Process for QR Transformation using a CORDIC Processor

8) A process for transformation of an H matrix to an upper right triangular matrix using a CORDIC processor having a VECTOR mode for rotating (X,Y) to (X′,Y′) and finding angle A such that said Y′ has a greatly reduced magnitude compared to said Y, the CORDIC processor also having a ROTATE mode for rotating (X,Y) to (X′,Y′) by said angle A, the process operative on the elements of a matrix having m rows and n columns, said process having the steps:

• a first step of identifying a submatrix initially set to said m rows and n columns;
• for each row of said submatrix, a second step of applying the real and imaginary parts of a left column element as (X,Y) respectively in said VECTOR mode and replacing said (X,Y) element with the rotated value (X′,Y′), using the resulting angle A from said VECTOR mode in one or more subsequent ROTATE mode operations to rotate and replace each said remaining (X,Y) element of said row with (X′,Y′) formed from the real and imaginary components, respectively, of each said rotated element;
• a second step of performing a cross-row transformation on pairs formed from adjacent rows, starting from a bottom pair of adjacent rows and thereafter each successive pair of adjacent rows above until no adjacent cross-row transformation row pairs of said submatrix remain, said cross-row transformation comprising applying to said CORDIC processor in VECTOR mode an (X,Y) input where X is the real component of an upper row of a particular column of said adjacent row pair and Y is the corresponding real component, respectively, of a lower row of a particular column of said adjacent row pair, generating transformed elements (X′,Y′) which replaces the corresponding previous elements (X,Y) and also generates a cross-row transformation angle B, said angle B subsequently used in said ROTATE mode to transform each of said remaining columns (X,Y) where X is either the real or imaginary component of an upper row of a particular column of said adjacent row pair and Y is the corresponding real or imaginary component, respectively, of a lower row of a particular column of said adjacent row pair, generating transformed element (X′,Y′) which replaces the corresponding previous elements (X,Y) of said adjacent row pairs;
• a third step of repeating said first and second step on a new submatrix formed from a previous submatrix by successively reducing the number of columns and rows by one until said second step is operative on a single element.

17) A process for a CORDIC processor having a VECTOR mode whereby an input (X,Y) generates an output (X′,Y′) formed from the rotation of (X,Y) until said Y′ is substantially zero, the CORDIC processor generating an angle A in said VECTOR mode, the CORDIC processor also having a ROTATE mode whereby an (X,Y) input is rotated by an angle A to form an output (X′,Y′), said process operative on an H matrix having a plurality of rows and columns, the process having the steps:

• for each said row of said H matrix, a first step of rotating the first column element of each said row in said VECTOR mode where said X is a real component and said Y is an imaginary component of said column element, thereby producing a rotation angle A for said row, thereafter rotating each said remaining element of said row in said ROTATE mode and using said rotation angle A of said previous VECTOR mode, said second step causing each original element of said matrix to be replaced by a corresponding transformed element;
• for each pair of adjacent rows of said H matrix, said adjacent rows formed from a lower row and an adjacent upper row, performing a cross-row transformation, performing a second step whereby said CORDIC processor accepts X as the real component of said upper row and Y as the real component of said lower row, thereafter rotating said (X,Y) to (X′,Y′), generating an angle A, and replacing the original element X with transformed X′ and Y with transformed Y′, said cross-row transformation thereafter forming an input (X,Y) from said lower row and said upper row of said adjacent rows and containing either real or imaginary components from an upper row particular column and lower row same particular column, rotating said (X,Y) in said ROTATE mode by said cross-row transformation angle A to form (X′,Y′), thereafter replacing the corresponding elements X with X′, and Y with Y′, until all such columns of a particular adjacent row pair are completed;
• a third step of repeating said first step and said second step on a submatrix of a transformed H matrix having a decreasing number of columns and rows until a main diagonal containing only real elements remains.