Performance Analysis of V-BLAST with Optimum Power Allocation

Transcription

1 Perforance Analyss of V-BLAST wth Optu Power Allocaton Vctora Kostna, Sergey Loyka School of Inforaton Technology and Engneerng, Unversty of Ottawa, 6 Lous Pasteur, Ottawa, Canada, KN 6N5 E-al: sergey.loyka@eee.org Abstract Coprehensve perforance analyss of the unordered V-BLAST algorth wth varous power allocaton strateges s presented, whch akes use of analytcal tools and resorts to Monte-Carlo sulatons for valdaton purposes only. Hgh-SNR approxatons for the zed average block and total error rates are gven. The SNR gan of zaton s rgorously defned and studed usng analytcal tools, ncludng lower and upper bounds, hgh and low SNR approxatons. The gan s upper bounded by the nuber of transtters, for any odulaton forat and any type of fadng. Ths upper bound s acheved at hgh SNR by the consdered zaton strateges. Whle the average zaton s less coplex than the nstantaneous one, ts perforance s alost as good at hgh SNR. A easure of robustness of the zed algorth s ntroduced and evaluated, ncludng copact closed-for approxatons. The zed algorth s shown to be robust to perturbatons n ndvdual and total transt powers. Based on the algorth robustness, a pre-set power allocaton s suggested as a lowcoplexty alternatve to the other zaton strateges, whch exhbts only a nor loss n perforance over the practcal SNR range. I. INTRODUCTION The V-BLAST algorth [] has attracted n recent years sgnfcant attenton as a sgnal processng strategy n the MIMO recever due to ts relatve splcty and also the ablty to acheve, under certan condtons, the full MIMO capacty. Unfortunately, the algorth has a few drawbacks as well. The al orderng procedure s coputatonallydeandng, whch s a ltaton for soe applcatons. Snce the successve nterference cancellaton s used, lower detecton steps have on average a saller SNR and thus produce ore errors, whch further propagate to hgher steps [5]-[7] so that the overall error perforance ay be not satsfactory, especally f no codng s used. A popular approach to prove the error perforance of the V-BLAST algorth s to decrease the error rates at lower steps by eployng a non-unfor power allocaton aong the transtters. Several technques have been reported that fnd the transt (Tx) power allocaton that nzes the nstantaneous (.e. for gven channel realzaton) total error rate (TBER) of the V-BLAST, wth or wthout the al orderng [3][4]. The approxate solutons for the The TBER s defned as the error rate at the output strea to whch all the ndvdual sub-streas are erged after the detecton (see [7] for ore detals). Thus, t takes nto account the actual nuber of errors at the transtted sybol vector. The block error rate (BLER) s defned as the probablty to have at least one error at the detected Tx sybol vector [7]. It does not take nto account the actual nuber of errors, but only the fact of ther presence. nstantaneous Tx power allocaton have also been found, based on varous approxatons. In [], the nstantaneous BLER (rather than the TBER) s consdered as an zaton crteron, and the u Tx power allocaton s found nuercally for the V-BLAST wth two transtters. Although the nstantaneous power allocaton technques proposed n []- [4] do deonstrate a few db perforance proveent over the orgnal (unzed) V-BLAST, they also add consderably to the syste coplexty, snce new feedback sesson and power reallocaton are needed each te the channel atrx changes; the nstantaneous per-strea (transtter) SNRs also need to be sent to the Tx end. A less coplex approach s to use an average rather than nstantaneous zaton,.e. the u power allocaton s found based on the average error rate (BLER or TBER) [5],[8]-[]. Snce ths gnores the sall-scale fadng, only occasonal feedback sectons and power reallocatons are requred, when the average SNR changes, and only the average SNR needs to be fed back to the Tx end. Perforance evaluaton of the zed systes has been done n []-[5],[] through sulatons, by coparng zed and non-zed error rate curves, and t was noted that the u power allocaton gves a few db gan n ters of the SNR. In ths paper, we present analytcal perforance evaluaton of the zed syste va a rgorous defnton of the SNR gan of the zaton and va a easure of robustness, n addton to the tradtonal error rate analyss. Upper and lower bounds on the SNR gan are gven, whch hold for any odulaton and any type of fadng. Specfcally, t s shown that the SNR gan cannot exceed (the nuber of transtters). Ths upper bound s acheved at hgh SNR. The lower bound s approached by the SNR gan at low SNR. Addtonal propertes of the gan, ncludng copact hgh and low-snr approxatons, are also gven. These results are suarzed n Theores -5 and Corollares n Secton V. The pact of perturbatons n the ndvdual and total Tx powers on the perforance of the zed syste s studed usng a easure of robustness and relyng on the generc prncples of convex zaton []. It s deonstrated that the zed syste s robust to such perturbatons, whch also ndcates that the closed-for approxatons for the u power allocaton can be used wthout notceable loss n the perforance. Based on ths, a pre-set power allocaton s suggested as a low-coplexty alternatve to other zaton strateges. Due to the robustness of the proposed power allocaton, t s expected that a sgnfcant porton of the X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.

