Simultaneous Spectrum Sensing and Data Reception for Cognitive

Simultaneous Spectrum Sensing and Data Reception for Cognitive
1
Simultaneous Spectrum Sensing and Data
Reception for Cognitive Spatial Multiplexing
Distributed Systems
arXiv:1609.07711v1 [cs.IT] 25 Sep 2016
Nikolaos I. Miridakis, Theodoros A. Tsiftsis, Senior Member, IEEE,
George C. Alexandropoulos, Senior Member, IEEE,
and Mérouane Debbah, Fellow, IEEE
Abstract
A multi-user cognitive (secondary) radio system is considered, where the spatial multiplexing mode
of operation is implemented amongst the nodes, under the presence of multiple primary transmissions.
The secondary receiver carries out minimum mean-squared error (MMSE) detection to effectively decode
the secondary data streams, while it performs spectrum sensing at the remaining signal to capture the
presence of primary activity or not. New analytical closed-form expressions regarding some important
system measures are obtained, namely, the outage and detection probabilities; the transmission power of
the secondary nodes; the probability of unexpected interference at the primary nodes; and the detection
efficiency with the aid of the area under the receive operating characteristics curve. The realistic scenarios
of channel fading time variation and channel estimation errors are encountered for the derived results.
Finally, the enclosed numerical results verify the accuracy of the proposed framework, while some
useful engineering insights are also revealed, such as the key role of the detection accuracy to the
N. I. Miridakis is with the Department of Computer Engineering, Piraeus University of Applied Sciences, 12244 Aegaleo,
Greece (e-mail: nikozm@unipi.gr).
T. A. Tsiftsis is with the Department of Electrical Engineering, Technological Educational Institute of Central Greece,
35100 Lamia, Greece and with the School of Engineering, Nazarbayev University, 010000 Astana, Kazakhstan (e-mails:
tsiftsis@teiste.gr, theodoros.tsiftsis@nu.edu.kz).
G.
France
C.
Alexandropoulos
Research
Center,
and
M.
Huawei
Debbah
are
Technologies
{george.alexandropoulos,merouane.debbah}@huawei.com).
with
the
France,
Mathematical
92100
and
Algorithmic
Boulogne-Billancourt,
Sciences
France
Lab,
(e-mails:
2
overall performance and the impact of transmission power from the secondary nodes to the primary
system.
Index Terms
Cognitive radio, detection probability, imperfect channel estimation, minimum mean-squared error
(MMSE), outage probability, spatial multiplexing, spectrum sensing.
I. I NTRODUCTION
Cognitive radio (CR) has emerged as one of the most promising technologies to resolve
the issue of spectrum scarcity, caused by the escalating growth in wireless data traffic of nextgeneration networks [1]. One of the principal requirements of CR is the effectiveness of spectrum
sharing performed by secondary (unlicensed) nodes, which is expected to intelligently mitigate
any harmful interference caused to the primary (licensed) network nodes. This requirement is
directly related to the accuracy of spectrum sensing techniques, reflecting the reliable detection
of primary transmission(s).
On the other hand, placing multiple antennas on each cognitive node represents a fruitful option
since the system capacity in terms of data rate can be greatly enhanced. Spatial multiplexing
represents one of the most prominent techniques used for multiple input-multiple output (MIMO)
transmission systems [2]. For computational savings at the receiver side, there has been a prime
interest in the class of linear detectors, such as zero-forcing (ZF) and minimum mean-squared
error (MMSE). It is widely known that MMSE outperforms ZF, especially in low-to-medium
signal-to-noise (SNR) regions, at the cost of a slightly higher computational burden, since the
noise variance is required in this case. In addition, when MIMO technology is combined with
distributed antenna systems (DAS), the so-called distributed-MIMO (D-MIMO) transmission is
emerged. The success behind D-MIMO relies on the multiplexing gains, which are produced by
the classical MIMO transmission, and the diversity gains, which are manifested from the use of
DAS [3].
Due to the complementary benefits of CR and D-MIMO, the cognitive (D-)MIMO systems
are of paramount research interest nowadays, e.g., see [4]–[7] and references therein.
3
A. Related Work and Motivation
The performance of spectrum sensing, i.e., the accuracy of the detection method used by the
cognitive system plays a key role to the performance of both the primary and secondary network.
It acts as an important tool for finding idle spectrum instances (the so-called spectrum holes [8])
to efficiently deliver cognitive data, while protecting the communication quality of the primary
service at the same time. Several spectrum sensing approaches have been proposed so far to
preserve transparency of CR networks, which can be categorized into two main types; quiet [9]
and active [10].
The quiet spectrum sensing type is the conventional approach in which each potential cognitive
transmitter first senses the spectrum for a fixed time-sensing duration and then transmits its data
in the remaining time, when it senses the channel as idle. The main problem of this approach
is the capacity reduction in terms of cognitive data transmission within a given frame duration.
Moreover, the detection accuracy is questionable by adopting the quiet type, since the sensingtime duration is rather limited (i.e., only a fraction of the entire frame duration) and, hence, the
required number of sensing samples is constrained.
In order to overcome this problem, the more sophisticated active sensing type has been
proposed. The idea behind this approach relies on the improvement of the former shortages
produced by quiet sensing. In particular, a simultaneous spectrum sensing and data transmission
technique was proposed in [11], where the receiver first cancels the secondary data using
interference cancellation and then senses the remaining signal for the presence or absence of
a primary activity. However, the scenario of a single transmitter-receiver pair for the cognitive
system was considered in [11] with the presence of only one primary node, a rather infeasible
condition for practical applications. Other active sensing techniques for multi-user cognitive
systems were proposed in [10] and [12]. In both studies, it was assumed that some secondary
nodes transmit while others perform spectrum sensing. In the case of a primary signal detection,
the latter nodes inform the former ones about the primary activity to stop their transmissions.
Nevertheless, several problems arise by following these methods; more spectrum resources are
required because of the signaling overhead caused by the informing process, whereas extra
power resources are consumed from the sensing nodes during spectrum sensing and because of
transmitting their sensing reports.
4
More recently, authors in [13] and [14] proposed a spatial isolation technique on the antennas of
each cognitive node in a sense that some antennas are devoted for spectrum sensing while others
for data transmission. The main drawback of this approach is the large amount of self-interference
produced during spectrum sensing, which can not always be sufficiently canceled. Hardware
constraints and/or impairments represent an immediate obstacle, whereas an appropriate physical
distance between the sensing and transmitting antennas should be maintained (i.e., in the order
of 20 − 40cm [15], [16]), which is not always feasible or preferable for simple small-sized
equipment.
In addition, the concept of simultaneous data reception and spectrum sensing for singleantenna nodes was studied in [17], [18], while for multiple-antenna nodes in [19]. However,
these works used the central limit theorem to approximate the total received signal as a Gaussian
input (invoking the constraint of sufficiently large amount of received samples), whereas they
provided only semi-analytical and/or simulation results with respect to the system performance.
Capitalizing on the aforementioned observations, in this paper, a new simultaneous (active)
spectrum sensing and data transmission approach for CR networks is presented. The spectrum
sensing is performed at the secondary receiver upon the overall signal reception from multiple
secondary transmitters. The spatial multiplexing mode of operation is adopted, for the first
time, where all the potential secondary transmitters send their data streams simultaneously in
a given frame duration. Thus, the self-interference problem is tackled, since all antennas at
the receiver are used first for signal detection/decoding for the secondary data and then for
spectrum sensing in the same frame duration. The receiver utilizes the linear MMSE approach
to efficiently detect the secondary streams. Since the noise variance is, in principle, a requisite
for the MMSE detection/decoding, the optimum energy detector (ED) can be used for the
following spectrum sensing process (which also requires the knowledge of the noise variance).
However, since the spectrum sensing is implemented at the receiver, it is possible that a primary
activity in the vicinity of one or more secondary transmitters may not be sensed by the receiver,
mainly due to the different link distances and/or independent signal propagation losses. To avoid
the latter hidden terminal problem, a distributed power allocation scheme is implemented by
each secondary transmitter, upon signal transmission. Based on this scheme, each secondary
transmitter appropriately adjusts its power in order not to cause any harmful interference to the
potentially active primary node(s), preserving transparency of the secondary activity.
5
Overall, the main benefits of this work are twofold: (a) an efficient tradeoff between sensing
