Performance of Eigenvalue-based Signal Detectors with Known and Unknown Noise Level

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1 Performance of Eigenvalue-base Signal Detectors with Known an Unknown oise Level Boaz aler Weizmann Institute of Science, Israel Feerico Penna Politecnico i Torino, Italy feerico.penna@polito.it Roberto Garello Politecnico i Torino, Italy garello@polito.it Abstract In this paper we consier signal etection in cognitive raio networks, uner a non-parametric, multi-sensor etection scenario, an compare the cases of known an unknown noise level. The analysis is focuse on two eigenvalue-base methos, namely Roy s largest root test, which requires knowlege of the noise variance, an the generalize likelihoo ratio test, which can be interprete as a test of the largest eigenvalue vs. a maximum-likelihoo estimate of the noise variance. The etection performance of the two consiere methos is expresse by close-form analytical formulas, shown to be accurate even for small number of sensors an samples. We then erive an expression of the gap between the two etectors in terms of the signal-to-noise ratio of the signal to be etecte, an we ientify critical settings where this gap is significant e.g., small number of sensors an signal strength). Our results thus provie a measure of the impact of noise level knowlege an highlight the importance of accurate noise estimation. I. ITRODUCTIO Spectrum sensing is a central issue in cognitive raio CR) systems [] [3] an has attracte great research interest in the last ecae. In particular, sensing techniques base on the eigenvalues of the receive sample covariance matrix see, for instance, [4] []) recently emerge as a promising solution, as they o not require a priori assumptions on the signal to be etecte, an typically outperform the popular energy etection metho when multiple sensors are available. Eigenvaluebase etection EBD) schemes can be further ivie into two categories: methos that assume knowlege of noise level referre to as semi-blin [4]), an methos that o not assume this knowlege blin ). Methos belonging to the first class provie better performance when the noise variance is known exactly, whereas blin methos are more robust to uncertain or varying noise level. In this paper, we analyze the performance of two nearlyoptimal etection criteria in their respective categories: for known noise variance, the largest eigenvalue test, or Roy s largest root test RLRT), originally propose in [] an introuce in CR by [9] erive as a blinly combine energy etector ); for unknown noise variance, the test of the largest eigenvalue ivie by trace of the covariance matrix, introuce in CR by [7] as a generalize likelihoo ratio test GLRT) an previously appeare in signal processing [] an statistics literature [3], [4]. The main contributions of our analysis are: i) erivation of novel expressions for the See Sec. III for a more rigorous efinition. false-alarm an etection probabilities of the GLRT, accurate for finite numbers of samples an receivers; ii) analytical expression of the performance gap between the two etectors. Base on these results, we show that in some scenarios large number of sensors, relatively high signal-to-noise ratio) the performance improvement of RLRT over GLRT is marginal, whereas in other settings few sensors, low signal-to-noise ratio) the gap is significant. In other wors, exact knowlege of the noise level may result, in some cases, in an increase etection capability of the orer of several Bs. The paper is organize as follows: the signal moel is introuce in Sec. II; the consiere etection methos are presente in Sec. III; a comparative performance analysis of RLRT an GLRT is provie in Sec. IV; the formulas erive analytically are valiate by numerical simulations in Sec. V; Sec. VI conclues the paper. II. MODEL We consier a multi-sensor etection setting, where the etector constructs its test statistic from K sensors receivers or antennas) an time samples. Letyn) = [y n)...y K n)] T be the K receive vector at time n, where the element y k n) is the iscrete baseban complex sample at receiver k. Uner H, the receive vector consists of K complex Gaussian noise samples with zero mean an variance σ v yn) H = vn) ) where vn) C K,σ vi K K ). Uner H, in contrast, the receive vector contains signal plus noise yn) H = xn)+vn) = hsn)+vn) ) where sn) is the transmitte signal sample, moele as a Gaussian ranom variable with zero mean an variance σs, an h is the K unknown complex channel vector. The channel is assume to be memoryless an constant uring the etection time. Uner H, we efine the SR at the receiver as ρ E xn) E vn) = σ s h K, 3) where enotes Eucliean L ) norm. The Gaussian signal assumption simplifies the mathematical analysis an, as far as etection performance is concerne, turns out to be a reasonable approximation also for igitally moulate signals e.g., 4/8-PSK, 6-QAM etc.) after pulse-shape filtering an non-coherent sampling.

