结合两个多相矩阵的任意长度线性相位超unit滤波器组晶格结构中起始块的新设计方法
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118 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—II: EXPRESS BRIEFS, VOL. 59, NO. 2, FEBRUARY 2012
A New Design Method of the Starting Block in
Lattice Structure of Arbitrary-Length Linear Phase
Paraunitary Filter Bank by Combining
Two Polyphase Matrices
Bodong Li, Xieping Gao, and Fen Xiao
Abstract—Constructing the starting block is the core of de-
signing an arbitrary-length linear phase paraunitary filter bank
(ALLPPUFB), and lattice structure is an efficient approach
of ALLPPUFB design. This brief proposes a new ALLPPUFB
starting block design by combining two polyphase matrices of
constrained-length linear phase paraunitary filter bank. The pro-
posed design is much easier to be understood than existing ones
due to the brief design process. Moreover, it might be modified and
used for unaddressed topics in neighboring fields, such as starting
block design for arbitrary-length oversampled linear phase parau-
nitary filter bank.
Index Terms—Arbitrary length, filter bank, lattice structure,
linear phase, paraunitary, starting block.
I. INTRODUCTION
L
INEAR PHASE paraunitary filter bank (LPPUFB) is of
great importance in signal processing. It is an efficient way
to design LPPUFB via lattice structure [1], [2], which factorizes
the associated polyphase matrix into a product of lower-order
building blocks. Those building blocks comprise starting block
and propagating block. Starting block produces the minimum-
order LPPUFB, and propagating blocks extend it to yield
higher-order LPPUFB. As illustrated in existing works, the key
of designing an LPPUFB is to construct its starting block.
A couple of primary and important works have been done on
the starting block design for arbitrary-length LPPUFB (ALLP-
PUFB) [3]–[7], i.e., LPPUFB with filter length KM + β,
where M is the decimation factor, M, K, β ∈ Z, and 0 ≤
β<M. Dai and Tran [8] studied the starting block for regular
ALLPPUFB, Tran and Nguyen [9] designed ALLPPUFB start-
ing block (ALLPPUFB-SB) for even M , and Ikehara et al. [10]
constructed the one for odd M. On the other hand, some new
and totally different ideas have been presented. For example,
Li et al. [11] published a reversible design of ALLPPUFB-SB,
which guarantees the completeness
1
of the starting block di-
Manuscript received April 15, 2011; revised July 17, 2011 and October 9,
2011; accepted December 5, 2011. Date of publication January 18, 2012; date
of current version February 23, 2012. This work was supported by the National
Natural Science Foundation of China under Grant 61172171. This paper was
recommended by Associate Editor Y. Yu.
The authors are with the College of Information Engineering and the
Key Laboratory of Intelligent Computing and Information Processing of the
Ministry of Education, Xiangtan University, Xiangtan 411105, China (e-mail:
xpgao@xtu.edu.cn).
Digital Object Identifier 10.1109/TCSII.2011.2180098
1
A lattice structure is said to be complete if it can represent all filter banks
satisfying certain properties such as linear phase and paraunitary properties
here [12].
rectly. Different from earlier designs of complete ALLPPUFB-
SB, extra proof of the completeness of the ALLPPUFB-SB is
not required.
A novel design of ALLPPUFB-SB has been proposed
and demonstrated in this brief. It is implemented by com-
bining two polyphase matrices. Both of the two matrices
produce constrained-length LPPUFB (CLLPPUFB) [13], i.e.,
LPPUFB with filter length KM
, where M
is the decima-
tion factor, and M
, K ∈ Z. The proposed new design here
is much easier to be understood than most available ones
[15] because of the brief design process. Moreover, it might
be used to handle unaddressed topics in neighboring fields.
One of such fields is arbitrary-length oversampled LPPUFB
(ALOLPPUFB) [14].
Notations: For a real number x, x and x denote the
ceiling and floor of it, respectively. Let m = M/2 and b =
β/2. Vector and matrices are written in boldfaced letters,
and subscripts will be given if their sizes are not clear from
the context. For a matrix A, A
T
represents its transpose. The
symbols I and J are reserved for identity and reversal matrices,
respectively. Aside from that, W
2m
, W
2m+1
, W
2m
(z), and
ˆ
I
M
denote special matrices as follows:
W
2m
=
I
m
I
m
I
m
−I
m
, W
2m+1
=
⎡
⎣
I
m
I
m
√
2
I
m
−I
m
⎤
⎦
W
2m
(z)=W
2m
I
m
z
−1
I
m
W
2m
,
ˆ
I
M
=
I
M
2
J
M
2
.
II. R
EVIEW
A filter bank with length of (KM + β) is an ALLP-
PUFB if and only if the associated M × M polyphase matrix
E(z) satisfies the paraunitary (PU) property E
T
(z
−1
)E(z)=I
and the linear phase property E(z)=z
−(K−1)
DE(z
−1
)
ˆ
J(z).
Here, D = diag(I
n
s
, −I
n
a
),
ˆ
J(z) = diag(z
−1
J
β
, J
M−β
), and
K − 1 is called the order of the system. Obviously, the
ALLPPUFB with β =0corresponds to CLLPPUFB.
As illustrated in [15], an ALLPPUFB can only be found in
three cases:
1) Case A: M ∈ even, β ∈ even, and K ∈ Z.
2) Case B: M ∈ odd, β ∈ even, and K ∈ odd.
3) Case C: M ∈ odd, β ∈ odd, and K ∈ even.
Furthermore, n
s
= n
a
for the first case, and n
s
= n
a
+1 for
the rest [15]. As similar to CLLPPUFB [13], an order-(K − 1)
1549-7747/$31.00 © 2012 IEEE
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