Long-Term Energy Constraints and Power Control in Cognitive Radio Networks
François Mériaux, Yezekael Hayel, Samson Lasaulce, Andrey Garnaev
aa r X i v : . [ c s . N I] J u l LONG-TERM ENERGY CONSTRAINTS AND POWER CONTROL IN COGNITIVE RADIONETWORKS
Franc¸ois M´eriaux , Yezekael Hayel , Samson Lasaulce , Andrey Garnaev L2S - CNRS - SUPELEC - Univ Paris-SudF-91192 Gif-sur-Yvette, France { meriaux,lasaulce } @lss.supelec.fr Lab. d’Informatique d’Avignon - Universit´e d'Avignon84911 Avignon - [email protected] V. I.Zubov Research Institute of Computational Mathematics & Control ProcessesSt Petersburg State University, Russia [email protected]
ABSTRACT
When a long-term energy constraint is imposed to a transmitter, theaverage energy-efficiency of a transmitter is, in general, not max-imized by always transmitting. In a cognitive radio context, thismeans that a secondary link can re-exploit the non-used time-slots.In the case where the secondary link is imposed to generate no in-terference on the primary link, a relevant issue is therefore to knowthe fraction of time-slots available to the secondary transmitter, de-pending on the system parameters. On the other hand, if the sec-ondary transmitter is modeled as a selfish and free player choosingits power control policy to maximize its average energy-efficiency,resulting primary and secondary signals are not necessarily orthog-onal and studying the corresponding Stackelberg game is relevantto know the outcome of this interactive situation in terms of powercontrol policies.
Index Terms — Cognitive radio, Energy-efficiency, Power con-trol, Primary user, Secondary user, Stackelberg games.
1. INTRODUCTION
One of the ideas of cognitive radio is to allow some wireless termi-nals, especially transmitters, to sense their environment in terms ofused spectrum and to react to it dynamically. The cognitive radioparadigm [1] has become more and more important to the wirelesscommunity since the release of the FCC report [2]. Indeed, cogni-tive radio corresponds to a good way of tackling the crucial prob-lem of spectrum congestion and increasing spectral efficiency. Morerecently, the main actors of the telecoms industry, namely carriers,manufacturers, and regulators have also realized the importance ofenergy aspects in wireless networks (see e.g., [3]) both at the net-work infrastructure and mobile terminal sides. There are many rea-sons for this and we will not provide them here. As far as this paperis concerned, the goal is to study the influence of long-term energyconstraints (e.g., the limited battery life typically) on power controlin networks where cognitive radios are involved. The performancecriterion which is considered for the terminal is derived from theone introduced by Goodman and Mandayam in [4]. Therein, the au- thors propose a distributed power control scheme for frequency non-selective block fading multiple access channels. For each block, aterminal aims at maximizing its individual energy-efficiency namely,the number of successfully decoded bits at the receiver per Jouleconsumed at the transmitter. Although, a power control maximizingsuch a performance metric is called energy-efficient, it does not takeinto account possible long-term energy constraints. Indeed, in [4]and related references (e.g., [5][6]), the terminals always transmit,which amounts to considering no constraints on the available (aver-age) energy. The goal of the present work is precisely to see howenergy constraints modify power control policies in a single-userchannel and in a cognitive radio channel. For the sake of simplic-ity, time-slotted communications are assumed.The