Linear time-invariant system
A good example of an LTI system is any electrical circuit consisting of resistors, capacitors, inductors and linear amplifiers.
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.
The defining properties of any LTI system are linearity and time invariance.
Similarly, a discrete-time linear time-invariant (or, more generally, "shift-invariant") system is defined as one operating in discrete time:
For all LTI systems, the eigenfunctions, and the basis functions of the transforms, are complex exponentials.
Most LTI systems are considered "easy" to analyze, at least compared to the time-varying and/or nonlinear case.
Most LTI system concepts are similar between the continuous-time and discrete-time (linear shift-invariant) cases.
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.
must satisfy Eq.1: And the time-invariance requirement is: In this notation, we can write the impulse response as
, can be represented by a continuum of time-shifted impulse functions, combined "linearly", as shown at Eq.1.
is dependent only on the parameter s. So the system's response is a scaled version of the input.
It is also possible to directly derive complex exponentials as eigenfunctions of LTI systems.
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems.
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.
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.
A system is causal if the output depends only on present and past, but not future inputs.
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.
Thus, for some bounded input, the output of the ideal low-pass filter is unbounded.
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,
Equivalently, the system's response to an impulse at n=0 is a "time" reversed copy of the unshifted weighting function.
The eigenfunction property of exponentials is very useful for both analysis and insight into LTI systems.
[clarification needed] Of particular interest are pure sinusoids; i.e. exponentials of the form
That is, The input-output characteristics of discrete-time LTI system are completely described by its impulse response
Non-causal (in time) systems can be defined and analyzed as above, but cannot be realized in real-time.
In the frequency domain, the region of convergence must contain the unit circle (i.e., the locus satisfying
