POWER ALLOCATION FOR CAPACITY MAXIMIZATION IN EIGEN-MIMO WITH OUTPUT SNR CONSTRAINT

Transcription

1 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 POWER AOCATION FOR CAPACITY MAXIMIZATION IN EIGEN-MIMO WIT OUTPUT CONSTRAINT S. Alreza Banan and Dena omayouneh Deartment of Electrcal and Comuter Engneerng, Unversty of Toronto, Toronto, ON, Canada Faculty of Scence and Engneerng, York Unversty, Toronto, ON, Canada ABSTRACT Water-fllng s the ower allocaton that maxmzes the arallel-channel egen-mimo nformatontheoretc caacty. owever, for ractcal desgns, desgnng for error erformance s also mortant. The overall uncoded error rate, n a system wth water-flled egen-channels usng dgtal modulaton, degrades through the errors n the data that uses the weaker egen-channels. Ths s because n water-fllng, a lower transmt ower s allocated (and a lower roorton of the overall caacty) to the weaker egenchannels. One general soluton s to dscard the weaker egen-channels, but ths does stll not allow smle control over the values of, or trade-off between, caacty and the error rate. In ths aer a desgn aroach s resented for the control of, and mrovement n, the overall error erformance, whle mantanng the ergodc caacty as hgh as oble. The aroach s drect; the egen-channel ower allocaton s otmzed for caacty maxmzaton constraned by a desred error erformance on the egenchannels va constrants on ther s. A useful feature s that the caacty and the error erformance can be managed through the egen-channel constrant. Statstcal smulatons wth Raylegh channels quantfy the sgnfcant SER mrovement comared to full egen-mimo (maxmum number of egen-channels) wth water-fllng, when usng a fxed modulaton set. Ths set comrses re-chosen constellatons for a constant number of egen-channels, and ths remans fxed for the all the channel realzatons,.e., wth tme for the tme-varyng channel. The mrovement n error erformance s at the exense of a small lo n caacty. KEYWORDS MIMO; water-fllng; convex otmzaton; egen-channel; ergodc caacty; MIMO caacty. INTRODUCTION Oen-loo multle-nut multle-outut (MIMO) systems have the ractcal advantage of not requrng channel knowledge at the transmtter. But ths ractcal advantage comes wth a erformance enalty because ont otmzaton between the transmtter and recever s not fully deloyed. When channel state nformaton (CSI) s avalable at the transmtter, as aumed n ths aer, MIMO erformance can be mroved based on a desred otmzaton crteron such as caacty, average error rate, etc. Several ontly otmal lnear recoder and decoder (PD) desgns, also called beamformers, based on erfect channel knowledge at the transmtter, are resented n the lterature. These are too numerous to lst here, but reresentatve examles are []-[]. The work n [] ncludes the desgn that maxmzes outut, whch s called domnant egenmode transmon. It DOI : 0.5/tmc.03.40

