2.1  Linear Time-Invariant systems

A quick lesson in systems theory and Linear Time-Invariant (LTI) systems

Linear Time-Invariant (LTI) systems are a special class of systems.

There are 2 properties that essentially define an LTI system:

• Linearity - if a level change (e.g. increase or decrease) is made to the Input to the system, the output level of the system changes (e.g. increases or decreases) by a similar proportional and predictable amount.

• Time Invariance - if a change to an Input to the system is delayed in time, the Output of the system, changes with the same amount of delay in time. 

LTI systems are generally well understood by scientists, physicists, mathematicians, signal processing specialists, and control system engineers. In contrast, non-linear systems and time-variant systems are typically extremely difficult to understand and often behave very chaotically. That is, non-LTI systems generally cannot be predicted or controlled.

For system engineers and many other specialists, the Output of LTI systems can typically be controlled by directly managing what is provided at the Input.

It is generally possible to mathematically determine the equations that define the system relationship between the Input and the Output of an LTI system and this relationship is called the Transfer Function. When an LTI system receives an Impulse Spike at its Input, the Output from the LTI system is very special as it contains the entire natural harmonic frequency response that defines the Transfer Function of the LTI system. In principle, if this Transfer Function is known it means the LTI system can be totally predictably controlled. 

System engineers also typically observe that very large LTI systems will take some amount of time to react and respond to a change in the Input, so there is often a time-delay with small initial changes growing before the maximum peak level at the Output is reached. This is because of the physical mass and energy that can be stored in a large LTI system. 

Now you know the basics of LTI systems, and it’s helpful to remember this.