9.1 • THE Z- TRANSFORM 347Therefore, the sequence x[nJ = 3-^1 (X(z)J is
x [nJ = 3-^1..
[
z2 l
(z-e't) (z-e-:")= Res[X(z)z"-^1 ,zi] + ResjX(z)z"-^1 ,z:i]
= ( 2 I - 2 i) e-. in• + (I 2 + 2 i) e-·-_,,.."
= cos( ~n) +sin( ~n).The following complex calculation can be used to find the coefficients of the
FIR filter equation( 1-e'" •z-I) ( 1-e-•z - h 1) =1-( e•+e-·- z-
(^1) " .) 1
Hence the FIR filter equation is y jn] = K(x[nJ - v'Zx[n - l ] + x[n - 2)), where
K is a constant (or gain factor).Remark 9.5 We leave it as an exercise to substitute x [n) = cos(~n) and x[nJ =
sin( in) into the right-hand side and verify that the output y[nJ becomes iden-
tically zero. •
9.1.6 First -O r der Difference Equations
The solution of difference equations is analogous to the solution of differential
equations. Consider the first-order homogeneous equation
yin+ l ] - ay[n ] = 0
where a is a constant. The following method is often used.
Trial solution methodUse the trial solution y[n] = c 1 rn, and substitute it into the preceding equation
to get c 1 r"+^1 - ac 1 r" = 0. Then divide through by r" and simplify to obtainr = a. The general solution to the difference equation is
Familiar models of difference equations are given in Table 9.3.