How Does Probabilistic Constellation Shaping Control Ranging Sidelobe Levels in ISAC?

I. INTRODUCTION

• Random communication payloads make sensing behavior vary across transmissions, while large sidelobes can mask nearby weak targets.Because payload symbols carry information, they cannot generally be designed as fixed sensing references.Source: §I · p. 1
• Existing PCS adjusts average sensing behavior through the input distribution, but its codeword-level sidelobe guarantees and limits were uncharacterized.Prior work also identified finite-length rate loss under conventional constructions and a given sensing constraint.Source: §I · p. 1
• The paper asks when type-based PCS can precisely control codeword-wise ISLR and how practical schemes can retain favorable rate-ISLR tradeoffs.Source: §I · p. 1
• The contributions include a permutation-invariance criterion, modulation-basis results, sifted shifting for coherent integration, and quadrant-PAS shaping methods.The proposed practical methods are reported to reduce finite-length rate loss versus existing methods under identical ISLR constraints.Source: §I · p. 2

A. Model and performance metrics

• The paper formulates an ISAC channel with symbols mapped through a unitary modulation basis and evaluates ranging through A-ACF or P-ACF-based ISLR.It also considers mutual information as the communication design objective.Source: A. Model and performance metrics · p. 2

• Type-based PCS first selects a type according to a distribution, then draws uniformly from that type's codeword class.The type distribution can be specified directly or through constraints on a type function.Source: B. Type-based PCS · p. 3
• The framework includes conceptual forms of CCDM, MPDM, shell mapping, and ESS.Source: B. Type-based PCS · p. 3
• At fixed rate, selecting the best codewords from exponentially growing type classes is typically impractical, so the analysis assumes uniform trimming.Under this assumption, codewords are selected randomly and uniformly within each type class.Source: B. Type-based PCS · p. 3

C. Known results and problems

• Under constant-power and rotational-symmetry assumptions, prior results express EISL through the fourth-order moment, and OFDM minimizes P-ACF EISL among unitary bases.Earlier PCS methods used this relationship to adjust the EISL-rate tradeoff.Source: C. Known results and problems · p. 3
• The paper poses two questions: when type-based PCS can precisely control codeword-wise ISLR, and how practical designs can achieve favorable rate-ISLR tradeoffs.Source: C. Known results and problems · pp. 3–4

III. ISLR CONTROLLABILITY

• Type-based controllability is equivalent to permutation invariance of the codeword metric.This provides a test for whether ISLR can be controlled through types.Source: §III · p. 4

A. The P-ACF case

• For N ≥ 7, P-ACF ISLR is type-controllable exactly for a characterized restricted family of unitary bases, with OFDM as a special case.When the employed types have exactly zero mean, all such controllable bases yield the same ISLR values as OFDM.Source: A. The P-ACF case · p. 4

• The characterization for N ≤ 6 is feasible but omitted because the analysis is tedious and considered of limited practical value.Source: B. The A-ACF case · p. 4
• For N ≥ 4, no unitary modulation basis permits type-based control of A-ACF ISLR.The proof characterizes the permutation action as monomial before deriving the impossibility result.Source: B. The A-ACF case · p. 5
• Thus, A-ACF-based PCS can adjust expected sidelobe behavior but cannot guarantee codeword-wise ISLR.The paper therefore focuses subsequent work on the P-ACF case.Source: B. The A-ACF case · p. 5

IV. COHERENT INTEGRATION VIA SIFTED SHIFTING

• Sifted shifting selects a frequency-domain circular shift for each OFDM symbol as part of modified coherent integration.The method is presented with ISLR-controllable type-based PCS to provide a guaranteed 1/M P-ACF ISLR reduction.Source: §IV · p. 5
• A 1/M reduction in ensemble ISLR is obtained over M OFDM symbols without altering each symbol’s ISLR.The codeword-level ISLR invariance follows because circular shifts are permutations.Source: §IV · p. 5
• The sifted-shifting algorithm has O(MN log N) complexity and requires ⌈log2 N⌉ additional bits per OFDM symbol to signal each shift.Source: §IV · p. 6