time and data transmission time and its relevant computation is no longer an issue; and (b) the
self-interference problem is effectively mitigated, since the simultaneous transmission and spectrum sensing are implemented by different (i.e., sufficiently separated in terms of transmission
wavelength) nodes.
B. Contributions
The contributions of this work are summarized as follows:
•
A new mode-of-operation and protocol design for cognitive networks is presented and
analytically described. The novelty of this scheme relies on the fact that it uses the spatial
multiplexing transmission scheme, where multiple single-antenna secondary nodes may send
their streams simultaneously to a multiple-antenna secondary receiver, under the presence of
multiple primary nodes. Independent and non-identically distributed (i.n.i.d.) statistics are
considered, suitable for practical networking setups (i.e., different link-distances amongst
the primary and secondary nodes). To this end, the considered secondary system forms a
(virtual) D-MIMO infrastructure. The receiver simultaneously performs signal detection and
spectrum sensing in the same frame duration. Further, a distributed power allocation scheme
is applied on the involved secondary transmitters.
•
New analytical closed-form expressions are derived for some important system measures
when all signals undergo Rayleigh channel fading, namely, the outage and detection probabilities, the transmission power for each secondary node and the probability of unexpected
interference at the primary nodes.
•
As it is explicitly indicated in the upcoming analysis, the accuracy of the detection scheme
plays a key role to the system performance. Thereby, we further investigate the detection
performance with the aid of the receive operating characteristics (ROC) curves, and a solid
performance measure, the area under the ROC curve (AUC). A new exact closed-form
expression of AUC for the considered system is also obtained.
•
For the above derivations, the channel aging effect and channel estimation errors are both
considered. In other words, the analysis incorporates outdated channel state information
(CSI) and/or imperfect CSI for i.n.i.d. Rayleigh fading channels. The results are simplified
for the scenario of independent and identically distributed (i.i.d.) statistics.
6
C. Organization of the paper
The rest of this paper is organized as follows. This Section continues with some notational
definitions for the most important mathematical symbols used in the subsequent analysis. In
Section II, the considered system model and the proposed mode of operation are described in
detail. Key statistical derivations regarding the received signal-to-interference-plus-noise ratio
(SINR) are obtained in Section III. In Section IV, the considered system is thoroughly analyzed, whereas the aforementioned performance measures are obtained in closed form. Further,
important insights regarding the transmission power used by the secondary nodes are presented
in Section V. In Section VI, the proposed framework is validated and cross-compared with
simulation results, while some useful engineering insights are revealed. Finally, Section VII
concludes the paper.
Notation: Vectors and matrices are represented by lowercase bold typeface and uppercase bold
typeface letters, respectively. Also, X−1 is the inverse of X and xi denotes the ith coefficient
of x. A diagonal matrix with entries x1 , · · · , xn is defined as diag{xi }ni=1 . The superscripts (·)T
and (·)H denote transposition and Hermitian transposition, respectively, k · k corresponds to the
vector Euclidean norm, while | · | represents absolute (scalar) value. In addition, Iv stands for
d
the v × v identity matrix, E[·] is the expectation operator, = represents equality in probability
distributions and Pr[·] returns probability. Also, fX (·) and FX (·) represent probability density
function (PDF) and cumulative distribution function (CDF) of the random variable (RV) X,
respectively. Complex-valued Gaussian RVs with mean µ and variance σ 2 , while chi-squared
2
RVs with v degrees-of-freedom are denoted, respectively, as CN (µ, σ 2 ) and X2v
. Furthermore,
Γ(a) , (a − 1)! (with a ∈ N+ ) denotes the Gamma function [20, Eq. (8.310.1)], Γ(·., ·) is the
upper incomplete Gamma function [20, Eq. (8.350.2)], while (·)p is the Pochhammer symbol
with p ∈ N [20, p. xliii]. Further, J0 (·) represents the zeroth-order Bessel function of the first
kind [20, Eq. (8.441.1)], 1 F1 (·, ·; ·) denotes the Kummer’s confluent hypergeometric function
[20, Eq. (9.210.1)], 2 F1 (·, ·, ·; ·) denotes the Gauss hypergeometric function [20, Eq. (9.100)],
and Qν (·, ·) is the generalized νth order Marcum-Q function [21].
7
II. S YSTEM M ODEL
Consider a cognitive (secondary) communication system, which is consisted of mc singleantenna cognitive transmitters and a receiver equipped with N ≥ mc antennas1 operating under
the presence of mp single-antenna primary nodes. Notice that although N ≥ mc is a necessary
condition in order to capture the available degrees-of-freedom during the detection of the streams
from the cognitive transmitting nodes, it holds that N ≶ (mp + mc ). Moreover, i.n.i.d. Rayleigh
flat fading channels are assumed, reflecting non-equal distances among the involved nodes with
respect to the receiver, an appropriate condition for practical applications.
The spatial multiplexing mode of operation is implemented in the secondary system, where mc
independent data streams are simultaneously transmitted by the corresponding secondary nodes.
A suboptimal yet quite efficient detection scheme is adopted, the so-called linear MMSE, which
is performed at the secondary receiver.
Letting M , mp + mc , the received signal at the nth sample time-instance reads as
y[n] = Ĥ[n]s[n] + w[n],
(1)
where y[n] ∈ CN ×1 , Ĥ[n] ∈ CN ×M , s[n] ∈ CM ×1 and w[n] ∈ CN ×1 denote the received
signal, the estimated channel matrix, the transmitted signal and the additive white Gaussian noise
d
(AWGN), respectively. It holds that w = CN (0, N0 IN ) with N0 denoting the AWGN variance
and s = [s1 , . . . , smp , s1 , . . . , smc ]T with E[ssH ] = IM . In addition, Ĥ = [ĥ1 , . . . , ĥmp , ĥ1 , . . . , ĥmc ],
d
whereas ĥi = CN (0, βi IN ), for 1 ≤ i ≤ M, with βi , pi /(dωi i ), where pi , di , and ωi correspond
to the signal power, normalized estimated distance (with a reference distance equal to 1km) from
the receiver and path-loss exponent of the ith transmitter, respectively.
A. Protocol Description
The mode of operation for the considered cognitive system is constituted by three main phases;
namely, the training, data transmission and spectrum sensing phases, which are periodically
alternating.
In the training phase, all the involved nodes (i.e., primary and secondary transmitters) broadcast certain (orthogonal) pilot signals. The secondary receiver monitors the available spectrum
1
It follows from the subsequent analysis that the considered system is equivalent to the case when a single cognitive transmitter
is used equipped with mc antennas.
8
Probe Message
----Terminate
Transmissions
START
No
Training Phase
-----CSI
Acquisition
Idle
Primary
Network
Training
Signals
Possible Primary
activity
Idle
Secondary
activity
Spectrum
Sensing
No
P1
Yes
MMSE Spectrum
Detection Sensing
Yes
Initiate
Secondary
Transmission
Secondary
Network
Pm
SR
S1
p
(a) Training Phase
Smc
(b) Data Phase
(b)
(a)
Fig. 1: a) Flowchart of the proposed mode of operation at the secondary receiver; b) The
considered system configuration, where Si , Pj and SR stand for the ith secondary transmitter (with
1 ≤ i ≤ mc ), jth primary transmitter (with 1 ≤ j ≤ mp ) and secondary receiver, respectively.
resources in order to acquire the instantaneous channel gains from all the existing nodes (both
primary and secondary). Meanwhile, all the secondary transmitters also monitor the channel in
order to acquire channel gains between the primary nodes and themselves. This occurs in order
to appropriately modify their power, which will use for potential transmission in the subsequent
data phase. It is assumed that the channel remains constant during this phase. However, its status
may change in subsequent time instances.
Afterwards, the system enters the data phase, where the secondary nodes stay inactive for
one symbol-time duration. During this time period, the secondary receiver senses the spectrum
so as to capture the presence of a primary communication activity or not. In the former case,
no transmission activity is performed by the secondary transmitters (lack of triggering from
the secondary receiver in this case is interpreted as a busy spectrum notification to all the
transmitters). This procedure is repeated in every subsequent symbol-time duration, until the
receiver senses the spectrum idle. In the latter case, the receiver broadcasts a certain probe
message in order to initiate the secondary transmission(s). Hence, in the next symbol-time
instance, all active secondary nodes may simultaneously transmit their data streams. Upon the
overall signal reception, MMSE detection is performed at the secondary receiver and all data
streams are decoded concurrently.
After the removal of all secondary signals from the received signal, the spectrum sensing
phase is implemented (within the same symbol-time instance), where the receiver monitors the
9
remaining signal for the presence of a potential primary activity. If the remaining signal is sensed
idle (i.e., only the presence of noise), the same procedure keeps on (i.e., data transmissionspectrum sensing), until the next training phase. If at least one primary signal is detected at
the remaining signal, then the receiver immediately broadcasts another certain message in order
