Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

learn more… | top users | synonyms

0
votes
0answers
8 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
2
votes
1answer
13 views

Is the DTFT of a sampled Gaussian a positive function?

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in ...
0
votes
1answer
18 views

Find the Fourier Coefficients that minimize the error [duplicate]

I know that the coefficients that minimize the expression are the ones that make it's derivative 0. I have also expanded the whole expression and taken it's derivative, but still I can't figure out ...
0
votes
0answers
18 views

differentiation and integration of Fourier series.

If I have the fourier series of $|x|$ for $-l < x < l$ and I make it periodic with period $2l$ I get a cos series: $$ \frac{l}{2} ...
1
vote
0answers
8 views

Result obtained on deletion of finite number of Fourier Coefficients

I want to know the answer to the following question. If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier ...
0
votes
2answers
16 views

DTFT and its convergence

In the textbook "signals and systems", by prof. Simon Haykin, it says:   If $x[n]$ is not absolutely summable, but does satisfy square summable, then it can be shown that the following equation ...
0
votes
2answers
25 views

Find complex Fourier coefficients of $f(-x), f^*(x)$

For $f(-x)$ i have tried to replace the $k$ with $k'=-k$ but still i can't find any relationship between the coefficients. What could be a better way to approach this problem?
1
vote
1answer
26 views

Find the coefficients of the Fourier series that minimise the error.

I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme ...
-2
votes
0answers
19 views

Fourier coefficients [on hold]

Could you please tell me what is a general way to approach these kind of problems and maybe solve one of the above examples just for clarification?
0
votes
0answers
39 views

Why do sines and cosines form a basis, and can be considered a vector space?

Many times I've seen that Fourier series are justified because we are thinking that the set of all functions of the form $sin(ax)$ and $cos(ax)$ form a vector space. A function can therefore be ...
0
votes
1answer
30 views

Fourier series solution of the heat equation on $-2<x<2$

I have to solve the following boundary value problem: $u_t=u_{xx}$, $u(t,-2)=u(t,2)=0$ and $u(0,x)=f(x)$. I tried to solve the problem using the method of separation of variables. So assume ...
0
votes
1answer
16 views

Finding the value of a series using a known Fourier series

We are given the function $$f(x)=\begin{cases}1&\text{for }-\dfrac{\pi}{2}<x<\dfrac{\pi}{2}\\ 0&\text{for }\dfrac{\pi}{2}<x<\dfrac{3\pi}{2}\end{cases}$$ which I have found to have ...
1
vote
1answer
47 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
3
votes
1answer
50 views

Prove that $\{\sin x, \sin 2x, … , \sin nx\}$ is a linearly independent set

Prove that $\{\sin x, \sin 2x, ... , \sin nx\}$ is linearly independent. The short solution that I do not understand is as follow: For p and q are positive integer, we have $$ ...
2
votes
2answers
220 views

Why can the equality sign be used for Fourier series expansion of a discontinuous function?

Many of the Fourier series problems I deal with right now are with discontinuous functions. Many times the integrals involved have to be separated because there are discontinuities. However this is ...
0
votes
0answers
25 views

Fourier series convergence question from big Rudin.

I am working on some problems from the 3rd edition of Rudin's "Real and Complex Analysis" and I'm stumped on proving the following part from question #19 of chapter 5. Suppose $\lambda_n/\log n \to ...
0
votes
0answers
16 views

Exponential form of Fourier Series,

For a function of period $2L$ the exponential form of the fourier series is defined above. Why however is $|x|<L$ as opposed to $|x| \leq L$?
-1
votes
0answers
69 views

Fourier theorem proof

How do you prove the completeness of the trigonometric system? More precisely, how do you show that the trigonometric system ($\cos(nx)$, $\sin(nx)$, $\forall\; n\in \mathbb{N}_0$) provides a basis ...
1
vote
1answer
24 views

Is it possible to use a fourier series to make a sin wave with a wave length that is not in the fourier series?

This may seem backwards since a fourier series isn't typically used this way but I'm trying to prove whether or not the sum of sin and cos waves could produce a sin wave with a wave length that is not ...
1
vote
1answer
35 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
-1
votes
0answers
6 views

Relation between DFT and fourier transform on tempered distributions

Since tempered distributions can well capture the idea of discrete data, via sums of Dirac deltas, I wonder if there is a way to think of DFT as just the restriction of the analytic fourier transform ...
1
vote
1answer
23 views

fourier series sketching (by hand)

I calculated the Fourier Series representation of $f (x) = 1 − |x|$ on $−1 ≤ x ≤ 1$ and now I am asked to sketch the graph of the series on $−3 ≤ x ≤ 3$ by hand. How do I do this? I read through my ...
1
vote
1answer
53 views

Solve $\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$ for unknown function $f$

Let $g(\theta)$ be a known real-valued function with domain $[0, 2\pi]$. Given that: $$\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$$ How would I solve for the unknown real-valued function ...
-5
votes
0answers
26 views

Fourier problem with method of separation of variables

I'm trying to solve the following set of equations, using the method of separation of variables commonly found in textbooks on Fourier analysis (here's a sample) $u_{tt} = 3u_{xx}$ $u(0,t) = 0, ...
0
votes
1answer
20 views

Quick Fourier Series Question about Cn Integration

If I am given a function $$ f(x) = \left\{ \begin{array}{ll} 2 & \quad x \in (0,6) \\0 & \quad x\in(0,-6) \end{array} \right. $$ $I=(-6,6)$ and I want to ...
2
votes
1answer
83 views

