On the multivariate conditional probability density of a vector perturbed by Gaussian noise

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1 IEEE TRANS. ON INFORMATION THEORY VOL. X NO. XX XXXX On the multivariate conditional probability density of a vector perturbed by Gaussian noise Yuriy S. Shmaliy Senior Member IEEE Abstract This paper examines the joint conditional probability density function (pdf) of the main variables (envelope phase and their n- order time derivatives) of a time-varying random signal in the presence of additive Gaussian noise. The main variables are conditioned with respect to the given variables which are the amplitude phase and their derivatives of the signal alone. We prove a theorem stating that some of the conditional pdfs of the main variables do not depend on some of the given variables. This theorem together with Bayes s theorem can substantially simplify the derivations of conditional pdfs and give alternative forms of them. Both theorems can also help in finding reasonable approximations as we demonstrate for the phase and first time derivative of the envelope. Index Terms Joint probability density function Phase Random signal Time derivative of the envelope Time-varying vector. I. INTRODUCTION Statistical properties of an information bearing narrowband signal perturbed by noise have been studied for decades beginning with the early works of Rice. In spite of this as will become clear in the sequel one of the important problems still remains unsolved. Most generally a signal is represented in the form of s(t) S(t) cosω t + ϑ(t)] U c(t) cos ω t U s(t) sin ω t () where U c U cos ϑ U s U sin ϑ S(t) is an instantaneous power and ω is an angular carrier frequency. Here U(t) S(t) and ϑ(t) are the signal amplitude and phase respectively. A common case is that at a receiver () is perturbed by narrowband Gaussian noise whose model is ξ(t) A(t) cosω t + φ(t)] A c(t) cos ω t A s(t) sin ω t () where A c A cos φ and A s A sin φ are orthogonal low-pass stationary and zero-mean Gaussian processes with equal variances σ σ c σ s. Also it is supposed that ξ(t) is continuous and multiply differentiable. Both s(t) and ξ(t) are mixed at the receiver additively so that the signal becomes noisy v(t) V (t) cosω t + ϕ(t)] (3a) V c(t) cos ω t V s(t) sin ω t (3b) (U c + A c) cos ω t (U s + A s) sin ω t (3c) where V c V cos ϕ V s V sin ϕ V (t) is the envelope and ϕ(t) is the phase. Equating the amplitudes of the harmonic functions in (3b) and (3c) produces the Gaussian variables A c(t) V (t) cos ϕ(t) U(t) cos ϑ(t) (4) A s(t) V (t) sin ϕ(t) U(t) sin ϑ(t) (5) allowing the investigation of statistical properties of V ϕ and their n-order time derivatives. The procedure begins with the joint probability density function (pdf) of A c A s and their n-order time Manuscript received...; revised... Yu. S. Shmaliy is with the Guanajuato University FIMEE Salamanca Gto. Mexico ( shmaliy@salamanca.ugto.mx). Also with Kharkiv National University of Radio Electronics Ukraine. Fig.. Time behaviors of the cosine components U c and V c of s(t) and v(t) respectively: (I) unmodulated signal and (II) modulated signal. derivatives thereafter transferring to that for V ϕ and their n-order time derivatives. When the unnecessary variables are integrated out the pdf becomes marginal representing the required variable. By Shannon s philosophy ] the marginal pdf is commonly used to estimate differential entropy ] 3] at a fixed time instant of a given variable with noisy signaling