On the Use of High-Order Moment Matching to Approximate the Generalized-K Distribution by a Gamma Distribution

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1 On the Ue of High-Order Moent Matching to Approxiate the Generalized- Ditribution by a Gaa Ditribution Saad Al-Ahadi Departent of Syte & Coputer Engineering Carleton Univerity Ottawa Canada aahadi@ce.carleton.ca Hali Yanioeroglu Departent of Syte & Coputer Engineering Carleton Univerity Ottawa Canada hali@ce.carleton.ca Abtrac Copoite fading tae place in everal counication channel due to the rando variation of the local average power of the received ultipath-faded ignal. The generalized- (Gaa Gaa) probability denity function (PDF) ha been propoed recently to odel copoite fading in wirele channel. However further derivation uing the generalized- PDF have hown to be quite involved due to the coputational and analytical difficultie aociated with the ariing pecial function. In thi paper the approxiation of the generalized- PDF by a Gaa PDF uing the oent atching ethod i explored. A expected atching poitive and negative oent lead to a better approxiation in the upper and lower tail region repectively. However due to ariing liitation for all value of the ultipath fading and hadowing paraeter and the higher level of accuracy ought the ue of an adjutable for for the expreion of the approxiating Gaa PDF paraeter obtained by atching the firt two poitive oent i devied. The optial value of the adjutent factor for different integer and non-integer value of the fading and hadowing paraeter are given. The introduced approxiation ay iplify perforance analyi in ditributed antenna yte (DAS) networ MIMO ultihop relay networ radar and onar yte. eyword: copoite fading channel generalized- ditribution Gaa ditribution oent atching ethod negative and poitive oent aount of fading u of generalized- rando variable. I. INTRODUCTION Statitical odeling of counication channel play an eential role in the deign and analyi of different counication chee over uch channel. In wirele channel odeling the copoite fading phenoenon reulting fro the iultaneou occurrence of ultipath fading and hadowing i relevant to any counication proble uch a perforance analyi of ditributed antenna yte relay networ and radar yte. The Gaa-Gaa (generalized- ) odel wa firt introduced in the literature on the cattering in radar and onar yte [] and i recently propoed to odel copoite fading in wirele channel a a ubtitute to Thi wor i upported by ing Fahd univerity of Petroleu & Mineral. the analytically intractable lognoral-baed odel [-]. The theoretical and experiental validation for the ue of thi odel are preented in []. However further analyi uing the generalized- odel ha reulted in analytical and nuerical difficultie due to the pecial function encountered [5 ]. In [7] a region-wie approxiation of the generalized- PDF by the failiar Gaa probability denity function (PDF) wa introduced uing an adjutable for of the well-nown firt two oent atching ethod. Nuerical reult for different value of the copoite fading paraeter have hown that uch an approxiation can be ade accurate enough in both the lower and the upper tail region. In thi paper the approxiation of the generalized- PDF by a Gaa PDF i further explored by atching both the poitive and negative oent. The obtained reult have hown that atching higher order oent lead to a good approxiation up to a certain level of accuracy in both the upper and lower tail region and ay lead to lower and upper bound on the approxiated cuulative ditribution function (CDF). However atching higher order oent ha two ain liitation; (i) it reult in involved expreion that are difficult to olve and coplicated to draw inight fro and (ii) negative oent ay not exit for all value of the ultipath fading and hadowing paraeter. To bypa thee liitation the ue of the adjutable for for the expreion obtained by atching the firt two poitive oent a introduced in [7] i devied and the optial adjutent factor are coputed for relevant value of the copoite fading channel paraeter. II. THE GENERALIZED- COMPOSITE FADING MODEL The PDF of the intantaneou received power γ in a copoite fading channel can be expreed uing the generalized- odel a [ ] + + ( b x ) ( ) b pγ x x Γ( ) Γ( ) () x.5

