⚡ electrical engineering

Tracking performance of RLS algorithms in WSSUS channels

This paper presents a theoretical framework for analyzing and predicting the tracking performance of exponential and sliding-window Recursive Least Squares (RLS) algorithms in wide-sense stationary uncorrelated scattering (WSSUS) channels by deriving general mean square deviation formulas based on power spectral density moments, which are then validated through numerical examples across various channel models and algorithm extensions.

Original authors: Yuriy Zakharov, Lu Shen

Published 2026-08-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuriy Zakharov, Lu Shen

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to catch a butterfly with a net. If the butterfly sits perfectly still, catching it is easy. But if the butterfly is fluttering wildly, your net needs to be smart enough to predict where it will be a split second from now. This is the daily struggle of modern wireless communication. Every time you send a text or stream a video, your signal travels through the air, bouncing off buildings, trees, and cars. This journey is like a chaotic dance where the path the signal takes changes constantly. Engineers call these changing paths "time-varying channels." To keep your connection strong, computers use special math tools called "adaptive filters" to guess the current shape of the path and correct the signal in real-time. The big question is: how good is this guess? If the guess is too slow, the signal gets garbled; if it's too complex, the computer gets overwhelmed. Scientists have been trying to figure out the perfect balance for years, but the math to predict exactly how well these tools work in a messy, changing world has been notoriously difficult.

This paper dives into that messy world to build a better map for predicting how well these "smart nets" (specifically a type called Recursive Least Squares, or RLS) perform. The authors, Y. Zakharov and L. Shen, tackle a problem where previous maps were incomplete. They realized that while old formulas could predict how much noise messed up the signal, they missed a crucial part of the puzzle: the error caused simply by trying to fit a straight line to a curving road. The paper introduces a new way to calculate this "tracking error" by looking at the "speed" of the channel's changes, described by something called a Power Spectral Density (PSD). They test their new formulas against three different types of "butterfly movements": a uniform spread, a Jakes' pattern (common in mobile phones), and an autoregressive (AR) pattern.

The authors found that their new approach works like a high-precision ruler. They derived simple formulas that predict the "Mean Square Deviation" (MSD)—a fancy way of saying "how far off the guess is"—for different algorithms. They discovered that for standard algorithms, the error is a mix of noise and a "modeling error" (the error from using a simple model for a complex reality). However, for more advanced algorithms that use "delays" (looking slightly into the future or past to make a better guess), the modeling error becomes the dominant factor. By using a technique involving "Legendre polynomials" (which are just fancy mathematical shapes used to draw curves), they showed that these advanced algorithms can drastically reduce the error. In their simulations, the new formulas matched the computer results almost perfectly, with differences as small as 0.02 dB in some cases and never more than about 2.3 dB even in the trickiest scenarios.

The paper also explicitly argues against relying on older, simpler formulas for these advanced, delay-based algorithms. Previous methods often ignored the "modeling component" of the error because it was small for basic tools. But the authors show that when you use these super-smart, delay-based tools, that ignored error actually becomes the biggest problem. If you use the old formulas, you will underestimate how much error is actually there. They also clarify that their method works best when the channel changes slowly enough that a few terms of a mathematical "Taylor series" (a way of approximating curves) are sufficient. For extremely fast changes, the paper suggests that more terms would be needed for perfect accuracy, but for most practical scenarios, their simplified approach is spot on.

To visualize this, think of the channel as a wiggly snake moving across a screen. A basic algorithm is like a child trying to trace the snake with a pencil, but the child is only allowed to draw straight lines. The "approximation error" is the gap between the straight line and the snake's curve. The "modeling error" is the child's mistake in thinking the snake is a straight line. The authors' new math tells us exactly how big that gap will be based on how fast the snake is wiggling. They found that if the child is allowed to look a tiny bit ahead (a "delay"), they can draw a curve that fits the snake much better. But to do this, you have to account for the fact that the child's model is still an approximation. Their formulas act like a crystal ball, letting engineers know exactly how much "wiggle room" they have before the connection breaks, without needing to run thousands of slow computer simulations every time they change a setting.

The paper concludes that while they made some helpful simplifications—like treating the digital steps of the computer as a smooth, continuous flow of time—their results are robust. They validated their findings by running simulations where the "snake" moved at different speeds and followed different patterns. The match between their math and the simulation was so close that the difference was often less than the width of a single pixel on a graph. This means engineers can now use these simple formulas to design better communication systems, knowing exactly how their filters will behave in the real, wiggly world of wireless signals.

Technical Summary: Tracking Performance of RLS Algorithms in WSSUS Channels

Problem Statement
Recursive Least Squares (RLS) adaptive algorithms are extensively employed for estimating linear time-varying systems, such as communication channels. However, their tracking performance is contingent upon noise levels, the characteristics of channel time variations, and algorithm parameters. While existing literature provides performance analysis for RLS algorithms under specific conditions—often modeling channel variations as first-order Markov processes or utilizing integral expressions for the Mean Square Deviation (MSD) that are difficult to apply in practice—there is a gap in analyzing these algorithms for channels described by Wide-Sense Stationary Uncorrelated Scattering (WSSUS) random processes with arbitrary Power Spectral Densities (PSD). Furthermore, previous unified expressions for MSD often neglect the "least squares (LS) modelling component" of the error. This omission renders existing formulas inadequate for predicting the performance of advanced adaptive algorithms, such as non-causal RLS with delays or basis-expansion models (e.g., SRLS-L), where this modelling component becomes significant.

