Reliable wideband multichannel spectrum sensing using randomized sampling schemes

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Signal Processing

Signal Processing 90 (2010) 2232–2242

0165-16

doi:10.1

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Reliable wideband multichannel spectrum sensing usingrandomized sampling schemes

Bashar I. Ahmad �, Andrzej Tarczynski

University of Westminster, Department of Electronics, Communications and Software Engineering, London W1W 6UW, UK

a r t i c l e i n f o

Article history:

Received 12 November 2009

Accepted 5 February 2010Available online 16 February 2010

Keywords:

Digital alias-free signal processing

Nonuniform sampling

Spectral analysis and estimation

Fourier transform

84/$ - see front matter & 2010 Elsevier B.V. A

016/j.sigpro.2010.02.006

responding author. Tel.: +44 207 511 5000x36

4 207 511 5089.

ail address: [email protected] (B.I. Ahmad

a b s t r a c t

A wideband multichannel spectrum sensing approach that utilizes nonuniform

sampling and digital alias-free signal processing (DASP) to reliably sense the spectrum

using sampling rates well below the ones used in classical DSP is proposed. The

approach deploys a periodogram-type spectral analysis tool to estimate the spectrum of

the incoming signal from a finite number of its noisy nonuniformly distributed samples.

The statistical characteristics of the adopted estimator are analyzed and its accuracy is

assessed. It is demonstrated here that owing to the use of nonuniform sampling, the

sensing task can be carried out with the use of arbitrary low sampling rates. Most

importantly, general guidelines are provided on the required signal analysis time

window for a chosen sampling rate to guarantee sensing reliability within a particular

scenario. The extra requirement on such recommendations imposed by the presence of

noise is given. The analytical results are illustrated by numerical examples. This paper

establishes a new framework for multiband spectrum sensing where substantial saving

on the used sampling rates can be achieved.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Various spectrum sensing applications deal with anumber of signals of different sources sent over disjointspectral bands/channels i.e. multiband environment. Thetask of the sensing module is to detect meaningfulactivities within those bands, such as an ongoing transmis-sion or occurrence of some event within a predefined rangeof frequencies. The application areas include: astronomy[1,2], seismology [3], multiband communication systems[4,5] and many others. Several of these applications arecharacterised by low spectrum utilization i.e. only a smallproportion of the monitored channels are active at anygiven time instant. Typically, spectrum sensing in suchenvironments departs from the classical methods e.g.

ll rights reserved.

05;

).

power/energy detection within each channel and involvesestimating the spectrum of the incoming signal [4,6]. Thisapproach is adopted here where a periodogram-typespectrum analysis tool is deployed.

If no prior knowledge is available on the activity of theexamined spectral bands, the uniform sampling rate of thesensing device should exceed twice the total monitoredbandwidth regardless of the spectrum occupancy [7]. Failingto do so could result in aliasing and irresolvable detectionproblems. When dealing with low spectrum utilizationscenario, such sampling rates would exceed many times theLandau rate [8] which is the theoretically lowest samplingrate that permits full recovery of the sampled signal and isequal to twice the bandwidth of the concurrently activefrequency bands. For many bandpass and multiband signals,sampling at the Landau rate requires deployment of nonuni-form sampling as well as a prior knowledge of channelsactivity [9,10]. Exceeding Landau rate significantly mayindicate potential inefficiencies in using the sensing deviceresources such as power and/or deployment of high-cost, fast

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hardware capable of dealing with excessive sampling rates. Inthis paper, we demonstrate that even if the active channelsare unknown we can still detect them by the suitable useof arbitrary low-rate intentional nonuniform sampling(randomized sampling) and appropriate processing of thesignal—a methodology referred to as DASP. Few monographssuch as [11–14] give an overview on the topic.

1.1. Related work

Lomb–Scargle periodogram [1,2] and its recent moreefficient form [15] extended the classical periodogramspectral analysis tool to encompass arbitrary distributedsignal samples in order to detect discrete spectral compo-nents of the analyzed signal. In [16–18] consistent spectralestimation methods are studied for a number of alias-freerandomized sampling schemes where the signal isregarded as deterministic and in [19] these methods areutilized to perform spectrum sensing. In this paper, theproposed approach relies on the user’s ability to prescribethe sampling instants. Deliberately randomized, referred toby randomized for short, rather than arbitrary sampling isused to perform spectral analysis of a random multibandsignal disturbed by white noise. The latter assumption, i.e.the processed signal is random, postulates a generalstochastic framework that strengthens and generalizesthe preliminary results reported in [19].

Although the earliest papers on DASP-type algorithms[20–22] considered the problem of estimating the signal’spower spectral density (PSD) for various sampling schemes,they did not resolve the predicament of the estimator’sconsistency for a finite number of samples. This issue waspartially addressed by Masry in [23–25] where theexpressions for the estimator’s bias and variance areprovided under the assumption that the number ofprocessed samples tends to infinity. In this paper, theestimation of the exact signal’s PSD is not the objective andan estimate of a smoothed/windowed PSD that permitsdetection of the active spectral bands is sufficient. Here weassess the accuracy of the adopted estimator from a finiteset of samples in contrary to [23–25] where asymptoticconsistency expressions are given. It is noted that samplingschemes other than the ones examined in [20–25] areconsidered here and guidelines on detection dependabilityare provided. Sensing methods that entail computationallyexpensive algorithms e.g. [9,10], communication amongpossible multiple sensing devices e.g. [4,6] and the use oftransforms other than Fourier transform (FT) e.g. [26] areabandoned in this study.

