Compressed Wideband Spectrum Sensing with ... - Semantic Scholar
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JOURNAL OF NETWORKS, VOL. 9, NO. 4, APRIL 2014
Compressed Wideband Spectrum Sensing with Partially Known Occupancy Status by Weighted l1 Minimization Zha Song and Huang Jijun School of Electronic Science and Engineering, National University of Defense Technology, Changsha, China Email: [email protected], [email protected]
Li Ning Unit 77108, People's Liberation Army, Chengdu, China Email: [email protected]
Abstract—This paper considers the problem of compressed wideband spectrum sensing in wideband cognitive radio when partial occupancy status is known. While performing wideband spectrum sensing, incomplete prior information on the occupancy status may be obtained from coexisting narrowband detectors, collaborative sensing nodes or remote database. In this paper, we present a new optimization model in order to incorporate this prior information and then propose a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization to solve it. Numerical simulation results demonstrate that the use of partially known occupancy status leads to an improvement in detection performance and the proposed approach exploits such prior information effectively. As incorrect prior information is unavoidable in practical situation, cases in which the prior information is non-ideal are also investigated via simulations. Index Terms—Cognitive Radio; Wideband Spectrum Sensing; Compressed sensing; Partially Known Occupancy Status; Weighted l1 Minimization; Non-ideal Prior Information
I.
INTRODUCTION
Spectrum sensing, whose objectives are detecting signal of licensed users (LUs) and identifying the spectrum holes for dynamic spectrum access (DSA), is an important enabling technology for cognitive radio (CR), a leading choice for efficient utilization of spectrum resource [1]-[3]. Nowadays, wideband applications have received significant attentions since they not only offer high throughput, but also purport pronounced spectrum access opportunities for CR users. Meanwhile, wideband spectrum sensing (WSS) entails considerable challenges in practice, especially its very high signal acquisition costs [2]-[4]. Recently, compressed sensing (CS) theory [5]-[7] has been introduced to alleviate the heavy pressure on conventional analog to digital converter (ADC) technology in WSS by utilizing the low percentage of spectrum occupancy – a fact that motivates dynamic © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.4.866-873
spectrum access [8], [9]. Compressed sensing theory shows that sparse or compressible signal can be reconstructed from much fewer samples than that suggested by the Shannon-Nyquist sampling theorem. Although existing frameworks of compressed wideband spectrum sensing (CWSS) [10], [11] show powerful ability of reducing signal-acquisition complexity, they are quite vulnerable to noise and the performance of CWSS degrades severely when signal to noise ratio (SNR) is low. Different from traditional sparse signal recovery algorithms in which sparsity is the only prior information on signal characteristic, other forms of prior information about the signal’s structure, such as partial support knowledge [12]-[15], support probability [16], connected tree structure [17], block-sparsity structure [17], etc., have been introduced into the reconstruction process. It has been demonstrated that the further exploitation of signal model, in addition to sparsity, would give birth to recovery performance enhancement. While performing wideband spectrum sensing, partial occupancy status may be known from coexisting narrowband detectors, collaborative sensing nodes or remote database [2]. In this paper, we study how to exploit partially known occupancy status (PKOS) to improve the detection performance of CWSS. In order to incorporate this type of prior information, a new CWSS model is presented and then a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization is proposed to solve it. The key idea of our proposed approach, named CWSS with PKOS (CWSS-PKOS), is to separated the spectrum whose occupancy status is known into two disjoint parts, the part known to be occupied and that known to be unoccupied, then impose relative small weights on the former which encourage nonzero entries in the reconstructed signal and relative large weights on the latter which discourage nonzeros. Simulation results demonstrate that the introduction of partially known occupancy status provides an improvement in detection performance of compressed wideband spectrum sensing, and that the
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proposed CWSS-PKOS approach can exploit such prior information effectively. In addition, we exemplify cases in which the prior information is non-ideal, namely some prior information is incorrect, and such cases are more significant in practical situation than cases with ideal prior information. The remainder of the paper is organized as follows. The signal model and the spectrum sensing problem of interest are explained in Section II and a typical representative of the traditional compressed wideband spectrum sensing approach is provided in Section III. In order to incorporate partially known occupancy status, Section IV presents the proposed CWSS model and the particular weighting strategy. Performance of proposed approach is demonstrated by numerical experiments in Section V and we draw conclusions in Section VI. Throughout this paper, boldfaced characters denote matrices and vectors. The superscripts of · represent T
the operations of transpose. The notation r
k
denotes the
lk norm of the vector r . For a set S , S denotes its size (cardinality). II.
goal of spectrum sensing task is to determine which subbands are occupied and it is equivalent to detecting the support of sparse or compressible signal r f . For sparse case, the support of r f is defined as follow:
supp r f
domain discrete versions. The relationship between rt and r f is given by
r f FN rt
(1)
where FN is the N -point unitary discrete Fourier transform (DFT) matrix. Many investigations have shown that the radio spectrum is in a very low utilization ratio [1]-[4]. Recently a survey of a wide range of spectrum utilization across 6GHz of spectrum in some places of New York demonstrated that the maximum utilization was only 13.1% [8]. Therefore, it is reasonable, by using sparse or compressible structure, to model the received wideband signal of CR, at any given time and spatial region, which suggests that the received signal is inherently sparse or compressible in frequency domain, i.e. r f is a sparse or compressible signal. This is exactly the motivation for introducing CS theory into WSS. As to hierarchical access model [1], overlay spectrum sharing protocol is adopted in which CR user avoids transmitting at any occupied sub-bands. In this case, the
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(2)
f
where r f [i] is the i -th element of r f . If r f is a compressible signal, the concept should be replaced by a% -energy support [13], which is defined as
supp r f
1 i N : r [i]
2
f
(3)
where is the largest real value for which supp r f
contains at least a% of the signal energy. Therefore the true occupancy status d 0,1
N1
,
whose i -th element indicates whether the corresponding sub-band is occupied or not, is given by
1, i supp r f di i 1, 0, i supp r f
SIGNAL MODEL
Consider a slot-segment model [10], [11], [18] of wideband spectrum, where the monitored spectrum is divided into N non-overlapping narrowband sub-bands (also known as slots). Despite that the locations of these sub-bands are known in advance, their power spectral density (PSD) levels are unknown and dynamically varying, depending on whether they are occupied or not. Those temporarily unoccupied sub-bands are termed spectrum holes and they are available for opportunistic spectrum access by CR users. Suppose that the Nyquistrate discrete form of received wideband signal at CR is denoted by an N 1 vector rt and r f is its frequency-
1 i N : r [i] 0
,N
(4)
where di 1 means that the i -th sub-band is occupied, while di 0 means that the i -th sub-band is unoccupied. III.
