Optimal Power Control for Fading Channels with Arbitrary Input Distributions and Delay-Sensitive Traffic
aa r X i v : . [ c s . I T ] M a y Optimal Power Control for Fading Channels withArbitrary Input Distributions and Delay-SensitiveTraffic
Gozde Ozcan and M. Cenk Gursoy
Abstract
This paper presents the optimal power control policies maximizing the effective capacity achieved witharbitrary input distributions subject to an average power constraint and quality of service (QoS) requirements. Theanalysis leads to simplified expressions for the optimal power control strategies in the low power regime and twolimiting cases, i.e., extremely stringent QoS constraints and vanishing QoS constraints. In the low power regime,the energy efficiency (EE) performance with the constant-power scheme is also determined by characterizing boththe minimum energy per bit and wideband slope for arbitrary input signaling and general fading distributions.Subsequently, the results are specialized to Nakagami- m and Rician fading channels. Also, tradeoff betweenthe effective capacity and EE is studied by determining the optimal power control scheme that maximizes theeffective capacity subject to constraints on the minimum required EE and average transmission power. Circuitpower consumption is explicitly considered in the EE formulation. Through numerical results, the performancecomparison between constant-power and optimal power control schemes for different signal constellations andGaussian signals is carried out. The impact of QoS constraints, input distributions, fading severity, and averagetransmit power level on the proposed power control schemes, maximum achievable effective capacity and EE isevaluated. Index Terms
Effective capacity, energy efficiency, fading channel, low-power regime, mutual information, MMSE, optimalpower control, QoS constraints.
Gozde Ozcan was with the Department of Electrical Engineering and Computer Science, Syracuse University, and she is now withBroadcom Limited, Irvine, CA. M. Cenk Gursoy is with the Department of Electrical Engineering and Computer Science, SyracuseUniversity, Syracuse, NY, 13244 (e-mail: [email protected], [email protected]). . I
NTRODUCTION
Transmission power is one of the key factors in wireless communications since it is not only alimited resource but it also accounts for the significant portion of the total power consumption. Hence,a common type of resource adaptation is to efficiently vary the transmission power over time as afunction of the channel conditions in order to enhance the system performance. A legion of studies hasbeen conducted on power adaptation in wireless systems. It is well known that water-filling algorithmis the optimal power control policy, maximizing the spectral efficiency when the input is Gaussiandistributed and perfect channel side information (CSI) is available at the transmitter [1]. On the otherhand, Shannon capacity does not address quality of service (QoS) constraints in the form of constraintson buffer overflow probabilities or queueing delays.Many important wireless applications (e.g. mobile streaming/interactive video, voice over IP (VoIP),interactive gaming and mobile TV) require certain QoS guarantees for acceptable performance levelsat the end-user. In [2], effective capacity is proposed to serve as a suitable metric to quantify theperformance of wireless systems under statistical QoS constraints. In particular, effective capacityprovides the maximum throughput in the presence of limitations on the buffer-overflow/delay-violationprobabilities by capturing the asymptotic decay-rate of the buffer occupancy.The analysis and application of effective capacity in wireless systems have attracted growing interestin recent years. For instance, the authors in [3] first proposed the optimal power and rate adaptationschemes that maximize the effective capacity of a point-to-point wireless communication link. Then, theyconsidered multichannel communications and derived the optimal power control policy for multicarrierand multiple-input and multiple-output (MIMO) systems in [4]. The authors [5] formulated the energyefficiency (EE) by the ratio of effective capacity to the total power consumption including circuitpower, and determined the optimal power allocation for multicarrier systems over a frequency-selectivefading channel. The work in [6] mainly focused on energy-efficient power allocation for delay-sensitivemultimedia traffic in both low- and high-signal-to-noise ratio (SNR) regimes. Recently, the authors in[7] determined the QoS-driven optimal power control policy in closed-form to maximize the effectivecapacity subject to a minimum required EE level. The authors in [29] employed the notion of effective2apacity and analyzed the EE under QoS constraints in the low-power and wideband regimes bycharacterizing the minimum energy per bit and wideband slope. Also, the authors in [9] derivedthe minimum energy per bit and the wideband slope region for the dirty paper coding (DPC) andtime division multiple access (TDMA) schemes under heterogeneous QoS constraints. Additionally, theauthors in [10] obtained the effective capacity of correlated multiple-input single-output (MISO) channelsand further analyzed the performance in low- and high-SNR regimes. Moreover, the authors in [11]derived the asymptotic expression of the effective capacity in the low power regime for a Nakagami- m fading channel.The common assumption in the aforementioned works was that the input signal is Gaussian distributed.However, it may be difficult to realize Gaussian inputs in practice. Therefore, practical applications gen-erally employ inputs from discrete constellations such as pulse amplitude modulation (PAM), quadratureamplitude modulation (QAM) and phase-shift keying (PSK). Recently, the authors in [12] identified theoptimal power allocation scheme called mercury/water-filling for parallel channels with arbitrary inputdistributions subject to an average power constraint by using the relation between the mutual informationand minimum mean square error (MMSE). Subsequently, in [13], a low-complexity, suboptimal poweradaptation scheme was proposed in order to minimize the outage probability and maximize the ergodiccapacity for block-fading channels with arbitrary inputs subject to peak, average, and peak-to-averagepower constraints. The work in [14] mainly focused on the power allocation for Gaussian two-wayrelay channels with arbitrary signaling in the low and high power regimes. Despite recent interest in theperformance achieved with arbitrarily distributed input signals, most works have not incorporated QoSconsiderations into the analysis. Therefore, it is of significant interest to analyze the effective capacityachieved with arbitrarily distributed signals under statistical QoS constraints (imposed as limitationson buffer-overflow/delay-violation probabilities). Recently, we have considered Markovian sources andobtained the optimal power control schemes that maximize the EE of wireless transmissions with finitediscrete inputs [15]. Different from that work, we in this paper first derive the optimal power adaptationscheme that maximizes the throughput of delay-sensitive traffic (quantified by the effective capacity)achieved