ONE-DIMENSIONAL (1-D) two-channel FIR perfect-reconstruction

Transcription

1 3542 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 Eigenfilter Approach to the Design of One-Dimensional and Multidimensional Two-Channel Linear-Phase FIR Perfect Reconstruction Filter Banks Bhushan D. Patil, Pushkar G. Patwardhan, and Vikram M. Gadre Abstract We present an eigenfilter-based approach for the design of two-channel linear-phase FIR perfect-reconstruction (PR) filter banks. This approach can be used to design 1-D two-channel filter banks, as well as multidimensional nonseparable two-channel filter banks. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by an appropriate choice of the length of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a predesigned analysis filter, with the filter lengths satisfying certain conditions. We present several design examples to demonstrate the effectiveness of the method. Index Terms Eigen-filter design, least-square filter design, twochannel filter bank. I. INTRODUCTION ONE-DIMENSIONAL (1-D) two-channel FIR perfect-reconstruction (PR) filter banks have been extensively studied in the literature [1], [2]. In many applications, it is desirable that the filters in the filter bank have linear phase. The analysis and design of linear-phase PR FIR filter banks has received a lot of attention in the literature [1] [5]. Various methods for the design of 1-D linear-phase two-channel PR filter banks have been proposed. Design methods using lattice structures in the polyphase domain have been presented in [4] [6]. These methods rely on a lattice-structure parameterization of the polyphase matrix, and the lattice parameters are then chosen using some optimization method so that the filters approximate the desired passband shape. However, the objective function of the optimization is a nonlinear function of the parameters, so the optimization becomes difficult with increasing number of parameters (i.e., with increasing filter lengths). Design approaches based on spectral factorization are well known. In these approaches, a halfband product filter Manuscript received December 30, 2007; revised March 20, First published May 20, 2008; current version published December 12, This paper was recommended by Associate Editor Y.-P. Lin. The authors are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, Mumbai , India ( bhushanp@ee. iitb.ac.in). Digital Object Identifier /TCSI is first designed and is then factored into two filters, yielding the analysis and synthesis low-pass filters in the filter bank. Various time-domain optimization approaches have been presented [8] [10]. In these approaches, the filters in the filter bank are designed by a time-domain optimization of the filter coefficients (i.e., a direct optimization of the filter coefficients, without any parameterization) after imposing the PR constraints on the coefficients. The design of multidimensional (MD) nonseparable twochannel filter banks is more complicated than the 1-D case because of the lack of factorization theorems for general MD polynomials. Thus, the spectral-factorization-based approaches, which are commonly used for the design of 1-D two-channel filter banks, can no longer be used for the MD case. The method of transformations [13] [15] is a commonly used approach for the design of MD two-channel filter banks. This method divides the MD filter-bank design problem to a set of two independent problems: 1) that of designing a 1-D two-channel filter bank and 2) to design an MD transformation kernel satisfying certain constraints. The design of 2-D two-channel Quincunx filter banks has been considered in [15] [17], [28], and[29]. Design of MD nonseparable filter banks using lattice structures has been presented in [16] and [24]. However, unlike the 1-D case, the lattice structures are not complete. Furthermore, due to the considerations of the shape of the frequency passbands, the optimization of the lattice parameters becomes increasingly difficult with increasing number of filter coefficients. In this paper, we use a time-domain formulation of the PR problem and present an eigenfilter approach for the design of two-channel linear-phase PR filter banks. We use the term time-domain formulation in the 1-D as well as the MD case to refer to a formulation which directly uses the filter coefficients and does not use any other parameterization. This approach can be used for the design of 1-D, as well as nonseparable MD, two-channel filter banks. Although the eigenfilter method and its extensions have been effectively used for the design of 1-D and MD filters [11], [12], [20], [21], [25], the eigenfilter method has not been applied to the problem of filter-bank design. Our method consists of first designing the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be expressed as a set of linear constraints on the complementary-synthesis low-pass filter. We then design the complementary-synthesis filter by using the eigenfilter design method with linear constraints. We show that, by appropriately /$ IEEE

