What is Fourier coefficients: Definition and 71 Discussions
In mathematics, a Fourier series () is a periodic function composed of harmonically related sinusoids, combined by a weighted summation. With appropriate weights, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.
I could summarize this as
##\frac{1}{n\pi}## for ##n=1, 5, 9, \ldots##
##\frac{-1}{n\pi}## for ##n=3, 7, 11, \ldots##
and ##0## for all other ##n##.
How would I go about writing this in a single expression, with ##(-1)^{f(n)}## where ##f(n)## summarizes both cases above?
Here, ##\Phi(f_{x_n},f_{y_m})=|\mathscr{F(\phi(x,y))}|^2 ## is the Power Spectral Density of ##\phi(x,y)## and ##\mathscr{F}## is the Fourier transform operator.
Parseval's Theorem relates the phase ##\phi(x,y)## to the power spectral density ##\Phi(f_{x_n},f_{y_m})## by...
I have plotted the function for ##T=15## and ##\tau=T/30## below with the following code in Python:
import numpy as np
import matplotlib.pyplot as plt
def p(t,T,tau):
n=np.floor(t/T)
t=t-n*T
if t<(2*np.pi*tau):
p=np.sin(t/tau)
else:
p=0
return p...
I would try to determine whether ##p(t)## is even or odd. This would be so much easier if the values of ##\tau## and ##T## would be specified, but maybe it's possible to do without it, which I'd prefer. If for example ##\tau=1/2## and ##T=2\pi##, then ##p(t)=\sin{(2t)}## for ##0\leq t <\pi ##...
Hello everyone.
I have 4 samples of 50 elements from 4 unknown random variables obtained from a Karhunen-Loève decomposition using Matlab's pca (each one is a column of size 50 from the coefficient matrix). I am following the article SAMBA: Sparse Approximation of Moment-Based Arbitrary...
Consider the function ##f:[0,1]\rightarrow \mathbb{R}## given by
$$f(x)=x^2$$
(1) The Fourier coefficients of ##f## are given by
$$\hat{f}(0)=\int^1_0x^2dx=\Big[\frac{x^3}{3}\Big]^1_0=\frac{1}{3}$$
$$\hat{f}(k)=\int^1_0x^2e^{-2\pi i k x}dx$$
Can this second integral be evaluated?
Hi, I am interested in understanding the relationships between Fourier series and Fourier transform better. My goal is
1) Start with a set of ordered numbers representing Fourier coefficients. I chose to create 70 coefficients and set the first 30 to the value 1 and the remaining to zero.
2)...
Hi!
Situation: quasi-free electron in a 2D lattice, considering atomic potential V(r) = exp{-|r|/b} (r is the distance from the atom)
I'm trying to compute the first five energy gaps at point (10),
firstly I don't understand the meaning of calculated 5 energy gaps at one point and usually we...
Hey! :o
A real periodic signal with period $T_0=2$ has the Fourier coefficients $$X_k=\left [2/3, \ 1/3e^{j\pi/4}, \ 1/3e^{-i\pi/3}, \ 1/4e^{j\pi/12}, \ e^{-j\pi/8}\right ]$$ for $k=0,1,2,3,4$.
I want to calculate $\int_0^{T_0}x^2(t)\, dt$.
I have done the following:
It holds that...
ANY AND ALL HELP IS GREATLY APPRECIATED :smile:
I have found old posts for this question however after reading through them several times I am having a hard time knowing where to start.
I am happy with the sketch that the function is correctly drawn and is neither odd nor even. It's title is...
%My code:
%Type of signal: square
T = 40; %Period of the signal [s]
F=1/T; % fr
D = 23; % length of signal(duration)
dt=(D/T)*100;
N = 50; %Number of coefficientsw0 = 2*pi/T; %signal pulset1= 0:0.002:T; % original signal sampling
x1 = square((2*pi*F)*(t1),dt);%initial square signal
t2=...
Hey there!
I am current taking an introductory course on PDE's, and our professor hasn't really emphasized last part of solutions from separation of variables. Now its not strictly going to be on the exam, however I remember doing this with ease a few years back, but for some reason now I...
Given a function F(t)
$$ F(t) = \int_{-\infty}^{\infty} C(\omega)cos(\omega t) d \omega + \int_{-\infty}^{\infty} S(\omega)sin(\omega t) d \omega $$
I am looking for a proof of the following:
$$ \int_{-\infty}^{\infty} F^{2}(t) dt= 2\pi\int_{-\infty}^{\infty} (C^{2}(\omega) + S^{2}(\omega)) d...
Hi PF!
Unsure how to begin. Fourier transform of ##f## I've given as an equation. I'm not sure what is meant by Fourier coefficients. Fourier coefficients of what?
I am attempting to find the sine representation of cos 2x by letting
$$f(x) = \cos2x, x>0$$ and $$-\cos2x, x<0$$
Which is an odd function. Hence using $$b_n = \dfrac{2}{l} \int^\pi _0 f(x) \sin(\dfrac{n\pi x}{l})dx$$ I obtain $$b_n = \dfrac{2n}{\pi} \left( \dfrac{(-1)^n - 1}{4-n^2} \right)$$...
