Digital Signal Processing Reference
In-Depth Information
⎛
⎞
a
2
(
1
,
1
)
⎝
⎠
=
a
2
(
u
(
i
)
2
,
1
)
1
+
j
=
1
k
j
,
a
2
(
,
)
3
1
a
2
(
4
,
1
)
⎛
⎞
⎛
⎞
a
2
(
i
)
a
2
(
1
,
1
,
1
)
⎝
⎠
=
⎝
⎠
,
a
2
(
i
)
a
2
(
2
,
2
,
1
)
i
=
k
i
−
1
2
,
3
,
4
(3.14)
a
2
(
i
)
a
2
(
3
,
3
,
1
)
a
2
(
i
)
a
2
(
4
,
4
,
1
)
where
i
is the same as that in Equation (
3.12
). As we will discuss later, we choose
parameters
k
1
,
k
2
,
k
3
to maximize the coding gain. The choice of
k
1
,
k
2
,
k
3
will
complete the precoder design for Users 1 and 2 at time slot 1. Note that the designed
precoders
A
1
,
A
2
satisfy
A
1
||
2
F
A
2
||
2
F
1 and the signals of User 1 and User
2 will be transmitted along two orthogonal directions as shown in Fig.
3.2
.
In order to derive the orthogonality among Users 1, 2, 3 at time slot 1, we design
precoder
A
3
to satisfy the following properties:
||
=||
=
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
a
1
(
a
2
(
a
3
(
,
)
,
)
,
)
1
1
1
1
1
1
a
1
(
a
2
(
a
3
(
2
,
1
)
2
,
1
)
2
,
1
)
1. Complex vectors
H
1
,
H
2
,
H
3
are
a
1
(
a
2
(
a
3
(
3
,
1
)
3
,
1
)
3
,
1
)
a
1
(
a
2
(
a
3
(
4
,
1
)
4
,
1
)
4
,
1
)
orthogonal to each other.
2.
⎛
⎞
⎛
⎞
a
3
(
a
3
(
1
,
i
)
1
,
1
)
⎝
⎠
=
⎝
⎠
,
a
3
(
a
3
(
2
,
i
)
2
,
1
)
k
i
−
1
i
=
2
,
3
,
4
(3.15)
a
3
(
a
3
(
3
,
i
)
3
,
1
)
a
3
(
a
3
(
4
,
i
)
4
,
1
)
3. The Frobenius norm of complex matrix
A
3
is equal to 1.
In order to maximize the coding gain,
A
3
can be further chosen numerically such
that the norm of
H
3
A
3
is maximized. Similarly, for User 4, at time slot 1, in order to
derive the orthogonality as shown in Fig.
3.2
, we choose precoder
A
4
to satisfy the
following properties:
⎛
⎞
⎛
⎞
⎛
⎞
a
1
(
a
2
(
a
3
(
1
,
1
)
1
,
1
)
1
,
1
)
⎝
⎠
⎝
⎠
⎝
⎠
a
1
(
a
2
(
a
3
(
2
,
1
)
2
,
1
)
2
,
1
)
1. Complex vectors
H
1
,
H
2
,
H
3
,
a
1
(
a
2
(
a
3
(
3
,
1
)
3
,
1
)
3
,
1
)
a
1
(
a
2
(
a
3
(
4
,
1
)
4
,
1
)
4
,
1
)
⎛
⎞
a
4
(
1
,
1
)
⎝
⎠
a
4
(
2
,
1
)
H
4
are orthogonal to each other.
a
4
(
,
)
3
1
a
4
(
4
,
1
)
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