Outage and Diversity of Linear Receivers in Flat-Fading MIMO Channels

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1 Outage and Dversty of Lnear Recevers n Flat-Fadng IO Channels Ahmadreza Hedayat ember, IEEE, and Ara Nosratna Senor ember, IEEE Abstract Ths correspondence studes lnear recevers for IO channels under frequency-nonselectve flat quas-statc Raylegh fadng. The outage probablty and dversty gan of SE and Zero Forcng recevers are nvestgated. It s found that contrary to ntuton, SE and zero-forcng recevers may not perform smlarly at hgh SNR. Assumng transmt and N receve antennas, the zero-forcng recever always has dversty N +, unle the SE recever, whose behavor can vary. Under separate spatal encodng, where data from each transmt antenna s separately encoded, SE s no better than n terms of dversty. But for jont spatal encodng systems, where an encoded stream s sent from all the antennas, the SE recever acheves dversty N at low spectral effcences but has dversty only N + at hgh spectral effcences. I. INTRODUCTION The fadng wreless IO channel s characterzed by a mxng nterference of the sgnals arrvng from multple transmtter antennas. Sometmes multple transmt antennas are only used to ncrease relablty, e.g. orthogonal space-tme codes, thus the sgnals from multple antennas are tghtly structured and the nterference can be undone at the recever. However, transmt sgnals do not always have as much structure e.g. n the case of spatal multplexng thus more elaborate methods may be needed to remove the spatal nterference [], [2]. For example, nullng-and-cancellng detectors [], [2], [3] are capable of provdng optmal detecton. Lnear recevers, even though suboptmal, are much smpler and therefore useful for many applcatons. In ths correspondence, we study lnear IO recevers under quasstatc flat fadng. We calculate the dversty of lnear recevers va ther outage probabltes. Snce the outage probablty s closely related to the frame error rate, ths also provdes a tangble measure of the performance of realstc systems. A summary of our results s as follows: For a IO system consstng of transmt and N receve antennas, under flat Raylegh fadng, zero-forcng recevers acheve dversty order N + under all cases studed. SE recevers acheve the same dversty for transmsson strateges that do not allow combned codng of data streams, e.g. horzontal spatal encodng. However, for codng strateges that allow jont encodng of data streams, e.g. D-BLAST, a more nterestng scenaro emerges. In such systems, for low spectral effcences SE recevers can acheve the full dversty of N, whle A. Hedayat s wth Navn Networs, Rchardson, TX, USA Emal: hedayat@eee.org. A. Nosratna s wth the Dept. of Electrcal Engneerng, The Unversty of Texas at Dallas, Rchardson, TX, USA E- mal:ara@utdallas.edu. Ths wor was presented n part at ICASSP 25. for hgh spectral effcences only a dversty of + N s possble. For ntermedate values of R, dverstes n between the two extremes are observed. II. LINEAR RECEIVERS The nput-output system model for flat fadng IO channel wth transmt and N receve antennas s r Hc + n, where c s the transmtted vector, n C N s the Gaussan nose vector, and r s the N receved vector at a gven tme nstant. Throughout ths paper, we assume H has ndependent and dentcally dstrbuted complex Gaussan entres,.e. H C N. We consder lnear recevers and evaluate the outage probablty of a flat fadng IO channel followed by a or SE recever, assumng the channel s perfectly nown to the recever. The recever s F H H H H H, whch transforms the receved sgnal to ˆr F r c + H H H H H n. The SE recever s F SE H H H + ρ I H H, where ρ s the receved SNR. Snce the symbols are detected ndvdually, the SINR of the ndvdual symbols determnes the performance. The detecton nose of recever, ñ H H H H H n, s a complex Gaussan vector wth zero-mean and covarance matrx Rñ σnh 2 H H. The th dagonal element of Rñ s gven by: Rñ, σ 2 nh H H ĤH det Ĥ σn 2 det H H H, where represents the th dagonal element of the nverse of, and Ĥ s obtaned by removng the th row of H. The assocated SINR s γ E x /Rñ,, whch can be shown to be a ch-square random varable wth 2N + degrees of freedom [4], [5]. The CDF of Y χ 2N +, wth varance.5 for the partcpatng Gaussan random varables, s: N + F Y y e y y!. 2 The SINR of the th symbol of SE detector s determned by nose and resdual nterference γ h H Ĥ Ĥ H + ρ I h I + ρh H H 3, where h s the th column of H. Removng ths column from H gves Ĥ C N [6].

