Digital Signal and Image Processing Using MATLAB

Chapter 14

Appendix

A1    Fourier transform

Property 14.1 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 ;
• delay: the Fourier transform of x(t − t₀) is X(f)e^{−2j π f t₀};
• modulation: the Fourier transform of x(t)e^2jπf_ot is X(f − f₀);
• 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;

• 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, then the Fourier transform of the m-th derivative x^(m)(t) is (2jπf)^m X (f);
• if t ^m x (t) is summable, then the Fourier transform of (− 2jπt)^m x (t) is the m-th derivative X ^(m) (f).

A2    Discrete time Fourier transform ...