The Generalized Lapped Pseudo-Biorthogonal Transform - CiteSeerX

The Generalized Lapped Pseudo-Biorthogonal Transform: Oversampled Linear-Phase Perfect Reconstruction Filter Banks with Lattice Structures ∗ Toshihisa Tanaka and Yukihiko Yamashita May 1, 2003

Abstract We investigate a lattice structure for a special class of N-channel oversampled linear-phase perfect reconstruction filter banks with a decimation factor M smaller than N. We deal with systems in which all analysis and synthesis filters have the same FIR length and share the same center of symmetry. We provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filter banks. All filter coefficients are parameterized by rotation angles and positive values. The resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem. We consider next the case where we are given the GLPBT lattice structure with specific parameters and we a priori know the correlation matrix of noise which is added in the transform domain. In this case, we provide an alternative lattice structure which suppress the noise. We show that the proposed systems with the lattice structure cover a wide range of linear-phase perfect reconstruction filter banks. We also introduce a new cost function for oversampled filter bank design which can be obtained by generalizing the conventional coding gain. Finally, we exhibit several design examples and their properties.

EDICS: 2-FILB Permission to publish this abstract separately is granted. Corresponding author: Dr. T. Tanaka Laboratory for Advanced Brain Signal Processing Brain Science Institute, RIKEN 2-1, Hirosawa, Wako-shi, Saitama 351-0198 Japan Phone: +81-48-467-9665, FAX: +81-48-467-9694, E-mail: [email protected] ∗ Submitted to IEEE Trans. Signal Processing on February 16, 2002, Revised on May 1, 2003. This work was supported in part by JSPS Grant-in-Aid for JSPS Fellows 1210283. Toshihisa Tanaka is with the Laboratory for Advanced Brain Signal Processing, RIKEN Brain Science Institute, Japan. (E-mail: [email protected]) Yukihiko Yamashita is with the Graduate School of Science and Engineering, Tokyo Institute of Technology, Japan. (E-mail: [email protected])

1

1 Introduction Lapped transforms are powerful tools for signal and image processing, and have been investigated extensively [1–4]. Cassereau [5] and Malvar [1] pioneered the original class of lapped transforms called the lapped orthogonal transform (LOT). The LOT has been developed as a competitive alternative of traditional block transforms like the discrete cosine transform (DCT) [6] because of its extended basis functions, which overlap across traditional block boundaries. In other words, when the input signal is segmented into blocks of M samples, the LOT has M basis functions of length 2M and therefore generates M coefficients. Indeed, the LOT is a subclass of maximally decimated M-channel finite impulse response (FIR) paraunitary (PU) filter banks (FB’s). This fact was firstly pointed out by Vetterli and Le Gall [7]. From the viewpoint of PU FB’s, the basis functions of the LOT corresponds to the analysis and the synthesis filters. Moreover, Malvar provided a linear-phase (LP) solution for the LOT [1], which is essential for image processing and can be expressed by a lattice structure which offers fast implementation. In the FB community, a minimal lattice factorization for maximally decimated LP paraunitary filter banks (LPPUFB’s) has been found in [8]. Additionally, de Queiroz et al. investigated a formulation and a factorization for M-channel LPPUFB’s with filters of length L = K M as an extension of the LOT [3]. This extended class is called the generalized lapped orthogonal transform (GenLOT) [3, 9]. These LP FB’s are restricted to PU solutions which are in a subclass of maximally decimated Mchannel LP perfect reconstruction (PR) filter banks (LPPRFB’s). Recently, a general factorization for these LPPRFB’s has been developed in [4]. From the lapped transform perspective, this FB is called the generalized lapped biorthogonal transform (GLBT). It is shown that the lattice consists of nonsingular matrices and delays of which the number is minimal [4]. In all LPPRFB’s with the lattice structure in the literature, however, the number of channels is equal to the decimation factor. On the other hand, FB’s with a smaller decimation factor M than the number of channels N are called oversampled FB’s. In this paper, we make an attempt to construct N-channel (N > M) LPPRFB’s based on the lattice structure. Several studies on oversampled FB’s, which are related to redundant signal expansions [10–13], have been carried out [14, 15], and an application in quantization noise reduction has been proposed [16]. The oversampled FB’s have some advantages such as their improved design freedom and noise immunity [17,18]. However, these oversampled systems involve increased computational complexity. Therefore, oversampled DFT FB’s [14, 19–22] and oversampled cosine-modulated FB’s [17, 23] have been developed for a fast and efficient implementation by a factorization. These FB’s belong to a category of modulated FB’s. For application in image processing, the linear-phase property is very 2

significant. From this point of view, recently, a complete factorization of oversampled paraunitary (pseudo-orthogonal) FB’s 1 yielding LP filters has been proposed [24]. However, no work has been done on the general factorization of an oversampled LPPRFB in which the synthesis polyphase matrix is not restricted to the paraconjugate of the analysis polyphase matrix. In this paper, our goal is to establish lattice structures which can even represent the existing GenLOT and GLBT. After preliminaries, a class of oversampled LPPRFB’s is proposed in Section 3. From the lapped transform perspective, we call these FB’s the generalized lapped pseudo-biorthogonal transform (GLPBT). This fundamental factorization is further parameterized by applying the singular value decomposition (SVD) in Section 3.2. The SVD enables us to characterize all filters by rotation angles and positive real numbers. For odd N, the factorization can be established in a similar fashion as shown in Section 3.6. The relation between the GLPBT and the conventional lapped transforms are discussed in Section 3.8. We also consider in Section 4 the noise robust GLPBT which has the final block in the lattice suppressing noise added in the transform domain. We present some design examples in Section 5 and conclude this work in Section 6. This work can be regarded as a consequence of a generalization of [4] and [24], and can cover a wide range of possible LPPRFB’s, as summarized in Table 1.

