An Upper Bound for Limited Rate Feedback MIMO Capacity

Transcription

1 1 A Upper Boud for Limited Rate Feedbac MIMO Capacity Göha M Güvese, Studet Member, IEEE, ad A Özgür Yılmaz Abstract We develop a techique to upper boud the poitto-poit MIMO limited rate feedbac LRF capacity uder a wide class of vector quatizatio schemes The upper boud turs out to be tight ad ca also be used to obtai a absolute upper boud by usig a boudig distributio for Grassmaia beamformig The boudig techique ca be applied to other problems requirig the exact evaluatio of the expected value of matrix determiat Idex Terms Limited rate feedbac, MIMO, capacity boud, boudig distributio, vector quatizatio, reduced precodig, sigular value decompositio I INTRODUCTION Capacity gais promised by multi-iput multi-output MIMO systems ofte require a accurate owledge of the chael at trasmitter ad receiver sides especially i quest to capitalize these possible gais i practical systems A accuracy problem arises whe chael state iformatio CSI has to be trasmitted from the receiver to the trasmitter It is obvious that CSI caot be trasmitted with ifiite precisio A limited rate feedbac chael is usually available for this commuicatio ad this sets a limit for the accuracy of CSI at the trasmitter side It was show that the MIMO chael is iterferece limited whe the chael estimatio is imperfect [1] It was further observed i [1] that istataeous feedbac, eve if imperfect, gives large capacity gais i low SNR ad is useful i high SNR especially whe the umber of trasmit ateas t is larger tha the that of receive ateas r [] I [3], quatizatio rules ad correspodig quatizer desig criteria were proposed to be used i MISO multiple-iput sigleoutput ad MIMO chaels Quatizatio of beamformers were ivestigated uder a Grassmaia lie pacig framewor with regard to quatizatio codeboo size, capacity-snr loss, ad outage performace i [4], [5] We ivestigate the capacity of poit-to-poit MIMO chaels i this paper as opposed to the broadcast chael settigs i aforemetioed studies [6] Although the capacity is less affected by the lac of CSI o the trasmitter side at high SNR [7], its availability is very importat both at low SNR ad i desigig practical systems that ca operate close to the capacity as i adaptively modulated MIMO schemes [8] sice Mauscript received December, 7; revised May 1, 8; accepted July 1, 8 The associate editor coordiatig the review of this paper ad approvig it for publicatio was Dr Rohit Nabar This wor was supported i part by the Scietific ad Techological Research Coucil of Turey TUBITAK uder grat 14E7 The authors are with the Departmet of Electrical ad Electroics Egieerig, Middle East Techical Uiversity, Aara, Turey aoyilmaz@metuedutr, guvese@metuedutr the complex tas of joit detectio ad decodig is avoided Furthermore, the capacity is strictly smaller with t > r if o CSI is available at trasmitter [] We cocetrate o a fiite rate feedbac sceario i which precoders obtaied by the sigular value decompositio of the MIMO chael [9] are fed bac to the trasmitter side A capacity loss boud for covariace matrix based quatizatio was preseted i [1] ad a capacity loss boud was proposed i [11] for desigig matrix quatizatio based codeboos We herei focus o quatizig the colums of the precodig matrix obtaied from sigular value decompositio SVD The chael is quasi-parallelized by separately quatizig precoders ad well-ow adaptive modulatio ad codig techiques ca be utilized as stated i [] Covariace matrices geerated radomly with uiform distributio o the uit sphere are used i [1], that is, radom matrix quatizatio is studied O the other had, our mai cotributio i this paper is the derivatio of a capacity upper boud expressio that is valid for a wide rage of vector based quatizatio schemes The proposed upper boud turs out to be quite tight maily due to the exact evaluatio of the expected value of matrix determiat as opposed to similar studies usig Hadamard iequality to upper boud the determiat as i [1] ad usig approximate desity fuctio of determiat expressio ad partitio