The Impact of QoS Constraints on the Energy Efficiency of Fixed-Rate Wireless Transmissions

# The Impact of QoS Constraints on the Energy Efficiency of Fixed-Rate Wireless Transmissions

\authorblockNDeli Qiao, Mustafa Cenk Gursoy, and Senem Velipasalar The authors are with the Department of Electrical Engineering, University of Nebraska-Lincoln, Lincoln, NE, 68588 (e-mails: qdl726@bigred.unl.edu, gursoy@engr.unl.edu, velipasa@engr.unl.edu).This work was supported by the National Science Foundation under Grants CCF – 0546384 (CAREER) and CNS – 0834753.
###### Abstract

Transmission over wireless fading channels under quality of service (QoS) constraints is studied when only the receiver has channel side information. Being unaware of the channel conditions, transmitter is assumed to send the information at a fixed rate. Under these assumptions, a two-state (ON-OFF) transmission model is adopted, where information is transmitted reliably at a fixed rate in the ON state while no reliable transmission occurs in the OFF state. QoS limitations are imposed as constraints on buffer violation probabilities, and effective capacity formulation is used to identify the maximum throughput that a wireless channel can sustain while satisfying statistical QoS constraints. Energy efficiency is investigated by obtaining the bit energy required at zero spectral efficiency and the wideband slope in both wideband and low-power regimes assuming that the receiver has perfect channel side information (CSI). In the wideband regime, it is shown that the bit energy required at zero spectral efficiency is the minimum bit energy. A similar result is shown for a certain class of fading distributions in the low-power regime. In both wideband and low-power regimes, the increased energy requirements due to the presence of QoS constraints are quantified. Comparisons with variable-rate/fixed-power and variable-rate/variable-power cases are given.

Energy efficiency is further analyzed in the presence of channel uncertainties. The scenario in which a priori unknown fading coefficients are estimated at the receiver via minimum mean-square-error (MMSE) estimation with the aid of training symbols, is considered. The optimal fraction of power allocated to training is identified under QoS constraints. It is proven that the minimum bit energy in the low-power regime is attained at a certain nonzero power level below which bit energy increases without bound with vanishing power. Hence, it is shown that it is extremely energy inefficient to operate at very low power levels when the channel is only imperfectly known.

Index Terms: bit energy, channel estimation, effective capacity, energy efficiency, fading channels, fixed-rate transmission, imperfect channel knowledge, low-power regime, minimum bit energy, QoS constraints, spectral efficiency, wideband regime, wideband slope.

## I Introduction

The two key characteristics of wireless communications that most greatly impact system design and performance are 1) the randomly-varying channel conditions and 2) limited energy resources. In wireless systems, the power of the received signal fluctuates randomly over time due to mobility, changing environment, and multipath fading caused by the constructive and destructive superimposition of the multipath signal components [21]. These random changes in the received signal strength lead to variations in the instantaneous data rates that can be supported by the channel. In addition, mobile wireless systems can only be equipped with limited energy resources, and hence energy efficient operation is a crucial requirement in most cases.

To measure and compare the energy efficiencies of different systems and transmission schemes, one can choose as a metric the energy required to reliably send one bit of information. Information-theoretic studies show that energy-per-bit requirement is generally minimized, and hence the energy efficiency is maximized, if the system operates at low signal-to-noise ratio (SNR) levels and hence in the low-power or wideband regimes. Recently, Verdú in [1] has determined the minimum bit energy required for reliable communication over a general class of channels, and studied of the spectral efficiency–bit energy tradeoff in the wideband regime while also providing novel tools that are useful for analysis at low SNRs.

In many wireless communication systems, in addition to energy-efficient operation, satisfying certain quality of service (QoS) requirements is of paramount importance in providing acceptable performance and quality. For instance, in voice over IP (VoIP), interactive-video (e.g,. videoconferencing), and streaming-video applications in wireless systems, latency is a key QoS metric and should not exceed certain levels [22]. On the other hand, wireless channels, as described above, are characterized by random changes in the channel, and such volatile conditions present significant challenges in providing QoS guarantees. In most cases, statistical, rather than deterministic, QoS assurances can be given.

