A Parameter Estimation Algorithm for Propagation ... - IEEE Xplore

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X.Lu}@huawei.com. Abstract—In this contribution, a new algorithm derived based on a two-layer evidence framework is applied to estimating parameters in the ...
A Parameter Estimation Algorithm for Propagation Channels based on Two-Layer Evidence Framework Xuefeng Yin, Yuandong Hu, Zhen Zeng, Junhe Zhou, Meisong Tong

Zhimeng Zhong and Stan X. Lu

College of Electronics and Information Engineering Tongji University Shanghai, China [email protected]

Huawei Technology Company Xi An, China {zmzhong, Stan.X.Lu}@huawei.com

Abstract—In this contribution, a new algorithm derived based on a two-layer evidence framework is applied to estimating parameters in the generic stochastic channel models from measurement data. Different from conventional high-resolution parameter estimation algorithms, e.g. the space-alternating generalized expectation-maximization (SAGE), the method proposed is applicable to extracting both the parameters of multiple components in individual realizations of channel impulse responses and the statistical parameters of the wide-sense-stationary channel. Furthermore, the proposed two-layer evidence framework can be readily generalized to accommodate appropriate channel features of interest as certain prior information. Preliminary simulation results demonstrate the effectiveness of the proposed algorithm when being used to estimate the cumulative distribution function of delay spreads of propagation channels. Index Terms—Propagation channel, parameter estimation, stochastic channel modeling and evidence framework

I. I NTRODUCTION Recently, high-resolution parameter estimation algorithms, such as the space-alternating generalized expectationmaximization (SAGE) [1], and the Unitary ESPRIT [2], have been used to estimate the parameters of propagation paths between a transmitter (Tx) and a receiver (Rx) based on the multi-path generic model [3]. The estimated path parameters are used to characterize the channel dispersion in delay, Doppler frequency, direction of arrival, direction of departure, and polarizations and establish stochastic channel models [4]. However, many empirical settings used in the parameter estimation, such as the number of paths to estimate, dynamic range for the powers of valid paths, delay estimation range, angular parameter estimation range, etc., can have significant impact on parameter estimation, and deteriorate the accuracy of the models deduced. In this contribution, a new parameter algorithm is proposed which is derived based on a two-layer evidence framework (EF) by taking into account the parametric characteristics of components in individual channel realizations, and the generic features that the channel parameters follow. Compared with the existing estimation algorithms, the proposed method solves two layers of inferences, i.e. the parameter estimates for multiple channel components in instantaneous snapshots, and the estimates of parameters in known constraints or assumptions.

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II. T WO - LAYER EF AND THE MAXIMUM - A - POSTERIORI (MAP) PARAMETER ESTIMATOR The composite delay spread is an important large-scale parameter for stochastic channel modeling. For the ith snapshot of propagation channel, the composite delay spread στ,i is calculated as the second-central-moment of the power delay spectral density of the channel [4]. According to the widelyadopted channel models in the 3GPP standards [5], [6] and in the WINNER report [4], delay spread is a random variable following the lognormal distribution LN (¯ στ , κστ ) with σ ¯τ and κστ denoting respectively the mean and the standard deviation of log10 (στ,i ). In this contribution, we are interested at deriving an algorithm to estimate τi, , i = 1, . . . , I,  = 1, . . . , Li which yield the delay spreads complying with lognormal distribution. The multi-layer EF can be used to collect evidences of events, which in our case are the posteriori probabilities of channel parameters of interest. In this study, we use an EF with two layers, namely L1 and L2 , and present iterative maxa-posteriori (MAP) estimators for the channel parameters. For layer Li , i = 1, 2, the notation Hi is used to represent the predefined generic model which is considered to be an a-priori information, and θi for the parameters in the generic model. The MAP estimators of the model parameters θ = [θ1 , θ 2 ] can be obtained by maximizing the joint posterior probability p(θ|Y, H1 , H2 ), i.e. p(θ|Y, H1 , H2 ) = p(θ1 |Y, H1 )p(θ2 |θ 1 , H2 ),

(1)

where Y being the realizations of the received signals. By increasing p(θ1 |Y, H1 ) and p(θ 2 |θ1 , H2 ) iteratively, the maximum of p(θ|Y, H1 , H2 ) can be achieved. It is worth mentioning that some empirical settings in e.g. the SAGE algorithm, such as the number of paths to estimate, the dynamic range of the path power, can be considered as predefined parameters of the generic model and included into the EF. These parameters can be viewed as parts of the specular path generic model and present as conditions in the layer that the generic model resides. As an example where the path number D is considered as a parameter of the generic specular-path model, the first layer L1 can be defined to be the evidence of the path parameters,

