Tidskonvolution

Tidskonvolution, often translated as time convolution, is a mathematical operation used to describe how the output of a linear time-invariant (LTI) system changes over time when subjected to an input signal. It represents the process of combining an input signal with the system's impulse response to determine the system's output. The impulse response is the output of the system when the input is an impulse function, a signal that is infinitely short and has unit amplitude at time zero and zero everywhere else.

Mathematically, the time convolution of an input signal $x(t)$ and a system's impulse response $h(t)$ to produce the output signal $y(t)$ is defined by the integral:

$y(t) = \int_{-\infty}^{\infty} x(\tau) h(t - \tau) d\tau$

This integral essentially "slides" the time-reversed impulse response $h(t - \tau)$ across the input signal $x(\tau)$, multiplying them at each point in time and summing (integrating) the results. The result is the output signal $y(t)$ at time $t$.

In discrete time, the convolution operation is represented by a summation:

$y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n - k]$

where $x[k]$ is the discrete input signal, $h[n]$ is the discrete impulse response, and $y[n]$ is the discrete output signal. Time convolution is a fundamental concept in signal processing, systems theory, and many areas of physics and engineering, enabling the analysis and design of systems that respond to time-varying inputs. It is widely used in areas such as audio processing, image filtering, and control systems.