Explanation of some Fourier terms

what's the meaning or difference between :

- Fourier transform
- Fourier integral
- Discrete Fourier transform
- Fast Fourier transform
- Fourier series

Re: Fourier terms

Excluding the Fourier Integral, since I am not familiar with that.

They are all basically the same thing, in that they operate using the "Fourier Transform" principle to give you the spectral content of the signal. By the "Fourier Transform" principle, I mean a complex exponential weighting function.

The Fourier Series assumes that the given time-domain signal is periodic. The Fourier transform is a generalization of the Fourier Series such that the time-domain signal does not need to be periodic. It will give you the spectrum of any time-domain signal (provided the integral converges). If the time-domain signal is periodic, then the Fourier Transform breaks down into the Fourier Series.

The discrete-time Fourier transform (DTFT) is the adaptation of the Fourier Transform to work with discrete time signals. It does the same thing, it is just developed for discrete-time signals. The resulting function of frequency from the DTFT is a continuous function, however. The discrete-time signal could be a sampled version of any continuous-time signal.

The Discrete Fourier transform (DFT) is a Fourier Transform where both time and frequency are discrete. Computers can perform the DFT to get the spectrum of certain time-domain signals. The Fast Fourier Transform is a computationally efficient version of the DFT.

Hope this helps. This is the quickest, and roughest answer to your question i could give. There are entire books on Fourier Analysis.

Re: Fourier terms

hi all
assume f(t)=A0+A1 sin(wt)+...An sin(nwt)+...

∫sin(mwt) *sin(n(wt) dt=½∫[cos(n+m)wt +cos(n-m)wt] dt

for a priod (wt=0 to pi) ∫[cos(n+m)wt =0
∫cos(n-m)wt dt =0 n≠m
=1/τ n=m
it is because this term is priodic function

so ∫sin(nwt) f(t) dt =1/τ An wt=0 to pi
above equation known as Orthogonal transformation.


- Fourier transform :is a transformation form time domain to w domain
- Fourier integral :is a tool for Fourier transform at all
- Discrete Fourier transform :is Fourier transform for Discrete time wave form
- Fast Fourier transform : is Fourier transform for sampled (fixed priod) wave form
- Fourier series :is is Fourier transform for priodic wave form


be happy