The Fluctuating Two-Ray Fading Model: Statistical Characterization and Performance Analysis

Transcription

1 The Fluctuating Two-Ray Fading Model: Statitical Characterization and Performance Analyi Juan M. Romero-Jerez, F. Javier Lopez-Martinez, Joé F. Pari and Andrea J. Goldmith arxiv:6.563v [c.it] 7 May 7 Abtract We introduce the Fluctuating Two-Ray FTR fading model, a new tatitical channel model that conit of two fluctuating pecular component with random phae plu a diffue component. The FTR model arie a the natural generalization of the two-wave with diffue power TWDP fading model; thi generalization allow it two pecular component to exhibit a random amplitude fluctuation. Unlike the TWDP model, all the chief probability function of the FTR fading model PDF, CDF and MGF are expreed in cloed-form, having a functional form imilar to other tate-of-the-art fading model. We alo provide approximate cloed-form expreion for the PDF and CDF in term of a finite number of elementary function, which allow for a imple evaluation of thee tatitic to an arbitrary level of preciion. We how that the FTR fading model provide a much better fit than Rician fading for recent mallcale fading meaurement in 8 GHz outdoor millimeter-wave channel. Finally, the performance of wirele communication ytem over FTR fading i evaluated in term of the bit error rate and the outage capacity, and the interplay between the FTR fading model parameter and the ytem performance i dicued. Monte Carlo imulation have been carried out in order to validate the obtained theoretical expreion. Index Term Wirele channel modeling, envelope tatitic, moment generating function, multipath propagation, Rician fading, mall-cale fading, two-ray. I. INTRODUCTION The ue of millimeter-wave mmwave band to overcome the wirele pectrum hortage caued by the exponential increae in aggregate traffic i being embraced by emerging wirele tandard uch a 5G []. Thi ha led to ignificant reearch in mmwave radio communication in urban outdoor environment []. Much of thi reearch ha focued on mmwave channel modeling [3 6]. Mot of the tochatic channel model for mmwave communication aume Rayleigh or Rician ditribution for the mall-cale fading path amplitude in NLOS and LOS cenario, repectively. Very recently [7], the mall-cale fad- Thi work will be preented in part at IEEE Globecom 6. J. M. Romero- Jerez i with Departmento de Tecnología Eectrónica, Univeridad de Malaga - Campu de Excelencia Internacional Andalucía Tech., Malaga 97, Spain. Contact romero@dte.uma.e. F. J. Lopez-Martinez and J. F. Pari are with Departmento de Ingeniería de Comunicacione, Univeridad de Malaga - Campu de Excelencia Internacional Andalucía Tech., Malaga 97, Spain. A. Goldmith i with the Wirele Sytem Lab, Department of Electrical Engineering, Stanford Univerity, CA, USA. Thi work ha been funded by the Conejería de Economía, Innovación, Ciencia y Empleo of the Junta de Andalucía, the Spanih Government and the European Fund for Regional Development FEDER project P-TIC-79, P-TIC-838, TEC4-579-R, TEC3-47-R and TEC P Thi work ha been ubmitted to the IEEE for publication. Copyright may be tranferred without notice, after which thi verion may no longer be acceible. ing tatitic obtained from a 8 GHz outdoor meaurement campaign howed that Rician fading wa more uited than Rayleigh even in NLOS environment. However, a deeper look into the reult of [7] indicate that conventional fading model in the literature do not accurately model the random fluctuation uffered by the received ignal. In particular, the empirical CDF and PDF reported in [7] and [8], repectively, for different mmwave cenario exhibit a bimodality that cannot be captured even by generalized fading model [9 ]. We here propoe a new amplitude fading model, the Fluctuating Two-Ray FTR fading model, whoe tatitical ditribution capture the wide heterogeneity of random fluctuation a ignal experience in propagation environment with multiple catterer. The FTR model i a natural generalization of the two-wave with diffue power TWDP fading model propoed by Durgin, Rappaport and de Wolf [3]. In thi generalization, the contant-amplitude pecular wave randomly fluctuate. The incluion of an additional ource of randomne allow for a better characterization of the amplitude fluctuation experienced by the radio ignal, compared to the TWDP model which i indeed included a a particular cae of the FTR model. Remarkably, thi larger flexibility doe not come at the price of an increaed mathematical complexity, but intead facilitate the analytical characterization of thi new fading model. The benefit of uing the FTR fading model, which will be derived below, can be ummarized a follow: Depite being more general than the original TWDP model, the primary probability function CDF, PDF and MGF of the FTR model are given in cloed-form, The FTR fading ditribution i inherently bimodal, but alo include claical unimodal fading model like Rician, Nakagami-m, Hoyt and Rayleigh a particular cae; thu, it can be matched to a wider variety of propagation condition than conventional fading model, and 3 The FTR fading ditribution provide a much better fit than exiting fading model to field meaurement, for example the 8 GHz field meaurement recently reported in [7]. The remainder of thi paper i tructured a follow: the phyical jutification of the FTR fading model i introduced in Section II. Then, in Section III, the FTR fading model i tatitically characterized in term of it MGF, CDF and PDF. The empirical validation of our model i preented in Section IV by fitting the FTR fading model to mall-cale fading We alo note that the FTR model here propoed differ from the Generalized Two-Ray GTR model propoed in []. Unlike the TWDP model, the GTR model allow the phae ditribution of the pecular wave to be other than uniform, but the amplitude of the pecular component are till kept contant.

