Re: Differences between laplace transform, z transform and fourier transform
- From: Tim Wescott <tim@xxxxxxxxxxxxxxxx>
- Date: Tue, 14 Jul 2009 13:16:43 -0500
On Tue, 14 Jul 2009 06:58:13 -0500, somanath17 wrote:
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.
Thanks,
Soma
Yes an no.
Laplace transform: Yes, but it is also widely used for designing control
systems and for determining the stability characteristics of continuous-
time systems.
Z-transform: The sampled-time (_not_ digital, although it's widely used
in digital systems) version of the Laplace transform. Useful for all the
same things that the Laplace transform is useful for.
Note that by carefully defining the sampling process you can derive the z
transform from the Laplace transform in a way that is exact. The usual
derivation defines sampling as multiplication by a series of impulses of
infinite height and finite area; this gives everyone gas pains but is
generally a nice way of thinking of it. You can avoid impulses at the
expense of convenience (but not rigor, as far as I can tell).
Note also that by carefully defining the reconstruction process you can
derive the Laplace transform from the z transform, if you're so
inspired. You say ta-mah-toe, I say toe-may-toe.
Your definition of the Fourier transform sounds like it is specifically
referring to the discrete Fourier series (AKA FFT). Further, your
assertion that it always results in even data is generally incorrect,
although all the flavors of Fourier transform/series will yield even real
output if you present them with even real input. There are four flavors
of things called Fourier:
The 'real' Fourier transform has a continuous, infinite extent input and
a continuous, infinite extent output. It is an operation on continuous-
time signals to express them in the frequency domain. Because the time
variable drops out and a frequency variable comes in, all without losing
any information, it's a transform.
The Fourier series for continuous-time periodic signals has a continuous
finite-time (one cycle) input and a discrete (at the harmonics), infinite-
extent output. Used for analyzing (what else!) periodic signals.
The Fourier transform for sampled-time signals. This is _not_ the FFT,
as it takes a discretely sampled infinite extent input and coughs up a
continuous finite extent output. Like the z transform this can be
derived from the 'real' continuous-time infinite extent Fourier transform.
Properly, the Fourier series for discrete-time periodic signals (whose
period is restricted to an integer number of samples, naturally). This
is the one that can be turned into the FFT; it takes discrete, finite-
span input data and coughs up discrete, finite-span output data. Because
the computation is so efficient, the FFT is used extensively to generate
numerical approximations to 'real' Fourier transforms for infinite
extent, continuous-time signals. This is done by sampling and truncating
the signal (which is where the approximation happens), windowing (which
usually makes the approximation better), performing the FFT, then finally
interpreting the data in a way that makes sense given the sample rate and
length of the sampled signal.
--
www.wescottdesign.com
.
- Follow-Ups:
- References:
- Prev by Date: Re: Cycle Slip in Viterbi and Viterbi Non-linear Phase Estimator
- Next by Date: Re: Cycle Slip in Viterbi and Viterbi Non-linear Phase Estimator
- Previous by thread: Re: Differences between laplace transform, z transform and fourier transform
- Next by thread: Re: Differences between laplace transform, z transform and fourier transform
- Index(es):
Relevant Pages
|