fading and fluctuations in communication channels radars wireless systems etc. II. MULTIVARIATE CONDITIONAL PDF AND PROBLEM FORMULATION With zero signal the joint pdf representes Gaussian noise that was considered by many authors 4] 9]. If a signal exists U c and U s can be supposed to be either time-invariant (unmodulated) or timevarying (modulated). In view of that two different behaviors of a signal can be recognized as shown in Fig. for cosine components. In the most general case represented by the region (II) U c and U s are modulated and their n-order time derivatives at some point t commonly exist. If these amplitudes are random at t then U ϑ and their n-order time derivatives appear as given variables in the joint pdf 8] 3] 4] 6] so that the latter can be written as p(v V V... ϕ ϕ ϕ... U U Ü... ϑ ϑ ϑ...) (6) In the region (I) both U c and U s are constant although randomly valued. Therefore all of the time derivatives are removed in (6) from the given variables. This case was analyzed in 8] ] ]. If the marginal pdfs of each of the given variables p(u) p( U)... p(ϑ) p( ϑ)... are supposed to be known the joint unconditional pdf is obtained via (6) by... p(v V... ϕ ϕ...) p(v V... ϕ ϕ... U U... ϑ ϑ...) p(u)p( U)... p(ϑ)p( ϑ)... dud U... dϑd ϑ... (7) If s(t) is deterministic then each of the given variables U U... ϑ ϑ... has a delta-shaped pdf; that is p(u U... ϑ ϑ...) δ(u U )δ( U U ) δ(ϑ ϑ )δ( ϑ ϑ ). By the sifting property of the delta function δ(x) all of the deterministic quantities are commonly accounted for as coefficients in the joint pdf. Now assume that we have a complete general probabilistic picture of v(t) sketched by (6) and would like to obtain for example the marginal pdf of V. If we integrate out in (6) all other variables we formally get a marginal pdf that is conditional

2 IEEE TRANS. ON INFORMATION THEORY VOL. X NO. XX XXXX... p(v U U... ϑ ϑ...) p(v V... ϕ ϕ... U U... ϑ ϑ...) d V... dϕd ϕ... On the other hand we know that the envelope V has the Rice pdf p(v U) if the only given variable is U. Note that Rice derived his pdf for a time-invariant vector. To the best of our knowledge the pdf of V was never derived from (6). The question then arises whether the conditional pdf of V with respect to all the given variables has a form that is more general than the Rice p(v U)? Certainly a similar question can be asked for other variables. The problem now formulates as follows. Given the general joint conditional pdf (6) we would like to know whether the marginal pdf of each of the main variables conditioned on all the given variables depends only on a subset of the given variables. The rest of the paper is organized as follows. In Section III we derive a six-variable (V V V ϕ ϕ ϕ ) joint conditional pdf of a modulated random signal in the presence of Gaussian noise; this pdf does not seem to have been published before. By integrating out step-by-step the high-order time derivatives of main variables we arrive at an important rule that is generalized in the theorem of Section IV. As examples of applications in Section V we use the theorem to derive alternative and approximate pdf forms for ϕ and V. Finally concluding remarks are drawn in Section VI. III. SIX VARIABLE JOINT CONDITIONAL PDF Information bearing signals are typically characterized by the first three terms in the Taylor series expansion of the envelope and phase. In phase channels for instance of prime importance are statistical properties of the phase ϕ (or time delay) linear phase