Classification of systems

Continuous time and discrete-time systems

In a continuous time system, the inputs and the outputs are capable of changing at any time instant. Otherwise, the input and output signal values are of interest at any time instant. In discrete time systems or sampled systems, the signal values are of interest at discrete time instants only. Between these instants, the signals need not be defined.

Analog and digital systems

In an analog system, both the input and output signals are capable of having any value in a limited interval. Otherwise any signal value in a limited interval is of interest. In quantized systems, or digital systems, only a countable number of signal values is used.

Linear and nonlinear systems

In a strictly mathematical way, a system is linear, if and only if both properties of homogeneity and additivity hold. Mathematically, this can be written as

(199)\[\mathcal{H}\{\alpha x_{1}(t)+\beta x_{2}(t)\}=\alpha\mathcal{H}\{x_{1} (t)\}+\beta\mathcal{H}\{x_{2}(t)\}, \label{linear}\]

where \(\mathcal{H}\) is the system operator, \(\alpha\) \ and \(\beta\) are two scalars, and \(x_{1}(t)\) and \(x_{2}(t)\) are two excitations.

Instantaneous and dynamic systems

A system is called instantaneous if its responses, at any instant, only depend on the excitations at the same instant (not on past and future values). A system is called dynamic if a response at any time instant not only depends on the present input, but also on at least one of the past values of an excitation. A dynamic system is said to have a memory of length \(T\) if the output at time \(t\) is completely determined by the input values in the interval \((t-T,t)\).

Fixed and time-variant systems

A system is called fixed, time-invariant or stationary, if the system response does not depend on the time of the application of the excitation, when the system is assumed to be at rest prior to the application of the excitation. This implies that \(\mathcal{H}\) is not a function of time. This can be expressed as

(200)\[\text{if }\mathbf{Y}(t)=\mathcal{H}\{\mathbf{X}(t)\}\text{, then } \mathbf{Y}(t-\tau)=\mathcal{H}\{\mathbf{X}(t-\tau)\}\text{, for all } \tau\text{.} \label{fixed}\]

Otherwise, a system is stationary if the system properties do not change with time.

Stable and unstable systems

A system is said to be stable if and only if any bounded input signal, results in a bounded output signal. A bounded time function \(x(t)\) is one that never becomes infinite.

Causal systems

A causal system (physical or non-anticipatory system) is one whose responses to any input does not depend on any future value of the excitations.

Lumped and distributed systems

A dynamic system is said to be lumped if its behavior is governed by a set of ordinary differential (or difference) equations. Systems that require the use of partial differential equations to describe their behavior are said to be distributed. In this book, we will only deal with causal, analog, stationary, continuous time, lumped systems.