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Synchronization-Oriented Multi-Radius BMOCZ for Non-Coherent Short-Packet Communications
Hamoud Abdelbasset Hamidouche, Riadh Ouzane, Salah Eddine Zegrar, Huseyin Arslan
TL;DR
Non-coherent short-packet systems avoid channel estimation but face blind synchronization and detection under uncertainty. MR-BMOCZ uses structured multi-radius zeros to create synchronization signatures, and the paper reports lower CFO-estimation NMSE than the considered schemes alongside improved timing synchronization.
Problem
Avoiding channel knowledge shifts the short-packet system bottleneck toward blind synchronization and detection under uncertainty.
Source: §I · p. 1Method
MR-BMOCZ distributes information-bearing zeros across multiple radii, producing additional deterministic autocorrelation peaks used for blind synchronization.
Source: §I · p. 2Results
Across the entire SNR range, MR-BMOCZ achieves the lowest NMSE among the considered schemes, while the conclusion reports improved time synchronization performance.
Source: B. Normalized CFO performance · p. 11, §VII · p. 12Takeaways & Limitations
The multi-radius waveform reduces PAPR compared with BMOCZ-based schemes while retaining the non-coherent nature of conventional BMOCZ.
Source: §VII · p. 12
Abstract
from arXiv · showhide
Modulation on conjugate-reciprocal zeros (MOCZ) is a promising candidate for blind short-packet communication (SPC) in future wireless systems; however, its performance is sensitive to timing and frequency synchronization errors. In this paper, we propose a novel scheme named multi-radius binary MOCZ (MR-BMOCZ). This method introduces a new root placement strategy that enhances diversity in the autocorrelation domain. The resulting structure enhances the autocorrelation properties of the transmitted waveform, leading to improved timing offset (TO) estimation. Moreover, the multi-radius construction distributes the signal energy over a larger set of radii, hence reducing the peak-to-average power ratio (PAPR). The resulting waveform also exhibits a distinctive spectral structure that is exploited for carrier frequency offset (CFO) estimation and correction, yielding lower normalized mean square error (NMSE) compared with BMOCZ-based schemes. Simulation results over additive white Gaussian noise (AWGN) and fading channels demonstrate that MR-BMOCZ achieves superior synchronization performance while maintaining comparable bit error rate (BER) and block error rate (BLER) performance. Consequently, the proposed scheme offers a synchronization oriented extension of BMOCZ for robust non-coherent SPC.
I. Introduction
Non-coherent short-packet communication avoids pilot overhead but leaves blind synchronization and detection under uncertainty as central challenges. MR-BMOCZ addresses these challenges through structured multi-radius zero placement and additional autocorrelation diversity.
I. Introduction
- BMOCZ encodes information in z-transform zeros, where channel convolution becomes polynomial multiplication and received channel zeros coexist with information-bearing zeros.The paper describes this representation as enabling recovery without explicit channel estimation.Source: §I · pp. 1–2
- MR-BMOCZ distributes information-bearing zeros across multiple radius pairs, adding a design degree of freedom while preserving the BMOCZ structure.The transmitted waveform is represented as a product of lower-degree BMOCZ polynomials.Source: §I · p. 2
- The multi-radius structure creates additional deterministic autocorrelation peaks, providing a richer synchronization signature than conventional BMOCZ.The paper characterizes this structure in both the time and z-domains.Source: §I · p. 2
- The paper develops multipeak timing-metric and generalized Huffman-bracket techniques for timing-offset estimation, evaluated with BER and BLER simulations under timing errors.These techniques exploit the proposed autocorrelation diversity.Source: §I · p. 2
- The multi-radius construction produces a distinctive spectral structure for CFO estimation; the paper establishes an unambiguous-estimation condition and develops a generalized multi-radius DiZeT decoder.The decoder is evaluated through NMSE analysis.Source: §I · p. 2
II. PRELIMINARIES ON HUFFMAN-BMOCZ
- In the preliminaries, the transmit polynomial is factored into complex zeros, establishing a one-to-one mapping between signal coefficients and the zero constellation.The received signal is also represented in the z-domain because time-domain linear convolution corresponds to polynomial multiplication.Source: §II · p. 3
1) Key Property:
Huffman-BMOCZ has an impulse-like aperiodic autocorrelation property that supports timing estimation. MR-BMOCZ builds on this by combining multiple submessage-specific BMOCZ polynomials into a structured multi-radius constellation.