2 PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS 3543 choosing the lengths of the filters, we can ensure the existence of a solution to the constrained eigenfilter design problem for the complementary-synthesis filter. Thus, our approach gives an eigenfilter-based method of designing the complementary filter, given a predesigned analysis filter, with the filter lengths satisfying certain conditions. The paper is organized as follows. Section II gives a brief review of the eigenfilter method as it applies to the design of FIR filters. In Section III, we consider the case of 1-D twochannel filter banks. In Section III-A, we present a formulation of the 1-D two-channel FIR PR filter-bank design problem and cast it in an eigenfilter-design framework in Section III-B. We present design examples of 1-D filter banks in Section III-C. In Section IV, we present the extension of the formulation for the case of MD two-channel nonseparable filter banks, and in Section IV-A, we present the detailed formulation for the case of the 2-D nonseparable Quincunx filter banks and present design examples. Notation: Boldfaced lowercase letters are used to represent vectors, and bold-faced uppercase letters are used for matrices. denotes the transpose of. denotes the inverse of. denotes the determinant of the matrix. denotes the absolute value of the scalar. The following notation is required for Section IV which discusses the case of MD filter banks. A vector raised to a vector power gives a scalar defined as follows:, where and.a vector raised to a matrix power, where is a matrix, is defined as follows: is a vector whose th entry is, where,, are the columns of the matrix. denotes the set of all integer vectors. The lattice generated by a nonsingular integer matrix is denoted by and is defined as the set of all vectors of the form, where. is the set of integer vectors of the form, with. II. REVIEW OF THE EIGENFILTER DESIGN METHOD [11], [12] In this section, we briefly review the eigenfilter design method, as it applies to the design of zero-phase FIR filters [11], [12]. We will first review the 1-D case and then review the extensions of the method to the MD case. We then review the technique of imposing linear constraints on the eigenfilter design method (we refer to [11] and [12] for more details on the eigenfilter design method and its various extensions for the design of 1-D and MD filters). 1) Eigenfilter Method for the Design of 1-D FIR Filters: In the eigenfilter design method, the objective is to formulate the error function to be minimized in the form, where is a real, symmetric, and positive-definite matrix, and is a real vector. The goal is to find a vector which minimizes. For the design of 1-D FIR filters, the elements of the vector are related in some manner to the filter impulse response. The constraint is imposed to avoid trivial solutions. The error measure should be chosen to properly reflect the deviation of the passband and the stopband from the ideal values of the desired response. Once such an error measure is chosen, by the Rayleigh principle [11], [12], [26], the eigenvector associated with the smallest eigenvalue of the matrix minimizes the error. Consider a 1-D zero-phase low-pass FIR filter, where. We note that, within a delay factor, this is an odd-length linear-phase filter, with length. Since is zero-phase, we have. With this, the frequency response of takes the form Defining the vectors can be written as. The frequency response of the 1-D low-pass filter should approximate a desired response given by where and are the passband and stopband cutoff frequencies, respectively. With this desired response, the stopband error can be defined as can be written in the form, where is a real, symmetric, and positive-definite matrix. The passband error can be expressed as, where is the deviation of the response from the zero frequency response, which is given as. Thus, can be written in the form, where is a real, symmetric, and positive-definite matrix. The total error to be minimized is is a real, symmetric, and positive-definite matrix, as required for the eigenfilter formulation. With this, the eigenvector associated with the smallest eigenvalue of matrix minimizes the error. can then be used to obtain the filter coefficients using (1). (1) (2)

