Interpolating FIR filters

This page offers a brief explanation of interpolation FIR filters.

Example

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

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

This filter gives: $$\begin{gathered} [1 \\ 1 \\ 2 \\ 2 \\ 3 \\ 3 \\ 4 \\ 4 \\ 5 \\ 5 \\ 4 \\ 4 \\ 3 \\ 3 \\ 2 \\ 2 \\ 1 \\ 1] \end{gathered}$$. This is a hold function.

Filter 2: $$\begin{gathered} [0.5 \\ 1 \\ 0.5] \end{gathered}$$ (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: $$\begin{gathered} [.5 \\ 1.0 \\ 1.5 \\ 2.0 \\ 2.5 \\ 3.0 \\ 3.5 \\ 4.0 \\ 4.5 \\ 5.0 \\ 4.5 \\ 4.0 \\ 3.5 \\ 3.0 \\ 2.5 \\ 2.0 \\ 1.5 \\ 1.0 \\ 0.5] \end{gathered}$$. 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: $(L_h{L}_x-p{L}_x)/q$ 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.