2 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 transmts only va the strongest egenchannel, so the ower allocaton s trval and the desgn s very ractcal. Water-flled ower allocaton (.e., wth the total ower constraned) for caacty maxmzaton, s resented n [] and [3]. For Raylegh channels, the maorty of the avalable caacty s avalable by deloyng ust one or a few of the avalable egenchannels [4]. The otmal desgn that mnmzes the overall error rate allocates the ower on the egenchannels accordng to an nverse water-fllng olcy [5], [6]. A dfferent aroach based on nverse (or equvalently, mean squared estmaton error) between the nut and outut of each egenchannel s studed n [5], [7]. A general soluton s a weghted sum of nverse s and leads to a number of well-known solutons deendng on the choce of weghts [7]. The otmum desgn that maxmzes subect to a zero-forcng constrant s also consdered n [6]. An otmal PD desgn n [8] s for mnmzng the arwse error robablty (PEP) wth the total transmt ower constrant. Mnmzaton of the geometrc mean square error, defned as the determnant of the error covarance matrx, s treated n [9]. Caacty maxmzaton wth a eak ower constrant s treated n [0]. There are many other secalzed schemes (e.g. []-[4] wth channel estmaton and merfect feedback), but the basc rncles aear n dfferent forms from dfferent dsclnes (nformaton theory, sgnal roceng, communcatons theory and technques, adatve antennas, etc.). owever, there are ractcal shortcomngs wth each tye of otmzed PD desgn. For offerng nsght nto MIMO confguratons that are ractcal but can stll mantan hgh erformance, formulatons are requred that le between the lmtng caacty, and, for examle, the exstng constraned desgn examles of []-[]. As an examle, usng water-fllng n ractce (.e., wth dgtal modulatons), causes the uncoded error erformance to deterorate wth the weakest egenchannel. Although the weaker egenchannels contrbute weakly to the overall caacty, the large dfference between the average of the strongest egenchannel and the average on the lowest egenchannel can cause dsroortonate error rates acro the egen-channels. In ths sense, the overall error rate s domnated by the weakest egen-channel. For examle, n a 3 3 system wth water-fllng, the dfference between the average on the strongest egenchannel and the average on the lowest egenchannel s aroxmately 3.5 db when the average at each receve antenna s 0 db. Ths dfference ncreases to 6.8 db n a 4 4 system. As a result, full egen-mimo systems wth water-fllng (.e. wth maxmzed caacty wthout an error erformance constrant) have a weak uncoded SER erformance whch s may not be desrable n ractce. A referred system has good relablty and hgh throughut, smultaneously. One soluton s to deloy only the strong egen-channels, but t s stll not straghtforward to control the caacty and SER. Also, the hghest oble caacty cannot always be guaranteed for a desred outut SER erformance - t s well-known that these quanttes trade-off wth each other. For the smle examle of a MIMO wth the fxed modulaton set (see below) and a target overall SER of 3 0, the domnant egenmode transmon s not the scheme for the hghest caacty (seen va smulaton below, for 5 db ). In ths aer, the caacty and relablty (the uncoded SER), are determned together. The method s to constran the egenchannel s for relablty (maxmum error rate) and seek a ower allocaton over the egenchannels for the hghest caacty. Frst, an otmzaton roblem s addreed for maxmzng the caacty wth the total transmt ower constrant and a maxmum allowable reducton (lo of relatve to known average at each recever) mosed on the egenchannels - a quantty controlled by the system desgner. The otmal ower allocaton s obtaned va convex otmzatons wth nequalty constrants [6]. The use of convex otmzaton tself s not new t has become a standard tool n MIMO systems desgn over the last decade. Then, based on the at each recever and the maxmum allowable lo, the

3 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 best ower allocaton scheme s selected from the resented otmzaton, water-fllng and domnant egenmode transmon for the hghest caacty. One feature of the aroach s that the oston of the caacty between the bounds can be controlled by arorately settng the maxmum allowable lo. The smulaton results demonstrate an mrovement n the SER erformance over the case wth water-fllng, usng a fxed modulaton set n a MIMO system (a choce of QAM s agned to egenchannel and BPSK to egenchannel ). In general, fxed-set modulaton has dfferent constellatons agned to a fxed number of egenchannels, all the tme. Fxed-common modulaton, n whch the same constellaton s agned to all the egenchannels, s a secal case of fxed-set modulaton. In other words, unlke adatve modulaton, the modulaton over the ordered egenchannels does not change for dfferent channel realzatons. The advantage of the fxed-set modulaton over adatve modulaton s the reduced comlexty n both hardware and requred rotocol suort. The settng-u of adatve modulaton, artcularly n a large multuser network, can lead to a large caacty overhead n the system, e.g. [5]. Fnally, n dscung relablty va the error erformance of a ractcal system, the role of data codng s mortant, but t s not yet oble to otmze egen-mimo wth general codng. The rest of the aer s organzed as follows. Secton II descrbes the egen-mimo system model. The roblem of caacty maxmzaton wth constrant s addreed n secton III, wth the smulaton results resented n secton VI, and secton V s the concluson. The notaton s conventonal: vectors are lower case letters and matrces are n bold uer case, wth, T and * meanng conugate transose, transose, and comlex conugate, resectvely; I s the dentty matrx; and E denotes exectaton.. COSED-OOP MIMO SYSTEM MODE Consder a MIMO quas-statc, flat block-fadng channel wth M transmt and N receve antennas. The channel s modeled by a random (fadng) dstrbuton whch remans statc over a fadng block cycle, but becomes ndeendent acro dfferent blocks. A beamformer W, derved usng channel knowledge at the transmtter, s shown n Fg.. x W s Channel : v y Decoder Perfect Feedback Fgure. An deal, closed-loo MIMO system wth a transmt beamformer for usng channel knowledge whch has deal feedback from the recever. The deal feedback lnk s not art of the caacty calculaton for the forward lnk. At each symbol tme, the M data vector sgnal to be transmtted, x, s multled by the M M weght matrx W, before transmon. Wth erfect tmng, etc., the MIMO system s modeled n the usual way wth notaton y M Wx v M s v () 3