drift ϕ (Doppler frequency) and linear frequency drift ϕ (Doppler rate) 5] 6]. If a signal is randomly valued (by signalling fading and/or fluctuations) and perturbed by Gaussian noise at the receiver it is described by (6) having six main variables (V V V ϕ ϕ ϕ) and six given variables (U U Ü ϑ ϑ ϑ). Let us find this pdf. It is known 7] that at a fixed time instant the Gaussian variables A c da c/dt and A s da s/dt are not correlated with each other and also are not correlated with A c A s Ä c d A c/dt and Äs d A s/dt. However A c Ä c and A s Ä s are mutually correlated. Therefore for a symmetric one-sided power spectral density ( (PSD) S ξ (f) of the Gaussian noise the covariance matrix of A c A ) c Äc As A s Äs can be written as R() σ σ ρ σ σ ρ σ ρ σ ρ σ ρ σ ρ (4) σ ρ σ ρ (4) (8) where the time derivatives of the normalized envelope ρ(τ) R(τ)/σ of the correlation function of ξ(t) ρ d ρ(τ) dτ τ and ρ (4) d 4 ρ(τ)/dτ 4 τ are defined by 7] respectively ρ σ σ b b < ρ (4) σ σ b4 b >. Here σ and σ are variances of the first and second time derivatives of ξ(t) and the mth spectral moment of ξ(t) is calculated by 5] b m (π) m (f f ) m S ξ (f)df where m is an integer. By (8) the joint pdf of the Gaussian variables is written as p(a c A c Äc As A s Äs) p( A c)p( A s)p(a c Äc)p(As Äs) e σ g h(a s +A c )+q(ȧ s +Ȧ c ) ρ (Ä s +Ä c )+ ρ (AsÄs+AcÄc)] (πσ ) 3 ( g) (9) where g ρ(ρ (4) ρ ) σ (σ σ σ) 4 < σ 6 q ρ (4) ρ σ σ σ 4 > h ρ ρ (4) σ 4 σ σ <. σ 4 Using the well-known trick of transferring to the polar coordinates V ϕ and their time derivatives 4] we define the determinant of the Jacobian of the transformation (Ac A s A c A det ] s Äc Äs) (V ϕ V ϕ V V 3 ϕ) and derive the required pdf p(z ż z ϕ ϕ ϕ γ γ γ ϑ ϑ ϑ) π 3 ( ρ)(ρ (4) ρ ) ρ ρ (4) γ ρ(γ ϑ + γ + γ ϑ 4 +4 exp ρ(ρ (4) ρ ) ϑ ϑ γ γ + 4 ϑ γ ϑ γ γ) +(ρ (4) ρ )(γ ϑ + γ) + ρ ( γ γ γ ϑ ) ρ ρ (4) z + (ρ (4) ρ )ż + (ρ (4) 3 ρ )z ϕ ρ ( z + 4ż ϕ z z ϕ + 4zż ϕ ϕ + z ϕ + z ϕ 4 ) + ρ z z + ρ ( ρ γ ϑ ρ γ +ρ (4) γ)z + ρ( ρ γ γ + ϑ γ)(z ϕ z) + ρ ( γ ϑ + γ ϑ)(ż ϕ + z ϕ) (ρ (4) ρ ) exp (4) ρ(ρ (4) ρ ) γ ϑz ϕ (ρ ρ ) γż] cos(ϕ ϑ) + ρ ( γ ϑ + γ ϑ)z + ρ ( ρ γ + γ ϑ γ) (ż ϕ + z ϕ) + ρ ( γ ϑ + γ ϑ) ( z z ϕ ) (ρ (4) ρ ) γ ϑż +(ρ (4) ρ ) γz ϕ] sin(ϕ ϑ) () in which z V/ σ ż V / σ z V / σ γ(t) U (t) σ z 3 S(t) σ () is the instantaneous signal-to-noise ratio (SNR) γ U /σ and γ Ü /σ. Several particular cases of () have earlier been observed in the literature. The case of γ γ γ ϑ ϑ ϑ corresponding to zero signal returns us to the works of Rice 4] ] and some more recent papers 6] 8] 9] in which the unconditional joint pdf p(z ż z ϕ ϕ ϕ) of a narrowband Gaussian noise was of concern. The case of γ and ϑ ω was studied in 8] in the presence

3 IEEE TRANS. ON INFORMATION THEORY VOL. X NO. XX XXXX 3 of Gaussian noise with an arbitrary PSD shape. In 8] we also find the pdf for a symmetric one-sided noise PSD that is identical to the relevant degenerate version of (). For our purposes we now integrate out first z and ϕ and then ż and ϕ. Lemma : Given the six variable joint conditional pdf (). Integrating out ϕ and z yields the four variable joint conditional pdf p(z ż ϕ ϕ γ γ ϑ ϑ) z π ( ρ exp ρ ) ( ρ γ γ ϑ ] γ) exp ( ρ ) z ϕ ρ z + ż ( γ ϑz ϕ + γż ρ γz) cos(ϕ ϑ) +( γz ϕ γ ϑż) sin(ϕ ϑ). () Proof: First integrate () over ϕ from to. Thereafter by an identity ( ) e px ±qx π q dx p exp p > (3) 4p integrate over z from to and arrive at () that was earlier derived in 4]. Lemma : Given the four variable conditional pdf (). Integrating out ϕ and ż leads to the bivariate