3 3544 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 the PR condition for the filter bank can be written as [1] [5] (3) Fig. 1. One-dimensional two-channel filter bank. 2) Eigenfilter Method for the Design of MD FIR Filters: The eigenfilter method can be extended to the case of MD zero-phase FIR filters [12]. Consider an -dimensional zero-phase low-pass FIR filter, where is an vector. Since is zero-phase, we have. With this, the frequency response of takes the form, where is the set of integer vectors corresponding to the indexes of the independent coefficients of the zero-phase filter.now, by imposing some ordering on the independent coefficients of, we can form a vector. With this, can be written as, where is a vector consisting of elements of the form. With this vectorization, the design of the -dimensional zero-phase low-pass FIR filter can be formulated as an eigenfilter design problem, similar to the formulation done earlier for the 1-D case. 3) Imposing Linear Constraints on the FIR Eigenfilters: Linear constraints of the form, where is a matrix having constant elements and is a vector having constant elements, can be imposed on the FIR eigenfilter design [11], [12]. Note that, here, is the coefficient vector corresponding to the filter to be designed. The constraint can be expressed in the following form (we refer to [12] for details):, where, where is the reference frequency which we choose to be the zero frequency for the case of low-pass filters. The linear constraints can be imposed as follows: if and only if lies in the null space of. Any such can be expressed as, where is a rectangular unitary matrix whose columns form an orthonormal basis for the null space of and is any arbitrary vector. Using this, it can be shown that, after imposing linear constraints, the error function to be minimized can be written as. Therefore, the optimal is the eigenvector corresponding to the smallest eigenvalue of. Once we find the optimal, the optimal is given by. For the PR condition in (3), any constant value instead of two could be used. We use the constant value of two in this paper. Defining the product filter as, (3) states that should be a halfband filter, i.e., the coefficients of corresponding to the terms with even powers, other than the origin, should be zero. This can be written as where is the inverse -transform of. is the convolution of and Let be of length and be of length, i.e., Using (6) in (5), we have Note that, since and are zero-phase, it follows that is also zero-phase. Thus, we have only considered positive values for in (7). The length of is, i.e.,, for. But since is also zero-phase,. We would like to note that the filters and can be made causal by having delay factors and in the zerophase and, respectively. This results in a delay factor in, which retains PR with a delay factor in the right-hand side of (3). However, in this paper, we will assume zero-phase filters for convenience. Using (7) in (4) gives (4) (5) (6) (7) III. DESIGN OF 1-D TWO-CHANNEL LINEAR-PHASE FIR PR FILTER BANKS In this section, we present the design of 1-D linear-phase PR filter banks using the eigenfilter approach. A. Problem Formulation A two-channel 1-D filter bank is shown in Fig. 1. In this paper, we will assume that and are zerophase, i.e., and. We note that, within a delay factor, this corresponds to odd-length filters [3]. With the following choice of the two high-pass filters: where denotes the highest integer less than. Thus, to design a PR two-channel filter bank, we need to design zero-phase and, such that (8) is satisfied. (8)

