Hadamard Measurement Matrix for Compressive Signal Representation
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Solution Overview
Problem
Existing methods for capturing and processing analogue signals are inefficient, as they require sampling at twice the Nyquist frequency and subsequent lossy compression, which discards redundant information.
Innovation Solution
A method that uses a measurement matrix dependent on a Hadamard or generalized Hadamard matrix and a representation matrix to select a smaller number of measurements, maximizing incoherence and enabling efficient signal representation through compressive sampling.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If traditional sampling at twice the Nyquist frequency is used, then the signal can be captured completely, but the number of measurements is large and redundant information is generated
Solution Approach 1:
The patent combines sampling and compression into a single operation by using a measurement matrix that is the product of a Hadamard matrix and a representation matrix. This merging allows the system to capture essential signal information while simultaneously compressing it, eliminating the need for separate sampling and compression steps and reducing the total number of measurements required.
Solution Approach 2:
The patent changes the measurement parameters by using a predetermined measurement matrix constructed from Hadamard matrices rather than traditional sampling methods. This parameter change enables the system to obtain a sufficient number of measurements at a lower rate, achieving both signal capture completeness and measurement reduction.
2Quantity of substance
If lossy compression is applied after sampling, then the signal size is reduced, but redundant information is discarded
Solution Approach 1:
The patent performs compression action preliminarily by incorporating the compression function into the sampling measurement matrix itself. By designing the measurement matrix as the product of a Hadamard matrix and a representation matrix, the system performs compression during the sampling process rather than as a separate subsequent step, thereby preserving essential information while achieving compression.
3Productivity
If a predetermined measurement matrix dependent on Hadamard matrix and representation matrix is used, then measurement efficiency is improved, but the device complexity increases
Solution Approach 1:
The patent segments the measurement matrix construction into distinct components: a Hadamard matrix and a representation matrix. This segmentation allows each component to be optimized independently and combined through simple matrix multiplication, reducing the overall complexity compared to designing a single complex measurement matrix from scratch.
Solution Approach 2:
The Hadamard matrix serves multiple functions in the system: it provides the measurement basis, enables incoherent sampling, and contributes to the compression function. This multi-functionality reduces the need for separate components and simplifies the overall device architecture despite the advanced measurement approach.
Data Source
AI summary
A method for determining a representation (y) of a signal (s) comprise selecting a predetermined number (m) of row vectors (v1, . . . , vm) from a predetermined measurement matrix (M). The predetermined measurement matrix (M) is predetermined dependent on a product of a predetermined Hadamard matrix or generalized Hadamard matrix (H) and a predetermined representation matrix(B). The predetermined representation matrix (B) represents a predetermined basis for the signal(s). The method further comprises determining a respective inner product of the signal (s) and each of the predetermined number (m) of selected row vectors (v1, . . . , vm) resulting in a predetermined number (m) of measurements (y1, . . . , ym) forming the representation (y) of the signal (s).