2 The receive samples are store by the etector in thek matrix Y [y)...y)] = hs+v 4) where s [s)...s)] is a signal vector an V [v)...v)] is a K noise matrix. The sample covariance matrix R is then efine as R Y Y H. 5) Let λ... λ K be the eigenvalues of R without loss of generality, sorte in ecreasing orer). In general, let T be the test statistic employe by the etector to istinguish between H an H : several possible test statistics will be examine throughout the paper. To make the ecision, the etector compares T against a pre-efine threshol t: if T > t it ecies for H, otherwise for H. As a consequence, the probability of false alarm is efine as an the probability of etection as P fa = PrT > t H ) 6) P = PrT > t H ). 7) Usually, the ecision threshol t is etermine as a function of the target false-alarm probability, to ensure constant falsealarm rate CFAR) etection. The corresponing etection probability or the misse-etection probability P m = P ) is also very important for CR networks where the interference cause by an opportunistic user to primary users must be very limite. As an example, the requirements impose by the WRA stanar [3] are P fa <. an P m <.. III. COSIDERED METHODS We restrict our analysis to non-parametric etection methos, i.e., which o not assume any prior knowlege about the signal to be etecte. We focus on the ifference in etection performance between the cases of known an unknown noise level σ v). A. Known oise Variance When testing a simple hypothesis H against a simple alternative H, in general, the most powerful test is given by the eyman-pearson P) likelihoo ratio [5]. In the consiere scenario, with unknown channel vector h, the eigenvalues of the sample covariance matrix R are sufficient statistics for the P test see [6] p., an [7] Sec.III-A), which can be written as LRT = pλ,,λ K H ) pλ,,λ K H ). 8) In the asymptotical regime, with given signal strength ρ an noise variance σ v, the above criterion can be shown [6], [7] to epen only on the largest eigenvalue λ ), i.e., it reuces to Roy s largest root test [], efine as T RLRT λ. 9) otice that energy etection ED), another commonly use criterion for known noise level, is suboptimal to RLRT in the sense of the eyman-pearson lemma. It can be written as T ED K K k=n= y k n) = Y F Kσ v where F enotes the Frobenius norm. Since Y F = try Y H ), we obtain that T ED = K K i= λ i. Therefore, asymptotically in, ED has reuce statistical power compare to 9) as it tests against the noise level the sum of all eigenvalues, instea of just λ. B. Unknown oise Variance When σ v is unknown, H an H are composite hypothesis an the P lemma oes not apply. A common proceure is the GLRT, obtaine from GLRT = sup h,σ v py H ) sup σ v py H ), ) which in our moel 3 is equivalent to see [8], Sec. II) ote that, since T GLRT = T GLRT λ ) KtrR). K i= λ K i i= = + λ i, λ λ the GLRT is equivalent up to a nonlinear monotonic transformation) to T GLRT = λ K K i= λ. ) i The enominator of T GLRT is the maximum-likelihoo ML) estimate of the noise variance assuming the presence of a signal [8], hence the GLRT can be interprete as a largest root test with an estimate ˆσ v instea of the true unknown) σ v. Remark: another popular etection criterion, propose in [5], is the eigenvalue ratio test T ERD = λ /λ K also calle maximum-minimum eigenvalue, or conition number test). Compare to ), this test is clearly suboptimal unlessk =. Base on the above consierations, RLRT an GLRT are taken as the reference etection methos for the cases of known an unknown noise variance, respectively. IV. DETECTIO PERFORMACE OF RLRT AD GLRT In [8] it is shown that, in the asymptotic regime,k with K/ fixe, the GLRT etection performance converges to that of RLRT. However, a natural question is: how ifferent is their performance for realistic values of K,? In other wors, what is the performance gap gaine in practical applications by exact knowlege of the noise level? To answer this 3 This expression of the GLRT is specific to the consiere moel single signal, unknown σ, unknown channel h), i.e., the same as in [8]. Other GLRTs for ifferent moels, e.g., unknown number of signals an generic signal covariance matrix, are erive in [].