4 PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS 3545 B. Design Method We observe that, given, (8) gives a set of linear equations for the unknown independent coefficients of, for. The set of linear (8) can be written in matrix notation as, where is the vector containing the variables, for of the linear equations, is the matrix containing the constant coefficients of the linear equations, and is an vector, whose first entry is two, and all other entries are zeros. The elements of the matrix are given by As an example, for and, the matrix is shown as the equation at the bottom of the page. As discussed in Section II, for the eigenfilter design formulation, the constraint can be written in the form, where the dimensions of are the same as. Thus, the set of linear equations in unknowns,, will have a nontrivial solution if the rank of the matrix,. Since, a nontrivial solution can be guaranteed if, i.e.,. We would like to note that, when, there exists a unique solution, i.e., the constraints completely determine the unknowns, and there are no free variables to optimize. Thus, we choose the values of and so that and so that we can use the eigenfilter design method to optimize the free variables. Thus, this suggests the following procedure for the design of the filters and. 1) Design zero-phase of length. We note that cannot be arbitrary. As shown in [4], a necessary and sufficient condition for to be an analysis filter of a PR filter-bank pair is that its two polyphase components be coprime (except for possible zeros at ). The techniques that we use in this paper to ensure that is valid (i.e., satisfies the earlier condition) are as follows. a) Design using the unconstrained eigenfilter method and, then, verify explicitly by factorization that its polyphase components do not have common (9) zeros. This technique works well, because, for a general 1-D filter, it is almost always true that its polyphase components are coprime. In fact, this holds for the filter design examples that we show in Section III-C as follows. b) Choose as the analysis filter of a known PR filter-bank pair. For example, in one of the design examples as follows, we choose as the analysis filter of the Daub97 biorthogonal filter bank [2]. Therefore, in such cases, is valid by design. c) Another simple way is to design as a halfband filter. Since one of the polyphase components is a constant (or a monomial, in general), a halfband is always valid. 2) Choose a such that. 3) Design zero-phase of length by imposing the linear constraints, where is as defined in (9), and This design can be done using the constrained eigenfilter method as described in Section II. As discussed earlier, with chosen as in 2) in the list, a solution is guaranteed to exist. In the earlier procedure, it is very easy to increase the length of the filters. We would also like to note that any FIR filter design method can be used to design, the only requirement being that should be valid, as discussed earlier. In fact, the earlier procedure can design the complementary filter,given any valid filter, within certain constraints on the lengths of the two filters. We would like to note that our proposed design method imposes constraints on the design of, whereas the design of is relatively mildly constrained (it needs to be valid). Thus, to achieve a certain frequency-response criteria (for example, stopband attenuation), the required filter length of will be larger than that of (with the same frequency-response criteria). C. Design Examples 1) Design Example-1: In this example, for, we use the nine-tap analysis low-pass filter from the Daub97 filter bank [2]. Given this, we design the complementary-synthesis filter with. In the eigenfilter error formulation of (2), we use,, and. The frequency-response plots of and are shown in

5 3546 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 Fig. 2. Plot of H (!) for Example-1. Fig. 4. Plot of H (!) for Example-2. Fig. 3. Plot of G (!) for Example-1. TABLE I COEFFICIENTS OFg [n]of DESIGN EXAMPLE 1 Figs. 2 and 3, respectively. The coefficients of are shown in Table I. 2) Design Example-2: For this example, we use the unconstrained eigenfilter method for the design of with Fig. 5. Plot of G (!) for Example-2., and after obtaining the coefficients of, we explicitly verify that it is valid. We then design with. Again, we use,, and. The frequency-response plots of the filters and are shown in Figs. 4 and 5, respectively. The coefficients of and are shown in Tables II and III, respectively. 3) Design Example-3: For this example, we design a halfband using the eigenfilter design method with. Note that, for the halfband, every even indexed coefficient except the origin is zero. We then choose for the design of. The frequency-response plots of the filters and are shown in Figs. 6 and 7, respectively. 4) Design Example-4: One of the advantages of using the eigenfilter method is the ease with which certain time- and frequency-domain constraints can be incorporated in the design [12]. In this example, we demonstrate that this flexibility of the eigenfilter method can be effectively used in the filter-bank

6 PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS 3547 TABLE II COEFFICIENTS OFh [n]of DESIGN EXAMPLE 2 TABLE III COEFFICIENTS OFg [n]of DESIGN EXAMPLE 2 Fig. 6. Plot of H (!) for Example-3. design. To demonstrate this, we impose the constraints corresponding to having zeros at in both of the filters and. For both the filters, we impose a third-order zero at. We design by using the eigenfilter method after imposing the zero constraints. Moreover, in the design of, the zero constraints are added to the PR constraints. Adding the zero constraints increased the number of constraints, and so the minimum required length of, i.e., the value of, increases accordingly. For this example, we use and for the design of and, respectively. As for the earlier examples, we use,, and. The frequency-response plot of the filters and are shown in Figs. 8 and 9, respectively. The coefficients of and are shown in Tables IV and V, respectively. IV. EXTENSIONS TO THE MD CASE We now extend the approach to the case of MD two-channel linear-phase filter banks. An -dimensional two-channel filter bank is shown in Fig. 10. Here, is the sampling matrix, which is a nonsingular integer matrix with. Denoting the integer Fig. 7. Plot of G (!) for Example-3. vectors in as and, we can assume without loss of generality that (the zero vector). Let the high-pass filters be chosen as [15], [22] (10) where, are the columns of, and the symbol denotes the elementwise product. With the choice of the high-pass filters as in (10), the PR condition is [15] (11)

