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where fH (x) is the joint pdf of the observed samples for The distribution of p conditioned on α is given by the Chi-hypothesis H. For a selected threshold λ, the detector declares Squared distribution, orH1 when L(x) ≥ λ, otherwise it declares H0 . The threshold α2can be computed by ﬁxing Pfa and inverting the cdf for the f (p|α) = (α2 p)N/2−1 exp{−α2 p/2}, (7) 2N/2 Γ(N/2)H0 (noise only) hypothesis. When noise and signal are both i.i.d. Gaussian, the energy and the marginal distribution f (p) therefore becomesof the signal, given by 1 ∞ f (p) = f (α)α2 (α2 p)N/2−1 exp{−α2 p/2} dα. N 2N/2 Γ(N/2) 0 (8) p= x2 , n (3) The idea of this paper is to choose a distribution for the inverse n=1 noise level f (α) that not only can be used to calibrate ais a sufﬁcient statistic. Detection based on p is known as practical system, but also has a simple form allowing (8) toenergy detection, and the distribution of p is given by the be derived in closed form.Chi-Squared distribution, allowing the required LRT threshold The lognormal distribution is often proposed for modelingand resulting detection performance to be computed in closed the variance of fading and noise processes, in which case f (σ) 2form. Given that the variance σ0 = Var(wn ) is known, the is expressed asenergy detector can eventually provide near-perfect detection 1 1if N is made large enough, even at very low SNR. fLN (σ) = √ exp − (log σ − µLN )2 /σLN . (9) 2 σ 2πσLN 2 Unfortunately, a practical system will only have an estimate 2of σ0 , and this imperfect knowledge is referred to as noise where µLN and σLN are the mean and standard deviation ofuncertainty (NU). The NU concept was identiﬁed and studied log σ. Expressed in dB units µLN = δµdB and σLN = δσdB ,in detail in [2], where noise variance is assumed to be conﬁned where δ = log(10)/20. Letting α = 1/σ, (9) can be 2 2to the interval [σlo , σhi ] but otherwise unknown. In this case, transformed toworst-case detection performance for the N-P detector can be 1 1 fLN (α) = √ exp − (log α + µLN )2 /σLN , 2computed by assuming ασLN 2π 2 (10) 2 2 σhi , under H0 , which differs from (9) only in the sign of µLN . A major draw- σ0 = 2 (4) σlo , under H1 , back of the lognormal distribution, however, is that closed- form analysis is often difﬁcult.thus providing the minimum separation of the H0 and H1 pdfs. For small levels of noise uncertainty, we consider a muchFor a given noise interval, as the SNR is lowered a threshold is simpler model, where f (α) is assumed to be Gaussian whichreached below which the worst-case energy detector exhibits is ﬁt to (10) using closed-form expressions for the mean andPd < Pfa regardless of the number of samples. This complete variance of (10) given bydetection failure is referred to as the SNR wall. 2 µα = E {α} = exp{−µLN + σLN /2}, (11) III. N OISE U NCERTAINTY M ODELING σα = Std{α} = 2 [exp(σLN ) − 1] exp(−2µLN + 2 σLN ), (12) The main idea of this paper is to overcome the SNR where Std(·) denotes standard deviation. The pdf f (α) is thenwall phenomenon by more detailed modeling of the noise given byuncertainty. In this work, noise and signal are modeled as conditional Gaussian processes where a single real sample xn 1 1 (α − µα )2 √ exp − , α > 0, f (α) = 2 σα2has the conditional distribution Cα 2πσα 0, otherwise, α f (xn |α) = √ exp{−α2 x2 /2}, n (5) (13) 2π where the rescaling constant Cα = erfc[−µα /( 2σα )]/2 2α = 1/σ, and σ 2 is the variance. Note that the choice of results from the truncation of the left tail of the Gaussian atusing α rather than σ as the modeled noise parameter in this α = 0 and erfc(·) is the complementary error function. Notework avoids having integration variables in the denominator, Cα ≈ 1 is omitted from later derivations, but it should bethus simplifying closed-form analysis. Given an i.i.d. process included if exact expressions are required.where α is ﬁxed for a short time consisting of N samples, the A. Single Sample: Marginal Distributionmarginal pdf of the vector x is Assuming the Gaussian model for f (α), the marginal distri- ∞ N bution of a single real sample is f (x) = 0 f (α)f (x|α)dα, 1 ∞ α2 f (x) = f (α)αN exp − x2 n dα, (6) or (2π)N/2 0 2 n=1 1 ∞ 2 2 2 2 f (x) = αe−α x /2 e−(α−µα ) /(2σα ) dα, (14)where f (α) is the pdf of the unknown noise parameter α. Since 2πσα 0the energy p = N x2 is a sufﬁcient decision statistic here, 1 ∞ 2 n=1 n = αe−[aα −bα+c] dα, (15)we concentrate on this parameter. 2πσα 0 288