2 field meaurement in the mmwave band. The performance of wirele communication ytem operating under FTR fading i analyzed in Section V, with aociated numerical reult given in Section VI. Our main concluion are ummarized in Section VIII. II. PRELIMINARIES AND CHANNEL MODEL The mall-cale fluctuation in the amplitude of a ignal tranmitted over a wirele channel can be modeled by the uperpoition of a et of N dominant wave, referred to a pecular component, to which other diffuely propagating wave are added [3]. Under thi model, the complex baeband voltage of a wirele channel experiencing multipath fading can be expreed a V r = N V n exp jφ n + X + jy, n= where V n exp jφ n repreent the n-th pecular component, which i aumed to have a contant amplitude V n and a uniformly ditributed random phae φ n, uch that φ n U[, π. Since the ditance travered by the propagating wave are typically order of magnitude greater than their wavelength, the random phae variable of each pecular component are aumed to be tatitically independent. On the other hand, X + jy i a complex Gauian random variable, uch that X, Y N, σ, repreenting the diffue received ignal component due to the combined reception of numerou weak, independently-phaed cattered wave. Thi Gauian model i baed on application of the central limit theorem to the um of thee numerou wave. The general model preented in include very important tatitical wirele channel model a particular cae. Thu, when N =, i.e., no pecular component i preent, the Rayleigh fading model i obtained, while for N =, a ingle dominant pecular component, we have the Rician fading model. The cae in which there are two dominant pecular component N = i uually referred to a the Two Wave with Diffue Power TWDP fading model or, alternatively, the Generalized Two-Ray fading model with Uniformly ditributed phae GTR-U []. Thi recently-developed model contain the aforementioned claical fading model a particular cae and accurately fit field meaurement in a variety of propagation cenario [3]. Unfortunately, the tatitical characterization of the TWDP fading model i much more complicated than that of claical fading model, a there are not known cloed-form expreion for the PDF and the CDF of the received ignal envelope. Notably, the MGF of the power envelope in TWDP fading wa recently derived in []. The pecular component in the general model in have contant amplitude. We mut here note that variation in the amplitude of the dominant pecular component, often aociated with the LOS propagation, have been conidered in ome pecific cenario and validated with field meaurement: thee are the cae of the Rician hadowed fading model [4] which generalize the Rician fading model, or the κ-µ hadowed fading model introduced in [] a a generalization of Yacoub κ-µ fading model. However, while the word hadowing wa ued when the model [, 4] were introduced, thee model hould not necearily be linked to the large-cale fading phenomena alo called hadowing, due to a complete or partial blockage by obtacle many time larger than the ignal wavelength. Intead, thee model reflect any amplitude fluctuation in the pecular wave e.g. ay variation in the propagation condition or fat moving catterer that take place over the time period of interet. Therefore, conidering the amplitude of the pecular component to be modulated by a Nakagami-m random variable with quared unit mean a in [, 4], we can write: N V r = ζvn exp jφ n + X + jy, n= where ζ i a unit-mean Gamma ditributed random variable with PDF f ζ u = mm u m e mu. 3 Γ m Note that we are conidering the ame fluctuation for the pecular component, which i actually a natural ituation in different wirele cenario. When the catterer are in the vicinity of the tranmitter and/or the receiver, the pecular component will travel alongide mot of the way, and the eventual channel fluctuation would affect them imultaneouly. Thi i, for example, the cae of the human body hadowing a the uer move. Alo, there i a number of caue of electromagnetic diturbance that will typically affect the peculat component imultaneouly, including ionopheric cintillation for atellite communication, udden change of the channel electromagnetic field due to natural e.g. olar activity or artificial e.g. motor ignition, electric power generator ource, etc. The wirele channel model given in -3 for the particular cae when N = correpond to the Rician hadowed fading model [4]. In the ret of thi paper, we will conider the cae when N = and will derive a tatitical decription of the reulting channel model. Thi model will be ubequently denoted a the Fluctuating Two-Ray FTR model, in order to indicate the preence of two pecular component with random phae for which their amplitude exhibit a random fluctuation. III. STATISTICAL CHARACTERIZATION OF THE FTR FADING MODEL Let u conider the complex baeband received ignal, which can be written a V r = ζv exp jφ + ζv exp jφ + X + jy. 4 Thi model i conveniently expreed in term of the parameter K and, defined a K = V + V σ, 5 = V V V + V. 6 The K parameter repreent the ratio of the average power of the dominant component to the power of the remaining

