Source-linked AI summary
Generalized Tan-Arlery-Rabaste-Lehmann-Ovarlez Lower Bound on Ambiguity Function of a Set of Sequences With Mismatched Filters
Shibsankar Das
TL;DR
The paper seeks lower bounds on maximum AF sidelobes within a desired LAZ for sets of unimodular sequences. It introduces associated mismatched filters and delay–Doppler weight vectors, allowing filter lengths to differ from sequence lengths. The resulting bound extends the Tan-Arlery-Rabaste-Lehmann-Ovarlez correlation bound and contains the Welch bound as a special case.
Problem
Zero AF sidelobes over the entire delay–Doppler plane are unavailable, motivating lower bounds for maximum sidelobes within a desired LAZ.
Method
The paper derives an AF lower bound using associated mismatched-filter sets and two weight vectors for delay and Doppler shifts.
Results
The proposed LAZ AF lower bound extends the Tan-Arlery-Rabaste-Lehmann-Ovarlez correlation bound and includes the Welch bound as a special case.
Takeaways & Limitations
The framework relates mismatched-filter AF sidelobe bounds to established correlation and Welch bounds through special cases.
Abstract
from arXiv · showhide
In this paper, a lower bound on the maximum ambiguity function (AF) sidelobes of a set of unimodular sequences is formulated for the desired low-ambiguity-zone (LAZ). Our main idea is to introduce a set of mismatched filters associated to a set of unimodular sequences and two weight vectors for the delay and Doppler shifts, respectively. The length of mismatched filter maybe different to the length of unimodular sequence. The proposed lower bound on the maximum AF sidelobes for the desired LAZ can be treated as an extension of Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound, published in 2020, which dealt with the conventional correlation of sequences.
I. INTRODUCTION
The paper addresses unavoidable ambiguity-function sidelobes by targeting low-sidelobe behavior within a local delay–Doppler region. It formulates a mismatched-filter lower bound for sets of unimodular sequences.
- Ideal zero sidelobes across the entire delay–Doppler plane are impossible for unimodular sequences because of the lower bound on AF volume.
- A low ambiguity zone (LAZ) is a local delay–Doppler region where low AF sidelobes are sought.
- Mismatched filters can achieve lower AF sidelobes within the desired LAZ than matched filters, with some loss in processing gain.
- The paper studies M unimodular sequences of length Lx and associated mismatched filters of length Ly, distinguishing auto-AF from cross-AF.
- Two weight vectors for delay and Doppler shifts are used to derive a lower bound on maximum AF sidelobes in the desired LAZ.
- The proposed bound includes existing lower bounds as special cases in the zero-Doppler cut, while the paper develops the definitions and formulation in subsequent sections.
B. The Maximum AAF Magnitude and Maximum CAF Magnitude Associated with the LAZ
The LAZ is defined as a bounded rectangle around the origin in the delay–Doppler domain, and maximum auto- and cross-AF magnitudes are evaluated within it.
- The LAZ Q contains delay shifts satisfying −Zx < τ < Zx and Doppler shifts satisfying −Zy < ν < Zy.
- The parameters Zx and Zy are determined in practical scenarios by the maximum delay shift and maximum Doppler frequency shift.
- The maximum AAF magnitude associated with Q is defined separately before defining the maximum AF magnitude over the LAZ.
III. THE GENERALIZED MISMATCHED FILTER BOUND
The generalized bound considers sequence and mismatched-filter sets that may have different lengths, while favoring self-association and suppressing cross-association.
- The paper uses M unimodular sequences of length Lx and M associated mismatched filters of length Ly.
- Each mismatched filter has length Ly = Lx + 2Ls, where Ls ∈ N.
- The design assumption is to maximize inner products between each sequence and its associated filter while minimizing inner products with other mismatched filters.
A. The AF Between a Set of Unimodular Sequences With Its Associated Mismatched Filter
The paper defines AFs between sequences and mismatched filters, then examines phase shifting and the resulting maximum sidelobe quantities within the LAZ.
- Periodic and aperiodic CAFs are defined between a length-Lx unimodular sequence and an associated length-Ly mismatched filter.
