1. 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

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

3. 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

4. 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)

5. Concepts Superposition Phase Periodic non-sinusoidal wave
Fourier Spectrum

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

7. Superposition (2) A B S t 3 2 1 -1 -2 0 + (-2) = -2 -31 -1 2 -2 3 -3 A B
0 + (-2) = -2
(+1.4) + (-1.4) = 0

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

9. 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

10. 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

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

12. 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

13. t = 0
t=0