Digital Signal Processing 9.4 Waveform Analysis 217
Part A | 9.4
time instant t to lie within an arbitrary real interval [a; b]
as follows
PrŒa Ä X.t/ Ä b D
b
Z
a
f X .x/ dx :
(9.78)
As well established by probability theory, the probability density function is, in turn, associated with
a cumulative distribution function (CDF) for random
signal X.t/ as follows
F X .x/ D PrŒX.t/ Ä x D
x
Z
1
f X .x # / dx # :
(9.79)
One can easily verify by combining (9.78) with (9.79)
that
PrŒa Ä X.t/ Ä b D F X .b/ F X .a/
(9.80)
holds. Finally and assuming certain smoothness conditions hold, the PDF can be calculated as the derivative
of the CDF for the same random variable, i. e.,
f X .x/ D
d
dx
F X .x/ :
(9.81)
For a random signal one can, in effect, define a probabilistic mean value as follows
hX.t/i D X .t/ ,
C1 Z
1
xf X.t/ .x/ dx :
(9.82)
Along the same lines a probabilistic correlation (cross
and/or autocorrelation) can be introduced
hX.t 1 /; Y.t 2 /i D R X;Y .t 1 ; t 2 / ;
,
C1 Z
1
C1 Z
1
xyf X.t1/;Y.t2/ .x; y/ dydx :
(9.83)
To properly define it, though, the joint probability density function needs to be introduced; as can be seen
below, it quantifies the probability of random signal X
assuming a value in an infinitesimal vicinity of x at instant t 1 and, jointly, random signal Y assuming a value
in an infinitesimal vicinity of y at instant t 2
f X.t1/;Y.t2 / .x; y/ dx dy
D PrŒ.x Ä X.t 1 / Ä x C dx/ and
.y Ä Y.t 2 / Ä y C dy/ :
(9.84)
In effect, the probability on the right-hand side in the
above is that of the joint event to find the values of
signals X and Y at certain time instants within an
arbitrary infinitesimal rectangle on the XY plane. Applying Bayes’ theorem on the joint probability of the
right-hand side of (9.84) one can derive the following
equations involving conditional probabilities
PrŒx Ä X.t 1 / Ä x C dxjy Ä Y.t 2 / Ä y C dy
PrŒy Ä Y.t 2 / Ä y C dy
D f X.t1/;Y.t2 / .x; y/ dx dy
D PrŒy Ä Y.t 2 / Ä y C dyjx Ä X.t 1 / Ä x C dx
PrŒx Ä X.t 1 / Ä x C dx :
(9.85)
In the case that random variables X and Y are independent (not to be confused with mutually exclusive
variables or events), then using (9.85) for the conditional probabilities one can see that the following holds
f X.t1/;Y.t2/ .x; y/ D f X.t1/ .x/ f Y.t2/ .y/
(9.86)
A very important category of random signals is that
of wide sense stationary (WSS). A WSS signal is one
that has: (a) constant, i. e., time invariant, probabilistic
mean, and, (b) autocorrelation solely depending on the
time shift (delay) D t 2 t 1 and not on the individual
time instants. In effect, for a random signal to be WSS
it must hold that
hX.t/i D hXi D X ;
(9.87)
hX.t/; X.t C /i D R X;X ../ :
(9.88)
In part, WSS signals are important because both the
probabilistic mean value and autocorrelation can be obtained as the time-wise mean and autocorrelation of
anyone of its instances, x.t/, introduced for waveforms
earlier. For this to be possible though, the random signal (stochastic process) X needs to be ergodic on top of
WSS. Indeed, for an ergodic process statistical (probabilistic) properties (such as its mean, correlation, and
variance) can be deduced from a single, sufficiently
long instance (sample realization) of the process. Therefore, for an ergodic WSS process, the following hold
hX.t/i D x ;
(9.89)
hX.t/; X.t C /i D R x;x ../ :
(9.90)
It should be noted, however, that for the above equations to hold exactly the integrals over time in (9.74)
