to the concept of âcontracted QoSâ according to pre- defined QoS classes. So, the algorithm we present at- tempts to achieve the target throughput class and if.
A distributed Power-Allocation and Signal-Shaping Game for Multiple-Antenna ad-hoc networks impaired by Multiple-Access Interference Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni, Roberto Cusani, Giuseppe Razzano {enzobac, biagi, pelcris, robby, razzano }@infocom.uniroma1.it ∗ Abstract
the resulting SINR γj measured at the output of the j-th receive antenna during the payload phase equates
This contribution considers the power control and signal-shaping problem for ad-hoc networks composed by Multi-Antenna non-cooperative terminals affected by spatially colored Multiple-Access Interference (MAI). The Multi-Antenna interference channel is modeled as a non cooperative and strategic game. Furthermore iterative power control and signal shaping algorithms are presented to efficiently achieve the Nash Equilibrium under Best Effort and ”contracted QoS” policies.
γj = P/(N0 + cjj ), 1 ≤ j ≤ r.
1. The System Modeling Let us consider the signal streams {φi (n) ∈ C1 , TL + Ttr + 1 ≤ n ≤ T }, 1 ≤ i ≤ t, to be radiated during the payload phase. The corresponding (sampled) signals {yj (n) ∈ C1 , TL + Ttr + 1 ≤ n ≤ T }, 1 ≤ j ≤ r, measured at the outputs of the receive antennas may be modelled as [1] 1
hji φi (n) + dj (n), yj (n) = √ t i=1 t
(1)
(with TL + Ttr + 1 ≤ n ≤ T, 1 ≤ j ≤ r) where the sequences dj (n) vj (n) + wj (n), 1 ≤ j ≤ r, account for the overall disturbances (e.g., MAI plus noise) experienced during the payload phase. Therefore, after assuming that these last meet the power constraint [2] 1
E{||φi (n)||2 } ≤ P, TL + Ttr + 1 ≤ n ≤ T, (2) t i=1 t
∗
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni, Roberto Cusani and Giuseppe Razzano are with INFO-COM Dept., University of Rome ”La Sapienza”, via Eudossiana 18, 00184 Rome, Italy. Ph. no. +39 06 44585466 FAX no. +39 06 4873330
0-7803-8815-1/04/$20.00 ©2004 IEEE
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(3)
Furthermore, from (1) we also deduce that the (r × 1) column vector y(n) [y1 (n)...yr (n)]T collecting the outputs of the r receive antennas over the n-th payload slot is linked to (tx1) column vector φ(n) [φ1 (n)...φt (n)]T of the corresponding signals radiated by the transmit node as in 1 y(n) = √ HT φ(n) + d(n), TL + Ttr + 1 ≤ n ≤ T, (4) t where {d(n) [d1 (n)...dr (n)]T , TL + Ttr + 1 ≤ n ≤ T } is the temporally white Gaussian sequence of disturbances with spatial covariance matrix still given by Kd (see [6]). Furthermore, directly from (2) it follows that the (t × t) spatial covariance matrix Rφ E{φ(n)φ(n)† } of the t-dimensional signal radiated during each slot must meet the power constraint T ra[Rφ ] ≡ E{φ(n)† φ(n)} ≤ tP, TL +Ttr +1 ≤ n ≤ T. (5) Finally, after stacking the Tpay observed vectors in (4) into the corresponding (Tpay r × 1) block vector T → − y yT (TL + Ttr + 1) ...yT (T ) , we may compact the Tpay relationships (4) in the following one: T → 1 − − → − → y = √ ITpay ⊗ H φ + d , t
(6)
where the (block) covariance matrix of the correspond→ − ing disturbance block vector in (6) d equates → →− − E{ d ( d )† } = ITpay ⊗ Kd
(7)
→ − while the block vector φ of the random signals transmitted during overall payload phase is constrained as in (see (2)) → →† − − (8) E{ φ φ } ≤ Tpay tP.
