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Fourier Analysis

Fourier series

Find the Fourier series in trigonometric form for f(t) = |sin(pi*t)|. Graph its power spectrum.

Fourier series by using MATLAB

(See attached file for full problem description) Find the Fourier Series coefficients for the signal (see the attached file). Use Matlab to plot the truncated Fourier series reconstruction for the signal, using the first 15 terms of the sum. Given an and bn, what would be the complex coefficients if you had instead calcula

Bessel and Legendre Series : Fourier-Legendre Expansions

8. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = 1/2(3x2- 1). If x = cosθ , then P0( cosθ ) = 1 and P1( cosθ ) = cos θ . Show that P2( cosθ ) = 1/4( 3cos2θ + 1 ). 9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (θ) = )of F(

Least-Squares Approximation and Mean Square Error

5. (a). Find the least squares approximation of sin(πx) over the interval [-1,1] by a polynomial of the form ao+ a1x+a2x². (b). Find the mean square error of the approximation. Note: Part (a) that is suppose to be sin(piex) and the polynomial is a sub 0, a sub 1, a sub 2. For some reason it wouldn't allow me t

Use Parseval's equality

We are using the book Methods of Real Analysis by Richard R. Goldberg (See attached file for full problem description) --- 12.5-2 Show that the Fourier series for is a) Use 12.5E to show that Fourier series at t=0 converges to . Deduce that 12.5E: Theorem. Let ( this

Fourier Series and Fourier Sine and Cosine Series

1.) Find fourier series of f(x)=4, x greater than -3 and less than 3 and 2.) Find fourier series of f(x) = x^2-x+3, x greater than -2 and less than 2 and 3.) Write the cosine and sine fourier series f(x)=x^2 for x greater than 0 and less than 2

Fourier Series

Please help ... what is the Fourier series expansion of f(t). It is a multiple answer question. (See attached file for full problem description)

Fourier Cosine Transform

The problem is from Fourier Cosine and Sine Transforms, and Passage from Fourier Integral to Laplace Transform: Solve using a cosine or sine transform. u'' - 9u =50e^-3x (0<x<infinty) u'(0) = 0, u(infinity)bounded

Sturm-Liouville expansion

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. 3. Expand... in terms of the eigenfunctions of the Sturm-Liouville problem. Please see attached.

Eigenfunctions - Expand the function

Please show each step of your solution. When you use theorems, definitions, etc., please include in your answer. Expand the function f(x) = {x^4, 0 <= x < 2 {0, 2 <= x <= pi in terms of the eigenfunctions of the given eigenvalue problem. Use computer software, such as the Maple int command, to evaluate the expan

Sturm Liouville problem

4. Use the results of Exercise 3 to recast each of the following differential equations in the Sturm-Liouville form (1a). Identify p(x), q(x), and w(x). (a) xy" + 5y' + lambda xy = 0 (b) y" + 2y' + xy + lambda x^2y = 0 (c) y" + y' + lambda y = 0 (d) y" - y' + lambda xy = 0 (e) x^2y" + xy' + lambda x^2y = 0 (f) y" + (cot

Fourier Transform

Find the Fourier transfrom of the following function: f(t) = te^(-2t), for t > 0

Fourier Series help

Find the complex Fourier series coefficients for the function x(t) depicted in the below figure Please see

Fourier Coefficients

Here, we have to find f(t) from the given value of Cn. I am not able to arrive at f(t)={3/[5-4cos(pit+pi/20)} despite many attempts. Please show me how to arrive at the final expected f(t) value.

Fourier Series and Convergence

Let f(x) ={0 -pi<x<0 {x 0<x<pi a) Compute the Fourier series for f on the interval [-pi, pi] b) Sketch the function to which the series converges. Please see the attached file for the fully formatted problems.

Fourier Series Sketched

f(x)= {0 -2<x<0 f(x+4) = f(x) {1 0<x<2 I have to find the Fourier series of the problem and sketch the graph of the function at 3 periods. I'm not sure if you need my text and what sections were covering or not so I'll just give it. "Elementary differential equations and boundary value prob

Gibbs Phenomenon and Fourier Series expansion

4. In this problem, you will devise a computer experiment to investigate Gibb's phenomenon, which is the presence of spurious oscillations in the graph of a truncated Fourier series near the places where the full Fourier series is discontinous. Choose any function you like that demonstrates Gibb's phenomenon. Your goal is to

Fourier Transform and Schrodinger Equation

Please view the attached file for the full description of the two questions being analyzed. Essentially, this posting is asking the following: Solve the Schrodinger equation with different potentials using the Fourier transform.

Evaluate: Complex Fourier Transform

Use the Fourier transform to solve the following differential equation: g" + 2g' + 5g = delta(x) Where delta is the dirac delta function (impulse). Please view the attached file for the full problem description.

Finding a Fourier Series

Suppose f(t) and g(t) are 2&#960; periodic functions with Fourier series representations {see attachment}. Find the Fourier series of {see attachment}.

Fourier Transforms

Use the Fourier transform to solve the one-dimensional wave equation. See attached file for full problem description.

Fourier Transform of the Signal

If the Fourier transform of the signal v(t) is v(w) = AT sinwt / wt then the energy contained in v(t) is a) (A^2)/2 b) A^2 c) (A^2)T d) (A^2)T/2

Complex Fourier Series

Please see the attached file for the fully formatted problem. Let Y: R --> R be the periodic function whose restriction to [0,1] is X (0,1/2) - X(1/2,1) Y is an odd function. S 1--> 0 Y(x) cos 2pi kx dx = S 1/2-->-1/2 Y(x) cos 2pi kx dx Vk Conclude the the complex Fourier Series...can be expressed in the form...

Eignevalues and Eigenvectors of the Fourier Transform

Problem attached. "Eigenvalues and Eigenvectors of the Fourier Transform" Recall that the Fourier transform F is a linear one-to-one transformation from L2 (?cc, cc) onto itself. Let .. be an element of L2(?cc,cc). Let..= , the Fourier transform of.., be defined by ..... It is clear that ..... are square-integrable fu

Stretching and Compressing Functions in Fourier Transform

If f(x) is a Gaussian with unit area - show that the scaled and stretched function 1/a * f(x/a) also has unit area - that's the hardest part. The other parts (along with a detailed explanation of this one) are in an attachment as both mathcad v.11 and in an html file - they're the same thing - but if you don't have mathcad y

Inner Product : Fourier Transform

Please see the attached file for the fully formatted problem. 2. Let p be a fixed and given square integrable function, i.e. 0 < S g(x)g(x) dx = ||g^2|| < inf The function g must vanish as |x| ---> inf. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x

Fourier Analysis Relationship Between Spectral Components

(a) Explain the relationship between the spectral components indicated above and the corresponding graph showing the motion of the surface of the motor plotted as a function of time. What do they represent? (b) Given the amplitudes indicated in the above diagram, copy and complete the table below for the amplitude of the vib