Analysis of Search Schemes in Cognitive Radio

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1 Analysis of earch chemes in Cognitive Raio ing uo an umit Roy Department of Electrical Engineering University of Washington, eattle, WA Abstract Cognitive Raio systems face an important challenge fast an reliable channel searching to enable seconary users to optimize available spectral resources. We revisit conventional urn moels for channel occupancy an analyze the performance of several search schemes in terms of the mean time to etection. In particular, we focus on correlate Markov moels for bin occupancy an highlight the performance of n-step serial search (n) algorithm Inex Terms Cognitive Raio, channel moel, search scheme I. ITRODUCTIO Due to the rapily increasing emans on wireless banwith, available spectral resources for new services are scarce, especially for - GHz ban. Recent measurements from FCC emonstrate that only 5-85% of this spectrum is utilize on (time) average []. These motivate Cognitive Raio (CR) [,] operation whereby seconary users are allowe to actively search an uire ile channels (unuse by primary users) provie they o not interfere with primary users rights, i.e. use the spectrum on a non-interfering basis. Fast an reliable channel searching is thus an important component for any CR network. Research on spectrum sensing has pai consierable attention to novel approaches base on cooperative sensing, ynamic spectrum access an spectral agility [5,6,7,8,]. Different sensing strategies are apropos to emerging new CR network architectures, leaing to the esire for a uniform metric for comparing ifferent search strategies. We use the mean etection time rawing on ranom urn occupancy moels [9] in this work. We seek to highlight the impact of ifferent a) channel availability (occupancy) moels an b) search strategies on the mean etection time for spectral scanning by seconary users; in particular, we investigate the ifferences between inepenent an Markov (correlate) occupancy moels. The rest of this paper is organize as follows. Detaile escriptions of channel moels an search schemes are provie in ections an, respectively. We present the analysis of mean-time to etection in ection 4. imulation results in ection 5 for each search strategy support the analysis an explore trae-offs among average etection time, power consumption an achievable probabilities of etection an false alarm. We conclue the paper in ection 6. II. CHAE MODE DECRIPTIO Each seconary user of a CR network attempts to sense the spectral ban compose of a -set of contiguous iscrete frequency omain channels. At a given instant, the average number of ile channels is (both an are large, an / << ). We assume that the etection probability (P ) an false alarm probability (P fa ) pertinent to the sensing of any channel is the same. In the event of a false alarm, we assume that the sensing process requires J sensing urations to recover an resume scanning, i.e. is the associate penalty. A. Ientical Inepenent Distribution (i.i.) Moel In an i.i. moel, each iscrete channel has the same probability / to be ile. We efine the binary variable O k to enote the status of the kth channel (k, enotes the channels in sequence) where () means that channel is busy (free). P( O k ), k,,... () The traitional i.i. moels are consiere as Ranom Occupancy. As shown in Fig., the ranomly available (i.i.) channels are scattere over the -set.

2 III. EARCH CHEME Fig. Ranom occupancy B. Correlate Markov Moel To moel channel clustering, we introuce Correlate Occupancy whereby the state O k for the k-th channel epens on those of it s neighbors. Using a first-orer Markov epenence between two contiguous channels, we introuce the conitional probabilities: ( ρ is transition probability) P ( O k O k + ) ρ (5) P ( O k O k + ) ρ (6) where the mean persistence time in each state is assume ientical. The moel is escribe via a Markov Chain in Fig., with a transition matrix: (eigenvalues are an ρ ) ρ ρ P ' (I) ρ ρ (ρ ) (ρ ) + ( k) 4( ρ) 4( ρ) P '' P' (II) k (ρ ) (ρ ) + 4( ρ) 4( ρ) A. Ranom earch Ranom search is a basic search strategy in CR: the seconary user ranomly picks up a channel to etect whether it is ile or not. If it is etecte to be busy, the user then resorts to searching another channel ranomly. The search stops when an ile channel is foun. B. erial earch In serial search, the seconary user searches channels in sequence till an ile channel is iscovere. The state iagram for serial search is escribe in Fig., where the seconary user chooses the initial channel ranomly to start searching. If an ile channel is not etecte in one pass, the search is again re-initialize. Fig. Correlate occupancy We let the expecte number of ile channels in this case equal which is the sum of the probabilities that each channel is ile. Assuming that the Markov Chain (I) starts from the status, it follows using (II) (>): P( Ok ) P''[,] k (7) (ρ ) 4( ρ) 4( ρ) ρ (8) ( ) Fig. erial search C. n-step erial earch This is the generalization of the stanar serial search with step size larger than. However, as shown in Fig.4, the step size n < to ensure that a cluster of free channels is not misse.

