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# A1 Fourier transform

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 L2();
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(tt0) is X(f)e–2jπft0;
modulation: the Fourier transform of x(t)e2jπf0t is X(ff0);
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:

(11.1)

and has X(f)Y(f) as its Fourier transform;
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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