# Composition of interpolation and decimation matrices

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 $$P that maps an $$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.

• ah, you're trying to do rational resampling. Fret not, rational resamplers use polyphase "magic" to work at the lower rate, so that you don't have to upsample by 127·128 first. Jan 18, 2020 at 11:32
• @MarcusMüller can the "magic" be explained as an optimization of the matrix composition? I have tried to simplify the matrix multiplications involved, but it becomes quite complicated. Jan 18, 2020 at 20:01