4 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 where y s the N receved sgnal vector, s the channel matrx wth ( ), ~ CN (0, ) s the known average at each receve antenna, and v ~ CN (0, ) s the addtve nose. Wth E xx I M and s Wx, the covarance matrx of the transmtted sgnal s gven by R E WW. From the total average transmt ower constrant, W satsfes W trr M F, so wth Wx s, we have E tr M. 3. EIGEN-MIMO CAPACITY MAXIMIZATION The caacty of a samle realzaton of the channel s frst addreed. The ergodc caacty s the ensemble average of the caacty acheved when the otmzaton s erformed for each realzaton of,.e. C E C. The nformaton theoretc caacty wth a fxed channel s defned as [3] C max I( s;y) max log deti N R () s s ( ) E tr M M where s (s) s the robablty densty functon of the vector s, and I ( s; y) s the mutual nformaton between s and y. The matrx wth rank r mnm, N can be rewrtten as, / UΛ V where V and U are the untary matrces contanng the corresondng nut and outut sngular vectors, resectvely and / Λ s a non-negatve N M dagonal matrx wth th dagonal element as / (the square root of th egenvalue). In addton the dagonal elements satsfy. As a result, VV Also, note that, and maxmzaton over ~ R C V R wth max E tr M log deti N Λ M snce for any non-negatve defnte matrx A, det A / V R VΛ /. (3) ~ RV s non-negatve defnte, and trr trr. Thus, the E tr M, can be over R ~ wth E tr ~ s ~ s M. Moreover, A, / ~ / ~ det I N Λ R Λ R, (4) M M, so wth the equalty for when R ~ s dagonal. Therefore ~, where R r C max log (5) E tr ~~ s s M M, and the otmum desgn corresonds to the on the egenchannels. The above results set the scene and are known. The remanng roblem s the otmal ower allocaton under constrant. 4

5 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November Otmal Power wth Constrant ere, the r strongest egenchannels are deloyed, and ther s are constraned as (db) ( (db) (db)) ;,..., (db) (6) where s the maxmum dfference between the on the egenchannels and the average at each receve antenna,. In other words, the aroach guarantees that the lo on each egenchannel s smaller than nequalty constrants, ( / M ) 0. Thus, n ths otmzaton roblem, there are (db)/0 M. The otmzaton roblem now becomes, n (6) as well as one equalty constrant, maxmze,,..., subect to f (,,..., ) 0 ( / M ) M log ;,..., ( / M ) (7) ) where the obectve for the maxmzaton,.e., log ( / M varable ;,...,. Equvalently, we can rewrte (7) as, s concave n the mnmze f (,,..., ),,..., subect to f (,,..., ) ( / M ) 0 M log ( / M ) 0 ;,..., (8) where f, f,..., f are convex real functons of ;,...,. In fact, (8) s a convex otmzaton roblem that ncludes nequalty constrants. A artcular nteror-ont algorthm called the logarthmc barrer method [6] s used here to solve the roblem (8). The frst ste s to rewrte (8), makng the nequalty constrants mlct n the obectve where mnmze f0 (,,..., ) g( f (,,..., )),,..., (9) subect to M g : s the ndcator functon for the nonostve reals, 0 ; u 0 g ( u). (0) ; u 0 The roblem (9) has no nequalty constrant, but ts obectve functon s not dfferentable, so descent methods (e.g, Newton s method, etc.) cannot be aled. The logarthmc barrer method aroxmates the ndcator functon, g, by the functon gˆ ( u) (/ t)log0( u) () 5

