- LTI system theory
**LTI system theory**or**linear time-invariant system theory**is a theory in the field ofelectrical engineering , specifically in circuits,signal processing , andcontrol theory , that investigates the response of a linear,time-invariant system to an arbitrary input signal. Though the standard independent variable is time, it could just as easily be space (as inimage processing and field theory) or some other coordinate. Thus an alternately used term is "linear translation-invariant". The term "linear shift-invariant" is the corresponding concept for adiscrete-time (sampled) system.**Overview**The defining properties of any linear time-invariant system are, of course, "linearity" and "time invariance":

* "Linearity" means that the relationship between the input and the output of the system satisfies the

superposition property . If the input to the system is the sum of two component signals:::$x(t)\; =\; c\_1\; x\_1(t)\; +\; c\_2\; x\_2(t)\; ,$:then the output of the system will be::$y(t)\; =\; c\_1\; y\_1(t)\; +\; c\_2\; y\_2(t)\; ,$:where $c\_1$ and $c\_2$ are constants, and $y\_k(t)$ is the output resulting from the sole input $x\_k(t)$.:It can be shown that, given this superposition property, the scaling property follows for any rational scalar. If the output due to input $x(t)$ is $y(t)$, then the output due to input $c\; x(t)$ is $c\; y(t).$

:Then, formally, a linear system is a system that exhibits the following property

**:**If the input of the system is::$x(t)\; =\; sum\_k\; c\_k\; x\_k(t)\; ,$:then the output of the system will be::$y(t)\; =\; sum\_k\; c\_k\; y\_k(t)\; ,$:for any constants $c\_k$ and where each $y\_k(t)$ is the output resulting from the sole input $x\_k(t)$.* "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 the "T" seconds. If the output due to input $x(t)$ is $y(t)$, then the output due to input $x(t-T)$ is $y(t-T)$. More specifically, an input affected by a time delay should effect a corresponding time delay in the output, hence time-invariant.

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 theconvolution of the input to the system with the system's impulse response. This method of analysis is often called the "time domain " point-of-view. The same result is true of discrete-time linear shift-invariant systems in which signals are discrete-time samples, and convolution is defined on sequences.Equivalently, any LTI system can be characterized in the "

frequency domain " by the system'stransfer function , which is theLaplace transform of the system's impulse response (orZ 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

eigenfunction s, and the basis functions of the transforms, are complex exponentials. This is, if the input to a system is the complex waveform $Aexp(\{st\})$ for some complex amplitude $A$ and complex frequency $s$, the output will be some complex constant times the input, say $Bexp(\{st\})$ for some new complex amplitude $B$. The ratio $B/A$ is the transfer function at frequency $s$.Because 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. 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 homogeneousdifferential equation with constant coefficients is an LTI system. Examples of such systems are electrical circuits made up ofresistor s,inductor s, andcapacitor s (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 bank s 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**Let the notation $\{x(u-\; au);\; u\}$ represent the function $x(u-\; au)$ with variable $u$ and constant $au$.

And let the shorter notation $\{x\},$ represent $\{x(u);\; u\}.$

A continuous-time system transforms an input function, $\{x\}$ into an output function, $\{y\}.$ In general, every value of the output can depend on every value of the input. Representing the transformation operator by $O$, we can write

**:**:$y(t)\; stackrel\{mathrm\{def\{=\}\; O\_t\{x\}.$ Note that unless the transform itself changes with

**t**, the output function is just constant, and the system is uninteresting. (Thus the subscript,**t**.) In a typical system,**y(t)**depends most heavily on the values of**x**that occurred near time**t**.For the special case of the

Dirac delta function , $x(t)\; =\; delta(t),$ the output function is the**impulse response:**:$h(t)\; stackrel\{mathrm\{def\{=\}\; O\_t\{delta(u);\; u\}.,$

