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*Jean Baptiste Joseph Fourier (1768 - 1830) laid a rock-solid foundation for signal analysis, when he claimed that all (continuously differentiable) signals can be represented as the sums of sines and cosines. | *Jean Baptiste Joseph Fourier (1768 - 1830) laid a rock-solid foundation for signal analysis, when he claimed that all (continuously differentiable) signals can be represented as the sums of sines and cosines. | ||
*It is hard to imagine the iPod generation without the work this great man did over 2 centuries ago. | *It is hard to imagine the iPod generation without the work this great man did over 2 centuries ago. | ||
| − | *However, the educated world (Electrical Engineers, :D) gradually evolved from | + | *However, the educated world (Electrical Engineers, :D) gradually evolved from their happy continuous perception of life, to the slightly scary, yet extremely promising world of discrete (a.k.a "digital") signals. |
*In this world, there is no such thing as continuity (obviously), and signals must be represented as discrete sets of zeros and ones. *These "jumps" would make Fourier very unhappy, because Fourier analysis starts breaking down (leakage, spectrogram uncertainty) when we make abrupt cut-offs and chop signals at will. | *In this world, there is no such thing as continuity (obviously), and signals must be represented as discrete sets of zeros and ones. *These "jumps" would make Fourier very unhappy, because Fourier analysis starts breaking down (leakage, spectrogram uncertainty) when we make abrupt cut-offs and chop signals at will. | ||
*Fundamentally, sines and cosines are infinite length and continuously differentiable, so to represent a jump (zero time or infinite frequency) one would technically need an infinite number of frequencies. Another concept that, albeit well defined on paper, but one that computers detest, is this whole concept of "infinity". | *Fundamentally, sines and cosines are infinite length and continuously differentiable, so to represent a jump (zero time or infinite frequency) one would technically need an infinite number of frequencies. Another concept that, albeit well defined on paper, but one that computers detest, is this whole concept of "infinity". | ||
Revision as of 23:45, 5 November 2009
==Page Under Construction==
Introduction to Wavelets
Taking Fourier's torch forward...
Background: Why Wavelets?
- I can bet a great deal of money, that as Electrical Engineers, the first person that comes to mind when someone says "SIGNAL PROCESSING" is Fourier.
- Jean Baptiste Joseph Fourier (1768 - 1830) laid a rock-solid foundation for signal analysis, when he claimed that all (continuously differentiable) signals can be represented as the sums of sines and cosines.
- It is hard to imagine the iPod generation without the work this great man did over 2 centuries ago.
- However, the educated world (Electrical Engineers, :D) gradually evolved from their happy continuous perception of life, to the slightly scary, yet extremely promising world of discrete (a.k.a "digital") signals.
- In this world, there is no such thing as continuity (obviously), and signals must be represented as discrete sets of zeros and ones. *These "jumps" would make Fourier very unhappy, because Fourier analysis starts breaking down (leakage, spectrogram uncertainty) when we make abrupt cut-offs and chop signals at will.
- Fundamentally, sines and cosines are infinite length and continuously differentiable, so to represent a jump (zero time or infinite frequency) one would technically need an infinite number of frequencies. Another concept that, albeit well defined on paper, but one that computers detest, is this whole concept of "infinity".
