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A Difference of WMA & EMA on a sinusoid becomes superposed?

  1. Jul 1, 2017 #1
    This is about signal processing, moving averages & superposed / standing waves. This is an online system: causal (univariate) time series analysis.

    Suppose you have a sinusoid of period n (i.e. n=40, so its frequency is 0.025). If you calculate a "weighted moving average" (WMA) on this sinusoid with a lookback-window equal to 1/4th the period of the sinusoid (i.e. 40/4 = 10), and from this WMA, you subtract an "exponential moving average" (EMA) with an alpha equal to 1 divided by 1/4th the period of the sinusoid (i.e. 1/10 = EMA alpha of 0.1), the resulting difference is perfectly "in phase" with the sinusoid. But this is only true if you do it for the n/4 setting. In fact, the result looks a lot like a superposed, standing wave: like in this image: ex.png

    This cannot be a coincidence. There must be some kind of "deeper" reason that I fail to understand: why is this only true, if you pick n/4 for the WMA, and its equivalent for the EMA, and if you compare this exact difference (WMA-EMA) with the input wave signal, they are exactly in-phase: they turn at the same moment, and reach 0 at the same moment in time. There is no phase difference. Why?

    Could it be related to the fact that a sine wave is made up of 4 identical pieces? (mirrored and inverted)?
    Or something about the sine & cosine and the unit circle?
    Or is it related to the lag / group delay of the WMA and EMA filters?
    Or to the fact that this difference (WMA-EMA) has negative filter weights (not common in causal time series analysis)?

    Why o why is this so.. I just know it's not a coincidence, there is a real explanation to this. Please help me because this intrigues me :) Thanks!
     
  2. jcsd
  3. Jul 1, 2017 #2

    mfb

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    Did you write down the formulas and see if it works out?
     
  4. Jul 2, 2017 #3
    I wrote the code for the image posted above myself in r. So yes, I did write down the formulas.
     
  5. Jul 2, 2017 #4

    mfb

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    That's not what I meant. Plug in a sine, subtract them, then simplify the expression, and see if you get a sine where you can calculate the phase.
     
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