The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)
MAXIMUM-MINIMUM EIGENVALUE DETECTION FOR COGNITIVE RADIO
Yonghong Zeng and Ying-Chang Liang
Institute for Infocomm Research, A*STAR
21 Heng Mui Keng Terrace, Singapore 119613
A BSTRACT
Sensing (signal detection) is a fundamental problem in cognitive radio. In this paper, a new method is proposed based on
the eigenvalues of the covariance matrix of the received signal. It is shown that the ratio of the maximum eigenvalue to
the minimum eigenvalue can be used to detect the signal existence. Based on some latest random matrix theories (RMT), we
can quantize the ratio and find the threshold. The probability
of false alarm is also found by using the RMT. The proposed
method overcomes the noise uncertainty difficulty while keeps
the advantages of the energy detection. The method can be
used for various sensing applications without knowledge of the
signal, the channel and noise power. Simulations based on randomly generated signals and captured ATSC DTV signals are
presented to verify the methods.
I.
I NTRODUCTION
Careful studies by FCC reveal that most of the allocated spectrum experiences low utilization. This motivates the concept of
opportunistic spectrum access that allows secondary networks
to borrow unused radio spectrum from primary licensed networks. The core technology behind opportunistic spectrum
access is “cognitive radio” [1]. Unique to cognitive radio is
its ability to sense the environment over huge swath of spectrum and adapt to it. That is, it is necessary to dynamically
detect the existence of signals of primary users. In December 2003, the FCC issued a Notice of Proposed Rule Making
that identifies cognitive radio as the candidate for implementing negotiated/opportunistic spectrum sharing [2]. In response
to this, from 2004, the IEEE has formed the 802.22 Working
Group to develop a standard for wireless regional area networks
(WRAN) based on cognitive radio technology.
As discussed above, sensing is a fundamental component
of cognitive radio. There are several factors which make
the sensing difficult. First, the signal-noise-ratio (SNR) may
be very low (lower than -20dB). Secondly, fading and multipath in wireless signal complicate the problem. Fading will
cause the signal power fluctuates dramatically (can be 10dB
or even higher), while unknown multipath will cause coherent
detection methods unreliable. Thirdly, noise/interference level
changes with time (noise power uncertainty) and the noise can
be non-Gaussian. There are two types of noise uncertainty:
receiver device noise uncertainty and environment noise uncertainty, which are caused by receiving devices or environment,
respectively [3, 4, 5]. Due to noise uncertainty, in practice, it is
very difficult (virtually impossible) to obtain the accurate noise
power.
There have been some sensing algorithms including the energy detection [6, 7, 3], the matched filtering (MF) [3] and cyc
1-4244-1144-0/07/$25.00°2007
IEEE
clostationary detection [8, 9]. These algorithms have different
requirements and advantages/disadvantages. Energy detection
is a major and basic method. Unlike other methods, energy
detection does not need any information of the signal to be detected and is robust to unknown dispersive channel. However,
energy detection is vulnerable to the noise power uncertainty
[7, 3, 4], because the method relies on the knowledge of accurate noise power. In practice, it is very difficult (virtually impossible) to obtain the accurate noise power. To overcome this
shortage, we propose a new method based on the eigenvalues of
the covariance matrix of the received signal. It is shown that the
ratio of the maximum eigenvalue to the minimum eigenvalue
can be used to detect the signal existence. Based on some latest
random matrix theories (RMT), we can quantize the ratio and
find the threshold. The probability of false alarm is also found
by using the RMT. The proposed method overcomes the noise
uncertainty difficulty while keeps the advantages of the energy
detection. The method can be used for various signal detection
applications without knowledge of the signal, the channel and
noise power. Simulations based on randomly generated signals
and captured ATSC DTV signals are presented to verify the
methods.
Some notations are used in the following: superscripts T
and † stand for transpose and Hermitian (transconjugate), respectively. Iq is the identity matrix of order q.
