The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
The discrete Fourier transform has many applications in science and engineering. For example, it is often used in digital signal processing applications such as voice recognition and image processing.
Ajay Kumar.Ph.D Research scholar at National Institute of Technology my mail id:-- ajaymodaliger@gmail.com
In this presentation slide i have Explained how to reducing Computational time complexity of Discrete Fourier transform(DFT) from O(n^2 ) to nlogn through Radix-2 .FFT Algorithm in this work i have also introduced how we can use Radix-2 FFT in encrypted signal processing application by considering homomarphic properties(RSA) of Paillier cryptosystem.
Fast Fourier transform is an extension of discrete Fourier transform, It is based on divide and conquer algorithm,it is of two types, decimation in time and decimation in frequency algorithm
Ajay Kumar.Ph.D Research scholar at National Institute of Technology my mail id:-- ajaymodaliger@gmail.com
In this presentation slide i have Explained how to reducing Computational time complexity of Discrete Fourier transform(DFT) from O(n^2 ) to nlogn through Radix-2 .FFT Algorithm in this work i have also introduced how we can use Radix-2 FFT in encrypted signal processing application by considering homomarphic properties(RSA) of Paillier cryptosystem.
Fast Fourier transform is an extension of discrete Fourier transform, It is based on divide and conquer algorithm,it is of two types, decimation in time and decimation in frequency algorithm
Signals and Systems is an introduction to analog and digital signal processing, a topic that forms an integral part of engineering systems in many diverse areas, including seismic data processing, communications, speech processing, image processing, defense electronics, consumer electronics, and consumer products.
A signal is a pattern of variation that carry information.
Signals are represented mathematically as a function of one or more independent variable
basic concept of signals
types of signals
system concepts
Cyclostationary analysis of polytime coded signals for lpi radarseSAT Journals
Abstract In Radars, an electromagnetic waveform will be sent, and an echo of the same signal will be received by the receiver. From this received signal, by extracting various parameters such as round trip delay, doppler frequency it is possible to find distance, speed, altitude, etc. However, nowadays as the technology increases, intruders are intercepting transmitted signal as it reaches them, and they will be extracting the characteristics and trying to modify them. So there is a need to develop a system whose signal cannot be identified by no cooperative intercept receivers. That is why LPI radars came into existence. In this paper a brief discussion on LPI radar and its modulation (Polytime code (PT1)), detection (Cyclostationary (DFSM & FAM) techniques such as DFSM, FAM are presented and compared with respect to computational complexity.
Keywords—LPI Radar, Polytime codes, Cyclostationary DFSM, and FAM
Embedded systems increasingly employ digital, analog and RF signals all of which are tightly synchronized in time. Debugging these systems is challenging in that one needs to measure a number of different signals in one or more domains (time, digital, frequency) and with tight time synchronization. This session will discuss how a digital oscilloscope can be used to effectively debug these systems, and some of the instrumentation considerations that go along with this.
Implementation of adaptive stft algorithm for lfm signalseSAT Journals
Abstract
Normally Time-Frequency analysis is done by sliding a window through the time domain data and computing the Fourier
Transform of the data within the window. The choice of the window length determines whether specular or resonant information
will be emphasized. A narrow window will isolate specular reflections but will not be wide enough to accommodate the slowly
varying global resonances; a wide window cannot temporally separate resonance and specular information. So we will adapt
window length according to changes in frequencies. In this case we are realizing the specifications of Linear Frequency
Modulation (LFM) signal.
Index Terms—LFM, FFT, DFT, STFT and ASTFT.
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.
Vaccine management system project report documentation..pdfKamal Acharya
The Division of Vaccine and Immunization is facing increasing difficulty monitoring vaccines and other commodities distribution once they have been distributed from the national stores. With the introduction of new vaccines, more challenges have been anticipated with this additions posing serious threat to the already over strained vaccine supply chain system in Kenya.
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.
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.
Democratizing Fuzzing at Scale by Abhishek Aryaabh.arya
Presented at NUS: Fuzzing and Software Security Summer School 2024
This keynote talks about the democratization of fuzzing at scale, highlighting the collaboration between open source communities, academia, and industry to advance the field of fuzzing. It delves into the history of fuzzing, the development of scalable fuzzing platforms, and the empowerment of community-driven research. The talk will further discuss recent advancements leveraging AI/ML and offer insights into the future evolution of the fuzzing landscape.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
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.
