A power allocation strategy is proposed for single-antenna overlay cognitive radio networks, in which the secondary user helps transmit the signal of the primary user while concurrently conveying its own signal by means of a superposition coding technique. The study commences by deriving analytical expressions for the bit error rates (BERs) of the primary and secondary users. A power allocation strategy is then proposed for minimizing the total power consumption of the two users while simultaneously satisfying their respective BER constraints. The analytical BER formulas are not convex, and hence the optimization process presents a significant challenge. Accordingly, two more tractable BER approximations for the primary and secondary users are proposed to transfer the non-convex problem into a convex one. The simulation results confirm the effectiveness of the proposed power allocation strategy under various channel environments.
Abkürzungen
BER
Bit error rate
CR
Cognitive radio
DF
Decode-and-forward
DPC
Dirty paper coding
MRC
Maximum ratio combining
QoS
Quality-of-service
SC
Superposition coding
1 Introduction
Cognitive radio (CR) transceiver design has received significant attention in recent years due to the ability it provides to make more efficient use of the spectrum resources [1‐3]. By assigning the CR users different priorities, i.e., primary users (or non-cognitive users), denoted as UA, and secondary users (or cognitive users), denoted as UB, both types of user are able to coexist in a neat way and fully utilize all of the spectrum resources. As a result, CR provides a highly promising solution for future wireless systems [4].
In developing CR networks, the aim is for UB to maximize its use of the spectrum resources without severely affecting the transmissions of UA. Existing CR schemes can be broadly classified as either interweave, underlay, or overlay [3, 5‐17]. For convenience in the following discussions, let TA and RA denote the primary transmitter and primary receiver, respectively, and let TB and RB denote the secondary transmitter and secondary receiver, respectively.
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In interweave networks, UB transmits its signal to the intended receiver only if the spectrum is not occupied by UA. The performance of the interweave CR schemes is thus dominated by the ability of the users to sense the spectrum occupancy [11, 12]. If the users fail to detect spectrum holes, performance degradation inevitably occurs. Thus, spectrum detection is a critical issue in interweave CR networks. Furthermore, each user consumes a distinct channel in the interweave CR model, and hence the capacity of the system to accommodate users is restricted. The underlay CR model improves spectrum utilization by allowing UB to convey its signal over the same channel as that used by UA provided that the transmissions of TB do not interfere with those of RA too severely. To minimize interference at RA, TB should be designed such that the power of its interference falls below a certain predefined threshold [2, 3, 13‐15]. However, the error performance of RB may still be unsatisfactory due to additional constraints imposed at RA. It is noted that this effect is particularly apparent when TB is located physically close to RA.
The overlay CR model is designed to overcome the poor error performance at RB by enabling TB to help UA relay its signal while concurrently transmitting its own signal. This assistive function can be realized using either some form of cooperative relaying technique or a superposition coding (SC) scheme [5, 14‐17]. For the case where TB is equipped with multiple antennas [6], the zero-forcing beamformer can be used to convey the signals of UA and UB concurrently using a random vector quantization feedback mechanism [5, 14]. In [7], the primary and secondary systems operate simultaneously using the space-time code originally developed for multiple-input multiple-output systems. More specifically, Alamouti code is applied at the secondary transmitter such that the signals received at the receiver can be decoupled and interference-free signals obtained. The overlay designs in [5‐7, 14, 18], rely on either precoding or the use of space-time coding at TB. However, for some applications such as sensor and body networks, allocating single antenna at each node can practically save power and cost [19, 20].
It was shown in [21] and [19] that through the assistance of TB the error performance of UA can be enhanced such that the required quality-of-service (QoS) is guaranteed. However, an efficient CR design is crucial when space diversity is not available at the transceiver pairs of UA and UB. Accordingly, this study focuses specifically on the problem of QoS-controlled overlay CR design with no space diversity. In developing the proposed overlay design, an assumption is made that the signals are transmitted in two phases, as shown in Fig. 1. In the first phase, TA broadcasts its signal to both RA and TB. In the second phase, TB performs decode-and-forward (DF) over the received signal from TA. Specifically, if TB fails to decode the received signal from TA, TB transmits only its own signal. However, if decoding is successful, TB transmits an SC-combined signal comprising the signals of both UA’s and UB’s signals to RA and RB with an appropriate power gain. RA combines the two signals received from TA and TB using a maximum ratio combining (MRC) technique and then decodes the signal. Similarly, RB decodes the SC-combined signal from TB.
