Wireless Synchronization Preamble Detection Scheme ... - Elsevier

Ingeniería Investigación y Tecnología, volumen XVI (número 3), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM (artículo arbitrado) doi: http://dx.doi.org/10.1016/j.riit.2015.05.011

Wireless Synchronization Preamble Detection Scheme Using Bispectra-Based Statistics in the Presence of Stationary Noise Esquema de detección del preámbulo de sincronización de los sistemas inalámbricos basado en la estadística del biespectro en presencia de ruido estacionario Aguilar-Torrentera Jorge State University of Nuevo León, Monterrey E-mail: [email protected]

Information on the article: received: March 2014, reevaluated: July 2014, accepted: November 2014

Abstract Higher-order statistical analysis has been applied in many areas of commu—’ŒŠ’˜—ȱœ¢œŽ–œǰȱœžŒ‘ȱŠœȱ‹•’—ȱŽšžŠ•’£Š’˜—ǰȱ™ŠĴŽ›—ȱ›ŽŒ˜—’’˜—ȱŠ—ȱŒ‘Š—nel estimation; among others. In this paper, wireless preamble sequences in additive noise are processed to reduce the signal to noise ratio of synchronization signals thereby improving interconnectivity. For such end, detection œ›žŒž›Žœȱ žœ’—ȱ –ŠŒ‘Žȱ ꕝŽ›’—ȱ ’—ȱ ‘Žȱ ˜›–ȱ ˜ȱ ™Šœœ’ŸŽȱ ’–™•Ž–Ž—Š’˜—ȱ along with cumulant-based processing are proposed. A framework based on ‹’œ™ŽŒ›Šȱ’œȱžœŽȱ˜ȱ˜‹Š’—ȱ‘Žȱ™›˜‹Š‹’•’¢ȱ˜ȱŽŽŒ’˜—ȱŒ˜—ę›–’—ȱœžŒŒŽœœž•ȱ synchronization in the presence of Gaussian and non-Gaussian noise. Signal bispectra and power of the tests are estimated over a number of samples within the length of standard wireless preamble signals.

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Resumen El análisis estadístico de alto orden ha sido aplicado en muchas áreas de los sistemas de comunicaciones, como igualación ciega, reconocimiento de patrones y estimación del canal; entre otros. En este artículo se procesan las secuencias del preámbulo mezcladas con muestras de ruido, con el objeto de reducir la relación señal a ruido de las señales de sincronización inalámbricas y por este medio mejorar la interconec’Ÿ’Šǯȱ Žȱ ™›˜™˜—Ž—ȱ Žœ›žŒž›Šœȱ Žȱ ŽŽŒŒ’à—ȱ ‹ŠœŠŠœȱ Ž—ȱ ꕝ›Š˜ȱ ŠŒ˜™•Š˜ȱ Ž—ȱ œžȱ forma pasiva conjuntamente con el procesamiento basado en cumulantes de alto orden. Se desarrolla un esquema basado en el biespectro para obtener la probabilidad de ŽŽŒŒ’à—ȱŒ˜—ę›–Š—˜ȱœ’—Œ›˜—’£ŠŒ’à—ȱŽ—ȱ™›ŽœŽ—Œ’ŠȱŽȱ›ž’˜ȱ Šžœ’Š—˜ȱ¢ȱ—˜Ȭ Šžœ’Šno. El biespectro de la señal y la prueba de hipótesis se estiman con un número de muestras cuya longitud es menor o igual que la longitud de las secuencias de sincronización en una trama.

Descriptores: ‡ ‡ ‡ ‡ ‡ ‡ ‡

sistemas inalámbricos filtro acoplado interferencia de banda amplia ruido no-Gaussiano biespectro ruido estacionario radio receptor

Wireless Synchronization Preamble Detection Scheme Using Bispectra-Based Statistics in the Presence of Stationary Noise

