Fourier series Definition and 706 Threads

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...

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...

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}...

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...

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...

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 Compute the power contained in the periodic signal x(t) = 10.0[cos(160.7*pi*t)]^4 Homework Equations The Attempt at a Solution Hey guys, I have just started Fourier Series and am struggling with this one. Without writing all my calculations, -I start with inverse Euler...

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...

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.

[SOLVED] Fourier Series Involving Hyperbolic Functions Hello everyone! Sorry if this isn't the appropriate board, but I couldn't think of which board would be more appropriate. I was running through some problems I have to do as practice for a test and I got stuck on one I'm 99% sure they'll...

Homework Statement See the second bullet point on this page: http://facweb.northseattle.edu/rjenne/e240w13flr/hwflr/feb21/e240w13hwfeb21.pdf Homework Equations So I know that fft(x) for a bunch of sample points x={x1, x2, ..., xn} returns the n Fourier coefficients for a function...

Homework Statement The signal g(t) is band limited to B Hz and is sampled by a periodic pulse train ##PT_{s}(t)## made up of a rectangular pulse of width ##1/8B## second (centered at the origin) repeating at the nyquist rate (2B pulses per second). Show that the sampled signal ##\bar{g}(t)##...

$$ -\sum_{n = 0}^{\infty}n^2\omega^2C_ne^{in\omega t} + 2\beta\sum_{n = 0}^{\infty}in\omega C_ne^{in\omega t} + \omega_0^2\sum_{n = 0}^{\infty}C_ne^{in\omega t} = \sum_{n = 0}^{\infty}f_ne^{in\omega t} $$ How can I justify removing the summations and solving for $C_n$? $$...

Hi all, How do I compute the Fourier series coefficients for unit periods for cos(pi)x, the interval is from -1/2 to 1/2. I know the formula but I am getting a wrong answer ?

Homework Statement what values does the Fourier series for f(t) converge to if t = 0 and t = 2? Homework Equations The Attempt at a Solution My answers the red rectangles for the even function t=0 >> 1 and t=2 -->1.5 and odd function t=0 >> 0 and t=2 -->1.5 because at t=0 is continuity...

So I know that sec(x) has period 2Pi, and it's even so I don't need to figure out coefficients for bn. Let's take the limits of the integral to go from -3/2 Pi to 1/2 Pi. How do I integrate sec(x) sin(nx) dx?! Am I on the right path? PS: I know that this doesn't satisfy the Dirichlet...

Homework Statement [SIZE="3"]these two functions will give the same Fourier series? because when I write the graph they look the same? Homework Equations The Attempt at a Solution in the picture thank you

Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated! Homework Statement Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate the coefficients of the orthogonal...

Somebody posted a question about Fourier series yesterday that got me thinking about an argument I heard some time before. If we have a (complex-valued) analytic function f, then any closed loop in the complex plane will be mapped by f to another closed loop. (If the loop doesn't enclose any...

Hi guys, I was studying the proof below and just can't figure out the the first highlighted step leads to the second and I was wondering if you guys can help me to fill that in. (: Thank you so much for your help in advance guys!

Homework Statement Third question of the day because this assignment is driving me crazy: Suppose that \left\{ f_{k} \right\} ^{k=1}_{\infty} is a sequence of Riemann integrable functions on the interval [0, 1] such that \int ^{0}_{1} |f_{k}(x) - f(x)|dx \rightarrow 0 as k \rightarrow...

Hi, The Fourier series can (among others) expressed in terms of sines and cosines with coefficients a_n and b_n and solely by sines using amplitudes A_n and phase \phi_n. I want to express the latter using a_n and b_n. Using a_n = A_n \sin(\phi_n) \\ b_n = A_n \cos(\phi_n) I...

The Fourier series of a delta train is supposedly (1/T) + (2/T ) Ʃcos(nωt) ... where T is period and ω=2*Pi/T ...but when I plot this, it doesn't give me just a spike towards positive infinity, but towards negative infinity as well (see attached pic), so this does not seem to converge to the...

Is there a way to "crudely" approximate PDEs with Fourier series? By saying crudely, I meant this way: Assuming I want a crude value for a differential equation using Taylor series; y' = x + y, y(0) = 1 i'd take a = 0 (since initially x = 0), y(a) = 1, y'(x) = x + y; y'(a)...

The problem statement: Obtain a Fourier Series Expression Form from the above graph: I can't post the graph, so I will describe it. It's a periodic function with period 1 and magnitude 5. The equation is the following: f(x) = -x, -1/2<x<1/2 I'm really stuck at trying to obtain a series...

Hey, thanks for taking the time to look ay my post (: I have attached a file which shows the question I am stuck on, and my attempt at working it out. My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...

semi urgent Fourier series question (small) Homework Statement Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet...

Homework Statement Please see picture attached Homework Equations The Attempt at a Solution ck = 1/T ∫ a-a x(t).e-jk2pit/T So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi but is is supposed to be: So x(t) = Ʃk=-∞∞ (sin(k.a.2pi/T).a-a e-jk2pit/T)/k.pi.2.a but I...

Hi, I have x(t) = 1/2 + cos(t) + cos(2t) so I can see that a0 = 1/2 and that it is an even function so there is no bn Also that T = 2pi so an = 2/2pi ∫02pi x(t).cos(nω0t) dt but when I integrate this I get an = 0 yet I've been told that the answer is x(t) = 1/2 + Ʃn = 12 cos(nω0t) which...