1.1 Notation The following conventions are adopted in terms of notation: Bold-faced characters are used to denote vectors and matrices. h f , gi inner product of two vectors f and g k fk

Euclidean norm of f

In

n × n identity matrix

Jn

n × n reversal matrix

0n

n × n null matrix

0m×n m × n null matrix We sometimes omit the subscript of these matrices if the size is obvious.

1

In the paper [24], the authors termed the proposed FB’s paraunitary, which means energy preserving. Therefore,

paraunitarity can be achieved both by maximally decimated and oversampled FB’s. From the algebraic point of view, however, a maximally decimated PU FB requires orthogonality of the polyphase matrix, but an oversampled PU FB does not. Moreover, the polyphase matrix for the synthesis FB does not give such an inverse as defined for a full-rank square matrix but a left-inverse, which will be introduced later. In order to distinguish those FB’s from maximally decimated PU FB’s, we will use the terminology pseudo-orthogonal for such oversampled FB’s.

3

AT

transposition of A

tr[ A]

trace of A

|A(z)|

determinant of A(z)

deg[A(z)]

degree of A(z)

2 Preliminaries 2.1 LP and PR Conditions for Oversampled FB’s When the channel number N is greater than the decimation factor M, that is, N > M, such a FB is called an oversampled FB. Throughout this paper, the polyphase matrices with respect to the analysis and the synthesis banks are written by E(z) and R(z), respectively. The polyphase matrix E(z) is of size N × M, and R(z) is of size M × N. Figure 1 illustrates the oversampled N-channel FB that is dealt with throughout this paper. This FB can be represented in terms of the corresponding polyphase matrices as shown in Fig. 2. Throughout this paper, we assume that all input samples are real, and therefore all filter coefficients are real. A FB system provides PR (with zero delay) if and only if R(z)E(z) = IM .

(1)

If R(z) is the paraconjugate of E(z), i.e., R(z) = ET (z−1 ), we call this system pseudo-orthogonal; otherwise we call it pseudo-biorthogonal. The special case where M = N gives paraunitary or orthogonal FB’s if it holds that R(z) = ET (z−1 ), and gives biorthogonal FB’s otherwise. It should be noted that the notion of pseudo-biorthogonality spans a very large space of PR FB’s. Let us consider an expression for the PR condition in the time domain, which is sometimes useful for understanding in a vector space. Let E(z) =

K−1 X

Ei z−i ,

(2)

i=0

where Ei is a matrix with no delay whose size is the same as E(z). Similarly, let R(z) =

K−1 X

Ri zi ,

(3)

i=0

where Ri is also a matrix with no delay. Substituting (2) and (3) into the PR condition (1), we obtain the equivalent condition in the time domain as follows [25]: K−1−s X i=0

Ri Ei+s =

K−1−s X i=0

Ri+s Ei = δ s I M , s = 0, . . . , K − 1, 4

(4)

where δs = 1 if s = 0; δs = 0 otherwise. In order that E(z) and R(z) has the LP property, it is required that E(z) = z−(K−1) DE(z−1 )J, R(z) = zK−1 J R(z−1 )D,

(5)

where D is the diagonal matrix whose entry is +1 when the corresponding filter is symmetric and −1 when the corresponding filter is antisymmetric.

2.2 Left-Inverses We introduce the notion of a left-inverse for further discussion. Definition 1 A matrix A is called left-invertible if there exists a matrix X such that X A = I.

(6)

Such a matrix X is denoted by A− and called a left-inverse of A. Keep in mind that given a matrix A, its left-inverse A− is not uniquely determined. For any A− , moreover, it does not hold that AA− = I in general. A left-inverse is included in a special class of pseudo (generalized) inverses [26]. Let A be a left-invertible matrix of size n × m. Then, n must be greater than or equal to m, i.e. n ≥ m, and rank( A) = m.

3 Generalized Lapped Pseudo-Biorthogonal Transform 3.1 Even-Channel GLPBT The generalized lapped pseudo-biorthogonal transform (GLPBT) is a class of oversampled LPPRFB’s and a natural extension of the existing lapped transforms with lattice structure as summarized in Table 1. Let i = 1, . . . , K − 1. Define the following matrix: 1 Gi (z) = ΦiWΛ(z)W, 2

(7)

where         IN/2  IN/2 IN/2   Ui 0  0 N/2  ,  , Λ(z) =   , W =  Φi =       0N/2 z−1 I N/2  IN/2 −I N/2 0 Vi 5

(8)

where Ui and V i are N/2 × N/2 nonsingular matrices; therefore, Gi (z) is FIR invertible. Moreover, notice that Gi (z) = z−1 DGi (z−1 )D,

(9)

where D is the diagonal matrix such that the upper N/2 and the lower N/2 entries are 1 and −1, respectively. Definition 2 The even-channel GLPBT is an oversampled LPPRFB defined by the N × M analysis and the M × N synthesis polyphase matrices with the factorization given as T −ˆ ˆ E(z) = E(z)Φ 0 S, R(z) = S Φ0 R(z),

(10)

ˆ and R(z) ˆ are given by respectively, where E(z) ˆ = E(z)

1 Y

ˆ Gi (z), R(z) =

i=K−1

K−1 Y

G−1 i (z),

(11)

i=1

where Φ0 and S are defined as follows: • if M is even,   U 0  Φ0 =   0N M × 2

2

   0 N × M  1  I M 2 2   , S = √  2 2  IM V0 

 J M   , 2  −J M 

2

(12)

2

where both U0 and V 0 are left-invertible matrices of size N/2 × M/2; • if M is odd,   U 0  Φ0 =   0 N M+1 2

×

2

0 N × M−1 2

V0

2

  I M−1   2  1    , S = √  0 2   I M−1 2

0 M−1 ×1 2 √ 2 0 M−1 ×1 2

 J M−1   2  0  ,  −J M−1 

(13)

