Short Notes: Fourier Series in Signals & Systems | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


F ourier Series
The F ourier Series is a fundamen tal concept in signal pro cessing that represen ts a p erio dic signal as a
sum of sin usoidal c omp onen ts (sines and cosines) with differen t frequencies, amplitudes, and phases. It is
widely used in the analysis of p erio dic signals in applications suc h as audio pro cessing, comm unications,
and con trol systems.
1. In tro duction to F ourier Series
A p erio d ic signal x(t) with p erio d T can b e expressed as a F ourier Series, whic h decomp oses the signal
in to a sum of harmonically related sin usoids. The frequencies of these sin usoids are in teger m ultiples of
the fundamen tal frequency f
0
=
1
T
. The F ourier Series pro vides a frequency-domain represen tation of
p e rio dic signals, enabling analysis of their sp ectral con ten t.
2. Mathematical Definition
F or a p erio dic signal x(t) with p erio d T , the F ourier Series is giv en b y:
x(t) = a
0
+
8
?
n=1
[
a
n
cos
(
2pnt
T
)
+b
n
sin
(
2pnt
T
)]
where:
• a
0
is the DC comp onen t (a v erage v alue of the signal),
• a
n
and b
n
are the F ourier co e?icien ts for the n -th harmonic,
•
2pn
T
= 2pnf
0
is the angular frequency of the n -th harmonic.
Alternativ ely , the F ourier Series can b e expressed in complex exp onen tial form:
x(t) =
8
?
n=-8
c
n
e
j
2pnt
T
where c
n
are the c omplex F ourier co e?icien ts, related to a
n
and b
n
b y:
c
n
=
a
n
-jb
n
2
, c
-n
=
a
n
+jb
n
2
, c
0
= a
0
3. F ourier Co e?icien ts
The F ourier co e?icien ts are calculated as follo ws:
• DC comp onen t:
a
0
=
1
T
?
T
x(t)dt
• Cosine co e?icien ts:
a
n
=
2
T
?
T
x(t)cos
(
2pnt
T
)
dt, n = 1,2,...
• Sine co e?icien ts:
b
n
=
2
T
?
T
x(t)sin
(
2pnt
T
)
dt, n = 1,2,...
1
Page 2


F ourier Series
The F ourier Series is a fundamen tal concept in signal pro cessing that represen ts a p erio dic signal as a
sum of sin usoidal c omp onen ts (sines and cosines) with differen t frequencies, amplitudes, and phases. It is
widely used in the analysis of p erio dic signals in applications suc h as audio pro cessing, comm unications,
and con trol systems.
1. In tro duction to F ourier Series
A p erio d ic signal x(t) with p erio d T can b e expressed as a F ourier Series, whic h decomp oses the signal
in to a sum of harmonically related sin usoids. The frequencies of these sin usoids are in teger m ultiples of
the fundamen tal frequency f
0
=
1
T
. The F ourier Series pro vides a frequency-domain represen tation of
p e rio dic signals, enabling analysis of their sp ectral con ten t.
2. Mathematical Definition
F or a p erio dic signal x(t) with p erio d T , the F ourier Series is giv en b y:
x(t) = a
0
+
8
?
n=1
[
a
n
cos
(
2pnt
T
)
+b
n
sin
(
2pnt
T
)]
where:
• a
0
is the DC comp onen t (a v erage v alue of the signal),
• a
n
and b
n
are the F ourier co e?icien ts for the n -th harmonic,
•
2pn
T
= 2pnf
0
is the angular frequency of the n -th harmonic.
Alternativ ely , the F ourier Series can b e expressed in complex exp onen tial form:
x(t) =
8
?
n=-8
c
n
e
j
2pnt
T
where c
n
are the c omplex F ourier co e?icien ts, related to a
n
and b
n
b y:
c
n
=
a
n
-jb
n
2
, c
-n
=
a
n
+jb
n
2
, c
0
= a
0
3. F ourier Co e?icien ts
The F ourier co e?icien ts are calculated as follo ws:
• DC comp onen t:
a
0
=
1
T
?
