Transform Function with Non Linearity

Question

I'm a newbie to Signal Processing - my apologies if this question is too obvious (I'm a financial trader trying to use DSP techniques).

For a linear filter:

$y[n] = (1-p) x[n]+p y[n-1]$

we can the write Transfer Function as

$H(z) = \frac{1-p}{1-pz^{-1}}$

But instead, suppose I have a non-linear filter:

$y[n] = y[n-1]+(1-p)\frac{1}{2}(x[n]e^{-2 y[n-1]}-1)$

How can I write a Transfer Function ?

For example, if I simply replace $y[n-1]$ by $y[n]z^{-1}$ and rewrite I get:

$H(z) = \frac{(1-p)\frac{1}{2}(x[n]e^{-2 z^{-1} y[n]}-1)}{1-z^{-1}}$

which is a function of $x[n]$ and $y[n]$ in addition to being a function of $z$?

Can this be written merely as a function of $z$?

Answer 1

The idea of the transfer function is roughly the following:

1. Break your input signal down into sine waves
2. Run the sine waves through the system
3. Assemble the output signal by summing up the output sine waves

This only works if systems output to a sine wave is a sine wave of the same frequency but with different amplitude and phase. That amplitude and phase modification is exactly the transfer function. However only LTI systems have the "one frequency in" "same frequencies out" property.

Non linear systems are typically "one frequency" "many frequencies" out, so simple transfer function doesn't work any more. There are advanced methods that can handle "one in" to "many out" and are based on sine waves, polynomial or Taylor series. See for example https://en.wikipedia.org/wiki/Volterra_series

Answer 2

Transfer function tries to describe a system using the Fourier Transform (More generally the Laplace Transform / Z Transform).

Yet in order for the Transform function to dependent on the frequency and amplitude only the input system must be LTI system.

Hence there is no Transfer Function in the Fourier Domain to your system which can be described the way you want.