E1C02 09/14/2010
13:35:17 Page 47
Signal Root-Mean-Square Value
Consider finding the magnitude of a constant effective current, I e , that would produce the same total
energy dissipation in the resistor as the time-varying current, I(t), over the time period t 1 to t 2 .
Assuming that the resistance, R, is constant, this current would be determined by equating
I e
ð Þ
2 R t 2 À t 1
ð
Þwith Equation 2.2 to yield
I e ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
t 2 À t 1
ð t 2
t 1
I t
ð Þ
½
2 dt
s
ð2:3Þ
This value of the current is called the root-mean-square (rms) value of the current. Based on this
reasoning, the rms value of any continuous analog variable y(t) over the time, t 2 À t 1 , is expressed as
y rms ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
t 2 À t 1
ð t 2
t 1
y
2
dt
s
ð2:4Þ
Discrete-Time or Digital Signals
A time-dependent analog signal, y(t), can be represented by a discrete set of N numbers over the time
period from t 1 to t 2 through the conversion
yðtÞ ! y rdt
ð Þ
f
g r ¼ 0; 1; . . . ; ðN À 1Þ
which uses the sampling convolution
y rdt
ð Þ
f
g¼ yðtÞd t À rdt
ð
Þ¼ y i
f g i ¼ 0; 1; 2; . . . ; ðN À 1Þ
Here, d t À rdt
ð
Þis the delayed unit impulse function, dt is the sample time increment between
each number, and Ndt ¼ t 2 À t 1 gives the total sample period over which the measurement of y(t)
takes place. The effect of discrete sampling on the original analog signal is demonstrated in Figure 2.6b
in which the analog signal has been replaced by y rdt
ð Þ
f
g , which represents N values of a discrete time
signal representing y(t).
For either a discrete time signal or a digital signal, the mean value can be estimated by the
discrete equivalent of Equation 2.1 as
y ¼
1
N
X NÀ1
i¼0
y i
ð2:5Þ
where each y i is a discrete number in the data set of y rdt
ð Þ
f
g. The mean approximates the static
component of the signal over the time interval t 1 to t 2 . The rms value can be estimated by the discrete
equivalent of Equation 2.4 as
y rms ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
N
X NÀ1
i¼0
y 2
i
v
u
u
t
ð2:6Þ
The rms value takes on additional physical significance when either the signal contains no DC
component or the DC component has been subtracted from the signal. The rms value of a signal
having a zero mean is a statistical measure of the magnitude of the fluctuations in the signal.
2.3 Signal Analysis 47
13:35:17 Page 47
Signal Root-Mean-Square Value
Consider finding the magnitude of a constant effective current, I e , that would produce the same total
energy dissipation in the resistor as the time-varying current, I(t), over the time period t 1 to t 2 .
Assuming that the resistance, R, is constant, this current would be determined by equating
I e
ð Þ
2 R t 2 À t 1
ð
Þwith Equation 2.2 to yield
I e ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
t 2 À t 1
ð t 2
t 1
I t
ð Þ
½
2 dt
s
ð2:3Þ
This value of the current is called the root-mean-square (rms) value of the current. Based on this
reasoning, the rms value of any continuous analog variable y(t) over the time, t 2 À t 1 , is expressed as
y rms ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
t 2 À t 1
ð t 2
t 1
y
2
dt
s
ð2:4Þ
Discrete-Time or Digital Signals
A time-dependent analog signal, y(t), can be represented by a discrete set of N numbers over the time
period from t 1 to t 2 through the conversion
yðtÞ ! y rdt
ð Þ
f
g r ¼ 0; 1; . . . ; ðN À 1Þ
which uses the sampling convolution
y rdt
ð Þ
f
g¼ yðtÞd t À rdt
ð
Þ¼ y i
f g i ¼ 0; 1; 2; . . . ; ðN À 1Þ
Here, d t À rdt
ð
Þis the delayed unit impulse function, dt is the sample time increment between
each number, and Ndt ¼ t 2 À t 1 gives the total sample period over which the measurement of y(t)
takes place. The effect of discrete sampling on the original analog signal is demonstrated in Figure 2.6b
in which the analog signal has been replaced by y rdt
ð Þ
f
g , which represents N values of a discrete time
signal representing y(t).
For either a discrete time signal or a digital signal, the mean value can be estimated by the
discrete equivalent of Equation 2.1 as
y ¼
1
N
X NÀ1
i¼0
y i
ð2:5Þ
where each y i is a discrete number in the data set of y rdt
ð Þ
f
g. The mean approximates the static
component of the signal over the time interval t 1 to t 2 . The rms value can be estimated by the discrete
equivalent of Equation 2.4 as
y rms ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1
N
X NÀ1
i¼0
y 2
i
v
u
u
t
ð2:6Þ
The rms value takes on additional physical significance when either the signal contains no DC
component or the DC component has been subtracted from the signal. The rms value of a signal
having a zero mean is a statistical measure of the magnitude of the fluctuations in the signal.
2.3 Signal Analysis 47