Articles of signal processing

What is the inverse z transform of 1/(z-1)^2?

I’d like to know how to calculate the inverse z transform of $\frac{1}{(z-1)^2}$ and the general case $\frac{1}{(z-a)^2}$

How can a unit step function be differentiable??

Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written ” When we differentiate a Unit Step Function, we get an Impulse function. ” But as far as I have read, a unit step function is […]

What is boxcar averaging?

This is an application in signal processing but what I don’t understand is how it’s done algorithmically. I’ve seen some stuff online but most of it is just pictures. I would like an example on some type of sample data such as [0 1 2 3 4 5 6 7 8 9 10] and if […]

Help in understanding a coding technique based on inverse mapping of a dynamical system

Based on paper titled : Simultaneous Arithmetic Coding and Encryption Using Chaotic Maps by Kwok-Wo Wong et. al I cannot understand how the data message is compressed. Instead of the forward iterations of the map, they apply the inverse of the map i.e., assuming the vector {s} is given, how can the compression to k […]

Help needed with the integral of an infinite series

Can you please help me with the integral of this series? I came across it in a signal processing paper and haven’t been able to figure out the solution myself. $$ \int\limits_{(n-1)T}^{nT}\left[\frac{2\pi}{T}\displaystyle\sum_{i=2}^{\infty}\left(\frac{TK}{2\pi}f(x)\right)^i\right]dx $$ given that: $T$ and $K$ are constants $ \int\limits_{(n-1)T}^{nT}Kf(x)dx = y[n] $ $ f(x) $ does not change significantly between $ (n-1)T […]

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, […]