TRADITIONAL COMPRESSED WIDEBAND SPECTRUM SENSING
According to CS theory, compressed measurements are in fact the linear projections of the received signal rt onto a M N measurement matrix 1T ,
, MT
T
with
M N , where 1 , , M are sensing waveforms. It makes sense that only M samples need to be measured instead of N samples. The compression ratio, which is defined as M N , reflects the reduced number of samples M collected at CR receiver, with reference to the number N needed in full-rate Nyquist sampling. The matrix format for collecting compressed measurements can be formulated as
yt rt FN1r f
(5)
where FN1 is the inverse DFT matrix. Notice that, the inverse problem of (5) is an underdetermined problem and the high-dimensional signal r f cannot be recovered from low-dimensional measurements yt . However, it is shown in [5]-[7] that the reconstruction problem will stop being 1 underdetermined if the matrix FN satisfies restricted isometry property (RIP). It is proved that random matrix whose entries are chosen according to a Gaussian distribution will satisfy the RIP with high probability provided M is sufficiently large. More significantly, the
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RIP will be preserved for if is Gaussian random matrix and is arbitrary orthonormal basis. The problem of reconstructing the spectrum estimate rˆ f can be formulated as a combination of a l0 norm minimization and a linear measurement fitting constraint. However, the exact solution of l0 norm minimization is NP-hard and it requires an intractable combinatorial search. There are mainly two practical and tractable alternatives [6], [7]: greedy algorithms, such as various matching pursuits and convex relaxation algorithms. Both of them have advantages and disadvantages when applied to different scenarios. A brief assessment of their differences would be that convex relaxation algorithms require fewer measurements while greedy algorithms have less computation complexity. An approximate solution that is largely used is obtained by solving the following convex relaxation leading to l1 -norm minimization problem.
arg min r f
rˆ f
1
rf
(6)
1 N f
y t F r
s.t.
s.t.
arg min r f
1
rf
1 N f
y t F r
2
(7)
where bounds the amount of noise energy in the compressed measurements. Note that both of (6) and (7) are convex optimization problem [19] and a number of Matlab toolboxes, such as CVX [20], SeDuMi [21], l1 Magic [22], can be used to efficiently solve the problem. After we get spectrum estimate rˆ f from (7), the energy detection [2], [3] can be used to make the decision on spectrum occupancy status. The i -element of the decision dˆ is made as follow:
1, if rˆf i dˆ i for , i 1, ˆf i 0, if r
,N
(8)
where rˆ f i denotes the i -element of rˆ f and is a decision threshold which is chosen according to the desired probability of false alarm. After simple thresholding, the spectrum holes for DSA can be clearly given. IV.
THE PROPOSED COMPRESSED WIDEBAND SPECTRUM SENSING
It has been demonstrated that further exploitation of signal model, in addition to sparsity, would give birth to recovery performance enhancement [13]-[17]. As mentioned above, traditional CWSS exploits only the prior that the received signal is sparse or compressible in frequency domain, and does not assume any additional © 2014 ACADEMY PUBLISHER
by Su . Thus, So and Su are disjoint and S So Su . Therefore, a new CWSS model incorporating partially known occupancy status can be formulated as follow:
rˆ f
In practice the received signal is inevitably polluted by noise. In this case, an conic constraint is required, i.e. the optimization problem in (6) needs to be changed to
rˆ f
knowledge about the unknown sparse spectrum. However in the practical implementation of wideband spectrum sensing, there could be other knowledge about the monitored wideband spectrum. Incorporating additional knowledge would potentially improve spectrum sensing performance. In this paper, partial occupancy status is assumed to be known in advance from coexisting narrowband detectors, other collaborative sensing nodes or remote database, and we study how to exploit such prior information to improve spectrum sensing performance. Now we consider the spectrum reconstruction of r f with partially known occupancy status from compressed measurements yt . The part of sub-bands, whose occupancy status is known and index set is denoted by S , can be divided into two parts: the known occupied part indexed by So and the known unoccupied part indexed
arg min r f
1
rf
1 N f
y t F r
s.t.
d[i ] 1 d[j ] 0
2
(9)
i So j Su
for for
where d is the true occupancy status. According to the relationship between occupancy status of each sub-band and the support of r f as stated in (4), the optimization problem in (9) can be reformulated as follow:
rˆ f s.t.
arg min r f
1
rf
1 N f
y t F r
2
So supp r f Su
supp r
(10)
f
Unfortunately, the support set of r f is a thresholding function of r f which is unknown before reconstruction because it is scenario dependent. However, according to the definitions of support set as stated in (2) and (3), it is easy to see that signal values on its support set are relatively large while those outside are relatively small, even close to zero. Besides, for weighted l1 -norm minimization problem (6) and (7) are special cases where the weights are equal to 1), it has been shown that large weights can be used to discourage nonzero entries in the recovered sparse signal while small weights encourage nonzero entries [23]. According to those analyses, a particular weighting strategy in weighted l1 minimization reconstruction algorithm is proposed, in which small weights are imposed on the known occupied sub-bands which encourage relatively large entries in the recovered signal,
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at the same time, large weights on the known unoccupied ones which encourage relative small entries. Problem stated in (10) then becomes
rˆ f
arg min Wr f yt FN1r f
s.t.
1
rf
2
where the weight matrix W diag w1 , w2
l , if wk s , if 1, if
(11)
, wN with
k Su k So for k 1, k S
,N
(12)
and l , s denote pre-specified large and small constant, respectively. When spectrum estimate rˆ f is obtained from (11), decision on spectrum occupancy status can be also made via thresholding in (8). It is noteworthy that the weighted formulation of l1 minimization as stated in (11) is still a convex programming in which the problem remains simple and it effectively exploits the prior information on partially known occupancy status. Apparently, problem (11) will be reduced to the traditional case without any prior information when Su , So . In addition, problem (11) will be reduced to modified basis pursuit denoising (modified BPDN [14]) when Su and s 0 . In modified BPDN, partial support is assumed to be available, and the sparseness-inducing ( l1 -norm) term only contains entries outside the known support. In essence, the main difference between modified BPDN and (11) is that modified BPDN does not consider any additional knowledge outside the support, while (11) does. V.
