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Set-Membership Adaptive Algorithms based on Time-Varying Error Bounds for Interference Suppression

Rodrigo C. de Lamare, Paulo S. R. Diniz

arXiv:1301.0097v1cs.ITmath.IT

TL;DR

In dynamic CDMA environments, specifying error bounds is difficult when noise, MAI, and ISI vary. The paper incorporates parameter and interference tracking into time-varying bounds for modified set-membership algorithms. Simulations report improved performance over previously reported techniques with fewer updates.

  • Problem

    Error-bound selection is difficult in dynamic wireless environments with time-varying noise, MAI, and ISI, and overbounding or underbounding can degrade performance.

    Source: §I · p. 1

  • Method

    The paper modifies SM-NLMS, SM-AP, and BEACON to use time-varying bounds incorporating parameter evolution and MAI/ISI power estimates.

    Source: §I · p. 2, §IV · p. 4

  • Results

    The proposed algorithms outperform previously reported techniques with fewer updates; for SM-AP with P = 3, PIDB saves up to 3 dB versus PDB and up to 4 dB versus fixed bounds at the same BER.

    Source: §VII · p. 9, C. BER Performance · p. 9

  • Takeaways & Limitations

    The algorithms may contribute to low-complexity receiver design for spread-spectrum systems.

    Source: §VII · p. 9

Abstract

from arXiv · show

This work presents set-membership adaptive algorithms based on time-varying error bounds for CDMA interference suppression. We introduce a modified family of set-membership adaptive algorithms for parameter estimation with time-varying error bounds. The algorithms considered include modified versions of the set-membership normalized least mean squares (SM-NLMS), the affine projection (SM-AP) and the bounding ellipsoidal adaptive constrained (BEACON) recursive least-squares technique. The important issue of error bound specification is addressed in a new framework that takes into account parameter estimation dependency, multi-access and inter-symbol interference for DS-CDMA communications. An algorithm for tracking and estimating the interference power is proposed and analyzed. This algorithm is then incorporated into the proposed time-varying error bound mechanisms. Computer simulations show that the proposed algorithms are capable of outperforming previously reported techniques with a significantly lower number of parameter updates and a reduced risk of overbounding or underbounding.

I. INTRODUCTION

Set-membership filtering can reduce update complexity, but its performance depends on error bounds that are difficult to specify in dynamic wireless environments. The paper develops time-varying-bound algorithms for DS-CDMA interference suppression, under a symbol-synchronous downlink model.

I. INTRODUCTION

  • Set-membership filtering uses a predetermined error bound to select parameter estimates with bounded filter-output errors, and data-selective updates can reduce complexity.Its objective-function modification addresses fast convergence and low misadjustment.Source: §I · p. 1
  • Dynamic noise, multiple-access interference, and intersymbol interference make error-bound selection difficult, increasing overbounding and underbounding risks that degrade performance.Prior time-varying-bound approaches also lacked mechanisms for tracking MAI and ISI and had not been applied to AP and LS-based techniques.Source: §I · p. 1
  • The proposed framework tracks parameter evolution and MAI/ISI power, incorporating these estimates into time-varying bounds for SM-NLMS, SM-AP, and BEACON.The algorithms are evaluated with adaptive linear multiuser receivers for DS-CDMA interference suppression.Source: §I · pp. 1–2

II. DS-CDMA SYSTEM MODEL AND LINEAR RECEIVERS

  • The model assumes a symbol-synchronous DS-CDMA downlink with K users, N chips per symbol, Lp paths, chip-rate delays, symbol-constant channels, and repeated spreading codes.The received vector has dimension M = N + Lp − 1; the model includes Gaussian noise and user-specific ISI.Source: §II · p. 2

III. SET-MEMBERSHIP ADAPTIVE ALGORITHMS WITH TIME-VARYING ERROR BOUNDS AND PROBLEM STATEMENT

Set-membership algorithms seek parameter estimates that satisfy an error constraint, updating only for data that require a change. This section extends SM-NLMS, SM-AP, and BEACON to general time-varying error bounds and derives an SM-NLMS update rule.

III. SET-MEMBERSHIP ADAPTIVE ALGORITHMS WITH TIME-VARYING ERROR BOUNDS AND PROBLEM STATEMENT

  • The feasibility set contains parameter vectors that yield estimates satisfying the error bound, and adaptive methods approximate it using membership sets formed from observations.BEACON approximates the exact membership set with ellipsoids, while SM-NLMS and SM-AP use projections based on constraint sets.Source: §III · p. 2
  • The paper modifies SM-NLMS, SM-AP, and BEACON to operate with general time-varying error bounds.The bound should adapt to characteristics of time-varying practical environments.Source: §III · p. 3

A. SM-NLMS Algorithm with Time-Varying Bounds

  • SM-NLMS derives its time-varying-bound update by solving a constrained optimization problem with Lagrange multipliers.The derivation sets gradients with respect to the next parameter vector and multiplier to zero.Source: A. SM-NLMS Algorithm with Time-Varying… · p. 3
  • For SM-NLMS, the update places the error on the closest feasibility-set boundary using the time-varying bound and its sign.The resulting data-dependent rule updates when the error magnitude exceeds the bound; otherwise, the update is zero.Source: A. SM-NLMS Algorithm with Time-Varying… · p. 3

B. SM-AP Algorithm with Time-Varying Bounds

The section describes time-varying-bound SM-AP and a BEACON recursion. SM-AP uses multiple data vectors, while the modified BEACON method is presented as a potential fast-converging alternative to stochastic-gradient algorithms.

