# 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.

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1. ### How to write sin(n*pi/2) as an expression (-1)^f(n)?

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?
2. ### Relationship between Fourier coefficients and power spectral density

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...
3. ### Comp Sci Plot periodic function with Fourier coefficients

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...
4. ### Characterize Fourier coefficients

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 ##...
5. ### A Calculation of Fourier coefficients using SAMBA methodology

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...
6. ### Finding the Fourier Coefficients of a Function

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?
7. ### MATLAB Turning Fourier coefficients into an interpolated freq domain function

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)...
8. ### Energy gaps for quasi-free electrons in a 2D lattice

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...
9. ### MHB Calculate the integral using the Fourier coefficients

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...
10. ### Comp Sci Fourier analysis & determination of Fourier Series

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...
11. ### Engineering Reconstruct a signal by determining the N Fourier Coefficients

%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=...
12. ### I Separation of variables - Getting the Fourier coefficients

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...

41. ### Fourier Coefficients of Continuous functions are square summable.

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...
42. ### MHB Proving Uniqueness of Fourier Coefficients for Continuous Periodic Functions

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...
43. ### Compute Sin Coeffs for f(x)=e^(-x^2) on [0,2π]

Compute the sine coefficients for f(x)=e^{-x^{2}} on the interval [0,2\pi]. Does this mean f(x+2\pi k)=f(x), k\in\mathbb{Z}? Can x\in[0,\infty)?
44. ### Fourier Coefficients of an Even Function

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...
45. ### Fourier Coefficients for asymmetric interval

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 =...
46. ### Basic question on fourier coefficients

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.
47. ### Finding Fourier Coefficients for a Function's Waveform

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 =...
48. ### Calculate RMS Voltage & Fourier Coefficients for 10Ω Load & 240V 50Hz

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...
49. ### Value of the infinite sum of fourier coefficients?

Homework Statement Calculate the exact value of the sum from minus infinity to infinity of Ck.Homework Equations Ck =...
50. ### Fourier coefficients in a discrete curve

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...