[LAU] [OTish] FFT and state-space representation

Fons Adriaensen fons at linuxaudio.org
Tue Aug 18 19:39:07 UTC 2015

On Tue, Aug 18, 2015 at 10:12:14AM +0200, F. Silvain wrote:

> I'm currently studying for one of my electircal engineering causes,
> which deals mainly with state-space representation of linear,
> time-invariant systems. But I have still failed to grasp the
> practical meaning of some of the values involved.
> I thought a practical example might help and I've turned to
> convolution, since that is a subject I am slightly familiar with.
> Could someone perhaps help me here? A link to such an example or
> explanatory article in LaTex form could be sufficient.

Convolution is probably a bad choice.

For digital systems, The 'state' of a LTI system is the information
it has to store between processing one sample and the next -- the
'memory' it needs to have. For each input sample, the output and 
the new values of the state will each be a linear combination of
the the current input and all the state variables.

For IIR filters the state is a number of values equal to the order
of the filter. For example, a second order IIR filter (a parametric
would be a typical example) needs to store two values. For each
input sample the output, and the new values of the two state values
will each be a linear combination of the input and the two state

So you can get some insight into how such a thing works by e.g.
make a 3D plot of the input (as x) and the two state variables
(as y and z).

For FIR filters (i.e. convolution) the state is as big as the
lenght of the IR you convolve with. If the IR you use has 1000
samples, then the output for each sample depends on the current
input and the past 999 samples - these 999 being the 'state'.
So that state is really too complex to be of any use - you can't
easily visualize or imagine how 1000 values interact to produce
some result.

State-space representation are mainly a tool to understand
how relatively simple LTI systems behave in a very abstract
sense. They can useful if you want to gain a better understanding
of the theory, but have very little relation to practical results.



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