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Published byJason Richardson Modified over 4 years ago

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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

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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:

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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:

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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

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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) :

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