# Multi-band RF Energy and Spectrum Harvesting in Cognitive Radio Networks

Ahmad Alsharoa, Nathan M. Neihart, Sang W. Kim, Ahmed E. Kamal
Iowa State University (ISU), Ames, Iowa, United States,
Email: {alsharoa, neihart, swkim<EMAIL_ADDRESS>
###### Abstract

This paper investigates a multi-band harvesting (EH) schemes under cognitive
radio interweave framework. All secondary users are considered as EH nodes
that are allowed to harvest energy from multiple bands of Radio Frequency (RF)
sources. A win-win framework is proposed, where SUs can sense the spectrum to
determine whether the spectrum is busy, and hence they may harvest from RF
energy, or if it is idle, and hence they can use it for transmission. Only a
subset of the SUs can sense in order to reduce sensing energy, and then
machine learning is used to characterize areas of harvesting and spectrum
usage. We formulate an optimization problem that jointly optimize number of
sensing samples and sensing threshold in order to minimize the sensing time
and hence maximize the amount of energy harvested. A near optimal solution is
proposed using Geometric Programming (GP) to optimally solve the problem in a
time-slotted period. Finally, an energy efficient approach based on multi-
class Support Vector Machine (SVM) is proposed by involving only training SUs
instead of all SUs.

##  I Introduction

With high demands of future generation wireless communication networks,
researchers have focused on proposing efficient and smart solutions for energy
and spectrum deficit problems. Prospective demands of future generation
wireless networks will require achieving around 1000 times higher data rates
and 10 times lower round-trip latency [1]. The profile ratio of the Internet
of Things (IoT) and the increasing number of mobile devices, are contributing
to these requirements. This is also coupled with IoT requirements of low
energy communication.

Cognitive radio is proposed as a novel solution to solve the spectrum deficit
problems. The basic idea of cognitive radio is that unlicensed or Secondary
Users (SUs) are allowed to utilize the spectrum of licensed users which are
also known as primary users (PUs) in an opportunistic manner [2, 3].

Energy harvesting (EH) is considered as one of the most robust methods to
mitigate the energy consumption problem and perpetuate the lifetime and
sustainability of wireless systems where around $30\%$ of the energy
expenditure of mobile devices is caused by wireless networking [4]. Radio
frequency (RF) EH, which is also known as wireless energy transfer, has been
introduced as an effective harvesting technique where energy is collected from
RF sources generated by other neighbor devices. nights [5].

Most of the previous works on RF EH focused on harvesting from a single band,
which limit the amount of the harvested energy [6]. Recently, few circuit
designs that use a single antenna [7, 8] or multiple antennas [9, 10] to
harvest from multiple band have been introduced in the literature, where the
multi-band concept increases the amount of harvested energy to the area ratio.
For instance, in [7], a dual-band circuit using single antenna has been
designed to harvest from 1.8GHz and 2.1GHz frequency bands. Another multi-band
circuit designating one wide antenna has been proposed in [8], where it
achieves a good efficiency around 40% for -15 dBm received power. While, in
[10] Keyrouz et.al designed multi-band RF with triple antennas using cascaded
rectifying stages. With -15 dBm received power, their proposed design achieves
efficiency around {45, 46, 25}% for {0.9, 1.8, 2.45} GHz, respectively. In
[9], the authors used multiple antennas for harvesting and showed that the
amount of harvested energy can be increased by harvesting from multiple bands
instead of a single band.

In this paper we investigate the classification problem of SUs, where SUs
perform sensing of the spectrum in order to determine 1) harvesting
geographical regions in which SUs in that region can harvest RF energy, and 2)
communication geographical regions in which SUs in that region can use the
spectrum without causing interference to the PUs. Furthermore, we propose an
efficient solution based on multi-class Support Vector Machine (SVM) [11].
This is therefore a win-win strategy for most of SUs because of the result of
sensing they either use the spectrum or harvest energy from RF signals. The
main contributions of this paper are summarized as follows

  * •

Investigating RF multi-band EH under cognitive radio framework, where SUs use
the harvested energy to perform sensing. For this reason, a time switching
protocol is adopted where SUs switch over time between sensing and EH.

  * •

Analyzing RF multi-band EH by using a wide antenna with multiple rectifying
stages.

  * •

Formulating a time-slotted optimization problem that aims to classify the
belonging regions of SUs and minimize the sensing time.

