The document discusses the Discrete Fourier Transform (DFT). It explains that while the discrete-time Fourier transform (DTFT) and z-transform are not numerically computable, the DFT avoids this issue. The DFT represents periodic sequences as a sum of complex exponentials with frequencies that are integer multiples of the fundamental frequency. It can be viewed as computing samples of the DTFT or z-transform at discrete frequency points, allowing numerical computation. The DFT provides a link between the time and frequency domain representations of a finite-length sequence.
Digital signal processing is a specialized microprocessor with its architecture optimized for operational needs of digital signal processing
Application's of DSP like STFT and Wavelet transform has been explained in detail with images.
Digital signal processing is a specialized microprocessor with its architecture optimized for operational needs of digital signal processing
Application's of DSP like STFT and Wavelet transform has been explained in detail with images.
Introduction to Angle Modulation, Types of Angle Modulation, Frequency Modulation and Phase Modulation Introduction, Generation of FM, Detection of FM, Frequency stereo Multiplexing, Applications, Difference between FM and PM.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
Introduction to Angle Modulation, Types of Angle Modulation, Frequency Modulation and Phase Modulation Introduction, Generation of FM, Detection of FM, Frequency stereo Multiplexing, Applications, Difference between FM and PM.
EXPERT SYSTEMS AND SOLUTIONS
Project Center For Research in Power Electronics and Power Systems
IEEE 2010 , IEEE 2011 BASED PROJECTS FOR FINAL YEAR STUDENTS OF B.E
Email: expertsyssol@gmail.com,
Cell: +919952749533, +918608603634
www.researchprojects.info
OMR, CHENNAI
IEEE based Projects For
Final year students of B.E in
EEE, ECE, EIE,CSE
M.E (Power Systems)
M.E (Applied Electronics)
M.E (Power Electronics)
Ph.D Electrical and Electronics.
Training
Students can assemble their hardware in our Research labs. Experts will be guiding the projects.
EXPERT GUIDANCE IN POWER SYSTEMS POWER ELECTRONICS
We provide guidance and codes for the for the following power systems areas.
1. Deregulated Systems,
2. Wind power Generation and Grid connection
3. Unit commitment
4. Economic Dispatch using AI methods
5. Voltage stability
6. FLC Control
7. Transformer Fault Identifications
8. SCADA - Power system Automation
we provide guidance and codes for the for the following power Electronics areas.
1. Three phase inverter and converters
2. Buck Boost Converter
3. Matrix Converter
4. Inverter and converter topologies
5. Fuzzy based control of Electric Drives.
6. Optimal design of Electrical Machines
7. BLDC and SR motor Drives
Low rank tensor approximation of probability density and characteristic funct...Alexander Litvinenko
Very often one has to deal with high-dimensional random variables (RVs). A high-dimensional RV can be described by its probability density (\pdf) and/or by the corresponding probability characteristic functions (\pcf), or by a function representation. Here the interest is mainly to compute characterisations like the entropy, or
relations between two distributions, like their Kullback-Leibler divergence, or more general measures such as $f$-divergences,
among others. These are all computed from the \pdf, which is often not available directly, and it is a computational challenge to even represent it in a numerically feasible fashion in case the dimension $d$ is even moderately large. It is an even stronger numerical challenge to then actually compute said characterisations in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose to represent the density by a high order tensor product, and approximate this in a low-rank format.
Computing f-Divergences and Distances of\\ High-Dimensional Probability Densi...Alexander Litvinenko
Talk presented on SIAM IS 2022 conference.
Very often, in the course of uncertainty quantification tasks or
data analysis, one has to deal with high-dimensional random variables (RVs)
(with values in $\Rd$). Just like any other RV,
a high-dimensional RV can be described by its probability density (\pdf) and/or
by the corresponding probability characteristic functions (\pcf),
or a more general representation as
a function of other, known, random variables.
Here the interest is mainly to compute characterisations like the entropy, the Kullback-Leibler, or more general
$f$-divergences. These are all computed from the \pdf, which is often not available directly,
and it is a computational challenge to even represent it in a numerically
feasible fashion in case the dimension $d$ is even moderately large. It
is an even stronger numerical challenge to then actually compute said characterisations
in the high-dimensional case.
In this regard, in order to achieve a computationally feasible task, we propose
to approximate density by a low-rank tensor.
An Optimized Transform for ECG Signal CompressionIDES Editor
A significant feature of the coming digital era is the
exponential increase in digital data, obtained from various
signals specially the biomedical signals such as
electrocardiogram (ECG), electroencephalogram (EEG),
electromyogram (EMG) etc. How to transmit or store these
signals efficiently becomes the most important issue. A digital
compression technique is often used to solve this problem.
