A SARS Method for Reliable Spectrum Sensing in Multiband Communication Systems

This paper introduces a novel method of spectrum sensing in communication systems that utilizes nonuniform sampling in conjunction with a suitable spectral analysis tool. It is referred to here as spectral analysis for randomized sampling (SARS). Owing to the deployment of nonuniform sampling, the proposed technique can accomplish the sensing task by using sampling rates well below the ones demanded by uniform-sampling-based digital signal processing (DSP). The effect of the cyclostationary nature of the incoming digital communication signal on the adequacy of the adopted periodogram-type estimator for the spectrum sensing operation is addressed. The statistical characteristics of the estimator are presented. General reliability conditions on the length of the required signal observation window, i.e., sensing time, for a chosen sampling rate or vice versa are provided amid a sought system performance. The impact of the presence of noise and processing transmissions with various power levels on the derived dependability recommendations is given. The analytical results are illustrated by numerical examples. This paper establishes a new framework for efficient spectrum sensing where considerable savings on the sampling rate and number of processed samples can be attained.


INTRODUCTION
Spectrum sensing entails scanning parts of the radio spectrum in search for a meaningful activity, such as a transmission or the occurrence of an event. Its techniques have lately received notable attention due to their crucial role in the emerging cognitive radio (CR) technology, i.e. by unveiling spectral holes for opportunistic spectrum access. Several reviews on the topic exist, e.g. [1][2][3][4]. This adds to the plethora of spectrum sensing application areas such as surveillance/interception [5], astronomy [6,7] and seismology [8]. Sensing methods that rely on nonparametric spectral analysis/estimation are regarded as efficient and adequate candidates for monitoring a wide frequency range consisting of a number of predefined nonoverlapping spectral subbands, without a priori knowledge of the signal's characteristics [1][2][3]. Such methods have clear advantages over single-band oriented ones, for instance those based on matched filtering or feature detecting that require the separation of the individual transmissions typically by tunable bandpass filtering [3]. In this paper, a multiband spectrum sensing technique that uses a periodogram-type spectral analysis tool to estimate the spectrum of the incoming signal from a finite set of its samples is adopted. This means has retained its popularity in several spectrum sensing studies, e.g. [9][10][11][12].
When uniform-sampling-based DSP is deployed, the sampling rate of the sensing device should exceed at least twice the total bandwidth of the monitored frequency range regardless of the spectrum occupancy [13]. Failing to do so results in aliasing and irresolvable detection problems. In the event of examining wide bandwidths such a constraint can pose a challenge to the system designer where a high sampling rate, high speed signal processing and treating large quantities of data are required [2,3]. Here, we demonstrate that we can detect the active spectral subbands by the suitable use of arbitrarily low rate nonuniform sampling and appropriate processing of the signal -a methodology referred to as digital alias-free signal processing (DASP). A few monographs, e.g. [14,15], give an overview on the topic.
Operating at low sampling rates can exploit the sensing device resources (such as power) more efficiently and/or avoid the possible need for a high-cost fast hardware. The main focus of this paper is to explore the possibility and benefits of employing the DASP methodology to conduct reliable detection in wideband communication systems.

A. Problem Formulation and Sensing Technique
Consider a communication system operating over L narrow nonoverlapping spectral subbands, each of them is of width C B . The total single-sided bandwidth that needs to be monitored by the system is y t x t n t = + . Our objective is to devise a method that is capable of scanning the overseen bandwidth B and identifying the active subbands. The algorithm should operate at sampling rates significantly lower than 2B which is the minimum rate (not always achievable) that could be used when uniform sampling is deployed [13].
Unlike methods that employ spectral analysis to estimate the subbands energy/power, e.g. classical energy detectors [9][10][11], the sensing procedure for each spectral subband comprises two steps: 1) estimating the magnitude of the signal spectrum at selected frequency point(s) and 2) comparing the magnitude(s) with pre-calculated threshold(s). Having a spectrograph that is relatively smooth would permit assessing fewer frequency points per system subband to determine its status. Here, we seek to inspect one frequency point per subband, i.e. L spectral points are calculated. The tackled sensing problem can be formulated as a conventional detection problem described by the following binary hypothesis testing 0, L k k f = are placed at the center of the subbands as shown in Section III. We emphasize that SARS aims at estimating a detectable frequency representation of the received signal and not its PSD.
The latter is defined as the Fourier transform (FT) of the signal's autocorrelation function [25].

