Majorization-Minimization-Based Direct Localization Using One-Bit Channel Measurements

Direct localization or direct position determination (DPD) can outperform the more traditional angle and delay estimation based approaches, yet being less used in practice due to the requirement of aggregating raw data or measurements to a single processing point. To reduce the network burden, this letter considers one-bit quantized channel response data, and proposes a majorization-minimization (MM) based one-bit DPD (MO-DPD) algorithm to localize an orthogonal frequency division multiplexing (OFDM) signal source. First, the one-bit DPD is formulated as a maximum likelihood (ML) estimation problem, which is then iteratively solved using the MM approach. The proposed MO-DPD avoids iteratively estimating any nuisance parameters, leading to high computational efficiency. The numerical results show that the MO-DPD outperforms the baseline one-bit ML solver in terms of computational load, while efficiently converging to one-bit Cramér-Rao lower bound (CRLB) over wide range of signal-to-noise ratios (SNRs). Furthermore, we show that no more than three iterations are required to achieve high accuracy.

TD-based methods always suffer from accuracy deterioration, especially at medium and low signal-to-noise ratio (SNR) levels [4].
The so-called direct position determination (DPD) is an alternative approach that withholds all useful information to estimate the user position directly from the received signal, without explicit angle and/or TD estimation, having thus capability to outperform the classical methods.DPD was first proposed by Weiss in [5], which showed a significant improvement in positioning accuracy.In general, in addition to improved accuracy, another major benefit of the DPD approach is that it is agnostic to waveform and BS hardware configurations, for example, the modulation scheme and the number of antennas in the BS [6], [7].Following [5], many extensions have been investigated, particularly for orthogonal frequency division multiplexing (OFDM) signals [8], [9], [10].However, despite superior performance, the popularity of DPD is still constrained by the following factors: (i) requirement of a central processing server, and (ii) huge network traffic needed to transmit raw measurements from the observing nodes to the central server.Therefore, developing DPD algorithms with feasible communication and computational requirements becomes an imperative task to facilitate emerging 6G applications.
One well-practiced option to develop lightweight algorithms is to apply low-bit quantization [11], [12], [13], including the extreme case of one-bit data [14], [15], [16].Using quantized observations in DPD is a promising option for saving network traffic [17], with the one-bit processing approach recently adopted in [18].Furthermore, Weiss and Wornell proposed a one-bit DPD algorithm in [19], building on the assumption of zero-mean Gaussian distributed observations.Later, another one-bit DPD algorithm was developed in [20], strictly based on the one-bit model, which works harmonically with the channel state information (CSI) via classical maximum likelihood (ML) estimation.However, [20] focused only on optimizing the thresholds for one-bit quantization while still relying on resolving the classical ML function for the position estimation.Such approach involves estimating the underlying channel or path attenuations (nuisance parameters) based on, e.g., gradient methods, which results in computational deficiency.
In this letter, we focus on the direct position estimation problem in an OFDM system, including static BSs and a single source.Each BS is equipped with only one antenna.The single antenna configuration is a latent indication that only the time delay information is implicitly used for position determination.Based on the majorization-minimization (MM) strategy, which is, in general, widely applied for solving non-linear optimization problems [21], [22], an iterative onebit ML DPD algorithm using zero threshold and appropriate initialization is proposed for single source position determination.In the proposed algorithm, the original ML search is  replaced by continually minimizing and updating the majorizing function.Moreover, only the position estimation requires the search operation in each iteration, while the nuisance parameters are derived from a closed-form solution, which helps to eliminate redundant calculations required in [20].In addition, the proposed MM one-bit DPD (MO-DPD) algorithm is shown to converge to the optimal position within a few iterations, reaching the one-bit Cramér-Rao lower bound (CRLB) over wide SNR range.Furthermore, the accuracy is shown to asymptotically tend towards the CRLB for DPD with unquantized data, when the number of involved subcarriers is high.Finally, the remarkable benefits in terms of processing complexity are explicitly shown against the prior art.
Notations: Vectors and matrices are represented in lowercase and upper-case bold fonts, respectively.The Euclidean norm is denoted as • while {x } and {x }, respectively, define the real and imaginary parts of x.Moreover, (•) * , (•) T , and (•) H denote the conjugate, transpose, and Hermitian transpose, respectively.Furthermore, | • | stands for the absolute value while denotes the Hadamard product.For a vector ν, [ν] i stands for the i-th element in ν.For a matrix X, [X] i,g and [X] g denote its (i, g)-th element and g-th column, respectively.Finally, j is the imaginary unit that verifies j 2 = −1, and Δ = is used for definitions.

