
In
system analysis, among other fields of study, a linear time-invariant (LTI) system is a
system
A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its open system (systems theory), environment, is described by its boundaries, str ...
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 of the system to an arbitrary input can be found directly using
convolution: where 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 ), 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
An electrical network is an interconnection of electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e. ...
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
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...
which has direct applications in
electrical circuit analysis and design,
signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, Scalar potential, potential fields, Seismic tomograph ...
and
filter design,
control theory
Control theory is a field of control engineering and applied mathematics that deals with the control system, control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the applic ...
,
mechanical engineering
Mechanical engineering is the study of physical machines and mechanism (engineering), mechanisms that may involve force and movement. It is an engineering branch that combines engineering physics and engineering mathematics, mathematics principl ...
,
image processing, the design of
measuring instruments of many sorts,
NMR spectroscopy, and many other technical areas where systems of
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable (mathematics), variable. As with any other DE, its unknown(s) consists of one (or more) Function (mathematic ...
s present themselves.
Overview
The defining properties of any LTI system are ''linearity'' and ''time invariance''.
* ''Linearity'' means that the relationship between the input
and the output
, both being regarded as functions, is a linear mapping: If
is a constant then the system output to
is
; if
is a further input with system output
then the output of the system to
is
, this applying for all choices of
'','' ''
'',
. 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
is
, then the output due to input
is
. 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
is simply the
convolution of the input to the system
with the system's impulse response
. 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:
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
In engineering, a transfer function (also known as system function or network function) of a system, sub-system, or component is a function (mathematics), mathematical function that mathematical model, models the system's output for each possible ...
, 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
for some complex amplitude
and complex frequency
, the output will be some complex constant times the input, say
for some new complex amplitude
. The ratio
is the transfer function at frequency
. 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
An electrical network is an interconnection of electrical components (e.g., batteries, resistors, inductors, capacitors, switches, transistors) or a model of such an interconnection, consisting of electrical elements (e.g., voltage so ...
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.
Continuous-time systems
Impulse response and convolution
The behavior of a linear, continuous-time, time-invariant system with input signal ''x''(''t'') and output signal ''y''(''t'') is described by the convolution integral:
:
where
is the system's response to an
impulse:
.
is therefore proportional to a weighted average of the input function
. The weighting function is
, simply shifted by amount
. As
changes, the weighting function emphasizes different parts of the input function. When
is zero for all negative
,
depends only on values of
prior to time
, and the system is said to be
causal.
To understand why the convolution produces the output of an LTI system, let the notation
represent the function
with variable
and constant
. And let the shorter notation
represent
. Then a continuous-time system transforms an input function,
into an output function,
. And in general, every value of the output can depend on every value of the input. This concept is represented by:
where
is the transformation operator for time
. In a typical system,
depends most heavily on the values of
that occurred near time
. Unless the transform itself changes with
, the output function is just constant, and the system is uninteresting.
For a linear system,
must satisfy :
And the time-invariance requirement is:
In this notation, we can write the impulse response as
Similarly:
:
Substituting this result into the convolution integral:
which has the form of the right side of for the case
and
then allows this continuation:
In summary, the input function,
, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at . The system's linearity property allows the system's response to be represented by the corresponding continuum of impulse
responses, combined in the same way. And the time-invariance property allows that combination to be represented by the convolution integral.
The mathematical operations above have a simple graphical simulation.
Exponentials as eigenfunctions
An
eigenfunction is a function for which the output of the operator is a scaled version of the same function. That is,
where ''f'' is the eigenfunction and
is the
eigenvalue, a constant.
The
exponential functions
, where
, are
eigenfunctions of a
linear,
time-invariant operator. A simple proof illustrates this concept. Suppose the input is
. The output of the system with impulse response
is then
which, by the commutative property of
convolution, is equivalent to
where the scalar
is dependent only on the parameter ''s''.
So the system's response is a scaled version of the input. In particular, for any
, the system output is the product of the input
and the constant
. Hence,
is an
eigenfunction of an LTI system, and the corresponding
eigenvalue is
.
Direct proof
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.
Let's set
some complex exponential and
a time-shifted version of it.
by linearity with respect to the constant
.
by time invariance of
.
So
. Setting
and renaming we get:
i.e. that a complex exponential
as input will give a complex exponential of same frequency as output.
