A Novel Decentralized Scheme for Cooperative Compressed Spectrum Sensing in Distributed Networks

Abstract

Compressed sensing (CS) recently turns out to be an effective approach to alleviate the sampling bottleneck in wideband spectrum sensing. However, the computation overhead incurred by compressed reconstruction is nontrivial, especially in a power-constrained cognitive radio (CR). Moreover, additional information, which is generally unavailable in practice, is needed in conventional CS-based wideband spectrum sensing schemes to improve the reconstruction quality as well as the detection performance. To address these issues, this paper proposes a novel decentralized scheme for cooperative compressed spectrum sensing in distributed CR networks. Our key observation is that the sparse signals are unnecessary to be reconstructed since the task of spectrum sensing is only interested in the spectrum occupancy status. The major novelty of the proposed scheme relates to the use of Karcher mean as a statistic indicating the spectrum occupancy status, thereby eliminating the compressed reconstruction stage and significantly reducing the computational complexity. Considering limited communication resources per CR, a decentralized implementation based on alternating direction method of multipliers is presented to calculate the Karcher mean via one-hop communications only. The superior performance of the proposed scheme is demonstrated by comparing with several existing decentralized schemes in terms of detection performance, communication overhead, and computational complexity.

1. Introduction

Spectrum sensing, whose objectives are detecting the presence of primary users (PUs) and identifying the spectrum holes for dynamic spectrum access, is an essential task for enabling cognitive radio technique, a leading choice for efficient utilization of spectrum resource [1]. Furthermore, wideband spectrum sensing has received significant attention since more spectrum access opportunities can be attained in wideband regime. However, spectrum sensing over a wide frequency band confronts several practical challenges [2, 3] and the major one lies in the wideband spectrum acquisition implementation.

Aiming at alleviating the heavy pressure on the conventional analog-to-digital converter (ADC) technology, CS theory [4, 5] has been recently introduced into the application of wideband spectrum sensing, which stems from the recognition that the radio spectrum is inherently sparse due to the low percentage of spectrum occupancy by active radios [1–3], a fact that motivates dynamic spectrum access. Different types of implementation structure for compressed wideband spectrum sensing have been developed in [6–8]. Since spectrum sensing on a per CR basis is quite susceptible to channel fading/shadowing and ambient noise, cooperative spectrum sensing (CSS) [3] that exploits the built-in spatial diversity among multiple CRs has been proposed for CR networks. Based on how cooperative CRs share their sensing data in the network, CSS can be classified into two categories: centralized CSS and decentralized CSS. In centralized CSS scheme, a fusion center (FC) is required to collect measurements from all CRs and to make centralized sensing decision. Centralized CSS schemes using CS theory are presented in [9, 10]. The reconstruction performance can be globally optimal, but the incurred power cost and communication workload in transmitting local information to the FC and conveying sensing results back to CRs are significantly high. Alternatively, decentralized schemes are quite attractive because of their low communication overhead. CS-based decentralized CSS schemes have been studied in [11, 12]. Here, CRs only communicate with their neighboring CRs within a short one-hop range to reduce the transmission power consumed during communications and converge to a unified decision on the spectrum occupancy by iterations, in the absence of a FC.

As stated above, the spectrum sensing process of most CS-based schemes is first acquiring compressed measurements, then reconstructing the signals involved, and lastly performing thresholding on the reconstructed signals to obtain the spectrum occupancy status [6–12]. CS theory shows excellent ability in reducing wideband signal acquisition cost. However, the computation overhead incurred by compressed reconstruction is nontrivial and makes the CS-based schemes difficult to implement, especially in resource-constrained scenarios. Moreover, additional information, such as the upper bound of ambient noise energy (for convex relaxation algorithms) or the sparsity order of received sparse signal (for greedy algorithms), is often needed in compressed reconstruction stage to improve the reconstruction quality as well as the detection performance. However, the additional information is usually beyond the ability to obtain in practice. In this paper, we focus on solving these two problems and propose a novel decentralized scheme for cooperative compressed spectrum sensing in distributed networks. In the proposed scheme, a compressed sensing mechanism is used at each CR by utilizing the inherent sparsity of the monitored wideband spectrum. More specifically, the received wideband signal of each CR is fed into a number of wideband filters and the outputs of the filters, which are actually the linear combinations of energies of the received signal in different channels, are served as compressed measurements. Our key observation is that the fundamental task of wideband spectrum sensing is not reconstructing the wideband signal, but determining the spectrum occupancy information. Therefore, the compressed reconstruction stage could be completely eliminated. To achieve this, we propose the use of Karcher mean of energy vectors of multiple CRs to estimate the spectrum occupancy directly from compressed measurements. The computational complexity is expected to be reduced significantly due to the avoidance of compressed reconstruction. Moreover, to save the limited communication resources per CR, a decentralized implementation based on alternating direction method of multipliers (ADMM) is presented to calculate the Karcher mean via one-hop communications only. The superior performance of the proposed decentralized scheme is testified by comparing with CS-based decentralized schemes reported in [11, 12], in terms of detection performance of the CR network, communication overhead, and computational complexity per CR. Moreover, the impacts of scheme parameters and system parameters are also investigated through simulations.

The remaining part of this paper is organized as follows. The signal model and the spectrum sensing problem of interest are introduced in Section 2. Section 3 presents the details of the proposed decentralized scheme to cooperative compressed spectrum sensing. Numerical simulations are given to demonstrate the performance of the proposed scheme in Section 4 and some conclusions are drawn in Section 5.

2. Signal Modeling

Suppose that the monitored wideband spectrum is divided into N nonoverlapping, equal-bandwidth channels whose center frequency and bandwidth are denoted by ${\left\{f_n\right\}}_{n=\mathrm{1}}^N$ and B, respectively. This division scheme of the monitored spectrum is known to all CRs. However, the power spectral density (PSD) of each channel is dynamically varying due to its occupancy status caused by the PUs. Those temporarily vacant channels are termed spectrum holes and are available for opportunistic spectrum access by CRs. We assume that I spatially distributed CRs collaboratively sense the wideband spectrum and each CR i is using a wideband antenna listening to the whole spectrum and providing each CR i with the wideband time-domain signal $r^{(i)}(t)$ , $i=\mathrm{1},\dots ,I$ .