cell approximatio i [11], [1] As a byproduct, a absolute upper boud to LRF MIMO capacity usig precodig based quatizatio is also herei derived by utilizig a boudig distributio for Grassmaia beamformig [13] The outlie is as follows The system model is explaied i Sectio II MIMO capacity expressios for LRF are obtaied i Sectio III A aalytical upper boud for the LRF MIMO capacity is derived i Sectio IV Two exemplary quatizatio schemes will be studied i Sectio V ad correspodig umerical results are preseted i Sectio VI The paper is cocluded with Sectio VII II SYSTEM MODEL The followig otatio is used throughout the mauscript Boldface lower ad upper-case letters deote colum vectors ad matrices, respectively Scalars are deoted by plai lower-case letters The superscript deotes the complex cojugate for scalars ad cojugate traspose for vectors ad matrices The absolute value of a scalar is show with The idetity matrix is show with I The trace operator ad determiat are deoted by tr ad, respectively The autocorrelatio matrix for a radom vector a is R a = E[aa ]

2 where E[ ] stads for the expected value operator The i, j th elemet of a matrix A is deoted by A i,j The geeral expressio for a poit-to-poit MIMO chael with r receive ateas ad t trasmit ateas is give by ỹ = H x + w, where ỹ is the received vector, H is the r t chael matrix, x is the trasmitted vector, ad w is the zero-mea circularly symmetric complex Gaussia ZMC- SCG white spatially ad temporally oise with ormalized variace 1 The chael matrix H is comprised of idepedet ZMCSCG radom variables with variace 1 Cosiderig a bloc fadig model, the chael matrix is assumed to be costat durig a coherece iterval sigificatly larger tha symbol duratio A fixed average power is allotted for each trasmissio which correspods to settig trr x P Sec 13 i [9] I the case that perfect chael iformatio is available both at the trasmitter ad receiver, sigular value decompositio SVD is applied to decompose the MIMO chael ito mi r, t parallel subchaels over which multiple streams may be trasmitted [14] The followig equivalet expressio is obtaied for the received vector whe SVD is performed to attai H = UDV : U ỹ = DV x + U w 1 The etries of D are tae to be decreasig without loss of geerality The trasmitted vector ca be writte i geeral as i x = PΛx where P is a precodig matrix, Λ is a diagoal matrix used to distribute power amog subchaels, ad x is the origial iformatio vector assumed to have R x = I mir, t If the precodig matrix is chose to be P = V, by the uitary property of the precodig matrix V V = I y = DΛx + w, where y = U ỹ ad w = U w Sice both D ad Λ are diagoal ad R w = I, the chael is decomposed ito parallel subchaels The capacity is achieved by Λ obtaied through the waterfillig procedure [14] with the costrait that trλ P We ote here that the colums of matrix V are isotropically distributed o the t dimesioal complex uit circle whe cosidered over the realizatios of H Whe there is oly a partial CSI i trasmitter due to fiite rate feedbac, oe has imperfect precodig ad power distributio matrices deoted by V f ad Λ f, respectively Eq ow becomes y = DV V f Λ f x + w 3 which suggests that subchaels ow iterfere with each other sice V V f I, i geeral We will ivestigate the capacity of LRF MIMO chaels based o 3 I order to reduce the rate of the feedbac chael, the idea of reduced precodig ca be used [15] I this scheme, the umber of beamformers used i a spatial multiplexig system is adaptively varied i order to miimize probability of symbol vector error or to maximize capacity by allocatig equal power Λ f = P I to selected subchaels [15], [11] Trasmittig oly the precodig vectors correspodig to the strogest subchaels will suffice to maximize commuicatio rate over MIMO chaels Thus, this strategy allows efficiet utilizatio of the feedbac bits by quatizig oly relevat precoders The aalytical boud for limited rate feedbac MIMO capacity to be obtaied i sectio IV ca be used to determie the umber of precoders to be used at each average SNR value i order to maximize the spectral