Spatial Multiplexing with MMSE Receivers in Ad Hoc Networks

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2011 proceedings

Spatial Multiplexing with MMSE Receivers in Ad Hoc Networks Raymond H. Y. Louie

Matthew R. McKay

Nihar Jindal

Iain B. Collings

University of Sydney Hong Kong University of University of Minnesota CSIRO Australia Science and Technology, Hong Kong USA Australia [email protected] [email protected] [email protected] [email protected]

Abstract—The performance of spatial multiplexing systems with linear minimum-mean-squared-error receivers is investigated in ad hoc networks. We present new exact closed-form expressions for the outage probability and transmission capacity. These expressions reveal that from a transmission capacity perspective, single-stream transmission is preferable over multistream transmission.

I. I NTRODUCTION Multiple antennas can offer significant performance improvements in wireless communication systems by providing higher data rates and more reliable links. A practical method which can achieve high data rates is to employ spatial multiplexing transmission in conjunction with low complexity linear receivers, such as the minimum-mean-squared-error (MMSE) receiver. The MMSE receiver is particularly important as it uses its receive degrees of freedom (DOF) to optimally trade off strengthening the energy of the desired signal of interest and reducing unwanted interference, such that the signal-to-interference-and-noise ratio (SINR) is maximized. Although MMSE receivers have been well investigated for simple network scenarios [1], a thorough characterization of their performance in more complicated network scenarios, such as ad hoc networks, is currently lacking. In this paper, we investigate spatial multiplexing systems with MMSE receivers in ad hoc networks. In particular, each transmitting node sends separate data streams to their corresponding receiver. At the receiver, a bank of linear MMSE filters are used to independently decode, without successive interference cancelation (SIC), each transmitted data stream. To facilitate this decoding, the transmitter-receiver channel and the interference covariance matrix are assumed to be known at each receiver. However, we note that no channel information is required at the transmitters. We consider a scenario where the transmitting nodes are spatially distributed according to a homogeneous Poisson point process (PPP) on an infinite 2-D plane. The transmission capacity will be investigated, which measures the maximum number of successful transmissions per unit area, assuming transmission at a fixed data rate, such that a target outage probability is attained for each data stream. We present two key contributions in this paper. First, we derive exact closed-form expressions for the outage probability and transmission capacity of spatial multiplexing systems with

a bank of linear MMSE receivers. Second, using these expressions, we show that from a transmission capacity perspective, single-stream transmission is preferable when the optimal linear MMSE processing strategy is employed. Prior work on single-stream transmission with multiple receive antennas in ad hoc networks and Poisson distributed transmitting nodes include [2–6], where spectral efficiency and transmission capacity scaling laws were presented for different receiver structures. In [2], receive antennas are used for spatial diversity to increase the desired signal power, while in [3], receive antennas are used to cancel interference from the strongest interferer nodes. In [4–6], for both sub-optimal and MMSE linear receivers, the transmission capacity was shown to scale linearly with the number of receive antennas. In this paper, we extend these prior contributions to derive new outage probability and transmission capacity scaling laws for arbitrary number of data streams using MMSE receivers. Multi-stream transmission with multiple receive antennas have been considered in [7–10]. In [7], MMSE receivers were used to reduce interference from all interfering nodes. However, analysis was limited to a per-link performance measure and the large antenna regime. In [8, 9], receive antennas were used to cancel interference from the corresponding transmitter, but not the interferers. For these papers, the transmission capacity as → 0 was shown to behave as1 o(). In [10], suboptimal partial zero forcing (PZF) receivers were considered, which were used to cancel interference from the strongest L−1 interferers. For these PZF receivers, thetransmission capacity 1 as → 0 was shown to behave as o L , thus indicating that for low outage probability operating values, PZF receivers achieves a higher transmission capacity than the receivers in [8, 9]. This transmission capacity result was used to show that single-stream transmission is preferable over multi-stream transmission. Our work differs from [10] by showing singlestream optimality for the optimal MMSE receiver. We will investigate the performance advantages of the MMSE receiver over the sub-optimal PZF receiver in this paper. II. S YSTEM M ODEL We consider an ad hoc network comprising of transmitterreceiver pairs, where each transmitter communicates to its 1 f (x)

