Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part I: Nash Equilibria

Optimal Linear Precoding Strategies for Wideband Non-Cooperative Systems based on Game Theory-Part I: Nash Equilibria

Gesualdo Scutari, Daniel P. Palomar, and Sergio Barbarossa
E-mail: aldo.scutari,sergio@infocom.uniroma1.it, palomar@ust.hk
Dpt. INFOCOM, Univ. of Rome “La Sapienza”, Via Eudossiana 18, 00184 Rome, Italy

Dpt. of Electronic and Computer Eng., Hong Kong Univ. of Science and Technology, Kowloon Hong Kong.
Submitted to IEEE Transactions on Signal Processing, September 22, 2005.
Revised March 14, 2007. Accepted June 5, 2007.thanks: This work was supported by the SURFACE project funded by the European Community under Contract IST-4-027187-STP-SURFACE.
Abstract

In this two-parts paper we propose a decentralized strategy, based on a game-theoretic formulation, to find out the optimal precoding/multiplexing matrices for a multipoint-to-multipoint communication system composed of a set of wideband links sharing the same physical resources, i.e., time and bandwidth. We assume, as optimality criterion, the achievement of a Nash equilibrium and consider two alternative optimization problems: 1) the competitive maximization of mutual information on each link, given constraints on the transmit power and on the spectral mask imposed by the radio spectrum regulatory bodies; and 2) the competitive maximization of the transmission rate, using finite order constellations, under the same constraints as above, plus a constraint on the average error probability. In Part I of the paper, we start by showing that the solution set of both noncooperative games is always nonempty and contains only pure strategies. Then, we prove that the optimal precoding/multiplexing scheme for both games leads to a channel diagonalizing structure, so that both matrix-valued problems can be recast in a simpler unified vector power control game, with no performance penalty. Thus, we study this simpler game and derive sufficient conditions ensuring the uniqueness of the Nash equilibrium. Interestingly, although derived under stronger constraints, incorporating for example spectral mask constraints, our uniqueness conditions have broader validity than previously known conditions. Finally, we assess the goodness of the proposed decentralized strategy by comparing its performance with the performance of a Pareto-optimal centralized scheme. To reach the Nash equilibria of the game, in Part II, we propose alternative distributed algorithms, along with their convergence conditions.

1 Introduction and Motivation

In this two-parts paper, we address the problem of finding the optimal precoding/multiplexing strategy for a multiuser system composed of a set of noncooperative wideband links, sharing the same physical resources, e.g., time and bandwidth. No multiplexing strategy is imposed a priori so that, in principle, each user interferes with each other. Moreover, to avoid excessive signaling and the need of coordination among users, we assume that encoding/decoding on each link is performed independently of the other links. Furthermore, no interference cancellation techniques are used and thus multiuser interference is treated as additive, albeit colored, noise. We consider block transmissions, as a general framework encompassing most current schemes like, e.g., CDMA or OFDM systems (it is also a capacity-lossless strategy for sufficiently large block length [1, 2]). Thus, each source transmits a coded vector

(1)

where is the information symbol vector and is the precoding matrix. Denoting with the channel matrix between source and destination , the sampled baseband block received by the -th destination is (dropping the block index)111For brevity of notation, we denote as source (destination) the source (destination) of link .

(2)

where is a zero-mean circularly symmetric complex Gaussian white noise vector with covariance matrix .222We consider only white noise for simplicity, but the extension to colored noise is straightforward along well-known guidelines. The second term on the right-hand side of (2) represents the Multi-User Interference (MUI) received by the -th destination and caused by the other active links. Treating MUI as additive noise, the estimated symbol vector at the -th receiver is

(3)

where is the receive matrix (linear equalizer) and denotes the decision operator that decides which symbol vector has been transmitted.