Introduction Wireless systems need to satisfy simultaneously the communication and computing demands through di쎛Ž—ȱ›Š’˜ȱ™›˜˜Œ˜•œǯȱ‘ŽȱŒž››Ž—ȱŽ—Ž—Œ¢ȱ’œȱ˜ Š›œȱ the usage of multiple-radio systems that preserve good performance through a range of synchronization techniques required for supporting coherent operation. Currently, orthogonal multicarrier frequency division multiplexing and multicarrier (code division multiple access) CDMA are considered for transmission wireless standards (Hanzo et al., 2003; Keller and Hanzo, 1996). In a commonly found environment surrounding wireless systems interference from radio systems crowding into small areas as well as incidental radiation ›˜–ȱ Ž•ŽŒ›’ŒŠ•ȱ ŽŸ’ŒŽœȱ Š›’œŽǯȱ ‘ŽœŽȱ œ˜ž›ŒŽœȱ ˜ȱ ’—Ž›Žrence, which are coupled to the radio receiver in the form of additive non-Gaussian noise, become a foremost impairment. Wireless systems are designed to ˜™Ž›ŠŽȱ –˜œȱ ŽěŽŒ’ŸŽ•¢ȱ ŠŠ’—œȱ ‘Žȱ žœžŠ••¢ȱ Šœœž–Žȱ additive white Gaussian noise (AWGN) and therefore non-Gaussian coupled to the radio receiver degrades greatly the interconnectivity. Additionally, the nonGaussian nature of the noise is strengthened by a large œ˜›ȱ ˜ȱ —ŽŠ›Ȭꎕȱ ’—Ž›Ž›Ž—ŒŽœȱ ›˜–ȱ Œ˜–™ž’—ȱ ™•Šforms that are currently integrated with wireless œž‹œ¢œŽ–œȱǻ•ŠĴŽ›¢ȱet al., 1999). In this article, the detection of wireless synchronization sequences using higher-order statistical analysis (HOSA) is proposed. –˜—ȱ˜‘Ž›ȱŠ™™•’ŒŠ’˜—œǰȱ ȱ’œȱŽěŽŒ’ŸŽȱ’—ȱŽŸŽ•˜ping algorithms to counteract channel distortion (Nikias and Petropulu, 1993) and also for detecting and classifying signals that features higher-order polyspec›Šȱ Ž—Ž›¢ȱ ǻ ’Š——Š”’œȱ Š—ȱ œŠœŠ—’œǰȱ ŗşşŖǼǯȱ Ž›Žǰȱ ˜›ȱ ‘Žȱꛜȱ’–Žǰȱ‘’‘Ž›Ȭ˜›Ž›ȱŒž–ž•Š—ȱ™›˜ŒŽœœ’—ȱ’œȱ™›˜posed to detect the preamble of wireless 802.11 standards. Synchronization of wireless systems is critical. Keller and Hanzo (1996) developed synchronization algorithms relying on the autocorrelation function between received and stored samples of synchronization frames. ‘Žȱ™Ž›’˜’Œ’¢ȱ˜ȱ‘Žȱœ¢—Œ‘›˜—’£Š’˜—ȱ™ŠĴŽ›—ȱ’—ȱŒ˜—œŽcutive frames allows frequency tracking and frame synŒ‘›˜—’£Š’˜—ǯȱ˜—Ž‘Ž•Žœœǰȱ‘ŽœŽȱž—Œ’˜—œȱŠ›Žȱ’—ĚžŽ—ŒŽȱ by additive Gaussian noise at lower signal-to-noise ratios (SNR’s) (Keller and Hanzo, 1996) and, consequently, wireless range is diminished. In dense networks preamble signals create interference that is handled in the form of Gaussian interference (Sridhara, 2007; Nagaraj et al., 2009). In Nagaraj et alǯȱǻŘŖŖşǼȱ’쎛Ž—’Š•ȱŒ˜››Ž•Š’˜—ȱ¢™Žȱ preamble detection sets time coherence to perform statis’ŒŠ•ȱ–Ž›’ŒȱŒ˜–™žŠ’˜—ȱŠ—ȱ̊ȱ‘Žȱ™›ŽœŽ—ŒŽȱ˜›ȱŠ‹œŽ—-

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ce of preamble at the receiver input. For our proposal, higher-order cumulant processing would be helpful in discriminating the presence of multiple wireless sysŽ–œǰȱ‘˜ ŽŸŽ›Dzȱ•ŠŽ—Œ¢ȱ˜ȱ–ŠŒ‘ŽȱꕝŽ›’—ȱ™›˜ŒŽŽ’—ȱ by cumulant-based processing needs to be addressed ŒŠ›Žž••¢ǯȱ‘’œȱ’œœžŽȱ’œȱ‹Ž¢˜—ȱ‘ŽȱœŒ˜™Žȱ˜ȱ‘Žȱ™›ŽœŽ—ȱ article. Here, receiver schemes that detect the presence of wireless systems are proposed and their performance is analyzed statistically considering stationary Gaussian and non-Gaussian interference. First, a receiver based on ™Šœœ’ŸŽȱ–ŠŒ‘ŽȱꕝŽ›’—ȱ˜›ȱœ¢—Œ‘›˜—’£Š’˜—ȱ˜ȱ ’›Ž•Žœœȱ œ¢œŽ–œȱ ’œȱ ’—›˜žŒŽǯȱ ‘Ž—ǰȱ ‘Žȱ Œ˜››Ž•Š’˜—ȱ ™›˜™Ž›’Žœȱ of the cumulants and the proposed detectors are analy£Žǯȱȱ‘Žȱ™›˜ŒŽœœ’—ȱ˜ȱ ’›Ž•Žœœȱ™›ŽŠ–‹•Žȱžœ’—ȱŠȱ–ŠŒ‘Žȱ ꕝŽ›ȱ ™›ŽŒŽŽȱ ‹¢ȱ ‘’‘Ž›Ȭ˜›Ž›ȱ Œž–ž•Š—ȱ Œ˜–Ȭ ™žŠ’˜—ȱ˜›ȱ’쎛Ž—ȱœŠ–™•Žȱœ’£ŽœȱŠ—ȱœ’—Š•Ȭ˜Ȭ—˜’œŽȱ ratios are investigated. Later, a hypothesis testing framework that obtains the preamble detection statistics for a given false alarm probability is developed. Finally, Œ˜—Œ•žœ’˜—œȱ˜ȱ‘Žȱ™›˜™˜œŽȱ–ŠŒ‘ŽȱꕝŽ›ȱœ›ŠŽ¢ȱŠ›Žȱ outlined.