In summary, it is vital for an important class of wireless systems to operate efficiently while also satisfying QoS requirements (e.g., latency, buffer violation probability). Information theory provides the ultimate performance limits and identifies the most efficient use of resources. However, information-theoretic studies and Shannon capacity formulation generally do not address delay and quality of service (QoS) constraints [2]. Recently, Wu and Negi in [3] defined the effective capacity as the maximum constant arrival rate that a given time-varying service process can support while providing statistical QoS guarantees. Effective capacity formulation uses the large deviations theory and incorporates the statistical QoS constraints by capturing the rate of decay of the buffer occupancy probability for large queue lengths. The analysis and application of effective capacity in various settings has attracted much interest recently (see e.g., [4][13] and references therein). For instance, Tang and Zhang in [6] considered the effective capacity when both the receiver and transmitter know the instantaneous channel gains, and derived the optimal power and rate adaptation technique that maximizes the system throughput under QoS constraints. These results are extended to multichannel communication systems in [7]. Liu et al. in [10] considered fixed-rate transmission schemes and analyzed the effective capacity and related resource requirements for Markov wireless channel models. In this work, the continuous-time Gilbert-Elliott channel with ON and OFF states is adopted as the channel model while assuming the fading coefficients as zero-mean Gaussian distributed. A study of cooperative networks operating under QoS constraints is provided in [11]. In [13], we have investigated the energy efficiency under QoS constraints by analyzing the normalized effective capacity (or equivalently the spectral efficiency) in the low-power and wideband regimes. We considered variable-rate/variable-power and variable-rate/fixed-power transmission schemes assuming the availability of channel side information at both the transmitter and receiver or only at the receiver.

In this paper, we consider a wireless communication scenario in which only the receiver has the channel side information, and the transmitter, not knowing the channel conditions, sends the information at a fixed-rate with fixed power. If the fixed-rate transmission cannot be supported by the channel, we assume that outage occurs and information has to be retransmitted. Similarly as in [10], we consider a channel model with ON and OFF states. In this scenario, we investigate the energy efficiency under QoS constraints in the low-power and wideband regimes by considering the bit energy requirement defined as average energy normalized by the effective capacity. Our analysis will initially be carried out under the assumption that the receiver has perfect channel information. Subsequently, we consider the scenario in which a priori unknown channel is estimated by the receiver with the assistance of training symbols, albeit only imperfectly.

The rest of the paper is organized as follows. Section II introduces the system model. In Section III, we briefly describe the notion of effective capacity and the spectral efficiency–bit energy tradeoff. Assuming the availability of the perfect channel knowledge at the receiver, we analyze the energy efficiency in the wideband and low-power regimes in Sections IV and V, respectively. In Section VI, we investigate the energy efficiency in the low-power regime when the receiver knows the channel only imperfectly. Finally, Section VII concludes the paper.

## Ii System Model

We consider a point-to-point wireless link in which there is one source and one destination. The system model is depicted in Figure 1. It is assumed that the source generates data sequences which are divided into frames of duration . These data frames are initially stored in the buffer before they are transmitted over the wireless channel. The discrete-time channel input-output relation in the symbol duration is given by

 y[i]=h[i]x[i]+n[i]i=1,2,…. (1)

where and denote the complex-valued channel input and output, respectively. We assume that the bandwidth available in the system is and the channel input is subject to the following average energy constraint: for all . Since the bandwidth is , symbol rate is assumed to be complex symbols per second, indicating that the average power of the system is constrained by . Above in (1), is a zero-mean, circularly symmetric, complex Gaussian random variable with variance . The additive Gaussian noise samples are assumed to form an independent and identically distributed (i.i.d.) sequence. Finally, denotes the channel fading coefficient, and is a stationary and ergodic discrete-time process. We denote the magnitude-square of the fading coefficients by .

