Approximate Analysis of Power Offset over Spatially Correlated MIMO Channels

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Commuicatios ad Network, 9, 5-3 doi:.36/c.9. Published Olie August 9 (http://www.scirp.

1 Commuicatios ad Network, 9, 5-3 doi:.36/c.9. Published Olie August 9 ( Approximate Aalysis of Power Offset over Spatially Correlated MIMO Chaels Guagwei YU, Xuzhe WANG School of Iformatio ad Commuicatio Egieerig, Beijig Uiversity of Posts ad elecommuicatios, Beijig, Chia {yuguagwei98, Gxitao}@gmail.com Abstract: Power offset is zero-order term i the capacity versus sigal-to-oise ratio curve. I this paper, approximate aalysis of power offset is preseted to describe MIMO system with uiform liear atea arrays of fixed legth. It is assumed that the umber of receive atea is larger tha that of trasmit atea. Spatially Correlated MIMO Chael is approximated by tri-diagoal toeplitz matrix. he of tri-diagoal toeplitz matrix, which is fitted by elemetary curve, is oe of the key factors related to power offset. Based o the curve fittig, the of tri-diagoal toeplitz matrix is mathematically tractable. Cosequetly, the expressio of local extreme poits ca be derived to optimize power offset. he simulatio results show that approximatio above is accurate i local extreme poits of power offset. he proposed expressio of local extreme poits is helpful to approach optimal power offset. Keywords: MIMO, multiplexig gai, power offset, high sr regio, toeplitz matrix. Itroductio Multiple-iput multiple-output (MIMO) system is widely used i wireless commuicatio to improve the performace. he spectral efficiecy of MIMO chael is much higher tha that over the covetioal sigal atea chael. he research of MIMO icludes two differet perspectives: the first oe cocers performace aalysis i terms of error performace of practical systems, the secod oe cocers the study of chael capacity. he desig of commuicatio schemes was maily cosidered i the former perspectives with the aid of theoretical aalysis ad simulatio [][], ad [3]. For the latter, importat parameters such as diversity gai [], multiplexig gai [5][6] (referred to degree of freedom i other literatures), ad power offset [7-9], ad [] were emphatically aalyzed i the high sigal-to-oise ratio (SN) regio. Furthermore, diversity ad multiplexig tradeoff has bee proposed i [][], ad [3], that is to say diversity ad multiplexig tradeoff ca be obtaied for a give multiple atea chael. It is worth poitig out i [] that the diversity ad multiplexig tradeoff is essetially the tradeoff betwee the error probability ad the data rate of a give system. he multiplexig gai is ot sufficiet to accurately characterize the property of MIMO capacity. A more accurate represetatio of high SN behavior i SN-capacity curve is provided by a affie approximatio to capacity, which icludes both the multiplexig gai (i.e. slope) ad power offset (i.e. zero-order term) [7]. igh SN power offset has bee aalyzed i [8] over multiple atea icea chaels. It was show i [8] that the impact of the icea factor at high SN regio could be coveietly quatified through the correspodig power offset. I [9], high SN power offset i multiple atea commuicatio was derived i detail. Achievable throughput was compared betwee the optimal strategy of dirty paper codig ad suboptimal liear precodig techiques (zero-forcig ad block diagoalizatio) i [] o applicatio of power offset. ece, power offset is a importat parameter i multiple atea commuicatio. I may practical eviromets, sigal correlatio amog the atea array exists due to the scatterig. Fadig correlatio ad its effect o the capacity of multielemet atea system has bee studied i []. ece, aalysis of spatially correlated MIMO chaels has bee aother topic i the past few years, which ecessitates the model of correlated chael. A geeral space-time correlatio model for MIMO systems i mobile fadig chael has bee preseted i [5]. he model i [5] was flexible ad mathematically tractable. I [6], the correlated model of [5] has bee used to ivestigate the capacity of spatially correlated MIMO ayleigh-fadig chaels. ecetly, applicatio of differet atea arrays i MIMO system has also bee exploited. With the cosideratio of physical costraits imposed by maximum size of the atea array, uiform liear array of fixed legth has bee used i [7] to aalyze the asymptotic capacity of MIMO systems. he uiform liear array of fixed legth is also used i this paper to aalyze the power offset of MIMO system over spatially correlated chael. O applicatio of curve fittig, the of tri-diagoal toeplitz matrix is mathematically tractable. Over spatially correlated cha- Copyright 9 Scies