2

where both U0 and V 0 are left-invertible matrices of sizes of N/2×(M+1)/2 and N/2×(M−1)/2, respectively. The fundamental factorization of the analysis bank E(z) is illustrated in Fig. 3. ˆ and R(z) ˆ Indeed, the pair of E(z) forms a maximally decimated LPPRFB belonging to the class proposed in [4]. Therefore, the GLPBT has the following properties similar to those of the LPPRFB [4]:

6

1. all analysis and synthesis filters are FIR with the same length L = K M and have the same center of symmetry; 2. it consists of N/2 symmetric filters and N/2 antisymmetric filters if N is even. These properties are easily verified by the fact that the analysis FB is described as the polyphase matrix of size N × M whose entries are z-polynomials of order K, and satisfies the LP property as in (5). The same argument holds in the synthesis. Remarks • The terminology pseudo-biorthogonal comes from a theory of pseudo-biorthogonal bases [10, 12], which is a particular class of frames [11, 13]. It can be shown that the filters of the analysis bank and the synthesis bank generate a frame or a pseudo-biorthogonal basis. • In the maximally decimated case, i.e., M = N, the second property above is the unique solution for the number of symmetric filters in LPPRFB’s [27, 28]. In the overcomplete case, however, the second property does not always hold in all possible oversampled LPPRFB’s [24]. Therefore, a factorization for another class of oversampled LPPRFB’s is an open problem. • A structure is said to be minimal if the number of delays used is equal to the degree of the transfer function [8]. The factorization in (10) gives a minimal realization, i.e., the number of delays required for its implementation is minimal. ˆ Proof of minimality: From Definition 2, since Φ0 S has no delay, we have deg[E(z)] = deg[E(z)]. It has been verified in [29] that for a causal square polyphase matrix having an anticausal inverse, the degree of the system is equal to the degree of its determinant. From (9) and (11), moreover, it holds that ˆ = E(z)

1 Y

−1

−(K−1)

(z DGi (z )D) = z

i=K−1

Then, we have

−1

  1   Y −1  ˆ −1 )D. Gi (z ) D = z−(K−1) D E(z D 

(14)

i=K−1

ˆ ˆ ˆ −1 )| |D|] = N(K − 1) − deg[ E(z)]. ˆ deg[ E(z)] = deg[|E(z)|] = deg[z−N(K−1) |D| | E(z

(15)

ˆ It follows that deg[E(z)] = N(K − 1)/2. This degree is equal to the total number of delays employed in the structure. Therefore, the factorization is minimal.

7

2

3.2 Parameterization of Each Block In this section, we parameterize Ui and V i in each building block Gi (z) with the lattice structure, leading fast implementation. The key technique to parameterization is the well-known singular value decomposition (SVD). By means of the SVD, any invertible matrix can be decomposed into two orthogonal matrices and one diagonal matrix consisting of positive parameters. Since an orthogonal ! n matrix of size n is completely characterized by rotation angles, the invertible matrix is parame2 terized by rotation angles and positive multipliers [4]. The complete parameterization by the SVD enables us to obtain all filter coefficients by solving an unconstrained optimization problem. However, Φ0 in the initial block is not invertible but left-invertible. As seen, a left-invertible matrix does not have the unique left-inverse. (This is a different point from what an invertible matrix has the unique inverse.) In the following, we provide a solution to this problem. The key technique is also the SVD parameterization.

3.3 Invertible Matrices Parameterization of invertible matrices with the SVD has been used in [4]. For i ≥ 1, the SVD decomposes every invertible matrix as Ui = Ui1 Γi Ui0 , where Ui0 and Ui1 are orthogonal matrices and Γi is a diagonal matrix consisting of positive values [30]. Similarly, the invertible Vi can be written as the product of orthogonal matrices Vi0 and V i1 and a diagonal matrix ∆i : V i = V i1 ∆i V i0 . −1 −1 T −1 T Their inverses U−1 i and V i are represented as the following factorized forms: Ui = Ui0 Γi U i1 and T −1 T V −1 i = V i0 ∆i V i1 . Consequently, Φi , i = 1, . . . , K − 1 can be further factorized [4] as

     Ui1 0   Γi 0   Ui0 0   ,          Φi =       0 V i0  0 V i1 0 ∆i

(16)

where all of the orthogonal matrices U ! i0 , Ui1 , V i0 , and V i1 are of size N/2 × N/2; and therefore each N/2 N(N − 2) matrix can be characterized by = rotations. The diagonal matrices Γi and ∆i are 2 8 parameterized by N/2 positive parameters each. Hence, the matrix Φi is parameterized by N2 /2 free parameters, which indeed agrees with the sum of the degrees of freedom of two N/2 × N/2 invertible (nonsingular) matrices.

3.4 Left-Invertible Matrices As mentioned above, an invertible matrix can be easily parameterized. However, the problem here is whether or not every left-invertible matrix can be parameterized. The key to solve the problem is the SVD. 8

Let us consider the left-invertible matrix U of size n × m, where n > m in the oversampled case.

Since the rank of the left-invertible matrix U is m, rank(UT U) = rank(UUT ) = m. This implies that both UT U and UUT have the same m positive eigenvalues λi > 0, i = 0, . . . , m − 1. Let U a be an n × m matrix such that ith column is an eigenvector of UUT with respect to λi . Similarly, let Ub be

an m × m matrix such that ith row is an eigenvector of UT U with respect to λi . Since both UT U

and UUT are real symmetric, we can choose columns of Ua and rows of Ub such that UTa Ua = Im

and Ub UTb = Im , respectively. If UT U and UUT have m distinct eigenvalues, the above orthogonal property holds automatically. Applying the SVD to U, we have the following decomposed form: U = U a ΓU b ,

(17)

√ where Γ is a diagonal matrix with positive parameters ! ! λi . Since the columns of Ua form an orthonorn n−m m(2n − m − 1) plane rotations [24, 31]. The mal system, Ua can be parameterized by − = 2 2 ! 2 m m(m − 1) orthogonal matrix Ub is characterized by = rotations. The factorizations of initial 2 2 blocks in the analysis bank are summarized in Figs. 5 and 6.

