 Research
 Open Access
 Published:
Efficient blind spectrum sensing for cognitive radio networks based on compressed sensing
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 306 (2012)
Abstract
Spectrum sensing is a key technique in cognitive radio networks (CRNs), which enables cognitive radio nodes to detect the unused spectrum holes for dynamic spectrum access. In practice, only a small part of spectrum is occupied by the primary users. Too high sampling rate can cause immense computational costs and sensing problem. Based on sparse representation of signals in the frequency domain, it is possible to exploit compressed sensing to transfer the sampling burden to the digital signal processor. In this article, an effective spectrum sensing approach is proposed for CRNs, which enables cognitive radio nodes to sense the blind spectrum at a subNyquist rate. Perfect reconstruction from fewer samples is achieved by a blind signal reconstruction algorithm which exploits ℓ_{ p }norm (0 < p < 1) minimization instead of ℓ_{1} or ℓ_{1}/ℓ_{2} mixed minimization that are commonly used in existing signal recovery schemes. Simulation results demonstrated that the ℓ_{ p }norm spectrum reconstruction scheme can be used to break through the bandwidth barrier of existing sampling schemes in CRNs.
Introduction
In cognitive radio networks (CRNs), spectrum sensing aims to identify the frequency support of a signal, which consists of spectrum intervals that the power of the signal exceeds that of noise[1]. Recently, many researchers have focused their attentions on spectrum sensing in CRNs, in which the cognitive radio (CR) nodes are able to perform the wideband spectrum sensing to detect the unoccupied frequency bands for temporal using. As a very promising technology in CRNs[2, 3], the compressed sensing theory can be used to alleviate the dynamic spectrum sensing problem by blindly detecting the spectrum holes[3–5].
The basic idea behind compressed sensing (CS in short) is to sample compressible signals at a lower rate than the traditional Nyquist, and then reconstruct these signals with compressed measurements[3]. In CS, the sampling and compression operations are combined into a low complexity compressed sampling[4], in which compressible signals can accurately be reconstructed from a set of random linear measurements by using nonlinear or convex reconstruction algorithms[6, 7]. Typically, the number of measurements in CS is much fewer than that in Nyquist sampling, thus leading to a significant reduction in sampling rates. Therefore, the requirements to analogtodigital converter resource can be reduced significantly, which is of great importance for wideband communication systems[4]. Previously, a lot of CSbased techniques have been proposed[3, 4]. The new CS theory is hoped to significantly reduce the sampling rate and computational costs at a CR node for compressible signals[8].
A compressible signal means that it can sparsely be represented in some basis, and can exactly be reconstructed only with a small set of random projections on an incoherent basis[8–10]. Recently, many research efforts have been done on random projection. The authors of[11] proposed a collaborative compressed spectrum sensing, where the compressed spectrum reconstruction is modeled with a gaussian process framework model. The authors of[12] investigate the problem of dynamic resource allocation in CRNs, where several CSbased techniques are used to detect occupied spectral bands from compressed measurements. For current CRNs, the CS has been used to alleviate the sampling bottleneck, which aims at decreasing the sampling rates for the acquisition of compressible signals[13, 14].
The CS techniques that have been used in spectrum sensing can be classified into two categories: (1) Convex relaxingbased methods, such as basis pursuit (BP)[15, 16] and Dantzig Selector[17]; (2) Greedy algorithmbased methods, such as matched pursuit (MP) and its variants[18]. Recently, many improved MPbased methods have been reported such as orthogonal matched pursuit[19], regularized orthogonal matched pursuit[20], compressive sampling matching pursuit[21], and so forth. Actually, the former or its variants can get higher reconstruction accuracy, however it may cause expensive computation costs. The greedy algorithmbased methods have less computing complex, however the reconstruction accuracy is limited compared with the convex programming. The basis pursuit denoising is commonly used in signal processing due to its additional denoising performance advantage[16, 22]. The advantages of the greedy algorithmbased approaches are fast, stable, uniform guarantees, however it requires a slightly stronger condition on the restricted isometry property (RIP) condition than first category[23].