Preamble bispectra Wireless transmission standards 802.11 specify message and synchronization frames for the Medium Access Control and Physical layers (Agere Systems Inc., 2004). Wireless standards 802.11a and 802.11b are based on the principles of Orthogonal Frequency Division Multiplexing (OFDM) and multicarrier code division multiple access (MC- CDMA) systems; respectively (Hanzo et alǯǰȱŘŖŖřǼǯȱ‘Žȱ›Š—œ–’ĴŽȱ›Š–Žȱ˜ȱ‘Žȱ ’›Ž•Žœœȱœ¢œtem consists of copies of the prescribed synchronization sequences, also named herein preambles, and the mesœŠŽœȱŠ›Žȱ›Š—œ–’ĴŽȱ’—ȱ‘Žȱ˜›–ȱ˜ȱȱœ¢–‹˜•œȱ˜›ȱ in a multi-carrier CDMA scheme. Figure 1 displays the ŞŖŘǯŗŗŠȱŠ—ȱŞŖŘǯŗŗ‹ȱ™›ŽŠ–‹•ŽœȱŠœȱŽę—Žȱ‹¢ȱ‘Žȱȱ œŠ—Š›ȱ ǻŽ›Žȱ ¢œŽ–œȱ —Œǯǰȱ ŘŖŖŚǼǯȱ ‘Žȱ ™›ŽŠ–‹•Žȱ ŞŖŘǯŗŗ‹ȱ’œȱŽěŽŒ’ŸŽ•¢ȱŠȱ™œŽž˜›Š—˜–ȱœŽšžŽ—ŒŽȱ ‘’•Žȱ the standard 802.11a consists of large and short periodicity terms corresponding to a set of frequency tones. ‘Žȱ˜Š•ȱ˜ȱ‘Žȱ™›ŽŠ–‹•Žȱ™›˜ŒŽœœ’—ȱŽŒ‘—’šžŽȱ’œȱ˜ȱ achieve an improvement on the receiver sensitivity. ReŒŽ’ŸŽ›ȱ œŽ—œ’’Ÿ’¢ȱ ’œȱ ’–™Š’›Žȱ ‹¢ȱ ’쎛Ž—ȱ Ž•ŽŒ›˜–Š—Ž’Œȱ Œ˜–™Š’‹’•’¢ȱ ǻǼȱ ŽěŽŒœǰȱ œžŒ‘ȱ Šœȱ ‹›˜Š‹Š—ȱ emissions (Bronaugh and Lambdin, 1988) that produce ’—Ž›Ž›Ž—ŒŽȱ˜ȱ—Šž›ŽȱœžĜŒ’Ž—•¢ȱ—˜—Ȭ Šžœœ’Š—ȱǻ’—Šres and Miranda, 1996). Some assumptions are as fo••˜ œǯȱ ’›œǰȱ —˜’œŽȱ ™›Ž ‘’Ž—’—ȱ ꕝŽ›ȱ Šȱ ‘Žȱ ’—™žȱ ’œȱ avoided and thus the ensuing receiver desense, and second, the preamble sequence has zero frequency and time misalignment along the frame structure which

Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM

Aguilar-Torrentera Jorge

IEEE 802.11a Short and Long Preamble (Real Part) 2 Period = 16 chips Period = 64 chips

IEEE 802.11b Standard (Real Part)

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allows to achieve large processing gain by the input –ŠŒ‘ŽȱꕝŽ›ȱǻǼǯ ‘Žȱ™›˜™˜œŽȱŽŽŒ’˜—ȱœŒ‘Ž–ŽœȱŠ›Žȱœ‘˜ —ȱ’—ȱ’ž›ŽȱŘǯȱ˜‘ȱœŒ‘Ž–ŽœȱŠ›Žȱ‹ŠœŽȱ˜—ȱ™Šœœ’ŸŽȱ–ŠŒ‘ŽȱꕝŽ›’—ȱ›ŽŠ•’£Š’˜—œȱ ‘Ž›Žȱ‘Žȱ–ŠŒ‘ŽȱꕝŽ›ȱž—Œ’˜—ȱ’œȱœŽȱ with the preamble sequence. If the preamble sequence ’œȱ™›ŽœŽ—ȱŠȱ‘Žȱ’—™žǰȱ‘Žȱ–ŠŒ‘ŽȱꕝŽ›ȱ˜ž™žœȱŠȱ£Žroth-lag pulse. If the channel is AWGN, the matched ꕝŽ›ȱœ˜•ž’˜—ȱ’—ȱ’ž›Žȱ؊ȱ’œȱ˜™’–Š•ȱ‹ŽŒŠžœŽȱ‘Žȱ˜ž™žȱ provides a signal-to-noise ratio (SNR) that minimizes ‘Žȱ ŽŽŒ’˜—ȱ Ž››˜›ȱ ™›˜‹Š‹’•’¢ȱ ǻ‘Š•Ž—ǰȱ ŗşŝŗǼǯȱ ‘Žȱ scheme in Figure 2b achieves cumulant computation using replicas of the preamble sequences stored in the processor. As shown later, a key advantage of the further cumulant-based processing is that the preamble detection ’œȱ–˜›Žȱ’—œŽ—œ’’ŸŽȱ˜ȱŠ’’ŸŽȱ—˜’œŽǯȱ‘Žȱ™›ŽŠ–‹•Žœȱ’—ȱ Figure 1 do not follow a Gaussian distribution given that their mean (which is equal to zero for both preamble sequences) and variance do not characterize completely the distributions. For the 802.11a sequence

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normalized with its r.m.s., the skewness; J = 0.2197 and kurtosis, N = 2.35, and for the normalized 802.11b sequence; J = 0.025 and N = 1.0. A more enlighten characteristic is the third-order cumulant function, c(n,m), of the preamble. Figure 3 displays the cumulant of the preamble 802.11a, which is approximately a two-dimensional impulse function of amplitude equal to the preamble skewness at zeroth lag, c(0,0) (Nikias and Pe›˜™ž•žǰȱŗşşřǼǯȱ‘Žȱ•Š›ŽȱŽ—Ž›¢ȱ˜ȱ‘Žȱ™ž•œŽȱ’œȱ™˜Ž—tially useful for preamble detection. ‘Žȱ™›ŽŠ–‹•Žȱ’œȱŠȱŽŽ›–’—’œ’Œȱœ’—Š•ȱŠ—ȱ’œȱ‘’›Ȭ order cumulant is computed simply as the averaged third-order correlation c p  n, m

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Consider the baseband receiver input signal is a summation samples comprising the preamble, p, non-Gaussian noise, P and Gaussian noise, K

Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM

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Wireless Synchronization Preamble Detection Scheme Using Bispectra-Based Statistics in the Presence of Stationary Noise

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Given N-samples of the preamble p  i i 0 ǰȱ ‘Žȱ ꗒŽȱ ’–™ž•œŽȱ›Žœ™˜—œŽȱǻǼȱꕝŽ›ȱ’œȱ–ŠŒ‘Žȱ˜ȱ‘Žȱ™›ŽŠ–ble sequence; i.e. h(i) = p(N – i) and the detector output is given by

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(10)