B. Signal Model
x t be the continuous-time signal transmitted over one of the system active subbands by a communication source that deploys a linear digital modulation scheme. It can be expressed by: It is in the interest of the forthcoming analysis to find certain first and second order moments of the processed signal. It can be easily checked that [ ] As a result, the autocorrelation function of (3) is  2  2  , , , Noting that (4) is time-varying, such processes are commonly regarded as wide sense cyclostationary including the cases when the symbol period is not an integer multiple of the carrier period [27].

III. STATISTICAL CHARACTERISTICS OF SARS
The adopted total random sampling is an alias-free sampling scheme whose behavior was investigated in [18,19]

A. Evaluation of the Estimator's Adequacy
In order to determine the appropriateness of (5) to the detection purpose, the expected value of the and then noting that the signal and AWGN are independent as well as The signal's weighted power within r T is First, we can write   2  2  *  *  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,   2  2  *  *  ,  ,  ,  ,  ,  , S m n f +∞ =−∞ ∑ (11) noting the bandpass nature of the propagation channel response over the -th m active subband and assuming , ( * X denotes the conjugate of a complex variable X ); (10) can be restated as . Substituting (11) into (12) leads to . Employing (14) and (15), (13) simplifies to such that , ( , ) i.e. within the identified guarded regions. It is noted that for WSS signals, the used estimator is a suitable

B. Estimator's Accuracy
The estimator ( , ) e r X f t can be reliably used for spectrum sensing only if the difference is relatively small for a single realization of ( ) x t , especially at the (2). Chebychev's inequality states that Λ( , ) r f t is directly related to the standard deviation of the estimator, i.e.
σ is the variable's variance and 0 κ > [28] . The variance of ( , ) e r X f t , i.e., should be evaluated in order to ensure the dependability of the SARS method. First, The phase-shift ( , ) r θ f t is chosen in a way that It does not alter the definition of the estimator in (5) since TRS r I f t is the sum of N independent random variables, thus according to the central limit theorem they can be assumed to be approximately normally distributed for large N . In practice, moderate values of N suffice for such an approximation [3]. As a result, 2 ( , ) TRS r X f t has approximately an unnormalised chi-squared distribution with two degrees of freedom [28] and the estimator ( , ) e r X f t variance is defined by The phase-shift in (20)- (28), whose role is to simplify the estimator's variance expression, is given by 2 2 ( , ) ( , ) ( , ) 0.5arccot 2 ( , ) (23) and (24) Equations (23)-(32) were derived in a similar manner to that of the WSS signals in [24]. , , according to the detection criterion in (2), i.e. one per monitored subband. From (25), where the term that includes the sinusoid represents a windowed Cosine transform of the signal's second moment plus a constant at frequency point 2 k f which is a high frequency outside the overseen frequency range. This is expected to be of a negligible value in comparison to { } Deciding the value of ( , ) r k η f t is important as it forms a substantial part of the estimator's variance. We have: ( , ) ( , ) 2 ( , )

n s t n w t w t e dt dt
which is the case for WSS signals [24]. To depict the impact of cyclostationarity on (35), we assume that the FT of the ( ) w t reduces to a Dirac delta ( ) δ f , i.e. very long time analysis window. Each of (38) and (39) emerges as  2  2  2  2  2  , , , This indicates that 2 ( , ) B . This is the case for any other linear modulation scheme that has only one branch, i.e. either in-phase or quadrature. Therefore, the accuracy of the spectrum estimator can be affected by the signal's cyclostationarity and any processing task that relies on the spectral analysis, e.g. spectrum sensing, should consider the possible presence of such phenomenon. In the following subsection, we give a numerical example to illustrate the estimator's response to processing two types of cyclostationary signals.   (43)

C. Numerical Example of the Estimator's Variance
The nonoverlapping signal segments are assumed to be uncorrelated in this study. Finding the value of K in (43) is essential to realize a dependable sensing strategy and quantify its constraints as well as complexity.

IV. MULTIBAND RELIABLE SPECTRUM SENSING
The reliability and robustness of the SARS technique is reflected by its ability to meet a sought system behaviour that is commonly expressed by the receiver operating characteristics. The ROC of each

A. Reliability Conditions
Distinctive ROC plots, i.e. , , , describing the desired multiband detection performance. Due to nonuniform sampling, the estimated spectrum suffers from smeared-aliasing defect present at all frequencies and embodies a form of the signal and noise powers as indicated by (8) [28]. Hence the

CDF of ˆ( )
e k X f can be assumed to be approximately equal to that of a normal distribution with the same mean and variance. This can be further justified by central limit theorem [28] and its accuracy is verified by the simulations in Section V. Accordingly, the CDF complement function mandates the ROC probabilities for a given threshold such that In order to use (48), we have to calculate in Section III. The noise power is denoted by The signal powers in (50)  We indicate each of those powers by "(0)" and "(1)" superscripts to signify 0,k H and 1,k H respectively.

