Can I use IIR or FIR to get a sliding mean?

Question

My output signal shall be computed by (meta code):

o = 0  # or some arbitrary initial value
for i in input:
    o = (o * 99 + i) / 100
    print o

I call this a "sliding mean", but maybe another term is established for this. (If so, mentioning this could help me researching this better ;-)

Is there a way to achieve this using the a / b coefficient arrays of IIRs or FIRs? What would they look like?

I'm aiming at a solution in Python / scipy.signal.

Or is there another established way to achieve this which I can find in such libraries like numpy, scipy, pandas, etc.?

Answer 1

It corresponds to an exponentially weighted moving average (EWMA) filter with parameter $\alpha = 0.01$.

The difference equation can be expressed as:

$$y(n)-(1-\alpha)y(n-1)=\alpha x(n)$$ $$y(n)-0.99y(n-1)=0.01 x(n)$$

Thus it's an IIR filter. You can easily get the coefficients from the difference equation.

$$Y(z)-0.99z^{-1}Y(z)=0.01X(z)$$ $$\frac{Y(z)}{X(z)}=\frac{0.01}{1-0.99z^{-1}}$$

Therefore, a = [1 ; -0.99] and b = 0.01.

Answer 2

It is also called exponential smoothing due to the fact that contributions from past values decay at an exponential rate.

When it is applied to sinusoids, the math is interesting because sinusoidal are also exponential. The filter introduces a frequency dependent phase lag when applied to a pure tone.

A clever trick is to also perform an exponential smoothing of your data in the reverse direction and to average the results. The forward lag cancels the backward lag and you get a smoothed version of your original signal with all phases intact. The amplitude of a pure tone is attenuated by a frequency dependent formula. Therefore, you can apply analysis techniques, such as using a DFT, and get better results since the smoothing acts as a noise reduction technique.

I cover the math for this in a blog article called "Exponential Smoothing with a Wrinkle".

Hope this helps.

Ced

Answer 3

I think it is called an exponential averager. It is also an order-1 IIR.

Your coefficients are a0 = 1, a1 = -0.99, b0 = 0.01, b1 =0

Answer 4

Since you've called it "sliding mean", I presume that's the functionality you actually want – and yes, that's actually a FIR filter, we'd call it "moving average".

A naive implementation of that would be [ 1.0 / L ] * L, so, an L long moving average implemented as FIR taps:

 fir_taps = [ 1.0 / L ] * L
 signal_out = numpy.convolve(signal_in, fir_taps)

For small L, this is a very sufficient method; for lengthier averages, a version where you only add the newest and subtract the oldest value might perform better.

Notice that both your IIR form (which we call EWMA, or single-pole IIR integrator) are low-pass filters. If you actually need a behaviour that suppresses high-speed signal components, then an averaging filter is not the best you can do!