efficiecy The idea of reduced precodig ad the utilizatio of feedbac for precoders are ot oly useful at low SNR values but also at high SNR especially for MIMO systems with t > r [] III CAPACITY WITH LIMITED RATE PRECODING Eq 3 ca be writte with a equivalet chael matrix H = DV V f as i y = HΛ f x + w 4 Notig that Λ f = P I, the capacity of this scheme which maes use of precoders is give by { C pre = E log det I + P } HH 5 ad ca be achieved with MMSE estimatio ad successive iterferece cacellatio [16] [18] where the equivalet chael matrix H ca be writte as H = DV V f d 1 d = d v 1 v v [ ] v1f v f The equivalet chael matrix H has its i, j th elemet as H i,j = d i vi v jf, where v i is the i th colum of V ad, v jf is the j th colum of V f Defiig V ij = vi v jf, oe ca evaluate HH i,j = d i d j =1 V i V j 6 7 Evaluatio of the capacity i 5 requires the probability distributios of V ij = vi v jf for i, j {1,,, } ad hece the quatizatio rule used for limited rate feedbac has to be specified A set of N f vectors {q 1,q,,q N f } geerated to costruct the quatizatio codeboo are defied where N f stads for the umber of feedbac bits per precodig vector Quatizatio vectors are legth- t complex vectors o the t - dimesioal complex uit circle ad the quatizatio while obtaiig the precodig vectors is determied by the followig rule used i LRF MIMO studies [4] [6], [19]: v if = arg max q j, j=1,, N f v i q j 8 There are two types of radom variables i 5 whose distributios ad depedece properties have to be determied i order to evaluate C pre First, the cumulative distributio fuctio cdf of V ii = v i v if is peculiar to the give quatizatio codeboo ad rule, ad we will ivestigate the cdf of V ii for two differet quatizatio methods i Sectio V Moreover, the

3 3 cdf for V ij s for i j is eeded I [4], [5], [19], the cdf of the squared absolute ier product betwee two isotropically distributed legth- t complex uit vectors is give as F t o x = 1 1 x t 1, x 1, x < 1, x > 1 The same result ad hece cdf hold for the case of oe fixed vector ad a isotropically distributed vector sice oe of them beig isotropically distributed is sufficiet for the result [4] Bearig i mid that vi v jf correspods to projectio of v i oto v jf, the followig holds by orthogoality of v i s i our problem 1 v v 1f = v v 1f vj v 1fv j 1 j=1 for =,, Defiig v 1f = v 1f 1 j=1 v j v 1fv j, the vector v 1f is i the ull space of v i s, i = 1,, 1, where the ull space has dimesio t +1 The squared orm of v 1f is v 1f v 1f = 1 1 j=1 v j v 1f = 1 1 j=1 V j1 Cosiderig the projectio of a fixed vector v oto a isotropically distributed vector v 1f which is of dimesio t +1, oe obtais the followig coditioal probability distributio fuctio for v v 1f by usig 9: F v v 1f x v i v 1f = a i, i = 1,, 1 = Fo t +1 x i=1 a i for =,, The expectatio i 5 is over the chael matrix H or equivaletly, over V ij s For a give chael realizatio, v i s are fixed ad the quatized precodig vectors v if s are chose idepedetly of each other by the rule give i 8 Hece, v if s are idepedet of each other o the coditio that v i s are give Oe should ote that V V f product ivolves V ij terms ad the phase of V ij becomes relevat i this case Recallig that the vectors are isotropically distributed, the phases of all the radom variables correspodig to the projectios of the precoders oto quatized precoders are idepedet ad uiformly distributed i [, π] sice the quatizatio rule i 8 is blid to multiplicatio of all the etries of a quatizatio vector by a complex umber α of uity amplitude as a b = αa b To summarize, it holds true that V ij has a uiformly distributed phase i, π ad it is idepedet of V i for all j ad V lj for all l i for give v i s sice owig v i oly is sufficiet to determie v if for the give rule i 8 Isotropical distributio implies that V ii = v i v if s for i = 1,, are idetically distributed ad idepedet radom variables Similarly, a correspodig distributio holds for V j = v v jf ad its cdf has a form idetical to that give i 11 IV A CAPACITY UPPER BOUND The capacity expressio i 5 is impractical