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Linear Coding of LTI Sources Over Vector Gaussian Channels: A Majorization Approach

Shihao Jin, Junhui Li, Shinji Hara, Wei Chen

arXiv:2608.29511v1cs.ITeess.SYmath.OC

TL;DR

The paper asks when LTI encoder-decoder pairs can transmit unstable discrete-time vector sources over feedback Gaussian channels while maintaining bounded estimation error covariance. It uses majorization and partial-order programming to characterize solvability and optimize power allocation, yielding constructive designs and water-filling procedures. The results show that transmission difficulty depends on both topological entropy and the evenness of the antistable poles’ log-magnitudes.

  • Problem

    The paper addresses the incomplete characterization of solvability and required power for LTI coding of discrete-time MIMO LTI sources over parallel Gaussian channels under individual or total power constraints.

  • Method

    The paper uses coupled majorization inequalities and partial-order programming to analyze feasibility, power allocation, and construction of LTI encoder-decoder pairs.

  • Results

    The paper establishes necessary and sufficient individual-constraint solvability conditions and derives minimum total power with analytic equal-noise and sequential unequal-noise water-filling procedures.

  • Takeaways & Limitations

    Transmission difficulty depends not only on topological entropy but also on how evenly the antistable poles’ log-magnitudes are distributed.

  • Takeaways & Limitations

    The analysis focuses on cyclic unstable sources; generalization to noncyclic sources remains under investigation.

Abstract

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We study the design of linear time-invariant (LTI) encoder-decoder pairs for transmitting the state of a discrete-time LTI vector source over power-constrained parallel Gaussian channels with feedback. Two types of power constraints are considered. Under individual subchannel power constraints, a necessary and sufficient condition for designing an encoder-decoder pair that achieves bounded estimation error covariance (EEC) is established via two coupled majorization inequalities involving the subchannel signal-to-noise ratios and the antistable poles of the source. Under total channel power constraint, we derive the minimum total power required for a feasible encoder-decoder design by exploiting partial-order progamming under majorization order. An analytical optimal power allocation is obtained for the case of equal noise variances, which admits a water-filling interpretation; for general noise case, a sequential water-filling algorithm is developed. Our results reveal that the difficulty of transmitting a discrete-time LTI source via LTI coding is governed not only by its topological entropy, but also by the evenness of the log-magnitudes of its antistable poles. The design methods for feasible encoder-decoder pairs are also provided.

I. INTRODUCTION

The paper studies LTI transmission of unstable discrete-time vector sources over parallel Gaussian channels with feedback, addressing solvability under individual and total power constraints. It develops majorization-based conditions, power allocation methods, and constructive encoder-decoder designs.

  • Motivation: Remote estimation of unstable sources requires sufficient communication resources to prevent estimation error covariance from diverging.The problem is relevant to networked control and scenarios involving communication disruption or interference.
  • Research problem: The paper addresses missing solvability conditions for matching parallel-channel qualities with unstable source poles under individual power constraints.It also targets the exact minimum total power and optimal allocation under a total power constraint.
  • Contributions: A necessary and sufficient solvability condition is expressed through two coupled majorization inequalities involving subchannel SNRs or capacities and unstable-pole log-magnitudes.The inequalities characterize how channel qualities should be matched with the source poles.
  • Contributions: For total power constraints, the paper derives the exact minimum power; equal-noise channels admit an analytic water-filling allocation, while unequal-noise channels use sequential water-filling.The total-power analysis exploits partial-order programming under majorization order.
  • Design: When solvability holds, the paper provides a systematic procedure for constructing an admissible LTI encoder-decoder pair.The formulation uses LTI encoder and decoder structures with feedback over parallel AWGN subchannels.
  • Problem formulation: The source model assumes unstable eigenvalues have geometric multiplicity 1, equivalently requiring the unstable Jordan blocks’ direct sum to be cyclic.The estimation error covariance is considered bounded when it converges to a positive semidefinite matrix.

C. Submodular Functions

This section introduces submodular functions and their associated polyhedral structures as preliminary tools for the paper’s majorization-based analysis.

  • Definitions: A set function f: 2^N → R is submodular when f(S) + f(T) ≥ f(S ∪ T) + f(S ∩ T) for all subsets S and T.The function satisfies f(∅) = 0.
  • Definitions: The submodular and base polyhedra are associated geometric sets defined from the submodular function.The supplied passage introduces these polyhedra but does not include their full defining expressions.
  • Useful lemma: For any b ∈ R^n, there exists x⋆ in the base polyhedron such that x⋆ + b is majorized by x + b for every x in that base polyhedron.This lemma supplies an extremal element useful for the subsequent analysis.

IV. LINEAR CODING WITH SUBCHANNEL POWER CONSTRAINTS

The section characterizes when LTI encoder-decoder pairs can achieve bounded estimation error under individual subchannel power constraints. Solvability is governed by coupled majorization relations linking subchannel powers and SNRs to the source’s antistable poles.

