Pseudocode implementing the competitively optimal cognitive access on the g-th uplink under the ‘contracted-QoS’ policy.

## 1. Introduction to the cognitive MIMO radio paradigm

Cognitive radio has recently shown its great potential for 4th-Generation wireless local access networks (i.e., 4GWLANs) and attracted wordwide interest in academics, industry and standardization activities (Glisic, 2006). The US Federal Communication Commission (FCC, 2003) has issued a notice of public rule-making and order regarding cognitive radio technologies. The Defense Advanced Research Project Agency (DARPA, 2001) launched the ‘next generation (XG) communication program’ to develop new technologies to dynamically manage the radio spectrum. The US Army has also been researching ‘adaptive spectrum exploitation’ (ASE) for real-time spectrum management in the battlefield (Lee, 2001). The US National Science Foundation (NSF) has recently launched the ‘Programmable Wireless Networking’ (ProWin) program focused on adaptive, agile, cognitive radios and networking. On the standardization side, the IEEE 802.22 working group is elaborating on the use of TV bands for spectral-agile wireless regional access networks (WRANs) (Cordiero et al., 2005; IEEE 802.22, 2005). Basically, cognitive radio is capable of ascertaining its operating environment and adapting to real-time conditions of its (localized) wireless channel, including the ability to sense spectrum-usage by neighbouring devices, change operating frequency band, adjust radiated power and modify transmission parameters (Mitola, 1999; Haykin, 2005; Mangold et al., 2004; Zheng & Cao, 2005). Such an intelligent and agile use of the spectrum resource is expected to improve the spectrum-access capability of emerging 4GWLANs (Glisic, 2006; Butler & Webb, 2006; IEEE 802.22, 2005; Chow et al., 2007).

However, wireless access capacity may be also improved significantly by resorting to multi-antenna (i.e., MIMO) architectures that allow exploiting the spatial dimension of the underlying access system via SDMA (Shad et al., 2001; Paulraj et al., 2003). From this point of view, the MIMO paradigm offers, indeed, several important capabilities (such as spatial degree of freedom, spatial diversity, multiple-access interference (MAI) mitigation) that may be suitably exploited for increasing the overall access efficiency of 4GWLANs (Glisic, 2006; Paulraj et al., 2003). However, in order to fully avail of these spatial capabilities offered by the MIMO paradigm in a flexible and agile way, a still open key-question concerns the design of asynchronous, scalable and self-configuring SDMA policies able to meet the QoS-requirements advanced by (battery-powered nomadic) users.

In principle, this flexibility may be attained by merging the cognitive paradigm with the MIMO one, thus giving rise to the cognitive MIMO radio paradigm. In fact, the cognitive concept directly leads to the development of distributed, scalable, and flexible radio access networking architectures by resorting to an opportunistic and (possibly) competitive real-time management of the available radio-resource, that, in turns, may be suitably modelled via the formal tool of Game Theory (Fudenberg & Tirole, 1991, Chaps 16, 24).

### 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.

First, the access strategy we present exploits both the spatial-dimension offered by the physical MIMO platform and the cognitive capability of the AN to maximize (in a competitive sense) the access throughput of each accessing radio. Interestingly, this goal is achieved without requiring radio synchronisation and/or peer-to-peer information-passing among the radio terminals (i.e., in a fully asynchronous and distributed way).

Second, the proposed access strategy allows to implement both best-effort and contracted-QoS access policies, and it may also provide for multiple QoS classes. Interestingly enough, we anticipate that, when the QoS-requirements advanced by the radios are no sustainable by the AN, thus, the proposed access algorithm is able to self-move the operating point of the whole AN (e.g., the operating NE point) to the nearest sustainable one. This means that, besides the radio terminals, the overall AN we go to develop is, indeed, cognitive and self-configuring. Thus, it may be considered as an instance of active access network (Glisic, 2006; Chap.16).

