Nearly Toeplitz Matrix Inversion via Perturbation and Iteration
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Solution Overview
Problem
Existing methods for inverting nearly Toeplitz matrices in adaptive wireless receivers are inefficient due to their high computational complexity, as they do not exploit the nearly Toeplitz structure, making them difficult to implement in real-time channel estimation and interference suppression tasks.
Innovation Solution
A method is introduced that involves determining a perturbation matrix to transform a nearly Toeplitz matrix into a Toeplitz matrix, allowing for efficient inversion using known Toeplitz matrix algorithms, with an iterative recursion to compute the inverse, exploiting the Toeplitz structure for faster convergence.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If conventional methods (Gaussian elimination, Cholesky decomposition, Gauss-Seidel algorithm) are used to invert nearly Toeplitz matrices, then the inversion can be performed, but the computational complexity is high and implementation efficiency is poor
Solution Approach 1:
The patent introduces a perturbation matrix E as an intermediary to transform the nearly Toeplitz matrix A into a true Toeplitz matrix T through the relationship A = T - E. This intermediary transformation allows the use of efficient Toeplitz matrix inversion algorithms while accounting for the near-Toeplitz structure, thereby reducing computational complexity compared to conventional methods that do not exploit this structure.
Solution Approach 2:
The patent changes the representation parameters by expressing the nearly Toeplitz matrix in terms of a Toeplitz matrix plus a perturbation matrix. This parameter transformation enables the application of specialized Toeplitz inversion algorithms that have lower computational complexity, converting the original difficult inversion problem into a more tractable form that can be solved efficiently.
2Reliability
If MLSE equalizer is used to reduce bit error rates, then performance is improved, but implementation complexity grows exponentially with channel estimate length
Solution Approach 1:
The patent inverts the conventional approach by using a simpler DFE equalizer structure combined with an MMSE prefilter that requires inverting a nearly Toeplitz matrix. Instead of using the complex MLSE equalizer that provides excellent performance but has exponential complexity, the patent finds an alternative path through prefiltering followed by simple sequential detection, achieving comparable performance with much lower complexity when the matrix inversion is performed efficiently.
3Device complexity
If DFE equalizer with MMSE prefilter is used to reduce complexity, then implementation complexity is reduced, but efficient inversion of nearly Toeplitz matrix is required for real-time operation
Solution Approach 1:
The patent uses the perturbation matrix E as an intermediary to enable real-time inversion of nearly Toeplitz matrices. By transforming the inversion problem into the Toeplitz domain where efficient algorithms exist, the patent achieves the real-time processing efficiency needed for practical DFE equalizer implementation while maintaining the lower complexity structure.
4Reliability
If whitening filter is adaptively computed using covariance method, then interference suppression is improved, but inversion of noise covariance matrix requires high computational resources
Solution Approach 1:
The patent changes the computational parameters by exploiting the nearly Toeplitz structure of the noise covariance matrix. Instead of performing general matrix inversion that requires O(n³) operations, the patent transforms the problem into the Toeplitz domain where specialized algorithms can exploit the structure to reduce computational resources while maintaining accurate interference suppression performance.
Data Source
AI summary
A method of computing an inversion (X) of a nearly Toeplitz n by n matrix (A). A perturbation matrix (E) is first determined such that the sum of the nearly Toeplitz matrix (A) and the perturbation matrix (E) is a Toeplitz matrix (T). The inversion is solved by solving the equation X=T−1(B+EX), where B is a vector or matrix of dimension n by m. An initial estimate X(0) is selected and estimates of the inversion X are iteratively computed through the recursion X(n−1)=T−1(B+EX(n)). The initial estimate X(0) may be equal to an inversion (T−1) of the Toeplitz matrix (T). The present invention may be utilized in a radio receiver to efficiently compute (1) a least-squares (LS) channel estimate, (2) minimum mean squared error (MMSE) prefilter coefficients for a decision feedback equalizer (DFE), or (3) an autoregressive (AR) noise-spectrum estimation from a finite number of observed noise samples.