ühikimpulssi

Ühikimpulssi, often translated as unit impulse or Dirac delta function in English, is a mathematical idealization representing a signal that is infinitely strong for an infinitesimal duration but whose integral over all time is unity. Physically, it can be thought of as a perfect, instantaneous spike. It is denoted by the symbol $\delta(t)$.

The defining characteristic of the unit impulse is its sampling property. When integrated with any continuous function $f(t)$ multiplied by $\delta(t-a)$, the result is simply the value of the function at point $a$, i.e., $\int_{-\infty}^{\infty} f(t)\delta(t-a) dt = f(a)$. This makes the unit impulse extremely useful in signal processing and system analysis for determining the response of a system to a brief, sharp input.

In the context of linear time-invariant (LTI) systems, the impulse response of a system, often denoted by $h(t)$, is the output of the system when the input is a unit impulse. Knowing the impulse response allows for the characterization of the entire system, as the output for any arbitrary input can be found by convolving the input signal with the system's impulse response. The concept of the unit impulse is fundamental to understanding concepts like system stability, causality, and frequency response. While not a true function in the traditional sense, it is rigorously defined within the theory of distributions.