2 theoretcally-predcted gan can also be acheved n practce. Analytcal results and conclusons are valdated va Monte- Carlo (MC) sulatons. II. SYSTEM MODEL We eploy the followng standard baseband dscrete-te syste odel, s = = [ r, r,... r ] T n r = HΑs + ξ = h α + ξ () where s = [ s, s,... s ] T and r are the vectors representng the Tx and Rx sybols respectvely, T denotes transposton, H = [ h, h,... h ] s the n atrx of the coplex channel gans between each Tx and each Rx antenna, where h denotes -th colun of H, whch are assued to be..d. Raylegh fadng unless otherwse ndcated, n and are the nubers of Rx and Tx antennas respectvely, n, ξ s the vector of crcularly-syetrc addtve whte Gaussan nose (AWGN), whch s ndependent and dentcally dstrbuted (..d.) n each recever, Α = dag ( α,, α ), where α s the power allocated to the -th transtter. For the regular (unzed) V-BLAST, the total power s dstrbuted unforly aong the transtters, α =α =... =α =. In the zed syste, α are chosen to nze the total BER or the BLER, ether average or nstantaneous. Snce we rely on the BLAST error rate perforance analyss n [5],[7]- [9], we also ad the sae basc assuptons. III. ERROR RATES AND OPTIMUM POWER ALLOCATION Closed-fro expressons for the BLER and TBER of the unordered V-BLAST can be found n [5],[7]-[]. For the sake of splcty and copleteness, we gve below ther hgh-snr approxatons only. The average BLER can be expressed as C PB( α ) Pe = = ( 4αγ ), () where P e s the average error rate at step condtoned on no errors at the prevous steps, γ s the average SNR, and Ck = k!/(!( k )!) are the bnoal coeffcents. The second equalty holds for BPSK odulaton, whch we assue below 3, unless otherwse ndcated. The average TBER wth u power allocaton can be approxated as ( ) C Pet ( α + ) = ( + ) Pe n + = = ( 4αγ ), (3) Based on these approxatons, closed-for expressons for u power allocaton have been obtaned [8][9]. When the average BLER s used as the perforance etrc, the u power allocaton s b α α, α, =,,, (4) = 4γ + soe of our results hold true for arbtrary channels 3 soe of our results hold true for arbtrary odulaton. where ( / ) /( + ) b = C (5) Note that α and α... α as γ,.e. ost of the power goes to st transtter at hgh SNR. Ths s consequence of the fact that st step error rate donates the whole error rate. When the average TBER s used as the perforance etrc, the u power allocaton s slar to that for the BLERbased zaton, so that (4) can be used wth b gven by ( + )( + ) ( + )( ) C n b = /( + ). (6) The sae tendency n power dstrbuton s also observed (ost power gong to st transtter). The approxatons n ()-(4) can be used to fnd the average error rates of the power-zed V-BLAST. Specfcally, based on () and (4), we observe that despte of the fact that the transtters to are allocated sall aounts of power (and, thus, the correspondng error rates ncrease copared to the unzed syste), the st step error rate stll donates the overall perforance, a P e, P e, (7) ( )( n+ + ) ( 4γ ) ( 4γ ) + where / n a = C b +, so that P e P e... P e >> >> >> at hgh SNR (the sae relatonshp as for the unzed syste n [7]). Thus, the average BLER can be approxated as P n B P e / 4 + γ, (8) Coparng (8) to the average BLER wthout zaton, PB Pe / ( 4γ ), we conclude that the power allocaton brngs log db SNR gan n ters of the average BLER. In a slar way, usng (3), the zed average TBER can be approxated as + + Pet Pe. (9) ( 4γ ) Coparng (9) to the unzed TBER [7] Pet ape /, where a > quantfes the effect of error propagaton (see [7][] for detals), we conclude that the u power allocaton brngs an SNR gan of a + <.e. less than that n ters of the average BLER., () IV. ROBUSTNESS OF THE OPTIMUM POWER ALLOCATION When the zaton algorth s pleented n a practcal syste, there are varous sources of naccuraces and perturbatons, whch ay affect ts perforance but whch X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.

