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# Difference between fourier series fourier transform discrete time fourier transform and DFT?

A Fourier series is a series of sine and cosine harmonics of a particular frequency.

For example sinf+icosf + 3 sin2f+ 5icos2f... where the successive terms are multiples of the fundamental frequency f. It is typical ( but as far as I know not required) that complex numbers are used.

A Fourier transform converts a time domain wave form (like a sound wave) into the coefficients of the corresponding Fourier series.

A DFT is a digital approximation to a Fourier transform, usually using something like the Cooley-Tuckey Fast Fourier Transform (FFT) for efficiency.

The underlying Fourier theorem is something like: Every bounded periodic continuous (needed to avoid Gibbs) function , or wave form, can be written as the sum of its Fourier series. i.e. It is a sum of sines and cosines

In otherwords, you take a wave form in the time domain like a sound wave and break it into its components (various frequencies) by the Fourier Transform. The results of the Transform are the coefficients of the Fourier series. The wave form of a voice converted to components (and perhaps a little more) is a voiceprint. Study guides

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## A number a power of a variable or a product of the two is a monomial while a polynomial is the of monomials

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