Finally, several numerical results corroborate the conclusion that the proposed cognitive and self-configuring AN is able to outperform conventional CSMACA/ OFDMA/CDMA/TDMA-based no cognitive access systems in terms of both aggregate access throughput and access capacity, especially when the MAI experienced at the AP side is strong.

### 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*Tra*[A] mean determinant and trace of the matrix_{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. The considered SDMA network-model

As previously anticipated, the application scenario we consider is the MIMO uplink of a packet-oriented WLAN, where *n* ≥ 2* multi-antenna battery-powered radios attempt to simultaneously access to a (possibly multi-antenna) AP (Fig. 1(a)). Since the accessing radios are assumed non-cooperative, we begin to focus on the MIMO uplink joining a single radio Tx to the AP, thus considering the signals radiated by all other *(n*-1)* accessing radios as (additive) MAI. The resulting (complex base-band equivalent) point-to-point MAI-impaired MIMO uplink is sketched in Fig. 1(b). Simply stated, this uplink is composed by an accessing radio Tx (equipped with *t ≥1* antennas) communicating to the AP (equipped with *r ≥1* antennas) via a MIMO channel impaired by both Rayleigh flat fading and additive MAI. Path gain *hji *from the transmit antenna *i* to the receive one *j* in Fig. 1(b) may be modelled as a complex zero-mean unit-variance proper complex random variable (r.v.) (Baccarelli & Biagi, 2003; Paulraj et al., 2003), and, for sufficiently spaced-apart antennas, the path gains *{hji**1 j r, 1 i t}* may be considered mutually uncorrelated (Paulraj et al., 2003). Furthermore, these gains *hji *may be also assumed time-invariant over *T ≥1* signalling periods, after which they change to new statistically independent values held for another *T* signalling period, and so on. The resulting ‘block-fading’ model well captures the main features of several frequency-hopping or packet based interleaved 4G systems, where each transmitted packet is detected independently of any other (Paulraj et al., 2003). Since the statistic features of the MAI in Fig. 1(b) depend on both WLAN topology and signals radiated by all accessing radios, it is reasonable to assume that these statistics remain constant over (at least) the whole packet duration. However, since both path gains *hji *and MAI statistics may change from a packet to another, we assume that radios and AP are not aware of them at the beginning of each transmitted packet, but they may exploit their cognitive capabilities to learn (i.e., acquire) them. For this purpose, according to the packet-oriented structure of the considered WLAN, we assume that the (coded and modulated) data-streams radiated in uplink by the transmit antennas of Fig. 1(b) are split into packets composed by T ≥1 slots, where the first *TL1 *slots are used by the receiver (i.e., the AP) to ‘learn’ the MAI statistics (Section 2.1), the second *TL2 *slots are employed to ‘learn’ the path gains of the forward MIMO channel (Section 2.2), and the last *Tpay = T-TL1-TL2 *slots convey payload data (Section 2.3).

### 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*Watt / Hz*) is the thermal noise level. The component {

at the beginning of each transmitted packet. Besides, since during this first learning phase the signal

As already pointed out in (Baccarelli et al., 2007), we anticipate that the performance effects of (possible) mismatches between actual

### 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

where the overall disturbance

where

Afterwards, observations

where

and*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

### 2.3. Payload phase

On the basis of the available

where the sequence*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

where*(j,j)-*th entry of the MAI covariance matrix*n-*th payload slot is linked to the

where*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

where the (block) covariance matrix of the corresponding disturbance (block) vector in (14)

while the average Euclidean squared norm of the block vector

## 3. Self-reconfiguration of the cognitive MIMO-radios

We pass now to introduce a suitable performance metric that allows the accessing radio of Fig. 1(b) to both ‘learn’ its current achieved performance and, possibly, self-reconfigure its access strategy for improving it. Specifically, the performance metric we consider is the Shannon capacity of the MIMO uplink, formally defined as

where

conveyed by MIMO uplink is the performance metric considered by the accessing cognitive radio Tx. About the analytic evaluation of

*Proposition 1*

Under the above reported assumptions about the considered access system, the

when at least one of the following conditions is met:

1. both*t* are large;

2.