3 were gnored n the dealstc analyss above. These ay nclude nuercal naccuraces of the zaton, naccurate or outdated estate of the average SNR, whch result n naccuraces n the u power allocaton coeffcents α. A robust algorth, whch s nsenstve to all these factors, s desred fro the practcal perspectve. In order to estate the pact of these factors on the syste perforance, let us ntroduce the easure of robustness δ (senstvty) of the average error rate (ether BLER or TBER) wth the u power allocaton, P = P( α ), to the changes n syste paraeter u, P/ P δ=, () u/ u where u ay represent the total Tx power, u = = α, or the power allocated to any of the transtters, u =α. The easure of robustness () s the rato of the noralzed varaton n the perforance P/ P to the noralzed varaton n the syste paraeter u/ u, whch causes ths perforance varaton. Note that the use of noralzed dfferences n the defnton s essental as t akes the easure to be ndependent of the scale. The algorth s robust to varatons n the syste paraeter u f relatvely sall change n u leads to relatvely sall change n the error rate P,.e. when δ s sall or oderate nuber. When both the perturbaton n the syste paraeter u and n the syste perforance P are sall enough, one can use the dervatves n () nstead of the fnte dfferences, P u δ δ =, () u P so that P / u deternes the algorth robustness, and δ serves as a easure of local robustness. It follows fro the Lagrange ultpler technque [] that P α = P u = λ, so that δ δ =λ u/ P, (3) where λ s the Lagrange ultpler evaluated at the u pont [9], whch s a part of the zaton proble soluton. Thus, the approprately noralzed λ s the easure of local senstvty 4 of the average error rate to varatons n the total or ndvdual Tx power. The noralzed varaton n the average error rate can be evaluated fro the noralzed varaton n the syste paraeter usng (3), P u u =δ δ, (4) P u u For the average BLER-based zaton at hgh SNR, usng the results n [8][9], the Lagrange ultpler can be approxated as, λ n n + ( 4γ ) +, (5) For sall varatons n the syste paraeter, u α, so 4 An extended dscusson of ths ssue n the general fraework of convex zaton can be found n []. that, usng (8)and (5), the robustness easure wth respect to the varatons n the total or st transtter power s δ, (6).e. equal to the dversty order of the syste. The algorth s locally robust as long as ( n ) s not too large; δ and consequently P / P u / u f n=. Ths result s a consequence of the fact that the hgh-snr average BLER s donated by the st step BER (see (8)) so that ts dversty order and hence the senstvty to the Tx power s nu when n= ; ncreasng ( n ) results n ncreasng dversty order and hence n ncreasng senstvty to the Tx power. Thus, the benefcal effect of hgher dversty order wth ore Rx antennas s accopaned by the negatve effect of hgher senstvty to varatons n syste paraeters. The robustness easure wth respect to α... α can be approxated as b δ, =..., (7) 4γ + Thus, the algorth s also robust n ters of α,, α at hgh SNR. Furtherore, hgher steps exhbt better robustness snce, coparng (6) and (7), δ >δ, =.... It should be noted that ths robustness of the algorth s an unexpected by-product, whch was not a goal of the orgnal desgn. For the TBER-based zaton, (6) and (7), and hence the conclusons above also hold true; b s gven by (6) n ths case. As an exaple, Fg. shows the average TBER versus α for x V-BLAST. When α s far away fro α, the slope of the curves s qute steep and deterned by the dversty order of the donatng step; thus, allocatng too lttle power to the st Tx ncreases the st step BER, akng t donant, whereas gvng too uch power to the st Tx boosts the nd step BER. Note that the slope s steeper n the doan of the donatng nd step BER, apparently because of ts hgher dversty order. But as the power allocaton algorth attepts to balance these two extrees and approaches α, the curves becoe very flat, confrng local (n the vcnty of α ) nsenstvty of the TBER to varatons n α. TBER P e donates here γ = db γ = db P e donates here α.5.5 α Fg.. Average TBER versus α for x V-BLAST wth BPSK odulaton. Thus, sall naccuraces n α do not affect the average X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.