‘Žȱ Œž–ž•Š—œȱ ˜ȱ ‘Žȱ ™›ŽŠ–‹•Žȱ Šž˜Œ˜››Ž•Š’˜—ǰȱ cyp ǻ ‘’Œ‘ȱ’œȱŽěŽŒ’ŸŽ•¢ȱ‘ŽȱŒž–ž•Š—œȱ˜ȱypȱ’ŸŽ—ȱ‹¢ȱšžŠtion 6), are determined in a similar fashion to the ™›ŽŠ–‹•Žœȱ ’—ȱ šžŠ’˜—ȱ ŗǯȱ ȱ ‘Ž˜›Ž’ŒŠ•ȱ ’—Ž›Žœȱ ’œȱ ‘Žȱ assumption that the random variables P(i) and K(i) are independent and identically distributed (i.d.d.) (Gian—Š”’œȱŠ—ȱœŠœŠ—’œǰȱŗşşŖǼǯȱ—Ž›ȱ‘’œȱŠœœž–™’˜—ǰȱc,K (j-i1) VK2G(j-i1) and c,P (j-i1) = VP2 G(j-i1) where VK2 and Vm2 are the variance of the processes and G(j-i1) is the 1-D delta function. On the other hand, the third-order cu–ž•Š—ȱ’—ȱšžŠ’˜—ȱŗŖȱc ,y(P K) depends only on the nonGaussian noise cumulant c,P (j1 i1,j2-i2) = JPG(j1-i 1,j2-i2) where JP is the skewness of the non-Gaussian noise and G(j1-i1,j2-i2Ǽȱ’œȱ‘ŽȱŘȬȱŽ•Šȱž—Œ’˜—ǯȱ‘Žȱ‘’›Ȭ˜›Ž›ȱŒžmulant of the Gaussian-noise c3,K(j1-i1,j2-i2) is equal to zero. Nonetheless, the result of the insensitivity to Šžœœ’Š—ȱ—˜’œŽȱ’—ȱšžŠ’˜—ȱŗŖȱ’œȱ—˜ȱœŠ’œęŽȱ˜›ȱŠȱꗒŽȱ number of samples but in the mean of a large number of independent records (Nikias and Petropulu, 1993; ’Š——Š”’œȱŠ—ȱœŠœŠ—’œǰȱŗşşŖǼǯȱȱȱ’œȱ ˜‘ȱ–Ž—’˜—’—ȱ that statistical analysis based on the bispectra shown in a following section presents asymptotic independence for lengths lower than the preamble size of interest, in agreement with results in Hinich and Wilson (1990).

'HWHFWRUSHUIRUPDQFH ˜ȱŽœȱ‹˜‘ȱŽŽŒ˜›œǰȱ‘Žȱ’—™žȱœŽšžŽ—ŒŽȱ˜ȱ‘Žȱ™›ŽŠ–ble is combined linearly with input noise sources of £Ž›˜ȱ–ŽŠ—ǯȱ‘ŽȱœŠ–™•Žȱ›ŠŽȱ˜ȱ‘Žȱ™›ŽŠ–‹•ŽȱœŽšžŽ—ŒŽœȱ shown in Figure 4 is inversely proportional to the Gausœ’Š—ȱ —˜’œŽȱ ‹Š— ’‘ǯȱ ŽŒŽ’ŸŽ›ȱ ̘˜›ȱ —˜’œŽȱ ’œȱ Žę—Žȱ for a given SNR. Non-Gaussian interference is included ‹¢ȱ œŽĴ’—ȱ Šȱ œ’—Š•Ȭ˜Ȭ’—Ž›Ž›Ž—ŒŽȱ ›Š’˜ȱ ǻǼȱ Š—ȱ ‘Žȱ same sampling rate. In our simulations samples of nonGaussian noise are produced by a random process cha›ŠŒŽ›’£Žȱ‹¢ȱŠȱ˜—˜›–Š•ȱ’œ›’‹ž’˜—ǯȱ‘ŽȱžœŽȱ˜ȱ‘’œȱ distribution facilitates testing the detectors as it allows simulating broadband stationary non-Gaussian interference which is often characterized by a large positive skewness and a long-tail distribution (Primak et al., 2004). Figure 5 displays the reconstruction of the 802.11a preamble achieved at SNR = 0 dB and SIR= 10 dB. It is seen that the third-order cumulant processing ˜ȱ‘Žȱ–ŠŒ‘ŽȱꕝŽ›ȱ˜ž™žȱ’–™›˜ŸŽœȱ™›ŽŠ–‹•Žȱ›ŽŒ˜—œ›žŒ’˜—ȱŒ˜–™Š›Žȱ ’‘ȱ‘Šȱ‹ŠœŽȱ˜—ȱ‘Žȱ–ŠŒ‘Žȱę•-

Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM

Aguilar-Torrentera Jorge

ter structure. It is apparent that OFDMA tones are best ›ŽŒ˜—œ›žŒŽǯȱ ˜–Žȱ —˜’œŽȱ Œ˜–™˜—Ž—œȱ Š›Žȱ ꕝŽ›Žȱ ˜žȱ which result from the high energy of the third-order cumulant. It is important to mention that at high SNR ratios (results not shown herein) the scheme based on the third-order cumulant cannot improve the preamble reŒ˜—œ›žŒ’˜—ȱ˜ȱŠ—¢ȱž›‘Ž›ȱŽ¡Ž—ǯȱ‘’œȱ’œȱžŽȱ˜ȱŠŒȱ‘Šȱ the preamble cumulant is not an ideal impulse function (Figure 3). However; the following statistical analysis of ‘Žȱ™›ŽŠ–‹•ŽȱŽŽŒ˜›ȱŒ˜—ę›–œȱ’œȱŽěŽŒ’ŸŽ—Žœœǯȱ Histogram Lognormal Distribution

