SINR Analysis of Opportunistic MIMO-SDMA Downlink Systems with Linear Combining

SINR Analysis of Opp ortunistic MIMO-SDMA Do wnlink Systems with Linear Comb ining Man-On Pun, V isa K oivunen a nd H. V incen t Poor Abstract — Opportunistic schedul ing (OS) schemes have been proposed prev iously by the authors for multi user MIMO-SDM A downlink sy stems with linear combining. In particular , it has been demonstrated that significant perfor mance impro vement can be achiev ed by incor porating low-complexity linear com bining techniques into the design of OS schemes for MIMO-SDMA. Howe ver , this previous analysis was perf ormed based on the effec- tive signal-to-in terference ratio (SIR), assumin g an in terference- limited scenario, which is typically a valid assumption i n SDMA- based systems. It was shown th at the limitin g distribution of the effective SIR is of th e Frechet type. Surprisingly , the corres ponding scaling laws were found to follo w ǫ log K with 0 < ǫ < 1 , rather t han the con ventional log log K form. Inspired by t his d ifference between the scaling law form s, in t his p aper a systema tic approa ch is dev eloped to derive asymptotic thr oughput and scaling la ws base d on signal-to- interference-noise ratio ( SINR) by uti lizing extreme value theory . The conv ergence of th e limitin g distribution of the effective SINR to the Gumbel type is established. The resulting scaling law is found to be gover ned by the conv en tional log log K f orm. These nov el results are validated by simulation results. The comparison of SIR and SINR-based an alysis suggests t hat the SIR-based analysis is more computationally effici ent for SDMA- based systems and it captu res the asymptotic system perf ormance with h igher fi delity . I . I N T RO D U C T I O N Opportu nistic scheduling (OS) has recently attracted consid- erable research interest as a promising technique to improve system throug hput by exploiting m ulti-user diversity with limited channel feedback [10]. Gen erally spe aking, existing OS schemes can be classified into two categories, nam ely the time-sharin g (T S) [ 10] and space-d ivision multiple acc ess- based (SDMA)-ba sed [9] OS schemes. In TS-OS, only the mobile terminal (MT ) with the best instantaneous c hannel condition s is scheduled in one slot, regardless of the number of beams em ployed by the base station ( BS). In contr ast, SDMA- based OS serves mu ltiple MTs simultan eously with multiple orthon ormal b eams in each slot. Den ote by M and N the number of tr ansmit a nd receive anten nas, respectively . It has been sh own recently that the sum-r ate of SDMA-based OS grows linearly with M whe reas that of T S-OS increa ses on ly linearly with min( M , N ) [8]. In addition to the more rapidly growing scaling law , SDMA-based OS is particularly attracti ve for practical systems with stringe nt latency requ irements. Man-On Pun and H. V incent Poor are with the Depa rtment of Electri cal Engineeri ng, Princeton Unive rsity , P rincet on, NJ 08544. V isa Koi vunen is with the Signal Processing Laboratory , Helsinki Unive r- sity of T echnology (HUT ), Finland. This researc h was supported in part by the Croucher Foundatio n under a post-doct oral fellowshi p, and in part by the U. S. National Science Foundati on under Grants ANI-03-38807 and CNS-06-25637. The SDMA-b ased OS in [ 9] was or iginally developed for systems with single- antenna MTs. For MTs with mu ltiple receive antennas, [9] pro poses to let each antenna comp ete fo r its d esired bea m as if it wer e an individual MT . As a result, each beam is assigned to a spe cific receive antenna of a chosen MT . Since signals received fro m the u ndesignated anten nas of a chosen MT are discarded, this leads to inef