On Capacity of Wireless Ad Hoc Networks with MIMO MMSE Receivers Jing Ma, Member, IEEE and Ying Jun (Angela) Zhang, Member, IEEE

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1 On Capacity of Wireless Ad Hoc Networs with MIMO MMSE Receivers Jing Ma, Meber, IEEE and Ying Jun (Angela) Zhang, Meber, IEEE Dept. of Inforation Engineering, The Chinese University of Hong ong, Hong ong Eail: Abstract - Widely adopted at hoe, business places, and hot spots, wireless ad-hoc networs are expected to provide broadband services parallel to their wired counterparts in near future. To address this need, MIMO (ultiple-input-ultiple-output) techniques, which are capable of offering several-fold increase in capacity, hold significant proise. Most previous wor on capacity analysis of ad-hoc networs is based on an iplicit assuption that each node has only one antenna. Core to the analysis therein is the characterization of a geoetric area, referred to as the exclusion region, which quantizes the aount of spatial resource occupied by a lin. When ultiple antennas are deployed at each node, however, ultiple lins can transit in the vicinity of each other siultaneously, as interference can now be suppressed by spatial signal processing. As such, a lin no longer exclusively occupies a geoetric area, aing the concept of exclusion region" not applicable any ore. This necessitates a revisit of the fundaental understanding of capacity of MIMO ad-hoc networs. In this paper, we investigate lin-layer throughput capacity of MIMO ad-hoc networs. In contrast to previous wor, the aount of spatial resource occupied by each lin is characterized by the actual interference it iposes on other lins, which is a function of the correlation between the spatial channels, the distance between lins, as well as the detection schee at the receivers. To calculate the lin-layer capacity, we first derive the probability distribution of post-detection SINR (signal to interference and noise ratio) at a receiver. The result is then used to calculate the nuber of active lins and the corresponding data rates that can be sustained within an area. Our analysis shows that there exists an optial active-lin density that axiizes the lin-layer throughput capacity. This will serve as a guideline for the design of ediu access protocols for MIMO ad-hoc networs. To the best of nowledge, this paper is the first attept to characterize the capacity of MIMO ad-hoc networs by considering the actual PHY-layer signal and interference odel. The results in this paper pave the way for further study on networ-layer transport capacity of ad-hoc networs with MIMO. ey words: Wireless ad hoc networs, MIMO, Networ capacity.

2 I. INTRODUCTION MIMO (Multi-Input Multi-Output) systes where ultiple antennas are deployed at both transitter and receiver, open up a new diension, i.e., space, to significantly iprove the spectral efficiency of wireless counication systes. Foschini and Telatar []-[] show that MIMO provides a linear growth of capacity with the nuber of antennas. Moreover, the extra degree of freedo, i.e., space, offered by ultiple antennas enables interference cancelation at receiving stations, which allows spectru to be reused ore aggressively [3]-[7]. On the other hand, MANET (obile ad hoc networ) is liely to play a ajor role in next-generation hoe networs and hot spots, thans to its siplicity, cost effectiveness, and siple reconfiguration [8]. One of the ajor challenges faced by MANET is the increasing deand for data-rate-intensive applications siilar to those in its wired counterpart. Deploying ultiple antennas at each node is a proising solution to iprove networ capacity and lin reliability required by these applications. To fully exploit the benefits of MIMO in ad hoc networs, it is essential to have a thorough understanding of the fundaental ipact of the use of MIMO on overall networ perforance. Most previous wor on capacity analysis of ad hoc networs is based on an iplicit assuption that only one antenna is used at each node [9]. Under this assuption, an active lin exclusively occupies a geoetric area, referred to as the exclusion region, to avoid collisions with other lins. The larger the exclusion region, the ore spatial resource a lin occupies, and the capacity of a networ is essentially deterined by the aount of spatial resource occupied by underlying lins. When ultiple antennas are deployed at each node, however, an active lin no longer exclusively occupies a spatial area, aing the original definition of exclusion region" not applicable any ore. In fact, ultiple adjacent lins can transit at the sae tie, as long as the utual interference can be suppressed by spatial signal processing. As such, previous wor on capacity analysis is not directly applicable to ad hoc networs with MIMO lins. The ipact of ultiple antennas on networ capacity was previously studied under different contexts. In [3]-[4], Zhang and Liew derived the upper and lower bounds on networ capacity when directional antennas with realizable generic patterns are used. It is well nown that directional antennas do not wor well in indoor and urban environents where a large nuber of local scatterers cause severe ultipath effects and large angular spread. Unfortunately, ost application scenarios of ad hoc networs perceive rich-scattering channels. As a result, the previous analysis based on directional antennas does not apply to general MIMO ad hoc networs. In [5], Chen et al studied the ergodic single-hop capacity of ad-hoc networs when single-user detection is eployed at each receiver. Though siple, single-user detectors