3 3 diffue multipath, jut like the Rician K parameter. On the other hand, i a parameter ranging from to expreing how imilar to each other are the average received power of the pecular component: when the magnitude of the two pecular component are equal, =, while in the abence of a econd component V = or V =, =. Note that = yield the Rician hadowed fading model. We will firt characterize the ditribution of the received power envelope aociated with the FTR fading model, or equivalently, the ditribution of the received ignal-to-noie ratio SNR. After paing through the multipath fading channel, the ignal will be affected by additive white Gauian noie AWGN with one-ided power pectral denity N. The tatitical characterization of the intantaneou SNR, here denoted a γ, i crucial for the analyi and deign of wirele communication ytem, a many performance metric in wirele communication are a function of the SNR. The received average SNR after tranmitting a ymbol with energy denity E undergoing a multipath fading channel a decribed in 4 will be { = E b /N E V r } = E b /N V + V + σ 7 = E b /N σ + K, where E{ } denote the expectation operator. With all the above definition, the chief probability function related to the FTR fading model can now be computed. A. MGF In the following lemma we how that, for the FTR fading model, it i poible to obtain the MGF of γ in cloed-form. Lemma : Let u conider the FTR fading model a decribed in 4-7. Then, the MGF of the received SNR γ will be given by M γ = mm + K + K m R m, k, ; m m + K m + K P m, R m, k, ; where R m, k, ; i a polynomial in defined a R m, k, ; = [ m + K K ] m + K m + K + m + K, and P µ i the Legendre function of the firt kind of degree µ, which can be calculated a P µ z = F µ, µ + ; ; z, given that 8 9 z <, where F i the Gau hypergeometric function [5, p ]. Proof: See Appendix I. TABLE I CONNECTIONS BETWEEN THE FTR FADING AND OTHER FADING MODELS IN THE LITERATURE. THE FTR FADING PARAMETERS ARE UNDERLINED TO AVOID CONFUSION WITH THE SPECIAL CASES. Channel One-ided Gauian Rayleigh Nakagami-q Hoyt Nakagami-m Rician Rician hadowed TWDP Two-Wave Fluctuating Two-Wave FTR Fading Parameter a =, b =, a =,, m =.5, m =, m = b =, =, m a =, = q q, =.5 m +K b {,K}, with q = +K+, m = =,, = m m =, = K, m =, = K, = m m =, = K, m =,, m =,, = m m The FTR fading model introduced here i well-uited to recreate the propagation condition in a wide variety of wirele cenario, ranging from very favorable one to wore-than Rayleigh fading. It alo include many important well-known tatitical fading model a particular cae, i.e., TWDP, Rician hadowed, Rician, Rayleigh, one-ided Gauian, Nakagamim and Nakagami-q Hoyt. The connection between the FTR fading model and the pecial cae included therein can eaily be validated uing the previou definition for K, and m, and i formally tated in Table I. Special attention i merited for the cae of Nakagami-q Hoyt fading, which can be een a a pecial cae of the FTR fading model in two different way. The firt one arie after pecializing the Rician hadowed model for m = / a indicated in [6]; however, a we will later ee, chooing the parameter m to be a poitive integer ha additional benefit in term of mathematical tractability. Thu, in the following corollary we how how the Nakagami-q Hoyt fading model can be obtained from the FTR fading model with m =. Corollary : For m =, the FTR fading model become the Nakagami-q Hoyt model with q = + K + K +. Proof: See Appendix II. Strikingly, the inherently non-circularly ymmetric Hoyt ditribution i obtained by adding two pecular component with uniformly ditributed phae and Rayleigh-ditributed random amplitude m = to a circularly ymmetric diffue component, for all q atifying. Setting q = or q = reduce to the one-ided Gauian and Rayleigh ditribution, repectively. Note that the Nakagami-q fading ditribution model cenario wore than Rayleigh deeper fade. The relationhip between, K and q i repreented in Fig..