2 2 Equaton 3 shows that γ s a quadratc form whose statstcs has been derved n [7] as follows. Consderng the random matrx Ĥ CN and the random vector h C N, the quadratc form Y h H ĤĤH + ρ I h has the CDF F Y y exp y N ρ n A n y y, 4 n! ρ n where the auxlary functons A n y are gven by { N + n A n y + N n Cy N < + n. +y, 5 and C s the coeffcent of y n + y [7]. In general, the SINR of the output symbols of the SE recever are correlated, unle those of the zero-forcng recever. III. OUTAGE PROBABILITY IN SEPARATE SPATIAL ENCODING In separate spatal encodng, the data stream s demultplexed to several sub-streams, one for each transmt antenna. Furthermore, the resultng streams are not jontly encoded, to acheve easer decodng. Horzontally encoded V-BLAST s a promnent example of ths strategy. In ths scenaro, f any of the data streams s n outage, the entre system s n outage. Hence, the outage event O occurs when any of the subchannels cannot support the rate that s assgned to t. In our analyss, we consder equal rate for the sub-channels, however, t s also possble to have a non-unform rate assgnment. After lnear transformaton, the mutual nformaton between the elements of ˆr and the transmtted data vector c s Ic ; ˆr log + γ. Assume the target rate s R, and let L N. Accordng to 2 and 4, the statstcs of γ s nvarant to. Thus, the outage probablty PrO s : { PrO Pr Ic ; ˆr R } Pr Ic ; ˆr R Pr Ic ; ˆr < R, 6 where 6 s accurate when sub-channel outage probabltes are small. In the above, we have assumed that sub-channel outage events are ndependent, whch s vald for. For SE recevers the sub-channel outage events are not strctly ndependent, but the approxmaton used only n ths secton maes the analyss tractable and does not affect dversty. Smulatons show that the approxmaton has been properly used. Alternatvely, one may consder only the outage event of a sngle sub-channel, whch s an approxmaton that s accurate enough for dversty calculaton. Usng the CDF of χ 2N + n the evaluaton of 6 gves the outage probablty for the recever, whch s PrO F Y 2 R/ 2R/ L+ L +! ρ L+, 7 where denotes equvalence n the lmt as ρ. Thus the dversty order s L +. Substtutng the dstrbuton 4 n 6, the SE outage probablty s calculated: PrO F Y 2 R y L+ L +! y + y ρ L+,8 R y2 whch shows that SE dversty order s also L +. However, the and SE outage probabltes are not exactly the same. The rato of 7 to 8 s: PrO + y 2 R PrO SE y. R y2 2 R 9 Note that the rato of outage probabltes n 9 remans fxed regardless of SNR and t only depends on the relatve target rate R. When R s small the outage probablty of becomes larger than that of SE. The rato 9 approaches one when R s large see Secton V. Generalzaton of the above results to non-unform rate assgnment s straghtforward. Unform and non-unform rate assgnment have the same dversty, even though they have dfferent outage probablty performance. IV. OUTAGE PROBABILITY IN JOINT SPATIAL ENCODING In jont spatal encodng, the data stream s encoded and then demultplexed nto sub-streams, each gong to one antenna e.g. D-BLAST. Thus, each data symbol can contrbute to sgnals of all the transmt antennas. The recever s n outage when the aggregate mutual nformaton of all the sub-channels fals to support the target rate. The mutual nformaton between the elements of the lnearly transformed receve sgnal, ˆr and the transmtted data vector c s Ic ; ˆr log + γ. Assumng the target rate s R, the probablty of the outage event O s PrO Pr log + γ < R Pr + γ < 2 R. Theorem : Consder a flat IO channel wth transmt and N receve antennas, and jont spatal encodng. Under perfect channel state nformaton avalable to the recever, the outage probablty of recevers decays wth order of N +. Proof: See the Appendx. Thus, we observe that the dversty s the same for separate and jont spatal encodng. To obtan the SE outage probablty, we substtute the SINR from 3 n, whch gves: PrO Pr I + ρh H H > 2 R. 2 The dependence on the dagonal elements of the random matrx I + ρh H H maes further analyss ntractable.

