## 9-2 Linear Time-Invariant Systems (嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙踝蕭嚙?

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:

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{•} 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.

Hint
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.

Audio Signal Processing and Recognition (音訊處理與辨識) 