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Energy efficiency in multiaccess fading channels under QoS constraints
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 136 (2012)
Abstract
In this article, transmission over multiaccess fading channels under qualityofservice (QoS) constraints is studied in the lowpower and wideband regimes. QoS constraints are imposed as limitations on the buffer violation probability. The effective capacity, which characterizes the maximum constant arrival rates in the presence of such statistical QoS constraints, is employed as the performance metric. A twouser multiaccess channel model is considered, and the minimum bit energy levels and wideband slope regions are characterized for different transmission and reception strategies, namely timedivision multipleaccess (TDMA), superposition coding with fixed decoding order, and superposition coding with variable decoding order. It is shown that the minimum received bit energies achieved by these different strategies are the same and independent of the QoS constraints in the lowpower regime, while they vary with the QoS constraints in the wideband regime. When wideband slope regions are considered, the suboptimality of TDMA with respect to superposition coding is proven in the lowpower regime. On the other hand, it is shown that TDMA in the wideband regime can interestingly outperform superposition coding with fixed decoding order. The impact of varying the decoding order at the receiver under certain assumptions is also investigated. Overall, energy efficiency of different transmission strategies under QoS constraints are analyzed and quantified.
1. Introduction
Energy efficiency is an important consideration in wireless systems and has been rigorously analyzed from an informationtheoretic perspective. In [1], Verdú has extensively studied the spectral efficiencybit energy tradeoff in the wideband regime, considering the Shannon capacity as the performance metric. For the Gaussian multiaccess channel, Caire et al. [2] have shown that time division multipleaccess (TDMA) is in general a suboptimal transmission scheme in the lowpower regime unless one considers the asymptotic scenario in which the power vanishes. It is also shown that fading channel makes the superposition strategy more favorable. In this analysis, Shannon capacity formulation is again adopted as the main performance metric. However, Shannon capacity does not quantify the performance in the presence of qualityofservice (QoS) limitations in the form of constraints on queueing delays or queue lengths. Indeed, most communication and informationtheoretic studies, while providing powerful results, do not generally concentrate on delay and QoS constraints [3].
At the same time, providing QoS guarantees is one of the key requirements in the development of next generation wireless communication networks since data traffic with multimedia content is expected to grow significantly and in realtime multimedia applications, such as voice over IP (VoIP) and interactivevideo (e.g., videoconferencing), latency is a key QoS metric. Many efforts have been made to incorporate the delay constraints in the performance analysis [4–7]. In [4], delay limited capacity has been proposed as a performance metric, which is defined as the rate that can be attained regardless of the values of the fading states. In [6], the tradeoff between the average transmission power and average delay has been analyzed by considering an optimization problem in which the weighted combination of the average power and average delay is minimized over transmission policies that determine the transmission rate by taking into account the arrival state, buffer occupancy, channel state jointly together. In [7], the longterm average capacity has been proposed to study the fading multiaccess channel in the wideband regime and the suboptimality of TDMA has been shown again.
In this article, we follow a different approach. We consider statistical QoS constraints and study the energy efficiency under such limitations. We adopt the notion of effective capacity [8], which can be seen as the maximum constant arrival rate that a given timevarying service process can support while providing statistical QoS guarantees. The analysis and application of effective capacity in various settings have attracted much interest recently (see e.g. [9–16] and references therein). For instance, related to this study, in [13, 14], energy efficiency is addressed in a singleuser setting when the wireless systems operate under buffer constraints and employ either adaptive or fixed transmission schemes for pointtopoint links. The effective capacity regions for multiaccess channel with different scheduling policies have been characterized in [16]. In that work, it has been found that TDMA and superposition coding with variable decoding with respect to channel states can outperform superposition strategy with fixed decoding. In this article, we consider the performance of TDMA and superposition strategy in the presence of statistical QoS constraints but concentrate on the lowSNR regime. More specifically, we employ the tools provided in [1, 2] to investigate the bit energy and wideband slope regions under QoS constraints in the lowpower and wideband regimes. The main contributions of this article are summarized in the following:

(1)
We show that different transmission and reception strategies do not affect the minimum bit energy levels required by each user. Additionally, we prove that while the minimum bit energies are independent of the QoS constraints in the low power regime, they vary with the QoS constraints in the wideband regime.

