This document provides a tutorial on the Fast Fourier Transform (FFT). It begins by explaining that the FFT is a faster version of the Discrete Fourier Transform (DFT) that takes a discrete signal in the time domain and transforms it into the discrete frequency domain. It then reviews other transforms taught in previous courses and explains how the DFT relates to the Discrete-Time Fourier Transform (DTFT). The document provides MATLAB examples demonstrating how to use the FFT function to compute the DFT of signals and understand the results. It concludes by discussing how to properly analyze a signal's spectrum using the FFT with MATLAB.
Discrete time signals are handled by discrete systems. The discrete time signals are dissected with the aid of Discrete Time Fourier Series (DTFS), Discrete Time Fourier Transform (DTFT), discrete fourier transform (DFT) and z-transform. Copy the link given below and paste it in new browser window to get more information on Determination of DTFT:- http://www.transtutors.com/homework-help/digital-signal-processing/frequency-analysis-dtft/determine-dtft-example.aspx
Discrete time signals are handled by discrete systems. The discrete time signals are dissected with the aid of Discrete Time Fourier Series (DTFS), Discrete Time Fourier Transform (DTFT), discrete fourier transform (DFT) and z-transform. Copy the link given below and paste it in new browser window to get more information on Determination of DTFT:- http://www.transtutors.com/homework-help/digital-signal-processing/frequency-analysis-dtft/determine-dtft-example.aspx
Interference Cancellation by Repeated Filtering in the Fractional Fourier Tra...IJERA Editor
The Fractional Fourier Transform (FrFT) is a useful tool that separates signals-of-interest (SOIs) from
interference and noise in non-stationary environments. This requires estimation of the rotational parameter „a‟ to
rotate the signal to a new domain along an axis „ta‟, in which the interference can best be filtered out. The value
of „a‟ is typically chosen as that which minimizes the mean-square error (MSE) between the desired SOI and its
estimate or that minimizes the overlap between the signal and noise, projected onto the axis „ta‟. In this paper, we
extend this concept to perform repeated filtering, in multiple FrFT domains to reduce the MSE further than can
be done with a single FrFT. We perform this solely using MSE as the metric by which to compute „a‟ at each
stage, thereby simplifying the approach and improving performance over conventional single stage FrFT
methods or methods based solely on the frequency domain filtering, such as the Fast Fourier transform (FFT).
We show that the proposed method improves the MSE two or three orders of magnitude over the conventional
methods using L ≤ 3 stages of FrFT filtering
Interference Cancellation by Repeated Filtering in the Fractional Fourier Tra...IJERA Editor
The Fractional Fourier Transform (FrFT) is a useful tool that separates signals-of-interest (SOIs) from
interference and noise in non-stationary environments. This requires estimation of the rotational parameter „a‟ to
rotate the signal to a new domain along an axis „ta‟, in which the interference can best be filtered out. The value
of „a‟ is typically chosen as that which minimizes the mean-square error (MSE) between the desired SOI and its
estimate or that minimizes the overlap between the signal and noise, projected onto the axis „ta‟. In this paper, we
extend this concept to perform repeated filtering, in multiple FrFT domains to reduce the MSE further than can
be done with a single FrFT. We perform this solely using MSE as the metric by which to compute „a‟ at each
stage, thereby simplifying the approach and improving performance over conventional single stage FrFT
methods or methods based solely on the frequency domain filtering, such as the Fast Fourier transform (FFT).