joint conditional pdf p(z ϕ γ ϑ) z γ+ γz cos(ϕ ϑ) π e z. (4) Proof: Integrate () by (3) over ϕ from to and thereafter over ż from to. Finally arrive at (4). Theorem : Given the joint conditional pdf (7) of the envelope z phase ϕ and their n-order time derivatives z (n) (t) dz (n) (t)/dt (n) and ϕ (n) (t) dϕ (n) (t)/dt (n) n N] respectively of a timevarying signal perturbed by Gaussian noise with randomly valued instantaneous SNR γ and phase ϑ and their n-order time derivatives γ (n) (t) dγ (n) (t)/dt (n) and ϑ (n) (t) dϑ (n) (t)/dt (n) n N] respectively. Then the conditional pdf... dz (N M+)... dz (N)... dϕ (N K+)... dϕ (N) } {{ } } {{ } M K p(z z ()... z (N) ϕ ϕ ()... ϕ (N) γ γ ()... γ (N) ϑ ϑ ()... ϑ (N) ) (8) does not depend on γ (N L+)... γ N and ϑ (N L+)... ϑ N where L min (M K). Proof: Let L M K. Substituting in (7) N by N L yields p(z z ()... z (N L) ϕ ϕ ()... ϕ (N L) γ γ ()... γ (N L) ϑ ϑ ()... ϑ (N L) ). (9) On the other hand by L M K (8) formally produces p(z z ()... z (N L) ϕ ϕ ()... ϕ (N L) γ γ ()... γ (N) ϑ ϑ ()... ϑ (N) ). () Now using the integral representation of the modified Bessel function of the first kind and zeroth order I (x) π π π ex cos(ϕ ϑ)dϕ and integrating (4) over ϕ from π to π lead to the conditional Rice pdf of the envelope ] p(z γ) ze z γ I ( γz) (5) that turns out to be independent on ϑ. On the other hand an identity ( xe px +qx dx + q π p p eq /4p Φ q p )] where Φ(x) π x e t / dt is the probability integral transforms (4) to the conditional Bennett s pdf of the random phase mod π ] p(ϕ γ ϑ) e γ γ π + sin (ϕ ϑ) ] e γ Φ γ cos(ϕ ϑ) cos(ϕ ϑ) (6) π that is conditional on both γ and ϑ. An important inference follows instantly. Integrating p(z ż z ϕ ϕ ϕ γ γ γ ϑ ϑ ϑ) over z and ϕ yields p(z ż ϕ ϕ γ γ ϑ ϑ) without γ and ϑ. Further integrating p(z ż ϕ ϕ γ γ ϑ ϑ ) over ż and ϕ produces p(z ϕ γ ϑ) without γ and ϑ. Note that the terms with γ ϑ γ and ϑ are compensated while integrating (). Below we extend this observation to the multivariate joint conditional pdf (6) proving an important theorem. IV. THEOREM Consider the multivariate joint conditional pdf () for the n-order time derivatives n N] of all of the main and given variables; that is p(z z ()... z (N) ϕ ϕ ()... ϕ (N) γ γ ()... γ (N) ϑ ϑ ()... ϑ (N) ). (7) An identity of (9) and () exists if and only if the variables γ (N L+)... γ N ϑ (N L+)... ϑ N vanish in () and the proof for M K is complete. The proof is supported by Lemma and Lemma. If K M then integrating out one of the excess variables z (n) or ϕ (n) does not necessarily remove the corresponding given variable and we still rely on the minimum of M and K. In other words the theorem states the following: In the joint pdf the maximum order of the time derivatives of given variables cannot exceed the maximum order of the time derivatives of main variables; that is for example p(ż ϕ γ γ ϑ ϑ) but not p(ż ϕ γ γ γ ϑ ϑ ϑ) p(z ϕ γ ϑ) but not p(z ϕ γ γ ϑ ϑ). (a) (b) In the marginal pdf the maximum order of the time derivatives of given variables cannot exceed the order of the time derivative of the main variable. For example p( ϕ γ γ ϑ ϑ) but not p( ϕ γ γ γ ϑ ϑ ϑ) (c) p(z γ) but not p(z γ γ ϑ ϑ) p(ϕ γ ϑ) but not p(ϕ γ γ ϑ ϑ). (d) (e) This means by extension that the Rice pdf p(z γ) and Bennett pdf p(ϕ γ ϑ) are both unique for any signal () and their marginal pdfs are explicitly specified by respectively p(ϕ) p(z) dγ π π dγ p(z γ)p(γ) dϑ p(ϕ γ ϑ)p(ϑ)p(γ) (a) (b) where p(γ) represents random variations in the SNR and p(ϑ) in the signal phase.