The above system model is sufficiently general to incorporate many cases of practical interest, such as: i) digital subscriber lines, where the matrices incorporate DFT precoding and power allocation, whereas the MUI is mainly caused by near-end cross talk [3]; ii) cellular radio, where the matrices contain the user codes within a given cell, whereas the MUI is essentially intercell interference [4]; iii) ad hoc wireless networks, where there is no central unit assigning the coding/multiplexing strategy to the users [5]. The I/O model in (2) is particularly appropriate for studying cognitive radio systems [6], where each user is allowed to re-use portions of the already assigned spectrum in an adaptive way, depending on the interference generated by other users. Many recent works have shown that considerable performance gain can be achieved by exploiting some kind of information at the transmitter side, either in single-user [2], [7]-[9] or in multiple access or broadcast scenarios (see, e.g. [10]). Here, we extend this idea to the system described above assuming that each destination has perfect knowledge of the channel from its source (but not of the channels from the interfering sources) and of the interference covariance matrix.

Within this setup, the system design consists on finding the optimal matrix set according to some performance measure. In this paper we focus on the following two optimization problems: P.1) the maximization of mutual information on each link, given constraints on the transmit power and on the spectral radiation mask; and P.2) the maximization of the transmission rate on each link, using finite order constellations, under the same constraints as above plus a constraint on the average (uncoded) error probability. The spectral mask constraints are useful to impose radiation limits over licensed bands, where it is possible to transmit but only with a spectral density below a specified value. Problem P.2 is motivated by the practical need of using discrete constellations, as opposed to Gaussian distributed symbols.

Both problems P.1 and P.2 are multi-objective optimization problems [11], as the (information/ transmission) rate achieved in each link constitutes a different single objective. Thus, in principle, the optimization of the transceivers requires a centralized computation (see, e.g., [12, 13] for a special case of problem P.1, with diagonal transmissions and no spectral mask constraints). This would entail a high complexity, a heavy signaling burden, and the need for coordination among the users. Conversely, our interest is focused on finding distributed algorithms to compute with no centralized control. To achieve this goal, we formulate the system design within a game theory framework. More specifically, we cast both problems P.1 and P.2 as strategic noncooperative (matrix-valued) games, where every link is a player that competes against the others by choosing its transceiver pair to maximize its own objective (payoff) function. This converts the original multi-objective optimization problem into a set of mutually coupled competitive single-objective optimization problems (the mutual coupling is precisely what makes the problem hard to solve). Within this perspective, we thus adopt, as optimality criterion, the achievement of a Nash equilibrium, i.e., the users’ strategy profile where every player is unilaterally optimum, in the sense that no player is willing to change its own strategy as this would cause a performance loss [14]-[16]. This criterion is certainly useful to devise decentralized coding strategies. However, the game theoretical formulation poses some fundamental questions: 1) Under which conditions does a NE exist and is unique? 2) What is the performance penalty resulting from the use of a decentralized strategy as opposed to the Pareto-optimal centralized approach? 3) How can the Nash equilibria be reached in a totally distributed way? 4) What can be said about the convergence conditions of distributed algorithms? In Part I of this two-part paper, we provide an answer to questions 1) and 2). The answer to questions 3) and 4) is given in Part II.

Because of the inherently competitive nature of a multi-user system, it is not surprising that game theory has been already adopted to solve many problems in communications. Current works in the field can be divided in two large classes, according to the kind of games dealt with: scalar and vector power control games. In scalar games, each user has only one degree of freedom to optimize, typically the transmit power or rate, and the solution has been provided in a very elegant framework, exploiting the theory of the so called standard functions [17]-[22]. The vector games are clearly more complicated, as each user has several degrees of freedom to optimize, like user codes or power allocation across frequency bins, and the approach based on the “standard” formulation of [17]-[19] is no longer valid.  A vector power control game was proposed in [23] to maximize the information rates (under constraints on the transmit power) of two users in a DSL system, modeled as a frequency-selective Gaussian interference channel. The problem was extended to an arbitrary number of users in [24]-[28]. Vector power control problem in flat-fading Gaussian interference channels was addressed in [29].

