O'Reilly logo

Digital Signal and Image Processing using MATLAB, Volume 1: Fundamentals, 2nd Edition by Gerard Blanchet, Maurice Charbit

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

Chapter 11

Appendix

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(images);
the Fourier transform is linear;
expansion/compression of time: the Fourier transform of x(at) is imagesX(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 images y)(t), is defined by:

(11.1) images

and has X(f)Y(f) as its Fourier transform;
likewise, the Fourier transform of x(t)y(t) is (X images Y)(f);
if x(t) is m times continuously differentiable and if its derivatives are summable up to the m-th order, ...

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required