B. SM-AP Algorithm with Time-Varying Bounds

  • The SM-AP algorithm is derived by substituting the optimization result into the bound constraint.The section first defines an observation matrix and desired-output vector comprising P outputs.Source: B. SM-AP Algorithm with Time-Varying… · p. 3
  • SM-AP has complexity UR × O(PM + 2KinvP^2), and using P data vectors generalizes SM-NLMS to increase convergence speed.Kinv is a factor required to invert a P × P matrix, and UR is the update rate.Source: B. SM-AP Algorithm with Time-Varying… · p. 3

C. BEACON Adaptive Algorithm with Time-Varying Bounds

  • Modified BEACON equips the constrained least-squares recursion with time-varying bounds and has potential for fast convergence relatively independent of data covariance eigenvalue spread.The comparison is with stochastic-gradient algorithms.Source: C. BEACON Adaptive Algorithm with… · p. 3

A. Parameter Dependent Bound

The parameter-dependent bound (PDB) uses parameter-vector evolution to set a bound for set-membership algorithms. The section also describes the BEACON optimization leading to its innovation check.

A. Parameter Dependent Bound

  • The PDB recursion computes an error bound for set-membership algorithms using the desired user's parameter-vector evolution.It is described as similar to an approach in.Source: A. Parameter Dependent Bound · p. 4
  • The forgetting factor β controls the time-averaged estimate of parameter evolution, while α is a tuning parameter.The variance term α||wk[i]||^2σ̂_v^2[i] represents the variance of the parameter vector's inner product with noise.Source: A. Parameter Dependent Bound · p. 4
  • Minimizing the BEACON cost function leads to the proposed algorithm's innovation check.The optimization solution uses the Lagrange multiplier λk[l].Source: A. Parameter Dependent Bound · p. 4

IV. ALGORITHMS FOR TIME-VARYING ERROR BOUNDS, INTERFERENCE ESTIMATION AND TRACKING

The section develops time-varying set-membership error bounds that account for parameter evolution and interference, alongside channel, amplitude, and interference estimation procedures. Its parameter- and interference-dependent bound uses a tracked estimate of interference power.

IV. ALGORITHMS FOR TIME-VARYING ERROR BOUNDS, INTERFERENCE ESTIMATION AND TRACKING

  • The proposed framework combines parameter-dependent bounds with estimated MAI and ISI power, using time-averaged recursions to track quantities for the adaptive bound.The recursions use forgetting-factor and weighting parameters to regulate time averages and selectively weight quantities.Source: §IV · p. 4, B. Parameter and Interference… · pp. 5–6

B. Parameter and Interference Dependent Bound

  • The interference estimator uses RAKE and linear receivers to subtract the desired signal, then uses the resulting difference to estimate MAI and ISI power.The method assumes the noise power is known at the receiver for the parameter-dependent bound.Source: B. Parameter and Interference… · pp. 4–5, §IV · p. 4
  • A stochastic-gradient channel estimator is derived from a cost function using an estimated channel to form the RAKE receiver.The channel estimate supports the receiver used in interference estimation and tracking.Source: B. Parameter and Interference… · p. 5
  • The receiver estimates user amplitudes to support tasks including interference cancellation and power control, and the interference-tracking procedure uses an amplitude estimate to subtract the desired signal.A simple stochastic-gradient algorithm is proposed for estimating user amplitude.Source: B. Parameter and Interference… · p. 5
  • The proposed parameter- and interference-dependent bound incorporates parameter dependency and estimated multiuser interference power into the time-varying bound.The interference-power estimate is formed from time averages of instantaneous values.Source: B. Parameter and Interference… · pp. 5–6

V. ANALYSIS OF THE ALGORITHMS

The analysis focuses on convergence properties of the channel, amplitude, and interference-power estimators in relation to their step-size parameters.

V. ANALYSIS OF THE ALGORITHMS

  • The analysis examines convergence of channel, amplitude, and interference-power estimators as a function of step sizes µh, µA, and β, respectively.Source: §V · p. 6

A. Channel Estimator

The channel-estimator analysis states conditions for asymptotic unbiasedness and convergence to the optimum channel estimate, based on the step size.

