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Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices
Accepted Manuscript Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices Jun-Zheng Jiang, Fang Zhou, Peng-Lang Shui, Shan Ouyang
PII: DOI: Reference:
S1051-2004(15)00184-0 http://dx.doi.org/10.1016/j.dsp.2015.05.012 YDSPR 1781
To appear in:
Digital Signal Processing
Please cite this article in press as: J.-Z. Jiang et al., Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices, Digital Signal Process. (2015), http://dx.doi.org/10.1016/j.dsp.2015.05.012
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Highlights
• The PR condition is derived for the 2D general DFT modulated filter banks. • Spatial domain condition of PR is derived to facility filter bank design. • 2D NPR DFT modulated filter banks with good overall performance are designed.
Theory and Design of Two-dimensional DFT Modulated Filter Bank with Arbitrary Modulation and Decimation Matrices Jun-Zheng Jiang, Fang Zhou∗, Peng-Lang Shui, and Shan Ouyang Abstract: It is well-known that two-dimensional (2D) filter bank is far removed from a straightforward extension of one-dimensional (1D) filter bank. There are many challenging problems on the theory and design methods for the 2D filter bank. Among these problems, the perfect-reconstruction (PR) theory of the 2D DFT modulated filter bank with arbitrary modulation and decimation matrices remains an unsolved difficulty, which is the focus of this paper. The necessary and sufficient condition for perfect reconstruction (PR) is derived by using the polyphase decomposition of the analysis and synthesis filters, as well as the fast implementation structure of the filter bank. Then, the PR condition in frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF), which is utilized to formulate the design problem into an unconstrained optimization problem. An efficient iterative algorithm is proposed to solve the problem. Numerical examples are included to verify the validity of the PR condition and the effectiveness of the design method. Index Terms: Nonlinear optimization, Perfect-reconstruction (PR), Two-dimensional DFT modulated filter bank.
Manuscript received September 3, 2014. This work was supported by the National Natural Science Foundation of China under grant 61261032 and Guangxi Natural Science Foundation under grants 2013GXNSFBA019264, 2013GXNSFFA019004. Jun-Zheng Jiang and Shan Ouyang are with the School of Information and Communication, Guilin University of Electronic Technology, Guilin, China.(e-mail: [email protected], [email protected], Telphone: 86-773-2290203, Fax: 86-773-2290203). Fang Zhou (Corresponding author) and Peng-Lang Shui are with the National Lab. of Radar Signal Processing, Xidian University, Xi’an, China (email: [email protected], [email protected], Telphone: 86-29-88201022-8611, Fax: 86-29-88236159). 1
1. INTRODUCTION Among general two-dimensional (2D) filter banks, 2D modulated filter banks are of particular interest due to their ease of design and implementation. (see [1] for fundamentals on 2D filter bank). Furthermore, 2D modulated filter banks can provide 2D fine frequency tiling and good directional selectivity. Generally, 2D modulated filter banks can be categorized into two classes: 2D cosine modulated filter banks (CMFBs) [2-9] and 2D DFT modulated filter banks (DMFBs) [10-15]. During the past two decades, the attention has to a large extent been paid to the theory and design of 2D CMFBs. The support permissibility and perfect reconstruction (PR) condition of the 2D two-parallelogram CMFBs were investigated in [2]. Nevertheless, the 2D two-parallelogram paraunitary CMFBs cannot provide high stopband attenuation due to lack of vertex permissibility. Numerically efficient algorithm was proposed for the design of the two-parallelogram CMFBs [8], which improves the stopband attenuation by relaxing the PR property. In [3], theory and design of 2D four-parallelogram CMFBs were exploited. In [6], 2D PR modulated filter banks with triangular supports were designed from a 2D complex prototype filter (PF) whose passband support is a triangle that is just half of a parallelepiped-shaped passband support defined by the sampling matrix. 2D oversampled PR CMFBs were considered in [7]. On the other hand, there has been increasing interest in the 2D DMFBs. 2D critically sampled DMFBs were preliminarily constructed from one-dimensional (1D) filters [1]. However, 2D critically sampled DMFBs with PR cannot behave high stopband attenuation due to the non-permissible configuration [2]. In [10], 2D oversampled separable DMFBs were
2
constructed by tensors of 1D DMFBs and provided good directional selectivity. However, for 2D separable filter banks, the degrees of design freedom are heavily restricted and the frequency tiling is limited to be rectangular. Compared with separable counterparts, 2D non-separable filter banks offer more flexibility and usually provide better overall performance [5]. Therefore, subsequent research mainly concentrates on 2D non-separable DMFBs. In [11], the permissibility of the 2D DMFBs was analyzed from a pictorial viewpoint. It was shown that the 2D DMFBs can have good PR property and high stopband attenuation only if the filter banks are fully oversampled, a necessary condition for the filter bank to possess good overall performance. Also, the bi-iterative second-order cone program (BI-SOCP) was utilized to design the filter banks with nearly PR (NPR) therein. In [13], the modified Newton’s method was