Week 15 projections, Haberman Chapter H10
Fourier Transformed my Cat
An objective of learning Fourier Transforms is to be able to laugh at
this joke, and then to explain it to your friends.
IKCD Fourier transformed my cat
Human Ear: Cochlea
The human ear cochlea has hair-like filaments which, when stimulated,
produce a signal which is transmitted to the aural portion of the brain.
The straightest spiral of the cochlea responds to low frequencies. Where
the spiral is of greatest curvature, it responds to high frequencies.
The cochlea converts time-dependent audio signals into frequency
signals. The conversion from time to frequency is the basis of the
FOURIER TRANSFORM. The cochlea is a biological system which performs a
Human ear cochlea hair structure (GIF)
To learn the meaning of the Fourier Transform is at the start similar to
deciphering the Klingon Alphabet.
Klingon Alphabet (JPEG)
Another Klingon Alphabet
Here's a look at the 2D Fourier Transforms of some English letters. The
unintelligible black and white patterns are not the entire transform, but
the image of the magnitude versus frequency (omega). The phase is not
shown. It's another Klingon alphabet.
Letters and their Fourier Transforms (GIF)
Stereo Equalizer and Fourier Series
Sliders on a stereo equalizer adjust the frequency of the signal (a
Fourier series). This is a hands-on way to visualize how Fourier series
and transforms supply frequency information. When you move a slider, it
tunes the magnitude of the corresponding term of matching frequency, in
the signal's Fourier series or transform. Example: Move slider 1200 HZ.
Then the Fourier series term sin(1200*2*Pi*t) has a re-tuned amplitude.
Remember: 2Pi/omega=Period, 1/Period=Frequency, HZ=cycles per second.
Stereo equalizer sliders and Fourier transform (PNG)
Approximate frequency ranges for music and voice (GIF)
Importance of Magnitude and Phase: 2D Transform of an Image
Magnitude and phase are provided by a 2D Fourier Transform. Both are
required to reconstruct an image.
Slide: Photo, magnitude, phase
Maple: magnitude, phase for the unit step (PDF)
Shannon Interpolation Example
Reconstruct a sample signal from samples.
PDF: Maple example