Thus in summary
According to Parseval's theorem: Equation (57) gives a conservative lower limit on the number of windows that need to be averaged as a function of the spectrum occupancy, average sampling rate, signal to noise ratio and the sought system performance. This recommendation can be used to decide the required average sampling rate for a number of estimate averages possibly imposed by practical constraints (e.g. latency) in a continuous processing environment. It is a clear indication of the trade-off between the sampling rate and the number of averages requested in relation to achieving reliable sensing. Equation (57) affirms that the sensing task can be reliably accomplished with arbitrarily low sampling rates at the expense of an infinitely long signal observation window. This confirms early results on DASP, e.g. [21,22], which were rather limited to PSD estimation for WSS signals.
The sensing process includes specifying the thresholds in (2) , , , , where the components of (58) and (59) can be computed according to (54) and (55) for 1, 2, k L = … . It is noted that correlated or overlapping signal windows scenario can be easily introduced into the SARS technique whenever the effect of correlation/overlapping on the variance reduction following averaging is known, e.g. Welch periodograms [29,30].

B. Randomised Versus Uniform Sampling
Spectrum sensing methods that employ periodogram-type estimators with uniform sampling to detect active transmissions via assessing spectral peak, e.g. [12], typically demand less estimate averaging compared to SARS which suffers from the smeared-aliasing defect. Following similar analysis/methodology to that of the TRS scheme, it can be shown that the number of estimate averages for the uniform-sampling-based algorithm is given by where US f is the uniform sampling rate and is proportional to the monitored bandwidth B to avoid the aliasing effects. Comparing the efficiency of both approaches based only on the sampling rates can be regarded as partial. The detection decision in both cases relies on calculating a form of discrete-time . This shows that as the spectrum occupancy decreases, the benefits of exploiting nonuniform sampling become more visible. Low spectrum utilization is faced in various applications, e.g. in CR networks it can be 15% or lower in certain bands [11]. It can be seen in Fig. 2 that the desired system performance was delivered with a sampling rate of 90MHz. Hence savings of around 60% on the sampling rate and more than 20% on the number of processed samples according to (62) and (63) were attained by using the proposed approach in this paper.

V. NUMERICAL EXAMPLES
At min K K = , the acquired probabilities match to a great extend the minimum specified ones. This confirms the reasonable conservativeness of the provided recommendations and that the assumptions undertaken in the conducted analysis did not have noticeable effects on the accuracy of the obtained results. Fig. 2b vindicates the effectiveness of the thresholding regime described by (60). If ˆ0.5 η = was chosen, i.e. the impact of signal's cyclostationarity was not recognised, the minimum number of estimate averages would be min 10 K = which would jeopardize the system response. Copyright (c) 2011 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.    It is clear from Fig. 5 that the pursued probabilities of the targeted subband were acquired by following the derived reliability recommendations. Besides, Fig. 5b shows that the condition in (48) is fulfilled for 16 K ≥ . However, this is not the case for the subband centered at 5 f ; shown in Fig. 6; that demands min 21 K K ≥ ≥ given its ROC probabilities and power level according to (57). This illustrates the compromising involved when a priority subband is specified by the user. To circumvent such cases, the user should survey the requisites for all system subbands and subsequently choose the combined K and α values that would meet all the desired , In general, the above numerical examples demonstrate that SARS can notably reduce the required sampling rates to perform wideband spectrum sensing and yet meet the predefined probabilities of detection and false alarm.

VI. CONCLUSIONS
In this paper, a multiband spectrum sensing method that is based on DASP methodology is proposed.
It uses a particular randomized sampling scheme along with appropriate processing to conduct reliable detection. This approach eliminates the adverse effect of aliasing that is inherently present when similar signal processing problems are solved with uniform-sampling-based techniques. The sampling rate is no longer related to the total bandwidth of the monitored subbands. In fact, it is shown in the paper that the sampling rate of the introduced spectrum sensing approach can be arbitrarily low. Taking into account the cyclostationary nature of communication signals, the reliability of the sensing procedure is formulated in terms of the average sampling rate, signal observation window 0 KT , signal to noise ratio, power levels of the active overseen subbands and the sought system performance. The provided dependability guideline can be employed as a tool to quantify the trade-off between the required sensing time (i.e. signal observation window) and sampling rate in a given scenario.
Comparing to methods based on uniform sampling, the proposed sensing technique offers substantial savings not only on the sampling rate but also on the total number of processed samples. The latter is