2 where Γ () i the Gaa function and are the Naagai ultipath fading and hadowing paraeter repectively. () i the odified Beel function of the econd ind and order ( ) b and i the ean of the local average power. The CDF of the generalized- rando variable (RV) and the oent generating function a obtained in [] contain pecial function naely the hyper-geoetric and the Whittaer function. Thee function have nuerical intabilitie and are analytically cuberoe to handle [5]. III. THE APPROXIMATION USING HIGHER ORDER MOMENTS The oent atching ethod i widely ued to fit ditribution to epirical data or to approxiate an analytically coplicated ditribution by a ore tractable one. In thi paper oent atching i ued to approxiate the generalized- ditribution by the ore tractable Gaa ditribution. The nth oent of the generalized- ditribution can be derived a [] E n [ X ] Γ ( + n) Γ( + n) Γ( ) Γ( ) n. () The nth oent of the Gaa ditribution whoe cale and hape paraeter are and repectively i []. E n Γ [ ] ( + n) X Γ ( ) n. () Now uing the expreion in () and () the firt and econd poitive oent of the generalized- ditribution and the approxiating Gaa ditribution can be atched a () ( + ) ( + ) ( + ). (5) Solving () and (5) for and reult in ( ) (-a) (-b) where i j and i j denote the cale and hape paraeter of the Gaa PDF obtained by atching the ith and the jth oent ( + ) ( + ) and Note that converge to a increae and vice vera. Siilarly atching the firt poitive oent a given in () and third poitive oent a ( + )( + )( ) (7) reult in where ( ( ) ) ( ) (-a) (-b) ( + ) ( + ). Note again that if and are large (i.e. and ) then a expected. Matching the econd poitive oent a in (5) and the third poitive oent a in (7) reult in + + (9-a) ( + ). (9-b) On the other hand negative oent a defined in [9] of the generalized- and the Gaa PDF can be expreed uing () and (). Matching again the firt poitive oent a in () and the firt negative oent a reult in where ( ) ( )( ) ( ) () (-a) (-b) +. Siilarly atching the firt poitive oent a in () E X a and the econd negative oent [ ] ( )( ) () reult in ( 9 + ( ) ) ( ) (-a) (-b) 9 +

3 where +. Finally atching the firt and the econd negative oent reult in + (-a) +. (-b) Now fro the expreion given in () to () the following ay be tated: The cale paraeter of the approxiating Gaa PDF obtained by atching the poitive oent i larger than the one obtained by atching negative oent. Since the negative oent characterize a ditribution at the origin [9] (the lower tail for a poitive RV) and the poitive oent characterize a ditribution at the upper tail we ay conclude that the generalized- PDF can be approxiated by a Gaa ditribution whoe cale and hape paraeter depend on the region of the PDF (CDF) of interet. A iilar region-wie (piece-wie) approxiation wa ued in [] to well-approxiate the u of lognoral RV by a ingle lognoral RV. Matching oent for n will lead to involved expreion a een in (9-a) and (9-b). Moreover not including the firt poitive oent in atching any two oent reult in an approxiating Gaa PDF that doe not have the ae ean a the approxiated generalized- PDF (i.e. the two PDF ay have different ean of the local power). Matching the negative oent i not poible for all value of and ince thee oent ay not exit a indicated in (-a) to (-b). The cale and hape paraeter of the approxiating Gaa ditribution are dependent on the fading paraeter in the ene that a and increae the difference between the predicted cale and hape paraeter decreae and hence the difference between the approxiating PDF (CDF) becoe all. So for all value of and/or (while and ) the difference between the two approxiating Gaa CDF ight be large enough to bound the approxiated CDF a een in Fig.. On the other hand atching the ae oent for large value of and doe not reult in a good approxiation if higher accuracy i ought a een in Fig. - ince the approxiating CDF are too cloe to each other. Note: in Fig. - the copleentary cuulative ditribution function (CCDF) and particularly the region