The original contributions of this paper with respect to the current literature on vector games [23]-[29] are listed next. We consider two alternative matrix-valued games, whereas in [23]-[27], [29] the authors studied a vector power control game which can be obtained from P.1 as a special case, when the diagonal transmission is imposed a priori and there are no spectral mask constraints. Problem P.2, at the best of the authors’ knowledge, is totally new. The matrix nature of the players’ strategies and the presence of spectral mask constraints make the analysis of both games P.1 and P.2 complicated and none of the results in [23]-[29] can be successfully applied. Our first contribution is to show that the solution set of both games is always nonempty and contains only pure (i.e., deterministic) strategies. More important, we prove that the diagonal transmission from each user through the channel eigenmodes (i.e., the frequency bins) is optimal, irrespective of the channel state, power budget, spectral mask constraints, and interference levels. This result yields a strong simplification of the original optimization, as it converts both complicated matrix-valued problems P.1 and P.2 into a simpler unified vector power control game, with no performance penalty. Interestingly, such a simpler vector game contains, as a special case, the game studied in [23]-[27], when the users are assumed to transmit with the same (transmit) power and no spectral mask constraints are imposed. The second important contribution of the paper is to provide sufficient conditions for the uniqueness of the NE of our vector power control game that have broader validity than those given in [23]-[27], [29] (without mask constraints) and, more recently, in [28] (including mask constraints). Our uniqueness condition, besides being valid in a broader context than those given in [23]-[29], exhibits also an interesting behavior not deducible from the cited papers: It is satisfied as soon as the interlink distance exceeds a critical value, almost irrespective of the channel frequency response. Finally, to assess the performance of the proposed game-theoretic approach, we compare the Nash equilibria of the game with the Pareto-optimal centralized solutions to the corresponding multi-objective optimization. We also show how to modify the original game in order to make the Nash equilibria of the modified game to coincide with the Pareto-optimal solutions. Not surprisingly, the Nash equilibria of the modified game can be reached at the price of a significant increase of signaling and coordination among the users.

The paper is organized as follows. In Section 2, the optimization problems P.1 and P.2 are formulated as strategic noncooperative games. Section 3 proves the optimality of the diagonal transmission and in Section 4 the conditions for the existence and uniqueness of the NE are derived. Section 5 gives a physical interpretation of the NE, with particular emphasis on the way each user allocates power across the available subchannels. Section 6 assesses the goodness of the NE by comparing the performance of the decentralized game-theoretic approach with the centralized Pareto-optimal solution. Numerical results are given in Section 7. Finally, in Section 8, the conclusions are drawn. Part of this work already appeared in [26, 27, 30, 31].

2 System Model and Problem Formulation

In this section we clarify the assumptions and constraints underlying the model (2) and we formulate the optimization problem addressed in this paper explicitly.

2.1 System model

Given the I/O system in (2), we make the following assumptions:

A.1 Neither user coordination nor interference cancellation is allowed; consequently encoding/decoding on each link is performed independently of the other links. Hence, the overall system in (2) is modeled as a vector Gaussian interference channel [34], where MUI is treated as additive colored noise;

A.2 Each channel is modeled as a FIR filter of maximum order and it is assumed to change sufficiently slowly to be considered fixed during the whole transmission, so that the information theoretical results are meaningful;

A.3 In the case of frequency selective channels, with maximum channel order , a cyclic prefix of length is incorporated on each transmitted block in (1);

A.4 A (quasi-) block synchronization among the users is assumed, so that all streams are parsed into blocks of equal length, having the same temporization, within an uncertainty at most equal to the cyclic prefix length;

A.5 The channel from each source to its own destination is known to the intended receiver, but not to the other terminals; an error-free estimate of MUI covariance matrix is supposed to be available at each receiver. Based on this information, each destination computes the optimal precoding matrix for its own link and transmits it back to its transmitter through a low (error-free) bit rate feedback channel.333In practice, both estimation and feedback are inevitably affected by errors. This scenario can be studied by extending our formulation to games with partial information [14, 15], but this goes beyond the scope of the present paper.