4 IEEE TRANS. ON INFORMATION THEORY VOL. X NO. XX XXXX 4 V. APPLICATIONS Below we apply Theorem to derive several alternative and approximate pdf forms for ϕ and ż. A. An Alternative Form of Bennett s pdf By Theorem and (d) an exhaustive form of the conditional Rice s pdf is p(z γ). By virtue of that the Bayes s theorem offers via (4) an alternative form of the conditional Bennett s pdf (6); that is p(ϕ z γ ϑ) p(z ϕ γ ϑ) p(z γ ϑ) p(z ϕ γ ϑ) p(z γ) πi γ cos(ϕ ϑ) (z γ) ez (3) in which z is Ricean (5). Equation (3) is the von Mises/Tikhonov conditional circular normal pdf 7] with a random parameter. Substituting z with its mean value 8] 8] π z(γ) ( ) F ; γ (4) where F (a b; x) is a degenerate hypergeometric function leads to an approximation p a(ϕ γ ϑ) πi e z γ cos(ϕ ϑ) (5) ( z γ) exhibiting a maximum fractional error p a(ϕ) p(ϕ)]/p(ϑ) of about 4.7 % at γ. The error decreases when γ. B. Alternative pdf forms for ż Brown showed in 9] and Isley proved in ] that the pdf of ż of an unmodulated signal with Gaussian noise is normal. However for the modulated and random signal the relevant pdf cannot be found in closed form. Instead Theorem offers several useful solutions. Let us first use (3) and integrate (4) over ϕ from to p(z ż ϕ γ γ ϑ ϑ) exp ( ρ ) z π π ρ exp ( ρ ϑ )γ γ ρ ż ρ z ( γż ρ γz) cos(ϕ ϑ) γ ϑż sin(ϕ ϑ) + ϑ γ γ sin (ϕ ϑ) + ( γ γ ϑ ) cos (ϕ ϑ) Then integrating (6) over ϕ from π to π by a series e ±a cos ϕ I (a) + (±) n I n(a) cos nϕ n ]. (6) yields p(z ż γ γ ϑ ϑ) +γ) ze (z exp ż + γ + γ ϑ ] π( ρ) ( ρ ) ε ni n(r )I n(r ) cos n(ψ ψ ) (7) n { if n where ε n if n > R r ( ρ ) R r ( ρ ) r ( γż ρ γz) + γ ϑ ż r 4 ϑ γ γ + ( γ γ ϑ ) ψ arcsin γ ϑż r and ψ arcsin ϑ γ γ r. Now by the Bayes s theorem we have p(z ż γ γ ϑ ϑ) p(z γ γ ϑ ϑ)p(ż z γ γ ϑ ϑ) and by Theorem and (d) we must replace p(z γ γ ϑ ϑ) by p(z γ). The conditional pdf of ż can then easily be found to be p(ż z γ γ ϑ ϑ) p(z ż γ γ ϑ ϑ) p(z γ) π( ρ)i( γz) exp ż + γ + γ ϑ ] ( ρ ) ε ni n(r )I n(r ) cos n(ψ ψ ) (8) n in which z is Ricean ((5). Like the case of (5) the mean value (4) offers for (8) a nice approximation p a(ż γ γ ϑ ϑ) π( ρ)i( γ z) exp ż + γ + γ ϑ ] ( ρ ) ε ni n(r )I n(r ) cos n(ψ ψ ) (9) n which maximum fractional error p a(ż) p(ż)]/p( ż) is several percent; that is about.9% at γ γ 3 ϑ.7 and ϑ. One can also exploit the fact that by Theorem and (b) the bivariate conditional pdf of z and ϕ is exhaustively represented by p(z ϕ γ ϑ). The Bayes s theorem then easily produces by (3) and (4) p(ż ϕ z ϕ γ γ ϑ ϑ) p(z ż ϕ ϕ γ γ ϑ ϑ) p(z ϕ γ γ ϑ ϑ) exp ( ρ ) p(z ż ϕ ϕ γ γ ϑ ϑ) p(z ϕ γ ϑ) z π( ρ) e γ ϑ + γ ρ ż + z ϕ ( γ ϑz ϕ + γż) cos(ϕ ϑ) +( γz ϕ γ ϑż) sin(ϕ ϑ). (3) Now integrating (3) over ϕ from to by () leads to the conditional normal pdf p(ż ϕ γ γ ϑ ϑ) ] ż γ + γ ϑ cos(ϕ ϑ ς) exp π( ρ) ( ρ ) (3) where ς arccos γ/( γ + γ ϑ ) with the variance ( ρ )/ and a random mean value. Surprisingly (3) does not depend on z. With γ ϑ the pdf (3) acquires a zero mean and readily degenerates to the normal pdf found by Brown 9] and Isley ]. The phase ϕ in (3) has the Bennett distribution (6) or approximately the von Mises/Tikhonov distribution (5). In the latter case the following alternative approximation can be useful π π p b (ż γ γ ϑ ϑ) πi ( z γ) π( ρ ) e + ż γ+γ cos(ϕ ϑ ς)] ϑ ρ z γ cos(ϕ ϑ) ( ρ ) dϕ (3) allowing like the case of (9) a maximum fractional error of several percent.

