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.

  • $\begingroup$ 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. $\endgroup$ – Marcus Müller Jan 18 at 11:32
  • $\begingroup$ @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. $\endgroup$ – user827822 Jan 18 at 20:01

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