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LTI System

A Linear Time-Invariant (LTI) system is a system that has two key properties:

1. It is linear,
   • so scaling the input has linear relationship to the scaling of the output
   • and the weighted sum of inputs is linear to the weighted sum of outputs.
2. It is time invariant, so shifting the input in time also shifts the output in time just the same.

These two properties give a class of systems that is widely applicable in data processing and modeling physical systems and that is easy to analyse, especially through the use of Transfer Functions.

Impulse Response

An LTI system can fully be described by its impulse response, so how the system acts on an infinitely short and strong impulse ([[Dirac]]). We can represent a signal as an infinite array of quick impulses scaled to the strength of the signal. Due to linearity and time-invariance, the output of the system is the weighted sum of impulse responses, where the weights are the amplitude of the input signal.

Each point \(\tau\) is modeled as the impulse response shifted to that point and scaled by the strength of the input signal \(x(\tau)\). Summing up all those weighted impulse responses over every infinitesimal time step gives us a [[Convolution]] of the input signal with the impulse response:

\[ y(t)=\int\limits_{-\infty}^{\infty}x(\tau)h(\tau-t)d\tau \]

As the convolution of time domain signals is equivalent to the product of their spectra, we can write

\[ \begin{array}{lll} & Y(s) &= X(s){\cdot}H(s) \\ \iff & H(s) &= \frac{Y(s)}{X(s)} \end{array} \]

giving us a definition for the Transfer Function.