3.5 Straightforward Choice for the Left-Inverse In order to parameterize a left-inverse, we adopt the product UTb Γ−1 UTa as a left-inverse matrix. It is easily confirmed that the product belongs to the collection of left-inverse matrices of U0 . To specify that it is a special element of the set of left-inverse matrices, we write it by U+0 , that is, U+0 = UTb Γ−1 UTa ,

(18)

which is indeed referred to as the Moore-Penrose (MP) pseudoinverse [26], which is uniquely determined with respect to a given matrix. Using this left-inverse, we can further factorize the N × M matrix Φ0 as in (12) and (13):      U01 0   Γ0 0   U00 0   ,          Φ0 =       0 V 00  0 ∆0 0 V 01

and the corresponding MP pseudoinverse is given by      UT   Γ−1 0   UT  0 0   0   01  , Φ+0 =  00 T   −1   T  0 V 00 0 ∆0 0 V 01

where if M is even,

• U00 and V 00 : M/2 × M/2 orthogonal matrices; 9

(19)

(20)

• Γ0 and ∆0 : M/2 × M/2 diagonal matrices; • U01 and V 01 : N/2 × M/2 matrices of which columns are orthonormal, and if M is odd, • U00 and V 00 : (M +1)/2×(M +1)/2 and (M −1)/2×(M −1)/2 orthogonal matrices, respectively; • Γ0 and ∆0 : (M + 1)/2 × (M + 1)/2 and (M − 1)/2 × (M − 1)/2 diagonal matrices, respectively; • U01 and V 01 : N/2 × (M + 1)/2 and N/2 × (M − 1)/2 matrices of which columns are orthonormal, respectively. Both for even M and for odd M, the number of free parameters for Φ0 is MN/2. Details of the initial building block are illustrated in Figs. 5 and 6. Now, we have obtained a special synthesis polyphase matrix which gives an alternative form as follows: K−1  Y   . R(z) = ST Φ+0  G−1 (z) i 

(21)

i=1

Actually, the MP pseudoinverse belongs to a subclass of the collection of left-inverses. For practical purpose, however, it may be effective and sufficient to adopt the MP pseudoinverse Φ+0 as a leftinverse, because it can be expressed by the lattice structure and has ability of noise suppression.

3.6 Odd-Channel GLPBT We give the factorization of the GLPBT in which the number of filters N is odd. The difference from the even case is that the order of Gi (z) is two. We provide only the results here. First of all, we define the invertible matrix Gi (z) for the odd-channel GLPBT as 1 Gi (z) = ΦiW o Λ0 (z)W o ΨiW o Λ1 (z)W o , 4

(22)

where       Q 0 0   IN/2 0   I N/2 i      Ui 0  √     , Ψ =  Φi =  , W = 2 0 0 0 q 0  ,  o  0  i      0 Vi    IN/2 0 −I N/2 0 0 Ri      IN/2 0   I N/2 0  0 0 N/2 N/2         −1 Λ0 (z) =  0 , Λ (z) = 1 0  1 z 0  ,  0        0N/2 0 z−1 IN/2 0N/2 0 z−1 I N/2 10

(23)

• V i , Qi , and Ri : • Ui : a

N+1 2

×

N−1 2

N+1 2

×

N−1 2

nonsingular matrices;

nonsingular matrix;

• q0 : a scalar. Definition 3 The odd-channel GLPBT is an oversampled LPPRFB defined by the N × M analysis and the M × N synthesis polyphase matrices with the factorization given as in (10), where for K odd, ˆ = GK−2 (z)GK−4 · · · G3 (z)G1 (z), R(z) = G−1 (z)G−1 · · · G−1 (z)G−1 (z). E(z) 1 3 K−4 K−2

(24)

Moreover, S is given as in (12) for M even and as in (13) for M odd, and Φ0 is defined as follows: • if M is even,   U 0  Φ0 =   0 N−1 M ×

 0 N+1 × M  2 2   ,  V

  U 0  Φ0 =   0 N−1 M+1

 0 N+1 × M−1  2 2   ,  V

2

2

(25)

0

where U0 and V 0 are left-invertible matrices of size (N + 1)/2 × M/2 and (N − 1)/2 × M/2, respectively; • if M is odd,

2

×

2

(26)

0

where both U0 and V 0 are left-invertible matrices of sizes of (N + 1)/2 × (M + 1)/2 and (N − 1)/2 × (M − 1)/2, respectively. Details of the initial building block are depicted in Figs. 7 and 8. The N-channel GLPBT consists of (N + 1)/2 symmetric and (N − 1)/2 antisymmetric filters. It may be shown in a manner similar to that for even N, that non-singular matrices Φi and Ψi can be parameterized with the SVD.

3.7 Normalized GLPBT Let hn and gn be the impulse responses of the nth analysis and synthesis filters, respectively. In the biorthogonal case (M = N), it is guaranteed that hhn , gn i = 1, for all n; however, in the pseudobiorthogonal case, generally, hhn , gn i , hhn′ , gn′ i, if n , n′ . This is an undesirable property when,

11

for example, we use the same quantizer with a fixed step-size for all channels, since a range of output values in each channel will be different from each other. Recall the perfect reconstruction condition R(z)E(z) = IM . We take the trace of this equation as K−1 K−1 N−1 X X X M = tr[I M ] = tr[R(z)E(z)] = tr[ Rk Ek ] = tr[ Ek Rk ] = hhn , gn i. k=0

k=0

(27)

n=0

We force here the inner product hhn , gn i to have the same value. Then, we define the normalized

filters h˜ n and g˜ n such that

h h˜ n , g˜ n i =

M . N

(28)

˜ n = cn hn , with cn = If we normalize gn such that k g˜ n k = k g˜ n′ k for all n, n′ , h˜ n is determined as h M . h hn , g ˜ n iN

In order to implement this normalization in the lattice structure, we only place multipliers

on each channel just after the final building block as depicted in Fig. 9.

