Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.

Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:

Additivity means that the response to the sum of multiple inputs is the sum of their individual responses. For inputs x₁(t) and x₂(t) producing outputs y₁(t) and y₂(t), respectively:

Combining homogeneity and additivity yields the equation of a linear system:

Linear systems include circuits with resistors, capacitors, and inductors that adhere to the superposition principle. Conversely, systems that fail to obey the linearity equation are classified as non-linear, such as rectifiers and diodes.

A causal system is one where the current response is independent of future input values. For instance, a car's movement cannot predict future driving actions, making it a causal system. In contrast, non-causal systems like ideal filters cannot be physically realized as they do not follow the principle of causality.

Dynamic systems exhibit memory, where the output depends on past and present inputs. An example is an electrical circuit with capacitors or inductors, where the present output is influenced by past inputs. Static systems, also known as memoryless or instantaneous systems, have outputs solely based on the present input, such as a simple resistor circuit where the output voltage is a direct function of the current input voltage.

Understanding these properties—linearity, causality, and memory—is essential for analyzing and designing systems in various engineering fields.