1Exploiting Interference Diversity for Event-BasedSpectrum Sensing

Arash Parsa, Amin Aminzadeh Gohari, and Anant Sahai

AbstractSpectrum sensing is a core problem incognitive radio. Detecting the presence/absence ofvery weak primary users with a single antenna canbe very difficult. Earlier it have been shown thatuncertainties in the environment result in SNRwalls that detectors cannot beat in a robust man-ner. Multiple antenna approaches show the po-tential of getting gains, but we show here thatfor a single user, multiple antenna detection stillmust suffer from an SNR Wall. The reason isthat the real world uncertainty in noise is dom-inated by the potential presence of an unknownnumber of low-powered interference sources in theexternal environment. The traditional approachto collaborative spectrum sensing attempts to usethe shadowing/multipath diversity across differentusers to boost the reliability of detection. We showhere that there is another kind of diversity that isalso available: interference diversity. This diversitycaptures the fact that these low-powered interfer-ence sources are local to individual users whereasthe primary user has a global footprint. To exploitthis diversity, we must shift our perspective fromexistence-based detection (whether the primary ispresent or not) to event-based detection (whetherthe primary has turned off or on). We study thisand explore the limits to this approach.

I. Introduction

SPECTRUM sensing at very low SNR regimes hasgained crucial importance in the context of cog-nitive radio. For example the required sensitivity inIEEE 802.22 standard is -116 dBm (-22 dB of signalto noise ratio) [1]. In addition to high sensitivity, thesecondary users need to be able to operate in differentsettings and environments. Uncertainty in the noisestatistics imposes fundamental limits on the sensitiv-ity of secondary users. Below a certain level of SNR,detectors will not be able to sense the existence of thesignal no matter how long they collect the data [2].This level is called the SNR wall of the correspondingdetector and partially quantifies the robustness of the

The authors are with the Department of Electrical Engi-neering and Computer Science at UC Berkeley. emails: {parsa,aminzade, sahai}@eecs.berkeley.edu

Fig. 1. Unintentional interferers (e.g. laptops in the fig-ure) are sensed locally while the primaries are sensed glob-ally.

system.For this sensing problem different algorithms such

as energy detection, matched filtering, cyclostation-ary feature detection [3] [4] [5] have been proposed.In this paper, we restrict ourselves to the scenarios inwhich one does not have knowledge about the featuresof the primary signal.One can categorize these algorithms into single,

non-cooperative multiple antenna and cooperativemultiple antenna systems. From the first two cate-gories we will briefly review radiometer and max-mineigenvalue detector [6] and discuss their fundamen-tal limitations. The last category of algorithms, i.e.cooperative sensing algorithms, aim to increase theperformance of detection of the overall system by us-ing a network of sensors and making a decision basedon the information gathered from different sensors.If sensors are experiencing independent shadowing ormultipath fading effects, cooperative approach provesto be effective [8][7].In this paper, we argue that one of the limiting

factors in many of the proposed sensing algorithmsis the choice of hypothesis. Many of the currentlyused algorithms choose to test the existence/absenseof the primary signal. We refer to these detectorsas existence based detectors. An equivalent set ofhypothesis is to test enterance/exit/idle of the pri-mary signal at any time instant. We refer to thesedetectors as event based detectors. When a setof events could be described using different sets ofhypotheses, the choice that one makes greatly in-

2TABLE IList of a few noncoherent detection algorithms

Detector CharacteristicsRadiometer single-antenna;

non-cooperativeMax-Min Eigenvalue [6] multi-antenna;

non-cooperativeBy Mishra et al. [7]By Ghasemi et al. [8] multi-antennaBy Matsui et al [9] cooperative

The proposed detector

fluences the structure and performance of the cor-responding hypothesis-testing algorithms. Whereasan existence/absense detector such as radiometerbases its decision on the expectation of different lev-els of energy under different hypothesis, an enter-ance/exit/idle detectors bases its decision on the ex-pectation of local surges/local drops/small changes inthe energy level of the received signal. A radiometerfires a false alarm when the average energy of the noiseis high whereas a single-antenna enterance/exit/idledetector fires a false alarm when there are local surgesor drops in the energy level of the noise. The latterevent could happen when for example a significantnoise interferer, such as a laptops leaking energy out-side its permitted bandwidth, turns on or off.Noise interferers are either unintended emitters

(e.g. a laptop leaking energy outside its permittedbandwidth during a communication, or electronic cir-cuits of a printer radiating electromagnetic waves dur-ing its activity) and intended emitters (other sec-ondary networks using the bandwidth when the pri-mary is absent). A sensor may have some knowledgeabout the possible turn on/turn offs of the intendedemitters, but there is little it can do about the un-intended emitters.In order to deal with the false-alarm events caused

by unintended interferers, the following cooperativemulti-sensor algorithm is proposed: when a primarysignal enters or exits the system, it does so in a globalscale causing many sensors to fire an alarm; howeverwhen an unintended emitters enters or exits the sys-tem, it does so in a geographically local scale trigger-ing only a few neighboring sensors. The system canthus use this Interference Diversity to distinguishbetween the two events by gathering information fromsensors placed at different physical locations(see fig-ure 1).

