L15-23+Notes

# L15-23+Notes - Barrett CS450 Winter 2011 LECTURES#15-23...

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Barrett – CS450 Winter 2011 LECTURES #15-23: Fourier Transforms (Ch. 4) 1. Fourier Synthesis and the Frequency Domain: • What is Fourier Synthesis? How do we represent/measure spatial frequency? • Sketch a high-pass filter. A band-pass filter. A low-pass filter. • An input signal can be decomposed into an infinite sum of infinitesimal sinusoids. • The FT of a sinusoidal function is and equally space impulse pair. • A linear system can be thought of as operation separately on the sinusoidal components of the input signal, which are summed at the output to form the output signal. 2. The Fourier Transform (FT) • Show how the FT represents a signal as a weighted sum of sines and cosines. What are the weights? • The FT as a "System" is Linear and Shift-Invariant. Explain why. Show it mathematically. • The FT of a sum of functions is the sum of their individual transforms (addition theorem). • Shifting the origin of a function introduces into its spectrum a phase shift that is linear with frequency and that alters the distribution of energy between the real and imaginary parts of the spectrum without changing the total energy (Shift Theorem). • Convolution of two functions corresponds to multiplication of their FTs (Convolution Theorem). • Narrowing a function broadens its FT and vice versa (Similarity Theorem). • The energy of a function (signal) is the same as that of its FT (spectrum). • The Transfer Function of a linear system can be determined as the ratio of its (measured) output spectrum to its (known) input spectrum. • If a function of two variables can be separated into a product of two functions of a single variable, then so can its FT.

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L15-23+Notes - Barrett CS450 Winter 2011 LECTURES#15-23...

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