Improved line-of-sight/non-line-of-sight classification methods for pulsed ultrawideband localisation

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1 Published in IET Communications Received on 22nd May 2013 Revised on 25th October 2013 Accepted on 7th November 2013 ISSN Improved line-of-sight/non-line-of-sight classification methods for pulsed ultrawideband localisation Arash Abbasi, Huaping Liu School of Electrical Engineering and Computer Science, Oregon State University, Corvallis, OR 971, USA Abstract: Classification of line-of-sight LOS) or non-los NLOS) propagation is critical for most pulsed ultrawideband localisation systems. In this letter, first, the authors propose a two-dimensional 2D) LOS/NLOS classification scheme that uses skewness of the channel impulse/pulse response, which has not been used by existing work. The log-likelihood ratio of the 2D LOS/NLOS classification is used to determine the weights for localisation. Then, the authors derive an iterative Gauss Newton method to solve the location of the target, which provides a higher accuracy than the Gauss Newton method used in existing work. Finally, they investigate improved localisation by incorporating the statistics of NLOS bias as a result of LOS/NLOS classification into the cost function, and derive the statistics of the corresponding errors of location estimates. The proposed 2D LOS/NLOS classification scheme is compared with an existing 3D scheme to demonstrate its effectiveness. 1 Introduction Extensive work has studied non-line-of-sight NLOS) propagation effects on ultrawideband UWB) localisation [1 16]. These efforts mainly address two areas: techniques to classify LOS/NLOS conditions [1, 2] and methods to mitigate the effects of NLOS [3, 4]. LOS/NLOS classification commonly uses the statistical of the channel impulse response CIR) or channel pulse response CPR) [5 8], or range estimation [9]. For example, kurtosis, root-mean square rms) delay spread, and mean excess delay are used in [5, 6]; cumulative distribution function of the CPR along with kurtosis is used in [7]; rms delay spread, kurtosis and maximum amplitude of the received signal are used in [8]. The time-of-arrival TOA) error models for LOS/NLOS channels [9] can also be used for channel classification. The impacts of NLOS condition on ranging and localisation systems are studied extensively in [8 16]. Various types of measurements have been used for NLOS effect mitigation: TOA [12], time difference of arrival TDOA) [13], or hybrid schemes such as TOA and angle of arrival AOA) [14], TDOA and AOA [15], and TOA and received signal strength [16]. This paper focuses on LOS/NLOS classification for pulsed UWB localisation. Although extensive work in this area exists, one channel statistical parameter which has not been used in existing work is the skewness of the CIR or CPR. Skewness is a measure of symmetry, or more precisely, the lack of symmetry in the probability density function PDF). A distribution, or data set, is symmetric if it looks the same to the left and right of the centre point; that is, if the bulk of the data is at the left and the right tail is longer, then the distribution is skewed right, or positively skewed; if the peak is towards the right and the left tail is longer, then it is said the distribution is skewed left, or negatively skewed [17]. Since skewness characterises the amount and direction of skew departure from horizontal symmetry), it could be an effective parameter for LOS/NLOS classification; if the skewness of the impulse response of a UWB channel is high, then the channel is more likely to be LOS. We propose a two-dimensional 2D) LOS/NLOS classification scheme that uses skewness and demonstrate its effectiveness. This approach is much simpler than existing 3D methods. In addition, we develop two areas of improvements for pulsed UWB localisation in this paper. First, we develop an improved position estimation scheme that uses the results of the channel classification and an iterative Gauss Newton algorithm to be derived. Second, we incorporate the statistics of the NLOS bias into the cost function, and derive the statistics of the corresponding location estimation errors. 2 Preliminaries 2.1 Statistical used in LOS/ NLOS classification A simplified model for UWB CIR is described in [18] as ht) = L 1 l=0 a ldt t l ), where a l is the complex multipath gain and t l denotes the delay of the lth path. Channel statistical which have been used in existing work for LOS/NLOS classification include: Kurtosis k): The kurtosis of ht) is defined as [ ] [ 4 ] 4 E ht) m h E ht) m h k = [ ] 2 2 = s E ht) m 4 h h 680

2 where μ h and σ h are the mean and standard deviation of ht), respectively [19]; Mean excess delay t m )[20]; rms delay spread t rms )[20]. 2.2 Statistical approach for channel classification NLOS classification can be modeled as a binary hypothesis test. Let hypothesis H 0 denote LOS and H 1 denote NLOS. One approach is to make a binary decision based on a measured channel profile. Given the transmitted signal st), the received signal is expressed as rt) = L 1 l=0 a lst t l ) + nt). The maximum-likelihood ML) hypothesis test is written as H 0 f r H0 r H 0 ) x f r H1 r H 1 ) 1) H 1 where r is the vector representation of signal rt) in the Hilbert space, which is spanned by a complete orthonormal set of basis functions. The conditional PDF is evaluated as [21] f r Hi r H i ) = = f ru Hi r, u H i )du f r u,hi r u, H i )f u Hi r, u H i )du, i = 0, 1 where θ represents a vector of such as path gains and delays. Evaluation of the multidimensional integration in 2) is computationally extensive. One way to resolve this problem is to treat θ as if its statistics were unknown, resulting in a composite hypothesis test. A common approach to solve the composite hypothesis test is to estimate θ under the assumptions H 0 and H 1, and use it as if θ were known. In this case, the decision rule is given by H 0 f r uh0 r û 0, H 0 x f r uh1 r û 1, H 1 H 1 where û i denotes the estimate of θ under hypothesis H i. If a ML estimate of θ is obtained, the hypothesis test becomes [21] max u H 0 f r uh0 r u, H 0 ) x 2) 3) max f r uh1 r u, H 1 ) 4) H 1 u The likelihood ratio LR) test can be performed for LOS/NLOS classification if some statistical information of the multipath channel is available. Kurtosis k), the mean excess delay t m ) and the rms delay spread t rms ) can be used to capture the amplitude and delay statistics for the LOS and NLOS classification scenarios. If a priori knowledge of the statistics of k, t m and t rms are available under LOS and NLOS scenarios in a certain environment, then a LR test can be performed for the hypothesis test. 3 Using skewness in channel identification For a certain channel realisation ht), skewness of ht) is defined as s = E [ ht) m h ) 3 ] [ E ht) m h ) 2 ] 3/2) = m3 s 3 5) where μ 3 is the third central moment of ht) [17]. The main contributions of this paper are as follows: 1. Developing a 2D LR test that uses skewness. 2. Deriving a new method for assigning weights in the weighted least-square WLS) algorithm based on the LR test. 3. Incorporating NLOS bias as a result of NLOS calcification for improved localisation D LR test Skewness can be added to k, t m and t rms that has been used by existing schemes as explained in Section 2.2. With skewness in the LR test, a suboptimal approach is to assume that all these are independent. Thus the joint PDF is expressed as f u H0 u H 0 ) = f k H0 k H 0 ) f trms H 0 t rms H 0 ) f tm H 0 t m H 0 ) f s H0 s H 0 ) f u H1 u H 1 ) = f k H1 k H 1 ) f trms H 1 t rms H 1 ) f tm H 1 t m H 1 ) f s H1 s H 1 ) where f k H0 k H 0 ), f k H1 k H 1 ), f tm H 0 t m H 0 ), f tm H 1 t m H 1 ), f trms H 0 t rms H 0 ), f trms H 1 t rms H 1 ), f s H0 s H 0 ) and f s H1 s H 1 ) are the PDFs of the k, t m, t rms and skewness for LOS and NLOS cases, respectively. The 4D LR test becomes Kt rms, t m, k, s) = f u H 0 u H 0 ) f u H1 u H 1 ) This 4D LR test is complex and computationally intensive. A scheme that uses skewness and t rms, forming a 2D test, is developed in this section. In this case, the suboptimal joint PDFs for LOS/NLOS classification becomes f u H0 u H 0 ) = f trms H 0 t rms H 0 ) f s H0 s H 0 ) f u H1 u H 1 ) = f trms H 1 t rms H 1 ) f s H1 s H 1 ) Accordingly, the LR test is expressed as Kt rms, s) = f u H 0 u H 0 ) f u H1 u H 1 ) 3.2 New method for assigning weights in the WLS algorithm The Euclidean distance between the ith base station BS) and the mobile node MN) is d i p) = x x i ) 2 + y y i ) 2 6) 7) 8) 9) 10) 681

