Transcript Document
Thursday, October 12, 2006 Last Class Fourier Transform (and Inverse Fourier Transform) Spectral Density (Power Spectrum) Convolution and Cross-correlation Discrete Fourier Analysis Nyquist Freq. (Highest Freq.) Lowest Frequency How to do Fourier Analysis (IDL, MATLAB) What is FFT?? What about the mean? and What if there is a trend? Go to the help! This Class • DFT • Aliasing example • Leakage and Tapering (Multi-tapering?) • Windowed Fourier Transforms, Wavelets Transforms • Applications (Filtering -Convolution and Spectral-, Spectral Coherency) DFT Useful derivation! with Fourier Transform Assume we have We sample for all to obtain a discrete representation Mathematically So Question: How well does Represents ? DFT…. 1) Use the (continuous) definition of Fourier transform DFT!!! 2) Use convolution Poisson’s Summation Formula DFT…. How well does Represents 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 ? Aliasing Example: Play around with the Following process (using Matlab or IDL) with 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). If : We are OK!! If we have aliasing!! Aliasing “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 We know that there is a lunar semi-diurnal tide with a 12.42 hours period!! Spectral Leakage 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 High frequencies in the spectrum of the signal Artificial discontinuities These frequencies could be much higher than the Nyquist frequency. It appears as if the energy at one frequency has leaked out into all the other frequencies. Numerical Example…. 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. DFT Taper DFT A sequence of real-valued constants (data taper) 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 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 Tapers (Window Functions) The idea behind tapering is to select sidelobes than so that the has smaller Hamming Hann (Hanning) Time domain Frequency domain 40 1 20 0 Magnitude (dB) Amplitude 0.8 0.6 0.4 -20 -40 -60 -80 -100 0.2 -120 0 10 20 30 Samples 40 50 60 -140 0 0.2 0.4 0.6 0.8 Normalized Frequency ( rad/sample) Multi-Tapering Use of multiple orthogonal tapers (dpss) Final Spectrum: Linear and Nonlinear combinations of individual ones See IDL and Matlab Code…. End