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Multistage Wiener filter aided MDL approach for wideband spectrum sensing in cognitive radio networks
Accepted Manuscript Regular paper Multistage Wiener Filter aided MDL Approach for Wideband Spectrum Sensing in Cognitive Radio Networks Haobo Qing, Heping Li, Nan Chu, Gang Liu, Yuanan Liu PII: DOI: Reference:
S1434-8411(16)30971-2 http://dx.doi.org/10.1016/j.aeue.2017.01.012 AEUE 51779
To appear in:
International Journal of Electronics and Communications
Received Date: Revised Date: Accepted Date:
26 October 2016 18 January 2017 18 January 2017
Please cite this article as: H. Qing, H. Li, N. Chu, G. Liu, Y. Liu, Multistage Wiener Filter aided MDL Approach for Wideband Spectrum Sensing in Cognitive Radio Networks, International Journal of Electronics and Communications (2017), doi: http://dx.doi.org/10.1016/j.aeue.2017.01.012
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Multistage Wiener Filter aided MDL Approach for Wideband Spectrum Sensing in Cognitive Radio Networks
Haobo Qing 1*, Heping Li1, Nan Chu 2, Gang Liu 3, Yuanan Liu4
1
The 29th Research Institute of China Electronics Technology Group Corporation, Chengdu,
China 2
China Internet Network Information Centre, Beijing, China
3
Key Laboratory of Information Coding and Transmission, Southwest Jiaotong University,
Chengdu, China 4
Beijing Key Laboratory of Work Safety Intelligent Monitoring, Beijing University of Posts
and Telecommunications, Beijing, China *
Email: [email protected]
Abstract: Wideband spectrum sensing in cognitive radio networks has gained significant research interests in the recent years. In the context of noise uncertainty, the noise variance has no access to accurate estimation which brings about an imprecise decision threshold. This paper investigates this subject from a novel perspective. Multistage Wiener filter (MSWF) is adopted in the conventional minimum description length (MDL) criterion to enhance the detection performance. By seeking the value that minimizes the MSWF aided MDL criterion, the number of occupied sub-channels in the wideband spectrum is determined. And then the accurate locations can be obtained based on the received energy of each channel. The
proposed scheme is robust to noise uncertainty since it requires no estimation of noise variance. Meanwhile, having no demand for the estimation of covariance matrix or its eigenvalue decomposition makes our approach computational attractive. In addition, the proposal partitions the array data into the cleaner signal and noise subspace components and can thereby improve the detection performance. Numerical results verify our approach and show that it is superior to other existing sensing algorithms in the previous works.
1. Introduction Cognitive radio Network (CRN) [1] is an emerging communication paradigm aiming at efficiently utilizing the scarce spectrum resources. It has now become a critical component for IEEE 802.22 standard for wireless regional area networks [2]. Since spectrum sensing is able to detect the spectrum holes, it is possible for secondary users (SUs) in CRNs to fulfil the dynamic spectrum access without causing harmful interference to the primary users (PUs). Among plenty of spectrum sensing algorithms, energy detection, matched filtering detection and cyclostationary detection are the most classical ones [3-5]. Cooperative spectrum sensing [6] can enhance the detection performance by combining the local information from multiple SUs using the above methods to jointly make the decision. Energy detection is the most widely applied sensing algorithm due to its flexibility and simplicity. The received signal energy is computed and then compared with a decision threshold, which involves with a preset false alarm probability. Energy detection has been proved to be optimal when there is no prior knowledge of PU signals but the noise variance. Nevertheless, it is quite vulnerable
to noise uncertainty and an inaccurate estimation of noise power will induce SNR wall, which severely deteriorates the detection performance [7]. 1.1. Related Work Wideband spectrum sensing provides an approach to sense multiple frequency bands [8]. The wideband spectrum can be divided into multiple nonoverlapping narrow subbands. Thus spectrum sensing can be carried out in each subband sequentially [9-11]. An early approach is to use a tunable narrowband bandpass filter at the radio frequency front-end to sense one narrow frequency band at a time [9]. Energy detection is carried out independently by performing narrow-band detection in an individual band each time [10]. Information theoretic criteria are applied to the spectrum sensing in the frequency domain, where the decision is made at each frequency band individually [11]. On the contrary, simultaneously detecting PU signals over multiple channels in a parallel manner is time-saving [12-16]. In order to jointly sense multiple frequency bands at a time, information theoretical criteria are applied to spectrum sensing as a model selection problem, which is robust to noise uncertainty [12-14]. The research in [12-13] judges the number of occupied channels firstly and then finds out the accurate locations of occupied channels as well as the vacant ones. A general approach that can be applied to any spectral representation is derived in [14] and the case for discrete Fourier transform is discussed in detail. A wavelet transform based method is come up with in [15] to estimate the power spectral density over a wide frequency range. Optimal multiband joint decision introduced in [16], where the threshold in each individual band is determined from wideband consideration, is made over multiple frequency bands by a bank of optimization problems.
1.2. Motivation and Principle Analysis This paper investigates wideband spectrum sensing for CRN from a novel perspective. Our approach searches over the whole spectrum bands simultaneously rather than an individual band each time. It first estimates the number of occupied channels and then determines the corresponding locations. The principle exploited in this paper is that the received energy of the occupied band is the superposition of signal and noise, whereas the energy of the vacant band is only contributed by noise. Thus energy of the occupied channel is larger than the vacant one. If there exists a prior knowledge that the number of occupied channels is K , those K channels with the largest energy are more likely to be the occupied ones. Therefore the estimation of the number of occupied channels becomes a critical issue. In this way, the wideband spectrum sensing problem is equivalent to the estimation of the number of occupied channels, which can refer to the issue of source number estimation. The authors in [17] originally applied Akaike information criterion (AIC) and minimum description length (MDL) to source enumeration. The number of signal sources in the information theoretical criteria is determined by minimizing the Kullback-Leibler distance between the hypothesized model and the observation data. In other words, for a given set of data and a class of probabilistic models, the model with the shortest description length should be selected as the best one. MDL is a consistent estimation while AIC yields an inconsistent estimation. Since AIC tends to overestimate the number of sources leading to a high false alarm probability, this paper focuses on the MDL criterion. Motivated by the idea of multistage orthogonal projection inherent in MSWF [18], this paper introduces a MSWF aided MDL approach to wideband spectrum sensing. The proposed scheme first partitions the
observation data into orthogonal signal and noise subspaces. Then, rather than the eigenvalues of covariance matrix, the variances of noise subspace components are utilized to encode the description length. Next, the number of occupied channels is attained by minimizing the description length. Eventually, each channel’s occupancy state is judged from its received energy. 1.3. Contribution The main contributions of this paper can be summarized as follows.
We put forward a MSWF aided MDL algorithm to perform wideband spectrum sensing. Since estimating noise power is no longer a necessity, our algorithm is not sensitive to the consequences arising from an inaccurate estimation. As a result, our approach is robust to noise uncertainty.
The MSWF aided MDL approach possesses the advantage of computational simplicity, particularly at a large number of antenna arrays or a small number of samples. This is due to the fact that the proposed method does not involve the estimation of covariance matrix or the eigenvalues decomposition. Actually, the former is replaced by some simpler vector-vector products while the latter is replaced by the variances of the noise subspace components, which can be directly generated in the procedures of multistage recursions.
Our scheme enjoys a more favourable performance than the original MDL criterion. Since the cross correlation between the observation data and reference signal is able to capture the signal information as well as suppress the additive noise, the cleaner signal and noise subspace components can be obtained when it is employed as the initial
information for the refinement procedure. As a result, the variances of the signal subspace components are well separated from the variances of the noise subspace components. In the meantime, the variances of the noise subspace components are clustered sufficiently closely. Therefore, our approach significantly reduces the likelihood of overestimating and underestimating the number of occupied channels, eventually leading to an improved performance. The following notations are used in this letter. Boldface letters are used to denote matrices and vectors. Superscripts (⋅)T , (⋅)H and (⋅)* stand for transpose, Hermitian transpose and complex conjugate, respectively. E [⋅] represents expectation operator. ⋅ denotes the vector norm. The identity matrix is shown by I .
2. System model In order to sense the wideband spectrum, the whole channel with the frequency bands from fl to fh is divided into Q nonoverlapping narrow bands. Each narrowband is with equal bandwidth W . K out of Q subbands are occupied by the PUs and the rest are the vacant ones. The illustration of the scenario is depicted in Figure 1. The spectrum usage is usually rather insufficient for CRNs and thus K < Q . Q − K Vacant Channels
K Occupied Channels
W fl
fh
1
2
3
Q − 2 Q −1
Q
Figure 1 Scenario illustration The binary hypothesis at subband i (1 ≤ i ≤ Q ) can be expressed as H0 : xi ( n ) = ε i ( n ) ,
(1)
H1: xi ( n ) = hi si ( n ) + ε i ( n ) ,
(2)
where H0 represents that the subband is idle and H1 denotes that the subband is busy. xi ( n ) is the received signal sample, si ( n ) stands for the transmitted PU signal obeying
independent Gaussian distribution with mean-zero and hi reflects the channel gain including the effects of multipath fading, path loss as well as time dispersion. ε i ( n ) is modeled as an independent additive white Gaussian noise (AWGN) with mean-zero and variance σ 2 . Under the assumption of AWGN, noise power spectral density is constant with frequency, so the noise powers are identical in different channels. It is assumed that the PU signal is uncorrelated with the noise. Consider that M uniform linear arrays are placed in the cognitive receiver. In order that the problem has a unique solution, K < M should be allowed throughout this paper due to the fact that PU signals occupy a subspace of dimension strictly smaller than the number of antenna arrays [19-20]. The received M × 1 signal vector from antenna arrays can be formulated as r ( n ) = r1 ( n ) ,L , rM ( n ) = ∑ i =1 s%i ( n ) ai + ε ( n ) = As% ( n ) + ε ( n ) , T
K
(3)
where ai = 1, exp ( j 2π dfi sin (θ i ) c ) ,L , exp ( j 2π ( M − 1) dfi sin (θi ) c )
T
.
