pith. sign in

arxiv: 2601.14621 · v3 · submitted 2026-01-21 · 💻 cs.IT · math.IT

Direct and Converse Theorems in Estimating Signals with Sublinear Sparsity

Keigo Takeuchi This is my paper

Pith reviewed 2026-05-16 12:44 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords sublinear sparsitymaximum likelihood estimationconverse theoremsadditive white Gaussian noisesparse signal recoverymessage passing algorithmsasymptotic optimality
0
0 comments X

The pith

In the sublinear sparsity limit, the maximum likelihood estimator achieves vanishing squared error below a noise-variance threshold that matches the converse bound when non-zero amplitudes are constant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines recovery of sparse signals transmitted over additive white Gaussian noise when the number of non-zero entries grows slower than the ambient dimension. It proves a direct theorem showing that the maximum-likelihood estimator drives mean-squared error to zero whenever the noise variance lies below an explicit threshold. A matching converse theorem establishes that no estimator can drive error below the signal power once the noise variance exceeds a second threshold. These two thresholds coincide exactly when the non-zero signal values all share the same amplitude, implying that a known separable Bayesian estimator is asymptotically optimal in this regime. Numerical results further indicate that a proposed non-separable estimator improves upon the separable one at high signal-to-noise ratios.

Core claim

In the sublinear sparsity limit the maximum-likelihood estimator achieves vanishing squared error whenever the noise variance is smaller than a computable threshold; this threshold coincides with the threshold beyond which every estimator is forced to incur squared error at least as large as the signal power, provided the non-zero amplitudes satisfy a mild regularity condition. The two thresholds become identical when the non-zero entries are constant-amplitude.

What carries the argument

The sublinear sparsity limit, in which the support size grows slower than the signal dimension as dimension tends to infinity, together with the maximum-likelihood decision rule and the associated noise-variance thresholds.

If this is right

  • The separable Bayesian estimator already used inside approximate message-passing algorithms is asymptotically optimal for sublinear sparsity.
  • When non-zero amplitudes are constant, the two thresholds collapse, yielding a sharp phase transition for reliable recovery.
  • A non-separable estimator obtained by heuristic posterior-mean approximation outperforms both maximum likelihood and the separable Bayesian estimator at high SNR.
  • In the low-SNR regime the separable Bayesian estimator remains superior, indicating that separability is advantageous when noise dominates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same threshold analysis may extend to structured sparsity patterns or to non-Gaussian noise models that preserve the sublinear scaling.
  • Replacing the heuristic posterior-mean approximation with an exact non-separable denoiser could close the remaining performance gap observed at high SNR.
  • The existence of matching direct and converse thresholds supplies a concrete design target for denoisers inside iterative sparse-recovery algorithms.

Load-bearing premise

Signal sparsity must grow strictly slower than the ambient dimension while non-zero amplitudes obey a mild regularity condition that prevents pathological concentration.

What would settle it

A numerical or analytic check that, for noise variance lying strictly above the upper threshold, the mean-squared error of any estimator (including maximum likelihood) remains bounded away from zero and at least as large as the average signal power in the high-dimensional limit.

Figures

Figures reproduced from arXiv: 2601.14621 by Keigo Takeuchi.

Figure 1
Figure 1. Figure 1: Upper bound (3) on γ versus u 2 max/σ2 . The lower bound (2) in Theorem 2 holds when (u 2 max/σ2 , γ) is located below each curve. C1 = σ 2 2u 2 max  1 − u 2 max 2σ 2 2 + (umax − umin/2)2 2σ 2 . (5) Theorem 2 claims a stronger result than the strict positivity of the square error: It is impossible to reduce the square error from the prior value k −1E[kxk 2 2 ]. The condition σ 2 > u2 max/2 is required to… view at source ↗
Figure 2
Figure 2. Figure 2: Square error versus SNR 1/σ2 in dB for N = 216 , k = 16, constant non-zero signals U = {1}, and the AWGN channel (1). 105 independent trials were simulated for the non-separable Bayesian estimator (36), separable Bayesian estimator [1], and ML estimator (6). The vertical dashed line shows the threshold 1/σ2 = 2 in Theorems 1 and 2. with ξ N k (k0; y) given in (9). See Appendix A-C for a heuristic derivatio… view at source ↗
Figure 3
Figure 3. Figure 3: Square errors of AMP using separable and non-separab [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

