Linear Time Invariant System

Created: 06/04/2020Last Updated: 07/06/2023


An LTI (Linear Time-Invariant) system is characterized by its output, Y, being linearly related to its input, X, and its output response Y being independent of time, T. This means that the system's response is a straight line relationship between input and output, and it remains consistent over time. However, one must know that the a linear line going through the origin is just one of the linear systems. A linear system does not have to be a straight line.

In Practice

Nonlinear systems are common and can be desired in certain applications, such as a RF mixer, where nonlinearity is intentionally introduced to achieve specific signal processing effects.

On the other hand, linear systems are generally preferred in many applications. For example, in an Analog-to-Digital Converter (ADC), a linear relationship between the input voltage and the resulting digital scale is desired. Any deviation or non-linearity in this conversion process is considered an error and can impact the accuracy of the conversion.

Regarding time invariance, it generally holds true for short-term durations. For instance, when a freshly built electronic filter is measured consecutively, its frequency response remains the same. However, over time, component aging can occur in the inductor and capacitor used in the analog filter design. As a result, the filter's frequency response will change, demonstrating that the system's response is no longer time-invariant.


Understanding the properties of LTI systems, as well as the implications of nonlinearity and time variance, is crucial in various areas of engineering and signal processing. It enables engineers to design and analyze systems accurately, ensuring the desired functionality and performance are achieved.