Digital Signal Processing Reference
In-Depth Information
This is one of the simpler models that we can consider; the current value of the out-
put signal,
y
[
n
], is a function of the current value and the last two values of the input
signal
x
[
n
] only, but is not a function of past output values.
We now consider a first-order discrete-time system model in which the current
value of output,
y
[
n
], is a function of the last value of output
y[n - 1]
and the cur-
rent value of the input
x
[
n
]:
y[n] = ay[n - 1] + bx[n].
(10.44)
The parameters
a
and
b
are constants; Equation (10.44) is a
linear difference equa-
tion with constant coefficients
. One of the coefficients is equal to unity, one is
a
, and
one is
b
.
To show that (10.44) is linear, we use superposition. Suppose that
y
i
[n]
is the
solution of (10.44) for the excitation
x
i
[n],
for
i = 1, 2.
By this, we mean that
y
i
[n] = ay
i
[n - 1] + bx
i
[n],
i = 1, 2.
(10.45)
We now show that the solution
(a
1
y
1
[n] + a
2
y
2
[n])
satisfies (10.44) for the excita-
tion
(a
1
x
1
[n] + a
2
x
2
[n]),
by direct substitution into (10.44):
(a
1
y
1
[n] + a
2
y
2
[n]) = a(a
1
y
1
[n - 1] + a
2
y
2
[n - 1]) + b(a
1
x
1
[n] + a
2
x
2
[n]).
This equation is rearranged to yield
a
1
(y
1
[n] - ay
1
[n - 1] - bx
1
[n]) + a
2
(y
2
[n] - ay
2
[n - 1] - bx
2
[n]) = 0.
(10.46)
Each term on the left side is equal to zero, from (10.45); hence, the difference equa-
tion (10.44) satisfies the principle of superposition and is linear. Also, in (10.44), we
replace
n
with
(n - n
0
),
yielding
y[n - n
0
] = ay[n - n
0
- 1] + bx[n - n
0
].
(10.47)
Thus, an excitation of
x[n - n
0
]
produces a response of
y[n - n
0
],
and (10.44) is
also time invariant.
A simple example of a linear difference equation with constant coefficients is
the first-order equation
y[n] = 0.6y[n - 1] + x[n].
The equation is first order because the current value of the dependent variable is an
explicit function of only the most recent preceding value of the dependent variable,
y[n - 1].
The general form of an
N
th-order linear difference equation with constant co-
efficients is, with
a
0
Z 0,
a
0
y[n] + a
1
y[n - 1] +
Á
+ a
N- 1
y[n - N + 1] + a
N
y[n - N]
= b
0
x[n] + b
1
x[n - 1] +
Á
+ b
M- 1
x[n - M + 1] + b
M
x[n - M],