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where respectively, followed by the substitution u = c1 α2 : c2 x2 1 2 a= + 2, (16) I1 = (−1)N −k αN −k e−c1 α dα, (35) 2 2σα 0 2 c1 c2 b = µα /σα , (17) (−1)N −k 2 = u(N −k−1)/2 e−u du, (36) c = µ2 /(2σα ). α 2 (18) 2cLk 1 0The integral is of the form of the error function, which can Γ(Lk , c1 c2 )Γ(Lk ) 2be obtained by completing the square, resulting in where Lk = (N + 1 − k)/2, and 2 x e−c3 e−c1 c2 π √ 1 f (x) = + erfc(− c1 c2 )c2 , (19) Γ(a, x) = e−t ta−1 dt (37) 4πσα c1 c1 Γ(a) 0 is the incomplete Gamma function. Similarly,where ∞ 2 1 I2 = αN −k e−c1 α dα = Γ(Lk ). (38) c1 = a, (20) 0 2cLk 1 c2 = µα /(σα x2 + 1), 2 (21) Combining results in 1 µ2α 1 N c3 = 2 1− 2 2 . (22) c0 e−c3 N ck 2 σα σα x + 1 f (p)= 2 Γ(Lk ) 1+(−1)N −k Γ Lk , c1 c2 . k cL k 2 2 1 k=0B. Multiple Samples: Energy Distribution (39) For multiple independent samples, we will consider only C. Comparison of Gaussian and Lognormalthe distribution of the energy p, which is a sufﬁcient statisticwhose distribution (8) becomes It is instructive to consider in what situations the Gaus- sian assumption for f (α) provides a reasonable model. Fig- 1 ure 1 plots f (α) side-by-side with f (x) (single sample) forf (p) = (23) 2N/2 Γ(N/2) µdB = 0 dB and different values of σdB ∈ {0.5 dB, 1.0 dB, ∞ 1 2 2 2 2.0 dB}. A log scale is used for f (x) to highlight the small × √ e−(α−µα ) /(2σα ) α2 (α2 p)N/2−1 e−α p/2 dα, 0 2πσα differences in the distribution tails. For small and moderate (24) levels of noise uncertainty, the Gaussian approximation for ∞ 2 f (α) is very close to the lognormal model. Also, for low noise = c0 αN e−(aα −bα+c) dα, (25) uncertainty, the small mismatch in f (α) results in negligible 0 ∞ 2 error in the marginal density f (x). For larger noise uncertainty, = c0 αN e−c1 (α−c2 ) −c3 dα, (26) signiﬁcant differences in the two models can be seen. 0 ∞ 2 IV. D ETECTION W ITH NU = c0 e−c3 (α + c2 )N e−c1 α dα, (27) −c2 In this section we demonstrate with a simple example how N ∞ having a model of the noise uncertainty can increase detection N k 2 = c0 e−c3 c2 αN −k e−c1 α dα, (28) performance and remove the SNR wall. In this example, k −c2 k=0 α is considered to be an unknown parameter following a I lognormal distribution with µdB =0 dB and σdB =1 dB, which is subsequently ﬁt using a Gaussian distribution. Signal andwhere noise variance are assumed to be equal (SNR=0 dB) and p 1 N = 20 samples are used for detection. a = c1 = + 2, (29) 2 2σα First, a worst-case analysis like that presented in [2] is 2 considered. Here, only bounds are set on the noise level, and b = µα /σα , (30) c= µ2 /(2σα ), 2 (31) the structure of the noise variation is ignored. It is assumed α N/2−1 that the worst-case values for α are µα ± 1.5σα , which is p conservative since the α will sometimes fall outside of these c0 = √ (32) 2 N/2 Γ(N/2) 2πσ α bounds. Figure 2 shows the Chi-Squared pdfs for the worst 2 c2 = µα /(σα p + 1) (33) case assumption (4), indicating that detection is not possible 1 µ2 1 since the H0 curve is actually to the right of the H1 curve. α c3 = 2 1− 2 . (34) Next, the structure of the noise error in (39) is taken into ac- 2 σα σα p + 1 count, producing the pdfs in Figure 3 and indicating sufﬁcientThe integral I = I1 + I2 can be evaluated by letting I1 and I2 separation for useful detection. Figure 4 shows the detectionbe the contribution from α on the negative and positive axes, performance from the worst case analysis and the case that 289