To overcome the sampling limitations of ADC hardware while acquiring wideband signal, compressed sensing mechanism is applied at each CR by using wideband frequency-selective filters [10, 13]. More specifically, each CR is provided with a random matrix ${\mathbf{\Phi }}^{(i)}$ of size $M\times N\hspace{0.17em}\hspace{0.17em}\left(M<N\right)$ and uses this matrix to generate M wideband frequency-selective filters ${\left\{H_m^{(i)}(f)\right\}}_{m=\mathrm{1}}^M$ . The frequency response of nth channel of ith CR's mth filter is given by

$$\begin{array}{cc}& H_m^{\left(i\right)}\left(f_n\right)={\mathbf{\Phi }}_{m,n}^{\left(i\right)},\hspace{1em}n=\mathrm{1},\dots ,N,\hspace{0.17em}\hspace{0.17em}m=\mathrm{1},\dots ,M,\end{array}$$

(1)

where ${\mathbf{\Phi }}_{m,n}^{(i)}$ denotes the entry in the mth row and nth column of the matrix ${\mathbf{\Phi }}^{(i)}$ . The random matrices can be generated once and stored in the CRs.

In order to mix different channel sensing information, we feed the received wideband signal into the frequency-selective filters and measure the energy of the output signal of each filter to get an $M\times \mathrm{1}$ energy vector ${\mathbf{y}}^{(i)}={\left[y_{\mathrm{1}}^{(i)},\dots ,y_M^{(i)}\right]}^T$ for CR i. Apparently, the energy at the output of ith CR's mth filter can be represented as

$$\begin{array}{cc}& y_m^{\left(i\right)}={\displaystyle \sum }_{n=\mathrm{1}}^N‍{\left|H_m^{\left(i\right)}\left(f_n\right)\right|}^{\mathrm{2}}{e}_n^{\left(i\right)},\end{array}$$

(2)

where $e_n^{(i)}={\int }_{f_n-B/\mathrm{2}}^{f_n+B/\mathrm{2}}‍{\left|\mathcal{F}\left\{r^{(i)}(t)\right\}\right|}^{\mathrm{2}}df$ denotes the portion of the received signal's energy in the nth channel of ith CR.

In the matrix form, the compressed measurements collected at CR i can be represented as

$$\begin{array}{cc}& {\mathbf{y}}^{\left(i\right)}={\widehat{\mathbf{\Phi }}}^{\left(i\right)}{\mathbf{e}}^{\left(i\right)},\end{array}$$

(3)

where ${\widehat{\mathbf{\Phi }}}^{(i)}=\left[\begin{array}{ccc}{\left|H_{\mathrm{1}}^{(i)}(f_{\mathrm{1}})\right|}^{\mathrm{2}}& \cdots & {\left|H_{\mathrm{1}}^{(i)}(f_N)\right|}^{\mathrm{2}}\\ ⋮& \ddots & ⋮\\ {\left|H_M^{(i)}(f_{\mathrm{1}})\right|}^{\mathrm{2}}& \cdots & {\left|H_M^{(i)}(f_N)\right|}^{\mathrm{2}}\end{array}\right]$ is an $M\times N$ matrix whose elements are square absolute values of the corresponding elements of the random matrix ${\mathbf{\Phi }}^{(i)}$ and ${\mathbf{e}}^{(i)}={\left[e_{\mathrm{1}}^{(i)},\dots ,e_N^{(i)}\right]}^T$ is the energy vector of the ith CR's received signal in different channels.

Many investigations show that the radio spectrum in a particular time and geographical area is in a very low utilization ratio [1–3] suggesting that ${\mathbf{e}}^{(i)}\hspace{0.17em}\hspace{0.17em}(i=\mathrm{1},\dots ,I)$ is sparse which is the exact motivation for introducing CS theory into wideband spectrum sensing. Moreover, the nonzero entries of ${\mathbf{e}}^{(i)}$ are usually in a large dynamic range [8, 10] as a result of the fading wireless environment. For the same channel, the energy may be too small to be detected at some CRs but relatively large at others; this may cause high miss detection probability for single CR, the reason why cooperation among multiple spatially distributed CRs is useful in combating many random factors, such as ambient noise and uncertain channel fading/shadowing. In addition to the sparsity at individual CR, it is worth noting that different CRs share the same nonzero support (known as joint sparsity structure [14, 15]) since all CRs are affected by the same PUs, provided that the channel does not experience deep fades [11], even though the nonzero values may be different at individual CRs.

Now, the goal of CS-based scheme to cooperative wideband spectrum sensing is to determine the spectrum occupancy status by using compressed measurements collected from all CRs and to detect spectrum holes for dynamic spectrum access.

3. The Proposed Scheme for Cooperative Compressed Spectrum Sensing

While CS-based schemes show powerful ability in breaking the sampling bottleneck in wideband spectrum sensing, the resulting increase in computation/complexity is nontrivial, especially in resource-constrained scenarios. Furthermore, additional information is often needed in the compressed reconstruction stage. However, this information is usually beyond the ability to obtain in practice.

Aiming to solve these problems, a novel decentralized scheme to cooperative compressed spectrum sensing is proposed in this section. The major novelty of this scheme relates to the use of the Karcher mean of the joint-sparse signals to infer the spectrum occupancy directly from the compressed measurements, without reconstructing the signals involved. Moreover, considering the limited communication resources, a decentralized implementation based on alternating direction method of multipliers (ADMM) [16, 17] is presented to calculate the Karcher mean via one-hop communications only.