We show that the proposed method improves the MSE two or three orders of magnitude over the conventional
methods using L ≤ 3 stages of FrFT filtering
Find the compact trigonometric Fourier series for the periodic signal.pdfarihantelectronics
Find the compact trigonometric Fourier series for the periodic signal x(t) and sketch the
amplitude and phase spectrum for first 4 frequency components. By inspection of the spectra,
sketch the exponential Fourier spectra. By inspection of spectra in part b), write the exponential
Fourier series for x(t)
Solution
ECE 3640 Lecture 4 – Fourier series: expansions of periodic functions. Objective: To build upon
the ideas from the previous lecture to learn about Fourier series, which are series representations
of periodic functions. Periodic signals and representations From the last lecture we learned how
functions can be represented as a series of other functions: f(t) = Xn k=1 ckik(t). We discussed
how certain classes of things can be built using certain kinds of basis functions. In this lecture we
will consider specifically functions that are periodic, and basic functions which are
trigonometric. Then the series is said to be a Fourier series. A signal f(t) is said to be periodic
with period T0 if f(t) = f(t + T0) for all t. Diagram on board. Note that this must be an everlasting
signal. Also note that, if we know one period of the signal we can find the rest of it by periodic
extension. The integral over a single period of the function is denoted by Z T0 f(t)dt. When
integrating over one period of a periodic function, it does not matter when we start. Usually it is
convenient to start at the beginning of a period. The building block functions that can be used to
build up periodic functions are themselves periodic: we will use the set of sinusoids. If the period
of f(t) is T0, let 0 = 2/T0. The frequency 0 is said to be the fundamental frequency; the
fundamental frequency is related to the period of the function. Furthermore, let F0 = 1/T0. We
will represent the function f(t) using the set of sinusoids i0(t) = cos(0t) = 1 i1(t) = cos(0t + 1)
i2(t) = cos(20t + 2) . . . Then, f(t) = C0 + X n=1 Cn cos(n0t + n) The frequency n0 is said to be
the nth harmonic of 0. Note that for each basis function associated with f(t) there are actually two
parameters: the amplitude Cn and the phase n. It will often turn out to be more useful to
represent the function using both sines and cosines. Note that we can write Cn cos(n0t + n) = Cn
cos(n) cos(n0t) Cn sin(n)sin(n0t). ECE 3640: Lecture 4 – Fourier series: expansions of periodic
functions. 2 Now let an = Cn cos n bn = Cn sin n Then Cn cos(n0t + n) = an cos(n0t) + bn
sin(n0t) Then the series representation can be f(t) = C0 + X n=1 Cn cos(n0t + n) = a0 + X n=1 an
cos(n0t) + bn sin(n0t) The first of these is the compact trigonometric Fourier series. The second
is the trigonometric Fourier series.. To go from one to the other use C0 = a0 Cn = p a 2 n + b 2 n
n = tan1 (bn/an). To complete the representation we must be able to compute the coefficients.
But this is the same sort of thing we did before. If we can show that the set of functions
{cos(n0t),sin(n0t)} is in fact an orthogonal set, then we can use the same.
We all have good and bad thoughts from time to time and situation to situation. We are bombarded daily with spiraling thoughts(both negative and positive) creating all-consuming feel , making us difficult to manage with associated suffering. Good thoughts are like our Mob Signal (Positive thought) amidst noise(negative thought) in the atmosphere. Negative thoughts like noise outweigh positive thoughts. These thoughts often create unwanted confusion, trouble, stress and frustration in our mind as well as chaos in our physical world. Negative thoughts are also known as “distorted thinking”.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
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The Indian economy is classified into different sectors to simplify the analysis and understanding of economic activities. For Class 10, it's essential to grasp the sectors of the Indian economy, understand their characteristics, and recognize their importance. This guide will provide detailed notes on the Sectors of the Indian Economy Class 10, using specific long-tail keywords to enhance comprehension.
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How to Split Bills in the Odoo 17 POS ModuleCeline George
Bills have a main role in point of sale procedure. It will help to track sales, handling payments and giving receipts to customers. Bill splitting also has an important role in POS. For example, If some friends come together for dinner and if they want to divide the bill then it is possible by POS bill splitting. This slide will show how to split bills in odoo 17 POS.
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Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
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How to Make a Field invisible in Odoo 17Celine George
It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
CLASS 11 CBSE B.St Project AIDS TO TRADE - INSURANCE
RES701 Research Methodology_FFT
1. University of Rhode Island Department of Electrical and Computer Engineering
ELE 436: Communication Systems
FFT Tutorial
1 Getting to Know the FFT
What is the FFT? FFT = Fast Fourier Transform. The FFT is a faster version of the Discrete
Fourier Transform (DFT). The FFT utilizes some clever algorithms to do the same thing as the
DTF, but in much less time.
