Digital Signal Processing Reference
In-Depth Information
snapshot
y
l
1
,
l
2
. Then
γ
l
1
,
l
2
and
µ
l
1
,
l
2
are related to
y
l
1
,
l
2
by unitary transformations
as follows:
S
g
(
l
1
γ
l
1
,
l
2
=
,
l
2
)
y
l
1
,
l
2
(6.12)
S
m
(
l
1
µ
l
1
,
l
2
=
,
l
2
)
y
l
1
,
l
2
,
(6.13)
S
g
(
l
1
S
m
(
l
1
,
,
×
×
−
where
l
2
) and
l
2
)are
M
1
M
2
g
l
1
,
l
2
and
M
1
M
2
(
M
1
M
2
g
l
1
,
l
2
) uni-
S
g
(
l
1
l
2
)
S
g
(
l
1
S
m
(
l
1
l
2
)
S
m
(
l
1
,
,
=
,
,
=
tary selection matrices such that
l
2
)
I
g
l
1
,
l
2
,
l
2
)
S
g
(
l
1
l
2
)
S
m
(
l
1
I
M
1
M
2
−
g
l
1
,
l
2
, and
,
,
l
2
)
=
0
g
l
1
,
l
2
×
(
M
1
M
2
−
g
l
1
,
l
2
)
.For example, let
M
1
=
3,
M
2
=
2, and let
,
y
l
1
,
l
2
Y
l
1
,
l
2
=
(6.14)
y
l
1
+
2
,
l
2
y
l
1
+
2
,
l
2
+
1
where each
indicates a missing sample. Then we have
g
l
1
,
l
2
=
3,
y
l
1
+
2
,
l
2
+
1
]
T
y
l
1
,
l
2
=
[
y
l
1
,
l
2
y
l
1
+
2
,
l
2
,
(6.15)
and
100
000
010
000
000
001
000
100
000
010
001
000
S
g
(
l
1
S
m
(
l
1
,
=
,
,
=
.
l
2
)
l
2
)
(6.16)
Because we have
=
S
g
(
l
1
l
2
)
y
l
1
,
l
2
l
2
)
S
g
(
l
1
S
m
(
l
1
l
2
)
S
m
(
l
1
y
l
1
,
l
2
,
,
l
2
)
+
,
,
S
g
(
l
1
S
m
(
l
1
l
2
)
γ
l
1
,
l
2
+
l
2
)
µ
l
1
,
l
2
=
,
,
(6.17)
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