Page 386 - Mechanical Engineers' Handbook (Volume 2)

8 Model Classifications 377

x(k 1) A(k)x(k) B(k)u(k)
y(k) C(k)x(k) D(k)u(k)
The mathematics of difference equations parallels that of differential equations in many
important respects. In general, the concepts applied to differential equations have direct
analogies for difference equations, although the mechanics of their implementation may vary
(see Ref. 11 for a development of dynamic modeling based on difference equations). One
important difference is that the general solution of nonlinear and time-varying difference
equations can usually be obtained through recursion. For example, consider the discrete
nonlinear model
y(k)
y(k 1)
1 y(k)
Recursive evaluation of the equation beginning with the initial condition y(0) yields
y(0)
y(1)
1 y(0)
y(1)
y(0)
y(0) y(0)
y(2) 1
1 y(1) 1 y(0) 1 y(0) 1 2y(0)
y(2) y(0)
y(3)
1 y(2) 1 3y(0)

the pattern of which reveals, by induction,

y(0)
y(k)
1 ky(0)
as the general solution.

Uniform Sampling
Uniform sampling is the most common mathematical approach to A/D conversion, that is,
to extracting the discrete-time approximation y*(k) of the form

y*(k) y(t kT)
from the continuous-time signal y(t), where T is a constant interval of time called the sam-
pling period. If the sampling period is too large, however, it may not be possible to represent
the continuous signal accurately. The sampling theorem guarantees that y(t) can be recon-
structed from the uniformly sampled values y*(k) if the sampling period satisfies the in-
equality

T
u
where is the highest frequency contained in the Fourier transform Y( )of y(t), that is, if
u
Y( ) 0 for all u
The Fourier transform of a signal is defined to be