2 6 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS el, the mai result of this paper is the derived expressio of local extreme poits of power offset, usig the uiform liear array of fixed legth. o the best of our kowledge, the mai result has ot bee preseted i other literatures. he rest parts are orgaized as follows. I Sectio, the basic defiitios of multiplexig gai ad power offset are preseted. System model ad correlatio model are give i Sectio 3. Approximate aalysis of power offset by fittig curve of tri-diagoal toeplitz matrix is put forward i Sectio. Sectio 5 shows the simulatio results. Fially, a brief coclusio is give i Sectio 6. Notatio: I the followig cotext, matrices ad vectors are deoted by boldface upper case symbols ad boldface lower case symbols, respectively. he traspose ad, re- ermitia traspose are deoted by ad spectively. he expected value is represeted by is used for the Euclidia orm.. Basic Defiitio E. I the high SN regio, the capacity of sigle-user MIMO system of coheret receptio is give by [8][9] CSN ( ) mi(, )log SN o() () where, deote the umber of trasmit ad receive atea elemets, respectively. SN is the sigal to oise ratio. he MIMO capacity i high SN regio is a liear fuctio of mi(, ), i.e. the statioary slope i SN-capacity curve. I other words, ay icrease i max(, ) is immaterial, i.e. without ay impact to the slope. he slope is the so-called maximum multiplexig gai (degree of freedom). If is fixed to a give value (geerally speakig, is less tha six, expressed as i the followig cotext), ad, the multiplexig gai equals to a fixed value without cosideratio of correlatio of chael respose ad the umber of receive atea elemets. So power offset is itroduced to compare the impact of chael property with the same value of multiplexig gai, ad () is replaced by [9] C( SN) S(log ( SN) L) o() SNdB S L o() 3dB where is multiplexig gai ad is power offset S i 3-dB uits. Whe SN,the multiplexig gai ad power offset ca be computed by S L () CSN ( ) lim (3) SN log ( SN ) L CSN ( ) lim log ( SN) SN S I the high SN regio, SN-capacity curve is approximately determied by multiplexig gai ad power offset. Most chaels, havig the same multiplexig gai, may have very differet capacities because of various values of the power offset. ece, power offset is a importat parameter to describe the capacity behavior i high SN regio. 3. System Model ad Correlatio Mode he system model ad chael correlatio model are defied i this sectio. For MIMO system, the geeral basebad model is give by () y x (5) where is the chael respose matrix, ad x [ x x x ] is the iput complex sigal vector whose spatial covariace matrix ormalized by the eergy per dimesio ca be expressed as Φ E( xx ) E[ x ] while y [ yyy ] ad [ are the ] output vector ad additive white Gaussia oise vector, respectively. Because of ormalizatio ad assumptio of isotropic iput, tr( Φ ) ad Φ I, where I is idetity matrix. Kroecker model [6] is used to describe the correlatio of chael respose. O the assumptio of rich scatterig eviromets ad havig o lie of sight, the correlated chael respose matrix is deoted by (6) Σ Σ Σ (7) where is Kroecker product, Σ ad Σ are correlated matrix of trasmit ad receive atea, respectively. So the chael respose matrix is give by Σ Σ (8) where is chael respose matrix whose elemets are idepedet ad idetical distributio radom variables. Without loss of geerality, we assume that the distace betwee trasmit atea elemets is large eough to eglect the correlatio at the trasmit ode. Cosequetly, without cosiderig the atea correlatio at the trasmit ode, the correlatio matrix Σ is idetity matrix, ad the receive atea correlatio model Σ is give by [5][6]. Copyright 9 Scies