  * •

Optimizing the number of sensing samples and sensing threshold for each RF
band and finding the SUs that can harvest from RF multi-band signals.

  * •

Proposing an energy efficient solution based on multi-class SVM to determine
the harvesting and communication regions of SUs.

##  II System Model

In this work, we investigate a time-slotted system of a finite period of time
slots $t=1,..,\Gamma$, each of equal duration $T$.

###  II-A Network Model

We consider a cognitive radio network consisting of $U$ PUs and $M$ time
synchronous EH SUs using RF signals. Each PU transmits its signal over one
frequency band. Without loss of generality, we assume that each SU can sense
energy from RF signals from $\mathbb{B}$ bands using multiple rectifying
stages.

Based on the received power at the SUs, each SU can be in one or more of three
regions as shown in Fig 1. The first one is Harvesting Region (HR), where the
received power at the SU is greater than a specific sensitivity threshold
$P_{\text{th}}$ which will allow energy to be harvested [12]. This threshold
is due to the limitation of RF-to-DC circuit sensitivity. The second region
called Communication Region (CR) contains the SUs that can not sense the PU
channel and therefore, the channel maybe used for communication. The middle
one is Inactive Region (IR) with no harvesting and no communication, where all
the SUs can sense the PU signal but the received power is not enough to
activate the harvesting circuit (i.e., the received power is less than
$P_{\text{th}}$). Note that, in IR, underlay cognitive radio can be used where
the SUs are allowed to transmit their signals under a certain primary
interference threshold [13], but the use of the spectrum is beyond the scope
of this paper. Note that it is possible for SU $m$ to be in more that one
region at the same time (e.g., SU $m$ can be in HR for bands $b=\\{1,3\\}$, in
IR for band $b=2$, and in CR for bands $b=\\{4,5\\}$). In this paper, we study
the sensing performance of multi-band system in order to classify the SUs
according to the three regions above.

###  II-B Channel Model

We denote by $h^{b}_{u,m,t}$ the channel gain during time slot $t$ between PU
$u$ and SU $m$ over band $b$ and is given by
$h^{b}_{u,m,t}=G_{t}G_{r}\left(\frac{c}{4\pi d_{u,m}f^{b}}\right)^{2}$, where
$d_{u,m}$ is the Euclidean distance between the PU and SU $m$ in meter. $c$
and $f^{b}$ are the speed of light in m/s and transmission frequency in Hz,
respectively. Without loss of generality, all channel gains are assumed to be
constant during the whole time slot of $T$ sec, i.e., during sensing and/or
harvesting times.

Figure 1: System model

###  II-C Spectrum Sensing

In this paper an energy detection method is used for spectrum sensing, where
energy detectors (EDs) have been exploited as an efficient technique in the
literature due to their simplicity, compatibility with any signal type, and
low implementation complexity [14]. A binary hypothesis testing can be built
to determine the PU status as follows: to detect PU channel, ED of SUs measure
the received signal energy for a number of samples, $\vartheta^{b}_{m,t}$, and
compares the cumulative energy to a threshold $\varepsilon^{b}_{m,t}$ for each
band $b$ as
$\sum\limits_{v=1}^{\vartheta^{b}_{m,t}}(y^{b}_{m,t}(v))^{2}\underset{\mathcal{H}_{0}}{\overset{\mathcal{H}_{1}}{\gtrless}}\varepsilon^{b}_{m,t}$,
where $(y^{b}_{m,t}(v))^{2}$ is the energy measured on sample $v$.
$\mathcal{H}_{0}$ represents the absence of the primary signal, i.e., the
received baseband complex signal contains only Additive White Gaussian Noise
(AWGN) and $\mathcal{H}_{1}$ represents the presence of the primary signal.
For a large enough number of samples $\bar{\vartheta}$, (i.e.,
$\vartheta^{b}_{m,t}\geq\bar{\vartheta}$) and normalized noise variance, the
probability of false alarm, $F^{b}_{m,t}$, and probability of detection,
$D^{b}_{m,t}$, can be respectively expressed as [15]

$\displaystyle F^{b}_{m,t}$
$\displaystyle=\mathcal{Q}\left[\left(\varepsilon^{b}_{m,t}-1\right)\sqrt{\vartheta^{b}_{m,t}}\textbf{
}\right]$ (1) $\displaystyle D^{b}_{m,t}$
$\displaystyle=\mathcal{Q}\left[\frac{\left(\varepsilon^{b}_{m,t}-\gamma^{b}_{m,t}-1\right)\sqrt{\vartheta^{b}_{m,t}}}{\gamma^{b}_{m,t}+1}\textbf{
}\right]$ (2)

where $\gamma^{b}_{m,t}$ and $\mathcal{Q}\left[\cdot\right]$ denote the
signal-to-noise ratio (SNR) of SUm received at band $b$ and the Q-function,
respectively.