3 do not exploit the interference cancelation capability of MIMO. As a result, the capacity calculated in [5] is far below the actual achievable capacity of MIMO ad hoc networs. In this paper, we investigate lin-layer throughput capacity, defined as the total data rate that can be successfully delivered through all single-hop lins within a unit area, of MIMO ad-hoc networs. In particular, MMSE (iniu ean square error), the optial linear ultiuser detection, is assued to be deployed at receiving nodes, as it has the highest interference suppression capability aong all linear detection schees including ZF (zero forcing) and single user detection [5]. In contrast to previous wor, we characterize the aount of spatial resource occupied by each lin by the actual interference active lins ipose on each other, taing into consideration the actual behavior of ultipath fading and MIMO systes. One of the ey challenges in this wor is the characterization of the distribution of post-detection SINR (signal to interference and noise ratio) when transitters are randoly located. SINR distribution of MMSE detectors was previously studied in [6]-[8], [9]. In [7] and [8], the authors proved asyptotic Norality of post-detection SINR for equal and non-equal interference power, respectively. In [9], Li et al iproved the accuracy by odeling SINR using a Gaa or a generalized Gaa distribution. All these papers assued that the signal strength of interfering data streas as detected at the receiver is deterinistic and nown. In ad-hoc networs, however, active nodes are randoly located, and hence the received interference power is also rando. Moreover, previous wor often assued that the nuber of interferers is saller than or coparable to the nuber of receive antennas. While the assuption is reasonable for traditional cellular networs, it is not true in ad hoc networs where the nuber of siultaneously transitting stations could be uch larger than the nuber of antennas at a receiving node. The ain contributions of this paper can be suarized as follows. We derive a closed-for expression for the distribution of received SINR of an active lin with axiu ratio transission and a ultiuser MMSE receiver. In contrast to [6]-[8], [9], signal power, interference power, and the nuber of interferers are all rando variables due to the randoness in the location and activity of transitting nodes. The analytical results are validated by nuerical siulations. Based on the SINR distribution, we analyze lin-layer throughput capacity, calculated as the su data rate that can be delivered by all lins within a unit area. In contrast to [5] where the data rate of each lin is represented by ergodic capacity, per-lin data rate is defined as Th =( Pout ) q in this paper, where q is the transission rate of each lin and Pout is the

4 probability of transission failure given q. This definition ore accurately reflects the characteristics of today's ad-hoc networs, where ultipath fading varies slowly copared with the transission tie of a pacet. Unlie the ean value analysis in [9], the PDF (probability density function) of SINR is needed in this paper to calculate, which aes the job here uch ore challenging. Our wor on lin-layer capacity paves the way for further study on networ-layer transport capacity of ad-hoc networs with MIMO lins. The analysis suggests that there exists an optial density of siultaneously transitting lins that axiizes the lin-layer capacity. Through nuerical study, we calculate the optial density under various scenarios. In real ipleentation, the optial active-lin density can be apped to an optial transission probability in the MAC (ediu access control) layer. This result serves as a guideline for the design of MAC protocols of next-generation ad-hoc networs with MIMO lins. Our analysis is based on the assuption that CSI (channel state inforation) fro all transitting nodes to the receiver of a tagged lin is available at the receiver. In practice, it is liely that a receiver only nows CSI fro a few neighboring lins. In this paper, we also study the lin-layer capacity and the corresponding optial lin density when only local CSI is available. The reainder of the paper is organized as follows. The syste and signal odels are presented in Section II. In Section III, we derive the distribution of post-detection SINR in ad hoc networs when MMSE receivers are deployed. The lin-layer throughput capacity is then given in Section IV. Nuerical exaples and discussions are presented in Section V, where we also study the ipact of partial CSI on the capacity and optial lin density. Finally, the paper is concluded in Section VI. P out II. SYSTEM MODEL We first describe the notation used in this paper for readers' convenience. Throughout the paper, scalars are given by noral letters, vectors by boldface lower case letters, and atrices by boldface upper case letters. Besides, the following notations are used. T X : Transpose operation X : Heritian transpose [ X ] ij : (, i j) th eleent of Tr( X ) : trace of atrix X X

