Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
z-Transform is for the analysis and synthesis of discrete-time control systems.The z transform in discrete-time systems play a similar role as the Laplace transform in continuous-time systems
The presentation covers sampling theorem, ideal sampling, flat top sampling, natural sampling, reconstruction of signals from samples, aliasing effect, zero order hold, upsampling, downsampling, and discrete time processing of continuous time signals.
Classification of signals and systems as well as their properties are given in the PPT .Examples related to types of signals and systems are also given .
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
Classification of signals and systems as well as their properties are given in the PPT .Examples related to types of signals and systems are also given .
This presentation describes the Fourier Transform used in different mathematical and physical applications.
The presentation is at an Undergraduate in Science (math, physics, engineering) level.
Please send comments and suggestions to improvements to solo.hermelin@gmail.com.
More presentations can be found at my website http://www.solohermelin.com.
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
WEBINAR ON FUNDAMENTALS OF DIGITAL IMAGE PROCESSING DURING COVID LOCK DOWN by by K.Vijay Anand , Associate Professor, Department of Electronics and Instrumentation Engineering , R.M.K Engineering College, Tamil Nadu , India
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Frequency deviation
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Signal Flow Graph, SFG and Mason Gain Formula, Example solved with Masson Gai...Waqas Afzal
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Mason Gain Formula
Example solved with Masson Gain Formula
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Why automatic controls is required
2. Process Variables
controlled variable, manipulated variable
3. Functions of Automatic Control
Measurement
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Correction
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System, Plant, Process, Controller, input, output, disturbance
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Natural, Manmade & Automatic control system
Open-Loop, Close-Loop control System
Linear Vs Nonlinear System
Time invariant vs Time variant
Continuous Data Vs Discrete Data System
Deterministic vs Stochastic System
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Accuracy, Sensitivity, noise, Bandwidth, Speed, Oscillations
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
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Discrete Time Fourier Transform
1. Discrete Time Fourier Transform
• Why Fourier Transform
• General Properties & Symmetry relations
• Formula and steps
• magnitude and phase spectra
• Convergence Condition
• mean-square convergence
• Gibbs phenomenon
• Direct Delta
• Energy Density Spectrum
2. Image Transforms
• Many times, image processing tasks can be
best performed in a domain other than the
spatial domain.
• Key steps
(1) Transform the image
(2) Carry the task(s) in the transformed domain.
(3) Apply inverse transform to return to the
spatial domain.
3. Why is FT Useful?
• Easier to remove undesirable frequencies in
the frequency domain.
• Faster to perform certain operations in the
frequency domain than in the spatial domain.
– i.e., using the Fast Fourier Transform (FFT)
4. Frequency Filtering: Main Steps
1. Take the FT of f(x):
2. Remove undesired frequencies:
3. Convert back to a signal:
x
6. FT - Definitions
• F(u) is a complex function:
• Magnitude of FT (or spectrum):
• Phase of FT:
• Magnitude-Phase representation:
• Power of f(x): P(u)=|F(u)|2=
7. Discrete-Time Fourier Transform
• Definition - The Discrete-Time Fourier
Transform (DTFT) of a sequence
x[n] is given by
• In general, is a complex function of
the real variable w and can be written as
)
( w
j
e
X
)
( w
j
e
X
n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(
)
(
)
(
)
( im
re
w
w
w j
j
j
e
X
j
e
X
e
X
8. Discrete-Time Fourier Transform
• and are, respectively, the
real and imaginary parts of , and are
real functions of w
• can alternately be expressed as
where
)
( w
j
e
X
)
( ω
re
j
e
X )
( ω
im
j
e
X
)
( w
j
e
X
)
(
)
(
)
( w
w
w j
j
j e
e
X
e
X
)}
(
arg{
)
( w
w
j
e
X
9. Discrete-Time Fourier Transform
• is called the magnitude function
• is called the phase function
• Both quantities are again real functions of w
• In many applications, the DTFT is called the
Fourier spectrum
• Likewise, and are called the
magnitude and phase spectra
)
( w
j
e
X
)
(w
)
( w
j
e
X )
(w
14. DTFT of unit Impulse Sequence
• Example - The DTFT of the unit sample
sequence d[n] is given by
1
]
0
[
]
[
)
(
d
d
w
w
n
j
n
e
n
n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(
15. DTFT of Causal Sequence
• Example - Consider the causal sequence
otherwise
0
0
1
]
[
,
1
],
[
]
[
n
n
n
n
x n
n
n
j
j
e
n
x
e
X ω
ω
]
[
)
(
w
w
w
0
]
[
)
(
n
n
j
n
n
n
j
n
j
e
e
n
e
X
w
w
j
e
n
n
j
e
1
1
0
)
(
16. DTFT of Causal Sequence
• The magnitude and phase of the DTFT
are shown below
)
5
.
