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Cooperative Compressed Sensing for Wide-BandSpectrum Detection with Sequential Measurements
Bin Gu, Zhen YangInstitute of Signal Processing and Transmission
Nanjing University of Posts and TelecommunicationsNanjing, 210003 China
Abstract Compressed sensing is a novel technology on signalinformation processing. It offers a new wide-band spectrumdetection scheme in cognitive radio. A major challenge of thisscheme is how to determinate the required measurements whilethe signal sparsity is not known a priori. This paper presents acooperative detection scheme based on sequential compressedsensing where sequential measurements are collected from the
analog-to-information converters. A novel cooperativecompressed sensing recovery algorithm named SSAMP is utilizedfor sequential compressed sensing in order to estimate thereconstruction errors and determinate the minimal number of required measurements. Once the fusion center obtains enoughmeasurements, the reconstruction spectrum sparse vectors arethen used to make a decision on spectrum occupancy.Simulations corroborate the effectiveness of the estimation anddetection performance of our cooperative scheme. Meanwhile,the performance of SSAMP algorithm and SOMP algorithm isevaluated by MSE and detection time.
Keywords-cognitive radio; wide-band spectrum detection; sequential compressed sensing; matching pursuit
I. INTRODUCTION In cognitive radio (CR) systems, secondary users (SUs)
need to sense the spectrum reliably to check whether it is beingused by primary users (PUs). However, spectrum sensing inCR can be a very challenging task due to the wide frequencybandwidth, potentially up to several GHz. Compressed sensing(CS) provides a way to sense sparse or compressible signalsefficiently [2], [3]. According to CS theories, the characteristicsof a discrete-time sparse signal can be completely captured bya number of projections over a random basis and reconstructedperfectly from these random projections. CS can be used as aframework to reduce the spectrum sensing rate in CR was firstintroduced in [4], where the authors first utilized CS to docoarse classification of the sparse spectrum at sub-Nyquist rate.Nevertheless, a high-speed ADC sampling at Nyquist rate isstill required when the received signal is wideband. And in [5],acquisition of the wide-band analog signal is performed usingan analog-to-information converter (AIC). An AIC directlyrelates to the idea of sampling at the information rate of thesignal. Practical approaches to the design of AIC have beenconsidered in [6]. However, as the sparsity level of the signal isoften not known a priori, it can be very challenging to estimatethe reconstruction error and choose the number of themeasurements in practical settings.
Another challenging question in compressed spectrumdetection is the development of fast reconstruction algorithmwith reliable accuracy and low computational complexity. Theorthogonal matching pursuit (OMP) [7] based on the idea of iterative greedy pursuit is one of the most popular recoveryalgorithms as it is faster and easier to implement. But itrequires more measurements for perfect reconstruction as itlacks provable reconstruction quality. More recently, greedyalgorithms such as the subspace pursuit (SP) [8] and thecompressive sampling matching pursuit (CoSaMP) [9]proposed by incorporating the idea of backtracking offercomparable theoretical reconstruction quality as that of the LPmethods and low reconstruction complexity. However, both theSP and the CoSaMP assume that the sparsity level K is known,whereas K may not be available in many practicalapplications. As a result, a new algorithm must be proposed totake the advantage of both OMP and CoSaMP.
In this paper, we consider a cooperative wind-bandspectrum detection scheme based on sequential compressedsensing where one is able to get measurements in sequencefrom AIC. We have a centralized fusion center to collectmeasurements and each individual SU acquires the same wide-band signal from the licensed system. A cooperative CSreconstruction algorithm named simultaneous sparsity adaptivematching pursuit (SSAMP) is proposed to performcomputations in between measurements to decide whetherenough measurements have been obtained when the sparsitylevel K is unknown. Finally, the reconstruction sparse vectorsare used to make a decision on spectrum occupancy.
The remainder of this paper is organized as follows. Sectionintroduces the CS background and the principle of
sequential CS. Section describes the spectrum detectionscheme based on sequential CS for single analog signals andthe cooperative scheme using SSAMP. Simulation results areshown in section and conclusions are made in section .
