Strategy of packet detection for burst-mode OFDM systems

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(1)IEICE Electronics Express, Vol.3, No.11, 249–256. Strategy of packet detection for burst-mode OFDM systems Chia-Sheng Penga) , Yuan-Shin Chuang, and Kuei-Ann Wen Institute of Electronics, National Chiao Tung University 1001 Ta Hsueh Road, Hsinchu, Taiwan 300, ROC a) cspeng.ee87g@nctu.edu.tw. Abstract: An efficient method on selecting threshold values according to minimax test for packet detection in burst-mode OFDM systems is proposed. Packet detection decides whether a packet is coming or not by comparing a threshold value in the wireless receiver. Related with sliding window size and SNR, the threshold value affects receiving performance including probabilities of false alarm and miss. The minimax test for detection based on empirical CDF and survival functions is proposed. Also the performances of two general used detection methods are surveyed and compared. Keywords: packet detection, frame detection, OFDM, WLAN Classification: Science and engineering for electronics References [1] IEEE, “Wireless LAN Medium Access Control and Physical Layer specifications: High-speed physical layer in the 5 GHz band,” P802.11a/D7.0, July 1999. [2] J. Terry and J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide, Sams, Indianapolis, 2002. [3] F. Manavi and Y. R. Shayan, “Implementation of OFDM modem for the physical layer of IEEE 802.11a standard based on Xilinx Virtex-II FPGA,” Proc. of VTC, vol. 3, pp. 1768–1772, Spring 2004. [4] H. L. Van Trees, Detection, Estimation, and Modulation Theory: Part I, John Wiley & Sons, New York, 1968. [5] E. T. Lee and J. W. Wang, Statistical Methods for Survival Data Analysis, John Wiley & Sons, New Jersey, 2003. [6] M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Handbook for Engineers and Scientists, Kluwer Academic Publishers, Boston, 2002. [7] J. Thomson, B. Baas, E. Cooper, and J. Gilbert, “An Integrated 802.11a Baseband and MAC Processor,” in Proc. of ISSCC, vol. 1, pp. 126–451, Feb. 2002.. c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. 249.

(2) IEICE Electronics Express, Vol.3, No.11, 249–256. 1. Introduction. Some burst-mode wireless communication systems such as WLAN and WPAN transmit information in each packet with three segments: preamble, header and data signal [1]. Ahead of any other operations in the receiver, packet detection shall decide whether a packet is coming or not. Some packet detection methods only use receiving signal power and easily suffer from a drawback: threshold values involving with received power and gain control in radio-frequency circuits [2]. For a specific communication system supporting repeated preambles, correlation properties can be used for packet detection and carrier synchronization in OFDM system, and furthermore these correlation values are divided by power value to eliminate problems of variant threshold [3]. The correlation and power functions used in the detection algorithm are defined as C(n) =. L−1 i=0. and P (n) =. ∗ rn+i rn+i+D ,. L−1 . |rn+i+D |2 ,. (1). (2). i=0. where rk implies kth complex-valued received sample, L is the sliding window size and D is the interval of two repeated preambles. Due to that C(n) is complex-valued and P (n) is real-valued, packet detection has two normalized functions according to practical implementation methods: M 2(n) =. |C(n)|2 , P (n)2. (3). M 1(n) =. |C(n)| . P (n). (4). and. c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. Derivative of the function M 1(n) needs an extra square root operation, whereas derivative of M 2(n) needs an extra square operation and greater precision representation in practice. In consideration of circuit implementation, these extra requirements are small enough compared to a full receiver. Thus we leave aside implement complexity and only consider which one performs better in the detection. On the other hand, the selected threshold values affect probabilities of detection and false alarm. We adopt a Bayes test, or called minimax test, to select a proper threshold value by minimizing the maximum possible risk according to different assumptions of hypothesis probability and risk [4]. Since M 1(n) and M 2(n) are random variables (RV) combined with multiple complex-valued RVs in numerators and denominators, their probability density function (PDF) are very difficult to be calculated in simple deterministic form. Moreover, because only cumulative distribution functions (CDF) are required in the minimax test, an empirical CDF can be used to estimate ideal CDF values from a statistical viewpoint. We adopt the product-limit (PL) method (Kaplan and Meier method) to. 250.