6 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 where t 0 s a arameter that sets the accuracy of the aroxmaton. ke g, the functon ĝ s convex and nondecreasng, and becomes for u 0. Unlke g, however, ĝ s dfferentable and ncreases to as u ncreases to zero. As t ncreases, the aroxmaton becomes more accurate. Substtutng ĝ for g n (9) gves mnmze f 0 (,,..., ) (/ t) log0 ( f (,,..., )),,..., () subect to M The obectve here s convex snce ( / t)log0( u) s convex and ncreasng n u, and t s dfferentable. The functon (,,..., ) log f 0( (,,..., )) s called the logarthmc barrer or log barrer for the roblem (). Its doman s the set of onts that satsfy ot ot ot the nequalty constrants of (8) strctly. Denotng (,,..., ) as the soluton of (), t s ot ot ot shown [6; age 563] that (,,..., ) s no more than /t-subotmal. Ths suggests a straghtforward method for solvng the orgnal roblem (8), wth a secfed accuracy by takng t /. The next ste s to solve the equalty constrant roblem (), by elmnatng the equalty constrant and then solve the resultng unconstraned roblem usng methods for unconstraned mnmzaton. (for examle) can be elmnated usng the arameterzaton, M. The reduced roblem s then mnmze (, 3,..., t log log 0,,..., ) ( / M ) ( M ) log ( / M ) ( / M ) ( M ) log ( / M ) 3 0 (3) where the obectve s now multled by t. Snce an affne functon of a convex functon s also convex, elmnatng equalty constrants reserves convexty. Moreover, the elmnaton of the equalty constrants nvolves lnear algebrac oeratons. Thus, (3) and () are equvalent. Snce ot ot ot (, 3,..., ) s dfferentable, a suffcent condton for a ont (, 3,..., ) to be otmal s ot ot ot (,,..., ) (4) 0 3 where (. ) denotes the gradent oerator. Thus solvng the unconstraned mnmzaton roblem (3) s the same as fndng a soluton of (4), whch s a set of equatons n the varables,,... 3,. There s no analytcal soluton to the otmalty roblem (4) and so the roblem s solved teratvely, for examle usng gradent descent: ( n) ( n) ( n) ( n) where T 3 ( n ) ( n) ( n ) (5) s the soluton vector ont at nth teraton and 6

7 Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 k, k,..., k t ( / M ) ( / M ) k log 0 ( / M ) M ( / M ) ( ) k k. (6) ( / M ) ( / M ) k ( / M )( M ) ( / M ) k k (n) In (5), s the ste sze at teraton n, chosen va a smle nexact lne search called a backtrackng lne search. In the termnology of teratve methods, the convergence of the gradent descent algorthm usng the backtrackng lne search s at least lnear [6]. The stong convergence crtera of an teratve algorthm such as (5) s usually of the form, where s small and ostve, as suggested by the subotmalty condton [6]. The stong crteron s often checked mmedately after the descent drecton,, s comuted. The method also requres a sutable startng ont (0). The obectve functon n (3) s convex only on the regon that the nequalty constrants f (,,..., ) 0;,..., are satsfed. As a result the obectve functon (4) s not convex over the whole work sace S (,,..., ) 0 M M of the teratve algorthm used here. Snce 3 ; the obectve functon (, 3,..., ) may have several local mnma, the work sace S s dvded nto D dstnct subsaces and the teratve algorthm s run wth dfferent startng ont canddates chosen from dfferent subsaces. For examle, wth unform grddng of the work sace and slttng the nterval for nto q equal, dstnct segments, we get, at most, D q subsaces. Then, the converged results are comared to see whch one s the global mnma. (c) Denotng d as the convergent ont aocated wth the startng ont chosen from the dth segment, the otmum ont allocaton s set as ot ot ot ot T ot ( c) T [ ] [ ot ] (7) ( c) ( c) wth arg mn ( ). (8) ot ( c) d d,..., D In ractce, MIMO systems wth N 4 are of mmedate nterest, and smulatons suggest that for any N 4, q N suffces to avod local convergence usng the above rocedure. Fnally, ot we get C log ( ) and the otmal beamformer M ot ~ R WW and R VR ot ot ~ ot / V as W V(R ). 3.. Selecton from Dfferent Power Allocaton Schemes d ot W s obtaned from In general, the resented otmzaton n (7) does not maxmze the ergodc caacty for all values of system arameters, and. (ths can be seen n the smulatons below.) The desgner may choose from dfferent ower allocaton schemes wth the metrc of ergodc caacty constraned by a maxmum allowable lo (n db). The desgner cks accordng to the desred caacty and SER system erformances,.e., caacty and SER are traded off through ths arameter. 7