paper is organized in two main parts. In Sec. 3 a single-user channel is considered. It is shown that maximizing an averageenergy-efficiency under a long-term energy constraint leads the ter-minal to not transmit on certain blocks. The probability that theterminal does not transmit is lower bounded. In a setting where aprimary transmitter has to control its power under energy-constraint,this probability matters since it corresponds to the fraction of avail-able time-slots which are re-exploitable by a secondary (cognitive)transmitter. In Sec. 3, the single-user channel model is sufficientsince the secondary link has to meet a zero interference constraint (itcan only exploit non-used time-slots). In Sec. 4, the secondary trans-mitter is assumed to be free to use all the time-slots. The technicaldifference between the primary and secondary transmitters is that theformer has to choose its power level in the first place while the latterobserves this level and react to it. The suited interaction model istherefore a Stackelberg game [7] where the primary and secondarytransmitters are respectively the leader and follower of the game.Sec. 5 provides numerical results which allow us to validate somederived results and compare the two cognitive settings (dependingwhether the secondary transmitter can generate non-orthogonal sig-nals). . GENERAL SYSTEM MODEL
In the whole paper the goal is to study a system comprising twotransmitter-receiver pairs. The signal model under consideration canbe described by a frequency non-selective block fading channel. Thesignals received by the two receivers write as: y = h x + h x + z y = h x + h x + z . (1)The channel gain of the link ij namely, h ij is assumed to be con-stant over each block or time-slot. The quantity g ij = | h ij | isassumed to be a continuous random variable having independent re-alizations and distributed according to the probability density func-tion φ ij ( g ij ) . The reception noises are zero-mean complex whiteGaussian noises with variance σ . The instantaneous power of thetransmitted signal x i on time-slot t is given by p i ( t ) = 1 N N X n =1 | x ( n ) | (2)where n is the symbol index and N the number of symbols per time-slot. For simplicity, transmissions are assumed to be time-slotted.Transmitter (resp. ), receiver (resp. ), link (resp. )will be respectively called primary (resp. secondary) transmitter,primary (resp. secondary) receiver, and (resp. secondary) primarylink. The main technical difference between the primary and the sec-ondary links is that the secondary transmitter can observe the powerlevels chosen by the primary transmitter but the converse does nothold. In this paper, two scenarios are investigated:• Scenario 1 (Sec. 3): the secondary transmitter is imposed tomeet a zero-interference constraint on the primary link. Sincethe primary and secondary signals are orthogonal, everythinghappens for the transmitter as if it was transmitting over asingle-user channel.• Scenario 2 (Sec. 4): this time, the secondary transmitter canuse all the time-slots and not only those not exploited by theprimary link. Primary and secondary signals are thereforenot orthogonal in general. In this framework, for each time-slot, the primary transmitter chooses its power level and isinformed that the secondary will observe and react to it in arational manner. A Stackelberg game formulation is proposedto study this interactive situation.
3. WHEN PRIMARY AND SECONDARY SIGNALS AREORTHOGONAL3.1. Optimal power control scheme for the primary transmitter
From the primary point of view, there is no interference and thesignal-to-noise plus interference ratio (SINR) coincides with thesignal-to-noise ratio (SNR):