For a linear system, $O$ must satisfy the relations

**:**:$O\_t\{x\_1(u)\; +\; x\_2(u);\; u\}\; =\; O\_t\{x\_1\}\; +\; O\_t\{x\_2\},$and

**:**:$O\_t\{lambda\; x(u);\; u\}\; =lambda\; O\_t\{x\}.,$

The time-invariance requirement is

**:**:$egin\{align\}O\_t\{x(u-\; au);\; u\}\; =\; y(t-\; au)\backslash stackrel\{mathrm\{def\{=\}\; O\_\{t-\; au\}\{x\}.,end\{align\}$In such a system, the impulse response, $\{h\},,$ characterizes the system completely. I.e., for any input function, the output function can be calculated in terms of the input and the impulse response. To see how that is done, consider the identity

**:**:$x(u)\; equiv\; int\_\{-infty\}^\{infty\}\; x(\; au)cdot\; delta(u-\; au)\; d\; au,,$

which is the "sifting property" of the delta function.

Therefore

**:**:$egin\{align\}y(t)\; =\; O\_t\{x\}\backslash =\; O\_tleft\{int\_\{-infty\}^\{infty\}\; x(\; au)cdot\; delta(u-\; au)\; d\; au;\; u\; ight\}.,end\{align\}$

The linearity condition allows this manipulation

**:**:$y(t)\; =\; int\_\{-infty\}^\{infty\}\; x(\; au)cdot\; O\_t\{delta(u-\; au);\; u\}\; d\; au.,$

And because of time-invariance, we may write

**:**:$egin\{align\}O\_t\{delta(u-\; au);\; u\}\; =O\_\{t-\; au\}\{delta(u);\; u\}\backslash stackrel\{mathrm\{def\{=\}\; h(t-\; au).,end\{align\}$

Therefore

**:**:

which is the familiar

. The operator $O\_t,$ can therefore be interpreted as proportional to a weighted average of the function $x(\; au).,$ The weighting function is $h(-\; au),,$ simply shifted by amount $t,.$ As $t,$ changes, the weighting function emphasizes different parts of the input function. Equivalently, the system's response to an impulse at $t=0$ is a time-reversed copy of the unshifted weighting function. When $h(\; au),$ is zero for all negative $au,,$ the system is said to be causal.convolution integral**Exponentials as eigenfunctions**An

eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols,:$mathcal\{H\}f\; =\; lambda\; f$,where "f" is the eigenfunction and $lambda$ is theeigenvalue , a constant.The

exponential function s $e^\{s\; t\}$, where $s\; in\; mathbb\{C\}$, areeigenfunction s of alinear ,time-invariant operator. A simple proof illustrates this concept.Suppose the input is $x(t)\; =\; e^\{s\; t\}$. The output of the system with impulse response $h(t)$ is then

:$int\_\{-infty\}^\{infty\}\; h(t\; -\; au)\; e^\{s\; au\}\; d\; au$

which is equivalent to the following by the commutative property of

convolution :$int\_\{-infty\}^\{infty\}\; h(\; au)\; ,\; e^\{s\; (t\; -\; au)\}\; ,\; d\; au$:$quad\; =\; e^\{s\; t\}\; int\_\{-infty\}^\{infty\}\; h(\; au)\; ,\; e^\{-s\; au\}\; ,\; d\; au$:$quad\; =\; e^\{s\; t\}\; H(s)$,where:$H(s)\; =\; int\_\{-infty\}^infty\; h(t)\; e^\{-s\; t\}\; d\; t$is dependent only on the parameter "s".

So, $e^\{s\; t\}$ is an

eigenfunction of an LTI system because the system response is the same as the input times the constant $H(s)$.**Fourier and Laplace transforms**The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The

Laplace transform :$H(s)\; =\; mathcal\{L\}\{h(t)\}\; =\; int\_\{-infty\}^infty\; h(t)\; e^\{-s\; t\}\; d\; t$

is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids, i.e. exponentials of the form $exp(\{j\; omega\; t\})$ where $omega\; in\; mathbb\{R\}$ and $j\; =\; sqrt\{-1\}$. These are generally called complex exponentials even though the argument is purely imaginary. The

