1. N

    Time-continuous signal Fourier transform or series

    Hello, i have a short understanding question on the subject of Fourier Transformation. My task is and here I would like to quote: "A periodic signal is given, calculate the continuous time Fourier transform". On the sketch at the task a rectangle is shown, which has its symmetry axis on the y...
  2. A

    Calculation of Fourier Transform Derivative d/dw (F{x(t)})=d/dw(X(w))

    Hello to my Math Fellows, Problem: I am looking for a way to calculate w-derivative of Fourier transform, d/dw (F{x(t)}), in terms of regular Fourier transform, X(w)=F{x(t)}. Definition Based Solution (not good enough): from I can find that w-derivative of Fourier transform for x(t) is...
  3. G

    Generalized Fourier Integral and Steepest descent path, Saddle point near the endpoin

    I am looking forward to solving the integration in the following equation with the assumption that $ka$ is very large \begin{align} H = 2jka\int_{-\pi/2}^{\pi/2}\cos{(\varphi-\phi)}e^{jka[\cos{\varphi}+\cos{(\varphi-\phi)}]}\ d\varphi \end{align} I used the steepest descent path method to...
  4. B

    How do I deal with a Fourier transform that contains derivatives?

    I want to calculate this: \frac{1}{2 \pi } \int_{- \infty}^{\infty} |X(ω)|^2 dω Should I just normally calculate the derivative inside of X(ω) and then use that final form of Χ(ω) to calculate the integral? Or can I use somehow the property which says that the Fourier transformation...
  5. S

    Fourier transform problem.

    I am having 2 problems which I do not even know where to start. I hope someone can help me. Problem 1) Let: \[ \mathcal{F}_N = \begin{bmatrix} 1 &1 &\cdots &1 \\ 1 &\omega &\cdots & \omega^{N-1} \\ \vdots &\vdots &\ddots &\vdots \\ 1 &\omega^{N-1} &\cdots &\omega^{(N-1)^2}...
  6. S

    Transformation of Equations - What is the deeper thing?

    Hi, If you have an ordinary differential equation (or equations), you can transform them under some conditions (integral, linear operator, ...) to algebraic equations. If you have a partial differential equation you may transform it to an ODE. This can be shown "easily" by doing Fourier...
  7. Z

    Requesting for some insights about Fourier transform basic

    I am studying the Fourier transform and trying to understand the following equation w is frequency.. t is time.. A(w) is a a signal amplified function delay(w) is a time delay function Trig Identity: sin( a + b ) = sin(a)cos(b) + sin(b)cos(a) Therefore: A(w)sin((wt + delay(w))) =...
  8. J

    Distribution - Fourier series

    Good day, I am trying to solve an exercise in the course of distribution theory and fourier analysis. I am new to the matter of using distribution in calculating, and I am thankful for any help to solve the following question: 1. Consider the $2\pi$ -periodic function $f(x)$ defined on...
  9. Z

    Proving Fourier property

    A function f has the property that f(x+\pi) = −f(x) for all x. Show that all its even Fourier coefficients are zero (i.e., a0 = a2 = a4 = a6 = . . . = 0, b2 = b4 = b6 = . . . = 0). Hint: Show that f must be periodic with period 2 \pi. Any tips would be much grateful. Thanks.
  10. Z

    General Fourier Series vs Fourier Cosine/Sine series

    I am currently learning the very impressive Fourier series. I am still new to this topic, but I am burning with a couple of questions... 1. Would the general form of Fourier series be enough to model any bit-wise or periodic curves? Why do we need Fourier cosine and sine series? What are they...
  11. G

    Fourier Analysis problem

    Hey folks, I don't get to the right result. I think that I've made an algebraic mistake somewhere. Checked it many times. Can anyone help me? Not sure if this is the right way to get help here. Please correct me if I am doing something wrong! Thanks!
  12. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

    1. The problem statement: Solve the following heat Eq. using FFCT: A metal bar of length L is at constant temperature of Uo, at t=0 the end x=L is suddenly given the constant temperature U1, and the end x=0 is insulated. Assuming that the surface of the bar is insulated, find the temperature...
  13. A

    Fourier Transform

    Hello everyone, am trying to solve this Fourier Trans. problem, here is the original solution >> Q/ How did he come up with this result and where is my mistake? All equation are in the above attached picture here is my attempt, part 1>>...
  14. D

    Fourier transform (application in electromagnetism)

    Hello everybody again! I hope the moderator will not angry for the fact that I am starting second thread today. If something's wrong- I can write my question in my first thread. I am examining the same text (actually it is compilation of very brief fragments) from the same author.... But this...
  15. N

    Question about Fourier transforms

    Hi all, I came up with the following question during a calculation. Suppose I have a function $f(x)$ defined only for $x>0$. Then the Fourier transform can be written as: \begin{eqnarray} f(p)=\int _{0}^{+\infty} dx e^{ipx}f(x)=\int _{-\infty}^{+\infty} dx e^{ipx}f(x)\theta(x) \end{eqnarray}...
  16. M

    integral of absolute value of a Fourier transform

    Hi guys, I am going to calculate the following integral: $$\int_0^{f_c+f_m} |Y(f)|^2\, df$$ where: $$Y(f)=\frac{\pi}{2} \alpha_m \sum_{l=1}^{L} \sqrt{g_l}\left [ e^{-j(\omega \tau_l - \theta_m)} \delta(\omega - \omega_0) + e^{-j(\omega \tau_l + \theta_m)} \delta(\omega + \omega_0)...
  17. R

    Fourier Series

    Unsure if this is the correct thread but I need assistance completing this: 𝑥(𝑡) = 𝑟 (1 − cos(𝑡 − 𝜓)) where r = radius 𝜓 = phase shift 1 revolution = 6.28 sec
  18. K

    Convolution Integral ...and not

    Dear Community, my goal is to calculate the following integral $$\mathcal{I} = \int_{-\infty }^{+\infty }\frac{f\left ( \mathbf{\vec{x}} \right )}{\left | \mathbf{\vec{c}}- \mathbf{\vec{x}} \right |}d^{3}x $$ in the particular case in which f\left ( \mathbf{\vec{x}} \right )=f\left ( x \right...
  19. U

    Fourier series solution

    How do I go about doing these questions on the Fourier series solution of differential equations? Thank you!!
  20. M

    Fourier series expansion without shifting to the given interval

    Given the function $f(x)=f(x+9)$ with $f(x)=x^2-11x+10$ The Fourier series expansion without shifting $f(x)$ on the working interval I will solve the Fourier series on the interval [1,10] thus the Period, P=9 and initial value, c=1 Calculating $a_0$ using the equation, $a_0= \frac{2}{P}...