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Explanation of laplace transform, z-transform and fourier

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leo83

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transforms

hey can anyone out there tell me what is the use of laplace transform, z-transform and fourier transform. ok i know we use it for various analysis like time domain and frequency domain. but why? and which is used when ?
 

Re: transforms

I am posting this in three posts because of pictures can not be displayed all if I attach them together in one post.

-------------------------------------------------------
I am including picture untitled1.jpg with various sources. I have dashed them but you still can see what is frequency of each source.

Lets simulate them for a marker called "SIGNAL"
 
Re: transforms

Now in TIME domain you get this waveform! Tell me something about it?
Figure in untitled2.jpg
What a mess in TIME domain!


If You simulate in FREQUENCY domain (Fourier one in untitled3.jpgfigure) you would see 4 impulses each with particual frequency. So signal is composed by 4 sinusoids.

Much clear! Much better in FREQUENCY domain!

Understand now why frequency domain is used? Each transform will be applied in various situations.

One of the most important tool in engineering and possiby the best of the best is Fourier Transform by Jean Baptiste Fourier. Viva La France! Hmm forgot it Laplace is also french.

Wow!
 
Re: transforms

Fourier gives frequency dependance of periodic signals, Laplace of non-periodic signals.
Z is used in discreete system control theory.
 

Re: transforms

pixel said:
Fourier gives frequency dependance of periodic signals, Laplace of non-periodic signals.
Z is used in discreete system control theory.
Click to expand...

Fourier series yes, but not the Fourier transform, and is not limited to periodic signals as Fourier Series is.
 

Re: transforms

djalli said:
Each transform will be applied in various situations.
Click to expand...
can u please say which is used in which situation? is there anything like only a particular one has to be used in a particular situation? what is the diff between the three.
 

Re: transforms

if these transforms are used for frequency domain analysis then why do u need three of them. i mean what is the diff between the three?
 

Re: transforms

The difference between the three transfroms lies in the value that sigma,in
(sigma+j*frequency),carries in sigma v/s frequency axis.

In these transforms frequency variation with respect to time is integrated out.
Check out GABOR transform in which time and frequency variations are intertwined.Such TF domain characteristics are much used in image compression.
 
Re: transforms

leo83 said:
if these transforms are used for frequency domain analysis then why do u need three of them. i mean what is the diff between the three?
Click to expand...

They are complements of each other.

Their difference (with no math!) is in method of approach in various situations.

For the case above the Fourier transform, Fast Fourier transform to be precise, was the best approach.

Fast Fourier transform speeds computations in less time. This is crucial today in era of high-speed communication/computations electronics. It is very efficient.

Can you do the same with Laplace? Of course. Can you do it efficiently as Fourier can? No!!!

There are situations where Laplace beats Fourier.

You will encounter these situations and it will be a habit to know what method to employ. Most engineering (electrical) situations will force you to love Fourier "and France". :oops: :oops:
 

Re: transforms

Laplace transform is more used, when you need transfer function.
Z can be used for some transition periods, and also for digital sytem analyzis.
 

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