Fourier transform $H(j\; omega)\; =\; mathcal\{F\}\{h(t)\}$ gives the eigenvalues for pure complex sinusoids. Both of $H(s)$ and $H(jomega)$ 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 can not 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 theWiener–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:$y(t)\; =\; (h*x)(t)\; =\; int\_\{-infty\}^infty\; h(t\; -\; au)\; x(\; au)\; d\; au$:$quad\; =\; mathcal\{L\}^\{-1\}\{H(s)X(s)\}.$

Not only is it often easier to do the transforms, multiplication, and inverse transform than the original convolution, but one can also gain insight into the behavior of the system from the system response. One can look at the modulus of the system function |"H"("s")| to see whether the input $exp(\{s\; t\})$ is "passed" (let through) the system or "rejected" or "attenuated" by the system (not let through).

**Examples**A simple example of an LTI operator is the

derivative ::$frac\{d\}\{dt\}\; left(\; c\_1\; x\_1(t)\; +\; c\_2\; x\_2(t)\; ight)\; =\; c\_1\; x\text{'}\_1(t)\; +\; c\_2\; x\text{'}\_2(t),$

:$frac\{d\}\{dt\}\; x(t-\; au)\; =\; x\text{'}(t-\; au).$

When the Laplace transform of the derivative is taken, it transforms to a simple multiplication by the Laplace variable s.:$mathcal\{L\}left\{frac\{d\}\{dt\}x(t)\; ight\}\; =\; s\; X(s)$That the derivative has such a simple Laplace transform partly explains the utility of the transform.

Another simple LTI operator is an averaging operator

:$mathcal\{A\}left\{x(t)\; ight\}\; =\; int\_\{t-a\}^\{t+a\}\; x(lambda)\; d\; lambda$.

It is linear because of the linearity of integration

:$mathcal\{A\}left\{c\_1\; x\_1(t)\; +\; c\_2\; x\_2(t)\; ight\}$

:$=\; int\_\{t-a\}^\{t+a\}\; left(\; c\_1\; x\_1(lambda)\; +\; c\_2\; x\_2(lambda)\; ight)\; d\; lambda$

:$=\; c\_1\; int\_\{t-a\}^\{t+a\}\; x\_1(lambda)\; d\; lambda\; +\; c\_2\; int\_\{t-a\}^\{t+a\}\; x\_2(lambda)\; d\; lambda$

:$=\; c\_1\; mathcal\{A\}left\{x\_1(t)\; ight\}\; +\; c\_2\; mathcal\{A\}left\{x\_2(t)\; ight\}$.

It is time invariant too

:$mathcal\{A\}left\{x(t-\; au)\; ight\}$

:$=\; int\_\{t-a\}^\{t+a\}\; x(lambda-\; au)\; d\; lambda$

:$=\; int\_\{(t-\; au)-a\}^\{(t-\; au)+a\}\; x(xi)\; d\; xi$

:$=\; mathcal\{A\}\{x\}(t-\; au)$.

Indeed, $mathcal\{A\}$ can be written as a convolution with the box function $Pi(t)$.

:$mathcal\{A\}left\{x(t)\; ight\}\; =\; int\_\{-infty\}^infty\; Pileft(frac\{lambda-t\}\{2a\}\; ight)\; x(lambda)\; d\; lambda$,

where the box function is

:$Pi(t)\; =\; left\{\; egin\{matrix\}\; 1\; |t|\; 1/2\; \backslash \; 0\; |t|\; 1/2\; end\{matrix\}\; ight.$.

**Important system properties**Some of the most important properties of a system are causality and stability. Causality is a necessity if the independent variable is time, but not all systems have time as an independent variable. For example, a system that processes still images does not need to be causal. Non-stable systems can be built and can be useful in many circumstances. Even non-real systems can be built and are very useful in many contexts.

