1.1. Previous works on competitive cognitive-inspired access

Specifically, distributed power-control in wireless access networks by exploiting Game Theory is tackled in (MacKenzie & Wicker, 2001). A game-theoretic approach is also pursued in (Palomar et al., 2003) for dealing with power-allocation in single-user MIMO systems in the presence of imperfect channel estimation. The paper (Altman & Altman, 2003) examines the general application of the super-modular games to the distributed power-control, while the contribution in (Goodman & Mandayam, 2000) considers an application scenario wherein the accessing radios attempt to maximize a suitable function of their throughput and leads to the conclusion that the underlying game converges to a Nash Equilibrium (NE). Later, (Saraydar et al., 2002) point out that the game considered in (Goodman & Mandayam, 2000) is, indeed, a super-modular game. The contribution in (Sung & Wong, 2003) focuses on a (somewhat modified) version of the single-cell distributed power-control game that is based on the concept of nonlinear group pricing. Applications of Game Theory to fair-efficient admission control are presented in (Virapanicharoen & Benjpolakaul, 2004), while the whole topic of competitive flow-control and distributed routing is discussed in (Kannan & Iyengar, 2004; Altman et al. 2002).

Game theoretic approaches for agile spectrum sharing have been recently developed in (Fattahi et al.,2007; Xing et al., 2007; Wang & Brown, 2007). Specifically, (Fattahi et al., 2007) focus on the design of a coordinated mechanism for allocating time and frequency slots to competing users that are capable to self-reconfigure their terminals via cross-layer strategies. (Xing et al., 2007) explore the price dynamics of a competitive market composed by (multiple) self-interested service providers that compete for potential customers by offering heterogeneous spectrum-agile networking technologies at different costs. (Wang & Brown, 2007) develop a public safety and commercial spectrum-sharing strategy that resorts to both agile network pricing and call admission control for the competitive maximization of the commercial revenue under low (e.g., upper limited) blocking-probability to public safety calls.

Overall, all the above cited works consider applications scenarios characterized by either single-antenna radios (i.e., they do not consider the spatial dimension), or single-user MIMO systems (i.e., they do not deal with the access problem), thus leaving open the question about potential access capability of the cognitive MIMO radio paradigm.

1.2. Proposed contributions and application scenarios

This chapter focuses on the competitive access in WLAN systems operating in an infrastructure-mode, where power-limited multi-antenna non-cooperative cognitive self-configuring radios attempt to join in uplink to a (possibly multi-antenna) access point (AP) (Fig. 1(a)). The target pursued by each radio is to maximize its own access information-throughput in the presence of both (Rayleigh-distributed) flat-fading and MAI induced by the other accessing terminals. For achieving this goal in a fully distributed and asynchronous way (i.e., in a non-cooperative way), each radio exploits its cognitive capability to ‘learn’ (i.e., estimate) both its own current MIMO faded uplink and the covariance matrix of the MAI induced by the other accessing radios. Thus, on the basis of these learned (i.e., acquired) context information, each radio proceeds to self-reconfigure its access policy by performing suitable power-allocation and (statistical) spatial-shaping of the radiated signals. This learning/self-configuring action is autonomously iterated by the accessing radios according to the general rules of the non-cooperative strategic games, until the overall Access Network (AN) converges to a stable operating state (e.g., the NE of the underlying game). We investigate the performance of this competitive and cognitive access policy under both best-effort and contracted-QoS access strategies.

Furthermore, we also consider the case when even the AP is cognitive, thus meaning that it attempts to learn and subtract the MAI contribution affecting the signal received by each accessing radio. Overall, the key-results of this work may be so summarized.

Figure 1.

a) The considered multi-antenna local AN (b) Zoomed block diagram of the up-down links joining the radio terminal Tx to the AP.

