0
votes
4answers
54 views

Why $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$?

I am working out a Fourier Series problem and I saw that the suggested solution used $\sin(n\pi) = 0$ and $\cos(n\pi)=(-1^n)$ to simply the expressions while finding the Fourier Coefficients ...
0
votes
1answer
58 views

Looking for a nice expression of these functions in terms of trig functions

I have come across three sinusoidal functions f1, f2, and f3 which, up to scaling and translation, are very close to each other. When normalized and plotted together, they are hard to tell apart. ...
2
votes
1answer
46 views

Fourier series of oscillation in form $\cos(2 \pi \frac{k}{T}+\phi)$

I would like to calculate the fourier coefficients of $\cos(2 \pi \frac{k}{T}+\phi)$ where $T \in \mathbb{N}$ is the period and is arbitrary but fixed, $k \in [1, N-1]$ is the number of oscillations ...
1
vote
2answers
35 views

trigonometric interpolation of a sampled signal

Given N sampled points, using the FFT we can get the Fourier transform of those N points $X_k$. With N/2 the Nyquist frequency and $X_0$ the DC value. Using the inverse we can then get back the ...
0
votes
1answer
52 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
2
votes
0answers
53 views

Proving periodicity of sine and cosine [duplicate]

If we define the sine and cosine functions by their Maclaurin expansions, how do we prove they are periodic with period $2\pi$?
1
vote
2answers
62 views

Find complex Fourier coefficients

let $f(x) = \sum^{10}_{m=1}(-1)^m \sin(2^m x)$. denote complex Fourier coefficients of $f(x)$ over $[-\pi, \pi]$ as $c_n = \frac{1}{2\pi} \int _{-\pi}^\pi f(x) e^{-inx}\,dx.$ ...
0
votes
1answer
52 views

Integral computation involving $\sin^2 x$

I am attempting to find the Fourier series for $\sin^2x$ but am getting stuck. For the value of $a_0$, I am trying to do it as follows: $$ \frac 1{2\pi}\int_{-\pi/2}^{\pi/2}(\frac 12 -\frac ...
1
vote
1answer
55 views

Fourier Series doubt

I have a doubt regarding the Fourier series usage in terms of the Fourier series formula, which has multiple variants and is quite complicated. EDIT: I would like to mention that this question (of ...
1
vote
1answer
49 views

Confusion regard Fourier Series formulae

EDIT: I need help today, please. It's very important for my homework. I need to understand this concept. Thank you! I have a doubt regarding the Fourier series formula. In one of my notes, it is ...
1
vote
1answer
38 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
0
votes
3answers
379 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
3
votes
1answer
203 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...
0
votes
1answer
80 views

Sinusoid sum of cosine and sine

I am studying Fourier series right now. I asked a question before of math.statckexchange regarding Fourier series. This question is related and hopefully quite simple: Generally Fourier series works ...
12
votes
1answer
317 views

Prove $\left|\sum_{k=2001}^{m}a_{k}\sin{(kx)}\right|\le 1+\pi $ ,$m\ge 2001,x\in R$

let $\{a_{n}\}$ is non-increasing postive sequence;show that if for $n\ge 2001,na_{n}\le 1$, then for any positive integer numbers $m\ge 2001,x\in R$, we have ...
0
votes
2answers
150 views

Conceptual question about Discrete Fourier Transform

On the wikipedia page for the discrete Fourier transform, the first sentence says: In mathematics, the discrete Fourier transform (DFT) converts a finite list of equally spaced samples of a ...
0
votes
2answers
344 views

Trapezoid rule over trigonometric polynomials

The question is regarding trapezoid rule applied on trigonometric polynomials Here is the question Show that the composite trapezoid rule over an equidistant partitioning with interval size $h = ...
1
vote
0answers
221 views

How can I find the compact trigonometric Fourier series from these signals?

I've been stuck on this for a while, but how exactly would I go about calculating the compact trigonometric Fourier series for both of these signals? I have a general formula down for it, but I just ...
1
vote
2answers
419 views

Proving that two periodic functions are orthogonal

Suppose we have a periodic function $f_K(\vec x)$. We want to show that $\int {f_K^*(\vec x) f_{K'}(\vec x) d\vec x} = \delta_{KK'}$, where the integration is over the period of $f(x)$. I know this ...
1
vote
1answer
88 views

Alternative complete bases for Fourier Series.