to coarsely finalize all the secondary transmissions. An appropriate ceiling on the transmission
power of the receiver is utilized in order not to cause unexpected co-channel interference to the
primary communication(s). Similarly, all the active secondary transmitters use a relevant ceiling
for their transmissions due to the same reason (explicit details on this ceiling are provided into
the next section). The basic lines of reasoning of the proposed scheme are sketched in Figs. 1a
and 1b.
It is noteworthy that the motivation behind the proposed system configuration relies on certain
conditions and/or limitations, which are viable in various realistic networking implementations.
More specifically, conventional CR services based on television white spaces may use the
traditional quiet spectrum sensing (without the training phase requirement). This is because
the number of TV channels is limited (approximately 50∼70, each with a bandwidth 6 − 8MHz,
within a total spectrum range between 54 − 862MHz [22]). In this spectrum range, spectrum
sensing time is indeed acceptable, whereas most IEEE 802.22 equipments are for indoor installation and, hence, their power consumption is not an actual problem [23]. Nonetheless, more
sophisticated CR services, such as the IEEE 1900.4 standard [24], are designed to use spectrum resources from multiple radio-access-technology (RAT) heterogeneous primary networking
systems, e.g., cellular systems. Consequently, spectrum range for these systems is emphatically
increased (e.g., 450MHz-3GHz), while the spectrum sensing time and the corresponding energy
cost are extremely increased in this case, thus, becoming non-efficient. To this end, training-based
signaling, which is, in principle, utilized for primary cellular configurations can be used from the
cognitive/secondary system to perform spectrum sensing and/or acquire important statistics, such
as channel gains and transmission powers of primary nodes/users. Besides, long-term evolution
(LTE) has initiated a CR-based operation quite recently [25], under the concept of licensedassisted access/licensed-shared access (LAA/LSA), which operates with the aid of training (pilot)
signaling [26]–[28].
10
B. Training Phase: Channel Estimation
During the training phase, M orthogonal pilot sequences (i.e., unique spatial signal signatures)
of length M symbols are assigned to the primary and cognitive nodes.2 Then, the received pilot
signal can be expressed as
Ytr [n] = Htr [n]Ψ + Wtr [n],
(2)
where Ytr [n] ∈ CN ×M , Htr [n] ∈ CN ×M , Ψ ∈ CM ×M and Wtr [n] ∈ CN ×M denote the received
signal, the channel matrix, the transmitted pilot signals and AWGN, respectively, all representing
the training phase. Also, the pilot signals are normalized satisfying E[ΨΨH ] = IM .
The MMSE channel estimate of hi [n], 1 ≤ i ≤ M, is given by [30, Eq. (10)] ĥi [n] =
−1 P
PM
M
βi N0 + j=1 βj
IN
j=1 hj [n] + wtr [n] , where wtr [n] is the AWGN at the ith channel
during the training phase. It is noteworthy that with MMSE channel estimation, the channel
estimate and the channel estimation error remain uncorrelated (i.e., due to the orthogonality
principle [31]). In particular, we have that
ĥi [n] = hi [n] + h̃i [n], 1 ≤ i ≤ M,
d
(3)
d
where hi = CN (0, (βi − β̂i )IN ) is the true channel fading of the ith transmitter and h̃i =
P
CN (0, β̂i IN ) denotes its corresponding estimation error with β̂i , βi2 /( M
j=1 βj + N0 ) [30, Eq.
(12)].
Except the channel estimation errors, the channel aging effect occurs in several practical
network setups. This is mainly because of the rapid channel variations during consecutive
sample time-instances, due to, e.g., user mobility and/or severe fast fading conditions. The
popular autoregressive (Jakes) model of a certain order [32], based on Gauss-Markov block
fading channel, can accurately capture the latter effect. More specifically, it holds that
ĥi [n] = αM ĥi [n − M] +
2
M
−1
X
m=0
αm ei [n − m],
(4)
In various network setups, primary users periodically transmit training signals intended for primary receivers to assist
them in channel estimation and/or synchronization [29, §12.3.1]. Building on this feature, secondary nodes can overhear these
transmissions to capture their own estimates amongst the primary nodes and themselves. The first step is to enable the training
process for the secondary nodes along with the primary ones. Doing so, the secondary receiver is able to acquire CSI statistics
from both networks.
11
where α , J0 (2πfD Ts ) with fD and Ts denoting the maximum Doppler shift and the symbol
PM −1 m
sampling period, respectively. Moreover, e′i ,
m=0 α ei [n − m] stands for the stationary
Gaussian channel error vector due to the time variation of the channel, which is uncorrelated
d
with hi [n − M], while e′i = CN (0, (1 − α2M )βi IN ). For the sake of mathematical simplicity
and without loss of generality, we assume that the channel remains unchanged over the time
period of training phase, while it may change during the subsequent data transmission phase.
Thus, adopting the autoregressive model of order one, (4) simplifies to
ĥi [n] = αĥi [n − 1] + ei [n].
(5)
Substituting (3) into (5), we have that3
ĥi = αhi + αh̃i + ei , gi + ǫi ,
d
(6)
d
where gi = CN (0, (βi − β̂i )α2 IN ) and ǫi = CN (0, α2 β̂i + (1 − α2 )βi )IN ).
It should be noted that the latter model in (6) combines both the channel aging effect and
the channel estimation error. Hence, by defining G , [g1 , . . . , gmp , g1 , . . . , gmc ] and E ,
[ǫ1 , . . . , ǫmp , ǫ1 , . . . , ǫmc ], (1) can be reformulated as
y = Gs + Es + w.
(7)
C. Data Transmission Phase: Signal Detection
Benefiting from the training phase whereby estimating the channel gains of all the signals,
the cognitive receiver proceeds with the detection/decoding of the simultaneously transmitted
streams from the mc cognitive nodes. The mean-squared error (MSE) of the ith received stream
(1 ≤ i ≤ mc ) is formed as
h
2 i
,
MSEi = E si − φH
y
i
(8)
where φi is the optimal weight vector.
Corollary 1: The optimal weight vector, which minimizes MSE of the ith received stream is
given by
φi =
3
p
H
βi C diag{βj }M
j=1 C + N0 IN
−1
ci ,
(9)
In what follows, the time-instance index n is dropped for ease of presentation, since all the involved random vectors are
mutually independent.
12
d
where C ∈ CN ×M and C = CN (0, IN ), while ci is its ith column vector.
Proof: The proof of (9) is relegated in Appendix A.
At the receiver, φH
i y is utilized for the detection of the ith transmitted stream, yielding
X
H
H
H
φH
zi = φH
y
=
φ
g
s
+
i
i
i gj sj + φi Es + φi w
i
i
j6=i
= A−1 gi
{z
|
H
,Pi
gi si + A−1 ǫi
}
|
H
gi s i +
X
H
H
φH
i gj sj + φi Es + φi w,
(10)
j6=i
{z
}
,Ri
H
where A , C diag{βj }M
j=1 C + N0 IN .
D. Spectrum Sensing
ED is the optimum detection method, since channel gains, signal, and noise variances are all
known (or estimated) [33]. In addition, the use of multiple antennas at the secondary receiver
can overcome the estimation uncertainty and improve the performance of spectrum sensing, by
exploiting many available observations in the spatial domain [34]. Let the remaining signal, after
decoding the mc secondary signals (thus, after removing their impact from the remaining signal),
be defined as r. Then, (7) becomes
r = Gp sp + Ep sp + w = Cp diag{
p
m
p
βi }i=1
sp + w,
(11)
where r ∈ CN ×1 , Gp ∈ CN ×mp , Ep ∈ CN ×mp , Cp ∈ CN ×mp and sp ∈ Cmp ×1 denote the
remaining received signal, the true channel matrix, the estimation error matrix, the equivalent
(joint) channel matrix and the transmitted signal from the primary nodes, respectively. Also,
d
Cp = CN (0, IN ).
In practice, perfect removal of the mc secondary signals (after decoding) may not always be
the case due to, e.g., hardware constraints and/or impairments at the secondary receiver. Hence,
this process may cause residual noise onto the remaining signal prior to spectrum sensing. In
this case, (11) becomes
r = Cp diag{
p
m
p
sp + w′ ,
βi }i=1
(12)
where w′ , w + wǫ with wǫ being the additive post-noise after the aforementioned imperfect
cancellation/removal. Assuming that wǫ is a zero-mean Gaussian distributed vector [33], [35],
13
d
we can model the post-noise as wǫ ∈ RN ×1 while wǫ = N (0, σǫ2 IN ), where σǫ2 denotes the
level of impact due to imperfect cancellation of the secondary signals. Typically, the value of
σǫ2 can be captured by the secondary receiver via measurements during operation [36] and/or is
predetermined from the system manufacturer. With known σǫ2 , the total noise w′ is modeled as
d
w′ = CN (0, N̂0 IN ) with N̂0 , N0 + σǫ2 .4
Thereby, the binary hypothesis test is formed as
TED ,
L−1
X
l=0
H1
kr(l)k2 ≶ λ,
(13)
H0
where L and λ denote the number of samples for the received signal and the energy threshold,
respectively. Moreover, the two hypotheses H0 and H1 correspond to the cases of no primary
signal transmission and at least one primary signal transmission, respectively. They are explicitly
defined by the structure of the received signal’s covariance matrix as
H0 : E[rrH ] = N0 IN , no signal is present
H1 : E[rrH ] = any positive semi-definite matrix.
III. S TATISTICS
OF
(14)
SINR
We commence by defining the SINR for each stream with its corresponding CDF with
respect to the cognitive communication performance, followed by the false alarm and detection
probabilities with respect to the spectrum sensing performance. Then, the unconditional outage
probability of the considered system is formulated in a closed form. Then, other important
system measures are also obtained in closed form, namely, AUC, the transmission power of
the secondary nodes, and the probability of unexpected co-channel interference at the primary
nodes.
d
Notice from (7) that (gi + ǫi )q=
√
CN (0, α2β̂i + (1 − α2 )βi IN ) = ( α2 β̂i + (1 − α2 )βi )ci . Hence, it follows that
s

