1
ContinuousTime Fourier Series
Notes for ECE 301 Signals and Systems Section 1, Fall 2011
Ilya Pollak
Purdue University
The concepts of orthogonal bases and projections can be extended to spaces
of CT signals. Determining whether a series representation converges (and if so,
what it converges to) is much more complicated than for Fniteduration or periodic
DT signals. We therefore will only consider one very important example–CT
±ourier series.
Consider the set of signals
L
2
(
T
) deFned as follows: it is the set of all periodic
complexvalued CT signals
s
(
t
) with period
T
for which
i
τ
+
T
τ

s
(
t
)

2
dt <
∞
,
where
τ
is an arbitrary real number–i.e., the integral is taken over any period. It
turns out that
L
2
(
T
) is a vector space (each vector in this case being a continuous
time signal). We deFne the inner product of two signals as follows:
a
s, g
A
=
i
τ
+
T
τ
s
(
t
)(
g
(
t
))
*
dt.
As in
C
N
,
s
and
g
being
orthogonal
still means
a
s, g
A
= 0. As in
C
N
, the
norm
of
a signal
s
is deFned as the square root of its inner product with itself:
b
s
b ≡
r
a
s, s
A
=
R
i
τ
+
T
τ

s
(
t
)

2
dt.
Note that the deFnition of
L
2
(
T
) guarantees that this quantity is Fnite for any
signal in
L
2
(
T
).
It turns out that the inFnite collection of
T
periodic complex exponentials
φ
k
(
t
) = exp
p
j
2
πkt
T
P
, k
= 0
,
±
1
,
±
2
, . . .
forms an orthogonal basis for
L
2
(
T
). In other words, these signals are pairwise
orthogonal (as shown below), and we can represent any
T
periodic CT signal
s
∈
L
2
(
T
) as a linear combination of these complex exponentials:
s
(
t
) =
∞
s
k
=
∞
a
k
φ
k
(
t
) =
∞
s
k
=
∞
a
k
exp
p
j
2
T
P
.
(1)
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The frst “=” sign in Eq. (1) needs careFul interpretation: unlike the fnite
duration DT case, the equality here is not pointwise. Instead, the equality is
understood in the Following sense:
v
v
v
v
v
s
−
M
s
k
=
N
a
k
φ
k
v
v
v
v
v
→
0
as
N
→ −∞
and
M
→ ∞
.
Nevertheless, the coe±cient Formula previously derived For orthogonal represen
tations in
C
N
, is still valid:
a
k
=
a
s, φ
k
A
a
φ
k
, φ
k
A
.
(2)
The inner product oF
φ
k
and
φ
i
is:
a
φ
k
, φ
i
A
=
i
τ
+
T
τ
exp
p
j
2
πkt
T
P
exp
p
−
j
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 Fall '06
 V."Ragu"Balakrishnan
 Fourier Series, Periodic function, Complex number, Orthonormal basis

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