Improving spectrum utilization efficiency is a fundamental goal of dynamic spectrum access technology. The definition of spectrum holes determines how to detect and exploit them. Current definitions of spectrum holes are ineffective in exploiting spatialtemporal spectrum holes. In this paper, a novel definition of spectrum holes is proposed, in which
throughput loss
indicates the impact of secondary users on primary users. The definition specifies spectrum holes, unifies the impact of secondary users on primary users and is effective exploiting spatialtemporal spectrum holes. Theoretical analysis and numerical simulations show that the new definition proposed in this paper significantly improves the spectrum utilization efficiency.
1. Introduction
R
adio frequency spectrum is a precious resource for wireless communication systems. With the rapid development of wireless communication services, there is an increasing demand for spectrum. Existing static spectrum allocations policies have resulted in the low utilization of spectrum
[1

4]
. The dynamic spectrum access (DSA) policy is proposed as an alternative to efficiently use radio spectrum
[5

8]
. In DSA, portions of the spectrum are allocated to one or more users, known as primary users (PUs). Spectrum use is not exclusively granted to these users, although they have a higher priority. Other users, referred to as secondary users (SUs), can also access the allocated spectrum, provided the PUs are properly protected. By doing so, the radio spectrum is reused opportunistically, thereby significantly improving the spectrum utilization
[5

8]
.
Cognitive radio (CR) technology is an important means to realize DSA. Users with cognitive functions can sense the spectrum environment using intelligent means. They can automatically search and use available spectrum, thereby realizing authorized spectrum sharing
[5

8]
. In
[9]
, a spectrum hole is defined as a (time, location, frequencyband)tuple that a secondary user can use, while maintaining interference with all primary systems within an acceptable level.
In order to reuse the spectrum, a secondary user must be able to reliably identify a spectrum hole. In the above definition of a spectrum hole, the key factor is the impact of SUs on PUs. For a temporal spectrum hole, the impact refers to the probability when PUs and SUs simultaneously use the same spectrum. This has been referred to as the collision probability
[10]
. The collision probability ignores the position relationship between SUs and PUs, which is essential for exploiting spatial spectrum opportunities. For a spatial spectrum hole, this impact refers to power which is received by primary receriver from secondary transmitters, referred to as interference
[11

13]
. It alternately refers to the outage probability of the primary receiver, which is a function of the interference. The interference and the outage probability only reflect the intensity of the interfering signal. They does not reflect the duration of the interference. It also cannot fully reflect the impact of SUs on PUs. In
[14]
, the Bayesian risk function was minimized by selecting the sensing threshold. This was a tradeoff between the risk of collision probability and secondary users losing communication opportunity.This tradeoff exploited both the temporal and spatial spectrum opportunity, and greatly improved the spectrum utilization. However, it lacks a theoretical basis. In
[15]
, a threeregion scheme for spacetime spectrum sensing and access was designed with one primary transmitter at the center. This includes the black, grey, and white region. However, the authors have not jointly considered the interference and collision probability, as above, which doesn’t fully reflect the impact of SUs on PUs.
In this paper, the average throughput loss percentage (hereafter referred to as throughput loss) of the PUs reflects the impact of SUs on PUs. The throughput loss is a comprehensive indicator, which contains the collision probability and intensity of interference. It unifies the impact of SUs on PUs in the spatialtemporal spectrum holes. Considering the throughput loss improves the spectrum utilization .In addition, the throughput loss is more intuitive, compared to Bayesian risk criteria. In this paper we design networks and channel models. Through theoretical analysis and numerical simulations, it is shown that the throughput loss is a more effective impact indicator than conventional indicators (e.g., collision probability, interference, and Bayesian risk) in terms of spectrum utilization.
The remainder of this paper is organized as follows. In section 2, we propose the spectrum holes definition with throughput loss as an impact indicator. In section 3, we introduce the networks and channel models to verify the definition. In section 4, we derive the optimal scheme of spectrum hole utilization in the models. In section 5, this definition is compared to conventional definitions of spectrum holes. In section 6, the numerical simulations and analysis are presented. In section 7, we conclude this research.
2. Spectrum Holes Definition
The definition of spectrum holes determines how to detect and exploit them. In this definition, the impact of SUs on PUs is a key factor. Conventional definitions of spectrum holes have many shortcomings. In this section, we first define spectrum holes with throughput loss as an impact indicator.
The average throughput loss percentage of PUs indicates the impact of SUs on PUs as
where
T_{p}
^{max}
is the maximum average throughput of PUs and
T_{p}^{ave}
is the average throughput of PUs. This can be predicted through channel models, provided a secondary transmitter (ST) is used.
