°
[0
°
°
]
G'In = GO,m, Gl,m, ... , GK-l,m
yin =[I1~,m, 'Yi~+l,m, ... , 'Yi~+K-l,m]
(18a,b)
Also, Rm is the KxK matrix
[
P Olml,lml
pO'ml+K_IJmll
o
... 0
Rm=:
:
P Olml,lml
pOlml+K-lJml
K-l . . . K-l
(19)
and 8Rm' 8y m , ARm' and AY m are semi-infinite matrices and vectors whose elements are:
[
pl~~KJml p~+K+IJml .... 1
8Rm = : :
(20)
°lml+KJml °1~+K+IJml
PK-l
PK-l
...
T
8ym = [l1ml+K,m, I1ml+K+l,m, ... J
(21)
[
P o Im-M/,Im-M/ pO Im-Ml+ 1 Jm-MI
pO Im+MI,lm+MI pO Im+MI+ 1 Jm+MI 1
o
0
... 0
0
...
AR = :
: :
:
m .
. .
.
P
o Im-Ml,lm-Mi pO Im-Ml+ 1 Jm-MI
pO Im+MI,lm+MI pO Im+MI+ 1 Jm+MI
K-l
K-l
... K-l
K-l
...
(22)
T
AYm = [l1m-Ml,m-M, 'Yim-Ml+l,m-M, ... , 'Yim+MI,m+M, I1m+MI+l,m+M, ... J
(23)
Clearly, 8ym contains harmonic coefficients of order m and degree greater than Iml + K - 1;
while AY m contains all coefficients with order greater than or equal to Ml2. The relative
domains for these coefficients are shown in Figure 1, for the special case that K = Ml2.
(For later reference, the spectral domain is also partitioned into regions A,B, and C.)
m, order
M/2
B, fry
K
n, degree
Fig.I. Spherical harmonic coefficients estimated using (17) are
indicated by the shaded areas. Aliases· are in the unshaded areas.
126