Editing System analysis

{{Short description|Field of electrical engineering}}
{{About|the field of electrical engineering|the interdisciplinary field|Systems analysis}}
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{{prose|date=August 2016}}
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'''System analysis''' in the field of [[electrical engineering]] characterizes electrical systems and their properties.  System analysis can be used to represent almost anything from population growth to audio speakers; electrical engineers often use it because of its direct relevance to many areas of their discipline, most notably [[signal processing]], [[Telecommunications|communication systems]] and [[Control theory|control systems]].

== Characterization of systems ==
A system is characterized by how it responds to input [[Electrical signal|signals]].  In general, a system has one or more input signals and one or more output signals.  Therefore, one natural characterization of systems is by how many inputs and outputs they have:
* ''[[Single-input single-output system|SISO]]''{{dash}}Single input, single output
* ''SIMO''{{dash}}Single input, multiple outputs
* ''MISO''{{dash}}Multiple inputs, single output
* ''[[Multiple-input multiple-output system|MIMO]]''{{dash}}Multiple inputs, multiple outputs

It is often useful (or necessary) to break up a system into smaller pieces for analysis.  Therefore, we can regard a SIMO system as multiple SISO systems (one for each output), and similarly for a MIMO system.  By far, the greatest amount of work in system analysis has been with SISO systems, although many parts inside SISO systems have multiple inputs (such as adders).

Signals can be [[Continuous signal|continuous]] or [[discrete signal|discrete]] in time, as well as continuous or discrete in the values they take at any given time:
* Signals that are continuous in time and continuous in value are known as ''[[analog signal]]s''.
* Signals that are discrete in time and discrete in value are known as ''[[Digital signal (signal processing)|digital signal]]s''.
* Signals that are discrete in time and continuous in value are called ''discrete-time signals''. [[Switched capacitor]] systems, for instance, are often used in integrated circuits.  The methods developed for analyzing discrete time signals and systems are usually applied to digital and analog signals and systems.
* Signals that are continuous in time and discrete in value are sometimes seen in the timing analysis of [[Digital circuit|logic circuits]] or [[PWM amplifier]]s, but have little to no use in system analysis.

With this categorization of signals, a system can then be characterized as to which type of signals it deals with:
* A system that has analog input and analog output is known as an ''analog system''.
* A system that has digital input and digital output is known as a ''digital system''.
* Systems with analog input and digital output or digital input and analog output are possible.  However, it is usually easiest to break these systems up for analysis into their analog and digital parts, as well as the necessary [[analog-to-digital converter|analog-to-digital]] or [[digital-to-analog converter]].

Another way to characterize systems is by whether their output at any given time depends only on the input at that time or perhaps on the input at some time in the past (or in the future!).
* ''Memoryless'' systems do not depend on any past input. In common usage memoryless systems are also independent of future inputs. An interesting consequence of this is that the impulse response of any memoryless system is itself a scaled impulse.
* Systems ''with memory'' do depend on past input.
* ''Causal'' systems do not depend on any future input.
* ''Non-causal'' or ''anticipatory'' systems do depend on future input. 
*:Note: It is not possible to physically realize a non-causal system operating in "real time".  However, from the standpoint of analysis, they are important for two reasons.  First, the ideal system for a given application is often a noncausal system, which although not physically possible can give insight into the design of a derived causal system to accomplish a similar purpose.  Second, there are instances when a system does not operate in "real time" but is rather simulated "off-line" by a computer, such as post-processing an audio or video recording.
*:Further, some non-causal systems can operate in pseudo-real time by introducing lag: if a system depends on input for 1 second in future, it can process in real time with 1 second lag.

Analog systems with memory may be further classified as ''lumped'' or ''distributed''.  The difference can be explained by considering the meaning of memory in a system.  Future output of a system with memory depends on future input and a number of state variables, such as values of the input or output at various times in the past.  If the number of state variables necessary to describe future output is finite, the system is lumped; if it is infinite, the system is distributed.

Finally, systems may be characterized by certain properties which facilitate their analysis:
* A system is ''[[linear system|linear]]'' if it has the superposition and scaling properties. A system that is not linear is ''[[non-linear]]''.
* If the output of a system does not depend explicitly on time, the system is said to be [[time-invariant system|time-invariant]]; otherwise it is [[time-variant system|time-variant]]<ref>{{Cite book |last1=Oppenheim |first1=Alan |title=Signals and Systems |last2=Willsky |first2=Alan |last3=Nawab |first3=S. |date=1996-08-06 |publisher=Pearson |isbn=978-0-13-814757-0 |edition=2nd |location=Upper Saddle River, NJ |language=English}}</ref>
* A system that will always produce the same output for a given input is said to be [[deterministic system|deterministic]].
* A system that will produce different outputs for a given input is said to be [[stochastic]].

There are many methods of analysis developed specifically for linear time-invariant (''LTI'') deterministic systems.  Unfortunately, in the case of analog systems, none of these properties are ever perfectly achieved.  Linearity implies that operation of a system can be scaled to arbitrarily large magnitudes, which is not possible.  By definition of time-invariance, it is violated by aging effects that can change the outputs of analog systems over time (usually years or even decades).  [[Thermal noise]] and other random phenomena ensure that the operation of any analog system will have some degree of stochastic behavior.  Despite these limitations, however, it is usually reasonable to assume that deviations from these ideals will be small.

== LTI systems ==
{{main|Linear time-invariant system}}
As mentioned above, there are many methods of analysis developed specifically for [[LTI system theory|Linear time-invariant systems]] (LTI systems).  This is due to their simplicity of specification.  An [[LTI system theory|LTI system]] is completely specified by its [[transfer function]] (which is a [[rational function]] for digital and lumped analog LTI systems).  Alternatively, we can think of an LTI system being completely specified by its [[frequency response]].  A third way to specify an LTI system is by its characteristic [[linear differential equation]] (for analog systems) or linear [[difference equation]] (for digital systems).  Which description is most useful depends on the application.

The distinction between [[Lumped-element model|lumped]] and [[Distributed-element model|distributed]] LTI systems is important.  A lumped LTI system is specified by a finite number of parameters, be it the [[zeros and poles]] of its transfer function, or the [[coefficient]]s of its differential equation, whereas specification of a distributed LTI system requires a complete [[Function (mathematics)|function]], or partial differential equations.

== See also ==

=== Important concepts in system analysis ===

* [[Linear time-invariant system]] theory
* [[Filter (signal processing)|Filter]] theory and [[Filter design]]
* [[Impulse response]]
** [[Infinite impulse response]] systems
** [[Finite impulse response]] systems
* [[Step response]]
* Transforms:
** [[Laplace transform]]
** [[Fourier transform]]: [[Continuous Fourier transform]] & [[Discrete Fourier transform]]
** [[Z-transform]]
* [[Transfer function]]
* [[Frequency response]]
* [[Poles and zeros]]
* [[Bode plot]]s
* [[Minimum phase]] transfer functions
* [[Linear phase]]
* [[Ordinary differential equations]] and [[Difference equation]]s
* [[Feedback]]
* [[BIBO stability|Stability]]
* [[Causality]]
* [[Steady-state]] and [[Transient state|transient]] behavior
* [[Limit cycle]]

=== Related fields ===
* [[Control system]] and [[control theory]]
* [[Digital signal processing]]
* [[Digital image processing]]
* [[Telecommunications]]

==References==
{{Reflist}}

{{DEFAULTSORT:System Analysis}}
[[Category:Electrical engineering]]
[[Category:Electronic engineering]]
[[Category:Digital signal processing]]
[[Category:Control theory]]