1) Key Property:
- Huffman-BMOCZ’s normalized autocorrelation has a dominant unit-magnitude central peak and two symmetric side peaks of magnitude η.This sparse structure supports robust synchronization and timing-offset estimation.Source: 1) Key Property · p. 3
A. Multi-radius BMOCZ Modulation
- The K-bit message is partitioned into submessages, each mapped to a polynomial through its corresponding zero constellation.Submessage lengths may differ across realizations, and their bit counts sum to K.Source: A. Multi-radius BMOCZ Modulation · pp. 3–4
- The scheme combines I independent zero constellations with different angular resolutions, allowing more flexible shaping than conventional single-constellation BMOCZ.It preserves the fundamental BMOCZ encoding principle.Source: A. Multi-radius BMOCZ Modulation · p. 4
- MR-BMOCZ forms the transmitted sequence by convolving multiple Huffman sequences, whose polynomial zero sets combine into a structured multi-radius constellation.The waveform is the product of the submessage polynomials.Source: A. Multi-radius BMOCZ Modulation · p. 4
B. Sub-polynomial angle design
Closely spaced zeros from different submessages can reduce inter-set distance and increase noise sensitivity. The proposed phase-offset design maximizes angular separation between radius groups while preserving each set’s radial placement.
B. Sub-polynomial angle design
- Closely spaced zeros from distinct submessages reduce minimum inter-set distance, potentially increasing noise sensitivity and degrading detection.The paper introduces controlled phase offsets to mitigate this issue.Source: B. Sub-polynomial angle design · p. 4
- For two radius groups, the relative phase offset is chosen to maximize minimum angular separation; for more groups, the procedure is applied recursively.The optimal rotation maximizes normalized minimum zero separation in Fig. 2.Source: B. Sub-polynomial angle design · p. 4, Figure 2 · p. 5
- The phase optimization adds separation between submessages without changing radial placement, preserving the optimal radial separation of each zero set.For I = 2, the paper illustrates the design with the first group’s phase set to zero.Source: B. Sub-polynomial angle design · p. 4
C. Receiver-Side Processing
MR-BMOCZ decoding applies a binary radial test independently across known radius groups, while its convolved Huffman autocorrelations create additional deterministic timing peaks. For K2 = 2K1, these peaks form a regular, symmetric signature, and their added autocorrelation energy provides more timing observations.
C. Receiver-Side Processing
- The receiver assumes known radius-group parameters, applies a binary radial test independently to each group, and concatenates the detected submessages without explicit channel estimation.The test compares degree-scaled z-transform magnitudes at reciprocal points.Source: C. Receiver-Side Processing · p. 5
A. Autocorrelation Structure
- Convolving multiple Huffman autocorrelations gives MR-BMOCZ more deterministic peaks than conventional BMOCZ, creating additional autocorrelation diversity.The peak indices depend on the selected radius parameters.Source: A. Autocorrelation Structure · pp. 5–6
- For two radii with K2 = 2K1, overlapping peaks produce a regular, symmetric autocorrelation signature that supports the timing template.The paper focuses on this two-radius configuration for its subsequent synchronization analysis.Source: A. Autocorrelation Structure · p. 6
- The additional deterministic autocorrelation energy provides more timing observations and improves synchronization reliability.For two radius groups, convolving the individual Huffman autocorrelations generates seven deterministic peak combinations.Source: A. Autocorrelation Structure · p. 6
B. Proposed Timing Offset Estimation
MR-BMOCZ timing estimation exploits a predictable lattice of autocorrelation peaks rather than only the two boundary side peaks used in conventional Huffman-BMOCZ. The proposed receiver correlates against the full peak template and uses a generalized bracket, with phase-shifted brackets averaged for arbitrary CFO.
