Generalized Phase Crossing Rate and Random FM Noise for Weibull Fading Channels

Daniel Benevides da Costa, Michel Daoud Yacoub, and Gustavo Fraindenraich State University of Campinas, School Electrical and Computer Engineering, Department of Communications, PO Box 6101, Campinas, SP, Brazil, 13083-970, Email: [daniel,michel,gf]@wisstek.org

Abstract— In this paper, exact closed-form expressions are within an arbitrary range, whose inferior and superior limits derived for the generalized phase crossing rate as well as for the are denoted here as r1 and r2, respectively. In addition, crossing statistics of random frequency modulation (FM) noise in the probability density functions of the signal envelope and Weibull fading channels. These expressions specialize to those of the Rayleigh case, which are known in the literature. In addition, random FM noise conditioned on the upward crossings of the the conditional probability density functions of the envelope and phase through a ﬁxed level are obtained. The expressions are the random FM noise conditioned on an arbitrary phase upward validated by specializing the general results to some particular crossing level are obtained. cases whose solutions are known. Index Terms— Crossing statistics, phase crossing rate, random FM noise, Weibull fading channels. II. PRELIMINARIES AND THE WEIBULL FADING MODEL

I. INTRODUCTION The Rayleigh fading envelope Rl and phase Θl are The Weibull distribution is an empirical distribution, which described by was first used as a statistical model for reliability analysis. 2 2 Rl = X + Y (1a) Its simplicity and flexibility soon paved its way to wireless communications applications. In [1], a physical model for p Y Θ = arctan (1b) a generalized fading distribution was proposed, in which l X Weibull appears as a special case. In [2], this model was where X and Y are independent zero-mean Gaussian used in order to obtain higher order statistics for the Weibull processes with identical variances σ2. The time derivative of environment. In essence, as proposed in [1], the fading model X and Y are, respectively, denoted as X˙ and Y˙ , which have for the Weibull distribution considers a signal composed of a 2 2 2 2 equal variances expressed as σ˙ = 2π fmσ , where fm is the cluster of multipath waves propagating in a non-homogeneous maximum Doppler shift in Hertz [6]. From (1), the Rayleigh environment. Within this cluster, the phases of the scattered complex signal can be expressed as Z = R exp(jΘ ). waves are random. The resulting envelope is obtained as a l l l Let R and Θ be random variables representing, respectively, non-linear function of the modulus of the sum of the multipath the envelope and phase of the Weibull signal. In the Weibull components. Whereas in [1] the non-linearity is assumed to fading model, the probability density function (PDF) p (r) affect the envelope, in the present work, both envelope and R and the cumulative distribution function (CDF) P (r) of the phase are considered to be influenced by it. R envelope are well-known statistics given by In wireless communication systems, the envelope and phase α 1 α of the received signal vary in a random manner because of αr − r p (r) = exp( ) (2a) multipath fading. It is therefore important to investigate the R Ω − Ω second order statistics of the fading channels. Besides the α statistics of the fading envelope, the statistical properties of r PR(r) = 1 exp( ) (2b) the phase process and its derivative are also of interest. For − − Ω instance, the phase behavior characterization is useful in the where α stands for the Weibull fading parameter (α > 0) and design of optimal carrier recovery schemes needed in the 1Ω = E(rα) = 2σ2. synchronization subsystem of coherent receivers [3]. Another According to [1], the resulting complex signal of a Weibull notable example involves the performance of FM receivers process is a non-linear process obtained not simply as the using a limiter-discriminator for detection, where random FM modulus of the sum of the multipath component, but as this spikes generated by phase jumps deteriorate the error-rate modulus to a certain given exponent. Suppose that such a non- performance [4]. Thus, the level crossing theory plays a central linearity is in the form of the power parameter α so that the role in the determination of the statistical properties of the resulting complex signal Z is channel phase and random FM noise [5]. 2/α 2/α (3) In this paper, we derive closed-form expressions for the sta- Z = Zl = Rl exp(j 2 Θl/α) tistics of random FM noise in Weibull fading channels. Closed- = R exp(j Θ) (4) form expressions are also derived for the phase crossing rate (PCR) conditioned on the fact that the fading envelope is 1E(·) denotes the expectation operator 2

