低信噪比环境下的多用户频谱感知研究
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收藏 513KB PDF 举报 "2013-Multiple primary user spectrum sensing in the low SNR regime"
这篇2013年的IEEE Transactions on Communications论文聚焦于在低信噪比(SNR)环境下的多用户频谱感知技术，这是认知无线电(Cognitive Radio, CR)网络中的一个重要研究领域。频谱感知是认知无线电的核心功能之一，它允许次级用户（secondary users, SUs）在不干扰主用户（primary users, PUs）的情况下，有效地利用空闲频段。
在介绍部分，论文指出在认知无线电网络中，动态频谱接入的关键是能够准确地检测是否存在主用户信号。当存在多个主用户和/或多径传播通道时，这种检测变得更加复杂。论文提出了一种在低SNR条件下无噪声不确定性检测器，该检测器在这一环境下具有最优性能。
作者Lu Wei、Prathapasinghe Dharmawansa和Olav Tirkkonen通过分析测试统计量的矩，得到了关键性能指标的简单且精确的解析公式。这些指标包括假警报概率和检测概率，以及接收机操作特性（Receiver Operating Characteristic, ROC）。他们给出了这些概率的封闭形式表达，使得在实际应用中更容易计算和理解。
论文的仿真设置显示，在相对较低的信噪比场景下，所提出的检测算法相比已知的其他算法具有显著的性能提升。这表明在低SNR条件下，多用户频谱感知的优化策略对于提高认知无线电网络的效率和可靠性至关重要。
"Locally Best Invariant Test"是一种用于处理非高斯噪声环境的检测方法，论文可能讨论了这种方法在多路径传播环境中的应用，因为多径传播通常会导致非高斯噪声。"Moment-based approximation"是指利用随机变量的矩来近似其分布，从而简化复杂的统计分析。
"Multiple primary users"和"multipath channels"的考虑使得问题更复杂，因为每个主用户可能有不同的信号强度和到达路径，这要求检测算法能同时处理多个信号源和多径衰落的影响。
这篇论文为认知无线电网络中的低SNR条件下的多用户频谱感知提供了理论基础和实用的解决方案，对于理解和改进认知无线电系统的性能具有重要价值。
1722 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 61, NO. 5, MAY 2013
B. Sensing Problem
Now the problem of interest is to use the data matrix X to
decide whether there are primary users.
3
By the assumptions in
the last subsection, the sample covariance matrix R = XX
†
of the received data matrix follows a complex Wishart dis-
tribution, denoted by R ∼W
K
(N,Σ). The corresponding
population covariance matrix calculated in the absence of
primary users, d enoted by hypothesis H
0
,is
H
0
: Σ := E[XX
†
]/N = σ
2
I
K
, (6)
and in the presence of primary users, denoted by hypothesis
H
1
,is
H
1
: Σ = σ
2
I
K
+
P
p=1
γ
p
h
p
h
†
p
. (7)
The received SNR of primary user p across the K sensors
is SNR
p
:= γ
p
||h
p
||
2
/σ
2
. These characterize the interference
level close to the primary transmitter from a transmission of
the secondary system, the control of which is the target of
dynamic spectrum management. The differences between the
population covariance matrices (6) and (7) can be explored to
detect the primary users. Declaring wrongly H
0
, or declaring
correctly H
1
, defines the false alarm probability P
fa
,and
the detection probability P
d
, respectively. Since the sample
covariance matrix R is a Wishart matrix, it is sufficient
statistics for the population covariance matrix Σ [15]. For
different assumptions on the number of primary users P ,and
on the knowledge of the noise power σ
2
, different test statistics
can be derived as functions of R.
When P ≥ 1 but not known a priori, the matrix
P
i=1
γ
i
h
i
h
†
i
in (7) is positive definite. Thus, the notation
A B is equivalent with A − B being positive definite, the
hypothesis test is
H
0
: Σ = σ
2
I
K
(8a)
H
1
: Σ σ
2
I
K
, (8b)
where the noise power σ
2
is assumed to be unknown. Essen-
tially, we are testing a null hypothesis Σ = σ
2
I
K
against all
the other possible alternatives of Σ, i.e. the hypothesis test
is blind to P . The corresponding GLR-optimal detector under
the hypotheses test (8) is based on the so-called Spherical Test
(ST)
T
ST
:=
|R|
1
K
tr(R)
K
=
K
i=1
λ
i
1
K
K
i=1
λ
i
K
, (9)
where |·| and tr(·) denote matrix determinant and
trace, respectively, and the ordered eigenvalues of R are
0 ≤ λ
K
≤ ...≤ λ
1
< ∞. In the context of spectrum sensing,
ST detection was first proposed in [10] and analyzed in [11].
Although the ST detector achieves good performance in
general, it is not the best detector in the low SNR regime.
3
This collaborative sensing scenario and the subsequent formulations are
more rele vant when the K sensors are in one device. For distributed
collaborating sensors, accurate time synchronization between devices and
communications to the fusion center become an issue given the limited
capability of the individual sensor.
For multiple primary users, a test statistics that is optimal in
detecting small deviations from H
0
is based on
T
J
:=
tr(R
2
)
tr(R)
2
=
K
i=1
λ
2
i
K
i=1
λ
i
2
. (10)
This test statistics was first considered by S. John [16]. A more
rigorous derivation of (10) can be found in [17, Eq. (1.2–1.7)],
where the resu lting test procedure is
T
J
H
1
≷
H
0
ζ, (11)
ζ being a threshold. It can be verified that the natural support
of T
J
is [1/K, 1]. The criterion under which John’s detector
is derived is known as the Locally Best Invariant (LBI)
criterion [17, Eq. (1.1)]. For every σ
2
and for every other test
T , there is a neighborhood of σ
2
I
K
such that T
J
achieves no
worse performance than T does [17], although the radius of
this neighborhood is not known. Considering (6) and (7) it is
clear that John’s test is optimal when
P
i=1
γ
i
h
i
h
†
i
, measured
in a suitable norm, e.g. the sum of its eigenvalues, is small.
This effectively requires that the SNRs are low. Note that
eigenvalue decomposition is not needed for John’s d etector
as opposed to most other eigenvalue based detectors. Finally,
it is worth noting that different detection techniques need to
be designed in a more general scenario of an arbitrary but
unknown noise covariance matrix. In this case, the detector
based on Roy’s statistics [18] turns out to be a good choice for
single-primary-user detection. For a similar scenario of arbi-
trary signal covariance matrix, the corresponding test statistics
may be derived following the lines of reasoning in [18].
III. P
ERFORMANCE ANALY S I S
In this section we derive closed-form expressions for the
moments of T
J
under both hypotheses. Based on the derived
results, we construct approximations to the distributions of
T
J
, which lead to analytical formulae for the false alarm
probability, the detection probability, as well as th e receiver
operating characteristic.
A. False Alarm Probability
Firstly, we study the moments of T
J
under H
0
, the first step
to which relies on the following lemma.
Lemma 1. Under H
0
the random variable
K
i=1
λ
i
2
is independent of the random variable
T
J
=
K
i=1
λ
2
i
K
i=1
λ
i
2
.
The proof of Lemma 1 is in Appendix A. Note that this
independence for the real Wishart case was proven in [21].
However, the method of proof there may not be applied to the
complex case, for which we have invoked a more generic ap-
proach based on the general polar coordinates transformation.
By virtue of Lemma 1, the m-th moment of T
J
under H
0
now
equals
E[T
m
J
]=E
K
i=1
λ
2
i
m
E
K
i=1
λ
i
2m
. (12)
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