Why is Fourier sine series of any function satisfying Dirichlet's theorem, not defined on the discontinuous points whereas we define it for Fourier cosine series? ex - sine series of f(x) = cosx, 0<=x<=∏ is defined on 0<x<∏ whereas cosine series of f(x) = sinx, 0<=x<=∏ is defined on 0<=x<=∏

$$ \sum_{n = 1}^{\infty}\sum_{m = 1}^{\infty}A_{nm}\sin\frac{n\pi x}{L}\sin\frac{m\pi y}{H} = -\frac{4}{\pi}\sum_{k = 1}^{\infty}\frac{1}{(2k-1)\sinh\frac{\pi(2k-1)H}{L}}\sin\frac{\pi(2k-1)x}{L}\sinh\frac{\pi(2k-1)y}{L} $$ If I start with x on the left, can I then end up with: $$...

If you have a function with countable discontinuities on an interval, I know that the Fourier series will converge to that function without those discontinuities. But how could you explain that formally? If the basis of the Fourier series span the space L^2[a,b], that would include functions...

can we simply truncate a Fourier series if it is divergent?? given a Fourier series of the form \sum_{n=0}^{\infty}\frac{cos(nx)}{\sqrt{n}} can i simply truncate this series up to some number finite N so i can get finite results ?? thanks.

Homework Statement Let f(x)=x, 0≤x≤p (a.) Compute the half-range sine series (b.) Use the series to show that 1-(1/3)+(1/5)+...=π/4 Homework Equations bn=(2/L)*int(from 0 to L) f(x)*sin(nπx/L) dx The Attempt at a Solution bn=(2/p)*int(from 0 to p) x*sin(nπx/p) dx Using...

Homework Statement Two similar problems, but once I find out how to do the first one, I can figure out how to do the second. My signals book tells me the answers to the following "Dn"s are: First problem: Dn = (1/∏) ∫ sin(t) * e^(-j2nt) dt = 2/(∏ (1-4n^2) ) if x(t) = rectified sin(t)...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to...

I am SO annoyed with this problem. Ready to jump out a window. Homework Statement Find the first three terms of the Fourier series that approximates f(θ) = tan(θ) from θ = -π/2 to π/2. The Attempt at a Solution So, I know that for an equation on [\frac{-b}{2}, \frac{b}{2}], to define the...

Homework Statement Given the function 10sin^2(10t) Find the fundamental frequency and period. Find the exponential and trigonometric coefficients of the Fourier Series. Homework Equations The Attempt at a Solution I really have no idea how to start this problem. The sin^2...

Hello, First post. I will attempt to use latex, something that involves me jabbing my keyboard with a pen since my \ key is missing. We have an assignment question which I have solved, but there is a deeper issue I don't understand. We are asked to find the complex Fourier series...

dealing with absolute functions that are limited always throws me off so let's consider this f(x)=|x| for -∏ ≤ x < ∏ f(t)= f(t+2∏) it's not too bad however finding the energy density is throwing me off a little.. the questions tend to be generally phrased as below: Find the energy...

Homework Statement Suppose, in turn, that the periodic function is symmetric or antisymmetric about the point x=a. Show that the Fourier series contains, respectively, only cos(k_{n}(x-a)) (including the a_0) or sin(k_{n}(x-a)) terms. 2. Homework Equations The Fourier expansion for the...

Suppose $f$ is continuous and periodic with period $2\pi$ on $(-\infty,\infty)$, and $f'$ exist and is in $\mathcal{P}\mathcal{C}[-\pi,\pi]$. Then $\sum\limits_{k = -\infty}^{\infty}\lvert A_k\rvert < \infty$.$f'$ has a Fourier series so let's call the coefficients $A_n'$. Then $f' =...

What is the statement of the mean square convergence of Fourier series?

Homework Statement Fx= sin2x for -pi<x<0 and 0 for 0<x<pi. Compute the coefficients of the Fourier series. Homework Equations The Attempt at a Solution I found Ao=0 even though I came up with a Fourier cosine series. There must be something wrong. And is that possible that I...

Homework Statement h(x)=\left\{\begin{matrix} 9+2x , 0<x<\pi\\ -9+2x , -pi<x<0 \end{matrix}\right. \\ Find \ the \ sum \ of \ the \ Fourier \ series \ for \ x=\frac{3\pi}{2} and\ x=\pi \\ The \ Fourier \ series \ is: \\ h(x)=9+\pi + \sum_{n=1}^{inf}...

Homework Statement How can i find the magnitude spectra of 2sin(4000*pi*t)*sin(46000*pi*t) The Attempt at a Solution im not sure how to go about this question, can someone please give me some help on what i should do. I know that for a square wave i can find the Fourier series...

Let $$ h(\theta) = \begin{cases} \frac{1}{2}(\theta + \pi), & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ \frac{1}{2}(\theta - \pi), & -\pi < \theta < 0 \end{cases} $$ How can I find the Fourier series without doing any integration?

What are rules for differentiating a Fourier series? For example, given $$ f = \frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\sin(2n-1)\theta}{2n-1} = \begin{cases} 1, & 0 < \theta < \pi\\ 0, & \theta = 0, \pm\pi\\ -1, & -\pi < \theta < 0 \end{cases} $$ Can this be differentiating term wise? If so...