Divergence-Minimizing Distribution Matching for Parallel Channels and Channels with Memory

I. INTRODUCTION

• Direct DMs approach a target distribution, while indirect sphere-shaping methods select codewords within a specified-energy sphere.CCDM is a direct example; shell mapping and ESS are indirect examples.Source: §I · p. 1
• Most DMs assume iid symbols, but channels with memory and parallel channels with unequal noise can require targets that are not iid.For parallel channels, targets may be independent yet non-identically distributed across channels.Source: §I · p. 1
• The paper extends DM results from iid to general distributions and develops a joint ESS algorithm for parallel AWGN channels.The theory is applied to parallel AWGN channels, with simulation results reported later.Source: §I · p. 1

A. Notation and Information Measures

• Sets use calligraphic letters, random variables use uppercase letters, and their realizations use lowercase letters.The actual and target pmfs are usually denoted P_A and Q_A, respectively.Source: A. Notation and Information Measures · p. 1

B. Distibution Matching

• A distribution matcher is an invertible mapping from k input bits to n output symbols that approximates target Q_A^n.Its rate is k/n, and matching quality is measured by D(P_A^n||Q_A^n).Source: B. Distibution Matching · p. 1
• With uniformly distributed input bits, each codeword in the DM codebook occurs with probability 2^-k.The codebook is the set of sequences generated by the matcher.Source: B. Distibution Matching · p. 1

A. Minimizing Divergence

• The minimizing integer count of sequences at the next self-information level is either zero or the maximum available count.The analysis considers a continuous count to locate the divergence maximum, then chooses an integer count.Source: A. Minimizing Divergence · p. 2
• The divergence-minimizing codebook contains the highest-probability sequences, with all sequences at a tied probability level either included or excluded together.The theorem characterizes the optimal codebook for a target distribution with memory.Source: A. Minimizing Divergence · p. 2

B. Divergence Scaling

• Innovation-driven processes include autoregressive and Markov processes within the Bernoulli-scheme framework.The paper gives A_i = A_(i-1) ⊕ Z_i, with iid Bernoulli Z_i and fixed known A_0, as an example.Source: B. Divergence Scaling · p. 2
• For an invertible mapping of an iid sequence, the optimal DM divergence grows as 1/2 log2 n.The target process may be generated from iid innovations through an invertible function.Source: B. Divergence Scaling · p. 2
• The proof uses data processing, with equality for an invertible mapping, and optimal codewords can be obtained by mapping the most likely innovation sequences.The mapping f is applied to the most likely z^n under Q_Z^n.Source: B. Divergence Scaling · p. 2

• Sphere shaping accepts sequences with energy at most Emax, and the induced distribution converges to a discrete Gaussian with the sampled distribution’s mean and variance.Source: §IV · p. 2
• For k bits, Emax is the smallest value for which the valid-sequence set contains at least 2^k sequences, and the codebook contains exactly 2^k.Source: §IV · p. 2
• For ASK, possible energy levels run from n to Emax in steps of 8; sphere shaping uses sequences inside or on the sphere, while CCDM uses codewords on its surface.Source: §IV · p. 2
• At large n, sphere shaping and CCDM converge in sequence selection and synthesized probability because sphere volume concentrates near the surface.Source: §IV · p. 2

• ESS builds an n+1-stage trellis whose paths represent valid sequences inside the sphere of radius √Emax.Source: A. Enumerative Sphere Shaping · p. 3
• The trellis path counts support indexing: Cover’s formula counts paths directed to lexicographically smaller symbols.Source: A. Enumerative Sphere Shaping · p. 3
• Shaping interprets k input bits as an index, selects amplitudes stage by stage, and outputs a sequence at rate k/n bits per symbol.Source: A. Enumerative Sphere Shaping · p. 3
• Deshaping takes a sequence as input and recursively computes its index, outputting the initial index i1.Source: A. Enumerative Sphere Shaping · p. 3

B. Storage and Computational Complexity

• ESS requires O(n^3) bits of trellis storage and about n bit operations per output symbol; bounded precision can reduce storage to O(n^2) bits and computation to a mantissa-dependent constant.Source: B. Storage and Computational Complexity · p. 3

A. Channel Model and Power Allocation

• For L parallel AWGN channels, Yℓ = hℓXℓ + Zℓ, with known hℓ and independent zero-mean unit-variance Gaussian noise; the channels are used n times under a block power constraint.Source: A. Channel Model and Power Allocation · p. 3
• The channel input is Xℓ = Sℓ∆ℓAℓ, where Sℓ is an independent uniform sign and Aℓ is the amplitude.Source: A. Channel Model and Power Allocation · p. 3