3.1. Compressed WSS Based on Karcher Mean

The Karcher mean of joint-sparse signal ensembles ${\left\{{\mathbf{e}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ can be obtained from the compressed measurements ${\left\{{\mathbf{y}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ as follows:

$$\begin{array}{cc}& \widehat{\mathbf{e}}=\underset{\widehat{\mathbf{e}}}{\min }{\displaystyle \sum }_{i=\mathrm{1}}^I‍w^{\left(i\right)}{d}^{\mathrm{2}}\left({\widehat{\mathbf{\Phi }}}^{\left(i\right)}\widehat{\mathbf{e}},{\mathbf{y}}^{\left(i\right)}\right),\end{array}$$

(4)

where the weight $w^{(i)}>\mathrm{0}$ controls the emphasis given to the ith CR and $d({\widehat{\mathbf{\Phi }}}^{(i)}\widehat{\mathbf{e}},{\mathbf{y}}^{(i)})$ denotes the distance function between the vectors ${\widehat{\mathbf{\Phi }}}^{(i)}\widehat{\mathbf{e}}$ and ${\mathbf{y}}^{(i)}$ .

The optimization problem in (4) can be viewed as a minimization of the sum of the squared distances in low-dimensional spaces between the compressed measurements and the Karcher mean. To explain the reason why the Karcher mean is introduced into the application of wideband spectrum sensing, we analyze the relationship between the Karcher mean in (4) and the common support of joint-sparse signal ensembles ${\left\{{\mathbf{e}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ . Without loss of generality, we assume that the distance function in (4) is Euclidean distance (i.e., $d({\widehat{\mathbf{\Phi }}}^{(i)}\widehat{\mathbf{e}},{\mathbf{y}}^{(i)})={‖{\widehat{\mathbf{\Phi }}}^{(i)}\widehat{\mathbf{e}}-{\mathbf{y}}^{(i)}‖}_{\mathrm{2}}$ ) and then (4) is equivalent to the following optimization problem:

$$\begin{array}{cc}& \widehat{\mathbf{e}}=\underset{\widehat{\mathbf{e}}}{\min }{‖\widehat{\mathbf{\Phi }}\widehat{\mathbf{e}}-\mathbf{y}‖}_{\mathrm{2}}^{\mathrm{2}},\end{array}$$

(5)

where

$$\begin{array}{cc}& \widehat{\mathbf{\Phi }}=\left[\begin{array}{c}\sqrt{w^{\left(\mathrm{1}\right)}}{\widehat{\mathbf{\Phi }}}^{\left(\mathrm{1}\right)}\\ ⋮\\ \sqrt{w^{\left(I\right)}}{\widehat{\mathbf{\Phi }}}^{\left(I\right)}\end{array}\right]\in R^{MI\times N},\\ \mathbf{y}=\left[\begin{array}{c}\sqrt{w^{\left(\mathrm{1}\right)}}{\mathbf{y}}^{\left(\mathrm{1}\right)}\\ ⋮\\ \sqrt{w^{\left(I\right)}}{\mathbf{y}}^{\left(I\right)}\end{array}\right]\in R^{MI\times \mathrm{1}}.\end{array}$$

(6)

Since the sensing matrices ${\left\{{\widehat{\mathbf{\Phi }}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ are generated randomly, the condition $MI\ge N$ can guarantee that the matrix $\widehat{\mathbf{\Phi }}$ is a full column rank matrix. Then the least square solution of (5) can be given by

$$\begin{array}{cc}& \widehat{\mathit{e}}={\left({\widehat{\mathbf{\Phi }}}^T\widehat{\mathbf{\Phi }}\right)}^{-\mathrm{1}}{\widehat{\mathbf{\Phi }}}^T\mathbf{y}.\end{array}$$

(7)

Using the matrix form of compressed measurements in (3) and recalling the definition of $\widehat{\mathbf{\Phi }}$ and y, (7) can be reformulated as follows:

$$\begin{array}{cc}& \widehat{\mathbf{e}}={\displaystyle \sum }_{i=\mathrm{1}}^I‍{\mathbf{A}}^{-\mathrm{1}}{\mathbf{B}}^{\left(i\right)}{\mathbf{e}}^{\left(i\right)},\end{array}$$

(8)

where the matrices ${\mathbf{B}}^{(i)},\mathrm{\hspace{0.17em}\hspace{0.17em}}\mathbf{A}\in R^{N\times N}$ are defined as ${\mathbf{B}}^{(i)}=w^{(i)}{\widehat{\mathbf{\Phi }}}^{(i{)}^T}{\widehat{\mathbf{\Phi }}}^{(i)}$ and $\mathbf{A}={\sum }_{i=\mathrm{1}}^I{\mathbf{B}}^{(i)}$ . Due to the incoherence [5] between any two columns of the sensing matrix ${\widehat{\mathbf{\Phi }}}^{(i)},\forall i$ , symmetric matrices ${\mathbf{B}}^{(i)},\mathrm{\hspace{0.17em}\hspace{0.17em}}\forall i$ , have the following properties: the diagonal entries are positive and for every row of the matrices, the magnitude of the diagonal entry is much larger than the magnitudes of all the other (nondiagonal) entries in that row. So it is with A and ${\mathbf{A}}^{-\mathrm{1}}$ . Therefore, for signals sharing the same support and sign, this mean result keeps the value in the common support while it depresses the values in the other locations. This is the reason why we propose determining spectrum occupancy by calculating a Karcher mean.

After obtaining the Karcher mean $\widehat{\mathbf{e}}$ from (4), the energy detection can be used to make decision on spectrum occupancy status. The nth element of the decision $\widehat{\mathbf{d}}\in {\left\{\mathrm{0,1}\right\}}^{N\times \mathrm{1}}$ is made as follows:

$$\begin{array}{l}{\widehat{d}}_n=\left\{\begin{array}{ll}1,\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\left|{\widehat{e}}_n\right|\ge \eta ,\hfill \\ 0,\hfill & \text{if}\hspace{0.17em}\hspace{0.17em}\left|{\widehat{e}}_n\right|<\eta ,\hfill \end{array}\right.\\ \phantom{hhhh}\text{for}\hspace{0.17em}\hspace{0.17em}n=\mathrm{1},\dots ,N,\end{array}$$

(9)

where ${\widehat{d}}_n=\mathrm{1}$ means that the nth channel is determined to be occupied, while ${\widehat{d}}_n=\mathrm{0}$ means it is determined to be unoccupied. η 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. In this way, we can benefit from the CS technique in alleviating the heavy pressure of sampling and avoid massive computations incurred by compressed reconstruction.

