Step size in numerical differentiation

Question

I get position information and a corresponding timestamp from a motion tracking system (for a rigid body) at 120 Hz. The position is in sub-millimeter precision, but I'm not too sure about the time stamp, I can get it as floating point number in seconds from the motion tracking software. To get the velocity, I use the difference between two samples divided by the $\Delta t$ of the two samples:

$\dot{\mathbf{x}} = \dfrac{\mathbf{x}[k] - \mathbf{x}[k-1]}{t[k]-t[k-1]}$.

The result looks fine, but a bit noisy at times. A realized that I get much smoother results when I choose the differentiation step $h$ larger, e.g. $h=10$:

$\dot{\mathbf{x}} = \dfrac{\mathbf{x}[k] - \mathbf{x}[k-h]}{t[k]-t[k-h]}$.

On the other hand, peaks in the velocity signal begin to fade if I choose $h$ too large. Unfortunately, I didn't figure out why I get a smoother signal with a bigger step $h$. Does someone have a hint? Is there a general rule which differentiation step size is optimal with respect to smoothness vs. "accuracy"?

This is a sample plot of one velocity component (blue: step size 1, red: step size 10):

Answer 1

This answer valid only if $\Delta{t} = \mathbf{t}[k] - \mathbf{t}[k-1]$ is a constant. Then you can rewrite your equation as: $$\dot{\mathbf{x}} = \dfrac{\mathbf{x}[k] - \mathbf{x}[k-1]}{\Delta{t}}$$

Consider:

$$ \dot{\mathbf{x}}_l = \dfrac{1}{h}\sum_{i=1}^{h}\dot{\mathbf{x}_i} = \dfrac{(\mathbf{x}[k] - \mathbf{x}[k-1])+(\mathbf{x}[k-1] - \mathbf{x}[k-2])+\dotsb+(\mathbf{x}[k-h+1] - \mathbf{x}[k-h])}{h\Delta{t}} = \bigg(\dfrac{\mathbf{x}[k] - \mathbf{x}[k-h]}{h\Delta{t}}\bigg) $$

$$ h\Delta{t} = \mathbf{t}[k] - \mathbf{t}[k-h] $$

Here $\dot{\mathbf{x}}_i$ is the $i^{th}$ sample of the reading and passing it through a moving average filter (which is a low pass filter) you can obtain $\dot{\mathbf{x}_l}$. So $\dot{\mathbf{x}_l}$ is smooth as it is a low pass signal. When you increase the value of $h$ you can minimize the bandwidth of $\dot{\mathbf{x}_l}$. So when you increase the value of $h$ the result is getting more smoother. So peaks begin to fade(Peaks means high frequency components).

As I know there isn't a generalize way to determine $h$ to get the result smoother and accurate. You have to choose appropriate $h$ by a trial and error or if you know the transfer function of the sensor, you can use that to determine an appropriate value for $h$.