SIMULATION RESULTS
In this section, simulation results are provided to illustrate performance of the proposed CWSS-PKOS approach with ideal and non-ideal prior information. First, the simulation setup and relevant performance metrics are described. Second, the performance of proposed approach with ideal prior information is evaluated by comparing with the traditional CWSS approach without any prior information as reference approach. Third, we consider cases in which some incorrect prior information exists in the known occupied part So and the known unoccupied part Su respectively. A. Simulation Setup and Performance Metrics Consider a monitored wide band partitioned into N 128 equal-bandwidth sub-bands and 24 among all sub-bands are randomly occupied by LUs, that is
supp r f
in
24 . Suppose that the amplitudes of elements supp r are generated according to Rayleigh f
distribution. The received signal is corrupted by additive white Gaussian noise (AWGN). The signal to noise ratio (SNR) is defined as the ratio of the average received
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signal to noise power over the entire wideband spectrum and is set to vary from -20dB to 30dB. For compressed sensing, the number of compressed samples M is set to range from 36 to 96. As to selecting parameter of weight matrix in (11), we set l 103 to discourage nonzero entries in the reconstructed signal, and s 0 to encourage nonzero entries which can also reduce the number of variables in objective function of (11) and then reduce the computational complexity of solving problem (11). For notational simplicity, we introduce N o and N u to denote the size of the known occupied part So and the known unoccupied part Su
respectively. That is,
No So and Nu Su . For the spectrum hole detection problem, performance metrics of interest are the probabilities of detection Pd and false alarm Pfa , which are evaluated by comparing the estimated occupancy status dˆ with the true occupancy status d over all sub-bands, as follows: d&dˆ 0 Pd d0
Pfa
(~ d) & dˆ 0
~d
0
where & and ~ denote bitwise AND and bitwise NOT respectively. The l0 norm function here is used to calculate the number of nonzero elements. B. CWSS-PKOS with Ideal Prior Information In this subsection, we consider cases in which prior information is ideal, namely all components of So and
Su are correct. To show the improvement in detection performance obtained by incorporating partially known occupancy status and to verify the effectiveness of CWSS-PKOS in exploiting such prior information, variations of detection probability Pd versus SNR and compression ratio are plotted in Fig. 1 and 2 respectively. In the following figures, each point on the curve is the average of the detection probabilities over 800 Monte Carlo trials. In addition, the opportunities for DSA will lose if the desired false alarm probability is set to be too small while determining threshold for making the decision on spectrum occupancy status. Therefore, we apply the receiver operation characteristic (ROC) curve to illustrate the tradeoff between the probabilities of detection and false alarm. ROC curves for Pfa 0.01, 0.11 are given in Fig. 3 since this regime of false alarm probability is of practical interest for achieving rational opportunistic throughput in CR. It has been shown in Section IV that the proposed CWSSPKOS approach will be reduced to the traditional CWSS without any prior information when Su , So . Therefore, in the following part, curves for
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No 0, Nu 0 depict the detection performance of traditional CWSS. As shown in Fig. 1, 2 and 3, detection probabilities of proposed CWSS-PKOS when partial occupancy status is available (curve 2, 3 and 4) are greater than that of traditional CWSS (curve 1). This suggests that incorporating partially known occupancy status, both the known occupied part and the known unoccupied part, indeed enables improvement in detection performance. Moreover, it is seen in Fig. 1 that compared to the improvement in high SNR regime, improvement in low SNR regime is much larger. This is because at high SNRs, such as 25dB and 30dB, the recovered signal obtained by solving (11), even when without any prior information, is quite closed to the original one. In simulations, there are 24 occupied sub-bands and 104 unoccupied sub-bands, thus No 12 and Nu 52 represent 50% of the occupied and unoccupied sub-bands respectively. In Fig. 1, Fig. 2 and Fig. 3, we can see that there is a big gap between curve 2 and curve 3. It suggests that, for a same proportion of occupied and unoccupied sub-bands, performance improvement provided by the former is larger than that provided by the latter. 1 0.9
0.7
subspaces, where C n, r denotes the number of combinations of ’ n ’ things selected ’ r ’ at a time. If the occupied sub-bands are partially known such that 12 subbands are known to be occupied, we only need to search for solutions in C 116,12 possible 12-dimensional subspaces. If 52 sub-bands are known to be unoccupied, we need to search for solutions in C (76, 24) possible 24dimensional subspaces. Clearly, the search space is reduced as long as occupied or unoccupied sub-bands are partially known. For a same proportion of occupied and unoccupied sub-bands, the reduction is even more for the former since the number of unoccupied sub-bands is more than that of occupied ones. This reduction enables reduction in number of measurements required to achieve a given CS reconstruction quality or enables improvement in the quality of CS reconstruction given the same number of measurements, therefore gives birth to the improvement in detection performance of CWSS.
0.6 0.5
0.85
0.4
0.8
0.3
0.75
Curve 1 (No=0, Nu=0)
0.2
Curve 2 (No=12, Nu=0)
0.1
Curve 3 (No=0, Nu=52)
0 -20
Curve 4 (No=12, Nu=52) -10
0 10 Signal to noise rate (dB)
20
30
Figure 1. Detection probability versus SNR for various combinations of N o and N u ( 0.5 , Pfa 0.01 )
Probability of detection, Pd
Probability of detection, Pd
0.8
Here we give an explanation for those improvements introduced by PKOS in qualitative analysis. It is wellknown that the solution of traditional CS reconstruction is the one with the minimum number of nonzeros among infinite data-consistency candidates. When partial occupancy status is known in advance, the candidates will be restricted in a signal space smaller than that in traditional CS. Under the given simulation conditions stated in subsection V-A, traditional CS needs to search for solutions in C 128, 40 possible 24-dimensional
0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.01 0.02
0.8
Curve 2 (No=12, Nu=0) Curve 3 (No=0, Nu=52) Curve 4 (No=12, Nu=52 0.04 0.06 0.08 Probability of false alarm, Pfa
0.1
0.11
Figure 3. Receiver Operating Characteristic (ROC) curves for various combinations of N o and N u (SNR=0dB, compression rate 0.5 )
0.7 Probability of detection, Pd
Curve 1 (No=0, Nu=0)
0.6
C. CWSS-PKOS with Non-Ideal Prior Information on So
0.5 0.4 0.3
Curve 1 (No=0, Nu=0) Curve 2 (No=12, Nu=0)
0.2 0.1
Curve 3 (No=0, Nu=52) Curve 4 (No=12, Nu=52) 0.3
0.4
0.5 0.6 Compression rate,
0.7
Figure 2. Detection probability versus compression ratio for various combinations of N o and N u (SNR=0dB, Pfa 0.01 )
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In the procedure of acquiring partial occupancy status, especially from collaborative sensing nodes, incorrect prior information is unavoidable. In this subsection, we consider cases in which prior information on the known occupied part So is non-ideal, namely some sub-bands corresponding to So are unoccupied. To facilitate analysis, we assume Su in this subsection. Let N o , c and N o, w denote the size of correct and incorrect prior information on So , respectively. Thus, No No,c No, w .
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ratio for various combinations of N o , c and N o, w . It is shown in curve 2, 3, 4 and 5 that, as the size of incorrect prior information increases, the curves N o, w corresponding to detection performance are shifted to the bottom, which indicates a consistent degradation in detection performance when more incorrect prior information exist in So . In Fig. 4 and 5, it is noteworthy that the detection probabilities plotted in curve 2, 3 and 4 are greater than that plotted in curve 1 for all value of SNR and compression ratio , indicating an improvement in regard to the traditional CWSS provided the size of correct prior information is larger than that of incorrect prior information. This observation is corroborated by the comparison between curve 5 and curve 1 in Fig. 4 and 5. Comparing curve 1 and 5 in Fig. 4, it is interesting that, even when most of the sub-bands corresponding to So are unoccupied, the improvement in detection performance is still existed at relatively low SNR. 1 0.9
D. CWSS-PKOS with Non-Ideal Prior Information on Su In this subsection, we exemplify cases in which prior information on the known unoccupied part Su is nonideal, namely some sub-bands corresponding to Su are occupied. Similar to the above subsection, we set So in this subsection and use N u , c and Nu , w to denote the size of correct and incorrect prior information on Su , respectively. 1 0.9 0.8 0.7 0.6 0.5 0.4 Curve 1 (No=0, Nu=0)
0.3
Curve 2 (Nu,c=26, Nu,w=0)
0.7
0.2
Curve 3 (Nu,c=26, Nu,w=1)
0.6
0.1
Curve 4 (Nu,c=26, Nu,w=2)
0.5
0 -20
Curve 5 (Nu,c=26, Nu,w=4) -10
0 10 Signal to noise rate (dB)
0.4 Curve 1 (No=0, Nu=0)
0.3
Curve 3 (No,c=8, No,w=2)
0.1
Curve 4 (No,c=8, No,w=6)
30
Curve 5 (No,c=8, No,w=10) -10
0 10 Signal to noise rate (dB)
20
30 0.8
Figure 4. Detection probability versus SNR for various combinations of N o ,c and N o ,w ( 0.5 , Pfa 0.01 and Nu 0 )
0.65 0.6 0.55 0.5
0.7
0.6
0.5 Curve 1 (No=0, Nu=0)
0.4
Curve 2 (Nu,c=26, Nu,w=0) Curve 3 (Nu,c=26, Nu,w=1)
0.3
Curve 4 (Nu,c=26, Nu,w=2)
0.45
0.2
0.4 0.35 Curve 1 (No=0, Nu=0)
0.3
Curve 2 (No,c=8, No,w=0)
0.25
Curve 3 (No,c=8, No,w=2)
0.2
Curve 4 (No,c=8, No,w=6) Curve 5 (No,c=8, No,w=10) 0.3
20
Figure 6. Detection probability versus SNR for various combinations of N u ,c and Nu ,w ( 0.5 , Pfa 0.01 and No 0 )
Curve 2 (No,c=8, No,w=0)
0.2
0 -20
Probability of detection, Pd
larger than N o, w .
