How to determine the parameter alpha of a Complementary Filter?

问题

I know that the Complementary Filter has the functions of both LPF and HPF. But I think my understanding on the principal behind it is still unclear.

I am quite new on digital signal processing, and maybe some very fundamental explanations will help a lot.

Say I have a Complementary Filter as follows:

y = a * y + (1 - a) * x

Then my parameter a may be calculated by

a = time_constant / (time_constant + sample_period),

where the sample_period is simply the reciprocal of the sampling_frequency.

The time_constant seems to be at my own choice.

My Questions:

1. What is the theory behind this calculation?
2. How do we choose the time_constant properly?

_{Note: I also posted this question on robotics, as the answers there are likely to be slightly different in emphasis.}

回答1:

What is the theory behind this calculation?

For a human readable introduction I would recommend:
The Balance Filter: A Simple Solution for Integrating Accelerometer and Gyroscope Measurements for a Balancing Platform.

How do we choose the time_constant properly?

Intuitively, the time_constant is the boundary between trusting the high-pass and the low-pass filter part. For shorter times than the time_constant you trust the high-pass filter part more, and for longer times you trust the low-pass part more.

Usually, you have some experience with the physical process you are dealing with and you can guess at least the order of magnitude of the time_constant. For example, if you are fusing accelerometer and gyro data (which I assume you do based on your other question), something between 0.5-1 second is a reasonable first guess. Then, you start tuning your filter by analyzing its performance on actual data and adjusting a accordingly.

来源：https://stackoverflow.com/questions/18095785/how-to-determine-the-parameter-alpha-of-a-complementary-filter