In this paper, we initially consider the scenario in which the receiver has perfect channel side information and hence perfectly knows the instantaneous values of while the transmitter has no such knowledge. Subsequently, we will analyze the effect of imperfect channel knowledge at the receiver. When the receiver perfectly knows the channel conditions, the instantaneous channel capacity with channel gain is

 C[i]=Blog2(1+\footnotesize{SNR}z[i]) bits/s (2)

where is the average transmitted signal-to-noise ratio. Since the transmitter is unaware of the channel conditions, information is transmitted at a fixed rate of bits/s. When , the channel is considered to be in the ON state and reliable communication is achieved at this rate. If, on the other hand, , outage occurs. In this case, channel is in the OFF state and reliable communication at the rate of bits/s cannot be attained. Hence, effective data rate is zero and information has to be resent. We assume that a simple automatic repeat request (ARQ) mechanism is incorporated in the communication protocol to acknowledge the reception of data and to ensure that the erroneous data is retransmitted [10].

Fig. 2 depicts the two-state transmission model together with the transition probabilities. In this paper, we assume that the channel fading coefficients stay constant over the frame duration . Hence, the state transitions occur at every seconds. Now, the probability of staying in the ON state, , is defined as follows111The formulation in (3) assumes as before that the symbol rate is symbols/s and hence we have symbols in a duration of seconds.:

 p22 =P{rα∣∣z[i]>α} (3)

where

 α=2rB−1\footnotesize{SNR}. (4)

Note that depends on the joint distribution of . For the Rayleigh fading channel, the joint density function of the fading amplitudes can be obtained in closed-form [16]. In this paper, with the goal of simplifying the analysis and providing results for arbitrary fading distributions, we assume that fading realizations are independent for each frame222This assumption also enables us to compare the results of this paper with those in [13] in which variable-rate/variable-power and variable-rate/fixed-power transmission schemes are studied for block fading channels.. Hence, we basically consider a block-fading channel model. Note that in block-fading channels, the duration over which the fading coefficients stay constant can be varied to model fast or slow fading scenarios.

Under the block fading assumption, we now have . Similarly, the other transition probabilities become

 p11 =p21=P{z≤α}=∫α0pz(z)dz (5) p22 =p12=P{z>α}=∫∞αpz(z)dz (6)

where is the probability density function of . We finally note that bits are successfully transmitted and received in the ON state, while the effective transmission rate in the OFF state is zero.

## Iii Preliminaries – Effective Capacity and Spectral Efficiency-Bit Energy Tradeoff

In [3], Wu and Negi defined the effective capacity as the maximum constant arrival rate333For time-varying arrival rates, effective capacity specifies the effective bandwidth of the arrival process that can be supported by the channel. that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent . If we define as the stationary queue length, then is the decay rate of the tail distribution of the queue length :

 limq→∞logP(Q≥q)q=−θ. (7)

Therefore, for large , we have the following approximation for the buffer violation probability: . Hence, while larger corresponds to more strict QoS constraints, smaller implies looser QoS guarantees. Moreover, if denotes the steady-state delay experienced in the buffer, then it is shown in [12] that for constant arrival rates. This result provides a link between the buffer and delay violation probabilities. In the above formulation, is some positive constant, , and is the source arrival rate.

Now, the effective capacity for a given QoS exponent is obtained from

 −limt→∞1θtlogeE{e−θS[t]}def=−Λ(−θ)θ (8)

where is the time-accumulated service process and denote the discrete-time, stationary and ergodic stochastic service process. Note that in the model we consider, depending on the channel state being ON or OFF, respectively. In [14] and [15, Section 7.2, Example 7.2.7], it is shown that for such an ON-OFF model, we have

 (9)

Using the formulation in (9) and noting that in our model, we express the effective capacity normalized by the frame duration and bandwidth , or equivalently spectral efficiency in bits/s/Hz, for a given statistical QoS constraint , as

 RE(\footnotesize{SNR},θ)=1TBmaxr≥0{−Λ(−θ)θ} =maxr≥0{−1θTBloge(p11+p22e−θTr)} (10) =maxr≥0{−1θTBloge(1−P{z>α}(1−e−θTr))} (11) =−1θTBloge(1−P{z>αopt}(1−e−θTropt))bits/s/Hz (12)

where is the maximum fixed transmission rate that solves (11) and . Note that both and are functions of SNR and .

