Performance Bounds for Detect and Avoid Signal Sensing

Transcription

1 Perfrmance unds fr Detect and Avid Signal Sensing Sam Reisenfeld Real-ime Infrmatin etwrks, University f echnlgy, Sydney, radway, SW 007, Australia samr@uts.edu.au Abstract Detect and Avid (DAA) is a Cgnitive Radi technique fr a primary user and secndary users sharing the same spectrum. he DAA spectral sensing system must prvide bth lw decisin delay and highly reliable decisins n the presence/absence f primary user transmissin. hese results prvide upper and lwer bunds n the decisin errr rate perfrmance f sensing systems as a functin f SR (signal-tnise rati) and the number f samples used in the sensing decisin. he bunds represent benchmarks n the perfrmances f practical detectin systems and give insight int the trade-ff between sensing perfrmance and decisin delay. I. IRODUCIO Cgnitive Radi may prvide spectrum sharing capabilities amng a primary user, wh has pririty in the use f the spectrum, and secndary users. In this cntext, a simple cexistence mechanism is prvided by the s-called Detect and Avid (DAA) peratin []. asically, a secndary user may use the spectral band during times that it is unccupied by the primary user. Hwever, the secndary user must quickly and accurately sense the presence f the primary user and avid transmissin when the primary user is transmitting. A fundamental issue arises regarding hw much primary user signal structure shuld be used in DAA signal prcessing. he simplest frm f detect and avid signal prcessing uses energy sensing. he mst cmplex frm f detect and avid signal prcessing uses wavefrm sensing, which utilies the cmplete detail f the primary user wavefrm, including mdulated data. he decisin errr perfrmance f any practical detect and avid signal sensing is upper bunded by the energy detectr errr perfrmance and lwer bunded by the wavefrm detectr errr perfrmance. herefre these bunds expressed as a functin f signal t nise rati and the number f time samples used in a primary user detectin decisin f are great practical interest in the design f cgnitive radi systems. he difference between the upper and lwer bunds is als imprtant in the determinatin f hw much primary user signal structure t use in DAA prcessing. It is assumed that the cmplex envelpe f the channel utput is time sampled at a rate f Rs cmplex samples per secnd and that the samples are statistically independent. = samples are used fr the primary user detectin, d where, d is the decisin delay time in secnds and is the primary user signal bandpass bandwidth in Hert. Using the Sampling herem, Rs = samples per secnd. herefre, fr a primary user signal with bandwidth, is directly prprtinal t the signal bservatin time. Fr successful DAA peratin, it is critically imprtant that the presence and absence f the primary user signal is quickly and accurately detected by the secndary user receivers. Sensing delay must be traded ff against sensing accuracy in the system design. he relatinship between the number f time samples and signal-t-nise rati required t achieve a particular level f sensing perfrmance was presented in Reisenfeld and Maggi []. hese results apply t any mdulatin type, including cnstant envelpe mdulatin and Gaussian signal mdulatin. Gaussian signal mdulatin, in which time samples f the mdulated signal are Gaussian randm variables, mdels OM. here are many cases f practical interest in Cgnitive Radi in which the primary user mdulatin is OM. he results in Reisenfeld and Maggi[] are summaried in this paper. his paper als prvides new simulatin results relating t the maximum likelihd signal sensing errr prbability fr Gaussian mdulatin, such as OM. he results are cmpared t the analytically derived errr prbability expressins fr bth energy sensing and wavefrm sensing. II. SPECRAL SESIG MODEL he mdel described by ang [3] fr spectral sensing was t prcess either a time dmain sequence f channel utput cmplex envelpe time samples r a frequency dmain set f FF cefficients. he detectin f the primary spectrum user signal can be characteried by the binary hypthesis test, where H is the hypthesis representing the primary signal nt being present and H is the hypthesis representing the primary signal being present. A er mean, additive white Gaussian nise channel is assumed. he receiver prcesses the channel utput t btain discrete samples f the cmplex envelpe r, alternatively, FF utput cmplex cefficients. he number f cmplex channel utput samples that are prcessed fr a detectin decisin is. herefre, H : y( n) = ( n), n=,,..., 0 H : y( n) = x( n) + ( n), n =,,..., where, yn ( ) are time samples f the cmplex envelpe f the channel utput, xn ( ) are time samples f cmplex envelpe f the primary user signal, and n ( ) is a sequence f statistically independent, er mean, cmplex Gaussian randm variables with variance equal t. () /09/$ IEEE 3 CgAR 09

