Formula Sheets: Fourier Transform in Signals & Systems | Signals and Systems - Electrical Engineering (EE) PDF Download

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Formula Sheet for Fourier Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Transform: Represents a non-periodic signal in the frequency domain.
• Continuous-Time Signal: x(t), Fourier Transform X(f) or X(j?).
• Discrete-Time Signal: x[n], Discrete-Time Fourier Transform (DTFT) X(e
j? ).
2. Continuous-Time Fourier Transform (CTFT)
• Analysis (Forward Transform):
X(f) =
?
8
-8
x(t)e
-j2pft
dt
X(j?) =
?
8
-8
x(t)e
-j?t
dt, ? = 2pf
• Synthesis (Inverse Transform):
x(t) =
?
8
-8
X(f)e
j2pft
df
x(t) =
1
2p
?
8
-8
X(j?)e
j?t
d?
3. Discrete-Time Fourier Transform (DTFT)
• Analysis:
X(e
j? ) =
8
?
n=-8
x[n]e
-j? n
• Synthesis:
x[n] =
1
2p
?
p
-p
X(e
j? )e
j? n
d? • Periodicity: X(e
j(?+2 p)
) = X(e
j? ).
4. Properties of CTFT
• Linearity:
ax
1
(t)+bx
2
(t)? aX
1
(j?)+bX
2
(j?)
• Time Shifting:
x(t-t
0
)? X(j?)e
-j?t
0
• Frequency Shifting:
x(t)e
j?
0
t
? X(j(?-?
0
))
1
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Formula Sheet for Fourier Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Transform: Represents a non-periodic signal in the frequency domain.
• Continuous-Time Signal: x(t), Fourier Transform X(f) or X(j?).
• Discrete-Time Signal: x[n], Discrete-Time Fourier Transform (DTFT) X(e
j? ).
2. Continuous-Time Fourier Transform (CTFT)
• Analysis (Forward Transform):
X(f) =
?
8
-8
x(t)e
-j2pft
dt
X(j?) =
?
8
-8
x(t)e
-j?t
dt, ? = 2pf
• Synthesis (Inverse Transform):
x(t) =
?
8
-8
X(f)e
j2pft
df
x(t) =
1
2p
?
8
-8
X(j?)e
j?t
d?
3. Discrete-Time Fourier Transform (DTFT)
• Analysis:
X(e
j? ) =
8
?
n=-8
x[n]e
-j? n
• Synthesis:
x[n] =
1
2p
?
p
-p
X(e
j? )e
j? n
d? • Periodicity: X(e
j(?+2 p)
) = X(e
j? ).
4. Properties of CTFT
• Linearity:
ax
1
(t)+bx
2
(t)? aX
1
(j?)+bX
2
(j?)
• Time Shifting:
x(t-t
0
)? X(j?)e
-j?t
0
• Frequency Shifting:
x(t)e
j?
0
t
? X(j(?-?
0
))
1
• Time Scaling:
x(at)?
1
|a|
X
(
j?
a
)
• Convolution:
x(t)*h(t)? X(j?)H(j?)
• Multiplication:
x(t)h(t)?
1
2p
X(j?)*H(j?)
• Di?erentiation:
dx(t)
dt
? j?X(j?)
• Integration:
?
t
-8
x(t)dt ?
X(j?)
j?
+pX(0)d(?)
5. Properties of DTFT
• Linearity:
ax
1
[n]+bx
2
[n]? aX
1
(e
j? )+bX
2
(e
j? )
• Time Shifting:
x[n-n
0
]? X(e
j? )e
-j? n
0
• Frequency Shifting:
x[n]e
j? 0
n
? X(e
j(? -? 0
)
)
• Convolution:
x[n]*h[n]? X(e
j? )H(e
j? )
• Multiplication:
x[n]h[n]?
1
2p
X(e
j? )*H(e
j? )
• Di?erence:
x[n]-x[n-1]? (1-e
-j? )X(e
j? )
6. Parsevals Theorem
• Continuous-Time:
?
8
-8
|x(t)|
2
dt =
?
8
-8
|X(f)|
2
df =
1
2p
?
8
-8
|X(j?)|
2
d?