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7 0 (a) 0.5 dB Gauss 6 LogNorm -10 5 log f (x) 4 -20 f (α) 3 -30 2 MC 1 -40 Gauss No NU 0 -50 0.7 0.8 0.9 1 1.1 1.2 1.3 -10 -5 0 5 10 α x 3.5 0 (b) 1.0 dB Gauss 3 LogNorm -10 2.5 log f (x) 2 -20 f (α) 1.5 -30 1 MC 0.5 -40 Gauss No NU 0 -50 0.6 0.8 1 1.2 1.4 1.6 -10 -5 0 5 10 α x 1.8 0 1.6 (c) 2.0 dB Gauss LogNorm 1.4 -10 1.2 log f (x) -20 f (α) 1 0.8 0.6 -30 0.4 MC -40 Gauss 0.2 0 No NU 0 0.5 1 1.5 2 2.5 -50 -10 -5 0 5 10 α xFig. 1. A comparison of the lognormal distribution on α and a Gaussian approximation of the same distribution for different levels of noise uncertainty:σdB ∈ {0.5 dB, 1.0 dB, 2.0 dB}. The noise uncertainty pdf f (α) is plotted on the left, and the corresponding marginal single sample pdf f (x) is plottedon the right for each level of noise uncertainty. MC gives the results of Monte-Carlo simulations of the exact distribution with lognormal NU, compared withthe Gaussian NU approximation (Gauss), and no noise uncertainty (No NU).exploits the noise error pdf. In the worst-case analysis, the the primary is present or not. However, since most of the noiseSNR wall has clearly been crossed, since Pd < Pfa . On the in a true receiver comes from the front-end low-noise-ampliﬁerother hand, exploiting the known statistics of the noise error (LNA), the simple architecture depicted in Figure 5(a) can beallows useful detection even when the exact noise level is used for noise calibration. To learn the noise distribution, theuncertain. cognitive radio node periodically switches the receive channel away from the antenna to the matched termination to sample V. N OISE C ALIBRATION M EASUREMENT and learn the noise distribution. In this section we present the results of an experiment that This idea was tested using the experimental setup showntests the possibility of energy detection at low SNR using schematically in Figure 5(b). The setup employs a custompractical hardware. As indicated in [2], noise calibration can be multiple-input multiple-output (MIMO) channel sounder thatdifﬁcult in traditional wireless receivers where it is unknown if is basically equivalent to that presented in [4], with the 290
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0.06 1 fH0 (p) 0.9 0.05 fH1 (p) 0.8 0.04 0.7 0.6f (p) Pd 0.03 0.5 0.4 0.02 0.3 0.01 0.2 Modeled NU 0.1 Worst Case NU 0 0 0 20 40 60 80 100 120 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 p PfaFig. 2. Distribution of f (p) for noise (H0 ) and signal plus noise (H1 ) Fig. 4. Probability of detection Pd versus probability of false alarm Pfa forassuming a worst-case model on the noise variation the worst-case assumption and the proposed NU model 0.06 fH0 (p) 0.05 fH1 (p) 0.04f (p) 0.03 0.02 0.01 0 0 20 40 60 80 100 120 pFig. 3. Distribution of f (p) for noise (H0 ) and signal plus noise (H1 ) usingthe proposed Gaussian NU modelexception that custom FPGA-based data acquisition is usedin the present system. The transmit (TX) node simulates the primary user, where abaseband Gaussian signal with a ﬂat W = 20 MHz bandwidthis generated in 100 µs frames with the arbitrary waveformgenerator (AWG), up-converted to 2.55 GHz, power ampliﬁedto 23 dBm, and fed to either the active transmit channel (TX1)or a matched load (TX0). The channel is a simple direct cable Fig. 5. Measurement setup for experimental study: (a) envisioned cognitiveconnection from the transmitter to receiver, where different radio employing noise calibration, (b) channel sounder based acquisitionﬁxed attenuators are inserted giving loss L and producing system for experiment, (c) acquisition frame structuredifferent SNR levels at the receiver. The receive (RX) node simulates the cognitive radio thatemploys a switch to feed its single receive chain either from Four phases are used in each record to probe all four switchthe channel (RX1) or from a matched load (RX0). The receive combinations. Within a single phase, the channel is acquiredchain consists of a 40 dB wideband LNA, down-conversion for T = 100 µs followed by a delay of TD , where TD = 0 canto a 50 MHz IF, followed by FPGA-based fs = 200 MS/s be used for back-to-back acquisition. During