The spectrum sensing in CRNs faces three main technical challenges: (1) The sampling rate, too high sampling rate may cause very high cost of signal processing and storage; (2) The design of radio frontend is very difficult, the computation intensive energy or feature detection operations are applied in many existing spectrum sensing methods. However, by using CSbased approach, the spectrum detection can be simplified; and (3) highspeed DSP that operates at or above the Nyquist rate is used in conventional spectrum estimation, which may cause failure of exactly signals reconstruction because of the high requirement on spectrum sensing timing windows[3].
In this article, we aim at developing an effective CSbased spectrum sensing approach at affordable complexity. First, we take a multicoset scheme to decompose the spectrum in CRNs. One of the goals of each CR is to effectively detect the unused spectrum holes while the spectrum sparsity is known a priori for the dynamic spectrum access of CRs. The cognitive spectrum sensing is decomposed into two stages: spectrum sensing and spectrum reconstruction. In spectrum sensing, the sensing time to find the spectrum holes is critical for the ‘cognition’ of the CRs. On the other hand, the spectrum recovery requires better antinoise performance. In order to cope with these challenges, we focus our works on the following issues:

1.
A CSbased spectrum sensing scheme is proposed which can adaptively sense the blind occupied bands with a sampling rate lower than that of Nyquist;

2.
In the spectrum reconstruction, we proposed an improved block sparse signal model, in which an approximate ℓ _{ p }norm (0 < p < 1) minimization is used to improve the reconstruction quality and speed spectrum.

3.
To further enhance the performance and reconstruction speed, an iterative weighted scheme is proposed to approximate the ℓ _{ p }norm optimization problem, by doing this the convergence speed can be enhanced in reconstruction.
Notation: For vectors/matrices the superscript, T, denotes transpose. A_{l,k} represents the (l,k)th element of a matrix A. ∥x∥ denotes the ℓ_{2}norm of vector x. In general, ∥x∥_{ p } denotes the ℓ_{ p }norm of x that is defined as${\parallel \mathbf{x}\parallel}_{p}={\left(\sum _{i=1}^{N}\right{\mathbf{x}}_{i}{}^{p})}^{1/p}$. The common notations that summarized in Table1 is used in this article.
System models
Signal model
In practice, a signal can always be sparsely or nearsparsely represented on a transform domain[24]. An Nlength signal x can easily be described as
where the N × 1 vector θ is the Ksparse representation of x, and K is the number of nonzero elements of θ (K ≪ N). Ψ = {ψ_{1},ψ_{2},…,ψ_{ N }} is an N × N transform matrix, and ψ_{ i }(i = 1 … N) is the similarly sampled basis function.
For a time window as t ∈ [0,T_{0}], N samples are necessary for Nyquist theorem to exactly reconstruct the power spectrum density.
where F is the discretetime Fourier transform (DFT) matrix and r_{ t } is the sample vector of a realvalue signal r(t) which needs to be reconstructed from samples x.
As mentioned above, the CS is able to accurately reconstruct signals only with a small portion of samples with size of M (M ≪ N)
in which y denotes an Mlength measurement vector, and Φ is the measurement matrix. The spectrum of r(t) can accurately be reconstructed when the measurement y is available. Because M ≪ N so the sample rate can be reduced significantly. Here, we aim at developing a spectrum sensing scheme with fewer nonadaptive measurements where Φ is well presented.
Each CR node is able to classify and estimate spectrum of a signal r(t) by using the sample set x[3]. Suppose that the total frequency range is available as B Hz, then each CR node periodically senses the spectrum environment to find spectrum holes for opportunistic use[25]. Following assumptions are made:

1.
The frequency boundaries are known to the CR and the bandwidth of the spectrum bands occupied by each CR is much less than B.

2.
The number of bands Q is known and their location are unknown to the CR nodes. In a time burst, the locations and the number of bands Q keep unchanged but may vary for different time bursts.

3.
The signal power spectrum density (PSD) over each spectrum subband B _{ n }is smooth, however the PSD over two neighboring bands are independent.
Signal sampling model and problem formulation
For a timecontinuous bandlimited signal r(t), its Fourier transform can be calculated as X(f) = x = Ψ θ, which is piecewise continuous in frequency f . We assume that the r(t) is a bandlimited signal in the nonzero frequencydomain support$\mathcal{F}$, and X(f) = 0 for$f\mathit{\notin}\mathcal{F}$.