Methodology
The paper proposes a novel analytical approach to evaluate the tracking performance of RLS algorithms (specifically Exponential RLS and Sliding-Window RLS) in WSSUS channels. The methodology involves the following key steps:

Channel Modeling: The channel is modeled as a discrete-time system with time-varying impulse response taps derived from continuous-time, zero-mean, stationary Gaussian random processes. The time variations are characterized by a normalized PSD, $G_h(\omega)$ , and spectral moments ( $\lambda_{2p}$ ). Three specific PSDs are analyzed as examples: Uniform, Jakes', and Autoregressive (AR).
Error Decomposition: The Normalized Mean Square Deviation (NMSD) is decomposed into three distinct components:
- Noise Component ( $\epsilon_n$ ): Dependent on the signal-to-noise ratio (SNR) and algorithm parameters (window length or forgetting factor).
- Modelling Component ( $\epsilon_m$ ): Arising from the mismatch between the LS signal model (which assumes time-invariance or specific basis functions) and the actual time-varying channel. This component is treated as an equivalent stationary noise derived from the "nuisance" parameters not covered by the LS model.
- Approximation Component ( $\epsilon_a$ ): Resulting from the error in estimating the channel impulse response at a specific time instant due to the finite observation window and delay.
Analytical Derivation: Using Taylor series expansions of the channel taps and spectral moments, the authors derive general formulas for the modelling and approximation components.
- For SRLS, the observation window is rectangular, and Legendre polynomials are used as basis functions.
- For ERLS, the window is exponential, and Laguerre polynomials are utilized.
- The analysis extends to delayed algorithms (dSRLS, dERLS) where the estimate is assigned to a time instant $t_0$ relative to the last sample, and the SRLS-L algorithm, which uses Legendre polynomials to approximate channel variations.
Validation: Theoretical results are validated through numerical simulations where channel realizations are generated to exactly match the predefined PSDs, ensuring statistical accuracy.

Key Contributions

General Analytical Framework: The paper introduces an approach to analyze RLS tracking performance in WSSUS channels using spectral moments of the PSD, applicable to arbitrary PSDs.
General Formulas: Simple, closed-form expressions for the NMSD are derived for classical RLS, delayed RLS (dSRLS, dERLS), and the SRLS-L algorithm. These formulas explicitly separate the noise, modelling, and approximation error components.
Specific PSD Applications: The general formulas are specified for Uniform, Jakes', and AR PSDs, providing practical expressions for common channel models.
Optimal Delay Identification: The analysis identifies optimal delays for delayed algorithms (e.g., $t_0 = T/2$ for dSRLS and $t_0 = \alpha^{-1}$ for dERLS) that minimize the approximation error by eliminating the first-order derivative term in the error expansion.
Modelling Component Inclusion: Unlike previous works, this approach accounts for the LS modelling component, which is crucial for accurately predicting the performance of algorithms with improved tracking capabilities (like non-causal filters).

Results
Numerical simulations demonstrate a strong agreement between the derived analytical formulas and simulation results:

Accuracy: For classical algorithms (SRLS, ERLS), the discrepancy between theory and simulation is minimal (often < 0.03 dB).
Delayed Algorithms: For dSRLS and dERLS, the theoretical predictions remain accurate, with maximum discrepancies generally under 0.5 dB, except in specific cases of very fast channel variations where higher-order Taylor series terms might be required.
SRLS-L Performance: The SRLS-L algorithm, which incorporates basis expansion, shows significantly improved tracking performance (approx. 24 dB improvement over dSRLS at optimal delay) due to the reduction in modelling error.
PSD Impact: The results confirm that the NMSD depends on the shape of the PSD (e.g., Jakes' PSD yields higher error than Uniform PSD for the same bandwidth) but is independent of the Power Delay Profile (PDP).

Significance and Claims
The paper claims that the proposed approach provides a unified and practical tool for predicting the tracking performance of various RLS algorithms in time-varying channels. By explicitly accounting for the modelling error component, the derived formulas are applicable to advanced non-causal and basis-expansion algorithms where traditional methods fail. The authors emphasize that while the analysis relies on certain idealizations—such as replacing discrete sums with continuous integrals, assuming tap decoupling, and approximating non-stationary modelling noise as stationary—these simplifications yield simple yet accurate formulas. The work serves as a foundation for optimizing algorithm parameters (such as window length, forgetting factor, and delay) to balance noise and tracking errors in practical communication systems. Future work is identified as necessary to rigorously validate the underlying idealizations and assumptions.

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