4 4 q K = K = K = K = K Fig.. Connection between the FTR and the Nakagami-q fading model parameter, with m =. We ee that for low value of K, only thoe value of q cloer to are poible for any. A K grow, we oberve that the whole range of q [, ] can be attained with q +. With the MGF in cloed-form, we now how that the PDF and CDF of the FTR fading ditribution can alo be obtained in cloed-form, provided that the parameter m i retricted to take poitive integer value i.e., m Z +. B. PDF and CDF When the parameter m take integer value, the MGF of the SNR in the FTR fading model can be calculated a a finite um of elementary term. Thi i baed on the fact that, for m an integer, the Legendre function in the MGF given in 8 ha an integer degree, thu becoming a Legendre polynomial. A Legendre polynomial of degree n can be written a [5, p ] P n z = n/ n q Cq n z n q, 3 q= where i the floor function and Cq n i a coefficient given by Cq n n n q n q! = = q n q! n q! n q!. 4 We will make ue of 3 to compute cloed-form expreion for the PDF and CDF of the power envelope for the FTR fading model in 5 and 6, repectively, which will be demontrated in the next lemma. Note that the PDF and CDF of the received ignal envelope can be eaily derived from 5 and 6 by a imple change of variable. Specifically, through a change of variable we get f r r = rf γ r and F r r = F γ r, with in 5 and 6 replaced by Ω = E{r }. Lemma : When m Z +, the PDF and CDF of the SNR γ in a FTR fading channel can be expreed in term of the confluent hypergeometric function Φ defined in [7, p. 34, 8], a given, repectively, in 5 and 6. Proof: See Appendix III. Note that depite requiring the evaluation of a confluent hypergeometric function, the PDF and CDF of the FTR fading model can be expreed in term of a well-known function in communication theory. In fact, the Φ function alo make an appearance in the CDF of common fading model uch a Rician hadowed or κ-µ hadowed [, 8]. Moreover, thi function can be efficiently evaluated uing an invere Laplace tranform a decribed in [9, Appendix 9B]. Thu, the evaluation of the FTR ditribution function doe not poe any additional challenge compared to other tate-of-the-art fading model. In the following lemma, we alo preent a family of approximate PDF and CDF for the FTR fading model, which are given in term of a finite um of exponential function and power. Thu, it evaluation become a imple a evaluating the well-known Gamma ditribution aociated with the quared envelope in the Nakagami-m fading model. Lemma 3: When m Z +, the PDF and CDF of the SNR γ in a FTR fading channel can be approximated by a finite um of elementary function, a given, in 7 and 9 repectively, where M > K, β = K+ and the coefficient α i and δ i are defined in 63 and 64 in the Appendix IV. Proof: See Appendix IV. In the next et of figure Fig. to 7, we tudy the effect of the FTR fading model parameter K, and m on the hape of the PDF. Specifically, the received ignal envelope PDF f r r and the power envelope f γ γ are repreented in order to better illutrate the veratility of the FTR fading model. For the approximated reult, M = K + ha been conidered in every cae. Monte Carlo imulation have been carried out in order to validate the depicted function, but they are not repreented in thee figure a the imulated value are inditinguihable from the exact reult. Similarly to the TWDP fading model, the FTR fading model i inherently bimodal; uch bimodality i dominated by the parameter K and. Specifically, and large value of K yield a more pronounced bimodality; thi correpond to the wore-than- Rayleigh fading cae. The additional parameter m moothen uch bimodality a m decreae; converely, a m, the FTR fading model reduce to the TWDP fading model. IV. EMPIRICAL VALIDATION In the previou ection, we have introduced the FTR fading model and derived it relevant tatitic. We will now how it uitability for modeling mall-cale fading in mmwave wirele link. We ue the empirical reult preented in [7] to validate the FTR fading model in the context of mallcale fading modeling of mmwave outdoor communication in the 8 GHz band. Detail on the pecific meaurement configuration can be found in [7]. A modified verion of the Kolmogorov-Smirnov KS tatitic ha been ued to define the error factor ɛ that quantifie the The bimodality of the ditribution i clearly identified by the appearance of two maxima in it PDF; thi would be tranlated into everal tranition from concavity to convexity i.e., inflection point in the CDF.