The information throughput to be maximized
− → ˆ → ˆ 1 sup I − , TG (H) y ; φ |H Tpay T ra[R ]≤P t φ
allocation ∆. Note that, according to [6] the power allocation is performed as (9)
conveyed by the MIMO channel for Gaussian input sigˆ so nals generally falls below channel capacity C(H), ˆ ˆ that we have TG (H) ≤ C(H). We have (see [6])
− → ˆ → y ; φ |H I − =
1 −1/2 ˆ T ˆ ∗ K−1/2 +σ 2 P K−1 Rφ H = Tpay lg det Ir + Kd H ε d d t
σ 2 Tpay −1 ∗ (Kd ) ⊗ Rφ (10) − lg det Irt + ε t when both Tpay and t are large; The throughput can be evaluated via the following formula (see [6]) ˆ = TG (H)
s
σ2 P + lg(1 + αm P (m)) lg 1 + ε µm m=1 m=1 r
−
r 1
Tpay
lg 1 + βl P (m)
(11)
l=1
1.1. Power allocation for the QoS ”contracted” and Best Effort policies In this subsection the algorithms for the spatial power allocation and spatial shaping under QoS contracted and Best Effort policies are considered. Before proceeding, some remarks about considered QoS are needed. We consider the QoS from the information throughput point of view. Thus, in place of guaranteed QoS, it is more reasonable, indeed, to resort to the concept of ”contracted QoS” according to predefined QoS classes. So, the algorithm we present attempts to achieve the target throughput class and if this class is not achievable (due to interference) the algorithm attempts to achieve the next lower QoS class (less throughput). From this point of view, the Best Effort strategy is a particular case of contracted QoS where the number of classes approaches infinity. The algorithm for achieving the maximal throughput over the g-th link under the above introduced contracted QoS policy is reported in Table I. It is important to underline that the algorithms here presented are based on games theory concepts. The steps from 0 to 11 are set-up procedures and eigen/singular values computations. Step 12 verifies that the game is playable (existence of the Nash Equilibrium) while steps 13 2 > and 14 set {m = 1, ..., s : km up ρ, set I(ρ)
1+
σε2 P µm
t ρ
P (m) =
+ σε2 T ra[K−1 d ] } and step size for power
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1 βmin L − 1+ 2βmin
{βmin L}2 + 4βmin ρ −
1 rρβmin − αm αm Tpay
σ 2 P t 2 + σε2 T ra[K−1 when km > 1+ ε ] , d µm ρ
,
(12)
being βmin min{βl , l = 1, .., r} and L 1 − r 1 Tpay ρ − αm and the spatial shaping is performed according to Rφ (opt) = UA diag{P (1), ...P (s), 0t−s }U†A , (13) where UA is derived by SVD of matrix ˆ ∗ K−1/2 Ud AH d The condition at step 15 assures that the power meets the constrain (8) and steps from 16 to 18 perform the power allocation and signal spatial shaping. In the steps from 18 to 22 the convergence of the algorithm in terms of spatial shaping covariance matrix is checked and in step 23 the maximized information throughput is evaluated for the g-th link. Finally, in step 24 the algorithm check if the achieved throughput is compliant with the QoS requirements. If it is compliant the game stops; otherwise the algorithm reduces the power and restart game. If the throughput is below the requested one, the game restarts with a target throughput that is below the previous requested (from (z)-QoS class to (z-1)-QoS class). It may be remarked that the power allocation algorithm is fully distributed scalable and asynchronous.
2. Numerical Results and Conclusions Fig.1 shows how two pairs of users achieve different throughput by considering squared topology and different values of shadowing factor χ2 (see [6]). Each N stands for Nash Equilibrium. Fig.2 Shows the throughput regions and the comparison with square one indicating orthogonal access (i.e. TDMA) The proposed power spatial shaping algorithm based on Games Theory has shown that it is possible to achieve the Nash Equilibrium in a fully distributed ad-hoc network. In addition, this algorithm assure the achieveability of maximum information throughput. Finally we show the capacity regions and these last give the region of allowable N+1-ple of user rates.
