4 ms·

I'm not a DSP person, but I thought the term "transfer function" was more typically used in reference to Laplace transforms of linear time-invariant dynamical s

by blt 6y ago

I'm not a DSP person, but I thought the term "transfer function" was more typically used in reference to Laplace transforms of linear time-invariant dynamical systems. Is it also common to use it for state-space mappings like the ones in this article?

radarsat1 6y ago
You make a good point. Indeed a transfer function is usually used to characterise linear systems. However, non-linear systems also have transfer functions, it's just that they are not scale invariant. Transfer function technically is just the "function" characterising the relationship between input and output. In a time-invariant linear system this is the ratio of the input spectrum to output spectrum, and is invariant to the scale of the input, and so can be characterised by a matrix multiplication (ie a linear transformation). For non-linear systems such as distortion, you now have a time-varying linear transfer function, a gain, depending on the amplitude of the input. So overall it is non-linear, and must be characterised by a function (the distortion curve), not by a single linear operation. One of the consequences of this is that it's a lot harder to go between the time domain and the frequency domain when dealing with non-linear transformations, so frequency analysis becomes more difficult -- but not impossible, depending on the system. For instance Chebyshev distortion can be analysed in the frequency domain. So, non-linear transfer functions can be seen as arbitrary functions, or can be seen as linear transfer functions parameterised by some parameter that varies non-linearly with the input (eg a gain modulated by a non-linear function of the input amplitude -- distortion). This second view is called "linearization" where you model a non-linear system with a local linear model, based on the assumption that non-linearities are smooth when viewed "close up", and is needed to apply certain linear control methods to non-linear systems.
TheOtherHobbes 6y ago
Yes, these are straight lookup tables/mappings, not transfer functions, which use complex math - and s-plane to z-plane transforms for DSP - and can implement various filter shapes. This is a good entry level guide to distortion, but it misses a few points. The first is that any non-linear mapping will introduce aliasing which folds back below the sample rate. The more clipping, the more aliasing you get. In commercial DSP you deal with this by oversampling, applying the distortion, and then filtering and downsampling to remove the aliasing. If you don't do that, the sound that comes out is nasty and crude with none of the creamy warmth of the real thing. The other is that high-quality amp distortion emulations - Kemper, etc - are more like real transfer function emulations. They're also level-dependent, so the mapping that comes out isn't one-size-fits all and responds to dynamics. That doesn't mean delay/echo/reverb, it means a network of very short delays characterised by a Volterra series and/or some form of convolution which models the dynamic tone characteristics of a real guitar amp.
- cochne 6y ago
  Would the non-linear mapping actually produce aliasing though, since it's in the discrete time domain? My understanding is that you only get aliasing when converting from analog to digital. While the discrete time sequence does map to higher frequency components (imaging) I would think this would need be filtered in the analog domain (the speaker's D2A converter)

by blt 6y ago

radarsat1 6y ago

You make a good point. Indeed a transfer function is usually used to characterise linear systems. However, non-linear systems also have transfer functions, it's just that they are not scale invariant. Transfer function technically is just the "function" characterising the relationship between input and output. In a time-invariant linear system this is the ratio of the input spectrum to output spectrum, and is invariant to the scale of the input, and so can be characterised by a matrix multiplication (ie a linear transformation). For non-linear systems such as distortion, you now have a time-varying linear transfer function, a gain, depending on the amplitude of the input. So overall it is non-linear, and must be characterised by a function (the distortion curve), not by a single linear operation. One of the consequences of this is that it's a lot harder to go between the time domain and the frequency domain when dealing with non-linear transformations, so frequency analysis becomes more difficult -- but not impossible, depending on the system. For instance Chebyshev distortion can be analysed in the frequency domain. So, non-linear transfer functions can be seen as arbitrary functions, or can be seen as linear transfer functions parameterised by some parameter that varies non-linearly with the input (eg a gain modulated by a non-linear function of the input amplitude -- distortion). This second view is called "linearization" where you model a non-linear system with a local linear model, based on the assumption that non-linearities are smooth when viewed "close up", and is needed to apply certain linear control methods to non-linear systems.

TheOtherHobbes 6y ago

Yes, these are straight lookup tables/mappings, not transfer functions, which use complex math - and s-plane to z-plane transforms for DSP - and can implement various filter shapes. This is a good entry level guide to distortion, but it misses a few points. The first is that any non-linear mapping will introduce aliasing which folds back below the sample rate. The more clipping, the more aliasing you get. In commercial DSP you deal with this by oversampling, applying the distortion, and then filtering and downsampling to remove the aliasing. If you don't do that, the sound that comes out is nasty and crude with none of the creamy warmth of the real thing. The other is that high-quality amp distortion emulations - Kemper, etc - are more like real transfer function emulations. They're also level-dependent, so the mapping that comes out isn't one-size-fits all and responds to dynamics. That doesn't mean delay/echo/reverb, it means a network of very short delays characterised by a Volterra series and/or some form of convolution which models the dynamic tone characteristics of a real guitar amp.

cochne 6y ago
Would the non-linear mapping actually produce aliasing though, since it's in the discrete time domain? My understanding is that you only get aliasing when converting from analog to digital. While the discrete time sequence does map to higher frequency components (imaging) I would think this would need be filtered in the analog domain (the speaker's D2A converter)