Low-Pass Filters

Low-Pass Filters

Low-pass filters are basically an upgrade from the Averaging filters. If you are not familiar with averaging filter, may be it is a good idea to go through them first.

Usually, the noise is in high frequency bands. The low-pass filter is designed to pass-through low frequency signals, while blocking high frequency signals, hence the name “low-pass filter”.

The moving average filter has the same weight for all measurements. However, in reality, the most recent measurements have more to say about the current state. The low-pass filter overcome this by using a heavier weight on the most recent data.

The first order low-pass filter can be written as follows with 0 < α < 1.

x_k = α · x_{k - 1} + (1 - α) · x_k

This is a bit similar to average filter, but α can be set arbitrarily. Higher α (closer to 1) means less noise, but lags, while lower α (closer to 0) means less lag, but noisy.

The first order low-pass filter in Laplace domain is:

Y(s) / X(s) = K · (1 / (𝜏s + 1))


Examples


Low-pass filter
Comparison between the low-pass filter (α=0.75) and the moving average filter: the low-pass filter (n=5) has a lesser lag compared to the moving average filter, bit more susceptible to the noise.



Low-pass filter comparison
Comparison between low-pass filter with different α values: high α value (green line with α=0.9) smoothens out the data more but has a lag, low α value (red line with α=0.4) has a lesser lag but affected more by the noise.


See Also

References

  1. Kim, P. (2011). Kalman Filter for Beginners: with MATLAB Examples. CreateSpace Independent Publishing Platform

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