Template:Short description Template:About Template:More citations needed In systems theory, 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 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.
DefinitionEdit
A general deterministic system can be described by an operator, Template:Math, that maps an input, Template:Math, as a function of Template:Mvar to an output, Template:Math, a type of black box description.
A system is linear if and only if it satisfies the superposition principle, or equivalently both the additivity and homogeneity properties, without restrictions (that is, for all inputs, all scaling constants and all time.)<ref name="Phillips_2008">Template:Cite book</ref><ref name="Bessai_2005">Template:Cite book</ref><ref name="Alkin_2014">Template:Cite book</ref><ref name="Nahvi_2014">Template:Cite book</ref>
The superposition principle means that a linear combination of inputs to the system produces a linear combination of the individual zero-state outputs (that is, outputs setting the initial conditions to zero) corresponding to the individual inputs.<ref name="Sundararajan_2008">Template:Cite book</ref><ref name="Roberts_2018">Template:Cite book</ref>
In a system that satisfies the homogeneity property, scaling the input always results in scaling the zero-state response by the same factor.<ref name="Roberts_2018" /> In a system that satisfies the additivity property, adding two inputs always results in adding the corresponding two zero-state responses due to the individual inputs.<ref name="Roberts_2018" />
Mathematically, for a continuous-time system, given two arbitrary inputs <math display="block">\begin{align} x_1(t) \\ x_2(t) \end{align}</math> as well as their respective zero-state outputs <math display="block">\begin{align} y_1(t) &= H \left \{ x_1(t) \right \} \\ y_2(t) &= H \left \{ x_2(t) \right \} \end{align} </math> then a linear system must satisfy <math display="block">\alpha y_1(t) + \beta y_2(t) = H \left \{ \alpha x_1(t) + \beta x_2(t) \right \} </math> for any scalar values Template:Mvar and Template:Mvar, for any input signals Template:Math and Template:Math, and for all time Template:Mvar.
The system is then defined by the equation Template:Math, where Template:Math is some arbitrary function of time, and Template:Math is the system state. Given Template:Math and Template:Nowrap the system can be solved for Template:Nowrap
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 Template:Math 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.
The previous definition of a linear system is applicable to SISO (single-input single-output) systems. For MIMO (multiple-input multiple-output) systems, input and output signal vectors (<math>{\mathbf x}_1(t)</math>, <math>{\mathbf x}_2(t)</math>, <math>{\mathbf y}_1(t)</math>, <math>{\mathbf y}_2(t)</math>) are considered instead of input and output signals (<math>x_1(t)</math>, <math>x_2(t)</math>, <math>y_1(t)</math>, <math>y_2(t)</math>.)<ref name="Bessai_2005" /><ref name="Nahvi_2014" />
This definition of a linear system is analogous to the definition of a linear differential equation in calculus, and a linear transformation in linear algebra.
ExamplesEdit
A simple harmonic oscillator obeys the differential equation: <math display="block">m \frac{d^2(x)}{dt^2} = -kx.</math>
If <math display="block">H(x(t)) = m \frac{d^2(x(t))}{dt^2} + kx(t),</math> then Template:Math is a linear operator. Letting Template:Nowrap we can rewrite the differential equation as Template:Nowrap which shows that a simple harmonic oscillator is a linear system.
Other examples of linear systems include those described by <math>y(t) = k \, x(t)</math>, <math>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math>, <math>y(t) = k \, \int_{-\infty}^{t}x(\tau) \mathrm d\tau</math>, and any system described by ordinary linear differential equations.<ref name="Nahvi_2014" /> Systems described by <math>y(t) = k</math>, <math>y(t) = k \, x(t) + k_0</math>, <math>y(t) = \sin{[x(t)]}</math>, <math>y(t) = \cos{[x(t)]}</math>, <math>y(t) = x^2(t)</math>, <math display="inline">y(t) = \sqrt{x(t)}</math>, <math>y(t) = |x(t)|</math>, and a system with odd-symmetry output consisting of a linear region and a saturation (constant) region, are non-linear because they don't always satisfy the superposition principle.<ref name="DeerghaRao_2018">Template:Cite book</ref><ref name="Chen_2004">Template:Cite book</ref><ref name="ElAliKarim_2008">Template:Cite book</ref><ref name="Apte_2016">Template:Cite book</ref>
The output versus input graph of a linear system need not be a straight line through the origin. For example, consider a system described by <math>y(t) = k \, \frac{\mathrm dx(t)}{\mathrm dt}</math> (such as a constant-capacitance capacitor or a constant-inductance inductor). It is linear because it satisfies the superposition principle. However, when the input is a sinusoid, the output is also a sinusoid, and so its output-input plot is an ellipse centered at the origin rather than a straight line passing through the origin.