We aim to exactly reconstruct r(t) from a set of samples based on following constraints: (1) r(t) is blind which means the locations of bands of r(t) are not available in signal acquisition and reconstruction; (2) The sampling rate should be minimal for exactly reconstruction; (3) The signal can be reconstructed with a high probability even when ambient additive and white noise involved.
In Nyquist sampling, the sample sequence x(nT) contains all the information about r(t). However, in CSbased sampling the uniform grid is divided into L consecutive sample blocks, which can implemented be by a constant set C. Assume the length of C is q and C includes the indices of q samples in each block ( e.g., q_{ i } denote the i th element). Then, the sequence of samples can be represented as
For a set C we have 0 ≤ c_{1} < c_{2} < ⋯ < c_{ q }≤ L − 1, it is easy to understand that the sampling rate is q/L·r_{NYQ}(r_{ NYQ } is the Nyquist sampling rate). Clearly, due to q < L the sampling rate is less than r_{NYQ}.
Having the estimation of frequency vector r_{ f }, it is necessary to detect the number and location of occupied bands of signal r(t). We use a modulated wideband converter (MWC)[5], which aims at sampling wideband sparse signal at a rate lower than that of Nyquist. Figure1 shows the MWC scheme.
Actually, the DFT${X}_{{c}_{i}}\left(f\right)$ of${x}_{{c}_{i}}\left[n\right]$ can be obtained according to Equation (5)
here 1 ≤ i ≤ q and$f\in \mathcal{F}$. Let x(f) denote vector${X}_{{c}_{i}}\left(f\right),({c}_{i}\in C)$ in frequency domain, then Equation (5) can be rewritten as
in which y(f) denotes a vector with length of q. It should be noted that the i th element of y(f) is${X}_{{c}_{i}}\left(f\right)$, and the matrix A is defined by
where the vector x(f) contains L unknown components for each$f\in \mathcal{F}$
in which the the multicoset sampling pattern C includes all sampling time offsets, which are distinct and positive values less than L[1]. It is crucial to properly select L, q, and C such that x(f) can be reconstructed from Equation (6). In this case, the multicoset sampling can provide the average sampling rate as$\frac{q}{\mathit{\text{LT}}}$ and the spectrum space can be extracted to bands with bandwidth$\frac{1}{\mathit{\text{LT}}}$.
Sequence x(f) is sparse since its Fourier property over frequency domain. However, under many scenarios the support set I(x(f)) is not available. Fortunately, it is possible to find a unique sparsest solution, and the authors of[7] have proved that if x_{ s }(f) is a solution for y(f) = Ax(f) when ∥x_{ s }(f) ∥_{0} ≤ σ (A)/2, then x_{ s }(f) is the unique solution. Here, σ(A) is the column rank of A. It is evident that the signal can perfectly be reconstructed when x(f) is σ(A)/2sparse.
Actually, the sparse level of A is related to the sampling coset pattern[1]. Because in multicoset strategy the value of Q B, and T are available, thus the signals can be reconstructed with a high probability as a CS problem. For a signal$x\left(t\right)\in \mathcal{M}$, if L ≥ 1/BT, C is a universal pattern, and q ≥ 2Q, then for every$f\in \mathcal{F}$, it is clear that vector x(f) is the unique solution of Equation (6).
Compressed spectrum sensing
For a normal signal, it is not difficult to find a sparse representation in a certain space[26]. Actually, signals involved in CRNs have been proved sparse in the frequency domain[26]. So, it is possible to find the unoccupied spectrum in CRNs with compressed spectrum sensing with a rate lower than Nyquist.
Let r denote the frequency response vector of signal r(t) that can be obtained by y = Φ F^{−1}r, in which F is the Fourier transform matrix and Φ is the measurement matrix.
