**Property 11.1** *The main properties of the DFT are listed below:*

– *X*(*f*) *is bounded, continuous, tends towards* 0 *at infinity and belongs to L*_{2}();

– *the Fourier transform is linear;*

– *expansion/compression of time: the Fourier transform of x*(*at*) *is* *X*(*f*/*a*)*;*

– *delay: the Fourier transform of x*(*t* – *t*_{0}) *is X*(*f*)*e*^{–2jπft0}*;*

– *modulation: the Fourier transform of x*(*t*)*e*^{2jπf0t} *is X*(*f* – *f*_{0})*;*

– *conjugation: the Fourier transform of x** (*t*) *is X**(–*f*). *Therefore, if the signal x*(*t*) *is real, X*(*f*) = *X**(–*f*). *This property is said to be of* hermitian symmetry*;*

– *if the signal x*(*t*) *is real and even, X*(*f*) *is real and even;*

– *if the signal is purely imaginary and odd, X*(*f*) *is purely imaginary and odd;*

– *the convolution product, written* (*x* *y*)(*t*), *is defined by:*
*and has X*(*f*)*Y*(*f*) *as its Fourier transform;*

(11.1)

– *likewise, the Fourier transform of x*(*t*)*y*(*t*) *is* (*X* *Y*)(*f*)*;*

– *if x*(*t*) *is m times continuously differentiable and if its derivatives are summable up to the m-th order, ...*

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