Linear systems: general

Principle

System analysis is the branch of engineering, mostly electrical engineering that characterizes electrical systems and their properties. Many of the methods of system analysis can be applied to non-electrical systems, e.g. mechanical and acoustic systems and in the theory of flow of liquids.

The most simple and by far most applied linear system in calculations and in technology (electronic filters etc.) is the Linear first order system. It is very basic and its understanding is a necessity to understand the working of more complicated systems. More complicated systems (filters) are often a combination of several linear first order systems (see the mathematically more advanced contribution  System analysis).

Linear systems can be distinguished in many ways.

Input and output

A system is characterized by how it responds to input signals. A system can have one or more input signals and one or more output signals. Therefore, a system can be characterized by its number of inputs and outputs. Single input with single output is most common. Another possibility is for instance  multiple inputs (e.g. visual and auditory) and single output (eye blink). Here and in other contributions about system analysis, a system is supposed to be a single-input-single-output system. By far, the greatest amount of work in system analysis has been done with these systems, although many parts (e.g. a resistance or capacity in an electric circuit) have multiple inputs.

It is often useful (or necessary) to break up a system into smaller pieces for analysis (see System analysis).

Since in physics and engineering systems process signals, properties of signals should first be defined before continuing the description of systems.

Analog and digital, continuous and discrete

Signals can be continuous (e.g. the EEG) or discrete in time (the time series of R peaks in the ECG or by approximation an action potential), as well as 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.

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 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!).

Memory and causality

Memory-less systems do not depend on any past input and 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. It is not possible to physically realize a non-causal system operating in "real time". However, they can be simulated "off-line" by a computer. They can give insight into the design of a causal system.

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 coefficients (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. In practice, most system in science as well as in medicine can be considered or approximated as lumped.

Linear and non-linear, superposition

A system is linear if the superposition principle holds. This means:

input  → system → output

   a     → system →    A

   a     → system →    B

 a+b   → system →  A+B

The principle also implies the scaling property: multiplying input a with k yields as output kA.

Further, it holds that when the input is a single sine wave with frequency f, the output solely comprises a sine wave with the same frequency f.

A linear system can be described by a linear differential equation. A system that behaves not linear is non-linear.

Time invariant and time variant

If the output of a system with equal input does not depend explicitly on time (hour of day, season etc.), the system is said to be time-invariant; otherwise it is time-variant. So, time invariant systems have time independent properties. But when, in terms of  for instance an electric system one or more parts (e.g. a resistance) behaves noisy or changes otherwise in time (e.g. with ambient temperature, e.g. the thermistor, a temperature dependent resistor) then the system is time variant. Time-invariance is violated by aging effects that can change the outputs of analog systems over time (usually years or even decades).

In general, system theory considers linear time-invariant systems, the LTI systems.

Biological systems, described in terms of linear system analysis, are usually considered to be time-invariant when the time scale of investigation is restricted.

Deterministic and stochastic

An LTI system that will always exactly produce the same output for a given input is said to be deterministic. There are many methods of analysis developed specifically for LTI deterministic systems.

An  LTI system that will produce slightly different outputs for a given input is said to be stochastic. Of coarse the average of many outputs should be the same as the output of its deterministic version. Its stochastic nature is caused by properties (e.g. a resistance or capacitor) that change in some unpredictable way in time. This means that the coefficients of the underlying differential equation are not constants. Thermal noise and other random phenomena ensure that the operation of any analog system, in fact every biological 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.

The electric analogue

LTI systems are more easy to handle when they are studied as their electric analogues. To study them they are split, if possible,  in a number of basic systems, be it first and second order systems. They can be arranged in parallel or serially.

First order systems always comprise an ideal resistance (the physical realization is an Ohmic resistor), and in addition one ideal capacitance (in practice a capacitor) or an ideal self-inductance (in practice a coil). Resistors are not ideal, they also have a small inductance and small capacity, and in the same way capacitors and coils are not ideal, but generally these imperfections can be ignored.

Second order systems comprise a resistance, capacitance and inductance. The electric behavior of these “building blocks” of linear systems is described in Electrical resistance, capacitance and inductance.   

Application

Applications can be found in all branches of science and technology, and consequently also in (bio)medicine, but also in econometrics and social sciences.

More Info

As mentioned above, there are many methods of analysis developed specifically for LTI systems. This is due to their simplicity of specification. An LTI system is completely specified by its transfer function (which is a rational function for digital and lumped analog LTI systems). The transfer function (H) is actually the solution of the linear differential equation (for analog systems) or linear difference equation (for digital systems). However this solution is a function of frequency. The solution in the time domain (h(τ)) is the alternative description. This solution is very useful for two specific input signals, the unit impulse function (infinite at time zero and zero at any other time, with integral 1) and its integral, the unit step function.

The transfer function is not a function of the real frequency ω, but of the imaginary frequency jω. So, H(jω) is a function in the complex jω plane. To simplify notation, often Laplace notation, so H(s), is used. From the transfer function the amplitude and phase characteristic can be derived. Together they give the frequency response. Which description, frequency or time domain, is most useful depends on the application.

The distinction between lumped and distributed LTI systems is important. A lumped LTI system is specified by a finite number of parameters, be it the zeros and poles of H(jω) (see System analysis) or the coefficients of its differential equation, whereas specification of a distributed LTI system requires a complete function (series expansion). Many bio-electric systems (brain, nerve axon) are actually distributed systems, but often they can be approximated by lumped systems.