Fourier series coefficients
Consider an audio signal given by
s(t) = sin(440´
2p t) + sin(550 ´
2p t) + sin(660´
2p t).
This is a major triad in a non-well-tempered
scale. The first tone is A-440. The third is approximately E, with
a frequency 3/2 that of A-440. The middle term is approximately C sharp,
with a frequency 5/4 that of A-440. It is these simple frequency relationships
that result in a pleasant sound. We choose the non-well-tempered scale
because it makes it much easier to construct a Fourier series expansion
for this waveform.
To construct the Fourier series expansion, we can follow these steps:
-
Find p, the period. The period is the smallest number p such
that s(t) = s(t - p).
To do this, note that
sin(2pft) = sin(2pf
(t - p))
if f p is an integer. Thus, we want to find the smallest p
such that 440p, 550p, and 660p are all integers. Equivalently,
we want to find the largest fundamental frequency f0
such that 440/ f0, 550/ f0, and 660/
f0 are all integers. Such an f0 is called
the greatest common divisor of 440, 550, and 660. This can be computed
using the gcd function in Matlab. In this case, however, we can
do it in our heads, observing that f0 = 110.
-
Find A0, the constant term. By inspection, there is no
constant component in s(t), only sinusoidal components, so
A0 = 0.
-
Find A1, the fundamental term. By inspection, there is
no component at 110 Hz, so A1 = 0.
-
Find A2, the first harmonic. By inspection, there is
no component at 220 Hz, so A2 = 0.
-
Find A3. By inspection, there is no component at 330
Hz, so A3 = 0.
-
Find A4. There is a component at 440 Hz, sin(440´
2p t). We need to find A4
and f 4 such that A4cos(440´
2p t +f 4)
= sin(440´ 2p
t). By inspection, f 4 = - p
/2 and A4 = 1.
-
Similarly determine that A5 = A6 =
1, f 5 = f
6 = - p /2, and all other terms are
zero.
Putting this all together, the Fourier series expansion can be written
where w 0 = 220p.
Clearly this method for determining the Fourier series coefficients
is tedious and error prone, and will only work for simple signals. We will
see much better techniques.
Exercise
Determine the fundamental frequency and the Fourier series coefficients
for the well-tempered major triad,
s(t) = sin(440´
2p t) + sin(554 ´
2p t) + sin(659´
2p t).