Chapter-4Synthesis and Analysis of

Complex Waves

Fourier Synthesis: The process of combining harmonics to form a complex wave.

Fourier Analysis: Determining the harmonic content of a complex wave.

Synthesis of Complex Waves

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Synthesis of Complex Waves

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=17

Synthesis of Complex Waves

http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=17

SummaryThe shape of the complex wave is determined by:(1) the number and relative amplitudes of the component harmonics.(2) the phases of the higher harmonics, relative to the fundamental.

The tone quality or timbre is affected by moderate changes in the amplitude of the higher harmonic but is hardly affected at all by rather large changes in the relative phases of the two harmonics.

Fourier Synthesis of a Triangular Wave

Fourier Synthesis of a Square Wave

Fourier Synthesis of a Sawtooth Wave

Fourier Synthesis of a Pulse Train

Harmonic Amplitudes for Sine, Triangle, Square, Sawtooth,

and Pulse

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Periodicity and Fundamental

4.2 Fourier Analysis and Fourier SpectraSince the tone quality of a complex wave is determined primarily by the amplitudes of the harmonics, it is useful to display the harmonic content graphically.

Fourier spectrum displays the harmonic content of complex waves.

Fourier Spectra

Fourier Spectra

Wave form and Fourier spectrum of the note C5 = 523.25 Hz, played on

an alto recorder

Wave form and Fourier spectrum of the note B3

b = 233.08 Hz, played on a clarinet

Wave form and Fourier spectrum of the note B4 = 493.88 Hz, played on

a violin

Wave form and Fourier spectrum of the note G3 = 196.00 Hz, played on

a tenor krummhorn