An important subclass of difference equations is the set of linear constant coefficient difference equations. These equations are of the form
C
y
(
n
)
=
f
(
n
)
C
y
(
n
)
=
f
(
n
)
(2)where CC is a difference operator of the form given
C
=
c
N
D
N
+
c
N

1
D
N

1
+
.
.
.
+
c
1
D
+
c
0
C
=
c
N
D
N
+
c
N

1
D
N

1
+
.
.
.
+
c
1
D
+
c
0
(3)in which DD is the first difference operator
D
(
y
(
n
)
)
=
y
(
n
)

y
(
n

1
)
.
D
(
y
(
n
)
)
=
y
(
n
)

y
(
n

1
)
.
(4)Note that operators of this type satisfy the linearity conditions, and c0,...,cnc0,...,cn are real constants.
However, Equation 2 can easily be written as a linear constant coefficient recurrence equation without difference operators. Conversely, linear constant coefficient recurrence equations can also be written in the form of a difference equation, so the two types of equations are different representations of the same relationship. Although we will still call them linear constant coefficient difference equations in this course, we typically will not write them using difference operators. Instead, we will write them in the simpler recurrence relation form
∑
k
=
0
N
a
k
y
(
n

k
)
=
∑
k
=
0
M
b
k
x
(
n

k
)
∑
k
=
0
N
a
k
y
(
n

k
)
=
∑
k
=
0
M
b
k
x
(
n

k
)
(5)where xx is the input to the system and yy is the output. This can be rearranged to find y(n)y(n) as
y
(
n
)
=
1
a
0

∑
k
=
1
N
a
k
y
(
n

k
)
+
∑
k
=
0
M
b
k
x
(
n

k
)
y
(
n
)
=
1
a
0

∑
k
=
1
N
a
k
y
(
n

k
)
+
∑
k
=
0
M
b
k
x
(
n

k
)
(6)The forms provided by Equation 5 and Equation 6 will be used in the remainder of this course.
A similar concept for continuous time setting, differential equations, is discussed in the chapter on time domain analysis of continuous time systems. There are many parallels between the discussion of linear constant coefficient ordinary differential equations and linear constant coefficient differece equations.
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