Our work can be viewed in the context of changedetection algorithms. Veeravalli [10] and Mei [11]study the application of quickest detection theory todecentralized decision systems. The quickest detec-tion [12] [13] refers to real-time detection of abruptchanges in the distribution of observed signals asquick as possible. The analysis and results based onthis method mostly depends on the assumption thatdistributions are known and fixed after and before thetransition points. Furthermore, Li et. al. [14] applythe theory of quickest detection to problem of sensingprimary signals in cognitive radio. Authors assumenoise distribution is constant and known while thereis some uncertainty about the amplitude and phase ofthe primary signal. Since the current proposed meth-ods using quickest detection are based on knowledgeof distributions they fail to address and capture theproblem of noise uncertainty and robustness. Specifi-cally many of the changes in the distribution of mea-sured signal could be a result of entrance/exit of in-terferers and do not necessarily comply to any prob-abilistic model.This paper is organized as follows. In Section II,

sources of noise uncertainty are mentioned. As a pre-lude to its following sections, Section III reviews acouple of existence based sensing algorithms. Therole of certain factors such as external interferers inimposing fundamental limits on the performance ofthis class of algorithms is highlighted. In Section IV,the new cooperative sensing algorithm is introducedand analyzed at an intuitive level; the rigorous formu-lation and analysis is moved to appendix B. SectionV includes the simulation results. Lastly, appendix Aincludes the proof for a result on the SNR Wall forthe max-min eigenvalue detector.

II. Noise Uncertainty

The sources of noise uncertainty can be decom-posed into two categories: internal noise and externalnoise. An internal noise is a noise that is not inher-ently present in the environment but is generated bythe detector itself. An external noise on the otherhand is generated by interferers in the environmentand is picked up by the detector.While the uncertainty in the variance of the aggre-

gate internal noise can be significantly reduced by cal-ibrating the device in a controlled environment. Theexternal noise coming from unintended emitters cannot be calibrated because it depends on the environ-ment in which the device is operating. The existenceand the structure of the signals emitted by these in-

Parsa, Aminzadeh and Sahai: EXPLOITING INTERFERENCE DIVERSITY FOR EVENT-BASED SPECTRUM SENSING3

terferers varies with location and time, and is thus notknown apriori. The maximum received energy fromthe interferers is a lower bound on the noise uncer-tainty. Intuitively the overall uncertainty is equal tothe maximum number of significant interferers thatmight co-exist in an environment times the averageenergy of a typical significant interferer.

III. Existence Based Detectors

In this section, we briefly review a class of detectorsthat are based on the following two hypothesis:H0: only noise x[n] = w[n]H1: signal + noise x[n] = w[n] + r[n]

where x[n], w[n] and r[n] are the received signal, theinput noise and the primary signal at time n. We referto these detectors as existence based or H0 H1detectors.The algorithms based on the above two hypothe-

sis are usually designed for a fixed observation win-dow. One hopes that by increasing the length of ob-servation window, the algorithm will have a betterperformance. However the SNR wall analysis haveshown that there are fundamental limitations to per-formance of the algorithm in presence of noise uncer-tainty [2]. The fundamental limits of these detectors,such as SNRWalls, are usually analyzed under the as-sumption that the detection time window approachesinfinity.As a prelude to the Event Based Detectors dis-

cussed in the next section, we review a couple of fac-tors imposing limits on the performance of ExistenceBased detection algorithms. Of particular impor-tance are the external interferers: in a practical en-vironment there are interferers that may fool a de-tector who is looking for weak primary signals. Thegoal of the following subsections is partly to highlightthe role of interferers in the performance of detectionalgorithms at low SNRs. We make this effect explicitin two particular detection algorithms: energy detec-tor (radiometer), and maximum-minimum eigenvaluedetector. The former detector has only one antennawhile the latter has multiple antenna.

A. Single Antenna Energy Detection: Radiometer

An energy detector (radiometer) measures the av-erage received energy in a time window, and makes adecision by comparing it with a fixed threshold.First assume that the variance of noise is perfectly

known. Then no matter how weak the primary sig-nal is, as long as the observation window, N , is largeenough, the detector will be able to identify it with

Fig. 2. Noise variance has a range from min noise energyto max noise energy. When the signal is added, thisinterval shifts to higher values. If the signal is not strongenough and these two intervals collide, there would be noguarantee of success. Otherwise the detector is able todistinguish the two cases if it observes long enough.

