1. Fourier’s Theorem

2. Fourier’s Theorem
‘any periodic (or regularly repeating) wave, however complicated, can be described in terms of an infinite number of sine waves (of various amplitudes and phases) added together’

3. Fourier’s Equation
the sine and cosine parts deal with phases of partials
coefficients (give energies of partials)
represents the fundamental frequency of the waveform
represents time
number of the harmonic
add all harmonics from n=1 to n= infinity

4. Angular Frequency 1 = angular frequency (in radians)
means the same as:
but only works for sine waves
(one cycle of a sine wave in radians = 2π)

5. Analysis and Synthesis
Fourier analysis involves taking a wave and breaking it up into it’s constituent components
Fourier synthesis involves constructing a wave by adding up sine waves

6. Fourier Analysis/Synthesis
Fourier analysis and synthesis are used a lot in D.S.P.
For example, to apply effects to digital audio

7. Fourier Synthesis Examples

8. Creating a Square Wave
T/2 T
3T/2
-1.5 -1
-0.5
0.5 1
1.5
time, t
v(t)

9. Coefficient Values for a Square Wave

10. Insert Coefficients insert these co-efficients:
into Fourier’s equation:
to get:

11. Calculate Terms at Each Time (t)
e.g. at t = T1/4 :
when n=1: sin 1t = sin (2/T1 * T1/4) = sin 2/4
= sin /2 = sin 90 = 1
 when n=3: 1/3 sin (31t) = 1/3 sin (3 * 2/T1 * T1/4) = 1/3 sin 6/4
= 1/3 sin 3/2 = 1/3 sin 270 = 1/3 * -1 = -1/3
when n=5: 1/5 sin (51t) = 1/5 sin (5 * 2/T1 * T1/4) = 1/5 sin 10/4
= 1/5 sin 5/2 = 1/5 sin 450 = 1/5 * 1 = 1/5

12. Calculating Terms Cont…
according to the formula multiply each value by 4/
this makes:
first term (n= 1) = 1*4/ = (at t= T1/4)
second term (n=3) = -1/3 * 4/ = (at t = T1/4)
third term (n = 5) = 1/5 * 4/ = (at t = T1/4)

13. Plot Values 4/ (1/n sin (n1t) time, t
1.273
-0.424
0.255
T/2 T
3T/2
-1.5 -1
-0.5
0.5 1
1.5
4/ (1/n sin (n1t)
A plot of the first 5 terms (all t values).

14. Add Waves Together v(t) time, t
T/2 T
3T/2
-1.5 -1
-0.5
0.5 1
1.5
time, t
v(t)
A plot of the first five terms all added together.

15. Add More Terms v(t) time, t
T/2 T
3T/2
-1.5 -1
-0.5
0.5 1
1.5
time, t
v(t)
A plot of the first twenty terms all added together.

16. Square Wave Freq. Domain Plot
0.2
0.4
0.6
0.8 1
1.2
1.4
frequency 1
31
71
51
111
91
131
amplitude

17. Practicalities
practical synthesis does not require an infinite number of sine waves
DiscoDSPs Vertigo has 256 partials
VirSins Cube has 512

18. Sawtooth Wave T 2T
-1.5 -1
-0.5
0.5 1
1.5
time, t
v(t)

19. Sawtooth - Freq. Domain Plot
Coefficients
frequency
relative value 1
21
41
31
61
51
71

20. Triangle Wave T 2T
-1.5 -1
-0.5
0.5 1
1.5
time, t
v(t)

21. Triangle – Freq. Domain Plot
Coefficients
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9 1
31
71
51
111
91
131
frequency
amplitude