2
2
α β̂i + (1 − α )βi 
A−1 ǫi = A−1 
φi ,
βi
4
d
βi ci and, thus, it is straightforward to show that ǫi =
(15)
In what follows, for ease of presentation and without loss of generality, N̂0 = N0 is assumed (which implies that w′ = w).
On the other hand, when the variance of post-noise wǫ is not available, exact closed formulations for the detection and false
alarm probabilities are not feasible; yet, current ones (presented in the following section) can serve as a performance benchmark
or upper performance bounds.
14
while based on (15), we have from (10) that
H
d
Pi = (βi − β̂i )α2 A−1 ci ci .
(16)
Using the above methodology, it also holds from (10) that
′(i)
H
Ri = φH
i CIM s + φi w,
(17)
′(i)
where IM is a special diagonal matrix, which is formed as

p M


diag
βj j=1,j6=i


′(i)
IM ,
q



 α2 β̂i + (1 − α2 )βi , for the ith position.
(18)
′(i)
H
2 H
Thereby, since E[Ri RH
i ] = φi (C(IM ) C + N0 IN )φi , the SINR of the ith transmitted stream
reads as
Pi2
H
(βi − β̂i )α2 (A−1 ci ) ci
2
2
H
(βi − β̂i )α2 (A−1 ci ) ci
=
′(i) 2 H
′(i) 2 H
−1
φH
(C(I
βi (A−1 ci )H
)
C
+
N
I
)φ
0
N
i
i
M
i (C(IM ) C + N0 IN ) (A ci )


2
2
H
2
(β
(βi − β̂i )α2 (A−1 ci ) ci
i − β̂i )α
−1

 H
=
≈
ci C diag{βj }M
CH + N0 IN
ci .

j=1
H
βi
βi (A−1 ci )i ci
SINRi =
E[Ri RH
i ]
=
(19)
′(i)
The approximation stage in the latter expression is formed by assuming that (C(IM )2 CH +
H
N0 IN ) ≈ C diag{βj }M
j=1 C + N0 IN = A. It becomes exact in the case when perfect CSI
conditions occur.5
Based on Woodbury’s identity [37, Eq. (2.1.5)], (19) can alternatively be expressed as

2 
2
 (βi − β̂i )α  Φi
,
SINRi = 

βi
1 + Φi
where Φi ,
cH
i
(20)
−1
P
M
H
K diag{βj }j=1 K + N0 IN
ci , while K , j6=i cj . The form of (20) is
j6=i
preferable than (19) for further analysis, because ci and K are statistically independent.
5
As indicated from the numerical results provided in Section VI, the approximation error remains negligible in moderate
channel estimation error conditions.
15
Lemma 1: The CDF of SINR for the ith transmitted stream for a system with M simultaneous
transmitting nodes and N receive antennas, while 1 ≤ i ≤ mc , is presented in a closed form as


(N ×M )
(N ×M )