Fig. 3. Single antenna energy detector fails to distinguishweak primary from interferer.

arbitrarily high probability. In the case of radiome-ter, we can model the noise uncertainty by assum-ing that noise variance is known up to an interval.The length of this uncertainty interval determinesthe SNR wall for the radiometer (see figure 2).This analysis indicates that in terms of robustnessone might prefer to have a larger average noise en-ergy and less uncertainty rather than small averagenoise energy with much more uncertainty.As mentioned in Section II, the SNR wall of the

radiometer is no less than the maximum number ofsignificant interferers that might co-exist in an envi-ronment times the average energy of a typical signif-icant interferer (see figure 3).

B. Max-Min Eigenvalue Detection

In this section, we specifically discuss a multi an-tenna sensing algorithm known as Max-Min Eigen-value Detector introduced by Liang and Zeng in [6]and highlight the effect of external interferers on theperformance of this detector.Liang and Zeng assume noise to be white and uni-

form (in terms of power) over different antennas. Theauthors argue that if primary signal does not exist,the correlation between samples at different anten-

4nas and/or different times will be zero. On the otherhand, if primary signal is present, some of these cor-relations will not be zero. Therefore the detector candistinguish between these two scenarios by measuringthe deviation of a covariance matrix from the identitymatrix in terms of the ratio of maximum and mini-mum eigenvalues.With the assumption of white and uniform (in

terms of power) noise over different antennas and/orat different times, the max-min eigenvalue detectorcan overcome noise uncertainty. However, in a real-istic scenario interferers are a major source of noiseuncertainty and can not be ignored. Being gener-ated by unknown sources such as electronic devicesleaking outside their permitted bandwidth, the unin-tended interference noise may not be white and uni-form (in terms of power) over different antennas. Inother words from the perspective of max-min eigen-value detector there is no difference between a veryweak primary and an interferer. In the appendix wemake this point more explicit by proving the follow-ing theorem on the SNR wall of this detector (seeAppendix for notations):Theorem A.1. For any positive , there exists a

threshold such that whenever the primary powerP is less than , the detector can not guarantee aprobability of miss-detection less than 1 . Further-more, given the probability of false alarm PrFA, isbounded from below by (1)

2

c1(+1)where c1 =O(1 ) and

= Thf(PrFA).As we saw above, many of the known detectors

are vulnerable to external interferers masquerading asweak primaries and vice versa. In Section IV, we pro-pose an event-based detection algorithm for dealingwith the negative effect of external interferers.

IV. Event Based Detection

As discussed in Section III, sensing algorithmsbased on existence/absence (H0H1) hypothesis arevulnerable to external interferers and uncertaintyabout the environment. The H0H1 hypothesis set-ting does no allow the sensing algorithm to exploit thefact that primary signals are sometimes present andsometimes absent. Intuitively, the transition betweenthe existence/absence states of the primary signal cre-ates rapid changes in the level of energy at the sensors.We refer to these changes as energy edges; if thereis a surge in the energy level, we call it a positiveedge, otherwise we call it a negative edge. Forevery time instance, we define three types of event

0 20 40 60 80 1002

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eive

d En

ergy

at a

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sor

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Fig. 4. Two sample noisy signal observation from thefirst noise model described in the High Level IntuitionSection. Radiometer is very sensitive to uncertainty aboutthe level of energy.

hypotheses:H[n] : when there is a negative edge at time n;H+[n] : when there is a positive edge at time n;HI [n] : otherwise.

The new set of hypothesis may be thought of as thefirst derivative of the previous hypotheses. Knowingthe location of these events is equivalent to know-ing the existence/absence of the primary signals. Inthis part of the paper we show that by changing ourhypotheses from sensing the level of energy to sens-ing the energy edges, we will be able to significantlyimprove our performance. Since this kind of sensingalgorithm aims at detecting the enterance/exit of theprimary signals and infer their existence from this in-formation, we call it event based sensing.

A. High Level Intuition

In this section, we would like to demonstrate thecore ideas behind the new detection algorithm.We start from a simplistic scenario in which the

noise power is unknown but constant over time. Alsoassume that when the primary enters/exits, the re-ceived power changes instantaneously. Furthermoreassume that in the time scale of interest, the primarychanges its state at most once. If the signal power isless than the noise uncertainty (that is the range ofthe noise power) the radiometer fails. On the otherhand, no matter how weak the primary signal is, anedge detection algorithm will be able to sense the pri-mary if the observation window is long enough; andtherefore no SNR Wall exists (see figure 4).Next, consider the less restricted scenario in which

the noise is comprised of different interferers each ofwhich have constant power over time but are allowedto enter/exit (or turn on/off)1. Furthermore, we as-1 For instance this noise model may be used in the following

