Reduced complexity transform for a low-frequency-effects channel
The computational resources that are needed to apply a transform-based filterbank to a limited-bandwidth audio signals are reduced by performing an integrated process of combining real-valued input data into complex-valued data and applying a short transform to the complex-valued data, applying a bank of very short transforms to the output of the integrated process, and deriving a sequence of real-valued output data from the outputs of the bank of very short transforms.
1. A method performed by an audio signal processing device, for decoding an encoded digital audio signal, wherein the method comprises:
receiving, by the audio signal processing device, the encoded digital audio signal;
decoding, by the audio signal processing device, the received encoded digital audio signal to generate a block of real-valued transform coefficients, wherein the block has a quantity K of real-valued transform coefficients of which only a number L of the real-valued transform coefficients represent spectral components of a limited-bandwidth audio signal, ½ L<M<K, and M is a power of two;
transforming, by the audio signal processing device, the block of real-valued transform coefficients to generate a sequence of N real-valued signal samples representing temporal components of the limited-bandwidth audio signal, wherein N=2·K; and
outputting, by the audio signal processing device, the real-valued signal samples representing temporal components of the limited-bandwidth audio signal;
wherein the transforming comprises:
applying, by the audio signal processing device, a first-transform of length R to either the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, or complex-valued coefficients derived from the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, wherein
R
=
M
P
and P is a power of two;
applying, by the audio signal processing device, a bank of Q second transforms of length P to outputs of the first transform; and
deriving, by the audio signal processing device, the sequence of N real-valued signal samples from outputs of the bank of second transforms.
2. The method of claim 1 , wherein:
each of the second transforms is equivalent to performing calculations expressed as
x
′
(
n
+
Q
·
m
)
=
∑
p
=
0
P
-
1
U
(
n
,
p
)
·
(
W
P
)
p
·
m
for
0
≤
n
<
Q
and
0
≤
m
<
P
;
the sequence of real-valued signal samples is derived from the outputs of the bank of second transforms by performing calculations equivalent to
y
(
n
)
=
(
Re
[
x
′
(
n
)
]
+
j
·
Im
[
x
′
(
n
)
]
)
·
(
-
cos
(
2
π
(
8
n
+
1
)
8
N
)
-
j
·
sin
(
2
π
(
8
n
+
1
)
8
N
)
)
;
where x′ represents the outputs of the second transform;
U(n,p)=a kernel function of the first transform;
W
P
=
e
j
(
2
π
P
)
;
y(n) represents intermediate signal samples;
Re[x′(n)]=real part of x′(n);
Im[x′(n)]=imaginary part of x′(n);
j=imaginary operator equal to √{square root over (−1)}; and
m, n and p are indices used in calculations.
3. The method of claim 1 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
]
=
∑
r
=
0
R
-
1
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
∑
r
=
0
R
-
1
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X′ represents the complex-valued coefficients;
R
=
M
P
;
and
r is an index used in calculations.
4. The method of claim 1 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
v
-
1
)
·
cos
(
I
(
v
,
n
)
)
-
j
·
X
(
N
2
-
2
v
-
1
)
·
sin
(
I
(
v
,
n
)
)
-
j
·
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
+
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
]
+
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
u
-
1
)
·
cos
(
I
(
u
,
n
)
)
-
j
·
X
(
N
2
-
2
u
-
1
)
·
sin
(
I
(
u
,
n
)
)
-
j
·
X
(
2
u
)
·
cos
(
I
(
u
,
n
)
)
+
X
(
2
u
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.
5. The method of claim 1 , wherein the first transform is equivalent to performing calculations expressed as
U
R
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
cos
(
I
(
u
,
n
)
)
]
and
U
I
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.
6. The method of claim 1 , further comprising generating the limited-bandwidth audio signal using the real-valued signal samples.
7. An audio signal processing device for decoding an encoded digital audio signal, wherein the audio signal processing device comprises one or more processors configured to:
receive the encoded digital audio signal
decode the received encoded digital audio signal to generate a block of real-valued transform coefficients, wherein the block has a quantity K of real-valued transform coefficients of which only a number L of the real-valued transform coefficients represent spectral components of a limited-bandwidth audio signal, ½ L<M<K, and M is a power of two;
transform the block of real-valued transform coefficients to generate a sequence of N real-valued signal samples representing temporal components of the limited-bandwidth audio signal, wherein N=2·K, and
outputting the real-valued signal samples representing temporal components of the limited-bandwidth audio signal;
wherein the transforming comprises:
applying a first-transform of length R to either the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, or complex-valued coefficients derived from the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, wherein
R
=
M
P
and P is a power of two;
applying a bank of Q second transforms of length P to outputs of the first transform; and
deriving the sequence of N real-valued signal samples from outputs of the bank of second transforms.
8. The audio signal processing device of claim 7 , wherein:
each of the second transforms is equivalent to performing calculations expressed as
x
′
(
n
+
Q
·
m
)
=
∑
p
=
0
P
-
1
U
(
n
,
p
)
·
(
W
P
)
p
·
m
for
0
≤
n
<
Q
and
0
≤
m
<
P
;
the sequence of real-valued signal samples is derived from the outputs of the bank of second transforms by performing calculations equivalent to
y
(
n
)
=
(
Re
[
x
′
(
n
)
]
+
j
·
Im
[
x
′
(
n
)
]
)
·
(
-
cos
(
2
π
(
8
n
+
1
)
8
N
)
-
j
·
sin
(
2
π
(
8
n
+
1
)
8
N
)
)
;
where x′ represents the outputs of the second transform;
U(n,p)=a kernel function of the first transform;
W
P
=
e
j
·
(
2
π
P
)
;
y(n) represents intermediate signal samples;
Re[x′(n)]=real part of x′(n);
Im[x′(n)]=imaginary part of x′(n);
j=imaginary operator equal to √{square root over (−1)}; and
m, n and p are indices used in calculations.
