th Euopean Signal Pocessing Confeence (EUSIPCO Spectum Sensing Using Enegy etectos with Pefomance Computation Capabilities Luca Rugini, Paolo Banelli epatment of Engineeing Univesity of Peugia Peugia, Italy {luca.ugini,paolo.banelli}@unipg.it Geet Leus Faculty of Elect. Eng., Math. and Compute Science elft Univesity of Technology elft, The ethelands g.j.t.leus@tudelft.nl Abstact We focus on the pefomance of the enegy detecto fo cognitive adio applications. Ou aim is to incopoate, into the enegy detecto, low-complexity algoithms that compute the pefomance of the detecto itself. The main paametes of inteest ae the pobability of detection and the equied numbe of samples. Since the exact pefomance analysis involves complicated functions of two vaiables, such as the egulaized lowe incomplete Gamma function, we intoduce new low-complexity appoximations based on algebaic tansfomations of the onedimensional Gaussian Q-function. The numeical compaison of the poposed appoximations with the exact analysis highlights the good accuacy of the low-complexity computation appoach. I. ITROUCTIO In cognitive adio netwoks, spectum sensing plays a cucial ole, since it allows to detemine if a given fequency band is available o not fo the signal tansmission of a seconday (unlicensed use [], []. Among the vaious citeia fo spectum sensing, enegy detection is pehaps the most famous [], [], and is paticulaly appopiate fo low-complexity applications [3]. This pape examines the possible incopoation of some pefomance computation capabilities into an enegy detecto (E. This way, an E can be able to self-estimate its own pobability of detection and to automatically select the sample size that is equied fo a given pefomance level. Although the exact pefomance of the E is well established [], the mathematical elation among the vaious pefomance paametes usually involves complicated functions of two vaiables [], [5], such as the incomplete Gamma function [], whose implementation into a low-complexity device is pohibitively expensive. As a consequence, some appoximations have been poposed in the liteatue [5], [7] [], with the aim of educing the computational complexity while peseving a sufficient accuacy. These appoximations diffe in two main aspects. Fist, the assumed model fo the signal of the pimay (licensed use may be eithe deteministic [5], [8] [], o andom Gaussian [5], [9] o unknown [7]. Heein, we focus on the andom Gaussian model fo the pimay signal [], [5], [9]. Second, diffeent functions have been poposed fo appoximating the pobability density function of the test statistic [5], [7] []. The conventional way to appoximate the test statistic of the E invokes the cental limit theoem and consides the decision vaiable as Gaussian [], [5]. Although this appoach woks quite well when the numbe of samples is lage, the accuacy of the Gaussian appoximation wosens significantly when the sample size is low [9]. ote that the educed samplesize scenaio is of pactical inteest fo spectum sensing applications, because the sample size is constained by timebandwidth poduct consideations. As a consequence of the educed accuacy of the Gaussian appoximation, [9] poposed an impoved appoximation based on the cube tansfomation of a Gaussian andom vaiable. This tansfomation belongs to a wide class of appoximations, known as powe (o oot tansfomations [] [3], whee the powe exponent can be diffeent fom that used in [9]. This pape investigates powe tansfomations with geneic exponents in the context of low-complexity appoximation of the pefomance of the E. In addition, linea combinations of powe tansfomations ae also consideed [3], [], because of thei potentially inceased accuacy. Specifically, this pape deives new closed-fom expessions fo the pobability of detection as a function of the pobability of false alam, of the signal-to-noise atio (SR and of the sample size, fo powe tansfomations and fo suitable linea combinations of powe tansfomations. In addition, in the context of powe tansfomations, new closed-fom expessions fo the equied sample size ae also deived (as a function of the pobability of detection, of the pobability of false alam, and of the SR. Fo linea combinations of powe tansfomations, we popose a fast algoithm that exactly finds the equied sample size with logaithmic complexity. In ode to validate the accuacy of the poposed appoximations, we include some numeical esults that highlight the supeio accuacy of the linea combination appoaches. II. EERGY ETECTOR A. System Model We conside a cognitive adio netwok, whee seconday (unlicensed uses pefom spectum sensing in a peselected fequency band in ode to detect the possible pesence of one o moe pimay (licensed uses. We assume that the aggegate signal of the pimay uses is andom and zeo-mean Gaussian distibuted, and that the noise at the input of the seconday-use eceive is zeo-mean Gaussian as well. Afte baseband convesion and sampling, the complex-valued eceived signal y = [ y,..., y ] T can be expessed by 978--998-5-7//$3. IEEE 8