This paper proposed a comparative study of transform based
approach for ECG signal compression. Adaptive threshold is
used on the transformed coefficients. The algorithm is tested
for 10 different records from MIT-BIH arrhythmia database
and obtained percentage root mean difference as around
0.528 to 0.584% for compression ratio of 18.963:1 to 23.011:1
for DWT. Among DFT, DCT and DWT techniques, DWT has
been proven to be very efficient for ECG signal coding.
Further improvement in the CR is possible by efficient
entropy coding.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
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.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
2. Introduction
The discrete-time Fourier transform (DTFT) provided the frequency-
domain (ω) representation for absolutely summable sequences.
The z-transform provided a generalized frequency-domain (z)
representation for arbitrary sequences.
These transforms have two features in common.
First, the transforms are defined for infinite-length sequences.
Second, and the most important, they are functions of continuous variables
(ω or z).
In other words, the discrete-time Fourier transform and the z-transform
are not numerically computable transforms.
The Discrete Fourier Transform (DFT) avoids the two problems
mentioned and is a numerically computable transform that is suitable
for computer implementation.
2
4. Periodic Sequences
Let 𝑥(𝑛) is a periodic sequence with period N-Samples (the fundamental
period of the sequence).
Notation: a sequence with period N is satisfying the condition
where r is any integer.
4
),...1(),...,1(),0(),1(),...,1(),0(...,)(~ NxxxNxxxnx
x(n) x(n)
)(~)(~ rNnxnx
5. Periodic Sequences
From Fourier analysis we know that the periodic functions can be synthesized
as a linear combination of complex exponentials whose frequencies are
multiples (or harmonics) of the fundamental frequency (which in our case is
2π/N).
The discrete version of the Fourier Series can be written as
where { 𝑋(𝑘), 𝑘 = 0, ± 1, . . . , ∞} are called the discrete Fourier series
coefficients.
Note That, for integer values of m, we have
(it is called Twiddle Factor)
5
k
kn
N
k
N
kn
j
k
kn
N
j
k WkX
N
ekX
N
eXnx )(
~1
)(
~1
)(~ 2
2
nmNk
N
N
nmNk
j
N
kn
j
kn
N WeeW )(
)(
22
6. Periodic Sequences
As a result, the summation in the Discrete Fourier Series (DFS) should
contain only N terms:
The Harmonics are
6
1
0
)(
~1
)(~
N
k
kn
NWkX
N
nx
knNj
k ene )/2(
)(
,2,1,0 k
7. Inverse DFS
The DFS coefficients are given by
Proof
7
1
0
1
0
2
)(~)(~)(
~ N
k
kn
N
N
k
N
kn
j
WnxenxkX
DFSInverse
)(
~
)()(
~
)(~
1
)(
~
)(~
)(
~1
)(~
1
0
1
0
2
1
0
1
0
)(
21
0
2
1
0
21
0
21
0
2
kXkpPXenx
e
N
PXenx
eePX
N
enx
N
P
N
k
N
kn
j
N
P
N
k
N
nkP
jN
k
N
kn
j
N
k
N
kn
jN
P
N
Pn
jN
k
N
kn
j
8. Synthesis and Analysis (DFS-Pairs)
8
Notation
)/2( Nj
N eW
Synthesis
1
0
)(
~1
)(~
N
k
kn
NWkX
N
nx
Analysis
)(
~
)(~ kXnx DFS
Both have
Period N
1
0
)(~)(
~ N
k
kn
NWnxKX
12. DFS vs. FT
12
)(~ nx
0 N nN
0 n
)(nx
0
)()(
n
njj
enxeX
1
0
)(
N
n
nj
enx
1
0
)/2(
)(~)(
~ N
n
knNj
enxkX
Nk
j
eXkX
/2
)()(
~
13. Example
13
0 1 2 3 4 5 6 7 8 9 n
0 1 2 3 4 5 6 7 8 9 n
)(~ nx
)(nx
)2/sin(
)2/5sin(
1
1
)( 2
54
0
j
j
j
n
njj
e
e
e
eeX
)10/sin(
)2/sin(
)()(
~ )10/4(
10/2 k
k
eeXkX kj
k
j
14. Example
0 1 2 3 4 5 6 7 8 9 n
0 1 2 3 4 5 6 7 8 9 n
)(~ nx
)(nx
)2/sin(
)2/5sin(
1
1
)( 2
54
0
j
j
j
n
njj
e
e
e
eeX
14
)10/sin(
)2/sin(
)()(
~ )10/4(
10/2 k
k
eeXkX kj
k
j
15. 0 1 2 3 4 5 6 7 8 9 n
0 1 2 3 4 5 6 7 8 9 n
)(~ nx
)(nx
Example
15
)2/sin(
)2/5sin(
1
1
)( 2
54
0
j
j
j
n
njj
e
e
e
eeX
)10/sin(
)2/sin(
)()(
~ )10/4(
10/2 k
k
eeXkX kj
k
j
16. Relation To The Z-transform
Let 𝑥(𝑛) be a finite-duration sequence of duration 𝑁 such that
Then we can compute its z-transform:
Now we construct a periodic sequence 𝑥 𝑛 by periodically repeating
𝑥(𝑛) with period 𝑁, that is,
The DFS of 𝑥 𝑛 is given by
16
Elsewhere
NnNonzero
nx
,0
)1(0,
)(
1
0
)(z)(
N
n
n
znxX
Elsewhere
Nnnx
nx
,0
)1(0),(~
)(
1
0
2
)(k)(
~ N
n
n
k
N
j
enxX
k
N
j
ez
zXX 2
)(k)(
~
17. Relation To The Z-transform
which means that the DFS 𝑿(𝑘) represents 𝑁 evenly spaced samples of
the z-transform 𝑋(𝑧) around the unit circle.
