高效MDCT/MDST计算结构

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"快速计算MDCT/MDST的高效算法结构" 本文主要介绍了一种新的快速计算方法，用于改进离散余弦变换（MDCT）和改进离散正弦变换（MDST）。这种新算法在正向和反向计算中都保持了相同的高效性，它源于对高效实现时域 aliasing 取消（TDAC）分析/合成 MDCT/MDST 滤波器银行的系统性构建。MDCT 和 MDST 在音频编码，特别是 MPEG 音频编码中扮演着重要角色。 MDCT 和 MDST 是数字信号处理中的关键变换工具，它们常用于音频压缩，如 MP3 编码。MDCT 是一种半重叠变换，能够在减少数据冗余的同时保持信号的大部分信息，从而实现高效的编码。MDST 是 MDCT 的一种变体，尤其适用于具有对称特性的信号处理。这两者都是通过窗口函数和离散傅里叶变换来操作信号的。该文章详细阐述了这个新的快速计算结构，它优化了 TDAC 过程中的计算效率。TDAC 是 MDCT 和 MDST 的核心部分，它通过消除时间域中的混叠现象来提高信号处理的精度。新算法的设计考虑了计算的并行性和存储效率，以减少运算量和内存需求，这对于实时音频处理和嵌入式系统的应用至关重要。作者 Vladimir Britanak 和 Huibert J. Lincklaen Arriens 分别来自斯洛伐克科学院和代尔夫特理工大学，他们在信息学和电气工程领域有深厚的背景。文章经过多次修订，最终在2009年被接受发表，显示了研究的严谨性和实用性。关键词涵盖了 MDCT、MDST、调制重叠变换（Modulated Lapped Transform）、调制复重叠变换（Modulated Complex Lapped Transform）以及快速计算结构，这些都是与音频信号处理和编码密切相关的技术。这项工作为 MDCT 和 MDST 的高效实现提供了新的途径，对于优化音频编码算法，尤其是在资源有限的环境中的应用，具有重要意义。通过采用这种新的计算结构，开发者可以设计出更快、更节省资源的音频处理软件和硬件，进一步推动音频技术的发展。

simpliﬁed form as [25,47]

ﬃﬃﬃﬃ

N1

n¼0

sin

2n þ 1 þ



ð2k þ 1Þ



k ¼ 0; 1; ...;

 1, (9)

MDST

ﬃﬃﬃﬃ

ðN=2Þ1

k¼0

sin

2n þ 1 þ



ð2k þ 1Þ



n ¼ 0; 1; ...; N  1. (10)

The MDCT and MDST basis vectors form the MCLT [47],

whose real part corresponds to the MDCT or MLT and

imaginary part is the MDST. In contrast to discrete unitary

transforms, the MDCT and MDST have a special property:

the original input data sequence cannot be perfectly

reconstructed from a single block of MDCT and MDST

coefﬁcients.

MDCT/MDST sequences fc

g=fs

g possess even anti-

symmetry/symmetry properties given by [18,33]

Nk1

¼c

; s

Nk1

¼ s

; k ¼ 0; 1; ...;

 1, (11)

while the time domain aliased data sequences

MDCT

g=f

MDST

g recovered by the backward MDCT/MDST

possess the following symmetries [18,33]:

MDCT

¼

MDCT

ðN=2Þ1n

;

MDCT

ðN=2Þþn

MDCT

N1n

MDST

ðN=2Þ1n

;

MDST

ðN=2Þþn

¼

MDST

N1n

n ¼ 0; 1; ...;

 1. (12)

The symmetry properties (11) and (12) can be simply

veriﬁed by the proper substitution into Eqs. (7)–(10). We

note that the symmetry properties (11) and (12) also hold

if the input data sequence fx

g is windowed. From (11) it

follows that only N=2 coefﬁcients are unique in the MDCT

and MDST sequences. Further, it can be easily seen that

the time domain aliased data sequences f

MDCT

g=f

MDST

exhibit two local symmetries (odd/even symmetry in the

ﬁrst half and even/odd symmetry in the second half).

