Hélice 2 de HPV 16 L1 Lineal

Owing to the isomorphism between augmented complex vectors and bivariate real vectors, and the duality analysis for stochastic gradient filters [19], we next show that the ACKF algorithm has a dual bivariate real-valued Kalman filter (RKF). This duality can be ex- ploited to reduce the computational complexity of ACKF in hardware implementations. A complex vector z =zr+jzi ∈Cq has a composite bivariate real representation in R2q

of the form za=   z z∗   =   I jI I −jI    | {z } ≡Jz   zr zi    | {z } =zr (3.32)

whereIis the identity matrix (with appropriate dimensions), and the invertible orthogonal mapping4 _J

z : C2q → R2q is such that J−z1 = 12JHz [20][21]. Based on this isomorphism,

4_{For a vectorz}

the real bivariate state space corresponding to the augmented complex state space in (3.5) is given by xr_n = Fr_n−1xr_n−1+wr_n yr_n = Hr_nxr_n+v_nr (3.33a) where xr_n=J−_x1xa_n, y_nr =J_y−1ya_n,Fr_n−1=J_x−1Fa_n−1Jx,Hr_n=J−_y1Ha_nJx,w_nr =J−_x1wa_n and vr

n=J−y1vna. In a similar manner, the real valued covariance matrices ofwrn and vrn take

the corresponding forms

Qr_n = E{w_nrw_nrH}=J−_x1Qa_nJ−_xH Rr_n = E_{vr_nvrH_{n }}=J−_y1Ra_nJ−_yH

Next the ACKF and its dual RKF are shown to have the same performance. As- suming that ACKF is initiated at time (n₋1), with initial statex_ba

n−1|n−1and MSE matrix

Ma

n−1|n−1, the corresponding dual RKF initialisation is given by

b

x_nr_−1|n−1 = J_x−1_bxa_n−1|n−1

Mr_n−1|n−1 = J−_x1Ma_n−1|n−1J−_xH (3.34) It is now straightforward to show that the state and MSE matrix predictions of the Kalman filters are also related as

b

x_nr_|n−1 = J_x−1_bxa_n|n−1

3.1 The Augmented Complex Kalman Filter (ACKF) 53

and that the Kalman gains are related as

Ga_n = Ma_n|n−1HaH_n [H_naMa_n|n−1HaH_n +Ra_n]−1

= JxMrn|n−1JHxJy−HHrHn JHx[JyHrnJ−x1JxMrn|n−1JHxJ−xHHrHn JHy +JyRrnJHy]−1 = JxMr_n|n−1HrHn [HrnMnr|n−1HrHn +Rrn]−1J−y1

= JxGrnJ−y1 (3.36)

Consequently, for the state estimates x_ba

n|n and bxrn|n we have

b

xr_n|n = x_br_n|n−1+G_nr(yr_n−Hr_nx_br_n|n−1)

= J−_x1_bxa_n|n−1+J−_x1Ga_nJy(ynr−HrnJx−1bxan|n−1)

= J−_x1_bxa_n|n (3.37)

while, the MSE matrices are related as

Mr_n|n=J−_x1Ma_n|nJ−_xH (3.38) Observe that based on the expression in (3.37), the state estimates _bxa_n|n and _bxr_n|n are equivalent and are related by an invertible linear mapping. To show that ACKF and its dual real valued bivariate Kalman filter achieve the same mean square error (MSE), recall that the MSE for the real valued Kalman filter is given by

ǫr_n= tr{Mr_n|n} (3.39)

where the symbol tr_{·} denotes the matrix trace operator. Similarly, the mean square error corresponding to the augmented MSE matrix Ma

n|n is given by the trace of (3.38),

that is

tr{Ma_n|n} = tr{JxMr_n|nJHx} = tr{Mr_n|nJH_xJx}

where the expression JH

x = 2J−x1 was utilised. At first, this result is misleading as it suggests that ACKF achieves twice the error of its dual real valued KF. However, this is because the error term is counted twice by the trace ofMa

n|n, owing to the block diagonal

structure of the augmented MSE covariance matrix, and hence needs to be halved to compute the true augmented MSE, that is

ǫa_n = 1 2tr{M

a

n|n}=ǫrn

Remark #5: The ACKF and the its dual bivariate RKF are equivalent forms of the same state space model. They achieve the identical state estimates and MSEs at every time instant, regardless of the propriety of the processed signals.

By utilising the bivariate RKF, the computational complexity of ACKF is reduced, whereby the number of additions and multiplications required are approximately halved and quartered, respectively.