In order to know if my signal is increasing or decreasing, I'm using the discrete derivative $y[n] = x[n] - x[n-1]$ or a smoothed version of it (for example Exponential Weight Moving Average of $y[n]$ that gives a "smooth" version of the derivative).

Now to make further analysis, I also want to consider the discrete acceleration:

$$z[n] = x[n] - 2 x[n-1] + x[n-2]$$

But as the signal is sometimes a bit noisy, I wanted to study a smoothed version of this acceleration. I thought about :

  • Standard Moving average of $z$ : $SMA(z[n], 10)$. Easy computation shows that $SMA(z[n], 10) = \frac{1}{10} (x[n]-x[n-1]-x[n-10]+x[n-11])$

    It's not so good because it uses too few values of x[k]...

  • Exponential Moving average of $z$ : $EMA(z[n], 10)$. Computations shows that: $EMA(z[n], 10) \approx 0.09 (x[n] - x[n-1])$ and this is proportionnal to ... derivative !

    Indeed, all the next coefficients are negligible $\ll 10^{-3}$.

    This is strange: I expected the the Exponential Moving Average of the acceleration to be smoother but it turns out that it nearly uses only 2 values : $x[n]$ and $x[n-1]$. (See below the impulse reponse of EMA of acceleration).


How to get a "smoothed" version of discrete acceleration, as $A[n] = a_0 x[n] + a_1 x[n-1] + ... + a_k x[n-k]$ ?

Remark: Impulse reponse of EMA of discrete acceleration:

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  • $\begingroup$ What EMA with what $\alpha$'s did you use to smooth the derivative ? $\endgroup$
    – Gilles
    Jan 12, 2016 at 11:31
  • 1
    $\begingroup$ You should have a look into Savitzky-Golay filters. $\endgroup$
    – Matt L.
    Jan 12, 2016 at 11:37
  • $\begingroup$ @Gilles : I used $\alpha = 1/(1+10)=1/11$ to smooth the derivative. When using same $\alpha$ with EMA to smooth the acceleration, I get this weird result : smoothed acceleration $\approx c (x[n] - x[n-1])$ ! $\endgroup$
    – Basj
    Jan 12, 2016 at 11:58
  • $\begingroup$ @MattL. : I didn't know. Why specifically these filters, and not another? (Kalman) etc. Can you develop into an answer? $\endgroup$
    – Basj
    Jan 12, 2016 at 12:00
  • $\begingroup$ Savitzky-Golay filters work very well for smoothing data, or smoothing derivatives of data. Have a look at this answer (this is not about derivatives though). This question is very similar to yours. $\endgroup$
    – Matt L.
    Jan 12, 2016 at 13:17


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