Re: Differences between laplace transform, z transform and fourier transform
- From: "Me" <invalid@xxxxxxxxxxxxxxx>
- Date: Tue, 14 Jul 2009 17:09:14 +0100
Sorry, I just realised that I went into gabble mode and did not actually
answer the
questions that you put, but I hope it helped your understanding!
"Me" <invalid@xxxxxxxxxxxxxxx> wrote in message
news:h3iae6$m5m$1@xxxxxxxxxxxxxxxxxxxx
1. Start with the Fourier Series for repetitive waveforms, based around
sin & cos.
This is the fundamental theory underlying all the others.
2. Develop this into the exponential form of the Fourier Series.
3. The extend to the Fourier Transform for non-repetitive finite
"transient" waveforms.
4. Then you hit the difficulty that this does not always converge, a
requirement of the
Dirichlet conditions.
5. So you multiply by an arbitrary decaying exponential, e^(-ct), with "c"
never
explicitly defined but with the understanding that it is always big enough
to bring
about convergence.
6. (5) is essentiallythe Laplace Transform.
7. Some Laplace Transforms do not converge, but all the ones which you
will
encounter during your training period will converge.
8. Then you come into the digital world where all hell breaks loose and
you
are asked to undertake some flights of fancy which contradict all the
maths that
you will have studied up until now.
9. The train of sampled digital pulses is related to the Unit Impulse, and
you
now need a transform to model Delayed Unit Impulses.
10. The Laplace Transform of a time shift T is e^(sT) where T is the time
duration of
each sample and also the time between samples. Note that T is a fixed
constant and
not the continuous variable t.
11. e^(sT) is messy to write down, and like all mathematicians, we are
lazy and so
we seek a compact way to write it down. We use the substitution Z = e^(sT)
and so
we end up with the Z Transform, but it is really just another way of
writing down Laplace
Transforms which in their turn are a fudged Fourier Transform to bring
about convergence.
12. e^(sT) is of course the exponential form of a complex number (Remember
Lesson 2 of
your complex variable theory?) and can be represented as a vector of unit
magnitude centred
at the origin. Dependent upon the value of T, it traces out a circle which
is where your
Unit Circle (about which you seemed to be confused) came in!
Voila! Zat is Cointreau!
(Quotation from a Brit TV Ad from 20+ years ago)
"somanath17" <somanath.k17@xxxxxxxxx> wrote in message
news:u6mdnWwkqdVI78HXnZ2dnUVZ_t6dnZ2d@xxxxxxxxxxxxxxx
Hi All,
I have studied three diff kinds of transforms, The laplace transform, the
z transform and the fourier transform. As per my understanding the usage
of
the above transforms are:
Laplace Transforms are used primarily in continuous signal studies, more
so in realizing the analog circuit equivalent and is widely used in the
study of transient behaviors of systems.
The Z transform is the digital equivalent of a Laplace transform and is
used for steady state analysis and is used to realize the digital
circuits
for digital systems.
The Fourier transform is a particular case of z-transform, i.e
z-transform
evaluated on a unit circle and is also used in digital signals and is
more
so used to in spectrum analysis and calculating the energy density as
Fourier transforms always result in even signals and are used for
calculating the energy of the signal.
Is my understanding correct. What more technical differences exist and
where do all these differences find their application. Would be really
helpful if someone can give an understanding of this and provide links
where i can look up for the same.
.
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