System analysis

System analysis is the branch of electrical engineering that characterizes electrical systems and their properties. Although many of the methods of system analysis can be applied to non-electrical systems, it is a subject often studied by electrical engineers because it has direct relevance to many other areas of their discipline, most notably signal processing.

A system is characterized by how it responds to input 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:

• SISO (Single Input, Single Output)
• SIMO (Single Output, Multiple Outputs)
• MISO (Multiple Inputs, Single Output)
• MIMO (Multiple Inputs, Multiple Outputs)

It is often useful 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 or 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 signals.
• Signals that are discrete in time and discrete in value are known as digital signals.
• Signals that are discrete in time and continuous in value are called discrete-time signals. While important mathematically, systems that process discrete time signals are difficult to physically realize. The methods developed for analyzing discrete time signals and systems are usually applied to digital signals and systems.

A system can 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 an 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.
• Systems with memory do depend on past input.
• Causal systems do not depend on any future input.
• Non-causal systems do depend on future input. Note: It is not possible to physically realize a non-causal system. 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 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.

Yet another way to characterize systems is by certain properties which facilitate their analysis:

• A system is linear if it obeys the following relation: For any input signals x₁ and x₂ and their corresponding output signals y₁ and y₂, and any real constants α and β, the output corresponding to αx₁ + βx₂ is αy₁ + βy₂. In other words, adding two input signals together adds their outputs and multiplying an input signal by a constant multiplies the output by the same constant.
• 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. This may be expressed mathematically as follows: If the input signal x(t) produces an output y(t), then x(t + a) produces the output y(t + a).
• Any system for which the above relation does not hold is said to be time-varying.
• A system that will always produce the same output for a given input is said to be 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, 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. Time-invariance 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.

Some important concepts in system analysis are the transfer function, feedback and stability, frequency response, steady-state and transient behavior, filters, and noise.