Formula Sheet: Fourier Series in Signals & Systems- 1 | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


Formula Sheet for Fourier Series (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Series: Represents a periodic signal as a sum of sinusoids.
• Periodic Signal: x(t) = x(t + T), period T, fundamental frequency f
0
=
1
T
,
?
0
=
2p
T
.
• Types: Trigonometric, exponential, and complex forms.
2. Continuous-Time Fourier Series
2.1 Exponential Form
• Synthesis:
x(t) =
8
?
k=-8
c
k
e
jk?
0
t
• Analysis:
c
k
=
1
T
?
T
x(t)e
-jk?
0
t
dt
where c
k
: Complex Fourier coe?cients, ?
0
=
2p
T
.
• Properties of Coe?cients:
– Real signal: c
-k
= c
*
k
(conjugate symmetry).
– Even signal: c
k
real and even.
– Odd signal: c
k
imaginary and odd.
2.2 Trigonometric Form
• Synthesis:
x(t) = a
0
+
8
?
k=1
[a
k
cos(k?
0
t)+b
k
sin(k?
0
t)]
• Analysis:
a
0
=
1
T
?
T
x(t)dt
a
k
=
2
T
?
T
x(t)cos(k?
0
t)dt, b
k
=
2
T
?
T
x(t)sin(k?
0
t)dt
• Relation to Exponential Form:
c
0
= a
0
, c
k
=
a
k
-jb
k
2
, c
-k
=
a
k
+jb
k
2
1
Page 2


Formula Sheet for Fourier Series (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Series: Represents a periodic signal as a sum of sinusoids.
• Periodic Signal: x(t) = x(t + T), period T, fundamental frequency f
0
=
1
T
,
?
0
=
2p
T
.
• Types: Trigonometric, exponential, and complex forms.
2. Continuous-Time Fourier Series
2.1 Exponential Form
• Synthesis:
x(t) =
8
?
k=-8
c
k
e
jk?
0
t
• Analysis:
c
k
=
1
T
?
T
x(t)e
-jk?
0
t
dt
where c
k
: Complex Fourier coe?cients, ?
0
=
2p
T
.
• Properties of Coe?cients:
– Real signal: c
-k
= c
*
k
(conjugate symmetry).
– Even signal: c
k
real and even.
– Odd signal: c
k
imaginary and odd.
2.2 Trigonometric Form
• Synthesis:
x(t) = a
0
+
8
?
k=1
[a
k
cos(k?
0
t)+b
k
sin(k?
0
t)]
• Analysis:
a
0
=
1
T
?
T
x(t)dt
a
k
=
2
T
?
T
x(t)cos(k?
0
t)dt, b
k
=
2
T
?
T
x(t)sin(k?
0
t)dt
• Relation to Exponential Form:
c
0
= a
0
, c
k
=
a
k
-jb
k
2
, c
-k
=
a
k
+jb
k
2
1
3. Discrete-Time Fourier Series
• Periodic Signal: x[n] = x[n+N], period N, fundamental frequency ? 0
=
2p
N
.
• Synthesis:
x[n] =
N-1
?
k=0
c
k
e
jk? 0
n
• Analysis:
c
k
=
1
N
N-1
?
n=0
x[n]e
-jk? 0
n
• Periodicity of Coe?cients: c
k+N
= c
k
.
• Properties:
– Real signal: c
N-k
= c
*
k
.
– Even signal: c
k
real and even.
– Odd signal: c
k
imaginary and odd.
4. Properties of Fourier Series
• Linearity:
ax
1
(t)+bx
2
(t)? ac
1k
+bc
2k
• Time Shifting:
x(t-t
0
)? c
k
e
-jk?
0
t
0
, x[n-n
0
]? c
k
e
-jk? 0
n
0
• Frequency Shifting:
x(t)e
jm?
0
t
? c
k-m
, x[n]e
jm? 0
n
? c
k-m
• Time Reversal:
x(-t)? c
-k
, x[-n]? c
-k
• Convolution:
x(t)*h(t)? Tc
k
d
k
, x[n]*h[n]? Nc
k
d
k
5. Parsevals Theorem
• Continuous-Time:
1
T
?
T
|x(t)|
2
dt =
8
?
k=-8
|c
k
|
2
• Discrete-Time:
1
N
N-1
?
n=0
|x[n]|
2
=
N-1
?
k=0
|c
k
|
2
2
Page 3