Parsa, Aminzadeh and Sahai: EXPLOITING INTERFERENCE DIVERSITY FOR EVENT-BASED SPECTRUM SENSING5

sume that whenever an interferer enters or exists,the energy level changes instantaneously. Under thismodel, any single antenna edge detector will also failsince it can not distinguish between the entrance/exitof primary and that of interferers. To solve this prob-lem, first we make the key observation that in manyrealistic scenarios interferers are short-range whileprimary signals are long-range (local-global assump-tion). Based on this assumption we propose a coop-erative sensing scheme that can overcome the inter-ference problem. The local-global assumption meansthat enterance/exit of the primary signal is sensed bymost of the sensor nodes while interferers will onlyaffect one or at most a few of the sensors. Thereforethe system can detect the change in the status of pri-mary whenever many sensors simultaneously declareso. It is important to note that we are proposing thissensing algorithm for a very low SNR regime. We willprove that as long as we do not deviate significantlyfrom the local-global assumption, under certain con-ditions, the proposed algorithm is able to correctlydetect weak primary signals.

B. Formulation and Analysis

In this section we explain the detection algorithmand its analysis at an intuitive level. For a more rig-orous formulation and analysis of the system see Ap-pendix B.Consider a network of sensors geographically dis-

tributed in an area. Assume that the sensors areconnected to a Central Unit (CU) through commu-nication links that have limited capacity. Each sen-sor node transmits its belief about a possible changein the status of primaries (i.e. whether a primaryhas entered (or exited), or that nobody has changedits state). At each time instant, the CU will decidewhether the primary has changed its state by, roughlyspeaking, taking a majority vote of the received mes-sages from the sensors (see appendix B for a rigorousformulation).The objective of a sensor is to detect the changes

in the level of energy due to entrance/exit of the pri-mary. One possible scheme is to compare the mea-sured energy levels of the first half and second halfof a shifting observation window. The goal of eachsensor node is to provide the CU with as much infor-

scenario: assume the interferer is a 802.11 wireless router leak-ing very weak signals out of its assigned bandwidth because ofimperfect filters. Because of On/Off nature of the packet tran-sitions there will be an entrance/exit pattern of interference inthe band of interest.

Fig. 5. Adaptive adjustment of the threshold at the sensor, is the threshold, R is the allowed data rate

mation about the changes in the energy level of theenvironment as possible. In reality, because of lim-ited resources, every sensor is forced to send only themost significant changes. Inevitably the sensors haveto choose some threshold and send an alarm to thecentral unit when the energy changes by more thanthis threshold. The length of the observation win-dows is small enough to ensure that the changes inthe energy levels are not coming from changes in thechannel statistics. Figure 10 depicts a sample out-put of this edge detector to the entrance of primarysignal.Having a lower threshold means sending more

alarms on average and vice versa. Therefore lim-ited data rate between the sensors and central unitimposes a lower bound on . If the links are wire-less, meeting these lower bounds imply saturating allthe links by using all of the available bandwidth andpower.Figure 5 provides an adaptive approach for setting

at each sensor node. Figure 7 shows a simulationresult on the tradeoff between and probability offalse alarm under a specific noise model.In the following paragraphs we explain the results

in Appendix B at an intuitive level. We first analyzethe behavior of the algorithm under the assumptionthat the network consists of a large number of nodesand that the capacity of the links between each sensornodes and the CU is some constant R.The Appendix quantifies the local-global assump-

tion in mathematical terms. The local-global assump-tion imposes certain constraints on the noise signalsobserved at different nodes. It is important to notethat the noise signals may not have statistical descrip-tion and can vary arbitrarily in time. The constraintsimposed are therefore not on the statistical distribu-tions of the noise signals but on their aggregate aver-age behavior in time or in space.In a large network the chance of an interferer ex-

6iting the system at some node of the network is thesame as the chance of an interferer entering the net-work. The local-global assumption states that theinterferers are short-range. This means that any en-trance or exit of an interferer affects only a smallnumber of sensor nodes. As long as our assumptionshold, one can expect that when status of the primariesdo not change, the number of sensors experiencing asurge in their energy level should be roughly equalto the number of sensors experiencing a drop in theirenergy level. This intuition is formally expressed asthe second constraint on noise in definition B.5.1.The local-global assumption not only states that

the interferers are local, but also that the primariesare global. This means that a change in the status of aprimary is heard by many sensors. Roughly speaking,when the primary signal is not very weak, its entranceshould increase the number of the sensors who experi-ence a rapid increase in their energy level. Similarly, aprimary leaving the system should increase the num-ber of sensors who experience a rapid decrease in theirenergy level. This intuition is formally expressed asthe second constraint on the noise in definition B.5.2.Finally, the first constraint in definition B.5.1. im-

plies that when the primaries do not enter or exit,the chosen threshold is sufficiently high so that thenumber of alarms generated on average over time byeach sensor node is bounded.Below, we mention a few examples of the noise sig-

nals that satisfy the constraints of definition B.5.1 anddefinition B.5.2 for appropriate choice of parameters: When the noise (including interference noise)at different nodes are white, independently andidentically distributed (i.i.d.) according to an un-known but continuous density function (such asGaussian).