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)LJXUH'LVWULEXWLRQRIWKHQRQ*DXVVLDQQRLVHXVHGLQ VLPXODWLRQV

FFT Preamble 802.11a 1

'HVHQVHOHYHOV ‘Žȱ ŒŠ™Š‹’•’¢ȱ ˜ȱ ŽŽŒ’—ȱ ‘Žȱ ™›ŽŠ–‹•Žȱ ’œȱ Š—Š•¢£Žȱ statistically using the corresponding schemes in Figure Ŝǯȱ ‘Žȱ –ŠŒ‘Žȱ ꕝŽ›ȱ ˜ž™žȱ ’œȱ ™›˜ŒŽœœŽȱ ˜ȱ Žœ’–ŠŽȱ both the preamble variance (energy detection) and the preamble bispectrum from which decision rules for hypothesis testing are derived. ‘ŽȱŽ –Š—ȬŽŠ›œ˜—ȱŒ›’Ž›’˜—ȱ’œȱžœŽȱ˜ȱ˜‹Š’—ȱ‘Žȱ probability of detection as it does not involve knowledge on apriori probabilities of the preamble transmission ǻ ‘’Œ‘ȱ’œȱ™Ž›’˜’Œȱ’—ȱ‘Žȱ›Š—œ–’ĴŽȱ›Š–ŽœǼǯȱ¢ȱžœ’—ȱ this criterion, one has to admit errors in the hypothesis Žœ’—ǯȱ—ȱ Ž››˜›ȱ ˜ȱ ‘Žȱ ꛜȱ ”’—ȱ ’œȱ ‘Žȱ ™›˜‹Š‹’•’¢ȱ ˜ȱ false alarm which is the event to saying the preamble is present at the input when there is none. A binary decision is made by computing the likehood ratio, O(x), which is constrained by a threshold level, lD that satisꎜȱŒŽ›Š’—ȱŠ•œŽȱŠ•Š›–ȱ™›˜‹Š‹’•’¢ǰȱD(Whalen, 1971). If O(x) tOD for a single observation x, the choice is to say ‘Žȱ ™›ŽŠ–‹•Žȱ ’œȱ ™›ŽœŽ—ǯȱ ‘Žȱ Œ‘˜’ŒŽȱ ‹Ž ŽŽ—ȱ ŸŠ›’Š—ŒŽȱ and bispectra depends on the nature of input noise; a decision rule based on the variance (or energy of the preamble) is suboptimal when the input noise is nonGaussian while bispectra can deal with both types of noise, straightforwardly. Consider the input noise is Gaussian. For the energy detector in Figure 6, the pulse obtained at the matched ꕝŽ›ȱ˜ž™žȱ’œȱ‘ŽȱŸŠ›’Š—ŒŽȱǻ™›ŽŠ–‹•ŽȱŽ—Ž›¢Ǽȱ ‘’Œ‘ȱ’œȱ compared against a threshold level, Vǯȱ ‘Žȱ ŽŒ’œ’˜—ȱ rule given by O(x) tOD is established by comparing the output pulse with the threshold level given by

0.5 T

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V n ln OD  12 ³ p 2  t dt

(11)

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Third-Order Cumulant

where Vn is the variance of the (colored) noise at the –ŠŒ‘ŽȱꕝŽ›ȱ˜ž™žȱŠ—ȱT is the preamble duration. Figure 7 shows the probability of detection using řŘŖȱŠ—ȱŜŚŖȱœŠ–™•Žȱ™˜’—œȱ˜ȱ‘Žȱ™›ŽŠ–‹•ŽȱŞŖŘǯŗŗŠǯȱ‘Žȱ probability of detection is obtained as an average over a

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Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM

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Wireless Synchronization Preamble Detection Scheme Using Bispectra-Based Statistics in the Presence of Stationary Noise

CH  f i , f k

2'N2 N B  N  f i , f k  Bn  f i , f k

2

S y N  f j S y N  f k S y( N )  f j  k (12)

where B(N)(fj, fk) is an estimator of the input bispectrum, B(fj, fk), for a sample size of N, Bn(fj, fk) is the bispectrum of the noise assuming that we have a noise-only sample Šȱ‘Žȱ–ŠŒ‘ŽȱꕝŽ›ȱ˜ž™žDzȱSy(N) is the estimated power spectrum density comprising the samples of the pream‹•ŽȱŠ—ȱŠ’’ŸŽȱ—˜’œŽȱŠȱ‘Žȱ›ŽšžŽ—Œ’Žœȱǿfk} . 'N is the ‹Š— ’‘ȱ˜ȱ‘Žȱ‹’œ™ŽŒ›ž–ǯȱœ’—ȱ‘ŽȱŠ™™›˜¡’–Š’˜—ȱ 'N # 1/—N Hinich and Wilson (1990) give a sampled bispectrum in the form of independent random varia‹•Žœȱ Žę—Žȱ Šȱ ‘Žȱ ‹’›ŽšžŽ—Œ¢ȱ ™Š’›ȱ ǻfi, fk) and then CH(fj, fk) can be considered as a chi-square variable with 2 degrees of freedom. ‘Žȱ‹’œ™ŽŒ›ž–ȱ˜ȱ‘Žȱ—˜’œŽȱBn(fj, fk) is estimated in absence of the preamble. Since noise is stationary; its bispectrum can be estimated at the beginning of the simulations and out-of-line computation of the input bispectrum B(fj, fkǼǯȱ‘Žȱž—Œ’˜—ȱCH(fj, fkǼȱ’ŸŽ—ȱ’—ȱšžŠ’˜—ȱ 12 is a random variable with a non-central parameter ‘Šȱ’œȱŽę—Žȱ‹¢ȱȱ‘Žȱ™›ŽŠ–‹•Žȱ‹’œ™ŽŒ›ž–ȱŠ—ȱŒ˜—œ’dered as the mean value of the random variable. On the ˜‘Ž›ȱ‘Š—ǰȱ‘Žȱ™˜ Ž›ȱ˜ȱ‘ŽȱŽœȱ’œȱŽę—Žȱ‹¢ȱ‘Žȱœ”Ž ness of the distribution (Hinich and Wilson, 1990),  ‘’Œ‘ȱ’œȱ˜‹Š’—Žȱ‹¢ȱ›Ž™•ŠŒ’—ȱ‘Žȱ’쎛Ž—ŒŽȱ’—ȱšžŠtion 12, B(N)(fj, fk)  Bn(fj, fk) by the bispectra of the preamble, Byp(fj, fk) /  f i , f k 2'N2 N Byp  f i , f k