COLLEGE BUS MANAGEMENT SYSTEM PROJECT REPORT.pdfKamal Acharya
The College Bus Management system is completely developed by Visual Basic .NET Version. The application is connect with most secured database language MS SQL Server. The application is develop by using best combination of front-end and back-end languages. The application is totally design like flat user interface. This flat user interface is more attractive user interface in 2017. The application is gives more important to the system functionality. The application is to manage the student’s details, driver’s details, bus details, bus route details, bus fees details and more. The application has only one unit for admin. The admin can manage the entire application. The admin can login into the application by using username and password of the admin. The application is develop for big and small colleges. It is more user friendly for non-computer person. Even they can easily learn how to manage the application within hours. The application is more secure by the admin. The system will give an effective output for the VB.Net and SQL Server given as input to the system. The compiled java program given as input to the system, after scanning the program will generate different reports. The application generates the report for users. The admin can view and download the report of the data. The application deliver the excel format reports. Because, excel formatted reports is very easy to understand the income and expense of the college bus. This application is mainly develop for windows operating system users. In 2017, 73% of people enterprises are using windows operating system. So the application will easily install for all the windows operating system users. The application-developed size is very low. The application consumes very low space in disk. Therefore, the user can allocate very minimum local disk space for this application.
2. Signals
In the fields of communications, signal processing, and in electrical engineering
more generally, a signal is any time‐varying or spatial‐varying quantity
This variable(quantity) changes in time
• Speech or audio signal: A sound amplitude that varies in time
• Temperature readings at different hours of a day
• Stock price changes over days
• Etc• Etc.
Signals can be classified by continues‐time signal and discrete‐time signal:
• A discrete signal or discrete‐time signal is a time series, perhaps a signal that
h b l d f ti ti i lhas been sampled from a continuous‐time signal
• A digital signal is a discrete‐time signal that takes on only a discrete set of
values 1
Continuous Time Signal
1
Discrete Time Signal
-0.5
0
0.5
f(t)
-0.5
0
0.5
f[n]
0 10 20 30 40
-1
Time (sec)
0 10 20 30 40
-1
n
2
3. Periodic Signal
periodic signal and non‐periodic signal:
1
Periodic Signal
1
Non-Periodic Signal
0 10 20 30 40
-1
0
f(t)
Time (sec)
0 10 20 30 40
-1
0
f[n]
nn
• Period T: The minimum interval on which
a signal repeats
• Fundamental frequency: f0=1/TFundamental frequency: f0 1/T
• Harmonic frequencies: kf0
• Any periodic signal can be approximated
by a sum of many sinusoids at harmonic frequencies of the signal(kf0) with y y q g ( f0)
appropriate amplitude and phase
• Instead of using sinusoid signals, mathematically, we can use the complex
exponential functions with both positive and negative harmonic frequencies
)cos()sin()exp( tjttj Euler formula:
3
4. Time‐Frequency Analysis
• A signal has one or more frequencies in it, and can be viewed from
two different standpoints: Time domain and Frequency domain
Time Domian (Banded Wren Song)
0
1
Amplitude
Time Domian (Banded Wren Song)
1
2
Power
Frequency Domain
0 2 4 6 8
x 10
4
-1
Sample Number
A
0 200 400 600 800 1000 1200
0
Frequency (Hz)
Time‐domain figure: how a signal changes over time
Why frequency domain analysis?
g g g
Frequency‐domain figure: how much of the signal lies within each given
frequency band over a range of frequencies
Why frequency domain analysis?