When the noise signal at each node has two com-ponents: 1. independent white noise at differ-ent sensor nodes, identically distributed accord-ing to an unknown continuous density function;2. Independent Poisson processes at differentnodes with an unknown and possibly varying, butbounded rate modeling the entrance/exit of theinterferers. The Poisson processes modeling theexit of the interferers appear with negative signand are expected to on average cancel out theeffect of Poisson processes modeling the entranceof the interferers.

If the power of the primary is very low, it will not beable to excite enough receivers and hence the detec-

Fig. 6. In this example, CU is placed at the top of thesensor nodes and at equidistance from all of them in theair

tion algorithm will fail. If the average energy of theprimary signal at the sensor nodes is above a certainlimit, the algorithm should however succeed. Theo-rem B.5.1. and its corollary formulates this intuition.Roughly speaking, the minimum energy level re-

quired for detection is determined by the followingfactor: the lower the energy of the primary signal,the more sensitive the sensors need to be in detectingthe changes in the energy level. But increasing thesensitivity of the sensor nodes, makes them vulnera-ble to entrance/exit of weaker interferers. This wouldnot only increase the number of false alarms generatedby each sensor node (and thus creating difficulty com-municating them to the CU) but more importantlygradually invalidates the local-global assumption. Astrong interferer may be heard by a larger group ofsensor nodes. It should be however noted that the de-tection algorithm is robust in terms of the validity ofthe local-global assumption: for a large network, thesecond constraints of definition B.5.1 remains valid aslong as each interferer affects only a sub-linear num-ber of sensors i.e. the fraction of affected sensors isrelatively small to the total number of sensors.

Computation in the air method:

Above, we assumed R1 = R2 = ... = Rms = R al-lowing the total rate, R1 +R2 + ...+Rms approachinfinity. This is not a realistic assumption. We how-ever note that the CU only needs to know the totalnumber of positive and the total number of negativealarms rather than the precise identity of those nodeswho send positive or negative alarms.2 Consider thefollowing strategy:

2 This is reminiscent of works which consider distributed com-putation combined with wireless communication (see for exam-ple [15]). It can be shown that separation between the commu-nication stage and the computation stage turns is not always theoptimal strategy, and a combined communication-computationscheme can improve the performance.

Parsa, Aminzadeh and Sahai: EXPLOITING INTERFERENCE DIVERSITY FOR EVENT-BASED SPECTRUM SENSING7

The sensor nodes divide their time resources intoconsecutive intervals. In odd intervals, those termi-nals who want to transmit a positive alarm simul-taneously transmit signals of power P while in theeven intervals those terminals who want to transmita negative alarm simultaneously transmit signals ofpower P . These communications are taking place ina control channel which is different from the chan-nel which needs to be sensed. The central unit, CU ,measures the energy of the received signal (which isa superposition of the transmitted signals) in each ofthe intervals. These average energies are used to ap-proximate the total number of sensors who are firingalarm; in particular the difference in the energy levelsof consecutive odd and even intervals provides an es-timate of the difference between the number of nodeswho transmit positive alarms, and those who transmitnegative alarms.Consider the simplistic model 3 shown in Figure 6

where CU is equidistance from all of sensors and isexperiencing the same channel to all of them. Thismakes sure that the sensor nodes are contributingequally to the total power received at the CU . If fraction of the ms nodes are transmitting signals atpower P , the expected value of the received energy atthe CU equals .P.Gav+Pn where Gav is the averagegain from the sensor nodes to the CU and Pn is theaverage energy of the noise. Therefore the CU canestimate based on the average received energy. Inthe general scenario when the CU is placed on theground, the channels between the CU and differentsensor nodes would be different. And CU would re-ceive a weighted average of the transmitted signalsfrom different sensors. Under the condition that thechannels from the sensors to the CU are different,if the sensor nodes have some knowledge about thestrength of their channel to the CU , they can adjusttheir power correspondingly. Otherwise a more de-tailed investigation is required.

V. Simulations

The simulation results of this section are based onthe following model: we consider a set of sensors uni-formly distributed inside an square. At every point

3 The result can be probably extended to the following model:assume that we divide the sensor nodes into the set of greennodes and red nodes. When the green nodes are talking, thered nodes are listening. The red nodes estimate the number ofgreen nodes who believe that the primary has entered/existedand thus effectively playing the role of CU . We have thus ineffect planted the CU inside the sensor node themselves.

