DFT-based Beamforming Weight-Vector Codebook Design for Spatially Correlated Channels in the Unitary Precoding Aided Multiuser Downlink

Transcription

1 DFT-based Beamformng Weght-Vector Codeboo Desgn for Spatally Correlated Channels n the Untary Precodng Aded Multuser Downln Du Yang Le-Lang Yang and Lajos Hanzo School of ECS Unversty of Southampton SO7 BJ UK Tel: ; Fax: E-mal: dy5rllylh@ecssotonacu; Abstract The DFT-based beamformng weght-vector codeboo s consdered as an effectve desgn for spatally correlated channels In ths paper we demonstrate that when the antenna elements are unformly spaced as well as lnearly arranged and the channels are spatally correlated the codewords n a DFTbased beaformng weght-vector codeboo approxmately match the dstrbuton of the optmal beamformng weght-vectors As a result the DFT-based codeboo s ndeed effectve Furthermore we also demonstrate that f the antenna elements are unformly spaced and crculary arranged the statstcal dstrbuton of the optmal beamformng weght-vectors becomes dfferent We wll demonstrate that n ths scenaro the DFT-based codeboo wll no longer outperform the Grassmannan codeboo whch has not been shown n prevous studes Fnally an algorthm s proposed for constructng the DFT-based precodng matrx whch outperforms the conventonal algorthm by ensurng the orthogonalty of the precodng matrx Index Terms DFT-based codeboo spatal correlated channel untary precodng multuser downln I INTRODUCTION In ths paper we employ the Untary Precodng (UP) preprocessng scheme proposed for the Long-Term-Evoluton (LTE) standard ] n a multuser DL system havng multple antenna aded Base Statons (BS) and sngle antenna aded Moble Termnals (MTs) The Channel State Informaton (CSI) requred by the UP scheme ncludes the nowledge of the untary beamformng weght-vector and the channel qualty For a Frequency-Dvson-Duplex (FDD) system where the Dowln (DL) and the Upln (UL) channels occupy dfferent frequency bands and have dfferent CSI the most common way of acqurng CSI at the transmtter sde s to estmate the CSI at the recever to quantse the estmated CSI usng a predefned codeboo and then to feed bac the ndex of the chosen CSI codeword to the transmtter through a bandlmted and potentally error-nfected feedbac channel Hence the desgn of the CSI quantzaton codeboo s crucal for the success of any preprocessng technque The quantzer desgn for the beamformng weght-vector s challengng because t s a mult-dmensonal varable Substantal research studes have been dedcated to ths area and several solutons have been proposed ] 5] The Grassmannan lne-pacng codeboo The fnancal support of the EPSRC UK under the auspces of the UK- Inda Centre of Excellence n Wreless Communcatons and that of the EU s Optmx project s gratefully acnowledged was proposed n ] for a beamformng aded system communcatng over spatally ndependent channels whle the Grassmannan subspace pacng codeboo was proposed n ] for a spatal multplexng asssted system For spatally correlated channels a modfed Grassmannan-based codeboo desgn was proposed n 5] However the LTE standard favours the Dscrete Fourer Transform (DFT) based codeboo proposed n ] 7] for ts smplcty whose beamformng weght-vector codewords are actually permuted columns of a DFT matrx However to the best of our nowledge there s no study n the open lterature on nvestgatng the reason why the DFTbased codeboo s effectve for spatally correlated channels The most detaled explanaton we found n the open lterature s n ] where the author argued that the ampltude dfference between the dfferent antenna elements s reduced by spatally correlated fadng hence the constant-modulus DFT codeboo mposes a lower CSI quantzaton error than the nonconstant-modulus Grassmannan codeboo In ths paper we demonstrate that the DFT-based codeboo hosts the Unformly spaced and Lnearly arranged Antenna Elements (UL-AEs ) Normalzed Impulse Responses (NIRs) to a Sngle Resolvable Path (SRP) By explotng the propertes of ths UL-AE NIR-SRP a benefcal scheme s proposed for constructng an orthogonal DFT-based precodng matrx Moreover we demonstrate that for UL-AEs - provded that the channel s spatally correlated - the beamformng weght-vectors n a DFT-based codeboo approxmately match the dstrbuton of the optmal beamformng weght-vectors As a result the DFT-based codeboo s ndeed effectve However we wll demonstrate that for Unformly spaced and Crculary arranged Antenna Elements (UC-AEs) the statstcal dstrbuton of the optmal beamformng weght-vectors becomes dfferent and hence the DFT-based codeboo wll no longer outperform the Grassmannan codeboo even for hgh spatal correlatons Ths problem has not been addressed n prevous studes where smplfed statstcal channel models were used Ths paper s organsed as follows In Secton II the UP preprocessng scheme the DFT-based beamformng weghtvector codeboo and the UL-AE NIR-SRP are ntroduced In Secton III the relatonshp between the beamformng weghtvector n a DFT-based codeboo and the UL-AE NIR-SRP s presented followed by the desgn of an algorthm for constructng an orthogonal precodng matrx In Secton IV the //$ IEEE

2 reasons behnd the effcency of the DFT-based beamformng weght-vector codeboo desgned for UL-AEs are hghlghted whch are contrasted to ts neffcency for UC-AEs Our smulaton results are provded n Secton V followed by our conclusons II PRELIMINARIES A The Untary Precodng Aded Mult-User System As llustrated n Fg a BS equpped wth downln (DL) antennas communcates wth K sngle-antenna aded MTs whch are unformly dspersed over the cell The wreless channels are assumed to be quas-statonary flat Raylegh fadng e they have a constant envelope durng each transmtted bloc The smplfyng assumpton of perfect power control s used hence no path-loss and shadowng are consdered It s also assumed that the total transmt power s equal to unty and that the nose varance N s the same for all the K users The beamformng weght-vector feedbac channel s dealzed e t s assumed to be both error-free and delay-free The Shannon-Hartley law the hghest supportable rate of the th user equals to Γ ( j )=log ( + γ ) In practce Γ wll be quantzed usng another codeboo and fed bac to the BS together wth the above mentoned PMI and PVI In ths study we assumed that usng an arbtrary modulaton and codng scheme s feasble n order to smplfy our study Havng receved the feedbac nformaton the precodng matrx W s and the assocated ntended users are selected wth the ad of the followng equaton n order to maxmse the achevable sum-rate of the B users W s = () argmax N B j= max K Γ (γ ( j ))δ( j j ) wherewehaveδ ( j j )=for ( j) =( j ) and zero otherwse The number of supported users s equal to B when the number of actve users K s suffcently hgh but t may be less than B when K s small Fnally the nformaton sgnals of the B supported users are preprocessed by the precodng matrx W s and they are transmtted smultaneously usng the transmt antennas CB s s s s B s w () w (j) w (B) W Schedulng x x x x h h h h ( j) Γ( j) MT e jarg(ht w(j) ) MT data det & dem MTB h codeword selecton CB: {W =w () w(b) ]; ; W N =w () N w(b) N ]}w(j) C Fg A mult-user system equpped wth transmt antennas at the BS and a sngle antenna at the MTs usng the untary precodng aded transmsson scheme beamformng weght-vector codeboo CB consstng of N precodng matrces W N] s employed at both the BS and the MTs Each precodng matrx W conssts of B untary precodng vectors w (j) w (j) C w (j)h w (j) =Inths study we assumed that B and the precodng vectors of a gven precodng matrx are orthogonal to each other e we have w (j)h w () =for B The th MT frst estmates ts own Channel Impulse Response (CIR) h wth the ad of the receved plot symbols and then selects the preferred Precodng Matrx Index (PMI) as well as the Precodng Vector Index (PVI) j from a prestored codeboo CB n order to maxmse ts own receved SINR γ expressed as ] ( j )=argmax