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Due to the signal transmission protocol used in the considered overlay CR network, the error performances of UA and UB are inherently affected by both the transmitted powers of TA and TB and the power allocation factor used in the SC scheme. To evaluate the respective effects of these factors on the error performance, this study commences by analyzing the BERs of UA and UB, where the BER is expressed as a function of the transmitted powers of UA and UB and the power allocation ratio applied in the SC scheme. A power allocation strategy for the SC scheme is then proposed to minimize the total power consumption of the two users while simultaneously satisfying their respective BER constraints (i.e., QoS requirements). The analytical BER formulations are not convex, and hence determining the optimal solution for the power allocation ratio is challenging, even for numerical methods. To address this problem, two more tractable BER formulas are proposed for UA’s and UB’s BER. The optimum power allocation factor is then determined by solving these two approximations with the proposed non-iterative and iterative approaches. The validity of the proposed power allocation strategy under various channel link conditions is demonstrated by means of numerical simulations.
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The remainder of this paper is organized as follows. Section 2 introduces the signal model of the overlay transceiver structure and derives the related analytical BER formulations. The approximated BER expressions are then formulated. Section 3 describes the proposed optimal power allocation strategy based on the approximated BER expressions. Section 4 presents and discusses the simulation results. Finally, Section 5 provides some brief concluding remarks.
2 System model and BER performance
2.1 System model
Figure 1 shows the overlay spectrum-sharing system considered in the present study. Note that each node is equipped with only a single antenna. The received signals \({y_{R_{A}}^{\left (1\right) }}\in {\mathbb {C}}\) and \({ y_{T_{B}}^{\left (1\right) }}\in {\mathbb {C}}\) at RA and TB in the first phase can be expressed respectively as
In (1) and (2), sA denotes the signal transmitted by UA. Moreover, \({h_{T_{A},R_{A}}}\sim \mathcal {G}_{0}\left ({\sigma _{T_{A},R_{A}}^{2}}\right) \) and \({h_{T_{A},T_{B}}}\sim \mathcal {G}_{0}\left ({\sigma _{T_{A},T_{B}}^{2}}\right) \) are the channel gains of links TA-to- RA and TA-to- TB, respectively, with variance \(\sigma _{T_{A},R_{A}}^{2}\) and \(\sigma _{T_{A},T_{B}}^{2}\). Here, \( \mathcal {G}_{0}(\sigma ^{2})\) denotes a zero-mean complex Gaussian distribution with variance σ2; finally, \({n_{R_{A}}^{\left (1\right) }}\sim \mathcal {G}_{0}\left ({\sigma _{n,R_{A}}^{2}}\right) \) and \({ n_{T_{B}}^{\left (1\right) }}\sim \mathcal {G}_{0}\left ({\sigma _{n,T_{B}}^{2}}\right) \) denote the noise received at RA and TB, respectively, with variance \({\sigma _{n,R_{A}}^{2}}\) and \({\sigma _{n,T_{B}}^{2}}\); and PA is the transmitted power of TA.