3 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS 7 I d jsi( ) d ij ij a J p, Σ (, i j) (9) ç è - ø I ( ) where [, ) is the so-called AOA(agle of arrival), [, ) is average value of AOA, dij is ormalized distace betwee atea array elemets, that is to say d ij d ij, d he factual distace betwee i ad j ij is t atea array elemet, i s wave legth, I () is -order modified Bessel fuctio. It is obvious that Σ (, i j ) is real symmetric toeplitz matrix. O the assumptio of isotropic scatterig, the correlatio matrix at the receive ode is simplified to Σ (, i j) J ( d ) () where J () is order Bessel fuctio, ad Σ (, i j ) ca be simplified to tri-diagoal toeplitz matrix through the simulatio results illustrated by Figure ad Figure. where Σ ij () æ L ö = L () where L is the fixed legth of receive atea array ormalized by. I this paper, L is assumed to be i the rage from to. I the Figure ad, power offset approximatio based o tri-diagoal, five-diagoal, ad seve-diagoal toeplitz matrix is obtaied versus the umber of receive atea with various value of L. Furthermore, power offset over idepedet chael is also provided as lower boud for umerical compariso. It ca be see from Figure ad that tri-diagoal toeplitz matrix ca be used to approximate the correlatio model at the receive ode.. Approximate Aalysis of Power Offset Depedig o the discussio i Sectio 3, tri-diagoal toeplitz matrix ca be used to approximate the correlatio model at the receive ode. I this sectio, the curve of tri-diagoal toeplitz matrix is fitted by a elemetary fuctio, which simplifies the approximate aalysis of power offset. Expressio of power offset for MIMO system over correlated chael is derived i [9], ad the expressio is give by L log E(l det( W Λ )) W l log det( Λ P ) (3) power offset L=5 Nt= tri-diagoal app. five-diagoal app. seve-diagoal app. o correlatio Figure. Approximate aalysis of power offset based o 3, 5, ad 7 diagoal toeplitz matrix with L 5, Copyright 9 Scies

4 8 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS power offset L=6 Nt= tri-diagoal app. five-diagoal app. seve-diagoal app. o correlatio Figure. Approximate aalysis of power offset based o 3, 5, ad 7 diagoal toeplitz matrix with L 6, where Λ ad Λ are diagoal matrix whose elemets are eigevalues of Σ ad Σ, respectively. P is also diagoal matrix whose elemets are eigevalues of ormalized spatial covariace matrix Φ, so P is idetity matrix ad / lo g P) de t(λ is costat zero. Λ ca be derived from (). It is obvious that Λ is full rak matrix, ad the (3) is simplified to L log ( f( ) g( )) () where f ( ) ad g( ) ca be described by Because W W with probability, E(l det( W W)) f( ) (5) l l det( Λ ) g ( ) l (6) is wishart matrix ad osigular f ( ) is reduced to [9] f( ) E(logedet( W W)) l l l ( l) (7) where () is digamma fuctio ad the defiitio of digamma fuctio is ( ) (8) l l where is Euler-Mascheroi costat, ad lim loge.577 (9) l l Whe it comes to fuctio g( ), the derivatio of g( ) ca be expressed as g ( ) ldet( Λ) ldet( Σ ) () l l It ca be foud from () that g( ) is oly determied by the of Σ, ad the of Σ ca be com- order tri-diagoal toeplitz matrix puted by []. det Σ () With the aid of umerical computatio ad mathematical aalysis, the curve of the of tri-diagoal toeplitz matrix ca be fit by some tractable elemetary Copyright 9 Scies