FSINRi (x) = FΦi
where
(N ×M )
FΦi
(y) =1 − exp −
×
N0
y
βi
M
X
"X
N
x
((βi −β̂i
−
(i − 1)!
X
j=N −i+1 1≤n1 <···<nj ≤M
j6=i
M Y
1+
βn1 βn2
βi βi
βn
y
βi
n=1
n6=i
)
2
−x
βi
( Nβi0 y)i−1
i=1
)α2

,
(21)
N
X
i=max{1,N −M +1}
···
βnj
βi
(N ×M )
FΦi
(y)
N0
= 1 − exp − y
β
N
X
( Nβ0 y)i−1
i=1
(i − 1)!
−
i=max{1,N −M +2}
(i − 1)!
#
.
N
X
i−1
yj
(22)
When β1 = β2 = · · · = βM , β, (22) reduces to
"
N0
y
βi
N0
y
β
i−1
M
−1
X
j
y #
M −1
j
j=N −i+1
(i − 1)!(1 + y)M −1
.
(23)
Proof: The proof is provided in Appendix B.
The CDF in (23), implies identical channel fading conditions for all the nodes (i.e., equal
distances with regards to the receiver), which is a rather infeasible scenario. Nonetheless, it can
be used as a performance benchmark and/or a good approximation when β1 ≈ β2 ≈ · · · ≈ βM .
IV. P ERFORMANCE M ETRICS
A. Detection Probability
It suffices to show that in the case of H1 hypothesis, even if only the weakest signal is present,
TED > λ should hold. The latter condition can be modeled as
p
rmin = βmin cmin smin + w,
(24)
where rmin represents the remaining received signal, when only the primary node with the
weakest channel gain (at the secondary receiver) is active. The transmitted signal from the
16
2
corresponding primary node is defined as6 smin with E[smin sH
min ] = σp . Also,
√
βmin cmin satisfies
m
p
that βmin kcmin k2 = min{βmin kcp,i k2 }i=1
, where cp,i represents the ith column vector of Cp .
Notice that a Gaussian vector is isotropically distributed, i.e., it remains Gaussian distributed
√
d
even if its norm is under some constraint [38, Theorem 1.5.5]. Thus, βmin cmin = CN (0, βminIN )
and βmin kcmin k2 is the minimum of mp non-identical χ22N RVs.
Lemma 2: The PDF of Y , βmin kcmin k2 is presented in a closed-form as
fY (x) =
mp N −1
X
X
s=1 t1 =0
t1 6=ts
···
N
−1
X
tmp =0
tmp 6=ts
−tmp
β1−t1 · · · βs−N · · · βmp
t1 ! · · · tmp !Γ(N)
Pmp
t +N −1
l=1 l
x l6=s
exp −
mp
X
t=1
Proof: The CDF of Y stems as
2
2
Pr[Y < x] = 1 − Pr[β1 kc1 k > x] · · · Pr[βmp cmp > x] .
1
βt
! !
x .
(25)
(26)
Using the standard complementary CDF of a χ22N RV into the previous expression yields
mp Γ N, x
Y
βt
FY (x) = 1 −
.
(27)
Γ(N)
t=1
By differentiating (27), it holds that
fY (x) =
mp xN −1 exp − x
mp Γ N, x
X
Y
βs
βt
s=1
Γ(N)βsN
t=1
t6=s
Γ(N)
.
(28)
Further, expanding Γ(., .) in terms of finite sum series according to [20, Eq. (8.352.4)], (25) is
obtained.
The detection probability is defined as Pd , Pr[TED |H1 > λ]. In the case of ED, it is given
by [39, Eq. (63)]
s
Pd (λ) = QN L 
6
2Lσp2 Y
,
N0
r

λ 
.
N0
(29)
In general, the signal variance can be estimated by the sample variance for sufficiently large number of samples as σp2 ≈
P
PL−1
PL−1
2
2
2
2
(1/L) L−1
l=0 |smin (l)| − ((1/L)
l=0 smin (l)) . If the sample mean goes to zero, then σp ≈ (1/L)
l=0 |smin (l)| .
17
Corollary 2: The unconditional detection probability of the considered system with N receive
antennas and mp active primary nodes is presented in a closed form as


s
r
mp N −1
m
m
−t
p
p
N
−1
m
XX
X β −t1 · · · β −N · · · βmp p X
2Lσp2
λ X 1
(N ×mp )
1
s
tl + N, NL,
Pd
(λ) =
F
,
,
···
,
t1 ! · · · tmp !Γ(N)
N0
N0 t=1 βt
t =0
s=1 t =0
1
t1 6=ts
l=1
l6=s
mp
tmp 6=ts
(30)
where
2 2m
2
− b2
a b Γ(k) exp
Γ(k)Γ(m,
+
F (k, m, a, b, p) ,
pk Γ(m)
m!pk 2m (a2 + 2p)
a2 b2
.
× 1 F1 l + 1, m + 1; 2
2a + 4p
{z
}
|
b2
)
2
k−1 X
l=0
2p
2
a + 2p
l
(31)
T
Proof: Based on (29), the unconditional detection probability is evaluated as
s
r 
Z ∞
2
2Lσp x
λ 
Pd (λ) =
QN L 
,
fY (x)dx.
N0
N0
0
Plugging (25) into (32), integrals of the following form appear
Z ∞
√
xk−1 exp(−px)Qm (a x, b)dx, {a, b, m, p, k} ≥ 0.
(32)
(33)
0
Fortunately, such integrals were analytically evaluated in [40, Eq. (12)]. Thus, using the latter
result into (32) and after performing some straightforward manipulations, (30) arises.
At this point, it should be stated that when the first two arguments of 1 F1 (., .; .) are nonnegative integers, this expression can be relaxed to finite sum series including simple elementary
functions, according to [41, Eq. (7.11.1.10)]. In fact, this is the case presented in (31), returning
only simple elementary functions, which reads as