**Causality**A system is causal if the output depends only on present and past inputs. A necessary and sufficient condition for causality is

:$h(t)\; =\; 0\; quad\; forall\; t\; <\; 0,$

where $h(t)$ is the impulse response. It is not possible in general to determine causality from the Laplace transform, because the inverse transform is not unique. When a

region of convergence is specified, then causality can be determined.**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:$|x(t)|\_infty\; <\; infty$

leads to an output satisfying

:$|y(t)|\_infty\; <\; infty$

(that is, a finite maximum absolute value of $x(t)$ implies a finite maximum absolute value of $y(t)$), then the system is stable. A necessary and sufficient condition is that $h(t)$, the impulse response, is in L

^{1}(has a finite L^{1}norm)::$|h(t)|\_1\; =\; int\_\{-infty\}^infty\; |h(t)|\; dt\; <\; infty.$

In the frequency domain, the

region of convergence must contain the imaginary axis $s=jomega$.As an example, the ideal

low-pass filter with impulse response equal to asinc 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 $t\; <\; 0,$ and equal to a sinusoid at thecut-off frequency for $t\; >\; 0,$, 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.

Formally, the DT signals studied are almost always uniformly sampled versions of CT signals. If $x(t)$ is a CT signal, then an

analog to digital converter will transform it to the DT signal**:**:$x\; [n]\; stackrel\{mathrm\{def\{=\}\; x(nT)\; qquad\; forall\; ,\; n\; in\; mathbb\{Z\},$

where "T" is the sampling period. It is very important to limit the range of frequencies in the input signal for faithful representation in the DT signal, since then the

sampling theorem guarantees that no information about the CT signal is lost. A DT signal can only contain a frequency range of $1/(2T)$; other frequencies are aliased to the same range.**Impulse response and convolution**Let $\{x\; [m-k]\; ;\; m\}$ represent the sequence $\{x\; [m-k]\; ;\; mbox\{for\; all\; integer\; values\; of\; m\}\}$.

And let the shorter notation $\{x\},$ represent $\{x\; [m]\; ;\; m\}.$

A discrete system transforms an input sequence, $\{x\}$ into an output sequence, $\{y\}.$ In general, every element of the output can depend on every element of the input. Representing the transformation operator by $O$, we can write

**:**:$y\; [n]\; stackrel\{mathrm\{def\{=\}\; O\_n\{x\}.$ 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 , $x\; [m]\; =\; delta\; [m]\; ,$ the output sequence is the**impulse response:**:$h\; [n]\; stackrel\{mathrm\{def\{=\}\; O\_n\{delta\; [m]\; ;\; m\}.,$

For a linear system, $O$ must satisfy the relations

**:**:$O\_n\{x\_1\; [m]\; +\; x\_2\; [m]\; ;\; m\}\; =\; O\_n\{x\_1\}\; +\; O\_n\{x\_2\},$and

**:**:$O\_n\{lambda\; x\; [m]\; ;\; m\}\; =lambda\; O\_n\{x\}.,$

The time-invariance requirement is

**:**:$egin\{align\}O\_n\{x\; [m-k]\; ;\; m\}\; =\; y\; [n-k]\; \backslash stackrel\{mathrm\{def\{=\}\; O\_\{n-k\}\{x\}.,end\{align\}$In such a system, the impulse response, $\{h\},,$ characterizes the system completely. I.e., 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

**:**:$x\; [m]\; equiv\; sum\_\{k=-infty\}^\{infty\}\; x\; [k]\; cdot\; delta\; [m-k]\; .,$

which is the "sifting property" of the delta function.

