On the Average Crossing Rates in Selection Diversity

Transcription

1 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) On the Average Crossing Rates in Selection Diversity Hong Zhang, Student Member, IEEE, and Ali Abdi, Member, IEEE Abstract This letter presents new results on the average crossing rate of the combined signal of a selection diversity in Rayleigh fading channels. Exact closed-form expressions are derived for the inphase zero crossing rate, inphase rate of maxima, phase zero crossing rate, and the instantaneous frequency zero crossing rate of the output of the selection combiner. The utility of the new theoretical formulas, validated by Monte Carlo simulations, are briey discussed as well. Index Terms Zero crossing rate, selection combining, Rayleigh channels, fading channels, instantaneous frequency. I. INTRODUCTION Diversity combining techniques are often used to combat the effect of fading and selection combining (SC) is one of the simplest diversity methods []. It is well nown that the average crossing rate represents the fading rate and effectively quanties the impact of the diversity combiner on channel uctuations []. Another application is speed/doppler estimation which is important for many applications such as handoff, adaptive modulation and equalization, power control, etc. [] [7] [] []. However, to the best of our nowledge, only the envelope level crossing rate (ELCR) in diversity systems has been considered so far [] [3] [3] [4]. In this This wor was presented in part at IEEE Global Telecommunications Conference, Dallas, TX, 4. H. Zhang and A. Abdi are with Center for Wireless Communications and Signal Processing Research (CWCSPR), Department of Electrical and Computer Engineering, New Jersey Institute of Technology, Newar, NJ 7 USA ( s: hz7@njit.edu, ali.abdi@njit.edu) November 6, 5

2 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) letter, closed-form expressions are derived which demonstrate the impact of SC on the uctuation rates of the inphase component, the phase, and the instantaneous frequency in Rayleigh fading channels. Monte Carlo simulation results are provided to verify the theoretical expressions as well. Finally, application of these results to speed estimation is briey discussed. II. SIGNAL AND CHANNEL MODELS Consider a noisy Rayleigh frequency-at fading channel and a combiner with L branches. The received lowpass complex envelope at the i-th branch is z i (t) = h i (t) + n i (t) ; i = ; ; :::; L; () where the independent zero-mean complex Gaussian processes h i (t) and n i (t) represent the channel gain (assuming a pilot has been transmitted) and the additive noise, respectively. In the Cartesian coordinates we have where j = z i (t) = x i (t) + jy i (t); (), and x i (t) and y i (t) are the inphase and quadrature components, respectively. Using the polar representation we obtain z i (t) = r i (t) exp f j i (t)g ; (3) where r i (t) and i (t) are the envelope and phase of z i (t), dened by q r i (t) = x i (t) + y i (t); tan i(t) = y i (t)=x i (t): (4) The autocorrelation function of z i (t) is dened by C zi (t) = E[z i (t)z i (t + )]= with as the complex conjugate. We also need the n-th spectral moment of z i (t), b i;n, n = ; ; ; :::; given by [4] b i;n = () n Z f n S zi (f)df = dn C zi () j n d n ; (5) = in which S zi (f) is the power spectral density of z i (t). As an example, assume the bandlimiting receive lter on each branch, with B RX as the bandwidth, has a non-zero response only over the range f D < f < f D, where f D is the maximum Doppler frequency. Then for Clare's isotropic scattering model [] with signal power P and white noise with N = as the two-sided power spectral density on each branch, respectively, we obtain C zi () = P J (f D )= + N B RX sinc(b RX )=, where J (:) is the zero-th order Bessel function of the rst ind, B RX = November 6, 5