to be used i practical system desig sice it eeds the distributio of 9 V ij s ad the expectatio over V ij s distributios We will obtai a very tight aalytical upper boud for capacity of the -precoder scheme ad this aalytical boud eeds oly the expectatios E 11 = E V 11 ad E 1 = E V 1 to be evaluated The expected values will be deoted with E ij = E V ij ad E ii s are the same for all i s ad ca be evaluated for a give quatizatio codeboo Whe the value of E 11 is give, the value of E 1 ca be calculated easily as follows Due to the isotropical distributio of v i s, E ij is same as E 1 for all i j The value of E 1 ca be foud i terms of E 11 by usig 11 as = E[ V 1 ] = E [ E[ V 1 V 11 ] ] = xf V1 x V 11 = af V11 adxda 1 a x t 1 x t 3 dx f V11 1 a 1 a ada = 1 a t 1 f V 11 ada = 1 E 11 t 1 1 This result i 1 is ituitive sice, after projectig v 1f o v 1, t 1 orthoormal vectors {v,v 3,,v t } are left Due to isotropical distributio, the power i the part of v 1f orthogoal to v 1 is distributed equally betwee t 1 orthoormal vectors i average The capacity for the precoder scheme ca be upper bouded by usig Jese s iequality such that E {log I + P } { HH log E I + P } HH 13 The capacity boud of the -precoder case give i 13 ca be writte i terms of E 11 ad E 1 by usig the result of the lemma give i Appedix as C pre log 1 + j 1 S j S, j j 1 =1 S P P E [ d a1d a d a ] E a1j 1 E a j j S, j j 1,,j 1 14 where E aij i = E [ V aij i ] { E11 = E [ V = 11 ] if a i = j i E 1 = E [ V 1 ], if a i j i 15 S = {1,,, }, S = {a 1,,a }; ad P is the set cotaiig all possible, combiatios of S The term j 1 S j S, j j 1,,j 1 E a 1j 1 E a j does ot deped o which elemet combiatio a 1, a,, a chose from set P is used i 14 sice j i s for i = 1,,, are chose from set S = {1,,, } Usig this fact, oe ca further simplify the capacity boud give i 14 by selectig

4 4 a 1, a,, a as 1,,, such that The secod method is the oe that maximizes the capacity { } by icreasig the projectio power, amely V P 11 By defiig C pre log 1 + E d a1 d a d a a radom variable z as z = 1 v1v 1f, we ca say that for =1 S P a good beamformer, the expected value of z has to be as close as possible to zero A upper boud for the cdf of z, F z z, E 1j1 E j 16 is give i [13] such that F z z F z z for z 1 ad j 1 S j S, j j 1 j S, j j 1,,j 1 E { S P d a1 d a d a } 1 s for = 1,, are required to evaluate the capacity boud i 16 ad oe does ot N f z t 1 1 t 1, z < F eed to calculate E [ z z = N f 1 19 d a1 d a d a ] 1 t 1 for all possible, 1, z N f combiatios a 1,, a from set S separately Istead, the A good beamformer shall try to come as close as possible to expectatio of the sum of all possible joit momets are this distributio which we refer to as the boudig distributio ecessary Hece, oly the expectatio of the sum of all possible joit momets E { S P d a1 d a d a } hereafter [13] For a hypothetical quatizatio method that has to attais this boudig distributio, we ca evaluate the E 11 be foud This ca be foud by usig Mote Carlo simulatios ad E 1 values i order to costruct a upper boud to the or aalytically by usig curretly available results for the joit capacity of limited rate feedbac MIMO scheme that oe of ordered momets i a closed form preseted i [] ad this the quatizatio methods ca exceed E 11 ca be foud as maes the proposed boud completely aalytical give below As will be see i Sectio VI, the capacity boud i 16 is very tight ad ca be used for practical purposes With the help of this aalytical boud, oe ca desig a adaptive E 11 = 1 E[z] = 1 N t 1 f 1 t t 1 t N f MIMO system that ca chage the umber of precoders ad ad E the feedbac rate accordig to the curret average SNR value 1 by 1 The value of E 11 obtaied by this boudig distributio i is the maximum value of E The boud give i 16 is valid for may vector quatizatio 11 that ca methods as log as V ij = vi v be achieved amog all quatizatio codeboos usig the jf has a distributio such that quatizatio rule i 8 for give N the phase of V ij is uiformly distributed i, π which f ad t values is idepedet of V i for all j ad V lj for all