As α increases, the severity of the fading decreases. For the special case of α = 2, (3) reduces to the well-known Rayleigh 0,07 complex process. 0,06 α/fm = 0.1, 0.2, ..., 1 Because the non-linearity is in the form of a power, the resulting envelope is observed as the modulus of the multipath 0,05 Rayleigh component Rl to the power of 2/α > 0. As seen ) ˙

θ 0,04 before, this non-linearity also affects the phase, so that the ( ˙ Θ resulting phase is obtained as a linear function of the phase p 0,03 Rayleigh component Θl. Then, from (3) and (4), the relation between the Weibull and the Rayleigh processes is given by 0,02

2 α R = Rl (5a) 0,01 PSfrag replacements 2Θ 0,00 Θ = l (5b) -80 -60 -40 -20 0 20 40 60 80 α θ˙

III. STATISTICS OF RANDOM FM NOISE The random nature of the time-varying phase of the fading signal, denoted as Θ(˙ t), causes a phenomenon known as Fig. 1. PDF of Θ˙ for different values of α/fm random FM noise [6]. For the purpose of deriving the statistics of random FM noise and the analysis of the crossing statistics, ˙ ˙ the joint PDF pR,R˙ ,Θ,Θ˙ (r, r˙, θ, θ) of the processes R(t), R(t), Θ(t), and Θ(˙ t) is required. This joint PDF can be obtained 3 2 2 2 from the joint PDF Rayleigh ˙ , which 2α 4 α θ˙ − pRl,R˙l,Θ1,Θ˙l (rl, r˙l, θl, θl) ˙ pΘ˙ (θ) = 2 2 + 2 (10) is given in [6]. By means of (5) and [6, Eq. 1.3-33], and σ σ˙ σ σ˙ ! following the standard statistical procedure of transformation By setting α = 2, (10) reduces to the case Rayleigh fading of variates, the Weibull joint PDF can be expressed as given in [6, Eq. 1.4-1], as expected. It follows from (10) that the CDF of Θ(˙ t) is obtained in an exact manner as rα 1 rα ˙ 1 pR,R˙ ,Θ,Θ˙ (r, r˙, θ, θ) = exp 2 2 4π2σ2σ˙ 2 −2 σ2 1 αθ˙ 4 α2θ˙ − P (θ˙ ) = 1 + 0 + 0 (11) 2 α 2 2 α 2 ˙2 Θ˙ 0 2 2 α r − r˙ r α θ 1 2  σ˙ σ σ˙ !  + 2 + 2 (6) 4σ˙ 4σ˙ !# × J | | Again, by setting α = 2, (11) reduces to [6, Eq. 1.4-4]. The where J is the Jacobian of the transformation given by fact that p ˙ (θ˙) is an even function in θ˙ leads to E Θ˙ (t) = 0 −α Θ 16r2 | | { } ˙ α4 . By substituting J in (6), the joint PDF of the Furthermore, it can be verified that the second moment of Θ(t) envelope R and the phase|Θ|, and their respective time deriv- equals atives, can be derived as ˙ 2 ∞ ˙2 ˙ ˙ 4 2α 2 E Θ(t) = θ pΘ˙ (θ)dθ = (12) ˙ α r − { } ∞ p ˙ ˙ (r, r˙, θ, θ) = Z−∞ R,R,Θ,Θ 64π2σ2σ˙ 2 Thus, the variance of Θ(˙ t), defined as σ2 = E Θ(˙ t)2 α 2 2 2 2 2 ˙2 2 2 Θ˙ r − α σ r˙ + r θ + 4r σ˙ ˙ 2 { } − exp E Θ(t) , is infinite. × − 8σ2σ˙ 2  { } ˙ For illustration purpose, Figs. 1 and 2 sketch pΘ˙ (θ) and ˙ ˙  (7) PΘ˙ (θ) for several values of α/fm. In Fig. 1, note that the Θ become determinist when α/fm tends to infinity. From Fig. where r 0, < r˙ < , 2π θ < 2π , and < ≥ −∞ −∞ − α ≤ α −∞ 2, as α/fm increases, more quickly the CDF tends towards to θ˙ < . Note that (7) does not depend on the phase θ. ∞ the unitary value. Performing the integrating in (7) with respect to r˙, the joint ˙ PDF pR,Θ,Θ˙ (r, θ, θ) can be directly obtained as