4 error rate sgnfcantly. Ths hnts at the concluson that the approxate closed-for α wll result n alost the sae average error rate as the accurate nuercal one. Nuercal (MC-based) analyss confrs ths expectaton. It should also be ponted out that the choce of the zaton crtera (BLER or TBER) does not affect sgnfcantly the fnal result ether [8][9]. Thus, BLER or TBER can be used equally well as a perforance crteron for zaton. Sall senstvty of the BLER/TBER to α suggests even further splfcaton n the zaton algorth: snce α changes slowly wth the SNR (see (4)), we can pck up only one fxed (pre-set) value of α and stll get perforance proveent for a wde range of γ. Such splfed algorth does not requre any feedback at all, and yet, as Fg. deonstrates, t attans alost the sae perforance as the dynacally zed syste. In ths exaple, the 3x3 V- BLAST wth = [ ] α.6.4 T s consdered, and ts perforance s very close to the zed V-BLAST n the range of γ = 35 db. TBER - Unzed Optzed α=[,.6,.4] SNR [db] Fg.. TBER of 3x3 V-BLAST wth BPSK odulaton for pre-set (fxed) power allocaton. It should be noted that t s the robustness of the algorth that s responsble for sall dfference between nstantaneous and average zaton at hgh SNR observed n [8][9]. The robustness consdered above s local robustness,.e. for sall varatons n the vcnty of the unperturbed values of the syste paraeter. When varatons are not sall, the fnte dfferences n () cannot be accurately approxated by the dervatves and the approxatons n (), (3), (6), (7) ay not be accurate. In such a case, one has to consder a easure of global robustness. To ths end, let us consder the average error rate of the perturbed syste P ( α ; u+ u), where u s not necessarly sall and u s the total Tx power. Let α u denote the u power allocaton of the perturbed syste, so that the u allocaton for the unperturbed syste s α = α u=, the zed average error rate of the unperturbed syste s P( α ; u), and the zed average error rate of the perturbed syste s ( α u ; + ) P u u. Fro the general theory of convex zaton [], the last two quanttes are related by the followng global nequalty, u P = P( α ; u+ u) P( α ; u) λ u, (8) where λ s evaluated at u =, and the equalty s acheved for u =. It follows that f u s postve,.e. the total Tx power s ncreased, the al value of P decreases by no ore than λ u ; f u s negatve,.e. total Tx power s decreased, the al value of P s guaranteed to ncrease by at least λ u. Dvdng (8) by P, one obtans P/ P δ u/ u, (9) where δ s gven by (3). Therefore, δ, whch was ntroduced as a easure of local robustness n (3), also serves as a easure of global robustness n (9). Snce P, u ay be postve as well as negatve (they always have opposte sgn), we re-wrte (9) n the for whch ncludes only postve ters, P/ P P/ P δ, u > ; δ, u < () u/ u u/ u δ gves upper and lower bounds on the noralzed varaton n the error rate due to any (not necessarly sall) varaton u n the total Tx power, for postve and negatve u (.e. ncreasng and decreasng the total Tx power), respectvely. Thus, when the total Tx power s ncreased by u, the average error rate decreases by not ore than δ P u/ u; when the total Tx power s decreased by u, the average error rate ncreases by at least δ P u / u, so that the postve effect never exceeds the negatve one. Snce the nequalty n (9) transfors nto the approxate equalty for sall perturbatons (see (4)), these two effects are equal n that case. Clearly, the Lagrange ultpler λ plays a key role not only n the local, but also n the global robustness. Snce t s not known n closed-for for arbtrary SNR, the hgh-snr