Interpolating FIR filters
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This page offers a brief explanation of interpolation FIR filters.
Example
Assume we start with the sample . Padding with zeros gives: . Let's apply 2 filters.
![{\displaystyle \ [1\ 2\ 3\ 4\ 3\ 2\ 1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c247a1674769d5f62d1a4898b5a0f9c15766e4f7)
![{\displaystyle \ [1\ 0\ 2\ 0\ 3\ 0\ 4\ 0\ 5\ 0\ 3\ 0\ 2\ 0\ 1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646ca3ae635fcbf735b35a68a5200537e01b8d38)
Filter 1: (also written as ).
![{\displaystyle \ [1\ 1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42f57e332af08133eb0597cf5aca06ef29caf12e)
This filter gives: . This is a hold function.
![{\displaystyle \ [1\ 1\ 2\ 2\ 3\ 3\ 4\ 4\ 5\ 5\ 4\ 4\ 3\ 3\ 2\ 2\ 1\ 1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76844ebac00f0689ff5bf08dd77f4b05fac57cc2)
Filter 2: (also written as
![{\displaystyle \ [0.5\ 1\ 0.5]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56f33be878d1a7d3549116043c5f66c1ece936ad)
This filter gives: . This is a linear interpolater.
![{\displaystyle \ [.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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82d055c53a3f172924b172c49ecb63d84a26c3fe)
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: where and are the lengths of (the impulse response of the FIR filter) and (the original signal), respectively.
![{\displaystyle \ h[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/510ff22bd387c23a8194c0b2cee7d630f48d2659)
![{\displaystyle \ x[n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98ced457fbdfb2c87a5cb5375b403a1c1c8212f6)