these separate $U$ and $V$ matrices?
The advantage in avoiding the intermediate $M$-dimensional space is because, if I want to resample a $127$-length vector to a $128$-length vector, I have $N=127$, $P=128$, but $M=127\times128 = 16256$, which would require a huge amount of extra computation.
