Fourier Definition and 1000 Threads

Hello, Something I have some time wondering and still couldn't find the answer is to this question: if there is some relation between the Spectrum (functional analysis) and the Frequency spectrum in Fourier Analysis. Now that I think about it there seems to be a casuality the use of the...

I have been studying Fourier Optics and I have a basic conceptual question. I understand the mathematics of how to perform Fourier Transforms however the part of this topic I seem to have missed is why the action of a lens on light is the same as performing a Fourier Transform on the functional...

Homework Statement The function f(x) is defined by: f(x) = -1 when \pi < x <0 and 0 when 0<x<\pi Show that \sum^{∞}_{0}\frac{(-1)^n}{2n+1}=\frac{\pi}{4} Homework Equations Fourier series for a function of period 2\pi = a_{0} + \sum^{∞}_{1}a_{n}cos(nx) + b_{n}sin(nx)...

Hi all, I'm a complete novice when it comes to describing images in frequency space and i understand that it is a way of representing images as being composed of a series of sinusoids. So a horizontal striped pattern with a single spatial frequency would have a magnitude image in frequency...

In a rigorous mathematical course I am talking, it seems to make a difference when I am given a function f and need to write its Fourier series, whether it is defined on [0,2∏] or [0,2∏). What difference does it make for my series whether it is an open or a closed interval?

Homework Statement Calculate the following integral: \int_{0}^{2\pi}(\sum_{k=0}^{\infty} \frac{\cos(kx)}{3^k})^2 dx Homework Equations Parseval's identity: \frac{1}{2 \pi} \int_{-\pi}^{\pi} {|f(x)|^2 dx} = \sum_{n=0}^{\infty} {|a_n|^2+|b_n|^2} Where a_n, and b_n are the trigonometric...

... ... I am curious to know why we have to multiply with e^{-j\omega t} in Fourier transform? What is the purpose of this? I have heard somewhere that the transform is merely a change of variables from one set of coordinates to another. I would like to know more about this. Can you help me?

Homework Statement f(x) = 1, 0<x<1 Extend f(x) t generate an even function P(x) and find Fourier coefficients Homework Equations an = 2/T ∫ P(x)cos(2nx/T) dx The Attempt at a Solution P(x) = 1, -1<x<1 0, -2<x<-1 , 1<x<2 even function so b0 = 0 Average of P(x)...

Hey guys! if anyone can help me I guess it is you! :) I'm trying to find the Fourier Series demonstration to continuous and periodic functions. I don't understand why people keep using X(jw) and X[e^jw] and even sometimes X(w) and X(f) If anyone can help me I'm really not understanding that...

Homework Statement Find the inverse Fourier transform of F(jω) = cos(4ω + pi/3)Homework Equations δ(t) <--> 1 δ(t - to) <--> exp(-j*ωo*t) cos(x) = 1/2 (exp(jx) + exp(-jx))The Attempt at a Solution So first I turned the given equation into its complex form using Euler's Formula. F(jω) = 1/2...

Hello Physics Forums community, I'm afraid I really need a hand in understanding Why are the Fourier Series for continuous and periodic signals using diferent notation of the Fourier Series for discrete and periodic Signals. I have been following the book " Signals and Systems " by Alan V...

Fourier evaluation of sum HELP Homework Statement Consider the signal: f(t) = |sint|, -pi/2 < t < pi/2 where f(t) = f(t+pi) Homework Equations Fourier. The Attempt at a Solution I determined the General Fourier Series representation for f(t) below: 2/pi +4/pi +...

First, I've had to find the Fourier series of F(t) = |sin(t)|, which I've calculated as f(t) = \frac{2}{\pi} + \sum_{n=1}^{\infty}\frac{4cos(2nt)}{\pi-4\pi n^2} I'm pretty sure that's right, but now I need to evaluate the sum using the above Fourier series...

can you use Fourier transform to find a moving average on a data set? so, you do a Fourier transform on your one dimensional data set. next remove high order harmonics from FT result. do reverse Fourier transform on new FT result. And, vola! smoothed out data set.

Homework Statement Find the Fourier coefficients for the function *Should be a piecewise function, not sure how to write one in [itex /itex] tags* f(x) = |x|, |x| < 1, 1, 1≤|x|< 2; f(x+4) = f(x) and Find the Fourier series for f(x) = cos1/2\pi x, -1≤x<1...

In my physics book there is an example of making a square wave by "simply" summing up a few cosine waves. The book says these first three waves are the first three Fourier components of a square wave, yet when I sum the three wave functions up, I get something way off; as does my calculator...

Hi, I was wondering what would the Fourier transform of a signal like below give: s(t) = sin(2πt*10) ; t in [0s,5s] = sin(2πt*20) ; t in [5s,10s] I certainly did not expect it to give me 2 sharp peaks at frequencies 10Hz and 20Hz - because I understand that the addition of...

