For any given discrete-time signal x[n], we can send it to a system to obtain the output signal y[n], with the following mathematical notations:
y[n] = L{x[n]}
In other words, the input to the sysmte is a function x[n], n = 0∼∞, while the output is also a function y[n], n = 0∼∞。
If the system, denoted by L{•}, satisfies the following equations, it is called linear:
If L{x[n]} = y[n], then L{kx[n]} = ky[n].
If L{x1[n]} = y1[n] and L{x2[n]}= y2[n], then L{x1[n] + x2[n]} = y1[n] + y2[n].
The above equations can be reduced to a single one:
If L{x1[n]} = y1[n] and L{x2[n]}= y2[n], then L{ax1[n] + bx2[n]} = ay1[n] + by2[n], for all constants a and b.
The above equation is referred to as the superposition principle. Namely, if a system satifies the superposition principle, then it is a linear system.
If L{•} satisties the following equation, it is called time-invariant:
If L{x[n]} = y[n], then L{x[n-k]} = y[n-k], for all k ≧ 0.
If a system is linear and time-invariant, we call it a linear time-invariant system, or LTI sytem for short.
For the rest of this book, we should assume all the systems under discussion are LTI systems.