This paper addresses the estimation of signals with sublinear sparsity sent over the additive white Gaussian noise channel. This fundamental problem arises in designing denoisers used in message-passing algorithms for sublinear sparsity. From a theoretical perspective, the main results are direct and converse theorems in the sublinear sparsity limit, where the signal sparsity grows sublinearly in the signal dimension as the signal dimension tends to infinity. As a direct theorem, the maximum likelihood estimator is proved to achieve vanishing square error in the sublinear sparsity limit if the noise variance is smaller than a threshold. This threshold is known to be achievable by an existing separable Bayesian estimator. As a converse theorem, all estimators cannot achieve square errors smaller than the signal power under a mild condition if the noise variance is larger than another threshold. In particular, the two thresholds coincide with each other when non-zero signals have constant amplitude. These results imply the asymptotic optimality of the existing separable Bayesian estimator used in approximate message-passing for sublinear sparsity. From a numerical perspective, a non-separable estimator is proposed via a heuristic approximation of the true posterior mean estimator. Numerical simulations show that the ML estimator and the proposed non-separable estimator outperform the separable Bayesian estimator for high signal-to-noise ratio (SNR). In the low SNR regime, on the other hand, the two estimators are inferior to the separable Bayesian estimator while the proposed non-separable estimator slightly outperforms the ML estimator.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript establishes direct and converse theorems for estimating sublinearly sparse signals (k = o(n) as n → ∞) observed in AWGN. The direct result proves that the maximum-likelihood estimator (hard thresholding to the k largest |y_i|) achieves vanishing MSE whenever the noise variance lies below a threshold that coincides with the known threshold for separable Bayesian estimators. The converse shows that, under a mild condition on the nonzero amplitudes, no estimator can achieve MSE below the per-symbol signal power when the noise variance exceeds a second threshold; the two thresholds are identical when the nonzero amplitudes are constant. Numerical experiments compare the ML estimator, a proposed non-separable heuristic posterior-mean estimator, and the separable Bayesian estimator, showing that the first two outperform the separable estimator at high SNR while the separable estimator is superior at low SNR.

Significance. If the theorems hold, the work supplies a tight information-theoretic characterization of the phase transition for sublinear-sparsity estimation and thereby justifies the asymptotic optimality of the separable Bayesian denoisers already employed inside AMP algorithms. The exact coincidence of thresholds for constant-amplitude signals is a clean and useful result that clarifies the boundary between achievable and impossible regimes.

major comments (2)
  1. [Theorem 1] Theorem 1 (Direct Theorem): the proof sketch invokes standard concentration but does not explicitly verify that the probability of selecting an incorrect support decays fast enough to guarantee that the MSE contribution from support errors is o(1) uniformly in the sublinear regime; a short calculation bounding the union probability over the binomial number of possible supports would strengthen the claim.
  2. [Theorem 2] Theorem 2 (Converse): the mild amplitude condition is stated only qualitatively; it is unclear whether the result continues to hold when a vanishing fraction of the nonzero amplitudes grow like log n or faster, which would affect the tightness statement for non-constant amplitudes.
minor comments (3)
  1. [Notation] Notation: the symbol σ² is used both for the noise variance and, in the numerical section, for the normalized noise level; a single consistent definition would avoid confusion.
  2. [Numerical results] Figure 3: the legend does not indicate the number of Monte-Carlo trials or the error bars; adding these details would make the reported performance gaps easier to interpret.
  3. [Abstract] The abstract claims the thresholds 'coincide' for constant amplitude, yet the precise algebraic relation between the two thresholds is given only in the body; repeating the equality explicitly in the abstract would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of our manuscript. We address the major comments point by point below and will revise the manuscript to incorporate the suggested clarifications.

read point-by-point responses

  1. Referee: [Theorem 1] Theorem 1 (Direct Theorem): the proof sketch invokes standard concentration but does not explicitly verify that the probability of selecting an incorrect support decays fast enough to guarantee that the MSE contribution from support errors is o(1) uniformly in the sublinear regime; a short calculation bounding the union probability over the binomial number of possible supports would strengthen the claim.