post-processing,data-acquisition. For this experiment, the raw IF samples are only Ns samples within each acquisition window are used,stored, passed to a PC, down-converted to complex baseband, thus spanning time Ts = Ns /fs in order to simulate differentand ﬁltered (20 MHz bandwidth) using MATLAB before integration windows in a cognitive radio energy detector. Weperforming energy detection. will denote the nth ﬁltered complex-baseband sample, of the A total of M data records are acquired during each mea- kth phase, in the mth record as xm,k,n . The four phases aresurement, where the mth record is depicted in Figure 5(c). denoted symbolically as k ∈{TX1 RX1, TX0 RX1, TX1 RX0, 291
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2 1 1.8 TX1 RX1 800 6400 1.6 TX0 RX1 0.8Prob. Density 1.4 TX1 RX0 TX0 RX0 100 1.2 0.6 Pd 1 0.8 0.4 0.6 0.4 0.2 0.2 0 0 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 Norm. Energy PfaFig. 6. Empirical pdfs (solid lines) for the four measurement phases Fig. 7. Probability of detection (Pd ) versus probability of false alarm (Pfa )with parameters: TD =0, L=110 dB (ρ = −6 dB), Ns = 100, and for TD = 0, L = 110 dB (ρ = −6 dB), and Ns ∈{100,800,6400}.M = 800 realizations. Fitted stationary Chi-Squared distributions are alsoshown (dashed lines). 2 1.8 TX1 RX1 1.6 TX0 RX1TX0 RX0}. Prob. Density 1.4 TX1 RX0 The energy in each record is computed using a variable TX0 RX0window size Ns according to simple integration, or 1.2 1 N1 +Ns 0.8 pm,k = |xm,k,n |2 , (40) 0.6 n=N1 +1 0.4where N1 = 50 samples are always skipped at the beginning 0.2of each frame to avoid artifacts from the switching operations. 0Empirical distributions of the noise and signal plus noise 0 0.5 1 1.5 2 2.5 3energy are ﬁnally computed with a histogram using the M Norm. Energyenergy snapshots in (40). Empirical signal and noise pdfs are also compared with ideal Fig. 8. Empirical pdfs (solid lines) for the four measurement phases withChi-Squared pdfs with N = 2W Ns /fs degrees of freedom parameters: TD =24 ms, L=110 dB (ρ = −6 dB), Ns = 100, and M =with sample variance estimated according to 600 realizations. Fitted stationary Chi-Squared distributions are also shown (dashed lines). M 2 1 σk = pm,k . (41) Ns M m=1 measuring the channel with the transmitter not present (TX0The SNR (ρ) in dB is estimated at an attenuation level of RX1). Finally, good separation of the pdfs for the H0 phaseL =80 dB (high SNR) using and H1 phase (TX1 RX1) is demonstrated, indicating that 2 useful detection with the direct noise measurement is possible. σTX1RX1 ρ(L = 80 dB) = 10 log10 2 , (42) Figure 7 plots probability of detection Pd versus probability σTX1RX0 of false alarm Pfa for the same case, computed directlyand SNR at lower attenuation levels is computed using ρ(L) = from the empirical pdfs, where phase TX1 RX0 is used toρ(L = 80 dB) + 80 − L. Note that for convenience, pdfs estimate hypothesis H0 . The result shows that when the noiseare normalized with respect to the expected noise energy distribution is measured, near perfect detection is possible if 2σTX1RX0 Ns . the sample size is made large enough. Figure 6 plots empirical noise/signal pdfs for the four phases Figure 8 plots the empirical pdfs of the four phases forcompared with Chi-Squared pdfs for TD = 0 (back-to-back ac- the same case, but with a longer delay between acquisitionsquisition), L = 110 dB (ρ = −6 dB), and Ns = 100 samples TD = 24 ms. Although we expected that increased noise(N = 40). First, the simple Chi-Squared distribution (no noise uncertainty would result and spread the pdfs for the longeruncertainty) provides a good ﬁt to the empirical pdfs for all acquisition time of 4(TD + T )M = 58 s, they still exhibitcases, suggesting that the noise parameter α is fairly constant an excellent ﬁt to the simple Chi-Squared distribution withover the total acquisition time of 4T M = 320 ms. Also, there no noise uncertainty. Possible reasons that the noise processis no apparent difference in the energy pdfs for the three noise- is so stable is that an instrument-grade LNA is used and theonly (H0 ) phases, indicating that having the RX connected to temperature in the channel sounder was likely very constant.a matched load (TX1 RX0 and TX0 RX0) is equivalent to In the future, we intend to study inexpensive consumer-grade 292