(4)
θi is the angle of arrival of the i th PU signal si ( n ) relative to the array broadside. fi
represents the central frequency of the PU signals, c denotes the wave speed and d stands for the antenna spacing. A = [a1 , a 2 ,L , a K ] is taken as the array response matrix with ai the steering vector. When the i th PU is not transmitting, the i th column of A is excluded. ε ( n ) = ε1 ( n ) , ε 2 ( n ) ,L , ε M ( n )
s% ( n ) = s%1 ( n ) , s%2 ( n ) ,L , s%K ( n )
T
T
is the M × 1 additive noise vector in the antenna arrays. denotes the received PU signal vector suffering from channel
response, i.e. s%i ( n ) = hi si ( n ) .The covariance matrix of the received signals r ( n ) can be represented as Cr = E r ( n ) r H ( n ) = AE s% ( n ) s% H ( n ) A H + σ 2 I ,
(5)
2
H
= AC s A + σ I
where Cs = E s% ( n ) s% H ( n ) denotes the covariance matrix of received PU signal vector. In practical situations, however, the covariance matrix can only be estimated from a finite set of sampling numbers, which is called sample covariance matrix and is represented as ˆ = 1 C r N
N
∑ r ( n )r ( n ) , H
(6)
n =1
where N denotes the number of samples. Note that the sample covariance matrix approaches its ideal form as the number of samples tends to be infinite.
3. Proposed wideband spectrum sensing method 3.1. Conventional MDL Method
The classical MDL criterion [17] tells that given a set of observation data X = { x (1) , L , x ( N )}
and a family of probability densities f ( X Θ k ) with Θk denoting an
unknown parameter vector, the model with the minimum code length will fit the data best. More specifically, the shortest description length can be evaluated quantitatively as ˆ + 1 φ log N , MDLk = − log f X Θ k 2
(
)
(7)
where Θˆ k is the maximum likelihood (ML) estimate of Θ k and φ is the number of free adjusted parameters in Θk . Since the observations are regarded as independent Gaussian distributions with mean-zero, we can eventually obtain the form of MDL criterion as follows through a series of derivations [17] M 1 ( M −k ) ∏ λi MDLk = − log i = k +1 M 1 M − k ∑ λi i = k +1
(M −k ) N
1 + k ( 2M − k ) log N , 2
(8)
where λi represents the eigenvalues of sample covariance matrix Cˆ r . The estimated number of sources, i.e. the number of occupied channels, is determined as the value of k ∈ {0,L , M − 1} for which the MDL is minimized.
We can see that the traditional MDL criterion suffers from a heavy burden of computational complexity since it requires the estimation of covariance matrix and eigenvalue decomposition. Therefore, the conventional MDL criterion is unsuitable in some practical situations, especially when the number of sources needs to be tracked in an online manner. To reduce the complexity, an MSWF aided MDL scheme is proposed in this paper. Details are given below. 3.2. MSWF aided MDL Scheme
Before introducing the new MDL method, we take another look at the uniform linear arrays in the cognitive receiver, i.e., Eq. (3). We can say that the first antenna array generates the reference signal d 0 ( n ) = r1 ( n ) = s%T ( n ) 1 + ε1 ( n )
(9)
where 1 = [1,1,L,1]T , while the remaining ones produce the new observation data r0 ( n ) = r2 ( n ) , r3 ( n ) ,L , rM ( n ) = A M −1 (θ ) s% ( n ) + ε 0 ( n )
T
,
(10)
= ∑ i =1 s%i ( n ) a M −1 (θ i ) + ε 0 ( n ) K
T
where ε 0 ( n ) = ε 2 ( n ) , ε 3 ( n ) ,L , ε M ( n ) . A M −1 (θ ) and a M −1 (θi ) are composed of the last M − 1 rows of A (θ ) and a (θ i ) , respectively. Physically, they can be viewed as the effect causing by removing one antenna array, which sacrifice one array gain but still contain the information of all the PU signals. The cross-correlation between the reference signal and the observation data is R r0 d0 = E r0 ( n ) d 0* ( n ) = A M −1 (θ ) C s 1 .
(11)
It is indicated that the cross-correlation is capable of capturing the signal information. Meanwhile, having no noise term in it means that the additive noise has been efficiently cancelled. In this sense, the cross-correlation can be regarded as a matched filter, which extracts the signal from the background noise. A normalized form can be obtained as
H1 =
R r0 d0 R r0 d0
.
(12)
With the above initial information, now we proceed to the multistage recursion procedure similar to the MSWF [21]. Given a reference signal d 0 ( n ) , the MSWF partitions
the observation data r0 ( n ) with the normalized matched filter H1 into two orthogonal subspaces: the desired signal d i ( n ) and the observation data ri ( n ) at the i th stage. The desired signal d i ( n ) is attained by filtering the observation data ri −1 ( n ) with the matched filter H i . The observation data ri ( n ) is partitioned stage by stage. A set of detailed formulas are shown as follows d i ( n ) = H iH ri −1 ( n )
(13)
ri ( n ) = ri −1 ( n ) − Hi d i ( n ) = ri −1 ( n ) − H i H iH ri −1 ( n )
(14)
= Bi ri −1 ( n )
Hi =
R ri −1di −1 R ri −1di −1
=
E ri −1 ( n ) d i*−1 ( n )
(15)
E ri −1 ( n ) d i*−1 ( n )
where Bi = I M −1 − Hi HiH spans the nullspace of R r
i −1 d i−1
and H i , i.e. Bi R r
i−1 d i −1
= 0 and Bi Hi = 0 .
Physically, the blocking matrix Bi annihilates the signal components in the subspace of the cross-correlation R r
i −1 d i−1
and the matched filter H i . An example of a four-stage MSWF block
diagram is presented in Fig. 2, where ei ( n ) is the estimation errors and ωi is the scalar weights, defined as
ωi =
E d i −1 ( n ) ei∗ ( n )
(16)
2 E ei ( n )
ei −1 ( n ) = d i −1 ( n ) − ωi∗ ei ( n )
(17)
d0 ( n )
r0 ( n )
H1
B1
+
d1 ( n ) r1 ( n )
H2
B2
+
d2 ( n) r2 ( n )
H3
+
d3 ( n ) +
ω4
H4
B3
r3 ( n )
d4 ( n )
−
∑
e3 ( n )
∑ ω3 −
e2 ( n )
∑ ω2
−
e1 ( n )
∑ ω1
−
e0 ( n )
Figure 2: The nested chain of a four-stage MSWF
The signal and noise subspaces can be spanned by the matched filter bank in the following manner.
where S and N
S = span {H1 , H 2 ,L , H K }
(18)
N = span {H K +1 , H K + 2, L , H M }
(19)
denote the signal subspace and noise subspace, respectively. As a
consequence, by filtering the initial observation data r0 ( n ) with the matched filter bank S and N , the components in the signal and noise subspaces can be obtained, respectively T
rs ( n ) = d1 ( n ) , d 2 ( n ) ,L , d K ( n ) = TsH r0 ( n ) T
rn ( n ) = d K +1 ( n ) , d K + 2 ( n ) ,L , d M ( n ) = TnH r0 ( n )
(20) (21)
in which the filter bank matrix is defined as K −1 Ts = H1 , B1H 2 ,L , ∏ Bi H K = [ H1 , H 2 ,L , H K ] i =1
(22)
K +1 M −1 K Tn = ∏ Bi H K +1 , ∏ Bi H K + 2 ,L , ∏ Bi H M = [ H K +1 , H K + 2 , L , H M ] i =1 i =1 i =1
(23)
The second equations hold due to the orthogonality between the matched filters, namely H iH H j = 0 ( i ≠ j ) ,
which has been proved in [22] through an induction manner. Accordingly,
the covariance matrix of noise subspace components can be described as Cn = E rn ( n ) rnH ( n )
(24)
N ˆ = 1 C rn ( n ) rnH ( n ) ∑ n N n =1
(25)
with the sample form
The elements in the aforementioned covariance matrix of noise subspace can be further derived as E d i ( n ) d ∗j ( n ) = E H iH ri −1 ( n ) r jH−1 ( n ) H j = H iH E ri −1 ( n ) r jH−1 ( n ) H j H 1 1 H = H E ∏ B k r0 ( n ) r0 ( n ) ∏ Bl H j k =i −1 l = j −1
(26)
H i
H
1 1 = H iH ∏ B k E r0 ( n ) r0H ( n ) ∏ B l H j k = i −1 l = j −1 H
1 1 = H I M −1 − ∑ H k H kH ( A M −1 (θ ) R s A MH −1 (θ ) + σ 2 I M −1 ) I M −1 − ∑ H l H lH H j k =i −1 l = j −1 H i
= H iH A M −1 (θ ) R s A MH −1 (θ ) H j + σ 2 H iH H j
Owing to the orthogonality as well as the normalization between matched filters, namely H iH H j = 0 ( i ≠ j )
and H iH H j = 1 ( i = j ) ,
and the orthogonality between noise subspace and
steering matrix, it can be drawn that E d i ( n ) d ∗j ( n ) = 0 ( i, j = K + 1, K + 2, L , M , i ≠ j )
E d i ( n ) di∗ ( n ) = σ 2
( i = K + 1, K + 2,L, M )
(27) (28)
In other words, the last M − K desired signals of the MSWF are uncorrelated with each other and their variances equal to the noise variance. Consequently, the covariance matrix of noise subspace components is a diagonal matrix with the following form Cn = diag δ d2K +1 , δ d2K +2 ,L , δ d2M
(29)
where δ d2 = E di ( n ) d i∗ ( n ) . It is quite important to note that the first K desired signals of the i
MSWF do not meet the aforementioned equations since the orthogonality between signal subspace and steering matrix does not hold.
Following the idea of MSWF, [18] has shown that the variances of noise subspace components can replace the eigenvalues of covariance matrix as for the calculation of MDL criterion. Substituting the variances of noise subspace components into Eq. (8), an MSWF aided MDL criterion is obtained as M ˆ2 1 (M −k ) ∏ δ di MDL _ MSWFk = − log i = k +1 M 1 ˆ2 M − k ∑ δ di i = k +1
( )
( M −k ) N
1 + k ( 2 M − k ) log N 2
(30)
N in which δˆd2 = 1 N ∑ n =1 di ( n )di* ( n ) is the ML estimate of δ d2 = E di ( n ) d i* ( n ) . The number of i
i
occupied channels is thereby Kˆ = arg
min
k = 0,1,L, M −1
MDL _ MSWFk .
(31)
With the information of the estimated number of occupied channels, we now set out to find the accurate positions of the occupied channels in the wideband CRNs. The received energy of each channel is computed as ϕi =
1 N
N
∑ x ( n) i
2
i = 1, 2,L , Q .