• Discrete-Time:
8
?
n=-8
|x[n]|
2
=
1
2p
?
p
-p
|X(e
j? )|
2
d? 2
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Formula Sheet for Fourier Transform (Signals and
Systems) – GATE
1. Basic Concepts
• Fourier Transform: Represents a non-periodic signal in the frequency domain.
• Continuous-Time Signal: x(t), Fourier Transform X(f) or X(j?).
• Discrete-Time Signal: x[n], Discrete-Time Fourier Transform (DTFT) X(e
j? ).
2. Continuous-Time Fourier Transform (CTFT)
• Analysis (Forward Transform):
X(f) =
?
8
-8
x(t)e
-j2pft
dt
X(j?) =
?
8
-8
x(t)e
-j?t
dt, ? = 2pf
• Synthesis (Inverse Transform):
x(t) =
?
8
-8
X(f)e
j2pft
df
x(t) =
1
2p
?
8
-8
X(j?)e
j?t
d?
3. Discrete-Time Fourier Transform (DTFT)
• Analysis:
X(e
j? ) =
8
?
n=-8
x[n]e
-j? n
• Synthesis:
x[n] =
1
2p
?
p
-p
X(e
j? )e
j? n
d? • Periodicity: X(e
j(?+2 p)
) = X(e
j? ).
4. Properties of CTFT
• Linearity:
ax
1
(t)+bx
2
(t)? aX
1
(j?)+bX
2
(j?)
• Time Shifting:
x(t-t
0
)? X(j?)e
-j?t
0
• Frequency Shifting:
x(t)e
j?
0
t
? X(j(?-?
0
))
1
• Time Scaling:
x(at)?
1
|a|
X
(
j?
a
)
• Convolution:
x(t)*h(t)? X(j?)H(j?)
• Multiplication:
x(t)h(t)?
1
2p
X(j?)*H(j?)
• Di?erentiation:
dx(t)
dt
? j?X(j?)
• Integration:
?
t
-8
x(t)dt ?
X(j?)
j?
+pX(0)d(?)
5. Properties of DTFT
• Linearity:
ax
1
[n]+bx
2
[n]? aX
1
(e
j? )+bX
2
(e
j? )
• Time Shifting:
x[n-n
0
]? X(e
j? )e
-j? n
0
• Frequency Shifting:
x[n]e
j? 0
n
? X(e
j(? -? 0
)
)
• Convolution:
x[n]*h[n]? X(e
j? )H(e
j? )
• Multiplication:
x[n]h[n]?
1
2p
X(e
j? )*H(e
j? )
• Di?erence:
x[n]-x[n-1]? (1-e
-j? )X(e
j? )
6. Parsevals Theorem
• Continuous-Time:
?
8
-8
|x(t)|
2
dt =
?
8
-8
|X(f)|
2
df =
1
2p
?
8
-8
|X(j?)|
2
d?
• Discrete-Time:
8
?
n=-8
|x[n]|
2
=
1
2p
?
p
-p
|X(e
j? )|
2
d? 2
7. Common Fourier Transform Pairs
• Impulse:
d(t)? 1, d[n]? 1
• Unit Step:
u(t)?
1
j?
+pd(?), u[n]?
1
1-e
-j? +p
8
?
k=-8
d(? -2pk)
• Rectangular Pulse:
rect
(
t
t
)
? tsinc(ft), sinc(u) =
sin(pu)
pu
• Exponential Decay:
e
-at
u(t)?
1
a+j?
, a > 0
8. Symmetry Properties
• Real Signal (CTFT):
X(j?) = X
*
(-j?), |X(j?)| even, ?X(j?) odd
• Real Signal (DTFT):
X(e
j? ) = X
*
(e
-j? )
• Even Signal: X(j?) real and even.
• Odd Signal: X(j?) imaginary and odd.
9. Convergence Conditions
• CTFT: Signal must be absolutely integrable:
?
8
-8
|x(t)|dt <8
or have ?nite energy.
• DTFT: Signal must be absolutely summable:
8
?
n=-8
|x[n]| <8
10. Design Considerations
• Frequency Analysis: Use Fourier transform to analyze signal spectrum.
• Filtering: Convolution in time domain corresponds to multiplication in frequency
domain.
• Applications: Signal processing, communication systems, audio/image analysis.
• NumericalComputation: UseFFT(FastFourierTransform)fordiscretesignals.
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