A. Channel Estimator

  • The SG channel estimator is asymptotically unbiased and converges to hopt when µh satisfies the stated condition.The analysis relates the condition to a geometric-series ratio whose magnitude must be less than 1.Source: A. Channel Estimator · p. 6

B. Amplitude Estimator

The amplitude-estimator analysis gives a step-size condition for asymptotic unbiased convergence and notes that the step size must be tuned for convergence and tracking.

B. Amplitude Estimator

C. Interference Estimation and Tracking Algorithm

The analysis frames interference estimation and tracking as a general recursive mechanism for time-varying bounds. It examines mean-square convergence to an optimal estimate under statistical-independence and step-size conditions.

C. Interference Estimation and Tracking Algorithm

  • The bound can be parameter-dependent (PDB) or parameter- and interference-dependent (PIDB), with its range adjusted for good convergence and tracking.The quantity Po[i] represents either type of bound.Source: C. Interference Estimation and… · pp. 6–7
  • The analysis defines γk,opt as the optimal estimate for γk[i] and seeks mean-square convergence to that value under conditions on β.The recursion's error is defined relative to the optimal estimate.Source: C. Interference Estimation and… · p. 7
  • The estimator's minimum MSE, |ek,opt[i]|^2, is the error term used in the recursion analysis.The expected-value analysis assumes this term is statistically independent of εγ[i].Source: C. Interference Estimation and… · p. 7
  • The general recursion for interference estimation and tracking converges asymptotically when the step size β satisfies the stated condition.The convergence analysis treats the recursion in terms of mean-squared error and assumes statistical independence between its terms.Source: C. Interference Estimation and… · pp. 6–7

VI. SIMULATIONS

The simulations compare proposed time-varying-bound SM-NLMS, SM-AP, and BEACON algorithms with existing methods in DS-CDMA scenarios. The proposed methods track interference power accurately and show faster SINR convergence with lower update rates in the reported comparisons.

VI. SIMULATIONS

  • The simulations compare NLMS, AP, and BEACON families, including proposed PDB and PIDB variants where specified.The proposed SM-NLMS, SM-AP, and BEACON variants are compared with their existing counterparts.Source: §VI · p. 7
  • Experiments use BPSK, length-31 Gold sequences, three-path fading channels, 200 training symbols, and averages over 100 independent runs.The interferer powers follow a log-normal distribution with a 3 dB standard deviation.Source: §VI · p. 7

A. Interference Estimation and Tracking

  • The interference-estimation algorithms track MAI and ISI power in dynamic environments, with average estimation error no greater than 5% of the estimated power level.The comparison is between estimated and actual interference power levels in Fig. 2.Source: A. Interference Estimation and Tracking · p. 7, Figure 2 · p. 8

B. SINR Performance

  • The SINR comparisons assess convergence at about the same SINR level and record update rate as a complexity measure.The reported evaluations cover modified SM-NLMS, SM-AP, and BEACON techniques.Source: B. SINR Performance · p. 8
  • Proposed SM-NLMS with PIDB converges fastest, followed by PDB, SM-NLMS-Huang, conventional SM-NLMS, and NLMS; both proposed variants have markedly lower update rates than conventional SM-NLMS.The comparisons assess SINR convergence and update rate as a complexity measure.Source: B. SINR Performance · p. 8
  • The proposed SM-AP and BEACON methods show faster convergence, better steady-state SINR, and lower update rates than NLMS despite their higher complexity.These results corroborate the pattern reported for the NLMS algorithms.Source: B. SINR Performance · p. 8

C. BER Performance

In the BER experiments, proposed time-varying bounds improve BEACON and AP performance while reducing update rates. The conclusion highlights the lower update count as relevant to receivers with limited battery and processing resources.

C. BER Performance

  • The BER evaluation uses 1500-symbol packets, with receivers trained on 200 symbols before switching to decision-directed adaptation and channel tracking.Source: C. BER Performance · p. 8
  • Across a wide range of fading rates, BEACON-PDB and BEACON-PIDB improve BER over original BEACON and RLS while reducing update rate.As fading becomes more hostile, the algorithms' BER performance approaches one another.Source: C. BER Performance · pp. 8–9, Figure 6 · p. 9
  • At equal BER, SM-AP with PIDB saves up to 3 dB versus PDB and 4 dB versus fixed bounds, and accommodates up to four more users than PDB.The comparison uses AP algorithms with P = 3.Source: C. BER Performance · p. 9, Figure 7 · p. 9

VII. CONCLUSIONS

  • The conclusion states that the proposed algorithms combine time-varying error bounds with parameter-dependency and MAI/ISI power tracking, and simulations show fewer updates than previously reported techniques.The authors identify potential relevance to spread-spectrum and future MIMO receivers, including cellular, ad hoc, and sensor networks with limited resources.Source: §VII · p. 9
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