N j B γ ( j) () h T = argmax w(j) N j B BN + n j ht w(n) where ( ) represents the conjugate of a complex-valued scalar/vector/matrx Ths selecton crteron suggests that the MTs assume that all other beams n the precodng matrx W -except for w (j) -wll be scheduled for other users at an dentcal transmt power of B Consequently accordng to the ŝ CB B DFT-based Beamformng Weght-Vector Codeboo The -by- DFT matrx W equals to ] W = (3) e jπ e jπ jπ e e jπ e jπ jπ e e jπ ( ) e jπ ( ) e jπ ( ) e jπ ( ) e jπ ( ) jπ ( ) e The frst rows of W wth a normalzaton factor of consttute the DFT-based beamformng weght-vector codeboo whose codeword w s expressed as w(l) = e jπ l e jπ l e jπ ( ) l ] T () l = Moreover for constructng precodng matrx t s defned n ] that the jth beamformng weght-vector n the th precodng matrx w (j) s calculated usng Equaton () by settng l = j N+ However the orthogonalty of the constructed precodng matrx cannot be guaranteed usng ths algorthm C UL-AE NIR-SRP As llustrated n Fg the transmt AEs are arranged n a lne wth a spacng of λ c where represents the dstance between the adjacent antennas normalsed by the wavelength λ c of the carrer frequency For a sngle resolvable path havng an Angle of Departure (AOD) of θ the CIR h s s formulated as 9] h s (θ) =

3 BS Θ t Θ t cluster cluster MT Lnear Antennas λ c Crcular Antennas λ c Fg Physcal channel model for unformly spaced lnearly allocated antenna elements and unformly spaced crcularly allocated antenna elements ae jφ e jπ Δtcosθ e jπ ( ) Δtcosθ ] T where a and φ represent the sgnal ampltude and the phase rotaton respectvely Hence the UL-AEs normalzed mpulse response to a sngle resolvable path s expressed as e UL AEs (θ) = h s (θ) h s (θ) = e jπ cosθ e jπ ( ) Δtcosθ ] T III CONSTRUCTION OF AN ORTHOGONAL DFT-BASED PRECODING MATRIX RELYING ON THE RELATIONSHIP BETWEEN w (j) AND e UL AEs By comparng Equatons () and (5) t may be shown that f l =Δ c cosθ wehave: w(l) = ] e jπl e jπ(n T t )l () = = e UL AEs(θ) e jπ( cosθ) (5) e jπ( )( cosθ) ] T Hence the beamformng weght-vectors n a DFT-based codeboo host the UL-AE NIR-SRP One of the mportant propertes of e UL AEs stated n 9] s that the absolute value of ts nner product g e = e UL AEs (θ ) H e UL AEs (θ ) equals to sn(π(δtcosθ Δtcosθ)) sn(π(cosθ cosθ )) Furthermore g e =when we have cosθ cosθ = = Physcally ths mples that a beam steered to θ wll completely suppress any sgnal arrvng from θ as long as cosθ cosθ = By substtutng cosθ for l the suffcent condton of havng two orthogonal beamformng weght-vectors w s obtaned as follows: If two beamformng weght-vectors n a DFT-based codeboo w (l ) and w (l ) l l ] are orthogonal to each other then we have l Δ = l l = = Consequently an algorthm of constructng an orthogonal DFT-based precodng matrx s proposed and llustrated n Table I usng Matlab code The dea s to frst calculate an approprate perodc value T and then to assgn the adjacent beamformng codewords of a gven precodng matrx accordng to the relatonshp of x (j) = T Ths algorthm s applcable for B For x (j+) = nb n N the same codeboo s generated by usng ths algorthm and by usng the one proposed n ] descrbed n Secton II-B snce they employ the same value of T TABLE I THE IMPROVED DFT CODEBOOK GENERATION ALGORITHM T = floor(/b)/; %floor(x) = x for =:N end x = ; for j =:B l = x + T (j ); w (j) = w(l); %w(x) s expressed n Equaton () end IV EFFICIENCY OF THE DFT-BASED BEAMFORMING WEIGHT-VECTOR CODEBOOK Based on quantzaton theory the DFT-based beamformng weght-vector codeboo can be consdered as an effcent desgn f t assgns ts quantzaton levels accordng to the statstcal dstrbuton of the optmal beamformng weght-vectors w opt so as to mnmze the quantzaton error ] Moreover the optmal weght-vector w opt n our study s equvalent to the normalsed CIR w opt = h h whch maxmzes the recever s SINR as well as the resultant sum-rate accordng to Equaton () The statstcal dstrbuton of w opt depends on