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Wireless Networks 12, 105–117, 2006 2006 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Asymptotically Optimal Power Control Policies for MMSE Multiuser Detector ZVI ROSBERG Department of Communication Systems Engineering, Ben Gurion University, Beer-Sheva, 84105, Israel

Abstract. Asymptotically optimal uplink transmission power control policies are derived for synchronous and asynchronous CDMA time varying fading channels with multiple user classes, random spreading sequences and MMSE multiuser detector. One optimal policy minimizes the outage probability subject to maximum average power budget and maximum transmission power constraints. Another minimizes the average power budget subject to maximum outage probability and maximum transmission power constraints. If the channel states are available, we show that both optimal policies are channel inversion threshold cut-off policies and express them explicitly. When channel states are estimated, we show how to transform the optimal policies into estimator-based policies while controlling the degradation of their performance measures . Finally, we compare between the performance of the optimal policies and other policies in an environment with Lognormal and Rayleigh fading. Keywords: power control, outage probability, CDMA, random spreading sequence, fading channels, MMSE multiuser detector

1. Introduction Code Division Multiple Access (CDMA) is a major building block for modern wireless networks such as IS-95, CDMA 2000 and W-CDMA. The technological advantages of CDMA such as decentralized control, graceful QoS degradation, natural facilitation of macro-diversity and soft handover, immunity to multipath fading and resistance to intentional or self-jamming have been long recognized. Nevertheless, the near-far problem is still one of the main problems in a CDMA cellular network. A near-far situation occurs when strong received user signals deteriorate weak received user signals due to non-orthogonal spreading sequences. The near-far problem can be addressed by two methods. One is transmission power control, a method employed in almost every modern wireless system, and the other is multiuser detection [18]. In [18], it has been shown that the optimal multiuser detector can drastically reduce the near-far problem but its computational complexity increases exponentially with the number of users. A suboptimal multiuser detector which has a blind adaptive implementation [5] and attractive computational complexity is the Linear Minimum Mean Square Error (L-MMSE) detector commonly referred to as MMSE. A blind adaptive implementation is desirable in cellular networks since it does not require information about the interfering users such as their spreading sequences and transmission powers. Another suboptimal multiuser detector which also has a blind adaptive implementation is the Decorrelator. However, the bit error rate of the Decorrelator is higher than that of the MMSE. The blind adaptive implementations of both detectors require a constant received power from every E-mail: [email protected]

active user. With this respect, power control is a necessity. Moreover, with MMSE multiuser and no power control, the near-far problem persists even without blind adaptive implementation. The advantages of the MMSE multiuser detector on one hand, and its need for power control on the other hand, has motivated this study. One conclusion of this study is that the optimal policies achieve constant received power and therefore can be jointly used with a blind adaptive implementation. In addition to solving the near-far problem, transmission power control prolongs mobile battery life and can be used to allocate different channel qualities to users having different Quality of Services (QoS) requirements. The latter is incorporated into our model by allowing multiple user classes. In this paper we consider the uplink transmission power control problem in a synchronous and asynchronous CDMA time varying fading channel with multiple user classes, random spreading sequences and MMSE multiuser detector. We derive two optimal constrained power control policies assuming channel state information is available and the system has a large number of users (K ) and a proportionally large processing gain (N ). One optimal policy minimizes the outage probability subject to maximum average power budget and maximum transmission power constraints. The other minimizes the average power budget subject to maximum outage probability and maximum transmission power constraints. We show that both optimal policies are threshold cut-off policies. Namely, at every channel state h, each optimal policy allocates transmission power inversely proportional to h and cuts off transmission power at some threshold h ∗ determined by the problem constraints. We also show how to compute the optimal thresholds and their corresponding outage probabilities analytically.

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Power control has been first studied [1,8,23] using a snapshot system, where all channel gains are fixed and known. Moreover, the power control has been defined by a centralized algorithm maximizing the minimum Signal to Interference Ratio (SIR) or attaining a common SIR target for all users. Within this framework, it has been shown that the optimal transmission powers are determined by solving the eigenvalues of non-negative matrices. A distributed algorithmic approach using the same snapshot model has been later taken in [3,7,22], where algorithms converging to a common SIR target has been derived. Those algorithms iteratively and distributively update the transmission powers based on measurable local information. A general framework for distributed uplink power control in a snapshot model that extends most of the earlier iterative distributed algorithms has been derived in [21]. In [15], the framework from [21] has been used to derive a distributed iterative algorithm in a snapshot model for a MMSE multiuser detector. The algorithm combines transmission power update with the computation of the MMSE receiver filter coefficients. Given that all users can achieve their SIR target, it has been shown that all the iterative algorithms in [3,7,15,21,22] converge to the minimum average power solution subject to constraints on each user SIR target. The shortcoming of those algorithms is that users which cannot achieve their SIR targets transmit with maximum power. In the model we study, the SIR targets may not be achievable at all time. Furthermore, when the number of users K approaches infinity while K /N is held constant, we express the optimal transmission power explicitly rather than algorithmically. An information theoretic approach to the power control problem has been taken in [4,11,12]. A single user fading channel with Channel Side Information (CSI) has been considered in [4]. It was shown there that the power allocation that maximizes the Shannon capacity subject to an upper bound constraint on the average transmission power over time is a “water-filling” policy in time. A multiaccess fading channel with K users and CSI has been considered in [11,12], where the objective has been to find the minimum average power policy that achieves the Shannon capacity region. It was shown there that there exists a vector of “power prices” λ = (λ1 , . . . , λ K ) such that for every vector of channel gains h = (h l , h 2 , . . . , h K ), the optimal power allocation corresponds to optimal successive decoding of the users in an order given by a permutation π satisfying λπ (1) / h π (1) ≥ · · · ≥ λπ (K ) / h π(K ) . The optimal powers were also found and expressed explicitly as a function of h and the required Shannon capacities. An algorithm to find the power prices λ was also proposed. In this paper we consider the channel capacities of a MMSE multiuser detector rather than the Shannon capacities. The remainder of the paper is organized as follows. In Section 2, we describe the system model and define the problem. Asymptotic optimal power controls based on channel

state information are derived in Section 3; and in Section 4, we show how to translate them into policies based on channel estimators. In Section 5, we compare the performance of the optimal power controls in a Lognormal and Rayleigh fading channels to those of Constant Transmit Power (CTP) control and Constant Received Power (CRP) control. Finally, conclusions and discussion are presented in Section 6.

