PART 2 : FOURIER SERIES
Objective :
1. To show that any periodic function (or signal) can be represented as a series of sinusoidal (or complex exponentials) function. 2. To show and to study hot to approximate periodic functions using a finite number of sinusoidal function and run the simulation using MATLAB.

Scope :
In experiment 1, students need to learn using MATLAB by connect it with Fourier series, where students must know how the output changes as higher order terms are added. Students must know to plot the graph. Besides, students must know to add instruction in appropriate line to plot frequencies versus coefficient for each wave form.

Equipment :
MATLAB software.

Experiment 1 : Fourier Series
Generally, student must know the basic concept of Fourier series. General form :
[pic]
Where :
[pic]
[pic]
For example, x(t) with the highest harmonic value = 2.
[pic]

Task 1 : Simulate Using MATLAB
Procedure :

Table 1 shows coefficient for complex exponential fourier series of half rectified sine wave with A=1, T=1.

|Wave form |Fourier Coefficients | |Half-rectified sine wave: |[pic] |

Table 2-1 : Complex Exponential Fourier Series of Half-rectified Sine Wave.

Execute program in Listing 1 using 4 inputs : [2 8 20 100].
Listing 1 :
• Program to give partial complex exponential fourier sums of a half rectified sine wave of unit amplitude in table 2-1.

clear
clc
k_max = input (‘Enter vector of highest harmonic values desired (even only) >>’); num_kmax = length (k_max);
for z = 1 : num_kmax
k = [-k_max (z) : k_max (z)];
L_k = length (k);
a_k = zeros (1, L_k);% Form vector of fourier series coefficients, a_k ((L_k+1)/2+1) = -0.25*j;% all odd-terms are zero except A(1)...

...Fourierseries
From Wikipedia, the free encyclopedia
Fourier transforms
Continuous Fourier transform
Fourierseries
Discrete-time Fourier transform
Discrete Fourier transform
Fourier analysis
Related transforms
The first four partial sums of the Fourierseries for a square wave
In mathematics, a Fourierseries (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourierseries is a branch of Fourier analysis.
Contents
[hide]
Definition[edit]
In this section, s(x) denotes a function of the real variable x, and s is integrable on an interval [x0, x0 + P], for real numbers x0 and P. We will attempt to represent s in that interval as an infinite sum, or series, of harmonically related sinusoidal functions. Outside the interval, the series is periodic with period P. It follows that if s also has that property, the approximation is valid on the entire real line. The case P = 2π is prominently featured in the literature, presumably because it affords a minor simplification, but at the expense of generality.
For integers N > 0, the...

...Abstract:
In 1965, Cooley and Turkey were two persons who discussed the FFT (Fast Fourier Transform) for the first time in history. In past years, researchers believed that a discrete Fourier transform can also be calculated and classified as FFT by using the Danielson-Lanczos lemma theorem. By using this theorem, this process is slower than other, as it is slightly tainted in speed due to the power of N (exponent of N) are not 2. Therefore, if the number of points i.e. N is not a power of two, then the transform will only gives you the sets of points matching to the prime factors of N [1]. FFT (Fast Fourier Transform) is a type of algorithm commonly known as discrete Fourier transform algorithm. This algorithm has much significance in the reduction of number of computations that governs for N points from the arithmetic expressions i.e. 2N2 to (2N log2 N), in this expression the ‘N’ is the number of computations and log2 is the logarithm having base 2. In Fourier analysis, if the above function is about to be transferred, not harmonically associated with the sampling frequency, then at this point, the reaction for this FFT behaves just like a sinc function i.e. commonly known as the sampling function, defined as a function used to rise the frequency in the signal processing and propagation, classified as Fourier transforms[4] . However, the other components such as integrated power and...

...ELE374 Fourier Analysis and Synthesis of Waveforms. By Anthony Njuguna EE08U122 – 080947424
Anthony Njuguna
ee08u122
Abstract Many applications in communication and systems are concerned with propagation of signals through networks. The resultant output signal is dependent on the properties of both the input signal and the processes acting on the signal. This is a laboratory Report will be focusing on using Fourierseries to analyze waveforms and the synthesis of waveforms. The report highlights Fourierseries analysis is a simple effective approach to analyse periodic signals. It will guide the reader through an experiment conducted in a lab to examine the effects of using Fourier analysis on signals and their waveforms. It should help the reader get a better understanding at signal analysis by looking at meaning of signal spectrum, how different waveforms have different spectrum but that standard results exist for standard waveforms. It should aid in the effects of spectrum limitations on the transmission of signals by looking at their bandwidth and bit rates.
1
Anthony Njuguna
ee08u122 Contents
Abstract...............................................................................................................................1 Introduction.........................................................................................................................3...

