On 6/23/14 11:58 AM, Andrew Simper wrote:
Ok, but where does
On 23 June 2014 22:59, robert bristow-johnson<r...@audioimagination.com> wrote:
On 6/23/14 10:50 AM, Andrew Simper wrote:
Ok, I'm still stumped here. Can someone please show me a reference to
how the bi-linear transform is created without using trapezoidal
integration?
not that you wanna hear from me, but the usual "derivation" in textbooks
goes something like this:
z = e^(sT)
= e^(sT/2) / e^(-sT/2)
=approx (1 + sT/2)/(1 - sT/2) (hence the "bilinear" approximation)
solving for s gets
s =approx 2/T * (z-1)/(z+1)
Ok, but what are the origins of that approximation? Where did it
actually come from. Looking at it in hindsight is fine, but doesn't
tell me anything.
i dunno where the insight to split up e^(sT) into half of it (the
"geometric" half) in the numerator and the reciprocal of the other half
in the denominator. i dunno who thought of that. when x is small, it's
commonly known that e^x =approx 1 + x . (they're the first two terms of
the Taylor series.)
a similar question could be asked about various implementation of
different Riemann summations to approximate the Riemann integral. who
first decided to split the difference between
b N-1
integral{ f(x) dx} = D * SUM{ f(a + nD) }
a n=0
and
b N
integral{ f(x) dx} = D * SUM{ f(a + nD) }
a n=1
where N*D = b-a in both cases?
splitting the difference between the two is the trapezoid rule of
integration.
--
r b-j r...@audioimagination.com
"Imagination is more important than knowledge."
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