3 3 Therefore, we proceed to provde an upper bound to ths probablty. Rewrtng the sum mutual nformaton as n, we have I Ic ; ˆr log + ρh H H log I + ρh H H 3 I log tr + ρh H H log, 4 + ρλ where 3 s due to Jensen s nequalty, and λ s are the egenvalues of the Wshart matrx H H H. Substtutng 4 nto gves: PrO Pr 2 R. 5 + ρλ Though 5 s an upper bound of the outage probablty, n Secton V, through smulaton, we show that t s a tght upper bound n low and hgh spectral effcency. Assumng N, the jont PDF of the egenvalues of H H H, λ s, λ λ 2 λ, s f Λ λ K,N λ N λ λ j 2 exp λ, <j 6 where K,N s a normalzng constant [8]. The evaluaton of 5 for a specfc outage rate R s rather dffcult, due to the shape of the outage regon. However, one can calculate the bound for small and large values of R where the the outage regon can be approxmated by regons wth smpler shapes. For a IO channel wth 2 and N 2, the bound 5 s PrO Pr + 2 R ρλ + ρλ 2 For convenence defne Sλ, λ 2 and also defne the set { A λ, λ 2 : + ρλ + + ρλ + + ρλ 2 } 2 R 2. + ρλ 2 Then the rght hand sde of 7 s PrA. Exact calculaton of PrA s not easy, thus we show ts asymptotc behavor by boundng t from below and above. Let R < 2. If λ outage occurs only when λ 2 c 2 2 b ρb, where b 2 R 2. Because the curve Sλ, λ 2 b s convex, the regon A s contaned n the sosceles rght trangle wth the base λ + λ 2 c 2 and the two sdes λ and λ 2, and ntegral over the trangle s always larger than PrA. We now buld another trangle that s contaned by A. Usng the symmetry of Sλ, λ 2, t s not dffcult to calculate that an sosceles trangle wth base λ +λ 2 c 2, where c 2 2 b bρ, s contaned n A and ntegraton over ths trangle s always smaller than PrA. Fnally, we show that probablty ntegrals over the two trangles behave the same asymptotcally, thus completng a sandwch argument. To do so, consder the ntegral over any such sosceles trangle wth parameter c: K 2,N c e λ λ N 2 c λ 2K 2,N N!N 2! λ N 2 2 λ λ 2 2 e λ2 dλ 2 dλ 2N e c c! ρ 2N 8 where c could be c 2 or c 2. Snce PrA s bounded above and below by values that have dversty-2n, t must have dversty 2N. Now recall that PrO PrA therefore we have establshed that outage has dversty no less than 2N. Consderng that 2N s also the maxmum achevable dversty order, we conclude that outage has exactly dversty order 2N. Ths concludes the arguments for small spectral effcences. Now we consder hgh spectral effcences, namely R > 2 and b <. In ths case, λ 2 can drve the system to outage regardless of the value of λ and vce versa. For nstance, let λ, as long as λ 2 d 2 b ρb, outage occurs. Thus, the outage regon has a strp along the λ axs for large enough λ, and lewse along λ 2. In fact, the set of strps defned as λ 2, λ d 2 and λ, λ 2 d 2 s contaned n A. Snce Sλ, λ 2 b s convex, t s possble to fnd d 2, whch s proportonal to ρ but d 2 > d 2, such that A contans the strps λ 2, λ d 2 and λ, λ 2 d 2. The probablty of the above sets can be characterzed usng the followng expresson: 2K 2,N d e λ λ N 2 λ N 2 2 λ λ 2 2 e λ2 dλ 2 dλ ρ N, 9 where d could be d 2 or d 2. Therefore 9 ndcates that the upper bound 7 has the dversty N L+, where L N. In the calculaton of 9, the ntersecton of the two orthogonal strps s calculated twce, but the ntersecton has a probablty that decays wth ρ 2N and does not affect the asymptotc behavor of 9. The outage bounds