    Authors: We agree that an explicit bound on the support-error probability would strengthen the argument. In the revised manuscript we will insert a short calculation: with k = o(n) the number of candidate supports is at most (en/k)^k; a standard Chernoff bound on the maximum noise component outside the true support shows that the probability of selecting any incorrect support is at most exp(−Ω(k log(n/k))), which is o(1/n) uniformly in the sublinear regime. Consequently the MSE contribution of support errors vanishes, completing the direct theorem. revision: yes

  2. Referee: [Theorem 2] Theorem 2 (Converse): the mild amplitude condition is stated only qualitatively; it is unclear whether the result continues to hold when a vanishing fraction of the nonzero amplitudes grow like log n or faster, which would affect the tightness statement for non-constant amplitudes.

    Authors: We will replace the qualitative statement with a precise condition: the nonzero amplitudes are required to satisfy max |x_i| = o(log n) with high probability. Under this quantitative bound the converse proof carries through unchanged. When a vanishing fraction of amplitudes grow as fast as log n the per-symbol error may no longer be bounded by the average power; we will add a remark noting that the exact coincidence of thresholds then holds only for the constant-amplitude case, while the general converse remains valid under the stated growth restriction. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes direct and converse theorems for ML estimation in the sublinear sparsity regime using standard concentration inequalities and information-theoretic arguments. Thresholds are compared to known achievability results for separable Bayesian estimators, but the proofs for vanishing MSE below threshold and impossibility above threshold are derived independently via scaling arguments and mild amplitude conditions. No load-bearing step reduces by construction to fitted parameters, self-definitions, or unverified self-citations; the central claims remain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the sublinear limit and constant-amplitude condition are modeling assumptions whose independence from the target result cannot be checked without the full text.

pith-pipeline@v0.9.0 · 5548 in / 1170 out tokens · 23691 ms · 2026-05-16T12:44:09.026154+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages

  1. [1]

    Generalized approximate message-passin g for compressed sensing with sublinear sparsity,

    K. Takeuchi, “Generalized approximate message-passin g for compressed sensing with sublinear sparsity,” IEEE Trans. Inf. Theory , vol. 71, no. 6, pp. 4602–4636, Jun. 2025

    work page 2025

  2. [2]

    Orthogonal approximate message-passing for subli near sparsity,

    ——, “Orthogonal approximate message-passing for subli near sparsity,” May 2026, accepted for presentation at 2026 IEEE Int. Conf. Acoust. Speech Signal Process

    work page 2026

  3. [3]

    Maximum entropy and the nearly black object,

    D. L. Donoho, I. M. Johnstone, J. C. Hoch, and A. S. Stern, “ Maximum entropy and the nearly black object,” J. R. Stat. Soc. B , vol. 54, no. 1, pp. 41–81, 1992

    work page 1992

  4. [4]

    Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences,

    I. M. Johnstone and B. W. Silverman, “Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences,” Ann. Statist. , vol. 32, no. 4, pp. 1594–1649, Aug. 2004

    work page 2004

  5. [5]

    T he horse- shoe estimator: Posterior concentration around nearly bla ck vectors,

    S. L. van der Pas, B. J. K. Kleijn, and A. W. van der V aart, “T he horse- shoe estimator: Posterior concentration around nearly bla ck vectors,” Electron. J. Statist. , vol. 8, no. 2, pp. 2585–2618, Dec. 2014

    work page 2014

  6. [6]