l i The quatizatio rule give i 8 is sufficiet but ot ecessary VI NUMERICAL RESULTS to meet this coditio Thus, the results of this sectio ca be applied to a wide variety of quatizatio schemes Moreover, the boud is also valid for o quatizatio cases ad we ca easily costruct a upper boud for the exact capacity of a MIMO system that uses of its strogest subchaels by simply settig E 11 = 1 ad E 1 = i 16 V BOUNDING DISTRIBUTION AND RVQ I this sectio, we will preset two exemplary quatizatio methods that will be used to produce some umerical results i sectio VI The first method is radom vector quatizatio RVQ i which a codeboo is geerated radomly ad the closest vector i the codeboo is coveyed to the trasmitter accordig to the rule give i 8 [19] The use of RVQ as a vector quatizatio scheme allows simpler aalysis ad ca be helpful i desigig practical limited rate feedbac MIMO systems The probability distributio fuctio of V 11 has bee readily obtaied i [4], [5], [19] as t F v 1 v 1f x = Fo x N f 17 The phase of V ij is idepedet of its magitude ad uiformly distributed i [, π] The expected value of a radom variable distributed with 17 is evaluated i [19] as E[ V 11 ] = E 11 = 1 N f B N f t,, 18 t 1 where Bx, y = ΓxΓy Γx+y is the Beta fuctio ad the Gamma fuctio is give by Γx = t x 1 e t dt while E 1 = E V 1 ca be calculated from 1 I this sectio, the capacity upper boud obtaied for LRF MIMO give i 16 is evaluated for RVQ ad the boudig distributio Radom variables are geerated by the iversio method [1] i simulatios ad placed i 5 to obtai ergodic capacity The radom variables V ii s are geerated first while the others are draw based o the distributio give i 11 ad each data poit is obtaied by geeratig 1, realizatios I Fig 1, 6 3 MIMO system capacity uder limited rate feedbac with N f = 8 is evaluated by usig RVQ ad the boudig distributio for ad 3-precoder schemes where the capacity boud give i 16 is also foud by usig E 11 ad E 1 values for these schemes It is see i the figure that the capacity upper boud is very tight It is 3-5 db away from the 6 3 LRF MIMO capacities It is further observed that the RVQ scheme is almost optimal sice RVQ ad boudig distributio capacities are quite close to each other There is a db differece betwee these two capacities ad hece we ca say that RVQ ca be used as a practical quatizatio techique that attai rates quite close to the capacity with tolerable N f values This also justifies its use i the literature for aalysis purposes Moreover, whe compared to the exact MIMO chael capacity obtaied with waterfillig WF that uses a short-term power costrait Sec 13 i [9], there is a 13 db loss i limited feedbac icremetal precodig scheme that uses RVQ with N f = 8 which is well predicted by the proposed boud at high SNR I Fig, RVQ is used as a quatizatio techique ad the capacities of precoder 8 4 MIMO schemes are evaluated with their correspodig bouds for N f = 1 ad =, 3,

5 5 5 Upper capacity boud usig boudig distributio with N f Upper capacity boud usig RVQ with N f Capacity simulatio usig boudig distributio with N f Upper capacity boud usig boudig distributio with N f =8 for precoders x4 MIMO capacity with WF, o quatizatio 8x4 equipower capacity Upper capacity boud usig RVQ with N f =1 for 4 precoders =1 for 4 precoders Upper capacity boud usig RVQ with N f =1 for 3 precoders Rate bps/hz 15 1 Upper capacity boud usig RVQ with N =8 for precoders f Capacity simulatio usig boudig distributio with N f =8 for precoders =8 for precoders 6x3 MIMO capacity with WF, o quatizatio Rate bps/hz 5 15 =1 for 3 precoders Upper capacity boud usig RVQ with N f =1 for precoders =1 for precoders Average SNR db Fig MIMO capacities ad upper bouds for ad 3-precoder LRF schemes RVQ ad boudig distributio are used with N f = Average SNR db Fig 8 4 MIMO capacities ad upper bouds for, 3 ad 4-precoder LRF schemes RVQ is used with N f = 1 ad 4 The equipower scheme, which correspods to capacity with o CSI at trasmitter ad allocates equal power amog the trasmit ateas, is also depicted for compariso It ca be