Digital Signal Processing Reference
In-Depth Information
M
is given by
y
(
t
), the three upward forces opposing the external downward
force
x
(
t
) are given by
spring
constant
k
wall
friction
r
y
(
t
)
M
d
2
y
M
inertial (or accelerating) force
F
i
=
d
t
2
;
(2.15a)
x
(
t
)
=
r
d
y
d
t
frictional (or damping) force
F
f
;
(2.15b)
(a)
spring (or restoring) force
F
s
=
ky
(
t
)
,
(2.15c)
.
My
(
t
)
ry
(
t
)
¨
ky
(
t
)
where
r
is the damping constant for the medium surrounding the mass. Apply-
ing Newton's third law of motion, the input-output relationship of the spring
damping system is given by
y
(
t
)
M
x
(
t
)
M
d
2
y
d
t
2
+
r
d
y
d
t
+
ky
(
t
)
=
x
(
t
)
,
(2.16)
(b)
which is a linear, constant-coefficient second-order differential equation.
Equation (2.16) describes the relationship between the applied force
x
(
t
) and
the resulting vertical displacement
y
(
t
). As in the case of the RLC circuit,
a second-order differential equation is used to model the mechanical spring
damper system.
Fig. 2.7. (a) Mechanical spring
damper system. (b) Free-body
diagram illustrating the
opposing forces acting on mass
M
of the mechanical spring
damping system.
2.1.6 Numerical differentiation and integration
Numerical methods are widely used in calculus for finding approximate values
of derivatives and definite integrals. Here, we present examples of differentiation
and integration of a CT function
x
(
t
). The systems representing integration and
differentiator are shown in Fig. 2.8. We show that the numerical approximations
of a CT differentiator and integrator lead to finite difference equations that are
frequently used to describe DT systems.
To discretize a derivative over a continuous interval [0
,
T
]
,
the time interval
T
is divided into intervals of duration
t
, resulting in the sampled values
x
(
k
t
)
for
k
=
0
,
1
,
2
,...,
K
, with
K
given by the ratio
T
/
t
. Using a single-step
backward finite-difference scheme, the time derivative can be approximated as
follows:
d
d
t
x
(
t
)
y
(
t
)
(a)
d
x
d
t
x
(
k
t
)
−
x
((
k
−
1)
t
)
t
≈
,
(2.17)
t
t
=
k
t
y
(
t
)
x
(
t
)
∫
d
t
which yields
0
x
(
k
t
)
−
x
((
k
−
1)
t
)
t
(b)
y
(
k
t
)
=
(2.18)
Fig. 2.8. Schematics of (a) a
differentiator and (b) an
integrator. Finite-difference
schemes are often used to
compute the values of
derivatives and finite integrals
numerically.
or,
y
(
k
t
)
=
C
1
(
x
(
k
t
)
−
x
((
k
−
1)
t
))
,
(2.19)
where
x
(
k
t
) is the sampled value of
x
(
t
)at
t
=
k
t
and
C
1
is a constant, equal
to 1
/
t
.
The CT signal
y
(
t
)
=
d
x
/
d
t
and represents the result of differentiation.
Search WWH ::
Custom Search