Probability of detection, Pd
Probability of detection, Pd
0.8
errors in the known occupied part So . This suggests that, in order to contain more correct prior information in So , more greedy strategy can be adopted while obtaining prior information on So , as long as N o , c is sufficiently
Probability of detection, Pd
Fig. 4 and 5 respectively illustrate the variations of probability of detection Pd versus SNR and compression
0.4
0.5 0.6 Compression ratio,
0.7
Figure 5. Detection probability versus for various combinations of N o ,c and N o ,w (SNR=0dB, Pfa 0.01 and Nu 0 )
According to the observations stated above, it is safe to conclude that the proposed CWSS approach is robust to © 2014 ACADEMY PUBLISHER
Curve 5 (Nu,c=26, Nu,w=4) 0.3
0.4
0.5 0.6 Compression ratio,
0.7
Figure 7. Detection probability versus for various combinations of Nu ,c and Nu ,w (SNR=5dB, Pfa 0.01 and No 0 )
Fig. 6 shows variations of detection probability versus SNR for different combinations of N u , c and Nu , w . For a given SNR, the probability of detection decreases as the size of incorrect prior information Nu , w increases, which indicates a consistent reduction in this quantity when more incorrect prior information exist in Su . By
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comparing cases ( Nu ,c 26 , Nu , w 1 ), ( Nu ,c 26 ,
Nu , w 2 ), ( Nu ,c 26 , Nu , w 4 ) with the case ( No 0 ,
Nu 0 ), it can be observed that detection performance degrades obviously even when a small amount of errors exist in Su , in comparison with that of traditional CWSS. Meanwhile this degeneration in detection performance occurs for all value of compression ratio, which is corroborated by variations of detection probability versus compression ratio for various combinations of N u , c and Nu , w depicted in Fig. 7. It can be seen from Fig. 6 and 7 that the detection performance of proposed approach is sensitive to errors in the known unoccupied part Su . This suggests that, in order to avoid containing incorrect prior information in Su , more conservative strategy should be adopted while obtaining Su . VI.
CONCLUSIONS
We studied the problem of compressed wideband spectrum sensing when partial occupancy status is known. In order to incorporate partially known occupancy status, a new CWSS model is presented and then a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization is proposed to solve the proposed model. Simulation results demonstrate that the introduction of partially known occupancy status, both the known occupied part and the known unoccupied part, enables improvement in detection performance, and the proposed CWSS-PKOS approach can exploit such prior information effectively. Moreover, performance improvement provided by partial occupied sub-bands is larger than that provided by the same proportion of the unoccupied sub-bands. It is also shown via simulations that the proposed approach is sensitive to errors in the known unoccupied part, however robust to errors in the known occupied part. These observations indicate that greedy and conservative strategy should be adopted to obtain prior information on the occupied and unoccupied sub-bands respectively. ACKNOWLEDGMENT This work was supported by the National Major Scientific and Technological Special Project (No. 2012ZX03006003-004). REFERENCES [1] Q. Zhao and B. M. Sadler, “A Survey of Dynamic Spectrum Access,” Signal Processing Magazine, IEEE, vol. 24, no. 3, pp. 79–89, 2007. [2] I. F. Akyildiz, F. Lo Brandon, and R. Balakrishnan, “Cooperative spectrum sensing in cognitive radio networks: A survey,” Physical Communication, vol. 2011, no. 4, pp. 40–62, 2011. [3] L. Lu, X. Zhou, U. Onunkwo, and G. Y. Li, “Ten years of research in spectrum sensing and sharing in cognitive radio,” Wireless Communications and Networking, EURASIP Journal on, pp. 1–16, 2012.
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[4] J. N. Laska, W. F. Bradley, T. W. Rondeau, K. E. Nolan, and B. Vigoda, “Compressive sensing for dynamic spectrum access networks: Techniques and tradeoffs,” in 2011 IEEE International Symposium on Dynamic Spectrum Access Networks DySPAN, 2011, pp. 156–163. [5] R. G. Baraniuk, “Compressive Sensing [Lecture Notes],” Signal Processing Magazine, IEEE, vol. 24, no. 4, pp. 118–121, 2007. [6] E. J. Candes and M. B. Wakin, “An Introduction To Compressive Sampling,” Signal Processing Magazine, IEEE, vol. 25, no. 2, pp. 21–30, 2008. [7] M. A. Davenport and M. F. Duarte, “Introduction to Compressed Sensing,” Electrical Engineering, vol. 93, no. 3, pp. 1–68, 2011. [8] M. Marcus, J. Burtle, and N. Mcneil, “Federal communications commission spectrum policy task force report of the spectrum efficiency working group,” Spectrum, vol. 1, no. 1, p. 37, 2002. [9] D. Cabric, S. M. Mishra, and R. W. Brodersen, “Implementation issues in spectrum sensing for cognitive radios,” in Conference Record of the Thirty Eighth Asilomar Conference on Signals Systems and Computers 2004, vol. 1, pp. 772–776. [10] Z. Tian, “Compressed Wideband Sensing in Cooperative Cognitive Radio Networks,” in IEEE Global Telecommunications Conference GLOBECOM, 2008, no. 1, pp. 1–5. [11] F. Zeng, C. Li, and Z. Tian, “Distributed compressive spectrum sensing in cooperative multihop cognitive networks,” Selected Topics in Signal Processing, IEEE Journal of, vol. 5, no. 1, pp. 37–48, 2011. [12] C. J. Miosso, R. von Borries, M. Argaez, L. Velazquez, C. Quintero, and C. M. Potes, “Compressive Sensing Reconstruction With Prior Information by Iteratively Reweighted Least-Squares,” Signal Processing, IEEE Transactions on, vol. 57, no. 6, pp. 2424–2431, 2009 [13] N. Vaswani and W. Lu, “Modified-CS: Modifying Compressive Sensing for Problems With Partially Known Support,” Signal Processing, IEEE Transactions on, vol. 58, no. 9, pp. 4595–4607, 2010. [14] W. Lu and N. Vaswani, “Modified Basis Pursuit Denoising (modified-BPDN) for noisy compressive sensing with partially known support,” in, IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP), 2010, pp. 3926–3929. [15] C. Miosso, R. von Borries, and J. Pierluissi, “Compressive Sensing with Prior Information-Requirements and Probabilities of Reconstruction in l1-Minimization,” Signal Processing, IEEE Transactions on, vol. 61, no. 9, pp. 2150– 2164, 2013. [16] J. Scarlett, J. S. Evans, and S. Dey, “Compressed Sensing With Prior Information: Information-Theoretic Limits and Practical Decoders,” Signal Processing, IEEE Transactions on, vol. 61, no. 2, pp. 427–439, 2013. [17] R. G. Baraniuk, V. Cevher, M. F. Duarte, and C. Hegde, “Model-Based Compressive Sensing,” Information Theory, IEEE Transactions on, vol. 56, no. 4, pp. 1982–2001, 2010. [18] Z. Quan, S. Cui, A. H. Sayed, and H. V Poor, “Optimal Multiband Joint Detection for Spectrum Sensing in Cognitive Radio Networks,” Signal Processing, IEEE Transactions on, vol. 57, no. 3, pp. 1128–1140, 2009. [19] S Boyd, L Vandenberghe, Convex Optimization. Cambridge University Press, New York, 2008. [20] M. Grant, S. Boyd, “CVX: Matlab Software for Disciplined Convex Programming,” http://cvxr.com/cvx/ [21] J. F. Sturm, “Using sedumi 1.02, a matlab toolbox for optimization over symmetric cones,” Optimization Methods and Software, vol. 11, no. 1, pp. 625–653, 1999.
