A discrete one-dimensional model of optical imaging looks like this: $$I(r) = \sum_i e_i P(r – r_i)$$ Here, the $e_i$ are point light sources at locations $r_i$ in the object and $P$ is a point spread function that blurs each point. We can assume that $P$ is even, non-negative and has a finite extent, ie […]

The question at hand is to use Parseval’s theorem to solve the following integral: $$\int_{-\infty}^{\infty} sinc^4 (kt) dt$$ I understand Parseval’s theorem to be: $$E_g = \int_{-\infty}^{\infty} g^2(t) = \int_{-\infty}^{\infty} |G(f)|^2 df $$ I began by doing the obvious and removing the squared such that: $$g^2(t) = Sinc^4 (kt)$$ $$g(t) = Sinc^2 (kt)$$ Following the […]

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as “Bk“. Meanwhile, cnx.org Signal Reconstruction says: The basis splines Bn are shown … as the order increases, the functions approach the Gaussian function, which is exactly B∞. but then says as the order increases, the cardinal basis splines approximate the […]

I am looking for ways to compute the subderivative of $ ||Au||_{L^{\infty}} $, as I want to solve the minimization problem of \begin{equation} \min\limits_u \quad \lambda ||Au||_{L^{\infty}} + \frac{1}{2} || u – f ||^2_{L^{2}} \quad. \end{equation} However I have no idea where to start. I tried looking at the Gateaux derivative to get an intuition […]

I am experiencing what I think is really simple confusion. Take $y(t) = \sin(2 \cdot \pi \cdot t \cdot\omega(t))$ and $\omega(t) = a \cdot t+b$ for $t \in [0,p)$ and let $\omega(t)$ have a periodic extension with period $p$. The values $a,b,p$ are parameters $\omega(t)$ looks like the blue function here with $\omega$ as such, […]

Given a set of discrete measurements in time $x_t, t \in \{0,\Delta t, 2\Delta t,\ldots,T-\Delta t,T\}$, what is the correct way to compute the discrete derivative $\dot x_t$. Is it more correct to take the difference with the previous value: $$\dot x_t = \frac{x_t-x_{t-1}}{\Delta t}$$ or with the next value in time: $$\dot x_t = […]

Say I pass 512 samples into my FFT My microphone spits out data at 10KHz, so this represents 1/20s. (So the lowest frequency FFT would pick up would be 40Hz). The FFT will return an array of 512 frequency bins – bin 0: [0 – 40Hz) – bin 1: [40 – 80Hz) etc So if […]

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using Mathematica, but it ran for several hours without producing any result.

I am sampling a continuous signal $x_c(t)$ that follows a triangle function in the time domain, meaning: $$x_c(t)=\left\{\begin{array}{rl}1-|t/a|,&|t|<|a|\\ 0,&\mbox{otherwise}\end{array}\right.$$ Parameter $a$ is an integer multiple of my sampling interval $T_0$ such that $a=kT_0$. Thus: $$x_d[n]=\left\{\begin{array}{rl}1-|n/k|,&|n|<|k|\\ 0,&\mbox{otherwise}\end{array}\right.$$ I am wondering if a discrete-time Fourier transform $x_d[n]$ exists in closed-form, like the continuous-time Fourier transform of $x_c(t)$. […]

Continuing from this rather silly trivial question factoring real valued filters into shorter complex ones, hoping this won’t be as trivial. If we modify it a bit: $$z_0 = e^{2\pi i / 8}$$ and $$\left(z_0^{[3k,2k]} * z_0^{[3k,-2k]}\right)$$ will for $k \in \{2,3,4\}$: $$k=2 \rightarrow \left[\begin{array}{ccc} -1&2i&1 \end{array}\right]$$ derivative in real part $$k=3 \rightarrow \left[\begin{array}{ccc} i&0&1 […]

Intereting Posts

Prob. 10 after Sec. 16 in Munkres' TOPOLOGY, 2nd edition: Which of these topologies is finer than which?
Thinking about $p^n = x^m + y^m$ where $p | m$
Inverse of $y=xe^x$
To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 … (2n-1)\over 2.4.6…(2n)$
find the inverse Laplace transform of complex function
$L^p(\mathbb{R})$ separable.
Operator norm of the inverse
A sub-additivity inequality
Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?
Elliptic regularization of the heat equation
Fraction field of the formal power series ring in finitely many variables
Are there any bases which represent all rationals in a finite number of digits?
Is every set countable according to some outer model?
Let $\pi$ denote a prime element in $\mathbb Z, \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
What is the coefficient of $x^{2k}$ in the $n$-th iterate, $f^{(n)}(x)$, if $f(x)=1+x^2$?