7 3548 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 TABLE V COEFFICIENTS OFg [n]of DESIGN EXAMPLE 4 Fig. 8. Plot of H (!) for Example-4. Fig. 10. MD two-channel filter bank. Note that is the MD convolution of and (13) Thus, from (12), we have the PR conditions as (14) Fig. 9. Plot of G (!) for Example-4. TABLE IV COEFFICIENTS OFh [n]of DESIGN EXAMPLE 4 By defining the product filter condition of (11) can be written as [15], the PR for for where and (12) where denotes the set of all integer vectors and is the MD inverse -transform of [23]. Equation (12) states that on all the nonzero points on the lattice. Now, given, (14) imposes a set of linear constraints on. By choosing the number of coefficients of appropriately, we can ensure that a solution to (13) exists. Thus, this problem can be cast into the framework of the eigenfilter design method with linear constraint, in a manner similar to that done for the 1-D case in Section III. We now present the specific formulation and design examples for the case of 2-D Quincunx filter banks. A similar formulation can also be done for the design of 3-D two-channel face-centered orthorhombic filter banks. A. Design of 2-D Quincunx Filter Banks Consider the 2-D Quincunx filter bank, whose sampling matrix is. Assume that the low-pass analysis and synthesis filters and are zero-phase. Note that, for the 2-D case,. Assume that the filters have a square region of support, as defined as follows: (15)

8 PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS 3549 Therefore, in this case, the product filter is given as follows: and. Then, we have (16) Since is zero-phase,, and a set of independent coefficients of are (17) Thus, there are independent coefficients of. Using this, we can rewrite (16) in terms of the independent coefficients of as follows: (18) Since and are both zero-phase, is also zero-phase, i.e.,. Thus, the independent set of equations are obtained by using the following values for, in (18): Now, from (12), we require that (19) The condition, where, are integers and, can also be written as follows: Thus, we have (20), shown at the bottom of this page. Let denote the number of locations, where, i.e., denotes the number of locations in the set where denotes the largest integer less than. From (18) and (20), we now have a set of equations in unknowns (which are the independent coefficients of ). Thus, by arranging the independent coefficients of in a vector, (20) can be written as (21) where is a matrix with rows and columns, which is obtained from (20), and is a vector formed from the unknowns (coefficients of ) of size. To ensure the existence of nontrivial solutions for (21), we require that the number of rows of are less than the number of columns (22) where is as given earlier. Thus, the overall procedure to design Quincunx filter banks is as follows. a) Design a 2-D FIR filter,,, with a diamond-shaped passband. Again, like the 1-D case, we note that cannot be arbitrary. It has been shown in [27] that a necessary and sufficient condition for an MD filter to be an analysis filter of a PR filter-bank pair is that its two polyphase components with respect to the subsampling matrix be coprime. The techniques b) and c) used for the 1-D case (see Section III-B) can also be used for the 2-D case (and for the MD case in general) to ensure that is valid (i.e., satisfies the earlier condition). We would like to note that the method a) of Section III-B, which works well for the 1-D case, is in general difficult for the 2-D case. In the design examples as follows, we present examples using both the techniques b) and c) to design. b) Choose such that (22) is satisfied. c) Use the eigenfilter design method with the formulation presented in this section to obtain the filter. We now present some design examples to demonstrate this method. 1) Design Example-5: For this example, we use the technique b) to obtain a valid analysis filter, i.e., we choose to be the analysis filter of a known PR Quincunx filter bank. We obtain by using the method (20)

