Complex number: z = x + jy = |z|ejθ
|z| =
q
x2 + y 2 ,
θ = arctan[ xy ]
Euler formulae:
ejθ = cos(θ) + j sin(θ)
ejθ + e−jθ
cos(θ) =
2
ejθ − e−jθ
sin(θ) =
2j
1
If a signal has both amplitude and phase it can
be expressed by a single complex number:
ejnω0t + e−jnω0t
cos(nω0t) =
2
ejnω0t − e−jnω0t
sin(nω0t) =
2j
2
• Complex Form of Fourier Series
For a real periodic function f (t) with period T ,
fundamental frequency f0
f (t) =
+∞
X
cnejnω0t
n=−∞
where
1
cn =
f (t)e−jnω0tdt
T T
is the ‘complex amplitude spectrum’.
Z
The coefficients are related to those in the
other forms of the series by
c0 = a 0 = A 0
cn =
1
1
(an − jbn) = Anejφn
2
2
for
n≥1
c−n = c∗n
Amplitude spectrum: |cn|
Phase spectrum: arg(cn)
3
Example: For the periodic rectangular wave
f (t) in the following Figure,
f(t)
A
T 0
4
T
4
2T
T
t
find the coefficients cn in the complex form of
Fourier series as given by (1),
f (t) =
∞
X
cnejnω0t
(1)
n=−∞
where ω0 = 2π
T is frequency (radian/s).
Note that for one period of the wave, e.g.
− T2 < t < T2 ,
f (t) =
(
A − T4 < t < T4
0 otherwise
(2)
4
Find cn.
1 T /2
cn =
f (t)e−jnω0tdt
T −T /2
←− Substitute Eq. (2) into left
Z
1 T /4
=
Ae−jnω0tdt
T −T /4
Z
1
−jnω
t
0
←− e
dt =
e−jnω0t + c
−jnω0
A
=
[e−jnω0(T /4) − ejnω0(T /4)]
T ∗ (−jnω0)
2π
←− ω0 =
T
A
=
[e−jn(π/2) − ejn(π/2)]
−j2πn
[ejnπ/2 − e−jnπ/2]
←−
= sin(nπ/2)
2j
A
=
sin(nπ/2)
(3)
nπ
Z
5
cn =
A
sin(nπ/2)
nπ
n
0
1
2
3
4
5
6
7
f
0
1
T
2
T
3
T
4
T
5
T
6
T
7
T
|cn|
A
2
A
π
0
A
3π
0
A
5π
0
A
7π
arg(cn)
0
0
0
−π
0
0
0
−π
6
Summary:
Having found cn, we simply have to find the
modulus to get the amplitude spectrum, and
the argument to get the phase.
The plots represent the distribution of cisoids,
in terms of the (positive and negative) harmonic number n. However it is to label the
axis as frequency, so that the lines appear at
intervals of 1/T .
7
Example: For the periodic rectangular wave
f (t) in the following Figure:
f(t)
A
0
T
2T
t
find the coefficients cn in the complex form of
Fourier series as given by (4),
f (t) =
∞
X
cnejnω0t
(4)
n=−∞
where ω0 = 2π
T is frequency (radian/s).
Note that for one period of the wave, e.g.
− T2 ≤ t < T2 ,
f (t) =
(
−A − T2 ≤ t < 0
A
0 ≤ t < T2
(5)
8
Find cn.
1 T /2
cn =
f (t)e−jnω0tdt
T −T /2
←− Substitute Eq. (5) into left
Z
Z T /2
1
1 0
=
−Ae−jnω0tdt +
Ae−jnω0tdt
T −T /2
T 0
Z
1
−jnω
t
0
dt =
e−jnω0t + c
←− e
−jnω0
A
=
[e−jnω0(0) − ejnω0(−T /2)]
T ∗ jnω0
A
+
[ejnω0(T /2) − e−jnω0(0)]
T ∗ (−jnω0)
2π 0
←− ω0 =
, e =1
T
A
A
−jnπ
]−
=
[1 − e
[ejnπ − 1]
j2πn
j2πn
[ejnπ + e−jnπ ]
←−
= cos(nπ)
2
A
A
=
−
cos(nπ)
jπn jnπ
A
(6)
= −j (1 − cos(nπ))
πn
Z
9
cn = −j
A
(1 − cos(nπ))
πn
n
0
1
2
3
4
5
6
7
f
0
1
T
2
T
3
T
4
T
5
T
6
T
7
T
|cn|
0
2A
π
0
2A
3π
0
2A
5π
0
2A
7π
arg(cn)
0
− π2
0
− π2
0
− π2
0
− π2
10