5
Power Spectral Density
of Standard Random
Processes — Part 1
5.1 INTRODUCTION
In Chapters 5 and 6 the power spectral density of commonly encountered
random processes are given in detail. Specifically, the power spectral density of
random processes associated with signaling, quantization, jitter, and shot noise
are discussed in this chapter, while the power spectral density associated with
sampling, quadrature amplitude modulation, random walks, and 1/ f noise, are
discussed in Chapter 6.
In this chapter, the random processes discussed have a general form that is
associated with signaling, and the terminology of a signaling random process
is introduced. The results associated with signaling random processes are used
in Chapter 7, to detail an approach for determining the power spectral density
of a random process after a nonlinear memoryless transformation.
5.2 SIGNALING RANDOM PROCESSES
As defined below, the signal form associated with signaling is such that
signaling random processes are found in models for a diverse range of physical
processes. For example, signaling random processes include baseband and
certain bandpass communication processes. The signal form of interest is that
of an information signal.
D:I S An information signal is one generated by
a sum of signals from a ‘‘signaling set,’’ where one signal is associated with each
138
Principles of Random Signal Analysis and Low Noise Design:
The Power Spectral Density and Its Applications.
Roy M. Howard
Copyright
¶
2002 John Wiley & Sons, Inc.
, ...,
,
, t) :
,
G
(
G
, t 9 (i 9 1)D),
G
+ S
, + E
(5.2)
where
G
+ S
and the vector (
,...,
,
) is an element of S
6
: S
;%;S
, (5.3)
The probability of any given signal from the signaling set is given by the
probability of the associated outcome from , that is,
P[(, t)] : P[] : p
A
P[(, t)"
AZA
M
A
M
>BA
] : P[+ [
M
,
M
; d]] : f
(
M
) d
countable case
uncountable case
(5.4)
where f
is the probability density function of the random variable for the
uncountable case. The probability associated with waveforms in E
6
are
,
f
,
(
, ...,
,
) d
...d
,
uncountable case
(5.5)
SIGNALING RANDOM PROCESSES
139
where p
A
A
,
and f
,
, respectively, are the joint probability of
(
) ... f
,
(
,
)(5.6)
Finally, the Fourier transform of a signaling waveform + E
, evaluated
over the interval (9-, -),is
(, f ) :
\
(, t)e\HLDR dt + S
(5.7)
5.2.0.1 Example — Standard Communication Signals Each outcome
of a signaling random process X is an individual signal, and with an
appropriate signaling set, is suitable for use in a communication system. For
the case of signaling at a constant rate r : 1/D, there are two standard
information signals defined on the interval [0, ND] according to
y(
, ...,
,
, t) :
, ...,
,
, t) :
,
G
(
G
, t 9 (i 9 1)D)
G
+ S
: +1, ..., M,
+ E
P[(
G
, t)] : P[
G
] : p
A
G
(5.9)
where E
: +(
G
A
t) sin(t/D)
0
t+ [0, D]
elsewhere
(5.11)
The plotted signal is x(1, 2, 2, 1, 1, t), for the case where A : 1, D : 1, and
f
A
: 4 Hz.
140
POWER SPECTRAL DENSITY OF STANDARD RANDOM PROCESSES — PART 1
1 2 3 4 5
−0.75
−0.5
−0.25
0
0.25
0.5
0.75
1
Time (Sec)
x(1, 2, 2, 1, 1, t)
Figure 5.1 Baseband signaling waveform, A : 1, D : 1, and f
c
: 4.
Note, if the duration of all signaling waveforms in the signaling set is kD
sec, then at any time t after the first k transient signaling intervals, there are
potentially k nonzero waveforms comprising the signal x.
5.2.0.2 Generality of Information Signal Form A broad class of signals
G
+ S
,
G
+ R (5.13)
where S
is the set of signals with bounded variation on the interval [0, D] and
which are zero outside this interval.
5.2.1 Power Spectral Density of a Signaling Random Process
The following theorem details the power spectral density of a signaling random
process.