with arbitrary input signaling subject to an average transmit power constraint. Then, we analyze3he proposed optimal power policy under extremely stringent QoS constraints and also vanishing QoSconstraints. Also, we analyze the performance with arbitrary input signaling in the low power regimeby characterizing the minimum energy per bit and wideband slope for general fading distributions. Inaddition, we provide a simple approximation for the optimal power control policy in the low powerregime. Finally, we consider the tradeoff between the effective capacity and EE by formulating theoptimization problem to maximize the effective capacity subject to constraints on the minimum requiredEE and average transmission power.The rest of the paper is organized as follows: Section II introduces the system model. Section IIIdescribes the notion of effective capacity. In Section IV, the optimal power control policy maximizingthe effective capacity achieved with arbitrary input distributions is derived. In Section V, the optimalpower control in limiting cases is analyzed. Section VI provides low-power regime analysis of theeffective capacity attained with constant-power scheme and the optimal power control. Before presentingthe numerical results in Section VIII, the optimal power control that maximizes the effective capacitysubject to a minimum EE constraint is obtained in Section VII. Finally, main concluding remarks areprovided in Section IX. II. S YSTEM M ODEL
In this paper, we consider a point-to-point wireless communication link between the transmitter andthe receiver over a flat fading channel. Hence, the received signal is given by y [ i ] = h [ i ] x [ i ] + n [ i ] i = 1 , , . . . (1)where x [ i ] and y [ i ] denote the transmitted and received signals, respectively, and n [ i ] is a zero-mean,circularly symmetric, complex Gaussian random variable with variance N B where B denotes thebandwidth. It is assumed that noise samples { n [ i ] } form an independent and identically distributed(i.i.d.) sequence. Also, h [ i ] represents the channel fading coefficient, and the channel power gain isdenoted by z [ i ] = | h [ i ] | .If the transmitter perfectly knows the instantaneous values of { h [ i ] } , it can adapt its transmission poweraccording to the channel conditions. Let P [ i ] denote the power allocated in the i th symbol duration.4hen, the instantenous received signal-to-noise ratio, SNR can be expressed as γ = P [ i ] z [ i ] N B . The averagetransmission power is constrained by ¯ P , i.e., E { P [ i ] } ≤ ¯ P , which is equivalent to E { µ [ i ] } ≤ SNR ,where µ [ i ] = P [ i ] N B and SNR = ¯ PN B . In the rest of the analysis, we omit the time index i for notationalbrevity. We express the transmitted signal x in terms of a normalized unit-power arbitrarily distributedinput signal s . Now, the received signal can be expressed as ˆ y = √ ρs + ˆ n, (2)where ρ = µz , jointly representing the channel gain and transmission and noise powers, and ˆ n is thenormalized Gaussian noise with unit variance. Let us define the input-output mutual information I ( ρ ) as I ( ρ ) = I ( s ; √ ρs + ˆ n ) . (3)For Gaussian input s , I ( ρ ) = log (1 + ρ ) , while for any input signal s belonging to a constellation X ,we have I ( ρ ) = log |X | − π |X |× X s ∈X Z log X s ′ ∈X e − ρ | s − s ′ | − √ ρ R{ ( s − s ′ ) ∗ ˆ n } ! e −| ˆ n | d ˆ n, (4)where R{} denotes the operator that takes the real part and the integral is evaluated in the complexplane . The relation between the mutual information and the minimum mean-square error (MMSE) isgiven by [17] ˙ I ( ρ ) = MMSE ( ρ ) log e , (5)which is used to derive the power control policy for independent and parallel channels [12]. Above, ˙ I ( . ) denotes the first derivative of the mutual information, I ( ρ ) , with respect to ρ . The MMSE estimate5MSE ( ρ ) = 1 − Z ∞−∞ (cid:16) e − ρ/ sinh (cid:0) p ρ φ (cid:1) + sinh (cid:0) p ρ φ (cid:1)(cid:17) e − ρ/ cosh (cid:0) p ρ φ (cid:1) + cosh (cid:0) p ρ φ (cid:1) e − φ − ρ/ √ π dφ. (9)of s is given by ˆ s (ˆ y, ρ ) = E { s | √ ρs + ˆ n } . (6)Then, the corresponding MMSE is MMSE ( ρ ) = E {| s − ˆ s (ˆ y, ρ ) | } . (7)It should be noted that MMSE ( . ) ∈ [0 , . When the input signal s is Gaussian, MMSE ( ρ ) = ρ . Onthe other hand, for any arbitrarily distributed signal s with a constellation X , we haveMMSE ( ρ ) = 1 − π |X | Z (cid:12)(cid:12) P s ∈X s e √ ρ R{ ˆ ys ∗ }− ρ | s | (cid:12)(cid:12) P s ∈X e √ ρ R{ ˆ ys ∗ }− ρ | s | e −| ˆ y | d ˆ y. (8)For a specific constellation such as 4-pulse amplitude modulation (4-PAM), MMSE is given by (9)on the next page [12]. The MMSE for 16-quadrature amplitude modulation (16-QAM) can be readilydetermined by using the MMSE of 4-PAM in (9) as follows:MMSE ( ρ ) = MMSE (cid:16) ρ (cid:17) . (10)As further special cases, binary phase-shift keying (BPSK) (or equivalently 2-PAM), and quadraturephase-shift keying (QPSK) (or equivalently 4-QAM), the above MMSE expressions can be furthersimplified as follows [12]:MMSE BPSK ( ρ ) = 1 − Z ∞−∞ tanh(2 √ ρφ ) e − ( φ −√ ρ ) √ π dφ, (11)MMSE QPSK ( ρ ) = MMSE
BPSK (cid:16) ρ (cid:17) . (12)It should be noted that the mutual information in (4) and MMSE expression in (8) can be easily computed6y decomposing them into two dimensional real integrals and applying Gauss-Hermite quadrature rules[18]. III. P RELIMINARIES
Before introducing the optimal power adaptation problem aiming at maximizing the effective capacityin the next section, we briefly review the notion of effective capacity. Based on the theory of largedeviations, effective capacity identifies the maximum constant arrival rate that can be supported by atime-varying service process while the buffer overflow probability decays exponentially fast asymptot-ically for large buffer thresholds. More specifically, effective capacity is the maximum constant arrivalrate in a queuing system with service process { R [ j ] } such that the stationary queue length Q satisfies lim q max →∞ − log Pr { Q ≥ q max } q max = θ (13)where q max denotes the buffer overflow threshold, Pr { Q ≥ q max } is the buffer overflow probability, and θ is called the QoS exponent. For a discrete-time stationary and ergodic service process { R [ j ] } , theeffective capacity is given by [2] [20] : C E ( SNR , θ ) = − lim t →∞ θt log E { e − θ P tj =1 R [ j ] } . (14)For large q max , the limit in (13) implies that the buffer overflow probability can be approximated as Pr { Q ≥ q max } ≈ e − θq max , (15)where θ characterizes the exponential decay rate of the buffer overflow probability. From the aboveapproximation, we can see that larger values of θ indicate more stringent QoS constraints since itimposes faster decay rate. Smaller θ reflects looser constraints. We note that the effective capacity C E ( SNR , θ ) is a function of both SNR and the QoS exponent θ . However, in order to simplifythe notation in the remainder of the paper, we henceforth express the effective capacity explicitly only in terms of SNR and denote it by C E ( SNR ) due to the fact the we generally conduct the analysis and obtain characterizations for given fixed θ .We further note that log (without an explicit base) denotes logarithm to the base of e (i.e., the natural logarithm) throughout the text.