II.
S ENSING MODEL
Assume that we are interested in the frequency band with central frequency fc and bandwidth W . We sample the received
signal at a sampling rate higher than the Nyquist rate. Assume
that there are M ≥ 1 receivers (antennas). The received discrete signal at receiver i is denoted by xi (n) (i = 1, 2, · · · , M ).
There are two hypothesises: (1) hypothesis H0 : there exists
only noise (no signal); (2) hypothesis H1 : there exist both noise
and signal. At hypothesis H0 , the received signal at receiver i
is
xi (n) = ηi (n), n = 0, 1, · · · ,
(1)
while at hypothesis H1 , the received signal at receiver i is
xi (n) =
Nij
P X
X
hij (k)sj (n − k) + ηi (n),
(2)
j=1 k=0
where sj (n) (j = 1, 2, · · · , P ) are P ≥ 1 source signals,
hij (k) is the channel response from source signal j to receiver
i, Nij is the order of channel hij (k), and ηi (n) is the noise
samples. Based on the received signals with little or no information on the source signals, channel responses and noise
power, a sensing algorithm should make a decision on the existence of signals. Let Pd be the probability of detection, that is,
The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)
at hypothesis H1 , the probability of the algorithm having detected the signal. Let Pf a be the probability of false alarm, that
is, at H0 , the probability of the algorithm having detected the
signal. Obviously, for a good detection algorithm, Pd should
be high and Pf a should be low. The requirements of the Pd
and Pf a depend on the applications.
III.
M AXIMUM - MINIMUM EIGENVALUE DETECTION
def
Letting Nj = max(Nij ), zero-padding hij (k) if necessary,
i
and defining
def
x(n) = [x1 (n), x2 (n), · · · , xM (n)]T ,
def
(3)
T
hj (n) = [h1j (n), h2j (n), · · · , hM j (n)] ,
(4)
def
η(n) = [η1 (n), η2 (n), · · · , ηM (n)]T ,
(5)
we can express (2) into vector form as
x(n) =
Nj
P X
X
where Rs is the statistical covariance matrix of the input signal,
Rs = E(ŝ(n)ŝ† (n)), ση2 is the variance of the noise, and IM L
is the identity matrix of order M L.
Let λ̂max and λ̂min be the maximum and minimum eigenvalue of R, and ρ̂max and ρ̂min be the maximum and minimum eigenvalue of HRs H† . Then λ̂max = ρ̂max + ση2 and
λ̂min = ρ̂min + ση2 . Obviously, ρ̂max = ρ̂min if and only if
HRs H† = δIM L , where is δ is a positive number. In practice,
when signal presents, it is very unlikely that HRs H† = δIM L
(due to dispersive channel and/or oversampling and/or correlation among the signal samples). Hence, if there is no signal, λ̂max /λ̂min = 1; otherwise, λ̂max /λ̂min > 1. The ratio
λ̂max /λ̂min can be used to detect the presence of signal. The
detection algorithm is summarized as follows.
Algorithm 1 Maximum-minimum eigenvalue (MME) detection
Step 1. Compute
hj (k)sj (n − k) + η(n), n = 0, 1, · · · . (6)
R(Ns ) =
j=1 k=0
def
=
[xT (n), xT (n − 1), · · · , xT (n − L + 1)]T ,
η̂(n)
def
=
[η T (n), η T (n − 1), · · · , η T (n − L + 1)]T ,
ŝ(n)
def
[s1 (n), · · · , s1 (n − N1 − L + 1), · · · ,
sP (n), · · · , sP (n − NP − L + 1)]T ,
=
(7)
we get
x̂(n) = Hŝ(n) + η̂(n),
def
where H is a M L × (N + P L) (N =
P
P
j=1
as
def 
Hj = 
hj (0) · · ·
..
.
0
···
H = [H1 , H2 , · · · , HP ],

hj (Nj ) · · ·
0

..