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[22] E. C and J. Romberg, “l1-magic: Recovery of sparse signals via convex programming,” 2005. [23] E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing Sparsity by Reweighted l1 Minimization,” Journal of Fourier Analysis and Applications, vol. 14, no. 5–6, pp. 877– 905, 2008.
Huang Jijun was born in 1970. He received the B.S. and M.S degrees in electromagnetic field and microwave technologies and the Ph.D. degree in electronic science and technology from the National University of Defense Technology (NUDT), Changsha, China, in 1993, 1997, and 2005, respectively. He is currently an Associate Professor in
Zha Song was born in 1987. He received the B.S. degree in communication and information system and M.S. degrees in signal and information processing from the Electronic Engineering institute (EEI), Hefei, China, in 2007 and 2010, respectively. He is now a Ph. D student in National University of Defense Technology (NUDT), Changsha, China. His current research interests include spectrum sensing, distributed compressed sensing.
NUDT. His current research interests include spectrum management and electromagnetic compatibility.
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Li Ning was born in 1983. He received the B.S. degree in communication and information system and M.S. degrees in military equipment from the Electronic Engineering institute (EEI), Hefei, China, in 2006 and 2010, respectively. He is now an Electronics Engineer in PLA Unit 77108, Chengdu, China.
JOURNAL OF NETWORKS, VOL. 9, NO. 4, APRIL 2014
Compressed Wideband Spectrum Sensing with Partially Known Occupancy Status by Weighted l1 Minimization Zha Song and Huang Jijun School of Electronic Science and Engineering, National University of Defense Technology, Changsha, China Email: [email protected], [email protected]
Li Ning Unit 77108, People's Liberation Army, Chengdu, China Email: [email protected]
Abstract—This paper considers the problem of compressed wideband spectrum sensing in wideband cognitive radio when partial occupancy status is known. While performing wideband spectrum sensing, incomplete prior information on the occupancy status may be obtained from coexisting narrowband detectors, collaborative sensing nodes or remote database. In this paper, we present a new optimization model in order to incorporate this prior information and then propose a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization to solve it. Numerical simulation results demonstrate that the use of partially known occupancy status leads to an improvement in detection performance and the proposed approach exploits such prior information effectively. As incorrect prior information is unavoidable in practical situation, cases in which the prior information is non-ideal are also investigated via simulations. Index Terms—Cognitive Radio; Wideband Spectrum Sensing; Compressed sensing; Partially Known Occupancy Status; Weighted l1 Minimization; Non-ideal Prior Information
I.
INTRODUCTION
Spectrum sensing, whose objectives are detecting signal of licensed users (LUs) and identifying the spectrum holes for dynamic spectrum access (DSA), is an important enabling technology for cognitive radio (CR), a leading choice for efficient utilization of spectrum resource [1]-[3]. Nowadays, wideband applications have received significant attentions since they not only offer high throughput, but also purport pronounced spectrum access opportunities for CR users. Meanwhile, wideband spectrum sensing (WSS) entails considerable challenges in practice, especially its very high signal acquisition costs [2]-[4]. Recently, compressed sensing (CS) theory [5]-[7] has been introduced to alleviate the heavy pressure on conventional analog to digital converter (ADC) technology in WSS by utilizing the low percentage of spectrum occupancy – a fact that motivates dynamic © 2014 ACADEMY PUBLISHER doi:10.4304/jnw.9.4.866-873
spectrum access [8], [9]. Compressed sensing theory shows that sparse or compressible signal can be reconstructed from much fewer samples than that suggested by the Shannon-Nyquist sampling theorem. Although existing frameworks of compressed wideband spectrum sensing (CWSS) [10], [11] show powerful ability of reducing signal-acquisition complexity, they are quite vulnerable to noise and the performance of CWSS degrades severely when signal to noise ratio (SNR) is low. Different from traditional sparse signal recovery algorithms in which sparsity is the only prior information on signal characteristic, other forms of prior information about the signal’s structure, such as partial support knowledge [12]-[15], support probability [16], connected tree structure [17], block-sparsity structure [17], etc., have been introduced into the reconstruction process. It has been demonstrated that the further exploitation of signal model, in addition to sparsity, would give birth to recovery performance enhancement. While performing wideband spectrum sensing, partial occupancy status may be known from coexisting narrowband detectors, collaborative sensing nodes or remote database [2]. In this paper, we study how to exploit partially known occupancy status (PKOS) to improve the detection performance of CWSS. In order to incorporate this type of prior information, a new CWSS model is presented and then a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization is proposed to solve it. The key idea of our proposed approach, named CWSS with PKOS (CWSS-PKOS), is to separated the spectrum whose occupancy status is known into two disjoint parts, the part known to be occupied and that known to be unoccupied, then impose relative small weights on the former which encourage nonzero entries in the reconstructed signal and relative large weights on the latter which discourage nonzeros. Simulation results demonstrate that the introduction of partially known occupancy status provides an improvement in detection performance of compressed wideband spectrum sensing, and that the
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proposed CWSS-PKOS approach can exploit such prior information effectively. In addition, we exemplify cases in which the prior information is non-ideal, namely some prior information is incorrect, and such cases are more significant in practical situation than cases with ideal prior information. The remainder of the paper is organized as follows. The signal model and the spectrum sensing problem of interest are explained in Section II and a typical representative of the traditional compressed wideband spectrum sensing approach is provided in Section III. In order to incorporate partially known occupancy status, Section IV presents the proposed CWSS model and the particular weighting strategy. Performance of proposed approach is demonstrated by numerical experiments in Section V and we draw conclusions in Section VI. Throughout this paper, boldfaced characters denote matrices and vectors. The superscripts of · represent T
the operations of transpose. The notation r
k
denotes the
lk norm of the vector r . For a set S , S denotes its size (cardinality). II.