5 IEEE TRANS. ON INFORMATION THEORY VOL. X NO. XX XXXX 5 VI. CONCLUDING REMARKS An important merit of the above-proved Theorem resides in the fact that it is fundamental for modulated and random narrowband signals perturbed by additive Gaussian noise with an arbitrary PSD shape. The theorem ascertains which of the given variables the joint and marginal pdfs of the main variables (envelope phase and their n-order time derivatives) can depend on. This theorem together with Bayes s theorem can substantially simplify the derivations of conditional pdfs and give alternative forms of them. A nice illustration is the normal conditional pdf with a random mean value derived for ż (3). Both theorems can also help in finding reasonable approximations as we demonstrated by deriving (9) and (3) for ż. ] C. T. Isley Probability distribution for the derivative of the envelope of signal and Gaussian noise IRE Trans. Inform. Theory vol. IT-6 pp Mar. 96. ACKNOWLEDGMENT The author would like to thank Dr. Charles Greenhall of the Jet Propulsion Laboratory (JPL) California Institute of Technology for assistance in reading and discussions of the results and two anonymous reviewers for valuable comments and remarks. REFERENCES ] C.E. Shannon A mathematical theory of communication Bell Syst. Tech. J. vol. 7 pp ] S.W. Golomb The information generating function of a probability distribution IEEE Trans. Inform. Theory vol. IT- pp Jen ] A.C.G. Verdugo Lazo and P.N. Rathie On the entropy of continuous probability distributions IEEE Trans. Inform. Theory vol. IT-4 no. pp. - Jen ] S.O. Rice Mathematical analysis of random noise in Selected Papers on Noise and Stochastic Processes. N. Wax Ed. New York: Dover 954 pp ] D. Middleton Spurious signals caused by noise in triggered circuits J. Applied Physics vol. 9 no. 9 pp Sep ] B.N. Zvyaghintsev Some statistical properties of the second derivative of the envelope of a normal random process Radio Eng. Elektron. Phys. vol. no. 4 pp Apr ] V.I. Tikhonov Statistical Radio Engineering Moscow: Sovetskoe Radio ] V.I. Tikhonov Nonlinear transformations of random processes Moscow: Radio i sviaz ] N.M. Blackman The distribution of local extrema of Gaussian noise and of its envelope IEEE Trans. Inform. Theory vol. 45. no. 6 pp. 5- Sep ] S.O. Rice Statistical properties of a sine wave plus random noise Bell Syst. Tech. J. vol. 7 no. pp Jan ] W.R. Bennett Methods of solving noise problems Proc. IRE vol. 44 pp May 956. ] V.V. Tsvetnov Statistical properties of signals and noises in twochannel phase systems Radiotekhn. vol. no. 5 pp. -9 May ] J. Salz and S. Stein Distribution of instantaneous frequency for signal plus noise IEEE Trans. Inform. Theory vol. IT- pp Oct ] N.G. Gatkin V.A. Garanin M.I. Karnovskiy L.G. Krasniy and N.I. Cherney Probability density of phase derivative of the sum of a modulated signal and Gaussian noise Radio Eng. Electron. Phys. vol. no. 8 pp. 3-9 Aug ] P. Bello Joint estimation of delay Doppler and Doppler rate IRE Trans. Inform. Theory vol. IT-6 no. 3 pp June 96. 6] Yu.S. Shmaliy Phase Errors of Passive Wireless SAW Sensors with Differential Measurement IEEE Sensors J. vol. 4 no. 6 pp Dec. 4. 7] Yu.S. Shmaliy Von Mises/Tikhonov-based distributions for systems with differential phase measurement Signal Processing vol. 85 no. 4 pp Apr. 5. 8] T. H. Park Moments of the generalized Rayleigh distribution Quart. Appl. Math. vol. 9 no. pp Jan ] W. M. Brown Some results on noise through circuits IRE Trans. Inform. Theory vol. IT-5 pp May 959.