  • Necessary and sufficient condition: Theorem 1 states that solvability is equivalent to the existence of auxiliary variables γ_i ≥1 satisfying two coupled majorization inequalities.The inequalities comprise an additive relation between squared pole magnitudes and γ_i, and a multiplicative relation involving logarithms and power-constrained channel terms.
  • Interpretation: More even distributions of subchannel SNRs or pole log-magnitudes facilitate solvability under the majorization conditions.The auxiliary variables couple these two distributional effects in the feasibility test.
  • Necessary and sufficient condition: Solvability verification is essentially convex and can be reduced, after ordering γ_i, to a geometric program transformable into convex form.This permits numerical verification using convex optimization methods.
  • Interpretation: If log|λ| ≼ log|λ̄|, then the feasible power region for λ contains that for λ̄, so a more evenly distributed pole profile is more favorable for transmission.The result is formalized by Proposition 2 and illustrated through two sources whose feasible regions are compared in Fig. 2.
  • Implications and scope: The difficulty of LTI transmission depends on both topological entropy and the evenness of antistable-pole log-magnitudes, rather than on their total sum alone.The paper notes that simpler sufficient corollaries are generally conservative, becoming necessary in the single-pole or single-subchannel cases.
  • Implications and scope: Theorem 1 sharpens prior sufficient conditions by providing an exact solvability characterization under prescribed subchannel powers.The corollaries remain easier to check but do not generally provide exact feasibility tests.

B. Solvability and Channel Capacity

The paper characterizes solvability under channel-capacity constraints using majorization relations and constructs feasible LTI encoder-decoder pairs. Total capacity exceeding topological entropy is necessary but generally insufficient unless pole magnitudes are sufficiently evenly distributed.

  • Individual subchannel constraints: Theorem 1 reduces individual-power solvability to feasibility over variables γ_i ≥ 1 satisfying coupled majorization conditions.These conditions connect channel qualities with the source's antistable eigenvalues.
  • Channel capacity: Total channel capacity must exceed the source topological entropy, but this requirement is generally insufficient for solvability.An example has C_total = 0.973 exceeding H(A) = 0.9 while the specified power allocation remains infeasible.
  • Channel capacity: C_total > H(A) is necessary and sufficient exactly when the log-magnitudes of the antistable poles satisfy the stated majorization condition.The condition automatically holds for equal-magnitude antistable poles and for a single subchannel.
  • Encoder-decoder design: When the majorization conditions hold, the sufficiency proof constructs an encoder-decoder pair with bounded estimation error covariance and satisfied subchannel power constraints.The design uses an isometry to distribute communication demands across subchannels and selects a sufficiently small positive ε.
  • Encoder-decoder design: The construction handles repeated eigenvalues by modifying the intermediate matrix design before repeating the isometry-based power-balancing argument.The modification addresses cases where the constructed matrix is noncyclic and therefore has a different Jordan canonical form.

A. Preliminary Analysis via Partial Order Programming under Majorization Order

The total-power problem is formulated as a partial-order program under majorization order to determine the minimum feasible transmission power. This formulation yields uniqueness, threshold characterizations, and water-filling-based allocation procedures.

  • Problem formulation: The minimum total power problem seeks a subchannel allocation that makes the majorization feasibility conditions hold while minimizing the sum of allocated powers.The problem is transformed into a partial-order program whose constraint is a majorization inequality.
  • POP analysis: The partial-order program has a unique optimal solution.This uniqueness result is established for the formulated POP problem.
  • POP analysis: The POP optimum determines the optimal value of the original power-allocation problem, which can be approached arbitrarily closely.The limiting allocation may assign vanishing power to surplus subchannels.
  • Feasibility threshold: Problem 2 is solvable if and only if the available total power exceeds the minimum threshold p⋆.For any p > p⋆, powers can be allocated so the majorization conditions hold and a feasible encoder-decoder pair can be constructed.
  • Power allocation: When more subchannels than antistable poles are available, the minimum power is approached using the subchannels with the lowest noise variances; remaining channels provide no further reduction.For equal noise variances, the analysis gives an analytic allocation with a water-filling interpretation; unequal variances use sequential water-filling.

B. The Case of Equal Subchannel Noise Variance: Analytic Solution and Water-Filling Interpretation

For equal noise variances, the optimal power-allocation solution has an analytic majorization-based form and admits a water-filling interpretation. The construction balances the workload across all subchannels, reducing total power relative to using only part of them.

  • Analytic solution: Theorem 3 gives the optimal solution to the partial-order program through the join of vectors under weak majorization.The join is the least feasible element under the relevant order.
  • Water-filling interpretation: The base profile contains baseline workloads from the largest antistable poles, while the remaining workload is distributed across subchannels.Water-filling raises components below the water level and leaves larger components unchanged.
  • Water-filling interpretation: The resulting allocation uses all subchannels when l≤na, and restricting transmission to part of them requires higher total power.All components of c⋆ are positive under this premise.
  • Example: The paper compares the exact minimum total power with the upper bound in for Example 3 with σ2 = 1.The comparison is shown in Fig. 5.
  • Example: In Example 3, equal allocation of the stabilizing workload across two subchannels gives the minimum total power p⋆= σ2(2|λ|3−2).The partition-based upper bound is strictly greater than p⋆, with the gap increasing as |λ| increases.