3 where p =[xy] T represents the unknown coordinates of the MN. Let the measured estimated) distance between the ith BS and the MN be modelled as { r i = d i + b i + n i, NLOS environment 11) d i + n i, LOS environment where b i is the non-negative NLOS bias, which is modelled as an exponential distribution with mean λ and n i is the Gaussian measurement noise [6]. In the WLS algorithm for localisation, the following cost function should be minimised [22] ep) r=1 2 b i d i p) r i 12) The weights β i s are chosen to reflect the reliability of the received signal at the ith BS. Therefore the coordinate estimate is determined as ˆp = arg min P ep) 13) The non-linear 2D optimisation problem given by 13) does not have a closed-form solution in general; in practice, an iterative approach is usually employed to find the solution. In this paper, an iterative Gauss Newton approach [23] is used for location estimation ˆp m = ˆp m 1 + H T ) 1H ˆp m 1 bh T ˆpm 1 ˆp m 1 b y H ˆp m 1 )ˆpm 1 14) with Jacobian matrix x x 1 y y 1 d 1 d 1 x x 2 y y 2 Hp) = p hp) = d 2 d 2.. x x N y y N d N d N 15) where p =[ / x) / y)] T and hp) isthesetofd i p) in 10). In [5], weights β i s are chosen to be g 1, if log 10 K i ) D 1 b i = g 2, g 3, if D 1, log 10 K i ) D 2 if log 10 K i ) D 2 16) where γ 1 < γ 2 < γ 3, and K i is the LR test result for the ith measurement. The γ 1 and γ 3 are the weights for the NLOS and LOS measurements, respectively; γ 2 is the weight for the ambiguity region when the log-lr, log 10 K i ), falls in between Δ 1 and Δ 2. The proposed algorithm quantises log 10 K i ) into N equal parts and assigns the weights more accurately based on the values of the log 10 K i ). A larger value of log 10 K i ) indicates that the channel between the BS and the MN has a higher probability to be LOS, and therefore a larger weight should be assigned in the WLS algorithm. The parameter β i in 12) is selected more accurately as g 1, if log 10 K i ) D 1 g 2, if D 1, log 10 K i ) D 2 b i =. g N, if log 10 K i ) D N 1 17) where N is the number of quantisation levels and K i is given by 9). 4 ML coordinate estimation incorporating LOS/NLOS classification results Instead of minimising the cost function in 12), a better optimisation is performed with respect to the unknown coordinates and the NLOS bias. After channel classification, the mean and variances of the ML coordinate estimate can be calculated. For simplicity, b i s in 11) are assumed to have the same exponential distribution, which means E[b i ]=λ, varb i )=λ 2 Also b i s and n i s are independent. Under these assumptions, the PDF of the error Fig. 1 PDF of the logarithm of the likelihood metric K for CM3 and CM4 a PDF of log 10 Kt rms, s) b PDF of log 10 Kt rms, t m, k) 682

4 can be written as [24] p ei = 1 ) l e 1/l) e i s 2 i /2l erfc s2 i le i 2 lsi 18) Table 1 Probability of uncertainty region Method sr Kre probability where λ is the mean of NLOS bias and s 2 i is the variance of the measurement noise. The joint conditional density function is expressed as where pr p, l) = N 1 exp r i d i ) 2 ) 2p si 2s 2 i { s i, 1 i N los s i = s 2 i + l 2, elsewhere { r i = r i, 1 i N los r i l, elsewhere 19) 20) 21) Taking logarithm of 19) and ignoring the constant that is irrelevant for minimisation, we have the following log-likelihood function Lr p, l) = log s i ) N The location of the ML is given by r=1 r i d i ) 2 2 s 2 i 22) ˆp = arg min p,l L p, l b) 23) For clarity of description, let us define Fp, l) = L p, l b ) 24) Expanding Fp, λ) atp 0 =[x 0 y 0 ], λ = λ 0, we have a 11 Dx + a 12 Dy + a 13 Dl a 14 a 21 Dx + a 22 Dy + a 23 Dl a 24 25) a 31 Dx + a 32 Dy + a Dl a 34 where {a ij } are given in the Appendix. Taking expectation over both sides of 25) yields a 11 E[Dx] + a 12 E[Dy] + a 13 E[Dl] E[a 14 ] a 21 E[Dx] + a 22 E[Dy] + a 23 E[Dl] E[a 24 ] a 31 E[Dx] + a 32 E[Dy] + a E[Dl] E[a 34 ] 26) Since E[a 14 ]=0,E[a 24 ] = 0 and E[a 34 ]=0,E[Δx]=0,E[Δy] =0,E[Δλ]=0. The variances of Δx, Δy and Δλ are obtained in the Appendix. 