3 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 3 f D, and sinc() = sin()=(). This results in b = P = + N B RX =, b = b 3 =, b = P fd + N BRX 3 =3, and b 4 = 3 4 P fd N BRX 5 =5, where the index i on b i;n is omitted, due to assumption of identically distributed branches. Let z(t), x(t), y(t), r(t), and (t) denote the complex envelope, inphase part, quadrature part, envelope, and the phase at the output of the SC, respectively. Then, based on the denition of an L-branch SC we have [] (t) = i (t) if r i (t) = max(fr (t)g L =); (6) where (t) fz(t); x(t); y(t); (t)g represents random process at the SC output and i (t) fz i (t); x i (t); y i (t); i (t)g corresponds to the i-th branch. In the next section we derive closedform expressions for the zero crossing rates (ZCRs) of x(t), (t), d(t)=dt (the instantaneous frequency), and the rate of maxima (ROM) of x(t). The results hold for a large class of fading correlations which have even-symmetric power spectra. III. NEW RESULTS ON THE AVERAGE CROSSING RATES Given a stationary real random process (t), N ( th ; T ) represents the number of times that the process crosses the threshold level th with positive (or negative) slope, over the time interval T. Also let M (T ) denote the number of maxima of the process (t), over the time interval T. Based on [5], the expected values of N ( th ; T ) and M (T ) can be calculated according to E[N ( th ; T )] = T E[M (T )] = T Z Z _f _ ( th ; _)d _; (7) f _ (; )d; (8) where f _ (; _) is the joint probability density function (PDF) of (t) and _(t), f _ ( _; ) is the joint PDF of _(t) and (t), and dot denotes differentiation with respect to t. In what follows, we derive closed-form expressions for four crossing rates in SC diversity systems: the inphase zero crossing rate (IZCR) E[N x (; T )]=T, the inphase rate of maxima (IROM) E[M x (T )]=T, the phase zero crossing rate (PZCR) E[N (; T )]=T, and the instantaneous frequency zero crossing rate (FZCR) E[N _ (; T )]=T. The branches are independent and identically distributed (iid) and odd-numbered spectral moments are zero due to the even symmetry of the power spectrum in each branch. November 6, 5

4 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 4 A. IZCR The covariance matrix of the three dimensional Gaussian random vector! V IZCR = [x i y i _x i ] tr, with tr as the transpose, at the i-th branch, is given by [4] 3 b A IZCR = 6 b ; (9) b where the subscript i of b i;n is dropped, since we have identically distributed branches. Clearly, the PDF of! V IZCR can be written as f ~VIZCR (x i ; y i ; _x i ) = Another random vector of interest is! W IZCR = [r i x i the PDF of! V IZCR in () as x exp i + yi + _x i b b : () () 3= b b = _x i ] tr, whose PDF can be determined from f ~WIZCR (r i ; x i ; _x i ) = X f ~VIZCR (x i ; y i ; _x i ) jj IZCR (x i ; y i ; _x i )j ; () where J IZCR (x i ; y i ; _x i ) = y i = p x i + y i is the Jacobian [5] of the transformation! V IZCR!! W IZCR. Based on (4), we then obtain the PDF of W! IZCR r r i exp i + _x i b b f ~WIZCR (r i ; x i ; _x i ) = p p : () b b r i x i According to (34) in the Appendix, the joint PDF of x(t) and _x(t) at the output of the L-branch SC can be shown to be ( + )r L _x XL Z L r exp f x _x (x; _x) = p exp ( ) b p dr ; (3) b b b = jxj r x m where = m!=[(m n)!n!]. Equation (3) can be simplied to n + f x _x (x; _x) = p _x L XL ( ) exp x L b exp p p : (4) b b b + = It is interesting to note that the inphase component of the SC complex envelope and its derivative at any time instant t are independent. Also, the distribution of _x(t) does not depend on L, and November 6, 5

5 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 5 still is Gaussian. By substituting (4) into (7) with x th =, we obtain the average zero crossing rate of the SC inphase component as E[N x (; T )] T = L r L b X b = L ( ) p : (5) + In the absence of noise, with isotropic scattering and L =, (5) reduces to the well-nown inphase zero crossing rate p f D = []. B. IROM Here we dene two random vectors! V IROM = [x i x i y i _x i ] tr and! W IROM = [r i _x i x i i ] tr. The covariance matrix of the Gaussian vector! V IROM is [4] 3 b b b b 4 A IROM = : (6) 6 4 b 7 5 b Then the PDF of! V IROM can be written as f ~VIROM (x i ; x i,y i ; _x i ) = exp ~ V tr IROM A IROM ~ V IROM 4 p det(a IROM ) ; (7) where det(:) is the determinant. The Jacobian of the transformation! V IROM!! W IROM is J IROM (x i ; x i ; y i ; _x i ) = p x i + y i = r i. So, we obtain the PDF of W! IROM as r i b 3 f ~WIROM (r i ; _x i ; x i, i ) = 4 p det(a IROM ) exp ri sin i b b x i b b r i x i cos i det(a IROM ) (b b b 4 ri b b _x i + b exp b 4 _x i ) : (8) det(a IROM ) Based on (34) in the Appendix, the joint PDF of _x(t) and x(t) at the output of the L-branch SC can be expressed as XL Z L f _xx ( _x; x) =L ( ) = = p _x exp b Z Z Z b b x cos exp B L p b B exp r f ~WIROM (r ; _x; x; ) exp b b XL L B x = r + r + b cos b b B d dr ( ) r dr d ; (9) November 6, 5

