5*sin (2 4t) 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
5*sin(2 4t) Amplitude = 5 Frequency = 4 Hz Sampling rate = 256 samples/second seconds Sampling duration = 1 second We are usually dealing with sampled data
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. Where t is the sample rate Frequencies higher than the Nyquist frequencies will be aliased to lower frequency
The Nyquist Frequency Where t is the sample rate Thus if t = 0.004 seconds, f Ny =
Fourier series: a weighted sum of sines and cosines Periodic functions and signals may be expanded into a series of sine and cosine functions
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 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.
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
The Fourier Transform The general problem is to find the coefficients: a 0, a 1, b 1, 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.
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).
Now integrate f(t) cos(i t) This is just the average of i periods of the cosine
Now integrate f(t) cos(i t) Use the identity If i=2 then the a 1 term =
What does this give us? And what about the other terms in the series?
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
Arbitrary period versus 2 Sometimes you’ll see the Fourier coefficients written as integrals from - to and
Exponential notation cos t is considered Re e i t where
The Fourier Transform A transform takes one function (or signal) and turns it into another function (or signal) Continuous Fourier Transform:
A transform takes one function (or signal) and turns it into another function (or signal) The Discrete Fourier Transform: The Fourier Transform
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.pdfhttp://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