Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,

Discrete-time systems have input and output signals at specific intervals, defined at distinct instants by difference equations. An example is a simple model for the balance in a bank account from month to month, represented as,

where B[n] is the balance at the month n, D[n] is the deposit, and W[n] is the withdrawal.

Systems can also be categorized as time-varying or time-invariant. A time-varying system has parameters that change over time. Continuous-time time-varying systems are described by time-varying differential equations, while discrete-time time-varying systems are defined by time-varying difference equations.

A time-invariant system exhibits identical time shifts in output signals when the input signal is time-shifted. Mathematically, if y(t) is the output for input x(t), then for a time-invariant system, y(t−t₀) is the output for input x(t−t₀). For example, an RC circuit is time-invariant if the resistance R and capacitance C values remain constant. If these values fluctuate, the system becomes time-variant, meaning the results will vary depending on when the experiment is conducted.

Understanding these distinctions in continuous-time and discrete-time systems, as well as time-varying and time-invariant systems, is crucial for analyzing and designing a wide range of engineering systems.