Hello,
I need help with question #2 c) from the following link (already LateX-formatted so I save some time...):
https://wiki.math.ntnu.no/_media/tma4135/2017h/tma4135_exo1_us29ngb.pdf
I do understand that the a0 for both expressions must be the same, but what about an and bn? I don't...
Homework Statement
Given a continuous non-periodic function, its Fourier transform is defined as:
$$f(x) = \int_{-\infty}^\infty c(k) e^{ikx} dk, \ \ \ \ \ \ \ \ \ \ \ \ \ c(k) = \frac{1}{2\pi} \int_{-\infty}^\infty f(x) e^{-ikx} dx$$
The problem is proving this is true by evaluating the...
Homework Statement
Find the Fourier spectrum ##C_k## of the following function and draw it's graph:
Homework Equations
3. The Attempt at a Solution [/B]
I know that the complex Fourier coefficient of a rectangular impulse ##U## on an interval ##[-\frac{\tau}{2}, \frac{\tau}{2}]## is ##C_k =...
Homework Statement
Consider the action of ##T_2## acting on ##M_k(\Gamma_{0}(N)) ##, and show that ##\theta^4(n)+16F ## and ##F(t)## are both eigenfunctions.
Functions are given by:
Homework Equations
For the Hecke Operators ##T_p## acting on ##M_k(\Gamma_{0}(N)) ##, the Hecke conguence...
If we have the normal KG scalar field expansion:
$$ \hat{\phi}(x^{\mu}) = \int \frac{d^{3}p}{(2\pi)^{3}\omega(\mathbf{p})} \big( \hat{a}(p)e^{-ip_{\mu}x^{\mu}}+\hat{a}^{\dagger}(p)e^{ip_{\mu}x^{\mu}} \big) $$
With ## \omega(\mathbf{p}) = \sqrt{|\mathbf{p}^{2}|+m^{2}}##
Then why do we associate...
Homework Statement
I am self studying an introductory quantum physics text by Marvin Chester Primer of Quantum Mechanics. I am stumped at a problem (1.10) on page 11. We are given
f(x) = \sqrt{ \frac{8}{3L} } cos^2 \left ( \frac {\pi}{L} x \right )
and asked to find its Fourier...
Homework Statement
I have ## f(t) = \sum\limits^{\infty}_0 a_{n} e^{2 \pi i n t} ## [1]
and ## g(t) = \sum\limits^{\infty}_0 b_{n} e^{ \pi i n t} ## [2]
I want to show that ##b_n = a _{2n} ##
Homework Equations
see above.
The Attempt at a Solution
[/B]
So obviously you want to use the...
Suppose that we have a 2\pi-periodic, integrable function f: \mathbb{R} \rightarrow \mathbb{R}, whose continuous Fourier coefficients \hat{f} are known. The convolution theorem tells us that:
$$\displaystyle \widehat{{f^2}} = \widehat{f \cdot f} = \hat{f} \ast \hat{f},$$
where \ast denotes the...
Hello
1. Homework Statement
Find the Fourrier coefficients in the annulus problem of the text.
uxx+uyy=0 in 0<a²<x²+y²<b²
u=g(θ) for x²+y²=a²
u=h(θ) for x²+y²=b²
Homework Equations
The solution is
The Attempt at a Solution
I have the solutions but when I solved it for...
Homework Statement
In my PDE course we have a homework question stating the following:
Let ϑ(x) = x in the interval [-pi, pi ]. Find its Fourier Coefficients.
Homework Equations
From my notes on this type of question:
a_o = 2c_o = 1/pi * integral from -pi to pi [f(x) dx]
a_n = c_n + c_(-n)...
Homework Statement
When ##f## and ##g## are ##2\pi##-periodic Riemann integrable functions define their convolution by
##(f*g)(x) = \frac{1}{2\pi} \int_0^{2\pi} f(y)g(x-y)dy##
Denoting Fourier coefficients by ##c_n(f)## show that ##c_n(f * g) = c_n(f)c_n(g)##.
Homework Equations
##c_n =...
Homework Statement
http://puu.sh/gGhdb.jpg
Solution:[/B]
http://puu.sh/gGh3E.jpg
Homework EquationsThe Attempt at a Solution
How did they get that solution for the Fourier coefficient? When I evaluate the integral I can only seem to get it to:
(1/-jk2π)[2*exp(-jkπt)-exp(-jk2πt)-1]
If a function f'(u) has Fourier coefficients anμ and bnμ, by integration one can make new coefficients Anμ ,Bnμ which include constants of integration.
My question how can I verify that :
Anμcos nτ + Bnμ sin nτ= -i/2 ((Bnμ) -Anμi) einτ- (Bnμ-iAnμ) e-inτ
I assume this is the complex form of...