5 5 f γ x = + K m Φ 4 m m + K K m m / q= q C m q + q m, m q, m q, m; ; m + K m + K x, m + K m + K + x, m + K m + K m + K m + K K + K x, x. m q 5 F γ x = + K m x Φ 4 m m + K K m m / q= q C m q + q m, m q, m q, m; ; m + K m + K x, m + K m + K + x, m + K m + K m + K m + K K + K x, x. m q 6 f γ x G m x; β, K = M i= F γ x H m x; β, K = α i {G m x; β, K δ i + G m x; β, K + δ i }, 7 m m βe βx m K+m m m K + m M i= m m n n= j= n= n Kβx K + m n n! ; 8 α i {H m x; β, K δ i + H m x; β, K + δ i } 9 m m n K n+j m n j m + K m +j βj n+j n + j!j! e β m m+k x x j goodne of fit between the empirical and theoretical CDF, denoted by ˆF r and F r repectively, i.e, ɛ max x log ˆF r x log F r x. Note that the CDF i ued in log-cale in order to outweigh the fit in thoe amplitude value cloer to zero, where the fading i more evere []. With the above definition, we mut highlight that a value of ɛ = can be interpreted a a difference of one order of magnitude between the empirical and theoretical CDF. In Fig. 8 and 9 we compare the et of meaurement correponding to the LOS and NLOS cro-polarized cenario decribed in [7, Fig. 6]. For thi et of meaurement, the empirical CDF lie within the theoretical CDF correponding to a Rician ditribution with value of K ranging from to 7 i.e. 3 to 8 db. According to the KS tatitic, the value of K that provide the bet fit to the Rician ditribution are KLOS Rice = 4.4 and KRice NLOS = 4.78 repectively. Such value of K yield an error factor value of ɛ Rice =.33 and ɛ Rice NLOS LOS =.357. Now, uing the propoed FTR fading model, we obtain the following et of parameter for the LOS and NLOS cae: FTR LOS = K = 8, =.5873, m = and FTR NLOS = K = 3.7, =.833, m =. Note that the parameter m play a key role in the goodne of fit, a it enable that the CDF can modify it concavity and convexity in order to better adjut the empirical data. For thee parameter, the error factor value obtained by the FTR fit are ɛ FTR.46 and ɛ FTR NLOS LOS = =.68. Thu, a remarkable improvement i attained when uing the FTR fading model intead of the impler Rician model. V. PERFORMANCE ANALYSIS OF WIRELESS COMMUNICATIONS SYSTEMS With the exact cloed-form expreion of the MGF, PDF and CDF for the SNR of the propoed FTR fading channel derived above, we can now calculate many performance metric of wirele communication ytem operating in channel following thi fading model. A an example of one application, we calculate the BER for a family of coherent modulation and the outage capacity. In both cae we will alo obtain exact aymptotic expreion for the high-snr regime.

6 6 frr m m = 3 m = m = 3 frr.5.5 =. =.5 = r Fig.. FTR ignal envelope ditribution for different value of m, with K = 5, =.9 and Ω =. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from 7. The cae m reduce to the TWDP fading ditribution [3] r Fig. 4. FTR ignal envelope ditribution for different value of, with K = 5, m = 5. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from 7..8 m m = 3 m = m = 3.8 =. =.5 =.9 fγγ.6.4 fγγ γ γ Fig. 3. FTR power envelope ditribution for different value of m, with K = 5, =.9 and =. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from 7. The cae m reduce to the TWDP fading ditribution [3]. Fig. 5. FTR power envelope ditribution for different value of, with K = 5, m = 5. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from 7. A. Average BER The average error rate can be calculated by averaging the conditional error probability CEP, i.e., the error rate under AWGN, over the output SNR, that i: P e = P E xf γ xdx, where P E x denote the CEP. Alternatively, integrating by part, the average error rate can be computed from the CDF a P e = P ExF γ xdx, 3 where P E γ i the firt order derivative of the CEP. The CEP for the bit error rate of many wirele communication ytem with coherent detection i determined by [] P E x = R r= α r Q βr x, 4 where Q i the Gau Q-function and {α r, β r } R r= are modulation-dependent contant. The derivative of 4 i given by R P βr βr x E x = α r 8πx e. 5 r= Introducing 6 and 5 into 3, and with the help of [7, p. 86, 43], a compact exact expreion of the average BER can be found, a given in 8, in term of the Lauricella function F D defined in [7, p. 33, 4].