3.8 Relation to the Conventional Lapped Transforms As seen previously, the GLPBT can represent a very large class of LPPRFB’s. If M = N, the GLPBT is identical to the GLBT [4]. If M < N and R(z) = ET (z−1 ), the GLPBT represents the lattice structure of LP PU (pseudo-orthogonal) FB’s proposed by Labeau et al. [24]. If M = N and R(z) = ET (z−1 ), the GLPBT can represent the GenLOT [3, 9] and the lattice for LPPUFB’s introduced by Soman et al. [8] Such a relation is illustrated in Fig. 10

4 Noise Robust GLPBT Recall here that a benefit of the overcomplete representation is noise suppressing properties due to increased design freedom. Unfortunately, the lattice structures developed in [24] and in the above disregard the effect of noise added to the transformed signal. The use of the MP pseudoinverse Φ+0 , which corresponds to the use of a transposition ΦT0 in the pseudo-orthogonal (paraunitary) case [24], seems to reduce the effect of additive noise because of its minimal norm and least square properties. However, this use provides the optimal noise suppression only if the noise component transformed ˆ is white. Generally, this is not white, even although the noise added to a signal transformed with R(z) with E(z) is white as depicted in Fig. 11. We consider in this section the case where we are given the GLPBT lattice structure with specific parameters and we a priori know the correlation matrix of noise which is added in the transform domain.

12

The problem here is to find the appropriate synthesis polyphase matrix X(z) that reduces noise for a given analysis bank E(z). Let fi ∈ R M and ni ∈ RN be time series of random vectors, where

i ∈ Z. This f i is transformed by E(z). Keep in mind that since z−k is the delay operator defined by PK−1 Ek f i−k . The signal E(z) f will be degraded by additive noise ni z−n f i = f i−n , we have E(z) f i = k=0 in the precess of transmission or storage. Therefore, the receiver or the observer obtains yi = E(z) f i + ni .

(29)

Note that E(z) provides overcomplete representation, and therefore there exist infinite number of synthesis polyphase matrices achieving PR. Let X(z) be determined by the submatrices Xi , that is, PK−1 Xi zk . Moreover, let X = [X0 · · · X K−1 ]. Then, the reconstructed signal fˆi is obtained as X(z) = k=0 fˆi = X(z)yi =

f i + X(z)ni .

(30)

The approximation problem to be solved is to minimize E nk fˆi − f i k2

(31)

under the PR condition X(z)E(z) = IM , where E n is the ensemble average on ni , . . . , ni+K−1 . Let n = [nTi · · · nTi+K−1 ]T , and let Q be the correlation matrix of n. Then, minimizing (31) is equivalent to minimizing the functional MSE[X] = E nkX(z)ni k2

  2

 ni 



 .. 



 = E n X  . 

 

  ni+K−1 = tr[XQXT ],

(32)

under the PR condition K−1−s X

Xi Ei+s =

i=0

K−1−s X i=0

Xi+s Ei = δ s I M , s = 0, . . . , K − 1.

(33)

We should use an iterative optimization technique to obtain the solution. Moreover, it is not guaranteed that the synthesis polyphase matrix X∗ (z) corresponding to the minimizer X∗ can be factored into a lattice structure, although the analysis bank E(z) is organized as the lattice structure. This may lead to difficulty in implementation.

13

Therefore, let us consider here the following alternative problem. Assume that in a fashion similar to R(z), a synthesis polyphase matrix X(z) is given by the form ˆ X(z) = ST Ξ R(z),

(34)

   ΞU 0   and Ξ and Ξ are left-inverses of U and V , respectively. Obviously, X(z) where Ξ =  U V 0 0  0 ΞV  attains PR. Then, the cost functional becomes MSE[ΞU , ΞV ] = E nkX(z)ni k2 2 ˆ = E nkST Ξ R(z)n ik

ˆ ··· R ˆ = E nkΞ [ R ] nk2 | 0 {z K−1} ˆ R

   

2

 ΞU 0   R ˆ    u  n

= E n



0 Ξ   R ˆ l 

V T

T

ˆ uQ R ˆ u ΞTU ] + tr[ΞV R ˆ lQR ˆ l ΞTV ], = tr[ΞU R

   R ˆ  P K−1 k ˆ ˆ k z and R ˆ =  u . It is easily verified that the PR condition is reduced to where R(z) = k=0 R R ˆl  ΞU U0 = ΞV V 0 = I M/2 .

(35)

(36)

Since ΞU and ΞV are determined independently, we can divide the above problem into two independent problems. Therefore, the minimization problem for ΞU results in the following: Minimize

ˆ uQR ˆ Tu ΞT ], tr[ΞU R U

subject to

ΞU U0 = I M/2 .

Similarly, the minimization problem for ΞV is as follows: T

Minimize

ˆ lQ R ˆ l ΞT ], tr[ΞV R V

subject to

ΞV V 0 = I M/2 .