I have some rather technical questions about the complex exponential form of the Fourier series: 1) What is the motivation behind the complex exponential form? Why not just use the real form (i.e. with sine and cosines)? 2) Surely the complex exponential form is an orthogonal set, i.e...

Hi At university, in a previous module the constant in front of the Fourier Transform was given as (1/2π), and the constant in front of the inverse F.T. was 1. However in a current module the lectrurer gives the constant as (1/√2∏), for both the F.T. and its inverse. Why is this? Thanks

Homework Statement I must calculate the Fourier Series of f(x) = 0, when -∏< x < 0 and f(x) = sinx, 0 < x < ∏ Homework Equations The Attempt at a Solution Using the formulae, I calculated a0 = 2/pi, an = [ (-1)^n + 1 ] / [ ∏(1 - n^2) ], and bn = 0, so my Fourier series goes...

Hello, Homework Statement A rope of mass M and length L is tend with tension T between two rings free to oscillate along a rod parallel to the y axis. Initially the rings are maintained at y=0 while we give to the rope a y(x,0)=dsin²(pix/L). Give the complete expression of motion of the rope in...

When creating a Fourier series for a function f(x), I consider whether the function is odd or even first. Yet, often these functions are in the positive region [0, L] . Since f(x) is only defined in this region, can I change the function to get a desired parity? By example, my concern...

Can anyone tell me if there exist analytical solution to the Fourier heat equation rhoCdt/dt= ∇.(k∇T) + S Thanks

Homework Statement I'm trying to find a Fourier series for the piecewise function where f(x)= 0 \in -\pi \leq x \leq 0 -1 \in 0 \leq x \leq \frac{\pi}{2} 1 \in \frac{\pi}{2} \leq x \leq \pi Homework Equations a_{n} = \frac{1}{\pi} \int_{0}^{2\pi}\cos(nx)y(x)\,dx b_{n} = \frac{1}{\pi}...

Hallo, I really don't understand Fourier transform. Do somebody know a good book for beginners? Something like Fourier transform for dummies or so? I need it just for physics. So it don't have to be to mathematical. ^^ THX

Fourier Transform on the "connected part" of QFT transition prob. Homework Statement Calculate ⟨0|T[ϕ(x₁)ϕ(x₂)ϕ(x₃)ϕ(x₄)]|0⟩ up to order λ from the generating functional Z[J] of λϕ⁴-theory. Using the connected part, derive the T-matrixelement for the reaction a(p₁) + a(p₂) → a(p₃) +...

Homework Statement Find the Fourier series representation of: f(t)={-t , -∏<t<0 f(t)={0 , 0<t<∏ This is a piecewise function. T=2∏ (the period) Homework Equations a_{0}=\frac{2}{T}*\int_0^T f(t),dt The Attempt at a Solution I need help only with calculating the DC...

Homework Statement In order to determine the characteristic function of a random variable defined by: Z = max(X,0) where X is any continuous rv, i need to prove that: F_{l,v}(g(l))=[ \phi_{X}(u+v)\phi_{X}(v) ] / (iv) where F_{l,v}(g(l)) is the Fourier transform of g(l) and...

The sine-cosine (SC) Fourier series: $$f(x) = \frac{A_0}{2} + \sum_{j=1}^{+\infty} A_j cos(jx) + \sum_{j=1}^{+\infty} B_jsin(jx) $$ This form can also be expanded into a complex exponential (CE) Fourier series of the form: $$ f(x) = \sum_{n=-\infty}^{+\infty} C_n e^{inx} $$ and vice versa...

Hello, Consider I have a linear time-invariant (LTI) system, with ##x(t)##, ##y(t)##, and ##h(t)##, as input, output, and impulse response functions, respectively. I have two choices to write the convolution integral to get ##y(t)##: $$ 1)\ \ \ y(t) = \int_{0}^{t} h(t-t')x(t')dt' $$ and...

Hi I'm doing Fourier analysis in my signals and system course and I'm looking at the solution to one basic problem but I'm having trouble understanding one step Can anyone explain to me why becomes From Eulers formula: http://i.imgur.com/1LtTiKX.png for example the Cosine in my problem. I...

IR and NIR spectroscopy usually employ Fourier transformation to separate the signal into individual wavelength, UV and Vis spectroscopy normally apply gratings for light dispersion (into individual wavelength). What is the cutoff wavelength, and why is so?

Homework Statement Find the Fourier transform of (1/p)e^{[(-pi*x^2)/p^2]} for any p > 0 Homework Equations The Fourier transform of e^{-pi*x^2} is e^{-pi*u^2}. The scaling property is given to be f(px) ----> (1/p)f(u/p) The Attempt at a Solution Using the information above, I got...