2 Alternatively, yn ( ) culd represent the FF utput cefficients resulting frm an input f a sequence f time samples f the cmplex envelpe f the channel utput. A. Energy Sensing he energy metric, S, is defined as, Define, S E[ S H ] 0 0 S E[ S H ], n= S = y( n) () μ =, (3) μ = (4) = Var[ S H ], (5) S S = Var[ S H ] (6) Als define the decisin threshld, S, such that the binary decisin rule fr spectrum sensing is: If S S, then decide H, If S < S, then decide H. hen tw perfrmance measures that characterie the detectin perfrmance are, P = Pr{false detectin}=pr{ S S H } (7) If P = Pr{missed detectin}=pr{ S < S H } (8) MD is sufficiently large, S is essentially Gaussian because f the Central Limit herem. assumptin, where, P S μ S 0 = Q S0 S = S Q μ S Under the Gaussian (9) (0) Qx ( ) = e d () π x Fr maximum likelihd detectin, P = P () the detectin threshld fr maximum likelihd detectin is, μs S + μ S S S = (3) + S Using the maximum likelihd decisin threshld, the sensing errr flr (SEF) is defined [3] as, MD S = = = S S SEF P Q μ μ S + S { } he signal sequence, x( n ) (4), may be mdeled as an ergdic randm prcess. As shwn in Reisenfeld and Maggi[], fr the energy detectin statistic defined in (), the evaluatin f (3), (4), (5), and (6) in (4) results in, SR SEF = Q, (5) + ( α ) SR + SR + ] where, 4 E x( n) α = E x( n) the signal-t-nise rati, SR, is defined as, (6) E x[ n] SR = (7) α is a parameter that characteries the shape f the prbability density functin f the magnitude f the cmplex envelpe f the mdulatin. Fr cnstant envelpe mdulatin, α =. If xn ( ) is Rayleigh distributed which is the case f Gaussian mdulatin, α =. Fr OM mdulatin, xn ( ) is asympttically Rayleigh distributed as the number f sub-carriers becmes large. Evaluating (9) and (0) fr the energy detectin results in, P = Q S (8) S ( SR+ ) = Q (9) ( α ) SR + SR + P and P MD may be traded ff fr fixed SR and by varying the decisin threshld, S.. Wavefrm Sensing Wavefrm sensing uses the cmplete functinal descriptin f the signal t be detected, including the data in the mdulated signal. Fr wavefrm detectin, the detectin statistic, S, is given by, * S = Re y( n) x ( n) (0) n= where, x * ( n ) is the cmplex cnjugate f xn ( ). In agreement with ang [3] fr the wavefrm sensing statistic given in (0), the evaluatin (3), (4), (5), and (6) in (4) results in the exact equatin fr SEF given by, SR SEF = Q () ( α ) SR + + Als, evaluating (9) and (0) fr wavefrm detectin results in, 33

3 P S = Q SR () S P = Pr{ S < S H } = Q ( SR, ) MD (5) S SR = Q (3) ( α ) SR + SR Equatins (5), (8), (9), (), () and (3) use the maximum likelihd threshld, S, given by (3) fr the ensemble f mdulated wavefrms. his threshld is fixed μ, μ,, and. by 0 III. EXAC MISSED DEECIO AD FALSE DEECIO PERFORMACE FOR EERGY SESIG AD α = he detectin errr perfrmance fr energy sensing in Sectin I used the assumptin that was sufficiently large such that detectin statistic was essentially Gaussian distributed. A cmparisn f (5) t the exact energy sensing SEF fr small is therefre imprtant. Under H, S has a central chi squared distributin with degrees f freedm [4], and under H with the assumptin f cnstant envelpe mdulatin, α = and S has a nn-central chi squared distributin with degrees f freedm [5]. Using these distributins, the exact perfrmance results fr energy detectin may be btained as, S S P = Pr{ S > S H} = e (4) k! k = 0 k where, Q ( a, b) is the generalied Marcum s Q functin m defined by, m x a + x m a x= b (6) Qm( a, b) = x e I ( ax) dx Iλ ( x) is the λ-rder mdified essel Functin f the first kind. III. HE SESIG ERROR FLOOR (SEF) PERFORMACE FOR EERGY SESIG FOR α = Fr α =, it is pssible t numerically find the maximum likelihd decisin threshld withut using the Gaussian apprximatin fr the decisin statistic. Since SEF = = P, equatins (4) and (5) can be used t btain the decisin threshld required fr the SEF. herefre, S S S Q (, ) SR = e (7) k = 0 k! he nrmalied threshld fr the SEF may be btained by S numerically slving (7) fr. his maximum likelihd threshld, which des nt utilie a Gaussian apprximatin, can then be used in t btain the SEF perfrmance. he threshld can either be used analytically in (4) and (5) r can be used in simulatin. It was verified that fr α = and = 0, there was clse agreement between the SEF btained using the Gaussian apprximatin and the SEF btained using the actual distributins f the decisin statistic. k 34