0
1
/(
1
)
( w
w
j
j
e
e
X
-3 -2 -1 0 1 2 3
0.5
1
1.5
2
w/
Magnitude
-3 -2 -1 0 1 2 3
-0.4
-0.2
0
0.2
0.4
0.6
w/
Phase
in
radians
17. Convergence Condition
• - An infinite series of the form
may or may not converge
• Consider the following approximation
w
w
n
n
j
j
e
n
x
e
X ]
[
)
(
w
w
K
K
n
n
j
j
K e
n
x
e
X ]
[
)
(
18. Convergence Condition
• Then for uniform convergence of ,
• If x[n] is an absolutely summable sequence, i.e., if
for all values of w
• Thus, the absolute summability of x[n] is a sufficient
condition for the existence of the DTFT
)
( w
j
e
X
0
)
(
)
(
lim
w
w
j
K
j
K
e
X
e
X
n
n
x ]
[
w
w
n
n
n
j
j
n
x
e
n
x
e
X ]
[
]
[
)
(
19. Convergence Condition
• Example - The sequence for
is absolutely summable as
and therefore its DTFT converges to
uniformly
]
[
]
[ n
n
x n
1
1
1
]
[
0
n
n
n
n
n
)
( w
j
e
X
)
1
/(
1 w
j
e
20. Convergence Condition
• Since
an absolutely summable sequence has always
a finite energy
• However, a finite-energy sequence is not
necessarily absolutely summable
,
]
[
]
[
2
2
n
n
n
x
n
x
21. • - The sequence
has a finite energy equal to
• However, x[n] is not absolutely summable since the
summation
does not converge.
Example
]
[n
x 0
0
1
1
n
n
n
,
,
/
6
1 2
1
2
n
x
n
E
1
1
1
1
n
n n
n
22. Mean Square Convergence
• To represent a finite energy sequence that is not
absolutely summable by a DTFT, it is necessary to
consider a mean-square convergence of
where
)
( w
j
e
X
0
)
(
)
(
lim
2
w
w
w
d
e
X
e
X j
K
j
K
w
w
K
K
n
n
j
j
K e
n
x
e
X ]
[
)
(
23. Mean Square Convergence
• Here, the total energy of the error
must approach zero at each value of w as K
goes to
• In such a case, the absolute value of the error
may not go to zero as K goes to and the
DTFT is no longer bounded
)
(
)
( w
w
j
K
j
e
X
e
X
)
(
)
( w
w
j
K
j
e
X
e
X
24. Example
• - Consider the DTFT
shown below
w
w
w
w
w
c
c
j
LP e
H
,
0
0
,
1
)
(
)
( w
j
LP e
H
c
w
0
1
c
w
w
25. Example
• The inverse DTFT of is given by
• The energy of is given by
• is a finite-energy sequence, but
it is not absolutely summable
)
( w
j
LP e
H
]
[n
hLP
,
sin
2
1
n
n
jn
e
jn
e c
n
j
n
j c
c
w
w
w
w
w
w
w
d
e
n
h
c
c
n
j
LP
2
1
]
[
n
w /
c
]
[n
hLP
26. Example
• As a result
does not uniformly converge to for all
values of w, but converges to in the
mean-square sense
)
( w
j
LP e
H
)
( w
j
LP e
H
w
w
w
K
K
n
n
j
c
K
K
n
n
j
LP e
n
n
e
n
h
sin
]
[
27. Example
• The mean-square convergence property of the
sequence can be further illustrated by
examining the plot of the function
for various values of K as shown next
w
w
w
K
K
n
n
j
c
j
K
LP e
n
n
e
H
sin
)
(
,
]
[n
hLP
29. Example