8 Ergodc caacty (bts er channel use) Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 The dfferent ower allocaton schemes for selecton are from: the resented otmzaton; the water-fllng schemes aled to dfferent number of egenchannels, ; and the domnant egenmode transmon. The selecton rocedure s as follows. For each value of, the best ower allocaton scheme, along wth the best choce for that acheves the hghest caacty, s selected from the Caacty lane. Ths selecton guarantees that the average lo on each egenchannel would be smaller than. 4. SIMUATION RESUTS To evaluate the erformance of the above aroach, a and a 3 3 MIMO system are smulated, but the formulaton s alcable to any N M MIMO system. In addton, throughout the smulatons, the theoretcal ergodc caactes of water-flled egen-mimo and domnant egenmode transmon are used as benchmarks. Fgure lots the ergodc caacty (n bts er channel use) versus the lo (db) for a 3 3 MIMO system usng the resented otmzaton aled to dfferent number of egenchannels,, wth 0dB. Recall that s the maxmum allowable lo relatve to the average at each receve antenna, mosed on the egenchannels. It can be verfed that for, the choce of that can be set wthn the resented otmzaton s lower bounded by the one obtaned wth equal s of all egenchannels, denoted as. The value of mn mn deends on varous system arameters such as r mn( M, N) and (e.g., for mn mn a 3 3 system, dB, and. 8 db whereas n a system mn 3.96dB). It s worth notng that n water-fllng schemes, a fxed reducton of s mosed on each egenchannel (ths cannot be controlled by the desgner) and snce the aocated lo on the weakest egenchannel s the largest lo of all the egenchannels, t can be set as a lower bound for the allowable maxmum lo n the water-fllng scheme. Ths sets (db) for the water fllng case. Ths s the reason that water-flled caacty curves are the straght lnes as seen n Fg.. The same stuaton also holds for the domnant egenmode transmon roosed algorthm wth three branches water-fllng wth three branches roosed algorthm wth two branches water-fllng wth two branches lo (db) Fgure. The ergodc caacty versus the lo (db) for a 3 3 MIMO system wth 0dB 8

9 Ergodc caacty (bts er channel use) Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November 03 (not shown n Fg. ). The domnant egenmode transmon has the (fxed) lowest lo of dom log ( ); max E max, among all schemes. 0 0 max The caacty results of the resented otmzaton fall behnd the otmal water-flled egen- MIMO caacty for the same. The reason s that, unlke the water-fllng scheme that dscards the weakest egenchannel f the ower allocated to t s negatve, all of the avalable egenchannels are used by the resented otmzaton all the tme. It s seen n Fg. that none of the ower allocaton schemes maxmze the ergodc caacty for all values of, and. The desgner can choose from a number of ower allocaton schemes (the resented otmzaton; the water-fllng schemes aled to dfferent number of egenchannels, ; and the domnant egenmode transmon) and the metrc for selecton s the hghest ergodc caacty wth an dom mn allowable maxmum lo on egenchannels. For, the domnant egenmode transmon s the only canddate for selecton (see Fg. ). Fgures 3 and 4 llustrate the maxmum ergodc caacty acheved va selecton, constraned wth dfferent choces of for 0 30dB n a 3 3 and a MIMO systems, resectvely. In general, for larger values of (larger lo n s allowed), a system wth hgher caacty s exected. As a useful feature, the oston of the caacty between uer and lower bounds can be controlled by the desgner through the choce of. Some other nterestng results are: makng use of more than one egenchannel, the caacty curves are uer and lower bounded by the otmal water-flled caacty usng all egenchannels ( mn( M, N) ) and mn the one obtaned va our aroach wth and ; n a system, egenmode transmon s the best for 6 db and values of ~ 3dB; and n a 3 3 MIMO system, t s oble to outerform the water-flled caacty aled to ust the two strongest egenchannels for any lo 0 db water-fllng = 3 = 8 db = 0 db = 6 db 5 0 = 0-3 db water-fllng, = 5 domnant egenmode mn = -.8 db transmon : (db) Fgure 3. The maxmum ergodc caacty acheved va selecton n a 33MIMO system constraned wth dfferent choces of 9