3.2. Decentralized Implementation via the ADMM

Computing the Karcher mean in a centralized manner requires a fusion centre to collect compressed measurements ${\left\{{\mathbf{y}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ and measurement matrices ${\left\{{\widehat{\mathbf{\Phi }}}^{(i)}\right\}}_{i=\mathrm{1}}^I$ from all CRs. Since the incurred power cost and communication overhead in transmitting local information to FC and conveying centralized result back to CRs are significantly high, a centralized approach is not always feasible, especially in distributed networks where no powerful computing center is available and where resources are limited. In this subsection, we aim to compute the Karcher mean in a decentralized manner. Following the idea of [17], an iterative decentralized implementation based on ADMM is presented to solve the optimization formulation expressed by (4). The decentralized implementation is performed using only one-hop local communications [11, 12] which can significantly reduce the communication resources consumed during sensing.

Assume that the distributed network containing I CRs can be described as an undirected graph $\mathbf{G}=\left(\mathbf{V},\zeta \right)$ , where CRs form the set of vertices $\mathbf{V}=\{\mathrm{1,2},\dots ,I\}$ and ζ is the set of edges with cardinality $\left|\zeta \right|=E$ . Each edge $(i,j)\in \zeta$ indicates that CR i and CR j are one-hop neighbors to each other, between which one-hop communication is allowed.

In order to solve (4) in a decentralized way, we let each CR i keep a local copy of the Karcher mean $\widehat{\mathbf{e}}$ , denoted as ${\mathbf{x}}^{(i)}\in R^{N\times \mathrm{1}}$ , and we let local copies of one-hop neighbors consent to the same value. If the network of CRs is connected, then the optimization problem (4) is clearly equivalent to

$$\begin{array}{l}\underset{\left\{{\mathbf{x}}^{\left(i\right)}\right\},\left\{{\alpha }^{(i,j)}\right\}}{\min }\hspace{1em}{\displaystyle \sum }_{i=\mathrm{1}}^I‍f_i\left({\mathbf{x}}^{\left(i\right)}\right)\\ \text{s}.\text{t}.\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathbf{x}}^{\left(i\right)}={\alpha }^{\left(i,j\right)},\hspace{1em}{\mathbf{x}}^{\left(j\right)}={\alpha }^{\left(i,j\right)},\hspace{1em}\forall \left(i,j\right)\in \zeta ,\end{array}$$

(10)

where the local objective function $f_i\left({\mathbf{x}}^{(i)}\right)=w^{(i)}{d}^{\mathrm{2}}({\widehat{\mathbf{\Phi }}}^{(i)}{\mathbf{x}}^{(i)},{\mathbf{y}}^{(i)})$ is only related to the local information at CR i and the auxiliary variable ${\alpha }^{(i,j)}\in R^{N\times \mathrm{1}}$ imposes a constraint that local copies at neighboring CRs i and j should be identical.

To reformulate the optimization problem (10) in the form that can be solved by ADMM [16], we define $\mathbf{x}\in R^{NI\times \mathrm{1}}$ as a vector concatenating all ${\mathbf{x}}^{(i)}$ and $\alpha \in R^{EN\times \mathrm{1}}$ as a vector concatenating all ${\alpha }^{(i,j)}$ and $f\left(\mathbf{x}\right)={\sum }_{i=\mathrm{1}}^I{f}_i\left({\mathbf{x}}^{(i)}\right)$ . Then (10) can be expressed as

$$\begin{array}{l}\underset{\mathbf{x},\alpha }{\min }\hspace{1em}f\left(\mathbf{x}\right)\\ \text{s}.\text{t}.\hspace{1em}\mathbf{\Psi }\mathbf{x}+\mathbf{\Theta }\alpha =\mathrm{0},\end{array}$$

(11)

where the matrices Ψ and Θ are defined as follows: $\mathit{\Psi }=\left[\begin{array}{c}{\mathit{\Psi }}_{\mathrm{1}}\\ {\mathit{\Psi }}_{\mathrm{2}}\end{array}\right]$ , where ${\mathit{\Psi }}_{\mathrm{1}},{\mathit{\Psi }}_{\mathrm{2}}\in R^{EN\times NI}$ are both composed of $E\times I$ blocks of $N\times N$ matrices. If $\left(i,j\right)\in \zeta$ and ${\alpha }^{(i,j)}$ is the qth block of α, the $\left(q,i\right)$ th block of ${\mathit{\Psi }}_{\mathrm{1}}$ and the $\left(q,j\right)$ th block of ${\mathit{\Psi }}_{\mathrm{2}}$ are identity matrices of size N; otherwise the corresponding blocks are zero matrices of size N. The matrix $\mathbf{\Theta }=\left[\begin{array}{c}-{\mathit{I}}_{EN}\\ -{\mathit{I}}_{EN}\end{array}\right]$ and ${\mathit{I}}_{EN}$ is an identity matrix of size $EN$ . Note that the matrix Ψ contains the information of the underlying network topology.

ADMM for problem (11) can be derived directly from the augmented Lagrangian. Consider

$$\begin{array}{cc}& {\ell }_{\rho }\left(\mathbf{x},\alpha ,\beta \right)=f\left(\mathbf{x}\right)+{\beta }^T\left(\mathit{\Psi }\mathbf{x}+\mathbf{\Theta }\alpha \right)+\left(\frac{\rho }{\mathrm{2}}\right){‖\mathit{\Psi }\mathbf{x}+\mathbf{\Theta }\alpha ‖}_{\mathrm{2}}^{\mathrm{2}},\end{array}$$

(12)

where $\beta \in R^{\mathrm{2}EN\times \mathrm{1}}$ is the Lagrange multiplier and ρ is a positive algorithm parameter.