5 Simulation and experimental results This section presents results from simulation that uses the IEEE a channel model and experiments conducted in a realistic environment. 5.1 Simulation results Simulation results are obtained based on the IEEE a Channel Model. The configuration is as follows. Six BS s are positioned at locations x 1 =[ 16, 10], x 2 =[ 14, 10], x 3 = [ 15.5, 11], x 4 = [15, 9.5], x 5 = [0.5, 11] and x 6 = [0.5, 10], and the MN locations are fixed at mn1 = [ 2, 2], mn2 = [2, 2], mn3 = [2, 2], mn4 = [ 2, 2] and mn5 = [0, 0] all in metres). AWGN n i in 11) has a variance of 0.3 m 2 and b i has an exponential distribution with a mean of 3 m. Two channel models, indoor office LOS environment CM3) and Table 2 RMSE of position estimates with different methods centimeter) MN position Method mn1 mn2 mn3 mn4 mn5 a between BS1 and MN kre sr wsr kre with known b between BS2 and MN kre sr wsr kre with known c between BS1, BS2 and MT kre sr wsr kre with known d between BS1, BS3 and MT kre sr wsr kre with known e between BS1, BS2, BS3 and MT kre sr wsr kre with known

5 indoor office NLOS environment CM4) [25], are considered. For simplicity, the various evaluated methods are called as: 1. kre: the method in [5], in which k, t m and t rms are utilised in channel classification. 2. sr: the 2D LOS/NLOS classification method proposed in this paper. 3. wsr: sr method for classification with the improved weight assignment method derived in this paper applied in localisation. 4. kre with known : the ideal case of kre assuming and NLOS biases are known. In this method, 23) is optimised to find the ML coordinate estimates. 5. sr with known : the ideal case of sr assuming and NLOS biases are known. The uncertainty region in channel classification for different methods is depicted in Fig. 1. The exact values of the probability of falling in the uncertainty region with these two methods is summarised in Table 1; it is found that the proposed 2D sr method reduces the probability of falling into the uncertainty region by about 35% compared to the 3D kre method. These results show that skewness is an effective statistical parameter for channel classification. The root-mean-square errors RMSEs) of position estimates given by 12) with the five methods listed above in different channel conditions are given in Table 2 all in centimeters); for all cases, the RMSEs with the proposed 2D sr method are lower than that with the 3D kre method. The proposed method wsr when β i s in 12) are chosen more accurately achieves higher accuracy than the 2D sr method. Also, when the statistics of NLOS errors, that is, λ in the cost function of the localisation error given by 12), are properly exploited, the position estimation RMSE decreases significantly compared with other methods. For more clarity, the results are showed in Fig. 2. The performances of various methods are also compared in terms of two following metrics [26] 1. Probability of false alarm, P FA : The probability that a LOS channel is declared i.e. H 0 is chosen) when the channel is in fact NLOS. 2. Probability of detection, P D : The probability that the channel is classified as LOS i.e. H 0 is chosen) when the channel is in fact LOS. Fig. 2 RMSE of position estimates with different methods a One BS-MS channel is in NLOS b Two BS-MS channels are in NLOS c Three BS-MS channels are in NLOS 684

