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Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Designs and System Identification
Venkatesh Saligrama
TL;DR
The paper addresses the need for aperiodic input sequences with good non-circular autocorrelation, including under unmodeled dynamics. It constructs higher order chirps from irrational numbers and shows polynomial worst-case autocorrelation decay alongside deterministic Toeplitz matrices with guaranteed RIP properties.
Problem
Applications require aperiodic input sequences with good non-circular autocorrelation, particularly when unmodeled dynamics affect the input.
Method
The paper constructs infinite-length higher order chirps using irrational numbers and applies continued-fraction and Diophantine-approximation results.
Results
The resulting sequences have polynomially decaying worst-case autocorrelation coefficients for every finite length, while associated Toeplitz matrices have guaranteed RIP properties.
Takeaways & Limitations
The construction provides a deterministic sequence family with uniformly decaying autocorrelation and deterministic Toeplitz matrices having guaranteed RIP properties.
Takeaways & Limitations
The derived RIP order appears small relative to what unstructured constructions can obtain, reflecting looseness in the guarantee.
Abstract
from arXiv · showhide
In this paper we present a new family of discrete sequences having "random like" uniformly decaying auto-correlation properties. The new class of infinite length sequences are higher order chirps constructed using irrational numbers. Exploiting results from the theory of continued fractions and diophantine approximations, we show that the class of sequences so formed has the property that the worst-case auto-correlation coefficients for every finite length sequence decays at a polynomial rate. These sequences display doppler immunity as well. We also show that Toeplitz matrices formed from such sequences satisfy restricted-isometry-property (RIP), a concept that has played a central role recently in Compressed Sensing applications. Compressed sensing has conventionally dealt with sensing matrices with arbitrary components. Nevertheless, such arbitrary sensing matrices are not appropriate for linear system identification and one must employ Toeplitz structured sensing matrices. Linear system identification plays a central role in a wide variety of applications such as channel estimation for multipath wireless systems as well as control system applications. Toeplitz matrices are also desirable on account of their filtering structure, which allows for fast implementation together with reduced storage requirements.
1 Introduction
The paper addresses the need for aperiodic, low-autocorrelation inputs and RIP guarantees for Toeplitz sensing matrices in sparse system identification and compressed sensing. It introduces deterministic sequences and shows that their Toeplitz matrices have RIP while retaining practical filtering advantages.
- Motivation: Aperiodic input sequences with uniformly decaying non-circular autocorrelation are sought to suppress unmodeled dynamics in system identification.The contribution of unmodeled dynamics to parametric error is tied to worst-case non-circular autocorrelation decay.
- Motivation: Sparse FIR reconstruction in multipath wireless and acoustic/RF systems motivates compressed sensing with Toeplitz-structured matrices.The system output is modeled as the convolution of inputs with sparse channel coefficients, represented by a Toeplitz matrix-vector product.
- Compressed sensing gap: RIP requires singular values of every k-column submatrix to remain close to one, enabling ℓ1 recovery for sufficiently small sparsity.Earlier deterministic constructions do not directly address the Toeplitz structure arising in system identification.
- Contribution: The paper designs deterministic Toeplitz matrices with guaranteed RIP properties for compressed sensing and system identification.This targets the structured sensing requirement that distinguishes sparse FIR identification from conventional compressed sensing.
- Sequence design: The proposed infinite sequences are higher-order chirps constructed from irrational numbers using continued-fraction and Diophantine-approximation results.Their worst-case autocorrelation coefficients for every finite-length sequence decay at a polynomial rate.
- Results and practical relevance: The resulting sequences exhibit Doppler immunity, while their Toeplitz matrices satisfy RIP and support fast implementation with reduced storage.Toeplitz structure is desirable because it provides a filtering implementation.
2 Problem Setup
The paper formulates linear system identification and sparse recovery around Toeplitz sensing matrices, emphasizing autocorrelation as a design criterion for reliable estimation. It then connects polynomially decaying worst-case autocorrelation to RIP guarantees for Toeplitz matrices.
- System identification: Linear system identification estimates a finite model from measured input, output, and noise, often separating modeled dynamics from residual unmodeled dynamics.The setup includes a system kernel, finite impulse-response models, and residual dynamics.