Knowing that $$\left\{ \sin\left(kx\right)\right\} _{k\in\mathbb{N}}$$ and $$\left\{ \cos\left(kx\right)\right\} _{k\in\mathbb{N\cup}\left\{ 0\right\} }$$ are complete systems in $L^2(0,\pi)$. How ...
1
vote
2answers
312 views

Trigonometric Identities and Fourier Series

I have the series: $$2+\sum_{m=1}^n 4(-1)^m\cos(m\pi x)$$ Here, $x\in (-1,1)$. I need to show that this equals some fraction with only cosine terms and $n$ (no $m$). Just looking for some ...
1
vote
1answer
734 views

Odd harmonics only in Fourier transform

If a function is even or odd, it implies that there are respectively only cos and sine terms in its Fourier expansion. But is there a condition for a function to have an expansion with only odd or ...
4
votes
1answer
102 views

Total variation of a Fourier series

Let $f(x) = f(x+2\pi)$ be a bounded real function given by the Fourier series of the form $$ f(x) = \sum_{k=1}^N a_k \sin(kx + \phi_k). $$ What is the total variation $V(f)$ of this function over one ...
9
votes
7answers
525 views

Why does this Fourier series have a finite number of terms?

I am learning about Fourier series in class and the basic form of a Fourier Series is $$a_{0}+\sum_{n=1}^{\infty} [a_{n}\cos(nx)+b_{n}\sin(nx)]$$ so a fourier series should have an infinity number ...
3
votes
2answers
97 views

Need help with this integral using trig identities

I am trying to integrate the function $$\int_{-\pi/2}^0 \sin(2x)\cos(nx) \, \mathrm{d} x.$$ My professor has an answer of $$\frac{-2\cos(\frac{n \pi}{2})+1}{n^{2}-4}.$$ When I do this problem, I ...
5
votes
1answer
133 views

A trigonometric identity

If one sees the simplification done in equation $5.3$ (bottom of page 29) of this paper it seems that a trigonometric identity has been invoked of the kind, $$\ln(2) + \sum _ {n=1} ^{\infty} ...
3
votes
2answers
133 views

Sum the series: $ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \cdots \ \text{ad inf}$

How do I sum the following series? $$ S = \frac{1}{2} \cdot \sin\alpha + \frac{1\cdot 3}{2 \cdot 4} \sin{2\alpha} + \frac{1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \sin{3\alpha} + \cdots \ \text{ad ...
1
vote
2answers
478 views

Please check my answer to $\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$

$$\sum_{i=1}^n \frac{\sin{(ix)}}{i} < 2\sqrt{\pi}$$ I have this answer, please let me know if there is a more beautiful proof. My answer: at first, we prove two inequalities: If $x\in ...
1
vote
1answer
271 views

Sum of sinusoidals (frequency/phase of acoustic beats)

I have a function that's the sum of two sinusoidals: $ A \cos(\Theta_1 + \omega_1 t) + B \cos(\Theta_2 + \omega_2 t) $. It basically forms an acoustic beat pattern. I need to find the frequency of ...
1
vote
1answer
299 views

expression for Dirichlet's kernel like sum

It is given in the book that the Dirichlet's kernel $D_n(t) = 1/2 + \sum\limits_{k=1}^{n} \cos(kt)$ is given as $\frac{\sin(n+1/2)t}{2\sin(t/2)}$. I'd like to know if there is any such expression for ...
1
vote
1answer
505 views

Relation between “harmonic form” and fourier series?

I am currently prepping for uni having been a few years out of the studying loop (programming as it happens). Anyway, I've been re-reading my A-level notes/exercises and looking through OpenCourseWare ...
6
votes
6answers
2k views

Example of a trigonometric series that is not fourier series?

My textbook doesn't give any example of this kind of series. Could you provide some? Trigonometric series is defined in wikipedia as : $A_{0}+\sum_{n=1}^{\infty}(A_{n} \cos{nx} + B_{n} \sin{nx})$ ...