Linear time-invariant system

In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of linearity and time-invariance; these terms are briefly defined in the overview below. These properties apply (exactly or approximately) to many important physical systems, in which case the response y(t) of the system to an arbitrary input x(t) can be found directly using convolution: y(t) = (x ∗ h)(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication). What's more, there are systematic methods for solving any such system (determining h(t)), whereas systems not meeting both properties are generally more difficult (or impossible) to solve analytically. A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.

Linear time-invariant system theory is also used in image processing, where the systems have spatial dimensions instead of, or in addition to, a temporal dimension. These systems may be referred to as linear translation-invariant to give the terminology the most general reach. In the case of generic discrete-time (i.e., sampled) systems, linear shift-invariant is the corresponding term. LTI system theory is an area of applied mathematics which has direct applications in electrical circuit analysis and design, signal processing and filter design, control theory, mechanical engineering, image processing, the design of measuring instruments of many sorts, NMR spectroscopy^{[citation needed]}, and many other technical areas where systems of ordinary differential equations present themselves.

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

Through these properties, it is reasoned that LTI systems can be characterized entirely by a single function called the system's impulse response, as, by superposition, any arbitrary signal can be expressed as a superposition of time-shifted impulses. The output of the system ${\displaystyle y(t)}$ is simply the convolution of the input to the system ${\displaystyle x(t)}$ with the system's impulse response ${\displaystyle h(t)}$. This is called a continuous time system. Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in discrete time: ${\displaystyle y_{i}=x_{i}*h_{i}}$ where y, x, and h are sequences and the convolution, in discrete time, uses a discrete summation rather than an integral.

LTI systems can also be characterized in the frequency domain by the system's transfer function, which for a continuous-time or discrete-time system is the Laplace transform or Z-transform of the system's impulse response, respectively. As a result of the properties of these transforms, the output of the system in the frequency domain is the product of the transfer function and the corresponding frequency-domain representation of the input. In other words, convolution in the time domain is equivalent to multiplication in the frequency domain.

For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials. As a result, if the input to a system is the complex waveform ${\displaystyle A_{s}e^{st}}$ for some complex amplitude ${\displaystyle A_{s}}$ and complex frequency ${\displaystyle s}$, the output will be some complex constant times the input, say ${\displaystyle B_{s}e^{st}}$ for some new complex amplitude ${\displaystyle B_{s}}$. The ratio ${\displaystyle B_{s}/A_{s}}$ is the transfer function at frequency ${\displaystyle s}$. The output signal will be shifted in phase and amplitude, but always with the same frequency upon reaching steady-state. LTI systems cannot produce frequency components that are not in the input.

LTI system theory is good at describing many important systems. Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or nonlinear case. Any system that can be modeled as a linear differential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up of resistors, inductors, and capacitors (RLC circuits). Ideal spring–mass–damper systems are also LTI systems, and are mathematically equivalent to RLC circuits.

Most LTI system concepts are similar between the continuous-time and discrete-time cases. In image processing, the time variable is replaced with two space variables, and the notion of time invariance is replaced by two-dimensional shift invariance. When analyzing filter banks and MIMO systems, it is often useful to consider vectors of signals. A linear system that is not time-invariant can be solved using other approaches such as the Green function method.