C. The General Case: Sequential Water Filling

With unequal subchannel noise variances, the optimal allocation is computed by sequential water-filling rather than the equal-noise analytic reduction. Each iteration incorporates another noise floor and balances the resulting effective levels.

  • Sequential water filling: The general unequal-noise problem is solved by the sequential water-filling algorithm in Theorem 4.The equal-noise case is recovered when the initial water-filling steps become trivial.
  • Algorithm: At each step, a new subchannel noise floor is added and the associated workload is poured over the combined base profile.The effective profile is updated and carried to the next iteration.
  • Algorithm: The algorithm performs one water-filling iteration for each of the l subchannels.Algorithm 1 explicitly loops over k from 1 to l and finds a water level at each iteration.
  • Algorithm: After l iterations, the resulting profile produces the optimal allocation c⋆.The construction proceeds recursively through the intermediate vectors d(k) and water levels h_k.
  • Optimality: The optimality proof establishes that c⋆ is weakly majorized by every feasible allocation after accounting for the noise profile.This identifies c⋆ as the least feasible solution under the relevant order.

VI. NUMERICAL EXAMPLES

The numerical examples construct encoder-decoder designs under individual and total power constraints and verify bounded estimation error covariance through error-system poles and simulations. The total-power example attains the predicted minimum using sequential water-filling.

  • Individual power constraints: Under individual constraints p1 = 20, p2 = 20, and p3 = 30, the selected subchannel parameters satisfy the solvability condition.The resulting encoder-decoder design has stable estimation-error poles and satisfies the subchannel power constraints in simulation.
  • Individual power constraints: The individual-constraint design places the estimation-error poles at e−1.2, e−1, e−0.55, and e−0.5.These are the reciprocals of the eigenvalues of A, confirming boundedness of the EEC.
  • Total power constraint: Sequential water-filling yields c⋆= 1.55 1.10 0.60′ and minimum required total power p⋆= 62.3851.The example then uses a power limit p = 62.5 > p⋆ to construct a feasible design.
  • Total power constraint: The total-power design has the same stable estimation-error poles and satisfies the total power constraint in 50,000 Monte Carlo simulations over T = 100 steps.The construction uses the optimal allocation c⋆ in the encoder design.
  • Conclusion: The paper concludes that individual constraints are characterized by coupled majorization inequalities, while total constraints admit analytic or sequential water-filling solutions.The corresponding coding design procedures are also provided.

APPENDIX

The appendix explains meet and join operations through cumulative-sum curves under majorization order. The meet is obtained from a pointwise minimum, whereas the join requires a concave envelope.

  • Cumulative-sum curves: For vectors with nonincreasing entries, cumulative-sum curves are concave and provide a geometric representation of majorization operations.The curves are formed by linearly interpolating cumulative sums.
  • Meet and join: The meet x∧y is the greatest cumulative-sum curve lying below both Cx and Cy.Because the pointwise minimum of concave functions remains concave, it can be obtained directly by interpolation.
  • Meet and join: The join x∨y is the least cumulative-sum curve lying above both Cx and Cy.Its construction generally requires the concave envelope of the pointwise maximum curve.
  • Example: Fig. 7 illustrates the computation of x∧y and x∨y for a concrete vector pair.The appendix gives the resulting meet and join vectors for the example.

B. Necessary Conditions for Problem 1

The section derives necessary conditions for bounded estimation error covariance and finite transmission power in LTI encoder-decoder systems. It shows that unstable source modes must be preserved in the encoder dynamics and remain controllable and observable through the resulting system.

  • An encoder-decoder pair with bounded EEC and finite transmission power requires a stabilizable and detectable encoder realization.
  • Every unstable eigenvalue of the source matrix A is also an eigenvalue of the encoder state matrix Ae, with no smaller algebraic multiplicity.
  • Finite transmission power requires all observable modes of the source-encoder cascade to be stable.
  • The encoder feedback dynamics Afb must be Schur stable; otherwise an unstable eigenvalue contradicts the bounded-EEC conditions.
  • A Sylvester-equation argument establishes the required rank property by showing that a nontrivial null space would contradict the full-column-rank condition on X.
  • The unstable eigenvalues of the source must be controllable in the reduced system, as required by the PBH test for bounded-EEC estimation.

C. Technical Lemmas 

The technical lemmas connect stabilizing Riccati solutions and eigenvalue structure to log-majorization. They also provide the water-filling-style majorization transformation used in the later analysis.

  • A semi-stabilizing Riccati solution exists with all eigenvalues of A − AXC′(I + CXC′)−1C in the closed unit disk.
  • The antistable eigenvalues of A are connected by log-majorization to the eigenvalues of I + CXC′.
  • The eigenvalues of I + Z′C′uCuZ are squares of the singular values of Z−1AuZ, enabling application of Lemma 4 to obtain the stated relation.
  • For vectors related by weak majorization, raising coordinates below a threshold h produces a transformed vector governed by the condition Σ_i max{0, h − x_i} = η.
  • The proof verifies the transformed vector’s partial-sum inequalities by contradiction over the indices at which coordinates are raised to h.
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