goal of spectrum sensing task is to determine which subbands are occupied and it is equivalent to detecting the support of sparse or compressible signal r f . For sparse case, the support of r f is defined as follow:
supp r f
domain discrete versions. The relationship between rt and r f is given by
r f FN rt
(1)
where FN is the N -point unitary discrete Fourier transform (DFT) matrix. Many investigations have shown that the radio spectrum is in a very low utilization ratio [1]-[4]. Recently a survey of a wide range of spectrum utilization across 6GHz of spectrum in some places of New York demonstrated that the maximum utilization was only 13.1% [8]. Therefore, it is reasonable, by using sparse or compressible structure, to model the received wideband signal of CR, at any given time and spatial region, which suggests that the received signal is inherently sparse or compressible in frequency domain, i.e. r f is a sparse or compressible signal. This is exactly the motivation for introducing CS theory into WSS. As to hierarchical access model [1], overlay spectrum sharing protocol is adopted in which CR user avoids transmitting at any occupied sub-bands. In this case, the
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(2)
f
where r f [i] is the i -th element of r f . If r f is a compressible signal, the concept should be replaced by a% -energy support [13], which is defined as
supp r f
1 i N : r [i]
2
f
(3)
where is the largest real value for which supp r f
contains at least a% of the signal energy. Therefore the true occupancy status d 0,1
N1
,
whose i -th element indicates whether the corresponding sub-band is occupied or not, is given by
1, i supp r f di i 1, 0, i supp r f
SIGNAL MODEL
Consider a slot-segment model [10], [11], [18] of wideband spectrum, where the monitored spectrum is divided into N non-overlapping narrowband sub-bands (also known as slots). Despite that the locations of these sub-bands are known in advance, their power spectral density (PSD) levels are unknown and dynamically varying, depending on whether they are occupied or not. Those temporarily unoccupied sub-bands are termed spectrum holes and they are available for opportunistic spectrum access by CR users. Suppose that the Nyquistrate discrete form of received wideband signal at CR is denoted by an N 1 vector rt and r f is its frequency-
1 i N : r [i] 0
,N
(4)
where di 1 means that the i -th sub-band is occupied, while di 0 means that the i -th sub-band is unoccupied. III.
TRADITIONAL COMPRESSED WIDEBAND SPECTRUM SENSING
According to CS theory, compressed measurements are in fact the linear projections of the received signal rt onto a M N measurement matrix 1T ,
, MT
T
with
M N , where 1 , , M are sensing waveforms. It makes sense that only M samples need to be measured instead of N samples. The compression ratio, which is defined as M N , reflects the reduced number of samples M collected at CR receiver, with reference to the number N needed in full-rate Nyquist sampling. The matrix format for collecting compressed measurements can be formulated as
yt rt FN1r f
(5)
where FN1 is the inverse DFT matrix. Notice that, the inverse problem of (5) is an underdetermined problem and the high-dimensional signal r f cannot be recovered from low-dimensional measurements yt . However, it is shown in [5]-[7] that the reconstruction problem will stop being 1 underdetermined if the matrix FN satisfies restricted isometry property (RIP). It is proved that random matrix whose entries are chosen according to a Gaussian distribution will satisfy the RIP with high probability provided M is sufficiently large. More significantly, the
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RIP will be preserved for if is Gaussian random matrix and is arbitrary orthonormal basis. The problem of reconstructing the spectrum estimate rˆ f can be formulated as a combination of a l0 norm minimization and a linear measurement fitting constraint. However, the exact solution of l0 norm minimization is NP-hard and it requires an intractable combinatorial search. There are mainly two practical and tractable alternatives [6], [7]: greedy algorithms, such as various matching pursuits and convex relaxation algorithms. Both of them have advantages and disadvantages when applied to different scenarios. A brief assessment of their differences would be that convex relaxation algorithms require fewer measurements while greedy algorithms have less computation complexity. An approximate solution that is largely used is obtained by solving the following convex relaxation leading to l1 -norm minimization problem.
arg min r f
rˆ f
1
rf
(6)
1 N f
y t F r
s.t.
s.t.
arg min r f
1
rf
1 N f
y t F r
2
(7)
where bounds the amount of noise energy in the compressed measurements. Note that both of (6) and (7) are convex optimization problem [19] and a number of Matlab toolboxes, such as CVX [20], SeDuMi [21], l1 Magic [22], can be used to efficiently solve the problem. After we get spectrum estimate rˆ f from (7), the energy detection [2], [3] can be used to make the decision on spectrum occupancy status. The i -element of the decision dˆ is made as follow:
1, if rˆf i dˆ i for , i 1, ˆf i 0, if r
,N
(8)
where rˆ f i denotes the i -element of rˆ f and is a decision threshold which is chosen according to the desired probability of false alarm. After simple thresholding, the spectrum holes for DSA can be clearly given. IV.
THE PROPOSED COMPRESSED WIDEBAND SPECTRUM SENSING
It has been demonstrated that further exploitation of signal model, in addition to sparsity, would give birth to recovery performance enhancement [13]-[17]. As mentioned above, traditional CWSS exploits only the prior that the received signal is sparse or compressible in frequency domain, and does not assume any additional © 2014 ACADEMY PUBLISHER
by Su . Thus, So and Su are disjoint and S So Su . Therefore, a new CWSS model incorporating partially known occupancy status can be formulated as follow:
rˆ f
In practice the received signal is inevitably polluted by noise. In this case, an conic constraint is required, i.e. the optimization problem in (6) needs to be changed to
rˆ f
knowledge about the unknown sparse spectrum. However in the practical implementation of wideband spectrum sensing, there could be other knowledge about the monitored wideband spectrum. Incorporating additional knowledge would potentially improve spectrum sensing performance. In this paper, partial occupancy status is assumed to be known in advance from coexisting narrowband detectors, other collaborative sensing nodes or remote database, and we study how to exploit such prior information to improve spectrum sensing performance. Now we consider the spectrum reconstruction of r f with partially known occupancy status from compressed measurements yt . The part of sub-bands, whose occupancy status is known and index set is denoted by S , can be divided into two parts: the known occupied part indexed by So and the known unoccupied part indexed
arg min r f
1
rf
1 N f
y t F r
s.t.
d[i ] 1 d[j ] 0
2
(9)
i So j Su
for for
where d is the true occupancy status. According to the relationship between occupancy status of each sub-band and the support of r f as stated in (4), the optimization problem in (9) can be reformulated as follow:
rˆ f s.t.
arg min r f
1
rf
1 N f
y t F r
2
So supp r f Su
supp r
(10)
f
Unfortunately, the support set of r f is a thresholding function of r f which is unknown before reconstruction because it is scenario dependent. However, according to the definitions of support set as stated in (2) and (3), it is easy to see that signal values on its support set are relatively large while those outside are relatively small, even close to zero. Besides, for weighted l1 -norm minimization problem (6) and (7) are special cases where the weights are equal to 1), it has been shown that large weights can be used to discourage nonzero entries in the recovered sparse signal while small weights encourage nonzero entries [23]. According to those analyses, a particular weighting strategy in weighted l1 minimization reconstruction algorithm is proposed, in which small weights are imposed on the known occupied sub-bands which encourage relatively large entries in the recovered signal,
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at the same time, large weights on the known unoccupied ones which encourage relative small entries. Problem stated in (10) then becomes
rˆ f
arg min Wr f yt FN1r f
s.t.
1
rf
2
where the weight matrix W diag w1 , w2
l , if wk s , if 1, if
(11)
, wN with
k Su k So for k 1, k S
,N
(12)
and l , s denote pre-specified large and small constant, respectively. When spectrum estimate rˆ f is obtained from (11), decision on spectrum occupancy status can be also made via thresholding in (8). It is noteworthy that the weighted formulation of l1 minimization as stated in (11) is still a convex programming in which the problem remains simple and it effectively exploits the prior information on partially known occupancy status. Apparently, problem (11) will be reduced to the traditional case without any prior information when Su , So . In addition, problem (11) will be reduced to modified basis pursuit denoising (modified BPDN [14]) when Su and s 0 . In modified BPDN, partial support is assumed to be available, and the sparseness-inducing ( l1 -norm) term only contains entries outside the known support. In essence, the main difference between modified BPDN and (11) is that modified BPDN does not consider any additional knowledge outside the support, while (11) does. V.