(2)
We determine that superposition coding with variable decoding order does not improve the performance in terms of slope region with respect to fixed decoding order in the low power regime, while it can achieve larger slope region in the wideband regime.

(3)
When wideband slope regions are considered, we show that TDMA is always suboptimal in the low power regime except the special case in which fading states are linearly dependent. On the other hand, TDMA in certain cases is demonstrated to perform better than superposition coding with fixed decoding order in the wideband regime. We also identify the condition for TDMA to be suboptimal in this regime.
The remainder of the article is organized as follows. In Section 2, the system model is briefly discussed. Section 3 presents some preliminaries on the analysis tools and effective capacity. The results in the lowpower regime are provided in Section 4. Section 5 presents the results in the wideband regime. Finally, Section 6 concludes this article.
2. System model
As shown in Figure 1, we consider an uplink scenario where M users with individual power and buffer constraints (i.e., QoS constraints) communicate with a single receiver. It is assumed that the transmitters generate data sequences which are divided into frames of duration T . These data frames are initially stored in the buffers before they are transmitted over the wireless channel. The discretetime signal at the receiver in the i th symbol duration is given by
where M is the number of users, X_{ j } [i] and h_{ j } [i] denote the complexvalued channel input and the fading coefficient of the j th user, respectively. We assume that {h_{ j } [i]}'s are jointly stationary and ergodic discretetime processes, and we denote the magnitudesquare of the fading coefficients by z_{ j } [i] = h_{ j } [i]^{2}. Let z = (z_{1}, z_{2}, ..., z_{ m } ) be the channel state vector. Above, n[i] is a zeromean, circularly symmetric, complex Gaussian random variable with variance $\mathbb{E}\left\{n\left[i\right]{}^{2}\right\}={N}_{0}$. The additive Gaussian noise samples {n[i]} are assumed to form an independent and identically distributed (i.i.d.) sequence. Finally, Y [i] denotes the received signal.
The channel input of user j is subject to an average energy constraint $\mathbb{E}\left\{{x}_{j}\left[i\right]{}^{2}\right\}\le {\stackrel{\u0304}{P}}_{j}/B$ for all j, where B is the bandwidth available in the system. Assuming that the symbol rate is B complex symbols per second, we can see that this formulation indicates that user j is subject to an average power constraint of ${\stackrel{\u0304}{P}}_{j}$. With these definitions, the average transmitted signal to noise ratio of user j is $\text{SNR}j=\frac{{\stackrel{\u0304}{P}}_{j}}{{N}_{0}B}$.
3. Preliminaries
3.1. Effective capacity region of the MAC channel
In [8], effective capacity is defined as the maximum constant arrival rate that a given service process can support in order to guarantee a statistical QoS requirement specified by the QoS exponent θ. The effective capacity is formulated as
where the expectation is with respect to $S\left[t\right]={\sum}_{i=1}^{t}s\left[i\right]$, which is the timeaccumulated service process, and {s[i], i = 1, 2, ...} denotes the discretetime stochastic service process. Effective capacity can be regarded as the maximum throughput of the system while the buffer violation probability is guaranteed to decay exponential fast with decay rate controlled by θ, i.e., the buffer violation probability behaves as $\text{Pr}\left\{Q>{Q}_{\text{max}}\right\}\approx {e}^{\theta {Q}_{\text{max}}}$ for large Q_{max} , where Q is stationary queue length.
We assume that the fading coefficients stay constant over the frame duration T and vary independently from one frame to another for each user. Hence, we basically consider a blockfading model. In this scenario, s[i] = T R[i], where R[i] is the instantaneous service rate in the i th frame duration [iT; (i+1)T). Then, the effective capacity in (2) can be expressed as
where R[i] is in general a function of the fading state z. (3) is obtained using the fact that instantaneous rates {R[i]} vary independently from one frame to another. It is interesting to note that as θ → 0 and hence QoS constraints relax, effective capacity approaches the ergodic capacity, i.e., $C\left(\theta \right)\to {\mathbb{E}}_{\mathbf{z}}\left\{R\left[i\right]\right\}$. On the other hand, as shown in [13], C (θ) converges to the delay limited capacity as θ grows without limit (i.e., θ → ∞) and QoS constraints become increasingly more strict. Therefore, effective capacity enables us to study the performance levels between the two extreme cases of delay limited capacity, which can be seen as a deterministic service guarantee or equivalently as a performance level attained under hard QoS limitations, and ergodic capacity, which is achieved in the absence of any QoS considerations.
