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Differential and Difference Equation Representations of LTI Systems
M
dk
dk
ak k y (t )   bk k x(t )

dt
dt
k 0
k 0
N
N
M
k 0
k 0
 ak y[n  k ]   bk x[n  k ]
( N , M ) order of the equation = # of energy storing devices in the system. Often N  M and the order is referred to
as N .
To solve equations of this kind it is required to have initial values coming from the past (memory) in general for an
order N system, N values are required.
It is very often used to select
t 0
or
n0
as the time instants to evaluate the initial conditions.
y[ N ], y[ N  1],..... y[1],
y (t ) t 0 ,
d
d2
dn
y (t ) , 2 y (t ) ,..... n y (t )
dt
dt
t 0  dt
t 0 
t 0 
Recursive Formulas
M
N
k 0
k 1
y[n]   bk x[n  k ]   ak y[n  k ]
An operator
L is said to be linear if it satisfies the two linearity conditions:
1)
2)
L(cy )  cL( y) for any function y and any constant c .
L( y1  y2 )  L( y1 )  L( y 2 ) for any two functions y1 and y 2 .
Otherwise it is said to be nonlinear.
L is an operator which can be expressed in the form L  L( y)  F ( x, y' , y' ' ,...., y (n ) ) where
n  1 is an integer and F is a function of (n+1) variables.
A differential operator
An operator
L
is called a linear differential operator if
It turns out that any linear differential operator
L
L
is a differential operator and if
L
is linear.
can in fact be expressed in the form
L( y)  P0 ( x) y  P1 ( x) y '  ......Pn ( x) y n
where P0 ,
operator.
P1 ,..... Pn
are functions of x , and conversely any operator of this form is a linear differential
A system y (t )  H {x(t )} described by a linear operator is a linear system. If
the system is time invariant
A system
P0 , P1 ,..... Pn
are not functions of t,
y(t )  H {x(t )} described by a linear differential operator is linear system with memory.
Example 2.15
n
1
Find the first two values of y[n] when x[n]    u[n] , and y[1]  1, y[2]  2
 2
1
y[n]  x[n]  2 x[n  1]  y[n  1]  y[n  2]
4
Solving Differential and Difference Equations
The homogeneous Equation
dk
ak k y (t )  0

dt
k 0
N
N
y (t )   ci e ri t
(h)
i 1
ri
are the
N solutions of the characteristic equation
N
a r
k 0
k
k
0
Discrete time LTI systems
N
a
k 0
k
y[n  k ]  0
N
y [n]   ci ri
(h)
n
i 1
ri
are the
N solutions of the characteristic equation
N
a r
k 0
Example 2.17
Determine the Homogeneous solution
k
N k
0
Example 2.18 First Order Recursive System
Find the homogeneous solution.
y[n]  y[n  1]  x[n]
Problem 2.16
d2
d
d
y(t )  5 y(t )  6 y(t )  2 x(t )  x(t )
2
dt
dt
dt
The Particular Solution
Example 2.20 RC Circuit Continued
x(t )  cos(0t )
Problem 2.16 (continued)
x(t )  e t
The Complete Solution
Example 2.18 First Order Recursive System
Find the Complete solution of:
y[n]  y[n  1]  x[n]
n
when
1
x[n]    u[n] and y[1]  8
 2
The Natural Response
Zero Input Response
Example 2.17
Determine the NaturalResponse
C  1, R  1 , and i(0)  2V
Example 2.25 First Order Recursive System
Find the natural response for y[1]  8
y[n] 
1
y[n  1]  x[n]
4
The forced Response
Zero State Response
Example 2.26 First Order Recursive System
n
1
Find the forced response for x[n]    u[n]
 2
1
y[n]  y[n  1]  x[n]
4
Example 2.17
Determine the Forced Response
C  1, R  1 , and x(t )  costu (t ) V
The Resonance Phenomenon