SNR( p ( g )) = g p ( g ) σ . (3)When using the notation p ( g ) instead of p ( t ) we implicitlymake appropriate ergodicity assumptions on g . The main purposeof this section is precisely to determine the optimal control function p ( g ) in the sense of the long-term energy efficiency, which is de-fined as follows: u ( p ( g )) = R Z + ∞ φ ( g ) f (SNR( p ( g ))) p ( g ) d g (4) where R is the transmission rate and f is an efficiency functionrepresenting the packet success rate f : R + → [0 , . The function f is assumed to possess the following properties:1. f is non-decreasing, C differentiable, f (0) = 0 , lim x → + ∞ f ( x ) =1 and there exists a unique inflection point x for f .2. f ′ is non-negative, f ′ (0) = lim x → + ∞ f ′ ( x ) = 0 . f ′ reaches itsmaximum for x .3. f ′′ is non-negative over [0 , x ] , negative over [ x , + ∞ [ . f (2) (0) = 0 , lim x → + ∞ f ′′ ( x ) = 0 − .These properties are verified by the two typical efficiency functionsavailable in the literature: f a ( x ) = (cid:26) e − ax ∀ x > if x = 0 (5)and f M ( x ) = (cid:0) − e − x (cid:1) M ∀ x ≥ . (6)The function f a , a ≥ has been introduced in [8][9] and corre-sponds to the case where the efficiency function equals one minus theoutage probability. On the other hand, f M , M ∈ N ∗ , correspondsto an empirical approximation of the packet success rate which wasalready used in [4].Compared to references [4][5][6], note that the user’s utilityis the average energy-efficiency and not the instantaneous energy-efficiency. This allows one to take into account the following energyconstraint: T Z + ∞ φ ( g ) p ( g )d g ≤ E (7)where T is the time-slot duration and E is the available energy forterminal . In order to find the optimal solution(s) for the powercontrol schemes, let us consider the Lagrangian L u . It writes as: L u = R Z + ∞ φ ( g ) f (SNR( p ( g ))) p ( g ) d g − λ ( T Z + ∞ φ ( g ) p ( g )d g − E ) . (8)It is ready to show that the optimal instantaneous signal-to-noiseratio (3) has to be the solution of ∂L u ∂p ( g ) = 0 : xf ′ ( x ) − f ( x ) = λT σ R g x . (9)Solving the above equation amounts to finding the zeros of F ( x ) = xf ′ ( x ) − f ( x ) − λTσ R g x . We have that F is C differentiable, F (0) = 0 , lim x → + ∞ F ( x ) = −∞ , and F ′ ( x ) = xf (2) ( x ) − λT σ R g x. (10)Then, ∃ ǫ, ∀ x ∈ ]0 , ǫ ] , F ′ ( x ) < . Considering the sign of F ′ , giventhe particular form of f (2) , two cases have to be considered.• If ∀ x , f ′′ ( x ) ≤ λTσ R g , F ′ is negative or null and F is de-creasing. Then is the only zero for F . If ∃ ( x , x ) , x < x st f ′′ ( x ) = f ′′ ( x ) = λTσ R g , and F ′ non-negative over [ x , x ] . F decreases over [0 , x ] , in-creases over [ x , x ] and decreases over [ x , + ∞ [ . Then F may have zero, one or two zeros different from .If F has one zero, it is and is the maximum for L u . If F hastwo zeros: and x ′ , L u is decreasing and is the maximum for L u . If F has three zeros: , x ′ and x ′ , L u decreases over [0 , x ′ ] ,increases over [ x ′ , x ′ ] and decreases over [ x ′ , + ∞ [ . The maximumfor L u is then or x .Assume SNR ∗ λ E ( g ) is the greatest solution of equation (9).Then an optimal power control scheme is given by: p ∗ ( g ) = σ g SNR ∗ λ E ( g ) (11)with SNR ∗ λ E ( g ) ≥ . Since E is fixed, the methodology con-sists in determining λ E , then a solution of (9) is determined nu-merically. Note that λ E is in bits/Joule . It can be interpreted as aminimal number of bits to transmit for Joule . The higher λ E is,the better the channel should be to be used. Remark (Capacity of fast fading channels).