6 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 6 where B = b b 4 b. Note that at any given time t, random variables _x and x are independent, and _x is Gaussian, as observed in (4), with zero mean and variance b. After substituting (9) into (8) and some lengthy calculations, nally we obtain the average rate of maxima of the SC inphase component E[M x (T )] = L XL ( L ( ) B b = p + T = ( + ) b b B + ( + )b b 3 b = ( + ) 3= 3 4B 3= Z F ; ; 5 9 ; ( + )B b cos >= d ; () 3b = b 5= cos >; where F (:; :; :; :) is the hypergeometric function [6]. scattering, () reduces to p 3f D =, the IROM derived in [7]. For L =, with noise-free isotropic C. PZCR The joint PDF of the vector W! P ZCR = [r i i i _ ] tr is given in [8] r f ~WP ZCR (r i ; i ; _ i exp ri + b ri b b _ i i ) = : () () 3= (b b ) = Substitution of () into (34) from the Appendix leads to f _(; ) _ = L r b XL L ( ) 4 b = + + b 3= : () _ b Interestingly, and _ are independent, is uniformly distributed over ( ; ], and the PDF of _ is consistent with the result given in [9]. By substituting () into (7), we obtain the average zero crossing rate of the SC phase E[N (; T )] T = L 4 r L b X b = L ( ) p : (3) + Note that for L = and the no-noise isotropic scattering scenario, (3) simplies to p f D =4, which is consistent with the result that one can derive from [8]. Since () does not depend on, (3) is valid for any _ th. To simplify the notation, _ th = is chosen in (3). For = and, we have y = r sin =. So, E[N y(; T )] = E[N (; T )] + E[N (; T )], as we always have r >. On the other hand, since () does not depend on, we get E[N (; T )] = E[N (; T )]. Therefore E[N (; T )] = E[N y(; T )]=, which agrees with the fact that (3) is half of (5). November 6, 5

7 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 7 D. FZCR We dene the random vector W! F ZCR = [r i i i _ i ] tr, whose PDF is given by [8] ( f ~WF ZCR (r i ; i ; _ i ; ri 3 ri i ) = () (b b B + 4b b _ exp + b _ b i ) i +!) i : b b B + 4b b _ i (4) As before, by substituting (4) into (34) in the Appendix and some simplications, we obtain the joint PDF of (t) _ and (t) at the output of the L-branch SC f _ ( ; _ L XL L ) = 4 b b B + 4b b _ ( ) = ( + + b _ b!) + : 3 (5) b b B + 4b b _ After substituting (5) into (7) with _ th =, we obtain the average zero crossing rate of the SC instantaneous frequency E[N _ (; T )] T = r b4 b ; (6) b b which interestingly, does not depend on L. For the noise-free and isotropic scattering case, (6) reduces to f D =, which for a single branch, can be derived from [8]. IV. SIMULATION RESULTS AND CONCLUSION To verify the four new crossing rate expressions, Monte Carlo simulations using the spectral method [] have been conducted. In each simulation, we have generated independent realizations of L iid zero-mean complex Gaussian processes, with complex samples per realization, over T = second. The simulated autocorrelation function at each branch is P J (f D )= with P =. As illustrated in Fig., there is a perfect agreement between the theoretical and simulation results. Note that the crossing rates are normalized by f D in Fig.. When there is no noise, it is easy to show that all these closed-form expressions are linear functions of f D, so that they can be used for speed estimation []. One also observes that some crossing rates of the combined signal do not necessarily decrease, 4 as the diversity order 3 By integrating (5) with respect to, the PDF of _ can be obtained, which agrees with the expression given in [9]. 4 The ELCR goes to zero as L! [3]. November 6, 5