Assumption A.1 is motivated by the need of finding solutions, possibly sub-optimal, but that can be obtained through simple distributed algorithms, that require no extra signaling among the users. This assumption is well motivated in many practical scenarios, where additional limitations such as decoder complexity, delay constraints, etc., may preclude the use of interference cancellation techniques. Assumption A.3 entails a rate loss by a factor , but it facilitates symbol recovery. For practical systems, is sufficiently large with respect to , so that the loss due to CP insertion is negligible. Observe that, thanks to the CP insertion, each matrix in (2) resulting after having discarded the guard interval at the receiver, is a Toeplitz circulant matrix. Thus, is diagonalized as , with denoting the normalized IFFT matrix, i.e., for and is a diagonal matrix, where is the frequency-response of the channel between source and destination , including the path-loss  with exponent and normalized fading with denoting the distance between transmitter and receiver

The physical constraints required by the applications are:

Co.1 Maximum transmit power for each transmitter, i.e.,

(4)

where is power in units of energy per transmitted symbol, and the symbols are assumed to be, without loss of generality (w.l.o.g.), zero-mean unit energy uncorrelated symbols, i.e., . Note that different symbols may be drawn from different constellations.

Co.2 Spectral mask constraint, i.e.,

(5)

where represents the maximum power user is allowed to allocate on the -th frequency bin. 444Observe that if we obtain the trivial solution Constraints in (5) are imposed by radio spectrum regulations and attempt to limit the amounts of interference generated by each transmitter over some specified frequency bands.

Co.3 Maximum tolerable (uncoded) symbol error rate (SER) on each link, i.e.,555Given the symbol error probability the Bit Error Rate (BER) can be approximately obtained from (using a Gray encoding to map the bits into the constellation points) as where is the number of bits per symbol, and is the constellation size.

(6)

where is the -th entry of given in (3). Another alternative approach to guarantee the required quality of service (QoS) of the system is to impose an upper bound constraint on the global average BER of each link, defined as . Interestingly, in [35] it was proved that equal BER constraints on each subchannel as given in (6), provide essentially the same performance of those obtained imposing a global average BER constraint, as the average BER is strongly dominated by the minimum of the BERs on the individual subchannels. Thus, for the rest of the paper we consider BER constraints as in (6).

2.2 Problem Formulation: Optimal Transceivers Design based on Game Theory

In this section we formulate the design of the transceiver pairs of system (2) within the framework of game theory, using as optimality criterion  the concept of NE [14]-[16]. We consider two classes of payoff functions, as detailed next.

2.2.1 Competitive maximization of mutual information

In this section we focus on the fundamental (theoretic) limits of system (2), under A.1-A.5, and consider the competitive maximization of information rate of each link, given constraints Co.1 and Co.2. Using A.1, the achievable information rate for user is computed as the maximum mutual information between the transmitted block and the received block , assuming the other received signals as additive (colored) noise. It is straightforward to see that a (pure or mixed strategy) NE is obtained if each user transmits using Gaussian signaling, with a proper precoder . In fact, for each user, given that all other users use Gaussian codebooks, the codebook that maximizes mutual information is also Gaussian [34]. Hence, given A.5, the mutual information for the -th user is [34]

(7)

where is the interference plus noise covariance matrix, observed by user , and is the set of all the precoding matrices, except the -th one. Observe that, for each link, we can always assume that the receiver is composed of an MMSE stage followed by some other stage, since the MMSE is capacity-lossless. Thus, w.l.o.g., we assume in the following that666It is straightforward to verify that the MMSE receiver in (8) is capacity-lossless by checking that, for each the mutual information (for a given set of ) after the equalizer is equal to (7).

(8)

Hence, the strategy of each player reduces to finding the optimal precoding that maximizes in (7), under constraints Co.1 and Co.2.  Stated in mathematical terms, we have the following strategic noncooperative game

(9)

where is the set of players (i.e., the links), is the payoff function of player given in (7), and is the set of admissible strategies (the precoding matrices) of player , defined as

(10)

The solutions to (9) are the well-known Nash equilibria, which are formally defined as follows.

Definition 1

A (pure) strategy profile  is a NE of game if

(11)

The definition of NE as given in (11) can be generalized to contain mixed strategies [14], i.e., the possibility of choosing a randomization over a set of pure strategies (the randomizations of different players are independent). Hence, the mixed extension of the strategic game is given by where denotes the set of the probability distributions over the set of pure strategies. In game , the strategy profile, for each player is the probability density function defined on and the payoff function is the expectation of defined in (7) taken over the mixed strategies of all the players A mixed strategy NE of a strategic game is defined as a NE of its mixed extension [14].

