Time constant and exponential forgetting factor in DSP
Post date: Aug 28, 2014 1:59:59 AM
In many adaptive DSP problems, one often encounters the following equation
where
is often set heuristically, says to be 0.98. Natural questions arise: What is the mathematical foundation behind it? And how could be set in a principled way?
To address the first question, it is worth noting that the above discrete equation actually has an analog version in the continuous domain [1].
The function here is the solution to a differential equations with the step function with amplitude as its forced input. It shows how a value, which is already governed by an
exponential decaying law with time constant , changes over time from the initial value if a step input is applied.
Once the first question is answered, the answer to the second question is quite straightforward, simply setting
will do the trick. Here is the desired time constant in samples, and is the time between new input, again in samples.
[1] https://en.wikipedia.org/wiki/Time_constant#Step_response_with_arbitrary_initial_conditions