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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. ISBN: 0-471-22617-3 signaling interval of D sec. With a signaling set E , an information signal has the form G ( G , t 9 (i 9 1)D) (5.1) where + E and G is an index variable which defines the signal from E that is associated with the ith signaling interval [(i 9 1)D, iD]. D:S R P A signaling random process X,is one whose ensemble consists of information signals. The ensemble E 6 charac- terizing such a random process for the interval [0, ND]is E 6 : x( , ., , , 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 , which is an index set to distinguish between waveforms in the ensemble. S 3 Z> for the countable case and S 3 R for the uncountable case. Equiv- alently, , ., , are the respective outcomes of N identically distributed random variables , ., , , and is used to denote any one of these. The sample space associated with is S . Associated with each element of the set S , or equivalently with each outcome of the random variable , is a signal, and this association defines the set or ensemble of signaling waveforms, E : E : +(, t): + 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 P[x( , ., , , t)] : P[ , ., , ] : p A A , countable case P[x( , ., , , t)" A G Z' G ]: ' % ' , f , ( , ., , ) d .d , uncountable case (5.5) SIGNALING RANDOM PROCESSES 139 where p A A , and f , , respectively, are the joint probability of , ., , for the countable case and the joint probability density function of . , for the uncountable case. For the independent case p A A , : p A .p A , f , ( , ., , ) : f ( ) . 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) : , G A A G (t 9 (i 9 1)D) G + S : +1, ., M, A A G + +A , ., A + , P[A G ] : P[ G ] : p A G (5.8) x( , ., , , 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 , t): G + S ,. The signaling waveforms are of a pulse form for baseband communication and A , ., A + are signaling amplitudes. The first signal defined above is one where a constant pulse shape is used, and the information is encoded through use of different amplitudes. The second is where different signaling waveforms are used to convey information. Clearly, the second form is more general and includes the first as a subcase. An example of the second signaling form is shown in Figure 5.1, where E :+(1, t), (2, t),, and (1, t) : A cos(2f A t) sin(t/D) 0 t+ [0, D] elsewhere (5.10) (2, t) : A sin(2f 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 can be written in an information signal form. To illustrate this, consider first, the fact that any bandlimited signal x can be written, on the interval (9-, -), in the information signal form (Gabel, 1987), x(t) : G\ x(iD) sinc t 9 iD D (5.12) Second, consider that any signal x with bounded variation can be written, on an interval [0, ND], in the information signal form according to x( , ., , , t) : , A A G (t 9 (i 9 1)D) A 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 )" 1 N sin(Nf /r) sin(f/r) ; 2r K G 1 9 i N Re[eHLG"D(R >G ( f ) 9 " ( f )")] (5.14) G 6 ( f ) : r"( f )" 9 r" ( f )";r" ( 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 \ (, f ) f () d uncountable case (5.16) "( f )" : A p A "(, f )" countable case \ "(, f )" f () dy uncountable case (5.17) R >G ( f ) : A A >G p A A >G ( , f )*( >G , f ) countable case \ \ ( , f )*( >G , f ) f >G ( , >G ) d d >G uncountable case (5.18) 142 POWER SPECTRAL DENSITY OF STANDARD RANDOM PROCESSES — PART 1 For the independent case R G >G ( f ) : " ( f )" and G 6 (ND, f ) : r"( f )" 9 r" ( f )";r" ( 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 form with the variance definition ( f ) : "( f )" 9 " ( f )" (5.21) however, the given forms best facilitate evaluation of the power spectral density. As the discrete and independent case, where there are M possible signaling waveforms, commonly occurs the following explicit expressions are useful: G 6 (ND, f ) : r + A p A "(, f )";r + A p A (, f ) 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 + A p A (, f ) L\ ( f 9 nr) The equations specified in Theorem 5.1 can be considerably simplified if the mean of the Fourier transform of the signaling waveforms , is zero. A sufficient condition for this is for the mean of the signaling waveforms to be zero, that is, (t) : 0 for t + (9-, -), where is defined, respectively, for the countable and uncountable cases according to (t) : A p A (, t) (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) : \ A p A (, t) e\HLDR dt : 0 SIGNALING RANDOM PROCESSES 143 The interchange of summation and integration in this equation is valid, according to the dominated convergence theorem, when there exists a function g+ L , such that + A p A (, t) : g(t) for all values of M + Z>. A typical case is where sup+"(, t)": + Z>, is bounded and integrable on the infinite interval, and for this case the interchange is valid. A similar argument can be used for the uncountable case. 5.2.1.2 Case 1: Mean of Signaling Waveforms is Zero For the case where the mean of the signaling waveforms is such that ( f ) : 0 for f + R, 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 >G ( f )] When : 0 and the signaling waveforms in different signaling intervals are independent, the simple result G 6 (ND, f ) : G 6 ( f ) : r"( f )" : G (D, f ) (5.27) holds, where G is the power spectral density of the random process defined by the ensemble E , as per Eq. (5.3), that is, G (D, f ) : 1 D A 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 uncountable case (5.29) where P[A A ] : P[] : p A P[A()" AZA M A M >BA ] : f ( M ) d (5.30) 144 POWER SPECTRAL DENSITY OF STANDARD RANDOM PROCESSES — PART 1 It then follows that ( f ) : ( f ) A 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()"f () d : "( f )""A" uncountable case (5.32) where, and "A", respectively, are the mean and mean square value of the signaling amplitudes. Further, R >G ( f ) : "( f )" A A >G p A A >G A A A * A >G countable case "( f )" \ \ A( )A*( >G ) f >G ( , >G ) d d >G uncountable case : "( f )"R >G (5.33) where, p A A >G is the joint probability of in the first signaling interval and >G in the 1 ; ith signaling interval, or equivalently, the joint probability of the amplitude A A in the first signaling interval, and the amplitude A A >G in the 1 ; ith signaling interval. Similarly, f >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 G 1 9 i N Re[eHLG"D(R >G 9 " ")] (5.34) G 6 ( f ) : r"( f )" "A" 9 " ";r" " L\ ( 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 >G : " ", it follows that G 6 (ND, f ) : r"( f )" "A" 9 " ";" " 1 N sin(Nf /r) sin(f/r) (5.37) G 6 ( f ) : r"( f )" "A" 9 " ";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 of the band used to measure spectral efficiency. The degree of spectral rolloff is a measure of the spectral spread and such spread impairs the ability of a receiver associated with an adjacent signal channel to recover a signal in that adjacent channel. The following examples give some insight into these issues, although they primarily illustrate the evaluation of the power spectral density of a signaling random process. 5.2.2.1 Example: Power Spectral Density of a Return to Zero Signal Consider the case of a signaling random process, defined for the interval 146 POWER SPECTRAL DENSITY OF STANDARD RANDOM PROCESSES — PART 1 D t 1 D ⁄ 2 φ(t) Figure 5.2 Pulse waveform. [0, ND] by the ensemble E 7 : y( , ., , , t) : , G 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/4, and R >G : " " for i . 1, it then follows from Eqs. (5.37) and (5.38), that the power spectral density is given by G 7 (ND, f ) : A 16r sinc f 2r 1 ; 1 N sin(Nf /r) sin(f/r) (5.42) G 7 ( f ) : 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 SIGNALING RANDOM PROCESSES 147
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