Maximum Likelihood Time Delay Estimation Based on Monte Carlo Importance Sampling in Multipath Environment

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Introduction
The time delay estimation problem has always been a hot topic in wireless communications and is widely applied in radar [1], sonar [2], wireless communication system [3], and other fields.In multipath environments, the time delay estimation schemes for single snapshot narrowband signal using super resolution algorithms show good estimation performance.Generally, the super resolution algorithms mainly include two categories: the subspace estimation algorithm and the maximum likelihood (ML) estimation algorithm.To be specific, the subspace estimation algorithms include multiple signal classification (MUSIC) algorithm [4,5], the root of MUSIC algorithm [6], and estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm [7].Under the condition of single snapshot, these algorithms adopt smoothing in frequency domain in order to make autocorrelation matrix transformed into singular matrix.As a consequence, the effective bandwidth becomes narrow, which will lead to the result that mean square error (MSE) of the time delay estimation cannot be close to the Cramér-Rao bound (CRB).
The ML estimator is an asymptotically best estimator and has the best estimation performance under the condition of the limited samples.Since the multidimensional likelihood function is a nonlinear function of time delays and has many local maxima, the exact ML estimate needs multidimensional grid search.However, the corresponding estimation accuracy is limited by the search interval, and the computational complexity increases exponentially with the increasing dimension.In order to reduce the complexity, the realization method of the ML estimation can adopt the iterative algorithm, such as the expectation maximization algorithm [8], but the iterative algorithm requires that the initial value must be close enough to the unknown parameters which will be estimated.Otherwise, the iterative algorithm will converge to local maxima of the likelihood function.In addition, the iterative algorithm uses multiple International Journal of Antennas and Propagation initial values to improve the performance.Accordingly, the iterative algorithm converges to the global maxima at the cost of high computational complexity.For this reason, literature [9] adopted Monte Carlo (MC) importance sampling to determine the ML estimation of time delay under the condition of no data assistance.This algorithm does not need iterative calculation, but it can only be applied to the single path scenario.Literature [10] used MC importance sampling to complete ML time delay estimation under the condition of multipath.The algorithm needs the known reference signal in frequency domain and cannot be directly applied to general time delay estimation model.The iterative expectation maximization algorithm was investigated [11].However, it is sensitive to initialization value and has the problem of converging to local optimal value.In this paper, the time delay likelihood function under the condition of multipath is deduced by using channel frequency response.The normalized pseudoprobability density function is established.The importance function (IF) is given according to the properties of the normalized pseudoprobability density function and is sampled by using MC method.The time delays can be estimated by calculating the mean of the samples.Finally, the simulation results present the performance comparisons of the proposed algorithm, MUSIC algorithm, and the grid search ML algorithm.
The symbols and the operators used in the paper are as follows: [⋅] Τ denotes a transpose; [⋅] * represents a conjugated matrix; [⋅] H represents a conjugate transpose; [⋅] means an expectation.

Signal Model
In the process of electromagnetic wave propagation, the radio channel impulse response under the condition of the multipath can be modeled as where   is the number of the multipath components,   = |  |   is the complex fading coefficient of the th multipath component, |  | presents the amplitude,   means the phase and obeys a uniform distribution (0, 2) [5],   denotes the time delay for the th multipath component, and  represents the Dirac delta function.
Let us take the Fourier transform of (1).Then, the channel frequency response can be represented as It is common practice that channel frequency response is used for time delay estimation.The discrete sampling of the channel frequency response in different systems can be obtained by different methods.For example, multicarrier demodulation technique is used in Orthogonal Frequency Division Multiplexing (OFDM) system, and the received signal deconvolution method is used in direct sequence spread spectrum system, and so forth.
The discrete measurement data is obtained by sampling the channel frequency response () at  equally spaced frequencies.Considering the impact of the additive white noise in the measurement process, we can express the sampled discrete channel frequency response as where  = 0, 1, . . .,  − 1,   is the carrier frequency, Δ denotes the frequency sampling interval, and () represents the additive white noise with the zero mean and variance  2 .The vector form of the signal model can be represented as where x = [(0), (1), . . ., ( − 1)] T and H = [(0), (1), . . ., ( − 1)] T are the channel frequency response estimation vector and frequency response vector, respectively, T contains the modified complex fading coefficients,   =    −2    , and w = [(0), (1), . . ., ( − 1)] T is the additive Gaussian white noise vector.
According to (4) and noise related assumptions, the likelihood function of a single snapshot for time delay estimation can be expressed as The ML time delay estimate τ can be expressed as Note that (x | , ) is the joint distribution function of both  and , and  is a quadratic function.The analytical expression of  with respect to  can be obtained by calculating partial derivatives: We substitute ( 7) into ( 5), take the logarithm, and remove the constant parts.Then, likelihood function with respect to  can be obtained as

Time Delay Estimation Algorithm Based on Importance Sampling
In ML time delay estimation algorithms, the computational complexity of the multidimensional grid search increases exponentially with the increase of the dimension, and estimation accuracy is limited by search intervals.The iterative algorithms require that the initial values be close to the estimated unknown parameters.Otherwise, they cannot guarantee the convergence to the global maximum.The MC algorithm converts the search of the global maxima into the expectations of a random variable.In the process of practical calculations, the expectation of a random variable can be replaced by the sample mean.Importance sampling is the most commonly used sampling method in classical MC methods and approaches the global maximum.Furthermore, the computational complexity does not increase exponentially with the increase of dimension of the likelihood function.The key of importance sampling is the selection of importance function.In order to reduce the estimation error, importance function should be similar to the original probability distribution.In addition, samples should be easily extracted from the importance function for implementation convenience.
In the following sections, the global maxima of likelihood function will be first introduced.Then, importance function and the random sampling method are derived.Finally, the algorithmic steps and the computational complexity analysis are presented.