It can be seen that Equation (9) is a nonconvex problem. According to the RIP constraint we have M ≥ c· M ·log(N/K) (c ∈ (0,1) is a universal constant). Equation (9) has a unique solution when Equation (10) holds
Actually, Equation (10) is a secondorder cone program and many software packages are available to solve this problem[9]. On the other hand, some variants of LASSO algorithm have been developed to deal with the noisy signals by minimizing the usual sum of squared errors.
where ε bounds the noise in signals. A number of convex optimization software packages have been developed to solve the LASSO problem, such as cvx, SeDumi, Yalmip, and so on[9]. Recently, the authors of[9] improved Equation (11) with a weighted scheme of LASSO
where signal r is separated into K subvectors, and the weight vector w = w_{1}w_{2},…,w_{ K }^{T}can be calculated according to p_{ i }—the subband power of the primary user existing in the i th subband as${w}_{i}=\frac{1}{{p}_{i}+\delta}$.
Previous methods take the sparsity of signals into consideration, and model the signal with CS theory in frequency domain based on ℓ_{1} or ℓ_{1}/ℓ_{2}mixed minimization.
Blind compressed spectrum sensing
In CRNs, it is a very challenging topic to design a spectrum blind samplingreconstruction system without knowing the locations of the bands. Actually, the bands occupied by different users may be discrete in CRNs, which makes it possible to design a spectrumblind reconstruction scheme by using CS based on a preceding multicoset model[27].
Let r_{ t } denote the Nyquist sample sequence of r(t), then we have y = Φ r_{ t }. It is possible to reconstruct the frequency response vector r_{ f } of r(t) from compressed samples y. On the other hand, we have r_{ f } = Ψ r_{ t }, in which Ψ is the DFT matrix. It is easy to understand that Equation (13) holds
As discussed in Section “System models”, the frequency response is sparse in CRNs, so Equation (13) can be solved with a twostep scheme: (1) use compressed measurements y to estimate the sparse frequency response r_{ f } (actually this is an illposed problem); (2) reconstruct signal r(t) according to the frequency response, which can be done by an inverse Fourier transfer.
It is easy to know that$\mathcal{F}\ll B$ which means r_{ f }is sparse in the frequency domain. In order to solve the first problem, it can be formatted by CS theory
In Section “Compressed spectrum sensing”, we have summarized several methods that proposed to solve this problem. For simplicity we use y and x to denote y(f) and x(f), respectively. Then rewrite Equation (6) as
in which A = F^{−1}ΦΨ, and F denotes the DFT matrix of compressed sample vector y.
Since x contains M unknown elements for each f , and
where X(·) is the Fourier transform of time shifted r(t). As analyzed above, in a CRN every band contributes only a few nonzero value, so x is a sparse vector which makes it possible to use CS theory to reconstruct spectrum of signals.
Let σ(A) denote the Kruskalrank of A[5]. When x is$\frac{\sigma \left(\mathbf{A}\right)}{2}$sparse then Equation (15) has a unique sparsest solution and the proof is given in[5]. It can be seen that reconstruction with high probability is possible for signal x that satisfies
As shown in Figure1, if the number of sampling cosets q is given, it can be proven that every signal$r\left(t\right)\in \mathcal{M}$ can perfectly be reconstructed by properly selecting parameters Q, B, and T.
Actually, if M ≤ 1/BT and q are greater than Q, then for every$f\in {\mathcal{F}}_{0}$ a unique Nsparse solution x is available according to Equation (15). According to Equation (16), it can be seen that x takes M values of X(f) by intervals of 1/MT, therefore the nonzero components are fewer than the number of bands in x(f). So we can say that x is Qsparse. Kruskalrank σ(A) = q, which implies that when q ≥ 2QB, x can perfectly be reconstructed without knowing any information about the locations of bands when$f\in {\mathcal{F}}_{0}$.
The blind reconstruction problem can be summarized as
A number of CS reconstruction algorithms are available to solve this problem, and many ℓ_{1}norm and ℓ_{1}/ℓ_{2}norm minimizationbased approaches have been proposed for the reconstruction of sparse signal. However, according to the original idea of compressive sensing, ℓ_{ p }norm minimization with p < 1 can improve the recovery performance for signals that are less sparse[24, 28, 29]. On the other hand, the ℓ_{ p }norm minimization offers good performance with reduced complexity.