5 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS 9 fuctio. Whe i (), det Σ ca be approximated by the fitted curve as follow: L h ( ) 5 5 J ( ) () he approximatio betwee the fitted curves ad curves of tri-diagoal toeplitz matrix are illustrated i Figure 3 ad Figure. From Figure 3 ad, we ca fid that the fittig curve ca exactly approach the extreme poits of the of tri-diagoal toeplitz matrix. owever, the expressio is much simpler. I other words, the further derivatio usig the fitted curve is tractable usig well-kow mathematical software such as WOLFAM MAEMAICA. I the followig cotext, power offset is aalyzed i detail. he approximate aalysis of power offset is based o the piecewise fuctio of. he coditio 5 i () is ecessary due to det Σ, so the umber of receiver atea ca ot be large tha L. Moreover, it shows from the coclusio i [] that the effect of correlatio ca be eglected whe the ormalized legth of receive atea array L is larger tha oe wave legth, that is to say L. ece, the cosidered rage of the umber of receive atea is L L. With the discussio above, whe L, the ormalized legth of receive atea array L is large tha oe wave legth ad the effect of correlatio ca be eglected, that is to say g ( ) ad the power offset is expressed as L log ( ) l l l (3) Whe it comes to the rage of L L, the ormalized legth of receive atea array L is smaller tha oe wave legth ad the effect of correlatio ca ot be eglected. I other words, g ( ). owever, for a give value of, the first-order derivative ( f ( ) assumed to be cotiuous) of f ( ) is obtaied by df ( ) ( l) d l l ( l) l l ' is () where () is trigamma fuctio. ece, it show from () that f ( ) approximately equal to the costat C (approximatio accuracy is related to the value of ). he power offset is expressed as L log g( ) C (5) of of tri-diagoal toeplitz matrix L= Figure 3. Approximatio of of tri-diagoal toeplitz matrix by curve fittig ad L 5 Copyright 9 Scies

6 3 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS of of tri-diagoal toeplitz matrix L= Figure. Approximatio of of tri-diagoal toeplitz matrix by curve fittig ad L 6 Without loss of geerality, the costat C ca be eglected i the approximate computatio. ece, the power offset i (5) is oly determied by g( ). From (), the curve of the of tri-diagoal toeplitz matrix ca be fitted by h ( ) 5, where L J ( ). he first order derivative ( h ( ) is assumed to be cotiuous) of h ( ) ca be computed by L L LJ ( ) J( ) dh( ) d ( ) (6) where L L, that is to say L. I the rage above, fuctio L J ( ) i (6) has two zero poits, i.e. L L P.8 ad P 5.5, ad L fuctio J( ) oly has oe zero poit, i.e. L P Cosequetly, h ( ) has three L extreme poits i the rage of, described by P, i,, 3. i Ad the, the secod-order derivative of h ( ) ca be computed by L L 8 LJ ( ) J( ) dh ( ) 3 d ( ) L 8 L J( ) ( ) L L L L J( ) J( ) J( ) ( ) (7) Usig (7), the secod-order derivative i poits P, i,, 3 is obtaied as follow: i ( ).5 P : (8) L dh d L.8 ( ) P : (9) L dh d L 5.5 ( ) 7.8 P : (3) L dh 3 d L With the discussio above, it is obvious that poit P ad P are local maximum poits of h ( ) ad poit Copyright 9 Scies

