Geol 491: Spectral Analysis - West Virginia Universitypages.geo.wvu.edu/~wilson/geol491/Lec1_Intro.pdf · 1/12/2011 1 Geol 491: Spectral Analysis [email protected] 1 2 1 0 2

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Geol 491: Spectral Analysis

[email protected]

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Purpose of the class

Explore ways of introducing some advanced mathematical concepts to students in such a way asmathematical concepts to students in such a way as

to increase their interest in higher level math.

To learn new and useful analytical tools

T h f !To have fun!

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Grade

• Computer lab assignments – 30%

• Lesson plan development (2 teams): initial and final• Lesson plan development (2 teams): initial and final drafts, 10% each for 20% of the semester grade

• Class presentation 30%

• Final report: revision of lesson plan with discussion of what additional activities you think would be useful to undertake. 20%

• Final report should include a brief half page to page discussion of what you got out of the class. Was it useful? Why or why not?

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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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

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seconds

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5*sin(2 4t)

We are usually dealing with sampled data

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5*sin(24t)

Amplitude = 5

Frequency = 4 Hz

Sampling rate = 256 samples/second

Sampling duration =

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-8

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seconds

g1 second

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2sin(28t), SR = 8.5 Hz

Faithful reproduction of the signal requires adequate sampling

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

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The Nyquist Frequency

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

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

1

2Nyft

Wh i h lWhere t is the sample rate

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

The Nyquist Frequency

Thus if t = 0.004 seconds, fNy =

1

2Nyft

Where t is the sample rate

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Fourier series: a weighted sum of sines and cosines

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

a t b t

a t b t

This applet is fun to play with & educational too.

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

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

An octave represents a doubling of the frequency.220H 440H d 880H l d t th d220Hz, 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/

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Fourier series

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

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( ) cos sinn nn

f t a a n t b n t

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

can be simulated using the curvaceous sinusoids.

Fourier series

• Try the excel file step2.xls

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( ) cos sinn nn

f t a a n t b n t

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The Fourier Transform

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

1( )

Tf t dt0 ( )f t dt

T What do you get? Looks like an average!

We’ll work through this on the board.

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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)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

... +...

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

a t i t b t i t

a t i t b t i t

cosia cos sin cos

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

Now integrate f(t) cos(it)

0 1 10 0( ) cos ( cos cos cos sin cos

cos 2 cos sin 2 cos

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

a t i t b t i t

2 2

3 3

cos 2 cos sin 2 cos

cos3 cos sin 3 cos

... +...

a t i t b t i t

a t i t b t i t

cos cos sin cos

... +... ) i ia i t i t b i t i t

dt

00cos 0

Ta i tdt This is just the average of i

periods of the cosine

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Now integrate f(t) cos(it)

10cos cos ?

Ta t i tdt U h id i

1 1cos cos cos( ) cos( )

2 2A B A B A B

Use the identity

If i=2 then the a1 term =

11 cos cos (cos 2 cos0)

aa t t t 1 cos cos (cos 2 cos0)

2a t t t

1 110 0 0cos cos cos 2 cos 0

2 2

T T Ta aa t tdt tdt dt

What does this give us?

110

0

cos cos 02

TT a

a 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

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In general to find the coefficients we do the following

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1( )

Ta f t dt

T

0

2( )cos

T

na f t n tdtT

2( )sin

T

nb f t n tdt

and

0( )n f

T 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

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1( )

2a f t dt

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( )cosna f t n tdt

and

1( )sinnb f t n tdt

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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)into another function (or signal)

• Continuous Fourier Transform:

dfefHth

dtethfH

ift

ift

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dfefHth

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

The Fourier Transform

into another function (or signal)

• The Discrete Fourier Transform:

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We’ll do some work with mp3 files. See http://soundmachine.gooddogie.com/sounds4.htm

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Tiger.mp3

Amplitude versus time on the sound track

tiger.mp3

Classic view

Spectral plots

Low to high frequency content in the sound file as a function of time

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Spectral filtering

Doppler shift

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Cut out specific parts of a sound file

Various spectral views under windows>classic, vertical, horizontal

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Change spectral display format in individual windows

Design spectral filters to see how sounds change when certain frequencies are removed. Try this with a

recording of your own voice

Try filtering everything out above 375 Hz

Get a view of the spectrum

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Lowpass filter

Highpass filter

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Some useful links

• http://www.falstad.com/fourier/– Fourier series java applet

htt // jh d / i l /• 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?

laugh2.mp3