(32)
n =1
As we have pointed out before, the occupied channels own higher received energy than the idle ones. Therefore, channels with the highest Kˆ received energy would be set as the occupied ones. In summary, the steps of the proposed MSWF aided MDL approach are described in Table 1. Table 1: The proposed MSWF aided MDL algorithm for spectrum sensing
Step 1: Initialize the MSWF and produce the reference signal
d0 ( n)
, observation data
r0 ( n )
, cross-correlation
and normalized matched filter
R r0d0
H1 ,
i.e., Eq.
(9)~(12); Step 2: Perform multistage recursions following the procedure of MSWF, and generate the desired signal
di ( n ) ,
and normalized matched filter
Hi ,
observation data
ri ( n ) ,
blocking matrix
Bi
i.e., Eq. (13)~(15);
Step 3: Partition the signal as well as noise subspace, and compute the variances of noise subspace components; Step 4: Construct the MSWF aided MDL criterion, i.e., Eq. (30); Step 5: Estimate the number of occupied channels, i.e., Eq. (31); Step 6: Select channels with the highest Kˆ received energy as the occupied ones.
3.3. Discussions of the Proposed Scheme
In our approach, several modeling functions under different hypotheses are constructed to denote the number of busy channels. The judgment of the true hypothesis is equivalent to finding the variable that minimizes these modeling functions, and thus the number of occupied channels is determined. Afterwards, the channels with maximum received energy are considered to be occupied by the PUs.
Distinct from the traditional spectrum sensing methods, the proposed scheme investigates this problem in a multistage recursion procedure inspired by the MSWF. The cross correlation between the reference signal and the observation data is exploited to capture the signal information. Note that it is capable of suppressing the background noise since
no noise item is involved. The cross correlation is then utilized to generate the signal and noise subspaces, which eventually leads to the enhanced performance.
Our approach is robust to noise uncertainty since it does not involve the estimation of noise variance. Thus the impact induced by an inaccurate estimation would be totally eliminated. Energy detection with noise uncertainty is provided in the following simulation section to validate the conclusion.
Note that the idea in this paper is different from interference alignment (IA), which is a promising technique for interference management and can be applied to spectrum sharing in CRN [23-24]. In IA, the channel state information (CSI) is required to construct a suitable precoding matrix, which should be found at each transmitter to constrain all interference into certain subspaces at each receiver. In this way, IA eliminates not only the interference to PUs but also the mutual interference among SUs. On the contrary, our approach focuses on the subject of spectrum sensing rather than spectrum sharing. The CSI is not needed to perform the detection. The feedback from the receiver to the transmitter is not required, either. 3.4. Complexity Analysis As we have pointed out, the proposed MSWF aided MDL scheme is more
computational attractive than the traditional MDL approach, particularly at a large number of antenna arrays or a small number of samples. This is because the traditional MDL principle, i.e., Eq. (8), resorts to the eigenvalues of sample covariance matrix. Hence, the estimation of sample covariance matrix and the eigenvalue decomposition are quite necessary. The former requires O ( M 2 N ) operations since its received M × 1 signal vector r ( n ) is averaged over
N samples, i.e., Eq. (6). The latter generally requires O ( M 3 ) operations due to a M × M
sample covariance matrix. On the other hand, the dominant computational cost of the proposed MSWF aided MDL scheme comes from the multistage recursion procedure, i.e., Eqs. (13)-(15), which requires vector-vector products. O ( M − 1) is needed for each sample and each recursion, thereby the total computational load is only around O ( ( M − 1) MN ) multiplications and additions. The aforementioned complexity analysis provides a brief overview. Now we proceed to a detailed description. Taking a further look at the two MDL criteria, i.e., Eq. (8) and Eq. (30), we can find that the main differences come from the eigenvalues λi and the estimated variances of desired signals δˆd2 . Since calculating the MDL criteria as well as determining i
the locations of occupied channels are similar for the two algorithms, we omit the complexity comparison for these steps in this subsection. Meanwhile,
we focus on the
multiplication/division (MD) and addition/subtraction (AS). Some simple operations, e.g., Hermitian transpose, complex conjugate, are neglected as well. The calculation of sample covariance matrix, i.e., Eq. (6), needs M 2 + M 2 N MDs as well as M 2 ( N − 1) ASs. The corresponding eigenvalue decomposition demands nearly 6M 3 MDs and 6M 3 ASs. These manipulations are complex. One complex MD operation needs 6 floating-point operations (flops) and one complex AS operation needs 2 flops. Thus the total number of flops needed in the conventional MDL algorithm is Flops _ MDL = 6 ( M 2 + M 2 N + 6 M 3 ) + 2 ( M 2 N − M 2 + 6 M 3 ) = 48 M 3 + 8 M 2 N + 4M 2
.
(33)
The initialization of the proposed MDL approach produces the reference signal d 0 ( n ) and the observation data r0 ( n ) . Calculating the cross-correlation R r d between the reference 0 0
signal and the observation data, i.e., Eq. (11), needs ( M − 1)( N + 1) MDs as well as
( M − 1)( N − 1) ASs. Accordingly, the matched filter M −2
H1
calls for 2 ( M − 1) MDs as well as
ASs. Then the desired signal d1 ( n ) at the 1th stage is attained through M − 1 MDs
and M − 2 ASs while its orthogonal component r1 ( n ) is obtained by M − 1 MDs and M − 1 ASs. In order to get di ( n )( i = 1, 2,L , M ) , M successive recursions have to be performed, which
indicates
that
ri ( n )( i = 1,2,L, M − 1) ,
calculating
R ri di ( i = 0,1,L, M − 1)
and
Hi ( i = 1,2,L, M ) becomes a necessity. In addition, computing the estimated variances of the
noise subspace components calls for M ( N + 1) MDs and M ( N − 1) ASs. Since these procedures are complex, the total number of flops needed in the proposed MDL algorithm is Flops _ MDL _ MSWF = 6 ( M ( M − 1)( N + 5 ) − ( M − 1) + M ( N + 1) ) +
2 ( M ( M − 1)( N + 2 ) + 2 M − ( M − 1) + M ( N − 1) ) . 2
(34)
2
= 8M N + 34M − 34 M + 8
To compare the complexity, we subtract Eq. (34) from Eq. (33) Flops _ MDL − Flops _ MDL _ MSWF
= ( 48M 3 + 8M 2 N + 4M 2 ) − ( 8M 2 N + 34 M 2 − 34 M + 8 ) = 48M 3 − 30 M 2 + 34 M − 8
.
(35)
= 2M 2 ( 24 M − 15 ) + 2 (17 M − 4 )
It can be readily seen from Eq. (35) that our proposed MDL scheme saves computational cost.
7
10
Flops
Traditional MDL Criterion Proposed MDL Scheme
6
10
5
10 100
200
300
400
500
600
700
800
900
1000
N
Figure 3: Complexity comparison versus the number of samples (M=16)
To explicitly illustrate the computational flops, we depict the complexity comparisons in the case of M = 16 in Fig. 3 and N = 100 in Fig. 4, respectively. Moreover, we define the ratio of complexity reduction as Φ=
Flops _ MDL − Flops _ MDL _ MSWF Flops _ MDL
,
(36)
which reflects the percentage of the flops saved by our proposed MDL algorithm compared to the original MDL criterion. Fig. 5 plots the ratio of complexity reduction versus the number of antenna arrays at various numbers of samples. We can observe from the figures that the proposed MDL scheme is more computationally efficient than the conventional one, particularly at a large number of antenna arrays or a small number of samples.
7
10
Traditional MDL Criterion Proposed MDL Scheme
6
Flops
10
5
10
4
10
4
8
12
16
20
24
28
32
M
Figure 4: Complexity comparison versus the number of antenna arrays (N=100)
1 N=100 N=500 N=1000
The Ratio of Complexity Reduction
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
4
8
12
16
20
24
28
M
Figure 5: The ratio of complexity reduction versus the number of antenna arrays
32
4. Numerical results Considering a wideband CRN with M = 16 uniform linear antenna arrays, there are Q = 20 channels in total to be detected among which K = 4 channels are occupied by the
PUs. Four PU signals in the occupied channels with central frequencies {565, 585, 615, 635 MHz} impinge upon the antenna arrays from distinct directions {-4°, 2°, 6°, 10°}, respectively. The antenna spacing is d = 0.25m . Fig. 6 depicts the normalized power spectrum of the PU signals for occupied channels. Note that PUs’ signal powers generally vary in various channels, and then SNR in the following simulations refers to the average SNR, namely, the ratio of average received signal power to noise power. One thousand Monte Carlo trials have been carried out independently in each realization. Detection probability Pd and false alarm probability Pf are two key parameters for performance evaluation. In IEEE 802.22 standard for CRNs [2], it is required that Pd ≥ 0.9 and Pf ≤ 0.1 . Since CRN aims at effectively utilizing the spectrum holes on the premise of not interfering PUs, the goal of our approach is to maximize the detection probability, and meanwhile, the false alarm probability maintains to an acceptable extent. The simulation parameters are summarized in Table 2.
1 0.9
Normalized Power Spectrum
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 540
560
580
600 620 f (MHz)
640
660
680
Figure 6: The normalized power spectrum of the PU signals for occupied channels Table 2: Simulation parameters Parameters
Setting
The number of subchannels (Q)
20
The number of occupied subchannel (K)
4
The number of antenna arrays (M)
16
Channel between PUs and SUs
AWGN, Rayleigh fading
Noise uncertainty
0,1,2 dB
Central frequency (f)
{565, 585, 615, 635 MHz}
Impinge direction ( θ )
{-4°, 2°, 6°, 10°}
Antenna spacing (d)
0.25m
Monte Carlo trials
1000
Scaling parameters
Pd, Pf
4.1. Detection Performance with Varying Channel Condition Firstly, this section validates the performance of our approach versus SNR. Energy detection is presented for comparison. There are two cases considered here for energy detection: the ideal one with false alarm probability setting to be 0.1 and energy detection with noise uncertainty. Fig. 7 shows the probability of detection versus SNR under the condition that the number of samples is 500, and the corresponding probability of false alarm is depicted in Table 3. Note that Pf is not related to the SNR since there is no signal, and thus, remains constant over various SNR. For the clarity of representation, the term ED is short for the ideal energy detection and ED (x dB) means energy detection with x dB noise uncertainty. It follows from the figure and the table that noise uncertainty severely deteriorates the detection performance of energy detection. Furthermore, the false alarm probability is intolerable since it far exceeds the acceptable range for CRNs. Although the ideal energy detection holds the best detection probability and a reasonable false alarm probability, however, we generally have no access to an accurate noise variance. That is to say, the ideal energy detection cannot be achieved. On the contrary, meeting the requirement of false alarm probability for CRNs, our approach surpasses energy detection with noise uncertainty in detection probability. This is also in accordance with the theoretic analysis given in the above section that our proposed scheme does not resort to the estimation of noise variance. Thereby noise uncertainty has no influence on our method while energy detection is greatly affected.
