14
Practical MATLAB
® Applications for Engineers
R.1.30 The analog unit parabolic function p K (t) u(t) is defi ned by
p t u t
K
t
t
K
t
K
K
( ) ( )
!
, ,
ϭ
ϭ
1
0
23
0
0
for
for
Ն
Ͻ
and
…
R.1.31 The discrete unit parabolic function p K (n) u(n) is defi ned by
p n u n
K
n
n
K
n
K
K
( ) ( )
!
, ,
ϭ
ϭ
1
0
23
0
0
for
for
Յ
Ͻ
and
…
R.1.32 Observe that
a. The unit ramp presents a sharp 45° corner at t = 0.
b. The unit parabolic function presents a smooth behavior at t = 0.
c. The unit step presents a discontinuity at t = 0.
R.1.33 The step, ramp, and parabolic functions are related by derivatives as follows:
a. (d/dt)[r(t)] = u(t)
b. d
__
dt
[p 2 (t)] = r(t)u(t)
c. (d/dt)[p a (t)] = p a−1 (t)
Observe that the fi rst relation makes sense for all t ≠ 0, since at t = 0 a discontinuity occurs, whereas the second and third relations hold for all t.
R.1.34 Note that, in general, the product f(t) times u(t) [f(t)u(t)] defi nes the composite function given by
f t u t
f t
t
t
( ) ( )
( )
ϭ
Ͻ
for Ն 0
0
0
for
R.1.35 A wide class of engineering systems employ sinusoidal and exponential* signals as
inputs. A real exponential analog signal is in general given by
f(t) = Ae
bt
where e = 2.7183 (Neperian constant) and A and b are in most cases real constants.
Observe that for f(t) = Ae
bt
,
a. f(t) is a decaying exponential function for b < 0.
b. f(t) is a growing exponential function for b > 0.
The coeffi cient b as exponent is referred to as the damping coeffi cient or constant.
In electric circuit theory, the damping constant is frequently given by b = 1/τ, where
τ is referred as the time constant of the network (see Chapter 2).
Note that the exponential function f(t) = Ae
bt repeats itself when differentiated
or integrated with respect to time, and constitutes the homogeneous solution of
* Recall that sinusoids are complex exponentials (Euler), see Chapter 4 of Practical MATLAB
® Basics for
Engineers.
CRC_47760_CH001.indd 14
CRC_47760_CH001.indd 14
7/25/2008 4:16:13 PM
7/25/2008 4:16:13 PM
Practical MATLAB
® Applications for Engineers
R.1.30 The analog unit parabolic function p K (t) u(t) is defi ned by
p t u t
K
t
t
K
t
K
K
( ) ( )
!
, ,
ϭ
ϭ
1
0
23
0
0
for
for
Ն
Ͻ
and
…
R.1.31 The discrete unit parabolic function p K (n) u(n) is defi ned by
p n u n
K
n
n
K
n
K
K
( ) ( )
!
, ,
ϭ
ϭ
1
0
23
0
0
for
for
Յ
Ͻ
and
…
R.1.32 Observe that
a. The unit ramp presents a sharp 45° corner at t = 0.
b. The unit parabolic function presents a smooth behavior at t = 0.
c. The unit step presents a discontinuity at t = 0.
R.1.33 The step, ramp, and parabolic functions are related by derivatives as follows:
a. (d/dt)[r(t)] = u(t)
b. d
__
dt
[p 2 (t)] = r(t)u(t)
c. (d/dt)[p a (t)] = p a−1 (t)
Observe that the fi rst relation makes sense for all t ≠ 0, since at t = 0 a discontinuity occurs, whereas the second and third relations hold for all t.
R.1.34 Note that, in general, the product f(t) times u(t) [f(t)u(t)] defi nes the composite function given by
f t u t
f t
t
t
( ) ( )
( )
ϭ
Ͻ
for Ն 0
0
0
for
R.1.35 A wide class of engineering systems employ sinusoidal and exponential* signals as
inputs. A real exponential analog signal is in general given by
f(t) = Ae
bt
where e = 2.7183 (Neperian constant) and A and b are in most cases real constants.
Observe that for f(t) = Ae
bt
,
a. f(t) is a decaying exponential function for b < 0.
b. f(t) is a growing exponential function for b > 0.
The coeffi cient b as exponent is referred to as the damping coeffi cient or constant.
In electric circuit theory, the damping constant is frequently given by b = 1/τ, where
τ is referred as the time constant of the network (see Chapter 2).
Note that the exponential function f(t) = Ae
bt repeats itself when differentiated
or integrated with respect to time, and constitutes the homogeneous solution of
* Recall that sinusoids are complex exponentials (Euler), see Chapter 4 of Practical MATLAB
® Basics for
Engineers.
CRC_47760_CH001.indd 14
CRC_47760_CH001.indd 14
7/25/2008 4:16:13 PM
7/25/2008 4:16:13 PM