The normalized effective capacity, , provides the maximum throughput under statistical QoS constraints in the fixed-rate transmission model. It can be easily shown that

 limθ→0RE(\footnotesize{SNR},θ)=maxr≥0rBP{z>α}. (13)

Hence, as the QoS requirements relax, the maximum constant arrival rate approaches the average transmission rate. On the other hand, for , in order to avoid violations of QoS constraints.

In this paper, we focus on the energy efficiency of wireless transmissions under the aforementioned statistical QoS limitations. Since energy efficient operation generally requires operation at low-SNR levels, our analysis throughout the paper is carried out in the low-SNR regime. In this regime, the tradeoff between the normalized effective capacity (i.e, spectral efficiency) and bit energy is a key tradeoff in understanding the energy efficiency, and is characterized by the bit energy at zero spectral efficiency and wideband slope provided, respectively, by

 EbN0∣∣∣R=0=lim\footnotesize{SNR}→0\footnotesize{SNR}RE(\footnotesize{SNR})=1˙RE(0) and S0=−2(˙RE(0))2¨RE(0)loge2 (14)

where and are the first and second derivatives with respect to SNR, respectively, of the function at zero SNR [1]. and provide a linear approximation of the spectral efficiency curve at low spectral efficiencies, i.e.,

 RE(EbN0)=S010log102(EbN0∣∣∣dB−EbN0∣∣∣R=0,dB)+ϵ (15)

where and . Moreover, is the minimum bit energy when the spectral efficiency is a non-decreasing concave function of SNR. Indeed, we show that when the channel is perfectly known at the receiver, in the wideband regime as . Moreover, we demonstrate that for Rayleigh and Nakagami fading channels (with integer fading parameter ) in the low-power regime as . On the other hand, for general treatment, we refer to the bit energy required as SNR vanishes as throughout the paper. As we shall see in Section VI that is not necessarily the minimum bit energy in a certain scenario of the imperfectly-known channel.

## Iv Energy Efficiency in the Wideband Regime

In this section, we consider the wideband regime in which the bandwidth is large. We assume that the average power is kept constant. Note that as the bandwidth increases, approaches zero and we operate in the low-SNR regime.

We first introduce the notation . Note that as , we have . Moreover, with this notation, the normalized effective capacity can be expressed as444Since the results in the paper are generally obtained for fixed but arbitrary , the normalized effective capacity is often expressed in the paper as instead of to avoid cumbersome expressions.

 RE(\footnotesize{SNR})=−ζθTloge(1−P{z>αopt}(1−e−θTropt)). (16)

Note that and are also in general dependent on and hence . The following result provides the expressions for the bit energy at zero spectral efficiency (i.e., as ) and the wideband slope, and characterize the spectral efficiency-bit energy tradeoff in the wideband regime.

###### Theorem 1

In the wideband regime, the bit energy at zero spectral efficiency, and wideband slope are given by

 EbN0∣∣∣R=0=−δloge2logeξand (17) S0=2ξlog2eξ(δα∗opt)2P{z>α∗opt}e−δα∗opt, (18)

respectively, where and . is defined as and satisfies

 δα∗opt=loge(1+δP{z>α∗opt}pz(α∗opt)). (19)

Proof: Assume that the Taylor series expansion of with respect to small is

 ropt=r∗opt+˙ropt(0)ζ+o(ζ) (20)

where and is the first derivative with respect to of evaluated at . From (4), we can find that

 αopt=2roptζ−1¯PζN0=r∗optloge2¯PN0+˙ropt(0)loge2+(r∗optloge2)22¯PN0ζ+o(ζ) (21)

from which we have as that

 α∗opt=r∗optloge2¯PN0 (22)

and that

 ˙αopt(0)=˙ropt(0)loge2+(r∗optloge2)22¯PN0 (23)

where is the first derivative with respect to of evaluated at . According to (22), . We now have

