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I have implemented a moving average filter (in Python) where I fill a list with values and average them. When new values arrive the oldest will be deleted.

Now I am wondering how much delay I get with this filtering.

I am sampling a sensor signal (voltage) with 1000 Hz and I have read often that the group delay (for my understanding that is the delay?) for a moving average filter for M values (window size) is (M-1)/2.

I can't really understand it because:

Lets say I have the a window size of 5 and following measured values arriving in my empty moving average filter:

  • 6x 1 Volt
  • 6x 3 Volt

    |1|1|1|1|1|1|3|3|3|3|3|3|

First I am waiting until my moving average filter is filled (until the fifth measured value).

Then I get these average values

|1|1|1|1|1| 1 |3|3|3|3|3|3|   1   [1,1,1,1,1]
|1|1|1|1|1|1| 3 |3|3|3|3|3|  7/5  [1,1,1,1,3] <- 3 Volt first time measured
|1|1|1|1|1|1|3| 3 |3|3|3|3|  9/5  [1,1,1,3,3]
|1|1|1|1|1|1|3|3| 3 |3|3|3| 11/5  [1,1,3,3,3]
|1|1|1|1|1|1|3|3|3| 3 |3|3| 13/5  [1,3,3,3,3]
|1|1|1|1|1|1|3|3|3|3| 3 |3| 15/5  [3,3,3,3,3] <- 3 Volt average

I would expect that my delay would be (M-1)/2 = (5-1)/2 = 2 measurements but I get my 3 volts in my moving average filter just after 4 more measurements (so delay M-1).

I would be very happy if someone could explain how I can determine my delay.

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"Group delay" isn't the delay between the change on the input and the first effect; it's the delay that a packet of oscillations of different frequencies experience.

In the case of a linear phase filter (and your moving average, its impulse response being symmetric, is linear phase) that is the "center" of the effects of a single impulse. So, it's the center between you seeing the first effect of your input change, and the last effect.

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If your window is realizable (causal, and it has to be as we cannot measure and use future values) you have to wait for whole window not the half of it.

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    $\begingroup$ but the group delay is half the window width. $\endgroup$ – robert bristow-johnson Aug 2 '17 at 19:14
  • $\begingroup$ Is my delay the full window size M or M-1? I think it's M-1. $\endgroup$ – ce_guy Aug 10 '17 at 14:27
  • $\begingroup$ It is M-1, thinking the extremes may help time to time. Consider your filter being a dirac delta function (A window of size 1), it causes no delay. $\endgroup$ – keoxkeox Aug 15 '17 at 21:01
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Here's another way to look at it:

group delay (what non-dsp people call "lag") is also the center of gravity (COG) of the weights of the average. For a Simple Moving Average (SMA), those weights look like a rectangle (length M, height 1/M), so the COG of the SMA is in the middle: that's why a 10-period SMA will have a 5 period lag. Another possible moving average would be the Weighted Moving Average: there the weights look like a right triangle (length M, height M/sum(1:M) ). The horizontal coordinate of the COG of a triangle is at 2/3rd the base, so the lag (group delay) of a WMA is 1/3 the length of the window it's calculated on.

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