the channel s propertes whch are dscussed as follows A Spatally Correlated Channels Usng UL-AEs For full spatally correlated channels usng UL-AEs w opt s equal to the UL-AE NIR-SRP Usng the physcal channel model characterzed n Fg the CIR h s formulated as h = J I j h s = J I ejφ() j a() j j e UL AEs (θ () j ) where I represents the number of resolvable paths n a cluster whle J denotes the number of clusters The θ () j of each path AOD has a Gaussan dstrbuton wth a mean value of Θ j and an angular spreadng varance of σ j The spatal correlaton ρ of the CIR h wll ncrease when the number of paths clusters and the angular spreadng varance decrease The channel exhbts full spatal correlaton assocated wth ρ =f J = I = and σ j = In ths case we have w opt = h h = h s h s = e UL AEs Based on Equaton (5) the vector e UL AEs can be represented as a functon of a real-valued scalar varable x expressed as e UL AEs (x) = e jπ x e ] jπ ( ) x T x = Δt cosθ Moreover e UL AEs (x) s a perodc functon wth a perod of T =for e jπ x = e jπ (x+) By settng β = mod(x T )x = cosθ x ]T = β ) a unque and unambguous one-to-one relatonshp s constructed between the varable β and the vector e UL AEs As a result we have the Probablty Densty Functon (PDF) p(e UL AEs )=p(f(β)]) = p(β) where the orgnal problem of fndng the dstrbuton of the optmal beamformng vectors w opt becomes equvalent to that of dentfyng the dstrbuton of the varable β Snce the users are unformly dspersed n a cell we have θ U( π)p(θ) = π Upon defnng Ω = cosθ the PDF of Ω becomes equal to p(ω) = π The PDF of the Ω

4 varable β recorded for =5; =and = 3 usng Monte-Claro smulaton s shown n Fg 3 In fact the curve correspondng to =respresents the shape of all PDFs when we have = N Fg 3 demonstrates that except for a few hgh-densty dstrbuton peas β s nearunformly dstrbuted n the range of ) Equaton () demonstrated that the beamformng weghtvector codewords n a DFT-based codeboo are also unformly dstrbuted over ) wth samples at dstances of Hence the DFT-based beamformng weght-vector codeboo has a PDF remnscent of the dstrbuton of the optmal beamformng vector of a fully spatally correlated channel Naturally when the spatal correlaton coeffcent ρ decreases the dstrbuton of β becomes less smlar to the dstrbuton of the optmal beamformng vector Consequently the DFT-based codeboo becomes less effectve B Spatally Correlated Channels Usng UC-AEs When antennas are equally spaced around a crcle wth a dstance of λ c the UC- AEs NIR-SRP s represented as e UC AEs (θ) = where Δ R = π jπδrcos(θ e N ) t sn( π ) ] T e jπδrcos(θ π( ) N ) t s the radus of the crcle Smlarly e UC AEs s the optmal beamformng weght-vector for fully spatally correlated channels usng UC-AEs However ts dstrbuton cannot be modelled by that of the same scalar varable β whch s unformly dstrbuted betwen and As a result the DFT-based beamformng weght-vector codeboo does not represent an effcent desgn for the UC-AEs The smulaton results of Secton V wll demonstrate that the performance of the DFT-based codeboo n fact becomes even worse than that of the Grassmannan codeboo Unfortunately the UC-AEs unt spatal sgnature e UC AEs cannot be expressed as a functon of a scalar varable such as θ or any combnaton of and θ As a result t s dffcult to capture and characterze ts dstrbuton However we have observed n our nformal nvestgatons that for =5 the beamwdth Φ s the same for any AOD Hence we proposed a quantzaton method whch unformly samples the angular range of θ π) wth a dsplacement equal to Φ = Consequently any angular samples wth an angular separaton of Φ are orthogonal wth respect to each other and can be used to construct orthogonal codeboos for B =In practce however more sophstcated desgns are needed for arbtrary values of B and V SIMULATION RESULTS Fg demonstrates the achevable throughput gan of the mproved DFT-based codeboo generated by the algorthm summarsed n Table I as well as the achevable