When TB receives the signal from TA in the first phase, it performs a DF operation on the received signal sA. If sA is decoded correctly, then TB combines sA and sB using SC1. However, if the decoding process fails, then TB simply transmits its own signal sB. Note that in the present study, it is assumed RA learns the result of the DF operation through a control signal. Thus, the received signal at RA and RB in the second phase are expressed respectively as
Here, \({h_{T_{B},R_{A}}}\sim \mathcal {G}_{0}\left ({\sigma _{T_{B},R_{A}}^{2} }\right) \) and \({h_{T_{B},R_{B}}}\sim \mathcal {G}_{0}\left ({\sigma _{T_{B},R_{B}}^{2}}\right) \) are the fading gains of links TB-to- RA and TB-to- RB respectively, with variance \({\sigma _{T_{B},R_{A}}^{2}}\) and \({\sigma _{T_{B},R_{B}}^{2}}\); \({n_{R_{A}}^{\left (2\right) }}\sim \mathcal {G}_{0}\left ({\sigma _{n,R_{A}}^{2}}\right) \) and \({ n_{R_{B}}^{\left (2\right) }}\sim \mathcal {G}_{0}\left ({\sigma _{n,R_{B}}^{2}}\right) \) denote the noise received at RA and RB, respectively, with variance \({\sigma _{n,R_{A}}^{2}}\) and \({\sigma _{n,R_{B}}^{2}}\); PB is the transmitted power of TB; and δ is the power allocation factor applied for combining signals sA and sB in the SC scheme. The larger δ is the larger portion of the composite signal allocating to sA.
2.2 BER performance
In evaluating the BER performance of the considered overlay CR scheme, the following discussions commence by deriving analytical formulas for the BERs in decoding sA at RA and sB at RB, respectively. Tight approximated BER expressions are then derived to facilitate the SC power allocation optimization process.
2.2.1 BER performance in decoding sA for UA
Decoding signal sA at RA involves two diverse cases, namely TB decodes sA correctly and TB decodes sA incorrectly. For the former case, TB transmits the SC-combined signal shown in (3) and (4) to RA and RB concurrently in the second phase. It is thus necessary to first decode sB correctly and then to subtract sB from \({y_{R_{A}}^{\left (2\right) }}\). Since sB is subtracted first, the SC power allocation factor δ is restricted to the interval [0,0.5] to avoid serious degradation in decoding sA. Then, MRC is adopted to combine the extracted signals for decoding sA from the two phases. Note that if RA fails to decode sB, MRC is not performed, and sA is decoded merely based on the signal \({ y_{R_{A}}^{\left (1\right) }}\). Here, we denote \(P_{e,s_{A}}^{R_{A},I}\) as the BER of sA at RA for this case.
For the latter case, TB incorrectly decodes sA, where RA detects sA through \({y_{R_{A}}^{\left (1\right) }}\). Denote \( P_{e,s_{A}}^{T_{B}}\) and \(P_{e,s_{A}}^{R_{A},II}\) as the BERs of sA at TB and RA, respectively. The overall average BER denoted as Pe,A can then be evaluated as
The three BER terms in (5), i.e., \(P_{e,s_{A}}^{T_{B}}\), \( P_{e,s_{A}}^{R_{A},I}\), and \(P_{e,s_{A}}^{R_{A},II}\) are separately derived in the following. Without loss of generality, for sA and sB, binary phase shift keying is adopted and an equal probability of the data symbol outcomes is assumed.
Since sA is corrupted only by the fading channel gain \(h_{T_{A},R_{A}}\), the conditioned BER can be expressed as [22]
The channel gains are assumed to be complex Gaussian distributed. As a result, \(P_{e,s_{A}}^{T_{B}}\) can be easily obtained by averaging \( P_{e,s_{A}}^{T_{B}}\left (h_{T_{A},T_{B}}\right) \) over \(h_{T_{A},T_{B}}\) to give [22]
where \({\gamma _{T_{A},R_{A}}}=\sigma _{T_{A},R_{A}}^{2}\)PA\(/\sigma _{n,R_{A}}^{2}\).
