simplified form as [25,47]
s
k
¼
ffiffiffiffi
4
N
r
X
N1
n¼0
x
n
sin
p
2N
2n þ 1 þ
N
2
ð2k þ 1Þ
,
k ¼ 0; 1; ...;
N
2
1, (9)
^
x
MDST
n
¼
ffiffiffiffi
4
N
r
X
ðN=2Þ1
k¼0
s
k
sin
p
2N
2n þ 1 þ
N
2
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
k
g=fs
k
g possess even anti-
symmetry/symmetry properties given by [18,33]
c
Nk1
¼c
k
; s
Nk1
¼ s
k
; k ¼ 0; 1; ...;
N
2
1, (11)
while the time domain aliased data sequences
f
^
x
MDCT
n
g=f
^
x
MDST
n
g recovered by the backward MDCT/MDST
possess the following symmetries [18,33]:
^
x
MDCT
n
¼
^
x
MDCT
ðN=2Þ1n
;
^
x
MDCT
ðN=2Þþn
¼
^
x
MDCT
N1n
,
^
x
MDST
n
¼
^
x
MDST
ðN=2Þ1n
;
^
x
MDST
ðN=2Þþn
¼
^
x
MDST
N1n
,
n ¼ 0; 1; ...;
N
4
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
n
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
^
x
MDCT
n
g=f
^
x
MDST
n
g
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
^
x
MDCT
n
and
^
x
MDCT
ðN=2Þþn
for n ¼ 0 ; 1; ...; ðN=4Þ1by
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
n
g is called a periodic
sequence if y
nþM
¼ y
n
, where M40 is called the period of
periodic sequence fy
n
g.
Definition 2. A data sequence fy
n
g is called an anti-
periodic sequence if y
nþM
¼y
n
, where M40 is the period
of anti-periodic sequence fy
n
g.
An anti-periodic sequence fy
n
g may be treated as a
periodic sequence with period 2M because y
nþ2M
¼
y
nþM
¼ y
n
. However, the properties of fy
n
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
t
ðcÞ
k;n
¼ cos
p
2N
2n þ 1 þ
N
2
ð2k þ 1Þ
,
t
ðsÞ
k;n
¼ sin
p
2N
2n þ 1 þ
N
2
ð2k þ 1Þ
, (13)
and substituting n þ N, and then n þ 2N for n into (13) we
obtain
t
ðcÞ
k;n
¼t
ðcÞ
k;nþN
; t
ðcÞ
k;n
¼ t
ðcÞ
k;nþ2N
,
t
ðsÞ
k;n
¼t
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:
t
ðcÞ
k;ðN=2Þ1n
¼t
ðcÞ
k;n
; t
ðcÞ
k;ðN=2Þþn
¼ t
ðcÞ
k;N1n
,
t
ðsÞ
k;ðN=2Þ1n
¼ t
ðsÞ
k;n
; t
ðsÞ
k;ðN=2Þþn
¼t
ðsÞ
k;N1n
; 8k,
n ¼ 0; 1; ...;
N
4
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
^
x
MDCT
n
g=f
^
x
MDST
n
g
given by (12).
2.2.3. Relation between MDCT and MDST
There exists a simple relation between the MDCT and
MDST given in [25]
c
ðN=2Þk1
¼ð1Þ
N=4
X
N1
n¼0
ð1Þ
n
x
n
sin
p
2N
2n þ 1 þ
N
2
ð2k þ 1Þ
,
k ¼ 0; 1; ...;
N
2
1. (16)
Also
s
ðN=2Þk1
¼ð1Þ
N=4
X
N1
n¼0
ð1Þ
n
x
n
cos
p
2N
2n þ 1 þ
N
2
ð2k þ 1Þ
,
k ¼ 0; 1; ...;
N
2
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