Weighted blind spectrum reconstruction (WBSR)
Smoothly approximation of ℓ_{0}norm problem
In CS theory, ℓ_{0}norm is the ideal minimization; however, it is an NPhard problem to find out the sparest solution. In general, ℓ_{1}norm minimization, the contour of ∥x∥_{1} = k, grows and touches the hyperplane Φ x = y, yielding a sparse solution as shown in Figure2a. Unfortunately, this may cause a special situation that when the hyperplane is parallel to the contour of ∥x∥_{1} = k, infinite solutions can be derived, which may cause the unreliability of the system, as shown in Figure2b.
On the other hand, the ℓ_{2}norm minimization may fail to work in CS. It is due to the fact that the contour of ℓ_{2}norm grows and touches the hyperplane, yielding an unsparse solution, which is no sense in solving our problem.
Our goal is to find out the sparest solution, which can be measured by using its ℓ_{0}norm pseudonorm. Unfortunately, the ℓ_{0}norm minimization problem is nonconvex with combinatorial complexity. An effective signal reconstruction strategy is to solve the ℓ_{ p }norm minimization problem as
in which 0 < p < 1. The ℓ_{ p }norm minimization problem is nonconvex. However, in ℓ_{ p }norm minimization when the contour of ∥x∥_{ p }grows and touches the hyperplane Φ x = y yields sparse solution, as shown in Figure3c. The possibility that the contour will touch the hyperplane at another point is eliminated.
In order to improve the convergence speed of the ∥x∥_{ p }problem, we build a differentiable and continuous function to approximate the ∥x∥_{0} problem as
in which σ is a very small constant that is used to guarantee the differentiability of Equation (20). It is easy to see that when x_{ i } is 0,$1{e}^{{x}_{i}^{2}/2{\sigma}^{2}}$ approximates to zero. When x_{ i } is a nonzero value, then$1{e}^{{x}_{i}^{2}/2{\sigma}^{2}}$ approximates to 1, so this function can approximate to ℓ_{0}norm problem smoothly.
In practice, the condition M ≥ c · K · log(N/K) is restrictive for signal spectrum reconstruction. Based on the analysis in above sections, we reformulate the blind reconstruction problem with ℓ_{ p }norm (0 < p < 1) minimization approach as
where ∥r_{ f }∥_{ p } is with 0 ≤ p < 1. For p < 1, Equation (21) becomes a nonconvex problem which has multiple solutions. If this problem can be solved with sufficient, then improved results can be available. We rewrite a smooth ℓ_{ p }norm of r_{ f } in Equation (22)
in which σ > 0 is a small constant. Equation (22) can be solved with steepestdescent approach. It can be seen that this optimization problem can offer accurate spectrum reconstruction performance with reduced computation complexity.
Signal reconstruction by ℓ_{ p }norm minimization
In this section, we will present an effective method to reconstruct signals by using ℓ_{ p } minimization. From the CS theory, the solutions of y = A r_{ f }can be represented as[29]
where${\mathbf{r}}_{f}^{\ast}$ is a special solution to Ar_{ f }= y, which can be calculated as${\mathbf{r}}_{f}^{\ast}={\mathbf{A}}^{T}{\left({\mathbf{AA}}^{T}\right)}^{1}$. V_{ r } is an N × (N − M) matrix and ξ is a N − M× 1 vector. V_{ r } can be calculated by QR decomposition of matrix A, then we can further rewritten the ℓ_{ p }norm problem as
where${\mathbf{v}}_{i}^{T}$ is the i th row of matrix V_{ r }. It is clear that Equation (24) is a differentiable function, so its gradient can be obtained[29], which reduce the problem size from N to N − M. So, a number of existing approaches are available to solve Equation (24) as an unconstrained optimization problem, such as quasiNewton[24], BroydenFletcherGoldfarbShanno (BFGS)[29] and so on.