###  II-D Energy Harvesting

Let us assume that $\delta^{b}_{m,t}$ is the harvesting sensitivity indicator
function which is equal to 1 (i.e., SU $m$ over band $b$ is in HR) if the
average received power over band $b$ greater than $P_{\text{th}}$ and 0
otherwise, and given as follows

$\small\delta^{b}_{m,t}=\left\\{\begin{array}[]{ll}1,&\left(\frac{\sum\limits_{v=1}^{\vartheta^{b}_{m,t}}P^{b}_{t}(v)h^{b}_{u,m,t}}{\vartheta^{b}_{m,t}}\geq
P_{\text{th}}\right)/\mathcal{H}_{1},\\\
0,&\hbox{otherwise}\end{array}\right.\normalsize$ (3)

where $P^{b}_{t}(v)$ is the transmit power of PU on band $b$ during time slot
$t$ of sample $v$. The condition
$(\sum_{v}^{\vartheta^{b}_{m,t}}P^{b}_{t}(v)h^{b}_{u,m,t})/\vartheta^{b}_{m,t}\geq
P_{\text{th}}$ is to ensure that the SUs is in HR.

Let us assume that $T_{s}$ is the sensing time for one sample. Hence, During
$(\vartheta^{b}_{m,t}T_{s})$ seconds, SU $m$ senses PU channel over band $b$
to determine the belonging region (i.e., HR, IR, CR). For the remaining time,
$\delta^{b}_{m,t}(T-\vartheta^{b}_{m,t}T_{s})$ seconds, SU $m$ can harvest if
it belongs to the HR. Otherwise, during
$(1-\delta^{b}_{m,t})(T-\vartheta^{b}_{m,t}T_{s})$ seconds, SU $m$ can
transmit its data if it belongs to CR, or keep silent if it belongs to IR as
shown in Fig. 2. Notice that, the choice of this $\vartheta^{b}_{m,t}$ affects
the performance of the SUs. In fact, decreasing $\vartheta^{b}_{m,t}$ allows
SU $m$ to harvest more energy that can be used for future sensing or
transmission. However, this will reduce the allocated time to perform sensing
and vice versa. Therefore, an optimal choice of $\vartheta^{b}_{m,t},\forall
m=1,..,M$ is required in order to enhance the overall system performance.

Figure 2: Energy harvesting scheme.

##  III Multi-band Energy Harvesting Scheme

Several RF harvester schemes can be used for multi-band EH. The main
difference lies in the RF bandpass filter design and number of antennas. In
this scheme, we consider that each SU is equipped with a single wideband
antenna feeding parallel rectifier circuits at dedicated bands as shown in
Fig. 3.

Figure 3: Multi-bands energy harvesting scheme

where the matching network is designed to match each parallel rectifier at the
dedicated frequencies. The total consumed sensing energy of SU $m$ during time
slot $t$, denoted by $S_{m,t}$ can be expressed as

$S_{m,t}=\sum\limits_{b=1}^{\mathbb{B}}\vartheta^{b}_{m,t}T_{s}P_{s},$ (4)

where $P_{s}$ is the sensing circuit power consumption per sample.

The total harvested energy of SU $m$ during time slot $t$, denoted by
$H_{m,t}$ can be written as

$H_{m,t}=\sum\limits_{b=1}^{\mathbb{B}}\delta^{b}_{m,t}\eta^{b}_{m,t}\left[(T-\vartheta^{b}_{m,t}T_{s})P^{b}_{t}h^{b}_{u,m,t}\right],$
(5)

where $\eta^{b}_{m,t}$ is the RF-to-DC efficiency and depends on rectifying
technology and operated frequency.The stored energy at the end of time slot
$t$ at SU $m$, denoted by $B_{m,t}$, is given as

$B_{m,t}=\max(B_{m,t-1}+H_{m,t}-S_{m,t},0),$ (6)

where $\max(x,0)$ takes the maximum value between $x$ and 0.

