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  • Does there exist a causal nonlinear, time-invariant system that maps each input function $k \cdot\cos(fx)$ to $k \cdot\cos(2fx)$ for all choices of $k, f \in \Bbb R$?

  • If so, can one represent this system via some sort of differential equation, or in discrete terms, as a nonlinear difference equation?

This would be similar to the second Chebyshev polynomial, but that only works for $k=1$.

Gilles
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Mike Battaglia
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  • what do you want? a pitch shifter? – robert bristow-johnson Aug 29 '16 at 23:26
  • No, I'm using this to model distortion. – Mike Battaglia Aug 29 '16 at 23:27
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    but you want to double the frequency of what? a sine wave? and you want to do that for any amplitude of sine wave? (if such is the case, then something like an AGC to make the amplitude going in constant and then a 2nd-order Chebyshev.) – robert bristow-johnson Aug 30 '16 at 02:22
  • That wouldn't be time-invariant, right? – Mike Battaglia Aug 30 '16 at 02:23
  • yeah it is. because the gain is dependent solely on the signal parameters. – robert bristow-johnson Aug 30 '16 at 04:27
  • You can fully specify a linear system by how a basis is mapped, but for a non-linear system this is not enough. Also, frequency doubling does not required non-linearity. So what makes you ask for a non-linear system? And what should happen if you have a superposition of sinusoids? – Jazzmaniac Aug 30 '16 at 12:57
  • @jazzmaniac - I'm trying to study these smooth time-invariant systems and want to see if the thing I asked is possible. Basically, I saw that Chebyshev polynomials only map cos(x) to cos(nx) if the gain is at unity. In contrast, the system y(t) = x(t) * x'(t) does almost what I want, but has a frequency-dependent gain. Taking the input signal and multiplying by a 90° phase shifted version, then phase shifting the product by another 90°, seems to do exactly what I want, except 90° phase shifts are tricky. I want a superposition of sinusoids to yield intermodulation distortion. – Mike Battaglia Aug 30 '16 at 14:00
  • Also, how could you do this in a linear system? I assume you mean a linear, time-variant system? – Mike Battaglia Aug 30 '16 at 14:01

1 Answers1

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As already suggested by Robert and Olli, a system that maps $x(t)=k\cos(2\pi f_0t)$ to $y(t)=k\cos(4\pi f_0 t)$ can be formalized as

$$y(t)=|x(t)|_{max}\left(2\left(\frac{x(t)}{|x(t)|_{max}}\right)^2-1\right)\tag{1}$$

which is a time-invariant non-linear system.

However, I doubt that this system works well (i.e., sounds good as a distortion effect) when applied to real-world (non-sinusoidal) input signals.

To get some ideas of digital implementations of distortion effects, take a look at this thesis and the references therein.

Matt L.
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