= o(g(x)) means limx→0

978-1-61284-231-8/11/$26.00 ©2011 IEEE

f (x) g(x)

= 0.


corresponding receiver in a point-to-point manner, treating all other transmissions as interference. The transmitting nodes are distributed spatially according to a homogeneous PPP of intensity λ (transmitting nodes per unit area) in R2 , and each receiving node is randomly placed at a distance d0 away from its corresponding transmitter. In this paper, we investigate network-wide performance. To characterize this performance, it is sufficient to focus on a typical transmitter-receiver pair, denoted by index 0, with the typical receiver located at the origin. The transmitting nodes, with the exception of the typical transmitter, constitute a marked PPP, which by Slivnyak’s theorem has the same distribution as the original PPP [11] (i.e., removing the typical transmitter from the transmit process has no effect). This is denoted by Φ = {(D , H ), ∈ N}, where D and H model the location and channel matrix respectively of the th transmitting node with respect to (w.r.t.) the typical receiver. The transmitted signals are attenuated by a factor 1/rα with distance r, where α > 2 is the path loss exponent. We consider a spatial multiplexing system where each transmitting node sends Nt independent data streams through Nt different antennas to its corresponding receiver, which is equipped with Nr antennas. Focusing on the kth stream, the received Nr × 1 signal vector at the typical receiver can be written as (a)

y0,k =

(b)

Nt

1 1 h0,k x0,k + h0,q x0,q α dα d 0 0 q=1,q=k Nt 1 + h,q x,q +n0,k |D |α q=1 D ∈Φ

(1)

(c)

where x,q is the symbol sent from the qth transmit antenna of the th transmitting node satisfying E[|x,q |2 ] = P , h,q d is the qth column of2 H ∼ CN Nr ,Nt (0Nr ×Nt , INr ) and d n0,k ∼ CN Nr ,1 (0Nr ×1 , N0 INr ) is the complex additive white Gaussian noise vector. We see in (1) that the received vector includes: (a) the desired data to be decoded, (b) the self interference from the typical transmitter and (c) the interference from the other transmitting nodes. To obtain an estimate for x0,k , we consider the use of MMSE linear receivers. The data estimate is thus given by x ˆ0,k = h†0,k R−1 0,k y0,k , from which the SINR can be written as SINR0,k =

γ † h0,k R−1 0,k h0,k dα 0

(2)

III. O UTAGE P ROBABILITY We consider the per-stream outage probability, defined for the kth stream as the probability that the mutual information for the kth stream lies below the data rate threshold Rk . At the receiver, the MMSE filter outputs are decoded independently. We assume the data rate thresholds for all streams are the same and equal to R. The outage probability for each stream can thus be written as FZ (z, λ) = Pr (SINR ≤ z)

(4)

