Understanding the Differences between Fourier Series and Fourier Transforms

[salil87]

Hi
Do Fourier Transforms give us actual amplitude/phase of the particular frequency (e^jωt) just like Fourier series?
Thanks
Salil

 

[rbj]

sorta, yes. but the inverse continuous Fourier transform is an integral not a summation like in the Fourier series. so the actual amplitude is proportional to the product of X(f) and the width of the sliver of spectrum df.

to compare, give the (inverse) Fourier integral a finite width (with the limits of the integral) and then represent that finite width integral with a Riemann summation and then you will be able to see the relationship between the inverse Fourier transform and the Fourier series. in a loose sense, they are the same thing.

 

[sophiecentaur]

The series is a Discrete process where the Transform is Continuous. The series can yield 'wrong' / misleading results if you ignore the basic rules.

 

[Anitha Sankar]

what is the Fourier transform of sum( Vhcos(hwt)) where h varies from 1 to infinity

 

[sophiecentaur]

what is the Fourier transform of sum( Vhcos(hwt)) where h varies from 1 to infinity

Hi
I assume that when you write Vh , the h is a suffix.
The transform will be a regular 'comb' of components at frequency w, hw, 2hw etc. with amplitdes given by the coefficients V. In fact, the original function is of a form that tells you the frequency spectrum just by 'observation'.

 

[Cecilia48]

http://www.infoocean.info/avatar2.jpg The series is a Discrete process where the Transform is Continuous.

 

[sophiecentaur]

I guess I meant "sum' as against 'integral'.
Is that better?