.
.
hj (0)
···
(8)
Nj ) matrix defined
def

(9)
(10)
def
=
1
Ns
L−1+N
X s
x̂(n)x̂† (n),
(11)
n=L
where Ns is the number of collected samples. If Ns is large,
based on the assumptions (A1-2), we can verify that
def
IV.
R(Ns ) ≈ R = E(x̂(n)x̂† (n)) = HRs H† + ση2 IM L ,
x̂(n)x̂† (n).
n=L
T HEORETIC ANALYSIS AND THE THRESHOLD
In practice, we only have finite number of samples available.
Hence, the sample covariance matrix R(Ns ) may be well away
from the statistical covariance matrices R. The eigenvalue distribution of R(Ns ) becomes very complicated [10, 11, 12, 13].
This makes the choice of the thresholds very difficult. At low
SNR, the performance of a sensing algorithm is very sensitive
to the threshold. Since we have no information on the signal
(actually we even do not know if there is signal or not), it is
difficult to set the threshold based on the Pd . Hence, usually
we choose the threshold based on the Pf a .
When there is no signal, R(Ns ) turns to Rη (Ns ), the sample
covariance matrix of the noise defined as,
hj (Nj )
The following assumptions for the statistical properties of
transmitted symbols and channel noise are assumed.
(A1) Noise is white.
(A2) Noise and transmitted signal are uncorrelated.
Let R(Ns ) be the sample covariance matrix of the received
signal, that is,
R(Ns )
L−1+N
X s
Step 2. Obtain the maximum and minimum eigenvalue of the
matrix R(Ns ), that is, λmax and λmin .
Step 3. Decision: if λmax > γλmin , signal exists (“yes” decision); otherwise, signal does not exist (“no” decision), where
γ > 1 is a threshold (to be given in the next section).
Considering L consecutive outputs and defining
x̂(n)
1
Ns
(12)
Rη (Ns ) =
1
Ns
L−1+N
X s
η̂(n)η̂ † (n).
(13)
n=L
Rη (Ns ) is a special Wishart random matrix [10]. The study of
the spectral (eigenvalue distributions) of a random matrix is a
very hot topic in recent years in mathematics as well as communication and physics. The joint probability density function
(PDF) of ordered eigenvalues of the random matrix Rη (Ns )
has been known for many years [10]. However, since the expression of the PDF is very complicated, no closed form expression has been found for the marginal PDF of ordered eigenvalues. Recently, I. M. Johnstone and K. Johansson have found
the distribution of the largest eigenvalue [11, 12] for real and
complex matrix, respectively, as described in the following theorems.
The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)
t
F1 (t)
-3.90
0.01
-3.18
0.05
-2.78
0.10
-1.91
0.30
-1.27
0.50
-0.59
0.70
0.45
0.90
0.98
0.95
2.02
0.99
Table 1: Numerical table for the Tracy-Widom distribution of order 1
Theorem 1. Assume
that the
√ noise2 is real. Let√A(Ns ) =
√
Ns
M L) and ν = ( Ns − 1 +
ση2 Rη (Ns ), µ = ( Ns − 1 +
√
1
1
L
1/3
√
√
= y
M L)( N −1 + M L ) . Assume that lim M
s
Ns →∞ Ns
λmax (A(Ns ))−µ
(0 < y < 1). Then
converges (with probability
ν
one) to the Tracy-Widom distribution of order 1 (W1 ) [14].
Theorem 2. Assume that the
Let
√ noise√is complex.
0
2
0
s
A(Ns ) = N
R
(N
),
µ
=
(
N
+
M
L)
and
ν
=
2
η
s
s
√ ση
√
1
1
M
L
1/3
( Ns + M L)( √N + √M L ) . Assume that lim Ns =
s
0
Ns →∞
s ))−µ
converges (with proby (0 < y < 1). Then λmax (A(N
ν0
ability one) to the Tracy-Widom distribution of order 2 (W2 )
[14].
