Signal Processing

Abstract

In Chapter 2, we showed how a digital signal can be defined as the solution of a linear time invariant difference equation. For example, the signal v where {v(k) = ha ^k; k ≥ 0} which is defined by the two parameters {h, a} can also be described as the initial condition response of the first order difference equation {v(k + 1) = av(k); v(0) = h}. More generally, the signal

$$ \upsilon (k) = {h_1}\lambda _1^k + {h_2}\lambda _2^k + \;...\; + {h_n}\lambda _n^k\,;\;k \geqslant 0 $$

which is represented by the 2n (possibly) complex parameters {h _ℓ , λ _ℓ } for 1 ≤ ℓ ≤ n can also be described as the initial condition response of an nth order difference equation.