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u(k – 1), u(k – 2), …). This solution technique is referred to as the convolution-
summation representation:

∞
yk() = ∑ w i u k i( ) ( − ) (2.28)
i=0

where the w(i) is the weighting function (or weight). Usually, the infinite sum
is reduced to a finite sum because the inputs with negative indexes are usu-
ally assumed to be zeros.
On the other hand, in the difference equation formulation of this class of
problems, the present output y(k) is expressed as a linear combination of the
present and m most recent inputs and of the n most recent outputs, specifically:

y(k) = b u(k) + b u(k – 1) + … + b u(k – m)
1
m
0
– a y(k – 1) – a y(k – 2) – … – a y(k – n) (2.29)
n
1
2
where, of course, n is the order of the difference equation. Elementary tech-
niques for solving this class of equations were introduced in Section 2.4.
However, the most powerful technique to directly solve the linear difference
equation with constant coefficients is, as pointed out earlier, the z-transform
technique.
Each of the above formulations of the input-output problem has distinct
advantages in different circumstances. The direct difference equation formu-
lation is the most amenable to numerical computations because of lower
computer memory requirements, while the convolution-summation tech-
nique has the advantage of being suitable for developing mathematical
proofs and finding general features for the difference equation.
Relating the parameters of the two formulations of this problem is usually
cumbersome without the z-transform technique. However, for first-order dif-
ference equations, this task is rather simple.

Example 2.4
Relate, for a first-order difference equation with constant coefficients, the sets
{a } and {b } with {w }.
n n n
Solution: The first-order difference equation is given by:

y(k) = b u(k) + b u(k – 1) – a y(k – 1)
0 1 1
where u(k) = 0 for all k negative. From the difference equation and the initial
conditions, we can directly write:

y(0) = b u(0)
0

© 2001 by CRC Press LLC