V. PRACTICAL ISAC-ORIENTED PCS DESIGN

• ISAC Pareto-optimal input distributions need not factorize into independent I/Q components, making standard PAS’s factorized design a mismatch.Source: §V · p. 6
• G-PAS embeds parity bits in APSK phase but is not compatible with QAM, and its equal-point ring structure may degrade communication performance.Source: §V · p. 6
• Quadrant PAS makes QAM compatible with PAS by shaping first-quadrant points and using uniform I/Q sign bits to indicate the quadrant.Source: §V · p. 6
• At finite blocklength, quadrant PAS’s larger first-quadrant support incurs greater rate loss than shaping a positive I/Q half-axis.The rate loss is measured by the gap between the reference-distribution entropy and the achievable matcher bits per blocklength.Source: §V · p. 6

A. CCDM

• With integer-valued types, enforcing CCDM’s hard ISLR constraint requires an extremely difficult combinatorial optimization; divergence-optimal N-type quantization instead yields one ISLR value that may violate the constraint.Source: A. CCDM · p. 6
• CCDM’s MI-maximizing reference distribution is quadrant-symmetric without further symmetry, and its larger support can increase finite-blocklength rate loss.Source: A. CCDM · pp. 6–7
• Shell degeneracy assigns equal probabilities to points of the same magnitude, but CCDM cannot exploit it efficiently when shell sizes differ.Practical CCDM therefore cannot reduce rate loss from the large quadrant set without inconvenient variable-length phase coding.Source: A. CCDM · p. 7

B. Joint hard-moment ESS

• Joint hard-moment ESS constrains sample moments and assigns uniform probabilities to legitimate codewords, whose marginal distribution approaches a maximal-entropy solution for sufficiently large N.Source: B. Joint hard-moment ESS · p. 7
• Replacing the ESS moment limits with the constrained-MI-optimal moments is proposed to improve achievable MI.The approach uses a two-dimensional search over the exponential-family parameters.Source: B. Joint hard-moment ESS · p. 7
• Exact joint hard-moment ESS is strictly ISLR-controllable because every permissible codeword satisfies the hard ISLR constraint.Source: B. Joint hard-moment ESS · p. 7

1) Exploiting shell degeneracy:

• JHM-ESS uses a weighted two-dimensional trellis in which each path represents a set of codewords whose size is the product of its edge weights.Source: 1) Exploiting shell degeneracy · p. 7
• Shell-degeneracy compression reduces trellis memory but cannot reduce finite-blocklength rate loss because it represents the same codebook.Source: 1) Exploiting shell degeneracy · p. 7

2) Trellis construction:

• JHM-ESS pruning handles coupled U and V constraints by bounding the minimum suffix V-increment for a given U-increment and pruning states above the threshold.Source: 2) Trellis construction · p. 8
• Finite-precision ESS implementation can also be applied to JHM-ESS, adding slight rate loss without affecting ISLR controllability.Source: 2) Trellis construction · p. 8

C. Scalarized ESS

• S-ESS merges the two constraints into one scalar cost, significantly reducing memory at the price of losing ISLR controllability.Source: C. Scalarized ESS · p. 8

3) Finite precision:

• In the large-N limit, S-ESS yields the same input distribution as JHM-ESS, but its half-space guarantee does not provide ISLR controllability.Source: 3) Finite precision · p. 8

1) Memory cost:

• CCDM memory is O(N), while JHM-ESS has cubic trellis memory that can be substantial for large constellations and long blocks.Source: 1) Memory cost · p. 8

2) Encoding and decoding time complexity:

• CCDM encoding and decoding take O(N|S|) per codeword, while ESS variants take O(NS), where S is the number of equal-magnitude shells.Source: 2) Encoding and decoding time complexity · p. 8

3) Sidelobe controllability:

• JHM-ESS guarantees a strict ISLR upper bound; CCDM fixes ISLR at a value constrained by integer type approximation, while S-ESS and 1D-ESS provide no such controllability.Source: 3) Sidelobe controllability · p. 9

4) Rate loss:

• JHM-ESS and S-ESS reduce rate loss versus CCDM by using multiple type classes, whereas 1D-ESS suffers severe rate loss from factorization under EISL constraints.Source: 4) Rate loss · p. 9

