Page 76 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS 65

where A is the eigenvalue of T associated with zn. Setting x[n] = zn in Eq. (2.391, we have

k= -m
= H(z)zn = Azn (2.51)
rn
where A = H(z) = x h[k] zPk (2.52)
k= -ra
Thus, the eigenvalue of a discrete-time LTI system associated with the eigenfunction zn is
given by H(z) which is a complex constant whose value is determined by the value of z via
Eq. (2.52). Note from Eq. (2.51) that y[O] = H(z) (see Prob. 1.45).
The above results underlie the definitions of the z-transform and discrete-time Fourier
transform which will be discussed in Chaps. 4 and 6.

2.9 SYSTEMS DESCRIBED BY DIFFERENCE EQUATIONS
The role of differential equations in describing continuous-time systems is played by
difference equations for discre te-time systems.

A. Linear Constant-Coefficient Difference Equations:

The discrete-time counterpart of the general differential equation (2.25) is the Nth-
order linear constant-coefficient difference equation given by
N M

where coefficients a, and b, are real constants. The order N refers to the largest delay of
y[n] in Eq. (2.53). An example of the class of linear constant-coefficient difference
equations is given in Chap. I (Prob. 1.37). Analogous to the continuous-time case, the
solution of Eq. (2.53) and all properties of systems, such as linearity, causality, and
time-invariance, can be developed following an approach that directly parallels the
discussion for differential equations. Again we emphasize that the system described by Eq.
(2.53) will be causal and LTI if the system is initially at rest.

B. Recursive Formulation:
An alternate and simpler approach is available for the solution of Eq. (2.53). Rear-
ranging Eq. (2.53) in the form

we obtain a formula to compute the output at time n in terms of the present input and the
previous values of the input and output. From Eq. (2.54) we see that the need for auxiliary
conditions is obvious and that to calculate y[n] starting at n = no, we must be given the
values of y[n,, - 11, y[no - 21,. . . , y[no - N] as well as the input x[n] for n 2 n,, - M. The
general form of Eq. (2.54) is called a recursiue equation since it specifies a recursive
procedure for determining the output in terms of the input and previous outputs. In the