Systems can be classified into following different categories in signals and systems because of their inherent properties:

- Order of the system
- Causal and non-causal systems
- Linear and Non-Linear Systems
- Fixed and Time-Varying Systems
- Lumped and Distributed parameter Systems
- Continuous-time and Discrete-time Systems
- Instantaneous and dynamic systems

Before proceeding to more detailed consideration of methods for solving the equations corresponding to the mathematical models of systems, it is essential to define more precisely some of the terms used in describing and specifying the systems. To this end it is useful to consider a very general mathematical model that encompasses a wide range of systems. Such a model for a system having an input x (t) and output y (t) is

\[\begin{matrix} {{a}_{n}}(t)\frac{{{d}^{n}}y}{d{{t}^{n}}}+{{a}_{n-1}}(t)\frac{{{d}^{n-1}}y}{d{{t}^{n-1}}}+\cdots +{{a}_{1}}(t)\frac{dy}{dt}+{{a}_{o}}(t)y= & {} & {} \\ {} & {} & \text{(1)} \\ {{b}_{m}}(t)\frac{{{d}^{m}}x}{d{{t}^{m}}}+{{b}_{m-1}}(t)\frac{{{d}^{m-1}}x}{d{{t}^{m-1}}}+\cdots +{{b}_{1}}(t)\frac{dx}{dt}+{{b}_{o}}(t)x & {} & {} \\\end{matrix}\]

The mathematical model given above expresses the behavior of the system in terms of a single nth-order differential equation.

**Order of the system**

The differential equation (1) is of the nth order since this is the highest-order derivative of the response to appear. The corresponding system is said to be nth order also.

**Causal and non-causal systems**

A causal system is one whose present response does not depend on future values of the input.

A non-causal system is one for which this condition is not assumed. Non-causal systems do not exist in the real world but can be approximated by the use of time delay, and they frequently occur in system analysis problems.

**Linear and Non-Linear Systems**

The system equation (1) represents a linear system , since all derivatives of the excitation (input) and response are raised to the first power only, and since there are no products of derivatives. One of the most important consequences of linearity is that superposition applies. In fact, this may be used as a definition of linearity. Specifically, if

${{y}_{1}}(t)=system\text{ }response\text{ }to\text{ }{{x}_{1}}(t)$

${{y}_{2}}(t)=system\text{ }response\text{ }to\text{ }{{x}_{2}}(t)$

And if

$a{{y}_{1}}(t)+b{{y}_{2}}(t)=system\text{ }response\text{ }to\text{ a}{{\text{x}}_{\text{1}}}\text{(t)+b}{{\text{x}}_{\text{2}}}\text{(t)}$

For all a, b, x_{1}(t), and x_{2}(t), then the system is linear. If this is not true, then the system is not linear.

In the case of nonlinear systems, it is not possible to write a general differential equation of finite order that can be used as the mathematical model for all systems. This is because there are many different ways in which nonlinearities can arise, and they cannot be all described mathematically in the same form. It is also important to remember that superposition does not apply in nonlinear systems.

A linear system usually results if none of the components in the system changes its characteristics as a function of the magnitude of the excitation (input) applied to it. In the case of an electrical system, this means that resistors, inductors, and capacitors do not change their values as the voltages across them or the currents through them change.

**Fixed and Time-Varying Systems**

Equation (1), as written, represents a time-varying system since the coefficients a_{i}(t) and b_{j}(t) are indicated as being functions of time. The analysis of time-varying devices is difficult since differential equations with non-constant coefficients cannot be solved except in special cases. The systems of greatest concern for the present discussion are characterized by a differential equation having constant coefficients. Such a system is known as fixed, time-invariant or stationary.

Fixed systems usually result when the physical components in the system, and the configuration in which they are connected, do not change with time. Most systems that are not exposed to a changing environment can be considered fixed unless they have been deliberately designed to be time-varying.

Time-varying system results when any of its components, or their manner of connection, do change with time. In many cases, this change is a result of environmental conditions.

**Lumped and Distributed parameter Systems**

Equation (1) represents a lumped-parameter system by virtue of being an ordinary differential equation. The implication of this designation is that the physical size of the system is of no concern since excitations (inputs) propagate through the system instantaneously. The assumption is usually valid if the largest physical dimension of the system is small compared with the wavelength of the highest significant frequency considered.

A distributed-parameter system is represented by a partial differential equation and generally has dimensions that are not small compared with the shortest wavelength of interest. Transmission lines, waveguides, antennas, and microwave tubes are typical examples of distributed-parameter electrical systems.

**Continuous-time and Discrete-time Systems**

Equation (1) represents a continuous time system by the virtue of being a differential equation rather than a difference equation. That is, the inputs and outputs are defined for all values of time rather than just for discrete values of time. Since time itself is inherently continuous, all physical systems are actually continuous-time systems.

However, there are situations in which one is interested solely in what happens at certain discrete instants of time. In many of these cases, the system contains a digital computer, which is performing certain specified computations and producing its answers at discrete time instants. If no change (in input or output) takes place between instants, then system analysis is simplified by considering the system to be discrete-time and having a mathematical model that is a difference equation.

**Instantaneous and dynamic systems**

An instantaneous is one in which the response at time t_{1} depends only upon the excitation (input) at time t_{1} and not upon any future or past values of the excitation. This may also be called a zero-memory or memoryless system. A typical example is a resistance network or nonlinear device without energy storage.

If the response does depend on past values of the excitation (input), then the system is said to be dynamic and to have memory. Any system that contains at least two different types of elements, one of which can store energy, is dynamic.

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