observed that the capacity bouds evaluated with the E 11 value of RVQ for N f = 1 is quite fie ad 15-4 db away from the ergodic capacity with =, 3, ad 4 Furthermore, it is see that the -precoder capacity with N f = 1 is better tha the equipower scheme up to 6dB, 3-precoder scheme is better up to 135 db The 4-precoder scheme that uses all the degrees of freedom i the system always has higher capacity tha the equipower scheme At high SNR, there is a 16 db differece betwee the 4-precoder ad equipower schemes Whe compared to the exact capacity with WF, there is a 14 db loss i limited feedbac icremetal precodig scheme at high SNR ad this ca be compesated by icreasig N f N f = 1 may seem to be large for practical systems but this large value is due to the large trasmit atea umber t = 8 I cotrast, a reasoable N f value is sufficiet to reach the capacity for small size MIMO system I case of limited feedbac, the losses are approximately equal to the SNR loss due to quatizatio 1 log 1 1 E 11 db Ituitively speaig, the istataeous effective SNR of the chael for sigle precoder case is P V 11 ad hece there occurs a 1 log 1 E 11 db SNR loss i the LRF sceario This holds approximately true for the geeral precoder scheme As N f icreases, E 11 asymptotically becomes 1 as observed i 18 ad so that the capacity of LRF MIMO approaches to the capacity with o quatizatio The capacity upper boud i 16 is maximized at E 11 = 1 ad E 1 = so a good quatizatio scheme should have a E 11 value which is as close to 1 as possible for a give umber of feedbac bits Amog the give vector quatizatio techiques ad N f, the best quatizatio codeboo is the oe that gives the highest E 11 value ad thus, it has the greatest capacity boud value i 16 As a result, we ca use the capacity boud give i 16 to evaluate the performace of differet quatizatio schemes For a give quatizatio scheme, it ca be used to determie the umber of precoders to be used at each average SNR value I other words, for a give MIMO system ad the quatizatio techique with N f value, oe calculates the E 11 value of the quatizatio ad the expectatio of the sum of possible joit momets E { S P d a1 d a d a } i 16 oly oce After that, these two values ca be used to costruct the boud i 16 easily With the help of this boud, it is possible to determie the SNR regios i which the umber of precoders to be used to maximize capacity are specified durig the operatio of the studied icremetal precodig scheme For our exemplary quatizatio i Fig, it is see that usig precoders up to db, usig 3 precoders betwee ad 5 db, ad usig 4 precoders above 5 db maximizes the capacity ad this strategy always results i sigificatly higher capacity tha the equipower scheme especially for t > r The mai coclusio is that the tightess of the proposed boud is established The upper boud ca be applied to may LRF MIMO schemes ad also to other problems where the exact evaluatio of the determiat expected value is eeded VII CONCLUSION We developed a tight upper boud to poit-to-poit LRF MIMO capacity that is valid for a large body of vector quatizatio schemes The umber of precoders used i a practical system ca be determied for each average SNR value based o the upper boud developed i this paper We furthermore evaluated the upper boud usig a boudig distributio from Grassmaia beamformig which resulted i the observatio that the simple RVQ techique performs quite close to capacity upper boud Perfect chael estimatio at the receiver is assumed i this study Future studies will iclude the cosideratio of imperfect chael estimatio ad its delayed trasmissio to the trasmitter side withi a limited rate feedbac sceario i a mobile system Also, practical quatizatio methods for MIMO systems will be ivestigated withi the framewor developed i this paper