The proposed definition of a spectrum holes is as follows:
Definition
: a spectrum hole is defined as a (time, location, frequencyband)tuple that a secondary user can use, while maintaining the throughput loss to all primary users in the specified frequency band within an acceptable level.
Choosing a suitable value for
k
is important. In a practical system, PUs cannot always work at the maximum average throughput. There is spare throughput against all kinds of interference. We choose a suitable value for
k
to guarantee QoS of the PUs and make the SUs attain the maximum throughput.
3. Network and Channel Models
 3.1 Networks Model
As shown in
Fig. 1
, we consider a network model consisting of a pair of fixed PUs (including a primary transmitter (PT) and a primary receiver (PR)) and a pair of SUs (including a secondary transmitter (ST) and a secondary receiver (SR)). The SUs opportunistically exploit a spectrum band licensed to the PUs. Assume that the PUs are rateadaptive digital communication systems, and occupy the spectrum band with a probability
q
.
Network model
In
Fig. 1
, PT is located in the origin of coordinates. The coordinates of PR, ST, and SR are labeled in the respective brackets. The angle between the communication direction from ST to SR and the Xaxis is
f
. The angle between the communication direction from PT to PR and the Xaxis is
q
.
r
is the distance from PT to ST.
r_{p}
and
r_{s}
denote the communication distance of the PUs and SUs, respectively.
d_{ps}
denotes the distance from PT to SR.
d_{sp}
denotes the distance from ST to PR.
In
Fig. 1
, the circles with radius
d
are the protected regions. Any active transmitter cannot be inside this region, to exclude the possibility that the receive signal and interference power reaches infinity. It's worth noting that the protected region of ST is used to exclude the possibility that the sensing receive power reaches infinity.
 3.2 Channel Model
Consider a wireless channel model with both largescale path loss and smallscale fading. Freespace path loss,
h_{PL}
, models the average power changing with distance. Rayleigh fading
h_{FD}
is adopted for smallscale variation. The channel model between any TxRx pair can then be written as follows:
where
h_{FD}
~
CN
(0.1) is a complex circular Gaussian random variable with independent real and imaginary parts with equal variance; and
where
A
is a constant dependent upon the frequency and transmitter/receiver antenna gain,
r
is the distance between the TxRx pair, and
a
is the path loss exponent. Without loss of generality, we normalize
A
=1 for simplicity and consider
a
³ 2, which is typical in practical applications.
 3.3 Signal and Sensing Models
To sense the primary transmission, ST must perform a hypothesis test between the following two hypotheses:
H
_{0}
(primary signal absent) and
H
_{1}
(primary signal present). This test is conducted as follows:
where
N
is the number of samples,
y
(
i
) is the received samples at ST,
x
(
i
)
CN
(0,
S
^{2}
_{x}
) is the signal received at ST from PT after path loss and fading, and
w
(
i
) is assumed to be a circularlysymmetric complex Gaussian random variable with mean zero and onesided power spectral density
S
^{2}
_{w}
, namely,
w
(
i
)
CN
(0,
S
^{2}
_{w}
) . Based on the channel model in (4),
S_{x}
^{2}
=
P_{p}h
^{2}
_{PL}d
where
P_{p}
is transmit power of PT. The primary signal
x
(
i
) is independent of the noise
w
(
i
).
The decision statistic is
. The probabilities of miss detection
p_{m}
=Pr{
H
_{0}
/
H
_{1}
} and false alarm
p_{f}
=Pr{
H
_{1}
/
H
_{0}
} are
[16]
:
where
g
is the signal to noise ratio (SNR) at ST,
g=P_{p}r^{a}S_{w}
^{2}
in the above channel model, and
e
is the decision threshold.
is the tail probability of the standard normal distribution.
4. Optimization of SpatialTemporal Spectrum
 4.1 Channel States
In DSA, for a temporal spectrum hole, SUs cannot simultaneously operate in the same channel with the PUs. Collision between the SUs and PUs may only occur due to sensing errors. But for a spatialtemporal spectrum hole, we allow for it if the distance from PR is far enough. So, we model the spectrum access process as a fourstate process, where state 0 means no user operates in the channel. State 1 means the PUs operate in the channel. State 2 means the SUs operate in the channel. State 3 means both the PUs and the SUs operate in the channel. The channel state set is
S
{0,1,2,3} . It is assumed that PUs occupy the spectrum band with a probability
q
and SUs opportunistically occupy it. The probability in every channel state is, respectively:
 4.2 Throughput Achieved in Every Channel State
In the system models described above, we assume the bandwidth is 1Hz. As the network model only consists of a pair of PUs and SUs, we replace the throughput with channel capacity
[17]
. The throughput of the PUs achieved in state1 is
The throughput of the SUs achieved in state2 is
where
s_{w}
^{2}
is power of the additive white Gaussian noise (AWGN),
P_{p}
is the transmission power for the PUs, and
P_{s}
is the transmission power for the SUs.