3 Fig.4 n-step serial search IV. AAYI OF AVERAGE DETECTIO TIME Each channel scanning step has two components: i. A fixe uration for the receiver to switch its sensing circuitry to the new channel; this is assume to be a constant value T c, regarless of magnitue of the jump in frequency omain at each scanning step, an largely epens on the circuit implementation. ii. The uration T sens for reaching a ecision on channel status (busy/ile); this is a function solely of the esire P -P fa which are assume constant for each channel. Thus the average etection (uisition) uration can be written as: T ( T + T ) (9) c sens where is the average number of steps in channel scanning prior to a success (i.e. etecting an ile channel). In our analysis below, we focus on the erivation of the average number of sensing steps neee prior to etection of an ile channel. A. ensing Duration In a typical energy etection moel, the receive signal samples are filtere to a etector banwith (B sense ), passe through a square law etector, an integrate for a set uration (T sense ) before the accumulate energy is compare with a ecision threshol (D t ). Here B sense is the banwith of one iscrete channel, which is set to MHz. From [], the following equations can approximate P an P fa of the etector: D t Tsense B sense ( + R ) Q( ) () Tsense B sense + R Dt Tsense B sense Pfa Q( ) () Tsense B sense The function Q(x) is efine as: Q τ / ( x) e τ () π x In this paper, R (P signal /P noise ) is set to B; P is.9 an P fa is.5. From ()-(), we can then obtain that: T sense 5. ms << ms Compare to the value of T c, the sensing uration T sense for each search step can be ignore in our work. B. erial earch uner Ranom Occupancy We efine a set of ranom variables {X k } as the location of the k-th free channel in the available spectrum. For instance, if the st ile channel is the 4th channel, then X 4. In Ranom Occupancy, X k ~U(,) an are mutually i.i.. P,P fa (Ieal): In this ieal scenario, the average number of etection steps is the mean istance from any starting point (nominally k) to the nearest one of these ranom variables of {X k }. et Y enote the steps require by serial search to reach an ile channel, then: Y min( X, X..., X ) () Thus the istribution of {Y} an the average steps are: F Y ( y) ( FX ( y)) (4) + ( / ) ( / ) yfy ( y) y [ ] (5) + For, an large: /( +) (6) P <, P fa > (on-ieal): We split the number of steps for serial search in this scenario into two parts: normal + punish, where normal an punish respectively enote the number steps to etect an ile channel base on P < an the penalty ue to non-zero false alarms in each search cycle. The probability that serial search successfully etects an ile channel is P / for each attempt. Therefore, it can be treate as the case of i.i.

4 variables {X k } that are uniformly istribute in (, /P ). + ( / ) ( / ) [ ] (7) normal + et y enote the number of steps in normal search an u enote the times of occurrence of false alarm uring each search cycle. It can be conclue that u follows Binomial Distribution: u~b(y,(-)p fa /). The average numbers of false alarm an penalty steps are fa u u / P uf( u) u f ( u y') f ( y') y' (8) normal punish J fa JPfa normal (9) Therefore, from (7)-(9), the average number of steps for success of serial search uner ranom occupancy is: + ( ) JPfa + ( / ) ( / ) [ ] + () ( ) JPfa + P ( + ) C. Ranom earch uner i.i. Moels In i.i. moels, the probability of each attempt for a seconary user to ranomly catch an ile channel is the same. imilar to the analysis in Part B of ection 4, we can conclue that the number of steps for ranom search follows a Geometric Distribution. Thereby its average value is given by: Pfa J ( + ( JPfa + ) ) () P P D. Ranom earch uner Correlate Occupancy In correlate occupancy, if there are m ile channels in the spectrum with probability p m, then we can obtain the average number of steps to success from (): Pfa J ( m) + E[ tep ile _ channel : m] () m If the st channel is assume to be busy, then the probability that all channels are unavailable PO ( O... O ) () PO ( ) PO ( O )...( PO O... O ) ρ either ranom nor serial earch can fin an ile channel in this situation. During the time the Markov Chain transitions to the next state, the number of search step is constant an set equal to. It can be observe from transition matrix (I) that: Y P P state change P P state unchange ρ P ρ (4) P (5) To calculate the probability of m ile channels, we consier the case that they occur as k groups. There will thus be typically state transitions.thus the probability that m ile channels appear in the Markov Chain (I) is: m P( m ile) C C P P (6) k k m k m state unchange state change (6) is base on the assumption that the number of ile channels is less than busy channels (m<-m). In case of m>-m, the probability of m ile channels in the spectrum is given by: m k C m k P ( m ile) C P P (7) k m+ state unchange state change Therefore, from ()-(7), the average number of steps is: ρ E[ E[ tep m ile]] + min( m) m k C k min( m) C k max( m) + ρ ( ρ) min( m) min( m, m), ma x( m) max( m, m) (8) E. erial earch uner Correlate Occupancy We efine four search states: tate enotes the successful etection of an ile channel; tate the unsuccessful etection of an ile channel; tate the successful etection of a busy channel; tate the false alarm of a busy channel. In any search algorithm, tate is an absorbing state, which means that the search is ene once an ile channel is successfully etecte. The transition iagram for this chain is shown in Fig.5, where the transition matrix P is: ρp ρ( ) ( ρ)( Pfa) ( ρ) Pfa P (III) ( ρ) ( ρ)( ) ρ( Pfa) ρp fa ( ρ) ( ρ)( ) ρ( Pfa) ρp fa If a group occurs at the beginning or en, the number of transitions equals -. We ignore this bounary conition, for,k large.

