3
The Power
Spectral Density
3.1 INTRODUCTION
The power spectral density is widely used to characterize random processes in
electronic and communication systems. One common application of the power
spectral density is to characterize the noise in a system. From such a
characterization the noise power, and hence, the system signal to noise ratio,
can be evaluated. This chapter gives a detailed justification of the two distinct,
but equivalent ways of defining the power spectral density. The first is via
decomposition, as given by the Fourier transform, of signals comprising
the random process; the second is through the Fourier transform of the
time averaged autocorrelation function of waveforms comprising the ran-
dom process. The first approach is used in later chapters and facilitates analysis
to a greater degree than the second. Finally, the relationship between the
power spectral density and autocorrelation function, as stated by the Wiener—
Khintchine theorem, is justified. A brief historical account of the development
of the theory underlying the power spectral density can be found in Gardner
(1988 pp. 12f ).
3.1.1 Relative Power Measures
In the following sections, the concepts of signal power and signal power
spectral density are introduced and used. Strictly speaking, the concepts are
that of relative signal power and relative signal power spectral density, as
signals typically have units that lead to relative, not absolute, power measures.
To simplify terminology, the word ‘‘relative’’ is dropped. The best justification
for the use of relative power measures, is the signal to noise ratio which is
defined as the signal power divided by the noise power. Provided both the
59
Principles of Random Signal Analysis and Low Noise Design:
The Power Spectral Density and Its Applications.
Roy M. Howard

\
G( f ) df (3.1)
This last requirement is consistent with the sum of the power in the constituent
waveforms equaling the total average power. In summary, a power spectral
density function G, should be such that
(1) G is a continuous function.
(2) G( f
V
) is proportional to the power of the constituent sinusoidal signals
with frequency f
V
.
(3) P

: 

\
G( f ) df.
The following subsections give details of a power spectral density function that
satisfies these three conditions or requirements.
3.2.2 Power Spectral Density of a Single Waveform
A natural basis for the power spectral density is the average power of a signal.
For an interval [0, T ] the average power of a signal x, by definition, is
P

(T ) :
1
T

2

c
0
c
1
c
2
c
3
2
Figure 3.1 Display of power in sinusoidal components of a signal.
−f
o
f
o
−2f
o
2f
o
−3f
o
3f
o
G(T, f )
f
X(T,0)
2
T
c
0
2

into this equation, that
P

(T ) : "a

";0.5


G
"a
G
";"b
G
":


G\
"c
G
":


G\
"X(T, if
M
)"
T 
(3.3)
where the last relationship follows from Eq. (2.137). As per Eq. (2.131), the
power associated with signal components with a frequency of if

\G
":"c
G
" and the display is symmetric with respect
to the vertical axis.
A problem with such a display is that the integral of the function defined by
the graph is zero. To overcome this problem an alternative display, based on
the relationship c
G
: X(T, if
M
)/T, can be constructed as shown in Figure 3.2.
DEFINITION
61
With such a graph the area under the defined function, by construction, equals
the average signal power.
The display in Figure 3.2 is consistent with writing the average power in the
form,
P

(T ) :


G\
"X(T, if
M
)"
T 
(3.4)
The interpretation of the graph in Figure 3.2 is as follows: The area under each

T

X

T,
f ; f
M
/2
f
M
f
M


9- : f : - (3.6)
With such a definition it follows that
P

(T ) :


