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?