2. System model and problem definition We consider a cellular system with symbol synchronous or asynchronous CDMA time varying fading channel and processing gain N. We focus on a single base station receiver that controls the transmission power of K intra and inter-cell users from J different classes. Each user class  j comprises K j = α j N  users, where α j is fixed and K = Jj=1 K j . In the first part of the paper we analyze the following synchronous channel model and then we extend the results to an asynchronous model. Following the convention in [18], the baseband synchronous received signal during the n-th symbol time interval in the presence of time variant fading and AWGN, is given by K   y(n) = xi (n)s i (n) pi (n)h i (n) + w(n), (1) i=1

where i is a user label. In (1), for every symbol n and user i, xi (n) is the symbol information with E[xi2 (n)] = 1, s i (n) is the unit√energy spreading sequence (signature) vector of length N , h i (n) is the random channel gain and pi (n) is the user transmission power. The vector w(n) of length N denotes the AWGN with zero mean and power spectral density σ 2 . The baseband model in (1) is appropriate for flat fading when sampling rate is not sufficiently fast to resolve multiple paths. As usual, we assume that for every user i, the channel gain power {h i (n) | n ≥ 0} is a stationary and ergodic process, and that all channel gain processes are independent and identically distributed (i.i.d.). Note however, that the process {h i (n) | n ≥ 0} could be time correlated. We model the spreading sequences (signatures) s i = √1 (si (1), si (2), . . . , si (N ))T by random sequences, where N si ( j) are assumed to be i.i.d. random variables with mean zero, variance one and bounded fourth moment. Such sequences accurately models pseudo-random number (PN) sequences used for some CDMA channels in IS-95, CDMA-2000 and WCDMA. The spreading sequences are chosen independently of the channel gains and once chosen they are known by the receiver and the transmitters. We assume that the receiver knows the fading state of every uplink channel during every symbol time n. Channel fading state estimation (which is a broad subject) is beyond the scope of this paper. Classical generic procedures are based on training sequences, channel sounding [14] and least square techniques [9]. Advanced and specialized

ASYMPTOTICALLY OPTIMAL POWER CONTROL POLICIES

techniques are also given [2,10,16]. In Section 4, we show how to transform policies based on channel state information into policies based on a channel estimator while maintaining the required outage probability and average transmission power. We focus on a MMSE multiuser detector from the reasons outlined above that have been further supported by [17,19]. Transmission power control is a major device to manage spectral resources in CDMA systems. Our objective is to derive and evaluate optimal constrained power control policies that have practical implementations. Since spreading sequences are known by each pair of communicating parties and synchronous CDMA is assumed in our first part, randomness is introduced only by the K -tuple of the uplink channel gains h(n) = (h 1 (n), h 2 (n), . . . , h K (n)) and the AWGN. For every n, h(n) = h = (h 1 , h 2 , . . . , h K ) will be referred to as the (system) state at time n. For every given power control policy, let p j (h) and γ j (h) denote the stationary transmission power and instantaneous received SIR, respectively, corresponding to a user from class j at state h. Also, for every specific user, let F(h) denote a common continuous stationary cumulative distribution function (cdf) with finite first and second moments defining the user channel gain distribution. Since uplink channel gains are assumed to be independent, the K cartesian product F K (h) is their joint cdf. Every user class corresponds to a communication service that dictates an average power budget P¯ j , a SIR target γ jt , and an outage probability  j < 1. A communication service j is supported at state h if {γ j (h) ≥ γ jt } and experiences outage, otherwise. Communication during supported states is possible and during outage it is not. In practice, mobile transmission power is bounded by a maximum transmission power depending on the system (e.g., 1 W att in DCS 1800 and PCS 1900, and 2 W att in GSM). Therefore, we constrain the user transmission power by a maximum transmission power denoted by Pmax . A stationary power control policy p j (h) is admissible for class j, if p j (h) ≤ Pmax and there is a set of states h, H j , satisfying the following requirements: (R1) SIR requirement: For every state h ∈ H j , γ j (h) ≥ γ jt . Namely, H j is the supported set for class j. (R2) Outage probability: P(H j ) ≥ 1− j , where P(H j ) =  K d h∈ H j F (h) is the probability not having an outage. (R3) Power budget: The average  transmission power does not exceed P¯ j . Namely, p j (h)d F K (h) ≤ P¯ j . A power control policy is admissible if it is admissible for every class j. For a given SIR target, the set H j above is referred to as the SIR target supported set. Note that by our definition, an admissible policy must satisfy p j (h) ≤ Pmax . For finite N and K , the users are tightly coupled and finding optimal power policies in this setting seems intractable. Therefore, as in [19,20], we consider a large system where

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K j = α j N  for every j and N → ∞. As will be noted below, for the interesting power control policies, this limiting regime provides excellent approximations for processing gains N ≥ 128. From implementation reasons, we further restrict our attention to stationary power control policies of the form p j (h) = p j (h j ), where h j is the uplink channel gain of the controlled user. In the asymptotic case, the optimal power for user j depends only on h j (see the interpretation after Corollary 1 below). For every power control policy p(h), denote by outage( p), its outage probability and by budget( p) its power budget. Finding the optimal power control policy in a large system with random spreading is greatly simplified by the results from [13] presented below.

3. Asymptotic power control For the sake of presentation we begin with a single user class. For every processing gain N and specific user, let ξ (N ) be the random SIR at the MMSE receiver determined by the received power of the user of interest, and the interference power introduced by the AWGN and the K − 1 other multiple access interfering users (MAI) combined. The following Theorem is due to Tse and Hanely, [13, Theorem 3.1]. Theorem 1 [Tse and Hanly [13, Theorem 3.1]]. If K = α N , the random sequence {ξ (N )} converges to ξ in probability as N → ∞, where ξ is the unique solution to the equation P ξ= (2)  p P . 2 σ + α E p P+ξ p Here, P is the received power and E p [·] is the expectation with respect to the distribution of the received power. A straightforward corollary of Theorem 1 and [13, Proposition 3.2] is the following. Let p(h) be a stationary power control policy and 1/β ∗ ( p) be the asymptotic interference power introduced by the AWGN and the K − 1 other MAI users combined, given that each user uses policy p(h). Corollary 1. The asymptotic interference power 1/β ∗ ( p) is the unique solution to the equation β ∗ ( p) =

σ 2 + α Eh



1

 h p(h) 1+β ∗ ( p) h p(h)

,

(3)

where E h [·] is the expectation with respect to the distribution of the wireless channel gain. Furthermore, for any β, β ∗ ( p) > β if and only if σ2

+ α Eh

1 

h p(h)  1+β h p(h)

> β.

