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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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
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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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Lecture 11 Fast Fourier Transform (FFT) Weinan E1‚2 and Tiejun Li2 1 Department of Mathematics‚ Princeton University‚ weinan@princeton.edu 2 School of Mathematical Sciences‚ Peking University‚ tieli@pku.edu.cn No.1 Science Building‚ 1575 Examples Fast Fourier Transform Outline Examples Fast Fourier Transform Applications Applications Examples Fast Fourier Transform Applications Signal processing Filtering: a polluted signal 1.5 1 0.5 0 −0.5 −1 −1.5 0 200 400 600 800
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2. Analysis of Signals Figure 2.45.: Approximate FTs of two bandlimited signals ¾º½ º Ì ÀÐ ÖØ ÌÖ Ò× ÓÖÑ The Hilbert transform of a function is by definition‚ H {x(t)} = xh (t) = ∞ x(τ ) dτ t −τ −∞ 1 π (2.171) which is the convolution of x(t) with 1/π t‚ H {x(t)} = xh (t) = x(t) ∗ 1 πt (2.172) if we take the FT of this convolution‚ Xh (ω ) = X (ω ) × F 1 πt (2.173) From Example 2.24‚ F {sgn(t)} = 2 jω (2.174) and using duality from
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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 = 1.5Hz ‚ repeat 4. fs
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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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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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and STFT | ELEM018: Advanced Transform Methods | | Arsal Javid | 12/1/2011 | [Type the abstract of the document here. The abstract is typically a short summary of the contents of the document. Type the abstract of the document here. The abstract is typically a short summary of the contents of the document.] | Section 2: The following code was used to calculate perform the DFT Function in Matlab: function sw = dft(st) % DFT - Discrete Fourier Transform M = length(st); N = M;
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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 Transforms‚ shifting Theorem‚ Transforms of derivatives and integrals‚ Solution of
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