• 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)
1
Example: Derive complex Fourier Series for
the rectangular form in the Figure below, and
the amplitude and phase spectrum.
Α
0
τ
1 τ
cn =
Ae−jnω0tdt
T 0
Z
Z
←−
Τ
e−jnω0tdt =
t
1
e−jnω0t + c
−jnω0
A
1
=
(
)[e−jnω0τ − e−jnω0×0]
T −jnω0
←− e0 = 1
A
1
=
(
)[e−jnω0τ − 1]
T −jnω0
2
cn =
=
=
=
=
A
1
(
)[e−jnω0 τ − 1]
T −jnω0
−jnω0 τ
−jnω0 τ
jnω0 τ
jnω0 τ
A
1
(
)[e 2 − 2 − e 2 + 2 ]
T −jnω0
←− ea+b = ea eb
jnω0 τ
−jnω0 τ
−jnω0 τ
jnω0 τ
A
1
(
)[e 2 e− 2 − e 2 e 2 ]
T −jnω0
−jnω0 τ
jnω0 τ
jnω0 τ
1
A
(
)e 2 [e− 2 − e 2 ]
T −jnω0
1
A
)e
(
T −jnω0
=
=
=
jnω0 τ
2
[−2j]
[e
jnω0 τ
2
−e
2j
−
jnω0 τ
2
]
jnω0 τ
nω0 τ
− e− 2 ]
←−
= sin[
]
2j
2
−jnω0 τ
2A
nω0 τ
(
)e 2 sin[
]
nω0 T
2
←− ω0 T = 2π
A −jnπτ
nπτ
e T sin[
]
nπ
T
A( Tτ ) −jnπτ
nπτ
T
e
sin[
]
nπ( Tτ )
T
sin[ nπτ
]
nτ
←− sinc[ ] = nπτT
T
[ T ]
Aτ
nτ −jnπτ
sinc[ ]e T
T
T
[e
=
−jnω0 τ
2
3
Note that the sinc function is given by
sin πx
sinc(x) =
πx
4
Aτ
nτ −jnπτ
cn =
sinc[ ]e T
T
T
nτ ],
|cn| = Aτ
sinc[
T
T
arg(cn) = − nπτ
T
6
4
τ=2.5
2
T=10
0
−2
−20
−15
−10
−5
2
0
time
5
10
15
20
1
0
1/T
−1
−2
π
1/τ
3/τ
2/τ
−1.5
−1
−0.5
0
frequency
0.5
1
1.5
2
−1.5
−1
−0.5
0
frequency
0.5
1
1.5
2
4
2
0
−2
−π
−4
−2
5
The harmonics are placed at intervals of 1/T ,
their envelop following the (modulus) of the
sinc function. A zero amplitude occurs whenT = 4, the
ever sinc( nτ
)
is
integral
so
with
T
τ
fourth, eighth, twelfth lines etc. are zero. These
zeros occurs at frequencies 1/τ , 2/t , 3/τ etc..
The repetition of the waveform produces lines
every 1/T Hz and the envelope of the spectrum
is determined by the shape of the waveform.
− jnπτ
T
is a phase term dependent on
The term e
the choice of origin and vanishes if the origin
is in chosen in the center of a pulse. In general
a shift of origin of θ in time produces a phase
term of e−jnω0θ in the corresponding spectrum.
6
•Useful deductions:
(i) For a given period T , the value of τ determines the distribution of power in the spectrum.
6
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
−1
−0.2
−2
−20
−10
0
time
10
20
−2
−1
0
frequency
1
2
−1
0
frequency
1
2
small τ
6
3
5
2.5
4
2
3
1.5
2
1
1
0
0.5
−1
−2
−20
0
−10
0
time
10
20
−2
large τ
7
(ii) For a given value of pulse width τ , the
period T similarly determines determines the
power distribution.
6
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
1/τ
2/τ
−1
−0.2
−2
−20
−10
0
time
10
20
−1.5
1/T
−1
−0.5
0
0.5
frequency
−1
−0.5
0
0.5
frequency
1
1.5
large T
6
1.8
5
1.6
4
1.4
1.2
3
1
2
0.8
0.6
1
0.4
0
0.2
0
−1
1/T
−0.2
−2
−20
−10
0
time
10
20
−1.5
1/τ
2/τ
1
small T
8
1.5
(iii) If we put T = τ , we get a constant (d.c)
level. |cn| is then given by Asinc(n), so a single
spectral line of height A occurs at zero frequency.
6
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
−1
−0.2
−2
−20
−10
0
time
10
20
−1.5
−1
−0.5
0
0.5
frequency
1
1.5
9
(iv) If we let the repetition period T become
very large, the line spacing 1/T becomes very
small. As T tends to infinity, the spacing tends
to zero and we get a continuous spectrum.
This is because f (t) becomes a finite energy
signal if T is infinite. Such signal has continuous spectra.
6
0.1
0.09
5
0.08
4
0.07
3
0.06
0.05
2
0.04
1
0.03
0
0.02
−1
−2
−20
0.01
−10
0
time
10
20
0
−2
−1
0
frequency
1
2
10
(v) Suppose we make τ small but keep the
pulse area Aτ constant. In the limit we get an
impulse of strength Aτ , and the spectrum will
simply be a set of lines of constants heights A
T.
5000
2
4500
1.8
4000
1.6
3500
1.4
3000
1.2
2500
1
2000
0.8
1500
0.6
1000
0.4
500
0.2
0
−20
−10
0
time
10
20
0
−4
−2
0
frequency
2
4
11
(vi) Finally, it is clear tha a single impulse will
have a constant but continuous spectrum.
6000
0.1
0.09
5000
0.08
0.07
4000
0.06
0.05
3000
0.04
2000
0.03
0.02
1000
0.01
0
−20
−10
0
time
10
20
0
−0.2
−0.1
0
frequency
0.1
0.2
12