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System analysis

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Field of electrical engineering

This article is about the field of electrical engineering. For the interdisciplinary field, seeSystems analysis.

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System analysis in the field ofelectrical 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 notablysignal processing,communication systems andcontrol systems.

Characterization of systems

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A system is characterized by how it responds to inputsignals. 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:

SISO – Single input, single output
SIMO – Single input, multiple outputs
MISO – Multiple inputs, single output
MIMO – 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 becontinuous ordiscrete 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 asanalog signals.
Signals that are discrete in time and discrete in value are known asdigital signals.
Signals that are discrete in time and continuous in value are calleddiscrete-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 oflogic circuits orPWM amplifiers, 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 ananalog system.
A system that has digital input and digital output is known as adigital 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 necessaryanalog-to-digital ordigital-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.
Systemswith memory do depend on past input.
Causal systems do not depend on any future input.
Non-causal oranticipatory 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 aslumped ordistributed. 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 islinear if it has the superposition and scaling properties. A system that is not linear isnon-linear.
If the output of a system does not depend explicitly on time, the system is said to betime-invariant; otherwise it istime-variant^[1]
A system that will always produce the same output for a given input is said to bedeterministic.
A system that will produce different outputs for a given input is said to bestochastic.

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

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Main article:Linear time-invariant system

As mentioned above, there are many methods of analysis developed specifically forLinear time-invariant systems (LTI systems). This is due to their simplicity of specification. AnLTI system is completely specified by itstransfer function (which is arational function for digital and lumped analog LTI systems). Alternatively, we can think of an LTI system being completely specified by itsfrequency response. A third way to specify an LTI system is by its characteristiclinear differential equation (for analog systems) or lineardifference equation (for digital systems). Which description is most useful depends on the application.

The distinction betweenlumped anddistributed LTI systems is important. A lumped LTI system is specified by a finite number of parameters, be it thezeros and poles of its transfer function, or thecoefficients of its differential equation, whereas specification of a distributed LTI system requires a completefunction, or partial differential equations.