7 7.8 K = K = K = 3.6 frr.4 Frr r Fig. 6. FTR ignal envelope ditribution for different value of K, with m = 5, =.9 and Ω =. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from r db about mean Meaured [7] Rician FTR Fig. 8. Empirical v theoretical CDF of the received ignal amplitude for LOS cenario. Parameter value are K Rice = 4.4 and K FTR = 8, =.5873, m =. Meaured data obtained from [7, Fig. 6, LOS]..8 K = K = K = 3.6 fγγ.4 Frr γ Fig. 7. FTR power envelope ditribution for different value of K, with m = 5, =.9 and =. Solid line correpond to the exact PDF derived from 5, marker correpond to the approximate PDF derived from r db about mean Meaured [7] Rician FTR Fig. 9. Empirical v theoretical CDF of the received ignal amplitude for NLOS cenario. Parameter value are K Rice = 4.78 and K FTR = 3.7, =.833, m =. Meaured data obtained from [7, Fig. 6, NLOS]. Although the derived BER expreion can be eaily computed uing the Euler form of the F D function, it doe not provide inight about the impact of the different ytem parameter on performance. We now preent an aymptotic, yet accurate, imple expreion of the error rate for the high SNR regime. Firt, note that the following equality hold: M γ = P m m m + K m + K K m m + K m + K K + o, 6 where we write a function ax a ox if lim x ax/x = and where the Legendre polynomial i calculated uing 3. Thu, performing a imilar approach to that in [, Propoition and 3], we obtain, after ome manipulation, the aymptotic expreion m m + K R P e m m + K K r= P m m + K m + K K α r β r 7,.

8 8 P e = + K m F 4 3 D m m + K K m m / q=, + q m, m q, m q, m; ; m + K β r m + K, m + K q C m q m + K m + K K β r m + K +, m + K β r m + K m q + K,. β r R r= α r β r 8 B. Outage capacity The intantaneou channel capacity per unit bandwidth conidering tranmit and receive antenna i well-known to be given by C = log + γ. 9 We define the outage capacity probability a the probability that the intantaneou channel capacity C fall below a predefined threhold R S given in term of rate per unit bandwidth, i.e., Therefore P out = P C < R S = P log + γ < R S. 3 P out = P γ < R S = F γ R S. 3 Thu, the outage capacity probability can be directly calculated from 6 pecialized for x = R S. Thi expreion i exact and hold for all SNR value; however, it offer little inight about the effect of parameter on performance. Fortunately, we can obtain a imple expreion in the high SNR regime a follow: From 6 and [, Propoition 5], the CDF of γ can be written a F γ x = P m m m + K m + K K m m + K x + o. m + K K 3 Therefore, the outage capacity probability can be approximated in the large SNR regime by P out m m + K m + K K m P m m + K R S,. m + K K VI. NUMERICAL RESULTS 33 In thi ection we preent figure howing numerical reult for the performance metric derived in the previou ection under different fading condition. All the reult hown here have been analytically obtained by the direct evaluation of the exact expreion derived in thi paper. Additionally, Monte Pe 3 4 =. = db m = m = 8 Fig.. Average BER of BPSK modulation v. average SNR for different value of m and. Parameter value K = 8. Marker correpond to Monte Carlo imulation. Carlo imulation have been performed to validate the obtained expreion, and are alo preented in the figure, howing an excellent agreement with the analytical reult. Fig. and how reult for, repectively, the average BER and the outage capacity probability a a function of the average SNR auming K = 8 and imilar =.9 and diimilar =. pecular component, a well a light m = 8 and trong m = fluctuation of thee component. For the average BER, binary phae-hift keying BPSK modulation i aumed, which can be obtained by etting R =, α = and β = in 8. For the outage capacity probability, a threhold R S = i aumed. Both preented metric actually how akin behavior: diimilar pecular component experiencing lighter fluctuation yield better performance, i.e. lower average BER and outage capacity probability. It i intereting to note that, for both metric, there i an inflection point in the log-log cale for =. and m = 8, which virtually diappear a m decreae. VII. FURTHER GENERALIZATIONS OF THE TWDP FADING MODEL The FTR model, a preented and analyzed in thi work, aume that the two pecular component are fully correlated and, therefore, i an appropriate model when both pecular

