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### The properties of the Discrete Fourier Transform

The Discrete Fourier Transform FN is a bijective transformation on periodic sequences such that
 FN,-1 = FN, = on

i.e. :

(FN-1(x))k = xj

Inside Xcas the discrete Fourier transform and it's inverse are denote by fft and ifft:
fft(x)= FN(x), ifft(x)= FN-1(x)
Definitions
Let x and y be two periodic sequences of period N.
• The Hadamard product (notation . ) is defined by:

(x . y)k = xkyk

• the convolution product (notation *) is defined by:

(x*y)k = xjyk-j

Properties :
 N*FN(x . y) = FN(x)*FN(y) FN(x*y) = FN(x) . FN(y)

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giac documentation written by Renée De Graeve and Bernard Parisse