Observe that for the payoff functions defined in (7), we can indeed limit ourselves to adopt pure strategies w.l.o.g., as we did in (9). Too see why, consider the mixed extension of in 9. For any player , we have

(12)

where . The inequality in (12) follows from the concavity of the function in [33] and from Jensen’s inequality [34]. Since the equality is reached if and only if reduces to a pure strategy (because of the strict concavity of in ), whatever the strategies of the other players are, every NE of the game is achieved using pure strategies.777This result was obtained independently in [29]-[31].

2.2.2 Competitive maximization of transmission rates

The optimality criterion chosen in the previous section requires the use of ideal Gaussian codebooks with a proper covariance matrix. In practice, Gaussian codes are substituted with simple (suboptimal) finite order signal constellations, such as Quadrature Amplitude Modulation (QAM) or Pulse Amplitude Modulation (PAM), and practical (yet suboptimal) coding schemes. Hence, in this section, we focus on the more practical case where the information bits are mapped onto constellations of finite size (with possibly different cardinality), and consider the optimization of the transceivers , in order to maximize the transmission rate on each link, under constraints Co.1 Co.3.

Given the signal model in (2), where now each vector is drawn from a set of finite-constellations , i.e., the transmission rate of each link is simply the number of transmitted bits per symbol, i.e.,

(13)

where denotes the size of constellation The (uncoded) average error probability of the -th link on the -th substream, as defined in (6), under the Gaussian assumption, can be analytically expressed, for any given set and as

(14)

where and are constants that depend on the signal constellation, is the -function [36], and is defined as

(15)

with where denotes the -th column of and (see, e.g., [7, 8]).

According to the constraints Co.3 in (6), because of (14), the optimal linear receiver for each user can be computed as the matrix maximizing simultaneously all the in (15), while keeping the set of precoding matrices and the constellations fixed. This leads to the well-known Wiener filter for as given in (8) [7, 8, 9], and the following expression for the s in (15):

(16)

Under the previous setup, each player has to choose the precoder and the constellations that maximize the transmission rate in (13), under constraints Co.1 Co.3. Since, for any given rate, the optimal combination of the constellations would require an exhaustive search over all the combinations that provide the desired rate, in the following we adopt, as in [9], the classical method to choose quasi-optimal combinations, based on the gap approximation [37, 38].888In our optimization we will use, as optimal solution, the continuous bit distribution obtained by the gap approximation, without considering the effect on the optimality of the granularity and the bit cap. The performance loss induced by these sources of distortion can be quantified using the approach given in [9]. As a result, the number of bits that can be transmitted over the substreams from the -th source, for a given family of constellations and a given error probability , is approximatively given by

(17)

where is defined in (16), and is the gap which depends only on the constellations and on For -QAM constellations, e.g., if the error probability in (14) is approximated by the resulting gap is [9].

In summary, the structure of the game is

(18)

where and are defined in (10) and (17), respectively. As in (9), in the following we focus on pure strategies only.

3 Optimality of the Channel-Diagonalizing Structure

We derive now the optimal set of precoding matrices for both games and and provide a unified reformulation of the original complicated games in a simpler equivalent form. The main result is summarized in the following theorem.

Theorem 1

An optimal solution to the matrix-valued games and is

(19)

where is the IFFT matrix, and with is the solution to the vector-valued game defined as

(20)

where and are the payoff function and the set of admissible strategies of user respectively, defined as

(21)

and

(22)

with

(23)

where and if is considered.

Proof. See Appendix A.  

Remark 1 Optimality of the diagonal transmission. According to Theorem 1, a NE of both games and is reached using, for each user, a diagonal transmission strategy through the channel eigenmodes (i.e., the frequency bins), irrespective of the channel realizations, power budget, spectral mask constraints and MUI. This result simplifies the original matrix-valued optimization problems (9) and (18), as the number of unknowns for each user reduces from (the original matrix to (the power allocation vector , with no performance loss.