1.3. Organization of the chapter

The remainder of this work is organized as follows. After the system modelling of Section 2, Section 3 deals with the evaluation of the conveyed information throughput in wireless access networks affected by MAI. In Section 4 the optimized power-allocation and interference mitigation algorithms are presented. Thus, after shortly reviewing in Section 5 some Game Theory essentials, in Section 6 we propose a game-inspired access policy. Actual performance of the proposed access strategy in terms of access throughput and access capacity is tested in Section 6, while some conclusive remarks are drawn in Section 7.

About the adopted notation, we anticipate that, capital letters indicate matrices, lower-case underlined symbols denote vectors, while characters overlined by arrow denote block-matrices and block-vectors. Furthermore, apexes * T † mean conjugation, transposition and conjugate-transposition respectively, while lowercase letters will be used for scalar quantities. Furthermore, det[A] and Tra[A] mean determinant and trace of the matrix A≜[a_1…a_m] . Finally, I_m is the (m×m) identity matrix, ║A║_E is the Euclidean norm of the matrix A, A⊗B is the Kronecker product of the matrix A by matrix B, 0_m is the m-dimensional zero-vector, lg denotes natural logarithm and δ(m,n) is the Kronecker delta.

2.1. First learning phase

During the first learning phase, no signal is radiated by the Tx radio of Fig. 1(b), so as to allow the AP to learn the statistics of the MAI induced by all other accessing radios (Fig. 1(a)). Thus, the r-dimensional (complex column) vector collecting the outputs of the receive antennas at the AP side over the n-th slot of this first learning phase may be modelled as

where N0 (Watt / Hz) is the thermal noise level. The component { v_˙(n) } in (1) accounts for the MAI induced by all other accessing radios, and thus it may be suitably modelled as a zero mean, temporally-white, spatially-colored Gaussian sequence (Paulraj et al., 2003), with covariance matrix Kv≜E{v_˙(n)(v_˙(n))†} constant over (at least) a packet-transmission interval. However, since Kv may change from a packet to another, it is reasonable to assume that both Tx and AP in Fig. 1(b) are not aware of the overall disturbance covariance matrix

at the beginning of each transmitted packet. Besides, since during this first learning phase the signal y_˙(n) in (1) received at the AP equates the MAI one d˙_(n) the AP may exploit its cognitive capabilities to learn Kd A very simple way to accomplish this task is suggested by the law of large numbers (Poor, 1994). By fact, this last guarantees that an unbiased and consistent (i.e., asymptotically exact) estimate K⌣d may be obtained via the following sample-average:

As already pointed out in (Baccarelli et al., 2007), we anticipate that the performance effects of (possible) mismatches between actual Kd and its estimate K⌣d are not so critical, so that, in the sequel, we directly assume K⌣d=Kd However, this assumption will be relaxed in Section 6.5, where the effects of possible mismatches are numerically tested.

2.2. Second learning phase

Goal of the second learning phase is to allow the accessing radio to perform the optimized shaping of the (deterministic) pilot streams {x˜i(n)TL1+1≤n≤TL1+TL21≤i≤t} to be used to estimate the (a priori unknown) (r×t) uplink gains hji Thus, the (sampled) signal y˜i(n) measured at the output of the j-th receive antenna of the AP node during this second learning phase may be modelled as (Paulraj et al., 2003)

where the overall disturbance d˜j(n) accounts for both MAI and thermal noise and exhibits the same statistics previously reported in (2). Therefore, the (TL2×r) observations in (4) may be recast into the (TL2×r) matrix Y˜≜[y˜_1…y˜_r] given by

where X˜≜[x˜_1…x˜_t] is the (TL2×t) matrix of the radiated pilot symbols in (4), H≜[h_1…h_r] is the (t×r) matrix composed by the uplink gains { hji } to be learned, and the (TL2×r) matrix D˜≜[d˜_1…d˜_r] collects the MAI-plus-noise terms d˜j(n) in (4).

