1. Geol 491: Spectral Analysis

2. Introduction to Fourier series and Fourier transforms
Fourier said that any single valued function could be reproduced as a sum of sines and cosines
5*sin (24t)
Amplitude = 5
Frequency = 4 Hz
seconds

3. We are usually dealing with sampled data
5*sin(24t)
Amplitude = 5
Frequency = 4 Hz
Sampling rate = 256 samples/second
Sampling duration =
1 second
seconds

4. 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,

5. Where t is the sample rate
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.
Where t is the sample rate
Frequencies higher than the Nyquist frequencies will be aliased to lower frequency

6. The Nyquist Frequency Thus if t = 0.004 seconds, fNy =
Where t is the sample rate

7. Fourier series: a weighted sum of sines and cosines
Periodic functions and signals may be expanded into a series of sine and cosine functions
Experiment with

8. This applet is fun to play with & educational too.

9. An octave represents a doubling of the frequency.
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.

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

11. Fourier series
The Fourier series can be expressed more compactly using summation notation
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.

12. Try the excel file step2.xls
Fourier series
Try the excel file step2.xls

13. This can be done with continuous functions or discrete functions
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

14. The Fourier Transform
The general problem is to find the coefficients: a0, a1, b1, etc.
Take the integral of f(t) from 0 to T (where T is 1/f).
Note =2/T
What do you get? Looks like an average!
We’ll work through this on the board.

15. 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(it) or sin(it).

16. Now integrate f(t) cos(it)
This is just the average of i periods of the cosine

17. Now integrate f(t) cos(it)
Use the identity
If i=2 then the a1 term =

18. And what about the other terms in the series?
What does this give us?
And what about the other terms in the series?

19. In general to find the coefficients we do the following
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

20. Arbitrary period versus 2
Sometimes you’ll see the Fourier coefficients written as integrals from - to 
and

21. cost is considered Re eit
Exponential notation
cost is considered Re eit

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

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

25. Tiger.mp3

26. Classic view

30. Windows > vertical control

31. Get a view of the spectrum
Try filtering everything out about 1000 Hz

32. Note how sounds change

33. Some useful links http://www.falstad.com/fourier/
Fourier series java applet
Collection of demonstrations about digital signal processing
FFT tutorial from National Instruments
Dictionary of DSP terms
Mathcad tutorial for exploring Fourier transforms of free-induction decay
This presentation