# 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 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:*(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, ...*Get *Digital Signal and Image Processing using MATLAB, Volume 1: Fundamentals, 2nd Edition* now with O’Reilly online learning.

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