9-2 Linear Time-Invariant Systems (��������������������?

Old Chinese version

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 = 0ۡC

If the system, denoted by L{E}, satisfies the following equations, it is called linear:

  1. If L{x[n]} = y[n], then L{kx[n]} = ky[n].
  2. 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{E} 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 simplicity, we shall assume the input signal x[n] = 0 when n < 0. In other words, x[n] is activated when n 0, thus y[n] is nonzero only when n 0.

For the rest of this book, we should assume all the systems under discussion are LTI systems.

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