• To decompose a complex signal into simpler parts to facilitate analysis
• Differential and difference equations and convolution operations in the
time domain become algebraic operations in the frequency domain
• Fast Algorithm (FFT)
4
5. Fourier TransformFourier Transform
We can go between the time domain and the frequency domain
by using a tool called Fourier transform
• A Fourier transform converts a signal in the time domain to the
frequency domain(spectrum)
A i F i f h f d i
y g f
• An inverse Fourier transform converts the frequency domain
components back into the original time domain signal
Continuous‐Time Fourier Transform:
dejFtf tj
)(
2
1
)(
dtetfjF tj
)()(
Continuous Time Fourier Transform:
Discrete‐Time Fourier Transform(DTFT):Discrete Time Fourier Transform(DTFT):
2
)(
2
1
][ deeXnx njj
n
njj
enxeX
][)(
5
6. Fourier Representation For Four Types of SignalsFourier Representation For Four Types of Signals
The signal with different time‐domain characteristics has different
frequency‐domain characteristics
1 Continues‐time periodic signal ‐‐‐> discrete non‐periodic
spectrum
q y
p
2 Continues‐time non‐periodic signal ‐‐‐> continues non‐periodic
spectrum
3 Discrete non‐periodic signal ‐‐‐> continues periodic spectrum3 Discrete non periodic signal > continues periodic spectrum
4 Discrete periodic signal ‐‐‐> discrete periodic spectrum
The last transformation between time‐domain and frequency is most q y
useful
The reason that discrete is associated with both time‐domain and frequency
domain is because computers can only take finite discrete time signalsp y g
6
7. Periodic SequencePeriodic Sequence
)(~)(~ kNnxnx , where k is integer
A periodic sequence with period N is defined as:
Periodic
, g
kn
N
j
kn
N eW
2
For example:
)()( NnknNkkn
WWW
(it is called Twiddle Factor)
Properties: Periodic )()( Nnk
N
nNk
N
kn
N WWW
Symmetric )()(*
)( nNk
N
nkN
N
kn
N
kn
N WWWW
Properties:
Orthogonal
other
rNnN
W
N
k
kn
N
0
1
0
For a periodic sequence with period N, only N samples
are independent. So that N sample in one period is enough to
represent the whole sequence
x(n)
represent the whole sequence
7
8. Discrete Fourier Series(DFS)Discrete Fourier Series(DFS)
Periodic signals may be expanded into a series of sine and
cosine functions
1
0
1
0
)(
~1
)(~
)(~)(
~
N
kn
N
N
n
kn
N
WkX
N
nx
WnxkX
cosine functions
))(
~
()(~
))(~()(
~
kXIDFSnx
nxDFSkX
0nN
is still a periodic sequence with period N in frequency
domain
)(
~
kX
The Fourier series for the discrete‐time periodic wave shown below:
1
Sequence x (in time domain)
0.2
Fourier Coeffients
0
0.5
Amplitude
0 4
-0.2
0
X
0 10 20 30 40
0
time
0 10 20 30 40
-0.4
8
9. Finite Length SequenceFinite Length Sequence
)(nx Nn 10
Real lift signal is generally a
fi i l h
If we periodic extend it by the period N then
rNnxnx )()(~
0
)(
)(
nx
nx
others
Nn 10
finite length sequence
If we periodic extend it by the period N, then
r
rNnxnx )()(
)(nx
)(~ nx
9
10. Relationship Between Finite Length Sequence
d P i di Sand Periodic Sequence
A periodic sequence is the periodic extension of a finite length
A finite length sequence is the principal value interval of the periodic
sequence
m
NnxrNnxnx ))(()()(~
A finite length sequence is the principal value interval of the periodic
sequence
)()(~)( nRnxnx N
0
1
)(nRN
others
Nn 10
Where
0 others
So that:
)()](
~
[)()(~)( nRkXIDFSnRnxnx NN
)()](~[)()(
~
)(
)()]([)()()(
nRnxDFSkRkXkX NN
NN
10
11. Discrete Fourier Transform(DFT)Discrete Fourier Transform(DFT)
• Using the Fourier series representation we have Discrete
Fourier Transform(DFT) for finite length signalFourier Transform(DFT) for finite length signal
• DFT can convert time‐domain discrete signal into frequency‐
domain discrete spectrum
Assume that we have a signal . Then the DFT of the
signal is a sequence for
1
0]}[{
N
nnx
][kX 1,,0 Nk
1
0
/2
][][
N
n
Njnk
enxkX
The Inverse Discrete Fourier Transform(IDFT):
1
0
/2
.1,,2,0,][
1
][
N
k
Njnk
NnekX
N
nx
Note that because MATLAB cannot use a zero or negativeNote that because MATLAB cannot use a zero or negative
indices, the index starts from 1 in MATLAB
11
12. DFT ExampleDFT Example
The DFT is widely used in the fields of spectral analysis,
acoustics medical imaging and telecommunicationsacoustics, medical imaging, and telecommunications.
