For a number of purposes, the use of complex-valued signals is a convenient mathematical shortcut, but these issues will be left to another lecture.

The definition of a linear system can be simplified as follows: For any two input signals *x _{1}(t)* and

This is sometimes called the ** superposition property**. Notice that by choosing

That is, the response of a linear system to zero input is zero output. This might be a bit confusing if you have studied linear differential equations, where zero input but nonzero initial conditions yield nonzero response. Of course there is an "initial time," usually *t = 0*, at which the initial conditions are imposed. In our setting, input signals and system responses are defined for all *t*. The differences can be reconciled by assuming, in the linear differential equation setting, that the input signal is zero for *t < 0*, the output signal also is zero for *t < 0*, and that the initial conditions are all zero. With these assumptions the linear differential equation is an LTI system in the sense we are considering.