7n addition, the delay-bound violation probability is characterized to decay exponentially and can beapproximated as [19] Pr { D ≥ D th } ≈ ϕ e − θC E ( SNR ) D th , (16)where D denotes the steady state queueing delay , D th represents the delay threshold, ϕ = Pr { Q > } is the probability that the buffer is nonempty, which can be approximated by the ratio of the averagearrival rate to the average service rate [20].When the service process { R [ j ] } is i.i.d., the effective capacity simplifies to C E ( SNR ) = − θ log( E { e − θR [ j ] } ) . (17)IV. O PTIMAL P OWER C ONTROL
Our goal is to derive the optimal power control policy that maximizes the effective capacity achievedwith an arbitrary input distribution, which can be found by solving the following optimization problem C opt E ( SNR ) = max µ ( θ,z ) − θT B log( E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) ) (18)subject to E { µ ( θ, z ) } ≤ SNR , (19)where T is the frame duration and the expectation E { . } is taken with respect to the channel powergain z . Above, C opt E ( SNR ) denotes the maximum effective capacity attained with the optimal powercontrol scheme and µ ( θ, z ) = P ( θ,z ) N B represents the instantaneous transmission power normalized by thenoise power, and both µ ( · ) and the power P ( · ) are expressed as functions of the QoS exponent θ andchannel power gain z . Moreover, the instantaneous transmission (or equivalently service) rate achievedwith an arbitrarily distributed input is formulated as proportional to the input-output mutual information I ( µ ( θ, z ) z ) .We first have the following characterization for the optimal power control policy. We note that in our setting the service process of the buffer, denoted by R in the effective capacity formula in (17), is the instantaneoustransmission rate over the fading channel, which depends on the type of input signal used for transmission. For instance, if the transmittedsignal is the capacity-achieving Gaussian signal, then R = T B log (cid:16) P ( θ,z ) zN B (cid:17) , which is the mutual information achieved by theGaussian input. For any other input, rate is proportional to the mutual information I achieved by this input. heorem 1: The optimal power control, denoted by µ opt ( θ, z ) , which maximizes the effective capacityin (18), is given by µ opt ( θ, z ) = , z ≤ αµ ∗ ( θ, z ) , z > α , (20)where µ ∗ ( θ, z ) is solution to e − θT B I ( µ ∗ ( θ,z ) z ) MMSE ( µ ∗ ( θ, z ) z ) z = α (21)and α satisfies Z ∞ α µ ∗ ( θ, z ) f ( z ) dz = SNR . (22)Above, f ( z ) is the probability density function (PDF) of the channel power gain z . Proof:
The mutual information I ( ρ ) is a concave function of ρ since its second derivative is negative,i.e., ¨ I ( ρ ) = − E { ( E {| s [ i ] − ˆ s [ i ] | (cid:12)(cid:12) ˆ y [ i ] } ) + | E { ( s [ i ] − ˆ s [ i ]) (cid:12)(cid:12) ˆ y [ i ] }| } < [21]. Subsequently, − θT B I ( ρ ) is a convex function for given values of θ , T , B , and e − θT B I ( µ ( θ,z ) z ) is a log-convex function of µ ,which takes non-negative values. Since expectation preserves log-convexity, E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) is alsolog-convex in µ [22]. This implies that log( E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) ) is a convex function of µ . Since thenegative of a convex function is concave, it follows that the objective function in (18) is concave in µ .Since logarithm is a monotonic increasing function, the optimal power control policy can be found bysolving the following minimization problem: min µ ( θ,z ) E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) (23)subject to E { µ ( θ, z ) } ≤ SNR . (24)Accordingly, we first write the expectations in (23) and (24) as integrals and then form the Lagrangian9s follows: L ( θ, z ) = Z ∞ e − θT B I ( µ ( θ,z ) z ) f ( z ) dz + λ (cid:16) Z ∞ µ ( θ, z ) f ( z ) dz − SNR (cid:17) . (25)Above, λ denotes the Lagrange multiplier. Setting the derivative of the Lagrangian with respect to µ ( θ, z ) equal to zero, we obtain ∂ L ( µ ( θ, z ) , λ ) ∂µ ( θ, z ) (cid:12)(cid:12)(cid:12)(cid:12) µ ( θ,z )= µ ∗ ( θ,z ) = 0= ⇒ (cid:16) λ − β e − θT B I ( µ ∗ ( θ,z ) z ) MMSE ( µ ∗ ( θ, z ) z ) z (cid:17) f ( z ) = 0 . (26)Above, we have used the relation between the mutual information and MMSE given in (5) and defined β = θT B log e . Let α = λβ . Rearranging the above expression inside the parentheses, we obtain theequation in (21) where α can be found from the average power constraint given in (22). (cid:3) Solving the equation in (21) does not result in a closed-form expression for µ ∗ ( θ, z ) . We next showthat the equation in (21) has at most one solution, denoted by µ ∗ ( θ, z ) . Hence numerical root findingmethods, e.g., bisection method, can efficiently determine µ ∗ ( θ, z ) [22]. Proposition 1:
The optimization problem in (18) has at most one solution.