Weighted approximation of ℓ_{ p }norm algorithm
Similar to ℓ_{1}norm or mixed ℓ_{1}/ℓ_{2}norm based reconstruction algorithms, ℓ_{ p }norm also brings the dependence on the power in each subband. In this section, we apply a weighted bands constraint to deal with this imbalance. For q ≥ 2NB, Equation (22) can be reduced to
in which${\mathbf{r}}_{f}^{\ast}\left(i\right)$ is the i th row of special solution and${\mathbf{v}}_{i}^{T}$ denotes the i th row of matrix V_{ r }. It can be seen that Equation (25) remains differentiable, so it can be easily solved with gradient descent method.
As analyzed in Section “System models”, in wideband spectrum sensing, the blind sensing problem can be formulated as Equation (9), where r_{1},r_{2},…,r_{ K }are K subsets of r with different length (blocklength), which corresponds to the bands of multicoset spectrum dividing scheme. It is clear that the power of each band may be different, which depends on the power of the primary user existing in the i th subband. So it can be reformulated as a weighted bands problem:
The weights w_{ i }, i = 1, …, K can be calculated according to the existing subband power (ESP) e_{ i },${w}_{i}=\frac{1}{{e}_{i}+\delta}$, here δ is a small constant (about 10^{−4}) which is used to guarantee w_{ i } to be nonconvex. The initial condition of the recursive relation is w_{ i }= 1, i ∈ {1, …, K}, which means that in the first step all the blocks are weighted equally.
With the increase of the iteration times, larger values of ESP in the i th subband are penalized lighter than smaller values of ESP. A threshold can be applied to terminate the iteration at the proper time, we have
in which${\mathbf{r}}_{f}^{l}$ is the estimated frequency spectrum at the l th iteration and ε bounds the iteration residual.
By this way, a compressible signal r_{ f } can effectively be reconstructed with measurement Ar_{ f }= y. The most innovation feature of WBSR is that it is able to perfectly reconstruct signals without prior information of the sparsity. Compared with previous methods, the advantages of WBSR also are fast, stable, and uniform guarantees.
Simulation
To evaluate the proposed method, we simulate the system on test signals contaminated by white Gaussian noise. We consider bands in the ISM bands with a frequency range from 2.4–2.4835 GHz. In general, 2.4–2.4835 GHz spectrum bands are shared by many wireless devices, such as home microwave oven, wireless sensor networks (Zigbee), WLANs (IEEE 802.11), Bluetooth devices (IEEE 802.15.1), cordless phones, wireless USB device, and so on. In CRNs, the primary users should be tolerant of ISM emission in these bands. The unlicensed second users are able to utilize these bands without causing interference to primary users.
In simulations, we consider a Bluetooth signal in ISM bands, and ignore the baseband protocol for Bluetooth chip. First, we evaluate the performance on 2,048 noisy Bluetooth signals, in which the noise is a white Gaussian noise process. The signal is constructed using the formula (QPSK):
For simplicity, the energy coefficients E = 1, the time offsets τ = 0.5μs, Q = 1, and B = 2 MHz. Because Bluetooth defines 79 channels, each channel being separated by 1MHz Bluetooth’s transmitted signals are spread across this 2.4 GHz band and the specification allows for 1600 frequency hops per second. In Bluetoothbased communications, because the information is spread across a number of frequency channels.
In simulation, we use ε as 10^{−4}. According to the basic idea of CS theory, the sampling is significantly reduced by 60%. To compare the results, we use BP and LASSO schemes under the same scenario. In Figure3 the comparison results are given. In each subfigure, the original signal and its spectrum are compared with the reconstructed signal and spectrum. The average squared errors are 0.3029, 0.0254, and 0.0087 for BP, LASSO, and WBSR, respectively, which are obtained according to Equation (29)
The performance of spectrum sensing of the proposed scheme is depicted in Figure4, where the estimated transmit power is shown. It is seen that the PUs start the transmission around t = 400 with 1 W of power. Note that the WBSR algorithm can correctly identify the presence of the primary user activities with the LASSO based scheme, which sets noise at 10 dB. It can be seen that the WBSR outperforms BP and LASSO based schemes in spectrum sensing and signal reconstruction.
Compared with the existing schemes as shown in Figure4, WBSR significantly improves the spectrum sensing speed by ignoring the edge detection phase. The proposed scheme only uses the energy detection to evaluate the weights for each iteration. Therefore, the proposed WBSR outperforms the existing BP and LASSObased scheme with a much smaller sensing sensitivity and higher accurate spectrum reconstruction for signals, so it might be a promising technique for wideband spectrum sensing in CRN.