k
l−m
2 2
2 2 X

(m−l)k − a2 b

2a
+4p
a
b

,
l≥m
exp 2a2 +4p

k!(m+1)k



k=0






m−l−1
X (l−m+1)k a22 b2 k
T =
(m−1)!(−m)
2a +4p

2 2 l+1
m

k!(1−m)k

l! a2 b

2a +4p

k=0


k !
l
2 2

X

(−l)k − a2 b
2
2

2a +4p
a b


, l<m
 − exp 2a2 +4p
k!(1−m)k
k=0
(34)
18
B. False Alarm Probability and Threshold Design
The scenario of a false alarm probability, namely, Pf (λ), can be modeled by Pf (λ) ,
Pr[TED |H0 > λ]. Under the H0 hypothesis, TED is the sum of the square of NL independent
d
and identically distributed Gaussian RVs with zero mean and variance N0 , i.e, TED = N0 χ22N L .
Hence, using the standard complementary CDF of a chi-square RV, it yields
λ
Γ NL, 2N0
Pf (λ) =
.
Γ(NL)
(35)
As it is obvious from (35), the false alarm probability is an offline operation, i.e., it is
independent from the instantaneous channel gain and the number of primary signals. Thus,
a convenient yet effective strategy is to select the optimum energy threshold using (35). Doing
so, it holds that
λ∗ = Pf−1 (τ ),
(36)
where λ∗ represents the optimum energy threshold for a predetermined target τ (on the false
alarm probability), while Pf−1 (.) denotes the inverse function of Pf (.), which can be efficiently
calculated by using well-known inverse algorithms, e.g., [42].
(N ×mp )
Afterwards, the online detection probability can be directly computed by calculating Pd
(λ∗ ),
using (30).
C. Outage Probability
Based on the above key analytical results, we are now in a position to formulate the outage
(i)
probability of the considered system. Outage probability of the ith stream (1 ≤ i ≤ mc ), Pout (γth ),
is defined as the probability that the SINR of the ith stream falls below a certain threshold value
γth , 2R − 1, where R stands for a given data transmission rate in bps/Hz.
Theorem: The outage probability of the ith stream (1 ≤ i ≤ mc ) is presented in a closed form
as
(i)
Pout (γth )
= (1 − Pf (λ
∗
(N ×m )
)) PAp {∅}FSIN Ric (γth )
M −j
j−mc
×
Y
d1 =1
PAp {d1 }
Y
d2 =1
M
X
(N ×(j−mc )) ∗
+
1 − Pd
(λ )
j=mc +1
(N ×j)
(1 − PAp {d2 }) FSIN Ri (γth ),
(37)
19
where PAp {∆} represents the probability that ∆ primary nodes are active, while PAp {∅} denotes
that there is no active primary node (i.e., an empty set) at the given time-instance.
Proof: The proof is given in Appendix C.
It is noteworthy that PAp {·} is directly related to the transmission arrival rate of each primary
node. For instance, a typical model used thoroughly in wireless systems for the distribution of
data traffic is the widely known Poisson process [43]. In this case, PAp {·} follows the inter-arrival
exponential distribution modeled as PAp {x} = exp(−vTs ), where v is the average transmission
arrival rate. Nevertheless, the analysis and/or the efficient modeling of transmission arrival rates
represents a research topic out of the scope of current work.
D. Area Under the ROC Curve
The accuracy of ED plays a crucial role to the outage probability, which is reflected on the
underlying detection and false alarm statistics. Due to this reason, we further investigate the
performance of ED using a more solid measure, the so-called AUC. The main benefit of AUC
is that it jointly evaluates the performance of both the detection and false alarm in the entire
energy threshold region.
More specifically, the conditional AUC (on a given channel gain) is defined as [44, Eq. (5)]
Z ∞
∂Pf (λ′ ) ′
dλ ,
(38)
AUC(Y) = −
Pd (λ′ )
∂λ′
0
where λ′ stands for the normalized energy threshold λ′ , λ/N0 .
Corollary 3: The conditional AUC of the considered ED scheme is presented in a closed form
as
Lσp2 Y
AUC(Y) = 1 − exp −
N0
NX
L−1
l=0
Lσp2 Y
(NL)l
.
1 F1 NL + l, NL;
l!2N L+l
2N0
(39)
Proof: The detailed proof is presented in Appendix D.
Averaging (39) over the PDF of Y, the unconditional (average) AUC is presented as follows.
Proposition: The unconditional AUC is given by
AUC =1 −
mp N −1
NX
L−1 X
l X
X
l=0
k=0 s=1 t1 =0
t1 6=ts
···
N
−1
X
tmp =0
tmp 6=ts
(NL)l β1−t1
· · · βs−N
−tm
· · · βmp p Γ
l!2N L+l t1 ! · · · tmp !Γ(N)
P
mp
Pmp
1
t=1 βt
+
l=1 tl + N
l6=s
P p
t +N
m
l=1 l
2
Lσp
l6=s
N0
20
Pmp
(−l)k NL − l=1 tl
l6=s
×
k!(NL)k
1−
Lσp2
2N0
k
2
Lσp
1
t=1 βt + N0
Pmp
t +l
l=1 l
Pmp
Lσp2
2N0
2
Lσp
1
t=1 βt + N0
Pmp
!k
!
.
(40)
l6=s
Proof: The proof is relegated in Appendix E.
V. I MPACT
OF THE
T RANSMISSION P OWER U SED
BY THE
S ECONDARY N ETWORK
A. Transmission Power of Secondary Nodes
First, we define the transmission power of the receiver in the case of the aforementioned
signaling process (c.f., Fig. 1a). Recall that in the case when the receiver senses the spectrum busy
(idle) by a primary transmission, upon an ongoing secondary communication, then it immediately
informs the secondary nodes to terminate (initiate) their transmissions using a certain probe
message. In order not to cause an additional co-channel interference to the potentially active
primary node(s), the power used for this message is appropriately upper bounded. Particularly,
it is defined as7
wth
pR = min pmax ,
QR
m
,
(41)
p
], wth denotes the outage power threshold of the primary service
where QR , E[maxi {kgi k2 }i=1
with regards to the secondary transmission(s), which is assumed as a predetermined parameter,
already known to all the secondary nodes, and pmax denotes the maximum achievable (unconstrained) power of the secondary system.
Corollary 4: The aforementioned transmission power at the receiver, pR , is expressed as
−1
1
QR
pR =
,
+
pmax
wth
(42)
where QR is given in a closed form by
mp
mp mp
QR =
X
X X (−1)l bN
R,i
i=1 l=0
l!Γ(N)
n =1
|1
···
{z
mp N −1
X
X
nl =1 k1 =0
}
n1 6=···6=nl ···6=l
7
···
N
−1
X
kl =0
l
Y
t=1
bR,kt
kt !
!
Pl
Γ N + t=1 kt + 1
N +Plt=1 kt +1 ,
Pl
bR,i + t=1 bR,nt
(43)
Note that pR is the fixed power of the secondary receiver, whereas it is determined by the QR statistic, which is computed
during the training phase. It can be updated in a per frame basis, i.e., in a consecutive training phase.
21
where bR,i , (βi − β̂i )α2 is a certain parameter corresponding to the link between the secondary
receiver and the ith primary node (1 ≤ i ≤ mp ).
Proof: The proof is provided in Appendix F.
Notice that pR is formed by using the channel estimates from the training phase. However,
since the secondary receiver has full awareness of the channel time-variation (i.e., known α),
(43) represents quite an efficient ceiling on the corresponding transmission power.8
The transmission power for all the secondary transmitters can be obtained quite similarly.
In particular, referring back to the structure of Htr = [h1 , . . . , hmp , h1 , . . . , hmc ] and Ψ =
[ψ 1 , . . . , ψ mp , ψ 1 , . . . , ψ mc ] from (2), each secondary transmitter sends its pilot in its corresponding symbol-time duration. Notice that the pilots from primary nodes are foregoing the ones
of the secondary nodes. Hence, each secondary transmitter can capture its channel response with
regards to every primary node, by monitoring the first mp pilots, during the training phase. Then,
using MMSE channel estimation (as explicitly described earlier), the jth transmission power at
the corresponding secondary node, pj , is determined by
−1
Qj
1
+
, 1 ≤ j ≤ mc ,
(44)
pj =
pmax wth
where Qj is directly obtained from (43), but denoting the jth secondary transmitter this time,
instead of the secondary receiver.9
In the remaining symbol-time duration of training phase, where the secondary pilot symbol
c
transmissions are sequentially established, {pj }m
j=1 are used to inform the secondary receiver
about the corresponding channel states.
B. Unexpected Co-channel Interference at the Primary Nodes
All the simultaneous secondary transmissions should not cause unexpected co-channel interference to any primary node. Thereby, the following condition should be satisfied
Ij ≤ wth , 1 ≤ j ≤ mp ,
8
(45)
We assume that the secondary system is not aware of the instantaneous transmit/receive status for each primary node at each
frame duration. Hence, QR is formulated so as to protect all the links between secondary receiver and primary nodes. In the
simplified scenario when the secondary receiver knows the exact primary receiver at each frame (or when it is fixed), then QR
is still obtained from (43) by setting mp = 1.
9
We use channel reciprocity between primary and secondary nodes in order to formulate the aforementioned transmission
powers in (41) and (44).
22
where Ij denotes the aggregate interfering power to the jth primary node from all the secondary
transmitters.
In order to analytically evaluate Ij , consider the case when the receiver senses the channel
busy during an ongoing multi-node (mc -fold) secondary transmission and then broadcasts its
termination signal back to the secondary transmitters. Doing so, the worst case scenario in
terms of unexpected co-channel interference includes the aggregate interfering power of mc + 1
signals. Assuming that the phases of the individual secondary signals fluctuate significantly, due
to mutually independent modulation, the latter aggregate interference can be efficiently formed
as an incoherent addition of the powers from mc + 1 signals [45], which is a suitable model for
practical applications [46]. Hence, for Rayleigh fading channels, each secondary signal power
follows the exponential distribution and, thus, Ij is distributed by [47, Eq. (5)]