From an algorithmic point of view this means that it is

sufﬁcient to compute only the time domain aliased

samples

MDCT

and

MDCT

ðN=2Þþn

for n ¼ 0 ; 1; ...; ðN=4Þ1by

the backward MDCT.

2.2.1. Periodicity and anti-periodicity of MDCT/MDST

transform kernels

Special kinds of data sequences, the periodic and anti-

periodic sequences, are fundamental notions in harmonic

analysis, convolution and correlation of signals. Now we

recall the deﬁnitions of periodic and anti-periodic

sequences [19].

Deﬁnition 1. A data sequence fy

g is called a periodic

sequence if y

nþM

¼ y

, where M40 is called the period of

periodic sequence fy

Deﬁnition 2. A data sequence fy

g is called an anti-

periodic sequence if y

nþM

¼y

, where M40 is the period

of anti-periodic sequence fy

An anti-periodic sequence fy

g may be treated as a

periodic sequence with period 2M because y

nþ2M

y

nþM

¼ y

. However, the properties of fy

g depend only

upon its values in one period M. The periodicity and anti-

periodicity of sequences are closely related to their

symmetry and anti-symmetry properties, respectively.

Properties (sums and products) of periodic and anti-

periodic sequences with a common period can be found in

[19].

Denoting the MDCT and MDST transform kernels,

respectively, as

ðcÞ

k;n

¼ cos

2n þ 1 þ



ð2k þ 1Þ



ðsÞ

k;n

¼ sin

2n þ 1 þ



ð2k þ 1Þ



, (13)

and substituting n þ N, and then n þ 2N for n into (13) we

obtain

ðcÞ

k;n

¼t

ðcÞ

k;nþN

; t

ðcÞ

k;n

¼ t

ðcÞ

k;nþ2N

ðsÞ

k;n

¼t

ðsÞ

k;nþN

; t

ðsÞ

k;n

¼ t

ðsÞ

k;nþ2N

; 8k. (14)

Eq. (14) implies that the MDCT and MDST transform

kernels are anti-periodic sequences with period N, and

periodic sequences with period 2 N.

2.2.2. Symmetry properties of MDCT/MDST basis vectors

For a given N, consider the MDCT and MDST transform

kernels given by (13). One can observe that the MDCT and

MDST basis vectors exhibit the following local symme-

tries:

ðcÞ

k;ðN=2Þ1n

¼t

ðcÞ

k;n

; t

ðcÞ

k;ðN=2Þþn

¼ t

ðcÞ

k;N1n

ðsÞ

k;ðN=2Þ1n

¼ t

ðsÞ

k;n

; t

ðsÞ

k;ðN=2Þþn

¼t

ðsÞ

k;N1n

; 8k,

n ¼ 0; 1; ...;

 1. (15)

The symmetry properties of the MDCT and MDST basis

vectors (15) can be simply veriﬁed by a proper substitu-

tion into Eq. (13). Note that they are quite similar to those

of the time domain aliased data sequences f

MDCT

g=f

MDST

given by (12).

2.2.3. Relation between MDCT and MDST

There exists a simple relation between the MDCT and

MDST given in [25]

ðN=2Þk1

¼ð1Þ

N=4

N1

n¼0

ð1Þ

sin

2n þ 1 þ



ð2k þ 1Þ



k ¼ 0; 1; ...;

 1. (16)

Also

ðN=2Þk1

¼ð1Þ

N=4

N1

n¼0

ð1Þ

cos

2n þ 1 þ



ð2k þ 1Þ



k ¼ 0; 1; ...;

 1, (17)

and this fact results in a simple method to compute the

MDST using a fast MDCT computational structure.

ARTICLE IN PRESS

Please cite this article as: V. Britanak, H.J.L. Arrie

ns, Fast computational structures for an efﬁcient implementation of the

complete TDAC analysis/synthesis MDCT/MDST ﬁlter banks, Signal Process. (2009), doi:10.1016/j.sigpro.2009.01.014

V. Britanak, H.J.L. Arrie

¨ns

/ Signal Processing ] (]]] ]) ]]]–]]]4

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