In this article, the computational complexity of the proposed algorithm can be measured in terms of the average CPU time that is obtained from a total of 40 trials for typical signals. Two kinds of signals with different lengths are used in simulations: one with 65535, another with 19600. All simulations are performed on a laptop with an Intel T5750 2 GHz processor and the CPU time was measured by the Matlab (version 2009b) commands tic and toc. Table2 depicts the simulation results, from which it can be seen that the compressed ratio (M/N) is about 40%. It is noted that the proposed WBSR is more efficient than other algorithms.
In simulations, we compared the proposed ℓ_{ p }norm minimization scheme with the most popular used ℓ_{1}(BP) and ℓ_{2}(LASSO) optimizations and the results can be found in Table2. It is clear that the proposed WBSR is converged much faster than BP and LASSO. Actually, in the existing works, blocksparse spectrum sensing is based on ℓ_{1}norm optimization, spectrumblind reconstruction and LASSOCWSS are based on ℓ_{2}norm optimization, respectively. In this simulation, when we use ℓ_{1}norm based optimization for a signal with 65,535 elements, the average CPU time used for reconstruction with 95% accuracy (or higher) is about 16.732 s. When ℓ_{2}norm is used, the average CPU time is 0.5361 s. However for WBSR, the average CPU time is only 0.2132 s that is much faster than the other two. Similar results can be obtained for signals with different length. In order to further demonstrate the performance of WBSR, we are working to implement the algorithm in practical platform and the results will be reported in the future.
Conclusion
In this article, we presented an approach that is able to reconstruct the blind bands signals for CRNs without knowing the bands location information, which can improve the spectrum sensing efficiency and reduce the sensing time. In addition, in order to further improve the performance of existing CSbased signal reconstruction algorithm and decrease the complexity, we proposed a weighted ℓ_{ p }norm (1 < p < 1) minimization problem to approximate the ℓ_{0}norm minimization problem, instead of ℓ_{1}norm or ℓ_{1}/ℓ_{2} mixed minimization in existing signal reconstruction schemes. Simulation results show that the proposed WBSR has a higher spectrum sensing sensitivity and accuracy, and improved reconstruction speed.
References
 1.
Mishali M, Eldar YC: Wideband spectrum sensing at subNyquist rates. http://arxiv.org/abs/1009.1305
 2.
FCC: Facilitating opportunities for flexible, efficient and reliable spectrum use employing cognitive radio technologies. FCC Report and Order FCC0557A1 2005, 266267.
 3.
Tian Z, Giannakis GB: Compressed sensing for wideband cognitive radios. IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, (Honolulu, Hawaii, USA, 2007), pp. 1357–1360
 4.
Agrawal T, Lakkundi V, Griffin A, Tsakalides P: Compressed sensing for OFDM UWB systems. IEEE Radio and Wireless Symposium, vol. 1(Phoenix, AZ, USA, 2011), pp. 190193
 5.
Mishali M, Eldar YC: From theory to practice: subNyquist sampling of sparse wideband analog signals. IEEE J. Sel. Topics Signal Process 2010, 4: 375391.
 6.
Mishali M, Eldar YC: Blind multiband signal reconstruction: compressed sensing for analog signals. IEEE Trans. Signal Process 2009, 57: 9931009.
 7.
Khalaf Z, Nafkha A, Palicot J: Blind spectrum detector for cognitive radio using compressed sensing. IEEE Global Telecommunications Conference (Globecom 2011) vol. 1 (San Diego, CA, USA, IEEE, 2011), pp. 1–5
 8.
Wang Y, Pandharipande A, Polo YL, Leus G: Distributed compressive wideband spectrum sensing. Information Theory and Applications Workshop vol. 1 (Huston, Texas, USA, IEEE, 2009), pp. 178–183
 9.
Liu Y, Wan Q: Generalized block sparse constraint based compressive wideband spectrum spectrum sensing for cognitive radio. http://arxiv.org/abs/1006.2055
 10.