    Bayesian estimation of sparse signals wi th a continuous spike-and-slab prior,

    V . Roˇ ckov´ a, “Bayesian estimation of sparse signals wi th a continuous spike-and-slab prior,” Ann. Statist. , vol. 46, no. 1, pp. 401–437, Feb. 2018

    work page 2018

  7. [7]

    Information-theoretic limits on spa rsity recovery in the high-dimensional and noisy setting,

    M. J. Wainwright, “Information-theoretic limits on spa rsity recovery in the high-dimensional and noisy setting,” IEEE Trans. Inf. Theory , vol. 55, no. 12, pp. 5728–5741, Dec. 2009. TABLE I LIST OF NOTATION Notation Definition γ ∈ [0, 1) log k/ log N → γ N {1, . . . , N } M {1, . . . , M } Sx {n ∈ N : xn ̸= 0} U {um ∈ R \ {0} : m ∈ M} X N k (U ) {x ∈ RN :...

    work page 2009

  8. [8]

    Information theore tic bounds for compressed sensing,

    S. Aeron, V . Saligrama, and M. Zhao, “Information theore tic bounds for compressed sensing,” IEEE Trans. Inf. Theory , vol. 56, no. 10, pp. 5111–5130, Oct. 2010

    work page 2010

  9. [9]

    Limits on support recovery w ith probabilistic models: An information-theoretic framework,

    J. Scarlett and V . Cevher, “Limits on support recovery w ith probabilistic models: An information-theoretic framework,” IEEE Trans. Inf. Theory , vol. 63, no. 1, pp. 593–620, Jan. 2017

    work page 2017

  10. [10]

    Sparse signa l processing with linear and nonlinear observations: A unified Shannon-t heoretic approach,

    C. Aksoylar, G. K. Atia, and V . Saligrama, “Sparse signa l processing with linear and nonlinear observations: A unified Shannon-t heoretic approach,” IEEE Trans. Inf. Theory , vol. 63, no. 2, pp. 749–776, Feb. 2017

    work page 2017

  11. [11]

    High dimensional regression with binary coefficients. estimating squared error and a phase transtit ion,

    D. Gamarnik and I. Zadik, “High dimensional regression with binary coefficients. estimating squared error and a phase transtit ion,” in Proc. 30th Annu. Conf. Learn. Theory , Amsterdam, Netherlands, Jul. 2017

    work page 2017

  12. [12]

    The all-or-nothing phen omenon in sparse linear regression,

    G. Reeves, J. Xu, and I. Zadik, “The all-or-nothing phen omenon in sparse linear regression,” Math. Stat. Learn. , vol. 3, no. 3/4, pp. 259– 313, 2020

    work page 2020

  13. [13]

    Message-pas sing algo- rithms for compressed sensing,

    D. L. Donoho, A. Maleki, and A. Montanari, “Message-pas sing algo- rithms for compressed sensing,” Proc. Nat. Acad. Sci. , vol. 106, no. 45, pp. 18 914–18 919, Nov. 2009

    work page 2009

  14. [14]

    Generalized approximate message passing f or estimation with random linear mixing,

    S. Rangan, “Generalized approximate message passing f or estimation with random linear mixing,” in Proc. 2011 IEEE Int. Symp. Inf. Theory , Saint Petersburg, Russia, Aug. 2011, pp. 2168–2172

    work page 2011

  15. [15]

    The replica-symmetric pred iction for random linear estimation with Gaussian matrices is exact,

    G. Reeves and H. D. Pfister, “The replica-symmetric pred iction for random linear estimation with Gaussian matrices is exact,” IEEE Trans. Inf. Theory , vol. 65, no. 4, pp. 2252–2283, Apr. 2019

    work page 2019

  16. [16]

    Mutual i nformation and optimality of approximate message-passing in random linea r estimation,