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y

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(N ) y

jk

(13)

where / is computed over the main dominion of the ‹’œ™ŽŒ›Šǯȱ ‘’œȱ ŸŠ›’Š‹•Žȱ ’œȱ ‘Ž—ȱ Šȱ œž––Š’˜—ȱ ˜ȱ K chisquare variables of 2K degrees of freedom, F22K. A conœ’œŽ—ȱ Žœ’–Š’˜—ȱ ˜ȱ ‘Žȱ ‹’œ™ŽŒ›ž–ȱ ’—ȱ šžŠ’˜—ȱ ŗřȱ provides the decision rule for the hypothesis testing. ‘Žȱ‘¢™˜‘Žœ’œȱŽœ’—ȱŠ—ȱ™˜ Ž›ȱ˜ȱ‘ŽȱŽœȱŠ›ŽȱŒ˜–puted using the Marcum’s Q function (Proakis, 1990). ‘’œȱ’œȱŽœŠ‹•’œ‘Žȱ‹¢ȱ›Ž“ŽŒ’—ȱ‘Žȱ‘¢™˜‘Žœ’œȱ‘Šȱ‘Žȱ input is noise only if / exceeds the threshold TD where Pr(F22K > TD) = Dǯȱ‘ŽȱŠ•˜›’‘–ȱ‘ŠȱŒ˜–™žŽœȱ‘Žȱ—˜—Ȭ centrality parameter includes the signal-to-noise power ratio, U(f) = Sy(f )/Sn(f ) at each of the three frequencies fj, fk and fj+kǰȱŠœȱ˜‹Š’—ŽȱŠȱ‘Žȱ–ŠŒ‘ŽȱꕝŽ›ȱ˜ž™žǯ ȱ‘Žȱ™›˜‹Š‹’•’¢ȱž—Œ’˜—ȱ’œȱŒ˜–™žŽȱ‘›˜ž‘ȱ‘Žȱ Generalized Marcum’s Q function of 2K degrees of freedom as a function of the F2 random variable obtai—Žȱ’—ȱšžŠ’˜—ȱŗŘȱŠœȱ Ž••ȱŠœȱ‘Žȱœ”Ž —Žœœȱ˜ȱ‘Žȱ’œ›’‹ž’˜—ȱ’—ȱšžŠ’˜—ȱŗřǯȱȱ—ȱŠ•˜›’‘–ȱ ŠœȱŽŸŽ•˜™Žȱ to compute the threshold TD for a given false alarm D. In this algorithm, Gaussian and non-Gaussian procesœŽœȱŠ›ŽȱŽ—Ž›ŠŽȱŠȱŽŠŒ‘ȱ’Ž›Š’˜—ȱǻ˜›ȱ˜‹œŽ›ŸŠ’˜—Ǽǯȱ‘Žȱ power of the noise and interference are set by SNR and ȱ™Š›Š–ŽŽ›œDzȱŠŒŒ˜›’—•¢ǯȱ‘Žȱ™›˜‹Š‹’•’¢ȱ˜ȱŽŽŒtion is then obtained as an average over a number of observed signals that ensures convergence. Figure 7 allows contrasting the statistical analyses of ‘ŽȱœŒ‘Ž–Žœȱœ‘˜ —ȱ’—ȱ’ž›ŽȱŜǯȱ‘ŽȱŽŽŒ’˜—ȱŒž›ŸŽœȱ‹Šsed on cumulant processing display a rapid increase with SNR which stems from the non-linear relationship ˜ȱ‘Žȱ—˜—ȬŒŽ—›Š•’¢ȱ™Š›Š–ŽŽ›ȱ ’‘ȱǯȱ‘’œȱ’œȱ’—ȱž••ȱ agreement with the detection bispectra analysis in Hinich and Wilson (1990). Figure 7 also indicates that the short-preamble can be detected at SNR > 10 dB, which is slightly lower than the ratio at which the energy detector can detect the preamble. Figure 7 also shows that an increase on the preamble length renders an improvement of about 4 dB with respect to the energy detector. ‘’œȱ›Žœž•ȱŽ–˜—œ›ŠŽœȱ‘Šȱ‘Žȱ™›ŽŠ–‹•ŽȱŠž˜Œ˜››Ž•Š1

Probability of detection

number of observed signals that ensures convergence. It is seen that for a detector based on the preamble energy, the probability of detection does not change substantially with the sample size. At a SNR equal to 8 dB both preambles give approximately a probability of detection equal to 1. A thoughtful analysis of the energy detection by Hinich and Wilson (1990) shows that a 3-dB increase in the receiver sensibility involves an increase on sample size by a factor of 4. For our wireless preamble scheme, such increase is well beyond the chip length of the synchronization preamble within a frame. ‘Žȱ ‹’œ™ŽŒ›ŠȬ‹ŠœŽȱ ŽŽŒ˜›ȱ ŒŠ—ȱ ŽŠ•ȱ  ’‘ȱ —˜—Ȭ Šžœœ’Š—ȱ—˜’œŽǯȱ‘’œȱŽŽŒ˜›ȱ™›ŽœŽ—œȱ‘Ž˜›Ž’ŒŠ•ȱŠ—ȱ implementational aspects that have been thoroughly investigated in Hinich and Wilson (1990) and (Nikias and Raghuveer, 1987); among other references. In order to present the preamble detection based on thirdorder cumulant, some relevant results are shown ‘Ž›Ž’—ǯȱ ‘Žȱ ™›˜‹Š‹’•’¢ȱ ˜ȱ ŽŽŒ’˜—ȱ ’œȱ Ž›’ŸŽȱ ‹¢ȱ Šking into account the statistical characteristics of the random variable

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Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM

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Aguilar-Torrentera Jorge

’˜—ȱž—Œ’˜—ȱ’œȱœžĜŒ’Ž—•¢ȱ—˜—Ȭ Šžœœ’Š—ȱŠ—ȱ‘Žȱ‹’œ™ŽŒtrum can detect it at a lower SNR than the energy detector could detect it with the same sample size.

