Databases Reference
In-Depth Information
To cancel out the aliasing in the second half of the block, we need
CAq
+
CBr
+
DAp
+
DBq
=
q
From this we can get the requirements for the transform:
CB
=
0
(64)
DA
=
0
(65)
CA
+
DB
=
I
(66)
Note that the same requirements will help cancel the aliasing in the first half of block
i
by
using the second half of the inverse transform of block
i
−
1. One selection that satisfies the
last condition is
1
2
(
CA
=
I
−
J
)
(67)
1
2
(
DB
=
I
+
J
)
(68)
where
J
is the counteridentity matrix.
The forward modified discrete transform is given by the following equation:
x
n
cos
2
N
−
1
N
(
1
2
)(
1
2
+
N
4
)
X
k
=
k
+
n
+
(69)
n
=
0
where
x
n
are the audio samples and
X
k
are the frequency coefficients. The inverse MDCT is
given by
N
2
−
X
k
cos
2
1
2
N
N
(
1
2
)(
1
2
+
N
4
)
y
n
=
k
+
n
+
(70)
k
=
0
or in terms of our matrix notation,
cos
2
N
(
1
2
)(
1
2
+
N
4
)
[
P
]
i
,
j
=
i
+
j
+
(71)
N
cos
2
2
N
(
1
2
)(
1
2
+
N
4
)
[
Q
]
i
,
j
=
i
+
j
+
(72)
It is easy to verify that, given a value of
N
, these matrices satisfy the conditions for alias
cancellation.
Thus, while the inverse transform for any one block will contain aliasing, by using the
inverse transform of neighboring blocks, the aliasing can be canceled. What about blocks that
do not have neighbors—that is, the first and last blocks? One way to resolve this problem is
to pad the sampled audio sequence with
N
2 zeros at the beginning and end of the sequence.
In practice, this is not necessary, because the data to be transformed is windowed prior to the
transform. For the first and last blocks, we use a special window that has the same effect as
introducing zeros. For information on the design of windows for the MDCT, see [
200
]. For
more on how the MDCT is used in audio compression techniques, see Chapter 16.
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