II. COMPRESSED SENSING WITH SEQUENTIALMEASUREMENTS
A. Compressed Sensing Background We consider a 1 N vector of discrete-time signal x that is
K -sparse or compressible in some N N basis matrix
This work was supported by National High Technology Research andDevelopment Program Grant No. 2009AA01Z241 and National NaturalScience Foundation No. 60971129
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with the vectors { }i as columns. Signal x can be expressedas
1
or N
i ii
v=
= x x = v (1)
where v has only K non-zero elements with K N = - .
III. COOPERATIVE W IDE -BAND SPECTRUMDETECTION
A. Compressed Spectrum Detection with Sequential MeasurementsWe begin by describing the CS acquisition and recovery
scheme for a single SU case. The analog signal ( ) f t issampled using an AIC [6]. In the ideal case, the AIC is a linearsystem that maps a continuous-time signal to a discretesequence of samples. We consider x as the discrete signal of
( ) f t and the wide-band CR network with a total of N Hz inthe frequency range. When the discrete form x occupies P channels, every spectrum channel contains / N P spectrumpoints. Suppose that all spectrum point in each channel havethe same power spectral density (PSD) level. Define the N N matrix
(1)
1 1/ 2
1 1/ 2
1 1/ 2
1 1/ 2
N
N N
-
= -
W
M M
M M (8)
where (1) N W is the level 1 discrete wavelet transform (DWT)matrix. Suppose that the wavelet level is L , and the level l DWT matrix is
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1
( 1)( ) ( 1) /(2 ) for 2,3, , .
ll
l l N N N l L
--
-
= =
W 0W W
0 1L (9)
The sparse vector and the signal vector are related as follows( ) L
N x = Fs = FW v (10)
in which v has only K non-zero elements. Therefore, thesignal is sparse because of the number of non-zero elementsK N (15)
Corresponsively, the false alarm probability f P will use ORrule to detect all the P channels.
01
1 (1 Pr( | )).P
p p f
pP V H l
== - - > (16)
B. Cooperative Compressed Spectrum DetectionIn practice, the common sparse supported among the J
SUs enables a fast iterative algorithm to recover all of thesignals jointly. Distributed versions of CS have beenconsidered in [10] in order to exploit the underlying correlationstructures in the measurements. Recovery in distributedcompressed sensing can be done using algorithms likesimultaneous orthogonal matching pursuit (SOMP). However,it has been observed that SOMP based on OMP lacks provablereconstruction quality. Meanwhile, CoSaMP offers comparabletheoretical reconstruction but need to the sparsity level. Apractical algorithm must be adaptive to sparsity level and
robust to noisy measurements so as to detect wide-bandspectrum at low SNR. In this paper, we proposed one suchalgorithm, called simultaneous compressive sampling matchingpursuit (SSAMP) which is partly based on SAMP [12].
Let ( ) j f t be the wide-band analog signal received at the
-th j SU. Each SU processes the received signal using theacquisition scheme to obtain a vector of the measurements, asin the CS acquisition step described in Section . For the -th j SU, denote the corresponding sequential measurements
, 0,1, 2 M nT j n+ =y L ; the minimum number M measurements
are sent to the fusion center for a start. The fusion centerapplies a SSAMP algorithm to jointly reconstruct the J received sparse vectors M j jv of the spectrum. When we recover
jv either exactly or to a given tolerance, the sparse vectors arethen used for wide-band spectrum detection. The structure of cooperative spectrum detection based on CS is shown in Fig. 1.
Figure 1. Structure of cooperative spectrum detection based on SSAMP
We define T as the column sub matrix of whosecolumns are listed in the set T and suppose that all the J SUsobtain M measurements. The procedure of SSAMP algorithmis described as follows:
21 1 1, , M M T M T + +
y y y L
2, , M M T M T J J J + +
y y y L
1v
J v
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Figure 3. Detection performance versus compression rate. (1000 completetrials with SNR=10dB)
We compare the SOMP and SSAMP algorithm in Fig. 4with MSE (Top) and detection time (Bottom) respectively. Wecan see that the signal spectrum recovery quality of our
SSAMP algorithm improves faster than SOMP algorithm asthe compression rate / M N increases, which means thatSSAMP has better anti-noise performance, especially when
/ M N is smaller than 0.25. Note that when 1F S = , SSAMPcan be roughly regarded as the (generalized) SOMP although itmay require two tests to achieve more accuracy [12]. Therunning time of SSAMP is a little more than SOMP when
/ M N is more than 0.25. When 1F S > , each stage in theSSAMP still uses a similar principle of the CoSaMP (e.g.