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Fig. 7. The tradeoff between threshold and probability offalse alarm

of the square lies an interferer that changes its status(i.e. turns on/turns off) with a fixed small probabil-ity. Around each sensor, we consider a smaller square.In this model, the interferers falling inside this squareare sensed by the sensor without any attenuation.Figure 7 shows the tradeoff between threshold and

probability of false alarm. As expected, a decrease inthe threshold increases the number of alarms gener-ated by each sensors when the signal is not present,and thus the higher chance of CU making a mistake.Figure 8 depicts the trade off between number of

sensors and the probability of false alarm. The goalof this simulation is two fold. Firstly, as long as thesensitivity area of sensors do not overlap, the alarmssent to CU are zero mean i.i.d. random variables.As we increase the number of sensors, their sum willbe within an interval around zero with an increas-ing probability. Therefore probability of false alarmwill be reduced. Secondly, we observe that increasingthe number of sensors in a closed area will eventuallymake them sensitive to common interferers. In otherwords, in a dense network, the alarms sent by thesensors will become correlated and the performancewould not be improved by the same trend.

Appendices

A. SNR Wall Analysis for Max-MinEigenvalue Detector

In this appendix, we prove that any interferer canimpose limits on the performance of the max-mineigenvalue detector [6]. In this section, we adopt thesame notation used in [6]. The authors implicitly as-

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Fig. 8. Trade-off between number of sensors and proba-bility of false alarm

sume that the coherence time of the multi-antennareceiver is infinity. We base our theorem on the sameassumption.

Since the ratiomax(Kx)min(Kx)

can not be equal to one due

to the existence of interferers even when the primarysignal is not present, the detector has to set a thresh-old and decide between the hypothesis H0 and the

hypothesis H1 based on 1[max(Kx)min(Kx)

> ]. The value

of would be uniquely determined by the probabilityof false-alarm.Definition A.1. Assuming any specific model

for noise, the threshold function Thf is the an-

swer to the following equation: Pr(max(Kx)min(Kx)

>

Thf(PrFA))= PrFA

Theorem A.1. For any positive , there exists athreshold such that whenever the primary powerP is less than , the detector can not guarantee aprobability of miss-detection less than 1 . Further-more, given the probability of false alarm PrFA, isbounded from below by (1)

2

c1(+1)where c1 =O(1 ) and

= Thf(PrFA).Proof: We prove the statement by contradiction.

Fix some PFA and some > 0 and assume that theguarantee of PMD < 1 exists. In particular, thesensor should have PMD < 1 when the noise on thechannels are white and uniform, and that Ks = P I.In this case,Kx = PHH

+2IML for some constant. Please note that intuitively if P is small here,from the perspective of the detector it behaves like

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imum

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enva

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+P*H

*HT

P

99% confidence intervalof maximum eigenvalue

Fig. 9. Maximum and minimum eigenvalues of I+pHHT

as a function of p, power of the signal. Minimum eigenvalueremains around one while maximum eigenvalue increaseslinearly. Here H is a random matrix. At p= 0, both eigen-values are one. The maximum eigenvalue increases rapidlywith the increase in p, while the minimum eigenvalue re-mains almost the same.

an interferer and thus the ratio between maximumand minimum eigenvalue should be close to one.Using the fact that for any matrixes A and B we

have: {max(A+B) max(A)+max(B);min(AB) min(A)max(B),

we can boundmax(Kx)min(Kx)

from above by

2+Pmax(HH)

2Pmax(HH) as long as 2 Pmax(HH)> 0.

In order to use Markov inequality in the above ex-pression, we first need to show that E[max(HH)] 0 and

2 + Pmax(HH)

2 Pmax(HH)

2 + c1P2 c1P

) 1

Now, if P < (1)2

c1(+1), one can conclude that

2+c1P

2c1P . This would in turn imply that

Pr(2 + Pmax(HH

)2 Pmax(HH)

) 1 ,

and therefore

PrMD = Pr(max(Kx)min(Kx)

) 1 ,

which is a contradiction.

B. Formulation and Analysis of theProposed Detection Algorithm

A. Model Configuration

Consider a network ofms sensors, one Central Unit(CU). The sensors are geographically distributed inan area. Furthermore each sensor has ma antennas.There are communication links between each sensorand the CU with rate constraints R1,R2, ...,Rms . Themessage set at each node is {positive alarm, negativealarm, idle}. A positive (or negative) alarm corre-sponds to the sensors belief that the primary has en-tered (or exited) while an idle message correspondsto the belief that primary has not changed its state.At each time instant, the CU will decide whether theprimary has changed its state based on the receivedmessages from the sensors. In addition, there is a la-tency constraint D at the CU meaning that the sys-tem needs to detect state transitions of the primarywithin delay D. Let T be the minimum coherencetime of all wireless communication channels. Assumethat the minimum inter-transition duration of the pri-mary signal is lower bounded by Ts, i.e. time betweenconsecutive transitions of the primary is at least Ts.We assume that there at most mt primaries. The

active primaries are assumed to generate uncorrelatedsignals at any particular time instance.For simplicity of analysis, we assume that whenever

the primary signal enters/exits, all the sensor nodesdetect, and transmit corresponding alarms which arereceived instantaneously by the CU . It should behowever noted that in a realistic scenario there will in-evitably be some delay for reasons such as the codingused by the sensor nodes, possible clock mismatchesetc.