7 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS 3 P 3 is local miimum poit. Furthermore, it is obvious from () that the extreme poits of h ( ) is same as oes of g( ). Cosequetly, i the rage of L L, the extreme poits of power offset i (5) ca be approximately computed by arg mi L, for local miimal poit (3) arg max L, for local maximum poit where P is defied as follows: ì pl é pl ùü P= í ï +, + ý ï, i =,,3 (3) ê P ú i P ïë û ê î i úïþ where ê ëxú û ad é êxù ú deote the maximum iteger smaller tha x ad miimum iteger larger tha x, respectively. Pi, i,, 3 is the extreme poits computed above. (3) ca be used to approximately compute the umber of receive atea, i order to achieve the optimal power offset. 5. Simulatio esults I the Figure ad, with the assumptio of, power offset is approximately aalyzed based o tri-diagoal, five-diagoal, ad seve-diagoal toeplitz versus the umber of receive atea, with L 5 i Figure ad L 6 i Figure. Furthermore, power offset i idepedet chael is also provided as lower boud for umerical compariso. he approximatio tred ca be foud from Figure ad. he tri-diagoal toeplitz matrix ca be assumably used to characterize the property of power offset over spatially correlated chael. ece, tri-diagoal toeplitz matrix ca be used to approximate the correlatio model at the receive ode. he approximatio of fittig the curve of tri-diagoal toeplitz matrix is illustrated i Figure 3 ad Figure to compare the differece betwee fitted curves ad curves, with L 5 i Figure 3 ad L 6 i Figure. From Figure 3 ad, we ca fid that the fittig curve ca exactly approach the extreme poits of the of tri-diagoal toeplitz matrix. owever, the expressio is mathematically tractable. With the same assumptio of, approximatio of power offset is show i Figure 5 ad 6, with L 5 i Figure 5 ad L 6 i Figure 6. From Figure 5 ad 6, we ca fid the positio of two local miimal poits ad oe local maximum poit. With the coditio of L 5 ad, the local extreme poits computed by (3) is 9 for local maximum poit ad 7, 5 for local miimal poits. With the coditio of L 6 ad, the local extreme poits computed by (3) is for local maximum poit ad 8, 6 for local miimal poits. he computatio results based o (3) coicide with the illustratio results i Figure 5 ad 6. ece, all the approximatio i this paper is reasoable, icludig approximatio of correlatio model ad approximatio of of tri-diagoal toeplitz matrix. power offsetl=5 Nt= o correlatio Figure 5. Power offset comparatio betwee accurate ad approximate of toeplitz matrix with L 5, Copyright 9 Scies

8 3 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS power offsetl=6 Nt= o correlatio Figure 6. Power offset comparatio betwee accurate ad approximate of toeplitz matrix with L 6, Similar results are preseted i Figure 7 ad 8, with the assumptio of ad L 5 i Figure 7 while L 8 i Figure 8. With the coditio of L 5 ad, the local extreme poits computed by (3) is 9 for local maximum poit ad 7, 5 for local miimal poits. With the coditio of L 8 ad, the local extreme poits computed by (3) is for local maximum poit ad, for local miimal poits. he computatio results based o (3) coicide with the illustratio results i Figure 7 ad 8. Cosequetly, it is preseted i Figure 5 ad 7 that the effect caused by the umber ca be eglected. I other words, for small value of ad i the rage of umber of receive atea L L, the assumptio of f ( ) C is teable with eglectable loss of precisio. he extreme poits of power offset is oly determied by g( ). 8 6 power offset L=5 Nt= o correlatio Nr Figure 7. Power offset comparatio betwee accurate ad approximate of toeplitz matrix with L 5, Copyright 9 Scies

