Outline  Fourier transforms (FT)  Forward and inverse  Discrete (DFT)  Fourier series  Properties of FT:  Symmetry and reciprocity  Scaling in time and space  Resolution in time (space) and frequency  FT’s of derivatives and time/space-shifted functions  The Dirac’s delta function

Sin/cos() or exp() forms of Fourier series  Note that the cos() and sin() basis in Fourier series can be replaced with a basis of complex exponential functions of positive and negative frequencies: where:  The e inx functions also form an orthogonal basis for all n  The Fourier series becomes simply:

Time- (or space-) frequency uncertainty relation  If we have a signal localized in time (space) within interval  T, then its frequency bandwidth  (  f ) is limited by:  This is known as the Heisenberg uncertainty relation in quantum mechanics or  For example, for a boxcar function B(t) of length  T in time, the spectrum equals:  The width of its main lobe is:

Dirac’s delta function  The “generalized function”  plays the role of identity matrix in integral transforms:   (x) can be viewed as an infinite spike of zero width at x = 0, so that:  Another useful way to look at  (x):  (x) is the Heavyside function:  (x) = 0 for x 0

Dirac’s delta function  Recalling the formula we had for the forward and inverse Fourier transforms: …we also see another useful form for  (y) (y = x - x here) :