\
G(T, f ) df (3.7)
which is the third requirement of a power spectral density function.
Such a power spectral density function G, satisfies requirements (2) and (3)
but is not a continuous function. Obtaining a continuous function for the
power spectral density is discussed in the next subsection.
3.2.3 A Continuous Power Spectral Density Function
The basis for obtaining a continuous waveform for the power spectral density
is Parseval’s relationship (Theorem 2.31):

f
x
f
2
G( T,f ) = X (T,f ) /T
G( T,f
x
)
Figure 3.3 Continuous power spectral density function based on Parseval’s relationship.
and a power spectral density function G, as per the following definition:
D:P S D The power spectral density of a signal
x, evaluated on the interval [0, T ], is defined according to
G(T, f ) :
"X(T, f )"
T
(3.10)
This power spectral density function is commonly called the periodogram (see
Gardner, 1988 p. 13) or sample spectral density (Jenkins, 1968 p. 211; Parzen,
1962 p. 109).
The power spectral density function, as defined by Eq. (3.10), has the form
shown in Figure 3.3 and it remains to show that it satisfies the three
requirements of a power spectral density function. To this end note, first, that
the integral of the power spectral density, by construction, equals the total
power. Second, the power spectral density is a continuous function as stated
by the following theorem (Champeney, 1987 p. 60).
T 3.1. C  P S D If x + L [0, T ] then
the power spectral density function G, defined by Eq. (3.10), is continuous with
respect to f for f + R.
Proof. This result can be proved by first proving that X(T, f ) is continuous
with respect to f + R when x + L [0, T ]. The proof is straightforward and is

o
(i − 1)f
o
(i + 1)f
o
f
x
= if
o
=
f
o
G(T, f ) =
2
T
X(T, f )
2
T
X(T, if
o
)
2
f
o
c
i
Figure 3.4 Step approximation to power spectral density function. The area under the two
levels associated with f :9if
o
and f : if

:
1
T
(3.11)
then the average power in the sinusoidal components of x with a frequency if
M
,
and on the interval [0, T ], is given by
"c
\G
";"c
G
": f
M
[G(T, 9if
M
) ; G(T, if
M
)] i + Z>
(3.12)
"c

": f
M
G(T,0) i : 0
Proof. Using the relationship c
G
: X(T, if
M
)/T a step approximation to G can

T

2

x(t) dt :
X(T,0)
T
(3.13)
64
THE POWER SPECTRAL DENSITY
and this implies the following relationships:
P

I
V
(T ) :
1
T

2

"
V
" dt : "
V
":
"X(T,0)"
T 
: "c



GD
M
\D
M

G(T, f ) df
(3.15)
: 2

GD
M
>D
M

GD
M
\D
M

G(T, f ) df
where the last equality in this equation only applies for real signals. How
accurate this approximation is depends on the nature of the signal under
consideration, and hence, G. The following example illustrates this point.
3.2.3.3 Example — Power Spectral Density of a Sinusoid Consider a
sinusoidal signal A sin(2f
A
t) on the interval [0, T ]. From Eq. (3.10) it follows,
after standard analysis, that the power spectral density can be written as
G(T, f ) :

sinc

N

f
f
A
; 1

(3.16)
where T : NT
A
with T
A
: 1/ f
A
. This power spectral density is shown in Figure
3.5 for the case where A : 1, T : 1/f
M
: 1, and f
A
: 4. Note that G(T, if
M
) : 0
as expected, except for the case when if
M
: f
A
. However, it is clearly evident
from this figure that

0 2 4 6 8
0
0.05
0.1
0.15
0.2
0.25
G(T, f )
Frequency (Hz)
Figure 3.5 Power spectral density of a sinusoid with a frequency of 4 Hz, an amplitude of
unity, and evaluated on a 1 sec interval (f
o
: 1).
3.3.1 Symmetry in Power Spectral Density
For the case where x is real it follows that G is an even function with respect
to f, that is, G(T, 9 f ) : G(T, f ). This result follows from Eq. (2.136) which
states:
X(T, 9 f ) : X*(T, f )
3.3.2 Resolution in Power Spectral Density
For a measurement interval of T seconds, the frequency resolution in the
power spectral density is f
M
: 1/T. Clearly, as T increases the resolution
increases. In fact, for any resolution f in frequency, there exists an interval
[0, T ], where T : 1/f, such that the rectangular areas of width f, centered
at the frequencies 9 f
V
and f
V
and with respective heights of G(T, 9f