ficient utilization of m ultiple r eceiv e anten nas. In [4], various lin ear com bining technique s exploiting sign als r eceiv ed by all recei ve antennas were pro posed. The en hanced effectiv e SINR is employed as a scheduling metric. Both analytical and simulation results in [4] have demon strated that the system sum-rate perform ance ca n be significantly improved by using such combining techniques. For instance, the o ptimal combinin g technique can p rovide over 40% sum-rate imp rovement compared to the selection combinin g techniq ue fo r M = 4 and N = 2 [4 ]. The the oretical analysis in [4] has been conducted based on SIR, assuming an interfer ence-limited environment. The resulting scaling laws h av e a distinctive form, i.e. ǫ log K with 0 < ǫ < 1 , which is very different from the conventional form log log K der i ved based on signal-to-noise-ratio (SNR) [10] or SINR [8 ], [9] in the literature. Similar results have been ind ependen tly developed for multicell systems in [2]. In this work , we introdu ce a systematic app roach for deriving asymptotic throu ghpu t a nd scaling laws u sing SINR. The propo sed approach stems from extreme v alue theo ry [3]. W e prove th at th e cu mulative distribution fu nctions (CDFs) of the effecti ve SINR ob tained with linear c ombining co n verge to the Gumbel-ty pe limiting distribution. Furthermo re, we show th at the SINR-based scaling laws for the proposed opportu nistic beamfor ming and sched uling sche mes follow the conv e ntional log log K fo rm. Thr ough compa rison between th e SIR an d SINR-based analysis, it is argu ed that the SIR-based analysis is more compu tationally efficient for SDMA-based systems, and subsequently more ef fectiv e in cap turing the high-order behavior of the asy mptotic system perf ormance . T o make compariso n with ou r previous SIR-based analy sis reported in [4], we concentrate on a practical system with M = 4 and N = 2 in this work. Howe ver, it should be emphasized that the analysis can be easily generalized for systems with arbitrar y M and N . Notation : V ectors and matr ices are denoted by boldface letters. k·k r epresents the Euc lidean no rm o f the enclosed vector and |·| denotes th e a mplitude of the enclo sed c omplex- valued quantity . I N is the N × N iden tity matrix. W e use E {·} for expectation. Fin ally , log an d ln are the logarithms to the base 2 and e , respectively . I I . S I G N A L M O D E L Base station a 1 a 2 a B Scheduler Data Buffer s 1 s 2 s B Beamfomer MT # 1 MT # 2 MT # K Mobile terminal # 1 Linear Combining SINR evaluation Data detection Feedback link Mobile terminal # K Linear Combining SINR evaluation Data detection Feedback link Fig. 1. A block diagram of the opportunisti c MIMO SDMA do wnlink s ystem under consideration . W e con sider the oppor tunistic MIMO-SDMA downlink system d epicted in Fig. 1 wh ere the BS is equipp ed w ith M transmit antennas and each of the K MTs h as N receive antennas with N ≤ M . Let { a m ; m = 1 , 2 , · · · , M } be a vector set containing M orth ornor mal b eamform ing vectors of length M . W e fo cus on a particular time slot d uring which a beamfo rming v ector set { a m } has been cho sen from a common codebook shared b y the BS and MTs. D uring the p -th slot, the tra nsmitted signal ca n be expressed as x ( p ) = M X m =1 a m s m ( p ) = As ( p ) , (1) where A = [ a 1 , a 2 , · · · , a M ] is the u nitary beamfo rming ma- trix with A H A = I M and s ( p ) = [ s 1 ( p ) , s 2 ( p ) , · · · , s M ( p )] T with E n | s m ( p ) | 2 o = 1 is th e da ta vector transmitted in the p -th slot. The correspond ing receiv ed signal by the k -th MT can be written a s y k ( p ) = √ ρ k H k ( p ) x ( p ) + n k ( p ) , (2) where H k is the channel gain