Afterwards, observations Y˜ in (5) are employed by the AP to ‘learn’ H via the computation of the corresponding minimum mean squared error (MMSE) matrix estimate H^≜E{H|Y˜} . Thus, at step n=TL1+TL2 (i.e., at the end of the second learning phase), this matrix estimate H^ is communicated back to the accessing radio via the service downlink of Fig. 1(b). A detailed analysis of the structure and performance of the estimator computing the set h^ji≜E{hji|Y˜} of the (r×t) MMSE estimates of the MIMO uplink H in (5) may be found in (Baccarelli et al., 2007; Biagi, 2006). For our purposes, it suffices to stress that the pilot matrix X˜ minimizing the total average squared estimation error:

where

and P˜ (Watt) in (7) is the average power available at Tx radio for the transmission of the pilot signals in (4). Furthermore, it may be also proved (Baccarelli et al., 2007) that, when X˜ meets the condition in (6), the resulting MMSE estimation errors { εji≜h^ji−hji } are mutually uncorrelated, zero-mean, and equidistributed proper complex Gaussian r.v.s, with variance

2.3. Payload phase

On the basis of the available Kd and H^ matrices and actual packet to be transmitted, the Tx radio of Fig. 1(b) suitable shapes the signal streams {ϕi(n)∈ℂ1 TL1+TL2+1≤n≤T1≤i≤t} to be radiated during the payload phase. Hence, the corresponding (sampled) signals {yj(n)∈ℂ1 TL1+TL2+1≤n≤T1≤j≤r} measured at the outputs of the receive antennas at the AP side may be modelled (Paulraj et al., 2003; Baccarelli et al., 2007) as

where the sequence dj(n)≜vj(n)+wj(n)1≤j≤r accounts for the overall disturbances (e.g., MAI plus thermal noise) experienced during the payload phase, β≜D-z takes care of the path loss over a distance of D meters and z is the corresponding path-loss exponent. Therefore, after assuming that the transmitted streams meet the (usual) average power constraint

the resulting SINR γj measured at the output of the j-th receive antenna at the AP side during the payload phase equates (equations (9), (10))

where cjj is the (j,j)-th entry of the MAI covariance matrix Kd in (2). Moreover, from (9) 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 the (t×1) column vector ϕ_(n)≜[ϕ1(n)…ϕt(n)]T of the corresponding signals radiated by the accessing radio as in

where {d_(n)≜[d1(n)…dr(n)]T,  TL1+TL2+1≤n≤T} is the temporally-white spatially-colored Gaussian sequence of disturbances with spatial covariance matrix still given by Kd . Furthermore, directly from (10) it follows that the (t×t) spatial covariance matrix Rϕ≜E{ϕ_(n)ϕ_(n)†} of the t-dimensional signal vector radiated during each by the Tx radio of Fig. 1(b) must meet the following power constraint:

Finally, after stacking the Tpay observed vectors in (12) into the corresponding (Tpayr×1) block vector y_→≜[y_T(TL1+TL2+1)…y_T(T)]T , we may compact the Tpay relationships (12) into the following one:

where the (block) covariance matrix of the corresponding disturbance (block) vector in (14) d_→≜[d_T(TL1+TL2+1)…d_T(T)]T is given by

while the average Euclidean squared norm of the block vector ϕ_→≜[ϕ_T(TL1+TL2+1)…ϕ_T(T)]T of the transmitted random signals is constrained as (equation (10))

3.1. Optimized cognitive access policy

In order to compute the supremum in (18), we must proceed to carry out the power-constrained maximization of ℛ(H^) . For this purpose, let us indicate with

the singular value decomposition (SVD) of the MAI spatial covariance matrix Kd , where

is the corresponding (r×r) diagonal matrix of the magnitude-ordered singular values of Kd . Thus, after introducing the (t×r) dummy matrix

accounting for the combined effects of the imperfect channel estimate H^ and the spatial MAI Kd , let us denote by

the corresponding SVD, where UA and VA are unitary matrices, while

is the (t×r) diagonal matrix collecting the s≜min{rt} magnitude-ordered singular values k1≥k2≥…≥ks0 of the matrix A. Finally, for future convenience, let us also introduce the following dummy positions:

In this way, it can be proved (Baccarelli & Biagi, 2003; Biagi, 2006; Baccarelli et al., 2007) that an application of the Kuhn-Tucker conditions (Gallagher, 1968) leads to the following optimized access strategy for the cognitive Tx radio.

