9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 361This can be rearranged as Y(z)(l + L:=l apz-P) = X (z) L~=O bqz-q a nd then
solved for the quotient H(z) = ~~:~. T he sequence h[n) = 3-^1 [H(z)) can be
used to construct a particular solut ion to (9-14), i.e., Yp[n] = 3-^1 [H(z)X(z)] =
h[n] * x[n). This solution can be expressed using the convolution sum as follows:
n
yp[n) = h[n) * x[n] = L h[n - i]x[i]. (9-18)
i = O
R e mark 9.7
This particular solution does not involve initial conditions for (9-14). We will
illustrate how to use convolution at t he end of this section. •
9.2.3 Difference Equatio n s wit h Initial Conditions
Often a difference equation involves only one input on t he right-hand side of
(9-14) and we write
y[n] + a 1 y[n - 1] + a 2 y [n -2] + ... + apy[n - P] = x[n),
then we could shift the index and use the form
y [n + P] + a 1 y[n+ P-1] + a2y[n + P - 2] + ... + apy[n] = x [n + P].
Consider the first-order linear constant coefficient difference equation (LCCDE)
y [n + 2] - 2ay[n + l] + by(n] = x [n + 2], (9- 19 )
with the init ial conditions y[O] = Yo and y[l ) = y , (and implicitly we have
x[O] = xo and x[l] = x 1 ).
Step (i) Using the time forward properties
3[y[n + 1]] = z(Y (z) -Yo),
3[y(n + 2]] = z^2 (Y(z) - Yo - y1z-^1 ), and
3[x[n + 2]] = z^2 (X(z) - xo - x 1 z-^1 ) ,
t ake t he z-transform of each term and get the equation
z^2 (Y(z) - Yo -y,z-^1 ) - 2az(Y (z) - Yo)+ bY(z) = z^2 (X(z) - xo - x1z-^1 ).
(9-20)