# Time derivative of signal - effect on noise distribution

I have an angular velocity measurement that has a certain amount of ripple that yields an histogram shaped like a gaussian curve. I quantify that ripple using a standard deviation. I am interested in knowing more about the angle ripple.

What would be the relationship between the two ripples?

My guess would be that the derivative (from angle to angular velocity) push noises at higher frequency and that the gaussian curve will get a little wider (bigger standard deviation). How can I validate that and what mathematics are involved here?

## 1 Answer

Let's call $$a[n]$$ the samples of the angular measurement. Then, an estimate of the angular velocity (in angle/sec) is given by discrete differences, i.e., $$v[n] = (a[n]-a[n-1])/t_0$$, where $$t_0$$ is the sampling time.

Let's assume a signal pluse noise process so that $$a[n] = a_0[n] + w[n]$$. Inserting, we obtain $$v[n] = v_0[n] + \bar{w}[n]$$, where $$\bar{w}[n] = (w[n]-w[n-1])/t_0$$. Certainly, if $$w[n]$$ is zero mean, so is $$\bar{w}[n]$$ You can now compute the variance of your effective noise samples $$\bar{w}[n]$$, namely $$\mathbb{E}\{\bar{w}[n]^2\} = \frac{1}{t_0^2}\left(\mathbb{E}\{\bar{w}[n]^2\} + \mathbb{E}\{\bar{w}[n-1]^2\} -2 \mathbb{E}\{\bar{w}[n]\bar{w}[n-1]\}\right).$$ If your noise samples $$w[n]$$ are i.i.d. and have constant variance $$\sigma_w^2$$, this leads to $$\mathbb{E}\{\bar{w}[n]^2\} = \frac{2\sigma_w^2}{t_0^2}$$. So there is a doubling of variance, but also some kind of rescaling. This depends of course on the unit of your velocity (whether it's angle/second or angle/sample, in the last case your variance is just $$2\sigma_w^2$$). NB: If they are correlated, this changes to $$2\sigma_w^2(1-\rho)$$, where $$\rho$$ is the correlation coefficient.

All this is true if the velocity is simply estimated from differences of adjacent samples. There are more elaborate schemes available, which would have an effect on the noise variance.

• Interesting, thank you. Actual measurements seems to give a ratio of variances somewhat near 0.740 in angle per sample. I would have expected a little closer to 1/2 – Pier-Yves Lessard Jun 21 '19 at 15:47
• This could be an indication that your noise process on the angles is not i.i.d., but there is some amount of temporal correlation (with an effective correlation coefficient of about 0.32). – Florian Jun 21 '19 at 15:59
• I think you are right about the noise not being iid. I can see an oscillating pattern on it that looks like a beating between the sampling clock of the adc and the sampling clock of the dac generating the angle signal. I learned something today! – Pier-Yves Lessard Jun 21 '19 at 20:37