II. PROBLEM FORMULATION
Consider a scenario with L synchronized BSs as illustrated in Fig. 1, with the vector u l ∈ R D denoting the position of the l-th BS.Note that D ∈ {2, 3} denotes the dimensionality of the scenario.We further assume that there is also a stationary source located at p 0 ∈ R D in the Cartesian coordinates, which transmits an OFDM signal with N subcarriers over bandwidth of B. Specifically, we consider the transmission of an OFDM pilot or reference signal, commonly consisting of constant modulus QPSK samples at the pilot subcarriers.The corresponding received signal reads where s(n) ∈ {e j π/4 , e j 3π/4 , e j 5π/4 , e j 7π/4 } denotes the known QPSK-based reference signal sample at subcarrier n, TD τ l (p 0 ) is given by τ l (p 0 ) Δ = u l − p 0 /c and c is the signal propagation speed which is equivalent to the speed of light.Furthermore, n ∈ {0, . . ., N − 1}, Δf = B /N denotes the spacing of adjacent subcarriers, and b l is an unknown complex coefficient that represents the path attenuation from the source to the l-th BS.Furthermore, noise wl (n) is independent of the source signal and follows a complex Gaussian distribution with zero mean and unknown variance σ 2 .
Using knowledge of the reference signal samples, the CSI at the n-th sub-carrier is estimated at the l-th BS through r l (n) = s * (n)r l (n).This results in the unquantized channel response measurement or observation model of the form where the effective noise w l (n) is still complex Gaussian with zero mean and unknown variance σ 2 .In general, such preprocessing can be considered as frequency domain matched filtering or correlation, while other matched filtering based preprocessing methods have been considered in DPD context, for example, in [20].
Next, r l (n) is further quantized via comparing with the threshold 0 in order to reduce the network traffic needed for aggregation of raw measurements.The corresponding quantized observation y l (n) can now be written as where the comparator Q(•) is defined as with The complex one-bit quantizer Q(•) preserves coarse phase information, serving as basis for the positioning task.
The quantized data {y l (n)} N n=0 can be written in a vector form as Then, the task is to estimate the location of the source p 0 using the quantized data {y l (n)} LN l=1,n=0 .