x
i
m
+1
c
X Y
 exp − pi q̄j,i
pi q̄j,i
, i = 1, . . . , mc , R
fIj (x) =

(pi q̄j,i −pk q̄j,k ) 
pi q̄j,i
i=1
k=1
(46)
k6=i
−ωi
where q̄j,i , dj,i
denotes the link distance between the jth primary and ith secondary node,
while R stands for the secondary receiver. Then, using the standard complementary CDF of
exponential RVs, the probability of unexpected co-channel interference at the jth primary node
is expressed as
Pr[Ij > wth ] =
Z
∞
wth
fIj (x)dx =
m
c +1
X
i=1



i
Y
k=1
k6=i


pi q̄j,i
exp
(pi q̄j,i −pk q̄j,k ) 
wth
.
−
pi q̄j,i
(47)
VI. N UMERICAL R ESULTS
In this section, analytical results are presented and cross-compared with Monte-Carlo simulations. There is a good match between all the analytical and the respective simulation results and,
hence, the accuracy of the proposed approach is verified. Henceforth, for notational simplicity
and without loss of generality, we assume a common path-loss exponent ω = 4, corresponding to
a classical macro-cell urban environment [43, Table 2.2], while we fix the probability of transmission for all the primary nodes PAp = 0.5. Also, we set α = 0.1, σp2 = 1 and pmax = 20dBm, while
all the primary nodes use pmax for their transmissions. Some of the following numerical results
are presented with respect to the input SNR of the primary nodes, referred as SNR , pmax /N0 .
23
1
0.95
L=4
0.9
d
P (λ*)
L=8
0.85
L = 16
0.8
N=2
N=4
0.75
0.7
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
*
Pf (λ )
Fig. 2: Analytical ROC curve of the considered scheme for mp = 4 with d1 = 0.31, d2 = 0.1,
d3 = 0.15, and d4 = 0.2.
1
0.999
0.998
mp = 2
0.997
*
Pd (λ )
mp = 4
0.996
0.995
mp = 6
0.994
0.993
L=8
L = 32
0.992
0.991 −2
10
−1
10
0
10
*
Pf (λ )
Fig. 3: Analytical ROC curve of the considered scheme for N = 2, various numbers of primary
m
p
nodes and identical link distances with respect to the secondary receiver, i.e., {di }i=1
= 0.1.
Figures 2 and 3 present the ROC curves for the scenario of non-identical and identical statistics,
respectively. Obviously, the performance of detection probability against false alarm probability
improves for higher number of receive antennas. This is further enhanced when the available
number of samples is increased. In addition, the presence of more primary users degrades the
detection performance, since adding more unknown primary signals would be indistinguishable
from noise. This result is in agreement with [48, Fig. 7].
Similar conclusions can be drawn from Fig. 4, where the AUC performance is presented as a
24
1
0.95
d = 0.7
d = 1.2
Unconditional AUC
0.9
L=8
0.85
0.8
L = 16
0.75
0.7
0.65
mp = 4
mp = 2
0.6
Simulation
0.55
0.5
−10
−8
−6
−4
−2
0
2
4
6
8
10
SNR (dB)
Fig. 4: Unconditional AUC vs. various SNR values of the primary nodes, considering identical
link distances, while N = 4.
1
N=8
0.95
N=4
0.9
N=2
0.8
d
P (λ*)
0.85
0.75
0.7
0.65
L=4
L=8
Simulation
0.6
0.55
−12
−10
−8
−6
−4
−2
0
SNR (dB)
m
p
=
Fig. 5: Detection probability vs. various input SNR values for the primary nodes, when {di }i=1
0.3, mp = 2 and Pf = 0.01.
means of a more concrete performance tool in the entire energy threshold region, not only for the
optimum λ⋆ . In fact, the unconditional (average) AUC performance is depicted against different
distances between the primary nodes and secondary receiver. It can be seen that the detection
accuracy is reduced for far-distanced links, as expected, due to the unavoidable propagation
attenuation on the received signals. Severe fading due to propagation losses results to noise-like
signals. On the other hand, increasing SNR and/or the number of available samples for sensing
result to a more accurate detection performance (i.e., AUC tends to unity). Additionally, the
25
0
10
N=2
−1
10
−2
i
M)
(γth)
F(N×
SINR
10
N=4
−3
10
−4
10
M=2
M=4
M=8
Simulation
−5
10
−6
10
−15
−10
−5
γth / SNR (dB)
0
5
Fig. 6: CDF of the received SINR of a N × M system vs. various values of the normalized
outage threshold, when {di }M
i=1 = 0.8 + 0.05i.
0
10
Pf (.) = 1%
−1
10
Outage Probability
−2
10
Pf (.) = 10%
−3
10
−4
10
N = mc = 2
N = mc = 4
−5
10
L=4
L=8
−6
10
−15
−10
−5
γth / SNR (dB)
0
5
Fig. 7: Outage probability of the considered scheme vs. various values of the normalized outage
threshold for identical link distances {di }M
i=1 = 0.8, while mp = 2.
presence of more receive antenna elements enhances the detection performance of the secondary
receiver, as indicated in Fig. 5. This occurs due to the increased spatial diversity for higher N
values, which is manifested by capturing many different spatial observations for the same sample
time-instance.
In Fig. 6, the CDF of the considered (virtual) N × M MIMO system is presented with nonidentical statistics, where the analytical curves are based on (21). As can be seen, the performance
improves for higher number of receive antennas with fixed number of simultaneously transmitting
26
−2
10
−4
Probability of Unexpected Interference
10
−6
10
d1 = 1.3
−8
10
d1 = 1
d1 = 0.7
−10
10
−12
10
−14
10
Analytical, mc = 2
Analytical, mc = 4
−16
10
Simulation
−18
10
−20
−18
−16
−14
−12
−10
−8
−6
−4
−2
0
wth / SNR (dB)
Fig. 8: Probability of unexpected interference to the primary nodes vs. various values of the
normalized outage threshold (for the primary service), when N = mp = 4.
nodes, because of the emerged diversity gain. On the other hand, the performance is reduced
for higher number of simultaneously transmitting nodes and a fixed number of receive antennas,
since adding more co-channel interfering signal power degrades the total SINR. Importantly,
the difference between the analytical curves and simulation points is rather marginal (there is a
difference due to the approximation stage in (19), yet it is rather negligible), which enhances
the efficiency of the proposed scheme.
Moreover, Fig. 7 demonstrates the total (unconditional) outage performance for some selected
system scenarios. It is obvious that the target on the false alarm probability (i.e., the efficiency
of detection scheme) and the available spatial DOFs play a key role to the outage probability.
Finally, Fig. 8 presents the probability of unexpected interference at the primary nodes. To
preserve non-symmetrical distances (i.e., non-identical statistics appropriate for practical setups)
let q̄j,i , d1 (0.01i+ 0.01j). Interestingly, the latter probability is reduced for small link-distances
between secondary and primary nodes and/or increased number of secondary transmitters. As
an illustrative example, the cases when d1 < 0.7 return negligible probability of unexpected
interference for practical applications (i.e., below 10−6 ).
VII. C ONCLUSION
A D-MIMO cognitive (secondary) system was investigated, which operates under the presence
of multiple primary nodes/users. A novel communication protocol was presented and evaluated
27
when the secondary receiver utilizes MMSE detection. New analytical expressions for important
performance metrics were derived in closed form, such as the outage and detection probabilities,
the unconditional AUC, and the impact of the transmission power from the secondary to the
primary system. It was demonstrated that the probability of unexpected interference to the
primary nodes remains quite low, by following the proposed guidelines, while the performance
of the secondary system is directly associated with the signal detection accuracy.
A PPENDIX
A. Derivation of (9)
From (8), it holds that
h
H i
H
H
H
H
= 1 + φH
MSEi = E si − φH
y
s
−
φ
y
i
i Aφi − si y φi − φi ysi
i
i
H −1 H
H −1
− (gi + ǫi )H A−1 (gi + ǫi ),
A φH
= 1 + φH
i − (gi + ǫi ) A
i − (gi + ǫi ) A
(A.1)
H
where A , E[yyH ] = C diag{βj }M
j=1 C +N0 IN represents the covariance matrix of the received
signal. Since only the first term of (A.1) depends on φi , the optimal solution that minimizes
p
MSEi is φi = A−1 (gi + ǫi ). Finally, noticing that (G + E) = C diag{ βj }M
j=1 , (9) can be easily
extracted.
B. Derivation of (21)
From (20), it holds that Pr[SINRi ≤ x] equals (21). Hence, the derivation of FΦi (·) is required
to obtained the CDF of SINR for the ith transmitted stream. To this end, FΦi (·) is derived in a
closed form as [49, Eq. (11)]
i−1
X
N Ai(y) N0 y
βi
N0
(N ×M )
,
FΦi
(y) = 1 − exp − y
βi
(i − 1)!
i=1
where Ai(y) = 1 when N ≥ M + i − 1, or
1+
Ai(y) ,
N −i
X
j=1
X
βn1 βn2
βi βi
1≤n1 <···<nj ≤M
M
Y
1+ ββn y
n=1
n6=i
i
···
βnj
βi
yj
(B.1)
28
when N < M +i−1. With the aid of [50, Eqs. (65) and (70)], a combined formation of the latter
expression can directly be derived in (22) and (23) for the non-identical and identical statistics,
correspondingly.
C. Derivation of (37)
Outage probability can be modeled by using the total probability theorem. Specifically, an
outage event may occur if one of the following conditions hold: (a) when there is no active
(transmitting) primary node, the receiver accurately senses the idle spectrum, and evaluates outage
probability under the presence of mc independent signals; (b) when there is a miss detection
event (i.e., the complement of detection probability) under the presence of one primary node
averaging over its related probability; or (c) when there is a miss detection event under the
presence of two primary nodes averaging over its related probability, and so on.
Condition (a) is explicitly defined in the first term of (37), while conditions (b), (c) and so on
are modeled by the second term of (37) involving nested finite sum series (corresponding to the
cases from mc + 1 to mc + mp total active transmitted streams). Using (21), (30), (35), and (36)
into (37), outage probability can be directly computed in a closed-form, concluding the proof.
D. Derivation of (39)
Plugging the first derivative of the false alarm probability ∂Pf (λ′ )/∂λ′ = λ′2N L−1 exp(−λ′2 /2)
/(2N L−1 Γ(NL)) and (29) into (38), we have that