Rath G, Guillemot C: Sparse approximation with an orthogonal complementary matiching pursuit algorithm. IEEE International Conference on Acoustics, Speech and Signal Processing vol.1 (Huston, Texas, USA, IEEE, 2009), pp. 178–183
 11.
Chen J, Huo X: Collaborative compressed spectrum sensing: what if spectrum is not sparse. Electron Lett 2011, 47: 519520. 10.1049/el.2011.0274
 12.
Tian Z, Leus G, Lottici V: Joint dynamic resource allocation and waveform adaptation for cognitive networks. IEEE J. Sel. Areas Commun 2011, 29: 443454.
 13.
Gueguen L, Sayrac B: Sensing in cognitive radio channels: a theoretical perspective. IEEE Trans. Wirel. Commun 2009, 8: 11941198.
 14.
Zhang W, Letaief K, Cooperative spectrum sensing with transmit and relay diversity in cognitive radio networks: IEEE Trans. Wirel. Commun. 2008, 7: 47614766.
 15.
Chen SS: Basis pursuit. PhD thesis. 1995.
 16.
Chen S, Donoho D, Saunders M: Atomic decomposition by basis pursuit. SIAM J. Sci. Comput 1999, 20: 3161.
 17.
Candes E, Tao T: The Dantzig selector: statistical estimation when p is much larger than n . Ann. Stat 2007, 34: 23132351.
 18.
Mallat SG, Zhang Z: Matching pursuit in a timefrequency dictionary. IEEE Trans. Signal Process 1993, 41: 33973415. 10.1109/78.258082
 19.
Tropp J, Gilbert AC: signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inf. Theory 2007, 53: 46554666.
 20.
Needell D, Vershynin R: Uniformuncertainty principle signal recovery via regularized orthogonal matching pursuit. Found Comput. Math 2009, 9: 317334. 10.1007/s1020800890313
 21.
Needell D, Tropp JA: CoSaMP: iterative signal recovery from incomplete and inaccurate samples. Harmonic Anal. Appl. Comput 2008, 26: 301321.
 22.
Tibshirani R: Regression shrinkage and selection via the LASSO. J. R. Stat. Soc. B 1996, 58: 267288.
 23.
Zhang T: Sparse recovery with orthogonal matching pursuit under RIP. IEEE Trans. Inf. Theory 2011, 57: 62156221.
 24.
Mohimani H, BabieZadeh M, Jutten C: A fast approach for overcomplete sparse decomposition based on smoothed l_{0}norm. IEEE Trans. Signal Process 2009, 57: 289301.
 25.
Li S, Wang X, Wang J: Manifold learningbased automatic signal identification in cognitive radio networks. IET Commun 2012, 6: 955963. 10.1049/ietcom.2010.0590
 26.
Tian Z, Tafesse Y, Sadler BM: Cyclic feature detectionwith subNyquist sampling for wideband spectrum sensing. IEEE J. Sel. Topics Signal Process 2012, 6: 5870.
 27.
Li S, Wang X: Polychromatic set theorybased spectrum access in cognitive radios. IET Commun 2012, 6: 909916. 10.1049/ietcom.2010.0566
 28.
Antoniou A, Lu WS: Practical Optimization: Algorithms and Engineering Applications. 2006.
 29.
Pant JK, Lu WS, Antoniou A: Reconstruction of sparse signals by minimizing a reweighted approximate l0norm in the null space of the measurement matrix. 53rd IEEE International Midwest Symposium on Circuits and Systems (MWSCAS) (vol. 1 Taipei, IEEE 2010), pp. 3325–3328
Acknowledgements
This study was partially supported by the Nation Science Foundation of China (NSFC) under grant numbers 81101118 and 60972038.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ original submitted files for images
Below are the links to the authors’ original submitted files for images.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, S., Wang, X., Zhou, X. et al. Efficient blind spectrum sensing for cognitive radio networks based on compressed sensing. J Wireless Com Network 2012, 306 (2012). https://doi.org/10.1186/168714992012306
Received:
Accepted:
Published:
Keywords
 Cognitive Radio
 Primary User
 Lasso
 Cognitive Radio Network
 Orthogonal Matched Pursuit