# Purple Noise Modeling / Differentiated White Noise (Kalman Filtering)

I am designing a Kalman Filter for a signal which features a certain kind of noise and I do not know how to model it properly in the filter. The noise is constructed from a white noise source, called $w(t)$ by taking the difference of the current noise-sample and the last noise sample ($\Delta t$ is the sample-time):

$\eta(t) = w(t) - w(t-\Delta t)$

From my current point of view this is basically high-pass filtered noise or some kind of differentiated white noise (I read purple noise somewhere) with a sinusiodal power spectral density.

Question 1: Do you agree on my thoughts above?

Question 2: How can this noise be modeled in terms of a Kalman Filter? Would a Markov-model be suitable? How to express the color of this noise in a Kalman Filter?

Regards

• Purple noise increases with frequency, not a sinusoidal PSD... – CMDoolittle Aug 19 '15 at 18:46
• Kalman filter assumes white noise. You can get around this by augmenting your state vector. The new state represents the output of your coloring filter due to a white input. – David Wurtz Aug 19 '15 at 21:07
• @CMDoolittle Within the nyquist frequency I can do a taylor series of the exact $sin^2$ PSD which eventually leads to a PSD proportional to $f^2$. Hence this is violet noise. Am I correct that a Gauss-Markov model can only be used for colored noise which has a low-pass frequency behavior, i.e. I cannot use it? – Elarion Aug 19 '15 at 22:04
• @DavidWurtz I know that white noise is assumed, however this only depends on the modeling as you mentioned. Also brown noise is easily modeled using augmentation. The thing is I have never seen a Kalman Filter which works on differentiated white noise. Is this even possible? – Elarion Aug 19 '15 at 22:06
• The differentiation you refer to is a FIR filter. This fits within the state space framework no problem. For example, you could define 3 states: a(t) = b(t-1), b(t) = w(t), and c(t) = b(t-1) - a(t-1), where w(t) is your white noise source. Now c(t) is differencing the white noise source. – David Wurtz Aug 19 '15 at 23:02