...SeriesFOURIERSERIES
Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourierseries analysis
q Table of contents q Begin Tutorial
c 2004 g.s.mcdonald@salford.ac.uk
Table of contents
1. 2. 3. 4. 5. 6. 7. Theory Exercises Answers Integrals Useful trig results Alternative notation Tips on using solutions Full worked solutions
Section 1: Theory
3
1. Theory
q A graph of periodic function f (x) that has period L exhibits the same pattern every L units along the x-axis, so that f (x + L) = f (x) for every value of x. If we know what the function looks like over one complete period, we can thus sketch a graph of the function over a wider interval of x (that may contain many periods)
f(x )
x
P E R IO D = L
Toc
Back
Section 1: Theory
4
q This property of repetition deﬁnes a fundamental spatial frequency k = 2π that can be used to give a ﬁrst approximation to L the periodic pattern f (x): f (x) c1 sin(kx + α1 ) = a1 cos(kx) + b1 sin(kx),
where symbols with subscript 1 are constants that determine the amplitude and phase of this ﬁrst approximation q A much better approximation of the periodic pattern f (x) can be built up by adding an appropriate combination of harmonics to this fundamental (sine-wave) pattern. For example, adding c2 sin(2kx + α2 ) = a2 cos(2kx) + b2 sin(2kx) c3 sin(3kx + α3 ) = a3 cos(3kx) + b3 sin(3kx) (the 2nd harmonic) (the 3rd harmonic)
Here,...

...FourierSeriesFourierseries started life as a method to solve problems about the ﬂow of heat through ordinary materials. It has grown so far that if you search our library’s data base for the keyword “Fourier” you will ﬁnd 425 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial diﬀerential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylor series is based on the idea that you can write a general function as an inﬁnite series of powers. The idea of Fourierseries is that you can write a function as an inﬁnite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now. Legendre polynomials are an important example of functions used for such expansions. An example: On the interval 0 < x < L the function x2 varies from 0 to L2 . It can be written as the series of cosines L2 4L2 + 2 x = 3 π
2 ∞ 1
(−1)n nπx cos 2 n L 2πx 1 3πx πx 1 − cos + cos − ··· L 4 L 9 L (1)
=...

...[pic] FourierSeries: Basic Results
[pic]
Recall that the mathematical expression
[pic]
is called a Fourierseries.
Since this expression deals with convergence, we start by defining a similar expression when the sum is finite.
Definition. A Fourier polynomial is an expression of the form
[pic]
which may rewritten as
[pic]
The constants a0, ai and bi, [pic], are called the coefficients of Fn(x).
The Fourier polynomials are [pic]-periodic functions. Using the trigonometric identities
[pic]
we can easily prove the integral formulas
(1)
for [pic], we have
[pic]
(2)
for m et n, we have
[pic]
(3)
for [pic], we have
[pic]
(4)
for [pic], we have
[pic]
Using the above formulas, we can easily deduce the following result:
Theorem. Let
[pic]
We have
[pic]
This theorem helps associate a Fourierseries to any [pic]-periodic function.
Definition. Let f(x) be a [pic]-periodic function which is integrable on [pic]. Set
[pic]
The trigonometric series
[pic]
is called the Fourierseries associated to the function f(x). We will use the notation
[pic]
Example. Find the Fourierseries of the function
[pic]
Answer. Since f(x) is odd, then an = 0,...

...Introduction to Time Series
January 16, 2012
• Instructor: Aditya Guntuboyina (aditya@stat.berkeley.edu)
• Lectures: 12:30 pm to 2 pm on Tuesdays and Thursdays at 160 Dwinelle
Hall.
• Oﬃce Hours: 10 am to 11 am on Tuesdays and Thursdays at 423 Evans
Hall.
• GSI: Brianna Heggeseth (bhirst@stat.berkeley.edu)
• GSI Lab Section: 10 am to 12 pm OR 12 pm to 2 pm on Fridays at 334
Evans Hall (The ﬁrst section will include a short Introduction to R).
• GSI Oﬃce Hours: TBA.
All course materials including lecture slides and assignments will be posted on
the course site at bSpace.
Short Description: A time series is a set of numerical observations, each one
being recorded at a speciﬁc time. Such data arise everywhere. This course aims
to teach you how to analyze time series data. There exist two approaches to
time series analysis: Time Domain approach and Frequency Domain approach.
Approximately, about 60% of the course will be on time domain methods and
40% on frequency domain methods.
Tentative List of Topics: Time Domain Methods: Tackling Trend and
Seasonality, Stationarity and Stationary ARMA models, ARIMA and Seasonal ARIMA models, State space models. Frequency Domain Methods: Periodogram, Spectral Density, Spectral Estimation.
Prerequisite: This course is intended for students who have taken at least one
elementary statistics course (e.g., 101) and one elementary probability course
(e.g., 134)....

...MA 1001: MATHEMATICS I
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3
Module 1 (12 hours)
Preliminary Calculus : Partial differentiation, Total differential and total derivative, Exact differentials, Chain
rule, Change of variables, Minima and Maxima of functions of two or more variables. Infinite Series : Notion of
convergence and divergence of infinite series, Ratio test, Comparison test, Raabe’s test, Root test, Series of
positive and negative terms, Idea of absolute convergence, Taylor’s and Maclaurin’s series.
Module 2 (17 hours)
First order ordinary differential equations: Methods of solution, Existence and uniqueness of solution,
Orthogonal Trajectories, Applications of first order differential equations. Linear second order equations:
Homogeneous linear equations with constant coefficients, fundamental system of solutions, Existence and
uniqueness conditions, Wronskian, Non homogeneous equations, Methods of Solutions, Applications.
Module 3 (13 hours)
Fourier Analysis : Periodic functions - Fourierseries, Functions of arbitrary period, Even and odd functions,
Half Range Expansions, Harmonic analysis, Complex FourierSeries, Fourier Integrals, Fourier Cosine and Sine
Transforms, Fourier Transforms.
Module 4 (14 hours)
Gamma functions and Beta functions, Definition and Properties. Laplace...