3 Pr[Detection] K=6, =5, ρ = B UKOW noise variance KOW noise variance. RLRT GLRT. ED ERD Pr[False Alarm] Fig.. Simulate receiver operating characteristics ROC) curves for etection methos with known an unknown noise variance. question, we compare the performance of the two etectors by eriving analytical expressions for the etection probability 7), given a false-alarm rate 6)P fa = α an a SRρ. Our goal is to express the etection probability 7) of the two methos, P = PrT > tα) H,ρ), 3) where tα) is the ecision threshol such that P fa = α. Since we are intereste in low false alarm probabilities, we assume α. A preliminary performance assessment is provie by Fig., which compares the four aforementione methos RLRT, GLRT, ED an ERD) in a typical scenario for CR applications: small number of samples an receivers K = 6, = 5) an a single signal with low SR ρ = B). These simulation results illustrate the performance gap between RLRT an GLRT, as well as the suboptimality of ED compare to RLRT scenario of known noise level) an of ERD compare to GLRT scenario of unknown noise level). A. RLRT ) Setting the threshol: For RLRT, the ecision threshol tα) can be approximate thanks to the property that uner H, an in the joint limit,k, the ranom variable T RLRT asymptotically follows a secon-orer Tracy-Wiom TW) istribution [9]: [ ] TRLRT µ Pr < s F s), with suitably chosen centering an scaling parameters µ = [K/) / +] 4) = /3 [K/) / +][K/) / +] /3 5) refine expressions of µ an, proviing an improve convergence rate, are given in []). Therefore, TRLRT µ α = PrT RLRT > t) = Pr > t µ ) ) t µ F. Hence an approximate expression for the threshol of RLRT is t RLRT α) µ+f α) 6) where F is the inverse of the TW c..f. ) Detection probability: Uner H, the asymptotic istribution of λ in the joint limit,k is characterize by a phase transition phenomenon []. In the case of a single signal, the critical etection threshol for,k can be expresse irectly in terms of the SR as [] ρ crit =. 7) K This expression can be refine by aing correction terms for finite,k cf. [7], Eq. 5). If the SR ρ is lower than the critical value, the limiting istribution of λ uner H is the same as that of the largest eigenvalue unerh, thus nullifying the statistical power of a largest eigenvalue test. If ρ > ρ crit, on the contrary, the istribution of T RLRT is asymptotically Gaussian [7], [], with [ ] λ E = +Kρ) + K ) 8) Kρ [ ] λ Var = Kρ+), 9) σ v up to O/ ) terms. Therefore, the etection probability can be expresse as [ P RLRT) tα) Q Kρ+ K )] Kρ ) where Qz) = π z e x / x is the stanar Gaussian tail probability function. B. GRLT ) Setting the threshol: Asymptotically, as both, K, the ranom variable T GLRT also follows a secon-orer TW istribution [8], hence in first approximation t GLRT α) t RLRT α). However, as escribe in [3], this approximation is not very accurate for tail probabilities of T GLRT for small values of K. In [3] the following improve expression was erive: [ TGLRT µ Pr ] < s F s) K ) µ F s). Hence, for a require false alarm probability α, TGLRT µ α = PrT GLRT > t) = Pr > t µ ) ) t µ F + ) ) µ t µ F. K )

4 The above equation can be numerically inverte to fin the require threshol t GLRT α). ) Detection probability: To erive an explicit approximate expression for the etection performance [ of the GLRT ] uner K H, we note that K j= λ j = K rewrite the GLRT ) as with tα) = λ > tα) K j= λ j K λ + K j= λ j an ) K K t GLRT α) t GLRTα). 3) Assuming the presence of a sufficiently strong signal ρ > ρ crit ), the largest sample eigenvalue is with high probability) ue to a signal whereas the remaining eigenvalues, λ,...,λ K, are ue to noise. Let Z K K j= enote their mean. As iscusse in [4]Eq.), asymptotically in, the ranom variable ) Z is Gaussian istribute with variance O K ), an with a mean value that is slightly biase ownwars: [ ] Z E = ) Kρ+ +O Kρ. 4) σ v We then recall that λ / is asymptotically Gaussian istribute with mean an variance given by 8) an 9). For a large number of sensors K ), the fluctuations of Z are relatively much smaller than those of λ, hence we can approximate ) as +Kρ) + K Kρ λ j ) + +Kρ [ ] Z η > tα) E, 5) where η,) is a stanar Gaussian ranom variable. Therefore, we conclue that [ P GLRT) Q tα) Kρ+ ) K )] Kρ Kρ. 6) of µ 4) an 5), the threshol t RLRT α) 6) can be written as ) K t RLRT α) = + + s α K /6 +O 8) where s α = F α) = O). A similar expression hols for t GLRT α), with s α replace by s α,k,, foun by inverting ) an, in general, also having a weak epenence on, K. Inserting these expressions into 7) gives that, for /Kρ), the ifference is roughly [ K +Kρ K + s α,k, s ] α s α,k, + +O K /6 K )K /6 ). 