17
Re(z)
Im(z)
z0
z1
z2
z3
z4
z5
z6
z7
1
N = 8
20. Recall of DTFT
DTFT is not suitable for DSP applications because
In DSP, we are able to compute the spectrum only at specific discrete values of ω
Any signal in any DSP application can be measured only in a finite number of points.
A finite signal measures at N-points:
Where y(n) are the measurements taken at N-points
20
Nn
Nnny
n
nx
,0
)1(0),(
00
)(
21. Computation of DFT
Recall of DTFT
The DFT can be computed by:
Truncate the summation so that it ranges over finite limits x[n] is a finite-length
sequence.
Discretize ω to ωk evaluate DTFT at a finite number of discrete frequencies.
For an N-point sequence, only N values of frequency samples of X(ejw) at N
distinct frequency points, are sufficient to determine x[n]
and X(ejw) uniquely.
So, by sampling the spectrum X(ω) in frequency domain
21
DFTenxX
N
X
N
n
N
kn
j
1
0
2
)(k)(
2
),X(kk)(
10, Nkk
22. Computation of DFT
The inverse DFT is given by:
Proof
22
IDFTekX
N
x
N
k
N
kn
j
1
0
2
)(
1
n)(
1
0
1
0
1
0
)(
2
1
0
21
0
2
)()()(n)(
1
)(n)(
)(
1
n)(
N
m
N
m
N
n
N
nmk
j
N
k
N
kn
jN
m
N
mk
j
nxnmmxx
e
N
mxx
eemx
N
x
24. Example
Find the discrete Fourier Transform of the following N-points discrete
time signal
Solution:
On the board
24
)1(,...),1(),0()( Nxxxnx
25. Nth Root of Unity
252
1
2
1
10)
19)
2
1
2
1
8)
7)
2
1
2
1
6)
15)
2
1
2
1
4)
3)
2
1
2
1
2)
11)
9
8
2
9
8
8
8
2
8
8
7
8
2
7
8
6
8
2
6
8
5
8
2
5
8
4
8
2
4
8
3
8
2
3
8
2
8
2
2
8
1
8
2
1
8
0
8
2
0
8
jeW
eW
jeW
jeW
jeW
eW
jeW
jeW
jeW
eW
j
j
j
j
j
j
j
j
j
j
1*
2/
2
)2/(
NN
k
N
k
N
k
N
Nk
N
k
N
Nk
N
WW
WW
WW
WW
N
j
N eW
2
26. Relation To The Z-transform
Let 𝑥(𝑛) be a finite-duration sequence of duration 𝑁 such that
Then we can compute its z-transform:
Now we construct a periodic sequence 𝑥 𝑛 by periodically repeating
𝑥(𝑛) with period 𝑁, that is,
The DFS of 𝑥 𝑛 is given by
26
Elsewhere
NnNonzero
nx
,0
)1(0,
)(
1
0
)(z)(
N
n
n
znxX
Elsewhere
Nnnx
nx
,0
)1(0),(~
)(
1
0
2
)(k)(
~ N
n
n
k
N
j
enxX
k
N
j
ez
zXX 2
)(k)(
~
27. Relation To The Z-transform
which means that the DFS 𝑿(𝑘) represents 𝑁 evenly spaced samples of
the z-transform 𝑋(𝑧) around the unit circle.