F K S= ). The running time of SSAMP is ( / ) F KMN SO whileSOMP is ( )KMN O [12]. As a result, the novel algorithmSSAMP can also reduce the detection time comparing to theSOMP.
Figure 4. (Top) Mean-square Estimation Errors of SOMP and SSAMPversus compression rate. (Bottom) Average detection time of SOMP andSSAMP versus compression rate. (1000 complete trials with SNR=10dB)
V. CONCLUSIONS In this paper, we presented a cooperative compressive
wide-band spectrum detection scheme based on sequentialcompressed sensing where one is able to get measurements insequence from AIC. A novel algorithm named SSAMP isutilized to reconstruct the sparse vectors and determinatewhether enough measurements have been obtained without anya priori assumption on signal sparsity level. When we recoverthe signal either exactly or to a given tolerance from thesmallest possible number of random measurements, thereconstruction spectrum sparse vectors are then used to make adecision on spectrum occupancy. The simulation result showsthat the estimated error reliably indicate the reconstructionerror after a small delay of T additional measurements and theperformance of cooperative detection scheme is much betterthan a single spectrum detection scheme. Performanceevaluation using MSE and detection time is showed that theSSAMP algorithm has better anti-noise characteristics andshorter detection time compared to SOMP algorithm.
REFERENCES [1] S. Haykin. Cognitive radio: brain-empowered wireless communications
. IEEE Journal on Selected Area in Communication, Vol.23, pp.201-220, 2005.
[2] E. Candes, J. Romberg, T. Tao. Robust uncertainty principles: Exactsignal reconstruction from highly incomplete frequency informationIEEE Trans. on Information Theory, Vol.52, pp.489-509, 2006.
[3] D. L. Donoho. Compressed Sensing . IEEE Trans. on InformationTheory, Vol.52, pp.1289-1306, 2006.
[4] Z. Tian, G. B. Giannakis. Compressed sensing for wideband cognitiveradios. International Conference on Acoustics, Speech, and SignalProcessing, Honolulu, HI, USA, pp.1357-1360, Apl.15-20, 2007,.
[5] Y. L. Polo, Y. Wang, A. Pandharipande, et al. Compressive Wide-BandSpectrum Sensing. International Conference on Acoustics, Speech, andSignal Processing, San Diego, CA, USA, pp.178-183, Feb. 8-13, 2009.
[6] J. Laska, S. Kirolos, Y. Massoud, et al. Random sampling for analog-to-digital information conversion of wideband signals. IEEE DallasCircuits and Systems Workshop, Richardson, TX, USA, pp119-122, Oct.2006.
[7] J. A. Tropp, A. C. Gilbert. Signal recovery from random measurementsvia orthogonal matching pursuit. IEEE Trans. on Information Theory,Vol.53, pp.4655-4666, 2007.
[8] W. Dai, O. Milenkovic. Subspace Pursuit for Compressive SensingSignal Reconstruction. IEEE Trans. on Information Theory, Vol.55,pp.2230-2249, 2009.
[9] D. Needell, J. A. Tropp. CoSaMP: Iterative signal recovery fromincomplete and inaccurate samples. Applied and ComputationalHarmonic Analysis, Vol.26, pp.301-321, 2009.
[10] M. F. Duarte, S. Sarvotham, D. Baron, et al. Distributed CompressedSensing of Jointly Sparse Signals. Asilomar Conference on Signals,Systems and Computers, Pacific Grove, CA, USA, pp.1537-1541, Oct.
28-Nov.1 2005.[11] D. M. Malioutov, S. R. Sanghavi, A. S. Willsky. Sequential
Compressed Sensing. IEEE Journal of Selected Topics in SignalProcessing, Vol.4, pp.435-444, 2010.
[12] T. T. Do, L. Gan, N. Nguyen, et al. Sparsity adaptive matching pursuitalgorithm for practical compressed sensing. Sparse Signals. AsilomarConference on Signals, Systems and Computers, Pacific Grove, CA,USA, pp.581-587, Oct. 28-Nov.1 2008.