where z = 2 − 1 is the SINR threshold. Note that we have dropped the subscript k and 0 from the SINR term as the perstream outage probability is the same for each stream at each receiving node. Before deriving an expression for the outage probability, we first introduce some notation and concepts from number theory. The integer partitions of a positive integer k are defined as the different ways of writing k as a sum of positive integers [12]. For example, the integer partitions of 4 are: i) 4, ii) 3+1, iii) 2+2, iv) 2+1+1 and v) 1+1+1+1. We denote hi,j,k as the ith summand of the jth integer partition of k, |hj,k | as the number of summands in the jth integer partition of k and |hk | as the number of integer partitions of k. For example, when k = 4, we have h2,3,4 = 2, h2,4,4 = 1, |h3,4 | = 2 and |h4 | = 5. It is also convenient to introduce the concept of nonrepeatable integer partitions, defined as the integer partitions without any repeated summands. For example, the nonrepeatable partitions of 4 are given by i) 4, ii) 3+1, iii) 2, iv) 2+1 and v) 1. We denote gi,j,k as the number of times the ith summand of the jth non-repeatable integer partition of k is repeated in the jth integer partition of k, and |gj,k | as the number of summands in the jth non-repeatable partition of k. For example, when k = 4, we have g1,3,4 = 2, g1,5,4 = 4 and |g3,4 | = 1. With these definitions, we present the following theorem for the outage probability3 . Theorem 1: The per-stream outage probability of spatial multiplexing systems with MMSE receivers is given by ⎛ α υ−1 ⎞ zdα zd0 N r −p r −1 − γ0 −ΘNt λ N γ e e ⎟ ⎜ FZ (z, λ) = 1 − ⎠× ⎝ N −1 t (1 + z) (υ − 1)! p=0 υ=1 R

|hp−q | Nt − 1 q z Ξj,p−q (−ΘNt λ)|hj,p−q | q j=1

min(p,Nt −1)

q=0

where

(5)

where

R0,k =

γ dα 0

Nt q=1,q=k

h0,q h†0,q + γ

|D |−α H H† + INr

D ∈Φ

(3) 2 The

and γ = NP0 is the transmit SNR. We assume that each receiving node has knowledge of the corresponding transmitter channel H0 and the interference (plus noise) covariance matrix R0,k . The practicalities of this assumption are discussed in [5].

d

notation X ∼ Y means that X is distributed as Y .

|hj,w | hi,j,w i=1

Ξj,w =

2 (Nt −k+1)(k−1− α )

k=1

|gj,w | υ=1

2 k(Nt + α −k)

(6)

gυ,j,w !

3 Note that the outage probability for the specific case where N = 1 was t recently derived independently in [6].


N =1 t

−1

−1

10

Transmission capacity, c(ε)

t

Outage probability, F(z,λ/N )

10

−2

10

N =2 t

−2

10

N =3 t

−3

10

N =4 t

N =1, N =2, N =3, N =4 t

t

t

t

−3

10

Monte−Carlo Analytical

Numerical Analytical

−4

10

−1

−4

10

10 Density, λ

−3

10

−2

10 Outage probability, ε

−1

10

Fig. 1. Outage probability vs. density for spatial multiplexing systems with MMSE receivers, and with Nr = 4, α = 4.6, z = 0 dB, γ = 20 dB and d0 = 1.

Fig. 2. Transmission capacity vs. outage probability for spatial multiplexing systems with MMSE receivers, and with Nr = 4, α = 4.5, z = 10 dB, and d0 = 1.

and

Theorem 2: The transmission capacity of spatial multiplexing systems with MMSE receivers as → FSU (z) is given by 1 1 Nt R SU − F (10) (z) + o ( − F (z)) c() = SU ΘNt Ω r where = N Nt , Ψp, is the set of all integer partitions of p with summands, · denotes the floor function and Nt −1 Nt −1 q Nr −1−q z (−1) q=0 1 p= j∈Ψp, Ξj,p q Ω= − . N −1 t ! (1 + z) (11)

ΘNt

2 2 2 α π (dα 0 z) Γ Nt + α Γ 1 − α . = Γ (Nt )

(7)