Calculating \(P_{e,s_{A}}^{R_{A},I}\) also involves two diverse cases, namely, sB is correctly decoded at RA in the second phase, or sB is incorrectly decoded. For the former case, if the received signal \({ y_{R_{A}}^{\left (2\right) }}\) subtracts the correctly decoded sB to be \(\tilde {y}{_{R_{A}}^{\left (2\right) }}\), then an MRC operation is optimally performed over the received signals \({y_{R_{A}}^{\left (1\right) }}\) and \( \tilde {y}{_{R_{A}}^{\left (2\right) }}\).2 Since sB needs to be correctly decoded first, sB requires a larger amount of power to be allocated, which means 0≤δ≤0.5. Otherwise, the error occurs even when the noise is absent. Let the BER of this case be denoted as \( P_{e,s_{A}}^{R_{A},I,\mathrm {(MRC)}}\). The corresponding BER conditioned on the channel gains required for decoding sA can be derived as
For the latter case, if sB is decoded incorrectly, we only use \( y_{R_{A}}^{\left (1\right) }\) to detect sA, and the corresponding BER is equal to \(P_{e,s_{A}}^{R_{A},II}\) as shown in (8). Let the BER of sB at RA be denoted as \(P_{e,s_{B}}^{R_{A},I}\) with the given channel gain \(h_{T_{B},R_{A}}\), \(P_{e,s_{B}}^{R_{A},I}\) can then be computed as
Here, we can observe that the formulation of (10) is more complicated than that of (6) since sB is decoded from a composite signal in (3), while sA is decoded from the received signal in (2). The BER in decoding sA at RA can be obtained with the assistance of (8)–(10) as
where \({\gamma _{T_{A},R_{A}}}=\sigma _{T_{A},R_{A}}^{2}\)PA\(/\sigma _{n,R_{A}}^{2}\) and \({\gamma _{T_{B},R_{A}}}=\sigma _{T_{B},R_{A}}^{2}\)PB\(/\sigma _{n,R_{A}}^{2}\). Finally, (5) can be calculated in the close form using the approximation given in Appendix A, where Pe,A is obtained as a function of δ, PA, and PB.
2.2.2 BER performance in decoding sB for UB
Determining the BER of UB also requires the consideration of two separate cases, namely TB decodes sA correctly and incorrectly. First, consider the case where TB decodes sA correctly. In this scenario, RB receives an SC-combined composite signal from TB in the second phase. Consequently, sB can be directly decoded from the composite signal \({y_{R_{B}}^{\left (2\right) }}\) using (4). As for the derivation of \(P_{e,s_{B}}^{R_{A},I}\left (h_{T_{B},R_{A}}\right) \) in (10), the conditional BER in decoding sB at RB, denoted as \(P_{e,s_{B}}^{R_{B},I}\left (h_{T_{B},R_{B}}\right) \), is given by
with \(\gamma {_{T_{B},R_{B}}}=\sigma _{T_{B},R_{B}}^{2}\)PB\(/\sigma _{n,R_{B}}^{2}\).
For the case where TB decodes sA incorrectly, TB merely transmits sB to RB. Thus, the BER in decoding sB, denoted as \(P_{e,s_{B}}^{R_{B},II}\), can then be calculated as
To facilitate the power allocation optimization process, this section derives two more tractable approximations for Pe,A and Pe,B, denoted as \(\widetilde {P}_{e,A}\) and \(\widetilde {P}_{e,B}\), respectively.
The BER Pe,A derived in (5) can be approximated as
The detailed derivation is provided in Appendix B.
Figure 2 compares the approximated BER curves with true BER curves for \(\sigma _{T_{A},T_{B}}^{2}/\sigma _{n,T_{B}}^{2}=5\) dB, \(\sigma _{T_{A},R_{A}}^{2}/\sigma _{n,R_{A}}^{2}=0\) dB, \(\sigma _{T_{B},R_{A}}^{2}/\sigma _{n,R_{A}}^{2}\,=\,10\) dB, \(\sigma _{T_{B},R_{B}}^{2}/\sigma _{n,R_{B}}^{2}\,=\,10\) dB, PA = PB = 1, and δ=0.1 and 0.3. As shown, the approximated BER values are in good agreement with the actual values; particularly in the high SNR region. Notably, the BER of UA has a diversity gain of two, whereas that of UB has a diversity gain of one. The lower diversity gain of UB arises since it only uses a link to transmit signals.
