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, and many other technical areas where systems of ordinary differential equations present themselves.

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

Linearity means that the relationship between the input x ( t ) {\displaystyle x(t)} and the output y ( t ) {\displaystyle y(t)} , both being regarded as functions, is a linear mapping: If a {\displaystyle a} is a constant then the system output to a x ( t ) {\displaystyle ax(t)} is a y ( t ) {\displaystyle ay(t)} ; if x ′ ( t ) {\displaystyle x'(t)} is a further input with system output y ′ ( t ) {\displaystyle y'(t)} then the output of the system to x ( t ) + x ′ ( t ) {\displaystyle x(t)+x'(t)} is y ( t ) + y ′ ( t ) {\displaystyle y(t)+y'(t)} , this applying for all choices of a {\displaystyle a} , x ( t ) {\displaystyle x(t)} , x ′ ( t ) {\displaystyle x'(t)} . The latter condition is often referred to as the superposition principle. Time invariance means that whether we apply an input to the system now or T seconds from now, the output will be identical except for a time delay of T seconds. That is, if the output due to input x ( t ) {\displaystyle x(t)} is y ( t ) {\displaystyle y(t)} , then the output due to input x ( t − T ) {\displaystyle x(t-T)} is y ( t − T ) {\displaystyle y(t-T)} . Hence, the system is time invariant because the output does not depend on the particular time the input is applied. 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 y ( t ) {\displaystyle y(t)} is simply the convolution of the input to the system x ( t ) {\displaystyle x(t)} with the system's impulse response h ( t ) {\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: y i = x i ∗ h i {\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.

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