Martin [12, p. 38] and Scargle [1] listed the deficienciesof periodogram spectral analysis tool in nonuniformsampling environment, namely inconsistency, leakage andsmeared-aliasing. However, in this study several measuresare taken, namely estimate averaging, tapering and alias-free sampling, to combat such drawbacks and appropriateperiodogram type of analysis to the handled problem i.e.spectrum sensing and not PSD estimation. Averaging anumber of periodograms to improve its accuracy has beenutilized long ago e.g. [27–29]. The number of averagesneeded is commonly overlooked and uniform sampling is

used. In this paper the number of averages that provides arequired level of detection reliability is sought and nonuni-form sampling is utilized.

1.2. Contribution

The contribution of this paper is twofold. First, it studiesthe statistical analysis of a periodogram-type spectralestimation method that uses total random sampling (TRS)scheme. The bias and the consistency of the adoptedestimator for a finite number of samples are evaluated.The DASP nature of the estimator is demonstrated in thecontext of identifying spectral components of the presentsignal. Second, the main contribution of this paper isdefining the sensing reliability in terms of the usedsampling rate, spectrum utilization and signal to noiseratio (SNR). A lower limit on the combined values of theaverage sampling rate and the number of estimate averagesis provided in order to guarantee detection reliability. Suchrecommendation provides a tool to assess the trade-offsbetween the needed sampling rate and the sensing-time aswell as a mean to inspect whether nonuniform sampling isbeneficial over the uniform case given a particular scenario.A decision making criterion on the channels activity is alsoproposed. Generalizing the obtained results to embracesampling schemes other than TRS is addressed.

The paper is organized as follows. In Section 2, thetackled problem is formulated and the adopted sensingmethodology is introduced. The deployed spectral analysistool, its statistical characteristics and suitability for thesensing task are presented/discussed in Section 3. InSection 4, sensing reliability recommendations are givenand the applicability of the results to other samplingschemes is discussed. The advantages of the proposedspectrum sensing method are demonstrated bynumerical examples in Section 5 and conclusions aredrawn in Section 6.

2. Problem formulation and sensing methodologyadopted

The considered system has L spectral bands, each ofthem with bandwidth BC : The total single-sided bandwidthof the monitored frequency range is B¼ LBC : The maximumnumber of simultaneously active channels and their jointbandwidth are given by LA and BA ¼ LABC ; respectively. Weassume low channel occupancy i.e. LA5L; central frequen-cies of all channels are known and the positions of activechannels are unknown beforehand. The main objective is todevise a method that is capable of scanning the monitoredbandwidth B and identifying which channels are active,if any. The algorithm should operate at sampling ratessignificantly less than 2B which is theoretically theminimum rate (not always achievable) that could be usedwhen bandpass sampling and classical DSP are deployed. Ifthe average sampling rate of the applied nonuniformsampling scheme is above Landau i.e. 2BA; it is possible toreconstruct the identified transmitted signals.

We recall that the detection approach adopted hererelies on estimating the spectrum of the signal. The sensing

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is made upon assessing the magnitude of the estimatedspectrum. The processed signal is assumed to be contami-nated with zero mean additive white Gaussian noise(AWGN) with a variance of s2

0: Hence, yðtnÞ ¼ xðtnÞþnðtnÞ

is the sum of the signal samples xðtnÞ and the added noisenðtnÞ: We perform spectral analysis via endorsing theperiodogram-type estimator given by

Xeðf Þ ¼N

ðN�1ÞmT0

N

XN

n ¼ 1

yðtnÞwðtnÞe�j2pftn

����������2

ð2:1Þ

where m is the energy of the deployed tapering/windowingfunction wðtÞ:

m¼Z t0þT0

t0

w2ðtÞdt ð2:2Þ

N is the number of the signal sample points tn’s and T0 isthe length of the signal analysis time window. Windowingis deployed to minimize leakage in the conducted spectralanalysis. A comprehensive review on windowing functionsis given in each of [30] and [31].

The sampling instants in (2.1) are placed inside the timeinterval ½t0; t0þT0� where t0 is the initial time instant of thewindow. The tapering function wðtÞ is aligned with thisinterval; for the simplicity of notations t0 is omitted from(2.1). In practice, a number of the estimates given by (2.1)need to be averaged using a moving window approach thatinvolves changing t0 and the repositioning of wðtÞ (seeSection 4). We demonstrate here that with the proper useof (2.1) we can identify with the required level ofconfidence which of the monitored channels are active.This involves selecting the length of the window T0; asuitable average sampling frequency and averaging thesufficient number of spectral estimates.

3. Properties of the spectrum estimator

The spectrum estimator proposed here exploits an alias-free randomized sampling scheme named TRS introducedin [16]. The sampling instants tn’s are independentidentically distributed (IID) random variables whose prob-ability distribution functions (PDFs) are given by

pðtÞ ¼1=T0; t 2 ½t0; t0þT0�

0 elsewhere

�ð3:1Þ

The processed signal is assumed to be bandlimited zeromean wide sense ergodic (WSE). In the following subsec-tion we show that (2.1) is an unbiased estimator of afrequency representation of the incoming signal that issuitable for the sensing task.