Fourier and Laplace transforms
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided
Laplace transform
is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form
where
and
). The
Fourier transform gives the eigenvalues for pure complex sinusoids. Both of
and
are called the ''system function'', ''system response'', or ''transfer function''.
The Laplace transform is usually used in the context of one-sided signals, i.e. signals that are zero for all values of ''t'' less than some value. Usually, this "start time" is set to zero, for convenience and without loss of generality, with the transform integral being taken from zero to infinity (the transform shown above with lower limit of integration of negative infinity is formally known as the
bilateral Laplace transform).
The Fourier transform is used for analyzing systems that process signals that are infinite in extent, such as modulated sinusoids, even though it cannot be directly applied to input and output signals that are not
square integrable. The Laplace transform actually works directly for these signals if they are zero before a start time, even if they are not square integrable, for stable systems. The Fourier transform is often applied to spectra of infinite signals via the
Wiener–Khinchin theorem even when Fourier transforms of the signals do not exist.
Due to the convolution property of both of these transforms, the convolution that gives the output of the system can be transformed to a multiplication in the transform domain, given signals for which the transforms exist
One can use the system response directly to determine how any particular frequency component is handled by a system with that Laplace transform. If we evaluate the system response (Laplace transform of the impulse response) at complex frequency , where , we obtain , ''H''(''s''), which is the system gain for frequency ''f''. The relative phase shift between the output and input for that frequency component is likewise given by arg(''H''(''s'')).
Examples
Important system properties
Some of the most important properties of a system are causality and stability. Causality is a necessity for a physical system whose independent variable is time, however this restriction is not present in other cases such as image processing.
Causality
A system is causal if the output depends only on present and past, but not future inputs. A necessary and sufficient condition for causality is
where
is the impulse response. It is not possible in general to determine causality from the
two-sided Laplace transform. However, when working in the time domain, one normally uses the
one-sided Laplace transform which requires causality.
Stability
A system is bounded-input, bounded-output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if every input satisfying
leads to an output satisfying
(that is, a finite
maximum absolute value of
implies a finite maximum absolute value of
), then the system is stable. A necessary and sufficient condition is that
, the impulse response, is in
L1 (has a finite L
1 norm):
In the frequency domain, the
region of convergence must contain the imaginary axis
.
As an example, the ideal
low-pass filter with impulse response equal to a
sinc function is not BIBO stable, because the sinc function does not have a finite L
1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for
and equal to a sinusoid at the
cut-off frequency for
, then the output will be unbounded for all times other than the zero crossings.
Discrete-time systems
Almost everything in continuous-time systems has a counterpart in discrete-time systems.
Discrete-time systems from continuous-time systems
In many contexts, a discrete time (DT) system is really part of a larger continuous time (CT) system. For example, a digital recording system takes an analog sound, digitizes it, possibly processes the digital signals, and plays back an analog sound for people to listen to.
In practical systems, DT signals obtained are usually uniformly sampled versions of CT signals. If
is a CT signal, then the
sampling circuit used before an
analog-to-digital converter will transform it to a DT signal:
where ''T'' is the
sampling period. Before sampling, the input signal is normally run through a so-called
Nyquist filter which removes frequencies above the "folding frequency" 1/(2T); this guarantees that no information in the filtered signal will be lost. Without filtering, any frequency component ''above'' the folding frequency (or
Nyquist frequency) is
aliased to a different frequency (thus distorting the original signal), since a DT signal can only support frequency components lower than the folding frequency.
Impulse response and convolution
Let
represent the sequence
And let the shorter notation
represent
A discrete system transforms an input sequence,
into an output sequence,
In general, every element of the output can depend on every element of the input. Representing the transformation operator by
, we can write:
Note that unless the transform itself changes with ''n'', the output sequence is just constant, and the system is uninteresting. (Thus the subscript, ''n''.) In a typical system, ''y''
'n''depends most heavily on the elements of ''x'' whose indices are near ''n''.
For the special case of the
Kronecker delta function,
the output sequence is the impulse response:
For a linear system,
must satisfy:
And the time-invariance requirement is:
In such a system, the impulse response,
, characterizes the system completely. That is, for any input sequence, the output sequence can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity:
which expresses
in terms of a sum of weighted delta functions.
Therefore:
where we have invoked for the case