SIGNALING RANDOM PROCESSES
141
T 5.1. P S D S R P
Assuming the effect of including components of the signaling waveform outside of
the interval [0, ND] is negligible, the dependency between signaling waveforms
depends on the difference between the location of the signaling intervals and not
on their absolute location, and the ith signaling waveform is independent of the
jth signaling waveform for "i 9 j"9m, then the power spectral density of the
random process X, defined by the ensemble as per Eq. (5.2), is
G
6
(ND, f ) : r"( f )" 9 r"
( f )";r"
( f )"
( f )"
L\
( f 9 nr)
; 2r
K
G
Re[eHLG"D(R
>G
( f ) 9 "
( f )")]
(5.15)
where r : 1/D and
( f ) :
A
p
A
(, f ) countable case
A
A
>G
p
A
A
>G
(
, f )*(
>G
, f ) countable case
\
\
(
, f )*(
>G
, f ) f
( f )"
1
N
sin(Nf /r)
sin(f/r)
(5.19)
G
6
( f ) : r"( f )" 9 r"
( f )";r"
( f )"
L\
( f 9 nr) (5.20)
Proof. The proof is given in Appendix 1.
5.2.1.1 Notes The results stated in the above theorem for the independent,
countable, and infinite interval case are consistent with those of van den Elzen
(1970).
The given expressions for G
6
(ND, f ) and G
6
( f ) can be written in a simpler
1
N
sin(Nf /r)
sin(f/r)
9 1
(5.22)
G
6
( f ) : r
+
A
p
A
"(, f )"9r
+
A
p
A
(, f )
(5.23)
; r
+
(t) :
\
(, t) f
() d (5.24)
When
(t) : 0 for t + (9-, -), it follows for the countable case that
( f ) :
A
p
A
(, f ) :
A
p
A
\
(, t)e\HLDR dt
(5.25)
the results given in Theorem 5.1, for the power spectral density of a signaling
random process, simplify to
G
6
(ND, f ) : r"( f )" ; 2r
K
G
1 9
i
N
Re[eHLG"DR
>G
( f )]
(5.26)
G
6
( f ) : r"( f )" ; 2r
K
G
Re[eHLG"DR
p
A
"(, f )" countable case
1
D
\
"(, f )"f
() d uncountable case
(5.28)
Consistent with Eq. (5.7), the contribution of the signaling waveform compo-
nents outside of the interval [0, D] are included in this power spectral density
definition.
5.2.1.3 Case 2: Information Encoded in Pulse Amplitudes Consider
the case where information is encoded in the pulse amplitudes, such that
(, t) :A()(t), and
E
:
A()(t): A() :
A
A
A()
+ Z>
+ R
countable case
p
A
A
A
:
( f ) countable case
( f )
\
A() f
() d :
( f ) uncountable case
(5.31)
"( f )" :
"( f )"
A
p
A
"A
A
":"( f )""A" countable case
"( f )"
A
>G
A
A
A
*
A
>G
countable case
"( f )"
\
\
A(
)A*(
>G
) f
>G
(
,
>G
is the joint probability density
function for amplitudes in the first and 1 ; ith signaling intervals. The
definition for R
>G
is obvious from this equation. With these definitions, it
follows that
G
6
(ND, f ) : r"( f )"
"A" 9 "
";"
"
1
N
sin(Nf /r)
sin(f/r)
; 2
K
( f 9 nr)
; 2
K
G
Re[eHLG"D(R
>G
9 "
")]
(5.35)
SIGNALING RANDOM PROCESSES
145
The variance definition
: "A" 9 "
" (5.36)
can simplify the form of these equations. Carlson (1986 pp. 388—389) gives
equivalent results.
For the independent case, where R
L\
( f 9 nr)
(5.38)
For the independent case, where the mean amplitude
is zero, the simpler
result follows:
G
6
(ND, f ) : G
6
( f ) : r"A" "( f )":r
"( f )" (5.39)
5.2.2 Examples and Spectral Issues for Communication Systems
The above theory has direct application to communication of information via
signaling waveforms, as the power spectral density contains the following
information. First, whether there are signal components in the transmitted
signal which do not convey information. Such components are periodic, show
up as impulses in the power spectral density, and indicate inefficient signaling.
