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Strictly speaking, the concepts arethat of relative signal power and relative signal power spectral density, assignals typically have units that lead to relative, not absolute, power mea

3 The Power Spectral Density

3.1 INTRODUCTION

The power spectral density is widely used to characterize random processes inelectronic and communication systems One common application of the powerspectral density is to characterize the noise in a system From such acharacterization 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 viadecomposition, as given by the Fourier transform, of signals comprisingthe random process; the second is through the Fourier transform of thetime 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 powerspectral density are introduced and used Strictly speaking, the concepts arethat of relative signal power and relative signal power spectral density, assignals typically have units that lead to relative, not absolute, power measures

To simplify terminology, the word ‘‘relative’’ is dropped The best justificationfor the use of relative power measures, is the signal to noise ratio which isdefined as the signal power divided by the noise power Provided both the

59

The Power Spectral Density and Its Applications.

Roy M Howard Copyright ¶ 2002 John Wiley & Sons, Inc.

ISBN: 0-471-22617-3

signal and noise have the same units, for example, watts or volts squared, itdoes not matter whether relative or absolute power measures are used Further,

in many electronic circuit applications a relative power measure is appropriate

as it is current and voltage levels, not power levels, that are of interest

3.2 DEFINITION

The approach detailed in this section is consistent with that of Priestley(1981

ch 4.3—4.8), Jenkins (1968 ch 6), and Peebles (1993 ch 7)

3.2.1 Characteristics of a Power Spectral Density

A power spectral density function, G, based on the standard sinusoidal or

complex exponential basis set should have the following characteristics First,

to facilitate analysis it should be a continuous signal Second, it should have

the interpretation that G( fV) is directly proportional to the power in the sinusoidal components of the signal with a frequency of fV Hz Third, this

proportionality should be such that the integral of the power spectral density

over all possible frequencies equals the average signal power denoted P
, that is,

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

2 2 2

2 2

where the last relationship follows from Eq (2.137) As per Eq (2.131), the

power associated with signal components with a frequency of ifM Hz, namely,

aG cos(2ifMt) and bG sin(2ifMt), is given by c\G;cG:(aG; bG)/2

Con-sistent with this result, Figure 3.1 represents one way to display the power inthe sinusoidal components of a single waveform, subject to the interpretation

that the power in the sinusoidal components with a frequency of ifM is the sum

of the values defined by the graph at frequencies ofNote, for a real signal 9ifM and ifM Hz.

c\G:cG 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 bythe graph is zero To overcome this problem an alternative display, based on

the relationship cG :X(T, ifM)/T, can be constructed as shown in Figure 3.2.

With such a graph the area under the defined function, by construction, equalsthe average signal power.

The display in Figure 3.2 is consistent with writing the average power in theform,

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 thepower 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):

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

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

omitted

Third, the last requirement of a power spectral density function is that

G(T, fV) should be proportional to the power in the constituent sinusoidal components that have a frequency of fV Hz This is not obviously the case, because a Fourier series decomposition on the interval [0, T ] only yields sinusoids with frequencies fM, 2fM, , where fM :1/T It may well be the case that fV is not an integer multiple of fM This issue is discussed in the following

subsection

3.2.3.1 Interpretation of Continuous Power Spectral Density Function

As a Fourier series decomposition of a signal on an interval [0, T ] yields sinusoidal components with frequencies fM, 2fM, it is reasonable to conclude

that G(T, fV) should only be interpreted for fV:ifM, i+Z> The problem then

is, how to interpret G(T, ifM) for some integer value of i The interpretation is

given in the following theorem

T 3.2 I  P S D If x + L [0, T ], and the power spectral density of x is defined according to

Proof Using the relationship cG :X(T, ifM)/T a step approximation to G can

be defined, as shown in Figure 3.4 and consistent with that shown in Figure3.2 With such a step approximation the area under each pair of levels with

width fM, centered at <ifM and with respective heights X(T, 9ifM)/T and

X(T, ifM)/T, equals c\G;cG, and hence, the power in the sinusoidal components with a frequency of ifM Hz.This theorem states that the power in the mean of a signal is given byfMG(T, 0) To confirm this, note that the mean V of the signal x on [0, T ] is

and this implies the following relationships:

3.2.3.2 Power as Area Under the Power Spectral Density Graph The

power in the sinusoidal components with a frequency ifM can be approximated

GDM\DM

G(T, f ) df

where the last equality in this equation only applies for real signals Howaccurate 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 fAt) on the interval [0, T ] From Eq (3.10) it follows,

after standard analysis, that the power spectral density can be written as

as expected, except for the case when ifM : fA However, it is clearly evident

from this figure that

0 2 4 6 80

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 fM:1/T Clearly, as T increases the resolution

increases In fact, for any resolution

[0, T ], where T

at the frequencies9 fV and fV and with respective heights of G(T, 9fV) and G(T, fV), equal the power in the sinusoidal components of the signal with a frequency fV The assumption here is that the frequency fV is some integer

multiple of the resolution

is interpreted as

3.3.3 Integrability of Power Spectral Density

An important property of the power spectral density function G, is that, in

general, it is integrable

T 3.3 I  P S D If x + L [0, T ] then G + L

Proof Given x + L [0, T ] it follows from Parseval’s relationship that



\X(T, f ) df

is finite which implies the integrability of G.