5 E( i ): Expectation Var( i ) : Variance A. Syste Description Consider an ad hoc networ as deonstrated in Fig., where obile nodes are uniforly distributed within an area. Each node is equipped with antennas. For siplicity, we ignore the edge effects and assue that each lin has the sae statistical characteristics. Without loss of generality, let Lin be the tagged lin. At a given tie, there are other lins sending data at the sae tie as Lin, resulting in co-channel interference. As a result, the data received by the tagged lin, given as follows, is a superposition of desired signal, interference, and noise. y = αphx + αphx + n () In the above, H is an channel atrix, representing the channel fading fro the transitter of the = th lin to the receiver of the tagged lin. Assuing a rich scattering environent and quasi-static Rayleigh flat fading channels, we can odel the eleents of as i.i.d. coplex Gaussian rando variables. Liewise, x denotes the transit signal vector of Lin ; p the transit power of Lin ; α the path loss fro the transitter of Lin to the receiver of the tagged lin; and n the AWGN (additive white Gaussian noise) with zero ean and unit variance. Note that the nuber of interferers,, is a rando variable depending on the transission probability of lins. Since the neighborhood observed by each lin is statistically identical, we assue that the interferers are randoly located within a disc of radius R centered at the tagged receiver, where R is the largest distance at which an interferer can cause non-negligible interference to the H receiver. Furtherore, let ε denote the iniu separation between interferers and the tagged receiver. Assue that ε is sall enough so that it does not affect the unifor distribution of nodes. Thus, the probability density function of the distance between a node and the tagged receiver is given by x fd ( x)= R ε Let c be the distance between the th transitter to the tagged receiver and θ be the path loss exponent. In particular, c is the length of the tagged lin. We can calculate the received power fro () the th interferer as

6 whose PDF is When θ =4, c θ αp =( ) α p (3) c / θ ( αp) c c θ c θ fα p ( x)= ( ) α ( )/ p x ( ) αp. (4) θ+ θ θ( R ε ) x R ε α pc c c fα p ( x)= ( ) p ( ) ( R ε ) x R ε 4 4 α 3/ x αp B. Maxiu Ratio Transission and MMSE Reception It was proved in [9] that SVD (singular value decoposition) based space-tie vector coding allows the collection of signal power in space and it is a theoretical eans to achieve high capacity for MIMO systes. By SVD, H can be decoposed into (5) H r λ, ju, jv, j j= =, (6) λ λ λ are the eigenvalues, and are the left singular vector and right where,,, r u, j singular vector, respectively, and r is the ran of H. Note that the left and right singular vectors have the sae distribution as noralized coplex Gaussian rando vectors []-[]. Liewise, the distribution of the square of the largest singular value cobination of eleentary Gaa densities: v, j λ is given by [7] as a finite linear, nn l+ l nx n x e f ( x)= g x>, (7) l! λ nl,, n= l= where gnl, are coputed and listed in [7] for ost antenna configurations of interest. The τ th oent of λ, is n n gnl, l+ τ n= l= nl! τ ( τ )! E[ ( λ,] ) =. (8) In an interference-liited environent such as ad hoc networs, an active lin should transit only one data strea at a tie to optiize the syste perforance []-[3]. In this case, the single data strea should be transitted on the largest singular ode of the channel for SNR (signal to noise ratio) axiization. Such schee, nown as MRT (axiu ratio transission), configures the transit

7 antenna weight using the right singular vector corresponding to the doinant singular value. For exaple, the transit antenna weight of Lin is. Siilarly, the transit beaforing vector of Lin, denoted by t, is the doinant singular vector of the channel atrix between its own transitter-receiver pair. Therefore, the received signal in () becoes v, y = αphv,b + αpht b + n = = α p λ u b + α p hˆ b + n (9),, = where y is a vector with the i th eleent being the received signal on the i th receive antenna. ˆ Denote Ht by h, whose eleents are still i.i.d. coplex Gaussian rando variables with zero ean and unit variance [], since t has unit nor and is independent of H. Define equivalent channel atrix and the transit power atrix as G as G =[ λ u, hˆ,, hˆ ],,, () P α p We can then rewrite (9) into a atrix for as α p =. / α p () y = GP b + n, () where b is [ b, b,, b ] T. Upon receiving the signal, the tagged receiver attepts to obtain an estiate of b fro the received signal y. Being the optial linear detector, MMSE detector iniizes the ean square error between b and its estiate. Specifically, the decision statistics b is obtained by linearly cobining the received signal vector as follows: To ipleent MRT, transitter-side CSI is needed. Transitter-side CSI is easily achievable in wireless networs with two-way counications. In case it is not available, rando antenna selection or space tie coding can be deployed instead of MRT. Our analysis can be easily extended to these cases with slight odification. Specifically, it is the distribution of λ that needs to be odified in the analysis.,

