Hadamard Measurement Matrix for Compressed Signal Sampling
Find Innovative SolutionsGenerate Solutions
Solution Overview
Problem
Existing methods for capturing and processing analogue signals are inefficient, particularly in compressing bandlimited signals, as they require sampling at twice the Nyquist frequency and involve lossy compression, which discards redundant information.
Innovation Solution
The method involves selecting row vectors from a measurement matrix that is a product of a Hadamard or generalized Hadamard matrix and a representation matrix, allowing for efficient and simple signal representation by determining inner products, which results in a compressed representation with fewer measurements than traditional methods.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If traditional sampling at twice the Nyquist frequency is used, then the signal can be captured, but the number of measurements is large and inefficient
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 allows the system to capture the essential information of the signal in fewer measurements while simultaneously compressing the data, eliminating the need for separate sampling and compression steps.
Solution Approach 2:
The patent changes the measurement parameters by using a structured measurement matrix based on Hadamard matrices rather than traditional uniform random sampling. This parameter change enables the system to achieve the same information capture with fewer measurements by exploiting the incoherence between the Hadamard matrix and the signal representation basis.
2Quantity of substance
If lossy compression is used to reduce data size, then the representation becomes more compact, but redundant information is discarded
Solution Approach 1:
The patent extracts only the essential information from the signal by using a measurement matrix that is maximally incoherent with the signal representation basis. This extraction process captures the most important signal components in fewer measurements without discarding redundant information, as the incoherence ensures that each measurement provides unique information about the signal.
3Quantity of substance
If a measurement matrix with maximal incoherence is used, then fewer measurements are needed, but the matrix construction becomes more complex
Solution Approach 1:
The patent performs preliminary action by pre-defining the measurement matrix as the product of a Hadamard matrix and a representation matrix. This pre-computation of the measurement matrix structure allows the system to achieve maximal incoherence without complex real-time calculations during the actual sampling process, as the matrix structure is established in advance.
4Reliability
If traditional separate sampling and compression steps are used, then the process is well-established, but the overall method becomes more complex and less efficient
Solution Approach 1:
The patent merges the separate sampling and compression steps into a single unified operation. By using a measurement matrix that inherently performs both sampling and compression simultaneously, the system reduces the number of processing steps while maintaining reliability, as the single step is based on well-established Hadamard matrix properties.
Data Source
Figure 1
Figure 2
Figure 3
AI summary
A method for determining a representation (y) of a signal (s)comprise selecting a predeter- mined 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).