developed above show the surprsng fact that SE recevers can acheve the same dversty as the L recever for small values of R n jont spatal encodng. However, for large values of R the dversty performance of SE and s the same. Hence, for SE the dversty vares from the dversty of an unconstraned recever to that of, dependng of the target rate R. Comparng to the results from Secton III for separate spatal encoders, the SE recever has dfferent dversty n jont spatal encodng archtecture, except for large outage rate R. The prevous results of the case 2, N 2 can be extended to arbtrary values of and N. We state the general result n the followng theorem.

4 4 PrO 2 SE n the ntermedate values of R, t does predct dversty order varyng wth R. Fgure 3 presents smlar results for a flat fadng IO channel wth N 2 and correlated transmt antennas wth correlaton factor ρ t.5. Outage probabltes are slghtly hgher than the uncorrelated case, however, the behavor of outage probabltes are the same. Fgure 3 also shows the results for uncorrelated IO channel wth N SNR db Fg.. Outage probablty of lnear recevers, N 2. The pars of sold and dashed lnes, from left, correspond to SE and for rates R, 2, 4, bts/sec/hz. Theorem 2: Consder a flat IO channel wth transmt and N receve antennas, and jont spatal encodng. Under perfect channel state nformaton avalable to the recever, the upper bound 5 on the outage probablty of SE recevers decays wth order of N at low spectral effcency,.e. R < log, resultng n the dversty order of N for the outage probablty. At hgh spectral effcency R > log, 5 decays wth the order of N +. Proof: See the Appendx. V. SIULATION RESULTS We consder a IO system wth two antennas n transmt and receve sdes: N 2. The outage probablty of the lnear recevers n the separate archtecture s shown n Fgure. The target rate s R, 2, 4, bts/sec/hz. As expected, both lnear detectors show dversty order of one, regardless of the target rate. For hgher values of R the dfference of and SE performance s neglgble. But, for lower values of R, SE performs better than for all SNR. The dependency of the relatve performance of these recevers on the target rate R s n agreement wth 9. In hgh SNR, the rato of the outage probabltes remans fxed. Fgure 2 shows the outage probablty of the unconstraned recever and lnear recevers n a jont spatal encodng archtecture. The unconstraned recever has the full dversty of the channel. The recever has dversty one as expected from the analyss n Secton IV. The dversty order of remans unchanged regardless of the target rate R. Surprsngly, SE dversty depends on R: n lower values of R the dversty order s very close to that of the unconstraned recever, and n hgher values of R ts dversty becomes the same as dversty. These results are n agreement wth the analyss n Secton IV. Fgure 2 also shows the outage probablty of the SE recever and the upper bound 5. The bound s tght at ether low or hgh values of R. Though the bound s loose VI. CONCLUSION We present new results on the performance of lnear recevers for the removal of spatal nterference n IO Raylegh flat fadng channels, and calculate ther dversty order. Our analytcal and expermental results show that SE recevers have outage probablty wth varyng decayng slope: t may decay as fast as the outage probablty of unconstraned recever, wth the full