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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 $z$?

Royi
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uday
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    Transfer functions only exist for [linear time invariant systems](https://en.wikipedia.org/wiki/LTI_system_theory). So in this case there is no transfer function. – fibonatic Jul 25 '15 at 02:21
  • The wiki article doesn't explain why the transfer function only exists for LTI systems – uday Jul 25 '15 at 03:36
  • 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 03:42
  • @Royi thanks for the reply. But I am not sure if your answer is correct , particularly the 2nd paragraph. If you can provide a derivation, that would be helpful. – uday Mar 20 '21 at 21:23
  • Derivation of what? For the argument that LTI system is required to be defined by the Transfer Function in Fourier Domain? – Royi Mar 20 '21 at 21:44
  • @uday, I added the reasoning and links for to see why you can't have a transfer function for your case. – Royi Mar 20 '21 at 21:49
  • Perhaps, I need to rephrase the question to make it easier to understand. As per the Wiki link, LTI simply is $y[n] = x[n] * h[n]$. But I don’t really care about it. I care about the question I asked, not an entirely different thing like LTI where the $h[n]$ is already pre-separated. – uday Mar 21 '21 at 06:24
  • Or to put it in other words, what I want to know is : how do I rewrite my non-linear filter as an LTI, or why it can’t be rewritten. I don’t need a link that gives a definition of LTI. – uday Mar 21 '21 at 06:29
  • @royi, apologies if my question was unclear. Really the help that I want is to know if there is a smart way to rewrite or restructure my non-linear equation in a linear form like LTI. Not really looking for the definition of LTI. – uday Mar 21 '21 at 06:33
  • @royi, for example, consider the case where $y[n]^2 = y[n-1]^2 + x[n]^2$. I can rewrite this as $y[n]^2 = x[n]^2 / (1 - z^{-1} )$. Here, I was still able to transform a non-linear expression to extract a transfer function $H(z) = 1/sqrt(1-z^{-1})$. – uday Mar 21 '21 at 06:48
  • No you can not. The system you derived isn't well defined. What would be the the root to choose from? If the system can be a ratio it means it is defined by multiplication which means it is a convolution in Time Domain. Namely it must be LTI. You can not achieve such thing as you wrote. – Royi Mar 21 '21 at 07:18

2 Answers2

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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.

Linear Time Invariant (LTI) System

By Convolution Theorem a convolution of LTI System can be described by a transfer function in Fourier Domain.

What do mean by a transfer function? We mean having $ H \left( \omega \right) $ which is independent of the input.
As for any output and input we can calculate the Fourier Transform and have the ratio of them as the Transfer Function. But only for LTI the result of this ratio will be constant.

Royi
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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 Volterra Series.

lennon310
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Hilmar
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    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