6 The channel detection threshold, T, can be calculated for the specific P FA in terms of the error function inverse as T = 2sH1 erf 1 1 2P FA ) + m H1 29) where m H1 and s H1 are the mean and standard deviation of the θ in H 1. P D is calculated as Fig. 3 Probability of detection against probability of false alarm P FA is calculated as P FA = +1 T f u H1 u H 1 )du 27) where f u H1 u H 1 ) is given by 6). Since the distributions of t rms, t m, k and skewness are modelled as log-normal, θ has also log-normal distribution. Thus +1 1 P FA = e u m H 1 ) 2 /2s 2 H 1 du T 2p sh1 ) = erf T m H1 28) 2 sh1 where erfx) = 2/ ) x p 0 e t2 dt. +1 P D = f u H0 u H 0 )du T +1 1 = e u m H 0 ) 2 /2s 2 H 0 du T 2p sh0 ) = erf T m H0 2 sh0 30) In the simulation, m H0, m H1, s H0 and s H1 ) for 3D LR test kre and 2D LR test sr methods are chosen as , , , ) and , , , ), respectively. Probability of detection P D against probability of false alarm P FA is depicted in Fig. 3. It is observed that the proposed 2D sr method has better performance in channel detection than the existing 3D kre method. Since the kre method requires extracting three statistical of the channel while the proposed method requires only two, the proposed method is much simpler. The computational complexity of the proposed 2D method is also lower than that of the 3D method, since only two statistical, rather than three, of the channel are needed for LR test. 5.2 Experimental results We have conducted extensive amount of experiments to verify the effectiveness of skewness for LOS/NLOS classification. Fig. 4 PDF of the logarithm of the likelihood metric K from experimental data a PDF of log 10 Kt rms, t m, k) b PDF of log 10 Kt rms, s) 685

7 5.2.1 Experimental environment and setup: The experiment took place in a vacant building with an approximate dimension of 76 m 40 m 7 m. There are some small metal poles inside this space, but the propagation will be LOS unless a transmitter receiver path is blocked intentionally. NLOS propagation scenarios in the experiment are created in two different ways: i) a person stands in front of the receiver and blocks the direct path; and ii) a metal object is placed right in front of the receiver to block the direct path. A pulsed UWB transmitter operating in the GHz frequency range is used as the transmitter. The transmitted signals under LOS and NLOS conditions are received by a set of antennas optimised for operation in this frequency range and then filtered and amplified, which is then sampled by a real-time sampling scope operating at 12.5 Gsps. The sampled data are transferred to a PC through Ethernet for further processing Results: Since the transmitted pulse shape is known and the pulse duration is very short <1 ns), instead of obtaining the statistical such as k, s, t m and t rms from the CIR, which can be done, a simpler but still effective and accurate method is to obtain these directly from the CPR as in some existing work [5 8]. LOS/NLOS classification results obtained by using 3D method k, t m and t rms ) and the proposed 2D LR method s and t rms ) are shown in Figs. 4a and b, respectively. The classification uncertainty region with the proposed 2D LR test is 18% less than that with the 3D LR test, demonstrating the effectiveness of using the skewness of the channel for LOS/NLOS classification. Note that the experimental environment does not fit exactly CM3 or CM4. Thus the difference between the simulation results in Fig. 1 and experimental results in Fig. 4 is expected. Nevertheless, the experimental results also verified the effectiveness of channel skewness for NLOS/LOS classification. 