spect_key Wrote:
can someone help me to explain the difference between Fourier Series,
Fourier Transform and Laplace Transform

-thanx


You can make a nice classification of these plus other
transformations based on their input and output domains:



---------------------------------
DOMAIN CLASSIFICATION CRITERIA:
---------------------------------

- domain is general (-\infty...+\infty) or causal
(0...+\infty)
- domain is infinite or finite (on a loop)
- domain is continious or discrete

-------------------------------------
INPUT DOMAIN IS GENERAL OR CAUSAL:
-------------------------------------

FOURIER VERSUS LAPLACE

The input domain is from -\infty ... +\infty for the
(general) Fourier
Transform while the causal range (0 ... +\infty) is used
for the
Laplace Transform.

The Laplace Transform has many engineering applications. It is
used to specify linear systems by their "impulse response":
The output of the system after receiving a Dirac delta input at t=0.
There can be no response at t0 if the system is causal.

Once the impulse response of a linear system is known then
the Laplace Transform can be used to determine the systems
response to many basic input stimuli simply by multiplying in
the frequency domain rather than performing a convolution
in the time domain.


------------------------------------------------------
INPUT DOMAIN IS FINITE (ON A LOOP) INSTEAD OF INFINITE
------------------------------------------------------

*Fourier Series Transform.*

If the input domain is either repetitive or if the input is on a
loop then it can be described by the Fourier Series Transform.
Its output domain is discrete instead of continuous. Only
multiples of the minimum frequency are allowed.

-----------------------------------------------------
OUTPUT DOMAIN IS FINITE (ON A LOOP) INSTEAD INFINITE
-----------------------------------------------------

*Z Transform*

If the input is discrete instead of continuous then the output
domain becomes finite (on a loop) The Z Transform can be
seen as the inverse Fourier Series Transform ( input and output
domain exchanged)

The Z Transform is used to do calculations on systems which
sample data at equidistant moments of time or space (Any real
system can only sample data at discrete points)

An input frequency to high for the sample rate will cause "aliasing":
It will appear as a lower frequency: The output "circles" one or
more times the finite output domain loop.

-----------------------------------------------------------
BOTH DOMAINS ARE FINITE (ON A LOOP) INSTEAD OF INFINITE
-----------------------------------------------------------

DISCRETE FOURIER TRANSFORM, DFT (ALSO FFT)

The DFT handles the Fourier Transform of a finite set of input
sampled at equidistant moments of time or space. Both input
and output domains are loops of discrete points.

A loop as input domain means that the input data is interpreted
as being repetitive. There is a minimum frequency which is equal
to the frequency of repetition. There is a maximum frequency and
aliasing occurs for frequencies above it.

A computed Fourier Transform is always a Discrete Fourier
Transform. A particular fast calculation method for the DFT
using the internal symmetries of the DFT is the so-called Fast
Fourier Transform (FFT)

----------------------------------------------------

Regards, Hans

------------------------------------------------------------------------
This post submitted through the LaTeX-enabled physicsforums.com
To view this post with LaTeX images:
http://www.physicsforums.com/showthr...188#post337038