Formula Sheet for Fourier Series (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Series: Represents a periodic signal as a sum of sinusoids.
• Periodic Signal: x(t) = x(t + T), period T, fundamental frequency f
0
=
1
T
,
?
0
=
2p
T
.
• Types: Trigonometric, exponential, and complex forms.
2. Continuous-Time Fourier Series
2.1 Exponential Form
• Synthesis:
x(t) =
8
?
k=-8
c
k
e
jk?
0
t
• Analysis:
c
k
=
1
T
?
T
x(t)e
-jk?
0
t
dt
where c
k
: Complex Fourier coe?cients, ?
0
=
2p
T
.
• Properties of Coe?cients:
– Real signal: c
-k
= c
*
k
(conjugate symmetry).
– Even signal: c
k
real and even.
– Odd signal: c
k
imaginary and odd.
2.2 Trigonometric Form
• Synthesis:
x(t) = a
0
+
8
?
k=1
[a
k
cos(k?
0
t)+b
k
sin(k?
0
t)]
• Analysis:
a
0
=
1
T
?
T
x(t)dt
a
k
=
2
T
?
T
x(t)cos(k?
0
t)dt, b
k
=
2
T
?
T
x(t)sin(k?
0
t)dt
• Relation to Exponential Form:
c
0
= a
0
, c
k
=
a
k
-jb
k
2
, c
-k
=
a
k
+jb
k
2
1
3. Discrete-Time Fourier Series
• Periodic Signal: x[n] = x[n+N], period N, fundamental frequency ? 0
=
2p
N
.
• Synthesis:
x[n] =
N-1
?
k=0
c
k
e
jk? 0
n
• Analysis:
c
k
=
1
N
N-1
?
n=0
x[n]e
-jk? 0
n
• Periodicity of Coe?cients: c
k+N
= c
k
.
• Properties:
– Real signal: c
N-k
= c
*
k
.
– Even signal: c
k
real and even.
– Odd signal: c
k
imaginary and odd.
4. Properties of Fourier Series
• Linearity:
ax
1
(t)+bx
2
(t)? ac
1k
+bc
2k
• Time Shifting:
x(t-t
0
)? c
k
e
-jk?
0
t
0
, x[n-n
0
]? c
k
e
-jk? 0
n
0
• Frequency Shifting:
x(t)e
jm?
0
t
? c
k-m
, x[n]e
jm? 0
n
? c
k-m
• Time Reversal:
x(-t)? c
-k
, x[-n]? c
-k
• Convolution:
x(t)*h(t)? Tc
k
d
k
, x[n]*h[n]? Nc
k
d
k
5. Parsevals Theorem
• Continuous-Time:
1
T
?
T
|x(t)|
2
dt =
8
?
k=-8
|c
k
|
2
• Discrete-Time:
1
N
N-1
?
n=0
|x[n]|
2
=
N-1
?
k=0
|c
k
|
2
2
6. Symmetry and Coe?cients
• Real and Even Signal: c
k
real, a
k
non-zero, b
k
= 0.
• Real and Odd Signal: c
k
imaginary, a
k
= 0, b
k
non-zero.
• Half-Wave Symmetry: x(t+T/2) =-x(t), only odd harmonics (c
k
= 0 for even
k).
7. Fourier Series of Common Signals
• Square Wave (period T, amplitude A):
c
k
=
A
pk
sin
(
kpt
T
)
, k?= 0, c
0
=
At
T
where t: Pulse width.
• Sawtooth Wave:
c
k
=
A
j2pk
, k?= 0, c
0
=
A
2
8. Gibbs Phenomenon
• E?ect: Overshoot ( 9%) near discontinuities in partial sum of Fourier series.
• Cause: Slow convergence of series at discontinuities.
9. Convergence Conditions
• Dirichlet Conditions:
– Signal has ?nite number of discontinuities in one period.
– Signal has ?nite number of extrema in one period.
– Signal is absolutely integrable:
?
T
|x(t)|dt <8.
10. Design Considerations
• Harmonic Analysis: Use Fourier coe?cients to analyze signal frequency content.
• Signal Reconstruction: Higher harmonics improve accuracy but increase com-
plexity.
• Applications: Signal processing, communication systems, ?lter design.
• Symmetry Exploitation: Use even/odd properties to simplify coe?cient calcu-
lations.
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