TABLE IINotations

Variable DescriptionCU Central Unitms Number of sensor nodesma Number of antennas at each nodemt Number of primaries

Ri (i= 1,2, ...,ms) Rate constraintsD Latency constraint at the CUTs lower bound on the minimum

inter-trans. duration of primariesT minimum coherence timesj Threshold at the jth sensorcu Threshold at the CU

j(.)[m] Output of the jth sensorDt(., ., .)[m] Output of the CU

B. Detection Algorithm at the Sensors

In this algorithm, the objective of a sensor is todetect the changes in the level of energy due to en-trance/exit of the primary. One possible scheme is tocompare the measured energy levels of the first halfand second half of a shifting observation window. Thelength of the observation windows is chosen to be lessthan the minimum coherence time, Tc, ensuring thatthe changes in the energy levels are not coming fromchanges in the channel statistics. Other constraintson the duration of the observation window are thenetworks latency constraint, D, and the minimuminter-transition duration, Ts. The latter constraintguarantees that the primary signal is active duringthe second half of some observation window.Definition B.2.1. For any positive , the output

of the detection algorithm at the jth sensor j() isdefined as follows: assuming that the received signalat the ith antenna of the jth sensor (1 i ma) isxi,j [.], the sensor transmits:

j()[m] =

Positive alarm if E1,j [m]E2,j [m]> Negative alarm if E1,j [m]E2,j [m]

10

30 40 50 60 70 800.5

0

0.5

1

1.5R

ecei

ved

Sign

al a

t the

Sen

sor

0 5 10 15 20

10.5

00.5

1

Edge

Det

ecto

r Filte

r

30 40 50 60 70 80 905

0

5

10

Out

put

Fig. 10. The output of the edge detector filter will beapproximately zero unless there is a surge or drop in thereceived energy level.

noisy environment later.

C. Detection Algorithm at the Central Unit

At any time instance m, the central unit receivessignals from the sensor nodes. It seems intuitive forthe CU to take the majority vote. We proceed withthis detecting strategy.Definition B.3.1. For any positive cu, the output

of the detection algorithm at the CU is defined asfollows:Dt( ,cu,R )[m] =

H+ :1ms

msj=11[j(j)[m] = Positive alarm]

1ms

msj=11[j(j)[m] = Negative alarm]> cu

H :1ms

msj=11[j(j)[m] = Positive alarm]

1ms

msj=11[j(j)[m] = Negative alarm]

Parsa, Aminzadeh and Sahai: EXPLOITING INTERFERENCE DIVERSITY FOR EVENT-BASED SPECTRUM SENSING11

We therefore have: limma E1,j [m] =1N0

N01n=0 E,j [mnN0 1],

limmaE2,j [m] =1N0

N01n=0 E,j [mn].

Observation B4.1. Since 2N0 Tc, j,z[n] doesnot change within an observation window. We canthus draw the conclusion that if the status of aprimary does not change throughout the observationwindow, it does not contribute to the differencelimmaj()[m] = limma(E1,j [m]E2,j [m]). In the following subsections, we analyze the de-

tection algorithm under different configurations. Inthe first subsection, we analyze the asymptotic be-havior of the algorithm under the assumption thatR1 = R2 = ...= Rms = R. We will prove that the de-tector has no SNR-Wall when ma,ms as longas the noise is following from a certain model. Theproblem with this configuration is however that thetotal rate, R1+R2+ ...+Rms approaches infinity asms.In the next subsection, we justify the choice of R1 =

R2 = ... = Rms = R arguing that the CU only needsto know the total number of positive and the totalnumber of negative alarms; CU is only interested inthis function of the transmitted messages, and doesnot need to know the identity of the nodes that aretransmitting positive or negative alarms.