3 3α 1 2 2 2 ˙ IV. GENERALIZED PCR ˙ α r − 1 α 4 α θ pR,Θ,Θ˙ (r, θ, θ) = 3 exp r 2 + 16√2π 2 σ2σ˙ "−8 σ σ˙2 !# (8) The usual (unconditioned) PCR, denoted by NΘ(θ0), is defined as the average number of upward (or downward) In a similar manner, we found that crossings per second at a specified phase level θ0. This 3 2 definition can be extended to a general case, in which the α2 4 α2θ˙2 − p ˙ (θ, θ˙) = + (9) crossing rate of the phase is conditioned on the arbitrary range Θ,Θ 2πσ2σ˙ σ2 σ˙ 2 ! r1 and r2 of the fading envelope. Thus, a general expression 3

same result of the Rayleigh case. Although this leads to a 1,2 same result, (9) is different of the Rayleigh case. Thus, for the

α/fm = 0.2, 0.3, ..., 1 unconditional case, it can be observed that the non-linearity, 1,0 expressed for the α parameter, does not affect the phase process, although the crossing rate of the envelope process 0,8 [7] be affected. To the best of the authors’ knowledge, this ) ˙ θ

( has never been reported in the literature. ˙ Θ 0,6 P

0,4

0,2 V. CONDITIONAL PDFS OF R(t) AND Θ(˙ t)

PSfrag replacements 0,0 -80 -60 -40 -20 0 20 40 60 80 θ˙ This section relates the conditional PDFs of the envelope and the random FM noise Θ(˙ t) conditioned on the upward crossings events occurring at an arbitrary phase crossing 2 level + θ0 . These conditional PDFs allows us to describe the statistics ˙ + Fig. 2. CDF of Θ˙ for different values of α/fm of R(t) and Θ(t) at the instants when Θ(t) = θ0 . In the following, p + (r0) and p ˙ + (θ˙0) denote, respectively, the R θ0 Θ θ0 | | + conditional PDF of R(t) given the phase crossing level θ0 and ˙ + for PCR can be presented as the conditional PDF of Θ(t) given the phase crossing level θ0 . To obtain the PDFs described above, it is necessary to attain ∞ ˙ ˙ ˙ NΘ R(θ0; r1, r2) = θpΘ,Θ˙ ;R(θ, θ r1 < R < r2)dθ the joint conditional CDF P ˙ + (r0, θ˙0) which is defined in | | R,Θ θ0 Z0 | r2 ˙ ˙ ˙ [8] and [9] as ∞ θp Θ Θ˙ (r, θ0, θ)dθdr r1 0 R, , ˙ = (13) r0 θ0 dr θ˙p (r, θ+, θ˙)dθ˙ R R PR(r2) PR(r1) 0 0 R,Θ,Θ˙ 0 − P ˙ + (r0, θ˙0) = R,Θ θ0 + Substituting (2b) and (8) into (13), and carrying out some ma- | R R NΘ(θ0 ) nipulations, the authors have found an exact and closed-form r0 0, θ˙0 0 (17) expression for generalized PCR in Weibull fading channels ≥ ≥ ˙ Replacing (8) and (16) in (17), PR,Θ˙ θ+ (r0, θ0) follows. For α/2 α/2 | 0 σ˙ r1 r2 the sake of simplicity this expression will be omitted. NΘ(θ0; r1, r2) = erf , 4πσ √2σ √2σ ˙ ! From the joint conditional CDF PR,Θ˙ θ+ (r0, θ0), the desired | 0 conditional PDF pR θ+ (r0) is obtained as 1 | 0 α α (14) ×  r1 r2  d exp 2 exp 2 2σ 2σ p + (r0) = P Θ˙ + (r0, ) − − − R θ0 dr R, θ0 | α | ∞  b  1 where erf(a, b) = 2 exp( t2)dt. Note that (14) is αr 2 − rα √π a − = 0 exp 0 (18) independent to the specific phase level θ0. √2πσ −2σ2 R In particular, for α = 2, we obtain the generalized PCR for ˙ Following a similar rationale, the conditional PDF pΘ˙ + (θ0) the Rayleigh fading channel, which can be expressed as θ0 can be expressed as | σ˙ r1 r2 NΘ(θ0; r1, r2) = erf , d p ˙ + (θ˙0) = P ˙ + ( , θ˙0) 4πσ √2σ √2σ Θ θ0 R,Θ θ0 | dθ˙0 | ∞ 3 1 2 ˙ 2 ˙ 2 − 2 2 2 (15) 2α θ0 4 α θ0 ×  r1 r2  = + (19) exp 2 exp 2 2 2 2 − 2σ − − 2σ σ σ˙ σ σ˙ !   After obtaining the generalized PCR, let us particularized Figs. 3 and 4 outline the conditional PDFs for several fading this statistic for the specific case (unconditioned) in which parameters (α = 1, 2, 3, 4, 5). In Fig. 3, note that it reduces r1 = 0 and r2 = . Then, applying this approach to (14), we ∞ to a Gaussian function, which is verify in (18) when α = obtain 2. Furthermore, observe that R and Θ˙ (presented in Fig. 4) become determinist when α tends to infinity. σ˙ N (θ ) = (16) Θ 0 4πσ