##  IV Problem Formulation

In this section, the optimization problem that minimizes the total sensing
energy of the SUs is formulated by taking into consideration the detection and
false alarm probabilities constraints in addition to SUs battery constraints.

The false alarm and detection probabilities need to satisfy the following
conditions:

$F^{b}_{m,t}\leq\bar{F},\quad D^{b}_{m,t}\geq\bar{D},\quad\forall m,\forall
t,\forall b,$ (7)

where $\bar{F}$ and $\bar{D}$ are the false alarm and detection thresholds. By
substituting (1) and (2) in the above equations and with simple manipulations
(7) can be written, respectively, as

$\frac{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\mathcal{Q}^{-1}\left[\bar{F}\right]+\vartheta^{b}_{m,t}}\leq
1,\forall m,\forall t,\forall b,$ (8)
$\frac{(\gamma^{b}_{m,t}+1)\mathcal{Q}^{-1}\left[\bar{D}\right]+(\gamma^{b}_{m,t}+1)(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}\leq
1,\forall m,\forall t,\forall b.$ (9)

Therefore, our optimization problem can be formulated as

$\displaystyle\underset{\varepsilon^{b}_{m,t}\geq
0,\vartheta^{b}_{m,t}\in\mathbb{R^{+}}}{\text{minimize}}\quad\quad\sum\limits_{m=1}^{M}\sum\limits_{b=1}^{\mathbb{B}}\vartheta^{b}_{m,t}T_{s}P_{s}$
(10) subject to:
$\displaystyle\frac{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\mathcal{Q}^{-1}\left[\bar{F}\right]+\vartheta^{b}_{m,t}}\leq
1,\quad\forall m,\forall b,$ (11)
$\displaystyle\frac{(\gamma^{b}_{m,t}+1)\mathcal{Q}^{-1}\left[\bar{D}\right]+(\gamma^{b}_{m,t}+1)(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}\leq
1,\quad\forall m,\forall b,$ (12)
$\displaystyle\sum\limits_{b=1}^{\mathbb{B}}\vartheta^{b}_{m,t}T_{s}P_{s}\leq
B_{m,t-1},\quad\forall m,$ (13)
$\displaystyle\bar{\vartheta}\leq\vartheta^{b}_{m,t}\leq\frac{T}{T_{s}},\quad\forall
m,\forall b,$ (14)

where, constraints (11) and (12) represent the false alarm and detection
probabilities constraints, respectively. Constraint (13) represents battery
causality constraint. Constraint (14) represents the number of samples
constraint.

##  V Joint Optimization Solution based on Geometric Programming

Notice that the formulated optimization problem in (10)-(14) is a mixed-
integer non-linear problem (MINP). In order to simplify the problem, we
propose to proceed with a joint-optimization approach where we optimize
$\vartheta^{b}_{m,t}$ and $\varepsilon^{b}_{m,t}$ jointly using Geometric
Programming (GP). To do this, we firstly relax the variable
$\vartheta^{b}_{m,t}$ and make it continuous. After obtaining an optimal value
of $\vartheta^{b}_{m,t}$, we can obtain the closest upper integer value, which
does not negatively affect the system performance. Furthermore, we apply a
successive convex approximation (SCA) approach to transform the non-convex
problem into a sequence of relaxed convex subproblems [16].

###  V-A Geometric Programming Method

The standard form of GP is defined as the minimization of a posynomial
function subject to inequality posynomial constraints and equality monomial
constraints as given below:

$\displaystyle\underset{\boldsymbol{z}}{\text{minimize}}\quad\quad f_{0}(z)$
(15) subject to: $\displaystyle f_{l}(z)\leq 1,\quad\forall l=1,\cdots,L,$
(16)
$\displaystyle\tilde{f}_{\tilde{l}}(z)=1,\quad\forall\tilde{l}=1,\cdots,\tilde{L},$
(17)

where $f_{l}(z)$, $l=0,\cdots,L$, are posynomials and
$\tilde{f}_{\tilde{l}}(z)$, $\tilde{l}=1,\cdots,\tilde{L}$ are monomials. A
monomial is defined as a function $f:$ R${}^{n}_{++}$ $\rightarrow$ R as
follows:$f(z)=\check{s}\ z_{1}^{s_{1}}z_{2}^{s_{2}}...\ z_{N}^{s_{N}}$, where
the multiplicative constant $\check{s}\geq 0$, and the exponential constants
$s_{n}$ $\in$ R, $n=1,...,N$. A posynomial is a non-negative sum of monomials.