Proof:
We first rewrite the equation in (21) by using the relation in (5) as follows: g (cid:0) ( µ ( θ, z ) z ) (cid:1) = e − θT B I ( µ ( θ,z ) z ) ˙ I ( µ ( θ, z ) z ) z log 2 − α. (27)Then, differentiating g (cid:0) ( µ ( θ, z ) z ) (cid:1) with respect to ( µ ( θ, z ) z ) results in ˙ g (cid:0) ( µ ( θ, z ) z ) (cid:1) =e − θT B I ( µ ( θ,z ) z ) z log 2 × (cid:16) − θT B ( ˙ I ( µ ( θ, z ) z )) + ¨ I ( µ ( θ, z ) z ) (cid:17) . (28)Since ¨ I ( ρ ) < , the first derivative of g (cid:0) ( µ ( θ, z ) z ) (cid:1) is always negative, i.e., ˙ g (cid:0) ( µ ( θ, z ) z ) (cid:1) < . Hence,using Rolle’s theorem [23], the equation in (21) cannot have more than one root. It is easily seen thatwhen µ ( θ, z ) = 0 , g (cid:0) ( µ ( θ, z ) z ) (cid:1) = z − α , which is greater than when z > α . As µ ( θ, z ) → ∞ , the firstterm in (27) is since ˙ I ( µ ( θ, z ) z ) = 0 by using the relation in (5). As a result, g (cid:0) ( µ ( θ, z ) z ) (cid:1) = − α ,10ABLE I Algorithm 1
Proposed power control algorithm for the effective capacity maximization with arbitrarilydistributed inputs under an average power constraint
1: Initialization: µ h ( θ, z ) = µ h, init , µ l ( θ, z ) = µ l, init , ε > , δ > , ζ > , α (0) = α init repeat n ← repeat
5: update µ ∗ ( θ, z ) = ( µ h ( θ, z ) + µ l ( θ, z ))
6: if g ( µ ∗ ( θ, z )) g ( µ h ( θ, z )) < (where g ( . ) is defined in (27)), then7: µ l ( θ, z ) ← µ ∗ ( θ, z )
8: else if g ( µ ∗ ( θ, z )) g ( µ l ( θ, z )) < , then9: µ h ( θ, z ) ← µ ∗ ( θ, z )
10: end if11: until | g ( µ ∗ ( θ, z )) | < ε
12: update α using the projection subgradient method as follows13: α ( n +1) = (cid:2) α ( n ) − ζ ( SNR − E { µ ∗ ( θ, z ) } ) (cid:3) + n ← n + 1 until | α ( n ) ( SNR − E { µ ∗ ( θ, z ) } ) | ≤ δ which is less than or equal to zero by definition of α . Hence, we can conclude that there exists a uniqueoptimal power policy for z > α . (cid:3) Therefore, there exists unique optimal power level denoted by µ ∗ ( θ, z ) . (cid:3) In Table I, the proposed power control algorithm that maximizes the effective capacity with an arbitraryinput distribution subject to an average power constraint is summarized, where α in (21) is determinedby using the projected subgradient method. In this method, α is updated iteratively according to thesubgradient direction until convergence as follows: α ( n +1) = h α ( n ) − ζ (cid:0) SNR − E { µ ∗ ( θ, z ) } (cid:1)i + (29)where [ x ] + = max { , x } , n is the iteration index and ζ is the step size. When ζ is chosen to be constant,it was shown that the subgradient method is guaranteed to converge to the optimal value within a smallrange [24]. Remark 1:
When the input signal is Gaussian, we have MMSE ( ρ ) = ρ and I ( ρ ) = log (1 + ρ ) .11ubstituting these expressions into (21), we can see that the optimal power control policy reduces to µ opt ( θ, z ) = z ≤ α, α β +1 z ββ +1 − z z > α, (30)which has exactly the same structure as given in [3].V. O PTIMAL P OWER C ONTROL IN A SYMPTOTIC C ASES
In this section, we analyze two limiting cases of the proposed optimal power control, in particular,when the system is subject to extremely stringent QoS constraints (i.e., as θ → ∞ ) and vanishing QoSconstraints (i.e., as θ → ), respectively. A. Optimal Power Control under Extremely Stringent QoS Constraints
Asymptotically, when θ → ∞ , the system is subject to increasingly stringent QoS constraints andhence it cannot tolerate any delay. In this case, the transmitter maintains a fixed transmission rate andthe optimal power control for extremely stringent QoS constraints is known from [3] to be given bythe total channel inversion scheme as follows: µ opt ( z ) = C z , (31)where the constant C can be found by satisfying the average transmit power constraint with equality.In particular, Z ∞ C z f ( z ) dz = SNR . In Nakagami- m fading channel, the channel power gain is distributed according to the Gamma distri-bution f ( z ) = z m − Γ( m ) (cid:16) m Ω (cid:17) m e − m Ω z for m ≥ . , (32)12here m is the fading parameter, Ω is the average fading power and Γ( x ) is the Gamma function [25,eq. 8.310.1]. In this case, C is given by C = SNR E n z o = SNR Ω( m − m m > m ≤ . (33)It should be noted that Nakagami- m fading can model different fading conditions, e.g. including Rayleighfading (i.e., m = 1 ) and one-sided Gaussian fading (i.e., m = 0 . ) as special cases. Also, Nakagami- m fading distribution is commonly used to characterize the received signal in urban radio [26] andindoor-mobile multipath propagation environments [27]. Remark 2:
The power control policy under very stringent QoS constraints in (31) is the same regardlessof the signaling distribution while the effective capacity depends on the input distribution throughmutual information expression. More specifically, with channel inversion power control policy, themutual information becomes a constant, independent of the channel fading, i.e., I ( µ ( θ, z ) z ) = I ( C ) = I (cid:18) SNR E { z } (cid:19) . Therefore, the effective capacity achieved with this policy can be expressed as C E ( SNR ) = − θT B log( E (cid:8) e − θT B I ( C ) (cid:9) )= − θT B log(e − θT B I ( C ) )= I ( C ) = I SNR E (cid:8) z (cid:9) ! which can be regarded as the delay-limited rate achieved with a given input. For instance, with Gaussiansignaling, we have the delay-limited capacity I ( C ) = log (1 + C ) = log (cid:18) SNR E { z } (cid:19) [29]. B. Optimal Power Control under vanishing QoS Constraints As θ → , QoS constraints eventually vanish, and hence the system can tolerate arbitrarily longdelays. In this case, the effective capacity is equivalent to the achievable (mutual information) rate with13nite discrete inputs. Subsequently, the optimization problem is expressed as max µ ( z ) E {I ( µ ( θ, z )) } (34)subject to E { µ ( θ, z ) } ≤ SNR . (35)By following similar steps as in the proof of Theorem 1, the optimal power control policy is given by µ opt ( z ) = 1 z MMSE − (cid:16) min n , η log ( e ) z o(cid:17) . (36)Above, MMSE − ( . ) ∈ [0 , ∞ ) denotes the inverse MMSE function and the Lagrange multiplier, η canbe found by inserting the proposed power control into the power constraint in (35) and satisfying thisconstraint with equality as follows: Z ∞ η log2( e ) µ opt ( z ) f ( z ) dz = SNR . (37) Remark 3:
