EEE 226: Signals and Systems
Lecture Notes # 7
Dr. Aykut Hocanın
Department of Electrical and Electronics Engineering
Eastern Mediterranean University
April 11, 2002
Please read section 2.4 of the textbook.
1
Causal LTI Systems Described by Difference and
Differential Equations
dy(t)
+ 2y(t) = x(t)
(1)
dt
The equation above provides an implicit specification of a system. The differential equation describes a constraint between the input and the output of
the system but auxiliary conditions are needed to completely characterize the
system. The initially at rest condition leads to LTI systems.
The complete solution of a differential/difference system consists of the particular solution and the homogenous solution.
y(t) = yp (t) + yh (t)
(2)
In order to find yp (t), we look for the forced response. The homogenous solution
is the natural response of the system. The generalized constant coefficient
differential/difference equations are described by:
N
X
k=0
and
N
X
ak
M
dk x(t)
dk y(t) X
=
b
k
dtk
dtk
k=0
ak y[n − k] =
k=0
M
X
bk x[n − k]
(3)
(4)
k=0
The difference equations may be recursive or non-recursive.
• Recursive equations depend on previous values of input and output.
• Non-recursive equations do not contain the previous values of the output.
Therefore no auxiliary conditions are needed.
1
EEE 226 Signals and Systems
Example 1
Dr. Aykut Hocanın
2
dy(t)
− 7y(t) = x(t)
dt
let x(t) = Ke5t u(t) where K ∈ R.
y(t) = yp (t) + yh (t)
where yp (t) = Y e5t (because of the nature of the input) and yh is the solution
of dy(t)
dt − 7y(t) = 0.
5Y e5t − 7Y e5t = Ke5t t > 0
K
−2Y e5t = Ke5t ⇒ Y = −
2
and
K
yp (t) = − e5t t > 0
2
st
Let yh (t) = Ae ,
sAest − 7Aest = 0
(s − 7)Aest = 0 ⇒ s = 7
yh (t) = Ae7t t > 0
Finally,
y(t) = −
K 5t
e + Ae7t t > 0
2
Applying the initial conditions, y(0) = 0, we get A =
y(t) =
Example 2
K
2
K 7t
[e − e5t ]u(t)
2
1
y[n] − y[n − 1] = x[n] y[0] = 0 n ≤ 0
2
for an input x[n] = Kδ[n].