Linear Shift-Invariant systems:

Linear Shift-Invariant systems, called LSI systems for short, form a very important class of practical systems, and hence are of interest to us. They are also referred to as Linear Time-Invariant systems, in case the independent variable for the input and output signals is time. Remember that linearity means that is y₁(t) and y₂(t) are responses of the system to signals x₁(t) and x₂(t) respectively, then the response to ax₁(t) + bx₂(t) is ay₁(t) + by₂(t).

Shift invariance implies that the response of the system to x₁(t - t₀) is given by y₁(t - t₀) for all values of t and t₀. Linear systems are of interest to us for primarily two reasons: first, several real-life systems can be well approximated by linear systems. Second, linear systems come with several properties which make their analysis simple. Similarly, shift-invariant systems allow us to use simpler math to analyse the system. As we proceed with our analysis, we will point out cases where some results (which are rather intuitive) are valid for only LSI systems.