Second, how spectrally efficient the signaling scheme is in terms of the level of
information transmitted in the frequency band containing the majority of
signal energy. The usual measure here is the number of bits of information per
Hz of bandwidth. A greater degree of spectral efficiency allows a greater
number of signal or information channels in a specified frequency band. Third,
the degree of spectral rolloff associated with the residual signal energy outside
A
A
G
(t9(i91)D),
G
+ +1, 2,, A
: 0, A
: A
(5.40)
and the pulse waveform has the form shown in Figure 5.2. Signaling with
such a waveform is called ‘‘return to zero’’ (RZ) signaling. The Fourier
transform of is
( f ) :
1
2r
sinc
f
2r
e\HLDP r : 1/D (5.41)
Assuming independent and equally probable amplitudes, such that
:
A/2, A : A/2,
A
16r
sinc
f
2r
;
A
16
sinc
f
2r
L\
( f 9 nr) (5.43)
The power spectral density is plotted in Figure 5.3 for the case of N : 256,
D : 1, r : 1, and A : 1. Clearly evident in this figure is the continuous sinc
squared form and the discrete ‘‘impulsive’’ components. For the case where
A : 1, the power in the impulsive components at frequencies 0, r,2r,3r, ...is
1/16, 0, 1/4, 0, 1/36, . . . . In Figure 5.4, the power spectral density for the
infinite interval is plotted using logarithmic scaling.
The following can be inferred from these power spectral density graphs.
First, the impulses in the spectrum are wasted power as far as communication
of information is concerned, and thus, RZ signaling is inefficient signaling. The
impulsive components, however, may facilitate synchronization and data
recovery at the receiver. Second, the signaling pulse is relatively narrow, which
,
, t) :
,
G
(
G
, t9(i91)D), + E
,
G
+ +91, 0, 1,
(5.44)
where
E
: +(
G
, t): (
G
, t) :
G
p(, t 9 D/2),
G
+ +91, 0, 1,, (5.45)
p(, t) : A sinc
t
D
A
r
" f ":
r
2
(1 9 )
A
r
cos
2
" f "9r(19)/2
r
r
2
(19) -"f ":
r
2
(1;)
0 " f ".
r
2
(1 ; )
(5.47)
where r : 1/D. P is shown in Figure 5.5 for the cases of : 0.5 and : 1.0.
The parameter is the rolloff factor and is such that 0 :-1. The graph of
p(, t) is shown in Figure 5.6 for the cases of : 0.5 and : 1.0.
β = 0.5
P(β, f )
Frequency (Hz)
β = 1
Figure 5.5 Raised cosine spectrum for the case where r : A : 1.
−2
−1 0 1 2 3
0
0.2
0.4
0.6
0.8
1
Time (Sec)
p(β, t)
β = 1
β = 0.5
Figure 5.6 Inverse Fourier transform of raised cosine spectrum for the case where D : A : 1.
where
"( f )" :
A\
p
A
"(, f )"
(5.49)
R
1
Time (Sec)
x(1, 0, 1, 1, 1, 1, 0, 0, 1, 0, t)
Figure 5.7 Bipolar signaling waveform where pulses associated with a raised cosine spectrum
have been used. : 1, D : 1, A : 1, and data is
+
1, 0, 1, 1, 1, 1, 0, 0, 1, 0
,
.
TABLE 5.1 Possible Outcomes for Two Consecutive Signaling Intervals
Data Signaling Waveforms Probabilities
00 0, 0 p
00
: 0.25
01 0, (91, t )or0,(1, t) p
0,91
: p
01
: 0.125
10 (91, t ), 0 or (1, t ), 0 p
91,0
: p
10
: 0.125
11 (91, t), (1, t )or(1, t), (91, t ) p
91,1
: p
1,91
: 0.125
In these equations, p
1 9
1 9
1
N
cos(2f/r)
(5.50)
G
6
( f ) : r sin(f/r)"P(, f )" (5.51)
where the relationship sin(A) : 0.5 9 0.5 cos(2A) has been used. The power
SIGNALING RANDOM PROCESSES
151
0.2 0.4 0.6 0.8 1
0.1
0.2
0.3
0.4
0.5
0.6
β = 1
Frequency (Hz)
G
X
∞
( f )
β = 0.5
A
−A
∆
y
i
ε
i
x
i
Figure 5.9 Illustration of quantization error with a 2-bit DAC (4 levels).
that in the ith sample interval [(i 9 1)D, iD], the constant level y
G
: x
G
;
G
is
generated. The model is one of an additive error to an ideal signal. In general,
the actual levels in a DAC will vary from device to device because of
manufacturing tolerances and will vary with device age, etc. Accordingly, it is
appropriate to consider an infinite ensemble of DACs, where each is driven by
the same sample values, such that in the i th sample interval
G
is independent
of x
G
when considered across the ensemble. From the nature of quantization, it
follows that
G
takes on values with a uniform distribution, from the interval
G
+
9
2
,
2
(5.52)
where is a pulse function defined according to
(t) :
1
0
0 - t : D
elsewhere
"( f )" :
1
r
sinc
f
r
(5.53)
and x
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