Lab 6: Sound Analysis Fourier Synthesis Fourier Analysis Two Sine Waves of the Same Frequency Building a Square Wave from Sine Waves Does One Hear Phase? Fourier Analysis Fourier Analysis of Sine Waves Fourier Spectrum of the Square Wave Fourier Analysis of Your Voice online manual printed manual Red – use tabletop devises Magenta – use tabletop devises and java applet Blue – use java applet

Waveform (Wave Shape) Is it voice or musical instrument?

Sound of Musical Instruments Waveform superposition of harmonics amplitude (A) and period (T) quality of sound indiscernible Fourier spectrum Fourier components (harmonics) amplitude (A) and frequency ( f ) unique to each instrument

Fourier Theorem “Any periodic function of period T can be expressed as a sum of sine curves whose frequencies are multiples of f = 1/T and whose amplitude and phase are suitably chosen.” (textbook)

Concepts Superposition Phase Periodic non-sinusoidal wave Fourier Spectrum

Superposition (1) t S 1 -1 2 -2 3 -3 A B

Superposition (2) A B S t 3 2 1 -1 -2 0 + (-2) = -2 -3 1 -1 2 -2 3 -3 A B 0 + (-2) = -2 (+1.4) + (-1.4) = 0

Superposition (3) t S 1 -1 2 -2 3 -3 A B

Sine function: S = sin(360 · f · t) Phase of Sine Wave t S 90 180 270 0 1 -1 Sine function: S = sin(360 · f · t) Phase angle “(360 · f · t)” 0 90 180 270 360 or 0 Value of sine S 1 -1

Phase Shift Phase shift  : phase of sine curve at t = 0 180 0 90 A 270 0 Phase shift  : phase of sine curve at t = 0 t S A 0 t S 180 t S 90

Two Sine Waves of Same Frequency In phase t S 1 -1 180° out of phase t S 1 -1 180°

Fourier Synthesis + + + Superposition (Fourier synthesis) S: A periodic wave having the frequency f1 and period T fn: Frequency of the nth mode that is an integer multiple of f1 (= n·f1) fn: Phase of the nth mode at t = 0 t t t t + + + t Superposition (Fourier synthesis) Fourier analysis

t = 0 t=0