- System identification: Periodic inputs reveal only linear combinations of system coefficients, so individual model coefficients cannot be determined regardless of period length.The unmodeled dynamics can couple with the model-set dynamics in the worst case.
- Autocorrelation design: Worst-case aperiodic autocorrelation asymptotically approaching zero is a sufficient condition for estimating an optimal finitely parameterized model.This criterion motivates designing input sequences with uniformly small autocorrelation.
- Toeplitz RIP: Toeplitz matrix correlation coefficients are autocorrelation coefficients, and Gershgorin-based eigenvalue bounds yield sufficient RIP conditions.Steady-state Toeplitz structure produces time-dependent autocorrelations, so the analysis takes the worst case across relevant matrices.
- Compressed sensing: Sparse FIR identification connects the problem to compressed sensing, where the restricted isometry property supports sparse recovery using sensing matrices.The paper defines RIP through eigenvalue bounds on correlation matrices formed from selected columns.
- Toeplitz RIP: A Toeplitz matrix has RIP order q = O(n^γ) when its generating sequence satisfies the PDACF property with decay coefficient γ.The paper also notes that PDACF may not hold for an arbitrary number of columns n.
3 Sequence Design
The paper constructs higher-order chirp sequences from irrational numbers and analyzes their aperiodic autocorrelation using continued-fraction properties. The resulting sequences have polynomially decaying autocorrelation, including for third-order chirps and quadratic irrational parameters.
- Higher-order chirp sequences are shown to be PDACF sequences.
- Autocorrelation Bounds: For third-order chirps, the paper establishes the PDACF property for the autocorrelation function.
- The construction includes complex-valued higher-order chirps, with corresponding properties applying to their real and imaginary parts.
- Autocorrelation Bounds: The proof bounds autocorrelation by decomposing integer shifts through convergents and controlling how close their irrational multiples can approach integers.
- Continued Fractions: Continued-fraction convergents provide rational approximations whose even and odd subsequences approach an irrational number from opposite sides.
4 Matrices with RIP Property
The paper quantifies the RIP of Toeplitz matrices generated by the designed sequences by combining autocorrelation bounds with number-theoretic counting arguments. It derives an RIP order for the construction and examines the golden-ratio case and related quadratic irrationals.
- The paper analyzes the RIP property of Toeplitz sensing matrices generated by the sequence designs.
- Main Result: The resulting Toeplitz construction satisfies RIP of order (λn, n, n^1/4/λ^0.5).
- Golden-Ratio Construction: For sufficiently large n, the HOC construction with α equal to the golden ratio satisfies the stated RIP property.
- RIP Analysis: The autocorrelation analysis separates contributions by decay-rate regions and shows that only relatively few terms have the slowest decay.
- Proof Strategy: The proof controls slow-decay contributions using quadratic-irrational approximation bounds, Ostrowski representations, and divisor-count estimates.
- Golden-Ratio Construction: The golden ratio provides a moderate improvement in the RIP property, linked to the scaling of large-type integers between large values.
5 Discussion
The discussion highlights practical properties of the Toeplitz construction, including real-signal applicability, Doppler resilience, sequential processing, and low-memory generation. It also describes numerical condition-number experiments and notes limitations in the derived RIP order and bounding technique.
- Limitations: The derived RIP order appears small relative to what unstructured constructions can obtain.The authors attribute this looseness to the bounding technique used.
- Limitations: The bounding technique introduces looseness, motivating further confirmation of the authors’ assessment.The discussion explicitly characterizes the looseness as an inherent consequence of the bounding technique.
- Practical properties: Higher-order chirp properties for complex-valued signals also hold for their real and imaginary parts.This extends the stated signal properties beyond complex-valued representations.
- Practical properties: Doppler resilience follows because higher-order-chirp autocorrelation properties are unaffected by constant frequency shifts.The discussion links the resilience directly to the autocorrelation behavior under frequency shifts.
- Implementation: Adding a new row preserves the RIP property, enabling sequential processing of the constructed Toeplitz matrices.The new matrix is formed by concatenating the previous matrix with the added row.
- Implementation: The sequences and matrices can be generated with relatively little memory.This is presented as an implementation and practical advantage of the construction.