4 Fig.. he SEF perfrmance fr energy sensing and wavefrm sensing as a functin f SR in d fr = 00. he slid lines are simulated results and the dashed lines are analytic results. Fig.. he SEF perfrmance fr energy sensing and wavefrm sensing as a functin f SR in d fr = 000. he slid lines are simulated results and the dashed lines are analytic results. Fig. 3. he theretically required as a functin f SR in d t achieve a 3 SEF equal t 0, fr α =. he energy sensing results used the assumptin that the detectin statistic is Gaussian distributed. Fig. 4. he theretically required as a functin f SR in d t achieve a 5 SEF equal t 0 fr α = he energy sensing results used the assumptin that the detectin statistic is Gaussian distributed. IV. UMERICAL EXAMPLES In this Sectin, specific numerical examples f the results f Sectins I-III are given. he SEF was btained as a functin f SR fr α =, which is cnstant envelpe mdulatin, and α =, which is Gaussian mdulatin, by bth analysis and cmputer simulatin. Results are given fr bth energy sensing and wavefrm sensing. Figures and shw the results fr = 00 and =000, respectively, fr the energy sensing and wavefrm sensing cases. Fr energy sensing, with = 00, there is reasnably clse agreement between the analysis and 5 simulatin fr SEF 0. Fr the =000 energy sensing case and fr all values f fr wavefrm sensing there is excellent agreement between analysis and simulatin. Figures 3 and 4 shw the theretically required as a functin f SR in d t achieve SEF equal t and 0, respectively, fr α =. Figure 5 shws the theretical SEF perfrmance as a functin f SR in d, fr = 0 and α =. th the Gaussian apprximatin and the exact prbability density functin cases are shwn. Figures 6 and 7 shw the SEF perfrmance as a functin f SR in d, fr = 0 and α =, which is the Gaussian signal case, fr energy sensing and wavefrm sensing, respectively. Fr energy sensing in Figure 6, there is gd 35

5 Fig. 5. he SEF as a functin f SR in d fr the energy detectin with = 0 and α =. Curves fr bth the cases f the Gaussian apprximatin and exact results are shwn. Fig. 6. he SEF perfrmance fr energy sensing as a functin f SR in d fr = 0 and α =. he slid line is an analytic result and the dashed line is a simulated result. Fig. 7. he SEF perfrmance fr wavefrm sensing as a functin f SR in d fr = 0 and α =. he slid line is an analytic result and the dashed line is a simulated result. Fig. 8. he SEF perfrmance fr energy sensing as a functin f SR in d fr = 00 and α =. he slid line is an analytic result and the dashed line is a simulated result. agreement between thery and simulatin fr SEF Fr high SR, the large difference between thery and simulatin is due t the detectin statistic nt being Gaussian distributed and the decisin threshld nt being ptimied. he asympttic cnvergence f the SEF t a nn-er value fr increasing SR is verified in Figures 6 and 7. Fr Figure 7, which is wavefrm sensing, there is reasnable agreement between thery and simulatin. Figures 8 and 9 shw the SEF perfrmance as a functin f SR in d, fr = 00 and α =, which is the Gaussian signal case, fr energy sensing and wavefrm sensing, respectively. Fr larger values f, the detectin statistic tends twards a Gaussian distributin because f the Central Limit herem. here is gd agreement between thery and simulatin. V. COCLUSIO his paper has presented results n the relatinship between the detectin perfrmance and decisin delay time fr bth energy sensing and wavefrm sensing. he tradeff is parameteried n, the number f time samples per decisin. Increasing bth increases the DAA decisin delay time and decreases the SEF. herefre, needs t be selected in a systematic tradeff fr which the Cgnitive Radi interference mitigatin is ptimied. he perfrmances f all DAA systems are bunded by the energy and wavefrm sensing perfrmances. he SEF 36

6 results prvide benchmarks fr f practical spectral sensing systems. Fig. 9. he SEF perfrmance fr wavefrm sensing as a functin f SR in d fr = 00 and α =. he slid line is an analytic result and the dashed line is a simulated result. he results are general and apply t any mdulatin type. he cases, α =, fr cnstant envelpe mdulatin, and α =, fr Gaussian mdulatin, such as OM, are f special interest. Fr these cases, the analytical and simulatin results were cmpared. here is clse agreement in the SEF perfrmance btained frm analysis and simulatin REFERECES [] R Giulian, F Maenga, J.H. Pabl, and I.A. enede, Perfrmance f cperative and nn-cperative detect and avid prcedures fr UW, in Prc. 6 IS Mbile and Wireless Cmmunicatin Summit, July 007, pp. -5. [] S. Reisenfeld and G.M. Maggi, Detect and avid fr UW-WiMedia: Perfrmance bunds f signal sensing, in Prc. Internatinal Cnference n Advanced echnlgies fr Cmmunicatins, AC008, 6-9 Octber 008. [3] H. ang, Sme physical layer issues f wideband cgnitive radi systems, in Prc. DySPA 005, First Internatinal Cnference n Dynamic Spectrum Access Systems, 8- vember, 005, pp [4] H.L. Van rees, Detectin, Estimatin, and Linear Mdulatin hery, Part, ew Yrk: Wiley, 00. [5] J.G. Prakis, Digital Cmmunicatins, 4 th Editin, ew Yrk: McGraw- Hill k Cmpany,