Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Designs and System Identification (0806.4958v2)

Abstract: 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.

• The paper presents higher-order chirp sequences based on irrational numbers that achieve polynomial decay in auto-correlation.
• It demonstrates that Toeplitz matrices formed from these sequences satisfy the restricted isometry property for reliable sparse recovery.
• The design offers practical advantages in system identification, enhancing channel estimation and real-time control applications.

Deterministic Designs with Deterministic Guarantees: Toeplitz Compressed Sensing Matrices, Sequence Design, and System Identification

The paper introduces a novel class of deterministic sequences exhibiting properties beneficial for compressed sensing (CS) and linear system identification. These sequences manifest uniformly decaying auto-correlation properties, thereby addressing critical requirements in communication, radar, and control systems. The sequences are constructed using higher-order chirps based on irrational numbers, leveraging the theory of continued fractions and Diophantine approximations.

Key Contributions

1. Sequence Design: The paper presents higher-order chirp sequences, denoted as HOCs, based on irrational numbers which ensure polynomial decay of auto-correlation coefficients. These sequences are Doppler resilient and exhibit robust properties under Doppler shifts.
2. Toeplitz Matrices with RIP: The formation of Toeplitz matrices from the designed sequences is demonstrated to satisfy the restricted-isometry property (RIP). The RIP of order $\mathcal{O}(n^{1/4})$ ensures that these matrices are suitable for sparse recovery in compressed sensing.
3. Application to System Identification: Special emphasis is placed on the applicability of these Toeplitz matrices in linear system identification, crucial for applications such as channel estimation in multipath wireless communications and control systems.

Theoretical Foundations

The deterministic sequences developed in the paper showcase uniformly decaying auto-correlation properties through polynomial decay rates. The auto-correlation decay is fundamentally linked to the choice of irrational numbers used in sequence construction. Employing results from continued fractions and Diophantine approximations, the sequences ensure that the worst-case auto-correlation coefficients decay at a rate defined by a polynomial function. Specifically, for quadratic irrational numbers, the decay rate is significant, making it feasible to meet stringent requirements in system identification and beyond.

Practical Implications

The practical implications are profound, particularly in the field of CS and system identification:

• Compressed Sensing: The RIP of Toeplitz matrices derived from these sequences implies they can be effectively used in CS to recover sparse signals from under-sampled data. This is especially important in scenarios where rapid data acquisition and storage efficiency are paramount.
• System Identification: The Toeplitz structure inherently supports efficient implementation and storage, facilitating real-time processing in control systems and dynamic environments such as wireless communications. The polynomial decay of auto-correlation aids in mitigating noise and unmodeled dynamics, essential for robust system characterization.

Numerical Results and Real-World Impact

The paper's simulation results align with theoretical predictions, confirming that higher-order chirp sequences significantly outperform traditional pseudo-random binary sequences (PRBS) and sine-sweeps in terms of auto-correlation properties. The Monte Carlo simulations underline the robustness of the proposed Toeplitz matrices, showing favorable condition numbers, thus validating their practical applicability in large-scale problems.

Future Directions in AI

The research opens several future avenues:

• Further Optimization: Exploring sequences derived from higher-order algebraic numbers or transcendental numbers could provide even more favorable auto-correlation properties.
• Extended Applications: While the focus is on linear system identification, the deterministic sequences and structured matrices could be adapted for non-linear system identification and other domains requiring efficient sparse representations.
• Integration with Advanced AI Techniques: Leveraging these deterministic designs in combination with advanced AI techniques such as deep learning could enhance the efficiency and robustness of AI systems in tasks involving high-dimensional data.

In conclusion, the paper offers significant advancements in the design of deterministic sequences and structured matrices with provable guarantees, showing considerable promise for compressed sensing and system identification. These contributions are likely to impact both theoretical research and practical implementations, fostering innovation in fields reliant on efficient signal processing and data acquisition methods.