VI. NUMERICAL RESULTS

• JHM-ESS CDFs end at the prescribed reference level, CCDM has one quantized ISLR value, and S-ESS and 1D-ESS retain upper tails.CCDM's mean ISLR is not strictly controllable across different κoff values.Source: §VI · p. 9
• After 100 sifted-shifting integrations, all ISLR CDFs shift left, and the observed improvement can substantially exceed the guaranteed 1/M reduction.The method preserves individual type-based guarantees and bounds integrated ISLR by γ/Meff.Source: §VI · p. 9, Figure 2 · p. 10, §VII · p. 11
• Across the simulated blocklength range, two-dimensional ESS has the smallest rate losses, with S-ESS slightly better than JHM-ESS and CCDM and 1D-ESS worse.The reported ordering reflects more available codewords for S-ESS, CCDM's single-type restriction, and 1D-ESS's additional factorization constraint.Source: §VI · pp. 9–10, Figure 3 · p. 10
• The two-dimensional ESS schemes have nearly identical uncoded rate trends versus mean ISLR, while CCDM leads empirical MI and 1D-ESS is less efficient.The comparison uses N = 300 and SNR=14dB; the plotted tradeoff also considers maximum ISLR.Source: §VI · p. 10, Figure 4 · p. 10
• At blocklength 648, κoff = −10dB, and code rate 2/3, JHM-ESS and S-ESS operate at closely comparable Eb/N0, while 1D-ESS is shifted farther right.CCDM incurs only a modest displacement; the comparison is for coded schemes with LDPC codes.Source: §VI · p. 10, Figure 5 · p. 11
• At BLER=0.01, S-ESS has the smallest gap from the finite-blocklength rate bound, while 64APSK G-PAS is comparable at high SNR but lower at low SNR.At κoff = −10dB, 64PSK even outperforms the MI bound for factorized input distributions.Source: §VI · p. 11, Figure 6 · p. 11

VII. CONCLUSIONS

• For N ⩾7, type-based P-ACF ISLR control is characterized by permutation invariance; no unitary basis permits type-based A-ACF ISLR control for N ⩾4.OFDM is a principal case in the resulting P-ACF family, and zero-mean types yield the same ISLR as OFDM.Source: §VII · p. 11
• The two ESS schemes achieve smaller finite-length rate loss than CCDM and factorized 1D-ESS under the same ISLR constraint.JHM-ESS imposes hard moment constraints for maximum-ISLR control, whereas S-ESS scalarizes the moments and controls ensemble behavior.Source: §VII · p. 11

A. Proof of Proposition 1

• Type-based quantities are constant across codewords in a type class because those codewords differ only by permutations.The proof defines the quantity on a type class using any representative codeword.Source: A. Proof of Proposition 1 · p. 11

B. Proof of Lemma 1

• A unitary transformation preserving the stated ℓ4 isometry must be monomial.The proof obtains this from the vanishing cross-products in each column, which leaves at most one nonzero entry per column.Source: B. Proof of Lemma 1 · pp. 11–12

C. Proof of Theorem 1

• The proof of Theorem 1 analyzes the orbital conditions separately, using Condition (a)'s group action and the associated centralizer argument.The supplied proof passages develop the Condition (a) branch through a unitary matrix commuting with permutation matrices.Source: C. Proof of Theorem 1 · p. 12, 1) Condition (a) · p. 12

1) Condition (a):

• Under Condition (a), the symmetric-group action on the lines is a natural permutation representation, leading to a centralizer characterization.The proof uses the invariant vector and adjusts line phases before identifying the action as a permutation representation.Source: 1) Condition (a) · p. 12

2) Condition (b):

• For N ≥ 5, Condition (b) is impossible because its orbital size requires an image of size at least N − 1, contradicting the homomorphism bound.The proof places the alternating group in the homomorphism kernel before deriving the incompatible image-size requirement.Source: 2) Condition (b) · p. 12

D. Proof of Lemma 2

• A sequence has zero aggregate sidelobe energy only when it has a single nonzero entry.With two or more nonzero entries, the autocorrelation at the lag between the extreme nonzero indices is nonzero.Source: D. Proof of Lemma 2 · p. 12

E. Proof of Lemma 3

• Preserving aggregate sidelobe energy under the stated unitary setting reduces to preserving the fourth-power norm, with the index permutation restricted to identity or reversal.The proof derives the restriction from a bijection whose integer shift must be ±1.Source: E. Proof of Lemma 3 · p. 12

F. Proof of Lemma 4

• For the identity permutation component, four permitted index permutations yield the same monomial; the reversal case follows by altering the indices.The proof treats identity and reversal as the two permutation components.Source: F. Proof of Lemma 4 · p. 13

G. Proof of Theorem 2

• For N ≥ 4, the non-Abelian alternating group cannot map injectively into the Abelian group required by the stated condition.The proof uses this obstruction to rule out the condition.Source: G. Proof of Theorem 2 · p. 13