Linear system

This article is about systems as studied in systems theory. For a set of linear equations, see System of linear equations. For the concept in algebraic geometry, see Linear system of divisors.

A linear system is a mathematical model of a system based on the use of a linear operator. Linear systems typically exhibit features and properties that are much simpler than the general, nonlinear case. As a mathematical abstraction or idealization, linear systems find important applications in automatic control theory, signal processing, and telecommunications. For example, the propagation medium for wireless communication systems can often be modeled by linear systems.

Definition

A general deterministic system can be described by an operator, $H$, that maps an input, $x(t)$, as a function of $t$ to an output, $y(t)$, a type of black box description. Linear systems satisfy the property of superposition. Given two valid inputs

$x_1(t) \,$

$x_2(t) \,$

as well as their respective outputs

$y_1(t) = H \left \{ x_1(t) \right \}$

$y_2(t) = H \left \{ x_2(t) \right \}$

then a linear system must satisfy

$\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \}$

for any scalar values $\alpha \,$ and $\beta \,$.

The system is then defined by the equation $H(x(t)) = y(t)$, where $y(t)$ is some arbitrary function of time, and $x(t)$ is the system state. Given $y(t)$ and $H$, $x(t)$ can be solved for. For example, a simple harmonic oscillator obeys the differential equation:

$m \frac{d^2(x)}{dt^2} = -kx$.

If

$H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t)$,

then $H$ is a linear operator. Letting $y(t) = 0$, we can rewrite the differential equation as $H(x(t)) = y(t)$, which shows that a simple harmonic oscillator is a linear system.

The behavior of the resulting system subjected to a complex input can be described as a sum of responses to simpler inputs. In nonlinear systems, there is no such relation. This mathematical property makes the solution of modelling equations simpler than many nonlinear systems. For time-invariant systems this is the basis of the impulse response or the frequency response methods (see LTI system theory), which describe a general input function $x(t)$ in terms of unit impulses or frequency components.

Typical differential equations of linear time-invariant systems are well adapted to analysis using the Laplace transform in the continuous case, and the Z-transform in the discrete case (especially in computer implementations).

Another perspective is that solutions to linear systems comprise a system of functions which act like vectors in the geometric sense.

A common use of linear models is to describe a nonlinear system by linearization. This is usually done for mathematical convenience.

Time-varying impulse response

The time-varying impulse response h(t₂,t₁) of a linear system is defined as the response of the system at time t = t₂ to a single impulse applied at time t = t₁. In other words, if the input x(t) to a linear system is

$x(t) = \delta(t-t_1) \,$

where δ(t) represents the Dirac delta function, and the corresponding response y(t) of the system is

$y(t) |_{t=t_2} = h(t_2,t_1) \,$

then the function h(t₂,t₁) is the time-varying impulse response of the system.

Time-varying convolution integral

Continuous time

The output of any continuous time linear system is related to the input by the time-varying convolution integral:

$y(t) = \int_{-\infty}^{\infty} h(t,s) x(s) ds$

or, equivalently,

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

where

$s = t-\tau$

Discrete time

The output of any discrete time linear system is related to the input by the time-varying convolution sum:

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

or equivalently,

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

where

$k = n-m \,$

represents the lag time between the stimulus at time m and the response at time n.

Causality

A linear system is causal if and only if the system's time varying impulse response is identically zero whenever the time t of the response is earlier than the time s of the stimulus. In other words, for a causal system, the following condition must hold:

$h(t,s) = 0\text{ for }t < s \,$

See also

• Linear system of divisors in algebraic geometry
• Shift invariant system
• LTI system theory
• Nonlinear system
• System analysis
• System of linear equations