It is noted that the solution of the above problem is equivalent to the BLUE (best linear unbiased estimator) [32]. Therefore, we obtain the solutions ˆ uQR ˆ Tu )+ U0 )+ UT ( Rˆ u Q R ˆ Tu )+ , Ξ∗U = (UT0 ( R 0 T

T

ˆ lQR ˆ l )+ V 0 )+ V T ( R ˆ lQ R ˆ l )+ . Ξ∗V = (V T0 ( R 0 14

(37)

T

T

ˆ uQR ˆ u )+ and ( R ˆ lQR ˆ l )+ can be In most cases, we can assume that Q is nonsingular, and therefore (R ˆ Tu )−1 and ( Rˆ l Q R ˆ Tl )−1 , respectively. replaced by (Rˆ u Q R In the general case, if white noise is added, the solution of the BLUE is provided by the MP pseudoinverse. We show however that in the GLPBT case, the choice of the MP pseudoinverse is generally incorrect. Assume that Q = σ2 I N where σ2 is the variance of the noise. The minimizer Ξ∗U yields Ξ∗U

T

T

ˆ uR ˆ u )+ U0 ]+ UT (σ2 R ˆ uR ˆ u )+ = [UT0 (σ2 R 0 + ˆ +u U0 ]+ ( R ˆ +u U0 )T R ˆ +u = [( Rˆ u U0 )T R +

+

ˆu, = ( Rˆ u U0 )+ R

(38)

where we used the fact that A+ = (AT A)+ AT . Therefore, the minimizer is not equal to U+0 in general 2.

ˆ = [ Eˆ T0 · · · Eˆ TK−1 ], In the pseudo-orthogonal (paraunitary) case, the PR condition implies that R

which yields that ˆR ˆT = R

K−1 X

T Eˆ k Eˆ k = I M .

(39)

k=0

Hence, we have Ξ∗U = U+0 = UT0 , Ξ∗V = V +0 = V T0 .

(40)

However, keep in mind that this straightforward solution does not always hold when Q , σ2 IN even in the pseudo-orthogonal case.

5 Design 5.1 Cost Functions and Design Examples Some design examples are provided in this section. A cost function to design a filter bank will depend on its application. We use here the combination of coding gain and stopband attenuation. The coding gain for overcomplete filter banks is different from that of maximally decimated filter banks. We 2

Indeed, this result is due to the fact that the synthesis filter bank is restricted to the lattice structure. In [18], the

ˆ authors derived the “global" minimum solution for the synthesis X(z) by defining the so-called “para-pseudoinverse" as −1 ˜ ˆ ˜ ˜ X(z) = [ E(z)E(z)] E(z), where E(z) is the paraconjugate of E(z). This inverse optimally suppresses the white subband −1 ˜ noise. However, the calculation of the inverse [E(z)E(z)] is generally a cumbersome task. Moreover, the inverse generally

becomes a polyphase matrix with IIR filters. Actually, in [18], they exhibited an approximation to the inverse with the Neumann series expansion.

15

derive the coding gain for the oversampled PR FB called the generalized coding gain in Appendix A. This cost function can be regarded as a generalization of the conventional coding gain (see Remarks in Appendix A). Although the conventional coding gain is independent of the amount of allocated bits, the coding gain for the oversampled case is not. As illustrated in Appendix A, we can separate the coding gain function JCG into two parts J1 and J2 : the first and the second terms depend on filter coefficients and the amount of bits, respectively. We use the first term J1 given as in (58), which is independent of bit rates, as a cost function in order to optimize free parameters. In order to avoid a trivial solution such that all filter coefficients are zero, which leads to an infinite coding gain, filter coefficients should be normalized as described in Section 3.7. In the maximally decimated case, this undesirable solution is never obtained because of the biorthogonal condition hhi , g j i = δi, j , i, j = 0, . . . , N −1, where δi, j is the Dirac. However, this constraint is not imposed on the overcomplete case. Let C be a correlation matrix of an input signal f . The variance of the nth subband channel output σ2yn can be obtained by using the correlation matrix as σ2yn = E f [|hhn , f i|2 ] = hhn , E f [ f f T ]hn i = hhn , Chn i,

(41)

under the assumption that every subband output has zero-mean. With the normalization, the cost J1 becomes Jˆ1 = 10 log10

M N

− N1 N−1 N−1 − N1 !2  Y Y  M M 2   h h˜ n , C h  ˜ n ik g˜ n k2  = 10 log10 hh , Ch ikg k . (42) n n n    N  n=0 hhn , gn iN n=0

This cost Jˆ1 can be regarded as an energy compaction cost of a given oversampled system under the constraint that all filters are normalized. Indeed, although the coding gain for a maximally decimated PR system, in which the normalization constraint for filters to be designed is implicitly involved, is defined as the SNR improvement against the PCM, it also provides how the system can concentrate the energy of an input signal into fewer coefficients. From the formulation of Jˆ1 , this characteristic is still preserved in the cost function Jˆ1 . Therefore, we can design a highly energy concentratable oversampled PR FB with normalized filters by maximizing this cost. We assume that the input signal is the first-order Markov process (AR(1) process) with the correlation coefficient ρ = 0.95. The stopband attenuation cost is frequently used for filter design. Minimization of this cost makes each filter a bandpass filter. The stopband attenuation costs for the analysis and the synthesis filter banks are respectively given as JA = JS =

N−1 XZ

n=0 ω∈Ωstopband N−1 XZ n=0

Wa (ω)|Hn (e jω )|2 dω,

(43)

W s (ω)|F n (e jω )|2 dω,

(44)

ω∈Ωstopband

16

where Hn (e jω ) and Fn (e jω ) are the frequency responses of the nth analysis filter and the nth synthesis filter, respectively, and Wa (ω) and Ws (ω) are weighting functions. The cost function to be used for design is defined as a linear combination of these three costs: J = α1 Jˆ1 + α2 JA + α3 JS .