Find the DTFT of: h[n]=(-1)^{n}\frac{sin(\frac{\pi}{2}n}{sin(\pi n} useful properties: x[n]y[n] --> X[Ω]*Y[Ω] \frac{sin(\frac{\pi}{2}n}{sin(\pi n} --> rect[\frac{2Ω}{\pi} I have no clue how to deal with the (-1)[itex]^{n}[\itex] the DTFT of that doesn't converge. . . any help...

Hello all, I'm working through a majority of the problems in "A First Course in Wavelets with Fourier Analysis" and have stumbled upon a problem I'm having difficulty with. Please view the PDF attachment, it shows the problem and what I have done with it so far. Once you have seen the...

define f(t)=|t|, t between - pi and pi. I have found the Fourier co-efficents of f and am now charged with showing that the infinite series of 1/(2m+1)^2 is equal to (pi^2)/8. Can I use the Fourier co-efficents?

Hi, (To cut a long story short, can I cancel the integrals in Eq. 6 to leave me with Eq. 7?) I am trying to follow the method for modelling the motion of a tethered bead from a couple of papers ("Te Velthuis, A. J. W. et al. (2010) Biophys. J. 99 1292–1302" and "Lansdorp, B. M., & Saleh, O...

At page 285 in Peskin and Schroeder's Introduction to quantum field theory the author defines the integration measure D\phi = \Pi_i d\phi(x_i) where space-time is being discretised into a square lattice of volume L^4. He proceeds by Fourier-transforming \phi(k_n) = \frac{1}{V} \sum_n e^{-i...

Homework Statement What is the percentage of power (out of the total power) contained up to the third harmonic (power in DC component, a1 , a-1 , a2 , a-2 , a3 , a-3 ) of the square waveform shown above? (the duty cycle = D = τ/T0= 0.5) Homework EquationsThe Attempt at a Solution Hey all...

Homework Statement Length of rod = 1 Initial Conditions: u(x,0)=sin(πx) Boundary conditions: u(0,t)=0 and u(1,t)=5. Alright I am supposed to find the temperature at all times, but I am curious about the setup of the problem itself. When x = 1, the boundary condition says...

Hi, I am learning Fourier transformation by my own. I am reading a book "Fourier Transformation" by R. Bracewell. In chapter 11, in examples of discrete Fourier transforms, it gives for N =2, {1 0} transforms to 1/2{1 1}. I can do this in MATLAB but I can't figure it out how to do it by hand...

is the interference pattern produced by a double slit a one dimensional phase/amplitude Fourier transform? and if you did a reverse Fourier transform on it would you get an image of the two slits?

I've been taught (in the context of Sturm-Liouville problems) that Fourier series can be explained using inner products and the idea of projection onto eigenfunctions in a Hilbert space. In those cases, the eigenvalues are infinite, but discrete. I'm now taking a quantum mechanics course, and...

Can someone please explain WHY the statement below is valid: s = σ + jω ; left hand side σ < 0 So it basically says if all the poles have negative real parts then we can directly substitute s = jω to get the Fourier transform. This doesn't make sense to me, does it make sense to you...

Homework Statement Problem 1.60. A frying pan is quickly heated on the stovetop to 200 C. It has an iron handle that is 20 cm long. Estimate how much time should pass before the end of the handle is too hot to grab with your bare hand. (Hint: The cross-sectional area of the handle doesn't...

Homework Statement Hello I'm learning Fourier transforms via the Stanford lecture series on Youtube. In the 6th lecture, the professor claims that the FT of a triangle function is the square of the sinc function. I'm trying to derive this, but I can't get my math to work out. Could someone...

This is A and B my friend is telling me that Co is actually 0 and I am getting 1/2 and i don't see exactly what I am doing wrong if i indeed am doing something wrong hopefully someone here can check this out and let me know exactly where i went wrong.. Thanks

If $$f(x)=x+1$$, expand $$f(x)$$ in Fourier series and hence show that $$\sum_{n=0}^\infty \frac{1}{(2n-1)^2}=\frac{\pi^2}{8}$$This question was set in an exam. I am in a position to try it if there is some interval say $$[-\pi \quad \pi]$$ or like that. But there is no interval in the...

1. http://imgur.com/UoUb27B 2. none? 3. not really sure what this question is asking. I thought that n=1 because its the fundamental frequency and the DC value should just be 120 V. I looked at some other questions and the answers were not found using that method.

Homework Statement See Attachment Homework Equations The Attempt at a Solution Ok so in a previous question I worked out fd = e-ipd*2*sinc(pa)/√(2∏), also worked out its Fourier transform if that helps. Now I really am stuck on the question, any guidance would be appreciated...