A Spectrum Sensing Method Based on Stratified Sampling Bashar I. Ahmad and Andrzej Tarczynski Department of Electronic, Communication and Software Engineering, University of Westminster 115 New Cavendish Street, London W1W 6UW email:
[email protected],
[email protected] Abstract—This paper introduces a multiband spectrum sensing technique that deploys nonuniform stratified sampling to perform reliable sensing using sampling rates well below the ones demanded by uniform-sampling-based DSP. Reliability guidelines are provided to ensure the credibility of the detection method amid a sought system performance. It is demonstrated that stratified sampling can be implemented in practice with finite resources - two interleaved analogue to digital converters suffice. This is a clear advantage over previously reported nonuniform sampling schemes e.g. total random sampling where an arbitrary number of its samples can be arbitrarily close.
I.
INTRODUCTION
Spectrum sensing entails unveiling a meaningful activity within a predefined range of frequencies e.g. an ongoing transmission or the occurrence of an event. It has various application areas e.g. astronomy and communication systems. The latter includes the emerging Cognitive Radio (CR) technology which triggered intensive research into robust/dependable sensing techniques e.g. [1-3]. In the scenario of monitoring a wideband frequency range consisting of a number of non-overlapping spectral bands and without prior knowledge of the signals’ characteristics, spectrum sensing methods that rely on nonparametric spectral analysis are regarded as adequate/efficient candidates [1-3]. This approach is endorsed here where the aforementioned scenario is studied i.e. Multiband Spectrum Sensing (MSS). Uniform-sampling-based DSP imposes a minimum sampling rate of twice the monitored frequency range despite the bands’ activity. Otherwise aliasing causes irresolvable detection problems [4]. In this paper, we show that we can reliably detect the active spectral bands by the suitable use of considerably low-rate nonuniform sampling (well below the uniform sampling counterpart) and appropriate processing of the signal – a methodology referred to as Digital Alias-free Signal Processing (DASP) [5]. Operating at low sampling rates, DASP can exploit the sensing device resources such as power more efficiently and/or avoid the possible need for a high cost fast hardware. The majority of the reported sensing algorithms at this time are uniform-sampling-based [1-3]. In [6, 7], we studied a MSS technique based on DASP using Total Random Sampling (TRS). Related work reported
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in the literature can be found therein. The TRS sample points tn ’s are Independent Identically Distributed (IID) random variables [8]. Their Probability Distribution Functions (PDFs) p (t ) take nonzero values only within the time analysis window T r = [ tr , tr + T0 ] . Let α = N / T0 be the average sampling rate where N is the number of the processed samples and βn = 1/(tn +1 − tn ) be the instantaneous sampling frequency that may change for every n . Although MSS can be carried out with arbitrarily low α when using TRS [6], any two or more sample points of a TRS sequence can be arbitrarily close i.e. βn can be infinitely high. This implies that TRS may require unlimited resources in terms of the Analogue to Digital Converter (ADC) speed. Early papers on DASP; which were rather limited to Power Spectral Density (PSD) estimation; used sampling schemes that suffered from similar defect e.g. additive Poisson sampling [5]. In this study, we exploit a form of stratified sampling that guarantees a minimum distance between any two samples by employing two interleaved ADCs to conduct dependably spectrum sensing. This approach is more suited for implementation in hardware and can deliver similar performance to that of TRS i.e. the impractical aspect of TRS is alleviated. The main focus here is to explore the possibility/benefits of adopting an implementable DASP technique to perform reliable MSS. II.
PROBLEM FOMULATION AND SENSING TECHNIQUE
Consider a communication system operating over L predefined non-overlapping spectral bands, each of them with bandwidth BC . The total single-sided monitored bandwidth is B = LBC . Whilst, the maximum number of concurrently
active bands at any particular point in time is LA i.e. the joint bandwidth of the active bands never exceeds BA = LA BC . Our objective is to devise a technique that is capable of scanning the overseen bandwidth B and identifying the active bands. The algorithm should operate at sampling rates significantly lower than 2B which is theoretically the lowest uniform sampling rate (not always achievable) that can be used [4]. The proposed method relies on estimating the spectrum of the incoming signal from a finite set of its noisy nonuniform
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distributed samples using a periodogram-type spectral analysis tool. It encompasses two steps: 1) estimating the spectrum of the received signal at certain frequency points and 2) comparing these estimates with a selected threshold(s). We note that estimating the signal’s exact PSD is not the objective and a frequency representation that permits detection of the active bands is sufficient. The tackled sensing problem can be represented by the following binary hypothesis testing:
where
H 0, k :
Xˆ e ( f k ) < γ k
H1, k :
Xˆ e ( f k ) ≥ γk
k = 1,2,… L
(1)
Xˆ e ( f ) is the estimated spectrum and γk is the
threshold.