8 where ( ) b = (3) V y V I + G PG GP + and I is a ( + ) ( + ) Identity atrix. With MMSE, the = post-detection SINR of the tagged lin can be calculated as [5] + SINR = ( + + ) I G PG where denotes the (,) th eleent of a atrix. The distribution of SINR was previously studied [], in [6]-[8], [9]. However, their wor assues that interference power (i.e.,, p (4) α for >) is deterinistic and nown. This assuption, however, is not applicable to ad hoc networs where interfering lins are randoly located. In this paper, we focus on MMSE receivers, for it achieves the optial perforance in ters of BER (bit error rate) or SINR aong all linear detectors. Our conclusions, however, can easily be extended to other suboptial detectors such as ZF (zero forcing) and single-user detection. Note that eqns. (3) and (4) have assued that the tagged receiver has the nowledge of hˆ for all. Although not realistic, this assuption allows us to investigate the fundaental liit of wireless MIMO networ capacity without taing into account ipleentation details. This assuption will later be reoved in Section V in Fig. 8 and Fig. 9, where the receiver only nows the CSI fro its neighboring interfering nodes. III. SINR DISTRIBUTION OF MMSE In this section, we derive the distribution of SINR in ad hoc networs when MMSE detection is deployed. To this end, we first copute the ean and variance of SINR in subsections III.A to III.E. The PDF of SINR is then presented in III.F. A. Siplified For of SINR We define and then have / G = P G p u p hˆ p hˆ. (5) = α λ,,, α,, α

9 where G is G with first colun reoved. α p λ α p λ u G GG G u G G,,, = λ,, Before going further, we first describe the following lea. Lea : Write a atrix A into a, a, A =, a, A,, (6) (7) where a, is the (,) th eleent of A, a, is the first colun of A with the first eleent reoved and A is A with the first colun and row reoved. Then,, By using Lea, (4) can be siplified as Denoting the SVD of ( ) [ A ] = a a ( A ) a. (8),,,,, SINR = ( I + + G G ) G, ( ) = α p λ α p λ u G I + G G G u. (9) as,,,, G =, WDZ where the i th diagonal eleent of D is d, we then can derive the SINR as i ( ) ( ( + ) ) SINR = α p λ α p λ u WDZ I + Z D DZ Z D W u,,,, = α pλ,u,w I I D D W u, = α p λ u W( I + DD ) W u,,, () = α p λ u Bu,,,, () where B is defined as WI ( + DD) W. B. Conditional Mean of SINR It is easy to see that B is deterinistic function of G and P. Given a channel realization, the conditional ean of SINR given B is

10 E(SINR B) = α p E( λ )E( u Bu ).,,, () Liewise, () i () i () i ( j) E( u,bu,) = E u, u,bii + E u, u, Bij i= i j () i () i ( j) ( u, ) ii + (,, ) i= i j = E B E u u B ( i) ( j) = Tr( B ) + E( u, u, ) Bij (3) i j ij where () i B ij is the ( i, j) th eleent of B, and u, is the i th eleent of u,. Since u, is a noralized coplex Gaussian rando vector as entioned in Section II, it is not difficult to prove that ( i) ( j) (,, ) E u u = i j. (4) Hence, we have the conditional expectation of SINR given B as shown in (5). C. Conditional Second Moent of SINR () i 4 Denoting E( u ) ( i ) as a, and ( ( + D ) W ) E(SINR ) = pe(,)tr α λ B W I D = E( )Tr (( I + DD ) ) p, α λ = α E( ) p λ, i= + di We now derive the conditional second oent of SINR given B., derive (6) fro (3). ( u Bu ) (5) () i ( j) E( u u ) ( i j) as, respectively, we first,, () i 4 () i ( j) () i ( j),, u, Bii Bii + u, u, BijBij + u, u, BiiBjj i= i j i j E ( ) = E( ) E( ) E( ) + i j, i j, i i, j j ( ) ( ) ( ) ( ),,,, i j i j E( u u u u ) B B () i 4 () i ( j) () i ( j) u, BiiBii + u, u, BijBij + u, u, Bii Bjj i= i j i j = E( ) E( ) E( ) ii ii + BB + B B ii ii i= i= = a B B a Tr( ) a Tr ( )Tr( ) a B B a