s
Z ∞
2
√
2Lσp Y
1
t
AUC(Y) = N L
QN L 
, t dt.
tN L−1 exp −
2 Γ(NL) 0
2
N0
(D.1)
A closed-form solution for the latter expression was reported in [40, Eq. (8)]. Thus, after some
simple manipulations, (39) is extracted.
E. Derivation of (40)
In principle, the unconditional AUC can be captured as AUC ,
R∞
0
AUC(x)fY (x)dx. Hence,
using (39) and (25), while utilizing [20, Eq. (7.621.4)], we have that
AUC =1 −
mp N −1
NX
L−1 X
X
l=0
s=1 t1 =0
t1 6=ts
···
N
−1
X
tmp =0
tmp 6=ts
(NL)l β1−t1
· · · βs−N
−tm
· · · βmp p Γ
l!2N L+l t1 ! · · · tmp !Γ(N)
P
Pmp
mp 1
t=1 βt
+
l=1 tl + N
l6=s
P p
t +N
m
l=1 l
Lσp2
l6=s
N0
29

mp
X

tl , NL;
× 2 F1 NL + l,
l=1
l6=s
Lσp2
2N0
P
mp
1
t=1 βt
+
Lσp2
N0


.
(E.1)
Using [51, Eq. (07.23.03.0145.01)] into (E.1), we arrive at (40).
F. Derivation of (42) and (43)
Regarding the derivation of (42) and recalling the Rayleigh fading condition, the PDF of the
SNR for the probe message transmitted from the receiver becomes

N x
N0 exp − p 0 X̄


max R
th

,
, QR < pwmax

pmax X̄R

fXR (x) =


N Q x

N0 QR exp − w0 X̄R

th
th R

, QR > pwmax
.
wth X̄R
(F.1)
where XR and X̄R denotes the instantaneous and average input SNR of the receiver. Hence, it
yields that

1
N
0 pmax +
FXR (x) = 1 − 1 − FXR |pmax (x) 1 − FXR | wth (x) = 1 − exp −
QR
X̄R
QR
wth

x
.
(F.2)
By differentiating (F.2), the corresponding PDF follows the classical exponential PDF with the
yielded transmission power pR as defined in (42).
Based on (7) and (11), we have that the actual channel matrix for the primary nodes can
√ mp
− Ep . Although the instantaneous values of E are not
be expressed as Gp = Cp diag{ βi }i=1
available, its distribution is known from (6), using MMSE channel estimation. It easily follows
that
q
mp
2
(βi − β̂i )α
Gp = Cp diag
.
d
(F.3)
i=1
Thus, using the standard PDF/CDF expressions for chi-squared RVs, the maximum squared
column norm of Gp is distributed as
f
mp
maxi {kgi k2 }i=1
(x) =
mp
X
fbR,i χ22N (x)
i=1
mp
=
FbR,i χ22N (x)
l=1
l6=i
N −1
Xx
i=1
mp
Y
x
exp − bR,i
bN
R,i Γ(N)
mp

N −1
Y
x X
1 − exp −
bR,i k=0
l=1
l6=i
x
bR,i
k!
k 

.
(F.4)
30
By invoking the product expansion identities [52, Eq. (6)], (F.4) becomes after some simple
manipulations
fmaxi {kgi k2 }mp (x) =
i=1
mp
mp mp
X
X (−1)l bN
X
R,i
i=1 l=0
l!Γ(N)
n =1
|1
···
{z
mp N −1
X
X
nl =1 k1 =0
}
···
N
−1
X
kl =0
l
Y
bR,k
kt !
t=1
t
!
n1 6=···6=nl ···6=l
× exp − bR,i +
Thereby, recognizing that Q =
is derived.
R∞
0
l
X
bR,nt
t=1
! !
Pl
x x
t=1
kt +N −1
.
(F.5)
xfmaxi {kgi k2 }mp (x)dx and utilizing [20, Eq. (3.381.4)], (43)
i=1
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