Note that for large Ns , µ and µ0 , ν and ν 0 are almost the
same, that is, the mean and variance for the largest eigenvalue
of real and complex matrix are almost the same. However, their
limit distributions are different.
Bai and Yin found the limit of the smallest eigenvalue [13]
as described in the following theorem.
L
Theorem 3. Assume that lim M
= y (0 < y < 1).
Ns →∞ Ns
√
2
2
Then lim λmin = ση (1 − y) .
Ns →∞
Based on the theorems, we have the following results:
λmax
≈
λmin
≈
√
ση2 p
( Ns + M L)2
Ns
√
ση2 p
( Ns − M L)2 .
Ns
(14)
(15)
The Tracy-Widom distributions were found by Tracy and
Widom (1996) as the limiting law of the largest eigenvalue of
certain random matrices [14]. Let F1 and F2 be the cumulative
distribution function (CDF) (sometimes simply called distribution function) of the Tracy-Widom distribution of order 1 and
order 2, respectively. There is no closed form expression for the
distribution functions. It is generally difficult to evaluate them.
Fortunately, based on numerical computation, there have been
tables for the functions [11]. Table 1 gives the values of F1 at
some points. It can be used to compute the F1−1 (y) at certain
points. For example, F1−1 (0.9) = 0.45, F1−1 (0.95) = 0.98.
Using the theories, we are ready to analyze the algorithms.
For real signal, the probability of false alarm of the MME de-
tection is
Ã
=P
ση2
Pf a = P (λmax > γλmin )
!
Ns
λmax (A(Ns )) > γλmin
³
´
p
√
≈ P λmax (A(Ns )) > γ( Ns − M L)2
Ã
!
√
√
λmax (A(Ns )) − µ
γ( Ns − M L)2 − µ
=P
>
ν
ν
à √
!
√
γ( Ns − M L)2 − µ
= 1 − F1
. (16)
ν
Hence we should choose the threshold such that
à √
!
√
γ( Ns − M L)2 − µ
1 − F1
= Pf a .
ν
This leads to
!
à √
√
γ( Ns − M L)2 − µ
F1
= 1 − Pf a ,
ν
or, equivalently,
√
√
γ( Ns − M L)2 − µ
= F1−1 (1 − Pf a ).
ν
(17)
(18)
(19)
From the definitions of µ and ν, we finally obtain the threshold
√
√
( Ns + M L)2
√
γ= √
( Ns − M L)2
Ã
!
√
√
( Ns + M L)−2/3 −1
· 1+
F1 (1 − Pf a ) .
(20)
(Ns M L)1/6
For complex signal, the only difference is that the function
F1 should be replaced by F2 , the CDF of the Tracy-Widom
distribution of order 2.
V.
S IMULATIONS
We require the probability of false alarm Pf a 6 0.1. Then
the threshold is found based on the formulae in Section IV.
For comparison, we also simulate the energy detection for the
same system. Energy detection needs the noise power as a priori. Unfortunately, in practice, the noise power uncertainty always presents [7, 3]. Due to the noise uncertainty, the estimated
(or assumed) noise power may be different from the real noise
power. Let the estimated noise power be σ̂η2 = αση2 . We define
the noise uncertainty factor (in dB) as B = max{10 log10 α}.
It is assumed that α (in dB) is evenly distributed in an interval
The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)
method
Pf a
EG-2 dB
0.497
EG-1.5 dB
0.496
EG-1 dB
0.490
EG-0.5 dB
0.470
EG-0dB
0.103
MME
0.077
Table 2: Probabilities of false alarm (M = 4, P = 2)
1
0.9
Probability of detection
0.9
0.8
Probability of detection
EG−2dB
EG−1.5dB
EG−1dB
EG−0.5dB
EG−0dB
MME
1
EG−2dB
EG−1.5dB
EG−1dB
EG−0.5dB
EG−0dB
MME
0.7
0.6
0.5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.4
0.8
−20
−15
−10
−5
0
1
1.2
1.4
1.6
Number of samples
5
SNR (dB)
Figure 1: Probability of detection (M = 4, P = 2)
[−B, B] [3, 5]. In practice, the noise uncertainty factor of receiving device is normally 1 to 2 dB [3, 5]. The threshold for
the energy detection is given in [3]. In the following, all the
results are averaged over 2000 Monte Carlo realizations.