(4)

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Although non-strict inequalities are used in [22, Proposition 3.2], strict inequalities (which is needed below) follow from the uniqueness of β ∗ . The expression in (3) can be interpreted as follows. The interference and noise power in a system with a large number of users (and a proportional large processing gain) employing some power control policy is approximately deterministic. Clearly, the actual interference power depends on the employed policy. In [13], it has been demonstrated that for policies p(h) with constant received power and N = 128, 1/β(N ) is 1 − 2dB away from the asymptotic value 1/β ∗ ( p). This observation is meaningful since the optimal policies, as will be shown below, do have constant received power. Hereinafter we replace β(N ) by β ∗ ( p). Consequently, using policy p(h), the instantaneous SIR in a user uplink channel with gain h is given by γ p∗ (h) = h p(h) β ∗ ( p).

(5)

In the following, we will derive the asymptotic optimal power control policies with respect to either one of the two optimality criteria defined below. In cases where outage is more severe than power budget, it is sensible to use the following notion of optimality. Given a SIR target γ t and an outage probability , an admissible policy p(h) is optimal outage-constrained, if it minimizes the average transmission power over all admissible policies. That is, it solves the optimization program: min E h [ p(h)] p(h)

subject to E h [I{h | γ p∗ (h) < γ t }] ≤ ,

(6) (7)

where I{·} is the set indicator function. In cases where power budget is more important than outage, it is sensible to constrain the power budget and minimize the outage probability. That is, given a SIR target γ t and ¯ an admissible policy is optimal powera power budget P, constrained, if it minimizes the outage probability over all admissible policies. That is, it solves the optimization program: min E h [I{h | γ p (h) < γ t }] p(h)

¯ subject to E h [ p(h)] ≤ P.

(8) (9)

3.1. Cut-off policies A cut-off policy is a policy that cuts off the transmission power when the SIR target is either not attainable or the required power is too high. The following special cut-off policies will be shown to form an optimal set of policies with respect to both optimality criteria defined above. For every SIR target γ t and admissible power control policy p(h), define R ∗ ( p) = γ t /β ∗ ( p).

(10)

Given an outage probability  > 0 and a SIR target γ t , the specially structured cut-off policies are defined by p H (h) =

R∗( p H ) I{h | h ∈ H }, h

(11)

where R ∗ ( p H ) = γ t /β ∗ ( p H ) ≤ h Pmax and H is a SIR target supported set satisfying P(H ) ≥ 1 − . Note that R ∗ in (11) depends on the employed policy as β ∗ does, and equals the minimum power required to attain the instantaneous SIR target. The cut-off policies in (11), cut off the transmission power when the channel gain is not in the SIR target supported set H . Also, note that H is now a set of scalar uplink gains and not of K -tuples. The following policy improvement Lemma shows that we can restrict attention to cut-off policies with a specific structure defined in (11). Lemma 1. The set of power control policies { p(h)}, where ∗ p(h) ∈ {0, R h( p) }, is optimal with respect to both optimality criteria. Proof. Let p (0) (h) be an admissible policy. For every n ≥ 1, define the following sequence of general cut-off policies: 

  R ∗ p (n−1) R ∗ p (n−1) (n) (n−1) p (h) = I h|p . (h) ≥ h h (12) Note that the cut-off policies in (12) do not have the special structure as defined in (11). First, we show by induction on n that if p (n−1) (h) is admissible, then policy p (n) (h) is also admissible and satisfies:   p (n) (h) ≤ p (n−1) (h), ∀ h; R ∗ p (n) ≤ R ∗ p (n−1) ; (13)   budget p (n) ≤ budget p (n−1) ;   outage p (n) ≤ outage p (n−1) .

(14)

The proofs for n = 1 and for the induction step are identical. Let n ≥ 1. The definition in (12) implies that p (n) (h) ≤ p (n−1) (h), which further implies  1 β ∗ p (n) =   (n) 2 σ + α E h 1+β ∗ (hp(n)p ) h(h)p(n) (h) ≥

σ2

+ α Eh



1

 h p (n−1) (h) 1+β ∗ ( p (n) ) h p (n−1) (h)

.

(15)

The first equality above follows from (3) of Corollary 1 and the inequality is implied by p (n) (h) ≤ p (n−1) (h). By (4) of Corollary 1, (15) implies that β ∗ ( p (n) ) ≥ ∗ (n−1) β (p ). Otherwise, a contradiction to the inequality in (15) would result. From (10), we also have   R ∗ p (n) ≤ R ∗ p (n−1) . (16)

ASYMPTOTICALLY OPTIMAL POWER CONTROL POLICIES

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Let h be a channel gain in the supported set of policy ∗ (n−1) p (n−1) (h). Then, (5) implies that p (n−1) (h) ≥ R ( ph ) . Combining it with (16) and the definition of p (n) (h) we obtain,   R ∗ p (n) R ∗ p (n−1) p (n) (h) = ≥ , h h ∀ h in the supported set of p (n−1) (h).

The following Lemma provides a closed form expression for R ∗ ( p H ) for every cut-off policy p H (h), which facilitates the derivation of the optimal policies. Lemma 3. For every given SIR target γ t and feasible cut-off policy p H (h), R∗( p H ) =

(n)

By (5) again, h is also in the supported set of p (h). Therefore, outage( p (n) ) ≤ outage( p (n−1) ). Furthermore, p (n) (h) ≤ p (n−1) (h) implies that budget (n) ( p ) ≤ budget( p (n−1) ) and p (n) (h) ≤ Pmax . Hence, p (n) (h) is admissible and outperforms p (n−1) (h) with respect to outage probability and power budget, which completes the proof of (13) and (14). Consequently, { p (n) (h)} is a positive non-increasing (component-wise) sequence of functions and therefore converges to a function p ∗ (h). Since {R ∗ ( p (n) )} is also a positive non-increasing sequence, it also converges, and by (12), we have limn→∞ R ∗ ( p (n) ) = R ∗ ( p ∗ ). Furthermore, p ∗ (h) is an admissible cut-off power control policy conforming the special structure defined in (11), and it outperforms p(h) with respect to outage probability and power budget. The following Lemma shows that without loss of generality we can restrict attention to policies p(h) satisfying the equality outage( p) = . ¯ then the set of power Lemma 2. Given a triplet (γ t , , P), control policy { p(h)}, where outage( p) = , is optimal with respect to both optimality criteria. H

Proof. By Lemma 1, without loss of generality, let p (h) be a cut-off policy as defined in (11), where P(H ) > 1 − . Since F(h) h for which the is continuous,

there is a positive

set H = H \ [0, h ] satisfies P(H ) = 1 − . Define the following initial improvement policy p (0) (h) =

R ∗ (P H ) I{h | h ∈ H }. h

The definitions of P H (h) and H imply that p (0) (h) is admissible and satisfies p (0) (h) ≤ P H (h). Furthermore, budget( p (0) ) ≤ budget(P H ) and outage( p (0) ) ≤ outage(P H ). For every n ≥ 1, define the following sequence of general cut-off policies:  R ∗ p (n−1) (n) p (h) = (17) I{h | h ∈ H }. h As in the proof of Lemma 1, the sequence of policies in (17)

converges to the admissible policy P H (h), which outperH forms P (h) with respect to both optimality criteria. Fur thermore, outage(P H ) = .