Conclusions

3UHDPEOHGHWHFWLRQLQQRQ*DXVVLDQQRLVH ‘Žȱ Ž—Ž›¢ȱ ŽŽŒ’˜—ȱ ˜ȱ ‘Žȱ ™›ŽŠ–‹•Žȱ œŽšžŽ—ŒŽȱ ’œȱ —˜ȱ optimal when non-Gaussian noise is present at the in™žǯȱ—ȱ‘Žȱ˜‘Ž›ȱ‘Š—ǰȱ‘Žȱ‹Ž—Žęȱ˜ȱ‘Žȱ™›ŽŠ–‹•ŽȱŽtection based on bispectra-based statistics is obtained when larger preamble lengths are used (for example 640 sample points, see Figure 7) and the preamble or its autocorrelation function is characterized by large a skewness. However, the preamble 802.11b and its autoŒ˜››Ž•Š’˜—ȱ ž—Œ’˜—ȱ ‘ŠŸŽȱ •˜ ȱ œ”Ž —Žœœȱ Œ˜ŽĜŒ’Ž—œǰȱ which results in a probability detection equal to 1 at higher SNR (>9 dB, see Figure 8 for the detector including MF). Notice that the dense level is slightly lower to that obtained by energy detection. Figure 8 also shows changes in desense levels when non-Gaussian noise of power equal śȱȱŠ›ŽȱŠŽȱŠȱ‘Žȱ’—™žǯȱ‘Žȱ›ŽžŒtion in the desense is about 1.7dB. Other scheme that reduces at some extent the receiver complexity is the one based on cumulant-processing without including the MF at the input and is referred herein to as the third-order cumulant scheme. ‘Žȱ’—™žȱœ’—Š•ȱ’œȱ‘Ž—ȱ™›˜ŒŽœœŽȱ’›ŽŒ•¢ȱ‹¢ȱ‘Žȱ‘’›Ȭ order cumulant processor that stores replicas of the preamble sequence. It is seen in Figure 8 that the thirdorder cumulant processor behaves poorly due to the low skewness of the preamble in both Gaussian and —˜—Ȭ Šžœœ’Š—ȱ—˜’œŽǯȱ‘ŽȱŽŽŒ’˜—ȱ•ŽŸŽ•ȱ›ŽžŒŽȱ‹¢ȱŘǯŞȱ dB when non-Gaussian noise of power equal to 5 dB is ŠŽȱŠȱ‘Žȱ’—™žǯȱ‘Žȱ›Žœž•œȱŒ˜—ę›–ȱ‘Šȱ‘Žȱ™›ŽŠ–ble autocorrelation function is best detected than the

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‘Žȱ›Š’˜ȱ›ŽŒŽ’ŸŽ›œȱ˜ȱ ’›Ž•Žœœȱœ¢œŽ–œȱ‘ŠȱŽ–™•˜¢ȱ ’deband transmission protocols are susceptible to nonGaussian interference. In this paper, the detection of wireless preamble sequences in additive Gaussian and non-Gaussian noise is investigated. It is seen that wireless preambles and their autocorrelation functions have œ’—’ęŒŠ—ȱ Ž—Ž›¢ȱ ’—ȱ ‘Ž’›ȱ ™˜•¢œ™ŽŒ›Šȱ  ‘’Œ‘ȱ ŒŠ—ȱ ŽĜŒ’Ž—•¢ȱ‹ŽȱŽŽŒŽȱžœ’—ȱ‘Žȱ™›˜™˜œŽȱœ›žŒž›Žœǯȱȱ œŠ—Š›ȱ ™›ŽŠ–‹•Žœȱ ‘ŠŸŽȱ ’쎛Ž—ȱ œ”Ž —Žœœȱ Œ˜ŽĜŒ’Ž—œǰȱ ‘’Œ‘ȱ•ŽŠœȱ˜ȱ’쎛Ž—ȱŽŽŒ’˜—ȱ™Ž›˜›–Š—ŒŽǯȱ Improved preamble detection for the standard 802.11a is observed in wireless synchronization regardless the nature of input noise. For instance; the structure based –ŠŒ‘Žȱ ꕝŽ›’—ȱ Š—ȱ Œž–ž•Š—Ȭ™›˜ŒŽœœ’—ȱ ’–™›˜ŸŽœȱ the receiver desense by approximately 4-dB compared  ’‘ȱ‘Šȱ˜‹Š’—Žȱ‹¢ȱžœ’—ȱŽ—Ž›¢ȱŽŽŒ’˜—ǯȱ‘ŽȱœŠtistics of the receiver show that the skewness of the preamble becomes higher than the energy of the noise as well as the energy in its third-order spectrum for non-Gaussian noise at the input. On the other hand, the preamble of the standard 802.11b and its correlation ž—Œ’˜—ȱ ‘Šœȱ •˜ Ž›ȱ œ”Ž —Žœœȱ Œ˜ŽĜŒ’Ž—ȱ  ‘’Œ‘ȱ ›Žœž•œȱ in lower performance when Gaussian noise only is present at the input. For both preambles, an increase of the sample size leads to improve desense levels. Statistically, the bispectra estimation shows convergence within the length of standard synchronization sequences  ‘’Œ‘ȱŒ˜ž•ȱ‹Žȱ‹Ž—ŽęŒ’Š•ȱ’—ȱ‘ŽȱŽŸŽ•˜™–Ž—ȱ˜ȱŠ•˜rithms with multiple wireless systems discrimination. ‘Žȱœ›žŒž›Žœȱ™›˜™˜œŽȱ’—ȱ‘’œȱ™Š™Ž›ȱ˜›–ȱŠȱ‹Šœ’œȱ˜›ȱ improving radio receiver immunity against stationary interference.