B. Proposed Timing Offset Estimation
- The radius-induced peak lattice gives MR-BMOCZ a predictable timing signature beyond the two boundary side peaks of conventional Huffman-BMOCZ.The selected radius partition controls the peak locations.Source: B. Proposed Timing Offset Estimation · p. 7
- Timing acquisition correlates the received waveform with the complete MR peak template, whose dominant autocorrelation peaks occur at known positions.The timing setup first considers zero CFO, then extends to simultaneous timing and frequency offsets.Source: B. Proposed Timing Offset Estimation · p. 7
- The timing bracket is matched to MR-BMOCZ's autocorrelation structure, and a four-tap MR Huffman bracket offers a low-complexity implementation.At the correct timing hypothesis, radius-induced peaks align with template positions and add coherently.Source: B. Proposed Timing Offset Estimation · p. 7
- For arbitrary CFO, the final timing metric averages brackets with uniformly spaced phase offsets to reduce sensitivity to unknown phase relations.The phase offsets are ψp = 2πp/P for p = 0, …, P − 1.Source: B. Proposed Timing Offset Estimation · p. 7
C. Proposed Carrier Frequency Offset Estimation
CFO rotates the received zeros away from their nominal angular grids, but MR-BMOCZ’s multiple radius groups provide a spectral and geometric signature for estimating it. The method searches compensated signals for multi-radius radial contrast, subject to the constellation’s rotational ambiguities.
C. Proposed Carrier Frequency Offset Estimation
- CFO produces a common angular drift that disrupts DiZeT detection at nominal constellation angles, while joint alignment across radius groups creates intrinsic ambiguities.Those rotations are indistinguishable when they leave the multi-radius zero constellation invariant.Source: C. Proposed Carrier Frequency Offset… · p. 7
- The constellation is invariant to CFO phase rotations ε = 2πq gcd(K1, . . . , KI), q ∈ Z, setting its minimum nonzero ambiguity period.The uniquely identifiable range is bounded by this rotational periodicity.Source: C. Proposed Carrier Frequency Offset… · pp. 7–8
- Under ε < εmin2, MR-BMOCZ’s dominant spectral peak can estimate CFO, and its unambiguous range εmin,MR exceeds conventional BMOCZ’s εmin,B, allowing larger frequency offsets to be uniquely identified.Source: C. Proposed Carrier Frequency Offset… · p. 8
- MR-BMOCZ estimates CFO by compensating candidate rotations and maximizing multi-radius radial contrast across the angular grids.The metric is constrained by several angular grids, so a candidate aligning one radius group may fail to align the others.Source: C. Proposed Carrier Frequency Offset… · p. 8
V. PERFORMANCE ANALYSIS
The performance analysis compares MR-BMOCZ with conventional BMOCZ across PAPR, noise robustness, and computational complexity.
V. PERFORMANCE ANALYSIS
- PAPR is one of the performance dimensions compared between MR-BMOCZ and conventional BMOCZ.Source: §V · p. 8
- Noise robustness is another dimension of the comparison between MR-BMOCZ and conventional BMOCZ.Source: §V · p. 8
- The analysis also compares the computational complexity of MR-BMOCZ and conventional BMOCZ.Source: §V · p. 8
A. PAPR Analysis
The section attributes lower PAPR to energy spread across the multi-radius waveform’s coefficients and analyzes how inter-radius geometry affects zero stability under noise.
A. PAPR Analysis
- Distributing signal generation across lower-order sub-polynomials spreads energy across convolutional coefficients, reducing large-amplitude peaks and PAPR versus conventional BMOCZ.The analysis derives waveform energy and edge-coefficient behavior for the product construction.Source: A. PAPR Analysis · p. 8
B. Sensitivity Analysis
- In AWGN without synchronization errors, noise perturbs information-bearing zeros, and their expected displacement informs DiZeT decision reliability.The analysis models the perturbation as zero-mean Gaussian and approximates its mean-squared value.Source: B. Sensitivity Analysis · p. 9
- Unlike conventional BMOCZ, MR-BMOCZ root stability depends on both intra-radius spacing and the geometric relationship between radius groups.The minimum inter-radius separation is a parameter governing detection performance.Source: B. Sensitivity Analysis · p. 9
- Larger minimum inter-radius separation reduces expected root displacement and improves DiZeT reliability; noisy zero clusters tighten with SNR but vary in spread across zeros.The unequal sensitivity follows from the cross-radius robustness factor.Source: B. Sensitivity Analysis · p. 9, Figure 6 · p. 9
C. Complexity Analysis
MR-BMOCZ constructs sub-polynomials independently and combines them by convolution; its receiver decoding complexity remains comparable to conventional BMOCZ when the channel is short relative to K.