σ2 γt 1−

α P(H ) γ t 1+γ t

.

(18)

Proof. From (3) it follows that for every cut-off policy P H (h), γt =

R∗( p H ) σ2 +

α R ∗ ( p H ) P(H ) 1+γ t

,

(19)

from which R ∗ ( p H ) is uniquely resolved by (18). Observe that for every given γ t , R ∗ ( p H ) = R ∗ (γ t , P(H )). That is, the limiting interference power of cut-off policies p H (h) depend only on P(H ) rather than on the exact set H of supported channel states. The observation above facilitates the proofs in the next two subsections, where we show that both optimal policies, for the outage constrained and for the power budget constrained problem, are threshold cut-off policies. It is worth nothing that these optimal policies are analogous to the “water-filling” policy maximizing Shannon capacity subject to the power budget constrained in a time variant single user fading channel [4]. 3.2. Outage constrained optimality In this subsection we derive the optimal policy for the optimization program (6) and (7). Given a SIR target γ t and an outage probability , Lemmas 1–3 imply that the optimal outage constrained policy is given by the supported set H that solves:  R ∗ (γ t , P(H )) min d F(h). (20) {H :P(H )=1−} h∈H h In the following Theorem, we show that the optimal outage-constrained policy is a threshold cut-off policy defined below. Theorem 2. If an admissible policy exists for , then the op∗ timal outage-constrained policy is the cut-off policy p H (h), ∗ H∗ ∗ ∗ where R ( p ) is given by (18), H = [h , ∞) and h ∗ is determined by  ∞ P(H ∗ ) = d F(h) = 1 − . (21) h∗

Proof. From Lemmas 1–3 and (20), it is sufficient to show that every admissible cut-off policy p H (h) with H = H ∗ and ∗ P(H ) = 1 −  is strictly worse than p H (h). Define the sets ∗ ∗ ∗ H1 = H \ {H ∩ H } and H1 = H \ {H ∗ ∩ H }. Since H =

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H ∗ , H1 and H1∗ are nonempty disjoint sets. Moreover, since P(H ∗ ) = P(H ) = 1 − , P(H1∗ ) = P(H1 ),

(22)

and by their definitions we have: H1 ∩ H ∗ = φ ; H1∗ ∩ H = φ and H1∗ ⊂ H ∗ . Therefore, h < h , for every h ∈ H1 and h ∈ H1∗ .

(23)

3.3. Power constrained optimality In this subsection, we derive the optimal policy for the optimization program (8) and (9). ¯ Lemmas 1–3 Given a SIR target γ t and a power budget P, ipmly that the optimal outage constrained policy is given by the supported set H that solves:

Also note that

{H :

H ∗ = {H \ H1 } ∪ H1∗ .



=  =  +



∗

h∈H1∗

p H (h)d F(h) + ∗

h∈H1∗

h∈{H \H1 }

< P(H1∗ ) 

∗

h∈{H \H1 }

p H (h)d F(h)

∗

R (γ , p(H )) d F(h) h t

R ∗ (γ t , p(H ∗ )) d F(h) h

R ∗ (γ t , P(H ∗ )) h min (H1∗ )

R ∗ (γ t , p(H ∗ )) d F(h) h h∈{H \H1 }  R ∗ (γ t , p(H )) R ∗ (γ t , P(H )) P(H1 ) + d F(h) h min (H1∗ ) h h∈{H \H1 }  R ∗ (γ t , p(H )) R ∗ (γ t , P(H ) P(H1 ) + d F(h) h sup (H1 ) h h∈{H \H1 }   R ∗ (γ t , p(H )) R ∗ (γ t , p(H )) dF(h)+ d F(h) h h h∈H1 h∈{H \H1 }  p H (h)d F(h). (25)

+ = < < =

h∈H

¯ p H (h)d F(h)≤ P}

P(H ).

(26)

(24)

To complete the proof, it is left to show that the average power ∗ of p H (h) is strictly less than that of p H (h). Let h min (H ) and h sup (H ) be the minimum and suprimum values, respectively, in the set H . The following holds true ∗ for the average power of policy p H (h):  ∗ p H (h)d F(h) h∈H ∗

min



In the following theorem we show that the optimal powerconstrained policy is also a threshold cut-off policy that differs from the optimal outage-constrained policy only by its threshold value. ¯ then the Theorem 3. If an admissible policy exists for P, ∗ optimal power-constrained policy is the cut-off policy p H (h), where H ∗ = [h ∗ , ∞) and the value of h ∗ is determined as follows. ˆ = [h, ˆ ∞) and hˆ be the solution to Let H  ∞ 1 ˆR(γ t , P( H ˆ )) ¯ d F(h) = P, (27) h ˆh ˆ. where Rˆ is given by (18) when replacing H with H ˆ max , then H ∗ = H ˆ and h ∗ = h. ˆ Otherwise, (i) If hˆ ≥ R/P ∗ ∗ ∗ (ii) h = R /Pmax , where R is the fixed point of (18) with H = [R ∗ /Pmax , ∞). In this case, the average power of ¯ the optimal policy is strictly less than P. Proof. From Theorem 2 it follows that for every policy p H (h) that attains an outage probability , there is a threshold cut-off policy with the same outage probability and at most the same average transmission power. Therefore, we may consider only threshold cut-off policies. (i) Suppose that the condition in case (i) of the theorem holds true. That is, the transmission power at the cut-off value is not greater than Pmax . We will show that in this case an admissible threshold cut-off policy with  p H (h)d F(h) < P¯ (28) h∈H

h∈H

The first equality is implied by (24). The second equality follows from (11) and (18). The next strict inequality is due to the continuity of F. The subsequent equality is a consequence of P(H ∗ ) = P(H ) and (22). The following inequality is a result of (23) and the continuity of F. The rest is straightforward. Observing that the first and last terms of (25) are the av∗ erage power of policy p H (h) and of p H (h), respectively, completes the proof. Note that the value of h ∗ as determined by (21) could be constrained by Pmax .