3.1. Targeted frequency representation

The estimator represented by (2.1) has three sources ofrandomness: signal, sample point process and AWGN. Theprobabilistic expectation should be taken in terms of thosethree stochastic processes. Given that the components ofthe summation (2.1) are independent with respect to thesample points, it is shown in Appendix A that the

conditional expectation of the estimator is given by

E½Xeðf ÞjxðtÞ� ¼N

ðN�1Þma

nZ t0þT0

t0

½xðtÞwðtÞ�2 dtþs20moþ

1

m jXW ðf Þj2

ð3:2Þ

where a¼N=T0 is the average sampling rate and

XW ðf Þ ¼

Z t0þT0

t0

xðtÞwðtÞe�j2pft dt ð3:3Þ

is the FT of the windowed xðtÞ: In the classical theory ofspectral estimation, jXW ðf Þj

2=m is a continuous form ofperiodogram where no sampling is involved. It is noted thatE½jXW ðf Þj

2� ¼FXðf Þ � jWðf Þj2, where FXðf Þ is the PSD of xðtÞ,

Wðf Þ is the FT of the windowing function and ‘‘*’’ denotesthe convolution operation. PSD is defined as the FT of theautocorrelation function of a wide sense stationary (WSS)signal [32]. As a result,

Cðf Þ ¼ E½Xeðf Þ� ¼N

ðN�1Þa fE½x2ðtÞ�þs2

0gþ1

mFXðf Þ � jWðf Þj2

ð3:4Þ

Away from asymptotic limits and in the context of theproblem considered in this paper where the pursued a isrelatively small and T0 is finite, the bias of the estimator interms of the windowed signal PSD i.e. the first term in (3.4) isconstant and frequency independent. Assuming that thesignal observation window period T0 is long enough, thetapered PSD forms an identifiable feature. Consequently, Cðf Þ

comprises of a detectable spectral feature i.e. ð1=mÞFXðf Þ �

jWðf Þj2 plus a constant offset. Hence, Xeðf Þ given by (2.1) is anunbiased estimator of Cðf Þ regardless of the sampling rate.Although the constant offset in (3.4) does not have a visibleeffect on the detectable feature of Cðf Þ, it deteriorates thespectrum dynamic range which can have a negative effect onthe spectrum sensing procedure as discussed in Section 4.

3.2. Length of the signal analysis window

The use of a long signal analysis window T0 results in ahigh resolution spectral analysis. The detection methodadopted here relies on sensing the magnitude of theestimated spectrum via assessing a number of spectralpoints to determine the channels status. Maintaining lowspectrum resolution by utilizing short signal time windowminimizes the number of needed frequency points or DFTpoints i.e. save on computations. We aspire to use onefrequency point per channel to detect any activity within.Using the theorem of minimum number of zero-crossings ofbandlimited signals as discussed in [14, p. 170] and exploi-ting the frequency-time duality characteristic [32] enable usto deploy the number of zero-crossings per channel as acriterion to describe the resolution of the spectrum. We notethat a zero-crossing is perceived as a notable fluctuation inthe spectrum magnitude. Having two crossings per activechannel is a reasonable assumption and as a result aguideline on the window period can be described by

T0Z1

BCð3:5Þ

Nonetheless, the signal analysis period T0 shouldensure that Cðf Þ exhibits distinguishable feature for the

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detection task. The chosen T0 should strike a balance betweenspectrum resolution and usefulness of the Cðf Þ. Experimentalresults showed that exceeding (3.5) a number of times wouldsuffice. In the following sections, T0 is considered as beingfixed i.e. predefined by the user.

3.3. Accuracy of the estimator

Although Xeðf Þ is an unbiased estimator of Cðf Þ, it will be asuitable tool for assessing the channels activity only if thedifference Dðf Þ ¼ jCðf Þ�Xeðf Þj is small. According to Cheby-chev’s inequality [33], which states that: PrfjX�E½X�jZ

esXgr1=e2 where sX is the standard deviation of a randomvariable X and e40; the difference Dðf Þ can be controlled bythe suitable reduction of the variance of Xeðf Þ: In this subsec-tion we derive an expression for the variance of Xeðf Þ given by

s2e ðf Þ ¼

N

ðN�1Þm

� �2

varT0

N

XN

n ¼ 1

yðtnÞwðtnÞe�j2pftn

����������2

8<:

9=; ð3:6Þ

We define

jXWSðf Þj2 ¼

T0

N

XN

n ¼ 1

yðtnÞwðtnÞe�j2pftn

����������2

¼ R2WSðf Þþ I2

WSðf Þ ð3:7Þ

where RWSðf Þ and IWSðf Þ represent the real and imaginaryparts of XWSðf Þ such that.

RWSðf Þ ¼T0

N

XN

n ¼ 1

yðtnÞwðtnÞcosð2pftnÞ ð3:8Þ

IWSðf Þ ¼T0

N

XN

n ¼ 1

yðtnÞwðtnÞsinð2pftnÞ ð3:9Þ

It can be noticed that each of RWSðf Þ and IWSðf Þ consist ofthe sum of N statistically independent random variables forevery f. According to the Central Limit theorem they can beassumed to have a normal distribution for large N [34](NZ20 is often perceived as sufficient in practice [4] whichis reasonable for the conducted analysis). It can beimmediately seen that each of (3.8) and (3.9) are of zeromean i.e. E½RWSðf Þ� ¼ E½IWSðf Þ� ¼ 0: The sum of the squares oftwo zero mean normally distributed random processesresults in un-normalized chi-squared distribution with twodegrees of freedom given that the two variables areindependent [34]. Since RWSðf Þ and IWSðf Þ are of normaldistribution, being uncorrelated at all frequencies issufficient to fulfill the independency condition. Theircorrelation is shown in Appendix B to be

rðf Þ ¼fE½x2ðtÞ�þs2

0gEWCSðf Þ

aþ

N�1

NE½RW ðf ÞIW ðf Þ� ð3:10Þ

where

EWCSðf Þ ¼

Z t0þT0

t0

w2ðtÞcosð2pftÞsinð2pftÞdt ð3:11Þ

and

E½RW ðf ÞIW ðf Þ� ¼

Z t0þT0

t0

Z t0 þT0

t0

RXðt1�t2Þwðt1Þwðt2Þ

�cosð2pft1Þsinð2pft2Þdt1 dt2 ð3:12Þ

whilst RW ðf Þ ¼R t0þT0

t0xðtÞwðtÞcosð2pftÞdt; IW ðf Þ ¼

R t0 þT0

t0xðtÞ

�wðtÞsinð2pftÞdt and RXðtÞ is the signal’s autocorrelation

function. It can be seen from (3.10) that RWSðf Þ and IWSðf Þ

are not independent for all f. Yet the two latter randomvariables can be replaced with independent ones withoutaltering Xeðf Þ since jXWSðf Þe