(45)

All design examples here were obtained by unconstrained nonlinear optimization, where we used the routines provided by MATLAB

3

version 6.1. To obtain oversampled PR FB’s consisting of

bandpass filters, we should choose M and N such that N is an integer multiple of M in order that adjacent-channel alias terms can be canceled with each other (see [33], p. 226). Figure 13 shows the filter coefficients and the corresponding frequency responses of an eight-channel GLPBT in which a decimation factor is four and all filters have length 16, i.e., L = 16, M = 4, and N = 8. These filters are optimized for stopband attenuation to design bandpass filters. The initial values for the optimization are given at random. The analysis and the synthesis filters seem almost the same, since the combination coefficients αi are chosen so that α1 = 0 and α2 = α3 . Figure 14 illustrates an eight-channel GLPBT (M, N, L) = (7, 8, 21). All filters depicted in Fig. 14 are designed by optimizing the cost function Jˆ1 with initial values which are optimized by minimizing stopband attenuation. As seen in these figures, the filters provide poor frequency selectivity compared to the FB shown in Fig. 13, since N is not an integer multiple of M. Coding gains attained in the pseudo-orthogonal (paraunitary) [24] and in the pseudo-biorthogonal cases are compared in Table 2. All the filters listed here are optimized by maximizing the cost Jˆ1 . In all examples, the GLPBT achieves higher coding gain than the pseudo-orthogonal transform. This is due to the increase of design freedom. In order to evaluate the effect of the optimization, let us consider the eight-channel oversampled LPPRFB in which two four-channel GLBT’s (M = N = 4) optimized for coding gain are connected in parallel as illustrated in Fig. 12. In this case, the additive noise in the figure will be random or quantization noise. We can formally calculate the value Jˆ1 of this doubled filter bank. Let JCG [GLBT] be coding gain of the GLBT. Similarly, let Jˆ1 [p-GLBT] be the Jˆ1 -cost of the parallel connection of two GLBT’s. Then, given a GLBT, we have the following: M 20 log10 2 + 10 log10 . Jˆ1 [p-GLBT] = JCG [GLBT] + N N

(46)

When we use the GLBT attaining coding gain of 8.85 dB, which is a subclass of the GLPBT, the resulting Jˆ1 -cost of this overcomplete system is 6.60 dB, which is much lower than that of the GLPBT 3

MATLAB is a trademark of The Math Works Inc.

17

(M, N, L) = (4, 8, 16) optimized for Jˆ1 listed in 2. It should be noted that although we can construct an oversampled LPPRFB by the parallel connection, it is much more effective to optimize free parameters in the GLPBT lattice structure in the energy packing sense.

5.2 GLPBT with the Noise Robust Building Block We confirm the effect of the noise robust GLPBT when the building block Φ+0 in the GLPBT is replaced by Ξ∗ . For comparison, we use two GLPBT’s designed in the above subsection. All filter banks consist of filters of length 16 (L = 16), have eight channels (N = 8), and downsample the filtered signal by four (M = 4). One is a bandpass filter bank which is optimized for the stopband attenuation cost (GLPBT 1) as shown in Fig. 13. This is almost a pseudo-orthogonal (paraunitary) filter bank. The other is the one of Jˆ1 = 15.31 dB in Tables 2 and 2 (GLPBT 2). This is pseudo-biorthogonal. We consider here two kinds of noise. Case 1 The noise has the same correlation among channels. The noise is characterized by a blockdiagonal matrix with the same entry, that is, Q = diag[Θ, . . . , Θ], | {z }

(47)

K

where

[Θ]i, j

     si i = j =    c i, j

(48)

We used {si } = {10, 20, 30, 40, 40, 30, 20, 10} and c = 3 in this test. Case 2 All output coefficients are transmitted through one channel. The correlation function of the noise has exponential decay, that is, [Q]i, j = e−λ|i− j| .

(49)

We set that λ = 0.1 and 0.5 in this test. For each case, we compare the minimal error produced by the noise robust GLPBT MSE[Ξ∗ ] with the error produced by the GLPBT MSE[Φ+0 ]. Table 3 shows the difference in SNR (dB), that is, −10(log10 MSE[Ξ∗ ] − log10 MSE[Φ+0 ]).

(50)

The system can achieve PR, and therefore we can compute the difference of SNR in average on noise for any input. The table says that the differences in Case 2 (λ = 0.1) are greater than those in the other cases. This may be due to the fact the correlation matrix of Case 2 approaches to the identity matrix when λ increases. 18

6 Conclusions We have introduced the minimal factorization of a special class of N-channel oversampled linearphase perfect reconstruction filter banks (LPPRFB’s). The analysis and synthesis filters yield a frame or a pseudo-biorthogonal basis; and therefore this filter bank is called the generalized lapped pseudobiorthogonal transform (GLPBT) from the lapped transform perspective. The factorized filter banks are characterized by elementary rotation angles and scalar multiplications. Therefore, the lattice structure can provide fast implementation and enables us to determine the filter coefficients by solving an unconstrained optimization problem. Furthermore, we have considered the lattice structure of the GLPBT in presence of noise. For design purpose, we have introduced a novel cost function for evaluating energy compaction by generalizing the conventional coding gain. We have also shown some design examples where the newly proposed cost function and the stopband attenuation cost are used. It is shown that the GLPBT involving the increase of design freedom leads to the improvement in optimizing the cost function. In the noisy case, given an analysis bank of the GLPBT and a correlation matrix of noise, the proposed synthesis bank suppresses the noise added in the transform domain. Experimental results show that this class of filter bank is more effective in the presence of colored noise than white or almost white noise. It should be noted that the theory in this paper covers a very large class of LPPRFB’s. A lot of conventional works can be regarded as a subclass of the proposed filter banks with the lattice structure. For future research, it is necessary to develop the factorization for all possible N-channel oversampled LPPRFB’s. An application in noise removal/signal enhancement is also an open problem.