can strictly be improved. Let H = [h, ∞) be a SIR target supported set that satisfies (28) and I () = [h − , h) be its left adjacent interval of length . For channel gains in the interval I (), the function P(I ()) R ∗ (γ t , P(H ) + P([h − , h))) h− bounds the average transmission power from above. Note that f (0) = 0 and from (18), f () is continuously increasing with . Hence, there ∗  is a positive ∗ H such that f ( ) attains the value P¯ − h∈H p (h)d F(h).  That is, f (∗ ) = P¯ − h∈H p H (h)d F(h) > 0. Let H = f () =

ASYMPTOTICALLY OPTIMAL POWER CONTROL POLICIES

111

H ∪ I (∗ ). Since I (∗ ) and H are disjoint, the power

ˆ. budget of policy P H (h) equals P¯ and therefore H = H

Thus, H is admissible and P(H ) > P(H ). That is,

p H (h) strictly improves p H (h), which completes the proof of case (i). (ii) Suppose that the condition in case (i) of the theorem is not satisfied. That is, the transmission power at the cut-off value is greater than Pmax . We will show that in this case

any admissible threshold cut-off policy p H (h) defined

∗ ∗ by H = [h , ∞) with h = R /Pmax and R as given in case (ii) of the theorem, is not optimal. First observe that h ≥ R ∗ /Pmax . Indeed, suppose in contradiction that h < R ∗ /Pmax . We will show

that the transmission power of p H (h) in the interval

∗ [h , R /Pmax ) exceeds Pmax . Since p H (h) is admissible,

∗ for every h ∈ [h , R /Pmax ),

p H (h) = R / h > R ∗ / h > Pmax ,

h ≥ R ∗ /Pmax .

Note that for case (i) of the Theorem 3, h ∗ and its corresponding R ∗ are uniquely determined since the left hand side ˆ If it is strictly less than R ∗ /Pmax , in (27) is decreasing with h. ∗ ∗ then the optimal h is R /Pmax , where R ∗ is the unique solution to (19). In the next subsection we show that for multiple user classes the optimal policies are also threshold cut-off policies but with class dependent threshold values. 3.4. Multiple user classes The asymptotic regime in the case of J user classes is a system with K j = α j N  and N → ∞. In this case, the fixed point equation to resolve β ∗ for a given stationary policy ( p1 (h 1 ), p2 (h 2 ), . . . , p J (h J )) becomes +

J i=1

αi



1 h pi (h) h 1+β ∗ h pi (h) d F(h)

β∗ = Substituting β ∗ with γ jt =

.

(30)

Equation (3) holds true also with class dependent distributions {Fi (h)}.

σ2 +

J

σ2 + γ jt R ∗j

J

1

i=1

Ri∗ P(Hi ) 1+β ∗ Ri∗

.

(32)

for every j in (32), we have

R ∗j

i=1

αi

αi

Ri∗ P(Hi ) 1+γit

,

j = 1, 2, . . . , J.

(33)

It is easy to verify that for every given SIR targets γ jt and sets H j , j = 1, 2, . . . , J , {R ∗j } are resolved from (33) by R ∗j =

Consequently, P(H ) < P([R ∗ /Pmax , ∞)), which completes the proof.

σ2

where R ∗j = γ jt /β ∗ and H j is a SIR target supported set that satisfies P(H j ) ≥ 1 −  j . As in the single class case, β ∗ and R ∗j depend on the employed policy. Since we do not reproduce the proofs in the multiple class and in the asynchronous cases, we omit this dependency from our notations below. Since all uplink channel gains are assumed to be independent, the multiple class case is not fundamentally different from the single class case. For any cut-off policy, (30) takes to form

(29)

where R is given by (18) when replacing H with H . The first inequality in (29) follows from the contradiction assumption and the fact that the left hand side of (18) increases with P(H ). The second inequality is the result of our contradiction assumption. Thus, the maximum transmission power constraint is violated and we do have

β∗ =

A cut-off policy in the multiple class case is defined by J power control functions ⎧ ∗ ⎨ Rj (31) p H j (h) = h , h ∈ H j ; ⎩0, otherwise,

γ jt σ 2 , J γ t P(H ) 1 − i=1 αi i 1+γ t i

j = 1, 2, . . . , J.

(34)

i

Similar to the single class case, R ∗j = R ∗j (γ t , P H ), where  γ t = (γ1t , γ2t , . . . , γ Jt ), P H = P(H1 ), P(H1 ), . . . , P(H J ) , and it increases with every P(H j ). Closely following the derivation in Subsection 3.1, it can be shown that for the multiple class case we may also restrict attention to cut-off policies with P(H j ) = 1 −  j , for every class j. Similarly, we also obtain the following extensions for the two optimal policies. Theorem 4. If an admissible policy exists for (1 , . . . ,  J ), then the optimal outage-constrained policy is a threshold cut∗ off policy defined by the power control functions p H j (h), j = ∗ ∗ 1, 2, . . . , J , where the R j s are given by (34), H j = [h ∗j , ∞) and each h ∗j is determined by  ∞ ∗ P(H j ) = d F(h) = 1 −  j . h ∗j

Theorem 5. If an admissible policy exists for ( P¯ 1 , . . . , P¯ J ), then the optimal power-constrained policy is the cut-off policy ∗ p H j (h), j = 1, 2, . . . , J , where H j∗ = [h ∗j , ∞) and the values of h ∗j are determined as follows. Let Hˆ j = [hˆ j , ∞) and hˆ j be the solution to  ∞1  t ˆ ˆ R j γ j , P( H j ) d F(h) = P¯ j , hˆ j h

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where Rˆ j is given by (34) after replacing all Hi with Hˆ i . (i) If hˆ j ≥ Rˆ j /Pmax , then H j∗ = Hˆ j and h ∗j = hˆ j . Otherwise, (ii) h ∗j = R ∗j /Pmax , where R ∗j is the fixed point of (34) with H j = [R ∗j /Pmax , ∞). In this case, the average power of the optimal policy is strictly less than P¯ j . In the next subsection we extend our results to asynchronous CDMA and show that the optimal policies remain threshold cut-off policies. 3.5. Asynchronous CDMA We consider the symbol asynchronous case model where a symbol of every interfering user is delayed by a fraction of symbol time τ with respect to the received symbol. We assume that the delay fractions of all interfering users are independent uniform random variables. We further assume that the receiver is chip-synchronized with every transmitter and has an observation window of a single symbol. Note that under these assumptions, a typical interfering user will have two consecutive symbols that interfere with the received symbol. From [6 Theorem 3.2], β ∗ is bounded below by β ∗ that is uniquely resolved by the following fixed point equation: β∗ =