m atrix between the BS and the k - th MT with ind ependen t and ide ntically-distributed (i.i.d.) Rayleigh-distributed complex entries. Furthermo re, the noise term n k ( p ) is m odeled as C N ( 0 , I N ) and ρ k is a co nstant r elated to the average receiv ed SNR g iv e n by E n ρ k k H k ( p ) x ( p ) k 2 o = ρ k M . T o keep our fo llowing analy sis tractable, we c oncentrate on a homog enous system with ρ k = ρ in this work . For notational simplicity , we dr op the temp oral ind ex p in the sequel. Furtherm ore, we refer to the SINR o btained by linearly combinin g signals fro m all recei ve antennas as the effective SINR in or der to d istinguish it fro m th e observed SINR without combining . I I I . O S W I T H L I N E A R C O M B I N I N G In this section, we br iefly review the beam forming an d scheduling schemes for MIMO-SDM A systems with linear combinin g techniqu es p roposed in [4]. As shown in Fig. 1, in the beginning of a time slot, each MT ev aluates the effectiv e SINR for ea ch beam by linearly combining the recei ved signals with one of the following thr ee combinin g techniq ues, n amely selection combin ing ( SC), maximu m ratio combining (MRC) and optimum combin ing (OC) before returning the informatio n about M effective SINRs to the BS. Note that OC perform s activ e inter ference supp ression by exploiting the interference structure, whereas MRC and SC simply intend to amplify the desired signal. It will be shown later that this characteristic interferen ce-suppr ession feature o f OC enables the sched uling scheme with OC to con siderably outperfo rm those with SC and MRC. Upon r eceiving the effecti ve SINR information from all MTs, the BS sch edules and starts d ata transmission to m ultiple MTs with the largest effective SINRs on different beams until the end of the current time slot. At each ch osen MT , received signals fr om all an tennas a re linea rly co mbined u sing one o f the above linea r co mbining techniques, followed by data detection . It is worth no ting th at th e pro bability o f awarding m ultiple beams to the same MT is rath er small, as the number o f MTs is large. Fu rthermo re, recall that the m inimum mean squared erro r ( MMSE) and zero -forcing ( ZF) r eceiv e r structures for MIMO r eceiv ers am ount to co mbiners u sing OC and MRC for each b eam, respectively . As a result, fo r an MT assigned with mu ltiple b eams, it can foc us on on e assigned beam at a time using the chosen combining technique while regarding all other beams as interfering sources. I V . S I N R A NA L Y S I S Define γ ∗ m = max ( γ 1 ,m , γ 2 ,m , · · · , γ K,m ) , for m = 1 , 2 , · · · , M . Assuming γ k,m for k = 1 , 2 , · · · , K , are i.i.d. with CDF F X ( x ) , the resulting average system thr oughp ut can be computed as [ 9]: C = E ( M X m =1 log (1 + γ ∗ m ) ) = M Z ∞ 0 log (1 + x ) d [ F X ( x )] K . (3) In th e fo llowing, we first d erive F X ( x ) based on dif- ferent linear combin ing tech niques before establishing their correspo nding limiting distributions, i.e. lim K →∞ [ F X ( x )] K . By exploiting the limiting distributions, we deri ve the asymp totic throug hput a nd the correspondin g scaling laws. In the sequel, we f ocus on a practical system with M = 4 and N = 2 . Howe ver, it has been sho wn in [5] that the analysis can be easily generalized for system s with arbitrary M and N . A. S election Combining (S C) W e begin with the selection combining. Denote by x the maximum of the two SINR values of the i -th beam pe rceived by the two antennas at the k -th MT . The CDF of x can be derived based on the results in [9] and reads F (SC) X ( x ) = " 1 − e − x/ρ (1 + x ) 3 # 2 . (4) Differentiating F (SC) X ( x ) with respect to x , we can o btain the