Method and apparatus for implementing sparse finite-impulse-response equalizers

An apparatus, comprising:

• at least one processor;
• and
• at least one memory including computer program code,
• the at least one memory and the computer program code configured, with the at least one processor, to cause the apparatus at least to
• receive a transmission;
• determine a sparsifying dictionary (Φ) that sparsely approximates a data vector (d) of the transmission by performing a fast Fourier transform and/or an inverse fast Fourier transform to decompose a dictionary matrix and the data vector (d);
• project the data vector (d) into a lower dimensional vector space;
• and
• configure a filter based on the determined sparsifying dictionary (Φ) by tuning a performance of the filter according to a predefined performance-complexity tradeoff level (ϵ),
• wherein the performing the fast Fourier transform and/or the inverse fast Fourier transform is performed on a circulant matrix that estimates a Toeplitz matrix,
• wherein the performing the fast Fourier transform and/or the inverse fast Fourier transform on the circulant matrix replaces performing Cholesky and Eigen factorization,
• wherein configuring the filter comprises configuring a finite impulse response filter,
• wherein the at least one memory and the computer program code are further configured, with the at least one processor, to cause the apparatus at least to sparse a time-domain impulse response of the finite impulse response filter,
• wherein the determining the sparsifying dictionary (Φ) comprises determining a sparsifying dictionary (Φ) that has a smallest worst-case coherence metric,
• wherein the worst-case coherence metric is a similarity measure between columns of a design matrix (K),
• wherein the smaller the worst-case coherence metric is, the less similar the columns are, and
• wherein the at least one memory and the computer program code configured, with the at least one processor, to cause the apparatus at least to
• jointly use the sparsifying dictionary (Φ), the data vector (d), and the design matrix (K)
• by passing the sparsifying dictionary (Φ), the data vector (d), and the design matrix (K) into an orthogonal matching pursuit greedy algorithm to obtain an estimate of an unknown vector (z), and
• by minimizing ∥K(Φz−d)∥22,
• wherein values of the predefined performance-complexity tradeoff level (ϵ) determine a sparsity level and complexity level, a number of nonzero active taps, and a performance level of the filter, and
• wherein the sparsifying dictionary (Φ) has the lowest coherence out of a plurality of sparsifying dictionaries.