approxaton n (5) can be used wth reasonable accuracy (for γ > db). V. SNR GAIN OF OPTIMUM POWER ALLOCATION In ths secton, we explore soe propertes of the SNR gan of zaton usng ostly analytcal technques, and use nuercal results only as the last resort. The analyss and conclusons below are vald for any odulaton forat, unless otherwse stated. The SNR gan of u power allocaton s defned as the dfference n the SNR requred to acheve the sae error rate n the zed and unzed systes,.e. fro P (,..., ) P(,..., ) α α = α α, () where P s the perforance crteron,.e.the BLER or the TBER, ether nstantaneous or average; the left-hand sde represents the zed error rate under the total power constrant = α =, the rght-hand sde represents the error rate for the unfor power allocaton, α =α, and the SNR gan s G =α. For the average zaton, the average error rate s used n (). For the nstantaneous zaton, one ay use both nstantaneous and average error rate n (). In the forer case, one obtans the nstantaneous gan of the X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.

5 nstantaneous zaton, and n the latter case, one obtans the average gan (.e. n ters of the average error rate) of the nstantaneous zaton. To be able to copare the nstantaneous and average zatons below, we use the average gan of the nstantaneous zaton G nst. It copares to the gan of the average zaton G av as follows: Gnst Gav,.e. the nstantaneous zaton s at least as good as the average one. We consder below the SNR gan defned n ters of the BLER and the TBER, and also copare the propertes of these two dfferent defntons, whch share any slartes. BLER SNR Gan of the Optu Power Allocaton: In ths secton, we consder the BLER-based zaton strateges and present unversal bounds on the BLER SNR gan, ether nstantaneous or average, whch hold for arbtrary odulaton and fadng. These results are further refned n the case of BPSK odulaton and Raylegh-fadng channel. Theore. The BLER SNR gan of u power allocaton, ether nstantaneous or average, for arbtrary odulaton and fadng, s bounded as follows G () Proof. The key to the proof s the fact that the BLER, ether nstantaneous or average, s a onotoncally decreasng functon n each arguent α,..., α (see () or [8][9] for arbtrary SNR), and the fact that P e s a onotoncally decreasng functon of the SNR. Based on ths fact and also on the nequalty P P B α,..., α B,...,, whch sply states that the zed syste s at least as good as the unzed one, the lower bound n () follows. Usng the onotonc-decreasng property of the BLER and the fact that α, whch follows fro the total power constrant, one concludes that PB( α,..., α ) PB(,..., ). Coparng ths nequalty wth the defnton of the gan n () n vew of the onotonc-decreasng property of the BLER, the upper bound follows. Q.E.D. Below we explore the sall-snr behavor of the SNR gan n ters of the average BLER, whch s related to the lower bound n Theore, for the BLER-based zaton and for a varety of odulaton forats. Theore. Sall-SNR behavor of the BLER SNR gan for the average BLER-based zaton s as follows: where Gav G as γ, (3) a Pe G =, a, coherent detecton = ( a ) α γ αγ = ax b Pe G =, b =, noncoherent detecton b ( ) α γ αγ = (4) and a, b are the coeffcents n st ter of MacLaurn s seres expanson of P e. The equalty n (3) s acheved,.e. G =, f and only f all a or all b are equal, for coherent and non-coherent detecton respectvely. Sketch of the proof: expand the BLER n MacLaurn s seres and use t to fnd the u power allocaton and the correspondng BLER; the gan s found va () (detaled proof s gven n the extended verson of ths