Articles of signal processing

Analysing an optics model in discrete and continuous forms

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 […]

Using Parseval's theorem to solve an integral

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 […]

What is a cardinal basis spline?

Wikipedia says: the normalized cardinal B-splines tend to the Gaussian function and writes them as “Bk“. Meanwhile, 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 […]

Subderivative of $ ||Au||_{L^{\infty}} $ to compute proximal operator

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 […]

Frequency Swept sine wave — chirp

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, […]

Correct way to calculate numeric derivative in discrete time?

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 = […]

Extracting exact frequencies from FFT output

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 […]

Fourier transform of $\left|\frac{\sin x}{x}\right|$

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.

DTFT of a triangle function in closed form

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 $x_c(t)$. […]

Complex filter factorizations – continued

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 […]