9 9 P C < RS 3 4 m = m = 8 =. = db Fig.. Outage capacity probability v. average SNR for different value of m and. Parameter value R S = and K = 8. Marker correpond to Monte Carlo imulation. component are affected by the ame catterer or electromagnetic diturbance. A more general model would conider a partial correlation between the pecular component. The analytical difficulty of uch model eem to be ignificantly higher than the one preented here. However, in the limit cae when both pecular component are independent and the complex baeband received ignal can be expreed a V r = V ζ exp jφ + V ζ exp jφ + X + jy, 34 where ζ and ζ are i.i.d. random variable whoe PDF i given in 3, it i poible to obtain the MGF of the received SNR in cloed-form. Lemma 4: Let u conider the fading model a decribed in 34. Then, the MGF of the received SNR γ will be given by M γ = + K + K F m, m; ; Km g + K 4m g K g 4m 4mKg + K g m, 35 where K and are defined, repectively, a in 5 and 6, and g i defined a g = + K. 36 Proof: See Appendix V. Thi model i an alternative generalization of the TWDP model which i different to the propoed FTR model, and can be applied when the pecular component follow very different path and are affected by different catterer. A thorough analyi of uch model i of great interet and i left for future work. VIII. CONCLUSIONS The FTR fading model wa introduced to characterize the tatitic of a received ignal with dominant pecular component along with random fluctuation about thoe component. A detailed tatitical characterization i preented, and cloed-form expreion of the PDF, CDF and MGF of the model are derived. Additionally, a the exact PDF and CDF are given in term of a confluent hypergeometric function, alternative approximated expreion for thee tatitic are given a finite ummation of elementary function, which allow for a imple performance analyi. A an example of application of the model, we have derived both exact and aymptotic expreion for the outage capacity and the BER for a family of modulation cheme. Both performance metric how that diimilar pecular component experiencing lighter fluctuation yield better performance. The propoed model i alo hown to cloely model mall-cale fading, which ha been exemplified in the context of mmwave communication, on which the fit to empirical meaurement in the 8 GHz band how great improvement over the Rician fading model. APPENDIX I PROOF OF LEMMA Let u conider the fading channel model given in 4 conditioned to a particular realization ζ = u of the random variable modeling the fluctuation of the pecular component. In thi cae, we can write V r ζ=u = uv exp jφ + uv exp jφ +X +jy 37 Thi correpond to the claical TWDP fading model where the amplitude of the pecular component are given by uv and uv, for which the following parameter can be defined: K u = uv + uv σ = u V + V σ, 38 u = uv uv uv + uv = V V V + V. 39 It i clear that thee parameter are related to thoe defined in 5 and 6 for the FTR fading model by K u = uk, 4 u =. 4 The conditional average SNR for the fading model decribed in 37 will be u = E b /N uv + uv + σ = E b /N σ 4 + K u. The MGF of the TWDP fading model wa hown in [] to be given in cloed-form a + K u M γu = + K u u exp K u u + K u u 43 u K u u I, + K u u where I i the zero-order modified Beel function of the firt kind. Thi MGF can be written in term of the K and parameter defined for the FTR model. Note that from 7 and 4 we can write, repectively, + K = E b /N σ, 44

10 + K u = u E b /N σ, 45 and equating 44 and 45 it i clear that + K u u = + K. 46 Now, taking into account 4, 4 and 46, we have +K u M γu = u +K u u exp K u +K u u K u I u +K u u +K = exp uk I uk, +K +K +K and therefore the conditional MGF can be written a 47 M γu = B e ua I u A, 48 where we have defined K A = + K, B = + K + K. 49 The MGF of the SNR of the FTR model can be obtained by averaging 48 over all poible realization u of the random variable ζ, i.e., M γ = = B mm Γ m M γu f ζ u du u m e um A I u A du. 5 The integral in 5 can be olved in cloed-form, a from [3, p. 96 8] we have t µ e βt I αtdt = Γ µ + θ µ P µ β/θ, 5 where θ = β α. Uing 5 to olve 5, after ome algebraic manipulation, 8 i obtained. APPENDIX II PROOF OF COROLLARY For m =, the Legendre function in the MGF given in 8 ha a zero degree. Taking into account that P z = for all z, we can write M γ = + K R, K, ;. 5 Conidering now that [ R, K, ; = + K K ] + K + + K [ = + K K + K + ], 53 and introducing 53 into 5 we obtain M γ =. 54 K +K + By noting that the MGF of the SNR in Nakagami-q Hoyt fading i given by and equating M Hoyt =, 55 4q +q + K + K = 4q + q, 56 the expreion given in for the q parameter i finally obtained. APPENDIX III PROOF OF LEMMA We note that the polynomial R m, k, ; defined in 9 can be factorized a R m, K, ; = [m + K m + K + ] [m + K m + K ]. 