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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 Copyright ¶ 2002 John Wiley & Sons, Inc. ISBN: 0-471-22617-3 signal and noise have the same units, for example, watts or volts squared, it does 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( f V ) is directly proportional to the power in the sinusoidal components of the signal with a frequency of f V 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, P : \ 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 "x(t)" dt (3.2) Assume that x is either piecewise smooth or of bounded variation. It then 60 THE POWER SPECTRAL DENSITY 2 2 2 2 2 −3f o −2f o −f o f o 2f o 3f o 2 f c −3 c −2 c −1 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 f o = X(T, −2f o ) T 2 c −2 2 f o = X(T, 2f o ) T 2 c 2 2 f o = Figure 3.2 A power spectral density function. T he shaded areas equal the power associated with sinusoidal components that have a frequency of 2f o Hz. follows, from substitution of the Fourier series for the signal x [see Eq. (2.126)] 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 M Hz, namely, a G cos(2if M t) and b G sin(2if M t), is given by "c \G ";"c G ":("a G ";"b G ")/2. Con- sistent with this result, Figure 3.1 represents one way to display the power in the sinusoidal components of a single waveform, subject to the interpretation that the power in the sinusoidal components with a frequency of if M is the sum of the values defined by the graph at frequencies of 9if M and if M Hz. Note, for a real signal "c \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 pair of levels of the graph associated with the frequencies 9if M and if M , equals the power in the sinusoidal waveforms with a frequency if M . Consistent with this graph, the power spectral density function, G, can be defined as G(T, f ) : "X(T, if M )" T if M 9 f M 2 - f : if M ; f M 2 (3.5) or, more generally, according to G(T, f ) : 1 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): 2 "x(t)" dt : \ "X(T, f )" df (3.8) Scaling both integrals by T yields P (T ) : 1 T 2 "x(t)" dt : \ "X(T, f )" T df : \ G(T, f ) df (3.9) 62 THE POWER SPECTRAL DENSITY 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 omitted. Third, the last requirement of a power spectral density function is that G(T, f V ) should be proportional to the power in the constituent sinusoidal components that have a frequency of f V Hz. This is not obviously the case, because a Fourier series decomposition on the interval [0, T ] only yields sinusoids with frequencies f M ,2f M , ., where f M : 1/T. It may well be the case that f V is not an integer multiple of f M . This issue is discussed in the following subsection. DEFINITION 63 f −(i + 1)f o −(i − 1)f o −if 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 o equals the power in the sinusoidal components of the signal with a frequency of if o Hz. 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 f M ,2f M , .it is reasonable to conclude that G(T, f V ) should only be interpreted for f V : if M , i + Z>. The problem then is, how to interpret G(T, if M ) 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 G(T, f ) : "X(T, f )" T f M : 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 be defined, as shown in Figure 3.4 and consistent with that shown in Figure 3.2. With such a step approximation the area under each pair of levels with width f M , centered at <if M and with respective heights "X(T, 9if M )"/T and "X(T, if M )"/T, equals "c \G ";"c G ", and hence, the power in the sinusoidal components with a frequency of if M Hz. This theorem states that the power in the mean of a signal is given by f M G(T,0). To confirm this, note that the mean V of the signal x on [0, T ]is given by V (T ) : 1 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 ": f M G(T,0) (3.14) 3.2.3.2 Power as Area Under the Power Spectral Density Graph The power in the sinusoidal components with a frequency if M can be approximated by the integral "c \G ";"c G " \GD M >D M \GD M \D M G(T, f ) df ; GD M >D M 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(2f 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 ) : AT 4 sinc N f f A 9 1 ; sinc N f f A ; 1 9 2 sinc N f f A 9 1 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 "c \G ";"c G ":2"c G ":0 " 2 GD M >D M GD M \D M G(T, f ) df i : 1, 2, 3, 5, 6, . f M : 1 (3.17) 3.3 PROPERTIES The following subsections detail basic properties of the defined power spectral density function. PROPERTIES 65 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 V ) and G(T, f V ), equal the power in the sinusoidal components of the signal with a frequency f V . The assumption here is that the frequency f V is some integer multiple of the resolution f. This result is illustrated in Figure 3.4 provided f M is interpreted as f. Note that, in general, G(T, f ) will vary with T. 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. 66 THE POWER SPECTRAL DENSITY 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, 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 can be rewritten as P : lim 2 \ G(T, f ) df : \ lim 2 G(T, f ) df (3.19) 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 the order of integration and limit operations are the Dominated and Monotone convergence theorems (Theorems 2.23 and 2.24). 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)" HZH M H M >BH ] : f 6 ( M ) d S 6 3 R uncountable case (3.22) The average power in an individual waveform from the ensemble evaluated over the interval [0, T ]is P (, T ) : 1 T 2 "x(, t)" dt (3.23) For the countable case it is convenient to use a subscript rather than an argument according to x(i, T ) : x G (T ) and P (i, T ) : P G (T ). The probabilities defined in Eq. (3.22) are the ‘‘natural’’ weighting factor to use in determining the average signal power according to P (T ) : G p 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 : \ 1 T 2 "x(, t)" dt f 6 () d uncountable case (3.25) Provided Parseval’s relationship can be applied, and the order of summa- tion/integration and integration can be interchanged, then the average signal power can be written as P (T ) : \ 1 T G p G "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) where X G (T, f ) : 2 x G (t)e\HLDR dt X(, T, f ) : 2 x(, t)e\HLDR dt (3.28) and G: R ; R is the power spectral density function for the random process X, on the interval [0, T ], and defined according to: 68 THE POWER SPECTRAL DENSITY
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