The proposedanalysis is reminiscent to the capacity determination of fast fad-ing single-user channels [10]. Two important differences betweenthis and our analysis are worth being emphasized. First, mathe-matically, the optimization problem under study is more generalthan the one of [10]. Indeed, if one makes the particular choice f (SNR( p ( g ))) = p log (1 + SNR( p ( g ))) , the optimal SNRis given by SNR ∗ ( p ( g )) = g λ E σ − , which corresponds to awater-filling solution (the SNR has to be non-negative). Second, thephysical interpretation of the average utility is different from the fastfading case. In the fast fading case, the power control is updated atthe symbol rate whereas in our case, it is updated at the time-slotfrequency namely, T . Indeed, in power control problems, what isupdated is the average power over a block or time-slot and assumingan average power constraint over several blocks or time-slots gen-erally does not make sense. However, from an energy perspectiveintroducing an average constraint is relevant. This comment is akind of subtle and characterizes our approach. As shown in the preceding section, time-slots are not used by the pri-mary link when the solution
SNR ∗ λ E ( g ) is negative. Therefore,the probability that this event occurs corresponds to the probabilityof having a free time-slot for the secondary link. It is thus relevantto evaluate Pr[SNR ∗ λ E ( g ) ≤ . At first glance, explicating thisprobability does not seem to be trivial. However, one can see fromthe preceding section that if max f ′′ ≤ λTσ R g , the function F hasno non-negative solutions except from , in which case there is nopower allocated to channel g . Based on this observation, the fol-lowing lower bound arises: Pr (cid:20) max f ′′ ≤ λT σ R g (cid:21) ≤ Pr[SNR ∗ λ E ( g ) ≤ . (12)Many simulations have shown that this lower bound is reason-ably tight, one of them is provided in the simulation section; whatmatters in this paper is to show that the fraction of available time-slots can be significant and the proposed lower bound ensuresto achieve at least the corresponding performance. To concludeon this point, note that in the case where f (SNR( p ( g ))) = p log (1 + SNR( p ( g ))) , the probability of having a free time-slot for the secondary link can be easily expressed and is givenby: Pr h SNR ∗ λ E ( g ) ≤ i = 1 − e − λE σ g (13)where g = E ( g ) . A similar analysis has been made to designa Shannon-rate efficient interference alignment technique for staticMIMO interference channels [11][12].
4. A STACKELBERG FORMULATION OF THENON-ORTHOGONAL CASE
We assume now that both transmitters are free to decide their powercontrol policy. However, there is still hierarchy in the system in thesense that, for each time-slot, the primary transmitter has to chooseits power level in the first place and the secondary transmitter (as-sumed to equipped with a cognitive radio) observes this level andreacts to it. This framework is exactly the one of a Stackelberg gamesince it is assumed that the primary transmitter (called the gameleader) knows it is observed by a rational player (the game follower).The SINR for the first transmitter/receiver pair is:
SINR ( p , p ) = p g σ + p g := γ , (14)where g is the channel gain between transmitter 2 and receiver 1.For the second transmitter/receiver pair, the SINR is: SINR ( p , p ) = p g σ + p g := γ , (15)where g is the channel gain between transmitter 1 and receiver2. Using this relation, we have the powers for transmitters 1 and 2depending on the SINRs: p = σ g γ + γ γ g g − αγ γ , and p = σ g γ + γ γ g g − αγ γ with α = g g g g . (16)A Stackelberg equilibrium is a vector ( p ∗ , p ∗ ) such that: p ∗ = arg max p u ( p , p ∗ ( p )) , (17)with ∀ p , p ∗ ( p ) = arg max p u ( p , p ) . (18)Note that the above expression implicitly assumes that the best-response of the follower is a singleton, which is effectively the casefor the problem under study. In our Stackelberg game, the utility u of the secondary transmitter/receiver pair depends on the powercontrol scheme p through the expression: ∀ p , u ( p , p ) = R Z + ∞ Z + ∞ φ ( g ) φ ( g ) f ( p g σ + p g ) p d g d g , (19)with the energy constraint: T Z + ∞ φ ( g ) p d g ≤ E . (20)In order to determine a Stackelberg equilibrium, we first have to ex-press the best response of the follower that is, the best power controlcheme for the secondary transmitter/receiver pair, given the longterm power control scheme of the primary transmitter/receiver pair.For a given p ( g ) , the Lagrangian L u of u is given by: L u ( p , p , λ ) = R Z + ∞ Z + ∞ φ ( g ) φ ( g ) f ( p g σ + p g ) p d g d g − λ ( T Z + ∞ φ ( g ) p d g − E ) . (21) Proposition 1 (Optimal SINR for the secondary transmitter) . Thesecondary transmitter has to tune its power level such that its SINRis the greatest zero of the following equation: xf ′ ( x ) − f ( x ) = λ T ( σ + p g ) R g x . (22)The proof is ready and follows the single-user case analysis,which is conducted in Sec. 3. The optimal power control scheme p ∗ of the secondary transmitter/receiver pair, depending on the powercontrol scheme p is given by: p ∗ ( p ) = σ + p g g x ( p ) , (23)where x ( p ) is the greatest solution of (22).Now, let us the consider the case of the primary transmitter. Proposition 2 (Optimal SINR for the primary transmitter) . The pri-mary transmitter has to tune its power level such that its SINR is thegreatest zero of the following equation: xf ′ ( x ) [1 − αx x − G ( x )] − f ( x ) = λ T σ R g g g x − αxx ! x , with G ( x ) = αx (1 + g g x ) x (1 − αx x ) R g λ Tσ f ′′ ( x ) − (1 + g g x ) . (24) Proof.
