Systematic Construction of Real Lapped Tight Frame Transforms.

Area of Science:

• Signal Processing
• Applied Mathematics
• Harmonic Analysis

Background:

• Lapped orthogonal transforms (LOTs) and harmonic tight frames are established signal processing tools.
• Real lapped tight frame transforms offer a redundant alternative to LOTs and an infinite-dimensional analog to harmonic tight frames.
• Efficient implementation and robustness are key considerations for practical transform design.

Purpose of the Study:

• To present a constructive algorithm for designing real lapped equal-norm tight frame transforms.
• To explore the relationship between these transforms and maximally robust frames.
• To introduce new classes of lapped orthogonal and tight frame transforms.

Main Methods:

• Construction of real lapped orthogonal transforms from discrete Fourier transform submatrices.
• Seeding these orthogonal transforms to generate real lapped tight frame transforms.
• Analysis of frame properties, specifically robustness to erasures.

Main Results:

• A novel constructive algorithm for real lapped equal-norm tight frame transforms.
• Identification of frames maximally robust to erasures within the constructed class.
• Generation of new families of lapped orthogonal and tight frame transforms.

Conclusions:

• The proposed algorithm successfully designs real lapped equal-norm tight frame transforms.
• The construction yields transforms with desirable properties, including robustness.
• This work expands the toolkit of efficient and robust signal processing transforms.