 EbN0∣∣∣R=0 =limζ→0¯PN0ζRE(ζ)=¯PN0˙RE(0)=−θT¯PN0loge(1−P{z>α∗opt% }(1−e−θTr∗opt))=−δloge2logeξ (24)

where is the derivative of with respect to at , , and . Therefore, we prove (17). Note that the second derivative , required in the computation of the wideband slope , can be obtained from

 ¨RE(0) =limζ→02RE(ζ)−˙RE(0)ζζ2 =limζ→021ζ(−1θTloge(1−P{z>αopt}(1−e−θTropt))+1θTloge(1−P{z>α∗%opt}(1−e−θTr∗opt))) =limζ→0−2θT(pz(αopt)˙αopt(ζ)(1−e−θTr% opt)−P{z>αopt}θTe−θTropt˙ropt(ζ))1−P{z>αopt}(1−e−θTropt) (25) =−2θT(pz(α∗opt)˙αopt(0)(1−e−θTr∗opt)−P{z>α∗opt}θTe−θTr∗opt˙ropt(0))1−P{z>α∗opt}(1−e−θTr∗opt) (26)

where . Above, (25) and (26) follow by using L’Hospital’s Rule and applying Leibniz Integral Rule [23].

Next, we derive an equality satisfied by . Consider the objective function in (11)

 −1θTBloge(1−P{z>α}(1−e−θTr)). (27)

It can easily be seen that both as and , this objective function approaches zero555Note that increases without bound with increasing .. Hence, (27) is maximized at a finite and nonzero value of at which the derivative of (27) with respect to is zero. Differentiating (27) with respect to and making it equal to zero leads to the equality that needs to be satisfied at the optimal value :

 2roptζpz(αopt)N0loge2¯P(1−e−θTropt)=θTe−θTroptP{z>αopt} (28)

where . For given , as the bandwidth increases (i.e., ), . Clearly, in the wideband regime. Because, otherwise, if and consequently , the left-hand-side of (28) becomes zero, while the right-hand-side is different from zero. So, employing (22) and taking the limit of both sides of (28) as , we can derive that

 pz(α∗opt)N0loge2¯P(1−e−θT¯PN0loge2α∗opt)=θTe−θT¯PN0loge2α∗optP{z>α∗opt} (29)

which, after rearranging, yields

 θT¯PN0loge2α∗opt=loge(1+θT¯PN0loge2P{z>α∗%opt}pz(α∗opt)). (30)

Denoting , we obtain the condition (19) stated in the theorem.

Combining (29) and (23) with (26) gives us

 ¨RE(0) =−N0log2e2θT¯Pr∗opt2pz(α∗opt)(1−e−θTr∗opt)1−P{z>α∗opt}(1−e−θTr∗opt)=−r∗opt2P{z>α∗opt}e−θTr∗optloge21−P{z>α∗opt}(1−e−θTr∗% opt) (31)

Substituting (31) and the expression for in (24) into (14), we obtain (18).

The following result shows that in the wideband regime, is the indeed the minimum bit energy.

###### Theorem 2

In the wideband regime, the bit energy required at zero spectral efficiency (i.e., bit energy required as or equivalently as ) is the minimum bit energy, i.e.,

 EbN0∣∣∣R=0=−δloge2logeξ=EbN0min. (32)

Proof: Since , we can show the result by proving that monotonically decreases with increasing , and hence achieves its maximum as . We first evaluate the first derivative of with respect to :

 d(RE(ζ)/ζ)dζ =−1θTpz(αopt)N0¯P2roptζ(˙roptζ+ropt)ζloge2−(2roptζ−1)ζ2(1−e−θTr%opt)−θTe−θTropt˙roptP(z>αopt)1−P(z>αopt)(1−e−θTropt) (33) =−pz(αopt)N0θT¯P2roptζroptζloge2−(2roptζ−1)ζ2 (34)

where (34) is obtained by using the equation that needs to be satisfied by and as shown in the proof of Theorem 1 in (28). Note that the probability density function for all . Hence, if for all and , then proving that is indeed a monotonically decreasing function of . Now, we denote and define . The first derivative of with respect to is , implying that is a monotonically increasing function. Since , we immediately conclude that for all . Hence, for all and , , and is achieved in the limit as .