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3 Optimal power allocation
3.1 Optimization of power allocation
The previous section has analyzed the BER performance of the primary and secondary users in the considered overlay CR network. Observing the analytical results in (5) and (15), it is seen that the BERs for UA and UB are closely related to the transmitted power, i.e., PA and PB and the SC power allocation factor δ. Taking the BER as the benchmark of the provided QoS, this section proposes a strategy for determining the SC allocation factor which minimizes the total power consumption of the two users, PA + PB, while simultaneously achieving Pe,A≤BERA and Pe,B≤BERB, where BERA and BERB are the predefined QoSs of the two users. The optimization problem can thus be formulated as
As shown in (5) and (15), the optimization constraints have the form of non-convex functions. Thus, the optimal solution of (18) is almost impossible to obtain, even when using numerical methods. Thus, to facilitate the optimization process, Pe,A and Pe,B are replaced respectively with \(\widetilde {P}_{e,A}\) in (16) and \( \widetilde {P}_{e,B}\) in (17). The optimization problem can then be reformulated as
The optimal solutions of the proposed optimization (P2) can be sophisticatedly solved though it is still not convex. We have to mention that the proposed approximations (16) and (17) are conservative approximations meaning that the approximated BERs are the upper bounds of the true BERs, and consequently the QoS requirements are always satisfied. Moreover, the BER approximations are tight when the channel quality of the link TA-to- RA or TB-to- RA is favorable, or equivalently the QoS requirements are assumed to be moderately low. For brevity, define \({\eta _{a}}=\sigma _{n,R_{A}}^{4}\sigma _{n,T_{B}}^{2}/\left ({16\sigma _{T_{A},R_{A}}^{2}\sigma _{T_{A},T_{B}}^{2}} \right) \), \({\eta _{b}}=\sigma _{n,R_{A}}^{4}/\left ({16\sigma _{T_{A},R_{A}}^{2}\sigma _{T_{B},R_{A}}^{2}}\right) \), \({\eta _{c}}=\sigma _{n,R_{A}}^{4}\sigma _{n,T_{B}}^{2}/\left (64\sigma _{T_{A},R_{A}}^{2}\right.\)\(\left.\sigma _{T_{A},T_{B}}^{2}\sigma _{T_{B},R_{A}}^{2}\right) \), \({\eta _{d}}=\sigma _{n,T_{B}}^{2}\sigma _{n,R_{B}}^{2}/\left ({16\sigma _{T_{A},T_{B}}^{2}\sigma _{T_{B},R_{B}}^{2}}\right) \), \({\eta _{e}}=\sigma _{n,R_{B}}^{2}/\left ({4\sigma _{T_{B},R_{B}}^{2}}\right) \).
To optimize (19), we first notice that \(\widetilde {P}_{e,A}\) and \(\widetilde {P}_{e,B}\) are monotonically decreasing with increasing PA and/or PB for any fixed δ. Hence, the minimum total transmitted power (PA+PB) can be achieved solely with \(\widetilde {P}_{e,A}\) and \(\widetilde {P}_{e,B}\) being satisfied by the minimum requirement of BER, i.e., \(\widetilde {P}_{e,A} = {\text {BER}_{A}}\) and \(\widetilde {P}_{e,B} = {\text {BER}_{B}}\). With these two equalities, (19) can be solved by optimizing the parameter δ only. Specifically, with \(\widetilde {P}_{e,B} = {\text {BER}_{B}}\), we have
where \(\cos \theta =\frac {{{f_{Q}}\left ({\delta }\right) }}{\sqrt {-{f_{R}} \left ({\delta }\right) }}\). It can be shown that \(-\frac {g_{A}(\delta)}{3}\) and \(2\sqrt {-f_{R}(\delta)}\cos \frac {1}{3}\theta \) are both convex in δ∈[ 0,0.5]. Thus, PA is convex in δ∈[0,0.5]. It is noted that (23) has only one solution which satisfies both constraints since the other two solutions either contradict the basic power assumption or the BER constraint, i.e., PA ≥0 or \(\widetilde {P}_{e,A}\geq 1\), respectively.
From the last equality, it follows that \({\eta _{e}}-\frac {{{\eta _{d}}}}{{{ P_{A}}}}=\frac {{\sigma _{n,R_{B}}^{2}}}{{4\sigma _{T_{B},R_{B}}^{2}}}\left ({ 1-\frac {{\sigma _{n,T_{B}}^{2}}}{{4{P_{A}}\sigma _{T_{A},T_{B}}^{2}}}} \right) \). Notably, PB is convex in δ∈[ 0,0.5] for any given PA\(\geq \frac {{\sigma _{n,T_{B}}^{2}}}{{4\sigma _{T_{A},T_{B}}^{2}}}\).