- At zero Doppler, the AF becomes the correlation function θm,n(τ).
- Phase shifting each mismatched filter does not affect AF sidelobe levels when the delay shift is τ = 0.
- The phase-shifting result is established by defining a phase-shifted filter and calculating its AF from the AF definition.
- The maximum AF magnitude between each sequence and its associated filter is considered over the LAZ and specified delay and Doppler ranges.
B. The Generalized Mismatched Filter Bound Associated with the LAZ
The section constructs weighted sequence and filter matrices for a prescribed LAZ, then derives Frobenius-norm bounds leading to a lower bound on maximum AF magnitude.
- The LAZ Q uses delay and Doppler dimensions satisfying 1 ≤ Zx ≤ Lx and 1 ≤ Zy ≤ Lx ≤ Ly.
- The construction forms Doppler-shifted sequence and mismatched-filter extensions of length Lx + Ly −1 and their associated circulant matrices.
- Two nonnegative weight vectors α and β, normalized to unit sums, represent the delay and Doppler shifts in the matrix construction.
- The matrices X and Y are used to derive upper and lower bounds on the Frobenius norm ||XYH||F.
- Theorem 1 gives a lower bound on maximum AF magnitude for any length-Lx unimodular sequence set and associated length-Ly mismatched-filter set.
C. Optimal Weight Vectors
The generalized bound depends on the delay and Doppler weight vectors, and equal weights minimize their squared norms to produce the optimal bound.
- The generalized lower bound is a function of the two weight vectors α and β.
- Larger αTα and βTβ produce a larger bound, so the vectors are optimized by minimizing these squared norms.
- The minimum values occur when α and β assign equal weights across their delay and Doppler entries.
- Corollary 1 states the resulting maximum-AF-magnitude lower bound for optimal weight vectors over the specified LAZ.
A. Relationship With the Well-Known Welch Bound [9]
The proposed generalized lower bound contains the Welch bound as a special zero-Doppler case.
- Setting Zx = 2Lx −1, Zy = 1, and uniform weight vectors reduces the Doppler shift to ν = 0.
- Under this specialization, the resulting expression is the well-known Welch bound.
B. Relationship With Tan-Arlery-Rabaste-Lehmann-Ovarlez Correlation Lower Bound [13]
The proposed lower bound reduces to the Tan-Arlery-Rabaste-Lehmann-Ovarlez correlation lower bound under a zero-Doppler special case. Setting Zx = Lx = K and Zy = 1 yields the stated bound K = min{MK, Lx + Ly −1}.
- Special case: Under Zx = Lx = K and Zy = 1, the proposed lower bound uses a delay weight vector α satisfying (10) and β = [1/Zy]T.This special case sets the Doppler dimension to one.
- Special case: With Zy = 1, the Doppler shift becomes ν = 0, so AF ϕm,m(τ, 0) equals the correlation function θm,m(τ).The reduction holds for each sequence m = 1, 2, · · ·, M.
- Result: The resulting expression is K = min{MK, Lx + Ly −1}, which is the Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound.
V. CONCLUSION
The paper develops a Theorem 1 lower bound on maximum AF sidelobes over a delay-Doppler LAZ using mismatched filters and separate delay and Doppler weights. It extends the earlier correlation bound, relates to the Welch bound, and identifies sequence-filter design meeting the bound as future work.
- V. CONCLUSION: Theorem 1 gives a lower bound on maximum AF sidelobes over the LAZ Q in the delay-Doppler domain.
- V. CONCLUSION: The main innovation is a set of mismatched filters associated with unimodular sequences and two weight vectors for delay and Doppler shifts.The filter length Ly may differ from the sequence length Lx.
- V. CONCLUSION: The proposed bound extends the Tan-Arlery-Rabaste-Lehmann-Ovarlez lower bound and depends on M, Lx, Ly, Zx, Zy, α, β, and AF values.
- V. CONCLUSION: The proposed lower bound has a demonstrated relationship with the Welch bound.
- V. CONCLUSION: Future work is to design unimodular sequences and associated mismatched filters that meet the proposed AF lower bound.