gmd.m.zip_GMD_GMD分解_GMD编码_MIMO预编码_几何均值分解

% Matlab implementation of the "Geometric Mean Decomposition" % version of Hager, December 3, 2003 % %A = U*S*V' is the singular value decomposition of A % U, V unitary, S diagonal matrix with nonnegative % diagonal entries in decreasing order % = Q*R*P' is the geometric mean decomposition of A % P, Q unitary, R real upper triangular with r_ii = % geometric mean of the positive singular values of A, % 1 <= i <= p, p = number of positive singular values % All singular values smaller than tol treated as zero function [Q, R, P] = gmd (U, S, V, tol) if ( nargin < 4 ) tol = 0 ; end [m n] = size (S) ; R = zeros (m, n) ; P = V ; Q = U ; d = diag (S) ; l = min (m, n) ; for p = l : -1 : 1 if ( d (p) >= tol ) break ; end end if ( p < 1 ) return ; end if ( p < 2 ) R (1, 1) = d (1) ; return ; end z = zeros (p-1, 1) ; large = 2 ; % largest diagonal element small = p ; % smallest diagonal element perm = [1 : p] ; % perm (i) = location in d of i-th largest entry invperm = [ 1 : p ] ; % maps diagonal entries to perm sigma_bar = (prod (d (1:p)))^(1/p) ; for k = 1 : p-1 flag = 0 ; if ( d (k) >= sigma_bar ) i = perm (small) ; small = small - 1 ; if ( d (i) >= sigma_bar ) flag = 1 ; end else i = perm (large) ; large = large + 1 ; if ( d (i) <= sigma_bar ) flag = 1 ; end end k1 = k + 1 ; if ( i ~= k1 ) % Apply permutation Pi of paper t = d (k1) ; % Interchange d (i) and d (k1) d (k1) = d (i) ; d (i) = t ; j = invperm (k1) ; % Update perm arrays perm (j) = i ; invperm (i) = j ; I = [ k1 i ] ; J = [ i k1 ] ; Q (:, I) = Q (:, J) ; % interchange columns i and k+1 P (:, I) = P (:, J) ; end delta1 = d (k) ; delta2 = d (k1) ; sq_delta1 = delta1^2 ; sq_delta2 = delta2^2 ; if ( flag ) c = 1 ; s = 0 ; else c = sqrt ((sigma_bar^2 - sq_delta2)/(sq_delta1 - sq_delta2)) ; s = sqrt (1-c*c) ; end d (k1) = delta1*delta2/sigma_bar ; % = y in paper z (k) = s*c*(sq_delta2 - sq_delta1)/sigma_bar ; % = x in paper R (k, k) = sigma_bar ; if ( k > 1 ) R (1:k-1, k) = z (1:k-1)*c ; % new column of R z (1:k-1) = -z (1:k-1)*s ; % new column of Z end G1 = [ c -s s c ] ; J = [ k k1 ] ; P (:, J) = P (:, J)*G1 ; % apply G1 to P G2 = (1/sigma_bar)*[ c*delta1 -s*delta2 s*delta2 c*delta1 ] ; Q (:, J) = Q (:, J)*G2 ; % apply G2 to Q end R (p, p) = sigma_bar ; R (1:p-1, p) = z ; H=Q*R*P'