σ2

+

J

αi

i=1

1 0

1 + B(h))d F(h) dτ

h (A(h)

, (35)

where A(h) =

τ h pi (h) 1 + τ β ∗ h pi (h)

B(h) =

(1 − τ ) h pi (h) . 1 + (1 − τ ) β ∗ h pi (h)

σ2

+

J i=1

αi

 1 τ 0

1

Ri∗ P(Hi ) 1+τ β ∗ Ri∗

+

(1−τ ) Ri∗ P(Hi ) dτ 1+(1−τ ) β ∗ Ri∗

.(36)

Integrating the definite integral in (36) by parts yields a simpler fixed-point equation of the form β∗ =

σ2 +

J i=1

1 2 αi P(Hi ) ∗ β



1−

ln(1+β ∗ Ri∗ ) β ∗ Ri∗

γ jt =

σ2 +

R ∗j γ jt

γ jt R ∗j

in (37) yields

R ∗j

J i=1

 2 αi P(Hi ) 1 −

ln(1+γit ) γit

.

(38)

As before, it is easy to verify that for every given SIR targets γ jt and sets H j , j = 1, 2, . . . , J , R ∗j is resolved from (38) by R ∗j =

1−

J i=1

γ jt σ 2

 2 α i P(Hi ) 1 −

ln(1+γit ) γit

.

(39)

As for the synchronous case above, R ∗j = R ∗j (γ t , P H ), and it increases with every P(H j ). Similarly, we also obtain the following extensions for the two optimal policies. Theorem 6. If an admissible policy exists for (1 , . . . ,  J ), then the optimal outage-constrained policy is closely approximated by a threshold cut-off policy defined by the power ∗ control functions p H j (h), j = 1, 2, . . . , J , where the R ∗j s are given by (39), H j∗ = [h ∗j , ∞) and each h ∗j is determined by  ∞ P(H j∗ ) = d F(h) = 1 −  j . h ∗j

Theorem 7. If an admissible policy exists for ( P¯ 1 , . . . , P¯ J ), then the optimal power-constrained policy is closely approxi∗ mated by a threshold cut-off policy p H j (h), j = 1, 2, . . . , J , where H j∗ = [h ∗j , ∞) and the values of h ∗j are determined as follows. Let Hˆ j = [hˆ j , ∞) and hˆ j be the solution to   ∞1 Rˆ j γ jt , P( Hˆ j ) d F(h) = P¯ j , hˆ j h where each Rˆ j is given by (39) with H j = Hˆ j .

Recall that 1/β ∗ is the asymptotic interference power. Thus 1/β ∗ is an upper bound to the asymptotic interference power in our asynchronous CDMA model. It has been also demonstrated in [6] that the difference between the bound β ∗ and the exact β ∗ is within a 0.5 dB over a wide range of α’s and reasonable noise levels. For the cut-off policies defined in (31), where Ri∗ = γit /β ∗ and Hi is the SIR target supported set that satisfies P(Hi ) ≥ 1 − i , (35) takes the form β∗ =

For every given j, replacing β ∗ with

.

(37)

(i) If hˆ j ≥ Rˆ j /Pmax , then H j∗ = Hˆ j and h ∗j = hˆ j . Otherwise, (ii) h ∗j = R ∗j /Pmax , where R ∗j is the fixed point of (39) with H j = [R ∗j /Pmax , ∞). In this case, the average power of the optimal policy is strictly less than P¯ j .

4. Estimator based policies To implement the optimal policies in practice, channel states {h} need to be estimated. In this section we show how to transform any cut-off threshold policy based on h into a policy based on an estimator hˆ while maintaining a desirable outage probability and power budget. For presentation clarity we focus on a single user class and synchronous channel. The extension to multiple user classes and asynchronous channel is straightforward. ˆ Suppose we are given an estimator hˆ satisfying P(| h−h |≤ h δ) = 1 for some 0 < δ < 1; or equivalently, P((1 − δ)h ≤ hˆ ≤ (1 + δ)h) = 1.

(40)

ASYMPTOTICALLY OPTIMAL POWER CONTROL POLICIES

113

∗

Let p H (h) be a cut-off threshold policy for some SIR target γ t , where R ∗ is given by (18) and H ∗ = [h ∗ , ∞). ∗ ˆ For every p H (h), define the following corresponding hbased suboptimal policy  (1−δ)R ∗ , hˆ ≥ (1 + δ)h ∗ ; H∗ ˆ hˆ pˆ (h) = (41) 0, otherwise. From (40) and (41) it follows that for every realization h ˆ p H ∗ (h) ≥ pˆ H ∗ (h). ˆ That is, and respective estimator value h, ∗ ˆ transfor every realization of a channel state, policy p H (h) mits with higher power compared with its corresponding pol∗ ˆ Thus, we may write icy pˆ H (h). ∗

∗

p H (h) ≥ pˆ H (h) for every h,

(42)

ˆ although pˆ (·) is a function of h. Since the channel fading distribution is known, R ∗ and ∗ ∗ ˆ is well defined and can H are known and policy pˆ H (h) be implemented in practice. In the next Theorem we derive a relation between the outage probabilities and the average power budgets of both policies. Denote by outage( p, γ t ) and budget( p, γ t ) the outage probability and average transmission power, respectively, given SIR target γ t and policy p(h). ∗

Theorem 8. For every cut-off threshold policy p H (h),   ∗ H∗ 1 − δ t outage pˆ , γ ≤ outage( p H , γ t ) 1+δ + F((1 + δ)h ∗ ) − F(h ∗ ) and

 budget pˆ  −

H∗

1−δ t γ , 1+δ

(1+δ)h ∗ h∗



(43)

∗

≤ budget( p H , γ t )

R∗ d F(h). h

(44)

Proof. Let β ∗ and βˆ be the fixed-point solutions to (3) when ∗ ∗ p(h) = p H (h) and p(h) = pˆ H (h), respectively. First, we ˆ That is, the interference will use (42) to show that β ∗ ≤ β. H∗ power generated with policy p (h) is greater than that with ∗ ˆ Define the transformations policy pˆ H (h). T (β) =

Tˆ (β) =

σ2 + α

σ2 + α



1 h p H ∗ (h) h 1+β h p H ∗ (h) d F(h)



1 h pˆ H ∗ (h) h 1+β h pˆ H ∗ (h) d F(h)

,

(45)

.