, that is,
P


: lim
2
1
T

2

"x(t)" dt : lim
2


\
"X(T, f )"
T
df : lim
2


\
G(T, f ) df
(3.18)
If it is possible to interchange the order of integration and limit operations in
the last equation, then P


3.4 RANDOM PROCESSES
Consider a random process X with ensemble
E
6
: +x: S
6
; [0, T ] ; C, (3.21)
RANDOM PROCESSES
67
and associated signal probabilities
P[x(i, t)] : P[x
G
(t)] : p
G
S
6
3 Z> countable case
P[x(, t)"
HZH
M
H
M
>BH
] : f
6
(
M
) d S
6
3 R uncountable case

G
P

G
(T ) :


G
p
G
1
T

2

"x
G
(t)" dt countable case (3.24)
P

(T ) :


\
P

(, T ) f
6
() d
:

"X
G
(T, f )" df :


\
G(T, f ) df countable case (3.26)
P

(T ) :


\

1
T


\
"X(, T, f )"f
6
() d

df
:


\
G(T, f ) df uncountable case
(3.27)

(T, f )":


G
p
G
G
G
(T, f ) countable case
1
T


\
"X(, T, f )"f
6
() d:


\
G(, T, f ) f
6
() d uncountable case
(3.29)
where G
G
(T, f ) and G(, T, f ), respectively, are the power spectral densities for
the ith and th signals in the countable and uncountable ensembles.
Clearly, the power spectral density of a random process is the weighted
average of the power spectral density of individual signals in the ensemble.

2
1
T

2

"x
G
(t)" dt (3.30)
By using Parseval’s relationship, interchanging the order of the summation and
limit operations, and then interchanging the order of the summation and
integration, the average power can be written according to
P


: lim
2


\
1
T


G
p
G
"X
G
(T, f )" df : lim

( f ) : lim
2
G(T, f ) (3.33)
RANDOM PROCESSES
69
–1
1
t
2D
6D
8D
D
p(t)
4D
x(1, −1, −1, 1, –1, 1, 1, –1, t)
Figure 3.6 One waveform from a binary digital random process on the interval [0, 8D].
3.4.2 Example — PSD of Binary Digital Random Process
Consider a binary digital random process X, defined by either a pulse p or its
negative 9p in each interval of D sec. On the interval [0, ND] the ensemble
for this random process is
E
6
:

x(

, ..., 
,
, t) :
,

I
: 1] : P[
I
:91] : 0.5, then the probability of any signal
from the ensemble is 1/2,. It follows from the definition of the power spectral
density function, that the power spectral density of X, evaluated on the interval
[0, ND], is given by
G
6
(ND, f ) :
1
ND

A

% 
A
,
1
2,
"X(

, ..., 
,
, ND, f )" 
G
+ +<1, (3.36)
To evaluate the power spectral density, the assumption is made that the pulse
function is zero outside the interval [0, D]. With this assumption, first note that
"X(

I
"P( f )"e\HLDG\I"D (3.37)
70
THE POWER SPECTRAL DENSITY
0.01
0.05
0.1
0.5 1
0.01
0.02
0.05
0.1
0.2
0.5
1
W = D/2 W = D
Frequency (Hz)
G
X
(ND, f )
Figure 3.7 Power spectral density of binary digital random process for the case where D : 1.
where P is the Fourier transform of p. Next, note that substitution of Eq. (3.37)
into Eq. (3.36) and interchanging the order of summation yields, for the second
term:
1
ND2,
,

G
,

10- t : W
0 elsewhere
P( f ) : W sinc( fW)e\HLD5
0 : W - D
(3.40)
3.4.3 Miscellaneous Issues
3.4.3.1 Nonstationary Random Processes The definition for the power
spectral density [see Eq. (3.29)] is valid for single waveforms, stationary
random processes, and nonstationary random processes, as it is based on
RANDOM PROCESSES
71