3.3.4 Power Spectral Density on Infinite Interval

Taking the limit as T tends toward infinity of the average power on the interval [0, T ] yields a definition for the average signal power on the interval (0, -),

denoted P
, that is,

If it is possible to interchange the order of integration and limit operations in

the last equation, then P

and a power spectral density function G, for the interval [0, -] can be

defined according to the following definition

D: P S D  I I

G( f ) : lim 2 G(T, f ) (3.20)Note, the standard results that dictate whether it is possible to interchange theorder of integration and limit operations are the Dominated and Monotoneconvergence theorems(Theorems 2.23 and 2.24)

3.4 RANDOM PROCESSES

Consider a random process X with ensemble

and associated signal probabilities

P[x(i, t)] : P[xG(t)] :pG

The average power in an individual waveform from the ensemble evaluated

over the interval [0, T ] is

P
(, T ) :1

T 2

For the countable case it is convenient to use a subscript rather than an

argument according to x(i, T )The probabilities defined in Eq.: xG(T ) and P
(i, T ) :P
G(T ).(3.22) are the ‘‘natural’’ weighting factor to

use in determining the average signal power according to

X, on the interval [0, T ], and defined according to:

3.4.1 Power Spectral Density on Infinite Interval

The countable case is considered here The analysis for the uncountable casefollows in an analogous manner

The average power of the random process on the interval [0,-], denoted

P
, by definition, is the weighted sum of the individual signal powers

comprising the random process X as T tends towards infinity, that is,

P
 : 

G pG lim 2

P
G (T ): 

G pG lim 2

where G: R;R is the power spectral density for the random process on the

infinite interval(0, -) and defined as:

D: P S D  I I

G( f ) : lim 2 G(T, f ) (3.33)

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

negative9p in each interval of D sec On the interval [0, ND] the ensemble

for this random process is

E6 :x( , , ,, t) : ,

I Ip(t 9(k 91)D), I+91, 1  (3.34)One of the 2, possible waveforms in this ensemble is shown in Figure 3.6 for the case of a rectangular pulse function of duration D/2 sec For the case where the pulses are independent from one interval of D sec to the next, the

probability of a signal from the ensemble is

G6(ND, f ) : ND1 

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

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 secondterm:

where, r : 1/D Note that the power spectral density in independent of the

interval being considered This is due to the fact that the pulse function is zero

outside the interval [0, D] The power spectral density is plotted in Figure 3.7

for the case where the pulse function is rectangular, that is,

average signal power over the interval [0, T ] However, care needs to be taken

when interpreting the power spectral density when the random process isnonstationary

3.4.3.2 Single-Sided Power Spectral Density When x is real, it follows

from Eq (2.136) that G is an even function with respect to f, that is, G(T, 9 f ) : G(T, f ) This leads some authors to define a single-sided power

spectral density function according to

GQQ(T, f ) :2G(T, f )

0

f 0

Such a definition is not used in subsequent analysis

3.4.3.3 Discrete Approximation for the Uncountable Case In analysis,

it is often convenient to replace the continuous random variable characterizing

the waveforms in the ensemble by N outcomes of a discrete random variable.

To this end, consider the random process X defined by the ensemble

where

Next, consider a random process X with a finite number of outcomes

defined by the ensemble

E6 :xG : [0, T ] ; C i + 1, , N

xG (t) : x(G, t) G+IG (3.44)whereIG ,G is a partition of R and

P[xG]: pG:'G f6() d (3.45)The power spectral density of X can then be defined according to

T 3.4 D A  U R P

Assume T is fixed, sup X(, T, f ):  + R, f + R is finite, and that all waveforms

in the uncountable ensemble, E6, are L ebesgue integrable on [0, T ] Further, assume that the signal defined by y( ) : x(, t)R  has bounded variation over all finite intervals of the form [ 9M, M], that is, kM(t) 0, such that for all finite

N, for all partitions IG ,G of [9M, M], and with G+IG, it is the case that

Proof The proof is given in Appendix 1.

3.5 EXISTENCE CRITERIA

Sufficient conditions for the validity of the definitions of G and G,

respec-tively, defined by Eqs.(3.29) and (3.33), are stated in the following theorem

T 3.5 C  E  P S D

(a) Finite Interval: If x + L [0, T ] for each signal in the ensemble and the average power, defined by Eqs (3.24) and (3.25) is finite, then the power spectral density function G, defined by Eq (3.29), is valid.

T he average power is guaranteed to be finite if there exists constants , kM, IM0, such that

i  IM pGP
G(T )  i >? kM countable case

  IM P
(, T ) f6()  >? kM uncountable case

(3.48)

If sup P
G(T ): i+Z> for the countable case and supP
(, T ): +R for the

uncountable case are finite, then the average power is guaranteed to be finite Note, it can be the case that limG P
G(T ) is infinite while the average power

P
(T ) is finite.

(b) Infinite Interval T he power spectral density over the infinite interval G,

as defined by Eq (3.33), is valid if first, x + L [0, T ] for each signal in the ensemble and for all T + R> Second, for the countable case, there exists a

sequence QG: i+Z> such that pGP
G(T ) QG for every i+Z>, and T +R> and

G QG is finite For the uncountable case, there exists a function Q +L , such

the countable case sup P
G(T ): i+Z>, T +R>