11 ( ) where the second equality is due to the fact that BB + B B + a a ii i= = a Tr( ) Tr ( )Tr( ) ( ) ( ( i ) ( j ) ( j ) ( i ) ) B B jj (6) E u u u u = i j, i j, i i, j j (7),,,, () i Since is a noralized coplex Gaussian rando vector, the PDF of u is u, f x x x (8) () ( ) = ( )( ),. i u,, Therefore, we have and () i E( u, ) =, () i 4 E( u, ) =. + ( ) (9) (3) () i We can then derive the pdf of u on the condition of ( j u ) = y as,, () i ( j) u, u, 3 x y f ( x y)= x y. y y (3) Then, we have E( u u ) () i ( j),, as follows. y () i ( j),, ( j) () i ( j) u u,, u, E( u u ) = xyf ( x y) f ( y) dxdy Fro eqns. (6) and (3), we have 3 y x y = y( )( y) x d y y xdy = (3) ( + ) E ( ) = Tr( ) Tr ( )Tr( ) ( + ) We are now ready to derive the second oent of ( u,bu, ) ( B B + B B ) = ( ) + + i= + di i= + d i (33) SINR as

12 ( u Bu ) E(SINR B) = α p E( λ )E ( ) 4,,, D. Mean of SINR In order to copute the ean of 4 p λ, + ( + ) i= + di i= + di = α E( ). (34) SINR, we first introduce the ethod of asyptotic analysis of rando atrix [8]. Definition of η transfor: Given a nonnegative rando variable χ, the η transfor is defined as ηχ ( γ)=e + γχ (35) Theore [8]: Let H be an atrix whose entries are i.i.d. coplex Gaussian variables with variance. Let T be a Heritian nonnegative rando atrix, independent of H, whose epirical eigenvalue distribution converges alost surely to a nonrando liit. The epirical eigenvalue distribution of HTH converges alost surely, as, with distribution whose η -transfor satisfies β, to a η β =, η ( γη) T where for siplicity we have abbreviated η ( γ)= η. η () HTH HTH (36) and ηt ( ) stand for the η transfor of the eigenvalues of HTH and T, respectively. Theore can be applied to find the epirical eigenvalue distribution of G G, i.e., the epirical distribution of d i. Rewrite the atrix as G G = G P G. (37) where P is P with first colun reoved. Then, fro Theore, the epirical eigenvalue distribution of G G converges alost surely to a distribution whose η transfor satisfies

13 where η ( γ) G G =, η ( γη ( γ )) P G G η ( ) P γ is the η transfor of the eigenvalue of P. We now begin to derive distribution of (38) η ( ) P γ. Since P is a diagonal atrix, the epirical eigenvalue is the distribution of its diagonal eleents of P in (3), we derive the η transfor of P as α p. Given the distribution of α p η P ( γ)=e + γαp c α pγ c αpγ c αpγ = tan tan R ε ε R Substituting (39) to (38), we have (39) c α p γη ( γ ) c α p γη ( γ ) G G G G η ( γ)= ( ( tan ) G G R ( ε ) ε tan c αp γη ( γ) G G ( )). (4) R We are now ready to derive the ean of SINR as ( ) E(SINR ) = E E(SINR ) B B Note that which can be coputed by (4) as = αpe( λ,)e i= + d. (4) i E = η i= + di G G (), (4) c α p η () c α p η () η ()= ( ( R ( ε ) ε G G G G tan ) G G

14 At last, we have the ean of tan SINR as c α pη ( R G G () )). (43) E(SINR ) = E( )E αp λ, i= + di E. Variance of SINR We now begin to derive the various of SINR. = α E( λ ) η ( p ), G G Var(SINR ) = E (Var(SINR )) Var(E(SINR )) B B + B E(Var(SINR B)), (44) (45) where the approxiation is due to the fact that becoes very large. Var(E(SINR B)) Var(SINR ) E B (Var(SINR B)) ( ) =EB E(SINR B) E (SINR B ) converges to zero as the ran of B 4 p λ B, + ( + ) i= + di i= + di = α E E( ) E( ) λ, i= + di 4 () i E( λ,) E ( λ,) α p E i= + d i 4 = α p(e( λ,) E ( λ,))e + d i 4 di = α p(e( λ,) E ( λ,))e ( + di + di ) ( )( ) = α p E( λ ) E ( λ ) η() + η (), (46) 4,, where approxiation (i ) is due to the following inequality:

15 where the equality holds if all ds i + d + d i= i i= i are equal. F. Probability Density Function of SINR. (47) The close-for PDF of SINR is nown to be difficult to derive. Fortunately, it can be seen fro () that the SINR is a suation of any positive ters. Therefore, a Gaa distribution can e used to approxiate the SINR according to central liit theore for causal functions [] as follows. e f x x x b Γ ( a) xb / a SINR ( )=, >, a (48) where = E (SINR ) / Var(SINR ), b = Var(SINR ) / E(SINR ), and Γ ( a) is the gaa function. a Then, the CDF (cuulative distribution function) of SINR is F x f x dx t e dt Γ( a) x x/ a t SINR ( )= SINR ( ) =. b (49) In Fig. and Fig. 3, we respectively plot the CDF of SINR with and interfering nodes when there are 4 antennas at each station. Assue that the length of the tagged lin, c, is noralized to and the average received SNR α p N is equal to db. The interfering lins are uniforly distributed within a disc with R =3. The iniu separation between the tagged receiver and interferers is ε =.. Fro the figures, we can see that although the analytical results coe fro asyptotic analysis, they atch the siulation results very well even with a sall nuber of antennas. IV. LIN-LAYER THROUGHPUT CAPACITY In this section, we investigate the lin-layer throughput capacity of wireless ad hoc networs, which is defined as the total data rate that can be successfully delivered through all single-hop lins per unit area. Assue that an active lin transits at data rate q. The transission is successful only when SINR at the receiver side is higher than a threshold, SINR, which is a function of q. To be ore specific, the th relationship between q and SINR th is defined as q = log (+ SINR ). (5) th Denote by P out the probability of transission failure of a lin, which is calculated fro the CDF of

16 SINR derived in the last section. P out =Pr(SINR <SINR ) th = F SINR (SINR th ) (5) Therefore, the throughput of a counicating lin is given by Th =( P ) q. (5) If the radius of the networ R is very large so that the edge effect is negligible, we can assue that each lin experiences hoogeneous channel and interference conditions. As a result, the throughput is the sae for all lins in the networ. When there are + active lins (one tagged lin and interfering lins) siultaneously transitting in the networ, we can evaluate the capacity of the networ as the suation of the throughput of all the lins. C ( + )=( + )( P ) q. (53) In wireless ad hoc networs, the nuber of active lins varies fro tie to tie due to the rando-access nature of lins. Assue that there are in total L lins per unit area and each lin transits with a probability out p t. Then, the average nuber of active lins is equal to out = ρπ R (54) where ρ = Lpt is the average nuber of active lins per unit area. When L is large, the nuber of active lins follows Poisson distribution. The probability of having given by + active lins in the networ is + e Pr( + ) =. ( + )! Finally, we have the lin-layer throughput capacity of the networ as = (55) C = C ( + )Pr( +). (56) π R V. SIMULATION AND NUMERICAL RESULTS As shown in the last section, lin-layer capacity of wireless networs heavily depends on the nuber of siultaneously active lins within a unit area. This section investigates the ipact of the density of active lins on the capacity through nuerical results. Moreover, the effect of incoplete channel state inforation is studied.

17 Siilar to Fig. and Fig. 3, we noralize the length of the tagged lin to and assue that the average SNR at the tagged receiver is db. The SNR threshold for the counication pair is db. For siplicity, assue that all transitters have the sae transission power. Around the tagged receiver, interfering lins are uniforly distributed in the space. The iniu separation between the tagged receiver and interferers is SINR th ε =.. Given an average density of active lins ρ, the nuber of active lins is randoly generated according to Poisson distribution in (56). We first validate the analytical results derived in the previous sections by coparing the with siulation results. In Fig. 4 and Fig. 5, the ean and second oent of SINR are plotted against average lin density when there are, 4, and 6 antennas at each node, respectively. It is not surprising that both ean and second oent decrease as the lin density increases. The figures show that our analytical results atch the siulations well. In Fig. 6, lin-layer throughput capacity defined in (56) is plotted against the density of active lins, ρ. Fro the figure, we can see that when the active-lin density is low, capacity increases with the nuber of active lins, as the interference can be well handled by ultiple antennas. However, when ρ exceeds a certain level, co-channel interference becoes so severe that lin-layer capacity starts to decrease. As expected, the optial density of active lins that axiizes lin-layer capacity increases with the nuber of antennas, for ore co-channel interference can be tolerated when there are a larger nuber of antennas at each station. Moreover, lin-layer capacity increases as the nuber of antennas increases. For exaple, the axial capacity for networs with, 4, and 6 antennas is about.5,.78, and.53 bps/hz/, respectively. A close observation of the figure reveals an interesting fact: The axial capacity increases faster than the nuber of antennas. In particular, the noralized axial capacity (noralized by the nuber of antennas) is equal to.5,.95, and.55 bps/hz/, respectively. This provides a strong incentive in deploying ultiple antennas in future wireless networs. In Fig. 7, the optial active-lin density, denoted by ρ, is plotted as a function of the nuber of antennas at each station. To validate the analysis, siulation results are also plotted in the figure. The figure shows that our analysis can accurately predict the optial density of active lins in wireless networs with MIMO lins. In wireless networs, active-lin density is directly related to the transission probability of existing lins, as shown in the last section. In traditional wireless networs, transission probability is usually

