Linear time-invariant system

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In system analysis, among other fields of study, a linear time-invariant system (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 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(t) ∗ h(t) where h(t) is called the system's impulse response and ∗ represents convolution (not to be confused with multiplication, as is frequently employed by the symbol in computer languages). 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.[2]

Block diagram illustrating the superposition principle and time invariance for a deterministic continuous-time SISO system. The system satisfies the superposition principle and is time-invariant if and only if for all time , for all real constants , and and for all inputs and .[1] Click image to expand it.

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[citation needed], and many other technical areas where systems of ordinary differential equations present themselves.

OverviewEdit

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.

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   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.

 
Relationship between the time domain and the frequency domain

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   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  .

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.

Continuous-time systemsEdit

Impulse response and convolutionEdit

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:[3]

   
        (using commutativity)

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 Eq.1:

 

 

 

 

 

(Eq.2)

And the time-invariance requirement is:

 

 

 

 

 

(Eq.3)

In this notation, we can write the impulse response as  

Similarly:

   
        (using Eq.3)

Substituting this result into the convolution integral:

 

which has the form of the right side of Eq.2 for the case   and  

Eq.2 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 Eq.1. 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.[4]

Exponentials as eigenfunctionsEdit

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 proofEdit

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 transformsEdit

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 s = , where ω = 2πf, 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)).

ExamplesEdit

  • A simple example of an LTI operator is the derivative.
    •     (i.e., it is linear)
    •     (i.e., it is time invariant)

    When the Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable s.

     
    That the derivative has such a simple Laplace transform partly explains the utility of the transform.
  • Another simple LTI operator is an averaging operator
     
    By the linearity of integration,
     
    it is linear. Additionally, because
     
    it is time invariant. In fact,   can be written as a convolution with the boxcar function  . That is,
     
    where the boxcar function
     

Important system propertiesEdit

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.

CausalityEdit

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.

StabilityEdit

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 L1 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 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   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.[dubious ]

Discrete-time systemsEdit

Almost everything in continuous-time systems has a counterpart in discrete-time systems.

Discrete-time systems from continuous-time systemsEdit

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 convolutionEdit

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:

 

 

 

 

 

(Eq.4)

And the time-invariance requirement is:

 

 

 

 

 

(Eq.5)

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 Eq.4 for the case   and  .

And because of Eq.5, we may write:

 

Therefore:

   
        (commutativity)

which is the familiar discrete convolution formula. The operator   can therefore be interpreted as proportional to a weighted average of the function x[k]. The weighting function is h[−k], simply shifted by amount n. As n changes, the weighting function emphasizes different parts of the input function. Equivalently, the system's response to an impulse at n=0 is a "time" reversed copy of the unshifted weighting function. When h[k] is zero for all negative k, the system is said to be causal.

Exponentials as eigenfunctionsEdit

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   is the eigenvalue, a constant.

The exponential functions  , where  , are eigenfunctions of a linear, time-invariant operator.   is the sampling interval, and  . A simple proof illustrates this concept.

Suppose the input is  . The output of the system with impulse response   is then

 

which is equivalent to the following by the commutative property of convolution

 
where
 
is dependent only on the parameter z.

So   is an eigenfunction of an LTI system because the system response is the same as the input times the constant  .

Z and discrete-time Fourier transformsEdit

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.[clarification needed] Of particular interest are pure sinusoids; i.e. exponentials of the form  , where  . These can also be written as   with  [clarification needed]. The discrete-time Fourier transform (DTFT)   gives the eigenvalues of pure sinusoids[clarification needed]. Both of   and   are called the system function, system response, or transfer function.

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.

ExamplesEdit

  • A simple example of an LTI operator is the delay operator  .
    •     (i.e., it is linear)
    •     (i.e., it is time invariant)

    The Z transform of the delay operator is a simple multiplication by z−1. That is,

     
  • Another simple LTI operator is the averaging operator
     
    Because of the linearity of sums,
     
    and so it is linear. Because,
     
    it is also time invariant.

Important system propertiesEdit

The input-output characteristics of discrete-time LTI system are completely described by its impulse response  . Two of the most important properties of a system are causality and stability. Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time. Unstable systems can also be analyzed and built, but are only useful as part of a larger system whose overall transfer function is stable.

CausalityEdit

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.[5] A necessary and sufficient condition for causality is

 
where   is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique[dubious ]. When a region of convergence is specified, then causality can be determined.

StabilityEdit

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   and   are finite), then the system is stable. A necessary and sufficient condition is that  , the impulse response, satisfies

 

In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus satisfying   for complex z).

NotesEdit

  1. ^ Bessai, Horst J. (2005). MIMO Signals and Systems. Springer. pp. 27–28. ISBN 0-387-23488-8.
  2. ^ Hespanha 2009, p. 78.
  3. ^ Crutchfield, p. 1. Welcome!
  4. ^ Crutchfield, p. 1. Exercises
  5. ^ Phillips 2007, p. 508.

See alsoEdit

ReferencesEdit

Further readingEdit

External linksEdit