3. all SINRs

### 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

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

is the corresponding

accounting for the combined effects of the imperfect channel estimate

the corresponding SVD, where

is the

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

where

is the (*m*-indexes fulfilling the inequality in (27). Finally, the corresponding optimized spatial correlation matrix

### 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*y(n)*. Since both*y(n)* (Poor, 1994), so that we may write

In turns, the

where the conditioning in (32) accounts for the availability of

The MSE performance of the estimator in (31) is dictated by the corresponding error covariance matrix

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

## 4. Some Game Theory essentials

In order to model the dynamic behaviour of the uplink-state of the access system of Fig. 1(a), we resort to the formal tool of the Game Theory. We shortly recall that a non-cooperative and strategic game

Therefore, since there is no cooperation among the players, it is important to ensure the dynamic stability of the overall game. A concept which relates to this issue is the so-called Nash Equilibrium (NE). Simply stated, a NE is an action profile

## 5. The proposed cognitive access game

Let us focus now on the uplink in Fig. 1(a) of a WLAN composed by

where

where the g-th MAI covariance matrix

### 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

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).*(nat/slot)* in Table 1 indicates the target throughput defining the Z-th QoS class.

0 | Set the target throughput |

1 | Initialize |

2 | fl(g)=1 |

3 | |

4 | |

5 | |

6 | Compute and sort the r eigenvalues of |

7 | Compute the SVD of |

8 | Sort the |

9 | |

10 | |

11 | |

12 | If |

13 | Set |

14 | Set the step size |

15 | While |

16 | Update |

17 | Update the set |

18 | Compute the powers and the covariance matrix via equation (26), (27), (30) |

19 | Set |

20 | If |

21 | then fl(g)= 0, else fl(g)= 1; |

22 | |

23 | Evaluate |

24 | If |

25 | If |

26 | If |

### 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

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.

For all k such that |

For all terminals |

Evaluate the MAI matrix |

Run the algorithm of Table 1 so |

to compute [ |

radiate the signal vector [ |

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. Cognitive MIMO access performance

Since WLANs are characterized by topology-depending time-varying MAI, in principle, an effective SDMA protocol should be able to adaptively exploit (in a combined way) the above mentioned flexibility characteristics of the MIMO physical layer by operating in a scalable and asynchronous way. Currently, Carrier Sense Multiple Access with Collision Avoidance (CSMA/CA) is ‘de-facto’ the MAC protocol considered for WLANs. Interestingly enough, a simple extension of CSMA/CA for MIMO links can be designed to provide an

To corroborate this claim, we simulated a Rayleigh-flat faded WLAN composed of

### 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

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

When optimized power-allocation and signal-shaping are also performed at the radios (that is,

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

### 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*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*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*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

### 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

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

### 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

### 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

where

## 7. Conclusion

In this work, we attempted to give a first insight into the possible capacity improvement in the local radio access arising from the exploitation of the cognitive MIMO radio paradigm. The game-inspired approach we have proposed exploits the cognitive capabilities of the overall AN to dynamically perform both adaptive spatial-beamforming at the radios and MAI-mitigation at the AP. The reported numerical results support the conclusion that the cognitive SDMA policy is able to outperform the more conventional no cognitive strategies based on orthogonal access, requiring neither peer-to-peer cooperation among the accessing radios nor radio synchronization.

## Notes

- This work has been partially supported by the Italian National Project: Wireless multiplatfOrm mimo active access netwoRks for QoS-demanding muLtimedia Delivery (WORLD), under grant number 2007R989S.
- The notation ug(ag,a-g) emphasizes that the g-th player controls only own action ag , but his achieved utility depends also on the actions a-g taken by all remaining players (MacKenzie & Wicker, 2001; Fudenberg & Tirole, 1991).
- In the sequel, we adopt the index g for indicating the system parameters of the g-th uplink Tg→AP (Fig. 1(a)).