1 0.9 0.8 0.7
Pd
0.6 0.5 0.4 0.3 Pd Pd Pd Pd
0.2 0.1 0 -15
-13
-11
-9
-7 SNR (dB)
-5
of ED of ED (1dB) of ED (2dB) of Proposed Scheme -3
-1
0
Figure 7: Detection probability of our approach and energy detection versus SNR (N=500)
Table 3: False alarm probability of various methods (N=500) ED
ED (1dB)
ED (2dB)
Proposed Scheme
0.1000
0.4028
0.3636
0.0920
4.2. Detection Performance with Different Timing Requirements Now this section draws attention to the impact of the number of samples by fixing SNR at -10 dB when the channel obeys Rayleigh distributions [25]. The detection probability and false alarm probability are depicted in Fig. 8. As is shown in the figure, when the number of samples exceeds a certain value, the detection performance of energy detection with noise uncertainty is no longer improved whereas the proposed scheme is still enhanced. Due to SNR wall, energy detection with noise uncertainty is even not consistent since it cannot reach
Pd = 1 as the number of samples grows. Moreover, the false alarm probability of energy
detection with noise uncertainty is not endurable for CRNs, particularly at a large sampling number. Our approach outperforms energy detection with noise uncertainty both in Pd and Pf under Rayleigh fading channel. 1 Pd of ED Pf of ED Pd of ED (1dB) Pf of ED (1dB) Pd of ED (2dB) Pf of ED (2dB) Pd of Proposed Scheme Pf of Proposed Scheme
0.9 0.8 0.7 Pd
Pd and Pf
0.6 0.5 0.4 0.3
Pf
0.2 0.1 0 50
500
1000
1500 N
2000
2500
3000
Figure 8: Detection probability and false alarm probability of our approach and energy detection versus the number of samples under Rayleigh fading channel (SNR=-10 dB)
4.3. Detection Performance against the Existing Algorithms To further demonstrate the performance of our approach, the MDL criteria based wideband spectrum sensing algorithms for CRNs in [12] are adopted for comparison. Note that our approach as well as the algorithms in [12] are the improvements of the classical MDL criterion, and the rationales of them are the same, namely to judge the number of occupied channels firstly and then find out the accurate locations of occupied channels as
well as the vacant ones. The first scheme in [12], noted as MDL criterion A, performs ML estimation on each hypotheses model, and then directly applies the MDL criterion into the estimation of the number of occupied channels, and finally finds out the indexes of occupied channels. The second scheme in [12], noted as MDL criterion B, however, utilizes the sorted sample power of each channel to replace the eigenvalues. In this way the ML estimation is avoided and the MDL criterion is redesigned. The detection probability versus SNR is drawn in Fig. 9 and Table 4 presents the false alarm probability. We can see that our approach reaches Pd = 0.9 at a lowest SNR. Although our Pf is worse, it is not pessimistic since it still satisfies the requirement of CRNs. Therefore, our approach possesses the best detection performance among the algorithms. As we mentioned before, this is because the first four matched filters H i ( i = 1, 2, 3, 4 ) are capable of capturing signal information while excluding a large portion of noise. On the contrary, their orthogonal complements H i ( i = 5, 6,L ,16 ) have the ability to mitigate the signal subspace components from the noisy data. Hence, the estimated variances of the first four desired signals δˆd2 (i = 1, 2,3, 4) , namely the powers of i
signals, are well separated from the estimated variances of the desired signals after the fourth stage δˆd2 ( i = 5, 6,L ,16 ) , namely the powers of noises. Simultaneously, the variances of the i
desired signals after the fourth stage are clustered sufficiently closely. Consequently, the cleaner signal subspace and noise subspace can be obtained. Thereby, the probabilities of overestimating and underestimating the number of occupied channels are greatly reduced. Thus, the proposed method is capable of estimating the number of occupied channels more precisely and achieving a better performance.
1 0.9 0.8 0.7
Pd
0.6 0.5 0.4 0.3 0.2
Pd of MDL Criterion A Pd of MDL Criterion B Pd of Proposed Scheme
0.1 0 -10
-8
-6
-4
-2 SNR (dB)
0
2
4
5
Figure 9: Detection probability of our approach and the existing MDL algorithms versus SNR (N=200)
Table 4: False alarm probability of various methods (N=200) MDL Criterion A
MDL Criterion B
Proposed Scheme
0.0018
0.0648
0.0953
5. Conclusions This paper proposes an MSWF aided MDL approach for wideband spectrum sensing in CRNs. The proposed method is able to generate signal and noise subspaces. The signal subspace captures the signal information and excludes a large portion of additive noise. On the contrary, the noise subspace is capable of attaining noise components, thereby achieving a
clean noise subspace. This implies that the variances of the signal subspace components are well separated from the variances of the noise subspace components. In the meantime, the variances of the noise subspace components are clustered sufficiently closely. Consequently, our approach is able to reduce the likelihood of overestimating and underestimating the number of occupied channels, and eventually enhance the detection performance. Due to the fact that there is no need to estimate the noise power, our approach is robust to noise uncertainty. Meanwhile, the proposed method reduces the computational complexity since it does not rely on the estimation of covariance matrix or its eigenvalue decomposition, particularly at a large number of antenna arrays or a small number of samples. The proposal’s performance is demonstrated via numerical results.
Acknowledgments The authors would like to thank editors and the anonymous reviewers for their careful review and constructive comments which improve the quality of this paper. This work was supported in part by the National Key Basic Research Program of China (973 Program, no. 2014CB339900), National Natural Science Foundation of China (no. 61571373), Important National Science and Technology Specific Projects (no. 2012ZX03003001-004), National High Technology Research and Development Research Program of China (863 Program).
References [1] Haykin S, Setoodeh P. Cognitive radio networks: the spectrum supply chain paradigm. IEEE Trans Cog Commun Net 2015; 1(1):3-28 [2] Stevenson CR, Chouinard G, Lei Z, Hu W, Shellhammer SJ, Caldwell W. IEEE 802.22: the first cognitive radio wireless regional area network standard. IEEE Commun Mag 2009; 47(1):130-8 [3] Yang L, Chen Z, Yin F. Cyclo-energy detector for spectrum sensing in cognitive radio. Int J Electron Commun (AEU) 2012; 66(1):89-92 [4] Sadeghi H, Azmi P, Arezumand H. Cyclostationarity-based cooperative spectrum sensing over imperfect reporting channels. Int J Electron Commun (AEU) 2012; 66(10):833-40 [5] Ren X, Chen C. Spectrum sensing algorithm based on sample variance in multi-antenna cognitive radio systems. Int J Electron Commun (AEU) 2016; 70(12):1601-9 [6] Chen X, Chen H, Meng W. Cooperative communications for cognitive radio networksfrom theory to applications. IEEE Commun Surveys & Tutorials 2014; 16(3): 1180-92 [7] Tandra R, Sahai A. SNR walls for signal detection. IEEE J Sel Top Sig Process 2008; 2(1): 4-17 [8] Sun H, Nallanathan A, Wang C-X, Chen Y. Wideband spectrum sensing for cognitive radio networks: a survey. IEEE Wirel Commun Mag 2013; 20(2): 74-81 [9] Sahai A, Cabric D. A tutorial on spectrum sensing: fundamental limits and practical challenges. Proc IEEE Int Symp Dynamic Spectrum Access Networks 2005:1-5
[10] Taherpour A, Gazor S, Nasiri-Kenari M. Invariant wideband spectrum sensing under unknown variances. IEEE Trans Wirel Commun 2009; 8(5): 2182-6 [11] Jing Y, Yang X, Ma L, Ma J, Niu B. Blind multiband spectrum sensing in cognitive radio network. Proc IEEE Int Conf Consumer Electron Commun Net (CECNet) 2012: 2442-5 [12] Liu S, Shen J, Zhang R, Zhang Z, Liu Y. Information theoretic criterion-based spectrum sensing for cognitive radio. IET Commun 2008; 2(6): 753-62 [13] Qing H, Liu Y, Xie G. Smart antennas aided wideband detection for spectrum sensing in cognitive radio networks. Electron Lett 2014; 50(7): 490-2 [14] Mariani A, Giorgetti A, Chiani M. Wideband spectrum sensing by model order selection. IEEE Trans Wirel Commun 2015; 14(12): 6710-21 [15] Tian Z, Giannakis GB. A wavelet approach to wideband spectrum sensing for cognitive radios. Proc Int Conf Cognitive Radio Oriented Wireless Networks and Commun (CROWNCOM) 2006: 1-5 [16] Quan Z, Cui S, Sayed AH, Poor HV. Optimal multiband joint detection for spectrum sensing in cognitive radio networks. IEEE Trans Signal Process 2009: 57 (3): 1128-40 [17] Wax M, Kailath T. Detection of signals by information theoretic criteria, IEEE Trans Acoust Speech Signal Process 1985, 33(2): 387-92 [18] Huang L, Wu S. Low-complexity MDL method for accurate source enumeration. IEEE Signal Proc Let 2007, 14(9): 581-4 [19] Zhang R, Lim TJ, Liang YC, Zeng Y. Multi-antenna based spectrum sensing for cognitive radios: a GLRT approach. IEEE Trans Commun 2010, 51(1): 84-8
[20] Qing H, Liu Y, Xie G, Liu K, Liu F. Blind multiband spectrum sensing for cognitive radio systems with smart antennas. IET Commun 2014, 8(6): 914-20 [21] Goldstein JS, Reed IS, Scharf LL. A multistage representation of the Wiener filter based on orthogonal projections. IEEE Trans Inform Theory 1998, 44(7): 2943-59 [22] Honig ML, Xiao W. Performance of reduced-rank linear interference suppression. IEEE Trans Inform Theory 2001, 47(5): 1928-46 [23] Zhao N, Yu F.R, Sun H, Li M. Adaptive power allocation schemes for spectrum sharing in interference alignment (IA)-based cognitive radio networks. IEEE Trans Veh Technol 2016, 65(5): 3700-14 [24] Li X, Zhao N, Sun Y, Yu F.R. Interference alignment based on antenna selection with imperfect channel state information in cognitive radio networks. IEEE Trans Veh Technol 2016, 65(7): 5497-511 [25] Kumar N, Bhatia V, Dixit D. Performance analysis of QAM in amplify-and-forward cooperative communication networks over Rayleigh fading channels. Int J Electron Commun (AEU) 2017; 72:86-94
S1434-8411(16)30971-2 http://dx.doi.org/10.1016/j.aeue.2017.01.012 AEUE 51779
To appear in:
International Journal of Electronics and Communications
Received Date: Revised Date: Accepted Date:
26 October 2016 18 January 2017 18 January 2017
Please cite this article as: H. Qing, H. Li, N. Chu, G. Liu, Y. Liu, Multistage Wiener Filter aided MDL Approach for Wideband Spectrum Sensing in Cognitive Radio Networks, International Journal of Electronics and Communications (2017), doi: http://dx.doi.org/10.1016/j.aeue.2017.01.012
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Multistage Wiener Filter aided MDL Approach for Wideband Spectrum Sensing in Cognitive Radio Networks
Haobo Qing 1*, Heping Li1, Nan Chu 2, Gang Liu 3, Yuanan Liu4
1
The 29th Research Institute of China Electronics Technology Group Corporation, Chengdu,
China 2
China Internet Network Information Centre, Beijing, China
3
Key Laboratory of Information Coding and Transmission, Southwest Jiaotong University,
Chengdu, China 4
Beijing Key Laboratory of Work Safety Intelligent Monitoring, Beijing University of Posts
and Telecommunications, Beijing, China *
Email: [email protected]
Abstract: Wideband spectrum sensing in cognitive radio networks has gained significant research interests in the recent years. In the context of noise uncertainty, the noise variance has no access to accurate estimation which brings about an imprecise decision threshold. This paper investigates this subject from a novel perspective. Multistage Wiener filter (MSWF) is adopted in the conventional minimum description length (MDL) criterion to enhance the detection performance. By seeking the value that minimizes the MSWF aided MDL criterion, the number of occupied sub-channels in the wideband spectrum is determined. And then the accurate locations can be obtained based on the received energy of each channel. The
proposed scheme is robust to noise uncertainty since it requires no estimation of noise variance. Meanwhile, having no demand for the estimation of covariance matrix or its eigenvalue decomposition makes our approach computational attractive. In addition, the proposal partitions the array data into the cleaner signal and noise subspace components and can thereby improve the detection performance. Numerical results verify our approach and show that it is superior to other existing sensing algorithms in the previous works.