10 Ergodc caacty (bts er channel use) Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November = 6 db = 4.5 db 0 water-fllng, = domnant egenmode transmon mn = 3.96 db (db) Fgure 4. The maxmum ergodc caacty acheved va selecton n a MIMO system constraned wth dfferent choces of Smulaton results are now resented for a MIMO system n whch egenchannel s deloyed wth ndeendent QAM and egenchannel s loaded wth BPSK. The overall SER erformance of the system s a measure for comarson among dfferent ower allocaton schemes and one defnton s r # ofcorrectlydetected symbols over egenchannel SER (9) r # of transmtted symbols over egenchannel The above formulaton allows the number of egenchannels n the summatons to be smaller than r, and n general, the actual number deends on the tye of ower allocaton scheme used. The overall SER results for a system s llustrated n Fg. 5. Also dected are the SER curves aocated wth each of the egenchannels when water-fllng s used. These reveal how the overall SER deterorates wth the weakest egenchannel. In fact, the full egen-mimo waterfllng scheme has the worst overall SER erformance among other ower allocaton schemes and the domnant egenmode transmon has the best. Ths s exected because of the trade-off between the caacty and SER erformances. Because of the selecton asect of the resented aroach, the SER curves are ece-wse dscontnuous, and several SER curves for dfferent values of may le together for some regons. For examle, n Fg. 5, the curves tagged wth 6dB, 4.5dB, and 3.96dB le on the SER result of domnant egenmode transmon for low to moderate values of. The romsng result s that the overall SER erformance s better than that of full egen-mimo water-fllng. For examle, n a system wth 4 db, the mrovement over water-fllng s almost 3 db n for moderate to hgh values of ( ~ 6dB ). Ths mrovement s at the exense of a lo of.9 bts er channel use n caacty comared to that of water-fllng. At low to moderate values of, the advantage s more emhaszed for SER~0-3, the equvalent mrovement s more than about 5 db. ere the selecton scheme chooses the domnant egenmode transmon as the best scheme 0

11 Symbol error rate Internatonal Journal of Informaton Technology, Modelng and Comutng (IJITMC) Vol.,No.4,November the overall SER of the MIMO system usng water-fllng 0 - = 6 db SER of egenchannel wth BPSK usng water-fllng = 4.5 db 0-3 domnant egenmode transmon wth QPSK 0-4 SER of egenchannel wth QPSK usng water-fllng = 3.96 db : (db) Fgure 5. The overall SER results of a system usng dfferent ower allocaton schemes among others for ~ 6dB and 4 db and there s only a small lo n caacty (smaller than bt er channel use for ~ 0dB ) relatve to that of water-fllng. 5. SUMMARY AND CONCUSIONS In ths aer, the otmal ower allocaton s formulated for the caacty maxmzaton wth the total nut ower constrant and a maxmum allowable lo on egenchannels,. The goal s to fnd a way to desgn an egen-mimo system that has good relablty, or error erformance, and hgh caacty, smultaneously. Comarson of the ergodc caactes from the resented otmzaton wth that from water-fllng wth an arorate number of deloyed egenchannels, and domnant egenmode transmon, enable selecton for otmal caacty for a chosen lo,. One feature of the aroach s that the SER erformance of the system and the oston of the caacty between the uer and lower bounds can be controlled by the desgner through the choce of. Promsng results are resented for the mrovement of SER erformance over than that of full water-flled egen-mimo, when usng smlfed archrectures such as a fxed-set of modulatons or fxed common modulaton acro the egen-channels. ere, a desgn can have a controllable and sgnfcantly better uncoded error erformance than full egen-mimo at the exense of a modest lo n caacty. In a ractcal stuaton, based on the tye of modulaton (and any codng) used, the desgner may translate the desrable erformance to a desred SER erformance n a system, and then agn accordng to the desred caacty and SER system erformances. REFERENCES [] A.Paulra, R.Nabar and D.Gore, Introducton to sace-tme wrele communcatons, Cambrdge unversty re, 003. [] G.Ralegh and J. Coff, Sato-Temoral Codng for Wrele Communcatons," IEEE Trans. Commun., Vol 46. No3, March 998.

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