18 selected according to the networ contention level. In this paper, we argue that the optial transission probability should be deterined by the characteristics of PHY-layer co-channel interference as well as the interference cancelation capability at each receiver. As Fig. 7 shows, the optial transission probability can be accurately calculated through our analysis. The observations in Fig. 6 and Fig. 7 serves as a guideline in designing the transission probability in wireless networs with MIMO lins. So far, we have assued CSI at each receiving node. That is, the receiver nows the channel atrices hˆ (see eqn. ()) fro all interfering nodes. In practice, however, it is difficult for a receiver to onitor the CSI on all lins. Hence, it would be interesting to investigate networ capacity in a ore practical scenario where only the CSI fro neighboring interferers is available. In Fig. 8, we assue that a receiving node only estiates the channel fro interferers that are located within distance fro the receiver. Interference fro other interferers is treated as noise. By restricting the channel-onitoring range, the coputational coplexity due to channel estiation and MMSE detection can be significantly reduced. The figure shows that the axiu throughput is slightly reduced fro.8bps / to.7 bps / when the channel-onitoring range is restricted to. Intuitively, the larger the channel-estiation range, the higher the capacity. In real ipleentation, one can trade off between coputational coplexity and achievable capacity. In this paper, we have assued that the optial linear detector, MMSE, is deployed at each receiver. In real systes, suboptial detectors such as zero-forcing (ZF) detector are also widely used due to the easy ipleentation. In the case of ZF, V in eqn. (3) satisfies + V = G, (57) + where G denotes the psudo inverse of atrix G. For coparison purpose, we investigate the lin-layer capacity when ZF detector is deployed in Fig. 9. Note that the nuber of interferences a ZF detector can handle is no ore than, where is the nuber of antennas. In the figure, we assue that the strongest interferences are canceled by the ZF detector. Siilar to the case of MMSE detector, the figure shows that there exists an optial active-lin density when ZF detector is deployed. However, the axiu capacity is reduced by ore than 3% copared with the MMSE detector. Due to the lower interference cancelation capability of ZF copared with MMSE, the optial lin density is also reduced. VI. CONCLUSION

19 In this paper, we have investigated the lin-layer throughput capacity of wireless ad hoc networs when ultiple antennas are deployed at each node. In contrast to previous wor where networ capacity is calculated as if each lin exclusively occupies a geoetric area, we have argued that it is indeed the characteristics of PHY-layer interference and the interference cancelation capability of receivers that deterines the networ capacity. This is especially true in networs with MIMO lins, where lins can transit siultaneously in the vicinity of each other, with co-channel interference being reduced via space-doain signal processing. One ey contribution of this wor is the characterization of distribution of post-detection SINR of MMSE receivers when the nuber and locations of interferers are rando. The PHY-layer SINR is then translated into MAC-layer throughput capacity in wireless ad hoc networs. We have shown that there exists an optial transission probability that axiizes networ throughput capacity. In particular, the optial transission probability is deterined by the nuber of antennas as well as the ultiuser detection schee deployed at each node. This observation serves as a guideline for the design of MAC protocols in future wireless ad-hoc networs with MIMO lins. References: [] G. J. Foschini and M. J. Gans, ``On liits of wireless counications in a fding environent when using ultiple antennas," Wireless Personal Coun.: luwer Acadeic Press, no. 6, pp , 998. [] E. Telatar, ``Capacity of ulti-antenna Gaussian channels," Eur. Trans. Teleco ETT, vol.,no. 6, pp , Nov [3] Q. H. Spencer, C. B. Peel, A. L. Swindlehurst, and M. Haardt, ``An introduction to the ulti-user MIMO downlin," IEEE Coun. Mag., pp. 6-67, Oct. 4. [4] G. Caire and S. Shaai, ``On the achievable throughput of a ultiantenna Gaussian broadcast channel,'' IEEE Trans. Inf. Theory, vol.49, pp , July 3. [5] P. Viswanath and D. Tse, ``Su capacity of the vector Gaussian broadcast channel and uplin-downlin duality," IEEE Trans. Inf. theory, vol. 49, pp. 9-9, Aug. 3. [6] W. Rhee and J. M. Cioffi, ``On the capacity of ultiuser wireless channels with ultiple antennas," IEEE Trans. Inf. Theory, vol. 49, pp , Oct. 3. [7] W. Yu, W. Rhee, S. Boyd, and J. M. Cioffi, ``Iterative water filling for Gaussian vector ultiple-access channels," IEEE Trans. Inf. Theory, vol. 5, no., pp. 45-5, Jan. 4. [8] G. Anastasi, M. Conti, and E. Gregori, IEEE 8. Ad Hoc Networs: Protocols, Perforance

