Lecture 8: Linear Prediction: Lattice filters

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1 1 Lecture 8: Linear Prediction: Lattice filters Overview New AR parametrization: Reflection coefficients; Fast computation of prediction errors; Direct and Inverse Lattice filters; Burg lattice parameter estimator; Gradient Adaptive Lattice filters;

2 Lecture 8 2 Lattice Predictors Order -Update Recursions for Prediction errors Since the predictors obey the recursive in order equations a m = a B m = a m a B m 1 + Γ m 0 a B m 1 + Γ a m 1 m 0 it is natural that prediction errors can be expressed in recursive in order forms. These forms results considering the recursions for the vector u m+1 (n) Combining the equations we obtain u m+1 (n) = u m+1 (n) = f m (n) = a T mu m+1 (n) = [ a T m 1 0 ] u m (n) u(n m) u(n) u m (n 1) u m (n) u(n m) = a T m 1u m (n) + Γ m (a B m 1) T u m (n 1) = = f m 1 (n) + Γ m b m 1 (n 1) + Γ m [ 0 (a B m 1 ) T ] u(n) u m (n 1) =

3 Lecture 8 3 b m (n) = (a B m) T u m+1 (n) = [ 0 (a B m 1) T ] = (a B m 1) T u m (n 1) + Γ m (a m 1 ) T u m (n) = b m 1 (n 1) + Γ m f m 1 (n) u(n) u m (n 1) + Γ m [ (am 1 ) T 0 ] u m (n) u(n m) = The order recursions of the errors can be represented as f m(n) b m (n) = 1 Γ m Γ m 1 f m 1 (n) b m 1 (n 1)

4 Lecture 8 4 f m (n) = f m 1 (n) + Γ m b m 1 (n 1) b m (n) = b m 1 (n 1) + Γ m f m 1 (n) Using the time shifting operator q 1, the prediction error recursions are given by f m(n) b m (n) = 1 Γ mq 1 Γ m q 1 f m 1(n) b m 1 (n) which can now be iterated for m = 1, 2,..., M to obtain f M(n) b M (n) = = 1 Γ Mq 1 Γ M q 1 1 Γ Mq 1 Γ M q 1 Γ M 1 Γ M 1 1 Γ M 1 q 1 q 1 1 Γ M 1 q 1 q Γ 1q 1... Γ 1 q 1 1 Γ 1q 1 Γ 1 q 1 f 0(n) b 0 (n) 1 1 u(n) Having available the reflexion coefficients, all prediction errors of order m = 1,..., M can be computed using the Lattice predictor, in 2M additions and 2M multiplications.

5 Lecture 8 5 Some characteristics of the Lattice predictor: 1. It is the most efficient structure for generating simultaneously the forward and backward prediction errors. 2. The lattice structure is modular: increasing the order of the filter requires adding only one extra module, leaving all other modules the same. 3. The various stages of a lattice are decoupled from each other in the following sense: The memory of the lattice (storing b 0 (n 1),..., b M 1 (n 1)) contains orthogonal variables, thus the information contained in u(n) is splitted in M pieces, which reduces gradually the redundancy of the signal. 4. The similar structure of the lattice filter stages makes the filter suitable for VLSI implementation.

6 Lecture 8 6 Lattice Inverse filters The basic equations for one stage of the lattice are f m (n) = f m 1 (n) + Γ m b m 1 (n 1) b m (n) = Γ m f m 1 (n) + b m 1 (n 1) (1) and simply rewriting the first equation f m 1 (n) = f m (n) Γ m b m 1 (n 1) b m (n) = Γ m f m 1 (n) + b m 1 (n 1) we obtain the basic stage of the Lattice inverse filter representation.

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8 Lecture 8 8 Joint process estimation Find the optimal (in MSE sense) filter recovering a desired signal d(n) from the signal u(n) not using directly the observations u(n), u(n 1),..., u(n m) as in FIR filtering but using instead the samples b 0 (n), b 1 (n),..., b M (n) which comes from the orthogonalization of u(n) using a lattice filter. The structure of the filter comprises two sections: one lattice predictor section with reflection coefficients Γ 1, Γ 2,..., Γ M, transforming the observations u(n), u(n 1),..., u(n m) into the sequence of uncorrelated errors b 0 (n).b 1 (n),..., b M (n); a multiple regression filter, with parameters γ 0, γ 1,..., γ M which uses as observations the samples b 0 (n).b 1 (n),..., b M (n) to compute the output of the filter y(n). Denoting we can write the optimal Wiener filter b(n) = [ b 0 (n) b 1 (n)... b M (n) ] T γ = [ γ 0 γ 1... γ M ] T γ = [Eb(n)b(n) T ] 1 Eb(n)d(n)