4
5
6
e
Time domain signal
For example:
0
1
2
3
Amplitude
)3,2,1,0(,4],6142[][ nNnx
nk
nkj
jnxenxkX )(][][][
3
0
2
3
0
0 0.5 1 1.5 2 2.5 3
-1
Time
nn 00
116)1(42]0[ X
jjjX 2361)4(2]1[ 10
12
Frequency domain signal
96)1()4(2]2[ X
jjjX 2361)4(2]3[
4
6
8
10
|X[k]|
0 0.5 1 1.5 2 2.5 3
0
2
Frequency 12
13. Fast Fourier Transform(FFT)Fast Fourier Transform(FFT)
• The Fast Fourier Transform does not refer to a new or different
type of Fourier transform. It refers to a very efficient algorithm foryp y g
computing the DFT
• The time taken to evaluate a DFT on a computer depends
principally on the number of multiplications involved. DFT needsprincipally on the number of multiplications involved. DFT needs
N2 multiplications. FFT only needs Nlog2(N)
• The central insight which leads to this algorithm is the
realization that a discrete Fourier transform of a sequence of Nrealization that a discrete Fourier transform of a sequence of N
points can be written in terms of two discrete Fourier transforms
of length N/2
• Thus if N is a power of two it is possible to recursively applyThus if N is a power of two, it is possible to recursively apply
this decomposition until we are left with discrete Fourier
transforms of single points
13
14. Fast Fourier Transform(cont.)Fast Fourier Transform(cont.)
Re‐writing
1
0
/2
][][
N
n
Njnk
enxkX
as
1
0
][][
N
n
nk
NWnxkX
It is easy to realize that the same values of are calculated many times as thenk
WIt is easy to realize that the same values of are calculated many times as the
computation proceeds
Using the symmetric property of the twiddle factor, we can save lots of computations
NW
111 NNN
)12()2(
)()(][][
12
)12(
12
2
1
0
1
0
1
0
WrxWrx
WnxWnxWnxkX
N
rk
N
N
kr
N
N
nodd
n
kn
N
N
neven
n
kn
N
N
n
nk
N
)()(
)()(
)12()2(
12
0
22
12
0
21
00
kXWkX
WrxWWrx
WrxWrx
k
N
r
kr
N
k
N
N
r
kr
N
r
N
r
N
)()( 21 kXWkX k
N
Thus the N‐point DFT can be obtained from two N/2‐point transforms, one on even
input data, and one on odd input data.
14
15. Introduction for MATLABIntroduction for MATLAB
MATLAB is a numerical computing environment developed by
MathWorks. MATLAB allows matrix manipulations, plotting ofp , p g
functions and data, and implementation of algorithms
Getting helpGetting help
You can get help by typing the commands help or lookfor at
the >> prompt, e.g.
>> help fft>> help fft
Arithmetic operators
Symbol Operation Example
+ Addition 3 1 + 9+ Addition 3.1 + 9
‐ Subtraction 6.2 – 5
* Multiplication 2 * 3
/ Division 5 / 2/ Division 5 / 2
^ Power 3^2
15
16. Data Representations in MATLABData Representations in MATLAB
Variables: Variables are defined as the assignment operator “=” . The syntax of
variable assignment is
i bl l ( i )variable name = a value (or an expression)
For example,
>> x = 5
x =
5
>> y = [3*7, pi/3]; % pi is in MATLAB
Vectors/Matrices MATLAB can create and manip late arra s of 1 ( ectors) 2
Vectors/Matrices: MATLAB can create and manipulate arrays of 1 (vectors), 2
(matrices), or more dimensions
row vectors: a = [1, 2, 3, 4] is a 1X4 matrix
column vectors: b = [5; 6; 7; 8; 9] is a 5X1 matrix, e.g.
>> A = [1 2 3; 7 8 9; 4 5 6]
A = 1 2 3
7 8 9
4 5 64 5 6
16
17. Mathematical Functions in MATLABMathematical Functions in MATLAB
MATLAB offers many predefined mathematical functions for
technical computing, e.g.p g, g
cos(x) Cosine abs(x) Absolute value
sin(x) Sine angle(x) Phase angle
exp(x) Exponential conj(x) Complex conjugate
Colon operator (:)
exp(x) Exponential conj(x) Complex conjugate
sqrt(x) Square root log(x) Natural logarithm
Colon operator (:)
Suppose we want to enter a vector x consisting of points
(0,0.1,0.2,0.3,…,5). We can use the command
>> x = 0:0.1:5;;
Most of the work you will do in MATLAB will be stored in files called
scripts, or m‐files, containing sequences of MATLAB commands to be
executed over and over again
17
18. Basic plotting in MATLABBasic plotting in MATLAB
MATLAB has an excellent set of graphic tools. Plotting a given data set or
the results of computation is possible with very few commands
The MATLAB command to plot a graph is plot(x,y), e.g.