Hi, friends! Let ##f:[a,b]\to\mathbb{C}## be an http://librarum.org/book/10022/173 periodic function and let its derivative be Lebesgue square-integrable ##f'\in L^2[a,b]##. I have read a proof (p. 413 here) by Kolmogorov and Fomin of the fact that its Fourier series uniformly converges to a...
hey pf!
okay, so if you've studied PDEs you know the value of a Fourier series, and the difficulty of determining a Fourier coefficient. my question relates to finding this coefficient. briefly, i'll define a Fourier series as f(x)=\sum_{n=0}^{\infty} A_n\cos\frac{n\pi x}{L}+B_n\sin\frac{n\pi...
Homework Statement
Let ##f## be a ##2\pi## periodic function. Let ##\hat{f}(n)## be the Fourier coefficient of ##f## defined by
$$
\hat{f}(n)=\frac{1}{2\pi}\int_{a}^{b}f(x)e^{-inx}dx.
$$
for ##n\in\mathbb{N}##. If ##\overline{\hat{f}(n)}=\hat{f}(-n)## show that ##f## is real valued.
The...
Homework Statement
f(t) a continuously differentiable function twice over the circle T1
cr its Fourier coefficients and σn(f,t) partial sum of Fejer.
a.Demonstrate that
http://imageshack.us/a/img94/5992/ds35.png b. Consider k as -n≤k≤n , using cr coefficients calculate...
I'm having trouble finding a definite answer to this question: When finding the Fourier series of a function is it always possible to find ##a_0## by first finding ##a_n## and just plugging in ##n=0##?
Homework Statement
Let \hat{u}_k the Fourier coefficients of 2-periodic function u(t)=t with t\in [0,2). Evaluate the sum of the serie:
\sum_{k=-\infty}^{\infty}\hat{u}_k e^{\pi i k t} for t= 2
Ok, I think there is a trick that I don't know...
\sum_{k=-\infty}^{\infty}\hat{u}_k...
Homework Statement
Find the Fourier coefficients for the function
*Should be a piecewise function, not sure how to write one in [itex /itex] tags*
f(x) =
|x|, |x| < 1,
1, 1≤|x|< 2;
f(x+4) = f(x)
and
Find the Fourier series for
f(x) = cos1/2\pi x, -1≤x<1...
Homework Statement
I've reached a relation but then I need to obtain the coefficients ##A_m## and ##B_m##'s, those are the only unknowns.
Here's the expression: ##\sum _{m=0}^\infty a^m [A_m \cos (m \theta ) + B_m \sin (m \theta )]=T_0\sin ^3 \theta##.
Homework Equations
Fourier...
For the wave equation I managed to get
the coefficient of f:
a1=2
and
the coefficient of g:
\frac{12pi}{2pi*2}=B2
Is these answers right, since my B2 does not match the answer I was given.
Thank you
Supposing $f$ is bounded and $A_n$ is given by 1-8, prove that $\sup_n|A_n|$ is finite.
$$
f(\theta) = \sum_{n = -\infty}^{\infty}A_ne^{in\theta}
$$
Since $f$ is bounded, $|f| < M = |z|\in\mathbb{C}$. Since it could be $\mathbb{C}$, $M$ would be the modulus correct?
We know that the modulus of...
Homework Statement
If C^1(\mathbb T) denotes the space of continuously differentiable functions on the circle and f \in C^1(\mathbb T) show that
\sum_{n\in\mathbb Z} n^2 |\hat f(n)|^2 < \infty
where \hat f(n) is the Fourier coefficient of f.
The Attempt at a Solution
Since f is...
Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$
I know this has to do with the...
Homework Statement
For a given periodic function F(x), the coefficients An of its Fourier expansion can be found using the formulas (Form1) and (Form2). Consider a periodic square pulse and verify that the Fourier coefficients are as claimed:
An =(\frac{2}{πn})sin(\frac{πan}{λ})
for n≥1 and...
Homework Statement
Fidn the Fourier expansion for f of period 2Pi that corresponds to y=x/3 on the interval [0,2Pi)
Im just a little confused about if I am setting up the integration properly. The asymmetric interval is kind of confusing me here.
The Attempt at a Solution
a0 = 1/Pi ∫ x/3 dx =...
If f(x) has a period of 2*pi and |f(x)-f(y)| <= c*|x-y|^a where a and c are positive constants, why are are n-th Fourier coefficients <= c*(pi/n)^a ?
Help or hints would be appreciated.
Homework Statement
For an even function, the Fourier series takes the form
^{\infty}_{n=0}\Sigma A_n cos(\frac{2\pi n x}{\lambda})
where \lambda is the wavelength of the function. In this problem you will see how to find the Fourier coefficients A_n.
a.) Prove that
A_0 =...
A single-phase ac voltage regulator has a resistive load of R= 10 ohms, and the input voltage is Vs = 240 V, 50 Hz. The delay angle of each of the Thyristors is ɑ = Π/2. Determine:
(a) The rms value of the output voltage.
(b) The Fourier coefficients of the fundamental, 11th and 13th current...
I'm struggling in an application of Fourier transform.here is my problem:
a series of points from experimental data plotted as a cruve. I'm planning to do a Fourier transform to see how smooth the curve is? my question is: is it possible/useful to calculate the Fourier coefficients? if yes, how...