8 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 8 increases, as shown by IROM and FZCR in Fig.. Furthermore, for a given Doppler, IROM, IZCR, and PZCR can be utilized to measure the uctuation rate of the combined signal versus the number of branches, whereas FZCR can not. that APPENDIX DERIVATION OF THE JOINT PDF OF THE COMBINER OUTPUT AND ITS DERIVATIVE According to (6), the SC diversity system produces the signal (t) and its derivative _(t) such (t) = i (t) and _(t) = _ i (t) if r i (t) = max(fr (t)g L =); (7) which means the output at time instant t is coming from the branch with the largest instantaneous envelope. Of course i may change as t changes. We dene the event e i = f : r i (t) = max(fr (t)g L = )g, i = ; ; :::; L, with the probability P (e i ), where represents a point of the sample space. Based on the total probability theorem [5], the joint distribution function of (t) and _(t) can be written as F (; _) = where F (; _je i ) = P ( i ; _ i _je i ). This results in F (; _) = With iid branches, (9) simplies to LX F (; _je i )P (e i ); (8) i= LX P ( i ; _ i _; e i ): (9) i= F (; _) = LP ( ; ; e ) = LP ( ; ; r < r ; r 3 < r ; :::; r L < r ) = L = L Z Z P ( ; ; r < r ; r 3 < r ; :::; r L < r jr )f r (r )dr P ( ; jr )P L (r < r jr )f r (r )dr : (3) Since the envelope of each branch is Rayleigh distributed, we obtain Z r r r P (r < r jr ) = exp r dr = exp ; (3) b b b November 6, 5

9 PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION) 9 which after substitution in (3) and using the binomial expansion, simplies (3) to XL Z L F (; _) = L ( ) exp r P ( b ; jr )f r (r )dr : (3) = By taing the derivative of (3) with respect to and _ we obtain the joint PDF of (t) and _(t) XL Z L f _ (; _) = L ( ) = exp r f _ b (; _jr )f r (r )dr ; (33) where f _ (:; :jr ) is the joint PDF of (t) and _(t), conditioned on r (t). Equation (33) can be written in the following more compact form XL Z L f _ (; _) = L ( ) = f r _ (r ; ; _) exp where f r _ (:; :; :) is the joint PDF of r (t), (t), and _ (t). r dr ; (34) b REFERENCES [] G. L. Stuber, Principles of Mobile Communication, nd ed., Boston, MA: Kluwer,. [] A. Abdi and M. Kaveh, Level crossing rate in terms of the characteristic function: A new approach for calculating the fading rate in diversity systems, IEEE Trans. Commun., vol. 5, pp ,. [3] X. Dong and N. C. Beaulieu, Average level crossing rate and average fade duration of selection diversity, IEEE Comm. Lett., vol. 5, pp ,. [4] A. Abdi, On the second derivative of a Gaussian process envelope, IEEE Trans. Inform. Theory, vol. 48, pp. 6-3,. [5] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed., Singapore: McGraw-Hill, 99. [6] I. S. Gradshteyn and I. M. Ryzhi, Table of Integrals, Series, and Products, 5th ed., A. Jeffery, Ed., San Diego, CA: Academic, 994. [7] A. Abdi, H. Zhang, and C. Tepedelenlioglu, Speed estimation techniques in cellular systems: Unied performance analysis, in Proc. IEEE Vehic. Technol. Conf., Orlando, FL, 3, pp [8] S. O. Rice, Statistical properties of a sine wave plus noise, Bell Syst. Tech. J., vol. 7, pp. 9-57, 948. [9] B. R. Davis, Random FM in mobile radio with diversity, IEEE Trans. Commun., vol.9, pp , 97. [] K. Acolatse and A. Abdi, Efcient simulation of space-time correlated MIMO mobile fading channels, in Proc. IEEE Vehic. Technol. Conf., Orlando, FL, 3, pp [] M. J. Chu and W. E. Star, Effect of mobile velocity on communications in fading channels, in IEEE Trans. Vehic. Technol., vol. 49, pp. -,. [] H. Zhang and A. Abdi, Mobile speed estimation using diversity combining in fading channels, in Proc. IEEE Global Telecommun. Conf., Dallas, TX, 4, pp [3] G. Fraidenraich, M. D. Yacoub, and J. S. S. Filho, Second-order statistics of maximal-ratio and equal-gain combining in Weibull Fading, IEEE Comm. Lett., vol. 9, pp , 5. November 6, 5

10 Normalized Crossing Rate PREPARED FOR IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (ST REVISION).5 IROM the IROM sim FZCR the FZCR sim IZCR the IZCR sim PZCR the PZCR sim L Fig.. Normalized average zero crossing rates versus the number of branches L. [4] N. C. Beaulieu and X. Dong, Level crossing rate and average fading duration of mrc and egc diversity in Ricean fading, IEEE Trans. Commun., vol.5, pp. 7-76, 3. November 6, 5