>> x = 0:pi/100:2*pi; 0.8
1
Sine function
p / p ;
>> y = sin(x);
>> plot(x,y)
0.2
0.4
0.6
x
MATLAB enables you to add axis
Labels and titles, e.g.
i
-0.4
-0.2
0
Sineofx
>> xlabel('x=0:2pi');
>> ylabel('Sine of x');
>> tile('Sine function') 0 1 2 3 4 5 6 7
-1
-0.8
-0.6
x=0:2x 0:2
18
19. Example 1: Sine WaveExample 1: Sine Wave
0.5
1
Sine Wave Signal
Fs = 150; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
f = 5; % Create a sine wave of f Hz.
i (2* i*t*f)
-0.5
0
Amplitude
x = sin(2*pi*t*f);
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that length(X)
is equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
0 0.2 0.4 0.6 0.8 1
-1
Time (s)
80
Power Spectrum of a Sine Wave
% FFT is symmetric, throw away second half
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% Frequency vector
f = (0:nfft/2-1)*Fs/nfft;
40
60
80
Power
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Sine Wave Signal');
xlabel('Time (s)');
l b l('A lit d ')
0 10 20 30 40 50 60 70 80
0
20
Frequency (Hz)
P
ylabel('Amplitude');
figure(2);
plot(f,mx);
title('Power Spectrum of a Sine Wave');
xlabel('Frequency (Hz)');
ylabel('Power');Frequency (Hz) ylabel( Power );
19
20. Example 2: Cosine WaveExample 2: Cosine Wave
Fs = 150; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
f = 5; % Create a sine wave of f Hz.
x = cos(2*pi*t*f);
0.5
1
Cosine Wave Signal
x = cos(2*pi*t*f);
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that length(X) is
equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
-0.5
0
Amplitude
y , y
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% Frequency vector
f = (0:nfft/2-1)*Fs/nfft;
0 0.2 0.4 0.6 0.8 1
-1
Time (s)
80
Power Spectrum of a Cosine Wave
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Sine Wave Signal');
xlabel('Time (s)');
ylabel('Amplitude');
40
60
80
Power
ylabel( Amplitude );
figure(2);
plot(f,mx);
title('Power Spectrum of a Sine Wave');
xlabel('Frequency (Hz)');
ylabel('Power');
0 10 20 30 40 50 60 70 80
0
20
Frequency (Hz)
P
yFrequency (Hz)
20
21. Example 3: Cosine Wave with Phase ShiftExample 3: Cosine Wave with Phase Shift
Fs = 150; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
f = 5; % Create a sine wave of f Hz.
pha = 1/3*pi; % phase shift
0.5
1
Cosine Wave Signal with Phase Shift
pha = 1/3*pi; % phase shift
x = cos(2*pi*t*f + pha);
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that length(X) is
equal to nfft
X = fft(x,nfft);
-0.5
0
Amplitude
( , )
% FFT is symmetric, throw away second half
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% Frequency vector
/ /
0 0.2 0.4 0.6 0.8 1
-1
Time (s)
80
Power Spectrum of a Cosine Wave Signal with Phase Shift
f = (0:nfft/2-1)*Fs/nfft;
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Sine Wave Signal');
xlabel('Time (s)');
40
60
80
Power
xlabel( Time (s) );
ylabel('Amplitude');
figure(2);
plot(f,mx);
title('Power Spectrum of a Sine Wave');
xlabel('Frequency (Hz)');
0 10 20 30 40 50 60 70 80
0
20
Frequency (Hz)
P
q y
ylabel('Power');
Frequency (Hz)
21
22. Example 4: Square WaveExample 4: Square Wave
0.5
1
Square Wave Signal Fs = 150; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
f = 5; % Create a sine wave of f Hz.
