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Reduced-Rank Adaptive Filtering Based on Joint Iterative Optimization of Adaptive Filters
Rodrigo C. de Lamare, Raimundo Sampaio-Neto
TL;DR
Large filters can require many samples for steady-state behavior, creating convergence and tracking challenges. The paper jointly optimizes full-rank adaptive filters forming a projection matrix with an adaptive reduced-rank filter, using MMSE designs and low-complexity NLMS implementation; CDMA interference-suppression results outperform existing schemes and approach optimal full-rank MMSE.
Problem
Large filters can require many samples to reach steady state, creating convergence and tracking challenges in dynamic interference scenarios.
Method
The scheme jointly iteratively optimizes a bank of full-rank adaptive filters forming the projection matrix and an adaptive reduced-rank filter, with MMSE designs and NLMS algorithms.
Results
CDMA interference-suppression simulations show performance significantly better than existing schemes and close to the optimal full-rank MMSE.
Takeaways & Limitations
The proposed reduced-rank scheme combines adaptive joint optimization with low-complexity implementation and strong interference-suppression performance.
Abstract
from arXiv · showhide
This letter proposes a novel adaptive reduced-rank filtering scheme based on joint iterative optimization of adaptive filters. The novel scheme consists of a joint iterative optimization of a bank of full-rank adaptive filters that forms the projection matrix and an adaptive reduced-rank filter that operates at the output of the bank of filters. We describe minimum mean-squared error (MMSE) expressions for the design of the projection matrix and the reduced-rank filter and low-complexity normalized least-mean squares (NLMS) adaptive algorithms for its efficient implementation. Simulations for an interference suppression application show that the proposed scheme outperforms in convergence and tracking the state-ofthe- art reduced-rank schemes at significantly lower complexity.
I. INTRODUCTION
Large adaptive filters can converge slowly and track poorly in interference, motivating reduced-rank methods and a lower-complexity adaptive scheme based on jointly optimized filters.
- Motivation: Large filters require many samples to reach steady state and can show poor convergence or tracking in dynamic interference scenarios.Conventional convergence speed is tied to filter length; even RLS may require data proportional to 2M in stationary environments.
- Motivation: Reduced-rank filtering reduces adaptive coefficients by projecting received vectors onto a lower-dimensional subspace.This approach is intended to extract important data features while improving convergence speed for large-filter problems.
- Related work: Existing reduced-rank techniques include eigen-decomposition methods, MWF, and AVF, but they suffer from high complexity and numerical problems.These limitations motivate an adaptive alternative with lower implementation cost.
- Proposed approach: The proposed scheme jointly iteratively optimizes a bank of full-rank adaptive filters forming the projection matrix and an adaptive reduced-rank output filter.The bank performs dimensionality reduction, while the reduced-rank filter estimates the desired signal.
- Proposed approach: The scheme estimates a scalar, updates filters for each successive observation, and uses low-complexity NLMS algorithms for adaptive implementation.The paper describes MMSE expressions for both the projection matrix and reduced-rank filter, and evaluates the method in CDMA interference suppression.
- Expected behavior: In practice, the proposed and existing reduced-rank techniques converge very fast because rough projection estimates can still provide suitable dimensionality reduction.The reduced-rank filter governs most of the overall scheme’s performance.
II. REDUCED-RANK MMSE PARAMETER ESTIMATION AND PROBLEM STATEMENT
The reduced-rank MMSE formulation projects an M-dimensional received vector into D dimensions and jointly designs the projection matrix and reduced-rank filter.
- MMSE formulation: The full-rank MMSE filter w minimizes the mean-squared-error cost function for the received data.The filter is an M-dimensional vector of coefficients.
- Reduced-rank filtering: The reduced-rank filter operates on the projected received vector to produce the filter output at each time instant.The reduced-rank parameters use D-dimensional quantities represented with bars.
- MMSE formulation: The reduced-rank MMSE design uses a reduced-rank covariance matrix and cross-correlation quantities derived from the full-rank received data.The formulation also identifies the variance of d(i) and conditions under which projection preserves the MMSE.
- Problem statement: The central design problem is efficiently or optimally choosing S_D so the observed received data are projected into a lower-dimensional representation.The subsequent section presents the proposed reduced-rank approach.
- Dimensionality reduction: A projection matrix S_D with dimensions M × D maps the received M × 1 vector r(i) to a reduced-rank D × 1 vector r̄(i).The projected vector becomes the input to a D-dimensional tapped-delay-line filter.
III. PROPOSED REDUCED-RANK SCHEME
The proposed scheme jointly optimizes a bank of full-rank adaptive filters forming a projection matrix and a reduced-rank filter estimating the desired signal. It iteratively exchanges information between these components to reduce input dimensionality and enable a simpler adaptive implementation than MWF.
- III. PROPOSED REDUCED-RANK SCHEME: SD(i) and w̄(i) are jointly optimized according to the MMSE criterion.The two filters are obtained by alternately fixing one and minimizing the cost function with respect to the other.
- III. PROPOSED REDUCED-RANK SCHEME: The projection matrix SD(i) is formed by D full-rank adaptive filters and processes an M × 1 data vector for dimensionality reduction.The reduced-rank filter w̄(i), with dimensions D×1, processes the reduced-rank vector r̄(i) to produce scalar estimate x(i).