(z)
0. Set the target throughput T RT H of the z-th QoS Classes ; (g) (g) 1. Initialize R (new) := R (old) = [0tg ×tg ]; φ φ 2. fl(g)=1; 3. TG (g) = 0; (g) 4. α(g) (P˜ Ttr /rg )T ra[(Kd )−1 ]; 2 (g) −1 5. σε (g) (1 + α /tg ) ; (g) 6. Compute and sort the r eigenvalues of Kd ; ∗ (g) −1 T ˆ ˆ 7. Compute the SVD of Hg K Hg ; d
20
(g)
(g)
N
(g) 2
12. if km
(g)
≥ (µm + P (g) σε2 (g))
2 σε (g)
rTpay
(g) (g) µmin µm
Capacity (bits/slot)
N
12
χ22=1
χ22=1
χ21=1 N
8 χ22=1
6 χ23=1
User 1 TDMA User 2 User 3 (no active until 90th iteration)
4
;
N
χ22=1
10
0
(g)
õmin tg
N
N
2
2 σε (g)Tpay
χ21=0 χ22=0
χ21=0.8
χ12=1
14
8. Sort the s min(r, t) eigenvalues (g) of Kd ; (g) (g) (g) (g) 9. αm µm km 2 /tg (µm + P (g) σε2 (g)); (g) (g) 2 10. βl σε (g)Tpay /µl tg ; (g)
χ21=0.6
16
(g) (g) {k1 2 , ..., ks 2 }
11. µmin min1≤l≤r {µm }, βmax
N
18
0
20
40
60 number of iterations
80
100
120
Figure 1. Capacity achieved with Best Effort Allocation algorithm
for all m
and fl(g)=1 { 13. Set ρ(g) := 0 and I(ρ(g) ) := ∅; 14. Set thestep size ∆; (g) 15. While P (m) < P t do (g) g m∈I(ρ )
0
40 36 32
{ 16. Update ρ(g) = ρ(g) + ∆; 17. Update the set I(ρ(g) ); 18. Compute the powers and the covariance matrix via eq.(12), (13),; (g) (g) 19. Set Ψ(g) := R (new) − R (old); φ φ (g) 20. If (||Ψ||2E ≤ 0.05||R (g) (old)||2E ) φ 21. then fl(g)=0, else fl(g)=1; (g) (g) 22. R (old) := R (new) φ φ
28 R2 (bits/slot)
χ2=0 χ2=0.3
24
χ2=0.4 20
χ2=0.5 χ2=0.7
16
TDMA
12 8 4 0 0
4
8
12
16
20
24
28
32
36
R1 (bits/slot)
}
23. Evaluate TG (g) via (11) for the g-th link; (z) 24. if TG (g) = T RT H stop; else (z) 25. if TG (g) > T RT H reduce the radiated power P (g) and go to Step 1, else (z) 26. if TG (g) < T RT H lower the target class to z-1 and go to Step 1; }
Table 1. A pseudo-code for the implementation of the power-allocation and signal-shaping algorithm for the g-th transmitter/receiver pair under the ”contracted QoS” policy.
References [1] E.Baccarelli, M.Biagi, ”Error Resistant Space-Time Coding for Emerging 4G-WLANs”, WCNC 2003 proc., March 2003, pp.72-77.
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Figure 2. Capacity Regions comparisons for different vales of χ2 TDMA and Proposed Access (2 users).
[2] T.L.Marzetta, B.M. Hochwald, ”Capacity of a mobile Multiple-Antenna Communication link in Rayleigh flat fading”, IEEE Trans. on Inf. Theory, vol.45, no.1, pp.139157, Jan.1999. [3] T.M.Cover, J.A.Thomas,Elements of Information Theory,Wiley,1991. [4] M.J.Osborne, A.Rubinstein,A course in game theory,MIT Press1994. [5] R.G.Gallagher, Information Theory and Reliable Communication, Wiley, 1968. [6] E.Baccarelli, M.Biagi, ”Optimized Power Allocation and Signal Shaping for Interference-Limited Multi-Antenna ”ad-hoc” Networks”, PWC2003 proc., Sept.2003, pp.138152.