6 6 APPENDIX: PROOF OF 14 Defie a matrix B = P HH with elemets B i,j = P d id j =1 V ivj ad aother matrix A = I + B For the capacity boud of the precoder MIMO scheme, the determiat of the matrix A is ecessary i evaluatig 14 ad it ca be foud by Leibiz formula []: deta = sgσ, 1 Σ S, =1,,! i=1 A i,σ i where σ i is the i th elemet of Σ which is the th elemet of the permutatio group S, ad S icludes all possible permutatios of the set S = {1,,, } There are! differet permutatios of S ad hece S is composed of! permutatios The fuctio sg of permutatios i the permutatio group S returs +1 or 1 for eve ad odd permutatios, respectively [3] Recallig that A ij equals B ij for i j ad 1 + B ij otherwise, oe ca write the determiat expressio i a compact form i terms of B ij s directly from 1 after a careful ispectio as deta = 1 + Σ l S, l=1,,! i=1 =1 S =a 1,,a P B ai,σ isg Σ l l, where P is the set cotaiig all possible, combiatios of {1,,, } S icludes all possible permutatios of S = {a 1, a,,a } ad there are! differet permutatios i S I the above expressio, the set of elemet combiatio from S = {1,,, } is determied first Cosiderig the Σ a1,,a P term i the summatio i, σ l i is the i th elemet of Σ l which is the l th elemet of the permutatio group S Recallig that B ij = P d id j =1 V iv j, oe ca put B ij ito deta expressio give i ad obtai the followig: P d a i d σ l i deta = 1 + =1 S P Σ l S i=1 V aimv σ l im m=1 sg Σ l 3 The above expressio ca be simplified as follows Defiig S = {a 1,, a } ad a partitio S1,S,, Sp S which are disjoit sets that satisfy S1 S Sp = S = {a 1,,a }, s i j is the jth elemet i Si so that s i j, j = 1,, Si, are the elemets belogig to S i where S i is the cardiality of Si for i = 1,,,p 1 p Although there are may terms i 3, the oly terms that have ozero mea are the oes all composed of squared forms V ij This is due to the reaso that V ij has a uiformly distributed phase i, π which is idepedet of V i for all j ad V lj for all l i This uiform phase distributio will result i a zero expected value for ay term which has a o-squared form of V ij Usig this fact ad after a few straightforward steps, the followig simplified expressio ca be obtaied from 3 after taig expectatio P E{detA} = E{1 + d a1d a d a =1 S P, t=1,,p S1,S,,S p [S ] p=1,, Σ l t S t S p i V s i jz i sg Σ l }, i=1 z i=1 j=1 4 where z 1 z z p, Σ l = Σ l 1,, Σ l p, ad St is the permutatio group that icludes all possible permutatios of the elemets i St I the above equatio, [S ] p=1,, is the set that icludes all possible p elemet partitios S 1, S,, S p of the set S for p = 1,, Therefore, the summatio S1,S,,S p [S ] p=1,, Σ l t S t, t=1,,p i 4 is over all differet permutatios of all possible S1, S,,S p partitios, ie, the summatio shows that p elemet partitio is chose from [S ] p=1,, first ad the, the secod summatio Σ l t S t, t=1,,p is tae over all differet permutatios of the chose partitio of S Actually, the summatio Σ l t S t, t=1,,p is the shorthad otatio of Σ l 1 S 1 Σ l S Σ l p S p ad sg Σ l taes values +1 or 1 depedig o whether Σ l = Σ l 1,, Σ l p is a eve or odd permutatio Lemma 1: The expressio give i 4 ca be simplified as E{detA} = E{1 + j 1 S j S, j j 1 =1 S P j S, j j 1,,j 1 P d a1d a d a V a1j 1 V a j } 5 Proof: The lemma suggests that the oly remaiig terms i 4 are the terms resultig from the partitio of S1,,S p with p = The oly possible partitio is the S1 = {a 1 }, S = {a },,S = {a }, ad Si = 1 for all i s Eq 4 reduces to 5 for p = ad z 1 z z Ay term icluded withi the summatio S1,S,,S p [S ] p=1,, 4 but ot i 5 ca also be writte as p S i i=1 j=1 V s i jz i Σ l t S t, t=1,,p i 6 for ay give z 1, z,, z p betwee 1 ad where z 1 z z p the terms with the same z i s are already i aother partitio of S s i j is the jth elemet of Si as defied before For the above term, at least oe of the Si s has cardiality Si greater or equal to for some i sice p <,