The resulting ADMM algorithm at iteration $t+\mathrm{1}$ consists of the following updates [16]:

$$\begin{array}{cc}& \mathbf{x}\left(t+\mathrm{1}\right):=\underset{\mathbf{x}}{\arg \hspace{0.17em}\min }{\ell }_{\rho }\left(\mathbf{x},\alpha \left(t\right),\beta \left(t\right)\right),\\ \alpha \left(t+\mathrm{1}\right):=\underset{\alpha }{\arg \hspace{0.17em}\min }{\ell }_{\rho }\left(\mathbf{x}\left(t+\mathrm{1}\right),\alpha ,\beta \left(t\right)\right),\\ \beta \left(t+\mathrm{1}\right)-\beta \left(t\right)-\rho \left(\mathit{\Psi }\mathbf{x}\left(t+\mathrm{1}\right)+\mathbf{\Theta }\alpha \left(t+\mathrm{1}\right)\right)=\mathrm{0}.\end{array}$$

(13)

Inspired by the recent work in [17], the updates in (13) can be simplified and distributed to CRs. If all the local objective functions are convex, $\mathbf{x}\left(t+\mathrm{1}\right)$ , $\alpha \left(t+\mathrm{1}\right)$ , and $\beta \left(t+\mathrm{1}\right)$ can be obtained by solving (14), (15), and (16), respectively. Consider

$$\begin{array}{cc}& \nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)+{\mathit{\Psi }}^T\beta \left(t\right)+\rho {\mathit{\Psi }}^T\left(\mathit{\Psi }\mathbf{x}\left(t+\mathrm{1}\right)+\mathbf{\Theta }\alpha \left(t\right)\right)=\mathrm{0},\end{array}$$

(14)

$$\begin{array}{cc}& {\mathbf{\Theta }}^T\beta \left(t\right)+\rho {\mathbf{\Theta }}^T\left(\mathit{\Psi }\mathbf{x}\left(t+\mathrm{1}\right)+\mathbf{\Theta }\alpha \left(t+\mathrm{1}\right)\right)=\mathrm{0},\end{array}$$

(15)

$$\begin{array}{cc}& \beta \left(t\right)=\beta \left(t+\mathrm{1}\right)-\rho \left(\mathit{\Psi }\mathbf{x}\left(t+\mathrm{1}\right)+\mathbf{\Theta }\alpha \left(t+\mathrm{1}\right)\right),\end{array}$$

(16)

where $\nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)$ denotes the gradient of $f\left(\mathbf{x}\right)$ at point $\mathbf{x}=\mathbf{x}\left(t+\mathrm{1}\right)$ if f is differentiable or the subgradient if f is nondifferentiable.

Using (16) to eliminate the term $\beta \left(t\right)$ in (14) and (15), we have

$$\begin{array}{cc}& \nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)+{\mathit{\Psi }}^T\beta \left(t+\mathrm{1}\right)+\rho {\mathit{\Psi }}^T\mathbf{\Theta }\left(\alpha \left(t\right)-\alpha \left(t+\mathrm{1}\right)\right)=\mathrm{0},\end{array}$$

(17)

$$\begin{array}{cc}& {\mathbf{\Theta }}^T\beta \left(t+\mathrm{1}\right)=\mathrm{0}.\end{array}$$

(18)

Splitting the variable $\beta \in R^{\mathrm{2}EN\times \mathrm{1}}$ into $\left[\begin{array}{c}\mu \\ \nu \end{array}\right]$ with $\mu ,\nu \in R^{EN\times \mathrm{1}}$ and recalling the definition of matrix Θ, (18) can be expressed as

$$\begin{array}{cc}& \mu \left(t+\mathrm{1}\right)=-\nu \left(t+\mathrm{1}\right).\end{array}$$

(19)

Recalling the definition of matrix Ψ and substituting (19) into (17), we obtain

$$\begin{array}{cc}& \nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)+{\mathit{\Psi }}_{-}\mu \left(t+\mathrm{1}\right)-\rho {\mathit{\Psi }}_{+}\left(\alpha \left(t\right)-\alpha \left(t+\mathrm{1}\right)\right)=\mathrm{0},\end{array}$$

(20)

where ${\mathit{\Psi }}_{-}={\mathit{\Psi }}_{\mathrm{1}}^T-{\mathit{\Psi }}_{\mathrm{2}}^T$ and ${\mathit{\Psi }}_{+}={\mathit{\Psi }}_{\mathrm{1}}^T+{\mathit{\Psi }}_{\mathrm{2}}^T$ .

With the splitting strategy of β, (16) splits into two equations related to variables μ and ν, respectively. Summing and subtracting these two equations result in

$$\begin{array}{cc}& \frac{\mathrm{1}}{\mathrm{2}}{\mathit{\Psi }}_{+}^T\mathbf{x}\left(t+\mathrm{1}\right)-\alpha \left(t+\mathrm{1}\right)=\mathrm{0},\\ \mu \left(t+\mathrm{1}\right)-\mu \left(t\right)-\frac{\rho }{\mathrm{2}}{\mathit{\Psi }}_{-}^T\mathbf{x}\left(t+\mathrm{1}\right)=\mathrm{0}.\end{array}$$

(21)

By using (21), we can eliminate the auxiliary variable α and the term $\mu \left(t+\mathrm{1}\right)$ in (20) and then obtain

$$\begin{array}{l}\nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)+{\mathbf{\Psi }}_{-}\mu \left(t\right)-\frac{\rho }{\mathrm{2}}{\mathbf{\Psi }}_{+}{\mathbf{\Psi }}_{+}^T\mathbf{x}\left(t\right)\\ \hspace{1em}+\frac{\rho }{\mathrm{2}}\left({\mathbf{\Psi }}_{-}{\mathbf{\Psi }}_{-}^T+{\mathbf{\Psi }}_{+}{\mathbf{\Psi }}_{+}^T\right)\mathbf{x}\left(t+\mathrm{1}\right)=\mathrm{0}.\end{array}$$

(22)

To simplify the expression, we introduce ${\mathbf{L}}_{+}={\mathit{\Psi }}_{+}{\mathit{\Psi }}_{+}^T$ , ${\mathbf{L}}_{-}={\mathit{\Psi }}_{-}{\mathit{\Psi }}_{-}^T$ , $\mathbf{D}=\left(\mathrm{1}/\mathrm{2}\right)\left({\mathbf{L}}_{+}+{\mathbf{L}}_{-}\right)$ , and a new multiplier $\lambda ={\mathit{\Psi }}_{-}\mu$ . Then updates in (14)–(16) reduce to

$$\begin{array}{cc}& \nabla f\left(\mathbf{x}\left(t+\mathrm{1}\right)\right)+\lambda \left(t\right)+\rho \mathbf{D}\mathbf{x}\left(t+\mathrm{1}\right)-\frac{\rho }{\mathrm{2}}{\mathbf{L}}_{+}\mathbf{x}\left(t\right)=\mathrm{0},\\ \lambda \left(t+\mathrm{1}\right)-\lambda \left(t\right)-\frac{\rho }{\mathrm{2}}{\mathbf{L}}_{-}\mathbf{x}\left(t+\mathrm{1}\right)=\mathrm{0}.\end{array}$$