Observe that the optimality of the diagonalizing structure was well known in the single-user case, when the optimization criterion is the maximization of mutual information and the constraint is the average transmit power [7]-[9], [35]. However, under the additional constraint on the spectral emission masks, the optimality of the diagonal transmission has never been proved, neither in a single-user nor in a multi-user competitive scenario. But, most interestingly, Theorem 1 proves the optimality of the diagonal transmission also for game  where each player maximizes the transmission rate, using finite order constellations, and under constraints on the spectral emission mask, transmit power, and average error probability. In such a case, the optimality of the channel-diagonalizing scheme was not at all clear. Previous works on this subject adopted the typical approach used in single-user MIMO systems [23]-[27]: They first imposed the diagonal transmission and then employed the gap approximation solution over the set of parallel subchannels. However, such a combination of channel diagonalization and gap approximation was not proved to be optimal. Conversely, Theorem 1 proves the optimality of this approach and it subsumes, as particular cases, the results of [23]-[27], corresponding to the simple case where there are no mask constraints.

It is also worth noticing that the optimality of the diagonalizing structure is a consequence of the property that all channel matrices, under assumptions A.2 and A.3, are diagonalized by the same matrix, i.e., the IFFT matrix . There is another interesting scenario where this property holds true: The case where all the channels are time-varying flat fading and the constraints are on the transmit power and on the maximum power that can be emitted over some specified time intervals (this is the dual version of the spectral mask constraint). In such a case, all channel matrices are diagonal and then it is trivial to see that they have a common diagonalizing matrix, i.e., the identity matrix. Applying duality arguments to Theorem 1, the optimal transmission strategy for each user is a sort of TDMA over a frame of time slots, where each user optimizes the power allocation across the time slots (possibly sharing time slots with the other users). Clearly, as opposed to the case considered in Theorem 1, in the time-selective case, the transmitter needs to have a non-causal knowledge of the channel variation. In practice, this kind of knowledge would require some sort of channel prediction.

According to Theorem 1, instead of considering the matrix-valued games and we may focus on the simpler vector game , with no performance loss. It is straightforward to see that a NE of both matrix-valued games exists if the solution set of is non empty. Moreover, the Nash equilibria of , if they exist, must satisfy the waterfilling solution for each user, i.e., the following system of nonlinear equations:

(24)

with the waterfilling operator defined as

(25)

where denotes the Euclidean projection of onto the interval 999The Euclidean projection  is defined as follows: , if , , if , and , if . and the water-level is chosen to satisfy the power constraint

Given the nonlinear system of equations (24), the fundamental questions are: i) Does a solution exist? ii) If a solution exists, is it unique? iii) How can such a solution be reached in a distributed way?

The answer to the first two questions is given in the forthcoming sections, whereas the study of distributed algorithms is addressed in Part II of this paper [32].

4 Existence and Uniqueness of NE

Before providing the conditions for the uniqueness of the NE of game we introduce the following intermediate definitions. Given game define as

(26)

where denotes the set (possibly) deprived of the carrier indices that user would never use as the best response set to any strategy used by the other users, for the given set of transmit power and propagation channels:

(27)

with defined in (25) and .

The study of game is addressed in the following theorem.

Theorem 2

Game  admits a nonempty solution set for any set of channels, spectral mask constraints and transmit power of the users. Furthermore, the NE is unique if

(C1)

where  is defined in (26) and  denotes the spectral radius101010The spectral radius of the matrix is defined as with denoting the spectrum of [50]. of

Proof. See Appendix B.  

We provide now alternative sufficient conditions for Theorem 2. To this end, we first introduce the matrix , defined as

(28)

with the convention that the maximum in (28) is zero if is empty. Then, we have the following corollary of Theorem 2.

Corollary 1

A sufficient condition for (C1) is:

(C2)

where is defined in (28).

To give additional insight into the physical interpretation of the conditions for the uniqueness of the NE, we introduce the following corollary.

Corollary 2

A sufficient condition for (C1) is given by one of the two following set of conditions:

(C3)
(C4)

where is any positive vector and is defined in (26).

Note that, as a by-product of the proof of Theorem 2, one can always choose in (C1)-(C4), i.e., without excluding any subcarrier. However, less stringent conditions are obtained by removing the unnecessary carriers, i.e., those carriers that, for a given power budget and interference levels, are never going to be used.