9 3550 IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I: REGULAR PAPERS, VOL. 55, NO. 11, DECEMBER 2008 Fig. 11. Plot of H (z ;z ) for Design Example-5. Fig. 13. Plot of G (z ;z ) for Design Example-6. Fig. 12. Plot of G (z ;z ) for Design Example-5. Fig. 14. Plot of H (z ;z ) for Design Example-7. of McClellan transformations [23] on the analysis low-pass filter of a 1-D PR filter bank. We use the following five-tap zero-phase 1-D analysis filter from the spline family of filter banks: Moreover, we use the following 2-D kernel as the transformation kernel: With this, we have. The plot of the filter obtained is shown in Fig. 11. For designing, we use, which satisfies (22). The plot of the filter obtained is shown in Fig ) Design Example-6: In this example, we use the same as Design Example-5. For, we use. The plot of obtained is shown in Fig. 13. Fig. 15. Plot of G (z ;z ) for Design Example-7. 3) Design Example-7: In this example, we use a Quincunx halfband filter as. The region of support of is, with. We then design using the earlier formulation with. The plots of and obtained are shown in Figs. 14 and 15, respectively.

10 PATIL et al.: EIGENFILTER APPROACH TO THE DESIGN OF 1-D AND MD FIR PR FILTER BANKS 3551 V. CONCLUSION In this paper, we presented an eigenfilter-based approach to the design of two-channel linear-phase PR filter banks. This method can be used to design 1-D as well as MD filter banks. We independently designed the low-pass analysis filter. Given the low-pass analysis filter, the PR conditions can be written as a set of linear constraints on the synthesis filter coefficients. We casted this problem of the design of the synthesis filter into an eigenfilter design problem with linear constraints. We presented detailed formulations of this method for the design of 1- and 2-D filter banks and presented several design examples. REFERENCES [1] P. P. Vaidyanathan, Multirate Systems and Filter Banks. Englewood Cliffs, NJ: Prentice-Hall, [2] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding. Englewood Cliffs, NJ: Prentice-Hall, [3] M. Vetterli and D. 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Patil received the B.E. degree in instrumentation engineering from Government College of Engineering, Jalgaon, India, and the M.E. degree in instrumentation engineering from Shri Guru Gobind Singhji Institute of Engineering and Technology, Nanded, Jalgaon. He is currently working toward the Ph.D. degree in the field of communication and signal processing at the Indian Institute of Technology Bombay, Mumbai, India. His research interests include wavelets and filter banks and their applications. Pushkar G. Patwardhan received the B.E. degree (with an award for achieving second rank in the college) in electronics engineering from K. J. Somaiya College of Engineering, University of Mumbai, Mumbai, India, in 1995 and the M.Tech. degree (with the Prof. G. N. Revankar Award for achieving second rank in the Electrical Engineering Department) in communications engineering and the Ph.D. degree from the Indian Institute of Technology Bombay, Mumbai, in 1997 and 2007, respectively. He is currently with India Design Center, Tensilica Inc., Pune, India. His primary research interests are in the area of multidimensional multirate systems, filter banks, and wavelets. Vikram M. Gadre received the B.Tech. and Ph.D. degrees in electrical engineering from the Indian Institute of Technology (IIT), New Delhi, India, in 1989 and 1994, respectively. He is currently a Professor with the Department of Electrical Engineering, IIT Bombay, Mumbai, India. His research interests broadly include communication and signal processing, with emphasis on multiresolution approaches. Dr. Gadre was the recipient of the President of India Gold Medal from IIT Delhi in 1989 and the Award for Excellence in Teaching from IIT Bombay in 1999 and He was also the recipient of the SVC Aiya Memorial Award and the Prof. K. Sreenivasan Medal from the Institution of Electronics and Telecommunication Engineers for contribution to education in Electronics and Telecommunication.