The power control policy in the absence of QoS constraints in (36) has the same structureof mercury/water-filling [12]. It is seen that the power level depends on the input distribution throughthe expression of inverse MMSE.VI. L OW -P OWER R EGIME A NALYSIS
In this section, we study the performance in the low-power regime achieved with arbitrary inputdistributions depending on the availability of CSI at the transmitter. In particular, we initially assumethat the transmitter does not have the knowledge of the channel conditions and only the receiver hasperfect CSI. In this setting, we consider constant-power transmissions and address the energy efficiencyin the low-power regime via the first and second derivatives of the effective capacity. Subsequently, weassume both the transmitter and receiver have perfect CSI and characterize the optimal power controlin this regime. 14 . Constant Power Transmissions
Here, we assume that only the receiver has perfect CSI, and hence the signal is sent with constantpower. In the low-power regime, EE can be characterized by the minimum energy per bit E b N min andwideband slope S [28]. First, energy per bit is defined as E b N = SNR C E ( SNR ) . (38)Consequently, the minimum energy per bit required for reliable communication under QoS constraintsis obtained from [28], [29] E b N = lim SNR → SNR C E ( SNR ) = 1˙ C E (0) , (39)where ˙ C E (0) denotes the first derivative of the effective capacity C E ( SNR ) with respect to SNR in the limitas
SNR vanishes. Correspondingly, at E b N min , S represents the linear growth of the spectral efficiencywith respect to E b N (in dB), which is obtained from [28], [29] S = −
2( ˙ C E (0)) ¨ C E (0) log 2 . (40)Above, ¨ C E (0) denotes the second derivative of C E ( SNR ) with respect to SNR in the limit as
SNR approaches zero. By using the minimum energy per bit in (39) and wideband slope expression in(40), throughput can be approximated as a linear function of the energy per bit (in dB) as follows: C E = S
10 log (2) (cid:18) E b N dB − E b N , dB (cid:19) + o (cid:18) E b N dB − E b N , dB (cid:19) , (41)where E b N dB = 10 log E b N is the energy per bit in dB, and o ( · ) denotes the terms vanishing faster thanthe linear term.We characterize these two important energy efficiency metrics in the low-power regime under QoSconstraints in the following result. Theorem 2:
The minimum energy per bit and wideband slope with arbitrary input distributions under15oS constraints for general fading distributions are given, respectively, by E b N = log 2 E { z } and S = 2( − ¨ I (0) log 2 + β ) E { z } ( E { z } ) − β , (42)where ¨ I (0) denotes the second derivative of the mutual information evaluated at SNR = 0 , and β = θT B log e . Proof:
We first express the mutual information achieved with arbitrary input distributions in the low-power regime as follows: I ( SNR z ) = SNR z log ( e ) + ¨ I (0)2 SNR z + o ( SNR ) . (43)Inserting the above expression into the effective capacity formulation, C E ( SNR ) , given in (18) andevaluating the first and second derivatives of C E ( SNR ) with respect to SNR at SNR = 0 results in ˙ C E (0) = E { z } log 2 (44) ¨ C E (0) = ( ¨ I (0) − β log e) E { z } + β log e( E { z } ) . (45)Further inserting the above expressions into those in (39) and (40), the minimum energy per bit andwideband slope expressions in (42) are readily obtained. (cid:3) From the above result, we immediately see that the same minimum energy per bit is achievedregardless of the signaling distribution and QoS constraints. On the other hand, the wideband slopedepends on both the input distribution through ¨ I (0) , and the QoS exponent, θ . More speficially, forquadrature symmetric constellations such as QPSK, -PSK or -QAM, we have ¨ I (0) = − log ( e ) while real-valued constellations such as BPSK and m -PAM lead to ¨ I (0) = − ( e ) [12]. Hence,even though they have the same minimum energy per bit, quadrature symmetric constellations havehigher wideband slopes compared to real-valued constellations, yielding higher EE.It should also be noted that we obtain the low power behavior of the mutual information exhibited bythe Gaussian input by setting ¨ I (0) = − log ( e ) in (43). Hence, substituting ¨ I (0) = − log ( e ) in (42),the minimum energy per bit and wideband slope expressions can be specialized to the case of Gaussian16nput, which leads the same formulations as in [29] under the assumption of perfect CSI only at thereceiver. Remark 4:
For a Nakagami- m fading channel, E { z } = Ω and E { z } = Ω (cid:0) m (cid:1) . Inserting theseexpressions into (42), the minimum energy per bit and wideband slope for a Nakagami- m fading channelcan be found, respectively, as E b N = log 2Ω , and S = 2 − (cid:0) m (cid:1) ¨ I (0) log 2 + βm . (46)We note that while the minimum bit energy depends only on the average fading power, Ω , the widebandslope is a function of Nakagami- m fading parameter, input distribution and QoS exponent, θ (throughthe term β = θT B log e ).When there exists a dominant line of sight component along the propagation path, the Rician fadingchannel is an accurate model. This type of fading typically occurs in microcellular (e.g., suburban land-mobile radio communication) [30] and picocellular environments (e.g., indoor communication) [31]. Inthis case, the pdf of the channel power gain is given by f ( z ) = (1 + K )e − K Ω e − ( K +1) z Ω I (cid:18) r K ( K + 1) z Ω (cid:19) for K, Ω ≥ , (47)where K denotes the Rician K -factor and I ( x ) represents the zero-th order modified Bessel functionof the first kind [25, eq. 8.405.1]. Remark 5:
By substituting E { z } = Ω and E { z } = (2+4 K + K )Ω( K +1) into (42), we obtain the minimumenergy per bit and wideband slope for the Rician fading channel as follows: E b N = log(2)Ω and S = 2( K + 1) − (2 + 4 K + K ) ¨ I (0) log(2) + (2 K + 1) β . (48)It can be easily verified that the wideband slope is an increasing function of the Rician K -factor. Also,similar to the Nakagami- m fading channel, the minimum energy per bit for the Rician fading channeldepends only on the average fading power, Ω . 17 . Optimal Power Control In this subsection, we assume that both the transmitter and receiver have perfect CSI. Below, weidentify the optimal power control policy in the low-power regime.