6 Conclusions We have developed a LOS/ classification scheme that uses skewness and rms delay spread of the CIR/CPR, and demonstrated its effectiveness. Compared with an existing 3D classification scheme that requires obtaining three, kurtosis, rms delay spread and mean excess delay, the proposed 2D scheme is simpler. The proposed 2D scheme also has a smaller classification ambiguity region than the 3D scheme; in a typical indoor channel, the probability of falling into the uncertainty region of channel classification is reduced by about 35% with the proposed scheme when IEEE a channel models are assumed. In experiments conducted in a large vacant building, we observed an 18% reduction in the classification uncertainty region with the proposed 2D method over a 3D method. We have also derived an improved localisation scheme that incorporates the statistics of NLOS bias as a result of LOS/ NLOS classification into the cost function, and derived the statistics of the corresponding errors of location estimates. Numerical results reveal that localisation RMSEs with the sr method are all lower than that with the kre method. Also, the weight-selection method derived in this paper significantly improves localisation performance, as evidenced by the RMSEs with sr and wsr methods. 7 References 1 Mucchi, L., Re, E.D., Landi, T.: Multi-level environment identification method for impulsive radio systems. Proc. IEEE Int. Conf. Ultra-Wideband ICUWB), September 2011, pp Decarli, N., Dardari, D., Gezici, S., D Amico, A.A.: LOS/NLOS detection for UWB signals: a comparative study using experimental data. Proc. IEEE Int. Symp. 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8 24 Yu, K., Guo, Y.J.: Improved positioning algorithms for NLOS environments, IEEE Trans. Veh. Technol., 2008, 57, 4), pp Karapistoli, E., Pavlidou, F.N., Gragopoulos, I., Tsetsinas, I.: An overview of the IEEE a standard, IEEE Commun. Mag., 2010, 48, pp Yang, S., Zhao, Q.: Probability distribution characterisation of fault detection delays and false alarms, IET Control Theory Appl., 2012, 6, 7), pp Appendix The in 25), {a ij }, are given as a 11 = F x p, l) x, a 12 = F x p, l) p=p 0,l=l 0 y a 13 = F xp, l) l, a 21 = F yp, l) p=p 0,l=l 0 x a 22 = F yp, l) y, a 23 = F yp, l) p=p 0,l=l 0 l a 31 = F lp, l) x, a 32 = F lp, l) p=p 0,l=l 0 y a = F lp, l) l p=p 0,l=l 0 a 14 = F x p 0, l 0 ) a 24 = F y p 0, l 0 ) p=p 0,da=l 0 p=p 0,l=l 0 p=p 0,l=l 0 p=p 0,l=l 0 F x p, l) x F x p, l) y F y p, l) y F y p, l) x F l p, l) x F l p, l) y F l p, l) l + N d i x x i ) 2 d 2 i d i r i ) x x i ) 2 d i r i ) d 2 i s2 i x x i )y y i ) r i s 2 i d3 i d i y y i ) 2 d 2 i d i r i ) y y i ) 2 d i r i ) d 2 i s2 i x x i )y y i ) r i s 2 i d3 i x x i ) ri 2 + s 2 i + lr i 2d i l 2 ) + 2d i l d i s 2 i + l 2 y y i ) r 2 i + s 2 i + lr i 2d i l 2 ) + 2d i l d i s 2 i + l 2 s 2 i + l 2 )2s 2 i r 2 i + 3l 2 + 2r i d i + 2lr i 2ld i d 2 ) i s 2 3 i + l 4l 2ls 2 i r i s 2 i lri 2 + l 3 + 2lr i d i +l 2 r i l 2 d i + d i s 2 i l i di 2 ) s 2 i + l ) where a 34 = F l p 0, l 0 ) F x p, l) F y p, l) F l p, l) = N d i r i )x x i ) 2 s 2 i d i r i )y y i ) 2 s 2 i 2ls 2 i r i s 2 i lr 2 i + l 3 + 2lr i d i + l 2 r i l 2 d i + d i s 2 i l i d 2 i s 2 i + l 2 ) 2 31) E[a i4 ], i = 1, 2, 3, in 26) are E[a 14 ]=0,E[a 24 ] = 0 and E[a 34 ] =0. If Δx, Δy and Δλ are uncorrelated, then the variances of ML coordinates and λ can be approximated. With some mathematical manipulations over 25), we have a 2 11E[D 2 x] + a 2 12E[D 2 y] + a 2 13E[D 2 l] E[f 14 ] a 2 21 E[D2 x] + a 2 22 E[D2 y] + a 2 23 E[D2 l] E[f 24 ] a 2 31E[D 2 x] + a 2 32E[D 2 y] + a 2 E[D 2 l] E[f 34 ] where {f ij } are written as see 34)) where x = x 0, y = y 0 and λ = λ 0. ) f 11 = E[a 2 14] f 23 = E[a 2 24] f = E[a 2 34] = N x x i ) 2 s 2 i d i y y i ) 2 s 2 i d i N j=n los +1 E 2ls2 i r i s 2 i lri 2 + l 3 + 2lr i d i + l 2 r i l 2 d i + d i s 2 i l i di 2 s 2 i + l 2 2 2ls 2 j r j s 2 j lrj 2 + l 3 + 2lr j d j + l 2 r j l 2 d j + d j s 2 j l j dj ) s 2 j + l 2 j 687

9 From 34), the approximate variance of the ML coordinate and the variance of λ can be obtained as f 14 a 2 12 a 2 13 f 24 a 2 22 a 2 23 var[d 2 f x] 34 a 2 32 a 2 a 2 11 a 2 12 a 2 13 a 2 21 a 2 22 a 2 23 a 2 31 a 2 32 a 2 a 2 11 f 14 a 2 13 a 2 21 f 24 a 2 23 var[d 2 a 2 31 f y] 34 a 2 a 2 11 a 2 12 a 2 13 a 2 21 a 2 22 a 2 23 a 2 31 a 2 32 a 2 a 2 11 a 2 12 f 14 a 2 21 a 2 22 f 24 var[d 2 a 2 31 a 2 32 f l] 34 a 2 11 a 2 12 a 2 13 a 2 21 a 2 22 a 2 23 a 2 31 a 2 32 a 2 35) 688