Ok, but what is the DFT? The DFT is extremely important in the area of frequency (spectrum)
analysis because it takes a discrete signal in the time domain and transforms that signal into its
discrete frequency domain representation. Without a discrete-time to discrete-frequency transform
we would not be able to compute the Fourier transform with a microprocessor or DSP based system.
It is the speed and discrete nature of the FFT that allows us to analyze a signal’s spectrum with
Matlab or in real-time on the SR770
2 Review of Transforms
Was the DFT or FFT something that was taught in ELE 313 or 314? No. If you took
ELE 313 and 314 you learned about the following transforms:
Laplace Transform: x(t) ⇔ X(s) where X(s) =
∞
−∞
x(t)e−stdt
Continuous-Time Fourier Transform: x(t) ⇔ X(jω) where X(jω) =
∞
−∞
x(t)e−jωtdt
z Transform: x[n] ⇔ X(z) where X(z) =
∞
n=−∞
x[n]z−n
Discrete-Time Fourier Transform: x[n] ⇔ X(ejΩ) where X(ejΩ) =
∞
n=−∞
x[n]e−jΩn
The Laplace transform is used to to find a pole/zero representation of a continuous-time signal or
system, x(t), in the s-plane. Similarly, The z transform is used to find a pole/zero representation
of a discrete-time signal or system, x[n], in the z-plane.
The continuous-time Fourier transform (CTFT) can be found by evaluating the Laplace trans-
form at s = jω. The discrete-time Fourier transform (DTFT) can be found by evaluating the z
transform at z = ejΩ.
1
2. 3 Understanding the DFT
How does the discrete Fourier transform relate to the other transforms? First of all, the
DFT is NOT the same as the DTFT. Both start with a discrete-time signal, but the DFT produces
a discrete frequency domain representation while the DTFT is continuous in the frequency domain.
These two transforms have much in common, however. It is therefore helpful to have a basic
understanding of the properties of the DTFT.
Periodicity: The DTFT, X(ejΩ), is periodic. One period extends from f = 0 to fs, where fs
is the sampling frequency. Taking advantage of this redundancy, The DFT is only defined in the
region between 0 and fs.
Symmetry: When the region between 0 and fs is examined, it can be seen that there is even
symmetry around the center point, 0.5fs, the Nyquist frequency. This symmetry adds redundant
information. Figure 1 shows the DFT (implemented with Matlab’s FFT function) of a cosine with
a frequency one tenth the sampling frequency. Note that the data between 0.5fs and fs is a mirror
image of the data between 0 and 0.5fs.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
2
4
6
8
10
12
frequency/f
s
Figure 1: Plot showing the symmetry of a DFT
2
3. 4 Matlab and the FFT
Matlab’s FFT function is an effective tool for computing the discrete Fourier transform of a signal.
The following code examples will help you to understand the details of using the FFT function.
Example 1: The typical syntax for computing the FFT of a signal is FFT(x,N) where x is the
signal, x[n], you wish to transform, and N is the number of points in the FFT. N must be at least
as large as the number of samples in x[n]. To demonstrate the effect of changing the value of N,
sythesize a cosine with 30 samples at 10 samples per period.
n = [0:29];
x = cos(2*pi*n/10);
Define 3 different values for N. Then take the transform of x[n] for each of the 3 values that were
defined. The abs function finds the magnitude of the transform, as we are not concered with
distinguishing between real and imaginary components.
N1 = 64;
N2 = 128;
N3 = 256;
X1 = abs(fft(x,N1));
X2 = abs(fft(x,N2));
X3 = abs(fft(x,N3));
The frequency scale begins at 0 and extends to N − 1 for an N-point FFT. We then normalize the
scale so that it extends from 0 to 1 − 1
N .
