Thursday, October 12, 2006
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Transcript of Thursday, October 12, 2006
Thursday, October 12, 2006
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!
• DFT
• Aliasing example
• Leakage and Tapering (Multi-tapering?)
• Windowed Fourier Transforms, Wavelets Transforms
• Applications (Filtering -Convolution and Spectral-, Spectral Coherency)
This Class
DFT
Assume we have with Fourier Transform
Useful derivation!
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)
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!!
“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!!
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 discontinuitiesThese 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….
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
DFT Taper 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)
Tapers (Window Functions)
10 20 30 40 50 600
0.2
0.4
0.6
0.8
1
Samples
Am
plit
ud
e
Time domain
0 0.2 0.4 0.6 0.8-140
-120
-100
-80
-60
-40
-20
0
20
40
Normalized Frequency ( rad/sample)
Ma
gn
itud
e (
dB
)
Frequency domain
The idea behind tapering is to select so that the has smaller sidelobes than
Hamming
Hann (Hanning)
Multi-Tapering
Use of multiple orthogonal tapers (dpss)
Final Spectrum: Linear and Nonlinear combinations of individual ones
End
See IDL and Matlab Code….