Therefore

**:**:$egin\{align\}y\; [n]\; =\; O\_n\{x\}\backslash =\; O\_nleft\{sum\_\{k=-infty\}^\{infty\}\; x\; [k]\; cdot\; delta\; [m-k]\; ;\; m\; ight\}.,end\{align\}$

The linearity condition allows this manipulation

**:**:$y\; [n]\; =\; sum\_\{k=-infty\}^\{infty\}\; x\; [k]\; cdot\; O\_n\{delta\; [m-k]\; ;\; m\}.,$

And because of time-invariance, we may write

**:**:$egin\{align\}O\_n\{delta\; [m-k]\; ;\; m\}\; =O\_\{n-k\}\{delta\; [m]\; ;\; m\}\backslash stackrel\{mathrm\{def\{=\}\; h\; [n-k]\; .,end\{align\}$

Therefore

**:**:

which is the familiar discrete convolution formula. The operator $O\_n,$ 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 eigenfunctions**An

eigenfunction is a function for which the output of the operator is the same function, just scaled by some amount. In symbols,:$mathcal\{H\}f\; =\; lambda\; f$,where "f" is the eigenfunction and $lambda$ is theeigenvalue , a constant.The

exponential function s $z^n\; =\; e^\{sT\; n\}$, where $n\; in\; mathbb\{Z\}$, areeigenfunction s of alinear ,time-invariant operator. $T\; in\; mathbb\{R\}$ is the sampling interval, and $z\; =\; e^\{sT\},\; z,s\; in\; mathbb\{C\}$. A simple proof illustrates this concept.Suppose the input is $x\; [n]\; =\; ,!z^n$. The output of the system with impulse response $h\; [n]$ is then

:$sum\_\{m=-infty\}^\{infty\}\; h\; [n-m]\; ,\; z^m$

which is equivalent to the following by the commutative property of

convolution :$sum\_\{m=-infty\}^\{infty\}\; h\; [m]\; ,\; z^\{(n\; -\; m)\}$:$quad\; =\; z^n\; sum\_\{m=-infty\}^\{infty\}\; h\; [m]\; ,\; z^\{-m\}$:$quad\; =\; z^n\; H(z)$,where:$H(z)\; =\; sum\_\{m=-infty\}^infty\; h\; [m]\; z^\{-m\}$is dependent only on the parameter "z".

So, $z^n$ is an

eigenfunction of an LTI system because the system response is the same as the input times the constant $H(z)$.**Z and discrete-time Fourier transforms**The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems. The

Z transform :$H(z)\; =\; mathcal\{Z\}\{h\; [n]\; \}\; =\; sum\_\{n=-infty\}^infty\; h\; [n]\; z^\{-n\}$is exactly the way to get the eigenvalues from the impulse response. Of particular interest are pure sinusoids, i.e. exponentials of the form $e^\{j\; omega\; n\}$, where $omega\; in\; mathbb\{R\}$. These can also be written as $z^n$ with $z\; =\; e^\{j\; omega\}$. These are generally called complex exponentials even though the argument is purely imaginary.TheDiscrete-time Fourier transform (DTFT) $H(e^\{j\; omega\})\; =\; mathcal\{F\}\{h\; [n]\; \}$gives the eigenvalues of pure sinusoids. Both of $H(z)$ and $H(e^\{jomega\})$ are called the "system function", "system response", or "transfer function"'.The Z 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. The Fourier transform is used for analyzing signals that are infinite in extent.

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.:$y\; [n]\; =\; (h*x)\; [n]\; =\; sum\_\{m=-infty\}^infty\; h\; [n-m]\; x\; [m]$:$quad\; =\; mathcal\{Z\}^\{-1\}\{H(z)X(z)\}$

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. One can look at the modulus of the system function "|H(z)|" to see whether the input $z^n$ is "passed" (let through) by the system, or "rejected" or "attenuated" by the system (not let through).

**Examples**A simple example of an LTI operator is the delay operator $D\{x\}\; [n]\; :=x\; [n-1]$.

:$D\; left(\; c\_1\; x\_1\; [n]\; +\; c\_2\; x\_2\; [n]\; ight)\; =\; c\_1\; x\_1\; [n-1]\; +\; c\_2\; x\_2\; [n-1]\; =\; c\_1\; Dx\_1\; [n]\; +\; c\_2\; Dx\_2\; [n]\; ,$

:$D\{x\; [n-m]\; \}\; =\; x\; [n-m-1]\; =\; x\; [(n-1)-m]\; =\; D\{x\}\; [n-m]\; .\; ,$

When the Z transform of the delay operator is taken, it transforms to a simple multiplication by z

^{-1}::$mathcal\{Z\}left\{Dx\; [n]\; ight\}\; =\; z^\{-1\}\; X(z).$

That the delay operator has such a simple Z transform partly explains the utility of the transform.