Part A | 9.4
time instant t to lie within an arbitrary real interval [a; b]
as follows
PrŒa Ä X.t/ Ä b D
b
Z
a
f X .x/ dx :
(9.78)
As well established by probability theory, the probability density function is, in turn, associated with
a cumulative distribution function (CDF) for random
signal X.t/ as follows
F X .x/ D PrŒX.t/ Ä x D
x
Z
1
f X .x # / dx # :
(9.79)
One can easily verify by combining (9.78) with (9.79)
that
PrŒa Ä X.t/ Ä b D F X .b/ F X .a/
(9.80)
holds. Finally and assuming certain smoothness conditions hold, the PDF can be calculated as the derivative
of the CDF for the same random variable, i. e.,
f X .x/ D
d
dx
F X .x/ :
(9.81)
For a random signal one can, in effect, define a probabilistic mean value as follows
hX.t/i D X .t/ ,
C1 Z
1
xf X.t/ .x/ dx :
(9.82)
Along the same lines a probabilistic correlation (cross
and/or autocorrelation) can be introduced
hX.t 1 /; Y.t 2 /i D R X;Y .t 1 ; t 2 / ;
,
C1 Z
1
C1 Z
1
xyf X.t1/;Y.t2/ .x; y/ dydx :
(9.83)
To properly define it, though, the joint probability density function needs to be introduced; as can be seen
below, it quantifies the probability of random signal X
assuming a value in an infinitesimal vicinity of x at instant t 1 and, jointly, random signal Y assuming a value
in an infinitesimal vicinity of y at instant t 2
f X.t1/;Y.t2 / .x; y/ dx dy
D PrŒ.x Ä X.t 1 / Ä x C dx/ and
.y Ä Y.t 2 / Ä y C dy/ :
(9.84)
In effect, the probability on the right-hand side in the
above is that of the joint event to find the values of
signals X and Y at certain time instants within an
arbitrary infinitesimal rectangle on the XY plane. Applying Bayes’ theorem on the joint probability of the
right-hand side of (9.84) one can derive the following
equations involving conditional probabilities
PrŒx Ä X.t 1 / Ä x C dxjy Ä Y.t 2 / Ä y C dy
PrŒy Ä Y.t 2 / Ä y C dy
D f X.t1/;Y.t2 / .x; y/ dx dy
D PrŒy Ä Y.t 2 / Ä y C dyjx Ä X.t 1 / Ä x C dx
PrŒx Ä X.t 1 / Ä x C dx :
(9.85)
In the case that random variables X and Y are independent (not to be confused with mutually exclusive
variables or events), then using (9.85) for the conditional probabilities one can see that the following holds
f X.t1/;Y.t2/ .x; y/ D f X.t1/ .x/ f Y.t2/ .y/
(9.86)
A very important category of random signals is that
of wide sense stationary (WSS). A WSS signal is one
that has: (a) constant, i. e., time invariant, probabilistic
mean, and, (b) autocorrelation solely depending on the
time shift (delay) D t 2 t 1 and not on the individual
time instants. In effect, for a random signal to be WSS
it must hold that
hX.t/i D hXi D X ;
(9.87)
hX.t/; X.t C /i D R X;X ../ :
(9.88)
In part, WSS signals are important because both the
probabilistic mean value and autocorrelation can be obtained as the time-wise mean and autocorrelation of
anyone of its instances, x.t/, introduced for waveforms
earlier. For this to be possible though, the random signal (stochastic process) X needs to be ergodic on top of
WSS. Indeed, for an ergodic process statistical (probabilistic) properties (such as its mean, correlation, and
variance) can be deduced from a single, sufficiently
long instance (sample realization) of the process. Therefore, for an ergodic WSS process, the following hold
hX.t/i D x ;
(9.89)
hX.t/; X.t C /i D R x;x ../ :
(9.90)
It should be noted, however, that for the above equations to hold exactly the integrals over time in (9.74)