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18 more likely to be observed. However, these results suggest 16 TX1 RX1 TX0 RX1 that detection at SNRs of -16 dB or lower should be possible 14 using simple noise calibration.Prob. Density TX1 RX0 12 TX0 RX0 10 VI. C ONCLUSION 8 Although noise uncertainty can be a severe impairment to 6 robust detection in cognitive radio at low SNR, this paper 4 has shown that the SNR wall effect can be overcome by proper speciﬁcation of the noise variation. A closed-form pdf 2 0 that assumes a Gaussian distribution for the inverse noise 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 standard deviation was derived and it was shown that the Norm. Energy model provides a good ﬁt to the more commonly assumed lognormal pdf for low to moderate noise uncertainty. A simulation example was presented, conﬁrming that by properlyFig. 9. Empirical pdfs (solid lines) for the four measurement phases withparameters: TD =24 ms, L = 120 dB (ρ = −16 dB), Ns = 6400, and modeling the noise uncertainty, the SNR wall phenomenon canM = 600 realizations. Fitted stationary Chi-Squared distributions are also be avoided, providing useful energy detection performance atshown (dashed lines). very low SNR. 1 Initial experimental measurements were also presented that explore energy detection performance in a true receiver using 0.8 practical hardware. Detection performance based on empir- ically measured pdfs indicated that useful detection down 6400 to at least -16 dB is possible with energy detection using 0.6 800 sufﬁcient integration time. Measured noise distributions overPd 100 0.4 short (0.3 s) and moderate (58 s) acquisition times showed negligible deviation from a Chi-Squared distribution, suggest- 0.2 ing that the noise level in our system is very stable and that detailed modeling of the noise uncertainty is unnecessary for sub-minute integration times. 0 0 0.2 0.4 0.6 0.8 1 Since only an expensive instrument-grade LNA was consid- Pfa ered in this work, future work will explore noise variation in low-cost commercial-grade ampliﬁers that may exhibit noise statistics that are much less stable. Additionally, due to existingFig. 10. Probability of detection (Pd ) versus probability of false alarm (Pfa )for TD = 0, L = 120 dB (ρ = −16 dB), and Ns ∈{100,800,6400} samples limitations of the system acquisition ﬁrmware, we could not explore the performance of very long acquisition times, which will also be the subject of future investigations.LNAs under varying environmental conditions, where noiseuncertainty is more likely to occur and noise calibration is R EFERENCESmore challenging. [1] Mitola, J., III and Maguire, G. Q., Jr., “Cognitive radio: Making software Figure 9 shows pdfs for the case of lower SNR L = 120 dB radios more personal,” IEEE Personal Commun. Magazine, vol. 6, pp.(ρ = −16 dB), TD = 24 ms, and Ns = 6400 samples. 13–18, Aug. 1999. [2] R. Tandra and A. Sahai, “SNR walls for signal detection,” IEEE J.Even though the separation of the pdfs is poorer in this case, Selected Topics Signal Processing, vol. 2, pp. 4–17, Feb. 2008.the detection performance plotted in Figure 10 indicates that [3] S. M. Kay, Fundamentals of Statistical Signal Processing: Detectionthe SNR wall effect is avoided, and useful detection is still Theory, vol. II, Prentice-Hall, 1998. [4] B. T. Maharaj, J. W. Wallace, M. A. Jensen, and L. P. Linde, “A low-possible with a long enough integration time. cost open-hardware wideband multiple-input multiple-output (MIMO) In the future, the system will be modiﬁed to allow longer wireless channel sounder,” IEEE Trans. Instrum. Meas., vol. 57, pp.integration times where the effects of noise uncertainty are 2283 – 2289, Oct. 2008. 293
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