Mathematical and Technical Detail

Signals and Linear Systems

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 x1(t) and x2(t) and any real constant a, the responses should satisfy

S [a x1(t) + x2(t)] = a S [x1(t)] + S [x2(t)]

This is sometimes called the superposition property. Notice that by choosing x2(t) to be x1(t) and a = - 1, we obtain

S [0] = - S [ x1(t)] + S [ x1(t)] = 0

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.