References

0.6 0.4

preamble itself, in agreement with (Giannakis and œŠœŠ—’œǰȱŗşşŖǼǯ

1.7 dB

2.8 dB

Gauss. noise, MF included Gauss. noise, Cum(p,p,s) SIR = 5 dB, MF included SIR = 5 dB, Cum(p,p,s) -10

-8

-6 SNR (dB)

-4

-2

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Ž›Žȱ¢œŽ–œȱ—Œǯȱ —ȱœ¢—Œȱ™›˜™˜œŠ•ȱŽŒ‘—’ŒŠ•ȱœ™ŽŒ’ęŒŠ’˜—ǰȱ™›’vate communication, August 2004. ›˜—Šž‘ȱ ǯȱ Š—ȱ Š–‹’—ȱ ǯȱ Electromagnetic interference test methodology and procedures, (handbook series on electromag—Ž’Œȱ’—Ž›Ž›Ž—ŒŽȱŠ—ȱŒ˜–™Š’‹’•’¢ȱŸ˜•ǯŜǼǰȱǰȱ—Ž›Ž›Ž—ŒŽȱ ˜—›˜•ȱŽŒ‘—˜•˜’Žœǰȱ—Œǯǰȱ ž—ǯȱŗşŞŞǯ ’Š——Š”’œȱ ǯȱŠ—ȱœŠœŠ—’œȱǯȱ’—Š•ȱŽŽŒ’˜—ȱŠ—ȱŒ•Šœœ’ęŒŠ’˜—ȱ žœ’—ȱ–ŠŒ‘ŽȱꕝŽ›’—ȱŠ—ȱ‘’‘Ž›ȱ˜›Ž›ȱœŠ’œ’Œœǯȱȱ›Š—saction on Acoustics, Speech and Signal Processing, volume řŞȱǻ’œœžŽȱŝǼǰȱ ž•¢ȱŗşşŖDZȱŗŘŞŚȬŗŘşŜǯ

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Š—£˜ȱǯǰȱû—œŽ›ȱǯǰȱ‘˜’ȱǯ ǯǰȱ Ž••Ž›ȱǯȱOFDM and MC-CDMA for broadband multiuser communications, WLANs and broadcastingǰȱؗȱŽǯǰȱȱ›Žœœǰȱ ǰȱŘŖŖřǯ Hinich, M. and Wilson G. Detection of non-gaussian signals in non-gaussian noise using the bispectrum. IEEE Transactions on Acustic, Speech, and Signal Processing, volume 38 (issue 7), ž•¢ȱŗşşŖDZȱŗŗŘŜȬŗŗřŗǯ

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›˜Š”’œȱ ǯȱ Digital communications, 4th ed., New York, Mc Graw Hill, 1990. •ŠĴŽ›¢ȱ ǰȱŽŠ•ȱ ǯǰȱž’ȱǯȱŽŠ›Ȭꎕȱ–ŽŠœž›Ž–Ž—œȱ˜ȱȱŽŸ’ces. IEEE Transactions on Electromagnetic Compatibility, volume 34 (issue 5), November 1999: 439-453. Sridhara V., Hweechul S., Bohacek S. Performance of 802.11b/g in ‘Žȱ’—Ž›Ž›Ž—ŒŽȱ•’–’Žȱ›Ž’–Žǰȱȱ—Ž›—Š’˜—Š•ȱ˜—Ž›Ž—ce on Computer Communications and Networks (XVI, Hono•ž•žȱ Š Š’’ǰȱžǯȱŘŖŖŝǼǰȱȱǰȱŘŖŖŝǰȱ™™ǯȱşŝşȬşŞŚǯ Whalen A. Detection of signals in noise, New York, Academic Press, 1971.

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About the author Jorge Aguilar-Torrentera. Received a B.Sc. degree in electronic engineering at MetropoliŠ—ȱ —’ŸŽ›œ’¢ȱ ŒŠ–™žœȱ £ŒŠ™˜ĵŠ•Œ˜ȱ ’—ȱ ŗşşŗǰȱ ‘’œȱ ǯŒǯȱ ’—ȱ Ž•ŽŒ›’ŒŠ•ȱ Ž—’—ŽŽ›’—ȱ ˜ŒžœŽȱ˜—ȱŒ˜––ž—’ŒŠ’˜—œǰȱŠȱȬȱŒŠ–™žœȱŽ¡’Œ˜ȱ’¢ȱ’—ȱŗşşŞǯȱȱ ‘ǯǯȱ’—ȱ˜™’ŒŠ•ȱŒ˜––ž—’ŒŠ’˜—ȱœ¢œŽ–œȱ Šœȱ˜‹Š’—Žȱ’—ȱŘŖŖŚȱŠȱ—’ŸŽ›œ’¢ȱ˜••Žge London. His research interests are in wireless systems, design of ultra-broad‹Š—ȱ ˜™’ŒŠ•ȱ ›ŽŒŽ’ŸŽ›œǰȱ Œ–Šȱ ˜›ȱ ꋎ›ȱ —Ž ˜›”œȱ Š—ȱ Œ˜–™žŽ›ȬŠ’Žȱ Žœ’—ȱ ˜ȱ microwave circuits.

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Ingeniería Investigación y Tecnología, volumen XVI (número 2), julio-septiembre 2015: 383-390 ISSN 1405-7743 FI-UNAM