1. Introduction Cognitive radio Network (CRN) [1] is an emerging communication paradigm aiming at efficiently utilizing the scarce spectrum resources. It has now become a critical component for IEEE 802.22 standard for wireless regional area networks [2]. Since spectrum sensing is able to detect the spectrum holes, it is possible for secondary users (SUs) in CRNs to fulfil the dynamic spectrum access without causing harmful interference to the primary users (PUs). Among plenty of spectrum sensing algorithms, energy detection, matched filtering detection and cyclostationary detection are the most classical ones [3-5]. Cooperative spectrum sensing [6] can enhance the detection performance by combining the local information from multiple SUs using the above methods to jointly make the decision. Energy detection is the most widely applied sensing algorithm due to its flexibility and simplicity. The received signal energy is computed and then compared with a decision threshold, which involves with a preset false alarm probability. Energy detection has been proved to be optimal when there is no prior knowledge of PU signals but the noise variance. Nevertheless, it is quite vulnerable
to noise uncertainty and an inaccurate estimation of noise power will induce SNR wall, which severely deteriorates the detection performance [7]. 1.1. Related Work Wideband spectrum sensing provides an approach to sense multiple frequency bands [8]. The wideband spectrum can be divided into multiple nonoverlapping narrow subbands. Thus spectrum sensing can be carried out in each subband sequentially [9-11]. An early approach is to use a tunable narrowband bandpass filter at the radio frequency front-end to sense one narrow frequency band at a time [9]. Energy detection is carried out independently by performing narrow-band detection in an individual band each time [10]. Information theoretic criteria are applied to the spectrum sensing in the frequency domain, where the decision is made at each frequency band individually [11]. On the contrary, simultaneously detecting PU signals over multiple channels in a parallel manner is time-saving [12-16]. In order to jointly sense multiple frequency bands at a time, information theoretical criteria are applied to spectrum sensing as a model selection problem, which is robust to noise uncertainty [12-14]. The research in [12-13] judges the number of occupied channels firstly and then finds out the accurate locations of occupied channels as well as the vacant ones. A general approach that can be applied to any spectral representation is derived in [14] and the case for discrete Fourier transform is discussed in detail. A wavelet transform based method is come up with in [15] to estimate the power spectral density over a wide frequency range. Optimal multiband joint decision introduced in [16], where the threshold in each individual band is determined from wideband consideration, is made over multiple frequency bands by a bank of optimization problems.
1.2. Motivation and Principle Analysis This paper investigates wideband spectrum sensing for CRN from a novel perspective. Our approach searches over the whole spectrum bands simultaneously rather than an individual band each time. It first estimates the number of occupied channels and then determines the corresponding locations. The principle exploited in this paper is that the received energy of the occupied band is the superposition of signal and noise, whereas the energy of the vacant band is only contributed by noise. Thus energy of the occupied channel is larger than the vacant one. If there exists a prior knowledge that the number of occupied channels is K , those K channels with the largest energy are more likely to be the occupied ones. Therefore the estimation of the number of occupied channels becomes a critical issue. In this way, the wideband spectrum sensing problem is equivalent to the estimation of the number of occupied channels, which can refer to the issue of source number estimation. The authors in [17] originally applied Akaike information criterion (AIC) and minimum description length (MDL) to source enumeration. The number of signal sources in the information theoretical criteria is determined by minimizing the Kullback-Leibler distance between the hypothesized model and the observation data. In other words, for a given set of data and a class of probabilistic models, the model with the shortest description length should be selected as the best one. MDL is a consistent estimation while AIC yields an inconsistent estimation. Since AIC tends to overestimate the number of sources leading to a high false alarm probability, this paper focuses on the MDL criterion. Motivated by the idea of multistage orthogonal projection inherent in MSWF [18], this paper introduces a MSWF aided MDL approach to wideband spectrum sensing. The proposed scheme first partitions the
observation data into orthogonal signal and noise subspaces. Then, rather than the eigenvalues of covariance matrix, the variances of noise subspace components are utilized to encode the description length. Next, the number of occupied channels is attained by minimizing the description length. Eventually, each channel’s occupancy state is judged from its received energy. 1.3. Contribution The main contributions of this paper can be summarized as follows.
We put forward a MSWF aided MDL algorithm to perform wideband spectrum sensing. Since estimating noise power is no longer a necessity, our algorithm is not sensitive to the consequences arising from an inaccurate estimation. As a result, our approach is robust to noise uncertainty.
The MSWF aided MDL approach possesses the advantage of computational simplicity, particularly at a large number of antenna arrays or a small number of samples. This is due to the fact that the proposed method does not involve the estimation of covariance matrix or the eigenvalues decomposition. Actually, the former is replaced by some simpler vector-vector products while the latter is replaced by the variances of the noise subspace components, which can be directly generated in the procedures of multistage recursions.
Our scheme enjoys a more favourable performance than the original MDL criterion. Since the cross correlation between the observation data and reference signal is able to capture the signal information as well as suppress the additive noise, the cleaner signal and noise subspace components can be obtained when it is employed as the initial
information for the refinement procedure. As a result, the variances of the signal subspace components are well separated from the variances of the noise subspace components. In the meantime, the variances of the noise subspace components are clustered sufficiently closely. Therefore, our approach significantly reduces the likelihood of overestimating and underestimating the number of occupied channels, eventually leading to an improved performance. The following notations are used in this letter. Boldface letters are used to denote matrices and vectors. Superscripts (⋅)T , (⋅)H and (⋅)* stand for transpose, Hermitian transpose and complex conjugate, respectively. E [⋅] represents expectation operator. ⋅ denotes the vector norm. The identity matrix is shown by I .
2. System model In order to sense the wideband spectrum, the whole channel with the frequency bands from fl to fh is divided into Q nonoverlapping narrow bands. Each narrowband is with equal bandwidth W . K out of Q subbands are occupied by the PUs and the rest are the vacant ones. The illustration of the scenario is depicted in Figure 1. The spectrum usage is usually rather insufficient for CRNs and thus K < Q . Q − K Vacant Channels
K Occupied Channels
W fl
fh
1
2
3
Q − 2 Q −1
Q
Figure 1 Scenario illustration The binary hypothesis at subband i (1 ≤ i ≤ Q ) can be expressed as H0 : xi ( n ) = ε i ( n ) ,
(1)
H1: xi ( n ) = hi si ( n ) + ε i ( n ) ,
(2)
where H0 represents that the subband is idle and H1 denotes that the subband is busy. xi ( n ) is the received signal sample, si ( n ) stands for the transmitted PU signal obeying
independent Gaussian distribution with mean-zero and hi reflects the channel gain including the effects of multipath fading, path loss as well as time dispersion. ε i ( n ) is modeled as an independent additive white Gaussian noise (AWGN) with mean-zero and variance σ 2 . Under the assumption of AWGN, noise power spectral density is constant with frequency, so the noise powers are identical in different channels. It is assumed that the PU signal is uncorrelated with the noise. Consider that M uniform linear arrays are placed in the cognitive receiver. In order that the problem has a unique solution, K < M should be allowed throughout this paper due to the fact that PU signals occupy a subspace of dimension strictly smaller than the number of antenna arrays [19-20]. The received M × 1 signal vector from antenna arrays can be formulated as r ( n ) = r1 ( n ) ,L , rM ( n ) = ∑ i =1 s%i ( n ) ai + ε ( n ) = As% ( n ) + ε ( n ) , T
K
(3)
where ai = 1, exp ( j 2π dfi sin (θ i ) c ) ,L , exp ( j 2π ( M − 1) dfi sin (θi ) c )
T
.
