# What is the definition of a "complementary filter"?

I'm writing my MSE thesis report on an inertial navigation problem I've been working on. In my work, I use a complementary filter to track the orientation of a device. Now, since my fellow students are likely not familiar with complementary filters or Kalman filters, I wish to give a brief description of what a "complementary filter" is, in general.

A definition, if you will.

So, what defines a complementary filter? Is it any filter on the form z = a * x + (1 - a) * y, where x and y are separate measurements of a single quantity?

It's fine to define it in terms of Kalman filters if that's appropriate.

• If you're using it, and can't define it, you've got pretty big problems for writing a MSE thesis. Aug 12, 2015 at 16:36
• I know how the filter I'm using works, so that's not a problem. I also know it's a complementary filter (I didn't develop the filter myself, it's from Mahony et al. 2008). I just want to describe in general what a complementary filter is, but I am unable to find any general definition (if there is any) online. Aug 12, 2015 at 16:40
• When you talk of filter, how can it be defined in terms of two constant weighing coefficient? Instead it should be defined in terms of first or second order transfer functions (as per normal literature on filters). Can anyone explain how exactly filtering occurs in case of Complementary filter? Thanks
– xyz
Oct 6, 2016 at 12:17

Usually, a complementary filter (like a complementary function) complements another filter. The two filters that are complementary to each other add to one. Or, at least, add to an all-pass filter (which is what Linkwitz-Riley crossovers do.

so either

$$H(f) + G(f) = 1$$

or

$$H(f) + G(f) = A(f)$$

for $H(f)$ and $G(f)$ being complements of each other and $|A(f)|=1$ is an APF.

• Oh, so in the case of Accelerometer and Gyroscope fusion, the term complementary refers to the fact that the low-pass filter on the accelerometer and the high-pass filter on the gyroscope complement eachother? I also heard a definition like "it's a steady state Kalman filter" - would that also be correct? Aug 13, 2015 at 7:13
• i never knew that Kalman filters had much to do with complementary filters. (wavelets and filterbanks surely do.) but i really just don't know. Aug 14, 2015 at 13:45

In support of robert's answer, here's a paper that explains it in a similar way.