Proposition 2 (Optimized access strategy)

Let the assumptions reported in Proposition 1 be fulfilled. Therefore, for m=s+1,…,t the powers {P * (m)} achieving the supremum in (18) vanish, while for m=1,…,s they equate

where λmin≜min{λl l=1…r} and L≜(1-(r/Tpay))ρ−(1/αm) . Furthermore, the non-negative scalar parameter ρ in (26), (27) is set to meet the following power-constraint:

where

is the ( ρ -dependent) set of m-indexes fulfilling the inequality in (27). Finally, the corresponding optimized spatial correlation matrix Rϕ(opt) of the radiated signals is aligned along the right eigenvectors of the matrix A in (22) as in

3.2. MAI-learning and MAI-mitigation at the AP

Before proceeding to detect the transmitted packet, the AP may attempt to estimate the MAI component d(n) affecting the received signal y(n) in (12), so as to subtract the computed MAI estimate d_^(n) from the received signal y(n). Since both Kd and H^ are available at the AP, this last is also aware of Rϕ(opt) (equation (30)), so that it may exploit both H^ and Rϕ(opt) to improve the MAI estimate’s accuracy. Specifically, being ϕ(n) and d_(n) Gaussian distributed, the MMSE estimate d_^(n)≜E{d_(n)|y_(n)} of the MAI is linear with respect to the observation y(n) (Poor, 1994), so that we may write

In turns, the (r×r) matrix F in (31) may be computed via an application of the (usual) Orthogonal Principle (Poor, 1994), that allows us to write

where the conditioning in (32) accounts for the availability of H^ and Rϕ(opt) at the AP side (e.g., the AP is cognitive). Hence, by developing the expectations in (32), after some (tedious) algebra, we arrive at the following final expression for the matrix F in (31):

The MSE performance of the estimator in (31) is dictated by the corresponding error covariance matrix Kdε≜E{(d^_(n)-d_(n))d^_(n)-d_(n)†|H^Rϕ(opt)} that equates (equations (31), (33))

Roughly speaking, equation (34) measures the performance improvement arising from the exploitation of the cognitive capability of the AP node. In particular, equation (33) points out that F vanishes for vanishing Kd , while F approaches Ir for large Kd . As a consequence, on the basis of equation (34), we conclude that actual effectiveness of the MAI estimation/subtraction procedure implemented by the AP tends to improve in the presence of strong MAI.

5.1. Competitive optimal SDMA

We pass now to detail the algorithm for the competitively optimal access under ‘contracted-QoS’ and best-effort policies when the AP of Fig. 1(a) performs MAI estimation/subtraction. Before proceeding, we point out that the concept of ‘contracted-QoS’ we go to introduce relies on multiple QoS classes defined in terms of requested, or desired, access throughput. Therefore, the resulting access algorithm we report in Table 1 attempts to achieve the target throughput TℛA(Z) desired by the g-th radio; and if this throughput is not achievable due to the MAI induced by the other accessing terminals, the algorithm tries to achieve the next lower QoS-class TℛA(Z−1) .

Furthermore, from the outset it also follows that the best-effort policy is an instance of the ‘contracted-QoS’ one, where the number of allowed QoS classes approach infinity. The algorithm for achieving the competitively maximal access throughput over the uplink is detailed in Table 1. It must be iteratively run by all accessing radios of Fig. 1(a). TℛA(Z) (nat/slot) in Table 1 indicates the target throughput defining the Z-th QoS class.