In general, GP in its standard form is a non-convex optimization problem,
because posynomials and monomials functions are non-convex functions. However,
with a logarithmic change of the variables, objective function, and constraint
functions, the optimization problem can be turned into an equivalent convex
form using the property that the logarithmic sum of exponential functions is
convex (see [17] for more details). In order to convert the optimization
problem formulated in (10)-(14) to a GP standard form, we propose to apply
approximation for only constraint (11). The single condensation method is
employed to convert this constraint to posynomial as described below:

######  Definition 1.

The single condensation method for GP involves upper bounds on the ratio of a
posynomial over a posynomial. It is applied to approximate a denominator
posynomial $g(z)$ to a monomial function, denoted by $\tilde{g}(z)$ and
leaving the numerator as a posynomial, using the arithmetic-geometric mean
inequality as a lower bound [16]. Given the value of $z$ at the iteration
$r-1$ of the SCA $z^{(r-1)}$, the posynomial $g$ that, by definition, has the
form $g(z)\triangleq\sum^{K}_{k=1}\mu_{k}(z)$, where $\mu_{k}(z)$ are
monomials, can be approximated as:

$g(z)\geq\tilde{g}(z)=\prod^{K}_{k=1}\left(\frac{\mu_{k}(z)}{\tilde{\mu}_{k}(z^{(r-1)})}\right)^{\tilde{\mu}_{k}(z^{(r-1)})},$
(18)

where $\tilde{\mu}_{k}(z^{(r-1)})=\frac{\mu_{k}(z^{(r-1)})}{g(z^{(r-1)})}$.
$K$ corresponds to the total number of monomials in $g(z)$.

It can be noticed that the nominator and denominator of (11) are posynomials,
however, the ratio is not necessary a posynomial. Therefore, in order to
convert constraint (11) to a posynomial, we propose to apply the single
condensation method given in Definition 1 to approximate the denominator
posynomial to a monomial function, where in this case $K=2$, therefor
constraint (11) can be given as follows

$\frac{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\tilde{g}(\vartheta^{b}_{m,t})}\leq
1,\quad\forall m,\forall b,$ (19)

It can be seen that constraint (19) is now a posynomial because a posynomial
over a monomial is a posynomial. The convergence proof is given in [18]. By
considering the approximations of (19), we can formulate the GP approximated
subproblem at the $i^{\text{th}}$ iteration of the SCA as follows:

$\displaystyle\underset{\varepsilon^{b}_{m,t},\vartheta^{b}_{m,t}\geq
0}{\text{minimize}}\quad\quad\sum\limits_{m=1}^{M}\sum\limits_{b=1}^{\mathbb{B}}\vartheta^{b}_{m,t}T_{s}P_{s}$
(20) subject to:
$\displaystyle\frac{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\tilde{g}(\vartheta^{b}_{m,t})}\leq
1,\quad\forall m,\forall b,$ (21)
$\displaystyle\frac{(\gamma^{b}_{m,t}+1)\mathcal{Q}^{-1}\left[\bar{D}\right]+(\gamma^{b}_{m,t}+1)(\vartheta^{b}_{m,t})^{\frac{1}{2}}}{\varepsilon^{b}_{m,t}(\vartheta^{b}_{m,t})^{\frac{1}{2}}}\leq
1,\quad\forall m,\forall b,$ (22)
$\displaystyle\frac{\sum\limits_{b=1}^{\mathbb{B}}\vartheta^{b}_{m,t}T_{s}P_{s}}{B_{m,t-1}}\leq
1\quad\forall m,$ (23) $\displaystyle\frac{T_{s}}{T}\vartheta^{b}_{m,t}\leq
1,\quad\forall m,\forall b,$ (24)
$\displaystyle\frac{\bar{\vartheta}}{\vartheta^{b}_{m,t}}\leq 1,\quad\forall
m,\forall b,$ (25)

We solve the problem in an online fashion (i.e., time slot by time slot) for
$t=1,..,\Gamma$. Two solutions are proposed: The first one namely ”all
sensing” solution where all secondary users perform sensing to determine to
which regions they belong to. This is an exhaustive solution and consumes a
lot of energy but gives an accurate characterization or HR, IR, and CR. The
second solution is based on SVM where training of some SUs make them perform
sensing in order to characterize the belonging region using multi-class
classification. The latter solution is proposed to save some sensing energy by
allowing only training SUs to perform sensing while the others keep quiet
until they know their classification, however, the all sensing solution has a
superior performance over the SVM solution in terms of accuracy.