jyðf Þj2 ¼ jXWSðf Þj2: We can write

jXWSðf Þj2 ¼ ~R

2

WSðf Þþ~I

2

WSðf Þ ð3:13Þ

where

~RWSðf Þ ¼T0

N

XN

n ¼ 1

yðtnÞwðtnÞcosð2pftn�yðf ÞÞ ð3:14Þ

~IWSðf Þ ¼T0

N

XN

n ¼ 1

yðtnÞwðtnÞsinð2pftn�yðf ÞÞ ð3:15Þ

The introduced delay is chosen such that E½ ~RWSðf Þ~IWSðf Þ� ¼ 0and is shown in Appendix B to be

yðf Þ ¼ 0:5 arccotE½R2

WSðf Þ��E½I2WSðf Þ�

2rðf Þ

� �ð3:16Þ

where

E½R2WSðf Þ� ¼

fE½x2ðtÞ�þs20gEWCðf Þ

aþ

N�1

NE½R2

W ðf Þ� ð3:17Þ

EWCðf Þ ¼

Z t0þT0

t0

½wðtÞcosð2pftÞ�2 dt ð3:18Þ

E½R2W ðf Þ� ¼

Z t0þT0

t0

Z t0 þT0

t0

RXðt1�t2Þwðt1Þwðt2Þ

�cosð2pft1Þcosð2pft2Þdt1 dt2 ð3:19Þ

E½I2WSðf Þ� ¼

fE½x2ðtÞ�þs20gEWSðf Þ

aþ

N�1

NE½I2

W ðf Þ� ð3:20Þ

EWSðf Þ ¼

Z t0 þT0

t0

½wðtÞsinð2pftÞ�2 dt ð3:21Þ

E½I2W ðf Þ� ¼

Z t0þT0

t0

Z t0 þT0

t0

RXðt1�t2Þwðt1Þwðt2Þsinð2pft1Þ

�sinð2pft2Þdt1 dt2 ð3:22Þ

The variance for each of ~RWSðf Þ and ~IWSðf Þ is needed in orderto obtain s2

e ðf Þ: It can be shown that E½ ~RWSðf Þ� ¼ E ½~IWSðf Þ� ¼ 0:

Hence, s2~RWSðf Þ ¼ E½ ~R

2

WSðf Þ� and s2~I WSðf Þ ¼ E½~I

2

WSðf Þ�: Similarly to

the case of RWSðf Þ and IWSðf Þ

s2~RWSðf Þ ¼

fE½x2ðtÞ�þs20g~EWCðf Þ

aþ

N�1

NE½ ~R

2

W ðf Þ� ð3:23Þ

s2~I WSðf Þ ¼

fE½x2ðtÞ�þs20g~EWSðf Þ

a þN�1

NE½~I

2

W ðf Þ� ð3:24Þ

The identities ~EWCðf Þ, E½ ~R2

W ðf Þ�,~EWSðf Þ and E½~I

2

W ðf Þ� aredefined by equations identical to (3.18), (3.19), (3.21) and(3.22) respectively where t is replaced with t�yðf Þ: As aresult, the variance given by (3.6) for the adopted estimatorcan be obtained using the characteristics of un-normalizedchi-squared random variable. Thus,

s2e ðf Þ ¼ 2

N

ðN�1Þm

� �2

½s4~RWSðf Þþs4

~I WSðf Þ� ð3:25Þ

Computing the variance can be seen as a complicatedprocess that demands knowledge of the signal’s PSD.

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However, the variance calculations are solely manipulatedin ensuring the detection reliability. Spectrum sensingapproach proposed here only requires calculating Xeðf Þ.

3.4. Numerical example of spectral analysis

In this example we examine a multiband systemcomprising of 160 channels ðL¼ 160Þ: Each of them has5 MHz bandwidth i.e. BC=5 MHz. The system channelsreside in f 2 ½0:76;1:56�GHz which is the analyzed rangeof frequencies. Four adjacent channels are assumed to beactive where a wide sense stationary (WSS) signal with aPSD defined by 1 for f 2 7 ½780;800�MHz and zero else-where is generated. A Blackman window of lengthT0 ¼ 0:2ms and a=700 MHz are used. AWGN is introducedwhere the SNR is 2 dB. Fig. 1 shows E½Xeðf Þ� defined by (3.4)and the statistical mean calculated from 10,000independent experiments. Fig. 2 depicts s2

e ðf Þ given by(3.23)–(3.25) and the mean squared error (MSE) attainedfrom the aforementioned experiments. As seen in Fig. 1,Cðf Þ poses itself as an adequate spectral representation forthe detection pursuit. If uniform sampling was deployed ata similar rate, a replica of the underlying continuous timesignal’s tapered PSD would have appeared at 1.49 GHzwithin the analyzed frequency range indistinguishablefrom the one present at 790 MHz. Identifying the activebands would have been only possible if the signal spectral

Fig. 1. E½Xeðf Þ� from equation (solid line) and from simulations (b

Fig. 2. Variance of Xeðf Þ from equation

support was known. On the other hand, TRS suppressedaliasing in its classical sense and allowed detecting thepresence of the signal despite the used sampling rate. Thisdemonstrates the alias-free nature of the adoptedestimator. However, further analysis is needed to ensurethe consistency of Xeðf Þ for a single signal realization orexperiment. Both Figs. 1 and 2 confirm that there is a goodmatch between theoretical analysis and simulations i.e.assumptions made in theoretical analysis did not affect theaccuracy of the results.

