[sdiy] Waveform analysis into non-sine components

[Magnus Danielson]

On 04/10/2012 11:29 PM, David G Dixon wrote:
>>> I see two things that make the Fourier basis very special:
>>> * It is the basis of eigenfunctions of LTI (linear time-invariant)
>>> systems. The LTI bit: Electronic circuits built with ideal R, L, C
>>> elements and linear gain elements (ideal op-amps, OTA,
>> etc...) are LTI
>>> systems - systems whose behavior is captured in a transfer function.
>>
>> This nicely segues into the Laplace transform:
>>
>> http://en.wikipedia.org/wiki/Laplace_transform
>>
>> which is where the 's' operator so often used in filter
>> design comes from.
>
> Fourier transforms are just Laplace transforms for cyclic responses, where
> the Laplace variable defined explicitly in terms of the frequency.

... which you achieve by setting the real part of the s variable to 0.

There is a peculiarity relating to integration range, where LaPlace 
covers the range 0 to infinity and Fourier covers the range from 
-infinity to infinity. Some objects wildly to equivalence due to this 
difference, but for Fourier signals, there is no difference to integrate 
from -infinity or 0 as long as you integrate to infinity, as the 
base-vectors looks the same regardless. For the LaPlace transform, the 
start at 0 is important since it relates to both static and transient 
responses, and the transient response cannot have any energy prior to 0 
due to causality, which is one of several important aspects for a linear 
system.

Cheers,
Magnus