Outline Fourier transforms FT Forward and inverse Discrete
- Slides: 5
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 � einx 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 DT, then its frequency bandwidth Dw (Df) is limited by: or example, for a boxcar function B(t) of length DT in time, the spectrum equals: � For � The width of its main lobe is: � This is known as the Heisenberg uncertainty relation in quantum mechanics
Dirac’s delta function “generalized function” d plays the role of identity matrix in integral transforms: � The � d(x) can be viewed as an infinite spike of zero width at x = 0, so that: � Another useful way to look at d(x): q(x) is the Heavyside function: q(x) = 0 for x < 0 and q(x) = 1 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 d(y) (y = x - x here):