Whilst H 0,k and H1,k hypotheses signify the
absence and presence of an activity in band k respectively. One frequency point per band is assessed in (1) to determine the band’s status. To make this approach works effectively, a smooth spectrograph is attained by deploying appropriately short analysis time window. Here we examine the scenario where all the active bands are of similar power levels and the Wide Sense Stationary (WSS) signal propagates via an Additive White Gaussian Noise (AWGN) channel where σ n 2 denotes the noise variance. III.
1 K −1 Xˆ e ( f ) = ∑ X e ( rT0 , f ) . K r =0
Finding the value of K in (3) along with the average sampling rate is essential to realize a dependable sensing strategy and quantify its constraints as well as complexity. The processed signal is assumed to be zero mean, bandlimited and WSS. Although communication signals are known to be of a cyclostationary nature, phase randomization is a widely adopted technique to stationarize the process when its cyclic frequency is not of an interest [11]. Below, we show that (3) is a suitable for the sensing task. B. Target Frequency Representation Since the summands in (2) are independent, it can be shown that: C ( f ) = E [ X e ( tr , f ) ] =
with Equal Partitions (SSEP) where S n is the width of the n-th subinterval. Figure 1 depicts a SSEP sequence.
Periodogram-type Estimator The adopted estimator of a detectable frequency representation of the processed signal is given by:
∑ y(tn ) w(tn )e− j 2πftn
2
(2)
n =1
where y (tn ) = x (tn ) + n(tn ) comprises of signal plus AWGN noise, w(t ) is the windowing function and μ = ∫
+
PS + σ n2 − ψ ( f ) (4) α
tr + T0
w(t )e − j 2 πft dt , Φ X ( f ) is the where PS = E ⎡⎣ x 2 (t ) ⎤⎦ , W ( f ) = ∫ t signal’s PSD and “*” denotes the convolution operation (see r
N
for example [6]). Whilst, ψ ( f ) = ∑ Φ X ( f ) * Vn ( f ) / μ and 2
n =1
Vn ( f ) = ∫ w(t )e− j 2 πft dt . We note that C ( f ) = E ⎡⎣ Xˆ e ( f ) ⎤⎦ . It is 2
given by the tapered signal’s PSD Φ X ( f ) * W ( f ) / μ plus two components introduced due to the deployment of nonuniform sampling; commonly referred to by smeared-aliasing [5]. The frequency independent component ( PS + σ n2 ) / α merely serves as amplitude offset at all frequencies. Whilst ψ ( f ) minimizes the latter’s effect in the vicinity of the signal’s band i.e. attenuates the smeared-aliasing at such frequencies. This results in improving the dynamic range of C ( f ) locally i.e. within the signal band, but the range deteriorates across the overall monitored bandwidth. To illustrate such phenomenon, consider a rectangular w(t ) i.e. :
A.
N
μ
2
noticed from (4) that C ( f ) consists of a detectable feature
divided into N non-overlapping subintervals: S1 , S2 ,… S N . Within each of them a sample with uniform PDF is taken. Optimizing the sizes of the subintervals to enhance the quality of the spectral estimation would normally require beforehand knowledge of the processed signal [9]. Consequently, here we consider the case where the subintervals are equal i.e. S n = T0 / N Stratified Sampling
1 μα 2
ΦX ( f ) * W ( f )
Sn
STRATIFIED SAMPLING
With stratified sampling the signal analysis window T r is
X e ( tr , f ) =
(3)
tr + T0 tr
w2 (t )dt .
