Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.

The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated using convolution. This property is crucial for analyzing and predicting system behavior.

Linear systems are often represented by linear differential equations, where coefficients may or may not be functions of time. If the coefficients are time-invariant, the system is represented by a linear, constant-coefficient differential equation (LCCDE). LCCDEs cover circuits with ideal components and a single independent source, with multiple sources permitted by the superposition principle.

An LTI system can modify the amplitude and phase of an input sinusoid or complex exponential signal without changing its frequency. This makes LTI systems essential tools for designing filters to remove noise from signals and images. By preserving the frequency content while adjusting amplitude and phase, LTI systems enable precise signal manipulation.

In summary, the principles of linearity, time-invariance, and superposition underpin the analysis and design of LTI systems, making them integral in various engineering applications, from signal processing to control systems.