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IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 1, JANUARY 2014
Lattice Structure for Regular Linear Phase Paraunitary Filter Bank With Odd Decimation Factor Bodong Li and Xieping Gao
Abstract—Linear phase paraunitary filter bank (LPPUFB) with regular property, i.e. regular LPPUFB (RLPPUFB), can be efficiently designed via lattice structure. The key to the design of such filter bank is to formulate an easy-to-handle regular condition in terms of the lattice structure of LPPUFB, and the solution for the case with even decimation factor has been published. In this letter, a method is proposed to design RLPPUFB with odd decimation factor. First, small part of the free parameters in the lattice structure of LPPUFB is discarded. Then, the lattice structure is equivalently converted to a new one for which the easy-to-handle regular condition can be easily obtained. In contrast to a simple method that discards about a half of the free parameters, the proposed method only discards small part of the free parameters and it therefore can produce better RLPPUFBs. Index Terms—Filter bank, lattice structure, linear phase, paraunitary, regular.
I. INTRODUCTION
L
INEAR PHASE PARAUNITARY FILTER BANK (LPPUFB) with regular property, i.e. regular LPPUFB (RLPPUFB), is particularly appropriate for image processing. RLPPUFB can be efficiently designed via lattice structure [1]–[4], and the key to the design of RLPPUFB is to formulate an easy-to-handle regular condition about the lattice structure of LPPUFB. Several works have been done on the design of RLPPUFB using lattice structure. In 2001, Oraintara et al. [4] constructed the RLPPUFB with even decimation factor, wherein the filter length is , is the decimation factor, is a positive integer. Two years later, Dai et al. [3] designed the RLPPUFB with odd decimation factor (as well as the one with even decimation factor) and with filter length no more than , where is an integer between 0 and . In 2009, Xu et al. [2] studied the RLPPUFB with arbitrary filter length . As to the case with even decimation factor, the LPPUFB with structurally regular property (i.e. RLPPUFB) was provided because the easy-to-handle regular condition can be easily obtained. Regarding the case with odd decimation factor, the
Manuscript received June 28, 2013; revised August 24, 2013; accepted September 29, 2013. Date of publication October 11, 2013; date of current version November 07, 2013. This work was supported by the NSFC under Grants 61172171 and 61302182 and by the Scientific Research Fund of Hunan Provincial Education Department under Grant 12C0398. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Chandra Sekhar Seelamantula. The authors are with the MOE Key Laboratory of Intelligent Computing & Information Processing and the College of Information Engineering, Xiangtan University, Xiangtan 411105, China (e-mail:
[email protected]). Digital Object Identifier 10.1109/LSP.2013.2285435
regular condition is not easy to handle (for even ) or even difficult to obtain (for odd ), the RLPPUFB was therefore not given. We find that at least for the odd-decimation-factor RLPPUFB with even , one can obtain an easy-to-handle regular condition by simply setting some of the free orthogonal matrices involved in the regular condition to be identity matrix. However, this will cause about a half of the involved free parameters discarded and as a result one loses the chance to get much better RLPPUFBs. To address this issue, this letter will describe a design of the lattice structure of RLPPUFB with odd decimation factor. The resulted LPPUFB is structurally regular, and the length of the involved filter bank can be arbitrary. The core of the design is to discard only small part (rather than many) of the free parameters included in the involved lattice structure of LPPUFB, and then we equivalently convert such lattice structure to be one for which the easy-to-handle regular condition can be easily obtained. The design examples show that the proposed method is superior to the simple method that discards about a half of the free parameters. Notations: Let and , where denotes the floor of the real number . Let represent the transpose of the matrix . The symbols , , , and are reserved for identity matrix, exchange matrix, null vector, and the vector with each entry to be 1, respectively, and the subscripts will be given if their sizes are not clear from the context. The symbol diag(.) denotes block diagonal matrix. Besides, , and
II. PRELIMINARIES A filter bank with filter length is called an LPPUFB if its polyphase matrix satisfies and , where , , and are the numbers of symmetric and antisymmetric filters respectively. Based upon lattice structure, such an LPPUFB with odd can be constructed by factorizing as below [2]. For the case with even ,
1070-9908 © 2013 IEEE
LI AND GAO: LATTICE STRUCTURE FOR REGULAR LINEAR PHASE PARAUNITARY FILTER BANK
where
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is limited to positive odd number,
where
As to the free orthogonal matrices employed in this case, , and are in size , , and , respectively. For the case with odd ,
where is restricted to positive even number, and identical to the one in the case with even . Besides,
,
is
where
Fig. 1. Lattice structure for LPPUFB with filter length , where and . (a) The one in [2]. (b) The simple one that sets . (c) The one that discards small part of the free parameters. (d) The one we used for regular LPPUFB.
