9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 371
y[11]
8
6
4
2
.. '^10 •^15 20 · "
-2
Figure 9.3 A typical solution to y[n + 2] - ~ ,/2y[n + l J + ~y[nJ = cos(~n).
t heorem) acos(IJ) + b sin(IJ) = ..Ja2 + b2cos(IJ + a.rctan(-~)). Therefore, the
steady state solution is
yp(n] = ( ~~
8
)
2
+ (~;)
2
cos ( ~n + a.rcta.n ( -~;/ ( ~~
8
)))
= vb cos (in+ arctan (~)).
Figure 9.3 illustrates that the output signa.l y (n ] t ends t o this limit as n-+oo; i.e.,
n~oo lim y (n ] = n -lim oo y,.[n) + ln~im oo yp[n) =^0 + Yp(n] = Yp[n).
Loosely speaking, for la.rge values of n, the va.lues of the input signal x[nj =
cos(~n) are amplified by the factor vb= 2.49 615 to prod uce t he values of the
output signal y (n ].
-------~EXERCISES FOR SECTION 9.2
- Solve the homogeneous difference equations.
(a) y[2 + n ] - 6y[l + n] + 8y(nJ = 0 with y[O] = 3, y(l l = 4.
(b) y[2 + n] -6y[l + n] + (}y(n] = 0 with y[OJ = 2, y [l ) = 3.
( c) y[2 + nl -6y[l + n] + lOy[n ] = 0 with y[O] = 2, y[l] = 4.
2. (a) Solve y[n + 2] + y(n] = 0 with y(O) = 1 and y(l] = O.
(b) Solve y(n + 2] + y(n) = 0 with y(O) = 0 an d y(l] = 1.
- Solve the homogeneous difference equations.