Another simple LTI operator is an averaging operator

:$mathcal\{A\}left\{x\; [n]\; ight\}\; =\; sum\_\{k=n-a\}^\{n+a\}\; x\; [k]$.

It is linear because of the linearity of sums:

:$mathcal\{A\}left\{c\_1\; x\_1\; [n]\; +\; c\_2\; x\_2\; [n]\; ight\}$

:$=\; sum\_\{k=n-a\}^\{n+a\}\; left(\; c\_1\; x\_1\; [k]\; +\; c\_2\; x\_2\; [k]\; ight)$

:$=\; c\_1\; sum\_\{k=n-a\}^\{n+a\}\; x\_1\; [k]\; +\; c\_2\; sum\_\{k=n-a\}^\{n+a\}\; x\_2\; [k]$

:$=\; c\_1\; mathcal\{A\}left\{x\_1\; [n]\; ight\}\; +\; c\_2\; mathcal\{A\}left\{x\_2\; [n]\; ight\}$.

It is time invariant too:

:$mathcal\{A\}left\{x\; [n-m]\; ight\}$

:$=\; sum\_\{k=n-a\}^\{n+a\}\; x\; [k-m]$

:$=\; sum\_\{k\text{'}=(n-m)-a\}^\{(n-m)+a\}\; x\; [k\text{'}]$

:$=\; mathcal\{A\}left\{x\; ight\}\; [n-m]$.

**Important system properties**Some of the most important properties of a system are causality and stability. Unlike CT systems, non-causal DT systems can be realized. It is trivial to make an acausal FIR system causal by adding delays. It is even possible to make acausal

IIR systems (See Vaidyanathan and Chen, 1995). Non-stable systems can be built and can be useful in many circumstances. Even non-real systems can be built and are very useful in many contexts.**Causality**A system is causal if the output depends only on present and past inputs. A necessary and sufficient condition for causality is

:$h\; [n]\; =\; 0\; forall\; n\; <\; 0,$

where $h\; [n]$ is the impulse response. It is not possible in general to determine causality from the Z transform, because the inverse transform is not unique. When a

region of convergence is specified, then causality can be determined.**Stability**A system is

**bounded input, bounded output stable**(BIBO stable) if, for every bounded input, the output is finite. Mathematically, if:$||x\; [n]\; ||\_infty\; <\; infty$

implies that

:$||y\; [n]\; ||\_infty\; <\; infty$

(that is, if bounded input implies bounded output, in the sense that the maximum absolute values of $x\; [n]$ and $y\; [n]$ are finite), then the system is stable. A necessary and sufficient condition is that $h\; [n]$, the impulse response, satisfies

:$||h\; [n]\; ||\_1\; =\; sum\_\{n\; =\; -infty\}^infty\; |h\; [n]\; |\; <\; infty.$

In the frequency domain, the

region of convergence must contain the unit circle $|z|=1$.**Notes****See also***

circulant matrix

*frequency response

*impulse response

*system analysis

*Green function **References***Boaz Porat: "A Course in Digital Signal Processing", Wiley, ISBN 0471149616

* cite journal

author=P. P. Vaidyanathan and T. Chen

title=Role of anticausal inverses in multirate filter banks -- Part I: system theoretic fundamentals

journal=IEEE Trans. Signal Proc.

month=May

year=1995

doi=10.1109/78.382395

volume=43

pages=1090* cite journal

author=P. P. Vaidyanathan and T. Chen

title=Role of anticausal inverses in multirate filter banks -- Part II: the FIR case, factorizations, and biorthogonal lapped transforms

journal=IEEE Trans. Signal Proc.

month=May

year=1995

doi=10.1109/78.382396

volume=43

pages=1103

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2010.*