9. The audio signal processing device of claim 7 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
]
=
∑
r
=
0
R
-
1
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
∑
r
=
0
R
-
1
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X′ represents the complex-valued coefficients;
R
=
M
P
;
and
r is an index used in calculations.
10. The audio signal processing device of claim 7 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
v
-
1
)
·
cos
(
I
(
v
,
n
)
)
-
j
·
X
(
N
2
-
2
v
-
1
)
·
sin
(
I
(
v
,
n
)
)
-
j
·
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
+
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
]
+
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
u
-
1
)
·
cos
(
I
(
u
,
n
)
)
-
j
·
X
(
N
2
-
2
u
-
1
)
·
sin
(
I
(
u
,
n
)
)
-
j
·
X
(
2
u
)
·
cos
(
I
(
u
,
n
)
)
+
X
(
2
u
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.
11. The audio signal processing device of claim 7 , wherein the first transform is equivalent to performing calculations expressed as
U
R
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
cos
(
I
(
u
,
n
)
)
]
and
U
I
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.
12. A non-transitory medium recording a program of instructions that is executable by an audio signal processing device, wherein the program of instructions cause the audio signal processing device to perform a method for decoding an encoded digital audio signal, wherein the method comprises:
receiving, by the audio signal processing device, the encoded digital audio signal;
decoding, by the audio signal processing device, the received encoded digital audio signal to generate a block of real-valued transform coefficients, wherein the block has a quantity K of real-valued transform coefficients of which only a number L of the real-valued transform coefficients represent spectral components of a limited-bandwidth audio signal, ½ L<M<K, and M is a power of two;
transforming, by the audio signal processing device, the block of real-valued transform coefficients to generate a sequence of N real-valued signal samples representing temporal components of the limited-bandwidth audio signal, wherein N=2·K; and
outputting, by the audio signal processing device, the real-valued signal samples representing temporal components of the limited-bandwidth audio signal;
wherein the transforming comprises:
applying, by the audio signal processing device, a first-transform of length R to either the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, or complex-valued coefficients derived from the L real-valued transform coefficients representing spectral components of the limited-bandwidth audio signal, wherein
R
=
M
P
and P is a power of two;
applying, by the audio signal processing device, a bank of Q second transforms of length P to outputs of the first transform; and
deriving, by the audio signal processing device, the sequence of N real-valued signal samples from outputs of the bank of second transforms.
13. The medium of claim 12 , wherein:
each of the second transforms is equivalent to performing calculations expressed as
x
′
(
n
+
Q
·
m
)
=
∑
p
=
0
P
-
1
U
(
n
,
p
)
·
(
W
P
)
p
·
m
for
0
≤
n
<
Q
and
0
≤
m
<
P
;
the sequence of real-valued signal samples is derived from the outputs of the bank of second transforms by performing calculations equivalent to
y
(
n
)
=
(
Re
[
x
′
(
n
)
]
+
j
·
Im
[
x
′
(
n
)
]
)
·
(
-
cos
(
2
π
(
8
n
+
1
)
8
N
)
-
j
·
sin
(
2
π
(
8
n
+
1
)
8
N
)
)
;
where x′ represents the outputs of the second transform;
U(n,p)=a kernel function of the first transform;
W
P
=
e
j
·
(
2
π
P
)
;
y(n) represents intermediate signal samples;
Re[x′(n)]=real part of x′(n);
Im[x′(n)]=imaginary part of x′(n);
j=imaginary operator equal to √{square root over (−1)}; and
m, n and p are indices used in calculations.
14. The medium of claim 12 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
]
=
∑
r
=
0
R
-
1
X
′
(
P
·
r
+
p
)
·
(
W
N
/
4
)
(
P
·
r
+
p
)
·
n
+
∑
r
=
0
R
-
1
X
′
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
(
W
N
/
4
)
(
N
4
-
M
+
(
P
·
r
+
p
)
)
·
n
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X′ represents the complex-valued coefficients;
R
=
M
P
;
and
r is an index used in calculations.
15. The medium of claim 12 , wherein the first transform is equivalent to performing calculations expressed as
U
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
v
-
1
)
·
cos
(
I
(
v
,
n
)
)
-
j
·
X
(
N
2
-
2
v
-
1
)
·
sin
(
I
(
v
,
n
)
)
-
j
·
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
+
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
]
+
∑
r
=
0
R
-
1
[
-
X
(
N
2
-
2
u
-
1
)
·
cos
(
I
(
u
,
n
)
)
-
j
·
X
(
N
2
-
2
u
-
1
)
·
sin
(
I
(
u
,
n
)
)
-
j
·
X
(
2
u
)
·
cos
(
I
(
u
,
n
)
)
+
X
(
2
u
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.
16. The medium of claim 12 , wherein the first transform is equivalent to performing calculations expressed as
U
R
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
X
(
2
v
)
·
sin
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
cos
(
I
(
u
,
n
)
)
]
and
U
I
(
n
,
p
)
=
∑
r
=
0
R
-
1
[
-
X
(
2
v
)
·
cos
(
I
(
v
,
n
)
)
-
X
(
2
M
-
2
v
-
1
)
·
sin
(
I
(
u
,
n
)
)
]
for
0
≤
n
<
Q
and
0
≤
p
<
P
;
where X represents the real-valued transform coefficients;
R
=
M
P
;
I
(
s
,
n
)
=
π
(
8
s
·
(
1
+
4
n
)
+
1
)
4
N
;
v
=
P
·
r
+
p
;
u
=
N
4
-
M
+
(
P
·
r
+
p
)
;
and
r is an index used in calculations.