We assume that the SUs and PUs can vary their data rate through a combination of adaptive modulation and coding. This allows the transmitter and receiver to employ the most reliable communication at the highest rate permitted, given the signaltointerferenceplusnoise ratio (SINR).We assume that they use random Gaussian codebooks. As a result, their transmitted signals can be treated as white Gaussian processes. The transmission of other users is treated as Gaussian noise. The maximal rate of users when the secondary user and the primary user share the spectrum can be represented as follows:
The throughput of the SUs achieved in state3 is
The throughput of the PUs achieved in state3 is
 4.3 Optimization of SpatialTemporal Spectrum for Proposed Definition
The average throughput of the SUs is
The average throughput of the PUs is
The maximum average throughput of the PUs is
T_{p}
^{max}
=
qR^{p}
_{1}
.
Maximizing the available spatialtemporal spectrum opportunity using the spectrum holes definition in this paper can be formulized as follows:
where 0 #
k
1 is a scaling factor that denotes the maximum average throughput loss ratio that is acceptable for PUs due to interference of the SUs. We first introduce the following proposition to solve this problem:
Proposition1
:
p_{f}
is a decreasing function of
p_{m}
.
Proof
: According to (7) and (8), we can obtain:
So,
Therefore
p_{f}
is a decreasing function of
p_{m}
. Proposition 1 is proved.
The objective function in equation (19) is a decreasing function of
p_{f}
.
We can obtain from the constraint condition (20):
where
p_{m}
^{*}
denotes the optimal miss detection probability. Because
p_{m}
is a decreasing function of
p_{f}
, the optimal false alarm probability is:
It is worth noting that
kR^{p}
_{1}
(
R^{p}
_{1}

R^{p}
_{3}
)
^{1}
> 1 can occur when
r
is large enough. However, miss detection probability is not greater than 1. In this case, we order
p_{m}
^{*}
=1 and
p_{f}
^{*}
=0 to maximize the SUs throughput.
The maximum average throughput of the SUs is:
5. Comparison of the Proposed and Conventional Definitions
As stated in the introduction, the interference only reflects the intensity of the interfering signal. It does not reflect the duration of the interference signal. So, it cannot fully reflect the impact of SUs on PUs. We only consider the definition based on the collision probability constraint and the Bayesian risk in this section.
 5.1 Comparison to the Collision Probability Constraint
We maximize the available spatialtemporal spectrum using the spectrum holes definition as follows
[10]
:
where
is the minimum miss detection probability. The inequality (28) constrains the collision probability between the PUs and the SUs. We easily obtain the optimal false alarm probability to maximize the SUs throughput as follows:
The maximum average throughput of the SUs is:
In this case, the maximum value of
k
is:
where
R^{p}
_{3min}
is the minimum value of
R^{p}
_{3}
in the case of
d_{sp}
=max(
r_{p}
sin
q
,
d
) . Substituting
k_{m}
into (24), we obtain
Substituting (32) into (26), we obtain the maximum average throughput of the SUs in the case of
k
=
k_{m}
in the proposed definition:
where
is obtained by substituting
into (25).
Subtracting (30) from (33), we obtain the following:
It is easy to prove (34) is greater than zero. That is, the definition of spectrum holes based on the throughput loss is more advantageous than the one based on the collision probability constraint in terms of spectrum utilization.
The traditional scheme uses (29), while the proposed scheme uses (25). The computing complexity of the proposed scheme is not much higher than that of the traditional scheme. However, the overhead in the proposed scheme is more than the traditional schemes due to the information required to transmit. The additional overhead is limited in smallscale network.
 5.2 Comparison to the Bayesian Criteria
Using Bayesian criterion, the optimization objective is defined as a risk function involving the miss detection and false alarm probabilities as follows
[14]
:
where
I_{ps}
is the interference power from the PT to the SR,
I_{sp}
is the interference power from the ST to the PR, and
L_{s}
is the signal power received by the SR. Both
p_{m}
and
p_{f}
are a function of the detection threshold
e. b
³ 1 is the penalty parameter to place a higher priority on the PUs link.
The sensing threshold is designed to minimize the risk in the different cases of location information:
and
Substituting (31) and (32) into (36), we obtain
where
We obtain the optimal detection threshold:
Substituting
e
^{*}
into (7) and (8), we obtain
p_{m}
(
e
^{*}
) and
p_{f}
(
e
^{*}
) . From (17), we obtain the optimal throughput of the SUs.
The traditional scheme uses (40), while the proposed scheme uses (25). The computing complexity of the proposed scheme is not much higher than the traditional scheme. Additionally, the overhead in the proposed scheme is nearly equivalent to the traditional schemes, as both needs to transmit more information.
