Interpolating FIR filters

This page offers a brief explanation of interpolation FIR filters.

Example

Assume we start with the sample $\ [1\ 2\ 3\ 4\ 3\ 2\ 1]$. Padding with zeros gives: $\ [1\ 0\ 2\ 0\ 3\ 0\ 4\ 0\ 5\ 0\ 3\ 0\ 2\ 0\ 1]$. Let's apply 2 filters.

Filter 1: $\ [1\ 1]$ (also written as $\ y(kT)=1.0\cdot x(kT)+1.0\cdot x(k-1)T$).

This filter gives: $\ [1\ 1\ 2\ 2\ 3\ 3\ 4\ 4\ 5\ 5\ 4\ 4\ 3\ 3\ 2\ 2\ 1\ 1]$. This is a hold function.

Filter 2: $\ [0.5\ 1\ 0.5]$ (also written as $\ y(kT)=0.5\cdot x(kT)+1.0\cdot x(k-1)T+0.5\cdot x(k-2)T$

This filter gives: $\ [.5\ 1.0\ 1.5\ 2.0\ 2.5\ 3.0\ 3.5\ 4.0\ 4.5\ 5.0\ 4.5\ 4.0\ 3.5\ 3.0\ 2.5\ 2.0\ 1.5\ 1.0\ 0.5]$. This is a linear interpolater.

Multiply/add Operations

I had a lot of trouble finding generic information about the number of multiply/add operations used in an interpolation FIR filter. I did find formula for the number of multiply/add operation used by the MATLAB function upfirdn, which upsamples, applies an FIR filter, and then downsamples. It is: $\ (L_{h}L_{x}-pL_{x})/q$ where $\ L_{h}$ and $\ L_{x}$ are the lengths of $\ h[n]$(the impulse response of the FIR filter) and $\ x[n]$(the original signal), respectively.