Suppose that Θ = (θ_{1}, ..., θ_{ M } ) is a vector composed of the QoS constraints of M users. Let ${\beta}_{j}=\frac{{\theta}_{j}TB}{{\text{log}}_{e}2}$, j = 1, 2, ..., M be the associated normalized QoS constraints. Also, let C(Θ) = (C_{1}(θ_{1}), ..., C _{ M } (θ_{ M } )) denote the vector of the normalized effective capacities.
In [16], the effective capacity regions of the multiaccess channel for different scheduling policies have been characterized. The effective capacity region achieved by TDMA is
where δ_{ j } is the fraction of time allocated to user j.
The effective capacity region achieved by superposition coding with fixed decoding order is given by
where τ_{ m } is the fraction of time allocated to a specific decoding order π_{ m } , ${R}_{{\pi}_{m}^{1}\left(j\right)}$ represents the maximal instantaneous service rate of user j at a given decoding order π_{ m } , which is given by
where ${\pi}_{m}^{1}$ is the inverse trace function of π_{ m } .
Decoding orders can be varied for each channel fading state z. Suppose the vector space ${\Re}_{+}^{M}$ of the possible values for z is partitioned into M! disjoint regions ${\left\{{\mathcal{Z}}_{m}\right\}}_{m=1}^{M!}$ with respect to decoding orders ${\left\{{\pi}_{m}\right\}}_{m=1}^{M!}$. Then, the maximum effective capacity that can be achieved by the j th user is
for j = 1, ..., M , where p_{ z } is the distribution function of z and ${R}_{{\pi}_{m}^{1}\left(j\right)}$ is given in (6).
3.2. Spectral efficiency vs. bit energy
If we denote the effective capacity normalized by bandwidth or equivalently the spectral efficiency in bits per second per Hertz by
then it can be easily seen that ${\frac{{E}_{b}}{{N}_{0}}}_{\text{min}}$ under QoS constraints can be obtained from [1]
Hence, energy efficiency improves as SNR diminishes and the minimum bit energy is attained as SNR vanishes. At ${\frac{{E}_{b}}{{N}_{0}}}_{\text{min}}$, the slope ${\mathcal{S}}_{0}$ of the spectral efficiency versus E_{ b } /N_{0} (in dB) curve is is called the wideband slope, and is defined as [1]
Considering the expression for normalized effective capacity, the wideband slope can be found from^{a}
where ${\dot{\mathcal{\text{C}}}}_{E}\left(0\right)$ and ${\stackrel{\u0308}{\mathcal{\text{C}}}}_{E}\left(0\right)$ are the first and second derivatives, respectively, of the function ${\mathcal{\text{C}}}_{E}\left(\text{SNR}\right)$ in bits/s/Hz at zero SNR [1]. The minimum bit energy ${\frac{{E}_{b}}{{N}_{0}}}_{\text{min}}$ and the wideband slope provide a linear approximation of the spectral efficiencybit energy curve at low SNR levels and enables us to characterize and quantify the energy efficiency in the lowSNR regime.
4. Energy efficiency in the lowpower regime
As described above, in order to transmit energy efficiently and achieve bit energy levels close to the minimum level, one needs to operate in the lowSNR regime in which either the power is low or bandwidth is large. In this section, we consider the lowpower regime. We concentrate on the twouser multiaccess channel. Below, we first note the maximum effective capacities attained through different transmission strategies described in in Section 1. Subsequently, we identify the corresponding minimum bit energies and the wideband slopes.
