30th NATIONAL RADIO SCIENCE CONFERENCE (NRSC 2013) April 16‐18, 2013, National Telecommunication Institute, Egypt
C9. A New Wideband Spectrum Sensing Technique based on Compressive Sensing, Wavelet Edge Detection and Parallel Processing Said E. El-Khamy1, Mina B. Abdel-Malek1,2, Sara H. Kamel1 1 Faculty of Engineering, Alexandria University, Alexandria 21544, Egypt 2 School of Engineering and Science, American University in Cairo, Cairo 11511, Egypt
[email protected],
[email protected],
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ABSTRACT In this paper parallel processing is employed to improve the performance of compressive sensing techniques for wideband spectrum edge detection. By taking different sets of random sub-Nyquist samples of the spectrum and processing them in parallel then combining the edges detected by using each set we can get a better estimate of the location of the occupied bands. The compressive sensing technique applied uses the wavelet transfrom to detect the spectrum edges. An implementation of multiscale wavelet sum in the form of matrix multiplication is presented in order to enable its application in a compressive sensing framework.
Keywords: Compressive Sensing, Sub-Nyquist Sampling, Multiscale Wavelet Sum, Wavelet Transform, Cognitive Radio, Parallel Processing.
I. INTRODUCTION Wideband communication systems require spectrum sensing techniques for efficient spectrum utilization. Cognitive Radio (CR) is incorporated in such systems to detect spectrum holes so that unlicensed secondary users can use the unoccupied bands and avoid other bands occupied by primary (authorized) users [1]. Spectrum sensing in general involves energy detection, but for wideband systems it is crucial to first determine the band locations. This is the main difference between multiband systems (in which the band locations are known to the receiver a priori) and wideband systems where edge detection is of utmost importance [2]. Traditional energy detection techniques use a series of FIR filters each tuned to a certain frequency, or alternatively - one tuneable band-pass filter to sense one band at a time [3], but this is only suitable for narrowband systems and will immensely increase the complexity if used in a wideband system. Other more intuitive techniques can be used for energy detection among which are the matched filter approach, cyclostationary feature detection, multi-taper spectrum estimation and quadrature filter banks [4]. One of the most effective methods for edge detection is wavelet analysis. Wavelets have proven to be a powerful tool for edge detection applications with their time-frequency localization properties and their efficiency in analysing singularities. Wavelet analysis has therefore been utilized in wideband spectrum sensing and cognitive radio applications [5], [6]. Tian and Giannakis developed a wavelet approach to wideband spectrum sensing in [5], in which the continuous wavelet transform of the spectrum signal and its gradients were used to locate the band edges. Determination of frequency band locations effectively is not the only challenge in wideband systems; the need for fast and reliable processing tools in CR-based systems is just as essential, which led to the usage of compressive sensing which allows us to work with sub-Nyquist sampling rates by exploiting the sparseness of the spectrum [7], [8]. In this paper we use a wavelet approach in compressive sensing to detect the spectrum edges, and to improve the system performance, parallel processing is used where each processor uses a high compression ratio and the results obtained from each of the parallel processing units is fused to produce a better estimate of the band locations. The paper is organized as follows. Section II discusses the wavelet approach to edge detection, followed by a review of the mathematical background of compressive sensing in section III. The fourth section presents a new implementation of multiscale wavelet sum in compressive sensing motivated by the definition of the discrete wavelet transform at different dyadic scales using filter banks. The new parallel processing method is described in section V. In section VI details of the energy detection method and performance metric are presented. Finally, the simulation results highlighting the improvement in performance due to the use of parallel processing are shown in section VII followed by conclusive comments in section VIII. 978-1-4673-6222-1/13/$31.00 ©2013 IEEE
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30th NATIONAL RADIO SCIENCE CONFERENCE (NRSC 2013) April 16‐18, 2013, National Telecommunication Institute, Egypt
II. EDGE D ETECTION USING WAVELETS The wavelet transform is an efficient method for detecting and analysing local singularities, thus it has been widely used for edge detection in images and texture classification [9]. In this paper, the wavelet approach is implemented to detect the irregularities in the power spectral density (PSD) of the wideband signal which correspond to the band edges. If the PSD of the observed signal is given by Sr(f), then the continuous wavelet transform (CWT) of the PSD at a scale s is then given by Ws S r ( f ) S r ( f ) s ( f )
(1)
where * denotes convolution and ϕs(f) is the mother wavelet function scaled by a factor s, and it is defined as 1 f s s
s ( f )
(2)
The modulus of the first derivative of the CWT is then obtained and the local maxima of the resulting function are calculated. The local maxima represent the location of the spectrum edges. Assuming that the frequency edges are denoted by fn and n = 0, 1, 2... Nf where Nf is the number of frequency boundaries. The estimate of the frequency edges can be formulated as follows:
fˆn maxima
f
WsS r ( f ) ,
f [ f0 , f N f ]
(3)
Once these are known, we proceed to detect the energy of the signal between the edges to determine which spectrum bands are occupied and which are free [5]. The energy detection step and the performance metric are to be explained later in detail. Another method is proposed in [5] using multiscale wavelet products. The product of J continuous wavelet transforms of the spectrum signal Sr(f) (with the wavelet functions restricted to dyadic scales; s = 2j; j = 0, 1, 2...) is given by J
W prod , J S r ( f ) Ws 2 j S r ( f )
(4)
j 1
The advantage of using multiscale wavelet products is that some local maxima may be induced by noise peaks, but these peaks don’t usually appear at every scale, so multiplication of the wavelet transform at different scales reduces these peaks. On the other hand, at low signal-to-noise ratio (SNR) values, some frequency boundaries may produce low peaks at a certain scales, and in that case multiplication may further reduce the magnitude of this peak, hence when evaluating the local maxima of the multiscale wavelet product, this peak will be incorrectly classified as noise. Another approach is to use multiscale wavelet sum, in which case the peaks produced by actual frequency boundaries are added up and thus reinforced and the problem of losing peaks by multiplication is averted. The sum of J CWTs of the spectrum signal Sr(f) is given by J
Wsum, J S r ( f ) Ws 2 j S r ( f )
(5)
j 1
III. COMPRESSIVE SENSING Cognitive radios in wideband systems need to rapidly and efficiently adjust their transmitter parameters to use the available bands (that are not occupied by primary users) in the spectrum [7]. Operating at Nyquist rates in such systems means that the CR needs to process a considerably large number of samples in order to dynamically sense the spectrum. Compressive sensing exploits the sparseness of the spectrum signal and enables the CR to operate
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30th NATIONAL RADIO SCIENCE CONFERENCE (NRSC 2013) April 16‐18, 2013, National Telecommunication Institute, Egypt efficiently at sub-Nyquist rates. A brief explanation of the mathematical background of compressive sensing is presented in this section [10], [11]. Consider the signal x represented by a vector of length N in ℝN. Any such signal may be written as a linear combination of N orthonormal vectors ψi; i = 1, 2... N; these vectors naturally form a basis for ℝN. Arranging these orthonormal basis vectors to form a matrix Ψ by placing them as columns we can write the signal x as N
x si i
or
x s
(6)
i 1
where s and si are the length-N vector of weighting coefficients and the elements of that vector respectively [10]. From the matrix equation in (5), we can say that s is the pre-image of x under the linear transformation x → Ψx, so basically x and s represent the same signal; x is in the time domain while s is in the Ψ-domain. Now, if s is a Ksparse signal, where K