5.2. Asynchronous implementation of the access game and Nash Equilibria

After assuming that the access algorithm of Table 1 is iteratively run (in a non-cooperative and possibly asynchronous way) by all accessing radios of Fig. 1(a), let be Vg≜{tg1tg2…} the set of times at which g-th radio executes this algorithm. Thus, after indicating by V≜{τ1τ2…} the resulting overall set of update times V1∪V2∪…∪Vn* sorted in the increasing order ( τi≤τi+1 ), the asynchronous and distributed implementation of the considered access game works according to the pseudocode of Table 2.

Hence, key-questions about the access game of Table 2 regard the existence, uniqueness and achievability of the corresponding NE. The following Proposition 3 (Biagi, 2006) gives insight about these questions.

Proposition 3

By referring to the access game of Table 2, let us assume that the following conditions are met:

Thus, the NE of the access game of Table 2 exists, is unique and it may be reached by moving from any starting point.

6.1. The achievable throughput regions

In this operating conditions, the (two-dimensional) set of accessing rates sustainable by the AN may be (formally) described by resorting to the concept of achievable rate-region (Baccarelli et al., 2005). In a nutshell, given a statistical description of the uplinks T1→AP , T2→AP , and under an assigned set of constraints on the available radio resources (e.g., available power, bandwidth, noise level, and so on), the corresponding achievable rate region of the access network is the closure of all 2-ple ( ℛ1ℛ2 ) of the access rates that may be simultaneously sustained by the network.

Unfortunately, apart from some partial contributions, no closed-form analytical formulas are available yet for the evaluation of the rate-region of MAI-impaired access networks (Baccarelli et al., 2005). Forced by this consideration, we resort to numerical tests and Fig. 2 reports the rate-regions we have numerically obtained for some different access strategies. The n*=2 radios of the simulated access network of Fig. 2 are equipped with t1=t2=4 transmit antennas, while r= 4 receive antennas are available at the AP. The innermost A -labelled region of Fig. 2 reports the sustainable access rates when the power available at the radios is evenly split over the transmit antennas (i.e. Rϕ(1) =Rϕ(2) =P4I4 in equation (39)) and, in addition, no channel estimate are available for MAI mitigation ( H^=0 in equation (33), so that the AP node is no cognitive). In this case, when both the radios attempt to access to the AP, the resulting maximal throughput is quite low and it is limited up to ℛ1=ℛ2=11(bit/slot) . However, when the AP performs reliable MAI estimation and subtraction (i.e. H^≡H in equation (33) and thus the AP node is perfectly cognitive), we obtain the (larger) rate-region ℬ of Fig. 2, that covers the above A -region and allows radios to simultaneously access at a maximal throughput of ℛ1=ℛ2=165(bit/slot) .

Figure 2.

Rate regions for some different accessing policies and systems setting.

When optimized power-allocation and signal-shaping are also performed at the radios (that is, Rϕ(g)  are computed as in equation (39)) but no reliable MAI-mitigation is performed by the AP, we obtain the rate-region C of Fig. 2 that allows radios to simultaneously access at a (maximal) throughput of ℛ1=ℛ2=21(bit/slot) . A comparison of D and ℰ regions of Fig. 2 gives insight on the ultimate effectiveness of the competitive SDMA game played by the cognitive radios and the MAI mitigation performed by the cognitive AP. Specifically, the D region of Fig. 2 collects the 2-ple ( ℛ1ℛ2 ) of achievable access rates when both the competitive SDMA game of Table 2 is carried out by the radios and a reliable MAI estimation/subtraction is performed by the AP. In this case, the maximal access throughput sustained by the AN approaches ℛ1=ℛ2=235(bit/slot) . This value differs, indeed, less than 6% from the ultimate one of ℛ1=ℛ2=25(bit/slot) bounding the corresponding ℰ region. This last refers to the ideal case when MAI-cancellation is perfect (i.e. d_^(n)≡d_(n) in equation (31)) and both radios are allowed to transmit over the whole working-time of the system.