Proof: See the Appendix. For a fixed number of data streams per unit area, we can determine the optimal number of data streams used for transmis sion by considering the outage probability FZ z, Nλt . Fig. 1 plots this outage probability vs. density for different numbers of transmit antennas. The ‘Analytical’ curves are based on (5), and clearly match the ‘Monte-Carlo’ simulated curves. We see that single-stream transmission always performs better than multi-stream transmission. In the next section, we analytically prove this is true using the transmission capacity framework for low outage probability operating values. IV. T RANSMISSION C APACITY We consider the transmission capacity, a measure of the number of successful transmissions per unit area, defined as Δ

c() = Nt λ()(1 − )R

(8)

where is the desired outage probability operating value and λ() is the contention density, defined as the inverse of = FZ (z, λ) taken w.r.t. λ. Before presenting the transmission capacity, it is convenient to first present the outage probability with no multi-node interference, computed as [1] F SU (z) = 1−

zdα − γ0

e (1 + z)Nt −1

N r −1 p=0

(9) Nt −1 ⎞ p−υ υ ⎠ . zp ⎝ γ υ! υ=max(0,p−Nt +1) dα ⎛

p

0

The transmission capacity is given in the following theorem.

Proof: The result is proven by taking a first order expansion of the outage probability in (5) around λ = 0, followed by substituting the resultant expression into (8). The full proof is omitted due to space limitations. By observing that the exponent of in (10) is a decreasing function of the number of data streams, we see that for low outage probability operating values, the transmission capacity is maximized when only a single data stream is used for transmission. This indicates that to maintain a desired outage for a fixed number of data streams per unit area Nt λ(), it is preferable to have a high density of single-stream transmissions rather than a low density of multi-stream transmissions when using the optimal MMSE receiver in ad hoc networks. Figs. 2 and 3 plot the transmission capacity vs. outage probability and path loss exponent respectively when thermal noise is neglected, i.e., γ → ∞. We observe in both figures that the transmission capacity is a decreasing function of the number of transmit antennas for all outage probabilities and path loss exponents. In Fig. 2, the ‘Analytical’ curves are plotted using (10), and closely match the ‘Numerical’ curves for outage probabilities as high as = 0.1, which were obtained by numerically taking the inverse of FZ (z, λ) w.r.t. λ, and substituting the resulting expression into (8). Fig.


−1

10

1 0.9 λ=0.1 Nt=1

0.8 t

Transmission capacity, c(ε)

10

Outage probability, F(z,λ/N )

−2

N =2 t

−3

10

Nt=3 −4

10

Nt=4 −5

10

0.7 0.6 λ=0.05

0.5 0.4 0.3 0.2 0.1

−6

10

2.2

2.4

2.6

2.8 3 3.2 3.4 Path loss exponent, α

3.6

3.8

0

4

Fig. 3. Transmission capacity vs. path loss exponent for spatial multiplexing systems with MMSE receivers, and with Nr = 4, z = 15 dB, d0 = 1 and = 0.001.

PZF [10] MMSE 0

10

20 30 SINR threshold, z dB

40

50

Fig. 4. Outage probability vs. SINR threshold, with Nt = 1, Nr = 4, α = 3 and d0 = 1.

0

10

V. C OMPARISON WITH THE PZF SCHEME IN [10] In this section, we investigate key performance differences between the MMSE receiver we consider in this paper, and the (sub-optimal) PZF receiver considered in [10]. In particular, the PZF receiver cancels a fixed and finite subset of “strong” interferers, whereas the MMSE receiver aims to reduce interference from all interferers when maximizing the SINR. It is thus clear that the optimal MMSE receiver will have better performance, regardless of the operating SNR. As there are an infinite number of interferers4 , the PZF receiver is incapable of completely canceling the interference from all interferers. The performance advantages of MMSE over PZF receivers can be seen in Figs. 4 and 5, which respectively plot the outage probability vs. SINR threshold z, and the transmission capacity vs. number of antennas Nr . For both figures, singlestream transmission is considered, i.e., Nt = 1, and thermal noise is neglected5 . All curves are generated by Monte-Carlo 4 Mathematically, this implies that the interference covariance matrix is full rank 5 Note that [10] only presented results with thermal noise neglected. When thermal noise is included, the MMSE receiver is expected to perform even better than the PZF receiver, because the MMSE receiver considers the effects of both interference and thermal noise when maximizing the SINR, in contrast to the PZF receiver which only considers the interference.

