BISWAJIT ADHIKARI_28101623024_PC EE 601A.pdf

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1. 1
TOPIC:-SamPlIng TheOrem
Department of Electrical Engineering
Gargi Memorial Institute of Technology
Name of the Student: BISWAJIT ADHIKARI
Roll Number: 28101623024
Registration Number:232810120222
Present Semester: 6
th
Course Name:
Digital Control System
Course Code:
PC-EE-601A

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INTRODUCTION

The "sampling" is one of the most fundamental concepts in
digital signal processing that enables the conversion of
continuous analog signals to discrete-time digital signals.

It establishes the theoretical foundation for the analog-to-
digital conversion process which is very important for
applications like sound recording, video processing,
wireless communication, and more
.

Since all computer controlled systems operate at discrete
times only, it is important to know the condition under
which a signal can be retrieved from its values at discrete
points. Nyquistexplored the key issue and Shannon gave
the complete solution which is known as Shannon’s
sampling theorem.

3. MATHEMATICAL MODELING OF
SAMPLING PROCESS

Sampling operation in sampled data and digital control system is used to
model either the sample and hold operation or the fact that the signal is
digitally coded.

If the sampler is used to represent S/H (Sample and Hold) and A/D
(Analog to Digital) operations, it may involve delays, finite sampling
duration and quantization errors.

On the other hand used to represent digitally coded data the model will be
much simpler.

Following are two popular sampling operations:
1. Single rate or periodic sampling
2. Multi-rate sampling

We would limit our discussions to periodic sampling only.
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4. MATHEMATICAL MODELING OF SAMPLING
PROCESS(continued….)
●
Finite plusewidth sampler
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5. MATHEMATICAL MODELING OF SAMPLING
PROCESS ( continued….)
The pulse duration is p second and sampling period is T second.Uniformrate
sampler is a linear devicewhich satisfies the principle of superposition.
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6. MATHEMATICAL MODELING OF SAMPLING
PROCESS ( continued….)
●
Frequency domain characteristics:
Since p(t) is a periodic function, it can be represented by a Fourier series, as
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7. SAMPLING THEOREM
Sampling theorem states that “A band limited signal () with (ɷ) = for
|| ≥ ɷcan be represented into and uniquely determined from its samples
() if the sampling frequency ≥ , where is the frequency.
(i.e) for signal recovery, the sampling frequency must be at least twice the
highest frequency present in the signal.cy component present in it.
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8. SAMPLING THEOREM(CONTINUED….)
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9. SAMPLING THEOREM(CONTINUED….)
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10. SAMPLING THEOREM(CONTINUED….)
For > 2The spectral replicates have a larger separation between them
known as guard band which makes process of filtering much easier and
effective. Even a non-ideal filter which does not have a sharp cut off can also
be used.
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11. SAMPLING THEOREM(CONTINUED….)
●
For =
There is no separation between the spectral replicates so no guard band exists
and () can be obtained from () by using only an ideal low pass filter
(LPF) with sharp cutoff.
•
For < 2
The low frequency component in overlap on high frequency
components of so that there is presence of distortion and cannot be
recovered from by using any filter. This distortion is called aliasing.
So we can conclude that the frequency spectrum of () is not overlapped
for − ≥ , therefore the Original signal can be recovered from the
sampled signal. For − < , the frequency spectrum will overlap
and hence the original signal cannot be recovered from the sampled signal.
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12. SAMPLING THEOREM(CONTINUED….)
●
Aliasing effect (or) fold over effect
It is defined as the phenomenon in which a high frequency component in the
frequency spectrum of signal takes identity of a lower frequency component
in the spectrum of the sampled signal.
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13. Data Reconstruction or Interpolati
on
●
NyquistRate
It is the theoretical minimum sampling rate at which a signal can be sampled
and still be reconstructed from its samples without any distortion
•Data Reconstruction or Interpolati
on
The process of obtaining analog signal () from the sampled signal () is
called data reconstruction or interpolation.
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14. Data Reconstruction or Interpolation
●
The reconstruction filter, which is assumed to be linear and time invariant,
has unit impulse response h(). The reconstruction filter, output () is
given by convolu
●
tionof () and h().
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15. ………THANK YOU……..
References:-
1)https://www.rcet.org.in/uploads/academics/rohini_65660789483.pdf
2)https://www.gvpcew.ac.in/LN-CSE-IT-22-32/EEE/4-Year/DCS.pdf
3)https://testbook.com/electrical-engineering/sampling-theorem-in-
signals-and-systems
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