Bessel's Inequality simplification

Let $f:[-\pi,\pi] \to \mathbb{R}$ be a piecewise smooth function with $\int_{-\pi}^{\pi}f(x)dx = 0$. Does anyone have ideas on how to apply Bessel's inequality to show that $\int_{-\pi}^{\pi} ...
1
vote
0answers
28 views

Fourier Series and Uniform Convergence

This question is an extension of Fourier series simplification I know wish to show $$(\frac{1}{\pi})\int_{-\pi}^{\pi}f^2(x) dx = a_{0}^{2}/2 + \sum_{n=1}^{\infty} (a_{n}^{2} + b_{n}^{2})$$ But, we no ...
0
votes
2answers
33 views

Using Complex Fourier Series to Find Real Coefficients

I am about to go insane with this problem, so I really hope some kind, kind soul out there can help me. I am trying to find the complex Fourier series of the following function and interval, and then ...
2
votes
3answers
72 views

Fourier series simplification

I want to show that $$\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)g(x)dx = \frac{a_0\alpha_0}{2} + \sum_{n=1}^{\infty} (a_n\alpha_n + b_n\beta_n)$$ where $f,g: [-\pi,\pi] \to \mathbb{R}$ are integral ...
1
vote
1answer
56 views

Series of exponential function

I had a thought today and I've tried to see if it is a thing. I'm certain it is a thing, I just don't know how to search for it. We have the Taylor series which is a summation of monomials: ...
0
votes
1answer
20 views

Why do we write the first term of the Fourier cosine series as $c_{0}/2$ instead of simply $c_0$?

The Fourier cosine series of some function $f(x)$ defined over the interval $[0, L]$is written as: $$f(x) = \sum_{k = 0}^{\infty} c_k\cos(\frac{k\pi}{L} x)$$ Where $c_k$ can be determined by the ...
2
votes
1answer
63 views

Writing a partial sum of Fourier series as an integral [closed]

Show that the partial sum in equation (3) may be written as:$$f_N(x)=\frac{2}{\pi}\int_{0}^x \frac{\sin(2Nt)}{\sin(t)}\,dt$$ Can someone please explain me how to show these 2 are equal? The first ...
1
vote
2answers
35 views

Fourier series question

I am just a beginner in Fourier series.How should I get start to tackle this question and show the partial sum has extrema? I have no clue to this question. Any help would be highly appreciated.
1
vote
1answer
65 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
0
votes
0answers
3 views

Estimating certain singular discrete sums

I want to estimate sums of the following form: $S^d(\alpha,\beta,l):= \sum_{k \in \mathbb{Z}^d, k \notin \{0,l\}} \frac{1}{|k|^\alpha} \cdot \frac{1}{|k-l|^\beta}$, where $l \in \mathbb{Z}^d$ and ...
0
votes
0answers
18 views

Double Fourier series for inhomogeneous BC

So the task is, that the following 2D eigenvalue problem on a unit square is given. \begin{equation} -\nabla^2M(x,y)=\lambda M(x,y),\quad 0<x<1,0<y<1\\ M(x,y)=0\quad \text{on the boundary ...
0
votes
1answer
16 views

Discrete Fourier Transform real f_j's

Could you help me show that if $$\hat{f}(k)=\frac{1}{N}\sum\limits_{j=0}^{N-1}f_j \exp\left(-i\frac{2\pi jk}{N}\right)$$ (k=0,1,...,N-1) is the Discrete Fourier Transform of $f_0, f_1,\ldots, ...
0
votes
0answers
22 views

Function approximation by various means

I know several ways to approximate a function: Taylor series, Fourier series, or polynomials, like e.g. Legendre polynomials. Is the only difference between those various methods the speed at which ...
0
votes
1answer
43 views

Partial Sum Fourier Series

Show that the partial sum $$f_N(x)=\frac{4}{π}\sum^N_{n=1}\frac{\sin((2n-1)x)}{2n-1}$$ may be written as $$f_N(x)=\frac{2}{π}\int_0^x\frac{\sin(2Nt)}{\sin(t)}\,dt$$ The original question is 'Sketch ...
1
vote
1answer
75 views

Fourier sine series expansion

The function $f(x)$ is defined as $$f(x)=1\qquad0<x<\pi$$ Sketch the odd extension and show that the Fourier sine series expansion is ...
0
votes
0answers
46 views

Proving Gibbs phenomenon using Dirichlet kernel

I am working on a problem$^{(1)}$ on using Dirichlet kernel to prove Gibbs phenomenon. It is a long proof broken down into 7 steps, and on each step I have to answer some questions. Long story short, ...
2
votes
0answers
28 views

Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge(fourier series coefficients)?

Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$ $S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f. Where ...
1
vote
2answers
55 views

Fourier Series Representation $e^{ax}$

a) Compute the full Fourier series representation of $f(x) = e^{ax}, −π ≤ x < π.$ b) By using the result of a) or otherwise determine the full Fourier series expansion for the function ...
0
votes
0answers
25 views

Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
0
votes
0answers
9 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
1
vote
1answer
20 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
0
votes
1answer
13 views

Test question regarding convergence of Fourier series

I'm preparing for a test and I have no clue how I should solve the following question. Let $f:\Bbb{R}\to\Bbb{R}$ be $2\pi \text{-periodic}$ function such that $f(0)=1$ and $$\forall ...
5
votes
2answers
40 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
1
vote
0answers
22 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
0
votes
1answer
21 views

Fourier series coefficients which do not approach to zero

I want to know whether there are a finite number of coefficients in a Fourier series of a periodic function (with period $P$), whose magnitude are above a certain threshold. Those coefficients can can ...