Formula Sheets: Sampling Theorem | Signals and Systems - Electrical Engineering (EE) PDF Download

 Page 1


Formula Sheet for Sampling Theorem (Signals and
Systems) – GATE
1. Basic Concepts
• Sampling: Converting a continuous-time signal x(t) to a discrete-time signal x[n].
• SamplingProcess: x[n] =x(nT
s
), whereT
s
: Samplingperiod,f
s
=
1
Ts
: Sampling
frequency.
• Nyquist-Shannon Sampling Theorem: A band-limited signal with bandwidth
B (highest frequency) can be perfectly reconstructed if sampled at f
s
= 2B.
2. Sampling Theorem
• Nyquist Rate: Minimum sampling frequency to avoid aliasing:
f
s
= 2f
max
where f
max
: Maximum frequency in the signal spectrum.
• Nyquist Frequency:
f
N
=
f
s
2
3. Sampling in Time Domain
• Sampled Signal:
x
s
(t) = x(t)·
8
?
n=-8
d(t-nT
s
) =
8
?
n=-8
x(nT
s
)d(t-nT
s
)
• Discrete-Time Signal:
x[n] =x(nT
s
)
4. Sampling in Frequency Domain
• Fourier Transform of Sampled Signal:
X
s
(f) =
1
T
s
8
?
k=-8
X(f -kf
s
)
where X(f): Fourier transform of x(t).
• Aliasing: Occurs when f
s
< 2f
max
, causing spectrum overlap.
1
Page 2


Formula Sheet for Sampling Theorem (Signals and
Systems) – GATE
1. Basic Concepts
• Sampling: Converting a continuous-time signal x(t) to a discrete-time signal x[n].
• SamplingProcess: x[n] =x(nT
s
), whereT
s
: Samplingperiod,f
s
=
1
Ts
: Sampling
frequency.
• Nyquist-Shannon Sampling Theorem: A band-limited signal with bandwidth
B (highest frequency) can be perfectly reconstructed if sampled at f
s
= 2B.
2. Sampling Theorem
• Nyquist Rate: Minimum sampling frequency to avoid aliasing:
f
s
= 2f
max
where f
max
: Maximum frequency in the signal spectrum.
• Nyquist Frequency:
f
N
=
f
s
2
3. Sampling in Time Domain
• Sampled Signal:
x
s
(t) = x(t)·
8
?
n=-8
d(t-nT
s
) =
8
?
n=-8
x(nT
s
)d(t-nT
s
)
• Discrete-Time Signal:
x[n] =x(nT
s
)
4. Sampling in Frequency Domain
• Fourier Transform of Sampled Signal:
X
s
(f) =
1
T
s
8
?
k=-8
X(f -kf
s
)
where X(f): Fourier transform of x(t).
• Aliasing: Occurs when f
s
< 2f
max
, causing spectrum overlap.
1
5. Signal Reconstruction
• Ideal Reconstruction Formula:
x(t) =
8
?
n=-8
x(nT
s
)·sinc
(
t-nT
s
T
s
)
where sinc(u) =
sin(pu)
pu
.
• Ideal Low-Pass Filter for Reconstruction:
H(f) =
?
?
?
T
s
if |f|=
fs
2
0 otherwise
• Impulse Response of Filter:
h(t) = sinc
(
t
T
s
)
6. Aliasing
• Condition for No Aliasing:
f
s
= 2f
max
• Aliasing E?ect: Frequencies f +kf
s
(for integer k) overlap, distorting the signal.
• Anti-AliasingFilter: Low-pass?lterwithcut-o?f
c
=
fs
2
appliedbeforesampling.
7. Sampling of Bandpass Signals
• Bandpass Signal: Bandwidth B, frequency range [f
L
,f
H
], where B =f
H
-f
L
.
• Minimum Sampling Frequency:
f
s
= 2B
• Undersampling: Possible if f
s
= 2B and f
H
/B is an integer, allowing frequency
translation.
8. Discrete-Time Fourier Transform (DTFT) of Sam-
pled Signal
• DTFT:
X(e
j? ) =
8
?
n=-8
x[n]e
-j? n
• Relation to Continuous-Time Spectrum:
X(e
j? ) =
1
T
s
8
?
k=-8
X
(
? -2pk
2pT
s
)
2
Page 3


