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¡ã¡Û, while the output is also a function y[n], n = 0¡ã¡Û¡CIf the system, denoted by L{¡E}, satisfies the following equations, it is called

linear:The above equations can be reduced to a single one:

- If L{x[n]} = y[n], then L{kx[n]} = ky[n].
- If L{x
_{1}[n]} = y_{1}[n] and L{x_{2}[n]}= y_{2}[n], then L{x_{1}[n] + x_{2}[n]} = y_{1}[n] + y_{2}[n].If L{x The above equation is referred to as the_{1}[n]} = y_{1}[n] and L{x_{2}[n]}= y_{2}[n], then L{ax_{1}[n] + bx_{2}[n]} = ay_{1}[n] + by_{2}[n], for all constants a and b.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 ¡Ù 0. If a system is linear and time-invariant, we call it alinear time-invariant system, orLTI sytemfor short.For the rest of this book, we should assume all the systems under discussion are LTI systems.

Audio Signal Processing and Recognition (µ°T³B²z»P¿ëÃÑ)