(23)

Here it is noteworthy that some starting conditions [17] are needed to guarantee the equivalence between the updates in (14)–(16) and those in (23). However those conditions can be satisfied by initializing $\mathbf{x}(\mathrm{0})={\mathrm{0}}^{NI\times \mathrm{1}}$ and $\lambda (\mathrm{0})={\mathrm{0}}^{NI\times \mathrm{1}}$ .

In (23), the introduced matrices ${\mathbf{L}}_{-},{\mathbf{L}}_{+},\mathbf{D}\in R^{NI\times NI}$ are related to the underlying network topology. With regard to the undirected graph G, D is the extended degree matrix, and ${\mathbf{L}}_{-}$ and ${\mathbf{L}}_{+}$ are the extended signed and signless Laplacian matrices, respectively. By “extended,” we mean the Kronecker product of the original definitions of these matrices [18] and $N\times N$ identity matrices ${\mathbf{I}}_N$ .

Note that $\mathbf{x}=\left[{\mathbf{x}}^{(\mathrm{1})};\dots ;{\mathbf{x}}^{(I)}\right]$ , where ${\mathbf{x}}^{(i)}\in R^{N\times \mathrm{1}}$ is the local solution of CR i, and $\lambda =\left[{\lambda }^{(\mathrm{1})};\dots ;{\lambda }^{(I)}\right]$ , where ${\lambda }^{(i)}\in R^{N\times \mathrm{1}}$ is the local Lagrange multiplier of CR i. According to the definitions of ${\mathbf{L}}_{-}$ , ${\mathbf{L}}_{+}$ , and D, the updates in (23) can be distributed to CRs and the x-update and λ-update at CR i can be represented as follows:

$$\begin{array}{l}\hspace{0.17em}\nabla f_i\left({\mathbf{x}}^{\left(i\right)}\left(t+\mathrm{1}\right)\right)+{\lambda }^{\left(i\right)}\left(t\right)+\rho \left|{\mathbf{V}}^{\left(i\right)}\right|{\mathbf{x}}^{\left(i\right)}\left(t+\mathrm{1}\right)\\ \hspace{1em}-\left(\frac{\rho }{\mathrm{2}}\right)\left(\left|{\mathbf{V}}^{\left(i\right)}\right|{\mathbf{x}}^{\left(i\right)}\left(t\right)+{\displaystyle \sum }_{j\in {\mathbf{V}}^{\left(i\right)}}‍{\mathbf{x}}^{\left(j\right)}\left(t\right)\right)=\mathrm{0},\\ \hspace{0.17em}{\lambda }^{\left(i\right)}\left(t+\mathrm{1}\right)={\lambda }^{\left(i\right)}\left(t\right)+\left(\frac{\rho }{\mathrm{2}}\right)\end{array}$$

(24)

$$\begin{array}{l}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}·\left(\left|{\mathbf{V}}^{\left(i\right)}\right|{\mathbf{x}}^{\left(i\right)}\left(t+\mathrm{1}\right)-{\displaystyle \sum }_{j\in {\mathbf{V}}^{\left(i\right)}}{\mathbf{x}}^{\left(j\right)}\left(t+\mathrm{1}\right)\right).\end{array}$$

(25)

As shown in (24) and (25), this algorithm is fully decentralized since the two updates only rely on local and neighboring information.

3.3. The Proposed Decentralized Scheme Based on Karcher Mean

Without loss of generality, suppose the distance function in (4) to be Euclidean distance in practice. Therefore, the local objective function of CR i is given by

$$\begin{array}{cc}& f_i\left({\mathbf{x}}^{\left(i\right)}\right)=w^{\left(i\right)}{‖{\widehat{\mathit{\Phi }}}^{\left(i\right)}{\mathbf{x}}^{\left(i\right)}-{\mathbf{y}}^{\left(i\right)}‖}_{\mathrm{2}}^{\mathrm{2}}\end{array}$$

(26)

and then the x-update in (24) can be expressed as

$$\begin{array}{l}{\mathbf{x}}^{\left(i\right)}\left(t+\mathrm{1}\right)={\left(\mathrm{2}{w}^{\left(i\right)}{\widehat{\mathbf{\Phi }}}^{(i{)}^T}{\widehat{\mathbf{\Phi }}}^{\left(i\right)}+\rho \left|{\mathbf{V}}^{\left(i\right)}\right|{\mathbf{I}}_N\right)}^{-\mathrm{1}}\\ \hspace{1em}\hspace{1em}·\left[\mathrm{2}{w}^{\left(i\right)}{\widehat{\mathbf{\Phi }}}^{(i{)}^T}{\mathbf{y}}^{\left(i\right)}-{\lambda }^{\left(i\right)}\left(t\right)+\left(\frac{\rho }{\mathrm{2}}\right)\phantom{{\displaystyle \sum }_{j\in V^{\left(i\right)}}‍}\right.\\ \phantom{hhhhhhh}·\left.\left(\left|{\mathbf{V}}^{\left(i\right)}\right|{\mathbf{x}}^{\left(i\right)}\left(t\right)+{\displaystyle \sum }_{j\in {\mathbf{V}}^{\left(i\right)}}{\mathbf{x}}^{\left(j\right)}\left(t\right)\right)\right].\end{array}$$

(27)

Based on the above studies, we propose a new decentralized scheme for cooperative compressed spectrum sensing in distributed networks which is summarized in Algorithm 1.

Algorithm 1 (the decentralized scheme for cooperative compressed spectrum sensing based on Karcher mean).

