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 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 consisting of resistors, capacitors, inductors and linear amplifiers.[1]
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.
The defining properties of any LTI system are linearity and time invariance.
x(t)
y(t)
a
ax(t)
ay(t)
x'(t)
y'(t)
x(t)+x'(t)
y(t)+y'(t)
a
x(t)
x'(t)
x(t)
y(t)
x(t-T)
y(t-T)
The fundamental result in LTI system theory is that any LTI system can be characterized entirely by a single function called the system's impulse response. The output of the system
y(t)
x(t)
h(t)
yi=xi*hi
LTI systems can also be characterized in the frequency domain by the system's transfer function, which is the Laplace transform of the system's impulse response (or Z transform in the case of discrete-time systems). 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 transform 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. This is, if the input to a system is the complex waveform
Asest
As
s
Bsest
Bs
Bs/As
s
Since sinusoids are a sum of complex exponentials with complex-conjugate frequencies, if the input to the system is a sinusoid, then the output of the system will also be a sinusoid, perhaps with a different amplitude and a different phase, 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 (linear shift-invariant) 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.
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:[2]
y(t)=(x*h)(t) | l{\stackrel{def | |||||||
=
x(\tau) ⋅ h(t-\tau)d\tau, |
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:
h(t-\tau) | l{\stackrel{def | |
=Ot\{\delta(u-\tau); u\}. |
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.[3]
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
λ
The exponential functions
Aes
A,s\inC
x(t)=Aes
h(t)
where the scalaris dependent only on the parameter s.
So the system's response is a scaled version of the input. In particular, for any
A,s\inC
Aest
H(s)
Aes
H(s)
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.
Let's set
v(t)=ei
va(t)=ei
H[va](t)=ei\omegaH[v](t)
ei
H[va](t)=H[v](t+a)
H
So
H[v](t+a)=eiH[v](t)
t=0
ei
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The one-sided Laplace transformis exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids (i.e., exponential functions of the form
ej
\omega\inR
jl{\stackrel{def
H(j\omega)=l{F}\{h(t)\}
H(s)
H(j\omega)
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)).
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.
See main article: Causal system. 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
h(t)
See main article: BIBO 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
x(t)
y(t)
h(t)
In the frequency domain, the region of convergence must contain the imaginary axis
s=j\omega
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 L1 norm. Thus, for some bounded input, the output of the ideal low-pass filter is unbounded. In particular, if the input is zero for
t<0
t>0
Almost everything in continuous-time systems has a counterpart in discrete-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
x(t)
Let
\{x[m-k]; m\}
\{x[m-k];forallintegervaluesofm\}.
And let the shorter notation
\{x\}
\{x[m]; m\}.
A discrete system transforms an input sequence,
\{x\}
\{y\}.
O
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,
x[m]=\delta[m],
For a linear system,
O
And the time-invariance requirement is:
In such a system, the impulse response,
\{h\}
which expresses
\{x\}
Therefore:
where we have invoked for the case
ck=x[k]
xk[m]=\delta[m-k]
And because of, we may write:
Therefore:
y[n] | =
x[k] ⋅ h[n-k] | |||||||
=
x[n-k] ⋅ h[k], |
which is the familiar discrete convolution formula. The operator
On
An eigenfunction is a function for which the output of the operator is the same function, scaled by some constant. In symbols,
where f is the eigenfunction and
λ
The exponential functions
zn=esT
n\inZ
T\inR
z=esT, z,s\inC
Suppose the input is
x[n]=zn
h[n]
which is equivalent to the following by the commutative property of convolutionwhereis dependent only on the parameter z.
So
zn
H(z)
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The Z transform
is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids; i.e. exponentials of the form
ej
\omega\inR
zn
z=ej
H(ej)=l{F}\{h[n]\}
H(z)
H(ej\omega)
Like the one-sided Laplace transform, the Z transform is usually used in the context of one-sided signals, i.e. signals that are zero for t<0. The discrete-time Fourier transform Fourier series may be used for analyzing periodic signals.
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. That is,
Just as with the Laplace transform transfer function in continuous-time system analysis, the Z transform makes it easier to analyze systems and gain insight into their behavior.
The input-output characteristics of discrete-time LTI system are completely described by its impulse response
h[n]
See main article: Causal system. A discrete-time LTI system is causal if the current value of the output depends on only the current value and past values of the input.[4] A necessary and sufficient condition for causality iswhere
h[n]
See main article: BIBO stability. A system is bounded input, bounded output stable (BIBO stable) if, for every bounded input, the output is finite. Mathematically, if
implies that
(that is, if bounded input implies bounded output, in the sense that the maximum absolute values of
x[n]
y[n]
h[n]
In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus satisfying
|z|=1