57 For the ake of compactne, let u define the following parameter; m + K a = m + K, a m + K = m + K +, m + K a 3 = m + K, a 4 = + K. 58 From 8, uing 3 and 57, the MGF of γ can be rewritten a M γ = a a 3 m a 4 m Cq m a a 3 a a m / q= m q +q m a 3 q a m q +q m a 4 m. 59 Taking into account that the PDF i related to the MGF by the invere Laplace tranform, i.e., f γ x = L [M γ ; x], 5 follow from 59 and the Laplace tranform pair given in [7, eq ]. On the other hand, 6 i obtained analogouly by conidering that F γ x = L [M γ /; x]. APPENDIX IV PROOF OF LEMMA 3 The exact expreion for the TWDP fading power envelope PDF ha integral form [3] fγ T γ = β exp { βγ} exp { K} 6 π π exp { K co θ} I γβk co θ dθ, where β = + K/. In order to circumvent thi iue, a family of PDF that approximate the exact PDF of the TWDP fading model wa given in [3, eq. 7]. Thee approximate

11 PDF for the TWDP power envelope are expreed in cloedform a: M f γ T α i γ {F γ; β, K δ i 6 where i= + F γ; β, K + δ i }, F γ; β, K β exp βγ exp KI γβk and the coefficient α i and δ i are given by i α i = M M i!i! M M δ i = co k= k i i iπ M u k + i du,, 6 63, 64 repectively. The number of term in the ummation i related to the value of K and ; a argued in [3], etting M > K uffice to cloely match the exact PDF in 6. Since 6 correpond to the PDF of a Rician power envelope, expreion 6 allow for approximating the TWDP ditribution in term of a mixture of M Rician ditribution. Following a imilar reaoning a in Appendix I, the approximate PDF for the FTR fading power envelope can be obtained by averaging 6 with K u = uk over all poible realization u of the random variable ζ, which follow a Gamma ditribution a indicated in 3. Thu, the mixture of Rician PDF in 6 averaged over a Gamma ditribution lead to the following approximate expreion of the FTR power envelope PDF: where f γ γ G m γ; β, K M i= = mm Γ m α i {G m γ; β, K δ i +G m γ; β, K + δ i }, 65 F γ; β, uk f ζ udu 66 = mm β K + m m e βγ F F γ; β, uk u m e mu du m, ;. Kβγ K + m where F, ; i the Kummer confluent hypergeometric function, and the ame tep a in [, App. A] have been ued to derive the lat equation. Thu, the FTR fading power envelope PDF correpond to a mixture of M Rician hadowed PDF [4], which i in coherence with the connection between the TWDP and Rician ditribution that exit in the abence of the additional fluctuation in the pecular wave here conidered. Finally, noting that for m Z + the Kummer hypergeometric function can be expreed in term of the Laguerre polynomial by uing [4, eq. 4] and the well-known Kummer tranformation, we have F m, ; z = e z m n= m z n n n!. 67 Thu, combining yield the cloed-form approximation for the PDF of the FTR fading power envelope in 7, in term of a finite um of exponential function and power. Direct integration of 65 yield the approximate expreion for the FTR fading power envelope CDF in 9. Strikingly, we mut note that the additional fluctuation introduced by the FTR fading model doe not caue any increae in mathematical complexity, but intead facilitate the mathematical tractability. APPENDIX V PROOF OF LEMMA 4 The MGF of γ can be found a M γ = M γu,u f ζ u f ζ u du du, where f ζi, i =,, i given by 3 and + K u,u M γu,u = + K u,u u,u exp with K u,u u,u + K u,u u,u u,u I K u,u u,u + K u,u u,u 68, 69 u,u = E b /N u V + u V + σ, 7 K u,u = u V + u V σ, 7 u,u = u u V V u V + u V. 7 Uing a imilar approach a the one in Appendix I, the double integral in 68 can be olved in cloed-form with the help of [3, p. 97 ] and [3, p. 5 ], yielding 35. REFERENCES [] F. Boccardi, R. W. Heath, A. Lozano, T. L. Marzetta, and P. Popovki, Five diruptive technology direction for 5G, IEEE Communication Magazine, vol. 5, no., pp. 74 8, February 4. [] T. S. Rappaport, S. Sun, R. Mayzu, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, Millimeter Wave Mobile Communication for 5G Cellular: It Will Work! IEEE Acce, vol., pp , 3. [3] M. R. Akdeniz, Y. Liu, M. K. Samimi, S. Sun, S. Rangan, T. S. Rappaport, and E. Erkip, Millimeter Wave Channel Modeling and Cellular Capacity Evaluation, IEEE Journal on Selected Area in Communication, vol. 3, no. 6, pp , June 4. [4] J. Kyröläinen, P. Kyöti, J. Meinilä, V. Nurmela, L. Rachkowki, A. Roivainen, and J. Ylitalo, Channel modelling for the fifth generation mobile communication, in Proc. 8th European Conference on Antenna and Propagation, EuCAP 4. [5] T. S. Rappaport, G. R. MacCartney, M. K. Samimi, and S. Sun, Wideband Millimeter-Wave Propagation Meaurement and Channel Model for Future Wirele Communication Sytem Deign, IEEE Tran. Comm., vol. 63, no. 9, pp , Sept 5.

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