Finally, by adding an auxiliary variable t along with the results in (30) and (31), (19) can be rewritten as the following optimization problem:
where gA(δ) and fR(δ) are functions of δ expressed in (24) and (28), respectively. The problem formulated in (32) is essentially a convex optimization problem. Thus, the solutions of PA and δ can be obtained via numerical methods, such as the simple steepest descent method, while PB can be obtained via (31).
3.2 Iterative approach
The previous subsection first simplifies the optimization problem (P1) to (P2) with two equalities (\(\widetilde {P}_{e,A} = {\text {BER}_{A}}\) and \(\widetilde {P}_{e,B} = {\text {BER}_{B}}\)), and finally presents the compact results in (P3). In this subsection, we intuitively provide an alternative approach to solve (P2) suboptimally. It is clear that when any two parameters are fixed as constants in (P2), the problem becomes a convex and only the residual parameter has to be optimized. Thus, we can divide (P2) into the following three problems.
Since (P4 − 1)–(P4 − 3) are convex problems, we can iteratively solve the solutions through the cvx tool [24]. It is noteworthy that the iterative approach can essentially obtain a local solution and is sensitive to the initial condition, while the proposed non-iterative approach has only one local solution which suggests it is also a global solution. Thus, it is expected that the performance of the non-iterative approach is superior to the iterative approach. More discussion is provided in the following section.
4 Simulation results
This section evaluates the performance of the proposed power allocation method under realistic overlay CR network conditions. In performing the simulations, it is assumed that all of the channel links suffer fading with independent and identically distributed (i.i.d.) Rayleigh-distributed amplitudes. Let \(\sigma _{n,T_{B}}^{2}=\sigma _{n,R_{B}}^{2}=\sigma _{n,R_{A}}^{2}=1\), BERA=10−3, and BERB=5×10−3. Furthermore, let three levels of channel quality be considered, namely \( \left (\sigma _{T_{A},R_{A}}^{2},\sigma _{T_{A},T_{B}}^{2},\sigma _{T_{B},R_{A}}^{2},\sigma _{T_{B},R_{B}}^{2}\right)\) = (0,5,10,10), (0,5,20,20), (0,10,30,30) dBm. For convenience, let the three cases be referred to as case 1, case 2, and case 3, respectively. Figure 3 shows the results for the variation of the power consumption with the power allocation factor. Note that the circles represent the solution obtained using the proposed method. It is seen that the circles are coincident with the point of minimum total power consumption in every case. In other words, the optimum δ obtained in (32) coincides perfectly with the numerical results. In addition, it is observed that the total power consumption in case 2 is less than that in case 1 since the channel quality in case 2 is better than that in case 1. Finally, it is seen that a larger value of δ results in a higher \(\sigma _{T_{B},R_{B}}^{2}\). This result is intuitively reasonable since when \(\sigma _{T_{B},R_{B}}^{2}\) increases, BERB is more easily satisfied, and hence a greater amount of power is allocated to UA in the SC scheme.