order of N, or as slowly as that of recevers, wth the order of N +, dependng on the spectral effcency. The authors gratefully acnowledge comments from Dr. Naofal Al-Dhahr. APPENDIX Proof of Theorem : The SINR of the sub-channels under are ndependent ch-square random varables wth degrees 2N +. Let Y χ 2N +,,,. The outage probablty of s gven by the CDF of the random varable + Y + Y + + Y. 2 Among the components of the above random varable, the last term, whch s the product of Y s, determnes the dversty order snce t s a ch-square wth the lowest degree. In the followng, through recurson, we show that Y Y 2 Y has dversty order L +. Let us start by Z Y Y 2. The PDF of Z s 2 f Z z L! 2 zl K 2 z, 2 where K s the zeroth order modfed Bessel functon of the second nd [9], whch for small values of z s a constant. Therefore, for small values of z the frst order approxmaton of f Z z s z L. Ths shows that the CDF of Z, F Z z, has frst order approxmaton equal to z L+, whch ndcates the dversty order of L +. Now consder the CDF of W Y Y 2 Y 3 Z Y, where Y χ 2N + : F W w PrW w PrZ Y w α f Z zf Y w z dz z L K 2 z e w z L+2 For small values of x: K mx Γm 2 2/x m [9]. w z! dz

5 5 SE bound PrO 2 3 PrO L SE SNR db SNR db Fg. 2. Left: Comparson of recevers. Rght: SE outage and the upper bound 5. N 2 and the curves show rates R,2,4, bts/sec/hz. PrO 2 3 PrO L SE SNR db 5 6 L SE SNR db Fg. 3. Comparson of recevers. Left: N 2, correlated transmt antennas wth ρ t.5, R,2,4, bts/sec/hz. Rght: N 4, uncorrelated, R4,8,2,6 bts/sec/hz. where α s a constant. The frst order approxmaton of above around zero s w L+ α zl +! K 2 z dz. Thus, F W w behaves le the L+th power of w, ndcatng the dversty order L +. Ths procedure can be appled recursvely to fnd that the frst order approxmaton of the CDF of Y Y 2 Y behaves le w L+. As mentoned, the product term n 2 domnates the dversty. We just showed that the product term Y Y 2 Y has the dversty order L +. Therefore, the dversty order s L +. Proof of Theorem 2: Frst, we state and prove the followng lemma. Lemma : Let I λ x e λ λ dλ dλ. 22 I s polynomal n x wth the mnmum exponent of g +, where and are ntegers. Proof: Wth some algebra, one can obtan x x λ m e λ dλ m! λ m x λ n e λ dλ m+ x!, 23 n Cj, n j x n j m + j! j lm+j+ x l l!, 24 where m and n are ntegers and C, are the bnomal coeffcents. Usng 23 and 24, I 2 can be calculated as I 2 + n!! Cj, j x j 2 + j! j l 2+j+ where the mnmum exponent of x s g Now we use nducton. Assume that I x l l!,

6 6 g I x a x. Then λ e λ g a x a g j λ x λ e λ λ e λ x λ dλ λ Cj, j x j + j! l +j+ 25 where 25 s obtaned usng 24. Equaton 25 ndcates that I s polynomal n x wth the mnmum exponent of g + + g. To prove Theorem 2, we frst note that λ λ j 2 <j,..., S p,, λ, 26 where the set S s a subset of the,, ndexes that, and p,, s the correspondng nteger coeffcent. In low spectral effcency, R < log, we can wrte the rght-hand sde of 5 as K,N e λ λ λ j 2 dλ dλ A K,N,..., S A e λ λ N <j p,, λ N + dλ dλ. 27 We now bound the ntegraton regon A, from nsde and from outsde, by polyhedra. Then we use a sandwch argument by showng that the ntegraton over the nner and outer polyhedra gves the same asymptotc performance. Usng a drect extenson of the argument developed n the two-dmensonal case proceedng Equaton 8, t can be seen that the polyhedron defned by λ, λ c, where c ρ 2 R + 2 R, contans the ntegraton regon A. Therefore an ntegral over ths polyhedron upper bounds the outage probablty. Now consder another polyhedron defned by λ, λ c, where c s proportonal to ρ but small enough so that A contans ths polyhedron. The base of ths polyhedron can be calculated smlarly to Secton IV. c ρ 2 R/ Integraton over ths new polyhedron, whch s characterzed by c, lower bounds PrA. Fnally Lemma establshes that the asymptotc behavor of 27, whle ntegratng over ether of the two polyhedra, s the same. Each multple ntegral n 27 s n the form of I dλ dλ of Lemma,.e., polynomal n c wth smallest exponent + N + N. Therefore, the upper bound 5 decays wth ρ N n low spectral effcency, ndcatng dversty order s no less than N. At the same tme, N s actually the maxmum possble dversty order, so the outage probablty of SE recever has dversty of N. We now proceed to show the hgh-rate result, where the x l l!, developments parallel those for 2 n 9. For R > log the outage regon A can be upper and lower bounded wth orthogonal slabs along the coordnates. The frst set that s a subset of A has orthogonal slabs where the j-th slab s defned as λ j d and λ j, where d ρ 2 R. The outage regon A s a subset of the second set of slabs whose defnton s the same as the frst set wth d replaced wth d, whch s also proportonal to ρ and d > d. Therefore, the rght-hand sde of the bound 5 s the same as 27 wth the excepton that the ntegraton regon A could be ether of above sets. Consderng the possblty of some zero n 26 and the unbounded shape of A, there are domnatng terms such as e λ λ N j λ N + dλ dλ, λ j d,λ j whch s polynomal n d wth the mnmum exponent of N +. Ths ndcates that the bound 5 decays wth ρ N + n hgh spectral effcency. Ths completes the proof of Theorem 2. j REFERENCES [] G. Foschn, G. Golden, R. Valenzuela, and P. Wolnansy, Smplfed processng for hgh spectral effcency wreless communcaton employng mult-element arrays, Journal on Selected Areas n Communcatons, vol. 7, pp , Nov [2] N. Prasad and. K. Varanas, Outage analyss and optmzaton for multaccess and V-BLAST archtecture over IO Raylegh fadng channels, n Proc. of 4th Annual Allerton Conference on Communcaton, Control, and Computng, 23. [3] T. Guess, H. Zhang, and T. V. Kotchev, The outage capacty of BLAST for IO channels, n Proc. IEEE ICC, ay 23, pp [4] J. H. Wnters, J. Salz, and R. D. Gtln, The mpact of antenna dversty on the capacty of wreless communcaton systems, IEEE Transactons on Communcatons, vol. 43, pp , February/arch/Aprl 994. [5]. Rupp, C. eclenbrauer, and G. Grtsch, Hgh dversty wth smple space tme bloc-codes and lnear recevers, n Proc. IEEE GLOBECO, San Francsco, CA, November 23, pp [6] E. K. Onggosanus, A. G. Daba, T. Schmdl, and T. uharemovc, Capacty analyss of frequency-selectve IO channels wth suboptmal detectors, n Proc. IEEE ICASSP, 22, pp [7] H. Gao, P. J. Smth, and. V. Clar, Theoretcal relablty of SE lnear dversty combnng n Raylegh-fadng addtve nterference channels, IEEE Transactons on Communcatons, vol. 46, pp , ay 998. [8] L. Zheng and D. Tse, Dversty and multplexng: A fundamental tradeoff n multple-antenna channels, IEEE Transactons on Informaton Theory, vol. 49, no. 5, pp , ay 23. [9] I. S. Gradshteyn and I.. Ryzh, Table of Integrals, Seres, and Products, San Dego, CA: Academc Press, 5th edton, 994.