Now, for the twouser TDMA, if we fix the fraction of time allocated to user 1 as δ ∈ [0, 1], the maximum effective capacities of the twousers in the TDMA region given by (4) become
and
respectively,
Next, consider superposition coding with fixed decoding order. We denote the ratio of the transmitterside signaltonoise ratios as $\lambda =\frac{\text{SN}{\text{R}}_{1}}{\text{SN}{\text{R}}_{2}}=\left(\frac{{\stackrel{\u0304}{P}}_{1}}{{N}_{0}B}\right)/\left(\frac{{\stackrel{\u0304}{P}}_{2}}{{N}_{0}B}\right)$. We assume that the value of this ratio is arbitrary but is kept fixed as SNR_{1} and SNR_{2} diminish in the lowSNR regime. Additionally, we let τ denote the fraction of time in which the decoding order (2, 1) is employed. Note that if the decoding order is (2, 1), the receiver first decodes the second user's signal in the presence of interference from first user's signal, and subsequently decodes the first user's signal with no interference. Note that the symmetric case occurs when the decoding order is (1, 2) in the remaining (1 τ) fraction of the time. When this strategy is used, the maximum effective capacities in the region described in (5) can now be expressed as
Finally, we turn our attention to superposition coding with variable decoding order. In this case, the decoding order depends on the fading coefficients (z_{1}, z_{2}). We define z_{2} = g(SNR_{1}) = g(λ SNR_{2}) as the partition function in the z_{1}  z_{2} space.^{b} Depending on which decoding order is employed in each region, we have different effective capacity expressions. If users are decoded in the order (1,2) when z_{2}< g(SNR_{1}) and are decoded in the order (2,1) when z_{2}> g(SNR_{1}), the effective capacities are given by
Similar effective capacity expressions can be derived if users are decoded in the order (2,1) if z_{2}< g(SNR_{1}) and decoded in the order (1,2) if z_{2}> g(SNR_{1}).
Assumption 1: Throughout the article, we consider the partition functions g(SNR_{1}) that satisfy the following properties:

(1)
g(0) is finite.

(2)
The first and second derivatives of g with respect to SNR_{1}, $\u0121\left(\text{SN}{\text{R}}_{1}\right)$ and $\stackrel{\u0308}{g}\left(\text{SN}{\text{R}}_{1}\right)$, exist. Moreover, $\u0121\left(0\right)$ and $\stackrel{\u0308}{g}\left(0\right)$ are finite.
As will be seen in the ensuing analysis, the finiteness assumptions above will serve as sufficient conditions to ensure that the derivatives of effective capacity in the limit as SNR vanishes are finite.
Denote $\frac{{E}_{b,i}}{{N}_{0}}=\frac{\text{SN}{\text{R}}_{i}}{{\mathcal{\text{C}}}_{i}}$ as the bit energy of user i = 1, 2. The received bit energy is
As the following result shows, the minimum received bit energies for the different strategies are the same.
Theorem 1: For all $\lambda =\frac{\text{SN}{\text{R}}_{1}}{\text{SN}{\text{R}}_{2}}$ and all g(z_{1}, SNR_{1}) satisfying the properties in Assumption 1, the minimum received bit energy for the multiaccess fading channel attained through TDMA, superposition coding with fixed decoding order, or superposition decoding with varying decoding order, is the same and is given by
Proof: See Appendix 1.
Remark 1: The result of Theorem 1 shows that different transmission strategies (e.g., TDMA or superposition coding) and different reception schemes (e.g., fixed or variable decoding orders) lead to the same fundamental limit on the minimum bit energy. Similarly as in [2], TDMA is optimally efficient in the asymptotic regime in which the signaltonoise ratio vanishes. More interestingly, we note that this result is obtained in the presence of QoS constraints. Additionally, the minimum bit energy is clearly independent of the QoS limitations parametrized by the QoS exponents θ_{1} and θ_{2}. Hence, the energy efficiency is not adversely affected by the buffer constraints in this asymptotic regime in which SNR → 0.
Remark 2: It can be easily shown using the effective capacity expressions provided in (4), (5), and (7) that the characterization in Theorem 1, i.e., the result that the minimum received energy per bit requirement for each user is  1.59 dB under QoS constraints, holds in a more general setting in which the number of users M ≥ 2.
Having shown that the minimum bit energies achieved by different transmission and reception strategies are the same for each user, we note that the wideband slope regions have become more interesting since they quantify the performance in the nonasymptotic regime in which SNRs are small but nonzero. With the analysis approach introduced in [2], we have the following results.
Theorem 2: The multiaccess slope region achieved by TDMA is given by
where
β_{1} = θ_{1}T B log_{2}e and β_{2} = θ_{2}T B log_{2}e.
Proof: See Appendix 2.
The following results provide the wideband slope expressions when superposition transmission is employed.