5 Fig.5 Transition iagram for search states ince the number of ile channels is out of a total of, it follows that: P { } P, P{ } ( ) P{ } ( Pfa), P{ } Pfa (9) If there is no penalty ue to a false alarm (J) the average number of steps to reach the ile channel is: k kp{ ( k)} P{ k} () For k k, it is necessary that in the first k steps the Markov Chain (III) has not entere the absorbing tate []. ince the first step of serial search might start from tate i: (i,,): P{ i k} i [ P{ i}( P P{ ( k ) i, + P k i} P{ i} ( k ) i. + P ( k ) i, )] () As a result of the property of absorbing state, we ientify the following sub- matrix P, enote as A below. ρ( ) ( ρ)( Pfa ) ( ρ) Pfa A ( ρ)( ) ρ( Pfa ) ρpfa (IV) ( ρ)( ) ρ( Pfa ) ρpfa Thereby the formula () can be transforme into i ( k ) ( k ) ( k ) i, + Ai. + Ai, ) P { k} ( A () Assuming that the matrix power series converges, the average number of etection steps without any false alarm penalty can be written as: k ( k ) A ( I A) k [ i () P{ k} P{ i}( + + )](4) Meanwhile, the matrix can be obtaine by (), an the specific calculation can be easily run by some computer software. We next incorporate the penalty ue to false alarm by letting the seconary user wait for another J steps to start the search algorithm again in the event of a false alarm of an ile channel. In such a case, the number of steps in tate will be multiplie by J. Therefore, the mean number of etection steps in the general case is given by [ P{ i}(, +. + J, )] (5) i i F. n-step erial earch uner Correlate Occupancy In n-step serial search, the transition matrix P is given by P (n), the n-step transition matrix of the original Markov Chain. Thus the average number of etection steps k k i kp{ ( k)} [ P{ i}( P' ( k ) i, k + P' i P{ ( k ) i. i, + P' i i. k} ( k ) i, )] i, (6) Assuming convergence of the matrix power series, we fin the average number of etection steps [ P{ i}(, +. + J, )] (7) where ' ' ' i i i i ( k ) ( n) ' A' ( I A ) (8) k V. IMUATIO REUT A. Experimental Environment We simulate the three search schemes mentione above to evaluate their performance. In correlate occupancy, the transition matrix parameter ρ is base on (8). The basic parameters of the simulation environment are: 5, P.9, P fa.5, J4. Matlab is use to generate iscrete channels, an the ata are gathere by running realizations for each experiment. B. Average umber of Detection tep The number of ile channels is altere to investigate the performance of each search algorithm, shown in Fig.6. We can observe that serial search requires less