Remark 2 Physical interpretation of uniqueness conditions. As expected, the uniqueness of NE is ensured if the links are sufficiently far apart from each other. In fact, from (C3)-(C4) for example, one infers that there exists a minimum distance beyond which the uniqueness of NE is guaranteed, corresponding to the maximum level of interference that may be tolerated by the users. Specifically, condition (C3) imposes a constraint on the maximum amount of interference that each receiver can tolerate; whereas (C4) introduces an upper bound on the maximum level of interference that each transmitter is allowed to generate. This result agrees with the fact that, as the MUI becomes negligible, the rates in (21) become decoupled and then the rate-maximization problem in (20) for each user admits a unique solution. But, the most interesting result coming from conditions (C1)-(C4) is that the uniqueness of the equilibrium is robust against the worst normalized channels in fact, the subchannels corresponding to the highest ratios (and, in particular, the subchannels where is vanishing) do not necessarily affect the uniqueness condition, as their carrier indices may not belong to the set .

Remark 3 Uniqueness condition and distributed algorithms. Interestingly, condition (C2), in addition to guarantee the uniqueness of the NE, is also responsible for the convergence of both simultaneous and sequential iterative waterfilling algorithms, proposed in Part II of the paper [32].


Remark 4 Comparison with previous conditions. Theorem 2 unifies and generalizes many existence and uniqueness results obtained in the literature [23]-[27], [29] for the special cases of game in (20). Specifically, in [23]-[27] a game as in (20) is studied, where all the players are assumed to have the same power budget and no spectral mask constraints are considered [i.e., ]. In [29] instead, the channel is assumed to be flat over the whole bandwidth. Interestingly, the conditions obtained in [23]-[27], [29] are more restrictive than (C1)-(C4), as shown in the following corollary of Theorem 2.111111We summarize the main results of [23]-[27] using our notation to facilitate the comparison.

Corollary 3

Sufficient conditions for (C3) are [23, 24, 27]

(C5)

or [25]

(C6)

In the case of flat-fading channels (i.e., , , condition (C3) becomes [29]

(29)

Recently, alternative sufficient conditions for the uniqueness of the NE of game were given in [28].121212We thank Prof. Facchinei, who kindly brought to our attention reference [28], after this paper was completed. Among all, an easy condition to be checked is the following:

(C7)

where is defined as in (26), with each

All the conditions above depend on the channel realizations and on the network topology through the distances Hence, there is a nonzero probability that they are not satisfied for a given set of channel realizations, drawn from a given probability space. In order to compare the goodness of the above conditions, we tested them over a set of channel impulse responses generated as vectors composed of i.i.d. complex Gaussian random variables with zero mean and unit variance. We plot in Figure 1 the probability that conditions (C1), (C5) and (C7) are satisfied versus the ratio , i.e., the normalized interlink distance. For the sake of simplicity, we assumed and We considered [Figure 1(a)] and [Figure 1(b)] active links. We tested our condition considering in (C1) a set obtained using the following worst case scenario. For each user , we build the worst possible interferer as the virtual node (denoted by ) that has a power budget equal to the sum of the transmit powers of all the other users (i.e., ) and channel between its own transmitter and receiver as the highest channel among all the interference channels with respect to receiver i.e., We build a set that includes the set defined in (27) using the following iterative procedure: For each subcarrier the virtual user distributes its own power () across the whole spectrum in order to facilitate user to use the subcarrier as much as possible. If, even under these circumstances, user is not going to use subcarrier because of its own power budget and then we are sure that index can not possibly belong to

(a)
Figure 1: Probability of (C1), (C5) and (C7) versus ; [subplot (a)], [subplot (b)],  

We can see, from Figure 1, that the probability that the NE is unique increases as the links become more and more separated of each other (i.e., the ratio increases). Furthermore, we can verify that, even having not considered the smallest possible set , as defined in (27), our condition (C1) has a much higher probability of being satisfied than (C5) and (C7). The main difference between our condition (C1) and (C5), (C7) is that (C1) exhibits a neat threshold behavior since it transits very rapidly from the non-uniqueness guarantee to the almost certain uniqueness, as the inter-user distance ratio increases by a small amount. This shows that the uniqueness condition (C1) depends, ultimately, on the interlink distance rather than on the channel realization. This represents the fundamental difference between our uniqueness condition and those given in the literature. As an example, for a system with links and probability of guaranteeing the uniqueness of the NE, condition (C1) requires whereas conditions (C5) and (C7) require and , respectively. Furthermore, this difference increases as the number of links increases.