The leader is optimizing his utility function u taking into ac-count this best response power control scheme of the follower trans-mitter/receiver pair. The SINR of the leader transmitter/receiver pair,when the follower transmitter/receiver pair uses his best responsepower control scheme, is given by: SINR ( p , p ∗ ( p )) = p g σ + p ∗ ( p ) g = p g σ (1 + g g x ( p )) + p g g g x ( p ) . (25)The derivative of the SINR of the leader is ∂γ ∂p ( p ) = g σ (1 + g g x ( p )) − p σ g g x ′ ( p ) − p g g g x ′ ( p )( σ (1 + g g x ( p )) + p g g g x ( p )) (26) Then we have p ∂γ ∂p ( p ) = γ ( p ) σ (1 + g g x ( p )) − p σ g g x ′ ( p ) − p g g g x ′ ( p ) σ (1 + g g x ( p )) + p g g g x ( p ) , = γ ( p ) − p g g g x ( p ) + p σ g g x ′ ( p ) + p g g g x ′ ( p ) σ (1 + g g x ( p )) + p g g g x ( p ) ! , = γ ( p ) − x ( p ) αγ ( p ) − ( σ + p g ) p g g x ′ ( p ) σ (1 + g g x ( p )) + p g g g x ( p ) ! , = γ ( p ) − x ( p ) αγ ( p ) − σ + p g g αx ′ ( p ) γ ( p ) ! (27) Taking the expression of x ( p ) we get: x ′ f ′ ( x ) + x x ′ f ′′ ( x ) − x ′ f ′ ( x ) =2 λ T ( σ + p g ) R g g x + 2 λ T ( σ + p g ) R g x x ′ , (28)which yields to: x ′ f ′′ ( x ) = 2 λ T ( σ + p g ) R g g x + 2 λ T ( σ + p g ) R g x ′ . (29)Then we get the derivative of x ( p ) : x ′ ( p ) = λ TR g ( σ + p g ) g x f ′′ ( x ) − λ TR g ( σ + p g ) . (30)Then we have: ( σ + p g ) x ′ ( p ) g = λ TR g ( σ + p g ) x f ′′ ( x ) − λ TR g ( σ + p g ) . (31)Taking the expression of the power of receiver/transmitter pair 1 de-pending on both SINRs, we get: σ + p g = σ g g γ − αγ γ ! , (32)Then ( σ + p g ) x ′ ( p ) g = (1 + g g γ ) x (1 − αγ γ ) R g λ Tσ f ′′ ( x ) − (1 + g g γ ) . (33)Then we have: p ∂γ ∂p ( p ) = γ − αx γ − αγ (1 + g g γ ) x (1 − αx γ ) Rg λ Tσ f ′′ ( x ) − (1 + g g γ ) (34)By denoting x the largest solution of this equation, the optimalpower control scheme of the leader at the equilibrium is given by: p ∗ g σ (1 + g g x ( p ∗ )) + p g g g x ( p ∗ ) = x . (35) . NUMERICAL RESULTS The following simulations are performed with the parameters: T =10 − s, R = R = 10 bits/s, σ = 10 − W, the channel gains g and g are assumed to follow a Rayleigh distribution of mean − , when needed, g and g are assumed to follow a Rayleighdistribution of mean − and the efficiency function used is f a ,defined in Sec. 3 with a = 0 . . Fig. 1 illustrates the influence of λ E on the energy constraint in a single-user case. When λ E is low,the optimal power control scheme is to transmit most of the time,thus the energy spent is high. On the contrary, when λ E increases,transmission will only occurs when the channel gain is good enough,resulting in a lower energy spent. After a certain threshold, the opti-mal scheme is not to transmit at all. −12 −10 −8 −6 −4 λ (bits/J²) E ( J ) Fig. 1 . Energy spent on duration T depending on λ E .In Fig. 2, we are in the context of Sec. 3.2. We compute theprobability per time-slot that the primary link is not used and wecompare it to its lower bound. It is interesting to note that this lower-bound is relatively tight to the exact probability.Fig. 3 compares the expected utilities of Stackelberg equilibrium(Sec. 4) and the orthogonal case (Sec. 3). As we could expect, theprimary link of the orthogonal case offers the best utility, but theorthogonal secondary link has the worst performance. The leaderand follower of the Stackelberg case have are much more similar interms of performance and are very clos to the performance of theprimary link which makes the Stackelberg case a very efficient andfair scenario for both links. Of course, like in the single-user case,after a threshold for λ , they do not transmit at all. λ (bits/J²) F r ee t i m e − s l o t p r obab ili t y Exact probabilityLower bound