III. MAJORIZATION-MINIMIZATION BASED MAXIMUM LIKELIHOOD ESTIMATION The likelihood function of {y
where T and Φ(•) denotes the cumulative distribution function (CDF) of the standard normal distribution, which is defined as Φ(x ) We can next define two matrices X R ∈ R N ×L and X I ∈ R N ×L for simplifying the presentation, with the items on the (n + 1)-th row and l-th column being of the form (8) where bl √ 2b l /σ.Thus, the ML estimation of p 0 can be achieved by minimizing the negative log form of (7), that is, Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where ξ [p T 0 , b1 , b2 , . . ., bL ] T and Since the cost functions of the DPD problem are always nonconvex [23], the implementation of ( 9) is generally based on a grid search over all possible instances of ξ .We are only interested in estimating p 0 among variables in ξ , while treating { bl } L l=1 as nuisance parameters.In this situation, directly searching for ξ is computationally inefficient and unnecessary, even if { bl } L l=1 can be estimated by the Newton method with guaranteed global optimality when p 0 is given.Therefore, to focus on the ML estimation of p 0 , we resort to the majorization-minimization approach in the following.
To this end, according to [22], the (i+1)-th MM iteration cycle for the estimation of ξ can be expressed as where Note that the equality in the above inequality happens when ξ = ξ i .Thus, the MM iteration cycle in (11) approaches the ML estimation of ξ as long as the initial value L( ξ 0 ) is smaller than all local minima in L( ξ ).Next, we recall the following lemma in [14] to construct a suitable majorizing function of L( ξ ).Lemma 1: Let h(x ) − ln(Φ(x )).With the second-order Taylor theorem [24], the following inequality holds for any where the equality happens when x = v and Applying the result of Lemma 1, an appropriate H ( ξ , ξ i ) can now be expressed as where , and first focus on the (n+1, l)-th cumulative term h(x ln , where the constant term h 0 = − 1 2 [h (x i ln )] 2 + h(x i ln ) has no impact on the minimization of the majorizing function.Furthermore, recalling (8), it can be derived that {y l (n)}x ln = { bl e −j 2πnΔf τ l (p 0 ) }. Thus, after defining the function g(x ) x − h (x ), h(x ln , x i ln ) can be rewritten by eliminating h 0 and the coefficient 1  2 , expressed next as h(x ln , Next, based on (17), the expression of h(X R ( ξ ), X R ( ξ i )) can be further simplified as where the (n +1)-th element of the vector g With ( 18) and ( 19), H ( ξ , ξ i ) can be finally expressed through the norm of the complex vector as where Note that in (20) the form of H ( ξ , ξ i ) is similar to the DPD cost function with unquantized data, which focuses on estimating the target position.Hence, referring to [25], the update of ξ i+1 in the proposed MO-DPD algorithm can be finally expressed as Importantly, one feasible way of obtaining the initial value of the parameter vector, ξ 0 , is to momentarily treat {y l } L l=1 as unquantized data.Thus, building on (22), this is expressed as It is also important to note that in both ( 22) and ( 23), bl can be estimated through a closed-form expression, and thus the gradient-based iteration utilized for example in [20] is avoided.Furthermore, the stopping criteria for MO-DPD iterations is defined through the position residual as pi+1 0 − pi 0 ≤ ε r .Here, ε r is a sufficiently small positive number which indicates the feasibility tolerance for the position residual.Next, before proceeding to the numerical results, we evaluate the baseline computational complexity of the proposed algorithm.Complexity is here measured by the order of the number of multiplications, while actual run time assessments will also be provided along the upcoming numerical results.From (22), it can be seen that the computational resource is mainly consumed by searching for the optimal position pi 0 in each MO-DPD step.Let N i denote the number of iterations and let N s stand for the number of grid points.Then, the total computational complexity of the proposed algorithm is O(N s N i LN ), while that of the ONEBIT-DPD reference method proposed in [20] is O(N s LNN g N d ).The latter is stemming from the fact that in [20], the attenuation factors are constantly estimated for each candidate position via N g -iterations gradient descent algorithm.Furthermore, from (13), it can be seen that gradient calculation requires intractable integral and exponent operations, resulting in N d multiplications.Thus, the proposed MO-DPD exhibits remarkable computational savings when N g N d is much larger than N i , which is commonly the case in actual scenarios.