###  V-B Solution 1: All Sensing Solution

The optimization problem formulated in (20)-(25) can be solved optimally at
each iteration of the SCA as given in Algorithm 1 using Gurobi/CVX interface
[19]. Each GP in the iteration loop (line 3-7) tries to improve the accuracy
of the approximations to a particular minimum in the original feasible region.
This is performed until no improvement in the objective function ($\chi$) is
made. A parameter, $\varpi\rightarrow 0$, is introduced to control the
accuracy of the algorithm convergence as follows:
$|\chi^{(r+1)}-\chi^{(r)}|\leq\varpi$.

Algorithm 1 SCA Algorithm

1: for $t=1\cdots\Gamma$ do

2: r=1.

3: Select a feasible initial value of
$\boldsymbol{z}^{(r)}=[\varepsilon^{b}_{m,t},\vartheta^{b}_{m,t}],\forall
m,\forall b$.

4: repeat

5: r=r+1.

6: Approximate the denominators using the arithmetic-geometric mean as
indicated in (18) using $\boldsymbol{z}^{(r-1)}$.

7: Solve the optimization problem using the interior-point method to determine
the new approximated solution
$\boldsymbol{z}^{(r)}=[\varepsilon^{b}_{m,t},\vartheta^{b}_{m,t}],\forall
m,\forall b$.

8: until Convergence ($|\chi^{(r+1)}-\chi^{(r)}|\leq\varpi$).

9: Compute the corresponding harvested energy and battery level using (5) and
(6), respectively.

10: end for

###  V-C Solution 2: SVM Solution

Several binary classification approachs can be extended to handle the SVM
multi-class. Two main types of approaches have been proposed for multi-class
SVM by combining several binary classifications namely One-vs-All (OVA) and
One-vs-One (OVO). The OVO approach considering all classes optimization in one
formulation, therefore, it leads to more accurate results than OVA approach.
However, OVO approach is more computational expensive than the OVA approach
because it leads to larger scale optimization problem [20]. In this paper we
propose to use OVO approach as suggested in [20]. This approach constructs
$Y(Y-1)/2$ SVM classifiers where each classifier trains on data from two
classes, where $Y$ is the number of classes. For training data from class $i$
and class $j$, we solve the following binary classification problem

$\displaystyle\underset{w^{ij},\beta^{ij},\zeta\tau^{ij}\geq
0}{\text{minimize}}\quad\quad\frac{1}{2}(w^{ij})^{T}w^{ij}+\bar{\zeta}\sum\limits_{\tau}\zeta_{\tau}^{ij}$
(26) subject to: $\displaystyle(w^{ij})^{T}\phi(x_{\tau})+\beta^{ij}\geq
1-\zeta_{\tau}^{ij},\quad\text{if}\quad Y_{\tau}=i,$ (27)
$\displaystyle(w^{ij})^{T}\phi(x_{\tau})+\beta^{ij}\leq-1+\zeta_{\tau}^{ij},\quad\text{if}\quad
Y_{\tau}=j,,$ (28)

where the training data $x$ are mapped into a higher dimensional space of
function $\phi(x)$. Minimizing $\frac{1}{2}(w^{ij})^{T}w^{ij}$ is equivalent
to minimizing $2/||w^{i,j}||$, the margin between two groups of data. The
penalty term $\bar{\zeta}\sum\limits_{\tau}\zeta_{\tau}^{ij}$ can reduce the
number of training errors when the data is not linearly separable. The work in
[21] suggests to use a voting approach called Max-Wins approach, hence, if
sign$((w^{ij})^{T}\phi(x)+\beta^{ij})$ gives $x$ is in class $i$, then the
vote for class $i$ increases by one to this class. Otherwise, class $j$ is
increased by one. Finally, $x$ can be predicted as the class with the largest
vote. Practically we solve the quadratic program formulated in (26)-(28) using
dual method where number of variables is equal to the number of the training
data in two classes [17].