About the degrading effects induced by possibly imperfect MIMO uplink estimates, these last have been investigated in depth, for example, in (Baccarelli & Biagi, 2003; Baccarelli & Biagi, 2004). So, in this context we limit to remark that the A -region of Fig. 2 represents the sustainable access rates when both radios and AP are fully unaware about actual values of the MIMO uplink gains (in other words, they are not cognitive and H^(1)=H^(2)=0 ), while the corresponding D -region captures the achievable access rates when both radios and AP are equipped with perfect MIMO uplink estimates (i.e. H^(1)=H(1)  and  H^(2)=H(2) ). A direct comparison of these regions leads to the conclusion that the penalty in the access throughput arising from full unawareness about MIMO uplink-gains may be, indeed, no negligible and of the order of 5–6 (bit/slot) per accessing radio. Obviously, this conclusion is somewhat mitigated by considering that long throughput-consuming learning phases may be required in order to acquire reliable uplink estimates (Section 3.2). The interested reader may refer to the works in (Baccarelli & Biagi, 2003; Baccarelli & Biagi, 2004; Baccarelli et al., 2007) and references therein for a detailed description of the optimal estimation accuracy - vs. - throughput trade off.

6.2. Cognitive SDMA Game-vs-CSMA/CA(s): a throughput comparison

The effectiveness of the proposed competitive SDMA policy for cognitive ANs may be also appreciated by comparing the above mentioned D -region against the one of Fig. 2. Specifically, this T -region of Fig. 2 reports the 2-ple ( ℛ1ℛ2 ) of achievable access rates when the access policy followed by the (no cognitive) multi-antenna radios is the CSMA/CA(s). In this regard, we explicitly stress that, to guarantee both collision-free and fair access, in the carried out numerical tests the CSMA/CA(s) policy we implemented schedules a single uplink at a time, and transmits over the scheduled uplink at the maximum allowed power P and over a 1/n*=1/2 -fraction of the overall system working time. Furthermore, the throughput loss due to the (possible) exchange of RTS/CTS packets is not accounted for. Thus, being the implemented CSMA/CA(s) scheme collision-free (that is, fully MAI-free), the corresponding T -region of Fig. 2 is the largest attainable by the CSMA/CA(s) policy. Now, an examination of Fig. 2 leads to the conclusion that the access region achieved by the CSMA/CA(s) policy is strictly included in the D -region obtained by the proposed competitive cognitive-based SDMA policy, at least in the considered MAI-limited access scenario. Since the implemented CSMA/CA(s) strategy is, indeed, an example of multi-antenna orthogonal access policy, we conclude that the rate-region of any multi-antenna orthogonal access strategy (such as TDMA, FDMA, CDMA) overlaps the T -region of Fig. 2. This supports the superiority of the proposed competitively optimal cognitive access strategy over the orthogonal no-cognitive ones, at least when the spatial-dimension offered by the AN may be efficiently exploited to perform both signal-beam forming at the accessing (cognitive) radios and MAI-mitigation at the (cognitive) AP.

6.3. Cognitive access capacity

The following Figs 3, 4 and 5 give insight into the performance of the proposed cognitive SDMA game with MAI-mitigation in terms of access capacity (i.e., in terms of maximum number of radios capable to simultaneously access the network at a common target throughput). The plots of Fig. 3 report the (numerically evaluated) system capacity at access throughput of 10, 15, 20 (bit/slot) when all radios are equipped with t=4 transmit antennas, while the AP performs MAI-mitigation via a number of receive antennas ranging from one to eight. An examination of these plots leads to two main conclusions. The first (quite obvious) one is that the access capacity decreases for increasing values of the target access throughput (the target QoS-level) demanded by the radios. The second (less trivial) one is that, at any target QoS level, the slope of the capacity curves of Fig. 3 grows for increasing r up to r=t=4 , and then it starves for rt=4 . Due to the iterative nature of the proposed access strategy of Table 2, we are not able to give formal (e.g., analytical) support to this behaviour. However, roughly speaking, it may be understood by noting that, under mild assumptions, the Shannon capacity (in bit/slot) of a point-to-point flat-faded MIMO channel scales as the min(r t) (Paulraj et al., 2003), so that no slope increment in the capacity curves is expected when rt . However, by increasing r beyond t, MAI mitigation capability of the AP still grows, so that the capacity curves of Fig. 3 increases (at a nearly constant rate) for growing values of r.

