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One example is analyse a music note, represented as a waveform, into a series of frequencies called the fundamental, second harmonic, third harmonic etc. This is useful in designing audio systems because an ordinary note, let's say middle-C, is at 260 Hz but played by an instrument the note also has harmonics that let you identify the instrument. An audio system has to reproduce the fundamental note but the harmonics also, otherwise the listener won't hear proper music.

A square wave with a period of 2pi has a Fourier series of 4/pi *(cos x - 1/3 cos 3x + 1/5 cos 5x . . . . ) and it can be integrated to give this series: 4/pi*(sin x -1/9 sin 3x + 1/25 sin 5x . . . ) which is obviously the series for a triangular wave, so the series shows that the upper harmonics are smaller.

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Q: Where is use of fourier analysis in real life?
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