Also, the output of a linear system can contain harmonics (and have a smaller fundamental frequency than the input) even when the input is a sinusoid. For example, consider a system described by <math>y(t) = (1.5 + \cos{(t)}) \, x(t)</math>. It is linear because it satisfies the superposition principle. However, when the input is a sinusoid of the form <math>x(t) = \cos{(3t)}</math>, using product-to-sum trigonometric identities it can be easily shown that the output is <math>y(t) = 1.5 \cos{(3t)} + 0.5 \cos{(2t)} + 0.5 \cos{(4t)}</math>, that is, the output doesn't consist only of sinusoids of same frequency as the input (Template:Nowrap), but instead also of sinusoids of frequencies Template:Nowrap and Template:Nowrap; furthermore, taking the least common multiple of the fundamental period of the sinusoids of the output, it can be shown the fundamental angular frequency of the output is Template:Nowrap, which is different than that of the input.
Time-varying impulse responseEdit
The time-varying impulse response Template:Math of a linear system is defined as the response of the system at time t = t2 to a single impulse applied at time Template:Nowrap In other words, if the input Template:Math to a linear system is <math display="block">x(t) = \delta(t - t_1)</math> where Template:Math represents the Dirac delta function, and the corresponding response Template:Math of the system is <math display="block">y(t=t_2) = h(t_2, t_1)</math> then the function Template:Math is the time-varying impulse response of the system. Since the system cannot respond before the input is applied the following causality condition must be satisfied: <math display="block"> h(t_2, t_1) = 0, t_2 < t_1</math>
The convolution integralEdit
The output of any general continuous-time linear system is related to the input by an integral which may be written over a doubly infinite range because of the causality condition: <math display="block"> y(t) = \int_{-\infty}^{t} h(t,t') x(t')dt' = \int_{-\infty}^{\infty} h(t,t') x(t') dt' </math>
If the properties of the system do not depend on the time at which it is operated then it is said to be time-invariant and Template:Mvar is a function only of the time difference Template:Math which is zero for Template:Math (namely Template:Math). By redefinition of Template:Mvar it is then possible to write the input-output relation equivalently in any of the ways, <math display="block"> y(t) = \int_{-\infty}^{t} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(t-t') x(t') dt' = \int_{-\infty}^{\infty} h(\tau) x(t-\tau) d \tau = \int_{0}^{\infty} h(\tau) x(t-\tau) d \tau </math>
Linear time-invariant systems are most commonly characterized by the Laplace transform of the impulse response function called the transfer function which is: <math display="block">H(s) =\int_0^\infty h(t) e^{-st}\, dt.</math>
In applications this is usually a rational algebraic function of Template:Mvar. Because Template:Math is zero for negative Template:Mvar, the integral may equally be written over the doubly infinite range and putting Template:Math follows the formula for the frequency response function: <math display="block"> H(i\omega) = \int_{-\infty}^{\infty} h(t) e^{-i\omega t} dt </math>
Discrete-time systemsEdit
The output of any discrete time linear system is related to the input by the time-varying convolution sum: <math display="block"> y[n] = \sum_{m =-\infty}^{n} { h[n,m] x[m] } = \sum_{m =-\infty}^{\infty} { h[n,m] x[m] }</math> or equivalently for a time-invariant system on redefining Template:Math, <math display="block"> y[n] = \sum_{k =0}^{\infty} { h[k] x[n-k] } = \sum_{k =-\infty}^{\infty} { h[k] x[n-k] }</math> where <math display="block"> k = n-m </math> represents the lag time between the stimulus at time m and the response at time n.
See alsoEdit
- Shift invariant system
- Linear control
- Linear time-invariant system
- Nonlinear system
- System analysis
- System of linear equations