# Mathematical Proof of Complementary Filter？

Having found some unofficial sources on Complementary Filter (Thousand Thoughts Sensor Fusion and The Balance Filter by Shane Colton), I wish to work out its rigorous mathematical proof.

The Complementary Filter, $$y=\alpha \times y+(1-\alpha) \times x$$ where $\alpha$ is the filter parameter, usually chosen to be ~0.98, is named as such, because effectively the filter highpasses $y$ and lowpasses $x$.

With this setup ($\alpha=0.98$), man claims that the filter can filter out the low-frequency part of $y$ and meanwhile the high-frequency part of $x$.

What is the mathematical proof here?

There is no rigorous proof so far available. All the sources simply take it as granted and validates it with experiment data, which is unacceptable in scientific validation.

• The definition of complementary in the original article is sloppy at best: One quote on Kalman filters is "I have no idea how it works. It’s mathematically complex, requiring some knowledge of linear algebra". Before you can proof something you need a crispy definition. Best I can tell is that "complementary" refers to the use of two different types of sensors which are combined in some way and it has little to do with the actual filters – Hilmar Sep 4 '13 at 11:34
• In addition to the good answer below, this type of filter is often called a leaky integrator. Previous questions that talk about it are here, here, and here. – Jason R Sep 4 '13 at 14:50
• @Hilmar thanks for the comment! It helps! – Sibbs Gambling Sep 11 '13 at 15:26
• @JasonR THank you for the resources pointed out! They really help! :) – Sibbs Gambling Sep 11 '13 at 15:27