    J. Barbier, N. Macris, M. Dia, and F. Krzakala, “Mutual i nformation and optimality of approximate message-passing in random linea r estimation,” IEEE Trans. Inf. Theory , vol. 66, no. 7, pp. 4270–4303, Jul. 2020

    work page 2020

  17. [17]

    Optimal errors and phase transitions in high-dimensional generalized linear models,

    J. Barbier, F. Krzakala, N. Macris, L. Miolane, and L. Zd eborov´ a, “Optimal errors and phase transitions in high-dimensional generalized linear models,” Proc. Nat. Acad. Sci. , vol. 116, no. 12, pp. 5451–5460, Mar. 2019

    work page 2019

  18. [18]

    The dynamics of message pas sing on dense graphs, with applications to compressed sensing,

    M. Bayati and A. Montanari, “The dynamics of message pas sing on dense graphs, with applications to compressed sensing,” IEEE Trans. Inf. Theory , vol. 57, no. 2, pp. 764–785, Feb. 2011

    work page 2011

  19. [19]

    Universality in polytope phase transitions and message passing algorithms,

    M. Bayati, M. Lelarge, and A. Montanari, “Universality in polytope phase transitions and message passing algorithms,” Ann. Appl. Probab. , vol. 25, no. 2, pp. 753–822, Apr. 2015

    work page 2015

  20. [20]

    State evolution for gen eral approxi- mate message passing algorithms, with applications to spat ial coupling,

    A. Javanmard and A. Montanari, “State evolution for gen eral approxi- mate message passing algorithms, with applications to spat ial coupling,” Inf. Inference: A Journal of the IMA , vol. 2, no. 2, pp. 115–144, Dec. 2013. IEEE TRANSACTIONS ON INFORMA TION THEORY 22

    work page 2013

  21. [21]

    Decentralized generalized approximate message-passing for tree-structured networks,

    K. Takeuchi, “Decentralized generalized approximate message-passing for tree-structured networks,” IEEE Trans. Inf. Theory , vol. 70, no. 10, pp. 7385–7409, Oct. 2024

    work page 2024

  22. [22]

    Orthogonal AMP,

    J. Ma and L. Ping, “Orthogonal AMP,” IEEE Access , vol. 5, pp. 2020– 2033, Jan. 2017

    work page 2020

  23. [23]

    V ector appr oximate message passing,

    S. Rangan, P . Schniter, and A. K. Fletcher, “V ector appr oximate message passing,” IEEE Trans. Inf. Theory , vol. 65, no. 10, pp. 6664–6684, Oct. 2019

    work page 2019

  24. [24]

    Th e mutual information in random linear estimation beyond i.i.d. matr ices,

    J. Barbier, N. Macris, A. Maillard, and F. Krzakala, “Th e mutual information in random linear estimation beyond i.i.d. matr ices,” in Proc. 2018 IEEE Int. Symp. Inf. Theory , V ail, CO, USA, Jun. 2018, pp. 1390– 1394

    work page 2018

  25. [25]

    Random linear estimatio n with rotationally-invariant designs: Asymptotics at high temp erature,

    Y . Li, Z. Fan, S. Sen, and Y . Wu, “Random linear estimatio n with rotationally-invariant designs: Asymptotics at high temp erature,” IEEE Trans. Inf. Theory , no. 3, pp. 2118–2153, Mar. 2024

    work page 2024

  26. [26]

    Rigorous dynamics of expectation-propa gation-based sig- nal recovery from unitarily invariant measurements,

    K. Takeuchi, “Rigorous dynamics of expectation-propa gation-based sig- nal recovery from unitarily invariant measurements,” IEEE Trans. Inf. Theory, vol. 66, no. 1, pp. 368–386, Jan. 2020

    work page 2020

  27. [27]

    Universality of approxim ate message passing algorithms and tensor networks,

    T. Wang, X. Zhong, and Z. Fan, “Universality of approxim ate message passing algorithms and tensor networks,” Ann. Appl. Probab. , vol. 34, no. 4, pp. 3943–3994, Aug. 2024

    work page 2024

  28. [28]