Theorem 3: For any $\lambda =\frac{\text{SN}{\text{R}}_{1}}{\text{SN}{\text{R}}_{2}}$, the multiaccess slope region achieved by the superposition coding with fixed decoding order is
where ${\mathcal{S}}_{1}^{\text{up}}$ and ${\mathcal{S}}_{2}^{\text{up}}$ are the same as defined in Theorem 2.
Proof: See Appendix 3.
Theorem 4: For any $\lambda =\frac{\text{SN}{\text{R}}_{1}}{\text{SN}{\text{R}}_{2}}$, and any g(SNR_{1}) satisfying the properties in Assumption 1, the multiaccess slope region achieved by superposition coding with variable decoding order is
where ${\mathcal{S}}_{1}^{\text{up}}$ and ${\mathcal{S}}_{2}^{\text{up}}$ are the same as defined in Theorem 2.
Proof: See Appendix 4.
Remark 3: Comparing (63) with (65) or (64) with (66) in the proof of Theorem 4 in Appendix 4, we see that different decoding orders do not change the wideband slope values for given user only if g(0) = z_{1}, i.e., the z_{1}z_{2} space is equally divided. One more interesting remark is that if we compare the third conditions in (21) and (22), we notice that fixed decoding order achieves the same performance as variable decoding order.
Remark 4: It is interesting to note in the above results that, unlike the minimum bit energy levels, the wideband slopes depend on the QoS exponents θ_{1} and θ_{2} through β_{1} and β_{2}. Indeed, as can be seen from the expressions of the upper bounds ${\mathcal{S}}_{1}^{\text{up}}$ and ${\mathcal{S}}_{2}^{\text{up}}$, the wideband slopes tend to diminish as QoS constraints become more stringent and θ_{1} and θ_{2} increase. Smaller slopes indicate that at a given energy per bit level greater than ${\frac{{E}_{b}^{r}}{{N}_{0\phantom{\rule{0.3em}{0ex}}}}}_{\text{min}}$, a smaller spectral efficiency is attained. Therefore, spectral efficiency degrades under more strict QoS constraints. Equivalently, to achieve the same level of spectral efficiency, higher energy per bit is required. Hence, from this perspective, a penalty in energy efficiency is experienced as buffer limitations become more stringent.
In the following result, we establish the suboptimality of TDMA.
Theorem 5: The wideband slope region of TDMA is inside the one attained with superposition coding.
Proof: We only need to consider the third conditions of (20) and (21). Substituting (58) and (59) into the lefthand side (LHS) of the third constraint in (20), we obtain
Comparing the sum of the last two terms with 1 (or more precisely subtracting 1 from the sum), we can write
We are interested in the numerator which is a quadratic function of the parameter τ . We note that the discriminant of this quadratic function satisfies
where the CauchySchwarz inequality ${\left(\mathbb{E}\left\{{z}_{1}{z}_{2}\right\}\right)}^{2}\le \mathbb{E}\left\{{z}_{1}^{2}\right\}\mathbb{E}\left\{{z}_{2}^{2}\right\}$ is used. Thus, the numerator of (24) is always nonnegative, i.e., the slope region achieved by TDMA is inside the one achieved by superposition coding. The equality holds only if z_{1} and z_{2} are linearly dependent. □
In Figure 2, we plot the slope regions in independent Rayleigh fading channels with variances $\mathbb{E}\left\{{z}_{1}\right\}=\mathbb{E}\left\{{z}_{2}\right\}=1$. We assume β_{1} = 1 and β_{2} = 2. From the figure, we immediately observe the suboptimality of TDMA compared with superposition coding.
5. Energy efficiency in the wideband regime
In this section, we consider the wideband regime in which the overall bandwidth of the system B is large. Let $\zeta =\frac{1}{B}$. Similar as in [13], we know that the minimum bit energy achieved in sparse multipath fading channels^{c} as B → ∞ (or equivalently ζ → 0) can be expressed as
To make the analysis more clear, below we first express the capacity expressions in (12)(17) as functions of ζ. (12) and (13) can be rewritten as
and
respectively.
