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Fourier Analysis

Proof of sum of a given infinite series of constants (closed form).

The sum of the infinite series, 1/2^2 - 2/3^2 + 3/4^2 - 4/5^2 + ... is given as pi^2/12 - log 2 on pages 64-65 in the book "Summation of Series" by L. B. W. Jolley, 2nd ed., 1961, Dover Pubs. Inc. (the ^ symbol denotes exponentiation in the above series and sum). For most of the series in his book, he lists a source (referen

Derivation of Fourier Transform of a Gaussian

From equation 6 of the attached, derive equation 7. In the expression for s(v), the natural log does not apply to the term [(v-vo)/(dv/2)]^2 , should just be ln(2). Yes, the limits from minus infinity to plus inifinity are adequate. Thanks for the help!

Fourier coefficients outputs

Fourier coefficients / b1, b2, b3, b4, b5... b11. -------------------------------------------------------------------------------- I have an output of an electronic device (full wave rectifier) that gives a sine wave with the negative part transposed symmetric to xx so that the function is always positive. I have to find the f

Matlab plots of the FFT of sequence

(See attached file for full problem description) For sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, so N=8 Using above x[n]: a) stem(x); b) Use the shift theorm to plot x delayed by 1, 4, 5, 6, and 8 samples, and plot the result for each. Remember the shift theorem says a delay by t0 seconds is equal to multiplying the spe

Discrete time Fourier transform of sequence and Matlab plot

Please see the attached file for full description. Calculate by hand the X(omega), DTFT of the sequence x[n]=[1 1 1 1 0 0 0 0] for n=0:7, zero else. Using Matlab, plot the real and imaginary components of your result for X(omega) for omega=0:0.01:2*pi, one plot for the real, one part for the imaginary. On the same plot

Fourier transform of functions

Using Fourier transforms where possible, derive the Fourier transforms of the following functions using the relationship: a.) f(x) = exp[i2po(x/lamda)sin(theta)] b.) f(x) = exp(- /ax/ ) See the attached file for full description.

Inverse Fourier transform,.,,

Find the inverse Fourier transform of each of the following Fourier transforms: X(x) = jw The answer I have is x[n] = (-1)^n / n (for n not equal to zero) 0 (for n = 0) I don't know how to get there.

Fourier Series of Signal

(See attached file for full problem description) Consider a periodic function f(x) with period L. Over one period, f(x) = sin(2*pi*x/L) over the interval -L/4 to L/4, f(x) = 0 over the intervals -L/2 to -L/4, and L/4 to L/2. Derive an expression for the nth Fourier series coefficient, an. In the Fourier series expansion

Fourier Transform of a Signal: Scaling and Time-Shift Property

Use tables and the scaling and time-shift property to find the Fourier transform of the signal below (which is zero for all other values of t than those shown, that is, it's not periodic) Plot the magnitude of the spectrum of this signal. Hint: Find the transform of the pulse centered around t=0, then use the time-shift p

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

Forurier Series and Coefficients

1.(a) Evaluate the Fourier coefficients ao, an, bn and the exponential coefficient Dn for the waveform shown below; do not use MATLAB or a calculator for integrations. (b) Plot 2 or 3 cycles of the Fourier series using MATLAB and verify whether the plot matches the given waveform. (c) Find C0 and Cn from the answer to part (a)

One Dimensional Heat Equation on a Finite Rod

Can someone please solve this heat equations with details on how to arrive at the solution.? Solve the heat equation: u_t = 3*u_xx with the following I.C/B.C: u(0,t) = u(L,t) = 0 u(x,0)=L*[1-cos(2*Pi*x/L)]

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

Find Fourier Series

Find Fourier series for f(x) = {-4 for x greater than/= -pi, and x less than/= 0 { 4 for x greater than/= 0, and less than/= pi

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

Find Fourier Series and Cosine and Sine Fourier 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 Coefficient of the Signal

Find the nth Fourier coefficient (in terms of the an's) of the signal y(t) defined by: y(t) = x(t+1) + x(1-t), for all t in <R>. Please see attachment for more readable form of question.

Finding Values Using Dirichlet's Theorem

For of the periodic functions , I need to find the value to which the Fourier series converges at x= 0 , Pi/2 , - Pi/2 , Pi , -Pi , 2Pi , - 2Pi Using Dirichlet's theorem (See attached files for full problem description).

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