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Recent Advances in Biomedical Engineering 2011 Part 4

Tham khảo tài liệu 'recent advances in biomedical engineering 2011 part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Multichannel analysis of EEG signal applied to sleep stage classification 109 The input signal u n is a totally unknown biological signal actually it is considered as inaccessible signal therefore the signal s n can be linearly predicted only approximately by 2 and it is defined as p s n - A k s n - k 5 k 1 Then the error between the actual value s n and the predicted value s n is given by p s_ n s n -s n s n A k s n - k 6 k 1 Since the assumption that the input u n is inaccessible the gain G does not participate in the linear prediction of the signal. And so it is irrelevant to determine a value for G. However 6 can be rewritten as p s n - A k s n - k s_ n 7 k 1 From 2 and 7 the following can be seen Gu n e_ n 8 Meaning the input signal is proportional to the error signal. From comparing 6 with 8 we get Gu n e_ n s n - s n 9 By squared Eq. 9 and taking the expectation we receive E Gu n 2 G2E u2 n E s2 n E s n -s n 21 10 The input u n is assumed to be a sequence of uncorrelated samples with zero mean and unit variance i.e. E u n 0 for all n and Var u n 1. The derived equation is E u 2 n 1 11 By placing 11 into 10 we receive G2 E e2 n E s n - s_ n 2 12 When 12 can be written as G2 E s2 n E s n - s n 2 E s n - s n s n - s n T E s n - s n s n - s n T 13 E s n s n - s n T - E s n s n - s n T From 13 and 9 we get E s n s n -s n T - E s n s n - s n T 14 E s n s n -s n T - E s n n 110 Recent Advances in Biomedical Engineering By the orthogonality principle the next expression is valid E s n eT n 0 15 The 12 14 and 15 yields G2 E s2 n EIs n s n - s n T 16 EIs n s n - s n EIs n sT n EIs n ST n Now by placing 5 into 16 we receive G2 E ff2 n EI s n sT n A k EI s n sT n k 17 k 1 When the autocorrelation matrix of lag i is defined as R i EI s n sT n i 18 Where every R i i 1 . p is a d X d matrix. By placing 18 into 17 we receive the estimation of residual error covariance matrix as follow G2 E I 2 n R 0 A k RT k 19 k 1 This expression will assist us in the forward MAR .