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….