ABSTRACT
A discrete-time causal linear time-varying system with single-input u(k) and single-output y(k) can be modeled by the input/output relationship
y(k) = k∑
j=−∞ h(k, j)u( j) (3.137)
where k is an integer-valued variable (the discrete-time index) and h(k, j) is the output response resulting from the unit pulse δ(k− j) (where δ(k− j) = 1 for k = j and = 0 for k = j) applied at time j. It is assumed that u(k) and/or h(k, j) is constrained so that the summation in Equation 3.137 is well defined. The system defined by Equation 3.137 is time invariant if and only if h(k, j) is a function of only the difference k− j, in which case Equation 3.137 reduces to the convolution relationship
y(k) = h(k) ∗ u(k) = k∑
j=−∞ h(k− j)u( j) (3.138)
where h(k− j) = h(k− j, 0). The system defined by Equation 3.138 is finite dimensional if the input u(k) and the output y(k) are
related by the nth-order difference equation
y(k+ n)+ n−1∑ i=0
ai(k)y(k+ i) = m∑
i=0 bi(k)u(k+ i) (3.139)
where m ≤ n and the ai(k) and the bi(k) are real-valued functions of the discrete-time variable k. The system given by Equation 3.139 is time invariant if and only if all coefficients in Equation 3.139 are constants, that is, ai(k) = ai and bi(k) = bi for all i, where ai and bi are constants.