Global Maxima of Likelihood Function.
In order to make the sample averages approximate the global maxima of corresponding parameters, we can make the distribution function more accurate to adopt the exponential for the likelihood function () [10], which makes the estimation more accurate.The exponential likelihood function is defined as where  0 is a constant.The different values of  0 have significant influence on the distribution characteristics of   0 ().If  0 is sufficiently large, then   0 () will approach the Dirac delta function.
According to literature [12], τ can be expressed as where  0 → ∞;   represents the th time delay search region.
Let us define the normalized pseudoprobability density function: Then, τ can be simplified into the following form: According to the principle of importance sampling, (12) can be rewritten as where () is the importance function.

Importance Function.
The choice of importance function () will affect the estimation accuracy in the proposed algorithm.() is selected as close as possible to (), and  should be easily sampled from ().
For the matrix V H V, where I   represents the   ×   identity matrix.Then, where  1 is a constant coefficient and Define the importance function as where The size of  1 determines the distribution of the importance function samples.The corresponding sample is monotonically decreasing with respect to  1 .As a consequence, the choice of  1 should be moderate.
The MC method is an effective calculation method for the integral.It recasts the definite integral as a mathematical expectation of a random variable.As long as one can realize the sampling of the random variable, it can be effectively solved.By substituting the sampling average for the integral using the MC importance sampling method, we can express τ as where   () is a random sample for the importance function
The closed-form expression for the inverse function  −1  () is not easy to derive.We can obtain the sampling equation of   () for the th sample by 3.4.Algorithm Flow.According to the above derivation and analysis, the processes of the proposed algorithm can be summarized as follows.

Cramér-Rao Bound
The CRB gives the lower bound of the mean square error of an unbiased estimator.The following model gives the corresponding CRB of time delay estimation.Firstly, we define the unknown parameter vector [ 2  T ] T .The log-likelihood function is given by ln  ≜ ln  (x | , ) Re and  Im represent the real and imaginary parts of , respectively; that is,  Re = Re() and  Im = Im().The respective partial derivatives of ln  with respect to  2 , Re(), Im(), and , can be obtained as where Therefore, the partial derivative of ln  with respect to   is where By (25), we can obtain the following results: The Fisher information matrix (FIM) is ( T ), where  =  ln / [ 2  T ] T .According to FIM and [13], the CRB is given by

Simulation Result and Performance Analysis
5.1.Simulation Result.Consider an OFDM wireless system where the number of multipaths is two and we analyze and compare the proposed algorithm with the MUSIC time delay estimation algorithm, the grid search ML time delay estimation algorithm, and the CRB [5].Finally, the computational complexities of the above algorithms are analyzed and compared.The simulation parameter set-up of the OFDM system is shown in Table 1.
To begin with, define the mean square error as where x indicates the parameter estimate value obtained in the th simulation,  denotes the true value of the corresponding parameter, and  represents the number of estimates.
Simulation 1. Compare the likelihood function and the exponential likelihood function with  0 = 1 and  0 = 100.
As shown in Figure 1, compared with the normalized likelihood function, the two-dimensional bend surface is The multipath power is unchanged in the simulation process.
As shown in Figure 2, the MSE of the proposed algorithm, MUSIC algorithm, and grid search ML algorithm all decrease with increasing SNR.The MSE of the proposed algorithm approaches the CRB algorithm and basically corresponds to the performance of grid search ML algorithm.The reason lies in that the importance sampling algorithm uses a weighted average of the sample to approach the global maxima of the objective function.The single snapshot MUSIC algorithm needs to use smoothing in frequency domain to make the autocorrelation matrix singular, which leads to the loss of effective bandwidth and the reduction of the estimation accuracy.

Algorithm Complexity.
In this paper, the computational complexity of the proposed algorithm is (  ( +  log 2 ) + ( 2   +  3  + 2  +  2  +   )).The computational complexity of MUSIC algorithm is ((/2) 2 ((/2) + 1)2+(/2) 3 +(/2)  ), where   is the number of candidate delay grid points.The computational complexity of the grid search ML algorithm for time delay estimation is (( 2  + 3  + 2  +  2  +   )    ).We can see that the computational complexity of the proposed algorithm is slightly higher than that of MUSIC algorithm but is significantly lower than that of grid search ML algorithm.

Conclusions
In the multipath wireless communication scenario, to alleviate the computational burden of the ML time delay estimation for single snapshot, we have proposed a ML time delay estimation algorithm based on Monte Carlo importance sampling.Meanwhile, we have introduced a normalized pseudoprobability density function, an importance function structure method, a random sampling method, and CRB of the model as well as the analysis of the computational complexity.The algorithm has taken sample for importance function and has used the weighted average of the samples to calculate the time delay estimation.The simulations have shown that the proposed algorithm can significantly reduce the computational complexity and obtain the approximate performance with the ML algorithm of grid search.

Table 1 :
The simulation parameter set-up of the OFDM system.The number of samples  = 3000, doing 100 Monte Carlo experiments and taking  0 = 100 and  1 = 1.