T
x(t)dt
• Cosine co e?icien ts:
a
n
=
2
T
?
T
x(t)cos
(
2pnt
T
)
dt, n = 1,2,...
• Sine co e?icien ts:
b
n
=
2
T
?
T
x(t)sin
(
2pnt
T
)
dt, n = 1,2,...
1
• Complex co e?icien ts:
c
n
=
1
T
?
T
x(t)e
-j
2pnt
T
dt, n = 0,±1,±2,...
The in tegrals are tak en o v er one p erio d T of the signal.
4. Pr op erties of F ourier Series
The F ourier Series has sev eral k ey prop erties:
1. Linearit y : If x(t) = ax
1
(t)+bx
2
(t) , then the F ourier co e?ic ien ts of x(t) are a times those of x
1
(t)
plus b times those of x
2
(t) .
2. Time Shifting : A time shift x(t-t
0
) mo difies the phase of the c omplex co e?icien ts:
c
n
? c
n
e
-j
2pnt
0
T
3. P arsev al’s Theorem : The a v erage p o w er of the signal is r elated to the F ourier co e?icien ts:
1
T
?
T
|x(t)|
2
dt =
8
?
n=-8
|c
n
|
2
4. Conjugate Symmetry : F or real-v alued signals, c
-n
= c
*
n
, where c
*
n
is the complex conjugate of
c
n
.
5. Con v ergence Conditions
The F ourier Series con v erges to x(t) under certain conditions (Diric hlet conditions):
• x(t) is p erio dic and absolutely in tegrable o v er one p erio d.
• x(t) has a finite n um b er of discon tin uities and extrema in one p erio d.
• A t discon tin uities, the series con v erges to the a v erage of the left and righ t limits.
6. Applications of F ourier Series
F ourier Series are used in v arious domains:
• Signal Analysis : T o decomp ose p erio dic signals (e.g., audio or electrical w a v eforms) in to their
frequency comp onen ts.
• Circuit Analysis : T o analyze the resp onse of linear systems to p erio dic inputs, suc h as in A C
circuits.
• Comm unications : T o design mo dulation sc hemes and filters for p erio dic signals.
• Vibration Analysis : T o study mec hanical sys tems with p erio dic motion.
7. Gibbs Phenomenon
When a F ourier Series appro ximates a signal with discon tin uities (e.g., a square w a v e), the partial sum
exhibits oscillations near the discon tin uities, kno wn as the Gibbs Phenomenon. These oscillations do not
diminish with more terms but con v erge to a fixed o v ersho ot (appro ximately 9
2
Page 3


F ourier Series
The F ourier Series is a fundamen tal concept in signal pro cessing that represen ts a p erio dic signal as a
sum of sin usoidal c omp onen ts (sines and cosines) with differen t frequencies, amplitudes, and phases. It is
widely used in the analysis of p erio dic signals in applications suc h as audio pro cessing, comm unications,
and con trol systems.
1. In tro duction to F ourier Series
A p erio d ic signal x(t) with p erio d T can b e expressed as a F ourier Series, whic h decomp oses the signal
in to a sum of harmonically related sin usoids. The frequencies of these sin usoids are in teger m ultiples of
the fundamen tal frequency f
0
=
1
T
. The F ourier Series pro vides a frequency-domain represen tation of
p e rio dic signals, enabling analysis of their sp ectral con ten t.
2. Mathematical Definition
F or a p erio dic signal x(t) with p erio d T , the F ourier Series is giv en b y:
x(t) = a
0
+
8
?
n=1
[
a
n
cos
(
2pnt
T
)
+b
n
sin
(
2pnt
T
)]
where:
• a
0
is the DC comp onen t (a v erage v alue of the signal),
• a
n
and b
n
are the F ourier co e?icien ts for the n -th harmonic,
•
2pn
T
= 2pnf
0
is the angular frequency of the n -th harmonic.
