In this section, the problems of joint bandwidth and power allocation with AF or DF relaying protocol are formulated and solved for different performance metrics. For convenience, we set{\gamma}_{0}=\frac{1}{{N}_{0}} in the subsequent discussions, and focus on the high regime of γ_{0} in AF relaying protocol. Therefore, the twohop sourcedestination link throughput for the k th relaying link with AF relaying protocol can be approximated as
{R}_{k,\text{SD}}^{\mathrm{AF}}\sim {W}_{k}{\text{log}}_{2}\left(1+\frac{{P}_{k}^{S}{P}_{k}^{R}{\left{h}_{k}^{\text{SR}}\right}^{2}{\left{h}_{k}^{\text{RD}}\right}^{2}{\gamma}_{0}}{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}{W}_{k}+{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}{W}_{k}}\right).
(5)
Moreover, in this section, we assume that relay SU will be able to execute DF relaying protocol if the data rate between source SU and relay SU is greater than zero. In other words, there is no decoding rate constraint in DF relaying protocol.
Sum throughput maximization
1) AF relaying protocol
For AF relaying protocol, the joint bandwidth and power allocation problem aiming at maximizing the sum throughput of the CR network can be formulated as follows:
Problem P1:
\begin{array}{ll}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{R}_{k,\text{SD}}^{\mathrm{AF}}\phantom{\rule{2em}{0ex}}\end{array}
(6)
\begin{array}{ll}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{W}_{k}\le W\phantom{\rule{2em}{0ex}}\end{array}
(7)
\begin{array}{l}\phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)\le {P}_{\text{th}}\phantom{\rule{2em}{0ex}}\end{array}
(8)
\begin{array}{l}\phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{\left{g}_{\mathrm{SP}}\right}^{2}{P}_{k}^{S}\le {I}_{\text{th}}\phantom{\rule{2em}{0ex}}\end{array}
(9)
\begin{array}{l}\phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{\left{g}_{k}^{\mathrm{RP}}\right}^{2}{P}_{k}^{R}\le {I}_{\text{th}}\phantom{\rule{2em}{0ex}}\end{array}
(10)
where P_{th} is the maximum total power that can be used for transmission. I_{th} is the maximum allowed interference to PU band.
Proposition 1
For any given power\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n S\n \n \n \n and\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n R\n \n \n \n (k = 1,2,…,K), the optimal bandwidth allocation of Problem P1 can be found as
{W}_{k}^{\ast}=W\frac{\frac{{P}_{k}^{S}{P}_{k}^{R}{\left{h}_{k}^{\text{SR}}\right}^{2}{\left{h}_{k}^{\text{RD}}\right}^{2}{\gamma}_{0}}{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}+{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}}{\sum _{k=1}^{K}\frac{{P}_{k}^{S}{P}_{k}^{R}{\left{h}_{k}^{\text{SR}}\right}^{2}{\left{h}_{k}^{\text{RD}}\right}^{2}{\gamma}_{0}}{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}+{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}}.
(11)
Proof
See Appendix 1. □
According to Proposition 1, Problem P1 can be equivalently written as
Problem P2:
\begin{array}{ll}\underset{{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}W{\text{log}}_{2}\left(1+\frac{1}{W}\sum _{k=1}^{K}\frac{{P}_{k}^{S}{P}_{k}^{R}{\left{h}_{k}^{\text{SR}}\right}^{2}{\left{h}_{k}^{\text{RD}}\right}^{2}{\gamma}_{0}}{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}+{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}\right)\phantom{\rule{2em}{0ex}}\end{array}
(12)
\begin{array}{cc}{\mathrm{s}}_{\xb7}{\mathrm{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}\left(8\right)\left(9\right)\left(10\right)\end{array}
(13)
Furthermore, Problem P2 is equivalent to
Problem P3:
\begin{array}{ll}\underset{{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}\frac{{P}_{k}^{S}{P}_{k}^{R}{\left{h}_{k}^{\text{SR}}\right}^{2}{\left{h}_{k}^{\text{RD}}\right}^{2}{\gamma}_{0}}{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}+{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}\phantom{\rule{2em}{0ex}}\end{array}
(14)
\begin{array}{cc}{\mathrm{s}}_{\xb7}{\mathrm{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}\left(8\right)\left(9\right)\left(10\right)\end{array}
(15)
Proposition 2
The objective function of Problem P3 is concave in\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n S\n \n \n \n and\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n R\n \n \n \n (k = 1,2,…,K).