##  VI Simulation Results

3 (a) Band 1, All sensing (b) Band 2, All sensing (c) Band 3, All sensing (d)
Band 1, 50% sensing (e) Band 2, 50% sensing (f) Band 3, 50% sensing Figure 4:
The performance of solution 1 versus solution 2
---

In this section, selected numerical results are provided to investigate the
benefits of RF multi-band EH under cognitive radio framework. We consider an
area of 40m$\times$40m with seven PUs transmit their signals over seven bands
$b=\\{0.9,1.24,1.56,1.78,2.19,2.46,2.68\\}$ GHz, respectively. Total number of
SUs is equal to $M=60$. We assume that the SUs are initially charged with
$B_{0}=P_{S}T$ mJ of energy. In Table I, we present the values of the
remaining parameters used in the simulations [15]

Table I:  System parameters Parameter | Value | Parameter | Value | Parameter | Value
---|---|---|---|---|---
$T_{s}$ ($\mu$s) | 1 |  $P_{s}$ (dBm) | 0 | $\eta^{b}_{m,t}$ | 0.45
$T$ (s) | $1$ |  $P_{\text{th}}$ (dBm) | $-20$ |  $c$ (m/s) | $3\times 10^{8}$
$\bar{F}$ | 0.1 | $\bar{D}$ | 0.9 |  $B_{\text{th}}$(mJ) | 1

Figure 5: The effect of choosing training data numbers

In Fig. 4 and Fig. 5, because of the paper space limitation, we only select
three bands out of seven bands (i.e., $b=\\{0.9,1.78,2.68\\}$ namely band 1,
band 2, and band 3, respectively) in order to validate our approach. Fig. 4,
the three regions (HR,IR, and CR) are shown for the two solutions provided in
Section V. $"x"$ symbol (usually near the boundaries) means that this location
of SU plays a significant role in determining the region, while $"o"$ symbol
has less significant role in characterizing the regions. For instance, Fig.
4-a, Fig. 4-b, and Fig. 4-c, show the regions for the three selected bands. It
can be deduced that the closest users to the PU can harvest RF energy and the
faraway users can transmit their signals without interference with the PU.
Also, it can be noticed that the HR for low frequency band (i.e., 0.9 GHz) is
greater than the HR for high frequency band (i.e., 2.68 GHz). This is can be
justified by knowing that the channel is inversely proportional with the
square of the operating frequency. On the other hand, Fig. 4-d, Fig. 4-e, and
Fig. 4-f, plot the three regions using random 50% of SUs as training data.
$"*"$ symbol corresponds to the unselected SUs. Note that, if $"*"$ SUs
located in the boundaries, then, it will significantly change characteristics
of the the regions. Hence, it will lead to inaccurate regions characteristic.

Fig. 5 plots the error for the selected bands versus number of training data
(i.e., percentage). It is shown that as the training date increased, the
accuracy increased. For instant, for band 2 (operated at 1.78 GHz), the error
can be reduced from 52% to around 8% by using only 20% instead 80% of the
training data. Therefore, the accuracy can be considered as an important
factor in our design. Moreover, this figure, compare between optimal solution:
solution 1 (all SUs sensing) when the training data is 100% and solution 2
(multi SVM) when the training data is less than 100%.

Figure 6: The effect of training data numbers

On the other hand, the total energy consumption is plotted versus the
percentage number of training data in Fig. 6 to show the trade off of
selecting the percentage of the training data. It is obvious to see that as
the number of training data increases, the energy consumption increases, this
is can be justified by the fact that more training data consume more energy
for sensing. For example, the consumed energy is almost doubled by considering
90% instead of 53% of the training data. Furthermore, based on optimization
problem given in (26)-(28) one can notice that the complexity increases as we
include more training data because it leads to a larger scale problem. Based
on the above discussion, we can design our selection of training data based on
three factor; 1- accuracy, 2- energy consumption, and 3- complexity. In other
words, given specific range of accuracy, energy consumption limits, and
complexity, we can find the optimal number of training data required to
satisfy these conditions.

##  VII Conclusion

In this paper, we introduced and solved a novel and new optimization problem
for multi-band EH schemes under cognitive radio interweave framework. We
proposed a win-win solution, where SUs can sense the spectrum to determine the
harvesting and communication geographical regions, therefore, they can take a
decision to harvest or transmit data based on the belonging region. Also, we
jointly optimized number of sensing samples and sensing threshold in order to
minimize the sensing energy. A near optimal solution based on GP has been
investigated. Furthermore, we have investigated an energy efficient multiple
classification approach based on SVM.

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