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10 Lecture 8 10 Relationship between Lattice parameters and optimal (direct) FIR filter parameters We found the autocorrelation matrix of backward errors to be and from b(n) = Lu(n) we found E[b(n)b(n) T ] = P P P M = D E[b(n)b(n) T ] = LE[u(n)u(n) T ]L T = LRL T = D We can now compute the optimal γ parameters as γ = [Eb(n)b(n)] 1 Eb(n)d(n) = D 1 Eb(n)d(n) = D 1 ELu(n)d(n) = D 1 Lp = D 1 LRw o Multiplying both sides with L T and recalling R 1 = L T D 1 L we obtain L T γ = w o Thus we have a one-to-one correspondence between the parameters of the optimal FIR filter, w o and the parameters of the optimal lattice filter. Burg estimation algorithm The optimum design of the lattice filter is a decoupled problem.

11 Lecture 8 11 At stage m the optimality criterion is: and using the stage m equations J m = E[f 2 m(n)] + E[b 2 m(n)] f m (n) = f m 1 (n) + Γ m b m 1 (n 1) b m (n) = b m 1 (n 1) + Γ m f m 1 (n) J m = E[f 2 m(n)] + E[b 2 m(n)] = E[(f m 1 (n) + Γ m b m 1 (n 1)) 2 ] + E[(b m 1 (n 1) + Γ m f m 1 (n)) 2 ] = E[(f 2 m 1(n) + b 2 m 1(n 1)](1 + Γ 2 m) + 4Γ m E[b m 1 (n 1)f m 1 (n)] Taking now the derivative with respect to Γ m of the above criterion we obtain and therefore d(j m ) dγ m = 2E[(f 2 m 1(n) + b 2 m 1(n 1)]Γ m + 4E[b m 1 (n 1)f m 1 (n)] = 0 Γ m = 2E[b m 1(n 1)f m 1 (n)] E[(fm 1(n)] 2 + E[b 2 m 1(n 1)] Replacing the expectation operator E with time average operator 1 N Nn=1 we obtain one direct way to estimate the parameters of the lattice filter, starting from the data available in lattice filter: Γ m = 2 N n=1 b m 1 (n 1)f m 1 (n) Nn=1 [(f 2 m 1(n) + b 2 m 1(n 1)] The parameters Γ 1,..., Γ M can be found solving first for Γ 1, then using Γ 1 to filter the data u(n) and obtain f 1 (n) and b 1 (n), then find the estimate of Γ 2...

12 Lecture 8 12 There are other possible estimators, but Burg estimator ensures the condition Γ < 1 which is required for the stability of the lattice filter. Gradient Adaptive Lattice Filters Imposing the same optimality criterion as in Burg method J m = E[f 2 m(n)] + E[b 2 m(n)] the gradient method applied to the lattice filter parameter at stage m is d(j m ) dγ m = 2E[f m (n)b m 1 (n 1) + f m 1 (n)b m (n)] and can be approximated (as usually in LMS algorithms) by ˆ J m 2[f m (n)b m 1 (n 1) + f m 1 (n)b m (n)] We obtain the updating equation for the parameter Γ m Γ m (n + 1) = Γ m (n) 1 2 µ m(n) ˆ J m = Γ m (n) µ m (n)(f m (n)b m 1 (n 1) + f m 1 (n)b m (n)) In order to normalize the adaptation step, the following value of µ m (n) was suggested where µ m (n) = 1 ξ m 1 (n) ξ m 1 (N) = N [(fm 1(i) 2 + b 2 m 1(i 1)] = ξ m 1 (N 1) + fm 1(N) 2 + b 2 m 1(N 1) i=1

13 Lecture 8 13 represents the total energy of forward and backward prediction errors. We can introduce a forgetting factor using ξ m 1 (n) = βξ m 1 (n 1) + (1 β)[f 2 m 1(n) + b 2 m 1(n 1)] with the forgetting factor close to 1, but 0 < β < 1 allowing to forget the old history, which may be irrelevant if the filtered signal is nonstationary.

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4 Reinforcement Learning Basic Algorithms

Learning in Complex Systems Spring 2011 Lecture Notes Nahum Shimkin 4 Reinforcement Learning Basic Algorithms 4.1 Introduction RL methods essentially deal with the solution of (optimal) control problems