(25) Fourier Analysis - Fourier Analysis Playing a musical...

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Fourier Analysis Playing a musical instrument, such as a guitar or an organ, generates a set of standing waves that cause a sympathetic oscillation in the surrounding air. Such an oscillation consists of a fundamental harmonic, whose frequency determines the pitch of the musical note heard by the listener, accompanied by a set of overtone harmonics that determine the timbre of the note. By definition, the oscillation frequencies of the overtone harmonics are integer multiples of that of the fundamental. Thus, we expect the pressure perturbation generated in a listener's ear to have the general form (339) where is the angular frequency of the fundamental (i.e., ) harmonic, and the and are the amplitudes and phases of the various harmonics. The preceding expression can also be written (340) where and . The function is periodic in time with period . In other words, for all . This follows because of the mathematical identities and , where is an integer. [Moreover, there is no for which for all .] Can any periodic waveform be represented as a linear superposition of sine and cosine waveforms, whose periods are integer subdivisions of that of the waveform, such as that specified in Equation ( 340 )? To put it another way, given an arbitrary periodic waveform , can we uniquely determine the constants and appearing in Equation ( 340 )? It turns out that we can. Incidentally, the decomposition of a periodic waveform into a linear superposition of sinusoidal waveforms is commonly known as Fourier analysis .
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