Fourier theory made easy (?)

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-6

-4

-2

0

2

4

6

8

5*sin (24t)

Amplitude = 5

Frequency = 4 Hz

seconds

A sine wave

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-6

-4

-2

0

2

4

6

8

5*sin(24t)

Amplitude = 5

Frequency = 4 Hz

Sampling rate = 256

samples/second

seconds

Sampling duration =

1 second

A sine wave signal

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-1.5

-1

-0.5

0

0.5

1

1.5

2

sin(28t), SR = 8.5 Hz

An undersampled signal

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.

http://www.falstad.com/fourier/j2/

Fourier series

• Periodic functions and signals may be

expanded into a series of sine and cosine

functions

The Fourier Transform

• A transform takes one function (or signal)

and turns it into another function (or signal)

The Fourier Transform

• A transform takes one function (or signal)

and turns it into another function (or signal)

• Continuous Fourier Transform:

close your eyes if you

don’t like integrals

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

1

0

2

1

0

2

1 N

n

Nikn

nk

N

k

Nikn

kn

eHN

h

ehH

Fast Fourier Transform

• The Fast Fourier Transform (FFT) is a very efficient algorithm for performing a discrete Fourier transform

• FFT principle first used by Gauss in 18??

• FFT algorithm published by Cooley & Tukey in 1965

• In 1969, the 2048 point analysis of a seismic trace took 13 ½ hours. Using the FFT, the same task on the same machine took 2.4 seconds!

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-1

0

1

2

0 20 40 60 80 100 1200

50

100

150

200

250

300

Famous Fourier Transforms

Sine wave

Delta function

Famous Fourier Transforms

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0.1

0.2

0.3

0.4

0.5

0 50 100 150 200 2500

1

2

3

4

5

6

Gaussian

Gaussian

Famous Fourier Transforms

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0

0.5

1

1.5

-100 -50 0 50 1000

1

2

3

4

5

6

Sinc function

Square wave

Famous Fourier Transforms

Sinc function

Square wave

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0

0.5

1

1.5

-100 -50 0 50 1000

1

2

3

4

5

6

Famous Fourier Transforms

Exponential

Lorentzian

0 50 100 150 200 2500

5

10

15

20

25

30

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

FFT of FID

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-1

0

1

2

0 20 40 60 80 100 1200

10

20

30

40

50

60

70

f = 8 Hz SR = 256 HzT2 = 0.5 s

2exp2sin

T

tfttF

FFT of FID

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-1

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1

2

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2

4

6

8

10

12

14

f = 8 HzSR = 256 HzT2 = 0.1 s

FFT of FID

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-1

0

1

2

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50

100

150

200

f = 8 Hz SR = 256 HzT2 = 2 s

Effect of changing sample rate

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10

20

30

40

50

60

70

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-1

0

1

2

0 10 20 30 40 50 600

5

10

15

20

25

30

35

f = 8 Hz T2 = 0.5 s

Effect of changing sample rate

0 10 20 30 40 50 600

10

20

30

40

50

60

70

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-1

0

1

2

0 10 20 30 40 50 600

5

10

15

20

25

30

35

SR = 256 HzSR = 128 Hz

f = 8 HzT2 = 0.5 s

Effect of changing sample rate

• Lowering the sample rate:

– Reduces the Nyquist frequency, which

– Reduces the maximum measurable frequency

– Does not affect the frequency resolution

Effect of changing sampling duration

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-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

10

20

30

40

50

60

70

f = 8 Hz T2 = .5 s

Effect of changing sampling duration

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-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

10

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40

50

60

70

ST = 2.0 sST = 1.0 s

f = 8 HzT2 = .5 s

Effect of changing sampling duration

• Reducing the sampling duration:

– Lowers the frequency resolution

– Does not affect the range of frequencies you

can measure

Effect of changing sampling duration

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-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

50

100

150

200

f = 8 Hz T2 = 2.0 s

Effect of changing sampling duration

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-1

0

1

2

0 2 4 6 8 10 12 14 16 18 200

2

4

6

8

10

12

14

ST = 2.0 sST = 1.0 s

f = 8 Hz T2 = 0.1 s

Measuring multiple frequencies

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-2

-1

0

1

2

3

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20

40

60

80

100

120

f1 = 80 Hz, T2

1 = 1 s

f2 = 90 Hz, T2

2 = .5 s

f3 = 100 Hz, T2

3 = 0.25 s

SR = 256 Hz

Measuring multiple frequencies

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-2

-1

0

1

2

3

0 20 40 60 80 100 1200

20

40

60

80

100

120

f1 = 80 Hz, T2

1 = 1 s

f2 = 90 Hz, T2

2 = .5 s

f3 = 200 Hz, T2

3 = 0.25 s

SR = 256 Hz

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