SIMULATION RESULTS
In this section, simulation results are provided to illustrate performance of the proposed CWSS-PKOS approach with ideal and non-ideal prior information. First, the simulation setup and relevant performance metrics are described. Second, the performance of proposed approach with ideal prior information is evaluated by comparing with the traditional CWSS approach without any prior information as reference approach. Third, we consider cases in which some incorrect prior information exists in the known occupied part So and the known unoccupied part Su respectively. A. Simulation Setup and Performance Metrics Consider a monitored wide band partitioned into N 128 equal-bandwidth sub-bands and 24 among all sub-bands are randomly occupied by LUs, that is
supp r f
in
24 . Suppose that the amplitudes of elements supp r are generated according to Rayleigh f
distribution. The received signal is corrupted by additive white Gaussian noise (AWGN). The signal to noise ratio (SNR) is defined as the ratio of the average received
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signal to noise power over the entire wideband spectrum and is set to vary from -20dB to 30dB. For compressed sensing, the number of compressed samples M is set to range from 36 to 96. As to selecting parameter of weight matrix in (11), we set l 103 to discourage nonzero entries in the reconstructed signal, and s 0 to encourage nonzero entries which can also reduce the number of variables in objective function of (11) and then reduce the computational complexity of solving problem (11). For notational simplicity, we introduce N o and N u to denote the size of the known occupied part So and the known unoccupied part Su
respectively. That is,
No So and Nu Su . For the spectrum hole detection problem, performance metrics of interest are the probabilities of detection Pd and false alarm Pfa , which are evaluated by comparing the estimated occupancy status dˆ with the true occupancy status d over all sub-bands, as follows: d&dˆ 0 Pd d0
Pfa
(~ d) & dˆ 0
~d
0
where & and ~ denote bitwise AND and bitwise NOT respectively. The l0 norm function here is used to calculate the number of nonzero elements. B. CWSS-PKOS with Ideal Prior Information In this subsection, we consider cases in which prior information is ideal, namely all components of So and
Su are correct. To show the improvement in detection performance obtained by incorporating partially known occupancy status and to verify the effectiveness of CWSS-PKOS in exploiting such prior information, variations of detection probability Pd versus SNR and compression ratio are plotted in Fig. 1 and 2 respectively. In the following figures, each point on the curve is the average of the detection probabilities over 800 Monte Carlo trials. In addition, the opportunities for DSA will lose if the desired false alarm probability is set to be too small while determining threshold for making the decision on spectrum occupancy status. Therefore, we apply the receiver operation characteristic (ROC) curve to illustrate the tradeoff between the probabilities of detection and false alarm. ROC curves for Pfa 0.01, 0.11 are given in Fig. 3 since this regime of false alarm probability is of practical interest for achieving rational opportunistic throughput in CR. It has been shown in Section IV that the proposed CWSSPKOS approach will be reduced to the traditional CWSS without any prior information when Su , So . Therefore, in the following part, curves for
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No 0, Nu 0 depict the detection performance of traditional CWSS. As shown in Fig. 1, 2 and 3, detection probabilities of proposed CWSS-PKOS when partial occupancy status is available (curve 2, 3 and 4) are greater than that of traditional CWSS (curve 1). This suggests that incorporating partially known occupancy status, both the known occupied part and the known unoccupied part, indeed enables improvement in detection performance. Moreover, it is seen in Fig. 1 that compared to the improvement in high SNR regime, improvement in low SNR regime is much larger. This is because at high SNRs, such as 25dB and 30dB, the recovered signal obtained by solving (11), even when without any prior information, is quite closed to the original one. In simulations, there are 24 occupied sub-bands and 104 unoccupied sub-bands, thus No 12 and Nu 52 represent 50% of the occupied and unoccupied sub-bands respectively. In Fig. 1, Fig. 2 and Fig. 3, we can see that there is a big gap between curve 2 and curve 3. It suggests that, for a same proportion of occupied and unoccupied sub-bands, performance improvement provided by the former is larger than that provided by the latter. 1 0.9
0.7
subspaces, where C n, r denotes the number of combinations of ’ n ’ things selected ’ r ’ at a time. If the occupied sub-bands are partially known such that 12 subbands are known to be occupied, we only need to search for solutions in C 116,12 possible 12-dimensional subspaces. If 52 sub-bands are known to be unoccupied, we need to search for solutions in C (76, 24) possible 24dimensional subspaces. Clearly, the search space is reduced as long as occupied or unoccupied sub-bands are partially known. For a same proportion of occupied and unoccupied sub-bands, the reduction is even more for the former since the number of unoccupied sub-bands is more than that of occupied ones. This reduction enables reduction in number of measurements required to achieve a given CS reconstruction quality or enables improvement in the quality of CS reconstruction given the same number of measurements, therefore gives birth to the improvement in detection performance of CWSS.
0.6 0.5
0.85
0.4
0.8
0.3
0.75
Curve 1 (No=0, Nu=0)
0.2
Curve 2 (No=12, Nu=0)
0.1
Curve 3 (No=0, Nu=52)
0 -20
Curve 4 (No=12, Nu=52) -10
0 10 Signal to noise rate (dB)
20
30
Figure 1. Detection probability versus SNR for various combinations of N o and N u ( 0.5 , Pfa 0.01 )
Probability of detection, Pd
Probability of detection, Pd
0.8
Here we give an explanation for those improvements introduced by PKOS in qualitative analysis. It is wellknown that the solution of traditional CS reconstruction is the one with the minimum number of nonzeros among infinite data-consistency candidates. When partial occupancy status is known in advance, the candidates will be restricted in a signal space smaller than that in traditional CS. Under the given simulation conditions stated in subsection V-A, traditional CS needs to search for solutions in C 128, 40 possible 24-dimensional
0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.01 0.02
0.8
Curve 2 (No=12, Nu=0) Curve 3 (No=0, Nu=52) Curve 4 (No=12, Nu=52 0.04 0.06 0.08 Probability of false alarm, Pfa
0.1
0.11
Figure 3. Receiver Operating Characteristic (ROC) curves for various combinations of N o and N u (SNR=0dB, compression rate 0.5 )
0.7 Probability of detection, Pd
Curve 1 (No=0, Nu=0)
0.6
C. CWSS-PKOS with Non-Ideal Prior Information on So
0.5 0.4 0.3
Curve 1 (No=0, Nu=0) Curve 2 (No=12, Nu=0)
0.2 0.1
Curve 3 (No=0, Nu=52) Curve 4 (No=12, Nu=52) 0.3
0.4
0.5 0.6 Compression rate,
0.7
Figure 2. Detection probability versus compression ratio for various combinations of N o and N u (SNR=0dB, Pfa 0.01 )
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In the procedure of acquiring partial occupancy status, especially from collaborative sensing nodes, incorrect prior information is unavoidable. In this subsection, we consider cases in which prior information on the known occupied part So is non-ideal, namely some sub-bands corresponding to So are unoccupied. To facilitate analysis, we assume Su in this subsection. Let N o , c and N o, w denote the size of correct and incorrect prior information on So , respectively. Thus, No No,c No, w .
