Physics 120 - David Kleinfeld - Spring 2016
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Notes on Fourier series of periodic functions
1.1
Background
Any temporal function can be represented by a multiplicity of basis sets. One
convenient representation for a function, V (t), is the representation in the frequency
domain. Here the function is assumed to exist for all of time, a not unreasonable
approximation for real signals in the steady state. Here we express V (t) in terms of
its continuous expansion in sines and cosines, which are most conveniently written
in their complex forms, i.e., sinx = (eix − e−ix )/(2i) and cosx = (eix + e−ix )/2. Then
∞
1 Z
dω Ṽ (ω) eiωt
V (t) =
2π
(1.1)
−∞
where Ṽ (ω) is a complex function that sets the contribution of different frequencies,
ω. The inverse transform is
Z∞
Ṽ (ω) =
dt V (t) e−iωt .
(1.2)
−∞
When V (t) is periodic with period T , so that V (t + T ) = V (t), we have
Z∞
dt V (t) e
−iωt
Z∞
=
−∞
dt V (t) e−iω(t+T )
(1.3)
−∞
so that
e−iωT = 1
(1.4)
and ω can only take on discrete values, i.e.,
4π
6π
2π
, ± , ± , ... .
(1.5)
T
T
T
We define ωo = 2π/T and write the Fourier expansion in a discrete form, i.e.,
ω = 0, ±
V (t) =
∞
X
c̃k eikωo t .
(1.6)
k=−∞
The c̃k are complex numbers defined by
+T /2
1 Z
c̃k =
dt V (t) e−ikωo t .
T
(1.7)
−T /2
Since we need to end up with sines and cosines, we expect that c̃k = c̃−k for c̃k real
and c̃k = −c̃−k for c̃k imaginary.
1
1.2
Sine waves
This is trivial case. We have:
1.3
c̃0
= 0
c̃±1
= ±
c̃±k; k≥2
= 0
(1.8)
1
2i
Square waves
Here V (0 < t < T /2) = -1 and V (T /2 < t < T ) = +1. We have:

c̃k
=
1

T
Z0
dt (−1) e−ikωo t +
+T
Z /2

dt (+1) e−ikωo t 

(1.9)
0
−T /2
i
1
T /2
− e−ikωo t |0−T /2 + e−ikωo t |0
−ikωo T
h
i
1
−1 + e+ikωo T /2 + e−ikωo T /2 − 1
=
−ikωo T
1
=
[2 − 2 cos (kωo T /2)] .
ikωo T
=
h
We recall that ωo T = 2π, so that
c̃k
2
[1 − cos (πk)]
(1.10)
ik2π
#
"
4
1
sin2 (πk/2)
=
π
2i
k
1
1
1
4
1
1
=
..., 0, − , 0, − , 0, −1, 0, +1, 0, + , 0, + , 0, ...
π
2i
5
3
3
5
=
and
4 ... − 15 e−i5ωo t − 31 e−i3ωo t − e−iωo t + eiωo t + 31 ei3ωo t + 15 ei5ωo t + ...
V (t) =
π
2i
4
1
1
=
sin(ωo t) + sin(3ωo t) + sin(5ωo t) + ...
(1.11)
π
3
5
!
The result is that the square wave is given by a weakly converging set of odd harmonics. Of potential interest, the smoother the function the faster the series for c̃k
converges, i. e., c̃k = constant for a period series of delta functions (comb), c̃k ∝
1/k for a square wave (as we saw), c̃k ∝ 1/k 2 for a triangular wave, etc.
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