4. Reliable sensing

In order for the spectrum sensing to be reliable, the peakamplitude(s) of the estimator’s outcome associated withthe active channel(s) must be significantly higher than thelevel of the estimated spectrum in the channels that are notactive. According to Chebychev’s inequality, the estimator’serror/inconsistency can be linked to the standard deviation.Based on the desired accuracy, the peak amplitude of Cðf Þ

that represents the active channel should be Z timesgreater than seðf Þ.

Inspecting seðf Þ given by (3.23)–(3.25), it can be easilyshown that it is nearly constant at frequencies where thereis no spectral activity. It exhibits a white-noise-like effectand is denoted by se;cont which is inversely proportional toa. On the other hand seðf Þ has its highest value where the

roken line). (a) An overview and (b) zoomed around 790 MHz.

(solid line) and MSE (broken line).

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signal is present and the substantial part of this inaccuracyis independent of the used sampling rate. A classicalmethod to reduce the latter error is to resort to averaginga number of Xeðf Þ estimates from K signal windows oflength T0: For simplicity non-overlapping signal segmentsscenario is considered in this study. The variance of theestimator is reduced by a factor of 1/K if the data in thenon-overlapping signal windows is uncorrelated. Other-wise the reduction is inversely proportional to the level ofcorrelation between these signal segments [27–29]. There-fore, a and K are the utilities available to minimize thepresent error level and ensure the dependability ofdetection. The non-overlapping data windows are assumedto be uncorrelated in accordance with the typically adoptedapproach in literature e.g. [27–29]. Consequently, thesensing procedure proposed here relies on averaging K

number of Xeðf Þ estimates over non-overlapping signalwindows of length T0 using

X̂ eðf Þ ¼1

K

XK

i ¼ 1

Xieðf Þ ð4:1Þ

4.1. Reliability recommendations

Let HU be the average peak amplitude of Cðf Þ of theactive channel(s) at frequency f0 while HL is the amplitudeof the constant factor in Cðf Þ i.e. N

�E½x2ðtÞ�þs2

0

�=ðN�1Þa:

Fig. 3 represents a model of the sought spectrum for abetter understanding of the reliability criterion where Cðf Þ

and the expected deviations of the estimate for a givensignal realization are shown. In general for the activechannel(s) to be detectable H¼HU�HL should be visiblyhigher than the error level i.e. HU�Zseðf0ÞbHLþZse;const;

hence

Hbðse;contþseðf0ÞÞ=ffiffiffiffiKp

ð4:2Þ

It is noticed that the window averaging effect is included in(4.2). This leads to

H24Z2�se;contþseðf0Þ

�2=K ð4:3Þ

where ZZ1 is a parameter that represents the desiredaccuracy according to Chebychev’s inequality. Noting thatE½ ~R

2

WSðf Þ�þE½~I2

WSðf Þ� ¼ mCðf Þ we can write

0:5m2C2ðf ÞrfE½ ~R2

WSðf Þ�g2

þfE½~I2

WSðf Þ�g2

rm2C2ðf Þ ð4:4Þ

Fig. 3. The targeted frequency representation of the processed signal.

Since between E½ ~R2

WSðf Þ� and E½~I2

WSðf Þ� the main differenceis a 0:5p phase shift, the lower bound of (4.4) isconsidered. It is assumed that ~EWC ðf0Þ ¼ 0:5T0þ0:5

R t0þT0

t0

cosð4pf0t�2yðf0ÞÞdt� 0:5T0 for a rectangular window i.e.m¼ T0 and similarly ~EWSðf0Þ � 0:5T0: This results in

s2e ðf0Þ ¼ 2

N

N�1

� �2 0:5ðPSþPNÞ2

a2þðN�1ÞðPSþPNÞ

Na HþðN�1Þ2

2N2H2

( )

ð4:5Þ

s2e;cont ¼

N

N�1

� �2

ðPSþPNÞ2

ð4:6Þ

seðf0Þse;cont ¼N

N�1

� �2ðN�1ÞðPSþPNÞ

Na HþðPSþPNÞ

2

a2

!

ð4:7Þ

where PS ¼ E½x2ðtÞ� and PN ¼ s20 denote the signal and the

noise powers respectively. Hence

H242Z2

K

N

N�1

� �2 2ðPSþPNÞ2

a2þ

2ðN�1ÞðPSþPNÞ

Na HþðN�1Þ2

2N2H2

!

ð4:8Þ

Provided that T0 is long enough such that ð1=mÞFXðf Þ �

jWðf Þj2 resembles to the signal’s PSD, the processed signalpower can be described byPSr2BAH: Due to theconservative nature of the recommendations, (4.8) leads to

K42Z2 N

N�1

� �2 8B2Að1þ2SNR�1þSNR�2Þ

a2þ

4BAðN�1Þð1þSNR�1Þ

Na

� �þZ2

ð4:9Þ

where SNR¼ PS=PN is the signal to noise ratio. It is notedthat K is an integer.

Formula (4.9) gives a conservative lower bound on thenumber of needed window averages which is a function ofthe channel occupancy, average sampling rate and signal tonoise ratio. This recommendation can also be used todecide on the needed average sampling rate given adecided number of estimate averages possibly imposedby practical constraints (e.g. latency) in a continuousprocessing environment. It is a clear indication of thetrade-off between the sampling rate and the number ofaverages needed in relation to achieving dependablesensing. According to (4.9), we can use arbitrary lowsampling rates for the sensing operation at the expense ofusing considerably long signal analysis window i.e. K can bearbitrarily large. This observation confirms/demonstratesresults presented in the first paper on DASP i.e. [20,21] butfor a finite N and signal observation window KT0: It is notedthat the deployment of tapering functions other thanrectangular has no effect on (4.9) and correlated as wellas overlapping signal windows can be easily incorporatedinto the analysis conducted above by using existing resultsin literature on the variance reductions e.g. [28].