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ratio for various combinations of N o , c and N o, w . It is shown in curve 2, 3, 4 and 5 that, as the size of incorrect prior information increases, the curves N o, w corresponding to detection performance are shifted to the bottom, which indicates a consistent degradation in detection performance when more incorrect prior information exist in So . In Fig. 4 and 5, it is noteworthy that the detection probabilities plotted in curve 2, 3 and 4 are greater than that plotted in curve 1 for all value of SNR and compression ratio , indicating an improvement in regard to the traditional CWSS provided the size of correct prior information is larger than that of incorrect prior information. This observation is corroborated by the comparison between curve 5 and curve 1 in Fig. 4 and 5. Comparing curve 1 and 5 in Fig. 4, it is interesting that, even when most of the sub-bands corresponding to So are unoccupied, the improvement in detection performance is still existed at relatively low SNR. 1 0.9
D. CWSS-PKOS with Non-Ideal Prior Information on Su In this subsection, we exemplify cases in which prior information on the known unoccupied part Su is nonideal, namely some sub-bands corresponding to Su are occupied. Similar to the above subsection, we set So in this subsection and use N u , c and Nu , w to denote the size of correct and incorrect prior information on Su , respectively. 1 0.9 0.8 0.7 0.6 0.5 0.4 Curve 1 (No=0, Nu=0)
0.3
Curve 2 (Nu,c=26, Nu,w=0)
0.7
0.2
Curve 3 (Nu,c=26, Nu,w=1)
0.6
0.1
Curve 4 (Nu,c=26, Nu,w=2)
0.5
0 -20
Curve 5 (Nu,c=26, Nu,w=4) -10
0 10 Signal to noise rate (dB)
0.4 Curve 1 (No=0, Nu=0)
0.3
Curve 3 (No,c=8, No,w=2)
0.1
Curve 4 (No,c=8, No,w=6)
30
Curve 5 (No,c=8, No,w=10) -10
0 10 Signal to noise rate (dB)
20
30 0.8
Figure 4. Detection probability versus SNR for various combinations of N o ,c and N o ,w ( 0.5 , Pfa 0.01 and Nu 0 )
0.65 0.6 0.55 0.5
0.7
0.6
0.5 Curve 1 (No=0, Nu=0)
0.4
Curve 2 (Nu,c=26, Nu,w=0) Curve 3 (Nu,c=26, Nu,w=1)
0.3
Curve 4 (Nu,c=26, Nu,w=2)
0.45
0.2
0.4 0.35 Curve 1 (No=0, Nu=0)
0.3
Curve 2 (No,c=8, No,w=0)
0.25
Curve 3 (No,c=8, No,w=2)
0.2
Curve 4 (No,c=8, No,w=6) Curve 5 (No,c=8, No,w=10) 0.3
20
Figure 6. Detection probability versus SNR for various combinations of N u ,c and Nu ,w ( 0.5 , Pfa 0.01 and No 0 )
Curve 2 (No,c=8, No,w=0)
0.2
0 -20
Probability of detection, Pd
larger than N o, w .
Probability of detection, Pd
Probability of detection, Pd
0.8
errors in the known occupied part So . This suggests that, in order to contain more correct prior information in So , more greedy strategy can be adopted while obtaining prior information on So , as long as N o , c is sufficiently
Probability of detection, Pd
Fig. 4 and 5 respectively illustrate the variations of probability of detection Pd versus SNR and compression
0.4
0.5 0.6 Compression ratio,
0.7
Figure 5. Detection probability versus for various combinations of N o ,c and N o ,w (SNR=0dB, Pfa 0.01 and Nu 0 )
According to the observations stated above, it is safe to conclude that the proposed CWSS approach is robust to © 2014 ACADEMY PUBLISHER
Curve 5 (Nu,c=26, Nu,w=4) 0.3
0.4
0.5 0.6 Compression ratio,
0.7
Figure 7. Detection probability versus for various combinations of Nu ,c and Nu ,w (SNR=5dB, Pfa 0.01 and No 0 )
Fig. 6 shows variations of detection probability versus SNR for different combinations of N u , c and Nu , w . For a given SNR, the probability of detection decreases as the size of incorrect prior information Nu , w increases, which indicates a consistent reduction in this quantity when more incorrect prior information exist in Su . By
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comparing cases ( Nu ,c 26 , Nu , w 1 ), ( Nu ,c 26 ,
Nu , w 2 ), ( Nu ,c 26 , Nu , w 4 ) with the case ( No 0 ,
Nu 0 ), it can be observed that detection performance degrades obviously even when a small amount of errors exist in Su , in comparison with that of traditional CWSS. Meanwhile this degeneration in detection performance occurs for all value of compression ratio, which is corroborated by variations of detection probability versus compression ratio for various combinations of N u , c and Nu , w depicted in Fig. 7. It can be seen from Fig. 6 and 7 that the detection performance of proposed approach is sensitive to errors in the known unoccupied part Su . This suggests that, in order to avoid containing incorrect prior information in Su , more conservative strategy should be adopted while obtaining Su . VI.
CONCLUSIONS
We studied the problem of compressed wideband spectrum sensing when partial occupancy status is known. In order to incorporate partially known occupancy status, a new CWSS model is presented and then a particular weighting strategy in the reconstruction algorithm based on weighted l1 minimization is proposed to solve the proposed model. Simulation results demonstrate that the introduction of partially known occupancy status, both the known occupied part and the known unoccupied part, enables improvement in detection performance, and the proposed CWSS-PKOS approach can exploit such prior information effectively. Moreover, performance improvement provided by partial occupied sub-bands is larger than that provided by the same proportion of the unoccupied sub-bands. It is also shown via simulations that the proposed approach is sensitive to errors in the known unoccupied part, however robust to errors in the known occupied part. These observations indicate that greedy and conservative strategy should be adopted to obtain prior information on the occupied and unoccupied sub-bands respectively. ACKNOWLEDGMENT This work was supported by the National Major Scientific and Technological Special Project (No. 2012ZX03006003-004). REFERENCES [1] Q. Zhao and B. M. Sadler, “A Survey of Dynamic Spectrum Access,” Signal Processing Magazine, IEEE, vol. 24, no. 3, pp. 79–89, 2007. [2] I. F. Akyildiz, F. Lo Brandon, and R. Balakrishnan, “Cooperative spectrum sensing in cognitive radio networks: A survey,” Physical Communication, vol. 2011, no. 4, pp. 40–62, 2011. [3] L. Lu, X. Zhou, U. Onunkwo, and G. Y. Li, “Ten years of research in spectrum sensing and sharing in cognitive radio,” Wireless Communications and Networking, EURASIP Journal on, pp. 1–16, 2012.
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Huang Jijun was born in 1970. He received the B.S. and M.S degrees in electromagnetic field and microwave technologies and the Ph.D. degree in electronic science and technology from the National University of Defense Technology (NUDT), Changsha, China, in 1993, 1997, and 2005, respectively. He is currently an Associate Professor in
Zha Song was born in 1987. He received the B.S. degree in communication and information system and M.S. degrees in signal and information processing from the Electronic Engineering institute (EEI), Hefei, China, in 2007 and 2010, respectively. He is now a Ph. D student in National University of Defense Technology (NUDT), Changsha, China. His current research interests include spectrum sensing, distributed compressed sensing.
NUDT. His current research interests include spectrum management and electromagnetic compatibility.
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Li Ning was born in 1983. He received the B.S. degree in communication and information system and M.S. degrees in military equipment from the Electronic Engineering institute (EEI), Hefei, China, in 2006 and 2010, respectively. He is now an Electronics Engineer in PLA Unit 77108, Chengdu, China.
