Spectrum sensing using low-complexity principal components for cognitive radios
© Idrees et al. 2015
Received: 31 October 2014
Accepted: 13 May 2015
Published: 26 June 2015
Principal component (PC) algorithm has recently been shown as a very accurate blind detection technique in comparison with other covariance-based detection algorithms. However, it also has a higher complexity owing to the computation of the eigenvectors. We propose a low-complexity Lanczos principal component (LPC) algorithm that utilizes Lanczos iterative method to compute the eigenvectors. In comparison with the PC algorithm, the proposed LPC algorithm offers significant reduction in complexity while giving a similar detection performance. Low-complexity LPC algorithm allows for the use of larger sized covariance matrix that further improves the detection performance. Maximum-minimum eigenvalue (MME) algorithm is also included in the comparison and it gives an inferior performance as compared to both PC and LPC algorithm. All the algorithms were tested with experimental data while using universal software radio peripheral (USRP) testbed that was controlled by GNU radio software.
Cognitive radio has the ability to communicate over the unused frequency spectrum intelligently and adaptively. Spectrum sensing in a cognitive radio (CR) is crucial in generating awareness about the radio environment . Blind detection methods such as covariance-based detection (CBD) algorithms enable signal detection in low signal-to-noise ratio (SNR) conditions without relying on the prior knowledge of the primary user’s (PU) signal. CBD techniques also overcome the issue of noise power uncertainty that exists in an energy detector [2, 3]. These methods use the covariance and variances of the received signal and do not require information about the noise variance. The performance of the CBD algorithms is associated with the number of samples involved in the detection. However, using a large number of samples also increases the sensing time and complexity [4–8]. Recently, principal component analysis (PCA) has been applied for spectrum sensing in cognitive radios [9–11]. The principal of dimension reduction has been used in  to devise PC algorithm that outperforms other CBD algorithms, such as Maximum-minimum eigenvalue (MME), maximum eigenvalue detection (MED), and energy with maximum eigenvalue (EME). PCA reduces the dimensionality of the data and retains the most significant components that account for the greatest variation of the original data [13–16]. However, the PC algorithm also has the highest complexity in comparison with other CBD techniques.
As envisaged in internet of things (IoT), the number of things (or devices) connected to the network might exceed the number of human users. The same idea also derives the research on 5G networks where the network capacity will be enhanced by a 1000 fold. Opportunistic spectrum access may help in this scenario where multiple overlaid devices try to access the spectrum. Spectrum sensing can help eradicate collisions and excessive contention delay experienced by dense node deployment. Such devices/sensors have embedded computing nature, hence energy efficiency is their major concern. Therefore, low complexity, energy efficient spectrum sensing algorithms are vital for implementation in such devices.
The detection performance of the PC algorithm is analysed under a low SNR (<−15 dB) scenario while varying the number of principal components included in the decision test statistic. The effect of dimension reduction on the sensing performance is also considered along with the complexity involved in each case.
A low-complexity LPC algorithm is proposed that employs an iterative approach to compute the principal components and achieves the similar detection performance as of the PC algorithm with a reduced complexity. This reduction in complexity saves the sensing time and improves the energy efficiency.
The performance of the proposed LPC algorithm is compared with the PC and the MME algorithms while using the actual signals. In addition, computational complexity of all the three algorithms (MME, PC and LPC) is also computed and compared mathematically and graphically.
All the algorithms has been evaluated under both single and multiple receive antenna system with actual wireless microphone signals. Experimental setup is established using USRP2 and GNU radio software.
2 System model, detection with MME and PCA
2.1 System model
Where H 0 is the null hypothesis that shows the absence of the PU’s signal. H 1 is the alternative hypothesis which indicates the presence of the signal. Probability of detection (P d ) and probability of false alarm (P fa ) characterizes the sensing performance. Where P d =Pr(H 1|H 1) and P fa =Pr(H 1|H 0).
PU’s signal exists if the ratio is greater than ψ, where ψ is the threshold set according to desired probability of false alarm. MME is a blind detection algorithm without noise uncertainty issue, but calculation of eigenvalues using conventional method is computationally intensive.
