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Thursday, October 12, 2006

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Fourier Transform (and Inverse Fourier Transform) Last Class How to do Fourier Analysis (IDL, MATLAB) What is FFT?? What about the mean? and What if there is a trend? Convolution and Cross-correlation Spectral Density (Power Spectrum) Discrete Fourier Analysis Nyquist Freq. (Highest Freq.) Lowest Frequency Go to the help!

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DFT Aliasing example Leakage and Tapering (Multi-tapering?) Windowed Fourier Transforms, Wavelets Transforms Applications (Filtering -Convolution and Spectral-, Spectral Coherency) This Class

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DFT Assume we have with Fourier Transform Useful derivation! We sample for all to obtain a discrete representation Mathematically So Question:How well doesRepresents ?

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DFT…. 1) Use the (continuous) definition of Fourier transform DFT!!! 2) Use convolution Poisson’s Summation Formula

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DFT…. How well doesRepresents ? The sum of all values of separated by frequency The proportionality is only achieved when the power vanishes for The Fourier transform of a sampled function will be the Fourier transform of the original continuous function only if the original function is bandlimited and is chosen to be small enough such that

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Aliasing Example: Play around with the Following process (using Matlab or IDL) What to do? Make sure the sampling rate is at least twice the highest frequency component present in the signal to be sampled (Sampling Theorem). with If : We are OK!! If we have aliasing!!

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“Professional” Example Aliasing is an elementary result, and it is pervasive in science. Those who do not understand it are condemned–as one can see in the literature–to sometimes foolish results (Wunsch, 2000). TOPEX/POSEIDON satellite altimeter Samples a fixed position on the earth with a return period Aliasing We know that there is a lunar semi-diurnal tide with a 12.42 hours period!!

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When DFT/FFT is used to find the frequency content of a signal, it is inherently assumed that the data that you have is a single period of a periodically repeating waveform Artificial discontinuities These frequencies could be much higher than the Nyquist frequency. Spectral Leakage High frequencies in the spectrum of the signal It appears as if the energy at one frequency has leaked out into all the other frequencies. Numerical Example….

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Tapering Spectral leakage cannot in general be eliminated completely, but its effects can be reduced by applying a tapered window function to the sampled signal. Sampled values of the signal are multiplied by a (window) function which tapers toward zero at either end. The sampled signal, rather than starting and stopping abruptly, "fades" in and out. This reduces the effect of the discontinuities where the mismatched sections of the signal join up DFTTaper DFT In a way, a data taper acts as a Filter. The window function filters out frequencies that appear due to discontinuities. So be careful with the variance!! There are many different data tapers A sequence of real-valued constants (data taper)

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Tapers (Window Functions) The idea behind tapering is to select so that the has smaller sidelobes than Hamming Hann (Hanning)

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Multi-Tapering Use of multiple orthogonal tapers (dpss) Final Spectrum: Linear and Nonlinear combinations of individual ones

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End See IDL and Matlab Code….

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Fourier Theory and its Application to Vision

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