Formula Sheet for Sampling Theorem (Signals and
Systems) – GATE
1. Basic Concepts
• Sampling: Converting a continuous-time signal x(t) to a discrete-time signal x[n].
• SamplingProcess: x[n] =x(nT
s
), whereT
s
: Samplingperiod,f
s
=
1
Ts
: Sampling
frequency.
• Nyquist-Shannon Sampling Theorem: A band-limited signal with bandwidth
B (highest frequency) can be perfectly reconstructed if sampled at f
s
= 2B.
2. Sampling Theorem
• Nyquist Rate: Minimum sampling frequency to avoid aliasing:
f
s
= 2f
max
where f
max
: Maximum frequency in the signal spectrum.
• Nyquist Frequency:
f
N
=
f
s
2
3. Sampling in Time Domain
• Sampled Signal:
x
s
(t) = x(t)·
8
?
n=-8
d(t-nT
s
) =
8
?
n=-8
x(nT
s
)d(t-nT
s
)
• Discrete-Time Signal:
x[n] =x(nT
s
)
4. Sampling in Frequency Domain
• Fourier Transform of Sampled Signal:
X
s
(f) =
1
T
s
8
?
k=-8
X(f -kf
s
)
where X(f): Fourier transform of x(t).
• Aliasing: Occurs when f
s
< 2f
max
, causing spectrum overlap.
1
5. Signal Reconstruction
• Ideal Reconstruction Formula:
x(t) =
8
?
n=-8
x(nT
s
)·sinc
(
t-nT
s
T
s
)
where sinc(u) =
sin(pu)
pu
.
• Ideal Low-Pass Filter for Reconstruction:
H(f) =
?
?
?
T
s
if |f|=
fs
2
0 otherwise
• Impulse Response of Filter:
h(t) = sinc
(
t
T
s
)
6. Aliasing
• Condition for No Aliasing:
f
s
= 2f
max
• Aliasing E?ect: Frequencies f +kf
s
(for integer k) overlap, distorting the signal.
• Anti-AliasingFilter: Low-pass?lterwithcut-o?f
c
=
fs
2
appliedbeforesampling.
7. Sampling of Bandpass Signals
• Bandpass Signal: Bandwidth B, frequency range [f
L
,f
H
], where B =f
H
-f
L
.
• Minimum Sampling Frequency:
f
s
= 2B
• Undersampling: Possible if f
s
= 2B and f
H
/B is an integer, allowing frequency
translation.
8. Discrete-Time Fourier Transform (DTFT) of Sam-
pled Signal
• DTFT:
X(e
j? ) =
8
?
n=-8
x[n]e
-j? n
• Relation to Continuous-Time Spectrum:
X(e
j? ) =
1
T
s
8
?
k=-8
X
(
? -2pk
2pT
s
)
2
9. Practical Sampling Considerations
• Sample-and-Hold: Approximates ideal sampling, introduces aperture e?ect.
• Quantization Noise: Error due to ?nite-bit representation in ADC.
• Oversampling: Sampling at f
s
» 2f
max
to simplify anti-aliasing ?lter design.
10. Design Considerations
• Sampling Rate Selection: Choose f
s
= 2f
max
to avoid aliasing.
• Anti-Aliasing Filter: Ensure cut-o? frequency =
fs
2
.
• Applications: Digitalsignalprocessing,communicationsystems,audio/videopro-
cessing.
• Reconstruction Filter: Use low-pass ?lter to remove high-frequency replicas.
3