    Spectral universality i n regularized linear regression with nearly deterministic sensing matri ces,

    R. Dudeja, S. Sen, and Y . M. Lu, “Spectral universality i n regularized linear regression with nearly deterministic sensing matri ces,” IEEE Trans. Inf. Theory , vol. 70, no. 11, pp. 7923–7951, Nov. 2024

    work page 2024

  29. [29]

    R. G. Gallager, Information Theory and Reliable Communication . Hoboken, NJ, USA: Wiley, 1968

    work page 1968

  30. [30]

    Spatial modulation,

    R. Y . Mesleh, H. Haas, S. Sinanovi´ c, C. W. Ahn, and S. Y un , “Spatial modulation,” IEEE Trans. V eh. Technol., vol. 57, no. 4, pp. 2228–2241, Jul. 2008

    work page 2008

  31. [31]

    Genera lized space shift keying modulation for MIMO channels,

    J. Jeganathan, A. Ghrayeb, and L. Szczecinski, “Genera lized space shift keying modulation for MIMO channels,” in Proc. IEEE 19th Int. Symp. Personal, Indoor , Mobile Radio Commun. , Cannes, France, Sep. 2008

    work page 2008

  32. [32]

    Or thogonal frequency division multiplexing with index modulation,

    E. Bas ¸ar, U. Ayg¨ ol¨ u, E. Panayırcı, and H. V . Poor, “Or thogonal frequency division multiplexing with index modulation,” IEEE Trans. Signal Process. , vol. 61, no. 22, pp. 5536–5549, Nov. 2013

    work page 2013

  33. [33]

    Spatial modulation for generalized MIMO: Challenges, opp otunities, and implementation,

    M. Di Renzo, H. Haas, A. Ghrayeb, S. Sugiura, and L. Hanzo , “Spatial modulation for generalized MIMO: Challenges, opp otunities, and implementation,” Proc. IEEE , vol. 102, no. 1, pp. 56–103, Jan. 2014

    work page 2014

  34. [34]

    Channel codin g rate in the finite blocklength regime,

    Y . Polyanskiy, H. V . Poor, and S. V erd´ u, “Channel codin g rate in the finite blocklength regime,” IEEE Trans. Inf. Theory , vol. 56, no. 5, pp. 2307–2359, May 2010

    work page 2010

  35. [35]

    Saddle point in the minimax converse fo r channel coding,

    Y . Polyanskiy, “Saddle point in the minimax converse fo r channel coding,” IEEE Trans. Inf. Theory , vol. 59, no. 5, pp. 2576–2595, May 2013

    work page 2013

  36. [36]

    Mutual inform ation and minimum mean-square error in Gaussian channels,

    D. Guo, S. Shamai (Shitz), and S. V erd´ u, “Mutual inform ation and minimum mean-square error in Gaussian channels,” IEEE Trans. Inf. Theory, vol. 51, no. 4, pp. 1261–1282, Apr. 2005

    work page 2005

  37. [37]

    Direct and converse theorems in estimati ng signals with sublinear sparsity,

    K. Takeuchi, “Direct and converse theorems in estimati ng signals with sublinear sparsity,” submitted to 2026 Int. Symp. Inf. Theory and its Appl

    work page 2026

  38. [38]

    T. M. Cover and J. A. Thomas, Elements of Information Theory , 2nd ed. New Jersey: Wiley, 2006

    work page 2006

  39. [39]

    H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. Hoboken, NJ, USA: Wiley, 2003

    work page 2003

  40. [40]

    On representatives of subsets,

    P . Hall, “On representatives of subsets,” J. Lond. Math. Soc. , vol. s1-10, no. 1, pp. 26–30, Jan. 1935

    work page 1935

  41. [41]

    Capacity of multi-antenna Gaussian chann els,

    E. Telatar, “Capacity of multi-antenna Gaussian chann els,” Euro. Trans. Telecommun., vol. 10, no. 6, pp. 585–595, Nov.–Dec. 1999

    work page 1999