Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling
Abstract
:1. Introduction
2. System Model
3. Joint Angle and Delay Estimation (JADE)
3.1. Derivation of the CLF
3.2. Overview/Summary of IS for ML DA JADE
- Step (1): we start by evaluating the periodogram in (17) at all grid points where denotes the set of points obtained over the interval with a uniform sampling step .
- Step (2): we evaluate the so-called joint pseudo-pdfs [11] set of values over the above AoA-TD grid as follows:
- Step (3): we evaluate the marginal pseudo-pdf of as follows:
- Step (4): for , we compute the pseudo-CDF of as follows:
- Step (5): For , we generate R realizations . Then, we apply a linear interpolation to obtain the rth TD realization:
- Step (6): We evaluate the conditional pseudo-pdf of given for as follows:
- Step (7): similarly to Step 4, we compute the conditional pseudo-CDF as:
- Step (8): Similarly to Step 5, we generate R realizations for . Then, we apply a linear interpolation to obtain the rth AoA realization:
3.3. GWO vs. Combining|Embedding IS (IS-GWO|GWOEIS)
- Step (1): first, the wolves’ positions are initialized in the space according to one of the following cases (a) or (b).
- Step (1.a) [GWO]: The conventional GWO initially places the wolves pack of individuals at random positions in the search space where and . Hence, it requires larger packs and longer hunting (iterations) to catch the prey, i.e., find the correct angles of arrival (AoAs) and time delays (TDs) without guaranteeing global convergence.
- Step (1.b) [IS-GWO or GWOEIS]: Instead of random initial placement, IS-GWO and GWOEIS position the wolves at , stemming from the realizations generated in (29). Hence, even with relatively less realizations, this still guarantees global convergence, and it would always provide good-enough rough initialization values to GWOEIS to make the latter converge much faster and more accurately with relatively less hunting iterations.
- Step (2): at each iteration over the hunt duration , is evaluated over each individual in the pack, and the fittest three that better minimize it are identified as , , and , respectively.
- Step (3): the so-called convergence factor guiding the hunt is updated according to one of the following cases (a) or (b).
- Step (3.a) [GWO or IS-GWO]: is simply set to decrease linearly from 2 to 0 over the hunt duration . Therefore, the positions of the wolves to converge to local minima.
- Step (3.b) [GWOEIS]: To improve and speed up convergence, instead of a common convergence factor, each realization or individual in the pack is assigned one of its own that accounts both for the AoA and TD pseudo-CDFs calculated in Steps (4) and (7) of the IS technique (cf. Section 3.2) as follows:
- Step (4): For each lead gray wolf , , or , we generate two random values, and in for the calculation of two update coefficients (or with respect to each realization or individual r in the pack), and according to one of the following cases (a) or (b).
- Step (4.a) [GWO or IS-GWO]:
- Step (4.b) [GWOEIS]:
- Step (5): Let denote the distance between the lead wolf * and the rth individual in the pack (or search agent) across the j-th dimension. The lead wolves’ positions are then updated with respect to each realization r in the pack through intermediate variables according to one of the following cases (a) or (b).
- Step (5.a) [GWO or IS-GWO]:
- Step (5.b) [GWOEIS]:
4. Simulation Results
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
AoA | Angle of Arrival |
AWGN | Additive White Gaussian Noise |
CDF | Cumulative Distribution Function |
CFR | Channel Frequency Response |
CLF | Concentrated Likelihood Function |
CRLB | Cramér–Rao Lower Bound |
DA | Data-Aided |
DE | Differential Evolution |
DoA | Direction of Arrival |
GWO | Gray Wolf Optimization |
GWOEIS | Gray Wolf Optimization Embedding Importance Sampling |
IS | Importance Sampling |
IS-DE | Importance Sampling–Differential Evolution |
IS-GWO | Importance Sampling–Gray Wolf Optimization |
JADE | Joint Angle and Delay Estimation |
LLF | Log-Likelihood Function |
LS | Least Squares |
ML | Maximum Likelihood |
MSE | Mean Square Error |
MUSIC | Multiple Signal Classification |
NDA | Non-Data-Aided |
NMSE | Normalized MSE |
OFDM | Orthogonal Frequency-Division Multiplexing |
Probability Density Function | |
RMSE | Root Mean Square Error |
SAGE | Space-Alternating Generalized Expectation |
SIMO | Single Input Multiple Output |
SNR | Signal-to-Noise Ratio |
TD | Time Delay |
UAV | Unmanned Aerial Vehicles |
UMP | Unitary Matrix Pencil |
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Abdelkhalek, M.; Ben Amor, S.; Affes, S. Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling. Sensors 2024, 24, 5821. https://doi.org/10.3390/s24175821
Abdelkhalek M, Ben Amor S, Affes S. Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling. Sensors. 2024; 24(17):5821. https://doi.org/10.3390/s24175821
Chicago/Turabian StyleAbdelkhalek, Maha, Souheib Ben Amor, and Sofiène Affes. 2024. "Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling" Sensors 24, no. 17: 5821. https://doi.org/10.3390/s24175821
APA StyleAbdelkhalek, M., Ben Amor, S., & Affes, S. (2024). Data-Aided Maximum Likelihood Joint Angle and Delay Estimator Over Orthogonal Frequency Division Multiplex Single-Input Multiple-Output Channels Based on New Gray Wolf Optimization Embedding Importance Sampling. Sensors, 24(17), 5821. https://doi.org/10.3390/s24175821