th Euopean Signal Pocessing Confeence (EUSIPCO y = α s+ w, ( whee s = [ s,..., s ] T epesents the pimay-use signal, assumed complex Gaussian with zeo mean and covaiance ΣS = σ SI, is the numbe of samples, w = [ w,..., w ] T is the seconday-use eceive noise, assumed complex Gaussian with zeo mean and covaiance ΣW = σ WI, independent fom the pimay-use signal s, and α {,} denotes the absence o pesence of s, efeed to as the H o H hypothesis, espectively. Using (, the E calculates the test statistic = = i= T( y y y ( and then compaes it to a theshold t : if T( y t the E decides that a pimay use is pesent, wheeas if T( y < t the E assumes that pimay uses ae absent. B. Exact Pefomance Analysis The exact pefomance of the E can be detemined by statistical analysis of the test (. Unde the H hypothesis, T ( y / σ W is a chi-squaed andom vaiable with degees of feedom (OF: this leads to a pobability of false alam [] i ( σ P = P{ T( y > t α = } = F t/, (3 W x/ ν F = [ Γ] ν e dν, ( whee Γ is the Gamma function []. Unde the H hypothesis, T ( y /( σs + σw is a chi-squaed andom vaiable with OF: hence, the pobability of detection is [] ( P = P{ T( y > t α = } = F t/ ( σ + σ. (5 S W By eliminating the theshold t fom (3 and (5, the eceive opeating chaacteistic (ROC is expessed by ( γ P = F ( + F ( P, ( whee γ = σs / σw is the SR and x = F ( p is the invese of p = F with espect to x. The ROC (, which summaizes the elation among the fou paametes P, P,, and γ, can be inveted in ode to find eithe the pobability of false alam P o the SR γ as a function of the othe thee paametes, as expessed by ( γ P = F ( + F ( P, (7 F ( P γ =. (8 F ( P Unfotunately, the equations ( (8 ae not suitable fo implementation into a low-complexity device, since they depend on complicated functions, such as the egulaized lowe incomplete Gamma function F ( x in ( and its invese F. Moeove, since both F ( x and F ae functions of two vaiables, stoing thei values into a lookup table (LUT would equie significant memoy ovehead. In addition, the equations ( (8 cannot be inveted with espect to : theefoe, if a sensing E device wants to calculate the minimum numbe of samples as a function of P, P, and γ, an iteative numeical appoach is equied fo the multiple evaluations of F. As a consequence, when a low-complexity device wants to automatically select eithe P, o P, o, some appoximations ae necessay. III. APPROXIMATE PERFORMACE COMPUTATIO A. Gaussian Appoximation In the spectum sensing liteatue, thee exist diffeent appoximations fo the pefomance of the E [5], [8] []. Fo both cases of deteministic signals and andom signals, the conventional appoach appoximates a chi-squaed andom vaiable with a Gaussian andom vaiable chaacteized by a suitable mean and vaiance [5], [8]. Using the statistical signal model of Section II.A, the conventional Gaussian appoximation coesponds to x F F = Q, (9 + / Qx ( = e ν dν. ( x π As summaized in [5] and [9], the Gaussian appoach of (9 poduces both simple appoximations of ( (8 and an analytical expession fo the equied sample size. Theefoe, these appoximated expessions can be easily implemented into a low-complexity E, povided that a LUT is available fo the evaluation of the one-dimensional Q-function in ( (and of its invese. On the othe hand, the accuacy of the Gaussian appoximation is quite low [5], especially fo small sample sizes [9]. otewothy, the sample size cannot be abitaily lage, due to time and bandwidth constaints. B. Powe Tansfomation In ode to incease the appoximation accuacy, [9] and [] discuss diffeent options that ae valid fo the cases of deteministic signals [9], [], and andom signals [9]. Specifically, the cube-of-gaussian appoach of [9] appoximates a chisquaed andom vaiable with the cube of a Gaussian andom vaiable with suitable mean and vaiance []. This cube-of- Gaussian appoach is quite pomising, since the obtained expessions ae moe accuate than the Gaussian appoximations, but with simila complexity [9]. In this section, we