• The method uses discrete Gaussian distributions to induce the amplitude distributions, with independent channel inputs.Source: B. Target Distributions · p. 3
• The weighted energy constraint uses Emax for the desired rate and weights wℓ = νℓ∆ℓ^2 to modify the powers aℓ,i^2.Source: B. Target Distributions · pp. 3–4
• Choosing ∆ℓ = ∆/hℓ gives the received constellations common spacing relative to noise; the weights are selected to minimize transmit power at target rate Rdm.Source: B. Target Distributions · p. 4

C. Joint Enumerative Sphere Shaping

• The target distribution receives a rounded weight for each channel so that the joint trellis has integer branch costs.Source: C. Joint Enumerative Sphere Shaping · p. 4
• J-ESS builds a trellis with L stages per time index, for Ln + 1 stages total, visiting channels round-robin and charging each amplitude branch w_ℓa^2.Rounded channel weights make the branch costs integers; node counts follow the same backward recursion as in the earlier ESS algorithm.Source: C. Joint Enumerative Sphere Shaping · p. 4
• The trellis uses the indicator e ≤ E_max for its terminal condition, while indexing, shaping, and deshaping remain unchanged.Source: C. Joint Enumerative Sphere Shaping · p. 4
• For L = n = 2, the example uses channel weights 1 and 4 with E_max = 30, admitting four valid amplitude sequences.The channel alphabets are {1, 3, 5} and {1, 3}, giving branch costs {1, 9, 25} and {4, 36}; storage and complexity follow Sec. IV-B with n replaced by Ln.Source: C. Joint Enumerative Sphere Shaping · p. 4

VI. SIMULATION RESULTS

• The example uses three channels with coefficients h_ℓ = 2, 1, 0.5 and weights w_ℓ = 1, 4, 16, carrying 32-, 16-, and 8-ASK.Source: §VI · p. 4

A. Rate Loss

• At n = 10, rate loss is 0.400, 0.118, and 0.063 bits per use for PDM, per-channel ESS, and J-ESS, respectively.At n = 50, the corresponding losses are 0.107, 0.044, and 0.019 bits per use; J-ESS reduces loss by factors of 5–6 versus PDM and 2–3 versus per-channel ESS.Source: Figure 3 · p. 4, A. Rate Loss · p. 4
• Joint shaping indexes one codebook over all Ln amplitudes, amortizing the finite-length penalty, and avoids PDM’s restriction to product distributions over bit levels.By Theorem 1, J-ESS is the divergence-minimizing code for the target distribution at that rate.Source: A. Rate Loss · p. 4

VI. SIMULATION RESULTS

• Fig. 4 presents FER comparisons between PDM and J-ESS.Source: Figure 4 · p. 5

B. Simulations with 5G-NR LDPC Codes

• J-ESS and PDM use the same information-bit count at each blocklength, but their average rates across blocklengths are not matched and drift by 0.11 bpcu.Because CCDM rate loss stays nearly constant in bits while Ln shrinks sixfold, curves at different blocklengths should not be compared.Source: B. Simulations with 5G-NR LDPC Codes · p. 5

C. Complexity

• For N = 132 amplitudes, the J-ESS trellis has about 2.7 × 10^6 nodes, each storing an integer of up to k ≈ 350 bits.Storage grows as O(N^3) bits, or O(N^2) bits with bounded precision; PDM uses four binary CCDMs in parallel with small storage and linear output-length complexity.Source: C. Complexity · p. 5

VII. CONCLUSIONS

• The conclusion states that Theorem 1 characterizes the divergence-minimizing DM for any Q_A^n by selecting sequences with the largest target probability.It also says Theorem 2 extends the 1/2 log2 n divergence scaling to innovation-driven processes.Source: §VII · p. 5

B. Simulations with 5G-NR LDPC Codes

• J-ESS gains carry over to coded performance with PAS and a 5G-NR LDPC code, while J-ESS has higher trellis complexity than PDM.The simulations use belief-propagation decoding with up to 100 iterations; Fig. 4 shows FERs.Source: §VII · p. 5, B. Simulations with 5G-NR LDPC Codes · p. 5

VII. CONCLUSIONS

• The conclusion characterizes J-ESS as reducing rate loss over PDM by a large factor for an illustrative channel and short blocklengths.The same conclusion identifies complexity reduction, for example through quantized ESS, as future work.Source: §VII · p. 5