correponding P X x. i hown to obtain ore illutrative reult for to ( ) the upper tail region. The notation μi j in thee figure denote that the ith and jth oent are atched. IV. THE MOMENT MATCHING METHOD WITH ADJUSTMENT In order to bypa the liitation explained before on the ue of the oent atching for higher order oent we ay conider an adjutable for for the paraeter of the approxiating Gaa PDF obtained by atching the firt two poitive oent ince (i) the expreion in (-a) and (-b) are iple and valid for any value of and and (ii) the firt poitive oent i included in the atching. The expreion in (-a) and (-b) ay be re-written a + + [ AF ] AF AFax (5-a) AF AFax. (5-b) AF In the above + + i the aount of fading (AF) in the copoite fading channel a derived in []. The value of AF i finite ince the allet value of ax and in real propagation channel are non-zero. However the approxiating PDF a given by (5-a) and (5-b) reult in a poor fit in the lower and upper tail region ince atching only the firt and econd oent yield a good fit only around the ean. To overcoe thi liitation we ay conider the following adjutable for of the expreion in (5-a) and (5- b) [ ε ] AF AF ε ε AF AF ax (-a) AF AFax ε ε AF. AF ε (-b) Since the aount of fading added to the cale paraeter of the approxiating Gaa PDF can not exceed the original aount of fading of the approxiated PDF (i.e. ε AF ) the value of ε hould be bounded a AF ε AF. Due to the fact that the relevant practical range of AF i fro zero (for non-fading channel) to around (for evere ultipath fading and hadowing condition where.5 and.5) the relevant range of the adjutent factor ε becoe ε. The optial value of the adjutent factor can be coputed through iniizing a nuerical eaure of the Saller value of and ay tae place in land obile atellite channel [].

4 difference between the approxiated and the approxiating PDF (CDF). A coon eaure i the abolute value of the difference between the approxiated and the approxiating PDF (CDF) that i iilar to the well-nown ologorov- Sirnov tet on the difference between the CDF of two continuou ditribution []. Alo in thi paper the CDF rather than the PDF are conidered ince the Gaa PDF goe to infinity near the origin for < [] which caue nuerical intabilitie in the lower tail region. The plot of the optial adjutent factor veru the ultipath fading and hadowing paraeter are hown in Fig. and 5 for value of both and ranging fro.5 to. It i oberved fro the plot that the adjutent factor decreae a either or both and increae. The decreae of the adjutent factor a both and increae i intereting ince it indicate that the product of two Gaa PDF can be approxiated by a Gaa PDF uing the ethod of atching the firt two oent. The aount of fading for can be expreed a AF ( + ) / which i approxiately / for oderate value of and it converge to zero for very large value of. However if a high level of accuracy i ought then the agnitude of the adjutent factor increae. Siilar plot can be obtained for other region of interet. The optial value of the adjutent factor can be tabulated a in Table I to be available for ue. V. ON APPROXIMATING THE PDF OF THE SUM OF GENERALIZED- RVS The ditribution of the u of N independent generalized- RV i needed to analyze the perforance of axial ratio cobining (MRC) the outage capacity and the ergodic capacity in copoite fading channel. The adjutable for of the firt two oent atching can be ued to approxiate the PDF of the u of N independent generalized- RV and the -D plot of ε veru and for each value of N can be obtained. Moreover it i found [] that the PDF of the u of N independent and identically ditributed (i.i.d.) generalized- RV i cloely approxiated by the PDF of another generalized- RV ζ whoe aount of fading i AFζ + + ; the correponding ζ and N N N ζ can be coputed by atching the AF and etting ς and ζ N. It can alo be hown that the ς PDF of the u of N generalized- RV whoe hadowing coponent are identically ditributed and fully correlated and ultipath coponent are i.i.d. i identical to the PDF of another generalized- RV whoe paraeter are u N and u and vice vera. Furtherore the approxiation in Section