 

Initialization. Each CR obtains its local objective function $f_i$ based on the local information via (26) and initializes local copy ${\mathbf{x}}^{(i)}(\mathrm{0})={\mathrm{0}}^{N\times \mathrm{1}}$ and local Lagrange multiplier vector ${\lambda }^{(i)}(\mathrm{0})={\mathrm{0}}^{N\times \mathrm{1}}$ , $\forall i$ . set parameter ρ  (>0) empirically and the maximum number of iterations $t_{\max }$ allowed, and a small value ε as the tolerable deviation in convergence.

Iteration

•  

Repeat the following:

• (a)

  All CRs broadcast ${\mathbf{x}}^{(i)}(t)$ to their one-hop neighbors in ${\mathbf{V}}^{(i)}$ , $\forall i$ ,

• (b)

  all CRs update ${\lambda }^{(i)}(t)$ via (25), $\forall i$ ,

• (c)

  all CRs update ${\mathbf{x}}^{(i)}(t+\mathrm{1})$ via (27), $\forall i$ ,

• (d)

  increase t by 1,

 

until $t>t_{\max }$ or ${‖{\mathbf{x}}^{(i)}\left(t+\mathrm{1}\right)-{\mathbf{x}}^{(i)}\left(t\right)‖}_{\mathrm{1}}<\epsilon$ , $\forall i$ .

Decision. Upon convergence, each CR obtains the Karcher mean $\widehat{\mathbf{e}}$    (i.e., ${\mathbf{x}}^{(i)}\left(\infty \right)=\widehat{\mathbf{e}}$ , $\forall i$ ) and then the binary spectrum occupancy decision $\widehat{\mathbf{d}}$ by thresholding $\widehat{\mathbf{e}}$ via (9), where the threshold is set to a small value corresponding to the tolerable false alarm probability.

As shown in Algorithm 1, additional information, such as the upper bound of ambient noise energy or the sparsity order of received sparse signal, is not needed. Therefore, the proposed scheme is more flexible than the traditional CS-based CSS schemes in practice.

4. Simulations

This section presents Monte Carlo simulation results that verify the effectiveness of the proposed scheme. First, the simulation setup and relevant performance metrics are described. Then we compare the detection performance of several CS-based decentralized schemes and investigate their communication overhead and computational complexity. Lastly, the impacts of scheme parameters and system parameters are also studied through simulations.

4.1. Simulation Setup and Performance Metrics

Consider a monitored wideband which is equally partitioned into $N=\mathrm{128}$ channels, among which 16 channels are randomly occupied by PUs. Each channel experiences frequency-selective fading, while the fading coefficient on each channel is time-invariant within each sensing period and is generated randomly according to Rayleigh fading distribution.

Unless explicitly stated otherwise, some parameters are set as follows. The number of Monte Carlo trials is set to be 300. For the distributed CR network, we set the number of CRs $I=\mathrm{15}$ and their connectivity is given in Figure 1. For parameters of the proposed scheme, we give the same emphasis to all CRs by setting $w^{(\mathrm{1})}=\cdots =w^{(I)}=\mathrm{1}$ and setting $\rho =\mathrm{0.6}$ , $\epsilon =\mathrm{1}{\mathrm{0}}^{-\mathrm{1}}$ , and $t_{\max }=\mathrm{200}$ . Though these settings for ρ and ε are certainly not necessarily optimal, they work well in the simulations which will be discussed in the following.

Figure 1

Connectivity of CR network with 15 nodes (both ends of each line segment are one-hop neighbors and local communications are allowed between them).

The signal to noise ratio (SNR) is defined as the ratio of the average received signal to noise power over the entire wideband spectrum. Suppose that the received signal at each CR is corrupted by additive white Gaussian noise (AWGN) and all CRs have the same received SNR. The compression ratio $\gamma =M/N$ reflects the reduced number of samples M collected at each CR receiver, with reference to the number N needed in full-rate Nyquist sampling. For the spectrum hole detection problem, performance metrics of interest are the probabilities of detection $P_d$ and false alarm $P_{\text{fa}}$ , which are evaluated by comparing the estimated occupancy status $\widehat{\mathbf{d}}$ with the true occupancy status ${\mathbf{d}}_{\text{ture}}$ over all channels, as follows:

$$\begin{array}{l}{P}_d=\frac{{‖{\mathbf{d}}_{\text{ture}}\&\widehat{\mathbf{d}}‖}_{\mathrm{0}}}{{‖{\mathbf{d}}_{\text{ture}}‖}_{\mathrm{0}}},\\ P_{\text{fa}}=\frac{{‖\left(~{\mathbf{d}}_{\text{ture}}\right)\&\widehat{\mathbf{d}}‖}_{\mathrm{0}}}{{‖~{\mathbf{d}}_{\text{ture}}‖}_{\mathrm{0}}},\end{array}$$

(28)

where & and ~ denote bitwise AND and bitwise NOT, respectively, and here ${\ell }_{\mathrm{0}}$ -norm function is used to calculate the number of nonzero elements.

4.2. Performance Comparison of Three Sensing Schemes

For the sake of contrastive analysis, two existing CS-based decentralized schemes are chosen as benchmarks. One is the fusion consensus scheme developed in [12], and the other is a consensus averaging scheme for decision fusion which is developed in [11]. In the former, based on the joint sparsity structure of the received signals, a mixed ${\ell }_{\mathrm{1,2}}$ -norm minimization problem [15] is iteratively solved by using a decentralized implementation and each CR obtains the spectrum occupancy based on the reconstructed average energy vector. In the latter, each CR makes local decision on spectrum occupancy based on the local reconstruction, and then the global decision, which is the average value of local decisions, is computed in a distributed manner using the consensus averaging technique [19].

Figure 2 depicts the detection performance of the three schemes. When the upper bound of ambient noise energy is unknown, our proposed scheme performs the best at all SNR. To illustrate the effectiveness of the proposed scheme, the performance of those two benchmark schemes with prior information on the upper bound of ambient noise energy is also given in Figure 2. As shown in Figure 2, the introduction of prior information on the upper bound can indeed improve the detection performance since it can improve the reconstruction performance. However, even if this prior information is available, those benchmark schemes cannot perform better than our proposed scheme which does not need this prior information.

