# Adaptive filters MNw by Ali H. Sayed

By Ali H. Sayed

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We can proceed to factor A1, which would lead to an expression of the form where 4 2 ) > 0 is the (0,O) entry of A1 and A2 > 0 is the Schur complement of 4 2 ) in A l . lVLi-l, where Ln-l is n x n lower-triangular and 2) is n x n diagonal with positive entries { u ( i ) } . The columns of C,-r are the successive leading columns of the Schur complements {Ai}, normalized by the inverses of the (0,O) entries of {A;}. The diagonal entries of D coincide with these (0,O) entries of the {A;}. where V'I2 is a diagonal matrix with the positive square-roots of If we define L.

N - 1,into an n x n upper-triangular matrix R, - IIToII qothl llnll dhz dhz ... , Q = [qo q1 . . qn-l] with q;q2 = &. , when H has rank n, all the diagonal entries of R will be positive. The factorization H = QR is referred to as the reduced QR decomposition of the matrix H ; in effect, the QR decomposition of a matrix simply amounts to the orthonormalization of its columns. 1 (Reduced QR decomposition) Given an N x n, n 5 N , matrix H = ho hl . h,-l It can be decomposed as H = QR, where R is 1.

L The figure shows the plots of the probability density functions of a Gaussian random variable z with mean 1 = 20, variance u; = 225 in the top plot, and variance ug = 4 in the bottom plot. where gz is called the standard deviation of 2. Recall further that the pdf of a random variable is useful in several respects. From Fig. l we observe that the smaller the variance of 2,the more concentrated its pdf is around its mean. 3) where a is a positive parameter that determines the mean and variance of o according to the expressions (see Prob.