D.1 Equal Rate At All Sensors

Assume that R1 = R2 = ... = Rms = R. We willdefine a class of noise functions under which the de-tection algorithm has arbitrarily low probability offalse alarm and miss-detection as ms and ma con-verge to infinity for arbitrarily weak primaries. Theconstraint imposed on the noise functions quantifiesthe local-global assumption according to which thenoise signal observed at different sensors are indepen-dent.Definition B.5.1. For any integer N0 and vec-

tors of positive real numbers and R , Noisemodel ( ,R,N0,ms,ma) is the set of noise func-tions wi,j [n] (1 i ma, 1 j ms, n =...,2,1,0,1,2, ...) satisfying the following two con-straints. Whenever primaries are absent, i.e. xi,j [.] =wi,j [.],1. j :limM 1M

Mm=11[j(j)[m] 6= Idle message]R

2. m :limms

{1ms

msj=1 1[j(j)[m] =

Positive alarm]1ms

msj=11[j(j)[m] = Negative alarm]

}= 0

Remark : The first constraint ensures that for thechosen threshold j , the uncoded transmission of thealarms generated by the noise signal when primariesare absent do not violate the rate constraints. Obser-vation B4.1 implies that when some of the primariesare present but do not change their state, the samealarms rate would be generated. Clearly increasingj makes enlarges the set of noise sequences that sat-isfy the first property. In the extreme case of j =,the constraint would be valid for all Rj . The sec-ond assumption is intuitively saying that noise is un-biased: as ms , the number of positive alarmsand negative alarms generated by the noise convergetogether. This phenomenon would happen if thenoise sequences are each unbiased and behave inde-pendently of each other. If the local-global assump-tion fails, the alarms generated by different nodes willno longer be independent of each other and this con-straint may no longer be valid.Definition B.5.2. For any constant c, Noise

model (c, ,N0,ms,ma) is the set of noise func-tions wi,j [n] (1 i ma, 1 j ms, n =...,2,1,0,1,2, ...) satisfying the following two con-straints. Whenever primaries are absent, i.e. xi,j [.] =wi,j [.], either of the following two conditions are sat-isfied:

1 = infm

limms

inf :jj for j=1,2,..ms and 1ms |

|1c{1ms

msj=1

[1[j(j)[m] = Positive alarm]

1[j(j)[m] = Positive alarm]]}

> 0;

or

2 = infm

limms

inf :jj for j=1,2,..ms and 1ms |

|1c{1ms

msj=1

[1[j(j)[m] = Negative alarm]

1[j(j)[m] = Negative alarm]]}

> 0.

Remark : Clearly for any j and any j j ,1[j(j)[m] = Positive alarm] 1[j(j)[m] = Positive alarm];

12

The first condition ensures that in the limit case,whenever the average gap between j and j is atleast a constant c, the rate of positive alarm is sensi-tive to changes in . The second condition has a sim-ilar interpretation. The reason for introducing thesetwo conditions is that intuitively when a primary en-ters the system, the chance of passing threshold ,and thus of Positive alarm, increases whereas thechance of passing , and thus of Negative alarm,decreases. This is because the effective threshold forPositive alarm from the perspective of noise de-creases, whereas the effective threshold for Positivealarm from the perspective of noise decreases.

Theorem B.5.1. The detection algorithmDt( , cu, R ) succeeds with probability one asma,ms if The detector is operating in an envi-ronment with a noise sequence from( ,R,N0)(c, ,N0,ms,ma);

The average power of the primary that has en-tered/existed the system over all antennas of thesystem is at least c;

cu < max(1, 2) where 1 and 2 are defined in(c, ,N0,ms,ma).

Proof. Assume that no primary changesits status in the interval [n N0 1, n + N0 1].The second property of ( ,R,N0) ensures thatlimmsDt( ,cu,R ) =HI .Next assume that the zth primary turns on at time

n. Since 2N0 + 1 is less than the minimum inter-transition duration of the primary signals, Ts, therecan be no other primary arriving or leaving in theinterval [nN0 1, n+N0 1]. Thus, other pri-maries do not contribute to limmaj()[m] ac-cording to observation B4.1. Since 2N0 + 1 is lessthan the coherence time, j,z[n] is constant duringthe interval [nN0 1, n+N0 1]; let j := j,z[u](for u [nN0 1,n+N0 1]).The best opportunity to detect the entrance of the

zth primary is at time n+N0 1, when the very be-ginning of the second half of the observation windowis the entrance time of the zth primary. The aver-age energy in the second part of the window is in-creased by jPz. Let j = j jPz, j = j + jPz.Since the average power of the primary that has en-tered/existed the system over all antennas of the sys-tem is at least c, we get 1ms

msj=1 jPz c. The condi-

tion imposed in (c, ,N0,ms,ma) ensures that theaverage number of positive alarms that the CU re-ceives is greater than the average number of nega-

tive alarms by at least max(1, 2). This means thatthe detectorDt( ,cu,R ) succeeds to detect the pri-mary.Corollary: The detector will be able to detect the

entrance/exit of primary signal with average powerpmin if noise signal belongs to the set:

cpmin(c, ,N0,ms,ma)

( ,R,N0)

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