Having analytically characterized the spectral efficiency–bit energy tradeoff in the wideband regime, we now provide numerical results to illustrate the theoretical findings. Fig. 3 plots the spectral efficiency curves as a function of the bit energy in the Rayleigh channel. In all the curves, we have . Moreover, we set ms in the numerical results throughout the paper. As predicted by the result of Theorem 2, in all cases in Fig. 3. It can be found that from which we obtain dB for , respectively. For the same set of values in the same sequence, we compute the wideband slope values as . We immediately observe that more stringent QoS constraints and hence higher values of lead to higher minimum bit energy values and also higher energy requirements at other nonzero spectral efficiencies. Fig. 4 provides the spectral efficiency curves for Nakagami- fading channels for different values of . In this figure, we set . For , we find that , , and , respectively. Note that as increases and hence the channel conditions improve, the minimum bit energy decreases and the wideband slope increases, improving the energy efficiency both at zero spectral efficiency and at nonzero but small spectral efficiency values. As , the performance approaches that of the unfaded additive Gaussian noise channel (AWGN) for which we have dB and [1].

## V Energy Efficiency in the Low-Power Regime

In this section, we investigate the spectral efficiency–bit energy tradeoff as the average power diminishes. We assume that the bandwidth allocated to the channel is fixed. Note that vanishes with decreasing , and we again operate in the low-SNR regime similarly as in Section IV. However, energy requirements in the low-power regime will be different from those in the wideband regime, because the arrival rates that can be supported get smaller with decreasing power in this regime.

The following result provides the expressions for the bit energy at zero spectral efficiency and the wideband slope.

###### Theorem 3

In the low-power regime, the bit energy at zero spectral efficiency and wideband slope are given by

 EbN0∣∣∣R=0=loge2α∗optP{z>α∗opt}and (35) S0=2P{z>α∗opt}1+β(1−P{z>α∗opt}), (36)

respectively, where is normalized QoS constraint. In the above formulation, is again defined as and satisfies

 α∗optpz(α∗opt)=P{z>α∗opt}. (37)

Proof: We first consider the Taylor series expansion of in the low-SNR regime:

 ropt=a\footnotesize{SNR}+b\footnotesize{SNR}2+o(\footnotesize{SNR}2) (38)

where and are real-valued constants. Substituting (38) into (4), we obtain the Taylor series expansion for :

 αopt=aloge2B+(bloge2B+a2log2e22B2)\footnotesize{SNR}+o(% \footnotesize{SNR}). (39)

¿From (39), we note that in the limit as , we have

 α∗opt=aloge2B. (40)

Next, we obtain the Taylor series expansion with respect to SNR for using the Leibniz Integral Rule [23]:

 P{z>αopt}=P{z>α∗opt}−(bloge2B+a2log2e22B2)pz(α∗opt)\footnotesize{SNR}+o(\footnotesize% {SNR}). (41)

Using (38), (39), and (41), we find the following series expansion for given in (12):

 RE(\footnotesize{SNR}) =−1θTBloge[1−(P{z>α∗opt}−(bloge2B+a2log2e22B2)pz(α∗opt)\footnotesize{SNR}+o(\footnotesize{SNR})) −1θT−1−(P{z>α∗%opt}×(θTa\footnotesize{SNR}+(θTb−(θTa)22)\footnotesize{SNR}2+o(\footnotesize{SNR}% 2))] (42)

Then, using (40), we immediately derive from (V) that

 ˙RE(0) =α∗optP{z>α∗opt% }loge2, (43) ¨RE(0) =−α∗opt3pz{α∗opt}loge2−θTBα∗opt2log2e2P{z>α∗opt}(1−P{z>α∗%opt}). (44)

