E1C02 09/14/2010
13:35:22 Page 72
NOMENCLATURE
f
frequency, in Hz (t
À1 )
k spring constant (mt
À2 )
m mass (m)
t
time (t)
y dependent variable
y m discrete data points
A amplitude
B amplitude
C amplitude
F force (mlt
À2 )
N total number of discrete data points; integer
T period (t)
U unit step function
Y Fourier transform of y
a angle (rad)
b angle (rad)
df frequency resolution (t
À1 )
dt sample time increment (t)
f phase angle (rad)
v circular frequency in rad/s (t
À1 )
PROBLEMS
2.1 Define the term ‘‘signal’’ as it relates to measurement systems. Provide two examples of static and
dynamic input signals to particular measurement systems.
2.2 List the important characteristics of input and output signals and define each.
2.3 Determine the average and rms values for the function
y t
ð Þ ¼ 25 þ 10 sin 6pt
over the time periods (a) 0 to 0.1 s, (b) 0.4 to 0.5 s, (c) 0 to 1/3 s, and (d) 0 to 20 s. Comment on the nature
and meaning of the results in terms of analysis of dynamic signals.
2.4 The following values are obtained by sampling two time-varying signals once every 0.4 s:
t
y 1 (t)
y 2 (t)
t
y 1 (t)
y 2 (t)
0
0
0
0.4
11.76
15.29
2.4
À11.76
À15.29
0.8
19.02
24.73
2.8
À19.02
À24.73
1.2
19.02
24.73
3.2
À19.02
À24.73
1.6
11.76
15.29
3.6
À11.76
À15.29
2.0
0
0
4.0
0
0
Determine the mean and the rms values for this discrete data. Discuss the significance of the rms
value in distinguishing these signals.
2.5 A moving average is an averaging technique that can be applied to an analog, discrete time, or digital
signal. A moving average is based on the concept of windowing, as illustrated in Figure 2.22. That
portion of the signal that lies inside the window is averaged and the average values plotted as a
function of time as the window moves across the signal. A 10-point moving average of the signal in
Figure 2.22 is plotted in Figure 2.23.
a. Discuss the effects of employing a moving average on the signal depicted in Figure 2.22.
b. Develop a computer-based algorithm for computing a moving average, and determine the effect of
the width of the averaging window on the signal described by
y t
ð Þ ¼ sin 5t þ cos 11t
72 Chapter 2 Static and Dynamic Characteristics of Signals
13:35:22 Page 72
NOMENCLATURE
f
frequency, in Hz (t
À1 )
k spring constant (mt
À2 )
m mass (m)
t
time (t)
y dependent variable
y m discrete data points
A amplitude
B amplitude
C amplitude
F force (mlt
À2 )
N total number of discrete data points; integer
T period (t)
U unit step function
Y Fourier transform of y
a angle (rad)
b angle (rad)
df frequency resolution (t
À1 )
dt sample time increment (t)
f phase angle (rad)
v circular frequency in rad/s (t
À1 )
PROBLEMS
2.1 Define the term ‘‘signal’’ as it relates to measurement systems. Provide two examples of static and
dynamic input signals to particular measurement systems.
2.2 List the important characteristics of input and output signals and define each.
2.3 Determine the average and rms values for the function
y t
ð Þ ¼ 25 þ 10 sin 6pt
over the time periods (a) 0 to 0.1 s, (b) 0.4 to 0.5 s, (c) 0 to 1/3 s, and (d) 0 to 20 s. Comment on the nature
and meaning of the results in terms of analysis of dynamic signals.
2.4 The following values are obtained by sampling two time-varying signals once every 0.4 s:
t
y 1 (t)
y 2 (t)
t
y 1 (t)
y 2 (t)
0
0
0
0.4
11.76
15.29
2.4
À11.76
À15.29
0.8
19.02
24.73
2.8
À19.02
À24.73
1.2
19.02
24.73
3.2
À19.02
À24.73
1.6
11.76
15.29
3.6
À11.76
À15.29
2.0
0
0
4.0
0
0
Determine the mean and the rms values for this discrete data. Discuss the significance of the rms
value in distinguishing these signals.
2.5 A moving average is an averaging technique that can be applied to an analog, discrete time, or digital
signal. A moving average is based on the concept of windowing, as illustrated in Figure 2.22. That
portion of the signal that lies inside the window is averaged and the average values plotted as a
function of time as the window moves across the signal. A 10-point moving average of the signal in
Figure 2.22 is plotted in Figure 2.23.
a. Discuss the effects of employing a moving average on the signal depicted in Figure 2.22.
b. Develop a computer-based algorithm for computing a moving average, and determine the effect of
the width of the averaging window on the signal described by
y t
ð Þ ¼ sin 5t þ cos 11t
72 Chapter 2 Static and Dynamic Characteristics of Signals