Typically, a K number of X e ( tr , f ) estimates are averaged to improve its accuracy. This evokes shifting T r i.e. changing
tr and repositioning/aligning of the tapering function w(t ) . Non-overlapping signal segments are examined here and are assumed to be uncorrelated e.g. Bartlett periodogram [10]. As a result, the variance of the estimator is reduced by a factor of 1/ K and the adopted sensing procedure is based on:
2 2 C ( f ) = ⎡ PS + σ n2 − Φ X ( f ) * sinc( f / α ) ⎤ / α + Φ X ( f ) * W ( f ) / μ . It ⎣ ⎦
shows that the dynamic range of the estimator’s expected 2 value depends on α where PS ≥ Φ X ( f ) * sinc( f / α ) . Hence, C ( f ) is a detectable frequency representation of the incoming signal that permits unveiling any activity within B . However, the average sampling rate of SSEP should be reasonably selected in order to ensure a conservative dynamic range of C ( f ) for the sensing pursuit. We recall that a short analysis time window is utilized to maintain low spectrum resolution in order to minimize the number of needed frequency points in the detection process; one per examined spectral band. This results in savings on computations. As discussed in [6], T0 ≥ 1/ BC offers a practical guideline for choosing the signal analysis window. It is noted that the use of a relatively short signal analysis window further justifies the stationarity assumption (pseudo-stationarity) of a processed communication signal [3].
403
≈
is presumed to be negligible. Provided a reasonable dynamic range of C ( f ) and endorsing a conservative approach to the sensing problem, it can be shown that (7) leads to:
Figure 1, A SSEP sampling sequence within
IV.
{
K ≥ 2 BA (1 + SNR −1 ) ⎡⎣Q −1 ( Δ ) − Q −1 ( ) ⎤⎦ α − Q −1 (
Tr
RELIABLE SENSING
The reliability of a sensing technique is reflected by its ability to meet a sought system behaviour that is commonly expressed by the Receiver’s Operating Characteristics (ROC). In this section we deploy the ROC to derive the pursued dependability conditions. A. Reliability Condition According to the central limit theorem for a large number of averaged windows K , Xˆ e ( f ) becomes normally distributed ( K ≥ 20 is often perceived as sufficient in practice [1]). It can be compactly written as: Xˆ e ( f k ) ∼ N ( m0 , σ 0 2 ) and Xˆ e ( f k ) ∼ N ( m1 ( f k ), σ12 ( f k ) ) for H 0, k and H1, k respectively.
Such an approximation is further strengthened by the fact that X e ( rT0 , f ) in (3) have approximately unnormalised chisquared distribution with two degrees of freedom [6] i.e. the requirement on K for the normality assumption can be relaxed as shown in the numerical example below. The subscriptions of the mean and the variance are discarded since the active bands are assumed to be of similar power levels. Using the detection decision described by (1), the probability of a false alarm in a particular band is given by :
{
}
Pf , k ( γ ) = Pr H1, k H 0, k = Q ⎡⎣( γ − m0 ) / σ 0 ⎤⎦
(5)
and the probability of correct detection is:
{
}
Pd , k ( f k , γ ) = Pr H1, k H1, k = Q ⎡⎣( γ − m1 ( f k ) ) / σ1 ( f k ) ⎤⎦
(6)
where Q ( z ) is the tail probability of a zero mean and unit variance normal distribution. Due to nonuniform sampling, the false alarm can be triggered not only by the present noise but also by the existing smeared-aliasing.
m1 ( f k ) − m0 ≥ Q −1 ( Δ ) σ 0 − Q −1 ( ) σ1 ( f k )
(7)
which defines the reliability condition of the sensing procedure. In order to utilize (7) to ensure fulfilling the detection probabilities, the mean and the variance of the estimator in (2) should be calculated. Due to the limited space of this paper, the variance calculations are omitted. However, var ⎡⎣ X e ( tr , f ) x(t ) ⎤⎦ for SSEP is always less than or equal to that of TRS [9]. Such a difference diminishes for low average sampling rate and/or short signal time windows which are the pursued values in this study. Assuming that the narrow active bands are located within a wide overseen frequency rage, the contribution of the smeared-aliasing reduction factor i.e. ψ ( f )
2
(8)
where SNR = PS / σ n2 . It is similar to the TRS case especially when N /( N − 1) ≈ 1 and a numerical example is shown below to verify the accuracy of this recommendation. Formula (8) gives a conservative guideline on the number of needed window averages which is a function of the spectrum occupancy, average sampling rate and signal to noise ratio. It is a clear indication of the trade-off between the sampling rate and the number of averages needed in relation to achieving dependable sensing. According to (8), we can use very low sampling rates for the sensing operation at the expense of using considerably long signal observation window i.e. KT0 . It is noted that for Pf , k ≤ Δ and Pd , k ≥ , an upper and lower bound of the threshold γk in (1) that would satisfy the sought performance can be provided given the mean and the variance of the estimator e.g. [1, 7]. Besides, correlated and overlapping signal windows can be easily incorporated into the analysis above by using existing results in the literature on variance reductions e.g. Welch periodograms [10]. B. Stratified Sampling with Two ADCs The SSEP is an implementable sampling scheme by using two interleaved ADCs. ADC 1 captures the samples in the subintervals S1 , S3 , S5 ,… and ADC 2 takes the samples in the rest of the subintervals i.e. S2 , S4 , S6 ,… . By doing so, the minimum distance between two samples captured by the same ADC is D = T0 / N (see Figure 1). The guarantee of such a minimum distance constrains the maximum instantaneous sampling frequency where 0.5α ≤ βn ≤ α . This makes the implementation of the scheme in hardware feasible unlike TRS or additive Poisson sampling where the instantaneous sampling frequency can be infinitely high. The capability of the available acquisition device(s) should be taken into account upon deciding on T0 , α and K according to (8).