x = square(2*pi*t*f);
-0.5
0
Amplitude
x = square(2*pi*t*f);
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that length(X) is
equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
0 0.2 0.4 0.6 0.8 1
-1
Time (s)
100
Power Spectrum of a Square Wave
y , y
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% Frequency vector
f = (0:nfft/2-1)*Fs/nfft;
40
60
80
100
Power
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Square Wave Signal');
xlabel('Time (s)');
ylabel('Amplitude');
0 20 40 60 80
0
20
40
Frequency (Hz)
ylabel( Amplitude );
figure(2);
plot(f,mx);
title('Power Spectrum of a Square Wave');
xlabel('Frequency (Hz)');
ylabel('Power');y
22
23. Example 5: Square PulseExample 5: Square Pulse
0.8
1
Square Pulse Signal
Fs = 150; % Sampling frequency
t = -0.5:1/Fs:0.5; % Time vector of 1 second
w = .2; % width of rectangle
x = rectpuls(t w); % Generate Square Pulse
0.2
0.4
0.6
Amplitude
x = rectpuls(t, w); % Generate Square Pulse
nfft = 512; % Length of FFT
% Take fft, padding with zeros so that length(X) is
equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
-0.5 0 0.5
0
Time (s)
30
Power Spectrum of a Square Pulse
y , y
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% Frequency vector
f = (0:nfft/2-1)*Fs/nfft;
15
20
25
Power
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Square Pulse Signal');
xlabel('Time (s)');
ylabel('Amplitude');
0 20 40 60 80
0
5
10
Frequency (Hz)
ylabel( Amplitude );
figure(2);
plot(f,mx);
title('Power Spectrum of a Square Pulse');
xlabel('Frequency (Hz)');
ylabel('Power');y
23
24. Example 6: Gaussian PulseExample 6: Gaussian Pulse
3
4
Gaussian Pulse Signal
Fs = 60; % Sampling frequency
t = -.5:1/Fs:.5;
x = 1/(sqrt(2*pi*0.01))*(exp(-t.^2/(2*0.01)));
nfft = 1024; % Length of FFT
1
2
Amplitude
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that
length(X) is equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
X = X(1:nfft/2);
-0.5 0 0.5
0
Time (s)
( )
% Take the magnitude of fft of x
mx = abs(X);
% This is an evenly spaced frequency vector
f = (0:nfft/2-1)*Fs/nfft;
% Generate the plot, title and labels.
40
50
60
Power Spectrum of a Gaussian Pulse
r
figure(1);
plot(t,x);
title('Gaussian Pulse Signal');
xlabel('Time (s)');
ylabel('Amplitude');
figure(2);
0
10
20
30
Power
figure(2);
plot(f,mx);
title('Power Spectrum of a Gaussian Pulse');
xlabel('Frequency (Hz)');
ylabel('Power');
0 5 10 15 20 25 30
0
Frequency (Hz)
24
25. Example 7: Exponential DecayExample 7: Exponential Decay
1.5
2
Exponential Decay Signal
Fs = 150; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
x = 2*exp(-5*t);
nfft 1024 % Length of FFT
0.5
1
Amplitude
nfft = 1024; % Length of FFT
% Take fft, padding with zeros so that
length(X) is equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second
half
0 0.2 0.4 0.6 0.8 1
0
Time (s)
P S t f E ti l D Si l
half
X = X(1:nfft/2);
% Take the magnitude of fft of x
mx = abs(X);
% This is an evenly spaced frequency
vector
40
50
60
70
Power Spectrum of Exponential Decay Signal
er
f = (0:nfft/2-1)*Fs/nfft;
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Exponential Decay Signal');
xlabel('Time (s)');
0 20 40 60 80
0
10
20
30
Powe
xlabel('Time (s)');
ylabel('Amplitude');
figure(2);
plot(f,mx);
title('Power Spectrum of Exponential
Decay Signal');
0 20 40 60 80
Frequency (Hz)
y g );
xlabel('Frequency (Hz)');
ylabel('Power');
25
26. Example 8: Chirp SignalExample 8: Chirp Signal
0.5
1
Chirp Signal
Fs = 200; % Sampling frequency
t = 0:1/Fs:1; % Time vector of 1 second
x = chirp(t,0,1,Fs/6);
nfft = 1024; % Length of FFT
-0.5
0
Amplitude
; g
% Take fft, padding with zeros so that
length(X) is equal to nfft
X = fft(x,nfft);
% FFT is symmetric, throw away second half
X = X(1:nfft/2);
0 0.2 0.4 0.6 0.8 1
-1
Time (s)
Power Spectrum of Chirp Signal
% Take the magnitude of fft of x
mx = abs(X);
% This is an evenly spaced frequency
vector
f = (0:nfft/2-1)*Fs/nfft;
% Generate the plot title and labels
15
20
25
p p g
wer
% Generate the plot, title and labels.
figure(1);
plot(t,x);
title('Chirp Signal');
xlabel('Time (s)');
ylabel('Amplitude');
0 20 40 60 80 100
0
5
10
Pow
y p
figure(2);
plot(f,mx);
title('Power Spectrum of Chirp Signal');
xlabel('Frequency (Hz)');
ylabel('Power');
Frequency (Hz)
26