For superposition coding with fixed decoding order, and fixed $\lambda =\frac{\text{SN}{\text{R}}_{1}}{\text{SN}{\text{R}}_{2}}=\frac{{\stackrel{\u0304}{P}}_{1}\zeta /{N}_{0}}{{\stackrel{\u0304}{P}}_{2}\zeta /{N}_{0}}=\frac{{\stackrel{\u0304}{P}}_{1}}{{\stackrel{\u0304}{P}}_{2}}$, (14) and (15) now become
Note that we can write g(SNR_{1}) as $g\left(\frac{{\stackrel{\u0304}{P}}_{1}\zeta}{{N}_{0}}\right)$, so similarly we can write (16) and (17) as functions of ζ
Then we immediately have the following result.
Theorem 6: For all g(SNR1) satisfying the properties in Assumption 1, the minimum bit energies for the twouser multiaccess fading channel in the wideband regime attained through TDMA, superposition coding with fixed decoding order, and superposition decoding with varying decoding order, depend on the individual QoS constraints at the users and are given by
respectively.
Proof: See Appendix 5.
Remark 5: As Theorem 6 shows, the same minimum bit energy is achieved through different transmission strategies (e.g., TDMA or superposition coding) and different reception schemes (e.g., fixed or variable decoding orders), and therefore TDMA is optimally energy efficient in the wideband regime as B → ∞. As before, Theorem 6 can be readily extended and similar expressions for the minimum energy per bit can be easily obtained for cases in which there are more than 2 users, i.e., M ≥ 2.
Remark 6: A stark difference from the result in Theorem 1 is that the minimum bit energy now varies with the specific QoS constraints at the users. When θ = 0, we can immediately show that the righthand sides of (33) and (34) become $\frac{{\text{log}}_{e}2}{\mathbb{E}\left\{{z}_{1}\right\}}$ and $\frac{{\text{log}}_{e}2}{\mathbb{E}\left\{{z}_{2}\right\}}$, respectively, which is equivalent to (19). For θ > 0, the energy efficiency is now adversely affected by the buffer constraints in the wideband regime.
Similarly as in Section 4, we next investigate the wideband slopes in order to quantify the performances and energy efficiencies of different transmission and reception methods in the nonasymptotic regime in which the bandwidth B is large but finite. We have the following results.
Theorem 7: In the wideband regime, the multiaccess slope region achieved by TDMA is given by
where
Proof: See Appendix 6.
Theorem 8: In the wideband regime, the multiaccess slope region achieved by superposition coding with fixed decoding order is
where ${\mathcal{S}}_{1}^{\text{up}}$ and ${\mathcal{S}}_{2}^{\text{up}}$ are defined in Theorem 7.
Proof: See Appendix 7.
Theorem 9: For any g(SNR1) satisfying the properties in Assumption 1, the multiaccess slope regions achieved by superposition coding with variable decoding order in the wideband regime are different for different decoding orders. The slope region is
if the decoding order is (1,2) when z_{2} < g(z_{1}, SNR_{1}), and the decoding order is (2,1) when z_{2} > g(z_{1}, SNR_{1}).
The slope region is
if the decoding order is (2,1) when z_{2} < g(z_{1}, SNR_{1}), and the decoding order is (1,2) when z_{2} > g(z_{1}, SNR_{1}).
Proof: See Appendix 8.
Remark 7: Unlike previous discussions, we have no closed form expression for the wideband slope region achieved by superposition coding with variable decoding order in the wideband regime. Another observation in the above result is that different decoding orders can result in different wideband slope regions.
Below we show the superiority of superposition coding with variable decoding compared with fixed decoding order.
Theorem 10: Superposition coding with variable decoding order achieves better performance in terms of wideband slope region with respect to superposition coding with fixed decoding order.
Proof: See Appendix 9.
In the following, we present the condition under which the suboptimality of TDMA compared with superposition coding with fixed decoding order can be established.
Theorem 11: If the following is satisfied
then the wideband slope region of TDMA is inside the one attained with superposition coding with fixed decoding order.