(46)

Observe that both transformations are uniformly bounded and increase with β. For every n ≥ 2, recursively define the composite transformations T (n) (β) = T (n−1) (T (β)), where T (1) (β) = T (β). Similarly for Tˆ (n) (β). Consequently, it follows from (3) that β ∗ = limn→∞ T (n) (0) and βˆ = limn→∞ Tˆ (n) (0). Moreover,

∗ ∗ since p H (h) ≥ pˆ H (h) implies that T (β) ≤ Tˆ (β) we also have

ˆ β ∗ ≤ β.

(47)

ˆ be the instantaneous SIR with policy Now, let γˆ t (h, h) pˆ (·), given that the channel state is h and the estimator ˆ For every hˆ ≥ (1 + δ)h ∗ we have, value is h. H∗

∗ ˆ ˆ = (1 − δ)R h β ≥ (1 − δ) R ∗ βˆ ≥ (1 − δ) R ∗ β ∗ γˆ t (h, h) (1 + δ) (1 + δ) hˆ (1 − δ) t = (48) γ . (1 + δ)

The first equality in (48) follows from (41). The next inequality follows from (40) and the subsequent inequality follows from (47). The last equality is merely the definition of a cutoff policy in (11). Thus, the SIR target supported set of policy ∗ pˆ H (·) contains the set [(1 + δ)h ∗ , ∞). The definition in (41) and the inequality in (48) immediately imply relation (43) of the theorem. Furthermore, along with (40), they also imply that  1−δ  budget pˆ , γt 1+δ   (1 − δ)R ∗ R∗ = d F(h) ≤ d F(h) hˆ (1+δ)h ∗ (1+δ)h ∗ h  ∗   (1+δ)h ∗ R ∗ = budget p H , γ t − d F(h), h h∗ which completes the proof. ∗ ˆ polTheorem 8 implies that the performance of the pˆ H (h) icy converges to the performance of its corresponding policy ∗ p H (h) as δ → 0. The performance bounds in Theorem 8 facilitate the design of a practical power control policy for any given SIR target and power budget. Thus, by taking an opti∗ mal policy p H (h) one can approximate its performance by ∗ ˆ policy. using its corresponding pˆ H (h)

5. Performance comparison To explore the advantage of the optimal policies we compare the optimal power constrained policy (OPT) with a single user class to other policies. One policy is the Constant Transmit Power (CTP) which does not require any information about the channel states. The other is the Constant Received Power (CRP) that requires the same information as the optimal policy requires. ¯ the CTP policy Given an average power constrained P, ¯ regardless of the channel transmits with power p(h) = P, state h. The CTP policy on the other hand, attempts to maintain a constant received power by allocating all the power budget P¯ across all channel states subject to p(h) ≤ Pmax . If the required transmission power exceeds Pmax , CRP transmits

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114

with power Pmax . Specifically, with CRP C C , if ≤ Pmax ; p(h) = h h Pmax , otherwise, where C is determined by  C/Pmax  Pmax d F(h) + C C/Pmax

0

1 ¯ d F(h) = P. h

The CRP policy differs from OPT by three aspects: (1) it does not cut-off the transmission power; (2) it utilizes the entire power budget with no attempt to minimize the average transmission power; and (3) the received power is determined as a function of P¯ and the channel state distribution F(h) rather than as a function of the SIR target. Basically, the CRP policy is similar to most power control policies used in practice. Obviously, the perfect information of the channel state is replaced with an adaptive estimator. Since all policies, CTP, CRP and OPT, are special cases of the set policies for which the asymptotic noise and interference power given in (3) apply, their outage probabilities can numerically be computed based on (5). In our examples, we assume a radio transmission frequency of 900 MHz (common for GSM and W-CDMA) and a channel gain subject to path loss with a path loss parameter n = 3 (typical for outdoor cellular environments), Lognormal shadow fading with a standard deviation of 8 dB and Rayleigh fading with mean 1. The cell radius is set to 1500 meters. We have compared also between the policies in an environment without Rayleigh fading (i.e., with path loss and Lognormal fading). Since the behavior is similar, we discuss the findings without presenting the figures. Figures 1–4 present the outage blocking probabilities as a function of the SIR target using policies CTP, CTP and OPT

for medium and high values of α (0.5 and 1.0) and two values of power budgets. Observe that CTP has tolerable outage probabilities only for α = 0.5, P¯ = 0.125 Watt and a SIR target below 8 dB. Without Rayleigh fading, CTP has tolerable outage probabilities up to 18 dB. Thus, CTP completely breaks for the low power budget regime. Another observation is that CTP cannot sustain high values of α. Namely, CTP cannot support a number of users that is in the order of the processing gain. Figures 1–4 further reveal that policies CRP and OPT achieve the same outage probabilities up to some SIR tar¯ From that threshold get threshold (that depends on α and P). and on, the outage probabilities of OPT continue to increase slowly and those of CRP jump to 1. The higher is α and the ¯ the smaller is the threshold. This behavior of CRP lower is P, is not surprising since its transmission power does not account for the SIR target. To study the effect of α on policies CRP and OPT we depict in figure 5 the outage blocking probabilities for multiple values of α’s and P¯ = 0.125 Watt as a function of the SIR target. In figure 6 we do the same with P¯ = 0.01 Watt. The notation α = Max in these figures stands for 0.99 + 1/γ t (which is slightly smaller than the upper limit of the maximum SIR target with an MMSE receiver). We observe that reducing the power budget from 0.125 Watt to 0.01 Watt does not affect OPT (for any α) but has a major effect on CRP. For example, if P¯ = 0.01 Watt and α ≥ 0.5, CRP has an outage probability of 1 for all SIR targets greater than 7 dB. Without Rayleigh fading this occurs for all α ≥ 1. The resistance of OPT to power budget reduction is contributed by the fact that with the power budgets taken so far, the optimal transmission power is constrained by Pmax and not by the power budgets. To find how OPT behaves in the range of very low power budgets we compute its outage probabilities for P¯ = 10−6 Watt (figure 7). For power budget values 1

1

CTP

Fading = Path loss, Lognorml & Rayleigh

0.9

0.9

Noise Power = 1e−011 Watt 0.8

0.7

Outage probability

Outage probability

Noise Power = 1e−011 Watt

K/N = 0.5

0.7

Fading = Path loss, Lognorml & Rayleigh

0.8

Avg Power = 0.125 Watt

0.6

0.5

0.4

Avg Power = 0.125 Watt K/N = 1.0

0.6

0.5

0.4 CRP

0.3

0.3 CRP

CTP 0.2

0.2

0.1

0

0.1

OPT & CRP

6

8

10

12

14

16

18

SIR Target (dB)

Figure 1. Outage probabilities using OPT, CTP and CRT with α = 0.5 and P¯ = 0.125 Watt.