4.2. Numerical examples of the derived limit

Assessing whether identity (4.3) is satisfied for a given Zis an effective way to value the legitimacy or the level ofconservativeness of the lower bound given by (4.9). Themultiband system described in the numerical example in

ARTICLE IN PRESS

Fig. 4. Reliability condition, (a) noise-free case and Kmin ¼ 20 and (b) SNR¼�1 dB and Kmin ¼ 25.

B.I. Ahmad, A. Tarczynski / Signal Processing 90 (2010) 2232–22422238

Section 3 is reexamined in this subsection. Again a WSSsignal with a PSD defined by 1 for f 2 7 ½780;800�MHZ andzero elsewhere is generated i.e. four adjacent activechannels where BA=20 MHz and f0=790 MHz. An averagesampling rate of a=800 MHz, T0 ¼ 0:2 us and Z¼ 4 (� 94%success rate) are used. According to (4.9), the minimumnumber of averaged estimates that leads to reliabledetection is KminZ20 for the noise free case andKminZ25 for SNR¼�1 dB: Fig. 4 compares the squareroot of both sides of (4.3) using simulations where in eachof the two shown plots 10,000 independent experimentsare conducted to approximate the probabilisticexpectations.

It is noticed from Fig. 4 that for KZKmin, condition (4.3)is satisfied. As K increases the region of additionalreliability widens. However, at K ¼ Kmin the error level insome cases e.g. noisy case in Fig. 4 can be marginally higherthan the signal spectral peak. This is due to the fact that thedata within the average estimates is not totally uncorre-lated as assumed. To avoid such situation, the user isadvised to use values that slightly exceed Kmin: Thisexample confirms the credibility of the derived limit givenby (4.9) and its reasonable level of conservativeness. In thefollowing subsection we propose an effective decisionmaking criterion for the activity of the systems spectralbands.

4.3. Detection threshold

In order to be able to detect the active channels viaassessing their spectral magnitude, a threshold level shouldbe set and any channel that have a spectral point(s) higherthan that threshold is regarded as being active. Thedetection procedure consists of three steps: (1) computeM number of frequency points across the monitoredchannels, (2) find spectral peaks and (3) compare thosepeaks with a threshold g. This postulates a binaryhypothesis testing problem given by

H0 : X̂ eðfkÞog;H1 : X̂ eðfkÞZg;

k¼ 1;2; . . . ;M ð4:10Þ

where H0 represents the absence of an activity in the channeli.e. the spectrum is made of the present noise and the

estimators’ inaccuracies while H1 depicts the presence of theactive channel(s). The frequencies fk ’s are the assessedspectral points where using the minimum number of thosefrequency points is preferred-one per channel i.e. M¼ L.

Setting the threshold g correctly is essential to thereliability of the detection procedure. A practical approachto determining g is to anticipate/decide the level of thepresent error in the spectral analysis of the processedsignal realization based on the accuracy of the adoptedestimator and the sought success rate i.e. Z. We recall thatthe estimator’s standard deviation is given by se;cont atfrequencies where there is no channel activity which isidentical to Cðf Þ amplitude at such frequencies. Hence, thedetection threshold can be expressed by

g¼ NðPSþPNÞ

ðN�1Þa 1þZffiffiffiffiKp

� �ð4:11Þ

In order to utilize (4.11), the signal and noise powers i.e.PSN ¼ PSþPN need to be calculated from the set of capturednoisy signal samples. The combined signal and noisepowers within each of the analyzed signal windows canbe calculated via

P_

SNðkÞ ¼

1

T0

XN

n ¼ 1

y2ð~tnÞDn; k¼ 1;2; . . . ;K ð4:12Þ

where ~tn is the set of N collected samples per windowarranged in an ascending order whilst Dn is the distantbetween two successive ~tn samples i.e. Dn ¼ ~tn�~tn�1:

Approximation (4.12) belongs to the subject of integrationapproximation over a finite interval. Now PSN can becalculated from the number of analysis signal timesegments via: P̂ðkÞ ¼ ð1=KÞ

PKk ¼ 1 PðkÞ: The averaging effect

is expected to eliminate/suppress the additional compo-nent in (4.12) i.e.

PNn ¼ 1 xð~tnÞnð~tnÞDn as E½xðtÞnðtÞ� ¼ 0: The

effectiveness of this thresholding approach is demon-strated in the next section. We note that the detectionthreshold given by (4.11) is chosen in such a way that thechances/probabilities of a false detection of a non-activechannel and that of a missed detection of an active band arebounded by the same limit i.e. 1/Z2. However, in someapplications the users may want to reduce one of theseprobabilities at the expense of increasing the other one.Such requirements can be accommodated in the proposed

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B.I. Ahmad, A. Tarczynski / Signal Processing 90 (2010) 2232–2242 2239

approach by using Chebychev’s inequality. For example ifthe probability of false detection must not exceed Pf, then Zin (4.11) should be replaced with Pf

�0.5.As commonly known, high spectrum dynamic range

eases the requirement on the spectral analysis device andforms a safety margin for any inaccuracies that might beincurred. The dynamic range of the pursued spectrum i.e.Cðf Þ can be set by choosing appropriate a. Due to practicalsensing issues discussed in Section 3, T0 is assumed to befixed and hence the spectrum dynamic range depends onthe number of samples per window i.e. N. Referring to Fig. 3and for a desired dynamic range, we can write:HU ZmHL

where m41: Adopting a conservative approach thenumber of samples needed to safeguard HZðm�1ÞHL

should comply with

NZ2BAðk�1ÞT0f1þSNR�1gþ1 ð4:13Þ

The guideline given by (4.13) although not necessary forreliable detection gives the user an indication of the neededaverage sampling rate if the spectrum dynamic range is anissue.