paper []). Ths result s vald for a varety of odulaton forats for whch the error rate adts MacLaurn s seres expanson n SNR or SNR about SNR=. In ost cases, the strct nequalty n (3) holds,.e. there s an SNR gan of zaton even at very low SNR, snce dfferent P e exhbt dfferent behavor so that the expanson coeffcents are also dfferent. As an exaple, for coherent BPSK and non-coherent BFSK, G =.7 db and.79 db respectvely, for x syste. Corollary.. The result n (3), (4) also apples to the nstantaneous gan of the nstantaneous zaton, n whch case P should be used n (4) nstead of P, and the e coeffcents a, b and hence the gan depend on the channel realzaton, as long as the dervatves n (4) exst and are not all equal to zero sultaneously. Corollary.. The average SNR gan of the nstantaneous zaton s also lower bounded by G n (4), Gnst G, because of Gnst Gav. Thus, we conclude that (3) holds for a varety of scenaros for BLER-based zaton.we now show that the upper bound n () s acheved at hgh SNR. Theore 3. Hgh-SNR behavor of the average BLER SNR gan, for both nstantaneous and average zatons usng ether BLER or TBER as an objectve, wth BPSK odulaton n Raylegh fadng channel, s as follows: G as γ (5) Proof. Fro the hgh SNR approxaton of the average BLER (8) and correspondng approxaton of the MRC unzed BLER, PB Pe = Pn + ( γ ) [7], the frst step donates for both unzed and zed systes. Usng ths n (), one obtans Gav as γ. Usng () and Gnst Gav, ths also holds for the average gan of the nstantaneous zaton, Gnst as γ. Q.E.D. Corollary 3.. Theore 3 also extends to any odulaton/fadng for whch the frst step error rate donates the average BLER at hgh SNR. Based on the dversty order arguent, ths condton should hold for ost odulaton forats n Raylegh-fadng channels. For the average BLER-based zaton wth BPSK odulaton n a Raylegh fadng channel, a hgh-snr approxaton of the average BLER SNR gan s gven by G Gav (6) + c / 34γ n, n, 3 where c = [ b + 3 ]/[ b ], G =. Ths approxaton follows along the lnes of the proof of Theore 3. Note that (6) reduces to (5) for γ, as t should be. The convergence to the upper bound n (5) s however slow, snce the convergence condton s n + 3 γ >>. It follows fro (6) that the BLER SNR gan of the average BLER-based zaton wth BPSK odulaton s an ncreasng functon of the average SNR n the hgh-snr range. e X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.

6 Nuercal evdence ndcates that ths also holds for low to nteredate SNR. Ths concluson s further renforced by the followng Theore. Theore 4. Under the total power constrant = α =u, where u s the total Tx power, the BLER SNR gan of BLERbased zaton n (), ether nstantaneous or average, s a onotoncally ncreasng functon of u: G λ = (7) u P α α α B Proof. otted due to the page lt (see []). TBER SNR Gan of the Optu Power Allocaton: In ths secton, we adapt the results of the prevous secton to the SNR gan defned n ters of the average TBER. Due to the page lt, we gve the results wthout proofs ([] gves detaled proofs). Theore holds, provded soe addtonal condtons are satsfed by the average TBER and the u power allocaton, whch are not very restrctve []. Theore stll holds for the average TBER SNR gan, wth the substtuton of Pe Pet n (4). Theore 3 s no longer vald,.e. the upper bound s never attaned f the gan s defned n ters of the TBER. Instead, the followng holds. Theore 5. Hgh SNR behavor of the average TBER SNR gan for the average zaton s as follows: Gav as γ, (8) G where G s gven by the left-hand sde of (). Thus, the proveent n average TBER s less than the upper-bound n (). The reason