(4)
θi is the angle of arrival of the i th PU signal si ( n ) relative to the array broadside. fi
represents the central frequency of the PU signals, c denotes the wave speed and d stands for the antenna spacing. A = [a1 , a 2 ,L , a K ] is taken as the array response matrix with ai the steering vector. When the i th PU is not transmitting, the i th column of A is excluded. ε ( n ) = ε1 ( n ) , ε 2 ( n ) ,L , ε M ( n )
s% ( n ) = s%1 ( n ) , s%2 ( n ) ,L , s%K ( n )
T
T
is the M × 1 additive noise vector in the antenna arrays. denotes the received PU signal vector suffering from channel
response, i.e. s%i ( n ) = hi si ( n ) .The covariance matrix of the received signals r ( n ) can be represented as Cr = E r ( n ) r H ( n ) = AE s% ( n ) s% H ( n ) A H + σ 2 I ,
(5)
2
H
= AC s A + σ I
where Cs = E s% ( n ) s% H ( n ) denotes the covariance matrix of received PU signal vector. In practical situations, however, the covariance matrix can only be estimated from a finite set of sampling numbers, which is called sample covariance matrix and is represented as ˆ = 1 C r N
N
∑ r ( n )r ( n ) , H
(6)
n =1
where N denotes the number of samples. Note that the sample covariance matrix approaches its ideal form as the number of samples tends to be infinite.
3. Proposed wideband spectrum sensing method 3.1. Conventional MDL Method
The classical MDL criterion [17] tells that given a set of observation data X = { x (1) , L , x ( N )}
and a family of probability densities f ( X Θ k ) with Θk denoting an
unknown parameter vector, the model with the minimum code length will fit the data best. More specifically, the shortest description length can be evaluated quantitatively as ˆ + 1 φ log N , MDLk = − log f X Θ k 2
(
)
(7)
where Θˆ k is the maximum likelihood (ML) estimate of Θ k and φ is the number of free adjusted parameters in Θk . Since the observations are regarded as independent Gaussian distributions with mean-zero, we can eventually obtain the form of MDL criterion as follows through a series of derivations [17] M 1 ( M −k ) ∏ λi MDLk = − log i = k +1 M 1 M − k ∑ λi i = k +1
(M −k ) N
1 + k ( 2M − k ) log N , 2
(8)
where λi represents the eigenvalues of sample covariance matrix Cˆ r . The estimated number of sources, i.e. the number of occupied channels, is determined as the value of k ∈ {0,L , M − 1} for which the MDL is minimized.
We can see that the traditional MDL criterion suffers from a heavy burden of computational complexity since it requires the estimation of covariance matrix and eigenvalue decomposition. Therefore, the conventional MDL criterion is unsuitable in some practical situations, especially when the number of sources needs to be tracked in an online manner. To reduce the complexity, an MSWF aided MDL scheme is proposed in this paper. Details are given below. 3.2. MSWF aided MDL Scheme
Before introducing the new MDL method, we take another look at the uniform linear arrays in the cognitive receiver, i.e., Eq. (3). We can say that the first antenna array generates the reference signal d 0 ( n ) = r1 ( n ) = s%T ( n ) 1 + ε1 ( n )
(9)
where 1 = [1,1,L,1]T , while the remaining ones produce the new observation data r0 ( n ) = r2 ( n ) , r3 ( n ) ,L , rM ( n ) = A M −1 (θ ) s% ( n ) + ε 0 ( n )
T
,
(10)
= ∑ i =1 s%i ( n ) a M −1 (θ i ) + ε 0 ( n ) K
T
where ε 0 ( n ) = ε 2 ( n ) , ε 3 ( n ) ,L , ε M ( n ) . A M −1 (θ ) and a M −1 (θi ) are composed of the last M − 1 rows of A (θ ) and a (θ i ) , respectively. Physically, they can be viewed as the effect causing by removing one antenna array, which sacrifice one array gain but still contain the information of all the PU signals. The cross-correlation between the reference signal and the observation data is R r0 d0 = E r0 ( n ) d 0* ( n ) = A M −1 (θ ) C s 1 .
(11)
It is indicated that the cross-correlation is capable of capturing the signal information. Meanwhile, having no noise term in it means that the additive noise has been efficiently cancelled. In this sense, the cross-correlation can be regarded as a matched filter, which extracts the signal from the background noise. A normalized form can be obtained as
H1 =
R r0 d0 R r0 d0
.
(12)
With the above initial information, now we proceed to the multistage recursion procedure similar to the MSWF [21]. Given a reference signal d 0 ( n ) , the MSWF partitions
the observation data r0 ( n ) with the normalized matched filter H1 into two orthogonal subspaces: the desired signal d i ( n ) and the observation data ri ( n ) at the i th stage. The desired signal d i ( n ) is attained by filtering the observation data ri −1 ( n ) with the matched filter H i . The observation data ri ( n ) is partitioned stage by stage. A set of detailed formulas are shown as follows d i ( n ) = H iH ri −1 ( n )
(13)
ri ( n ) = ri −1 ( n ) − Hi d i ( n ) = ri −1 ( n ) − H i H iH ri −1 ( n )
(14)
= Bi ri −1 ( n )
Hi =
R ri −1di −1 R ri −1di −1
=
E ri −1 ( n ) d i*−1 ( n )
(15)
E ri −1 ( n ) d i*−1 ( n )
where Bi = I M −1 − Hi HiH spans the nullspace of R r
i −1 d i−1
and H i , i.e. Bi R r
i−1 d i −1
= 0 and Bi Hi = 0 .
Physically, the blocking matrix Bi annihilates the signal components in the subspace of the cross-correlation R r
i −1 d i−1
and the matched filter H i . An example of a four-stage MSWF block
diagram is presented in Fig. 2, where ei ( n ) is the estimation errors and ωi is the scalar weights, defined as
ωi =
E d i −1 ( n ) ei∗ ( n )
(16)
2 E ei ( n )
ei −1 ( n ) = d i −1 ( n ) − ωi∗ ei ( n )
(17)
d0 ( n )
r0 ( n )
H1
B1
+
d1 ( n ) r1 ( n )
H2
B2
+
d2 ( n) r2 ( n )
H3
+
d3 ( n ) +
ω4
H4
B3
r3 ( n )
d4 ( n )
−
∑
e3 ( n )
∑ ω3 −
e2 ( n )
∑ ω2
−
e1 ( n )
∑ ω1
−
e0 ( n )
Figure 2: The nested chain of a four-stage MSWF
The signal and noise subspaces can be spanned by the matched filter bank in the following manner.
where S and N
S = span {H1 , H 2 ,L , H K }
(18)
N = span {H K +1 , H K + 2, L , H M }
(19)
denote the signal subspace and noise subspace, respectively. As a
consequence, by filtering the initial observation data r0 ( n ) with the matched filter bank S and N , the components in the signal and noise subspaces can be obtained, respectively T
rs ( n ) = d1 ( n ) , d 2 ( n ) ,L , d K ( n ) = TsH r0 ( n ) T
rn ( n ) = d K +1 ( n ) , d K + 2 ( n ) ,L , d M ( n ) = TnH r0 ( n )
(20) (21)
in which the filter bank matrix is defined as K −1 Ts = H1 , B1H 2 ,L , ∏ Bi H K = [ H1 , H 2 ,L , H K ] i =1
(22)
K +1 M −1 K Tn = ∏ Bi H K +1 , ∏ Bi H K + 2 ,L , ∏ Bi H M = [ H K +1 , H K + 2 , L , H M ] i =1 i =1 i =1
(23)
The second equations hold due to the orthogonality between the matched filters, namely H iH H j = 0 ( i ≠ j ) ,
which has been proved in [22] through an induction manner. Accordingly,
the covariance matrix of noise subspace components can be described as Cn = E rn ( n ) rnH ( n )
(24)
N ˆ = 1 C rn ( n ) rnH ( n ) ∑ n N n =1
(25)
with the sample form
The elements in the aforementioned covariance matrix of noise subspace can be further derived as E d i ( n ) d ∗j ( n ) = E H iH ri −1 ( n ) r jH−1 ( n ) H j = H iH E ri −1 ( n ) r jH−1 ( n ) H j H 1 1 H = H E ∏ B k r0 ( n ) r0 ( n ) ∏ Bl H j k =i −1 l = j −1
(26)
H i
H
1 1 = H iH ∏ B k E r0 ( n ) r0H ( n ) ∏ B l H j k = i −1 l = j −1 H
1 1 = H I M −1 − ∑ H k H kH ( A M −1 (θ ) R s A MH −1 (θ ) + σ 2 I M −1 ) I M −1 − ∑ H l H lH H j k =i −1 l = j −1 H i
= H iH A M −1 (θ ) R s A MH −1 (θ ) H j + σ 2 H iH H j
Owing to the orthogonality as well as the normalization between matched filters, namely H iH H j = 0 ( i ≠ j )
and H iH H j = 1 ( i = j ) ,
and the orthogonality between noise subspace and
steering matrix, it can be drawn that E d i ( n ) d ∗j ( n ) = 0 ( i, j = K + 1, K + 2, L , M , i ≠ j )
E d i ( n ) di∗ ( n ) = σ 2
( i = K + 1, K + 2,L, M )
(27) (28)
In other words, the last M − K desired signals of the MSWF are uncorrelated with each other and their variances equal to the noise variance. Consequently, the covariance matrix of noise subspace components is a diagonal matrix with the following form Cn = diag δ d2K +1 , δ d2K +2 ,L , δ d2M
(29)
where δ d2 = E di ( n ) d i∗ ( n ) . It is quite important to note that the first K desired signals of the i
MSWF do not meet the aforementioned equations since the orthogonality between signal subspace and steering matrix does not hold.
Following the idea of MSWF, [18] has shown that the variances of noise subspace components can replace the eigenvalues of covariance matrix as for the calculation of MDL criterion. Substituting the variances of noise subspace components into Eq. (8), an MSWF aided MDL criterion is obtained as M ˆ2 1 (M −k ) ∏ δ di MDL _ MSWFk = − log i = k +1 M 1 ˆ2 M − k ∑ δ di i = k +1
( )
( M −k ) N
1 + k ( 2 M − k ) log N 2
(30)
N in which δˆd2 = 1 N ∑ n =1 di ( n )di* ( n ) is the ML estimate of δ d2 = E di ( n ) d i* ( n ) . The number of i
i
occupied channels is thereby Kˆ = arg
min
k = 0,1,L, M −1
MDL _ MSWFk .
(31)
With the information of the estimated number of occupied channels, we now set out to find the accurate positions of the occupied channels in the wideband CRNs. The received energy of each channel is computed as ϕi =
1 N
N
∑ x ( n) i
2
i = 1, 2,L , Q .
(32)
n =1
As we have pointed out before, the occupied channels own higher received energy than the idle ones. Therefore, channels with the highest Kˆ received energy would be set as the occupied ones. In summary, the steps of the proposed MSWF aided MDL approach are described in Table 1. Table 1: The proposed MSWF aided MDL algorithm for spectrum sensing
Step 1: Initialize the MSWF and produce the reference signal
d0 ( n)
, observation data
r0 ( n )
, cross-correlation
and normalized matched filter
R r0d0
H1 ,
i.e., Eq.
