# Transform Function with Non Linearity

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 $z$?

• Transfer functions only exist for linear time invariant systems. So in this case there is no transfer function. – fibonatic Jul 25 '15 at 2:21
• The wiki article doesn't explain why the transfer function only exists for LTI systems – uday Jul 25 '15 at 3:36
• One example for a nonlinear continues system would be that for its frequency response to a signal of a single frequency, you would either also get other frequencies in the output and or the gain and phase of the frequency present in the output will change nonlinear when the amplitude of the input frequency is changed. It is a bit harder to prove this, since it would involve quite some mathematics. – fibonatic Jul 25 '15 at 3:42

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

• A system would also be nonlinear if, when you increase the amplitude of the sine wave by some constant, the amplitude of the sine wave of the output change with by an amount which is a nonlinear function of that constant. – fibonatic Jul 25 '15 at 14:06

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.