0

OPT & CRP

6

8

10

12

14

16

18

SIR Target (dB)

Figure 2. Outage probabilities using OPT, CTP and CRT with α = 1.0 and P¯ = 0.125 Watt.

ASYMPTOTICALLY OPTIMAL POWER CONTROL POLICIES

115 0.4

1

Fading = Path loss, Lognorml & Rayleigh

Fading = Path loss, Lognorml & Rayleigh

0.9

Avg Power = 0.125 Watt

Avg Power = 0.06 Watt

0.3

OPT Power Control

K/N = 0.5 Outage probability

0.7

Outage probability

Noise Power = 1e−011 Watt

0.35

Noise Power = 1e−011 Watt 0.8

0.6

0.5

0.4

CRP

0.25

0.2

0.15

CRP

0.3 CTP

0.1

0.2

K/N=Max K/N=0.75

0.05

K/N=0.5

K/N=1.0

0.1 OPT & CRP 0

0

6

8

10

12

14

16

18

6

8

10

Figure 3. Outage probabilities using OPT, CTP and CRT with α = 0.5 and P¯ = 0.06 Watt.

14

16

18

Figure 5. Outage probabilities using OPT and CRT with P¯ = 0.125 Watt and several values of α. 0.4

1

Fading = Path loss, Lognorml & Rayleigh CTP

0.9

Noise Power = 1e−011 Watt

0.35

Avg Power = 0.01 Watt

Fading = Path loss, Lognorml & Rayleigh

0.8

0.3

Noise Power = 1e−011 Watt 0.7

Avg Power = 0.06 Watt Outage probability

Outage probability

12

SIR Target (dB)

SIR Target (dB)

K/N = 1.0

0.6

0.5

0.4

OPT Power Control CRP

0.25 K/N=0.5 0.2

0.15

CRP 0.3

0.1 K/N=Max

0.2

K/N=0.75 0.05

0.1

K/N=0.5

OPT 0

0

K/N=1.0

6

8

10

12

14

16

6

8

10

12

14

16

18

SIR Target (dB)

18

SIR Target (dB)

Figure 4. Outage probabilities using OPT, CTP and CRT with α = 1.0 and P¯ = 0.06 Watt.

greater than 10−6 Watt we have observed that the optimal transmission power is still constrained by Pmax and its outage probabilities remain unchanged. Figure 7 reveals that for every α, the OPT policy maintains low outage probabilities that are slowly increasing with SIR target up to some threshold value that depends on α. From that threshold and on, where the optimal transmission power ¯ the outage probabilities increase with a is constrained by P, much higher rate. Thus, OPT is most appropriate for systems with a tight power budget. To study the behavior of OPT with respect to different fading channels and very low power budget we depict in figure 8 its outage probabilities with and without the Rayleigh fading component. It can be observed that without Rayleigh fading the power budget constraint is attained at a SIR target that

Figure 6. Outage probabilities using OPT and CRT with P¯ = 0.01 Watt and several values of α.

is 2–3 dB greater compared with the Rayleigh fading case. As far as outage probabilities are concerned, there is no significant difference when the power budget constraint is not attained.

6. Conclusion We studied the uplink transmission power control problem in a synchronous and asynchronous CDMA channel with multiclass users and random spreading sequences. Such system accurately model modern cellular networks such as IS-95, CDMA-2000 and W-CDMA. For a system with a large number of users (K ) and a proportionally processing gain (N ), we derived two types of optimal constrained power control policies assuming complete

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timator while maintaining a desirable outage probability and power budget. To explore the advantage of the optimal policies we compared the constrained policy that minimizes the outage probability (OPT) to two other policies: The Constant Transmission Power (CTP) and the Constant Received Power (CRP). The following main results were demonstrated in a system with path loss, Lognormal and Rayleigh fading.

0.4

Fading = Path loss, Lognorml & Rayleigh Noise Power = 1e−011 Watt

0.35

Avg Power = 0.000001 Watt

Outage probability

0.3

OPT Power Control

0.25

0.2

0.15 K/N=Max 0.1

0.05

0

K/N=1.0

6

K/N=0.5

K/N=0.75

8

10

12

14

16

18

SIR Target (dB)

Figure 7. Outage probabilities using OPT with P¯ = 10−6 Watt and several values of α.

0.4

Noise Power = 1e−011 Watt Avg Power = 0.000001 Watt

0.35

OPT w/o Rayleigh

Outage probability

0.3

OPT with Rayleigh

0.25

(1) CTP badly fails in the low power budget regime. (2) CTP cannot support a number of users that is in the order of the processing gain. (3) CRP and OPT have the same outage probabilities up to ¯ For some SIR target threshold that depends on α and P. SIR targets greater than that threshold, the outage probabilities of OPT increases slowly and those of CRP jump to 1. (4) Reducing the power budget from 0.125 Watt to 0.01 Watt has a drastic effect on CRP, whereas reducing it to as low as 10−5 Watt does no affect OPT at all. Only with a power budget of 10−6 Watt, the outage probabilities with OPT start increasing faster for SIR targets greater than 14 dB for α = 0.5; greater than 12 dB for α = 0.75; and greater than 9 dB for α = 1.0. (5) As long as the power budget constraint is not attained, the channel fading distribution has only a marginal effect on the OPT outage probabilities.

0.2

K/N =0.5

0.15 K/N =Max

K/N =0.75

K/N =1.0

References

0.1

0.05

0

6

8

10

12

14

16

18

SIR Target (dB)

Figure 8. Outage probabilities using OPT with and without Rayleigh fading component, P¯ = 10−6 Watt and several values of α.

channel state information. One optimal policy minimizes the outage probability subject to a maximum average power budget and a maximum transmission power. The other minimizes the average power budget subject to a maximum outage probability and a maximum transmission power. We proved that both optimal policies are threshold cutoff policies that allocate the transmission power as follows. At every channel state h, the optimal policy transmits with power that is inversely proportional to h, if within the problem constraints, or cuts off the transmission power, otherwise. We also showed how to compute the optimal policies outage probabilities analytically. We further showed how to transform the optimal policies (that are based on channel state information) into policies that are based on channel state es-

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