Linear time-invariant theory

Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas. It investigates the response of a linear and time-invariant system to an arbitrary input signal. Trajectories of these systems are commonly measured and tracked as they move through time (e.g., an acoustic waveform), but in applications like image processing and field theory, the LTI systems also have trajectories in spatial dimensions. Thus, these systems are also called linear translation-invariant to give the theory the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. A good example of LTI systems are electrical circuits that can be made up of resistors, capacitors, and inductors.

The defining properties of any LTI system are linearity and time invariance.

Input   ${\displaystyle \int _{-\infty }^{\infty }c_{\omega }\,x_{\omega }(t)\,\operatorname {d} \omega }$   produces output   ${\displaystyle \int _{-\infty }^{\infty }c_{\omega }\,y_{\omega }(t)\,\operatorname {d} \omega \,}$

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