What is fourier series/fourier transform?

[quietrain]

can anyone give me a simple explanation what Fourier analysis is all about? i have read wiki and it is a little confusing

what am i doing when i express a funciton as a series? what is the difference to a transform?

i know how to follow the steps of a Fourier series/transform, but i have no idea why i am doing it?

thanks!

 

[quietrain]

assuming i have a function f(x) = x

what does it mean to take the Fourier series?

what does it mean to take hte Fourier transform?

 

[mathman]

Fourier series are related to finite intervals, while Fourier transforms look at the whole real line. f(x)=x can be expanded into a Fourier series, but not a Fourier transform.

http://en.wikipedia.org/wiki/Fourier_series

http://en.wikipedia.org/wiki/Fourier_transform

 

[quietrain]

is Fourier series something like taylor expansion? where greater powers of the terms give better accuracy of the particular function?

 

[quietrain]

http://en.wikipedia.org/wiki/Fourier_transform

i refer you to the section "introduction" , where at the end of this section, they described the Fourier transform as decomposing an oscillating function into several sines and cosines.

if i understood correctly what the 4 graphs are saying, does it mean that when i Fourier transform the first graph (blue), i will get a transformed that is (green, when i sub 3 hz into the Fourier transformed expression) and (red when i sub in 5hz into the Fourier transformed expression)

and when i integrate these 2 expressions, i will get 0.5(green) and 0.000... (red)

so the value 0.5 and 0.0000... is akin to telling me the fraction of blue graph that oscillates at 3 hz is 0.5, while the fraction of blue graph that oscillates at 5hz is 0.0000...?

is this what Fourier transform is about?

thanks!

 

[paulfr]

A Fourier Series approximation of a periodic waveform = a sum of sines and cosines of varying amplitudes of multiples of the the fundamental frequency of that waveform. FS is a discrete series approximation.

A Fourier Transform essentially converts a time domain function to its frequency domain components. That is its spectral content. For example the FT of a sine wave would just be a single spike at the given frequency. If the sine wave has harmonics, as in a guitar or piano tone, then the FT would be a series of spikes; one each at the fundamental frequency and 3rd and 5th and 7th etc all with declining amplitudes.
FT is a continuous function approximation.

 

[quietrain]

A Fourier Series approximation of a periodic waveform = a sum of sines and cosines of varying amplitudes of multiples of the the fundamental frequency of that waveform.

A Fourier Transform essentially converts a time domain function to its frequency domain components. That is its spectral content. For example the FT of a sine wave would just be a single spike at the given frequency. If the sine wave has harmonics, as in a guitar or piano tone, then the FT would be a series of spikes; one each at the fundamental frequency and 3rd and 5th and 7th etc all with declining amplitudes.

oh..

so when i integrate over all frequency, does it have to = 1?

 

[paulfr]

oh.. so when i integrate over all frequency, does it have to = 1?

No, you are thinking of Probability Density functions. The area under that curve =1

For spectral content, the ENERGY of the spectra must equal the energy calculated from the time domain waveform.

 

[quietrain]

this is confusing

thanks!