8.3 Discrete-Time Systems 487
the first derivative of a continuous-time functionvc(t)as
dvc(t)
dt
≈
vc(t)−vc(t−Ts)
Ts
and its second derivative as
d^2 vc(t)
dt^2
=
ddvdtc(t)
dt
≈
d(vc(t)−vc(t−Ts))/Ts)
dt
≈
vc(t)− 2 vc(t−Ts)+vc(t− 2 Ts)
T^2 s
to obtain a second-order difference equation whent=nTs. Choosing a small value forTsprovides an
accurate approximation to the differential equation. Other transformations can be used. In Chapter 0
we indicated that approximating integrals by the trapezoidal rule gives thebilinear transformation,
which can also be used to change differential into difference equations.
Just as in the continuous-time case, the system being represented by the difference equation is not LTI
unless the initial conditions are zero and the input is causal. The Z-transform will, however, allow us
to find the complete response of the system even when the initial conditions are not zero. When the
initial conditions are not zero, just like in the continuous case, these systems areincrementally linear.
The complete response of a system represented by the difference equation can be shown to be
composed of azero-inputand azero-state responses—that is, ify[n] is the solution of the difference
Equation (8.31) with initial conditions not necessarily equal to zero, then
y[n]=yzi[n]+yzs[n] (8.32)
The componentyzi[n] is the response when the inputx[n] is set to zero, thus it is completely due to the
initial conditions. The responseyzs[n] is due to the input, as we set the initial conditions equal to zero.
The complete responsey[n] is thus seen as the superposition of these two responses. The Z-transform
provides an algebraic way to obtain the complete response, whether the initial conditions are zero or
not.
It is important, as in the continuous-time system, to differentiate the zero-input and the zero-state
responses from thetransientand thesteady-stateresponses.
8.3.3 The Convolution Sum
Leth[n]be the impulse response of an LTI discrete-time system, or the output of the system corresponding
to an impulseδ[n]as input and initial conditions (if needed) equal to zero. Using the generic representation of
the inputx[n]of the LTI system,
x[n]=
∑∞
k=−∞
x[k]δ[n−k] (8.33)
