Hi @Thien Hieu,
Thank you for your inquiry regarding mathematical references for Pilot Average and Virtual Pilot approaches in channel estimation. I understand your need for proper academic sources to cite these methodologies beyond the MathWorks documentation.
Based on my research of the literature, I can provide you with the following comprehensive reference framework:
Primary Mathematical Foundation:
The fundamental mathematical basis for pilot-assisted channel estimation stems from the least squares estimation theory. The key seminal paper you should reference is:
Van de Beek, J.-J., O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson
"On Channel Estimation in OFDM Systems." IEEE 45th Vehicular Technology Conference, Volume 2, 1995, pp. 815-819.
This paper establishes the mathematical foundation for pilot-based channel estimation using the least squares approach, where the channel estimate at pilot locations is computed as:
h-hat of k comma l equals y of k comma l divided by x of k comma l where y of k comma l represents received pilot symbols and x of k comma l represents known transmitted pilot symbols.
Pilot Averaging Mathematical Theory:
The pilot averaging approach is mathematically grounded in statistical signal processing theory for noise reduction. Key references include:
Kay, Steven M Fundamentals of Statistical Signal Processing: Estimation Theory, Volume 1. Prentice Hall, 1993. (Chapters 4-5 provide the theoretical framework for averaging estimators)
3GPP TS 36.141 “Evolved Universal Terrestrial Radio Access (E-UTRA); Base Station (BS) Conformance Testing." Annex F.3.4 - This technical specification, while not a research paper, formally defines the 'TestEVM' pilot averaging method used in LTE systems.
The mathematical principle involves computing: h-hat subscript avg of k equals one over the magnitude of S multiplied by the sum of h-hat of k comma l for all l in S
where S represents the set of pilot locations within the averaging window.
Virtual Pilot Theory and References:
The virtual pilot concept addresses the boundary interpolation problem in channel estimation. The mathematical approach involves extending the pilot grid using plane fitting techniques. Relevant references include:
Morelli, Michele, and Umberto Mengali. “A comparison of pilot-aided channel estimation methods for OFDM systems." IEEE Transactions on Signal Processing 49.12 (2001): 3065-3073.
Colieri, Stefano, et al. “A study of channel estimation in OFDM systems." IEEE 56th Vehicular Technology Conference Proceedings, Volume 2, 2002, pp. 894-898.
Li, Ye (Geoffrey), Leonard J. Cimini Jr., and Nelson R. Sollenberger “Robust channel estimation for OFDM systems with rapid dispersive fading channels." IEEE Transactions on Communications 46.7 (1998): 902-915.
Advanced Virtual Pilot Implementation:
For the specific virtual pilot algorithm described in the MathWorks documentation (involving 10 closest pilots, cross-product plane fitting), you should reference:
Agarwal, Ankit Ashok “Pilot Based Channel Estimation For 3GPP LTE downlink." Master's Thesis, University of Texas at Arlington, 2011. Available at: https://mavmatrix.uta.edu/electricaleng_theses/265/
Recent work on virtual pilots in MU-MIMO: Check the IEEE paper "Virtual Pilot-Based Channel Estimation and Multiuser Detection for Multiuser MIMO in LTE-Advanced" (IEEE Xplore Document ID: 7881966) for extended applications.
Mathematical Interpolation Framework: The interpolation methods (linear, cubic) used with virtual pilots are based on standard numerical analysis.
Reference:
Press, William H., et al. Numerical Recipes in C: The Art of Scientific Computing, 2nd Edition. Cambridge University Press, 1992. (Chapter 3: Interpolation and Extrapolation)
Recommendation for Your Citations:
For comprehensive coverage, I recommend structuring your references as follows:
- Theoretical Foundation: Van de Beek et al. (1995) for least squares channel estimation
- Pilot Averaging: Kay (1993) for statistical averaging theory + 3GPP TS 36.141 for LTE-specific implementation
- Virtual Pilots: Li et al. (1998) for boundary problem motivation + Morelli & Mengali (2001) for comparative analysis
- Implementation Details: Agarwal (2011) thesis for practical LTE implementation
Important Clarification on TS 36.141: You mentioned finding TS 36.141 referenced but being unable to locate the "papers that introduce these approaches." This is because TS 36.141 is a 3GPP technical specification document, not a research paper. It standardizes the implementation of channel estimation methods for LTE systems but doesn't introduce the underlying mathematical concepts - those come from the academic papers I've listed above. The pilot average and virtual pilot approaches evolved from decades of OFDM research, with the fundamental theories established in the 1990s academic literature, then later standardized by 3GPP for practical LTE implementation.
Therefore, you should cite both the original research papers for mathematical foundation and TS 36.141 as the implementation standard.
I hope this comprehensive reference list helps with your research. The combination of theoretical papers and implementation standards should provide you with both the mathematical rigor and practical context you're seeking.
References compiled from IEEE Xplore, academic databases, and 3GPP technical specifications. For the most current versions of 3GPP standards, please visit https://www.3gpp.org
