Efficient blind spectrum sensing for cognitive radio networks based on compressed sensing
- Shancang Li^{1}Email author,
- Xinheng Wang^{2},
- Xu Zhou^{3} and
- Jue Wang^{1}
https://doi.org/10.1186/1687-1499-2012-306
© Li et al.; licensee Springer. 2012
Received: 2 October 2011
Accepted: 23 July 2012
Published: 3 October 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 sub-Nyquist 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.
Keywords
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 analog-to-digital converter resource can be reduced significantly, which is of great importance for wideband communication systems[4]. Previously, a lot of CS-based 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 CS-based 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 relaxing-based methods, such as basis pursuit (BP)[15, 16] and Dantzig Selector[17]; (2) Greedy algorithm-based methods, such as matched pursuit (MP) and its variants[18]. Recently, many improved MP-based 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 algorithm-based 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 algorithm-based 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 front-end is very difficult, the computation intensive energy or feature detection operations are applied in many existing spectrum sensing methods. However, by using CS-based approach, the spectrum detection can be simplified; and (3) high-speed 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].
- 1.
A CS-based 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
x(t) | A continuous-time signal with finite energy |
X(f) | The Fourier transform of x(t) |
x[n] | Bounded energy sequence |
v | Vector |
v_{ i }, or v(i) | i th entry of v |
v(f) | Vector that depends on a continuous parameter f |
Φ | Matrix |
Φ _{ ik } | ik th entry of Φ |
Φ _{ i } | i th column of Φ |
r _{ t } | Samples vector of a signal at time domain |
r _{ f } | Samples vector of a signal at frequency domain |
$\mathcal{F}$ | Non-zero frequency-domain support |
q,L | q number of blocks and L denotes the number of all blocks |
B,B_{ i } | Bandwidth of sub-band, B_{ i } denotes the i-th sub-band |
Q | Threshold for blind spectrum reconstruction |
∥x∥_{0} | ℓ_{0}-norm of x |
∥x∥_{ p } | ℓ_{ p }-norm of x, which is defined as${\parallel \mathbf{x}\parallel}_{p}={\left(\sum _{i=1}^{N}\right|{\mathbf{x}}_{i}{|}^{p})}^{1/p}$ |
I(A) | Support of A |
$\mathcal{M}$ | Denote a set of signals in CRN |
System models
Signal model
where the N × 1 vector θ is the K-sparse 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.
where F is the discrete-time Fourier transform (DFT) matrix and r_{ t } is the sample vector of a real-value signal r(t) which needs to be reconstructed from samples x.
in which y denotes an M-length 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 non-adaptive measurements where Φ is well presented.
- 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 time-continuous band-limited 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 band-limited signal in the non-zero frequency-domain 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.
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}.
in which the the multi-coset 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 multi-coset 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)/2-sparse.
Actually, the sparse level of A is related to the sampling coset pattern[1]. Because in multi-coset 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.
where signal r is separated into K sub-vectors, 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 sampling-reconstruction 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 spectrum-blind reconstruction scheme by using CS based on a preceding multi-coset model[27].
As discussed in Section “System models”, the frequency response is sparse in CRNs, so Equation (13) can be solved with a two-step scheme: (1) use compressed measurements y to estimate the sparse frequency response r_{ f } (actually this is an ill-posed problem); (2) reconstruct signal r(t) according to the frequency response, which can be done by an inverse Fourier transfer.
in which A = F^{−1}ΦΨ, and F denotes the DFT matrix of compressed sample vector y.
where X(·) is the Fourier transform of time shifted r(t). As analyzed above, in a CRN every band contributes only a few non-zero value, so x is a sparse vector which makes it possible to use CS theory to reconstruct spectrum of signals.
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 N-sparse 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 non-zero components are fewer than the number of bands in x(f). So we can say that x is Q-sparse. Kruskal-rank σ(A) = q, which implies that when q ≥ 2QB, x can perfectly be reconstructed without knowing any information about the locations of bands when$f\in {\mathcal{F}}_{0}$.
A number of CS reconstruction algorithms are available to solve this problem, and many ℓ_{1}-norm and ℓ_{1}/ℓ_{2}-norm minimization-based 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
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 un-sparse solution, which is no sense in solving our problem.
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 non-zero 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 which σ > 0 is a small constant. Equation (22) can be solved with steepest-descent 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
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 quasi-Newton[24], Broyden-Fletcher-Goldfarb-Shanno (BFGS)[29] and so on.
Weighted approximation of ℓ_{ p }-norm algorithm
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.
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 non-convex. 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.
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.
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 Bluetooth-based communications, because the information is spread across a number of frequency channels.
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 LASSO-based 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.
Average CPU time over 40 runs for different algorithm
Algorithm | Signal length | M/N(%) | Average CPU time(s) |
---|---|---|---|
BP | 65535 | 39.85 | 16.732 |
LASSO | 65535 | 39.85 | 0.5361 |
WBSR | 65535 | 39.85 | 0.2132 |
BP | 19600 | 40 | 6.2863 |
LASSO | 19600 | 40 | 0.1260 |
WBSR | 19600 | 40 | 0.0806 |
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, block-sparse spectrum sensing is based on ℓ_{1}-norm optimization, spectrum-blind reconstruction and LASSO-CWSS 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 CS-based 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.
Declarations
Acknowledgements
This study was partially supported by the Nation Science Foundation of China (NSFC) under grant numbers 81101118 and 60972038.
Authors’ Affiliations
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