Figure 3.

Number of radios accessing the net under the guaranteed QoS policy for various r values.

Similar conclusions may be drawn after an examination of the plots of Fig. 4, that capture the effects induced by the number t of transmit antennas equipping each accessing radio. Interestingly, a comparison of the curves of Fig. 3 and 4 shows that, at any target access throughput, the capacity of the access system with t=m transmit antennas and r=n receive ones outperforms the corresponding capacity of the ‘dual’ access system with r=m and t=n . This behaviour supports the conclusion that, at least in the considered operating scenarios, the increment in the access capacity arising from more accurate transmit beam forming outperforms the corresponding capacity improvement due to more reliable MAI-mitigation. Finally, Fig. 5 reports the system capacity under the above mentioned ‘contracted-QoS’ access policy (Table 1), when the radios accept access throughput within 10% of the desired one. A comparison of the plots of Fig. 5 corroborates the conclusion that the ‘QoS-contracted’ access policy may be valuable in terms of system capacity, especially when the MAI-mitigation capability of the AP is limited (e.g., when the number r of the receive antennas is less than that of the transmit ones).

Figure 4.

Number of radios accessing the net under the guaranteed QoS policy for various t values.

Figure 5.

Number of radios accessing the net under the ‘contracted-QoS’ policy for various r values.

6.4. Self-convergence of the cognitive SDMA Game toward the nearest sustainable operating point

In actual application scenarios, the accessing radios are not aware in advance about the sustainable access regions of Fig. 2, neither these last may be analytically evaluated in closed-form (Baccarelli et al., 2005). Therefore, a key-question concerns the self-convergence of the operating point of the SDMA cognitive game of Table 2, when the initial access throughput (ℛ1(0)ℛ2(0)) demanded by the radios falls out of the corresponding access region of Fig. 6. It can be proved (Biagi, 2006, Appendix 3) that, under the best-effort policy, the operating point of the SDMA game of Table 2 is able to self-move from (ℛ1(0)ℛ2(0)) and to converge to the boundary point of the underlying access region at the minimum Euclidean distance from that starting point (see the dotted arrow of Fig. 6). Likewise, under the ‘contracted-QoS’ policy, the operating point self-moves from (ℛ1(0)ℛ2(0)) to the point of the QoS grid at the minimum Euclidean distance (see the dashed grid of Fig. 6).

Figure 6.

Access region of the proposed cognitive SDMA game of Table 2 with MAI-mitigation (t₁=t₂=4 and r=4).

6.5. Mismatches in the estimation of the MAI covariance matrix

To test the sensitivity of the proposed cognitive SDMA game on errors possibly affecting the estimated MAI covariance matrices K^d(g)  g=1…n* , we have perturbed actual matrices Kd(g)  g=1…n* , via randomly-generated (r×r) matrices G(g)  g=1…n* , which are composed by zero-mean proper complex unit-variance uncorrelated Gaussian elements. In doing so, we generated

where δ≜E{‖Kd(g)−K^d(g)‖E2}/‖Kd(g)‖E2 was set according to the desired average squared estimation errors affecting K^d(g) . Thus, after replacing the actual covariance matrices Kd(g) with the corresponding perturbed ones K^d(g) , we have played once again the SDMA Game of Table 2 for the same access scenario previously described in Section 6.1. The resulting access region (labelled as D∗ ) is reported in Fig. 7, together with the corresponding D -region previously drawn in Fig. 2 for the case of perfect MAI covariance-matrix estimates. An examination of the regions of Fig. 7 points out that the resulting access throughput-loss is limited up to 4%, even for δ values as high as 0.05.

Figure 7.

Access regions for perfectly estimated MAI-covariance matrices ( ) and with estimation errors (  ).