and we call the above equation as regular condition. While lattice structure is used for the design of RLPPUFB, we formulate the regular condition about the lattice structure of LPPUFB and solve the condition to obtain LPPUFB with structurally regular property (i.e. RLPPUFB). III. MAIN RESULT A. Even and we let for the convenience of the later discussion. Regarding the free orthogonal matrices used in this case, , , and are of sizes , , , and respectively. The term regular in this letter means order one regular. As shown in [2]–[4], a filter bank is regular if its polyphase matrix satisfies (1)
For this case, the lattice structure of LPPUFB is depicted in Fig. 1(a), and the regular condition in (1) can be simplified into (2)
which has been given in [2]. The authors in [2] did not propose how to parameterize the orthogonal matrices in (2), and it is
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IEEE SIGNAL PROCESSING LETTERS, VOL. 21, NO. 1, JANUARY 2014
difficult to parameterize such orthogonal matrices without previously setting some of the matrices to be constant matrices (or some of the involved free parameters to be constant values). If the matrices in (2) are set to be identity matrix, then one can obtain a simpler regular condition as below: (3) in (3) is easy to parameterize, simThe orthogonal matrix ilarly as in Theorem IV.1 in [4]. However, the lattice structure that sets [i.e. Fig. 1(b)] discards too many free parameters as described below. Each orthogonal matrix of size can be represented by Givens rotation angles and sign parameters [5], and we mainly consider Givens rotation angles for the convenience of the following discussion. Therefore the lattice structure in Fig. 1(b) discards Givens rotation angles compared with the one in Fig. 1(a). Actually this is about a half of the free parameters included in the lattice structure in Fig. 1(a), where the number of Givens rotation angles is . As a result, one loses the chance to obtain much better RLPPUFBs. As discussed previously, maintaining all the free parameters will lead to a regular condition difficult to handle; and discarding too many free parameters will yield filter banks not good enough though the corresponding regular condition is easy to handle. One may ask if we can obtain an easy-to-handle regular condition by discarding only small part of the free parameters; and the resulted regular filter bank can thus be much better. In this letter, we carry out such idea as follows. First, the free orthogonal matrices in Fig. 1(a) are set to be , where is a free orthogonal matrix of size . This only discards small part of the free parameters, i.e. Givens rotation angles. Now the lattice structure becomes the one in Fig. 1(c), where
Therefore, the regular condition in (1) turns to
B. Odd As to the lattice structure in this case [i.e. Fig. 2(a)], one find that it is difficult to obtain a regular condition in simple form due to complicated starting block [i.e. ] used in the related lattice structure. One can address this issue by simply setting and , and the lattice structure turns to the one in Fig. 2(b). However, this will discard too many free parameters. To handle this problem, we set the free orthogonal matrix in the starting block to be , and the free orthogonal matrix in to be . Now the lattice structure becomes the one in Fig. 2(c), where
Following the way in [6], the lattice structure in Fig. 2(c) is essentially equivalent to the one in Fig. 2(d), where
Then, the regular condition is also (3), the same as the regular condition for the case with even To obtain the design of RLPPUFB in this case, the design freedom discarded is also a little, as fewer as Givens rotation angles. IV. DESIGN EXAMPLES
which is still difficult to handle. Fortunately, based on the way in [6], the lattice structure in Fig. 1(c) is essentially equivalent to the one in Fig. 1(d), where
Now the regular condition is easy to obtain, also (3), wherein the orthogonal matrix is easy to parameterize as mentioned earlier. It is surprising that the regular condition for the proposed design is identical to the one that simply sets the matrices in (2) to be identity matrix. However, our design only discards Givens a little of the free parameters, as fewer as rotation angles compared with Givens rotation angles by the method which simply sets the matrices in (2) to be identity matrix.
Optimizing the free parameters in the obtained lattice structure for RLPPUFB, one can obtain more practical filter banks for signal processing. The optimization is carried out according to the stopband energy criterion, and the tool used is the Matlab function fminunc in the optimization toolbox. Besides, the minimal-order RLPPUFB is initialized with DCT initialization (see [7, p. 2182]); and the order-( ) one is initialized by an obtained order-( ) one. We expect that the lattice structure for comparison can also been initialized by a lower-order version of such lattice structure. However, those in Fig. 1(b) and Fig. 2(b) are not easy to be initialized in such manner, and we use similar lattice structures as described below. As to the case with even , the lattice structure for comparison is given in Fig. 4, which is similar to the one in Fig. 1(b). Regarding the case with odd , the lattice structure for comparison is shown in Fig. 5, which is similar to the one in Fig. 2(b).
LI AND GAO: LATTICE STRUCTURE FOR REGULAR LINEAR PHASE PARAUNITARY FILTER BANK
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TABLE I COMPARISON OF STOPBAND ENERGY BETWEEN THE SIMPLE METHOD AND THE PROPOSED METHOD
Table I gives the stopband energy of the optimized RLPPUFBs, where the 2nd column is associated with the RLPPUFBs described in the previous paragraph that are similar to those obtained by simply setting the free orthogonal matrices to be identity matrix, and the 3rd column corresponds to our method. As can be seen, for both cases of even and odd , our method produces filter banks with good frequency property and is superior to the one that simply sets the free orthogonal matrices to be identity matrix. Such result can also be seen from Fig. 3. Besides, one can see from Table I that, in the design examples the simple method does not produce better filter banks as the filter order (i.e. ) increases to more than three, and this issue is addressed by our design. V. CONCLUSION
Fig. 2. Lattice structure for LPPUFB with filter length , where and . (a) The one in [2]. (b) The simple one that sets . (c) The one that discards small part of the free parameters. (d) The one we used for regular LPPUFB.
RLPPUFB plays important role in image processing, and the lattice structure for such filter bank with odd decimation factor has not been extensively studied, although it provides more choices of RLPPUFBs. This letter presents a design of RLPPUFB lattice structure which is carried out by discarding small part of the free parameters. The proposed method performs better than the one that discards about a half of the free parameters, and can be used for odd-decimation-factor RLPPUFB with arbitrary filter length. ACKNOWLEDGMENT
Fig. 3. Magnitude response of RLPPUFBs with (a) Simple method. (b) Proposed method.
,
and
.
The authors would like to thank the associate editor and the peer reviewers for their constructive suggestions which significantly improve the quality of this letter. REFERENCES
Fig. 4. The lattice structure similar to the one in Fig. 1(b).
Fig. 5. The lattice structure similar to the one in Fig. 2(b).
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