A

Derivation of the Generalized Coding Gain

Coding gain for oversampled filter banks can be derived in a manner similar to that for maximally decimated ones. Recall that f i is a random vector with M successive samples such that fi = [ fi (0), . . . , fi (M− 1)]T , where fi (m) is a wide-sense stationary process with zero-mean. We write the variance of fi (m) as σ2f . As is well known in the bit allocation problem [34], when bm bits are allocated for fi (m), the MSE between the original sample and the quantized sample is given by c2−2bm σ2f , where c is a constant determined by the probability density function (pdf) of fi (m). Now, let us assign b¯ bits for ¯ bits and obtain the distortion for the PCM for fi : each element of f i . Then, we use totally B = bM ¯

DPCM = Mc2−2b σ2f .

(51)

Consider next the case where the transform coefficients are quantized. The quantization is usually 19

approximated by the additive noise model which is described as in (29). The quantized coefficients are synthesized by R(z). In a way similar to (32), therefore, the distortion in this case is given as E nk f i − fˆi k2 = tr[RQRT ],

(52)

where R = [R0 · · · RK−1 ]. Assume that the quantization noise is uncorrelated over channels. Then, we can write Q as Q = diag[Θ, . . . , Θ], | {z }

(53)

Θ = diag[σ2q0 , . . . σ2qN−1 ],

(54)

K−1

with

where σ2qn is the variance of the quantization error of nth channel. Therefore, the distortion becomes E nk f i − fˆi k2 = tr[

K−1 X k=0

N−1 X

k=0

σ2qn kgn k2

n=0 N−1 X

≃ c

K−1  X  T Rk Rk  Θ1/2 ] 

 1/2 

= tr[Θ

=

Rk ΘRTk ]

n=0

2−2bn σ2yn kgn k2 ,

(55)

where bn and σ2yn are respectively the amount of bits allocated to and the variance of the nth channel PN−1 bn , and a coefficient of coefficient yn . Here, we assume that B bits are totally used, that is, B = n=0

each channel is distributed according to the same pdf as an input coefficient. The optimal bit allocation

is obtained by applying the arithmetic/geometric mean inequality to the above equation [34]: N−1 1/N N−1 1/N N−1 X Y  Y  M¯  b 2 2 −2bn 2 2 −2b 2 2 −2    n σyn kgn k  c 2 σyn kgn k ≥ Nc  2 σyn kgn k  = Nc2 N  n=0

(56)

n=0

n=0

The right-hand side of the above inequality is the distortion of oversampled filter banks DOFB when bits are optimally assigned. Finally, coding gain is defined as the ratio [34] D JCG = 10 log10 PCM D   OFB 2   σ f   −2(1− MN )b¯ M = 10 log10 2   N QN−1 σ2 kg k2 1/N  y n n n=0 = J1 + J2 ,

20

(57)

where J1 J2

σ2f M = 10 log10 Q 1/N , N N−1 2 2 σ kg k y n n n=0  M ¯ b log10 2. = −20 1 − N

(58) (59)

We call the cost JCG the generalized coding gain. We can observe that although the coding gain JCG depends on a bit rate, the second term J2 is independent of signal and filter coefficients. Therefore, when we find filter coefficients by maximizing the coding gain, it is sufficient to use only J1 . Remarks ¯ Therefore, oversampling • The function J2 is linear and monotonely decreasing for a bit rate b. is more effective in coding gain at relatively lower bit rates under the assumption that the MSE between a quantized and an original sample is given as c2−2b σ2 with some constant c, where b is the amount of bits and σ2 is the variance of the original signal. • If a given filter bank is maximally decimated, that is, M = N, J2 vanishes, and JCG is identical to the conventional coding gain.

Acknowledgment The authors would like to thank Prof. A. Nishihara of Tokyo Institute of Technology for his comment on the GLPBT, which yielded the problem discussed in Section 4. We also thank the annonymous reviewers for their comments and suggestions which improved the quality of this paper.

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24

List of Tables 1

A summary of previous works for a lattice structure of an N-channel LPPRFB with decimation factor M: E(z) and R(z) denote the polyphase matrices of the analysis bank and the synthesis bank, respectively. . . . . . . . . . . . . . . . . . . . . . . .

2

Comparison of coding gains of the pseudo-orthogonal case (oversampled LPPUFB’s) and the pseudo-biorthogonal case (oversampled LPPRFB’s)

3

25

. . . . . . . . . . . . .

25

Difference in SNR (dB) between the noise robust GLPBT and the GLPBT . . . . . .

25

List of Figures 1

N-channel uniform filter bank. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

Polyphase representation: We assume the anti-causal delay chain on the synthesis side. 26

3

Fundamental factorization of the analysis polyphase matrix E(z) . . . . . . . . . . .

26

4

SVD-based factorization of Gi (z) in the analysis bank . . . . . . . . . . . . . . . . .

26

5

The initial block in the analysis bank when both M and N are even . . . . . . . . . .

27

6

The initial block in the analysis bank when M is odd and N is even

. . . . . . . . .

27

7

The initial block in the analysis bank when M is even and N is odd

. . . . . . . . .

27

8

The initial block in the analysis bank when both M and N are odd . . . . . . . . . .

28

9

Normalization of the GLPBT

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

28

10

The relation between the GLPBT and other lapped transforms . . . . . . . . . . . .

28

11

The noise robust GLPBT in subspaces: When noise added in the transform domain is white, the MP pseudoinverse may not optimally suppress the noise.

26

. . . . . . . . .

29

12

An oversampled LPPRFB organized by the parallel connection of two GLBT’s . . .

29

13

Design example for M = 4, N = 8, L = 16, which is optimized for stopband attenua-

14

tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

Design example for M = 7, N = 8, L = 21, which is optimized for Jˆ1

31

25

. . . . . . . .

Table 1: A summary of previous works for a lattice structure of an N-channel LPPRFB with decimation factor M: E(z) and R(z) denote the polyphase matrices of the analysis bank and the synthesis bank, respectively. R(z) = ET (z−1 )

R(z) , ET (z−1 )

M=N

Paraunitary or Orthogonal [8]

Biorthogonal [4]

M