correspo nding prob ability den sity f unction (PDF). f (SC) X ( x ) = 2 " 1 − e − x/ρ (1 + x ) 3 # (1 + x ) 1 ρ e − x/ρ + 3 e − x/ρ (1 + x ) 4 . (5) It is straightforward to show th at F (SC) X ( x ) an d f (SC) X ( x ) satisfy the follo wing e quation lim x →∞ 1 − F (SC) X ( x ) f (SC) X ( x ) = ρ > 0 , (6) which is the necessary and sufficient condition for the lim- iting distribution o f h F (SC) X ( x ) i K being of the Gumbel type [3]. Consequently , F X ( K ) ( x ) = [ F X ( x )] K conv erges to the following Gum bel-type d istribution [3]. F (SC) X ( K ) ( a (SC) K x + b (SC) K ) = e − e − x , x ≥ 0 (7) or equiv alently , F (SC) X ( K ) ( x ) = e − e − x a (SC) K + b (SC) K a (SC) K , x ≥ 0 , (8) where a (SC) K and b (SC) K are n ormalizing factors affecting the shape an d locatio n of the limitin g distribution, respectively . From extreme value theo ry , b (SC) K can be computed from the characteristic extreme of (4) as [3] 1 − F (SC) X ( b (SC) K ) = 1 K . (9) Since (9) is an exponen tial-linear equation of b (SC) K , it is non- trivial to ob tain the exact solution of b (SC) K in clo sed fo rm. Fortunately , since 1 − F (SC) X ( K ) monoto nically decreases from 1 to 0 wherea s 1 /K ∈ [1 , 0) for K = 1 , 2 , · · · , ∞ , ther e always exists a unique solution of (9). Thus, we can resor t to numerical metho ds to co mpute the n umerical solu tion of b (SC) K . It should be emphasized that 1 − F (SC) X ( b (SC) K ) ten ds to 1 as K approa ches infin ity , which implies that b (SC) K increases with K . Furthermo re, a (SC) K can be obtained from solving th e fo llow- ing equation. a (SC) K = F (SC) X − 1  1 − 1 K e  − b (SC) K . (10) Similar to b (SC) K , we can show th at there always exists a unique solution of a (SC) K . Therefore, the nu merical solution of a (SC) K can be found by r esorting to numerical me thods. Finally , the throughpu t obtained with SC can be comp uted by substituting (8) into ( 3) and reads C (SC) = 4 ln 2 Z ∞ 0 1 − e − e − x a (SC) K + b (SC) K a (SC) K 1 + x dx. (11) Let z = e − x a (SC) K and ξ = exp ( b (SC) K /a (SC) K ) . W e have x = − a (SC) K ln z an d dx = − a (SC) K z dz . Thus, (11) can be rewritten as C (SC) = 4 ln 2 Z 1 0 1 − e − z · ξ 1 − a (SC) K ln z · a (SC) K z dz , (12) = 4 ln 2 " Z 4 ξ 0 1 − e − z · ξ (1 − a (SC) K ln z ) a (SC) K z dz + Z 1 4 ξ a (SC) K dz (1 − a (SC) K ln z ) z # . (13) The limit of th e first ter m on the r ight-hand -side (R.H.S) of (13) b ecomes negligib ly small as lim K →∞ 4 ξ = 0 while the limit of the second ter m can be compu ted by exploitin g th e approx imation of a (SC) K ≈ ρ as follows. lim K →∞ 4 ln 2 Z 1 4 ξ a (SC) K dz (1 − a (SC) K ln z ) z = lim K →∞ 4 lo g ( b (SC) K ) . (14 ) Thus, the correspondin g scaling la w is gi ven by lim K →∞ C (SC) 4 lo g ( b (SC) K ) = 1 . (15) In particular, for ρ = 1 , we can a pprox imate b (SC) K and a (SC) K as b (SC) K ≈ ln 2 K − 2 ln (1 + ln 2 K ) , (16) a (SC) K ≈ 1 , (17) respectively . Subsequen tly , the scaling law can be written as fo llows. lim K →∞ C (SC) ρ =1 4 lo g (ln 2 K − 2 ln (1 + ln 2 K )) = 1 , (18) which stands f or a typical scalin g la w in the lo g lo g K form . B. Ma ximum Ratio Combining (MRC) The effectiv e SINR obtained with MRC ca n be expressed as a ratio of two r andom v ariables gi ven by x = z 1 /ρ + y , where z and y are χ 2 distributed random variables with 2 N and 2 M − 2 degrees o f freedo m correspondin g to th e instantaneo us signal power of the desired signal and the interfering signal, respectively . In pa rticular, for M = 4 and N = 2 , we ha ve [6], [7] f (MRC) X ( x ) = xe − x/ρ ρ 2 (1 + x ) 3 + 6 xe − x/ρ ρ (1 + x ) 4 + 12 xe − x/ρ (1 + x ) 5 (19) and the corresponding CDF can be expressed as F (MRC) X ( x ) = 1 − e − x/ρ (1 + x ) 3 − xe − x/ρ ρ (1 + x ) 3 − 3 xe − x/ρ (1 + x ) 4 . (20) It can be shown that lim x →∞ 1 − F (MRC) X ( x ) f (MRC) X ( x ) = ρ > 0 . (21) Therefo re, the limiting distrib ution of h F (MRC) X ( x ) i K is also of the Gumbel ty pe. Follo wing similar steps as in the p revious section , we ha ve C (MRC) = 4 ln 2 Z ∞ 0 1 − e − e − x a (MRC) K + b (MRC) K a (MRC) K 1 + x dx. (22) and lim K →∞ C (MRC) 4 lo g ( b (MRC) K ) = 1 , (23) where a (MRC) K and b (MRC) K are the corr esponding norm alizing factors. In p articular for ρ = 1 , we c an sho w that b (MRC) K ≈ ln 3 K − 2 ln (1 + ln K ) , (24) a (MRC) K ≈ 1 (25) and the scaling la w has th e following log log K f orm. lim K →∞ C (MRC) ρ =1 4 lo g (ln 3 K − 2 ln (1 + ln K )) = 1 . (26) C. Optimal Combining (OC) The CDF of the effectiv e SINR obtain ed OC using N receive antennas in th e pr esence of M − 1 interf ering sources has been derived in [1] . For M = 4 an d N = 2 , the correspo nding CDF takes th e following form . F (OC) X ( x ) = 1 − e − x/ρ (1 + x ) 3 − 3 xe − x/ρ (1 + x ) 3 − xe − x/ρ ρ (1 + x ) 3 , (27) and the corresponding PDF is f (OC) X ( x ) = xe − x/ρ ρ 2 (1 + x ) 4  (3 ρ + 1) x +  6 ρ 2 + 6 ρ + 1  . (28) Since lim x →∞ 1 − F (OC) X ( x ) f (OC) X ( x ) = ρ > 0 , the lim iting distribution of h F (OC) X ( x ) i K is also of the Gumbel type. Similar to the cases of SC a nd MRC, we can show that C (OC) = 4 ln 2 Z ∞ 0 1 − e − e − x a (OC) K + b (OC) K a (OC) K 1 + x dx. (29) and lim K →∞ C (OC) 4 lo g ( b (OC) K ) = 1 , (30) where b (OC) K and b (OC) K are the correspo nding nor malizing factors. In particular for ρ = 1 , we c an sho w that b (OC) K ≈ ln 4 K − 2 ln ln K, (31) a (OC) K ≈ 1 (32) and the sum-rate scale s like the fo llowing lo g lo g K f orm. lim K →∞ C (OC) ρ =1 4 lo g (ln 4 K − 2 ln ln K ) = 1 . (33) 0 50 100 150 200 250 300 350 400 450 500 2 4 6 8 10 12 14 16 Number of users Throughput bps/Hz Solving normalizing factors with numerical methods Analysis (OC) Simulation (OC) Analysis (MRC) Simulation (MRC) Analysis (SC) Simulation (SC) ρ =5 ρ =1 Fig. 2. Simulati on versus analyt ical results with numeric al normaliz ing fac tors for ρ = 1 , 5 . V . S I M U L AT I O N R E S U LT S In this section, simulation is per formed to confirm our SINR analysis derived in Sec. I V. Unless other wise specified, we set M = 4 and N = 2 . W e first compare the asympto tic throug hput shown in (1 1), (22) and (29) against the ir correspondin g simu lation resu lts. Figure 2 sho w s the asymptotic throu ghput curves using the numerical no rmalizing factors obtained b y numer ical m ethods for ρ = 1 and 5 . Inspection of Fig. 2 r ev e als that the an alytical results shown in (11), (22) and (29) are in ac cord with the simulation results. Despite that the asymp totic an alysis is achieved b y assuming a large K , Fig. 2 in dicates that th e asymptotic analysis is a lso very accur ate for smaller K v a lues. Furthermo re, Fig. 2 co nfirms tha t the scheduling scheme with OC can substantially ou tperfor m those with MRC a nd SC whereas th e im provement p rovided by MRC is more apparent in the presence of stron ger n oise. Th is is because th e scheme with OC is d esigned to maxim ize SINR whe reas MRC in tends to maximize SNR. 50 100 150 200 250 300 350 400 450 500 2 3 4 5 6 7 8 9 10 11 Number of MTs, K Throughput bps/Hz ρ =1 with approximated normalizing factors Analysis (OC) Simulation (OC) Analysis (MRC) Simulation (MRC) Analysis (SC) Simulation (SC) Fig. 3. Simulat ion versus analyt ical results with approxi mated normaliz ing fac tors for ρ = 1 . Next, rather than th e n umerical solutio ns, Fig. 3 depicts the av erage sum-r ates using the app r oximated norm alizing factors computed in (16), (24) and ( 31) toge ther with a K ≈ 1 for ρ = 1 . Since th e ap proxim ation expressions have been derived by a ssuming a large K , the analytical cu rves shown in Fig. 3 approa ch the simulated curves on ly when K becomes large. 