First, randomly generated signals are used. We consider
a system with 4 receivers/antennas (M = 4) and 2 signals
(P = 2). The channel orders are N1 = N2 = 4 (5 taps).
Assume that all the channel taps are independent with equal
power. The smoothing factor is chosen as L = 6. 50000 samples are used for computing the sample covariance matrix. The
probabilities of detection (Pd ) for the MME method and energy
detection (with or without noise uncertainty) are shown in Figure 1, where and in the following “EG-x dB” means the energy
detection with x-dB noise uncertainty. If the noise variance is
exactly known (B = 0), the energy detection is pretty good.
However, if there is noise uncertainty, the detection probability
of the energy detection is much worse than that of the proposed
method.
The Pf a results are shown in Table 2 (note that the Pf a is
not related to the SNR because there is no signal). The Pf a for
the proposed method and the energy detection without noise
uncertainty meet the requirement (Pf a 6 0.1), but the Pf a
for the energy detection with noise uncertainty far exceeds the
limit. This means that the energy detection is very unreliable
in practical situations with noise uncertainty.
To test the impact of the number of samples, we fix the SNR
at -15dB and vary the number of samples from 7000 to 22000.
Figure 2 and Figure 3 show the Pd and Pf a , respectively. It
is seen that the Pd of the proposed algorithm and the energy
detection without noise uncertainty increases with the number
of samples, while that of the energy detection with noise uncertainty almost has no change. This means that the noise uncer-
1.8
2
2.2
4
x 10
Figure 2: Probability of detection (M = 4, P = 2, SNR=-15
dB)
tainty problem cannot be solved by increasing the number of
samples. For the Pf a , all the algorithms do not change much
with the varying number of samples (note that the threshold is
set based on Pf a , Ns and L).
Secondly, we test the algorithms based on the captured
ATSC DTV signals [15]. The real DTV signals (field measurements) are collected at Washington D.C., USA. The data
rate of the vestigial sideband (VSB) DTV signal is 10.762 samples/s. The sampling rate at the receiver is two times that rate.
The multipath channel and the SNR of the received signal are
unknown. In order to use the signals for simulating the algorithms at very low SNR, we need to add white noises to obtain
the various SNR levels [5]. In the simulations, the smoothing factor is chosen as L = 16. The number of samples used
for each test is Ns = 100000 (corresponding to 4.65 mili seconds sampling time). The results are averaged over 1000 tests
(for different tests, different data samples and noise samples
are used). Figure 4 gives the Pd results based on the DTV signal file WAS-003/27/01 (the receiver is outside and 48.41 miles
from the DTV station) [15]. The Pf a results are shown in Table 3. The Pf a of MME is well below 0.1. But the Pf a for the
energy detection with noise uncertainty far exceeds 0.1.
In summary, all the simulations show that the proposed
method works well without using information of the signal,
the channel and noise power. The energy detection is not reliable (low probability of detection and high probability of false
alarm) when there is noise uncertainty.
VI.
C ONCLUSIONS
Method based on the maximum and minimum eigenvalues of
the sample covariance matrix of the received signal has been
The 18th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’07)
method
Pf a
EG-2 dB
0.495
EG-1.5 dB
0.494
EG-1 dB
0.490
EG-0.5 dB
0.479
EG-0dB
0.107
MME
0.081
Table 3: Probabilities of false alarm (for real DTV signal detection, M = P = 1)
proposed. Latest random matrix theories have been used to set
the thresholds and obtain the probability of false alarm. The
method can be used for various signal detection applications
without knowledge of the signal, the channel and noise power.