(9)~(12); Step 2: Perform multistage recursions following the procedure of MSWF, and generate the desired signal
di ( n ) ,
and normalized matched filter
Hi ,
observation data
ri ( n ) ,
blocking matrix
Bi
i.e., Eq. (13)~(15);
Step 3: Partition the signal as well as noise subspace, and compute the variances of noise subspace components; Step 4: Construct the MSWF aided MDL criterion, i.e., Eq. (30); Step 5: Estimate the number of occupied channels, i.e., Eq. (31); Step 6: Select channels with the highest Kˆ received energy as the occupied ones.
3.3. Discussions of the Proposed Scheme
In our approach, several modeling functions under different hypotheses are constructed to denote the number of busy channels. The judgment of the true hypothesis is equivalent to finding the variable that minimizes these modeling functions, and thus the number of occupied channels is determined. Afterwards, the channels with maximum received energy are considered to be occupied by the PUs.
Distinct from the traditional spectrum sensing methods, the proposed scheme investigates this problem in a multistage recursion procedure inspired by the MSWF. The cross correlation between the reference signal and the observation data is exploited to capture the signal information. Note that it is capable of suppressing the background noise since
no noise item is involved. The cross correlation is then utilized to generate the signal and noise subspaces, which eventually leads to the enhanced performance.
Our approach is robust to noise uncertainty since it does not involve the estimation of noise variance. Thus the impact induced by an inaccurate estimation would be totally eliminated. Energy detection with noise uncertainty is provided in the following simulation section to validate the conclusion.
Note that the idea in this paper is different from interference alignment (IA), which is a promising technique for interference management and can be applied to spectrum sharing in CRN [23-24]. In IA, the channel state information (CSI) is required to construct a suitable precoding matrix, which should be found at each transmitter to constrain all interference into certain subspaces at each receiver. In this way, IA eliminates not only the interference to PUs but also the mutual interference among SUs. On the contrary, our approach focuses on the subject of spectrum sensing rather than spectrum sharing. The CSI is not needed to perform the detection. The feedback from the receiver to the transmitter is not required, either. 3.4. Complexity Analysis As we have pointed out, the proposed MSWF aided MDL scheme is more
computational attractive than the traditional MDL approach, particularly at a large number of antenna arrays or a small number of samples. This is because the traditional MDL principle, i.e., Eq. (8), resorts to the eigenvalues of sample covariance matrix. Hence, the estimation of sample covariance matrix and the eigenvalue decomposition are quite necessary. The former requires O ( M 2 N ) operations since its received M × 1 signal vector r ( n ) is averaged over
N samples, i.e., Eq. (6). The latter generally requires O ( M 3 ) operations due to a M × M
sample covariance matrix. On the other hand, the dominant computational cost of the proposed MSWF aided MDL scheme comes from the multistage recursion procedure, i.e., Eqs. (13)-(15), which requires vector-vector products. O ( M − 1) is needed for each sample and each recursion, thereby the total computational load is only around O ( ( M − 1) MN ) multiplications and additions. The aforementioned complexity analysis provides a brief overview. Now we proceed to a detailed description. Taking a further look at the two MDL criteria, i.e., Eq. (8) and Eq. (30), we can find that the main differences come from the eigenvalues λi and the estimated variances of desired signals δˆd2 . Since calculating the MDL criteria as well as determining i
the locations of occupied channels are similar for the two algorithms, we omit the complexity comparison for these steps in this subsection. Meanwhile,
we focus on the
multiplication/division (MD) and addition/subtraction (AS). Some simple operations, e.g., Hermitian transpose, complex conjugate, are neglected as well. The calculation of sample covariance matrix, i.e., Eq. (6), needs M 2 + M 2 N MDs as well as M 2 ( N − 1) ASs. The corresponding eigenvalue decomposition demands nearly 6M 3 MDs and 6M 3 ASs. These manipulations are complex. One complex MD operation needs 6 floating-point operations (flops) and one complex AS operation needs 2 flops. Thus the total number of flops needed in the conventional MDL algorithm is Flops _ MDL = 6 ( M 2 + M 2 N + 6 M 3 ) + 2 ( M 2 N − M 2 + 6 M 3 ) = 48 M 3 + 8 M 2 N + 4M 2
.
(33)
The initialization of the proposed MDL approach produces the reference signal d 0 ( n ) and the observation data r0 ( n ) . Calculating the cross-correlation R r d between the reference 0 0
signal and the observation data, i.e., Eq. (11), needs ( M − 1)( N + 1) MDs as well as
( M − 1)( N − 1) ASs. Accordingly, the matched filter M −2
H1
calls for 2 ( M − 1) MDs as well as
ASs. Then the desired signal d1 ( n ) at the 1th stage is attained through M − 1 MDs
and M − 2 ASs while its orthogonal component r1 ( n ) is obtained by M − 1 MDs and M − 1 ASs. In order to get di ( n )( i = 1, 2,L , M ) , M successive recursions have to be performed, which
indicates
that
ri ( n )( i = 1,2,L, M − 1) ,
calculating
R ri di ( i = 0,1,L, M − 1)
and
Hi ( i = 1,2,L, M ) becomes a necessity. In addition, computing the estimated variances of the
noise subspace components calls for M ( N + 1) MDs and M ( N − 1) ASs. Since these procedures are complex, the total number of flops needed in the proposed MDL algorithm is Flops _ MDL _ MSWF = 6 ( M ( M − 1)( N + 5 ) − ( M − 1) + M ( N + 1) ) +
2 ( M ( M − 1)( N + 2 ) + 2 M − ( M − 1) + M ( N − 1) ) . 2
(34)
2
= 8M N + 34M − 34 M + 8
To compare the complexity, we subtract Eq. (34) from Eq. (33) Flops _ MDL − Flops _ MDL _ MSWF
= ( 48M 3 + 8M 2 N + 4M 2 ) − ( 8M 2 N + 34 M 2 − 34 M + 8 ) = 48M 3 − 30 M 2 + 34 M − 8
.
(35)
= 2M 2 ( 24 M − 15 ) + 2 (17 M − 4 )
It can be readily seen from Eq. (35) that our proposed MDL scheme saves computational cost.
7
10
Flops
Traditional MDL Criterion Proposed MDL Scheme
6
10
5
10 100
200
300
400
500
600
700
800
900
1000
N
Figure 3: Complexity comparison versus the number of samples (M=16)
To explicitly illustrate the computational flops, we depict the complexity comparisons in the case of M = 16 in Fig. 3 and N = 100 in Fig. 4, respectively. Moreover, we define the ratio of complexity reduction as Φ=
Flops _ MDL − Flops _ MDL _ MSWF Flops _ MDL
,
(36)
which reflects the percentage of the flops saved by our proposed MDL algorithm compared to the original MDL criterion. Fig. 5 plots the ratio of complexity reduction versus the number of antenna arrays at various numbers of samples. We can observe from the figures that the proposed MDL scheme is more computationally efficient than the conventional one, particularly at a large number of antenna arrays or a small number of samples.
7
10
Traditional MDL Criterion Proposed MDL Scheme
6
Flops
10
5
10
4
10
4
8
12
16
20
24
28
32
M
Figure 4: Complexity comparison versus the number of antenna arrays (N=100)
1 N=100 N=500 N=1000
The Ratio of Complexity Reduction
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
4
8
12
16
20
24
28
M
Figure 5: The ratio of complexity reduction versus the number of antenna arrays
32
4. Numerical results Considering a wideband CRN with M = 16 uniform linear antenna arrays, there are Q = 20 channels in total to be detected among which K = 4 channels are occupied by the
PUs. Four PU signals in the occupied channels with central frequencies {565, 585, 615, 635 MHz} impinge upon the antenna arrays from distinct directions {-4°, 2°, 6°, 10°}, respectively. The antenna spacing is d = 0.25m . Fig. 6 depicts the normalized power spectrum of the PU signals for occupied channels. Note that PUs’ signal powers generally vary in various channels, and then SNR in the following simulations refers to the average SNR, namely, the ratio of average received signal power to noise power. One thousand Monte Carlo trials have been carried out independently in each realization. Detection probability Pd and false alarm probability Pf are two key parameters for performance evaluation. In IEEE 802.22 standard for CRNs [2], it is required that Pd ≥ 0.9 and Pf ≤ 0.1 . Since CRN aims at effectively utilizing the spectrum holes on the premise of not interfering PUs, the goal of our approach is to maximize the detection probability, and meanwhile, the false alarm probability maintains to an acceptable extent. The simulation parameters are summarized in Table 2.
1 0.9
Normalized Power Spectrum
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 540
560
580
600 620 f (MHz)
640
660
680
Figure 6: The normalized power spectrum of the PU signals for occupied channels Table 2: Simulation parameters Parameters
Setting
The number of subchannels (Q)
20
The number of occupied subchannel (K)
4
The number of antenna arrays (M)
16
Channel between PUs and SUs
AWGN, Rayleigh fading
Noise uncertainty
0,1,2 dB
Central frequency (f)
{565, 585, 615, 635 MHz}
Impinge direction ( θ )
{-4°, 2°, 6°, 10°}
Antenna spacing (d)
0.25m
Monte Carlo trials
1000
Scaling parameters
Pd, Pf
4.1. Detection Performance with Varying Channel Condition Firstly, this section validates the performance of our approach versus SNR. Energy detection is presented for comparison. There are two cases considered here for energy detection: the ideal one with false alarm probability setting to be 0.1 and energy detection with noise uncertainty. Fig. 7 shows the probability of detection versus SNR under the condition that the number of samples is 500, and the corresponding probability of false alarm is depicted in Table 3. Note that Pf is not related to the SNR since there is no signal, and thus, remains constant over various SNR. For the clarity of representation, the term ED is short for the ideal energy detection and ED (x dB) means energy detection with x dB noise uncertainty. It follows from the figure and the table that noise uncertainty severely deteriorates the detection performance of energy detection. Furthermore, the false alarm probability is intolerable since it far exceeds the acceptable range for CRNs. Although the ideal energy detection holds the best detection probability and a reasonable false alarm probability, however, we generally have no access to an accurate noise variance. That is to say, the ideal energy detection cannot be achieved. On the contrary, meeting the requirement of false alarm probability for CRNs, our approach surpasses energy detection with noise uncertainty in detection probability. This is also in accordance with the theoretic analysis given in the above section that our proposed scheme does not resort to the estimation of noise variance. Thereby noise uncertainty has no influence on our method while energy detection is greatly affected.