6 etection time than ranom search uner ranom occupancy. From (8), it can be conclue that as the number of ile channels is ecrease, the transition probability increases to, in correlate occupancy. When the transition probability is larger than.97 (corresponing to the case that there are no more than 6 ile channels), the free an busy channels have a high probability of clustering. When the transition probability is below.85 (the proportion of ile channel is higher than.47) for correlate occupancy, the ifference in the mean number of steps require between ranom an serial search is small, as expecte. It can be also observe that the average number of steps sharply ecreases as the number of ile channels goes up (transition probability rises) in correlate occupancy. The larger values of transition probability correspon to higher probability of consecutive busy channels an significant increase in the average number of etection steps. With the increase in the transition probability, the average length of a cluster of consecutive ile channels increases, implying reuction in k. This in turn means that the approximations in (6) an (7) are less accurate an eviations among analysis an simulation results for both search schemes begin to show up in such scenario. C. Performance of n-step erial earch The performance of n-step serial search uner correlate occupancy is shown in Fig.7. Of particular interest is the cross-over of the mean etection time for ranom search vs n-step serial search. Consier n -step serial search; ranom search performs better when the transition probability is less than.94, while -step serial search ominates at the higher transition probability. Further, the mean etection time of n-step serial search ecreases with increasing n. D. Channel ensing (T sense ) Channel sensing is commonly performe using non-coherent energy etection; for a higher P an lower P fa, a receiver has to integrate the signal over a longer uration. For BBsenseMHz, RB, the relationship between T sense an varying P -P fa is shown in Fig.8 an Fig.9. We can observe that the value of T sense is typically much smaller than T c (switching uration) of ms, an was therefore neglecte in our computations. However, as T c or B sense (sensing banwith) becomes smaller, T sense will become a significant portion of the whole etection time. The impact of such variations will be explore in future. Decision Duration (ms) Decision Duration (ms) x Probability of Detection x - Fig.8 T sense for ifferent P False Alarm Probability Fig.9 T sense for ifferent P fa

7 Average umber of Detection tep Ranom Occu: erial earch Ranom Occu: Ranom earch Corre. Occu: erial each Ana. Corre. Occu: erial earch imu. Corre. Occu: Ranom earch Ana. Corre. Occu: Ranom earch imu. Corre. Occu: -step erial Ana Ratio of Available Channel (/) Fig.6 Average number of step for serial an ranom search Average umber of Detection tep Ranom earch erial earch: Analysis erial earch: imulation -step erial earch: Analysis -step erial earch: imualtion 4-step erial earch: Analysis 4-step erial earch: imulation 8-step erial earch: Analysis 8-step erial earch: imulation Transition Probability Fig.7 Performance of n-step serial search uner correlate occupancy

8 I. COCUIO In this paper, we investigate the performance of ranom an serial search schemes for inepenent an correlate occupancy channel moels. It is shown via analysis an simulation results that n-step serial search has lower etection time than ranom search. Furthermore, with the increase in the transition probability, performance in correlate occupancy channels to have the similar performance as clustering. The search schemes with less etecting time also have the high cost of energy, an our future work is to achieve a better equilibrium among these characters of CR network. [] R. Iyappan an. Roy, On Acquisition of Wieban Direct-equence prea pectrum ignals, IEEE Trans. On Wireless Commun., Vol.5, Issue 6, pp , June 6 []. Ross, Introuction to Probability Moels, Acaemic Press, 8 th ED, REFERECE [] FCC, pectrum Policy Task Force Report, ET Docket o.-55, ov. [] J. Mitola an G.Q. Maguire, Cognitive Raio: Making oftware Raios More Personal, IEEE Personal Communications, Vol.6, Issue 4, pp.-8, Aug. 999 []. Haykin, Cognitive Raio: Brain-Empowere Wireless Communications, IEEE J. elect. Areas Commun., Vol., Issue, pp.-, Feb. 5 [4] I.F. Akyiliz et al., ext generation/ynamic spectrum access/cognitive raio wireless networks: A survey, Computer etworks, Vol.5, Issue, pp.7-59, ep. 6 [5].M. Mishra et al., Cooperative ensing among Cognitive Raio, IEEE ICC 6, Vol.4, pp , June 6 [6] Y. Xing et al., Dynamic pectrum Access in Open pectrum Wireless etworks, IEEE J. elect. Areas Commun., Vol.4, Issue, pp.66-67, Mar. 6 [7] C.T. Chou et al., What an How Much to Gain by pectral Agility, submitte to IEEE/ACM Trans. On etworking, June 4 [8]. Mangol et al., pectrum Agile Raio: Raio Resource Measurements for Opportunistic pectrum Usage, IEEE Globecom 4, Vol.6, pp , ov. 4 [9] O. Milenkovic an K.J. Compton, Probabilistic Transforms for Combinatorial Urn Moels, Combinatorics, Probability an Computing, Vol., Issue 4-5, pp , Jul. 4 [] A. aleh an R. Valenzuela, A tatistical Moel for Inoor Multipath Propagation, IEEE J. elect. Areas Commun., Vol.5, Issue, pp.8-7, Feb. 987