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# Fourier Analysis

### Fourier Cosine Transforms

Please see attached file. Please show all steps in detail. In the two problems below find the Fourier Cosine Transform of the given f(x) and write f(x) as a Fourier integral. 1) -1, -Pi <x<0 f(x) = 1, 0<x<Pi 0, |x| > Pi 2) 2x+2a -a<x<0 f(x) = -2x+2a

### Mathematica : Fourier Series Expansions

Please help me in solving this problem. Please see attached file for full problem description. Using Mathematica and computer facilities express the function f(x) in terms of Fourier series expansion and show that the series converges as the number of terms increases: f(x) = x e-x/4 sin(x/3) -&#960; < x < &#960;

### Fourier Transform

Please see attachment. 1. What is the Fourier Transform for the convolution of sin(2t)*cos(2t). 2. Compute the inverse Fourier transform for X(w)= sin^2*3w 3. A continuous time signal x(t) has the Fourier transform X(w) = 1/jw+b where b is a constant. Determine the Fourier transform for v(t) = x*(5t-4)

### Periodic Functions: Fourier Sine and Cosine Series

Please see attachment. #1 for following periodic functions acting on the given interval Do the following: a) Sketch 4 periods of the given function of period b) Expand the function in a sine - cosine Fourier Series f(x) = 2-x, -2<x<2 #2) Expand the function in a sine-cosine series and

### Fourier Series of Even and Odd Functions

Find the Fourier series expansion of the functions: f(t) = 1 if Pi/3<|t|<2Pi/3 0 everywhere else f(t) = 1 Pi/3 < t < 2Pi/3 -1 -2Pi/3 < t < Pi/3 0 everywhere else In the interval [-Pi , Pi]

### Fourier Transforms and Wave Analysis

The question is the example on page 2 of the attachment (entitled 'Uniform Transducer'). it states that the center of the finger is at z'=L/4. I assume this is an arbitrary position. For Eq (2.4.6), the contribution from the left-hand finger is added. I'm not entirely sure how this equation is arrived at. It does not look like a

### Fourier Transforms : Time-Frequency Duality and Time Differentiation Property

Please see the attached file for the fully formatted problems. You don't have to worry about (iii).

### Solving a Fourier Transform

Suppose f(x) has the Fourier transform F(Ω). If a ≠ 0 show that f(ax) has the Fourier Transform 1/|a| F (Ω/a). Please see the attached file for the fully formatted problems.

### Exponential Fourier Series

Find the exponential Fourier series for x(t), y(t) and z(t). In each of three cases it is not necessary to do any integration. ω=2πf t = n/256 (t goes from 0 to 1 in increments of 1/256) x(t)= cos ω;t frequency= 2Hz y(t)= cos ωt frequency= 16Hz z(t)= the product of x(t) and y(t)

### Fourier Integral and Convergence

Given the function below: Expand the function in an Fourier integral and determine what this integral converges to. f(x) = xe^-|4x| keywords: integration, integrates, integrals, integrating, double, triple, multiple

### Fourier Transforms, Gaussian and Ricker Wavelets and Waveforms and Dirac Delta Function

Please see the attached files for the fully formatted problems.

### Fourier Series : Expansions and Differentiation

Let f (x) = |x| for x greater or equal to -1, less than or equal to +1 a) Write the Fourier series for f (x) on [-1,1]. b) Show that this series can be differentiated term by term to yield the Fourier expansion of f'(x) on [-1,1] c) Determine f'(x) and write it's Fourier series on [-1,1] d) Compare b and c. key

### Examples of Fourier series and sums of numerical series.

We use the Fourier expansions of certain poynomial functions to compute the sum of some useful numerical series. The formulas are quite general and give, at the end, the Fourier expansion of every polynomial function. By the way, these formulas can be also used for a numerical approximation of pi=3.14....

### Set of functions (Fourier Series and Signal Spaces)

Functions, Interval. See attached file for full problem description. Given the set of functions f1(t) = A1*exp(-t) f2(t) = A2*e^(-2t) Defined on the interval (0, infinity). (a) Find A1 such that f1(t) is normalized to unity on (0, infinity). Call this function PHI_1(t). (b) Find B such that PHI(t) and f2

### Fourier Transform Integrals

Using the Fourier transform integral, find Fourier transforms of the following signals. xa(t) = t *exp(-αt) * u(t), α > 0; xb(t) = t2 * u(t) * u(1 – t) xc(t) = exp(-αt) * u(t) * u(1 – t), α > 0;

### Fourier series, Fourier Transform and Partial Differential Equations

Please see the attached file for the fully formatted problems. ODE: 1. Solve ()'sinyxy=+. 2. Find the complete solution of the ODE ()()42212cosyyyx&#8722;&#8722;=. 3. Find the complete solution of the ODE ()46sinyy&#8722;=. 4. Find a second order ODE whose solution is a family of circle with arbitrary radius and center on t

### 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

### Applications of Fourier transform

Define a Fourier Transform. What are its properties and application areas ? Describe its application in signal processing.

### Find the fourier transform of a positive integer

Find the fourier transform of x^k, where k is a positive integer and x is a single real variable.

### 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 = cos&#952; , then P0( cos&#952; ) = 1 and P1( cos&#952; ) = cos &#952; . Show that P2( cos&#952; ) = 1/4( 3cos2&#952; + 1 ). 9. Use the results of problem 8, to find a Fourier-Legendre expansion ( F (&#952;) = )of F(