throughput of the conventonal DFT-based beamformng weght-vector codeboo generated accordng to Equaton () at SNR=dB when the AEs are unformly spaced and are arranged n a lne The throughput was calculated as B = log( + γ ) where γ represents the SINR of the th selected user defned n Equaton () usng the selecton crteron stated n Equaton () Four scenaros were consdered namely = 3ρ =B =N =; =3ρ =B =N =; = ρ = B = N = and = ρ = B =N = Four feedbac bts were used for all four scenaros snce the codeboo-sze was = The followng observatons can be made based on the smulaton results of Fg Frstly the achevable sum-rate ncreases wth the number of actve users as a beneft of the ncreased multuser dversty Secondly f = nb n N for example we have = B = then the conventonal scheme and the proposed scheme generate the same codeboo both representng orthogonal precodng matrces hence the same performances are acheved Thrdly f nb n Nfor example we have =3B =and =B = Then usng the codeboo generated by the proposed algorthm s capable of achevng a substantally hgher throughput snce the nter-user nterference s reduced when usng an orthogonal precodng matrx Fourthly the achevable throughput s sgnfcantly reduced when the spatal correlaton coeffcent s ρ = ( =J =I =σ j =5) because the DFTbased conventonal codeboo s less effectve for channels exhbtng a low spatal correlaton Fg 5 compared the achevable sum-rate of the K users for both the DFT-based codeboo and for the Grassmannan subspace pacng codeboo ] at SNR =db and = usng UL-AEs When the channel s fully spatally correlated as ndcated by ρ = the achevable sum-rate of the K users employng the DFT-based codeboo and B = N = s almost twce the achevable sum-rate of the Grassmannan subspace pacng codeboo havng the same codeboo sze By contrast when the spatal correlaton was decreased to ρ = ( = 5J = I = σ j = 5) the performance of the DFT-based codeboo degraded whle that of the Grassmannan subspace pacng codeboo mproved When the channel became spatally ndependent e we had ρ = the achevable rate of usng the Grassmannan subspace pacng codeboo and the one usng the DFT-based codeboo assocated wth =codewords was smlar However when the codeboo sze ncreased to B =N =the achevable sum-rate of the K users employng the Grassmannan subspace pacng codeboo was about bts/symbol hgher than that of the one usng the DFT-based codeboo havng the same codeboo sze Hence the followng conclusons may be obtaned Frstly the DFT-based codeboo havng a small sze consttutes an effectve desgn for both spatally correlated and ndependent channels Secondly the Grassmannan-based codeboo assocated wth a suffcently large codeboo sze s an effectve desgn alternatve for spatally ndependent channels Thrdly more quantzaton bts are requred for spatally ndependent channels because the beamformng vectors are more dssmlar Fg 5 also demonstrates that the achevable sum-rate of spatally ndependent channels usng the Grassmannan-based codeboo havng seven feedbac bts s sgnfcantly lower than that of the fully spatally correlated channel usng the DFT-based codeboo havng only four feedbac bts Ths ndcates that a BS can ether reduce the AE spacng to obtan an ncreased throughput or ncrease the AE spacng to acheve a dversty gan

5 Fg llustrates the achevable sum-rate of the K users for the DFT-based codeboo the Grassmannan-based codeboo and the proposed codeboo descrbed n Secton IV-B at SNR = db = = 5B = N = when the AEs are unformly spaced and arranged n a crcle When the channel s fully spatally correlated the achevable sum-rate usng the DFT-based codeboo becomes lower than that of the Grassmannan-based codeboo When usng the proposed codeboo a substantal throughput gan s obtaned Hence we conclude that the DFT-based codeboo s neffectve for UC-AEs When the spatal correlaton was decreased to ρ = 7 the achevable sum-rate of both the DFTbased codeboo and of the Grassmannan codeboo ncreased compared to the ρ =scenaro and ther dfference became subtle Ths s because the beamformng vectors dstrbuton becomes more dspersed across the entre -dmensonal unt sphere As a result both the DFT-based and the Grassmannanbased codeboos become more effectve VI CONCLUSION In ths paper we have analysed the DFT-based beamformng weght-vector codeboo employed n the untary precodng aded mult-user downln By recognsng the fact that the beamformng weght-vector n a DFT-based codeboo s consttuted by the UL-AE NIR-SRP an algorthm was proposed for constructng the correspondng DFT-based orthogonal precodng matrx It was demonstrated that the DFT-based codeboo s effectve for UL-AEs but not for UC- AEs The smulaton results of Fg 5 also demonstrated that for spatally ndependent channels the Grassmannan codeboo wll only outperform the DFT-based codeboo f the codeboo sze s suffcently large REFERENCES ] S Sesa I Touf and M Baer LTE: The UMTS Long Term Evoluton from Theory to Practce John Wley and Sons Ltd 9 ] D J Love R W Heath and T Strohmer Grassmannan beamformng for multple-nput multple-output wreless systems IEEE Transactons on Informaton Theory vol 9 no pp Oct 3 3] D Yang L L Yang and L Hanzo Performance of SDMA systems usng transmtter preprocessng based on nosy feedbac of vectorquantzed channel mpulse responses n Vehcular Technology Conference 7 VTC7-Sprng IEEE 5th Dubln Apr 7 pp 9 3 ] D J Love and R W Heath Lmted feedbac untary precodng for spatal multplexng systems IEEE Transactons on Informaton Theory vol 5 no pp Aug 5 5] V Raghavan R W Heath and A M Sayeed Systematc codeboo desgns for quantzed beamformng n correlated MIMO channels IEEE Journal on Selected Areas n Communcatons vol 5 no 7 pp 9 3 Sept 7 ] Results on zero-forcng MU-MIMO Freescale Semconductor Inc 3GPP TSG RAN WG R-75 Tech Rep 7 Onlne] Avalable: 7] D J Love and R W Heath Equal gan transmsson n multple-nput multple-output wreless systems vol pp Nov ] J Zhu J Lu X She and L Chen Investgaton on precodng technques n E-UTRA and proposed adaptve precodng scheme for MIMO systems n th Asa-Pacfc Conference on Communcatons Oct pp 5 9] D Tse and P Vswanath Fundamentals of wreless communcaton Cambrdge Unversty Press 5 ] A Gersho and R M Gray Vector Quantzaton and Sgnal Compresson Kluwer Academc Publshers 99 Fg 3 p(β) =5 = =/3 β The pdf of β when =5; =and = 3 respectvely Achevable sum rate (bts/sym) SNR=dB unformly spaced lnearly allocated antenna elements 3 5 Number of Actve Users K =3ρ=B=N=old scheme =3ρ=B=N=old scheme =3ρ=B=N=new scheme =3ρ=B=N=new scheme =ρ=b=n=old scheme =ρ=b=n=new scheme =ρ=b=n=old scheme =ρ=b=n=new scheme Fg Comparson of the conventonal DFT-based codeboo generated from Equaton () and the proposed mproved DFT-based codeboo generated by the algorthm summarsed n Table I n terms of the achevable sum-rate versus the number of actve users K when ) =3ρ =B =N =; ) =3ρ =B =N =; 3) =ρ =B =N =and ) =ρ=b =N = Achevable sum rate (bts/sym) SNR=dB = N r = unformly spaced lnearly allocated antenna elements ρ=dft basedb=n= ρ=grass sub B=N= ρ=dft based B=N= ρ= Grass sub B=N= ρ=dft basedb=n= ρ=grass sub B=N= ρ=dft basedb=n= ρ=grass subb=n= 3 5 Number of Actve Users K Fg 5 The achevable sum-rate versus the number of actve users K for spatally correlated channels assocated wth ρ = ρ = and spatally ndependent channels havng ρ = whenb = N = and usng ether the DFT codeboo or the Grassmannan subspace pacng codeboo respectvely SNR=dB = =5 B= N= unformly spaced crcularly allocated antenna elements Achevable sum rate (bts/sym) DFT based ρ= Grasss sub ρ= proposedρ= proposed ρ=7 DFT based ρ=7 Grass sub ρ=7 3 5 Number of Actve Users K Fg Comparson of the achevable sum-rate versus the number of actve users K usng the DFT-based codeboo the Grassmannan subspace pacng codeboo wth B = N = as well as the proposed codeboo wth B =N =9for spatal correlated channels assocated wth ρ =and ρ =7 respectvely when antenna elements are unformly spaced and allocated n a crcle