Alternativ ely , the F ourier Series can b e expressed in complex exp onen tial form:
x(t) =
8
?
n=-8
c
n
e
j
2pnt
T
where c
n
are the c omplex F ourier co e?icien ts, related to a
n
and b
n
b y:
c
n
=
a
n
-jb
n
2
, c
-n
=
a
n
+jb
n
2
, c
0
= a
0
3. F ourier Co e?icien ts
The F ourier co e?icien ts are calculated as follo ws:
• DC comp onen t:
a
0
=
1
T
?
T
x(t)dt
• Cosine co e?icien ts:
a
n
=
2
T
?
T
x(t)cos
(
2pnt
T
)
dt, n = 1,2,...
• Sine co e?icien ts:
b
n
=
2
T
?
T
x(t)sin
(
2pnt
T
)
dt, n = 1,2,...
1
• Complex co e?icien ts:
c
n
=
1
T
?
T
x(t)e
-j
2pnt
T
dt, n = 0,±1,±2,...
The in tegrals are tak en o v er one p erio d T of the signal.
4. Pr op erties of F ourier Series
The F ourier Series has sev eral k ey prop erties:
1. Linearit y : If x(t) = ax
1
(t)+bx
2
(t) , then the F ourier co e?ic ien ts of x(t) are a times those of x
1
(t)
plus b times those of x
2
(t) .
2. Time Shifting : A time shift x(t-t
0
) mo difies the phase of the c omplex co e?icien ts:
c
n
? c
n
e
-j
2pnt
0
T
3. P arsev al’s Theorem : The a v erage p o w er of the signal is r elated to the F ourier co e?icien ts:
1
T
?
T
|x(t)|
2
dt =
8
?
n=-8
|c
n
|
2
4. Conjugate Symmetry : F or real-v alued signals, c
-n
= c
*
n
, where c
*
n
is the complex conjugate of
c
n
.
5. Con v ergence Conditions
The F ourier Series con v erges to x(t) under certain conditions (Diric hlet conditions):
• x(t) is p erio dic and absolutely in tegrable o v er one p erio d.
• x(t) has a finite n um b er of discon tin uities and extrema in one p erio d.
• A t discon tin uities, the series con v erges to the a v erage of the left and righ t limits.
6. Applications of F ourier Series
F ourier Series are used in v arious domains:
• Signal Analysis : T o decomp ose p erio dic signals (e.g., audio or electrical w a v eforms) in to their
frequency comp onen ts.
• Circuit Analysis : T o analyze the resp onse of linear systems to p erio dic inputs, suc h as in A C
circuits.
• Comm unications : T o design mo dulation sc hemes and filters for p erio dic signals.
• Vibration Analysis : T o study mec hanical sys tems with p erio dic motion.
7. Gibbs Phenomenon
When a F ourier Series appro ximates a signal with discon tin uities (e.g., a square w a v e), the partial sum
exhibits oscillations near the discon tin uities, kno wn as the Gibbs Phenomenon. These oscillations do not
diminish with more terms but con v erge to a fixed o v ersho ot (appro ximately 9
2
8. Practical Considerations
• Finite T erms : In practice, only a finite n um b er of terms are used, leading to appro ximation
errors, esp ecially near discon tin uities.
• Symmetry Exploitation : Ev en or o dd symmetry in x(t) simplifies co e?icien t calculations (e.g.,
b
n
= 0 for ev en signals, a
n
= 0 for o dd signals).
• Numerical Computation : In digital systems, F ourier Series co e?icien ts are often computed
n umerically using algorithms lik e the F ast F ourier T ransform (FFT) for discrete s ignals.
9. Conclusion
The F ourier Series is a p o w erful to ol for represen ting p erio dic signals as sums of sin usoids, enabling
frequency-domain analysis in signals and systems. By pro viding insigh ts in to the sp ectral con ten t of
signals, it forms the foundation for man y signal pro cessing tec hniques and is essen tial for understanding
linear systems and their resp onses.
3