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Further simulations were performed to investigate the variation of the total power consumption associated to the fixed power allocation, the iterative approach, and the proposed design. Let \(\sigma _{T_{A},R_{A}}^{2}/{\sigma _{n}^{2}}\,=\,0\) dB, \(\sigma _{T_{A},T_{B}}^{2}/{\sigma _{n}^{2}}\,=\,15\, \text {dB}\), \(\sigma _{T_{B},R_{B}}^{2}/{\sigma _{n}^{2}}=\sigma _{T_{B},R_{A}}^{2}/\sigma _{n}^{2}={\sigma _{h}^{2}}/{\sigma _{n}^{2}},\)δ=0.1, 0.2, 0.3, 0.4. Figures 4 and 5 show the results obtained for the total power consumption and the resultant BERs of UB and UA of the proposed design, respectively. As shown in Fig. 4, for a given δ, the power consumption decreases as δ decreases. This result follows intuitively since the signal of UA is potentially conveyed through two links, i.e., TA-to- TB-to- RA and TA-to- RA, whereas the signal of UB is transmitted through only one link, i.e., TB-to- RB. Due to the fixed channel quality of TB-to- RB, TB is able to use approximated constant power consumption to maintain the BER at RB. A smaller value of δ indicates that UB offers a lower level of assistance. Thus, the power consumption at TA must be increased, and hence the total power consumption also increases. Since the optimum value of δ is obtained by the proposed SC allocation factor strategy, the minimal power consumption can also be obtained. The iterative approach performs worse than the proposed design since the iterative approach is a suboptimal method. Figure 5 shows the resultant BER corresponding to the proposed design (the curve related to the optimum δ) in Fig. 4. The results confirm that the resultant BERs are consistent with the required QoS constants. Since δ represents the power allocated to UA, the power of UA must be increased to enable RA to conduct MRC when δ is small, which ensures that RA satisfies the QoS constraint.
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A final series of simulations was performed to investigate the power consumption given different channel link qualities and QoS constraints. Let BERA=BERB=γ. Four different link qualities were considered, namely \(\left (\sigma _{T_{A},R_{A}}^{2},\sigma _{T_{A},T_{B}}^{2},\sigma _{T_{B},R_{A}}^{2},\sigma _{T_{B},R_{B}}^{2}\right)\) = (0,5,10,10), (0,5,20,20), (0,10,20,20), (0,15,30,30) dBm. As shown in Fig. 6, for a given link quality, the power consumption increases with a decreasing BER. Furthermore, for a constant BER, the power consumption given a channel link quality of (0,5,20,20) is almost identical to that for a channel link quality of (0,10,20,20). However, for channel link qualities of (0,5,10,10) and (0,5,20,20), a significant difference in the power consumption is observed. These findings are to be expected since, in the former case, only link TA-to- TB has a good channel quality. As a result, the power of UA is decreased, while that of UB is potentially increased under the SC scheme. By contrast, in the latter case, both links TB-to- RA and TB-to- RB have favorable channel quality. Consequently, the powers allocated to UA and UB are both decreased, with the result that the total power consumption is also decreased.
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5 Conclusions
This paper has considered the problem of optimizing the power allocation strategy in cooperative spectrum-sharing cognitive CR systems. Given the use of a SC scheme at TB, analytical expressions have been derived for the corresponding BERs at RA and RB. To determine the optimal SC power allocation factor, two more tractable BER expressions have been proposed for UA and UB. The allocation problem has been formulated as an optimization problem in which the aim is to minimize the total power consumption of the two users while simultaneously satisfying their BER constraints. Notably, the proposed BER approximations enable the optimization problem to be expressed as a convex formulation amenable to numerical processing. The simulation results have confirmed the ability of the proposed power allocation strategy to achieve the optimal power distribution over a range of realistic CR conditions.
6 Appendix A: Derivation of \(\widetilde {P}_{e,A}\)
The second term of (11), denoted as \(I_{1}^{R_{A}}\), can be computed as
Note that the approximation given in (45) is followed by ignoring the second term of (44) whenever \(\frac {{{P_{A}}\sigma _{T_{A},R_{A}}^{2}}}{{\sigma _{n,R_{A}}^{2}}}\) or \(\frac {{{P_{B}}\sigma _{T_{B},R_{A}}^{2}}}{{\sigma _{n,R_{A}}^{2}}}\) approaches infinity, i.e.,
which completes the derivation of \(\widetilde {P}_{e,B}\).
8 Endnotes
1 It is noteworthy that dirty paper coding (DPC) may be applied at TB to convey sA and sB which is theoretical optimal in a 2-user SISO Gaussian broadcasting channel system [25]. However, DPC relies on the perfect channel state information to achieve optimal performance and is vulnerable to channel estimation error. Thus, DPC is not considered herein for it violating our economic scenario where only the statistical channel variance is required.
2 If sB is not detected in advance, a MRC operation can not optimally be adopted to facilitate the design of power allocation.
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