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computational load by designing an appropriate vector quantization scheme with
almost any desired number of steering matrices in the look-up table, and (iii) we
can pre-calculate kernel weights to lower the computational load further (since the
steering matrices are fixed).
From (18), the elements of the spatial covariance matrix C i are given by the
steering parameters with the following equations:
θ j )= c 11 c 12
c 12 c 22
,
C j (
γ j ,
ρ j ,
(33)
with
γ j
θ j
ρ j cos 2
ρ 1
j
sin 2
c 11 =
θ j +
(34)
γ j
θ j
ρ 1
j
c 12 =
ρ j cos
θ j sin
θ j +
cos
θ j sin
(35)
γ j
θ j
ρ j sin 2
ρ 1
j
cos 2
c 22 =
θ j +
(36)
where
θ j is the orien-
tation angle parameter. Fig. 8 visualizes the relationship between the steering param-
eters and the values of the covariance matrix. Based on the above formulae, using a
pre-defined set of the scaling, elongation, and angle parameters, we can generate a
lookup table for covariance matrices, during an off-line stage.
During the on-line processing stage, we compute a naive covariance matrix C naive
i
γ j is the scaling parameter,
ρ j is the elongation parameter, and
(15) and then normalize C naive
i
so that the determinant of the normalized naive co-
variance matrix det( C naive
i
) equals 1 . 0:
C naive
i
1
C naive
i
γ i C naive
=
det ( C naive
i
=
,
(37)
i
)
where again
γ i is the scaling parameter. This normalization eliminates the scaling pa-
rameter from the look-up table and simplifies the relationship between the elements
of covariance matrices and the steering parameters, and allows us to reduce the size
of the table. Table 1 shows an example of a compact lookup table. When the elon-
gation parameter
ρ i of C i is smaller than 2.5, C i is quantized as an identity matrix
(i.e. the kernel spreads equally every direction). On the other hand, when
ρ i
2 . 5,
we quantize C i with 8 angles. Using C naive
, we obtain the closest covariance matrix
i
C i from the table. I.e.,
F ,
C i = arg min
ρ j ,
C naive
C (
ρ j ,
θ j )
(38)
i
θ j
· F is the Frobenius norm. The final matrix C i is given by:
where
C i =
γ i C i .
(39)
 
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