A note on convergence under dynamical thresholds with delays

An oscillating discontinuity is a type of discontinuity in which a function’s values fluctuate infinitely often as the input approaches a particular point. Unlike jump discontinuities, where the function suddenly shifts between two values, or infinite discontinuities, where the function diverges without bound, an oscillating discontinuity arises from rapid back-and-forth variation. Because the function never stabilizes toward a single value, no finite limit exists at that point.One of the most...

The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is a crucial tool in the analysis of discrete-time systems, but its convergence is limited to specific values of the complex variable z. This range of values, known as the Region of Convergence (ROC), is fundamental in determining the behavior and stability of a system or signal. The ROC defines the region in the complex plane where the z-transform converges, which can take various...

The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
Consider a decaying exponential signal that begins at a specific time. When deriving its Laplace transform, the time-domain variable is replaced with a complex variable. This substitution...

A sequence is a function defined on the natural numbers that assigns a value to each index. It can be understood as an ordered list of terms generated one after another. In mathematical analysis, an important question is whether the terms of a sequence approach a single real number as the index becomes very large. When this happens, the sequence is said to converge, and the value approached is called the limit. From a graphical perspective, convergence means that the plotted terms approach a...

A mass attached to a vertical spring can exhibit oscillatory motion as it moves above and below a central equilibrium point. In an ideal spring, the oscillations would continue indefinitely with constant amplitude. In a damped spring, however, resistive forces such as air resistance or internal friction gradually reduce the size of each swing. This behavior is often modeled by combining a sinusoidal function, which represents the repeated motion, with an exponential decay factor, which reduces...

In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...