Proof: We consider the third conditions in (35) and (36). Substituting (86) and (87) into the LHS of the third condition in (35), we have
So if the wideband slope region is inside the one attained with superposition coding with fixed decoding order, we must have the above value to be greater than 1 for all 0 ≤ τ ≤ 1. After subtracting 1 from (40), we can obtain
The first two terms of the multiplication are positive values. The minimum value of the third term which is a quadratic function of τ is achieved at $\tau =\frac{1}{2}$, and the minimum value is
Thus, we obtain the condition stated in (39) for TDMA to be suboptimal. □
Remark 8: It is interesting that if the condition (39) is not satisfied, TDMA can achieve some points outside the wideband slope region attained with superposition coding with fixed decoding order. This tells us that TDMA can be a better choice compared with superposition coding with fixed decoding order in some cases. As an additional point, we note that if, on the other hand, the condition in (39) is satisfied, TDMA performs worse than superposition coding with variable decoding order as well due to the characterization in Theorem 10.
In the numerical results, we plot the wideband slope regions for independent Rayleigh fading channels with variances $\mathbb{E}\left\{{z}_{1}\right\}=\mathbb{E}\left\{{z}_{2}\right\}=1$. We assume θ_{1} = 0.01, θ_{2} = 0.1, T = 2 ms. In Figure 3, we assume $\frac{{\stackrel{\u0304}{P}}_{1}}{{N}_{0}}=2\frac{{\stackrel{\u0304}{P}}_{2}}{{N}_{0}}=1{0}^{4}$. The LHS of (39) is 0.1009, while the righthand side is 0.1283. Hence, the inequality is satisfied. From the figure, we can see that TDMA is suboptimal compared with superposition coding. In Figure 4, we assume $\frac{{\stackrel{\u0304}{P}}_{1}}{{N}_{0}}=\frac{1}{2}\frac{{\stackrel{\u0304}{P}}_{2}}{{N}_{0}}=1{0}^{4}$. The LHS of (39) is 0.0131, while the righthand side is 0.006. Hence, the inequality is not satisfied. Confirming the above discussion, we can observe in the figure that TDMA indeed achieves points outside the slope region attained with superposition coding with fixed decoding order.
6. Conclusion
In this article, we have analyzed the energy efficiency of twouser multiaccess fading channels under QoS constraints by employing the effective capacity as a measure of the maximal throughput. We have characterized the minimum bit energy and the wideband slope regions for different transmission strategies. We have conducted our analysis in two regimes: lowpower regime and wideband regime. Through this analysis, we have shown the impact of QoS constraints on the energy efficiency of multiaccess fading channels. More specifically, we have found that the minimum bit energies are the same for each user when different transmission and reception techniques are employed. While these minimum values are equal those that can be attained in the absence of QoS constraints in the lowpower regime, we have shown that strictly higher bit energy values, which depend on the QoS constraints, are needed in the wideband regime. We have also seen that while TDMA is suboptimal in the lowpower regime when wideband slope regions are considered, it can outperform superposition coding with fixed decoding order in the wideband regime. Moreover, we have proven in the wideband regime that varying the decoding order can achieve larger slope region when compared with fixed decoding order for superposition coding. Numerical results validating our results are provided as well.
Appendix
1. Proof of Theorem 1
Consider the TDMA strategy. Taking the first derivative of the functions in (12) and (13) and letting SNR_{1} = 0, SNR_{2} = 0, we obtain
Substituting (43) and (44) into (9), we have
which imply (19) according to (18).
For the superposition coding with fixed decoding, evaluating the first derivative of (14) and (15) at SNR_{1} = 0 and SNR_{2} = 0, we immediately obtain
which again imply (19) taking into consideration (9) and (18).
Next, we prove the result for the variable decoding case. First, we consider (16) and (17) with the associated decoding order assignment. The first derivative of (16) can be expressed as
where ${\dot{\varphi}}_{1}$ is the first derivative of ϕ_{1}, which is defined as
Under the assumptions that g(0) and $\u0121\left(0\right)$ are finite, we can easily see from (49) that letting SNR_{1} = 0 leads to
Similarly, taking the first derivative of (17) and letting SNR_{2} = 0, we obtain
Applying the definitions (9) and (18), we prove (19) for this decoding order assignment. For the reverse decoding order assignment (i.e., users are decoded in the order (2,1) if z_{2} < g(SNR_{1}) and decoded in the order (1,2) if z_{2} > g(SNR_{1})), following similar steps, we again obtain the result in (19). □
2. Proof of Theorem 2
Taking the second derivatives of the functions in (12) and (13) and letting SNR_{1} = 0, SNR_{2} = 0, we obtain