4.4. Applicability of the adopted approach to other sampling

schemes

In this subsection we discuss the applicability of theproposed sensing approach to randomized sampling schemesother than TRS. The detailed analysis of such generalizationsalong with considering cyclostationary signals lieu of WSS/WSE will be reported in a separate paper.

In principle, any sampling scheme that producesvarfXeðf ÞjxðtÞg lower than or equal to that of TRS and itsE½Xeðf Þ� is a form of Cðf Þ given by (3.4) is deemed to beadequate for the proposed approach. This criterion fits acertain set up of stratified and stratified antitheticalsampling schemes reported in [17,18] where the samplingpoint process is defined by a uniformly distributed PDFwithin an equally partitioned signal window i.e. a form ofjittered random sampling (JRS). However, applying jittered-stratified sampling (JSS) and jittered-antithetical sampling(JAS) to Xeðf Þ in its current form should be approached with

Fig. 5. K versus a. Noise-free case (solid

caution as the spectrum dynamic range of the targetedspectrum can affect the sensing reliability for suchschemes. This is referred to the high smeared-aliasingsuppression ability of JRS, with jitter uniformly distributedwithin 1/a period, within the active spectrum band(s)[12,13]. The spectrum dynamic range is locally improvedwithin the frequency band of the active channel(s) but itdeteriorates across the overall monitored bandwidth.Hence, the expected value of the estimator no longerguarantees detectable feature for any a, provided that (4.9)is satisfied, as the spectrum dynamic range deteriorates asa or/and SNR drops. In this case (4.13) can be deployed toavoid such adversity of using JSS and JAS.

5. Numerical examples

Consider a multiband system comprising 50 channelsðL¼ 50Þ that are 2 MHz each (BC=2 MHz). The systemchannels are located in f 2 ½800;900�MHz: A rectangularwindow of width T0=2ms is used. A zero mean WSS signalis generated for each of the active bands with a PSD ofamplitude 1 in the passband and zero elsewhere. Channeloccupancy of 10% i.e. BA=10 MHz is assumed and all activechannels are of similar power levels. The sensing procedureis based on taking a single frequency point at the middle ofeach of the systems channels to assess its activity. Fig. 5shows the average sampling rate a versus the number ofaveraged estimates K given by (4.9) for a rate of success of96% i.e. Z¼ 5: The asterisks in this figure mark the tested avalues in Figs. 6 and 7. The latter two show the results ofthe sensing operation along with the chosen threshold levelfor each of TRS and JSS in noise-free environment and withSNR¼ 3 dB.

It is evident from Figs. 6 and 7 that the five active channelsare detectable using the adopted estimator and the recom-mendation given by (4.9) with an a as low as 30 MHz for TRSand JSS in the presence as well as absence of noise. In bothfigures, spectral sample points belonging to non-activechannels are below the set threshold level which illustratesthe effectiveness of the thresholding level defined by (4.11).For the JSS it is noticed that the set threshold is more

-line) and SNR of 3 dB (dashed-line).

ARTICLE IN PRESS

Fig. 6. X̂ eðf Þ in absence of noise. Threshold level (dashed line) and active channels (asterisks).

Xe(f)

Xe(f)

Xe(f)

Xe(f)

800 810 820 830 840 850 860 870 880 890 9000

0.4

0.8

1.2

1.6

2

2.4

2.8

0

0.4

0.8

1.2

1.6

2

2.4

0

0.4

0.8

1.2

1.6

0

0.4

0.8

1.2

1.6

f(MHz)800 810 820 830 840 850 860 870 880 890 900

f(MHz)

800 810 820 830 840 850 860 870 880 890 900f(MHz)

800 810 820 830 840 850 860 870 880 890 900f(MHz)

TRS : α = 30 MHz, K = 231 JSS : α = 30 MHz, K = 231

TRS : α =60 MHz, K = 102 JSS : α =60 MHz, K = 102

Fig. 7. X̂ eðf Þ with SNR=3 dB. Threshold level (dashed line) and active channels (asterisks).

B.I. Ahmad, A. Tarczynski / Signal Processing 90 (2010) 2232–22422240

conservative in comparison to the TRS counterpart due to thesmeared-aliasing suppression capability of JSS within and nearthe active band(s). It is clearly seen in Figs. 6 and 7 that as a or/

and SNR drops the spectrum dynamic range deterioratesdespite imposing extra requirement on K. In terms of spectrumsensing discrepancies between TRS and JSS, those numerical

ARTICLE IN PRESS

Fig. 8. X̂ eðf Þ with SNR=3 dB with all system channels being inactive.

B.I. Ahmad, A. Tarczynski / Signal Processing 90 (2010) 2232–2242 2241

experiments demonstrate that such differences are marginalfor noise-free and noisy signals provided a reasonablespectrum dynamic range. Further experimental results (notshown here) showed that the distribution of simultaneouslyactive channels across the scanned bandwidth does not hinderthe performance of the detection procedure.

An interesting case is to assess the detection precision incase none of the channels are active. Fig. 8 shows thedetection results for the two test cases shown in Fig. 5where SNR=3 dB in case all system channels are ideal/silentwhilst the system designer assumed 10% spectrumutilization. It is clear in Fig. 8 that the adopted spectrumsensing approach along with the thresholding procedure isimmune against such situations.