genealize the cube-of-gaussian appoach of [9] to accommodate diffeent powe exponents. The motivation fo this genealization lies in the statistical investigation done in [], which analyzes the Kullback-Leible (KL divegence of a powe-tansfomed chisquaed andom vaiable fom a Gaussian andom vaiable with suitable mean and vaiance: the esults of this analysis clealy show that the KL divegence is minimized fo powe exponents anging fom = 3 to =. A simila esult is obtained by [3], which compaes the cumulants of a powe-tansfomed chi-squaed andom vaiable with the cumulants of a Gaussian andom vaiable: if we match eithe the skewness o the kutosis of the two andom vaiables, we obtain intege powe exponents. As a consequence, heein we popose to appoximate a chi-squaed andom vaiable x, nomalized by the numbe of OF, with the th powe of a Gaussian andom vaiable, as expessed by x / m F F = Q, ( V whee m ( and V ( ae the mean and the vaiance of the Gaussian vaiable, expessed espectively by [], [3] 9
th Euopean Signal Pocessing Confeence (EUSIPCO m, V. ( Fo =, the appoximation ( ( coesponds to the conventional Gaussian appoximation (9, while fo = 3 coesponds to the cube-of-gaussian appoximation of [9]. Howeve, ( ( can be used also fo othe values of, which could also be nonintege. Fo the special cases = and =, othe appoximations fo the mean have been suggested in [] and [], as expessed by m (, m (. (3 We now deive new simplified expessions fo the pobability of detection P, the pobability of false alam P, and the equied SR γ. Fom (, we obtain F ( ( ( ( ( x F x = V Q x + m, ( which togethe with (, ( and (8 leads to Q ( P m P Q, + γ + γ V (5 m P Q + γq ( P + ( + γ, V ( ( ( + ( V Q P m γ. (7 V Q ( P + m ote that the esults in (7 ae valid only when the faction is lage than one. In addition, using ( ( and ( (, we can expess the sample size in closed fom as ( ( b + b +, (8 Q ( P. (9 b = b( γ, P, P = Q ( P + γ + γ Fo = and = 3, the expessions ( ( and (5 (9 coincide with those aleady obtained fo the conventional Gaussian appoximation [5] and fo the cube-of-gaussian appoximation [9], espectively. Theefoe, the poposed th powe tansfomation appoach genealizes the peviously poposed appoximations to a boade ange of values of the powe exponent. Fo the special cases = and = that use the mean (3 instead of that in (, the expession (8 is not valid and can be eplaced, espectively, by b + b, + +. ( Inteestingly, the expessions (5-( ae chaacteized by a low complexity, especially when is intege, since these expessions only equie few algebaic computations and one LUT fo the (invese Q-function. As a consequence, (5 ( can fom the basis fo incopoating some pefomance computation capabilities into a low-complexity E, in place of the complexity-demanding exact equations ( (8. The accuacy of the diffeent appoximations (5 ( will be evaluated in the numeical section. C. Linea Combination of Powe Tansfomations In ode to futhe educe the appoximation eo of the simplified pefomance analysis, we popose othe new tansfomations obtained by linealy combining diffeent powe tansfomations [3], []. Basically, this second poposed appoach appoximates as Gaussian the linea combination of diffeent powes of a chi-squaed andom vaiable. Aiming at good accuacy, suitable linea combinations can again be chosen by using KL divegence minimization appoaches o cumulant-matching methods. Aiming at low-complexity expessions, we focus on simple linea combinations obtained using few powe exponents and simple coefficients. Among the possible choices, two inteestingly simple expessions have been suggested in [3] and [], espectively: L, ( x /( + /( ( L 3,3, = x/ x/ 3 + x/, ( whee x is again a chi-squaed andom vaiable with OF; in ( (, L, ( x and (,3, x ae appoximated as Gaussian with mean and vaiance expessed espectively by 9 m, 5, V, (3 5 m,3, 8 V,3, 3 ( This coesponds to the appoximations L, m, F F = Q, (5 V, L,3, m,3, F F = Q. ( V,3, Since both ( and ( ae monotonic inceasing functions of x, the invese functions of (5 and ( exists, and, due to the simple fom of ( and (, these invese functions can be / found analytically. By substituting y = [ x/] in (, we obtain the quadatic equation L, = y + y, whose unique / positive solution fo y can be found as y = ( L, +. Hence, by ( and (5, we obtain F F, whee F = ( V, Q ( x + m, +, (7 and m ( and V (,, ae expessed by (3. Analogously, / by substituting z = [ x/] in (, we obtain the cubic 3 equation L,3, = z 3z + z, which has a unique eal solution fo z (since this cubic function is stictly inceasing. This eal solution can be calculated as in [] using Cadano s fomula, leading to 3 3,3,,3,,3,,3, z = + Δ ( L + a( L Δ( L a( L, (8 3 Δ ( L = [ a( L ] + (3 /, (9,3,,3, al ( = (L 5/8. (3,3,,3, Fom (, (, (8 (3, we obtain F F, whee
