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 Fourier series to analyze waveforms and the synthesis of waveforms. The report highlights
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[pic] Fourier Series: Basic Results [pic] Recall that the mathematical expression [pic] is called a Fourier series. 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
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quantization and PCM generation. Pre-lab Assignment : Given signal x(t) = sinc(t)‚ x(t). 1. Find out the Fourier transform of x(t)‚ X(f )‚ sketch them; 2. Find out the Nyquist sampling frequency of 3. Given sampling rate terms of fs ‚ write down the expression of the Fourier transform of xs (t) → Xs (f ) in fs = 1 Hz‚ sketch the sampled signal X(f ). xs (t) = x(kTs ) and the Fourier 4. Let sampling frequency transform of xs (t). fs = 2Hz ‚ repeat 4. fs = 0.5Hz ‚ repeat 4. fs = 1
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Fourier series From Wikipedia‚ the free encyclopedia Fourier transforms Continuous Fourier transform Fourier series Discrete-time Fourier transform Discrete Fourier transform Fourier analysis Related transforms The first four partial sums of the Fourier series for a square wave In mathematics‚ a Fourier series (English pronunciation: /ˈfɔərieɪ/) decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions‚ namely sines and
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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‚
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Existence and uniqueness conditions‚ Wronskian‚ Non homogeneous equations‚ Methods of Solutions‚ Applications. Module 3 (13 hours) Fourier Analysis : Periodic functions - Fourier series‚ Functions of arbitrary period‚ Even and odd functions‚ Half Range Expansions‚ Harmonic analysis‚ Complex Fourier Series‚ Fourier Integrals‚ Fourier Cosine and Sine Transforms‚ Fourier Transforms. Module 4 (14 hours) Gamma functions and Beta functions‚ Definition and Properties. Laplace Transforms‚ Inverse Laplace
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Series FOURIER SERIES Graham S McDonald A self-contained Tutorial Module for learning the technique of Fourier series 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
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Jiawei Huang 37154135 Fourier Transform Assignment 1. Fourier transform of sine wave (code): import numpy as np import matplotlib.pyplot as plt from scipy.fftpack import fft‚fftfreq dt = 0.01 time = np.arange(0‚5.‚dt) f_1 = 3. a_1 = 2.3 y = a_1*np.sin(2.*np.pi*time*f_1) plt.plot(time‚y) plt.xlabel("Time t [s]") plt.ylabel("Wave") plt.title("Wave Signal") plt.show() n = time.shape[-1] transform = (fft(y)[:n/2]) * 2./n frequency = fftfreq(n‚time[1]-time[0])[:n/2]
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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
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Fractional Fourier Transform Adhemar Bultheel and H´ctor E. Mart´ e ınez Sulbaran 1 Dept. of Computer Science‚ Celestijnenlaan 200A‚ B-3001 Leuven Abstract In this note we make a critical comparison of some matlab programs for the digital computation of the fractional Fourier transform that are freely available and we describe our own implementation that filters the best out of the existing ones. Two types of transforms are considered: First the fast approximate fractional Fourier transform algorithm
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