Observe that NΘ(θ0) is independent on the speciﬁc phase level 2The superscript + refer to the upward crossings. Note also that in upward θ0 and the power parameter α. Note also that (16) gives the crossings events 0 ≤ Θ(˙ t) < ∞. 4

[2] M. D. Yacoub, D. B. da Costa, U. S. Dias, and G. Fraindenraich. Joint Statistics for Two Correlated Weibull Variates. IEEE Ant. and Wireless 4 Prop. Letters, 4, May 2005. [3] D.T.Hess. Cycle slipping in a ﬁrst order phase locked loop. IEEE Trans. Commun. Technol., 35:pp. 255–260, 1968. [4] R.G.Vaughan. Signals in mobile communications: A review. IEEE Trans. Vehic. Technol., 35:133–145, Nov. 1986. 3

) [5] N. Youssef, W. Elbahri, M. Patzold, and S. Elasmi. On the crossing

r statistics of phase processes and random fm noise in nakagami-q mobile

(

+ fading channels. IEEE Trans. on Wireless Commun., 4(1):24–29, Jan.

θ | 2005.

R 2

p [6] W. C. Jakes. Microwave Mobile Communications. New York: Wiley, α = 1, 2, 3, 4, 5 1974. [7] N. C. Sagias, D. A.Zogas, G. K. Karagiannidis, and G. S. Tombras. Channel capacity and second-order statistics in weibull fading. 8:377– 1 379, Jun. 2004. [8] I. B. David and S. Shamai. On the rice model of noise in fm receivers. IEEE Trans. Commun., 34:1406–1419, Nov. 1988. PSfrag replacements [9] M. Kac and D. Slepian. Large excursions of gaussian processes. Ann. 0 Math. statist., 30:1215–1228, 1959. 0 1 2 3 4 r0

Fig. 3. Conditional PDF pR θ+ (r0) for several fading parameters | 0

0.5

0.4

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0 ˙ α = 1, 2, 3, 4, 5

(

θ 0.3

0.2

0.1

PSfrag replacements 0.0 0 2 4 6 8 θ˙0

˙ Fig. 4. Conditional PDF pΘ˙ θ+ (θ0) for several fading parameters | 0 VI. CONCLUSIONS In this letter, an exact and closed-form expression was derived for the generalized PCR in Weibull fading channels. In addition, statistics of the random FM noise as well as the PDFs of the envelope and random FM noise conditioned at an arbitrary phase level were obtained. These expressions were validated by specializing the general results for the Rayleigh case. It has been found that the unconditioned crossing rate of the phase process for the Weibull fading environment was the same as that of the Rayleigh one, although the joint PDF of the phase and its time derivative were different.

REFERENCES [1] M.D.Yacoub. The α−µ distribution: A general fading distribution. IEEE Inter. Symp. on Personal, Indoor and Mobile Radio Communications, PIMRC2002, 2:629–633, Sep. 2002.