th Euopean Signal Pocessing Confeence (EUSIPCO /3 3 F = + [ A( x, ] + (3 / + A( x, /3 3 [ Ax (, ] + (3/ Ax (,, (3 3 5 Ax (, = V,3, Q ( x + m,3,, (3 8 and m ( and V (,3,,3, ae expessed by (. The invese appoximation functions (7 and (3 (3 can be used in ode to deive appoximated expessions fo the pobability of detection P, the pobability of false alam P, and the equied SR γ. Hence, the ROC can be appoximated as [ BP (, ] BP (, m, P Q +, (33 ( + γ V, ( + γ [ V,] V, ( / BP (, = V Q ( P + m +, (3,, 3 [ CP (, ] [ CP (, ] CP (, + m,3, 3 ( + γ 3 ( + γ ( + γ P Q, (35 V,3, /3 3 CP (, = + [ A( P, ] + (3 / + A( P, /3 3 [ A( P, ] (3/ A( P, +, (3 whee m (, V (, m (, V ( and Ax (,,,,3,,3, ae expessed by (3, ( and (3. The equied SR can be appoximated as BP (, CP (, γ, γ. (37 BP (, CP (, The expessions (33 (37, despite being slightly longe than the coesponding expessions (5 and (7, only equie algebaic opeations that can be easily done with low-complexity pocessing, and hence ae suitable fo device implementation. In ode to find the equied numbe of samples, we should analytically invet (33 (37 with espect to. Fo the linea combination in (, this pocedue leads to a quatic equation and hence can be solved analytically, wheeas, fo the linea combination in (, since the degee of the esulting equation is lage than fou, an analytical solution is not guaanteed due to the Abel-Ruffini Theoem []. As a consequence, we popose a low-complexity iteative algoithm that finds the equied numbe of samples with O(log ( complexity. Basically, the main idea behind the poposed algoithm is explained in the following. When is too small, the theshold of the E is not able to simultaneously ensue both the equied pobability of detection P and the equied pobability of false alam P, fo a given SR γ. Hence, when is too small, a theshold that ensues the given P is necessaily geate than a theshold that ensues the given P. Theefoe, the iteative algoithm looks fo the minimum such that the P -based theshold is less than the P -based theshold: this way, the P -based theshold suely guaantees that the pobability of detection is equal to (o lage than P. The pseudocode of the poposed iteative algoithm is included in the following. Iteative Algoithm to find the sample-size. Set. Compute = F ( P 3. Compute = ( + γ F ( P. If Then Go-to Line 5 5. Else While >. Set 7. Compute = F ( P 8. Compute = ( + γ F ( P 9. End-of-While. If == Then Go-to Line 5. Else Set step /. Set s sign 3. While step >. Set step / 5. Set + ssign step. Compute = F ( P 7. Compute = ( + γ F ( P 8. If Then Set ssign 9. Else Set s sign. End-of-If. End-of-While. Set + ( ssign + / 3. End-of-If. End-of-If 5. End-of-Algoithm The above iteative algoithm finds the equied sample size using two steps. Fist, the algoithm calculates a powe-oftwo uppe bound on by means of the While loop of Lines 5 9; concuently, a lowe bound on is obtained as half the uppe bound. Second, the algoithm applies a bisection method to efine the value of between the two bounds, by means of the While loop of Lines 3. Within the iteative algoithm, ( and epesent the scaled vesions of the P -based and P -based thesholds. The algoithm is valid fo both the linea combinations ( and (, since the computation of F in the Lines, 3, 7, 8, and 7 can be pefomed using eithe the appoximation (7 o (3. It can be shown that the poposed iteative algoithm evaluates Fn exactly max{ log,} times, whee is the final solution (we omit the poof fo the sake of bevity. IV. UMERICAL COMPARISO We compae the accuacy of the poposed appoximations by means of numeical esults. Fig. shows the elative eo on the pobability of detection P, as a function of the pobability of false alam P, when the SR is γ = 9 db and the sample size is = 5. Among the powe tansfomations, the cube-of-gaussian appoximation of [9] ( = 3 yields the lowest eo. Also the linea combinations L, and L,3, give accuate esults, which ae bette than fo = 3 ; howeve, the elative eo fo = 3 stays below fo any value of P. On the contay, the appoximations with =, = and =, can be inaccuate, especially fo the conventional Gaussian appoach ( =. In case of = and =, we have used the means expessed by (3; howeve, we have veified that using the mean in ( gives simila esults.