for ths s the ncreased power of propagatng errors for the zed syste, due to hgher power gong to lower steps, copared to the unzed one. For exaple, the u power allocaton algorth gves ost of the power to the st Tx tryng to avod the errors at the st step. But f the error does occur at the st step, ts apltude s hgher than that for the unzed syste, whch akes the error propagaton effect ore severe. VI. EXAMPLES To llustrate the generc results above and to deonstrated ther valdty va Monte-Carlo sulatons, we consder the V-BLAST. It follows fro (5) and (8) that the SNR gan s G av = (3 db) and G av = 8 / 5 ( db) at hgh SNR, for the average BLER and TBER respectvely. Thus, whle the u power allocaton s nsenstve to the crtera (.e. ether BLER or TBER) [8][9], the SNR gan of zaton does depend on t, though not n a draatc way. Fg. 3 deonstrates the accuracy of the analytcal result n (6). Addtonally, we note that the average BLER gan of the nstantaneous zaton also tends to 3 db and attans ths bound, but uch faster than that of the average zaton. The SNR gan of the u power allocaton s alost the sae, at hgh SNR, as that of the al orderng procedure (see [6] for detals). The coputatonal coplexty, however, of the forer s uch less than that of the latter. Hence, the average power zaton can be used nstead of the al orderng wth roughly the sae perforance. Slar conclusons also hold n ters of the TBER SNR gan dB upper bound 3. Average BLER gan nstant.. (MC) aver.. (MC) closed for Average SNR, db Fg. 3. Average BLER SNR gan vs. SNR for x V-BLAST wth BPSK odulaton (MC- Monte-Carlo sulatons). VII. REFERENCES [] G.J. Foschn et al, Splfed Processng for Hgh Spectral Effcency Wreless Councaton Eployng Mult-Eleent Arrays, IEEE Journal on Selected Areas n Councatons, v. 7, N., pp , Nov [] R. Kalbas, D.D. Falconer, A.H. Banhashe, Optu power allocaton for a V-BLAST syste wth two antennas at the transtter, IEEE Councatons Letters, v. 9, N. 9, pp , Sep. 5 [3] S.H. Na et al, Transt Power Allocaton for a Modfed V- BLAST Syste, IEEE Trans. Co., v. 5, N. 7, pp.74-79, Jul. 4. [4] N. Wang, S.D. Blosten, Mnu BER Transt Power Allocaton for MIMO Spatal Multplexng Systes, 5 IEEE Internatonal Conference on Councatons, San Dego, Calforna USA, v. 4, pp. 8 86, May 5. [5] N. Prasad, M.K. Varanas, Analyss of decson feedback detecton for MIMO Raylegh-fadng channels and the zaton of power and rate allocatons, IEEE Trans. Infor. Theory, v. 5, N. 6, June 4 [6] S. Loyka, F. Gagnon, Perforance Analyss of the V-BLAST Algorth: an Analytcal Approach, IEEE Trans. Wreless Co., v. 3, N. 4, pp , July 4. [7] S. Loyka, F. Gagnon, V-BLAST wthout Optal Orderng: Analytcal Perforance Evaluaton for Raylegh Fadng Channels, IEEE Transactons on Councatons, v. 54, N. 6, pp. 9-, June 6. [8] V. Kostna, S. Loyka, On Optzaton of the V-BLAST Algorth, 6 IEEE Internatonal Zurch Senar on Councatons, Feb. 6, ETH Zurch, Swtzerland. [9] V. Kostna, S. Loyka, Transt Power Allocaton for the V- BLAST Algorth, 3rd Bennal Syposu on Councatons, Kngston, ON, Canada, May 6. [] J. Cho, Nullng and Cancellaton Detector for MIMO Channels and ts Applcaton to Multstage Recever for Coded Sgnals: Perforance and Optzaton, IEEE Trans. Wreless Councatons, v.5, N.5, pp. 7-6, May 6. [] S. Boyd, L. Vandenberghe, Convex Optzaton, Cabrdge Unversty Press, 4. [] V. Kostna, S. Loyka, On Optu Power Allocaton for the V-BLAST, IEEE Trans. Councatons, subtted, X/7/$5. 7 Crown Copyrght Ths full text paper was peer revewed at the drecton of IEEE Councatons Socety subject atter experts for publcaton n the IEEE GLOBECOM 7 proceedngs.