If uniform sampling is deployed the minimum bandpasssampling rate that would avoid aliasing within themonitored bandwidth is 225 MHz (bandpass sampling).Hence, more than 85% saving on the used sampling ratecould be achieved with the use of the proposed approach inthis paper. However, comparing sampling rates of theuniform and nonuniform cases can be misleading as thelatter case would typically demand more estimate averagesi.e. larger K. In the uniform sampling case, the number ofneeded averages is KUSZZ2 for noise-free environments(following dependability analogy described in Section 4).As a result, around 25% saving on the needed number ofsignal samples to perform sensing is obtained by deployingthe proposed detection method in those numerical experi-ments. Therefore, the proposed approach offers substantialsavings over uniform sampling in terms of the samplingrate and the number of processed samples for lowspectrum occupancy. Each scenario should be evaluatedindividually to assess benefits of nonuniform over uniformsampling in terms of the number of needed signal samples.

6. Conclusion

A method that utilizes DASP methodology to detect theactive channel(s) in a multiband environment is proposed.The adopted approach can use arbitrary low sampling rateswhich are independent of the width of the monitoredbandwidth or its central frequency. However in order topreserve the reconstructability of the detected signals, it isrecommended that the sampling rates exceed twice thetotal bandwidth of the concurrently active channels BA:

This feature compares favorably with uniform samplingbased spectrum sensing methods where the requiredsampling rates grow proportionally to the number of the

monitored channels. In the latter case the suitable rates areconstraint by B and its central frequency; the minimumpossible rate is 2B regardless of the spectrum occupancy.With the low spectrum utilization assumption, it isunambiguously clear that the use of the proposed methodwould bring substantial savings in terms of the samplingrate and number of processed samples compared to theclassical DSP. In fact for the proposed method extendingthe monitored bandwidth, assuming constant SNR e.g. thesampling is preceded by a filter to limit the noisebandwidth, would not impose any additional requirementson the needed sampling rates or estimate averagesprovided that the bandwidth of the concurrently activechannels does not change.

Maintaining the average sampling rates above Landauprompts researching into effective reconstruction algo-rithms of noisy nonuniformly sampled data. This can beutilized to recover the conveyed signals/messages or tomodel the detected channel(s) prior to extracting themsequentially to reveal weaker present component(s) in casethe power levels of the concurrently active bands varysignificantly. This can be a more computationally efficientextraction method in comparison to existing ones such asSECOEX [35]. Besides, various applications of spectrumsensing in multiband communication systems e.g. cogni-tive radio stimulate investigating the adequacy of theproposed approach to communication signals given theircyclostationary/cycloergodic nature.

Appendix A. Proof of (3.2)

We start with

E½Xeðf Þ� ¼T2

0

NðN�1Þm Eh��� XN

n ¼ 1

yðtnÞwðtnÞe�j2pftn

���2i

The estimator can be expressed by

Xeðf Þ ¼T2

0

NðN�1ÞmXN

n ¼ 1

y2ðtnÞw2ðtnÞ

(

þXN

n ¼ 1

XN

m ¼ 1man

yðtnÞwðtnÞyðtmÞwðtmÞe�j2pf ðtn�tmÞ

9=;

Given that the components in the double summation areIID random variables

E½Xeðf ÞjxðtÞ� ¼T2

0

NðN�1Þm fNE½y2ðtnÞw2ðtnÞ�

þNðN�1ÞE½yðtnÞwðtnÞe�j2pftn �E½yðtnÞwðtnÞe

j2pftn �g

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B.I. Ahmad, A. Tarczynski / Signal Processing 90 (2010) 2232–22422242

Since the signal and the zero mean AWGN are independent

E½Xeðf ÞjxðtÞ� ¼T2

0

NðN�1Þm NE½x2ðtnÞw2ðtnÞ�þs2

0m=aþNðN�1Þ

T20

jXW ðf Þj2

( )

ðA:1Þ

By calculating the expected values in (A.1) in terms ofsample points according to (3.1), the expression given by(3.2) is obtained.

Appendix B. Proof of (3.16)

Since RWSðf Þ and IWSðf Þ are zero mean, their correlation isgiven by pðf Þ ¼ E½RWSðf ÞIWSðf Þ�: First we split the doublesummation into two

RWSðf ÞIWSðf Þ ¼T2

0

N2

XN

n ¼ 1

y2ðtnÞw2ðtnÞcosð2pftnÞsinð2pftnÞ

þT2

0

N2

XN

n ¼ 1

XN

m ¼ 1man

yðtnÞwðtnÞyðtmÞwðtmÞcosð2pftnÞsinð2pftmÞ

Similar to the E½Xeðf ÞjxðtÞ� shown in Appendix A, weobtained

E½RWSðf ÞIWSðf ÞjxðtÞ�

¼

R t0þT0

t0fx2ðtÞþs2

0gw2ðtÞcosð2pftÞsinð2pftÞdt

a

þN�1

NRW ðf ÞIW ðf Þ ðB:1Þ

and subsequently (3.10). Following the delay introductionwe can write

~RWSðf Þ ¼ cosðyðf ÞÞRWSðf Þþsinðyðf ÞÞIWSðf Þ

~IWSðf Þ ¼�sinðyðf ÞÞRWSðf Þþcosðyðf ÞÞIWSðf Þ

The value of yðf Þ would be chosen such thatE½ ~RWSðf Þ~IWSðf Þ� ¼ 0, consequently

0:5 sinð2yðf ÞÞfE½I2WSðf Þ��E½R2

WSðf Þ�gþcosð2yðf ÞÞE½RWSðf ÞIWSðf Þ� ¼ 0

ðB:2Þ

By manipulating (B.2), we attain (3.16).

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