F1 = [0 : N1 - 1]/N1;
F2 = [0 : N2 - 1]/N2;
F3 = [0 : N3 - 1]/N3;
Plot each of the transforms one above the other.
subplot(3,1,1)
plot(F1,X1,’-x’),title(’N = 64’),axis([0 1 0 20])
subplot(3,1,2)
plot(F2,X2,’-x’),title(’N = 128’),axis([0 1 0 20])
subplot(3,1,3)
plot(F3,X3,’-x’),title(’N = 256’),axis([0 1 0 20])
Upon examining the plot (shown in figure 2) one can see that each of the transforms adheres to
the same shape, differing only in the number of samples used to approximate that shape. What
happens if N is the same as the number of samples in x[n]? To find out, set N1 = 30. What does
the resulting plot look like? Why does it look like this?
3
4. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
N = 64
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
N = 128
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
5
10
15
20
N = 256
Figure 2: FFT of a cosine for N = 64, 128, and 256
Example 2: In the the last example the length of x[n] was limited to 3 periods in length.
Now, let’s choose a large value for N (for a transform with many points), and vary the number of
repetitions of the fundamental period.
n = [0:29];
x1 = cos(2*pi*n/10); % 3 periods
x2 = [x1 x1]; % 6 periods
x3 = [x1 x1 x1]; % 9 periods
N = 2048;
X1 = abs(fft(x1,N));
X2 = abs(fft(x2,N));
X3 = abs(fft(x3,N));
F = [0:N-1]/N;
subplot(3,1,1)
plot(F,X1),title(’3 periods’),axis([0 1 0 50])
subplot(3,1,2)
plot(F,X2),title(’6 periods’),axis([0 1 0 50])
subplot(3,1,3)
plot(F,X3),title(’9 periods’),axis([0 1 0 50])
The previous code will produce three plots. The first plot, the transform of 3 periods of a cosine,
looks like the magnitude of 2 sincs with the center of the first sinc at 0.1fs and the second at 0.9fs.
4
5. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
3 periods
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
6 periods
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
10
20
30
40
50
9 periods
Figure 3: FFT of a cosine of 3, 6, and 9 periods
The second plot also has a sinc-like appearance, but its frequency is higher and it has a larger
magnitude at 0.1fs and 0.9fs. Similarly, the third plot has a larger sinc frequency and magnitude.
As x[n] is extended to an large number of periods, the sincs will begin to look more and more like
impulses.
But I thought a sinusoid transformed to an impulse, why do we have sincs in the
frequency domain? When the FFT is computed with an N larger than the number of samples
in x[n], it fills in the samples after x[n] with zeros. Example 2 had an x[n] that was 30 samples
long, but the FFT had an N = 2048. When Matlab computes the FFT, it automatically fills the
spaces from n = 30 to n = 2047 with zeros. This is like taking a sinusoid and mulitipying it with a
rectangular box of length 30. A multiplication of a box and a sinusoid in the time domain should
result in the convolution of a sinc with impulses in the frequency domain. Furthermore, increasing
the width of the box in the time domain should increase the frequency of the sinc in the frequency
domain. The previous Matlab experiment supports this conclusion.
5 Spectrum Analysis with the FFT and Matlab
The FFT does not directly give you the spectrum of a signal. As we have seen with the last two
experiments, the FFT can vary dramatically depending on the number of points (N) of the FFT,
and the number of periods of the signal that are represented. There is another problem as well.
The FFT contains information between 0 and fs, however, we know that the sampling frequency
5
6. must be at least twice the highest frequency component. Therefore, the signal’s spectrum should
be entirly below fs
2 , the Nyquist frequency.
Recall also that a real signal should have a transform magnitude that is symmetrical for for positive
and negative frequencies. So instead of having a spectrum that goes from 0 to fs, it would be more
appropriate to show the spectrum from −fs
2 to fs
2 . This can be accomplished by using Matlab’s
fftshift function as the following code demonstrates.
n = [0:149];
x1 = cos(2*pi*n/10);
N = 2048;
X = abs(fft(x1,N));
X = fftshift(X);
F = [-N/2:N/2-1]/N;
plot(F,X),
xlabel(’frequency / f s’)
−0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5
0
10
20
30
40
50
60
70
80
frequency / f
s
Figure 4: Approximate Spectrum of a Sinusoid with the FFT
6