th Euopean Signal Pocessing Confeence (EUSIPCO The same conclusions of Fig. ae confimed by Fig., which exhibits the pobability of detection P, as a function of the SR γ, when P =. and = 5. On the othe hand, Fig. 3 displays the (signed eo on the estimation of the equied sample size as a function of γ, when P =.99 and P =.. Again, the thee appoximations = 3, L, and L,3, poduce vey accuate estimates (with mino eos only, while the othe appoximations oveestimate (o undeestimate the equied numbe of samples. ote that esults simila to Figs. 3 would be obtained fo othe values of P o γ. V. COCLUSIOS We have poposed new appoximations fo enegy detection sensos with self-pefomance computation capabilities. The poposed linea combination appoaches, due to thei supeio accuacy and low complexity, ae suitable fo device implementation. Futue wok may include the effect caused by impefect estimation of the SR [5]. Relative eo on the pobability of detection - - -3 - = (Gaussian = = 3 = L, L,3, -5-5 - -3 - - Pobability of false alam P Figue. Pobability of detection vesus the pobability of false alam. REFERECES [] T Yücek and H. Aslan, A suvey of spectum sensing algoithms fo cognitive adio applications, IEEE Commun. Suveys Tuts., vol., no., pp. 3, 9. [] E. Axell, G. Leus, E. G. Lasson, and H. V. Poo, Spectum sensing fo cognitive adio: State-of-the-at and ecent advances, IEEE Signal Pocess. Mag., vol. 9, no. 3, pp., May. [3] I. Sobon, P. S. R. iniz, W. A. Matins, and M. Velez, Enegy detection technique fo adaptive spectum sensing, IEEE Tans. Commun., vol. 3, no. 3, pp. 7 7, Ma. 5. [] S. M. Kay, Fundamentals of Statistical Signal Pocessing: etection Theoy. Pentice-Hall, 998. [5] R. Uma, A. U. H. Sheikh, and M. eiche, Unveiling the hidden assumptions of enegy detecto based spectum sensing fo cognitive adios, IEEE Commun. Suveys Tuts., vol., no., pp 73 78,. [] M. Abamowitz and I. A. Stegun, Handbook of Mathematical Functions: with Fomulas, Gaphs, and Mathematical Tables. ove Publications, 97. [7] J. E. Salt and H. H. guyen, Pefomance pediction fo enegy detection of unknown signals, IEEE Tans. Veh. Technol., vol. 57, no., pp. 39 39, ov. 8. [8] S. Ciftci and M. Tolak, A compaison of enegy detectability models fo spectum sensing, in IEEE Global Telecommun. Conf. (GLOBECOM 8, ew Oleans, ov./ec. 8. [9] L. Rugini, P. Banelli, and G. Leus, Small sample size pefomance of the enegy detecto, IEEE Commun. Lett., vol. 7, no. 9, pp. 8 87, Sep. 3. [] V. R. S. Banjade, C. Tellambua, and H. Jiang, Appoximations fo pefomance of enegy detecto and p-nom detecto, IEEE Commun. Lett., vol. 9, no., pp. 78 8, Oct. 5. [] E. B. Wilson and M. M. Hilfety, The distibution of chi-squae, Poc. at. Acad. Sci., vol. 7, no., pp. 8 88, ov. 93. []. M. Hawkins and R. A. J. Wixley, A note on the tansfomation of chi-squaed vaiables to nomality, Ame. Statist., vol., pp. 9 98, ov. 98. [3] M.. Goia, On the fouth oot tansfomation of chi-squae, Austal. J. Statist., vol. 3, no., pp. 55, 99. [] L. Canal, A nomal appoximation fo the chi-squae distibution, Comput. Statist. ata Anal., vol. 8, no., pp. 83 88, 5. [5] A. Maiani, A. Giogetti and M. Chiani, Effects of noise powe estimation on enegy detection fo cognitive adio applications, IEEE Tans. Commun., vol. 59, no., pp. 3 3, ec.. Relative eo on the pobability of detection Eo on the sample size - - = (Gaussian = -3 = 3 = L, L,3, - -5 - -5 5 5 SR γ (db 8 - - - -8 Figue. Pobability of detection vesus the SR. = (Gaussian = = 3 = L, L,3, - -5 - -5 5 5 SR γ (db Figue 3. Sample size vesus the SR.