Proof
See Appendix Appendix 2. □
Obviously, the constraint conditions of Problem P3 are convex. Therefore, Problem P3 is a convex optimization problem. Some standard numerical algorithms for convex optimization can be used to find the optimal solution.
2) DF relaying protocol
For DF relaying protocol, the joint bandwidth and power allocation problem aiming at maximizing the sum throughput of the CR network can be formulated as follows:
Problem P4:
\begin{array}{cc}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{R}_{k,\text{SD}}^{\text{DF}}\end{array}
(16)
\begin{array}{cc}{\mathrm{s}}_{\xb7}{\mathrm{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}& \phantom{\rule{1em}{0ex}}\left(7\right)\left(8\right)\left(9\right)\left(10\right)\end{array}
(17)
Similar to[20], through introducing new variables T_{
k
}, Problem P4 can be equivalently written as
Problem P5:
\begin{array}{cc}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R},{T}_{k}}{\text{max}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{T}_{k}\end{array}
(18)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{1em}{0ex}}& \phantom{\rule{1em}{0ex}}{T}_{k}{R}_{k,\text{SR}}^{\text{DF}}\le 0,\phantom{\rule{1em}{0ex}}{T}_{k}{R}_{k,\text{RD}}^{\text{DF}}\le 0,\end{array}
\begin{array}{c}k=1,2,\dots ,K\end{array}
(19)
\begin{array}{c}\phantom{\rule{2em}{0ex}}\left(7\right)\left(8\right)\left(9\right)\left(10\right)\end{array}
(20)
It is obvious that\left(\right)close="">\n \n \n \n R\n \n \n k\n ,\n SR\n \n \n DF\n \n \n \n and\left(\right)close="">\n \n \n \n R\n \n \n k\n ,\n RD\n \n \n DF\n \n \n \n are joint concave functions of W_{
k
},\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n S\n \n \n \n, and\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n R\n \n \n \n. Therefore, Problem P5 is a convex optimization problem. According to the well known relationship on the harmonic mean
\frac{\frac{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}}{{W}_{k}{N}_{0}}\frac{{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}{{W}_{k}{N}_{0}}}{\frac{{P}_{k}^{S}{\left{h}_{k}^{\text{SR}}\right}^{2}}{{W}_{k}{N}_{0}}+\frac{{P}_{k}^{R}{\left{h}_{k}^{\text{RD}}\right}^{2}}{{W}_{k}{N}_{0}}}\le \text{min}\left\{\frac{\underset{k}{\overset{S}{P}}{\left{h}_{k}^{\text{SR}}\right}^{2}}{{W}_{k}{N}_{0}},\frac{\underset{k}{\overset{S}{P}}{\left{h}_{k}^{\text{SR}}\right}^{2}}{{W}_{k}{N}_{0}}\right\},
(21)
we can easy to show that the sum throughput in DF relaying protocol is superior to that in AF relaying protocol. This conclusion is also validated by latter numerical simulation results.
Power minimization with considering the fairness
In the problem of maximizing sum throughput, the fairness of relay SUs is not considered. In general, fairness could be defined in terms of different parameters of the system[21]. In this article, we focus on the fairness of power drain of relay SUs. When the differences of relay SUs’ channel power gains are large, it is possible that relay SUs with higher channel power gains will consume most of power. Relay SUs with lower channel power gains might not need to consume any power. This will result in a lower survival time of some relay SUs and the CR network. Moreover, by taking into consideration the limited transmit power of the CR network, we minimize the total transmit power of the CR network simultaneously.