(3) IEICE Electronics Express, Vol.3, No.11, 249–256. calculate empirical CDFs and use interpolation method to acquire any cumulative probability corresponding to different threshold values [5]. Most OFDM systems transmit through frequency-selective fading channels, which can be viewed as time-variant linear filters with random coefficients of amplitude, phase and delay. The received equivalent baseband signal can be viewed as xb (t) =. Np . αk (t) exp(−jθk (t))sb (t − τk (t)),. (5). k=1. where sb (t) is the transmitted baseband signal, αk , θk , τk are time-variant coefficients, and Np is the number of resolvable paths. After passing through the fading channel, the signal is disturbed at the receiver by additive white Gaussian noise (AWGN) with zero mean and variance σn2 . Then the instantaneous signal-to-noise ratio (SNR) is defined as γ(t) = E[|xb (t)|2 ]/σn2 .. (6). The received discrete samples after analog-to-digital converter are represented as the sum of xb (t) and AWGN both multiplied by a RF gain GRF and sampled by Ts : (7) rk = GRF [xb (kTs ) + wk ], The AWGN is given as wk = wI,k + jwQ,k and wI,k , wQ,k ∈ N (0, σn2 /2). The RF gain GRF will be canceled in (3) and (4), which is the main purpose of the normalization. Thus the detection function is no longer relative to GRF . Define two hypotheses for two detection conditions: . H1 : a packet has been transmitted . H0 : no packet has been transmitted. (8). Let NL (0, σx2 ) be an independent Gaussian vector whose components are independent Gaussian RVs with zero mean and equal variance σx2 . Define (i) (i) two complex-valued independent Gaussian vectors as W (i) = WI + jWQ , (i). (i). i = 1, 2 with WI and WQ ∈ NL (0, σn2 /2). Therefore, for H0 is true, the power function P (n) can be represented as the squared norm of the Gaussian vector: H0 is true: P (n) =. L i=1. . (2) 2. . 2 . . . (2) 2. . . (2) 2. |wn+i+D |2 = W (2) = WI + WQ . . ,. (9). (2) 2. where WI and WQ are central chi-square RVs with L degrees of freedom. Thus P (n) in (9) is a central chi-square RV with 2L degrees of freedom: c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. pP (n)|H0 (y|H0 ) =. 1 σn2 Γ(L). . y σn2. L−1. . exp −. . y , σn2. y ≥ 0,. (10). 251.

(4) IEICE Electronics Express, Vol.3, No.11, 249–256. where Γ(x) is a Gamma function [6]. For H0 is true, the correlation function C(n) can be viewed as an inner product of two complex-valued independent Gaussian vectors: H0 is true: C(n) =. L−1 . (1). = (WI. (1). = [WI. (1). i=0. ∗ wn+i wn+i+D = W (1) · W (2)∗ (1). (2). + jWQ ) · (WI (2). · WI. (1). (2). ,. − jWQ ) (2). (1). (2). + WQ · WQ ] + j[WQ · WI. (2). (3). (4). (1). (1). (11). (2). − WI. · WQ ]. (2). ≡ [KL + KL ] + j[KL − KL ] ≡ K2L + jK2L (i). where Kn is a RV as inner product of two independent Gaussian vectors with identical variance σ 2 = σn2 /2, whose PDF is given below (for n = 2m) [6]: . pK (x) =. 1 |x| exp − 2 2 σ Γ(m) σ. m−1 i=0. Γ(m + i) m+i 2 Γ(i + 1)Γ(m − i). . |x| σ2. m−1−i. . (12). For H1 is true, the indoor time-variant channel can be viewed static within D samples because of low Doppler frequency. Thus assume the two complexvalued signal vectors are almost the same: (1). Xb. (2) = [xb,n . . . xb,n+L−1 ]T ∼ = Xb, = [xb,n+D . . . xb,n+D+L−1 ]T ,. (13). Then P (n) given H1 is true can be represented as H1 is true: P (n) =. L . (2). |xb,n+i+D + wn+i+D |2 = Xb. 2 . + W (2) . i=1 (2) 2 (2) 2 (2) (2) = Xb,I + WI + Xb,Q + WQ . ,. (14). which can be viewed as a noncentral chi-square RV with 2L degrees of freedom: 1 pP (n)|H1 (y|H1 ) = 2 2σ. . y a2. (L−1)/2. . y + a2 exp − 2σ 2. ⎛. . IL−1 ⎝. ⎞. a2 y ⎠ , y ≥ 0, σ4 (15). where σ 2 = σn2 /2. (2) 2. (2) 2. (2) 2. a2 = Xb,I + Xb,Q = Xb =. L−1. i=0. |xb,n+i+D |2. .. (16). From (6) the noncentral parameter a2 can be represented by instantaneous SNR as a2 = γLσn2 . Therefore the noncentral chi-square RV in (15) is not related to the distribution of Xb , but only involved with SNR value and noise power instead. For H1 is true, the correlation C(n) is given as c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. 252.

