|MadSci Network: Physics|
The Fourier series was developed by Baron Jean-Baptiste-Joseph Fourier (1768-1830). Fourier was a French mathematician and administrator who also accompanied Napoleon on his expedition to Egypt. Fourier is best known for his contribution to the analysis of periodic functions and heat conduction in metals.
Information is not strictly encrypted in a Fourier spectrum. The Fourier transform can be used to transform a function from time domain to frequency domain or vice-versa. This transformation is can be done both ways easily, whereas encryption by its nature is easy to do one way but hard to do the other way. Basically the Fourier spectrum can tell us the strength of frequencies in a periodic functions, e.g. sound from a musical instrument. There is some really good information here .
Fourier concluded that any periodic function could be written as the sum of a (possibly infinite) set of sine and cosine functions. Specifically, the set of functions are of the form: c + a1sin(wt) + b1cos(wt) + a2sin(2wt) + b2cos(2wt) + ... aksin(kwt) + bkcos(kwt) The values a1, a2, b1, b2, ... ak, ... bk, and c are constants sometimes called Fourier coefficients. w is the angular frequency which is 2*pi*cycles per second, t is time in seconds. The values a1 and b1 represent the magnitude or strength of the fundamental frequency. The fundamental frequency is what we often call pitch. The other values of a and b represent the strength of the harmonic frequencies. Harmonic frequencies are exact integer multiples of the fundamental. For example, the A note near middle C on a piano is 440Hz (cycles per second), it may contain harmonics of 880Hz, 1320Hz, 1760Hz and so on. In general, natural sources of periodic functions (musical instruments) yield harmonics that are weaker than the fundamental, usually falling off to zero as the frequency goes up. It is possible to come up with functions that do not behave this way, but they are generally not found in nature.
The harmonics contribute to what musicians call timbre. A violin and a flute playing the same note have the same fundamental frequency (pitch), but they sound different to us. They sound different because their timbre, or harmonic content is different. Harmonics give musical instruments their unique characteristics. A flute generates relatively few harmonics, whereas a violin generates many. The amount of harmonic content depends heavily on the instrument and its construction, materials used etc.
In general, it is not easy to characterize what particular harmonics sound like. Musicians often say that even harmonics (2, 4, 6, 8) are more pleasing, whereas odd (3, 5, 7) harmonics are harsh. Considerable debate has surrounded the design of electric guitar amplifiers. Older tube amplifers are believed to produce cleaner, more pleasing distortion whereas modern transistor amplifiers produce harsh distortion. Indeed, both produce harmonics when overdriven, but tube designs yield both even and odd harmonics whereas symmetric transistor amps produce almost all odd harmonics.
Given an arbitrary function, we can compute the Fourier coefficients by integrating the product of the function and a series of sines, cosines, and 1. This is often covered in first year calculus classes. If we do this for a square wave, the magnitudes are 1, 0, 1/3, 0, 1/5, 0, 1/7 and so on. Notice that the even harmonics are zero, a perfect square wave contains only odd harmonics. Fourier analysis will yield both magnitude and phase information of each harmonic, but our ears can only detect the magnitude. There are an entire class of sounds that have different Fourier coefficients, but sound the same to us. A spectrum analyzer is an instrument that can perform this analysis in real time although they generally display only the total magnitude of harmonics and not the phase information.
Fourier analysis can be generalized to use functions other than just sines and cosines. In fact, any set of orthogonal functions that span a basis can be used. We could for example, choose a set of square waves as our basis functions instead of sines and cosines. The resulting transforms are called Wavelet transforms. Square wave basis functions are called Haar Wavelets. Wavelet transforms find practical uses in data compression. Some of the streaming technologies used on the Internet may use Wavelet transforms. Some of the most interesting Wavelets, like Daubechies are also fractals.
The Fourier transform is an extremely powerful mathematical tool used by scientists and engineers in many disciplines. There is no doubt that you will learn about it in great detail in many advanced mathematics classes.
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