9 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS power offset L=8 Nt= o correlatio Nr Figure 8. Power offset comparatio betwee accurate ad approximate of toeplitz matrix with L 8, 6. Coclusios Approximate aalysis of power offset over spatially correlated chael is proposed to optimize the umber of receive atea with atea array of fixed legth. Atea correlatio matrix is approximated by tri-diagoal toeplitz matrix. he of tri-diagoal toeplitz matrix is reduced to simple style by curve fittig with eglectable loss of precisio. Based o the approximatio above, the expressio of local extreme poits of power offset is derived. Simulatio results shows that the derived expressio is accurate i the local extreme poits of power offset, that is to say, all the approximatio i this paper is reasoable. EFEENCES [] AOK V, JAFAKANI, CALDEBANK A. Space-time block code from orthogoal desigs. IEEE rasactios o Iformatio heory, Jul. 999, 5(5): 567. [] BOUBAKE N, LEAIEF K B, MUC D. Performace of BLAS over frequecy-selective wireless commuicatio chales. IEEE rasactios o Commuicatio, Feb., 5(): [3] LOZANO A, PAPADIAS C. Layered space-time receivers for frequecy-selective wireless chaels. IEEE rasactios o Commuicatio, Ja., 5(): [] BAKI L, FOLIK J. Diversity gais i two-ray fadig chaels. IEEE rasactios o Wireless Commuicatios, Feb. 9, 8(): [5] POON A, BODESEN, SE D. N. C. Degrees of freedom i multiple-atea chaels: A sigal space approach. IEEE rasactios o Iformatio heory, Feb. 6, 5(): [6] JAFA S A, SAMAI S. Degrees of freedom regio of the MIMO X chael. IEEE rasactios o Iformatio heory, Ja. 8, 5(): 5-7. [7] JINDAL N. igh SN aalysis of MIMO broadcast chaels. ISI 5, 5, 3-3. [8] ULINO A M, LOZANO A, VEDU S. igh-sn power offset i multi-atea icea chaels. ICC 5, 5, [9] LOZANO A, ULINO A M, VEDU S. igh-sn power offset i multi-atea commuicatio. IEEE rasactios o Iformatio heory, Dec. 5, 5(): 3-5. [] LEE J, JINDAL N. igh SN aalysis for MIMO broadcast chaels: Dirty paper codig versus liear precodig. IEEE rasactios o Iformatio heory, Dec. 7, 53(): [] ZENG L Z, SE D N C. Diversity ad multiplexig: A fudametal tradeoff i multiple-atea chaels. IEEE rasactios o Iformatio heory, May 3, 9(5): [] SE D N C, VISWANA P, ZENG L Z. Diversity multiplexig tradeoff i multiple-access chaels. IEEE rasactios o Iformatio heory, Sep., 5(9): [3] WAGNE J, LIANG Y C, ZANG. O the balace of multiuser diversity ad spatial multiplexig gai i radom beamformig. IEEE rasactios o Wireless Commuicatio, Jul. 8, 7(7): [] SIU D, FOSCINI G J, GANS M J, E AL. Fadig correlatio ad its effect o the capacity of multielemet atea system. IEEE rasactios o Commuicatio, Mar., 8(3): Copyright 9 Scies

10 3 APPOXIMAE ANALYSIS OF POWE OFFSE OVE SPAIALLY COELAED MIMO CANNELS [5] ABDI A, KAVE M. A space-time correlatio model for multielemet atea systems i mobile fadig chaels. IEEE J. Select. Areas Commuicatio, Apr., : [6] CIANI M, WIN M Z, ZANELLA A. O the capacity of spatially correlated MIMO ayleigh Fadig chaels. IEEE rasactios o Iformatio heory, Oct. 3, 9(): [7] WEI S, GOECKEL D, JANASWAMY. O the asymptotic capacity of MIMO systems with atea arrays of fixed legth. IEEE rasactios o Wireless Commuicatios, Jul. 5, (): 68. [8] ELAA I E. Capacity of multi-atea Gaussia chael. Europe rasactios o elecommuicatio, Nov./Dec. 999, : [9] FOSCINI G J, GANS M J. O the limits of wireless commuicatios i a fadig eviromet whe usig multiple atea. Wireless Persoal Commuicatio, 998, 6(3): [] XU Z, ZANG K Y, LU Q. Fast algorithm based o toeplitz matrix. Xi a: Northwest Idustrial Uiversity, 999. [] ZUANG Z M, CAI C Q, E AL. Space-time correlatio aalysis for MIMO fadig chael based o geometric model. Joural of Shatou Uiversity, 8, 3(): Copyright 9 Scies

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of