5 Physical Interpretation of NE

In this section we provide a physical interpretation of the optimal power allocation corresponding to the NE in the limiting cases of low and high MUI.131313For the sake of notation, in this section we consider only the case in which but it is straightforward to see that our derivations can be easily generalized to the case of spectral mask constraints. To quantify what low and high interference mean, we introduce the SNR of link (denoted by ) and the Interference-to-Noise Ratio due to the interference received by destination and generated by source with (denoted by ), defined as and Using and the SINR in (23) can be rewritten as

(30)

Low interference case.  Consider the low interference case, i.e., the situation where the interference term in the denominator in (30) can be neglected. A sufficient condition to satisfy this assumption is that the links are sufficiently far apart from each other, i.e., ,. For sufficiently small and sufficiently large , condition (C1) is satisfied and, hence, by Theorem 1, the NE is unique. Also, by inspection of the waterfilling solution in (24), it is clear that under those conditions, for all and . This means that each source uses the whole bandwidth. Furthermore, it is well known that as the SNR increases, the waterfilling solution tends to a flat power allocation. In summary, we have the following result.

Proposition 1

Given game there exist sets of values and with and such that, for all and the NE of is unique (cf. Theorem 2) and all users share the whole available bandwidth. In addition, if then the optimal power allocation of each user tends to be flat over the whole bandwidth.

From Proposition 1, it turns out, as it could have been intuitively guessed, that when the interference is low, at the (unique) NE, every user transmits over the entire available spectrum (like a CDMA system), as in such a case nulling the interference would not be worth of the bandwidth reduction. As a numerical example, in Figure 2, we plot the optimal power spectral density (PSD) of a system composed of three links, for different values of the ratio . The results shown in Figure 2 have been obtained using the distributed algorithms described in Part II [32]. From Figure 2, we can check that, as the ratio increases, the optimal PSD tends to be flat, while satisfying the simultaneous waterfilling condition in (24).

High interference case. When for all and , the interference is the dominant contribution in the equivalent noise (thermal noise plus MUI) in the denominator of (30). In this case, game admits multiple Nash equilibria. An interesting class of these equilibria includes the FDMA solutions (called orthogonal Nash equilibria), occurring when the power spectra of different users are nonoverlapping. The characterization of these equilibria is given in the following.

Proposition 2

Given game for each let denote the set of subcarriers over which only user transmits. For any given there exists with each such that for all , game admits multiple orthogonal Nash equilibria. If, in addition, and are such that

(31)

and an orthogonal NE still exists, the subcarriers are allocated among the users according to

(32)

Proof. See Appendix C.  

The above proposition has an intuitive interpretation: When the interference is very high, the users self-organize themselves in order to remove the interference totally, i.e., using nonoverlapping bands. In this case, game may have multiple orthogonal Nash equilibria. For example, in the simple case of there are different orthogonal Nash equilibria, corresponding to all the permutations where each transmitter uses only one carrier. As the interference level decreases (i.e., the ’s), the NE becomes unique and in such a case, if an orthogonal equilibrium still exists, then the distribution of the subcarriers among the users must satisfy the rule given by (32). This strategy is similar, in principle to FDMA, but differently from standard FDMA, here each user is getting the “best” portion of the spectrum for itself. Interestingly, (32) is the generalization of the condition satisfied by the subcarrier allocation in the multiple access frequency-selective channel, where the optimization problem is the sum-rate maximization under a transmit power constraint [39]. In Figure 2(b), we show a numerical example of the optimal power allocation at NE, for a system with two active links, in the case of high interference.

(a) Optimal PSDs at NE in the low-interference case.
(b) Optimal PSDs at NE in the high-interference case.
Figure 2: Optimal PSDs at NE in the low-interference and high-interference cases: a) Solid, dashed, and dashed-dot line curves refer on the PSD obtained for ,, respectively, and dB; b) Solid and dashed lines refer to PSD of each link and PSD of the MUI plus thermal noise, normalized by the channel transfer function square modulus of the link, respectively;