1) AF relaying protocol
For AF relaying protocol, the joint bandwidth and power allocation problem aiming at minimize the total transmit power of the CR network can be formulated as follows:
Problem P6:
\begin{array}{cc}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{min}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)\end{array}
(22)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}{r}_{k}{R}_{k,\text{SD}}^{\mathrm{AF}}\le 0,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,K\end{array}
(23)
\begin{array}{c}\phantom{\rule{1em}{0ex}}{P}_{1}^{R}:{P}_{2}^{R}:\cdots :{P}_{K}^{R}={\alpha}_{1}:{\alpha}_{2}:\cdots :{\alpha}_{K}\end{array}
(24)
\begin{array}{c}\phantom{\rule{1em}{0ex}}\left(7\right)\left(8\right)\left(9\right)\left(10\right)\end{array}
(25)
where r_{
k
} is the minimum acceptable throughput for k th relaying link. {α_{1},α_{2},…,α_{
K
}} is the set of predetermined proportional constraints that are used to ensure fairness. In this article, we set α_{1} = α_{2} = ⋯ = α_{
K
} = 1.
Proposition 3
The function\left(\right)close="">\n \n \n \n R\n \n \n k\n ,\n SD\n \n \n AF\n \n \n \n of Problem P6 is concave in W_{
k
},\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n S\n \n \n \n, and\left(\right)close="">\n \n \n \n P\n \n \n k\n \n \n R\n \n \n \n (k = 1,2,…,K).
Proof
The proof is similar to Proposition 2, and is omitted for brevity in this article. □
According to Proposition 3, we can know that Problem P6 is also a convex optimization problem. The optimal solution can be efficiently obtained.
2) DF relaying protocol
For DF relaying protocol, The joint bandwidth and power allocation problem aiming at minimize the total transmit power of the CR network can be formulated as follows:
Problem P7:
\begin{array}{cc}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{min}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)\end{array}
(26)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}{r}_{k}{R}_{k,\text{SD}}^{\text{DF}}\le 0,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,K\end{array}
(27)
\begin{array}{c}\phantom{\rule{1em}{0ex}}{P}_{1}^{R}:{P}_{2}^{R}:\cdots :{P}_{K}^{R}={\alpha}_{1}:{\alpha}_{2}:\cdots :{\alpha}_{K}\end{array}
(28)
\begin{array}{c}\phantom{\rule{1em}{0ex}}\left(7\right)\left(8\right)\left(9\right)\left(10\right)\end{array}
(29)
The solution of Problem P7 is similar to the solution of Problem P6. In order to save space, the description will not be repeated again.
Energy efficient
1) AF relaying protocol
Energy efficiency in the CR network is another widely considered design objective. Therefore, for AF relaying protocol, the corresponding joint bandwidth and power allocation problem aiming at maximizing energy efficiency can be formulated as follows:
Problem P8:
\begin{array}{cc}f=\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\frac{\sum _{k=1}^{K}{R}_{k,\text{SD}}^{\mathrm{AF}}}{\sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)}\end{array}
(30)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}{r}_{k}{R}_{k,\text{SD}}^{\mathrm{AF}}\le 0,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,K\end{array}
(31)
\begin{array}{c}\phantom{\rule{1em}{0ex}}\left(7\right)\left(9\right)\left(10\right)\end{array}
(32)
Let Z denotes the set of a solution to Problem P8 and f_{
Z
} denotes the energy efficiency achieved by using the set Z. Thus, the set of the optimal solution is given by{Z}^{\ast}=\underset{Z}{\text{arg max}}{f}_{Z}. The optimization Problem P8 belongs to the FP problem, which is difficult to solve directly. Instead, we can transform the fractional programming to the parametric formulation, which allows convex optimization technology to be applied to find the optimal bandwidth and power allocation strategy. Similar to[14], the parametric formulation can be given as follows:
Problem P9:
\begin{array}{cc}g\left(\lambda \right)=\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\sum _{k=1}^{K}{R}_{k,\text{SD}}^{\mathrm{AF}}\lambda \sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)\end{array}
(33)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}\left(7\right)\left(9\right)\left(10\right)\left(31\right)\end{array}
(34)
where λ is a given value. Let Z(λ) denotes the set of a solution to Problem P9 and g_{Z(λ)}(λ) denotes the value achieved by using the set Z(λ). Thus, the set of the optimal solution for a given value of λ is given by{Z}^{\ast}\left(\lambda \right)=\underset{Z\left(\lambda \right)}{arg\; max}{g}_{Z\left(\lambda \right)}\left(\lambda \right). According to Proposition 3, we can know that Problem P9 is a convex optimization problem for a given value of λ.