MMSE Transmission capacity, c(ε)

2 indicates that the optimality of single-stream transmission is not just applicable to small outage probability operating values, but the whole range of outage probabilities considered, i.e. 0.0001 ≤ ≤ 0.8. Fig. 3 indicates that with all other parameters kept fixed, the transmission capacity increases with the path loss exponent. This is interesting since whilst the quality of the desired transmit-receive links is reduced by a higher path-loss factor, the effect of interference is also reduced. Our results indicate that this interference reduction has a more significant effect on the overall network performance.

−1

10

PZF [10]

−2

10

−3

10

2

2.5

3

3.5 4 4.5 5 Number of receive antennas, Nr

5.5

6

Fig. 5. Transmission capacity vs. number of receive antennas, with Nt = 1, = 0.01, α = 3, z = 1 dB and d0 = 1.

simulations. We observe that for both figures, the MMSE receiver significantly outperforms the PZF receiver. VI. C ONCLUSION We derived outage probability and transmission capacity expressions for spatial multiplexing systems with optimal MMSE linear receivers in random ad-hoc networks. These expressions are in closed-form, applying for arbitrary numbers of transmit and receive antennas. We showed that from a transmission capacity perspective, it is preferable to have a high density of single-stream transmissions than a low density of multi-stream transmissions when using the optimal MMSE receiver in ad hoc networks. A PPENDIX The outage probability, conditioned on xi = |Di |α < a, where xi are independent and identically uniformly distributed


with i = 1, . . . L, is given by [1] FZ|x1 ,...,xL (z, λ) = 1 − e−

zdα 0 γ

⎛

N r −1

⎜ ⎝

Nr −p zdα0 υ−1 υ=1

(υ − 1)!

p=0

×

γ

z p dαp 0 Ip (x1 , . . . , xL , λ)

⎞ ⎟ ⎠

(12)

where Ip (x1 , . . . , xL , λ) =

(1 +

Cp (x1 , . . . , xL , λ) L α −1 Nt i=1 (1 + d0 xi z) (13)

z)Nt −1

and Cp (x1 , . . . , xL , λ) is the coefficient of z p in the polynoL −1 Nt Nt −1 mial expansion of (1 + d−α 0 z) i=1 (1 + xi z) . By noting that the number of nodes follows a Poisson distribution with intensity λ, the expected value of Ip (x1 , . . . , xL , λ) w.r.t. x1 , . . . , xL is given by 2 L ∞ e−λπa α 2λπ α E [Ip (x1 , . . . , xL , λ)] = (1 + z)Nt −1 L! L=0 2 a a N + −1 L x t α i ... Cp (x1 , . . . , xL , λ) dx1 . . . dxL Nt (x + dα i 0 0 0 z) i=1 2 L min(p,Nt −1) ∞ Nt − 1 1 2λπ e−λπa α α = q (1 + z)Nt −1 dαq L! 0 q=0 L=0

|hp−q |

L! |gj,p | j=1 (L − |hj,p |)! =1 g,j,p ! a a |hj,p−q | L −h ... xi i,j,p−q 0

0

hi,j,p−q

k=1

Nt − k + 1 k

(14)

N + 2 −1

xi t α dx1 . . . dxL . Nt (xi + dα 0 z) i=1

i=1

To solve the integral in (14), it is convenient to define the following function: a Nt + 2 −ς−1 α x (15) Jς := α Nt dx 0 (x + d0 z) 2 2 −Nt Γ Nt + − ς = aNt + α −ς (dα 0 z) α a 2 2 ˜ × 2 F1 Nt , Nt + − ς; Nt + − ς + 1; − α α α d0 z subject to ς < Nt + α2 where 2 F˜1 (·; ·; ·) is the regularized generalized Gauss hypergeometric function [13]. Now substituting (15) into (14), we obtain 2 min(p,Nt −1) Nt − 1 e−λπa α E [Ip (x1 , . . . , xL , λ)] = q (1 + z)Nt −1 q=0 |hp−q | 1 × αq d0 j=1

|hj,p−q |

where ΔL =

∞ L=|hj,p−q |

i=1

hi,j,p−q

Nt −k+1 k=1 k |hj,p−q | g !J υ,j,p υ=1 0

Jhi,j,p−q

|gj,p |

2λπJ0 L α

(L − |hj,p−q |)!

