Geol 491: Spectral Analysis

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tom.wilson@mail.wvu.edu

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5*sin (24t)

Amplitude = 5

Frequency = 4 Hz

seconds

Fourier said that any single valued function could be reproduced as a sum of sines and cosines

Introduction to Fourier series and Fourier transforms

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5*sin(24t)

Amplitude = 5

Frequency = 4 Hz

Sampling rate = 256 samples/second

seconds

Sampling duration =1 second

We are usually dealing with sampled data

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2sin(28t), SR = 8.5 Hz

Faithful reproduction of the signal requires adequate sampling

If our sample rate isn’t high enough, then the output frequency will be lower than the input,

The Nyquist Frequency

• The Nyquist frequency is equal to one-half of the sampling frequency.

• The Nyquist frequency is the highest frequency that can be measured in a signal.

12Nyf

t

Where t is the sample rate

Frequencies higher than the Nyquist frequencies will be aliased to lower frequency

The Nyquist Frequency

12Nyf

t

Where t is the sample rate

Thus if t = 0.004 seconds, fNy =

Fourier series: a weighted sum of sines and cosines

• Periodic functions and signals may be expanded into a series of sine and cosine functions

0 1 1

2 2

3 3

( ) cos sin cos 2 sin 2 cos3 sin 3 ... +...

f t a a t b ta t b ta t b t

This applet is fun to play with & educational too.

Experiment with http://www.falstad.com/fourier/

Try making sounds by combining several harmonics (multiples of the fundamental frequency)

An octave represents a doubling of the frequency.220Hz, 440Hz and 880Hz played together produce a

“pleasant sound”Frequencies in the ratio of 3:2 represent a fifth and

are also considered pleasant to the ear.220, 660, 1980etc.

Pythagoras (530BC)

You can also observe how filtering of a broadband waveform will change audible waveform properties.

http://www.falstad.com/dfilter/

Fourier series

• The Fourier series can be expressed more compactly using summation notation

01

( ) cos sinn nn

f t a a n t b n t

You’ve seen from the forgoing example that right angle turns, drops, increases in the value of a function

can be simulated using the curvaceous sinusoids.

Fourier series• Try the excel file step2.xls

01

( ) cos sinn nn

f t a a n t b n t

The Fourier Transform

• A transform takes one function (or signal) in time and turns it into another function (or signal) in frequency

• This can be done with continuous functions or discrete functions

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( ) cos sinn nn

f t a a n t b n t

The Fourier Transform

• The general problem is to find the coefficients: a0, a1, b1, etc.

01

( ) cos sinn nn

f t a a n t b n t

Take the integral of f(t) from 0 to T (where T is 1/f).

Note =2/T

0

1 ( )T

f t dtT

What do you get? Looks like an average!

We’ll work through this on the board.

Getting the other Fourier coefficients

To get the other coefficients consider what happens when you multiply the terms in the

series by terms like cos(it) or sin(it).

0 1 1

2 2

3 3

( ) cos cos cos cos sin cos cos 2 cos sin 2 cos cos3 cos sin 3 cos ... +... cosi

f t i t a i t a t i t b t i ta t i t b t i ta t i t b t i t

a

cos sin cos

... +...ii t i t b i t i t

Now integrate f(t) cos(it)

0 1 10 0

2 2

3 3

( )cos ( cos cos cos sin cos

cos 2 cos sin 2 cos cos3 cos sin 3 cos ... +...

T Tf t i tdt a i t a t i t b t i t

a t i t b t i ta t i t b t i t

cos cos sin cos ... +... )

i ia i t i t b i t i tdt

00cos 0

Ta i tdt This is just the average of i

periods of the cosine

Now integrate f(t) cos(it)

10cos cos ?

Ta t i tdt

1 1cos cos cos( ) cos( )2 2

A B A B A B

Use the identity

If i=2 then the a1 term =

11 cos cos (cos 2 cos0)

2aa t t t

1 110 0 0cos cos cos 2 cos0

2 2T T Ta aa t tdt tdt dt

What does this give us?

110

0

cos cos 02

TT aa t tdt

And what about the other terms in the series?

2 220 0 0

cos 2 cos cos3 cos2 2

T T Ta aa t tdt tdt tdt

In general to find the coefficients we do the following

0 0

1 ( )T

a f t dtT

0

2 ( )cosT

na f t n tdtT

0

2 ( )sinT

nb f t n tdtT

and

The a’s and b’s are considered the amplitudes of the real and imaginary terms (cosine and sine) defining

individual frequency components in a signal

Arbitrary period versus 2

Sometimes you’ll see the Fourier coefficients written as integrals from - to

01 ( )

2a f t dt

1 ( )cosna f t n tdt

1 ( )sinnb f t n tdt

and

Exponential notation

cost is considered Re eit

cos sinn te t i t

where

The Fourier Transform• A transform takes one function (or signal) and turns it into another function (or signal)• Continuous Fourier Transform:

dfefHth

dtethfH

ift

ift

2

2

• A transform takes one function (or signal) and turns it into another function (or signal)• The Discrete Fourier Transform: The Fourier Transform

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Some useful links• http://www.falstad.com/fourier/

– Fourier series java applet• http://www.jhu.edu/~signals/

– Collection of demonstrations about digital signal processing• http://www.ni.com/events/tutorials/campus.htm

– FFT tutorial from National Instruments• http://www.cf.ac.uk/psych/CullingJ/dictionary.html

– Dictionary of DSP terms• http://jchemed.chem.wisc.edu/JCEWWW/Features/McadInChem/mcad008/FT4FreeIndDecay.pdf

– Mathcad tutorial for exploring Fourier transforms of free-induction decay• http://lcni.uoregon.edu/fft/fft.ppt

– This presentation

Meeting times?

Other questions?

laugh2.mp3