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Number of MTs, K Normalizing factors Numerical (OC) Approximation (OC) Numerical (MRC) Approximation (MRC) Numerical (SC) Approximation (SC) b K a K Fig. 4. Comparison of the numerical and approximated solutions of the normaliz ing factors for ρ = 1 . Finally , to inspect the app roximatio n accuracy o f (16), (2 4) and (31), Fig . 4 shows the nu merical and appr oximated nor- malizing factors as a fun ction of the n umber of MT s, K . Sinc e solving the exact solutions to th e n ormalizing factors inv o lves the linear-exponential fun ctions, it is in general non-trivial to obtain accurate closed -form expressions for the normalizin g factors, which comp romises th e a ccuracy of the subsequen tly derived scaling laws. V I . C O M PA R I S O N B E T W E E N S I R A N D S I N R A N A L Y S I S It is inter esting to com pare the SINR analysis d erived in this work with o ur p revious SIR analysis reported in [ 4]. 1.) On the one hand, it is easy to verify tha t the CDFs of the effecti ve SINR in (4), ( 20) an d (27) c on verge the correspo nding CDFs of the ef fe cti ve SIR reported in [4] as ρ tends to infinity , respectively . On the other hand, our SIR and SINR-based analysis suggests th at the limitin g distributions of the effecti ve SIR and SINR do n ot belon g to the same domain o f attraction . Instead , they are of the Frec het-type and Gumbel-ty pe, respectiv e ly . I t is natural to conjecture that th e limiting d istribution function of SINR migh t also conv e rge to the Frechet- type if the noise po we r becomes zero. Howev e r , our results reveal that th is intuition is not tru e. This is becau se that the limit o perator is not com mutative in g eneral. 2.) I t is gen erally more difficult to obtain the normalizin g fac- tors in the SINR analysis than the SIR analysis since the SINR- based analysis inv olves e xponen tial-type CDFs and requir es solving expo nential-linear equation s such as (9). Therefore, it is more computationally advantageou s to de riv e the scaling laws in th e SIR-based analysis compar ed to th e SINR-based analysis in the presence of strong interfer ence. 3.) When com puting the n ormalizing factor s in the SINR- based analysis, we have to carefu lly take into account the high-o rder terms in F X ( x ) . For instance, if the high- order terms in F X ( x ) in (4), (20) a nd (27) a re igno red, the resulting simplified CDFs for different sche mes will all lead to the same set of normalizing factors, i.e e − b K /ρ (1+ b K ) 3 = 1 K . Thus, the perf ormance of OS sch emes with different combinin g technique s cannot be distinguished based on their scaling laws. Since it is generally much easier to compute the normalizin g factors with high accuracy in the SIR-based analysis [ 4], we argue that the SI R-based scaling laws can better char acterize the actual perfor mance o f different OS schem es by focusing on the in terferenc e-limited scenario s. V I I . C O N C L U S I O N In this p aper, we have de velop ed a systematic appro ach to derive the SINR-based asymptotic through put and scaling laws for OS sch emes by utilizing extreme value theor y . I n particular, we have in vestigated the asymptotic throu ghput and scaling laws of the OS schemes prop osed fo r MIMO-SDMA systems with different linear com bining techn iques. Our ana lytical results hav e shown that the limiting distrib u tion of the effecti ve SINR is of the Gumbel ty pe a nd th e scaling laws follow the log lo g K form. 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