(5) IEICE Electronics Express, Vol.3, No.11, 249–256. H1 is true: C(n) = =. L−1 . (xb,n+i + wn+i )(xb,n+i+D + wn+i+D )∗. i=0 (1) (Xb (1). = Xb. (2). + W (1) ) · (Xb + W (2) )∗ (2)∗. · Xb. (1). + Xb. (2)∗. · W (2)∗ + Xb. (1). (1). (2). (1). (2). = γLσn2 + (Xb,I + jXb,Q ) · (WI (2). + (Xb,I − jXb,Q ) · (WI . (1). (2). = γLσn2 + Xb,I · WI (2). (2). (1). (2). (2). − jWQ ) ,. (1). + jWQ ) + W (1) · W (2)∗ (1). (2). (2). (17). (1). + Xb,Q · WQ + Xb,I · WI (1). + Xb,Q · WQ + K2L + j[Xb,Q · WI. · W (1) + W (1) · W (2)∗. (1). . (2). (2). (1). (2). (1). − Xb,I · WQ + Xb,I · WQ − Xb,Q · WI. (2). + K2L ] The inner product of a constant vector and a Gaussian vector is still a Gaussian RV with zero mean and variance equal to the original variance (1) (2) multiplied by the norm of the constant vector, e.g. the term Xb,I · WI in (1) 2. (17) is a Gaussian RV with variance equal to Xb,I · σn2 /2. Therefore the middle four terms in the real part of (17) integrate into a Gaussian RV with (1) 2. variance equal to Xb σn2 = γLσn4 . The first four terms in the imaginary part has the same variance. Thus (17) can be simplified as . (1). . . (2). . C(n) = Z (1) + K2L + j Z (2) + K2L ,. (18). where Z (1) ∈ N (γLσn2 , γLσn4 ) and Z (2) ∈ N (0, γLσn4 ). When SNR is large enough, C(n) is close to a complex-valued Gaussian RV. The PDF of image and imaginary parts in (18) can be acquired by joint PDF: ∞. pCI (n)|H1 (y|H1 ) =. −∞. pZ (1) (x)pK (1) (y − x)dx. 2L. (19). Although the PDFs of (1) and (2) for two hypotheses H0 and H1 are derived, the distribution of M 1(n) and M 2(n) still can not be derived because of the dependence between the numerator and denominator. But we can conclude from (10) (12) (15) (18) that the detection functions only involve with SNR and window size Linstead of received signals and channels. This helps to build a simulation of RVs and to clarify relative parameters.. 2. Empirical CDF and minimax test for threshold values. According to different window sizes and SNR values, we build a RV simulation with Ns tests for the detection functions M 1(n) and M 2(n) and use nonparametric method to estimate empirical CDFs. The PL method is used here to acquire CDF and survival functions [5]. Assume there are Ns samples observed and we sort them in ascending order such that s(1) ≤ s(2) . . . ≤ s(Ns −1) ≤ s(Ns ) . The survival function is given as: c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. Ns − i , i = 1, 2, . . . , Ns , fˆs (s(i) ) = fˆs (s(i−1) ) Ns − i + 1. (20). 253.