2.3 PC Algorithm
Here p ji is the ith element of jth PC and ψ is the detection threshold determined empirically at a desired probability of false alarm. PU exists if T>ψ.
2.4 Performance analysis of the PC algorithm
3 Proposed LPC algorithm
From the performance analysis of the PC algorithm, it is observed that the detection performance can be improved by including more PCs while using larger sized covariance matrix. However, doing so increases the complexity. In this section, we propose the LPC algorithm that uses the iterative method to compute the eigenvectors, required to generate PCs; the use of this approach significantly reduces the complexity. The proposed method performs much faster than the existing method, as it only computes the eigenvectors corresponding to the highest eigenvalues, whereas the direct method computes all the eigenvectors, thus wasting resources in computing in significant eignvectors. This approach is more efficient as it obviates the calculation of all the eignvectors and then sorting them at the end. Advance algorithms such as Lanczos and Arnoldi save this data and use the Gram-Schmidt process or Householder algorithm to reorthogonalize them into a basis spanning the Krylov subspace corresponding to the matrix. As the matrix size increases, the direct method becomes very slow, therefore not feasible practically, while the proposed method only calculates the desired eigenvectors via an iterative approach.
There are many iterative approaches like Arnoldi algorithm, Jacobi-Davidson algorithm and Lanczos algorithm . We found Lanczos algorithm appropriate for our scenario as it has the least convergence time as compared to other approaches . A disadvantage of this algorithm is that the number of iterations can be large. To cater for this issue, a variation of the Lanczos algorithm known as implicitly restarted Lanczos algorithm (IRLA) is used to compute the desired eigenvectors . Implicitly restart (IR) extracts the useful information from a large Krylove subspace and resolves the storage issue and the difficulties associated with the standard approach. IR does this by compressing the useful data into a fixed size k dimensional subspace. IRLA is summarized in Algorithm 1 . A is the symmetric matrix of interest v is the starting vector, T k ∈ℜ k×k is real, symmetric and tridiagonal with nonnegative subdiagonal elements. V k ∈C n×k (the columns of V k ) are the Lanczos vectors.
Calculate the covariance matrix as in (6).
Decompose the covariance matrix via implicitly restarted Lanczos algorithm as described in Algorithm 1.
Generate the principal components.
Calculate test statistic T as in (9).
Decide between H 1 and H 0 by comparing the T with a predetermined threshold (Empirically determined at the desired probability of false alarm).
3.1 Computational complexity comparison
As of the other covariance-based detection techniques, complexity of the PC algorithm also comprises of two major steps, one is the computation of the covariance matrix as in (6) and the other is the decomposition of the covariance matrix to calculate eigenvectors in our case. As the covariance matrix is a block Toeplitz and Hermitian, due to these properties of covariance matrix, we only need to evaluate its first block. Calculation of the covariance matrix requires (M 2 L N s ) multiplications and O(M 2 L(N s −1)) additions; here M is the number of receive antennas, L is the smoothing factor and N s is the number of samples . Eigen decomposition of the covariance matrix requires O(M 3 L 3) multiplications and additions .
4 Experimental setup
5 Results and discussions
Complexity is a major issue in blind signal detection algorithms that are based on a covariance matrix. The use of a large number of received samples increases the size of the covariance matrix and as a result the complexity. In this paper, we proposed a novel algorithm for blind signal detection, i.e. LPC algorithm that has an iterative nature which reduces the complexity and saves sensing time. LPC achieves the same detection performance as PC, yet its complexity is significantly less than the PC algorithm. Thus, LPC can be used in even low-powered devices for blind signal detection. The performance of LPC is compared with PC as well as MME algorithms. The proposed method gives the best sensing performance while reducing the complexity from O(L 3+L 2) to O(L). All the algorithms are tested with actual wireless microphone signals while using a USRP2 testbed and GNU radio software. In the future, these algorithms can be tested on a stand-alone platform for real-time performanceevaluation.
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