0.55
EG−2dB
EG−1.5dB
EG−1dB
EG−0.5dB
EG−0dB
MME
0.5
Probability of false alarm
0.45
0.4
R EFERENCES
[1] J. Mitola and G. Q. Maguire, “Cognitive radios: making software radios
more personal,” IEEE Personal Communications, vol. 6, no. 4, pp. 13–
18, 1999.
0.35
[2] FCC, “Facilitating opportunities for flexible, efficient, and reliable spectrum use employing cognitive radio technologies, notice of proposed
rule making and order,” in FCC 03-322, Dec. 2003.
0.3
0.25
[3] A. Sahai and D. Cabric, “Spectrum sensing: fundamental limits and
practical challenges,” in Proc. IEEE International Symposium on New
Frontiers in Dynamic Spectrum Access Networks (DySPAN), (Baltimore, MD), Nov. 2005.
0.2
0.15
0.1
0.05
0.8
1
1.2
1.4
1.6
1.8
2
Number of samples
2.2
[4] R. Tandra and A. Sahai, “Fundamental limits on detection in low SNR
under noise uncertainty,” in WirelessCom 2005, (Maui, HI), June 2005.
4
x 10
Figure 3: Probability of false alarm (M = 4, P = 2)
[5] S. Shellhammer, G. Chouinard, M. Muterspaugh, and M. Ghosh, Spectrum sensing simulation model. http://grouper.ieee.org/groups/802/22
/Meeting documents/2006 July/22-06-0028-07-0000-SpectrumSensing-Simulation-Model.doc, July 2006.
[6] H. Urkowitz, “Energy detection of unkown deterministic signals,” Proceedings of the IEEE, vol. 55, no. 4, pp. 523–531, 1967.
[7] A. Sonnenschein and P. M. Fishman, “Radiometric detection of spreadspectrum signals in noise of uncertainty power,” IEEE Trans. on
Aerospace and Electronic Systems, vol. 28, no. 3, pp. 654–660, 1992.
[8] W. A. Gardner, “Spectral correlation of modulated signals: part i-analog
modulation,” IEEE Trans. Communications, vol. 35, no. 6, pp. 584–595,
1987.
1
EG−2dB
EG−1.5dB
EG−1dB
EG−0.5dB
EG−0dB
MME
0.9
Probability of detection
0.8
[9] W. A. Gardner, W. A. Brown, III, and C.-K. Chen, “Spectral correlation
of modulated signals: part ii-digital modulation,” IEEE Trans. Communications, vol. 35, no. 6, pp. 595–601, 1987.
0.7
[10] A. M. Tulino and S. Verdú, Random Matrix Theory and Wireless Communications. Hanover, USA: now Publishers Inc., 2004.
0.6
[11] I. M. Johnstone, “On the distribution of the largest eigenvalue in principle components analysis,” The Annals of Statistics, vol. 29, no. 2,
pp. 295–327, 2001.
0.5
0.4
[12] K. Johansson, “Shape fluctuations and random matrices,” Comm. Math.
Phys., vol. 209, pp. 437–476, 2000.
0.3
0.2
−20
−15
−10
−5
0
5
SNR (dB)
Figure 4: Probability of detection (based on DTV signal WAS003/27/01, M = P = 1)
[13] Z. D. Bai, “Methodologies in spectral analysis of large dimensional random matrices, a review,” Statistica Sinica, vol. 9, pp. 611–677, 1999.
[14] C. A. Tracy and H. Widom, “On orthogonal and symplectic matrix ensembles,” Comm. Math. Phys., vol. 177, pp. 727–754, 1996.
[15] V. Tawil, 51 captured DTV signal. http://grouper.ieee.org/groups/802/22
/Meeting documents/2006 May/Informal Documents, May 2006.