Figure 2

Detection probability versus SNR for $P_{\text{fa}}=0.01$ and $\gamma =\mathrm{0.4}$ .

To evaluate the communication overhead, we introduce the metric of communication step. When all the CRs transmit a vector of size $N\times \mathrm{1}$ to its neighbors, one communication step occurs. Figure 3 shows the communication steps which are required in those schemes depicted in Figure 2. Apparently, the consensus averaging scheme has the lowest overhead at the expense of worst detection performance. Comparing our proposed scheme and the fusion consensus scheme in the case where the prior information is unavailable, our proposed scheme has much less communication steps and better detection performance. The communication steps of fusion consensus scheme can be reduced by the introduction of prior information on the upper bound of ambient noise energy. However, it cannot be less than that of our proposed scheme which does not need this prior information.

Figure 3

Comparison among various schemes in terms of communication overhead.

To evaluate computational complexity of each scheme, we use the metric of CPU time. All codes are written in MATLAB and tested in a computer with Processor Intel Pentium Dual-Core E5300 running at 2.6 GHz with 2 GB of RAM and a Windows-based operating system. Figure 4 shows the CPU time per CR of those schemes. Apparently our proposed scheme has the least computational complexity since it only involves some arithmetic operations shown in Algorithm 1 while other two schemes have to solve a series of convex optimization problems.

Figure 4

Comparison among various schemes in terms of computational complexity per CR.

In summary, Figures 2–4 demonstrate the superior performance of the proposed scheme in terms of detection performance, communication overhead, and computational complexity.

4.3. Sensitivity to the Choice of Scheme Parameters

Obviously, the scheme parameters of Algorithm 1, that is, ρ and ε, affect the performance of the proposed scheme. In this subsection, we measure the sensitivity of the proposed scheme to the choice of these parameters.

4.3.1. Sensitivity to the Choice of ρ

Figure 5 and Table 1 show the influence of the parameter on the detection performance and iterative time of the proposed scheme, respectively. As shown in Figure 5, while choosing in a proper range, different ρ can lead to similar detection performance since all these choices of ρ can lead to convergence of the proposed scheme. However, it can be seen from Table 1 that the iterative time varies widely for different ρ. Since more iteration means more communication overhead and computational complexity, parameter ρ has to be properly chosen.

Table 1

Iterations of proposed scheme for different ρ.

ρ	0.1	0.3	0.5	0.6	0.7	0.9
Iterations	135	63.42	45.11	44.87	45.51	46.79

Figure 5

ROC performance of the proposed scheme for different ρ (SNR = −5 dB; $\gamma =\mathrm{0.375}$ ).

4.3.2. Sensitivity to the Choice of ε

Figure 6 and Table 2 show the influence of the parameter ε on the detection performance and iterative time of the proposed scheme, respectively. It is shown in Figure 6 that different ε choosing in a proper range leads to similar detection performance. Meanwhile, the iterative time varies widely for different ε, which can be seen from Table 2. Smaller ε leads to stricter stopping criterion; therefore, more iteration times are needed to converge for the proposed scheme. As mentioned above, more iteration means more communication workload and computational complexity. Therefore, parameter ε is not needed to be set too small.

Table 2

Iterations of proposed scheme for different ε.

ε	$$\mathrm{1}{\mathrm{0}}^{-\mathrm{4}}$$	$$\mathrm{1}{\mathrm{0}}^{-\mathrm{2}}$$	$$\mathrm{1}{\mathrm{0}}^{-\mathrm{1}}$$	1	10
Iterations	111.7	66.11	44.77	25.09	9.618

Figure 6

ROC performance of the proposed scheme for different ε (SNR = −5 dB; $\gamma =\mathrm{0.375}$ ).

4.4. Impact of System Parameters

Parameters of CR networks, such as the number of cooperating CRs and the compression ratio, significantly affect the detection performance of the proposed scheme. In this subsection, we investigate the impact of those system parameters.

4.4.1. Performance versus Number of CRs

Cooperation among multiple spatially distributed CRs offers cooperative gain. It is an attractive and effective approach to combat multipath fading and shadowing and to mitigate the receiver uncertainty problem. Figure 7 depicts the ROC curves for various numbers of collaborating CRs, under the parameter setting as follows: SNR = −5 dB; $\gamma =0.375$ . Apparently, the probability of detection $P_d$ improves as I increases, for the same probability of false alarm $P_{\text{fa}}$ . This suggests that more collaborating CRs bring more cooperative gain.

Figure 7

ROC performance of the proposed scheme for different numbers of collaborating CRs (SNR = −5 dB; $\gamma =\mathrm{0.375}$ ).

4.4.2. Performance versus Compression Ratio

In the proposed scheme, compressed sensing technique is introduced to break the bottleneck of ADC technique in wideband regime, such as sampling rate, fidelity, and power consumption; however, this also incurs performance degradation. Figure 8 depicts the ROC performance for various compression ratios, under the parameter setting as follows: SNR = −5 dB; $I=15$ . As shown in Figure 8, for the same probability of false alarm $P_{\text{fa}}$ , the probability of detection $P_d$ improves as more compressed measurements are collected.

Figure 8

ROC performance of the proposed scheme for different compression ratios (SNR = −5 dB; $I=15$ ).

5. Conclusions

Recognizing the joint sparsity structure of received signals at CRs, this paper has developed a decentralized scheme for cooperative compressed spectrum sensing based on Karcher mean. The major novelty of this scheme relates to using the Karcher mean of joint-sparse signals to estimate the spectrum occupancy directly from compressed measurements collected at CRs, thereby eliminating the compressed reconstruction stage and significantly reducing the computational complexity. To save the limited resources per CR, a decentralized implementation based on alternating direction method of multipliers is presented to calculate the Karcher mean via one-hop communications only. Moreover, additional information, which is generally unavailable in practice, is not needed in the proposed scheme. The superior performance of the proposed scheme is demonstrated by comparing with several existing CS-based decentralized schemes in terms of detection performance, communication overhead, and computational complexity. Lastly, the robust performance of scheme parameters and the impacts of system parameters are also investigated through simulations.