Adaptive IIR, Kalman Filtering

Often signals are contaminated by high frequency noise where the spectrum of the noise is changing or not known. The cutoff of a fixed low pass filter is not known or might be changing so that we need to adapt the cut-off continuously. One example is a Kalman filter which maximises the predictability of the filtered signal. It’s an adaptive lowpass filter which increases the predictability of a signal because it smoothes it.

$$H(z)=\frac{b}{1-a{z}^{-1}}$$

The parameter $a$ determines the cutoff frequency. Frequency Response :

$$\left|H\left(e^{j\omega }\right)\right|=\left|\frac{b}{1-a{e}^{-j\omega }}\right|$$

Let’s re-interpret our low-pass filter in the time domain:

$$\underset{\text{actual estimate }}{\underbrace{y(n)}}=a(n)\underset{\text{previous estimate }}{\underbrace{y(n-1)}}+b(n)\underset{\text{current data sample }}{\underbrace{x(n)}}$$

We would like to have the best estimate for $y(n)$

$$p(n)=E\left[{\left(y(k)-y_{\text{real }}(k)\right)}^2\right]$$

We need to minimise $p$ which gives us equations for a and b which implements a Kalman filter.