IV can be ued to approxiate the PDF of the u of N generalized- RV by a ingle Gaa RV. VI. CONCLUSIONS In thi paper the approxiation of the generalized- ditribution by the ore tractable Gaa ditribution uing the oent atching ethod i invetigated. The obtained reult have hown that atching the poitive and negative oent can yield a good approxiation in both the lower and upper tail region. However the aociated liitation have led to the propoal of an adjuted for of the expreion of the paraeter of the approxiating Gaa PDF obtained by atching the firt two poitive oent. Thi iple and ufficiently accurate approxiation can be further ued to approxiate the PDF of the u of both independent and correlated generalized- RV. Extenion of thi wor ay include conidering atching non-integer oent and approxiating the PDF of the u of non-i.i.d. generalized- RV uing the propoed ethod. ACNOWLEDGMENT The author would lie to acnowledge the helpful dicuion with Sebatian Szyzowicz of Carleton Univerity during early tage of thi wor. REFERENCES [] D. J. Lewiny Nontationary probabilitic target and clutter cattering odel IEEE Tran. Antenna & Propag. vol. AP- no. pp. 9-9 May 9. [] P. M. Shanar Error rate in generalized hadowed fading channel Wirele Peronal Coun. vol. no. pp. - Feb.. [] P. S. Bitha N. C. Sagia P. T. Mathiopoulo G.. aragiannidi and A. A. Rontogianni On the perforance analyi of digital counication over generalized- fading channel IEEE Coun. Lett. vol. 5 no. pp May. [] P. M. Shanar Outage probabilitie of a MIMO chee in hadowed fading channel with icro- and acrodiverity reception IEEE Tran. Wirel. Coun. vol. 7 no. pp. 5-9 June. [5] M. Gu and D. A. Abraha Uing McDaniel odel to repreent non- Rayleigh reverberation IEEE Tran. Oceanic Eng. vol. pp July. [] P. S. Bitha P. T. Mathiopoulo and S. A. otopoulo Diverity reception over generalized- ( G) fading channel IEEE Tran.Wirel. Coun. vol. no. pp. - Dec. 7. [7] Saad Al-Ahadi and Hali Yanioeroglu On the approxiation of the generalized- PDF by a Gaa PDF uing the oent atching ethod in Proc. IEEE Wirele Coun. Networing Conf. (WCNC 9) pp. - April 9. [] A. Papouli Probability Rando Variable and Stochatic Procee nd ed. New Yor.: McGraw-Hill 9. [9] N. Creie A. S. Davi J. L. Fol and G.E. Policello The oentgenerating function and negative integer oent The Aerican Statitician vol. 5 no. pp. -5 Augut 9. [] J. C. S. Santo Filho P. Cardieri and M. D. Yacoub Highly accurate range-adaptive lognoral approxiation to lognoral u ditribution Electron Lett. vol. no. Mar.. [] A. Abdi W. C. Lau M.-S. Alouni and M. aveh A new iple odel for land obile atellite channel: firt- and econd order tatitic IEEE Tran. Wirel. Coun. vol. no. pp May. [] S. Al-Ahadi and H. Yanioeroglu On the ditribution of the u of generalized- rando variable in preparation.

5 log(p(x<x) - - the lower tail of the CDF μ- μ- μ log(p(x<x) - - the lower tail of the CDF μ- μ- μ log(p(xx) the upper tail of the CDF - μ- - - μ μ log(p(xx) the upper tail of the CDF - μ- - - μ μ Figure. The CDF correponding to the generalized- and the approxiating Gaa PDF for.5 and.5 uing the oent atching ethod where atched. μ i j denote that the ith and jth oent are Figure. The CDF correponding to the generalized- and the approxiating Gaa PDF for and uing the oent atching ethod. log(p(x<x) log(p(xx) - - the lower tail of the CDF μ- μ- μ the upper tail of the CDF - μ- - - μ μ Figure. The CDF correponding to the generalized- and the approxiating Gaa PDF for 7 and uing the oent atching ethod. The adjutent factor ε Figure. The plot of the adjutent factor that iniize the abolute value of the difference between the approxiated generalized- and the approxiating Gaa ditribution over the entire CDF.

6 The adjutent factor ε 5 Figure 5. The plot of the adjutent factor that iniize the abolute value of the difference between the approxiated generalized- and the approxiating Gaa ditribution for the lower tail (CDF<.). Table I The optial value of the adjutent factor that iniize the abolute value of the difference between the generalized- and the Gaa ditribution over the entire CDF for different value of and (refer to Fig. )