1 0.9 0.8 0.7
Pd
0.6 0.5 0.4 0.3 Pd Pd Pd Pd
0.2 0.1 0 -15
-13
-11
-9
-7 SNR (dB)
-5
of ED of ED (1dB) of ED (2dB) of Proposed Scheme -3
-1
0
Figure 7: Detection probability of our approach and energy detection versus SNR (N=500)
Table 3: False alarm probability of various methods (N=500) ED
ED (1dB)
ED (2dB)
Proposed Scheme
0.1000
0.4028
0.3636
0.0920
4.2. Detection Performance with Different Timing Requirements Now this section draws attention to the impact of the number of samples by fixing SNR at -10 dB when the channel obeys Rayleigh distributions [25]. The detection probability and false alarm probability are depicted in Fig. 8. As is shown in the figure, when the number of samples exceeds a certain value, the detection performance of energy detection with noise uncertainty is no longer improved whereas the proposed scheme is still enhanced. Due to SNR wall, energy detection with noise uncertainty is even not consistent since it cannot reach
Pd = 1 as the number of samples grows. Moreover, the false alarm probability of energy
detection with noise uncertainty is not endurable for CRNs, particularly at a large sampling number. Our approach outperforms energy detection with noise uncertainty both in Pd and Pf under Rayleigh fading channel. 1 Pd of ED Pf of ED Pd of ED (1dB) Pf of ED (1dB) Pd of ED (2dB) Pf of ED (2dB) Pd of Proposed Scheme Pf of Proposed Scheme
0.9 0.8 0.7 Pd
Pd and Pf
0.6 0.5 0.4 0.3
Pf
0.2 0.1 0 50
500
1000
1500 N
2000
2500
3000
Figure 8: Detection probability and false alarm probability of our approach and energy detection versus the number of samples under Rayleigh fading channel (SNR=-10 dB)
4.3. Detection Performance against the Existing Algorithms To further demonstrate the performance of our approach, the MDL criteria based wideband spectrum sensing algorithms for CRNs in [12] are adopted for comparison. Note that our approach as well as the algorithms in [12] are the improvements of the classical MDL criterion, and the rationales of them are the same, namely to judge the number of occupied channels firstly and then find out the accurate locations of occupied channels as
well as the vacant ones. The first scheme in [12], noted as MDL criterion A, performs ML estimation on each hypotheses model, and then directly applies the MDL criterion into the estimation of the number of occupied channels, and finally finds out the indexes of occupied channels. The second scheme in [12], noted as MDL criterion B, however, utilizes the sorted sample power of each channel to replace the eigenvalues. In this way the ML estimation is avoided and the MDL criterion is redesigned. The detection probability versus SNR is drawn in Fig. 9 and Table 4 presents the false alarm probability. We can see that our approach reaches Pd = 0.9 at a lowest SNR. Although our Pf is worse, it is not pessimistic since it still satisfies the requirement of CRNs. Therefore, our approach possesses the best detection performance among the algorithms. As we mentioned before, this is because the first four matched filters H i ( i = 1, 2, 3, 4 ) are capable of capturing signal information while excluding a large portion of noise. On the contrary, their orthogonal complements H i ( i = 5, 6,L ,16 ) have the ability to mitigate the signal subspace components from the noisy data. Hence, the estimated variances of the first four desired signals δˆd2 (i = 1, 2,3, 4) , namely the powers of i
signals, are well separated from the estimated variances of the desired signals after the fourth stage δˆd2 ( i = 5, 6,L ,16 ) , namely the powers of noises. Simultaneously, the variances of the i
desired signals after the fourth stage are clustered sufficiently closely. Consequently, the cleaner signal subspace and noise subspace can be obtained. Thereby, the probabilities of overestimating and underestimating the number of occupied channels are greatly reduced. Thus, the proposed method is capable of estimating the number of occupied channels more precisely and achieving a better performance.
1 0.9 0.8 0.7
Pd
0.6 0.5 0.4 0.3 0.2
Pd of MDL Criterion A Pd of MDL Criterion B Pd of Proposed Scheme
0.1 0 -10
-8
-6
-4
-2 SNR (dB)
0
2
4
5
Figure 9: Detection probability of our approach and the existing MDL algorithms versus SNR (N=200)
Table 4: False alarm probability of various methods (N=200) MDL Criterion A
MDL Criterion B
Proposed Scheme
0.0018
0.0648
0.0953
5. Conclusions This paper proposes an MSWF aided MDL approach for wideband spectrum sensing in CRNs. The proposed method is able to generate signal and noise subspaces. The signal subspace captures the signal information and excludes a large portion of additive noise. On the contrary, the noise subspace is capable of attaining noise components, thereby achieving a
clean noise subspace. This implies that the variances of the signal subspace components are well separated from the variances of the noise subspace components. In the meantime, the variances of the noise subspace components are clustered sufficiently closely. Consequently, our approach is able to reduce the likelihood of overestimating and underestimating the number of occupied channels, and eventually enhance the detection performance. Due to the fact that there is no need to estimate the noise power, our approach is robust to noise uncertainty. Meanwhile, the proposed method reduces the computational complexity since it does not rely on the estimation of covariance matrix or its eigenvalue decomposition, particularly at a large number of antenna arrays or a small number of samples. The proposal’s performance is demonstrated via numerical results.
Acknowledgments The authors would like to thank editors and the anonymous reviewers for their careful review and constructive comments which improve the quality of this paper. This work was supported in part by the National Key Basic Research Program of China (973 Program, no. 2014CB339900), National Natural Science Foundation of China (no. 61571373), Important National Science and Technology Specific Projects (no. 2012ZX03003001-004), National High Technology Research and Development Research Program of China (863 Program).
References [1] Haykin S, Setoodeh P. Cognitive radio networks: the spectrum supply chain paradigm. IEEE Trans Cog Commun Net 2015; 1(1):3-28 [2] Stevenson CR, Chouinard G, Lei Z, Hu W, Shellhammer SJ, Caldwell W. IEEE 802.22: the first cognitive radio wireless regional area network standard. IEEE Commun Mag 2009; 47(1):130-8 [3] Yang L, Chen Z, Yin F. Cyclo-energy detector for spectrum sensing in cognitive radio. Int J Electron Commun (AEU) 2012; 66(1):89-92 [4] Sadeghi H, Azmi P, Arezumand H. Cyclostationarity-based cooperative spectrum sensing over imperfect reporting channels. Int J Electron Commun (AEU) 2012; 66(10):833-40 [5] Ren X, Chen C. Spectrum sensing algorithm based on sample variance in multi-antenna cognitive radio systems. Int J Electron Commun (AEU) 2016; 70(12):1601-9 [6] Chen X, Chen H, Meng W. Cooperative communications for cognitive radio networksfrom theory to applications. IEEE Commun Surveys & Tutorials 2014; 16(3): 1180-92 [7] Tandra R, Sahai A. SNR walls for signal detection. IEEE J Sel Top Sig Process 2008; 2(1): 4-17 [8] Sun H, Nallanathan A, Wang C-X, Chen Y. Wideband spectrum sensing for cognitive radio networks: a survey. IEEE Wirel Commun Mag 2013; 20(2): 74-81 [9] Sahai A, Cabric D. A tutorial on spectrum sensing: fundamental limits and practical challenges. Proc IEEE Int Symp Dynamic Spectrum Access Networks 2005:1-5
[10] Taherpour A, Gazor S, Nasiri-Kenari M. Invariant wideband spectrum sensing under unknown variances. IEEE Trans Wirel Commun 2009; 8(5): 2182-6 [11] Jing Y, Yang X, Ma L, Ma J, Niu B. Blind multiband spectrum sensing in cognitive radio network. Proc IEEE Int Conf Consumer Electron Commun Net (CECNet) 2012: 2442-5 [12] Liu S, Shen J, Zhang R, Zhang Z, Liu Y. Information theoretic criterion-based spectrum sensing for cognitive radio. IET Commun 2008; 2(6): 753-62 [13] Qing H, Liu Y, Xie G. Smart antennas aided wideband detection for spectrum sensing in cognitive radio networks. Electron Lett 2014; 50(7): 490-2 [14] Mariani A, Giorgetti A, Chiani M. Wideband spectrum sensing by model order selection. IEEE Trans Wirel Commun 2015; 14(12): 6710-21 [15] Tian Z, Giannakis GB. A wavelet approach to wideband spectrum sensing for cognitive radios. Proc Int Conf Cognitive Radio Oriented Wireless Networks and Commun (CROWNCOM) 2006: 1-5 [16] Quan Z, Cui S, Sayed AH, Poor HV. Optimal multiband joint detection for spectrum sensing in cognitive radio networks. IEEE Trans Signal Process 2009: 57 (3): 1128-40 [17] Wax M, Kailath T. Detection of signals by information theoretic criteria, IEEE Trans Acoust Speech Signal Process 1985, 33(2): 387-92 [18] Huang L, Wu S. Low-complexity MDL method for accurate source enumeration. IEEE Signal Proc Let 2007, 14(9): 581-4 [19] Zhang R, Lim TJ, Liang YC, Zeng Y. Multi-antenna based spectrum sensing for cognitive radios: a GLRT approach. IEEE Trans Commun 2010, 51(1): 84-8
[20] Qing H, Liu Y, Xie G, Liu K, Liu F. Blind multiband spectrum sensing for cognitive radio systems with smart antennas. IET Commun 2014, 8(6): 914-20 [21] Goldstein JS, Reed IS, Scharf LL. A multistage representation of the Wiener filter based on orthogonal projections. IEEE Trans Inform Theory 1998, 44(7): 2943-59 [22] Honig ML, Xiao W. Performance of reduced-rank linear interference suppression. IEEE Trans Inform Theory 2001, 47(5): 1928-46 [23] Zhao N, Yu F.R, Sun H, Li M. Adaptive power allocation schemes for spectrum sharing in interference alignment (IA)-based cognitive radio networks. IEEE Trans Veh Technol 2016, 65(5): 3700-14 [24] Li X, Zhao N, Sun Y, Yu F.R. Interference alignment based on antenna selection with imperfect channel state information in cognitive radio networks. IEEE Trans Veh Technol 2016, 65(7): 5497-511 [25] Kumar N, Bhatia V, Dixit D. Performance analysis of QAM in amplify-and-forward cooperative communication networks over Rayleigh fading channels. Int J Electron Commun (AEU) 2017; 72:86-94