(6) IEICE Electronics Express, Vol.3, No.11, 249–256. where fˆs (s(0) ) = 1 is assumed. The precision of estimates in (20) are dependent on the number of samples Ns, i.e. the minimum precision is 1/Ns. For a specific value λ, fˆs (λ) can be acquired by linear interpolation: fˆs (λ) =. ⎧ ⎪ ⎨. 1, λ < s(0) interpolation, s(0) ≤ λ ≤ s(Ns ) . ⎪ ⎩ 0, λ > s(Ns ). (21). The CDF can be acquired from the survival function: fˆc (λ) = 1 − fˆs (λ).. (22). Figure 1 shows the empirical CDF and survival curves of M 1(n) for two hypotheses with Ns = 2×106 , L=16, 32, 64, and SNR=0, 2, 4, 6 dB, and also reveals that the survival curves of M 1(n) given H0 are only dependent on window size L. Once a threshold value λ is assigned, we denote some useful probabilities: ∞ pM |H (R|H0 )dR = fˆs|H (λ) PF = λ. ∞. PD =. λ. 0. 0. pM |H1 (R|H1 )dR =fˆc|H1 (λ) .. (23). PM = 1 − PD = 1 − fˆc|H1 (λ) Also we denote CF and CM as the costs of false alarm and miss, respectively, and P1 and P0 for the a priori probabilities of H1 and H0 . Then the Bayes risk function is given as [4]: RB (λ) = P0 CF PF (λ) + P1 CM PM (λ).. c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. (24). Fig. 1. Empirical CDF and survival curves of packet detection function M 1. 254.

(7) IEICE Electronics Express, Vol.3, No.11, 249–256. Fig. 2. Receiver operating characteristic and minimax operating points (L=16, SNR=0, 1, 2, . . . 6 dB) Figure 2 illustrates the receiver operating characteristic (ROC) of both detection functions M 1 and M 2. ROC reveals the relationship between PF and PD as λ varies. As SNR value decreases or window size increases, the ROC curves moves toward left-top, which implies higher PD and lower PF can be achieved. That the ROC curves of M 1 and M 2 are overlapping implies the performances of M 1 and M 2 are equal. Assume P1 = P0 , the minimax equation is (25) CF PF (λ) = CM PM (λ), which has solutions of λ corresponding to the intersection points for different CM and CF ratios in Figure 2. Another approach to find the threshold value of minimum risk is directly drawing the risk functions as shown in Figure 3. Obviously the minimum points move as conditions change. Thus it is important to assign CF , CM , P1 and P0 for the selection of threshold values. Regarded as a watchdog in the OFDM receivers, packet detection shall operate at lower SNR required by lowest data transmission rate. In wireless LAN OFDM systems, the SNR required for 6 Mbps for PER=0.1 is about 5 dB [7]. Therefore SNR values lower than 5 dB shall be considered in the design of packet detection.. 3 c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. Conclusions. The paper verifies that the distributions of both detection functions M 1 and M 2 are not related to channels and preamble signals, but only involved with SNR and the sliding window size. The performances of M 1 and M 2 are the 255.

(8) IEICE Electronics Express, Vol.3, No.11, 249–256. Fig. 3. Risk functions of M 1 under three different conditions (L=32) same according to ROC curves. A strategy to decide threshold values in the detection is proposed: 1) decide SNR and window size according to system requirement, 2) build RV simulations for two hypotheses H0 and H1 , 3) generate empirical CDF and survival functions, 4) select a priori probabilities of hypotheses and costs of false alarm and miss, 5) draw risk functions and find threshold values corresponding to minimum points. Sufficient performance can be obtained with the newly proposed packet detection strategy.. c . IEICE 2006. DOI: 10.1587/elex.3.249 Received April 21, 2006 Accepted May 10, 2006 Published June 10, 2006. 256.

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