7 7 ie, max 1 i p Si Note that the terms i 5 have Si = 1 for i = 1,,,p ad p = The terms i the form give i 6 origiate from the term give below i 4 S p i E V s i jz i sg Σ l 7 i=1 z i=1 j=1 Now focus o the summatio i 4 over permutatios Σ l t S t for t = 1,,p Fix z 1,,z p z 1 z z p madated by 6 ad all the Σ l 1, Σ l,,σ l p permutatios except for oe of the Σ l c with S c The, 6 ca be writte as S p i S c V s i jz i V s c jz c sg Σ l 8 i c j=1 j=1 alog with the permutatio sig By taig the summatio over differet Σ l c permutatios, oe ca get sg Σ l t, t = 1,,p ad t c Σ l c S c p S i i c j=1 V s i jz i S c V s c jz c sg Σ l c =, 9 j=1 sice there are equal umber of eve ad odd permutatios of Sc set if S c with opposite sigs [3] I other words, i the above equatio sgσ l c is +1 for S c! times ad 1 for S c! times agai Therefore, the expressio i 9 goes to zero ad this ca be doe for other Σ l 1,, Σ l p permutatios The terms preseted i 6 cacel each other i 4 ad 4 reduces to the equatio give i 5 QED The simplified form of the capacity boud i 5 is importat, sice E [ V a1j 1 V a j ] = E [ V a1j 1 ] E [ V a j ] 3 by the idepedece of V a1j 1,, V a j Sice the distributio of V aij i s are idetical ad same as the distributio of V 11 if a i = j i ad equal to the distributio of V 1 if a i j i, oe eed oly the expected values E [ V 11 ] = E 11 ad E [ V 1 ] = E 1 i order to calculate the expressio give i 5 [6] N Jidal, MIMO broadcast chaels with fiite rate feedbac, Proc IEEE Globecom, 5 [7] I E Telatar, Capacity of multi-atea Gaussia chaels, Europ Tras Telecommu, vol 1, pp , Nov/Dec 1999 [8] Z Zhou, B Vucetic, M Dohler, ad Y Li, MIMO systems with adaptive modulatio, IEEE Tras Vehicular Tech, vol 54, o 5, pp , Sept 5 [9] A Goldsmith, Wireless Commuicatios, Cambridge Uiversity Press, 5 [1] Amir D Dabbagh ad David J Love, Feedbac rate-capacity loss tradeoff for limited feedbac MIMO systems, IEEE Trasactios o Iformatio Theory, vol 5, o 5, pp 19, May 6 [11] Jue Chul Roh ad Bhasar D Rao, Desig ad aalysis of MIMO spatial multiplexig systems with quatized feedbac, IEEE Trasactios o Sigal Processig, vol 54, o 8, pp , August 6 [1] Wiroosa Satipach ad Michael L Hoig, Asymptotic performace of MIMO wireless chaels with limited feedbac, Military Commuicatios Coferece, 3 MILCOM 3 IEEE, vol 1, pp , October 3 [13] B Giaais S Zhou, Z Wag, Quatifyig the power loss whe trasmit beamformig relies o fiite-rate feedbac, IEEE Tras o Wireless Commuicatios, vol 4, o 4, pp , July 5 [14] EG Larsso ad P Stoica, Space-Time Bloc Codig for Wireless Commuicatios, Cambridge Uiversity Press, 3 [15] David J Love ad Robert W Heath Jr, Multi-mode precodig usig liear receivers for limited feedbac MIMO systems, 4 IEEE Iteratioal Coferece o Commuicatios, vol 1, pp , Jue 4 [16] GD Forey Jr, O the role of MMSE estimatio i approachig the iformatio-theoretic limits of liear Gaussia chaels: Shao meets Wieer, Proc 3 Allerto Cof, pp , Oct 3 [17] GD Forey Jr, Shao meets Wieer II: O MMSE estimatio i successive decodig schemes, Proc 4 Allerto Cof, 4 [18] P Stoica, Y Jiag, ad J Li, O MIMO chael capacity: A ituitive discussio, IEEE Sigal Processig Mag, pp 83 84, May 5 [19] C Au-Yeug ad DJ Love, O the performace of radom vector quatizatio limited feedbac beamformig i a MISO system, IEEE Tras Wireless Commuicatios, vol 6, o, pp , February 7 [] Si Ji, Matthew R McKay, Xiqi Gao, ad Iai B Colligs, MIMO multichael beamformig: Ser ad outage usig ew eigevalue distributios of complex ocetral wishart matrices, IEEE Trasactios o Commuicatio, vol 56, o 3, pp , March 8 [1] L Devroye, No-Uiform Radom Variate Geeratio, New Yor: Spriger-Verlag, 1986 [] H Campbell, Liear Algebra With Applicatios, Appleto Cetury Crofts, 1971 [3] Joh D Dixo ad Bria Mortimer, Permutatio Groups, Graduate Texts i Mathematics, Spriger-Verlag, 1996 REFERENCES [1] T Yoo, E Yoo, ad A Goldmisth, MIMO capacity with chael ucertaity: Does feedbac help?, Proc IEEE Globecom, pp 96 1, 4 [] E Biglieri, R Calderba, A Costatiides, A Goldsmith, A Paulraj, ad H Vicet Poor, MIMO Wireless Commuicatios, Cambridge Uiversity Press, 7 [3] JC Roh ad BD Rao, Chael feedbac quatizatio methods for MISO ad MIMO systems, Proc IEEE PMIRC, vol, pp 85 89, Sept 4 [4] DJ Love, RW Heath Jr, ad T Strohmer, Grassmaia beamformig for multiple-iput multiple-output wireless systems, IEEE Tras Iform Theory, vol 49, o 1, pp , Oct 3 [5] KK Muavilli, A Sabharwal, E Erip, ad B Aazhag, O beamformig with fiite rate feedbac i multiple-atea systems, IEEE Tras Iform Theory, vol 49, o 1, pp , Oct 3