Fig. 2 . Comparison of the exact probability of having free time-slotwith the proposed lower bound of this probability. λ (bits/J²) u t ili t y ( b i t s / J ) Leader utilityFollower utilityPrimary userSecondary user
Fig. 3 . Comparison of the expected utilities of Stackelberg equilib-rium and the orthogonal case depending on λ . In this particular case, λ = λ = λ .In particular, Fig. 4 shows the optimal power profile of the lead-ing transmitter w.r.t. the channels gains g and g when λ = 10 bits/J . It is clear that for low values of g , the optimal policy is notto transmit. Then we distinguish two zones of interest:• when both g and g are good, the transmitter uses most ofits power for a relatively high value of g ,• when only g is good, we can see that the transmitter usesmost of its power for a lower value of g as it is not likelyto facing interference from the following transmitter in thiszone.
6. CONCLUSION AND PERSPECTIVES
In this paper, it is shown how a long-term energy constraint modifiesthe behavior of a transmitter in terms of power control policy. In −15 −10 −5 −15 −10 −5 g g T r an s m i tt i ng po w e r o f t he l eade r ( W ) Fig. 4 . Power profile of the leading transmitter w.r.t. g and g inthe two-player Stackelberg case.contrast with related works such as [4][5][6], a transmitter does notalways transmit when it is subject to such a constraint. This showsthat when implementing its best power control policy, a primary linkdoes not exploit all the available time-slots. The probability of hav-ing a free time-slot for the secondary link can be lower bounded in areasonably tight manner and shown to be non-negligible in general.As a second step, a scenario where the secondary link can inter-fere on the primary link is analyzed. The problem is formulated as aStackelberg game where the primary transmitter is the leader and thesecondary transmitter is the follower. An equilibrium in this game isshown to exist for typical conditions on the efficiency function f ( x ) .Interestingly, the fact that the transmitters have a long-term energyconstraint can make the system more efficient since this incites usersto interfere less; indeed simulations show the existence of a valueof an energy budget which maximizes the users’s utilities. Whilethe power control schemes at the equilibrium can be determined,the corresponding equations have a drawback: the power controlscheme of a given user does not only rely on the knowledge of itsindividual channel gain but also on the other channel gains. Thisshows the relevance of improving the proposed work by designingmore distributed power control policies. Additionally, the proposedscenarios included one primary link and one secondary link. Whenseveral cognitive transmitters are present, there is a competition be-tween the secondary transmitters for exploiting the resources left bythe primary link.
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