Theorem 3:
The optimal power policy that maximizes the effective capacity with arbitrary inputdistributions in the low power regime is given by µ ∗ opt ( θ, z ) = z − α (cid:0) β − ¨ I (0) (cid:1) z . (49) Proof:
In the low power regime, MMSE behaves as [12]MMSE ( ρ ) = 1 + ¨ I (0) ρ + O ( ρ ) (50)which follows from the first-order Taylor series expansion of MMSE and the fact that the MMSE isproportional to the derivative of the mutual information [17]. Incorporating the above approximationinto (21), we have e − β R µ ( θ,z ) z (1+¨ I (0) ρ + O ( ρ )) dρ × (cid:16) I (0) µ ( θ, z ) z + O (cid:0) ( µ ( θ, z ) z ) (cid:1)(cid:17) z = α. (51)Via the first-order Taylor expansion of the above equation, we obtain (cid:16) − (cid:0) β − ¨ I (0) (cid:1) µ ( θ, z ) z + O (cid:0) ( µ ( θ, z ) z ) (cid:1)(cid:17) z = α. (52)Solving the above equation provides the optimal power policy in (49) where α is again found bysatisfying the average power constraint as in (22). (cid:3) For Nakagami- m fading channel, α can be determined as the solution of − m α Γ( m − , mα Ω ) + Ω m Γ( m − , mα Ω )Ω Γ( m ) = (cid:0) β − ¨ I (0) (cid:1) SNR , (53)where Γ( a, x ) denotes the upper incomplete gamma function [25, eq. 8.350.2].18II. O PTIMAL P OWER C ONTROL UNDER A M INIMUM
EE C
ONSTRAINT
In this section, we analyze the tradeoff between the EE and the effective capacity achieved witharbitrary input distributions by formulating the optimization problem to maximize the effective capacitysubject to minimum EE and average transmit power constraints. More specifically, the optimizationproblem is expressed as C opt E ( SNR ) = max µ ( θ,z ) − θT B log E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) (54)subject to − θT B log E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) N B ( ξ E { µ ( θ, z ) } + P c n ) ≥ EE min (55) E { µ ( θ, z ) } ≤ SNR , (56)where P c n represents the normalized circuit power, EE min denotes the minimum required EE, and ξ isthe power amplifier efficiency. In the following, we first derive the optimal power control subject to aminimum EE constraint in (55) and then address the average power constraint given in (56). Theorem 4:
The optimal power control policy maximizing the effective capacity achieved with anarbitrarily distributed input subject to a minimum EE constraint is obtained as µ opt ( θ, z ) = ˜ µ ( θ, z ) . (57)Above, ˜ µ ( θ, z ) is the solution to the equation e − θT B I (˜ µ ( θ,z ) z ) MMSE (˜ µ ( θ, z ) z ) z = ν EE min N B E { e − θT B I (˜ µ ( θ,z ) z ) } ξ (1 + ν ) log ( e ) (58)where the Lagrange multiplier ν can be found by solving the equation below: − θT B log E (cid:8) e − θT B I ( µ opt ( θ,z ) z ) (cid:9) − EE min N B (cid:16) ξ E { µ opt ( θ, z ) } + P c n (cid:17) = 0 . (59)19onsequently, the required SNR that satisfies the minimum EE is calculated as SNR ∗ = E { µ opt ( θ, z ) } . (60) Proof:
The objective function, C E ( SNR ) is concave in transmission power (as shown in the proofof Theorem 1) and total power consumption in the denominator of (55) is both affine and positive,hence the feasible set defined by (55), i.e., S = n µ : C E ( SNR ) − EE min N B (cid:16) ξ E { µ ( θ, z ) } + P c n (cid:17) ≥ o is a convex set. Therefore, the Karush-Kuhn-Tucker conditions are sufficient and necessary to find theoptimal solution. First, the minimum EE constraint in (55) can be rewritten as − θT B log E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9) − EE min N B (cid:16) ξ E { µ ( θ, z ) } + P c n (cid:17) ≥ . (61)Let us define ν as the Lagrange multiplier associated with the minimum EE constraint. Then, theLagrangian function is given by L ( P ( g, h ) , ν ) = (1 + ν ) (cid:16) − θT B log E (cid:8) e − θT B I ( µ ( θ,z ) z ) (cid:9)(cid:17) − ν EE min N B (cid:16) ξ E { µ ( θ, z ) } + P c n (cid:17) . (62)Differentiating the above Lagrangian function with respect to µ ( θ, z ) and and setting the derivative equalto zero, we obtain ∂ L ( µ ( θ, z ) , ν ) ∂µ ( θ, z ) (cid:12)(cid:12)(cid:12)(cid:12) µ ( θ,z )=˜ µ ( θ,z ) = (1 + ν ) log ( e ) MMSE (˜ µ ( θ, z ) z ) z e − θT B I (˜ µ ( θ,z ) z ) E { e − θT B I (˜ µ ( θ,z ) z ) }− ν EE min ξ N B = 0 . (63)Rearranging the terms in (63) leads to the desired result in (58) and the Lagrange multiplier ν can befound by solving the equation in (58) and then inserting the optimal power control into the minimumEE constraint. Consequently, the average transmission power is determined by substituting the optimalpower control in (57) into (60). (cid:3) Now, we incorporate the average
SNR constraint in (56) into the proposed power control in (57).20ore specifically, if
SNR < SNR ∗ and the maximum EE subject to the average SNR constraint in (56) isless than EE min , then the optimization problem is not feasible and the power level is set to zero, i.e., µ ∗ ( θ, z ) = 0 . Otherwise the optimal power control is found considering the following two cases: • If SNR ≥ SNR ∗ , SNR constraint is loose. In this case, the optimal power control is given by (57)where the minimum EE constraint is satisfied with equality. • If SNR < SNR ∗ and the maximum EE subject to the average SNR constraint in (56) is greater thanEE min , the minimum EE constraint does not have any effect on the maximum effective capacity.In this case, the optimal power control is determined by (20) where the average
SNR constraint issatisfied with equality.