IV. NUMERICAL RESULTS
We next evaluate the performance of the proposed MO-DPD and compare it with that of ONEBIT-DPD from [20], which is also based on ML estimation.Root mean square error (RMSE), expressed as is the estimated target position for the n Mc -th trial.Both DPD algorithms are also compared with the applicable one-bit CRLB [20], which reflects the theoretically achievable performance taken as a benchmark.The position tolerance ε r is set to 0.1 m.
Let L = 3 BSs be located at [0, 0] m, [500, 0] m and [0, 500] m, respectively, while a source located at [153,322] m transmits an OFDM signal with a bandwidth of B = 5 MHz.Then, the BSs obtain CSI through the known training sequence.The channel amplitude is randomly generated following the Rayleigh distribution, and the corresponding phase is selected from the uniform distribution U (0, 2π).Finally, the number of Monte Carlo trials is N Mc = 1000.
First, we demonstrate the capability of the proposed MO-DPD algorithm to avoid local minima, when properly initialized as shown in (23).To this end, Fig. 2(a) shows the RMSE convergence of MO-DPD versus the number of iterations when N = 128 and SNR = −5 dB.It can be seen that only 3 iterations are enough for the proposed algorithm to achieve the one-bit CRLB.Fig. 2(a) thus indicates that despite being an iterative system, the proposed algorithm is an efficient representation of ML estimation, being able to reach the global optimum when properly initialized through (23).Furthermore, to explicitly demonstrate the convergence rate of the proposed algorithm, Table I shows the averaged numbers of needed iterations versus SNR when N = 128, over a wide SNR range.It can be seen that the number of iterations is lower for the proposed method, compared to the reference ONEBIT-DPD method.Additionally, only two iterations are typically needed with the proposed method, while three iterations suffice with negative SNR of -5 dB.Table I thus validates and highlights the fast iteration capabilities of the proposed algorithm.
In the next experiment, while maintaining N = 128, we investigate the estimation performance of the proposed algorithm under different SNR levels.From Fig. 2(b), it can be seen that the RMSE of the proposed MO-DPD can achieve the one-bit CRLB for most SNR values.Only in the very low SNR region, i.e., SNR< −5 dB, the estimation deviation becomes notable.The results in Fig. 2(b) indicate that the proposed MO-DPD is an accurate implementation of the one-bit ML estimator for a wide SNR range of practical relevance.
The SNR is next fixed to 5 dB, and we examine the MO-DPD performance for different numbers of subcarriers N. From Fig. 3, it can be observed that the estimation accuracies of MO-DPD and ONEBIT-DPD show a mutually similar tendency.The RMSEs of both MO-DPD and ONEBIT-DPD reference method decrease as N increases while gradually tending towards the ∞-bits CRLB, which corresponds to DPD in the same scenario based on unquantized data.To this end, Fig. 3 shows that the proposed MO-DPD approaches the optimum position estimation performance when N grows, thus achieving similar positioning accuracy to conventional DPD algorithms, such as [25], despite observing only coarse one-bit data.This is a relevant practical finding since the subcarrier numbers in for example 5G cellular systems are in the order of 512-2048.
Finally, we compare the actual processing execution times of MO-DPD and ONEBIT-DPD on a given computing device1   while using the same parameters as in Fig. 3. To this end, Fig. 4 shows and compares the average processing times of the algorithms at each grid point.The number of iterations for both algorithms is fixed to two.It can be seen that the required processing time of the proposed MO-DPD is significantly lower than that of the ONEBIT-DPD.Furthermore, the gap increases dramatically with the increasing number of subcarriers.This is because ONEBIT-DPD needs to estimate the involved attenuation factors, which infers lot of additional processing.The results in Fig. 4 are well inline with the theoretical analysis at the end of Section III, while highlighting the remarkable computational efficiency of the proposed MO-DPD algorithm.
V. CONCLUSION In this letter, we proposed a computationally efficient Maximum Likelihood DPD algorithm with quantized onebit channel response data to locate an OFDM signal source.Using the majorization-minimization approach, the proposed MO-DPD iteratively estimates the source position, with all parameters being updated using a Least Squares function of position at each cycle.Thus, the gradient processing for nuisance parameter estimation required in conventional onebit DPD algorithms is eliminated, resulting in a substantial reduction in complexity.Furthermore, the numerical results show that the proposed MO-DPD algorithm achieves an accuracy similar to that of the state-of-the-art one-bit DPD techniques.The MO-DPD approach pushes thus the DPD concept closer towards practical utilization, by providing high localization accuracy with reduced complexity.
the main performance metric where N Mc is the number of Monte Carlo trials and p[n Mc ] 0

Fig. 2 .
Fig. 2. (a) Example RMSE convergence of the proposed MO-DPD versus the number of iterations.(b) RMSE position estimation performance of the proposed MO-DPD versus SNR while also showing the one-bit CRLB for reference.

Fig. 3 .
Fig. 3. Position estimation performance of different algorithms versus the number of subcarriers N at SNR = 5 dB.The CRLB without quantization is also shown for reference.

Fig. 4 .
Fig. 4. Averaged processing times of different algorithms versus the number of subcarriers N at SNR = 5 dB, and for fixed iteration count of two.

TABLE I AVERAGE
NUMBERS OF ITERATIONS ON EACH GRID POINT TO CONVERGE FOR DIFFERENT ALGORITHMS VERSUS SNR, N = 128