contains the delays τi, and the complex amplitudes αi, of the th path. Maximizing p(θ 1 |y, D, H1 ) can be solved by using the standard maximum likelihood (ML) method or the iterative approximation of the ML method, such as the SAGE algorithm [1]. The evidence in the second layer can be written as p(¯ στ , κστ |στ,i , i = 1, . . . , I, H2 ), where H2 is referred to as the lognormal distribution that the delay spreads usually follow. In our case, the evidence in the layer L2 is defined to be the inverse of the largest distance between the empirical cumulative probability function (CDF) F (στ ) of στ,i , i = 1, . . . , I and the CDF of LN (¯ στ , κστ ) with (¯ στ , κστ ) determined by fitting LN (¯ στ , κστ ) to F (στ ). When the lognormal distribution fits better to the empirical distribution, the evidence of Layer 2 increases. In this example, the value of D may influence the evidences in both layers. Thus, p(θ|Y, H1 , H2 ) needs to be maximized with respect to D. III. S IMULATION RESULT Simulations are conducted to evaluate the performance of the proposed algorithm. The number of paths D is a parameter adjustable when seeking for the maximum posterior probability of θ. In the simulation, 100 realizations of impulse response of a wide-sense-stationary (WSS) channel are generated. We found that the true delay spreads στ,i , i = 1, . . . , 100 do follow the lognormal distribution. In the two-layer EF, we use the lognormal distribution as the a-priori information in L2 . The two-layer EF is implemented for channel parameter estimation using an iterative algorithm with the following steps. Step 1. Specify initial value for the path number D Step 2. Estimate the parameters of individual components in multiple snapshots using the SAGE algorithm. Step 3 Compute the delay spread and calculate the empirical CDF of the delay spread Step 4 Conduct the hypothesis testing for the delay spreads following lognormal distribution if false then Change the value of D and go back to Step 2. end if if true then Output the final estimation results. end if Simulation results are depicted in Fig. 1, where the comparison among the true CDF of delay spread, the CDF obtained with the proposed algorithm, and the CDFs obtained with D = 50 and 100 is illustrated. It can be observed from Fig. 1 that the proposed algorithm provides the better estimate

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CDF F (στ,i )

i.e. p(θ1 |y, D, H1 ). Here, H1 is the specular path model, y = [y1 (t), y2 (t), . . . , yI (t); t ∈ [0, T )] represents the received signals in totally I snapshots, each snapshot lasting T seconds, and θ 1 = [αi, , τi, ;  = 1, . . . , Li , i = 1, . . . , I]

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Real synthetic data Obtained with D = 50 Obtained with D = 100 Proposed method (D = 295)

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Fig. 1. The CDFs of the true, estimated delay spreads and the theoretical CDF of a lognormal distribution fitted to the empirical CDF.

of the CDF of delay spread than the conventional methods with predefined D. The appropriate path number identified by using the algorithm equals 295. This result shows that the delay spread statistics can be accurately extracted by using the proposed EF-based parameter estimator. IV. C ONCLUSIONS A novel parameter estimation algorithm has been proposed which uses a two-layer evidence framework to maximize the joint posterior probability of both instantaneous and statistical channel parameters. Simulation results demonstrated that by using the proposed algorithm, the parameters of propagation paths are extracted in such a way that their statistics comply with the conditions specified in the framework. This algorithm can be generalized to including more evidence layers accommodating channel statistical feature and constraints of interest. ACKNOWLEDGEMENT This work is supported by the fundamental project of the Science and Technology Commission of Shanghai Municipality [10ZR1432700, Multidimensional power spectrum characterization and modeling for wide-band propagation channels], the fundamental research funds for the central universities [20090072120015, Time-Variant Channel Characterization, Parameter Estimation and Modeling], and China fundamental education basic research project [Polarization characterization of wireless propagation channel]. R EFERENCES [1] B. H. Fleury, M. Tschudin, R. Heddergott, D. Dahlhaus, and K. L. Pedersen, “Channel parameter estimation in mobile radio environments using the SAGE algorithm,” IEEE Journal on Selected Areas in Communications, vol. 17(3), no. 3, pp. 434–450, Mar. 1999. [2] J. Haardt, M.; Nossek, “Unitary esprit: how to obtain increased estimation accuracy with a reduced computational burden,” IEEE Transactions on Signal Processing, vol. 43, no. 5, pp. 1232 –1242, 1995. [3] A. Molisch, “A generic model for mimo wireless propagation channels in macro- and microcells,” IEEE Transactions on Signal Processing, vol. 52(1), pp. 61–71, Jan. 2004. [4] WINNER II Channel Models, IST-WINNER II Deliverable 1.1.2 v.1.2. Std., IST-WINNER2. Tech. Rep., 2007. [Online]. Available: http://www.ist-winner.corg/deliverables.html996 [5] Spatial channel model for Multiple Input Multiple Output (MIMO) simulations (Release 7), 3GPP TR25.996 V7.0.0 Std., 2007. [6] “3rd generation partnership project; technical specification group radio access network; further advancements for e-utra physical layer aspects (release 9),” 3GPP TR36.814 V9.0.0, 2007.