9) We remark that for α, the ifference s α,k, s α is negative but quite small even for a small number of sensors. For large the first term is thus the ominant one. From this term we see that, as expecte, the performance gap between Roy s largest root test an the GLRT ecreases with a larger number of sensors, for which we have a better noise estimate, but at a relatively slow rate of O/ K). In particular, for a practical number of sensors, this gap is non-negligible. For example, for etection of weak signals Kρ ) with an array of K = 4 sensors, at a fixe alarm of α = %, an over a wie range of values of, the ifference is about.4 stanar eviations. Even with K = 6 sensors, the ifference is still about.5 stanar eviations. The performance gap between RLRT an GLRT can be evaluate also in terms of SR neee by the two etectors to achieve the same P. Let us set P RLRT) = P GLRT) ; after some algebra an neglecting O/) terms, we obtain ρ GLRT tα) t RLRT α) + tα) t RLRT α). 3) K t RLRT α) ow, from Eq. 3) it follows that tα) > t RLRT α): in fact, even though t GLRT α) is slightly lower than t RLRT α), the term K )/K t GLRT α)) > is largely ominant for α. Therefore, ρ GLRT >, i.e., RLRT achieves the same etection performance of the GLRT for signals with a lower SR. umerical examples are shown in next section. C. Performance Gap between RLRT an GRLT Comparison of 6) with ) shows quantitatively the performance gain obtaine using RLRT, i.e., knowing exactly the noise level, instea of GLRT, i.e., estimating the noise level from the sample eigenvalues. The ifference in stanar eviations between the etection probabilities in the two cases can be expresse, after some algebraic manipulations, as +Kρ K K t GLRT α) t GLRTα) t RLRT α) ) +O ) 7) This expression can be further simplifie when /Kρ). Uner this conition, an recalling the expressions V. SIMULATIO RESULTS In Fig. we compare the etection performance of RLRT an GLRT with the theoretical formulas ) an 6), as a function of the SR ρ an for a given false alarm rate of α =.5%. Fig. 3 illustrates the SR gap between GLRT an RLRT, i.e., the extra SR neee by the GLRT to achieve the same etection probability of RLRT. The figure shows that the gap increases for i) small number of sensors an ii) low. signal strength, in accorance to the theoretical formula 3). These conclusions highlight the key role that the noise variance estimation has for spectrum sensing, especially in challenging scenarios.

5 Pr[etection] K = 6, = 8, α =.5%. RLRT GLRT. RLRT Theory GLRT Theory SR ρ Fig.. Comparison of etection performance curves of RLRT an GLRT methos, from simulation an analytical expressions ) an 6). ρ GLRT / [B] SR gap between RLRT an GLRT to achieve equal P at α = % = B Simul. = B Theory = 5 B Simul. = 5 B Theory umber of sensors K Fig. 3. SR gap for GLRT to achieve the same P as RLRT, as a function of K for ifferent levels of signal strength ). Simulation results vs. analytical approximation 3). Fixe false alarm rate α = %, =. VI. COCLUSIOS The performance of eigenvalue-base etection has been investigate in this paper, consiering conitions of known an unknown noise level. Close-form expressions of etection probability of two best-performing etectors in their respective classes have been erive. These results provie a performance evaluation of two important etection techniques in cognitive raio systems, an can be use for an accurate esign of spectrum sensing parameters number of sensors an samples) given a target false alarm/etection rate impose by stanars e.g., P =.9 at SR= B with P fa =.). In aition, the quantitative characterization of the gap between GLRT an RLRT gives insight into the impact of knowing the exact noise level, which results in a significant performance improvement especially when the number of sensors is low. Base on this analysis, further research will be carrie out towars hybri etection methos where auxiliary time slots may be use to obtain a refine noise variance estimate an thus reuce the gap between GLRT an RLRT. REFERECES [] J. Mitola an G. Q. Maguire, Cognitive raios: making software raios more personal, IEEE Personal Commun., vol. 6, no. 4, pp. 3-8, 999. [] S. Haykin, Cognitive raio: brain-empowere wireless communications, IEEE Trans. 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