27
Re(z)
Im(z)
z0
z1
z2
z3
z4
z5
z6
z7
1
N = 8
29. DFT - Matrix Formulation
xWX
Nx
x
x
x
WWW
WWW
WWW
NX
X
X
X
NN
N
N
N
N
N
N
NNN
N
NNN
]1[
]2[
]1[
]0[
1
1
1
1111
)1(
)2(
)1(
)0(
)1)(1()1(21
)1(242
121
29
30. Computation Complexity
To calculate the DFT of N-Points discrete time signal, we need:
(N-1)2 Complex Multiplications
N(N-1) Complex Additions.
30
31. IDFT - Matrix Formulation
XW
N
x
XWx
NX
X
X
X
WWW
WWW
WWW
N
Nx
x
x
x
NN
N
N
N
N
N
N
NNN
N
NNN
*
1
)1)(1()1(2)1(
)1(242
)1(21
1
]1[
]2[
]1[
]0[
1
1
1
1111
1
)1(
)2(
)1(
)0(
31
32. Matrix Formulation
Result: Inverse DFT is given by
which follows easily by checking WHW = WWH = NI, where I denotes the
identity matrix. Hermitian transpose:
Also, “*” denotes complex conjugation.
32
33. DFT Interpretation
DFT sample X(k) specifies the magnitude and phase angle of the kth spectral
component of x[n].
The amount of power that x[n] contains at a normalized frequency, fk, can be
determined from the power density spectrum defined as
33
)(spectrumPhase
|)(|spectrumMagnitude
kX
kX
10,
|)(|
)(
2
Nk
N
kX
kSN
34. Periodicity of DFT Spectrum
The DFT spectrum is periodic with period N (which is expected, since the DTFT
spectrum is periodic as well, but with period 2π).
34
)()(N)k(
)(N)k(
)(N)k(
2
2
1
0
2
1
0
)(
2
kXekXX
eenxX
enxX
nj
nj
N
n
N
nk
j
N
n
N
nNk
j
35. Example
Example: DFT of a rectangular pulse:
the rectangular pulse is “interpreted” by the DFT as a spectral line at
frequency ω = 0. DFT and DTFT of a rectangular pulse (N=5)
35
36. Zero Padding
What happens with the DFT of this rectangular pulse if we increase N by
zero padding:
where x(0) = · · · = x(M − 1) = 1. Hence, DFT is
36
0,...,0,0,0),1(,...),1(),0()( Mxxxnx
(N-M) Positions
37. Zero Padding
DFT and DTFT of a Rectangular Pulse with Zero Padding (N = 10, M = 5)
37
38. Zero Padding
Zero padding of analyzed sequence results in “approximating” its DTFT
better,
Zero padding cannot improve the resolution of spectral components,
because the resolution is “proportional” to 1/M rather than 1/N,
Zero padding is very important for fast DFT implementation (FFT).
38
39. Frequency Interval/Resolution
Frequency Interval/Resolution: DFT’s frequency resolution
and covered frequency interval
Frequency resolution is determined only by the length of the observation
interval, whereas the frequency interval is determined by the length of
sampling interval. Thus
Increase sampling rate =) expand frequency interval,
Increase observation time =) improve frequency resolution.
Question: Does zero padding alter the frequency resolution?
Answer: No, because resolution is determined by the length of observation
interval, and zero padding does not increase this length.
39
40. Example (DFT Resolution)
Example (DFT Resolution): Two complex exponentials with two close frequencies
F1 = 10 Hz and F2 = 12 Hz sampled with the sampling interval T = 0.02 seconds.
Consider various data lengths N = 10, 15, 30, 100 with zero padding to 512 points.
DFT with N=10 and zero padding to 512 points. Not resolved: F2−F1 = 2 Hz < 1/(NT) = 5 Hz.
40
41. Example (DFT Resolution)
DFT with N = 30 and zero padding to 512 points. Resolved: F2 − F1 = 2 Hz > 1/(NT)=1.7 Hz.
41
42. Example (DFT Resolution)
DFT with N=100 and zero padding to 512 points. Resolved: F2−F1 = 2 Hz > 1/(NT) = 0.5 Hz.
42
43. DSF VS DFT
DFS and DFT pairs are identical, except that
DFT is applied to finite sequence x(n),
DFS is applied to periodic sequence .
Conventional (continuous-time) FS vs. DFS
CFS represents a continuous periodic signal using an infinite number of
complex exponentials, whereas
DFS represents a discrete periodic signal using a finite number of complex
exponentials.
43
)(~ nx