
Linear systems 27
A fundamental res ult in Fourier analysis is that any periodic function f(t)
with period T can be wr itten as
f(t) = F
0
+
∞
X
n=1
[F
cn
cos(nω
0
t) + F
sn
sin(nω
0
t)] ,
where ω
0
= 2π/T , and
F
0
=
1
T
Z
T
f(t)dt ,
F
cn
=
2
T
Z
T
f(t) cos(nω
0
t)dt , F
sn
=
2
T
Z
T
f(t) sin(nω
0
t) .
Note that F
0
is the average value of f (t) over a single period.
Example 2.3
Consider the square wave signal
P
w
(t) =
1 if 0 < t ≤ T/2
0 if T/2 < t ≤ T
Using the formula above we can easily compute the Fourier coefficients
F
0
=
1
2
, F
cn
= 0 ∀n ∈ N , F
sn
=
2
nπ
if n is odd
0 if n is even
.
Therefore, the square wave can be written
P
w
(t) =
1
2
+
2
π
sin(ω
0
t) +
2
3π
sin(3ω
0
t) +
2
5π
sin(5ω
0
t) + ··· (2.13)
Fig. 2.4 shows different ...