Remark 6:
Inserting MMSE ( ρ ) = ρ and I ( ρ ) = log (1 + ρ ) into (58), the optimal power controlscheme for Gaussian distributed signal becomes µ ∗ ( θ, z ) = z ≤ γ γ β z β β − z z > γ , (64)which is in agreement with the result obtained in [7]. Above, γ = ν EE min N B E { e − θTB I ( µ ( θ,z ) z ) } ξ (1+ ν ) log ( e ) is the scaledLagrange multiplier, which can be found by inserting the above power control into (58) and solving thecorresponding equation for γ . VIII. N UMERICAL R ESULTS
In this section, we present numerical results to illustrate the proposed optimal power control policiesand the corresponding performance levels. Unless mentioned explicitly, we consider Nakagami- m fadingchannel with m = 1 (which corresponds to Rayleigh fading) in the simulations, and it is assumed that T B = 1 , Ω = 1 and average transmit power constraint, ¯ P = 0 dB. In the iterations, step size ζ ischosen as . , ε and δ are set to − .In Fig. 1, we plot the instantaneous power level as a function of the channel power gain z and theQoS exponent θ for both Gaussian and BPSK signals. As θ decreases, QoS constraint becomes looser.In this case, the power control for BPSK input has the structure of mercury/water-filling policy. In21 Channel power gain, z (dB)
QoS exponent, θ (dB) -5-10-1500.5143.532.521.5-20 I n s t an t aneou s t r an s m i ss i on po w e r , µ ( θ , z ) Water-filling Total channelinversion (a)
Channel power gain, z (dB)
QoS exponent, θ (dB) -1000.511.53.532.52-20 I n s t an t aneou s t r an s m i ss i on po w e r , µ ( θ , z ) Mercury/water-fillingTotal channel inversion (b)
Fig. 1: The instantaneous transmission power as a function of channel power gain, z and QoS exponent, θ for (a) Gaussian input; (b) BPSK input .particular, the power is allocated to the better channel up to capacity saturation and then extra poweris assigned to the worse channel. When the input is Gaussian, the power adaptation policy becomesthe water-filling scheme, with which more power is assigned to the better channel opportunistically,deviating from the mercury/water-filling policy. When θ increases and hence stricter QoS constraintsare imposed, the optimal power control policy becomes channel inversion for both inputs. QoS exponent, θ -3 -2 -1 M a x i m u m e ff e c t i v e c apa c i t y , C E op t ( S NR ) Fig. 2: Maximum effective capacity C opt E ( SNR ) vs. QoS exponent θ for Gaussian, BPSK, QPSK and16-QAM inputs.In Fig. 2, we display the effective capacity C opt E ( SNR ) as a function of the QoS exponent θ for Gaussian,22PSK, QPSK and 16-QAM inputs with SNR = 0 dB. It is observed that as θ increases, the effectivecapacity for all inputs decreases since the transmitter is subject to more stringent QoS constraints, whichresults in lower arrival rates hence lower effective capacity. It is also seen that Gaussian inputs alwaysachieve higher effective capacity. For large θ values, Gaussian input and QPSK exhibit nearly the sameperformance. Therefore, under strict QoS constraints, QPSK can be efficiently used in practical systemsrather than the Gaussian input which is difficult to implement. Average power constraint, ¯ P (dB) -20 -15 -10 -5 0 5 10 15 20 25 M a x i m u m e ff e c t i v e c apa c i t y , C E op t ( S NR ) Fig. 3: Maximum effective capacity C opt E ( SNR ) vs. average transmit power constraint ¯ P for QPSK input.In Fig. 3, we plot maximum effective capacity C opt E ( SNR ) as a function of average transmit powerconstraint ¯ P for QPSK input (i.e., in all the curves we assume that QPSK signaling is employed). QoSexponent θ is set to . . We compare the performances of the constant-power scheme, power controlassuming Gaussian input and the optimal power control assuming QPSK input. It is observed that as ¯ P increases, the effective capacity increases and then saturates due to the fact that the input is generatedfrom a finite discrete modulation. It is seen that the power control considering the true input distribution,in this case QPSK, achieves the highest effective capacity since the power control assuming Gaussianinput is not the optimal policy for the QPSK input, and constant-power transmission strategy does nottake advantage of favorable channel conditions. In addition, the performance gap between the optimalpower control considering the discrete constellation and power control assuming Gaussian input increasesat moderate SNR levels. Note that as shown in Theorem 3 with the expression in (49), the optimal power23ontrol policy in the low power regime depends on the type of input via ¨ I (0) . As remarked in SectionVI-A, we have ¨ I (0) = − log ( e ) for both QPSK and Gaussian inputs. Therefore, at low power levels,we have the same power control policy regardless of whether it is designed for the QPSK input orthe Gaussian input, and consequently the same effective capacity values are initially attained by thetwo power control policies in the low power regime as observed in Fig. 3. However, these two powercontrol policies are no longer similar as power levels increase, leading to the observed performance gapat moderate SNR/power levels. At the other extreme, when the transmit power is sufficiently high, thethroughput of QPSK saturates at 2 bits/symbol and expectedly, all curves eventually start converging. QoS exponent, θ (dB) -4 -3 -2 -1 M a x i m u m e ff e c t i v e c apa c i t y , C E op t ( S NR ) Fig. 4: Maximum effective capacity C opt E ( SNR ) vs. QoS exponent θ for QPSK input.In Fig. 4, maximum effective capacity C opt E ( SNR ) as a function of the QoS exponent θ for QPSKinput is illustrated when with SNR = 0 dB. We again consider the constant-power scheme, power controlassuming Gaussian input and the optimal power control assuming QPSK input. The constant-powerscheme has the worst performance with the lowest effective capacity for all values of θ . It is alsointeresting to note that the performance