1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

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

1. 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.

3. Solve the homogeneous difference equations.

(a.) Solve y[n + 2) -,/2y[n + 1] + y[n] = 0 with y(O] = 2 and y[lJ = ,/2.

(b) Solve y[n + 2J - ,/2y[n + 1] + y[n J = 0 with y(O] = 0 and y(l J = v'2.