=

2λπJ0 α

ΔL (16)

|hj,p−q | e

2λπJ0 α

.

(17)

To proceed, we take the limit as a → ∞ in (16), since we are considering an infinite plane. It is thus convenient to note the following two limit functions, 2 2λπ J0 e−λπa α lim exp a→∞ α 2 2 2 α λπ (dα 0 z) Γ Nt + α Γ 1 − α (18) = exp − Γ (Nt ) and k−1− 2 2λπ −ς α Jς = − (dα ΘNt λ . 0 z) 2 a→∞ α N + − k t α k=1 ς

lim

(19)

Substituting (18) and (19) into (16), and substituting the resultant expression into (12), we obtain the desired result. R EFERENCES [1] H. Gao, P. Smith, and M. V. Clark, “Theoretical reliability of MMSE linear diversity combining in Rayleigh-fading additive interference channels,” IEEE Trans. Commun., vol. 46, no. 5, pp. 666–672, May 1998. [2] A. M. Hunter, J. G. Andrews, and S. P. Weber, “Transmission capacity of ad hoc networks with spatial diversity,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5058–5071, Jul. 2008. [3] K. Huang, J. G. Andrews, R. W. Heath Jr., D. Guo, and R. A. Berry, “Spatial interference cancellation for multi-antenna mobile ad-hoc networks,” IEEE Trans. Inform. Theory, submitted. [Online]. Available: http://arxiv.org/abs/0807.1773v1 [4] N. Jindal, J. G. Andrews, and S. P. Weber, “Rethinking MIMO for wireless networks: Linear throughput increases with multiple receive antennas,” in Proc. of IEEE Int. Conf. on Commun. (ICC), Dresden, Germany, Jun. 2009, pp. 1–5. [5] ——, “Multi-antenna communication in ad hoc networks: Achieving MIMO gains with SIMO transmission,” IEEE Trans. Commun., submitted. [Online]. Available: http://arxiv.org/pdf/0809.5008v2 [6] O. B. S. Ali, C. Cardinal, and F. Gagnon, “Performance of optimum combining in a Poisson field of interferers and Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 9, no. 8, pp. 2461–2467, Aug. 2010. [7] S. Govindasamy, D. W. Bliss, and D. H. Staelin, “Spectral efficiency in single-hop ad-hoc wireless networks with interference using adaptive antenna arrays,” IEEE J. Select. Areas Commun., vol. 25, no. 7, pp. 1358–1369, Sep. 2007. [8] R. H. Y. Louie, M. R. McKay, and I. B. Collings, “Open-loop spatial multiplexing and diversity communications in ad hoc networks,” IEEE Trans. Inform. Theory, vol. 57, no. 1, pp. 317–344, Jan. 2011. [9] M. Kountouris and J. G. Andrews, “Transmission capacity scaling of SDMA in wireless ad hoc networks,” in Proc. of IEEE Info. Theory Workshop (ITW), Sicily, Italy, 2009, pp. 534–538. [10] R. Vaze and R. W. Heath Jr., “Transmission capacity of ad-hoc networks with multiple antennas using transmit stream adaptation and interference cancelation,” IEEE Trans. Inform. Theory, submitted. [Online]. Available: http://arxiv.org/pdf/0912.2630v1 [11] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Geometry and its Applications, 2nd ed. England: John Wiley and Sons, 1995. [12] G. E. Andrews and K. Eriksson, Integer Partitions. UK: Cambridge, 2004. [13] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York: Dover Publications, 1970.