4.8Stability of LTI Discrete-Time Systems

One way of characterizing the stability of a discrete-time system is by the way it responds to a bounded input. When the response remains bounded, the system is said to exhibit BIBO stability. The implications of BIBO stability on the system’s z-domain transfer function, impulse response (weighting sequence), and natural response will be explored.

Consider an nth-order LTI discrete-time system described by Equation 4.456 in the previous section. The z-domain transfer function is

H(z)=Y(z)U(z)=b0zn+b1zn−1+ ⋅ ⋅ ⋅ +bmzn−mzn+a1zn−1+ ⋅ ⋅ ⋅ +an−1z+an,    n ≥ m

(4.585)

Suppose the poles of H(z) are real and distinct. Then

Y(z)=H(z)U(z)=b0zn+b1zn−1+ ⋅ ⋅ ⋅ +bmzn−m(z−p1)(z−p2) ⋅ ⋅ ⋅ (z−pn)U(z)

(4.586) ...

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