20 and Open Issues, New Yor: IEEE Press?CWiley, 4. [9] P. Gupta and P. R. uar, ``The capacity of wireless networs," IEEE Trans. Inf. Theory, vol. 46, pp , Mar.. [] S. Toupis and A. J. Goldsith, ``Capacity regions for wireless ad hoc networs," IEEE Trans. Wireless Coun., vol., pp , Jul. 3. [] R. S. Blu, ``MIMO capacity with interfrence," IEEE J. Selected Area Coun., vol., no. 5, pp , June 3. [] R. S. Blu, ``On the capacity of cellular systes with MIMO," IEEE Co. Lett., vol. 6, no. 6, pp. 4-44, June,. [3] W. Choi and J.G. Andrews, ``On spatial ultiplexing in cellular MIMO-CDMA systes with linear receivers," in Proc. IEEE Int. Conf. Coun., vol. 4, pp. 77-8, May, 5. [4] Y. Togoz and B. D. Rao, ``Perforance analysis of axiu ratio transission based ulti-celluar MIMO systes," IEEE Trans. On Wireless Co., vol. 5, no., pp , Jan. 6. [5] B. Chen and M. J. Gans, ``MIMO counications in Ad Hoc networs," IEEE Trans. on Sig. Processing, vol. 54, no. 7, pp , June 6. [6] M. Zorzi, J. Zeidler, A. Anderson, B. Rao, J. Proais, A. L. Swindlehurst and M. Jensen, ``Cross-layer issues in MAC protocol design for MIMO ad hoc networs," IEEE Wireless Co., vol. 3, no. 4, pp.6-76, Aug. 6. [7] P. A. Dighe, R.. Malli, and S. S. Jauar, ``Analysis of transit receive diversity in Rayleigh fading," IEEE Trans. Coun., vol. 5, no. 4, pp , Apr. 3. [8] A. M. Tulino and S. Verdu, Rando Matrix Theory and Wireless Counications, Delft : Now, 4. [9] P. Li, D. Paul, R. Narasihan, and J. Cioffi, ``On the distribution of SINR for the MMSE MIMO receiver and perforance analysis," IEEE Trans. on Inf. Theory, Vol. 5, No., pp. 7-86, Jan. 6. [] R. J. Muirhead, Aspects of Multivariate Statistical Theory. Wiley, 98. [] N. R. Goodan, ``Statistical analysis based on a certain ultivariate coplex gaussian distribution (an introduction)," Annals of Matheatical Statistics, vol. 34, pp. 5-77, 963. [] A. Papoulis, The Fourier Integral and its Applications. New Yor: McGraw-Hill, 96. [3] J. Zhang and S. C. Liew, ``Capacity iproveent of wireless ad hoc networs with directional antennae," ACM MobiCo'5, Aug. 5. [4] J. Zhang and S. C. Liew, ``Capacity iproveent of wireless ad hoc networs with directional

21 antennae," IEEE VTC'6, vol., pp. 9-95, 6. [5] S. Verdu, Multiuser Detection, Cabridge University Press, Cabridge, U, 998. [6] H. V. Poor and S. Verdu, ``Probability of error in MMSE ultiuser detection," IEEE Trans. Inf. Theory, vol. 43, no. 3, pp , May 997. [7] D. N. C. Tse and O. Zeitouni, ``Linear ultiuser receivers in rando environents," IEEE Trans. Inf. Theory, vol. 46, no., pp. 7-88, Jan.. [8] D. Guo, S. Verdu, and L.. Rasussen, ``Asyptotic norality of linear ultiuser receiver outputs," IEEE Trans. Inf. Theory, vol. 48, no., pp , Dec.. [9] G. G. Raleigh and J. M. Cioffi, "Spatio-teporal coding for wireless counications," IEEE Trans. Coun., vol. 46, pp , March 998. Fig. : Syste odel

22 Fig. : CDF of SINR when there are interfering nodes Fig. 3: CDF of SINR when there are interfering nodes

23 Fig. 4: Mean of SINR Fig. 5: Second oent of SINR

24 Fig. 6: Throughput capacity vs. the density of interfering nodes Fig. 7: Optial active-lin density

25 Fig. 8: Throughput capacity with incoplete channel state inforation Fig. 9: Throughput capacity of ZF receivers

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