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PublicouTiago Canário Estrela Alterado mais de 8 anos atrás
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos 1 Adaptive & Array Signal Processing AASP Prof. Dr.-Ing. João Paulo C. Lustosa da Costa University of Brasília (UnB) Department of Electrical Engineering (ENE) Laboratory of Array Signal Processing PO Box 4386 Zip Code 70.919-970, Brasília - DF Homepage: http://www.pgea.unb.br/~lasp
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (1) Estimation error: 2 Mean Squared-error Cost Function
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (2) Mean Squared-error Cost Funtion 3
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (3) Steepest-descent algorithm Oldest method for optimization Using directly the Wiener-Hopf equations Computational difficulties due to the matrix inversion large number of taps high rate input data Successive correction of w(n) should lead to J min 4 The definition of the steepest descent equation is given by: Since Replacing the gradient in the steepest descent equation:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (4) Comparison: Wiener Solution vs Steepest Descent 5 For the Wiener Solution: R and p should be given. One matrix inversion is necessary to find the optimal w. For the Steepest Descent Solution: R and p should be given. No matrix inversion is necessary. However, a certain number of iterations is necessary until w to converge. The speed of convergence, i.e. The number of iterations until convergence, depends on the step size . If is greater than the optimal value, the algorithm will never converge. If is smaller than the optimal value, the algorithm will converge slowly.
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (4) Stability of the Steepest-descent Algorithm 6 The equation to update w(n) can be write in terms of a c(n) vector defined below: Therefore:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (5) Stability of the Steepest-descent Algorithm 7 Computing the Eigenvalue Decomposition (EVD) of R: We can write the update equation of c(n) as
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (6) Stability of the Steepest-descent Algorithm 8 For the k-th element of each vector, we have that: Assuming some initial value when n = 0: Therefore, we can rewrite the n-th element as a function of the initial value.
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (7) Stability of the Steepest-descent Algorithm 9 Hence, for the convergence:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Method of the Steepest Descent (8) Stability of the Steepest-descent Algorithm 10 With this upper and lower bound for , it is possible to find the optimal that allows a fast convergence.
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Least Mean Squares (LMS) vs Steepest-descent (1) 11 To find a good approximation of p and R, we would need a large number of samples. However, since in many applications only a few samples are available, we can compute instantaneous estimates of p and of R. We can replace the estimates of p and of R in the Mean Squared-error Cost Function below. By applying the gradient, we obtain:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Least Mean Squares (LMS) vs Steepest-descent (2) 12 Replacing the gradient in the steepest-descent expression:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Least Mean Squares (LMS) vs Steepest-descent (3) 13 We can summarize the LMS algorithm in three steps: 1st step - Compute the filter output: 2nd step – Estimation error: 3rd step – Tap-weight adaptation:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (1) 14 We consider that the estimation error is more affected by more recent samples than by older samples. n is the total number of samples, (n,i) is the weighting factor,and e(i) is the estimation error for the i-th sample. Using the defition of e(i): We define the weighting factor (n,i) in the following way:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (2) 15 Using the definition of the weighting factor (n,i) Considering n samples, we can extend the Wiener-Hopf equations as: The M-by-M correlation matrix (n) is defined as The M-by-1 cross-correlation vector z(n) is defined as
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (3) 16 Using the definition of the correlation matrix, we can rewrite it in a recursive way:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (4) 17 Analogously we can also compute the recursive form of the cross- correlation vector z(n): Matrix Inversion Lemma (also known as Woodburry´s identity) Given as certain matrix A defined as the inverse A of is given by Comparing A matrix with, we have that:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (5) 18 Applying the matrix inverse lemma We define: Using the definition of P(n) and K(n), we can rewrite the inverse of (n)
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (6) 19 Using the above definition of K(n), we have that: Comparing with the expression of P(n), we have that: Therefore:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (7) 20 Expanding z(n) using its recursive form: Computing the expression to update the tap-weight vector: Replacing the first P(n) with its recursive expression:
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (7) 21 Computing the expression to update the tap-weight vector: Replacing
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Universidade de Brasília Laboratório de Processamento de Sinais em Arranjos Recursive Least Mean Squares (RMS) (8) 22 We can summarize the RLS algorithm in the following steps:
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