In practice, the user typically specifies: Pf , k ≤ Δ and Pd , k ≥ . Given (5) and (6), we obtain:
)}
V.
NUMERICAL EXAMPLE
Consider a multiband system comprising of L = 20 bands where BC = 5 MHz. The system bands are located in f ∈ [1.35,1.45] GHz. A blackman window is employed where T0 = 0.8 us. 16QAM signals with maximum bandwidths and
similar power levels are transmitted over the active bands. A spectrum occupancy of 10% is assumed i.e. LA = 2 and BA = 10 MHz. A sampling rate α = 70 MHz is used and the SNR is -0.2 dB. For Pd ≥ 0.95 and Pf ≤ 0.08 for all the system bands, the needed estimate averages is K ≥ 12 according to (8). Figure 2 depicts both sides of dependability condition (7) for the minimum requested ( Pf , Pd ) . In Figure 3, we show the ROC of the adopted method for various values of K via
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simulations for a threshold sweep. In each of the plots 10000 independent experiments are used to approximate the statistical measures. Figure 2 and 3 confirms the moderate conservative nature of the given reliability conditions where the desired performance is achieved for K ≥ 12 upon satisfying (7). This also affirms that the assumptions undertaken did not have noticeable effect on the accuracy of results. Using a small number of window averages ( K < 20 ) i.e. weakening the normality assumption did not inflict noticeable error on the attained results. Further experimental results (not shown here) showed that the distribution of simultaneously active bands across the scanned bandwidth does not hinder the performance of the detection procedure. If uniform sampling is deployed the minimum valid bandpass sampling rate that would avoid aliasing within B is 224 MHz. Whilst, MSS with TRS would deliver similar performance to SSEP for similar K and α [6, 7]. Table 1 exhibits the sampling rates for MSS with various sampling schemes to demonstrate the benefits of SSEP with two ADCs. Uniform sampling employing two ADCs is considered for the comparison to be impartial. The sampling rates of uniform sampling and SSEP in Table 1 are given for each ADC. Using more than one acquisition device for TRS does not impose a finite βn i.e. one ADC is presumed.
By adopting the proposed technique in this paper around 67% saving on the sampling rate is achieved in comparison to uniform sampling. Besides, the instantaneous sampling frequency of SSEP is restricted to 70 MHz and is nearly 62% of that of the uniform sampling i.e. 38% saving. Most importunately, SSEP permitted performing dependable MSS using DASP methodology with an affordable maximum βn unlike TRS case where it can be infinity. It is unambiguously clear that MSS with SSEP scheme offers tangible benefits in terms of the sampling rates and eases notably the requirements of the data acquisition hardware. VI.
CONCLUSION
This paper formulates a new framework for spectrum sensing where significant savings on the sampling rates can be attained. This is owed to the utilization of implementable randomized sampling. Reliability recommendations are provided and a means to apply the endorsed sampling scheme via two interleaved ADCs is proposed. In order to preserve the reconstrutability of the detected signals, it is necessary that the sampling rates exceed twice the total bandwidth of the concurrently active bands BA . This feature compares favorably with uniform sampling based spectrum sensing methods especially in low spectrum occupancy environments i.e. BA
