Part A | 9.4
218 Part A Fundamentals
and (9.75) need to span the entire real line from minus
to plus infinity. If not, then (9.89) and (9.90) provide only an estimate of the mean and autocorrelation
values of the random signal; the accuracy of the estimate (probabilistically) increases as time parameter
T in the time-wise integrals of (9.74) and (9.75) grow
larger.
A final note is made regarding strictly or strongly
stationary processes versus WSS ones. A strictly stationary process is a stochastic process whose joint
probability distribution does not change when shifted
in time. This requirement is much stronger than just
mean and autocorrelation and, therefore, much harder
to meet or assume in practical situations; that is why in
the remaining text we will only employ WSS as well as
ergodicity.
9.4.2 Signal Power
and Power Spectral Density
For a waveform x.t/, i. e., a deterministic signal or an
instance of a random one, the following definitions are
given:
1. The time mean is the direct current (DC) offset
(component) of the waveform and j x j
2 is the DC
power of it.
2. Consider
R x;x .0/ D lim
T!1
8
<
:
1
2T
CT
Z
T
x
2
.t/ dt
9
=
;
0 : (9.91)
This is the (total averaged) power of the waveform.
3. Then consider
cov.x/ D R x;x .0/ j x j
2
0 :
(9.92)
This is the covariance or alternating current (AC)
power of the waveform. Also, Œcov.x/
1=2 is the
standard deviation or RMS (root-mean-square)
value of the waveform.
The use of term power here stands for electric
power, i. e., that the waveform is considered as voltage
across or electric current through a reference resistance
equal to 1 ohm.
Finally, the following notation for cross-correlation
(for two waveforms) or autocorrelation (for a single
waveform)
hy; xi D R y;x .0/ , lim
T!1
8
<
:
1
2T
CT
Z
T
y.t/x.t/ dt
9
=
;
:
(9.93)
In the case of two distinct waveforms hy; xi is also
identified as their dot product. In the case of a single
waveform hx; xi is its power.
In the frequency domain and using the two-sided
Fourier transform as introduced in (9.8), the (cross)
power spectral density (PSD) can be defined for a couple of waveforms as follows
S y;x .!/ , F fR y;x ../g D Y.!/X
.!/ :
(9.94)
Note that for a real waveform it holds that
X.!/ D F fx.t/g D
C1 Z
1
x.t/ exp.i!t/ dt
m
X
.!/ D
0
@
C1 Z
1
x.t/ exp.i!t/ dt
1
A
D
C1 Z
1
x
.t/ exp.Ci!t/ dt D X.!/ :
Furthermore
F fx.t/g D
C1 Z
1
x.t/ exp.i!.t// d.t/
D
C1 Z
1
x.t/ exp.i!t/ dt D X.!/ :
Therefore, for a real waveform it holds that
X
.!/ D F fx.t/g D X.!/ :
(9.95)
Then, using (9.94) one can directly derive the following
S x;y .!/ D S y;x .!/ D S
y;x .!/
, R x;y ../ D R y;x ./ :
(9.96)
Moreover, in the case of autocorrelation PSD obtains
the following form
S x;x .!/ , F fR x;x ../g D jX.!/j
2
0 and
S x;x .!/ D S x;x .!/ :
(9.97)
218 Part A Fundamentals
and (9.75) need to span the entire real line from minus
to plus infinity. If not, then (9.89) and (9.90) provide only an estimate of the mean and autocorrelation
values of the random signal; the accuracy of the estimate (probabilistically) increases as time parameter
T in the time-wise integrals of (9.74) and (9.75) grow
larger.
A final note is made regarding strictly or strongly
stationary processes versus WSS ones. A strictly stationary process is a stochastic process whose joint
probability distribution does not change when shifted
in time. This requirement is much stronger than just
mean and autocorrelation and, therefore, much harder
to meet or assume in practical situations; that is why in
the remaining text we will only employ WSS as well as
ergodicity.
9.4.2 Signal Power
and Power Spectral Density
For a waveform x.t/, i. e., a deterministic signal or an
instance of a random one, the following definitions are
given:
1. The time mean is the direct current (DC) offset
(component) of the waveform and j x j
2 is the DC
power of it.
2. Consider
R x;x .0/ D lim
T!1
8
<
:
1
2T
CT
Z
T
x
2
.t/ dt
9
=
;
0 : (9.91)
This is the (total averaged) power of the waveform.
3. Then consider
cov.x/ D R x;x .0/ j x j
2
0 :
(9.92)
This is the covariance or alternating current (AC)
power of the waveform. Also, Œcov.x/
1=2 is the
standard deviation or RMS (root-mean-square)
value of the waveform.
The use of term power here stands for electric
power, i. e., that the waveform is considered as voltage
across or electric current through a reference resistance
equal to 1 ohm.
Finally, the following notation for cross-correlation
(for two waveforms) or autocorrelation (for a single
waveform)
hy; xi D R y;x .0/ , lim
T!1
8
<
:
1
2T
CT
Z
T
y.t/x.t/ dt
9
=
;
:
(9.93)
In the case of two distinct waveforms hy; xi is also
identified as their dot product. In the case of a single
waveform hx; xi is its power.
In the frequency domain and using the two-sided
Fourier transform as introduced in (9.8), the (cross)
power spectral density (PSD) can be defined for a couple of waveforms as follows
S y;x .!/ , F fR y;x ../g D Y.!/X
.!/ :
(9.94)
Note that for a real waveform it holds that
X.!/ D F fx.t/g D
C1 Z
1
x.t/ exp.i!t/ dt
m
X
.!/ D
0
@
C1 Z
1
x.t/ exp.i!t/ dt
1
A
D
C1 Z
1
x
.t/ exp.Ci!t/ dt D X.!/ :
Furthermore
F fx.t/g D
C1 Z
1
x.t/ exp.i!.t// d.t/
D
C1 Z
1
x.t/ exp.i!t/ dt D X.!/ :
Therefore, for a real waveform it holds that
X
.!/ D F fx.t/g D X.!/ :
(9.95)
Then, using (9.94) one can directly derive the following
S x;y .!/ D S y;x .!/ D S
y;x .!/
, R x;y ../ D R y;x ./ :
(9.96)
Moreover, in the case of autocorrelation PSD obtains
the following form
S x;x .!/ , F fR x;x ../g D jX.!/j
2
0 and
S x;x .!/ D S x;x .!/ :
(9.97)