Similarly as in the discussion in the proof of Theorem 1 in Section IV, the optimal fixed-rate , akin to (28), should satisfy

 2ropt/Bpz(αopt)loge2B% \footnotesize{SNR}(1−e−θTropt)=θTe−θTroptP{z>αopt}. (45)

Taking the limits of both sides of (45) as and employing (38), we obtain

 apz(α∗opt)loge2B=P{z>α∗opt}. (46)

¿From (40), (46) simplifies to

 α∗optpz(α∗opt)=P{z>α∗opt}, (47)

proving the condition in (37). Moreover, using (47), the first term in the expression for in (44) becomes . Together with this change, evaluating the expressions in (14) with the results in (43) and (44), we obtain (35) and (36).

Next, we show that the equation (37) that needs to be satisfied by has a unique solution for a certain class of fading distributions.

###### Theorem 4

The equation has a unique solution when has a Gamma distribution with integer parameter , i.e., when the probability density function of is given by

 pz(z)=λnΓ(n)zn−1e−λz (48)

where is an integer, , and is the Gamma function [24].

Proof: See Appendix -A.

Remark: In the special case in which and is an integer, the Gamma density (48) becomes

 pz(z)=mmzm−1Γ(m)e−mz (49)

which is the probability density function of in Nakagami- fading channels (with integer )[21]. Moreover, when , we have the Rayleigh fading channel in which has an exponential distribution, i.e., . Therefore, the result of Theorem 4 applies for these channels.

Remark: Theorem 3 shows that the for any depends only on . From Theorem 4, we know that if has the Gamma density function given by (48), then is unique and hence is the same for all . We immediately conclude from these results that also has the same value for all and therefore does not depend on when has the distribution given in (48).

Moreover, using the results of Theorem 4 above and Theorem 2 in Section IV, we can further show that is the minimum bit energy. Note that this implies that the same minimum bit energy can be attained regardless of how strict the QoS constraint is. On the other hand, we note that the wideband slope in general varies with .

###### Corollary 1

In the low-power regime, when , the minimum bit energy is achieved as , i.e., . Moreover, if the probability density function of is in the form given in (48) then the minimum bit energy is achieved as , i.e. , for all .

Proof: Recall from (13) that in the limit as ,

 RE(\footnotesize{SNR},0)=limθ→0RE(\footnotesize{SNR},θ)=maxr≥0rBP{z>2rB−1\footnotesize{SNR}}. (50)

Since the optimization is performed over all , it can be easily seen that the above maximization problem can be recast as follows:

 RE(\footnotesize{SNR},0)=maxx≥0xP{z>2x−1\footnotesize{SNR}}. (51)

¿From (51), we note that depends on only through . Therefore, increasing has the same effect as decreasing . Hence, low-power and wideband regimes are equivalent when . Consequently, the result of Theorem 2, which shows that the minimum bit energy is achieved as , implies that the minimum bit energy is also achieved as .

Note that for . Therefore, the bit energy required when is larger than that required when . On the other hand, as we have proven in Theorem 4, is unique and the bit energy required as is the same for all when has a Gamma density in the form given in (48). Since the minimum bit energy in the case of is achieved as , and the same bit energy is attained for all , we immediately conclude that for all when has a Gamma distribution.

Next, we provide numerical results which confirm the theoretical conclusions and illustrate the impact of QoS constraints on the energy efficiency. We set Hz in the computations. Fig. 5 plots the spectral efficiency as a function of the bit energy for different values of in the Rayleigh fading channel (or equivalently Nakagami- fading channel with ) for which . In all cases in Fig. 5, we readily note that . Moreover, as predicted, the minimum bit energy is the same and is equal to the one achieved when there are no QoS constraints (i.e., when ). From the equation , we can find that in the Rayleigh channel for which . Hence, the minimum bit energy is dB. On the other hand, the wideband slopes are for