6. Numerical Simulation and Analysis
Numerical simulations in
Fig. 2
,
Fig. 3
, and
Fig. 4
were conducted using the conditions described in
Table 1
.
the Parameters and Their Value in the Following Numerical Simulations
the Parameters and Their Value in the Following Numerical Simulations
Throughput comparison of SUs
Throughput comparison of SUs vs. q
Throughput comparison of SUs vs. N
Fig. 2
shows that the throughput of the SUs for different definitions varies with the distance,
r
. It is worth noting the selection of the throughput loss percentage of PUs,
k
. Let the penalty parameter
b
=1 . We obtain the optimal threshold
e
^{*}
from (40). We then obtain
p_{m}
(
e
^{*}
) and
p_{f}
(
e
^{*}
) from (6) and (7). By substituting them into (17) we obtain the throughput of the SUs. At the same time, we calculate the throughput loss percentage
k
. With
r
varying from 0.1m to 20m, we select the maximum value of
k
as the throughput loss percentage. The throughput of the SUs is optimized as described in section4.3, with the maximum value of
k
.
is obtained from (31). The throughput of the SUs is optimized as described in section5.1 using
.
Fig. 2
shows that the throughput of the SUs for the proposed definition is greater than that of the definitions in
[10]
and
[14]
. As
r
increases, the advantage of the proposed definition is more obvious compared to other definitions. This is because the definition in
[10]
is for temporal spectrum holes. However, as
r
increases, the SNR decreases. The false probability increases to satisfy the relatively small collision probability. As a result, the SUs lose spatial spectrum. When
r
>10, the throughput sharply decreases. However, for the definition in
[14]
and the proposed definition, the SUs utilize both temporal and spatial spectrum holes. As
r
increases, the influence of ST on the PR decreases. We increase the collision probability P
_{3}
to utilize more spatial spectrum opportunity. There is obvious improvement in the throughput of the SUs. In
[14]
, the optimization objective is a risk function
R
, not the throughput of SUs. Therefore, compared with the Bayesian criterion, the throughput of the SUs increases significantly with the proposed definition.
Fig. 3
shows the throughput of SUs for different definitions varies with the probability
q
. As shown, the throughput of the SUs for all definitions decreases as
q
increases. This is because the increase in
q
results in less temporal spectrum holes for SUs to utilize. In this process, the throughput for the proposed definition decreases less than that of the other definitions. This is because, in the case of
r
=12 , the SUs mainly utilize spatial spectrum holes.
Fig. 4
shows the throughput of SUs for different definitions with various numbers of samples
N
. It is shown that the throughput of the SUs for all definitions vary as
N
increases. For the proposed definition, it nearly remains unchanged. For the definition in
[10]
, the throughput increases. However, for the definition in
[14]
, it decreases. This shows that the proposed definition is insensitive to sensing accuracy.
7. Conclusion
In this paper, throughput loss is used to indicate the impact of SUs on PUs. This specifies the definition of spectrum holes, and unifies the impact of SUs on PUs in the spatialtemporal spectrum holes. In a typical cognitive radio system, theoretical analysis and numerical simulations show that, when compared to previous definitions of spectrum holes, the proposed definition significantly improves the spectrum utilization, and is insensitive to sensing accuracy.
BIO
Xiaoqiang Li received the B. S. degree and the M. S. degree both from PLA University of Science and Technology, Nanjing, China in 1997 and 2007 respectively. He is currently pursuing for the Ph. D degree in Communications and Information System at the Institute of Communications Engineering, PLA University of Science and Technology, Nanjing, China. His current research interests include cognitive radio technology and digital communications.
Qi Zhou received the B. S. degree from PLA University of Science and Technology, Nanjing, China in 1997 and the M. S. degree from Tsinghua University, Beijing, China in 2005. He is now a senior engineer in research institute. His working is concerned with wireless communication and networks.
Hui Dai received his doctor degree in 2008 at PLA University of Science and Technology, Nanjing, China. He is currently an engineer in Institute of China Electronic System Engineering Corporation. His research focuses on wideband mobile communication technologies including MIMO, link adaptation, cooperative transmission technologies.
Jie Zhang received the Ph.D. degree from Naval Aeronautical Engineering University Yantai, China in 2006. He is now a senior engineer at Beijing institute of electronic system engineering. His research interests include command and control system, effectiveness evaluation, cognitive radio.
Ying Li received her B.S. and Ph. D degree in 2001 and 2006 respectively from National University of Defense Technology, Changsha, Hunan, all in communication engineering. From 2007, she is a research member in Institute of China Electronic System Engineering Company, Beijing. Her research interests include wireless network protocol and signal processing in communications, more specially, the areas of Multicarrier transmission, MIMO and cognitive radio.
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