• As can be seen from these plots,
independent of the value of K there are
ripples in the plot of around
both sides of the point
• The number of ripples increases as K
increases with the height of the largest
ripple remaining the same for all values of K
)
(
,
w
j
K
LP e
H
c
w
w
30. Gibbs Phenomenon
• As K goes to infinity, the condition
holds indicating the convergence of
to
• The oscillatory behavior of
approximating in the mean-square
sense at a point of discontinuity is known as
the Gibbs phenomenon
)
(
,
w
j
K
LP e
H
)
( w
j
LP e
H
0
)
(
)
(
lim
2
,
w
w
w
d
e
H
e
H j
K
LP
j
LP
K
)
(
,
w
j
K
LP e
H
)
( w
j
LP e
H
31. Dirac delta function d(w)
• The DTFT can also be defined for a certain
class of sequences which are neither
absolutely summable nor square summable
• Examples of such sequences are the unit step
sequence [n], the sinusoidal sequence
and the exponential sequence
• For this type of sequences, a DTFT
representation is possible using the Dirac delta
function d(w)
)
cos(
w n
o
n
A
32. Dirac delta function d(w)
• A Dirac delta function d(w) is a function of
w with infinite height, zero width, and unit
area
• It is the limiting form of a unit area pulse
function as goes to zero, satisfying
)
(w
p
w
w
d
w
w d
d
p )
(
)
(
lim
0
w
2
2
0
1
)
(w
p
33. Example
• Example - Consider the complex exponential
sequence
• Its DTFT is given by
where is an impulse function of w and
n
j o
e
n
x w
]
[
w
w
d
w
k
o
j
k
e
X )
2
(
2
)
(
)
(w
d
w
o
34. Example
• The function
is a periodic function of w with a period 2
and is called a periodic impulse train
• To verify that given above is indeed
the DTFT of we compute the
inverse DTFT of
w
w
d
w
k
o
j
k
e
X )
2
(
2
)
(
)
( w
j
e
X
n
j o
e
n
x w
]
[
)
( w
j
e
X
35. Example
• Thus
where we have used the sampling property of
the impulse function )
(w
d
w
w
w
d
w
k
n
j
o d
e
k
n
x )
2
(
2
2
1
]
[
n
j
n
j
o
o
e
d
e w
w
w
w
w
d
)
(
36. Energy Density Spectrum
• The total energy of a finite-energy sequence
g[n] is given by
• From Parseval’s relation given above we
observe that
n
g n
g
2
]
[
E
w
w
d
e
G
n
g j
n
g
2
2
2
1
)
(
]
[
E
37. Energy Density Spectrum
• The quantity
is called the energy density spectrum
• Therefore, the area under this curve in the
range divided by 2 is the energy
of the sequence
w
2
)
(
)
( w
w j
gg e
G
S
38. Example
• - Compute the energy of the sequence
• Here
where
w
n
n
n
n
h c
LP ,
sin
]
[
w
w
d
e
H
n
h j
LP
n
LP
2
2
)
(
2
1
]
[
w
w
w
w
w
c
c
j
LP e
H
,
0
0
,
1
)
(
39. Energy Density Spectrum
• Therefore
• Hence, is a finite-energy sequence
w
w
w
w
c
n
LP
c
c
d
n
h
2
1
]
[
2
]
[n
hLP