gap between the power control policies assuming Gaussianinput and QPSK input is initially large for small values of θ , and decreases as θ increases. This ismainly due to the fact that for higher values of θ , the power control scheme does not depend on theinput distribution and becomes total channel inversion.In Fig. 5, we plot the effective capacity as a function of energy per bit E b N dB for constant-power24 nergy per bit, Eb/N0 (dB)-2 -1 0 1 2 3 4 5 E ff e c t i v e c apa c i t y , C E ( S NR ) (a) Energy per bit, Eb/N0 (dB) -2 -1 0 1 2 3 4 5 E ff e c t i v e c apa c i t y , C E ( S NR ) (b) Fig. 5: Effective capacity vs. energy per bit, E b N dB for Gaussian, BPSK, QPSK and 16-QAM inputs (a) θ = 0 . and (b) θ = 1 . Energy per bit, Eb/N0 (dB) -2 -1 0 1 2 3 4 5 6 7 E ff e c t i v e c apa c i t y , C E ( S NR ) Fig. 6: Effective capacity vs. energy per bit, E b N dB for QPSK input in Rician fading channel.transmission when θ = 0 . and θ = 1 . We compare the performances of Gaussian, BPSK, QPSK and16-QAM inputs in the low power regime by analyzing the minimum energy per bit and wideband slopevalues. It is observed that all inputs achieve the same minimum energy per bit of − . dB while thewideband slope for BPSK is smaller than those of Gaussian, QPSK and 16-QAM inputs, which indicateslower EE for BPSK. Gaussian input achieves the highest EE among the inputs. We also consider thelinear approximation for the effective capacity in the low power regime given in (41) and the exact25nalytical effective capacity expression in (17). It is seen that the linear approximation for all inputsis tight at low SNR values or equivalently low values of E b /N (dB). Additionally, when we compareFig. 5a with Fig. 5b, we readily observe that the minimum energy per bit remains the same as QoSexponent θ changes from . to . On the other hand, wideband slope decreases with increasing θ ,which confirms the result in (46).In Fig. 6, we display effective capacity as a function of energy per bit, E b N dB for QPSK input. Weconsider Rician fading channel with different values of Rician K -factor (i.e., K = 0 dB and K = 5 dB). It is again observed that the linear approximation for the effective capacity in (41) and the exactanalytical effective capacity expression in (17) matches well at low SNR values. Minimum energy perbit does not get affected by the Rician K -factor. However, as K increases, EE increases as evidencedby the increased wideband slope. This observation is in agreement with the minimum energy per bitand wideband slope expressions in (48). Average transmit power constraint, ¯ P (dB) -40 -38 -36 -34 -32 -30 -28 -26 -24 -22 -20 M a x i m u m e ff e c t i v e c apa c i t y , C E op t ( S NR ) Fig. 7: Effective capacity vs. average transmit power constraint ¯ P for Gaussian, BPSK, QPSK and16-QAM inputs.In Fig. 7, we plot effective capacity as a function of the average transmit power constraint ¯ P forGaussian, BPSK, QPSK and 16-QAM inputs. We consider the proposed optimal power control in (20)and low-power approximation for the power control in (49). The figure validates the accuracy of theapproximation at low power levels. Also, decreasing ¯ P leads to lower effective capacity for all inputs.26 E gain (%)
80 82 84 86 88 90 92 94 96 98 100 E ff e c t i v e c apa c i t y , C E ( S NR ) Fig. 8: Effective capacity gain vs. EE gain for Gaussian, BPSK, QPSK and 16-QAM inputs.Finally, we analyze the tradeoff between effective capacity and EE. In particular, we display effectivecapacity as a function of EE gain ( % ) for Gaussian, BPSK, QPSK and 16-QAM inputs in Fig. 8. Weassume that QoS exponent θ = 0 . and average transmit power constraint ¯ P = 6 dB. The EE gainis determined as the ratio of the minimum required EE denoted by EE min to the maximum achievableEE. It is seen that the effective capacity decreases with increasing EE gain for all inputs, indicatingthat gains in energy efficiency is obtained at the expense of lower throughput. Again, Gaussian inputachieves the highest effective capacity. IX. C ONCLUSION
In this paper, we have derived the optimal power control policies in wireless fading channels witharbitrary input distributions under QoS constraints by employing the effective capacity as the throughputmetric. We have proposed a low-complexity optimal power control algorithm. We have analyzed twolimiting cases of the optimal power control. In particular, when QoS constraints vanish, the optimalpower allocation strategy converges to mercury/water-filling for finite discrete inputs and water-fillingfor Gaussian input, respectively. When QoS constraints are extremely stringent, the power controlbecomes the total channel inversion and no longer depends on the input distribution. It is observed thatGaussian input achieves the highest effective capacity among the inputs. Subsequently, we have analyzed27he performance with arbitrary signal constellations at low spectral efficiencies by characterizing theminimum energy per bit and wideband slope for general fading distributions. The results are specializedto Nakagami- m and Rician fading channels. We have shown that while the minimum energy per bitdoes not get affected by the input distribution, the wideband slope depends on both the QoS exponent,the input distribution and the fading parameter. We have determined the optimal power control policyin the low-power regime. The accuracy of the proposed power control is validated through numericalresults. Finally, we have studied the effective capacity and EE tradeoff for arbitrary input signaling. Inparticular, we have solved the optimization problem to maximize the effective capacity achieved witharbitrarily distributed inputs subject to constraints on the minimum required EE and average powerconstraint. R EFERENCES [1] A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,”
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