Therefore, the relationship between Problem P8 and Problem P9 is established as follows.
Proposition 4
\left(\right)close="">\n \n \n \n f\n \n \n \n \n Z\n \n \n \u2217\n \n \n \n \n =\n \n \n \lambda \n \n \n \u2217\n \n \n \n if and only if\left(\right)close="">\n \n \n \n g\n \n \n \n \n Z\n \n \n \u2217\n \n \n (\n \lambda \n )\n \n \n \n \n \n \n \lambda \n \n \n \u2217\n \n \n \n \n =\n 0\n \n.
Proof
The proof is similar to Proposition 1 in[14], and is omitted for brevity in this article. □
Proposition 5
\left(\right)close="">\n \n \n \n g\n \n \n \n \n Z\n \n \n \u2217\n \n \n (\n \lambda \n )\n \n \n (\n \lambda \n )\n \n is a monotonously decreasing function of λ.
Proof
The proof is similar to Proposition 2 in[14], and is omitted for brevity in this article. □
According to Proposition 4, we can know that if we can find an optimal λ^{∗} such that the optimal value\left(\right)close="">\n \n \n \n g\n \n \n \n \n Z\n \n \n \u2217\n \n \n (\n \lambda \n )\n \n \n \n \n \n \n \lambda \n \n \n \u2217\n \n \n \n \n \n of Problem P9 is 0, then the corresponding optimal solution of Problem P9 is also optimal for Problem P8. Furthermore, instead of using exhaustive search to identify the optimal λ^{∗}, we can use a more efficient bisection search method according to Proposition 5. Algorithm 1 gives the method for finding the optimal λ^{∗}. In Algorithm 1, the interval[{\lambda}_{\text{min}},{\lambda}_{\text{max}}] is selected to contain λ^{∗}, and ϵ is a predefined small constant.
Algorithm 1: Find the optimal λ^{∗}

1.
Given:
\lambda \in [{\lambda}_{\text{min}},{\lambda}_{\text{max}}]

2.
Repeat

a)
\lambda \leftarrow \left({\lambda}_{\text{min}}+{\lambda}_{\text{max}}\right)/2

b)
Solve convex optimization Problem P9 and get\left(\right)close="">\n \n \n \n g\n \n \n \n \n Z\n \n \n \u2217\n \n \n (\n \lambda \n )\n \n \n (\n \lambda \n )\n \n

c)
if\left(\right)close="">\n \n \n \n g\n \n \n \n \n Z\n \n \n \u2217\n \n \n (\n \lambda \n )\n \n \n (\n \lambda \n )\n \u2264\n 0\n \n, set{\lambda}_{\text{max}}\leftarrow \lambda
else set{\lambda}_{\text{min}}\leftarrow \lambda

3.
Until
{\lambda}_{\text{max}}{\lambda}_{\text{min}}\le \u03f5
(2) DF relaying protocol
For DF relaying protocol, the joint bandwidth and power allocation problem aiming at maximizing energy efficiency can be formulated as follows:
Problem P10:
\begin{array}{cc}\underset{{W}_{k},{P}_{k}^{S},{P}_{k}^{R}}{\text{max}}& \phantom{\rule{1em}{0ex}}\frac{\sum _{k=1}^{K}{R}_{k,\text{SD}}^{\text{DF}}}{\sum _{k=1}^{K}\left({P}_{k}^{S}+{P}_{k}^{R}\right)}\end{array}
(35)
\begin{array}{cc}{\text{s}}_{\xb7}{\text{t}}_{\xb7}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}& \phantom{\rule{1em}{0ex}}{r}_{k}{R}_{k,\text{SD}}^{\text{DF}}\le 0,\phantom{\rule{1em}{0ex}}k=1,2,\dots ,K\end{array}
(36)
\begin{array}{c}\phantom{\rule{1em}{0ex}}\left(7\right)\left(9\right)\left(10\right)\end{array}
(37)
The solution of Problem P10 is similar to the solution of Problem P8. In order to save space, the description will not be repeated again.