- III. PROPOSED REDUCED-RANK SCHEME: For D > 1, the full-rank filters compute a subspace projection while the reduced-rank filter estimates the desired signal.For D = 1, the scheme becomes conventional full-rank filtering with an additional gain-providing weight parameter.
- III. PROPOSED REDUCED-RANK SCHEME: The MMSE filter expressions are not closed-form solutions, so equations for the reduced-rank filter and projection matrix must be iterated from an initial guess.The rank D must be set by the designer to ensure appropriate performance.
- III. PROPOSED REDUCED-RANK SCHEME: Unlike MWF, the scheme exchanges information iteratively between the reduced-rank filter and projection matrix and provides a simpler adaptive implementation.The projection matrix reduces input-data dimension, whereas the reduced-rank filter estimates the desired signal.
IV. ADAPTIVE ALGORITHMS
This section describes an adaptive NLMS implementation of the proposed reduced-rank scheme, together with its convergence conditions and arithmetic complexity.
- IV. ADAPTIVE ALGORITHMS: The proposed reduced-rank scheme is implemented using adaptive NLMS algorithms.The section also addresses convergence conditions and computational complexity in arithmetic operations.
A. Adaptive Algorithms
The adaptive implementation jointly updates the projection matrix and reduced-rank filter using an iterative Lagrangian-based procedure. Its algorithms use time-varying normalized step sizes and have complexity O(DM).
- A. Adaptive Algorithms: A Lagrangian cost function jointly updates the projection matrix SD(i) and reduced-rank filter w̄(i).The formulation uses scalar Lagrange multipliers, the Frobenius norm, and the real-part operator.
- A. Adaptive Algorithms: The resulting jointly optimized and iterative NLMS algorithms use convergence factors µ0 and η0.The updates are obtained by solving the gradient equations and introducing these convergence factors.
- A. Adaptive Algorithms: The step sizes µ(i) and η(i) are time-varying normalized quantities based on received-data and reduced-rank-filter powers.The expressions normalize by rH(i)r(i), with η(i) also depending on w̄H(i)w̄(i).
- A. Adaptive Algorithms: The algorithms described in (16)-(17) have complexity O(DM).The projection matrix consists of D full-rank adaptive filters operating simultaneously with one reduced-rank adaptive filter and exchanging information.
B. Convergence Conditions
Convergence analysis jointly tracks errors in the reduced-rank filter and projection matrix because the optimization couples both filters. Stability depends on the spectral radius of the resulting system matrix.
- B. Convergence Conditions: The analysis defines parameter-estimation errors relative to the optimal reduced-rank filter and projection matrix.Because of joint optimization, both filters are considered together.
- B. Convergence Conditions: Taking expectations of the coupled error recursions yields a system relation involving the joint error dynamics.The derivation substitutes the error definitions into the adaptive updates and simplifies the resulting terms.
- B. Convergence Conditions: The stability of the algorithms depends on the spectral radius of matrix A.This condition follows from the derived mean-error system.
- B. Convergence Conditions: For convergence, the step sizes should be chosen so that the eigenvalues of AHA are less than one.
C. Computational Complexity
The proposed scheme has lower complexity than several advanced reduced-rank methods while requiring only a slight increase over full-rank NLMS when D << M.
- The comparison includes NLMS and RLS implementations of the proposed scheme, Full-rank filtering, MWF, and AVF.
- The proposed scheme is much simpler than Full-rank RLS, MWF, and AVF.
- Its complexity is slightly higher than Full-rank NLMS when D << M.
V. SIMULATIONS
Simulations evaluate the proposed reduced-rank scheme for CDMA interference suppression using SINR and BER across rank, received symbols, and mobile-channel conditions. Rank D = 4 is near full-rank MMSE, while convergence and tracking outperform existing approaches at lower complexity.
- The experiments assess CDMA interference suppression in an uplink BPSK DS-CDMA system with K users and N chips.
- D = 4 is the best rank for the proposed scheme and produces SINR very close to optimal full-rank MMSE.
- D is invariant to system size across different processing gains, yielding considerable computational savings.
- The proposed scheme converges much faster than the full-rank filter and is comparable to MWF-RLS and AVF at much lower complexity.
- In mobile communications, the proposed scheme has much better BER performance than existing approaches and adequately tracks the desired signal.
VI. CONCLUSIONS
The paper proposes a low-complexity NLMS reduced-rank scheme that jointly optimizes adaptive filters for projection and desired-signal estimation. In CDMA interference suppression, it outperforms existing schemes and approaches optimal full-rank MMSE.
- The proposed scheme jointly iteratively optimizes adaptive filters using a low-complexity NLMS implementation.
- Full-rank adaptive filters estimate the subspace projection, while a small reduced-rank filter estimates the desired signal.
- In CDMA interference suppression, the proposed scheme performs significantly better than existing schemes and remains close to optimal full-rank MMSE.
APPENDIX
The MMSE formulation constrains the reduced-rank solution to the subspace generated by the projection matrix. Although multiple solutions exist, studies indicate no local minima and initialization-insensitive performance under stability conditions.
- For a projection matrix S_D with D ≤ M, MMSE is achieved when the minimizing full-rank filter lies in Range{S_D}.
- The proposed optimization problem has multiple solutions.
- Studies indicate no local minima and performance insensitive to initialization when initial values maintain algorithm stability and do not annihilate the signal.