C. Complexity Analysis
- MR-BMOCZ builds sub-polynomials independently and combines their time-domain signals by convolution.The convolution cost depends on the degrees of the partial and next sub-polynomials.Source: C. Complexity Analysis · pp. 9–10
- For L << K, receiver decoding has time complexity O(K^2), equivalent to conventional BMOCZ.The stated MR-BMOCZ total is O(K(K + L)).Source: C. Complexity Analysis · p. 10
- With parallel hardware, receiver latency is O((K + L) max_i Ki), while transmitter latency is O(max_i K_i^2), versus O(K(K + L)) and O(K^2) conventionally.Source: C. Complexity Analysis · p. 10
VI. Simulation and Numerical Results
The simulations compare MR-BMOCZ with conventional BMOCZ in AWGN and fading channels, including tests with timing offsets. After timing correction, MR-BMOCZ is less sensitive to timing errors while retaining comparable error-rate performance under perfect synchronization.
VI. Simulation and Numerical Results
- The simulations compare MR-BMOCZ with conventional BMOCZ, using K1 = 7 and K2 = 14 by default and the DiZeT receiver.The evaluation first establishes baseline BER, then tests robustness to timing offsets against J-BMOCZ and S-BMOCZ.Source: §VI · p. 10
- Tests use AWGN and fading channels, vary timing offsets across the observation interval, and draw CFO ε from U[0, π 7).AWGN samples are independently drawn from CN(0, N0), and binary messages are selected uniformly.Source: §VI · p. 10
A. TO performance
- With random timing offsets, MR-BMOCZ approaches perfectly synchronized BER performance and achieves lower BLER in fading after correction, while conventional BMOCZ degrades substantially.Under perfect synchronization, the schemes have comparable BER and nearly identical BLER in AWGN and flat fading.Source: A. TO performance · p. 10
B. Normalized CFO performance
MR-BMOCZ achieves lower CFO-estimation NMSE than the compared schemes across the tested SNR range and shifts PAPR curves toward lower values. BER tests report a favorable phase rotation and broadly stable performance across tested packet lengths, with degradation as frequency-selective channel length increases.
C. PAPR performance
- MR-BMOCZ shifts PAPR CCDF curves toward lower values than BMOCZ, J-BMOCZ, and S-BMOCZ; balanced radius partitions yield the lowest PAPR.The multi-radius construction generates the waveform through convolution of lower-order Huffman sequences, distributing energy more uniformly.Source: C. PAPR performance · p. 11
B. Normalized CFO performance
- MR-BMOCZ has the lowest CFO-estimation NMSE across the tested SNR range versus S-BMOCZ and J-BMOCZ, with faster decay at low and moderate SNR.At high SNR, all schemes converge to a level set by search-grid resolution.Source: Figure 8 · p. 11, B. Normalized CFO performance · p. 11
D. BER performance
- The phase rotation in (13) achieves the lowest BER among the tested configurations.The passage relates larger angular distances to a lower probability of incorrect zero classification under noise.Source: Figure 10 · p. 12, D. BER performance · p. 12
- Across tested K values, BER remains relatively stable in AWGN and flat fading; in frequency-selective fading, both schemes degrade as channel length L increases.The passage reports only a minor MR-BMOCZ performance gap at larger K in AWGN and similar degradation for both schemes as L grows.Source: Figure 11 · p. 12, Figure 12 · p. 12, D. BER performance · p. 12
VII. Conclusion and Future Work
MR-BMOCZ extends BMOCZ with a structured multi-radius zero constellation that supports blind synchronization. Simulations report improved synchronization and reduced PAPR with comparable BER and BLER, while future work targets joint design optimization.
VII. Conclusion and Future Work
- MR-BMOCZ distributes information-bearing zeros across conjugate-reciprocal radii, producing deterministic autocorrelation peaks for blind synchronization and a more balanced waveform energy distribution.The paper reports reduced PAPR compared with BMOCZ-based schemes.Source: §VII · p. 12
- Simulations in AWGN and fading show improved timing synchronization and CFO-estimation accuracy, with comparable BER and BLER performance.The paper reports lower NMSE than conventional BMOCZ-based synchronization approaches.Source: §VII · p. 12
- Future work will jointly optimize radius allocation, message partitioning, and zero placement to improve detection.The proposed receiver preserves BMOCZ's non-coherent nature.Source: §VII · p. 12