presented to design the 2D DMFBs with single-prototype. The necessary and sufficient condition was derived for the 2D 2× oversampled DMFBs and 2D critically sampled modified DFTDFBs (MDMFBs) to be PR [12]. The equivalence between these two types of filter banks was established in the single-prototype case. A lifting-based method was proposed for designing the 2D 2× oversampled DMFBs with structurally PR and a numerical algorithm was presented to design the 2D critically sampled MDFTDFBs with NPR. In [14], the 2D critically-sampled MDFTDFBs were redesigned by an iterative algorithm, which suppressed the numerical algorithm both in performance and computational efficiency. However, the necessary and sufficient PR condition remains an unsolved problem for the 2D DMFBs with arbitrary modulation and decimation matrices, which heavily restricts the development of the filter bank. In this paper, we investigate the theory and design of the 2D DMFBs with arbitrary 3
modulation and decimation matrices. A necessary and sufficient condition for the PR of the filter banks is derived from the polyphase decomposition of the analysis and synthesis filters. Also, the fast implementation structure of the filter banks is constructed. To the best of our knowledge, the PR condition and fast structure have not been derived for the 2D DMFBs with arbitrary modulation and decimation matrices before. Then, the PR condition in frequency domain is converted into a set of quadratic equations with the PF (termed as spatial domain condition), which leads us to formulate the design of the filter banks into an unconstrained optimization problem whose objective function consists of the PR condition and frequency selectivity of the PF. By setting the gradient vector of the objective function to be zero, the PF is iteratively optimized with closed-form formula. The validity of the PR condition and the effectiveness of the design algorithm are verified by the numerical examples. This paper is organized as follows. In Section II, the general structure of 2D DMFBs is reviewed, the PR condition is derived and fast implementation structure is constructed. In Section ċ, the PR condition is transformed into quadratic equations with the PFs and the design algorithm is presented. In Section IV, several numerical examples are given. Finally, conclusion is presented in Section V. Notations: Most of the notations in this paper are the same as that in [1]. Some frequently used ones are recalled below: a) Boldfaced letters denote matrices or vectors. Superscript T , ∗ and † denote transpose, conjugate, and transpose-conjugate, respectively. Symbol «¬ x »¼ denotes the integer part of x . Matrix I K denotes K × K identity matrix.
4
b) The Fourier transform of 2D signal x(n) is denoted by X (Ȧ) with Ȧ = [ωx , ω y ]T . c) For a 2×2 nonsingular integer matrix
M , its coset N (M ) is defined by
N (M ) = {m k = Mx k , x k ∈ [0,1) 2 , k = 0,1," , M − 1} . The number of vectors in set N (M ) is equal to
M , which is the absolute value of the determinant of matrix M . The parallelepiped FPD(M )
is
defined
FPD(M ) = {Mx, x ∈ [0,1) 2 }
by
.
Region
SPD(π D−T ) ≡ {Ȧ ∈ R 2 : −π ≤ ȦT d1 , ȦT d 2 < π } for D = [d1 , d 2 ] .
2 TWO-DIMENSIONAL DFT MODULATED FILTER BANKS 2.1 Preliminary of 2D DFT modulated filter banks
As 1D DFT modulated filter banks [16, 17], 2D DMFBs are generated by one or two 2D lowpass PFs. Let the modulation matrix D1 be a 2 × 2 nonsingular integer matrix. And denote H (Ȧ) and G (Ȧ) as the lowpass analysis and synthesis PFs, respectively. All the analysis and synthesis filters H i (Ȧ ) and Gi (Ȧ) are DFT modulated versions of the PFs as H i (Ȧ ) = H (Ȧ − 2π D1−T u i ), u i ∈ N ( D1T ), Gi (Ȧ ) = G (Ȧ − 2π D1−T u i ), i = 0,1," , D1 − 1.
(1)
Fig.1. Structure of 2D DFT modulated filter bank
Assume D 2 as another 2 × 2 nonsingular integer matrix. Then, a 2D DFT modulated filter bank with the modulation matrix D1 and decimation matrix D2 is shown in Fig.1, where 5
X (Ȧ), Yi (Ȧ) and X (Ȧ) are the 2D Fourier transforms of the input signal, subband signals, and
output signal, respectively. The Fourier transforms of subband signals and the output signal are written as [11] Yi (Ȧ) =
1 D2
D2 −1
¦ H (D i
−T 2
(Ȧ − 2π v k )) X ( D2−T (Ȧ − 2π v k )) ,
(2)
k =0
X (Ȧ) = T0 (Ȧ) X (Ȧ) +
D2 −1
¦ T (Ȧ) X (Ȧ − 2π D k
−T 2
v k ), v k ∈ N ( DT2 ) ,
(3)
k =1
where Tk (Ȧ) =
1 D2
D1 −1
¦ H (Ȧ − 2π D i
−T 2
v k )Gi (Ȧ), k = 0," , D2 − 1 with v 0 = [0, 0]T . In addition, the
i =0
stopband, transition-band, and passband of 2D DMFBs are specified as ȍ s = {Ȧ ∈ [−π , π ) 2 and Ȧ ∉ SPD (π D−2 T )}, ȍt = {Ȧ ∈ SPD(π D−2 T ) and Ȧ ∉ SPD(π D1−T )},
(4)
ȍ p = SPD(π D ). −T 1
When ȍ p ⊂ SPD(π D−2 T ) , the filter bank is oversampled, in this case D1 > D2 . Further, if ȍ p ⊂ SPD(π D−2 T ) and there is no intersection between ȍ p and ȍ s , the filter bank is fully
oversampled. It is shown in [11] that only 2D fully oversampled DMFBs are permissible, that is, the filter banks can simultaneously possess small reconstruction error and high stopband attenuation. Nevertheless, the permissibility is just the necessary condition for the filter banks to possess good overall performance. In what follows, we derive the necessary and sufficient condition for the PR of the filter bank using the polyphase decomposition. 2.2 Perfect Reconstruction Condition
In this subsection, we derive a necessary and sufficient condition for the PR of the 2D DMFBs with arbitrary modulation and decimation matrices. Assume D as the least common right multiple of the modulation matrix D1 and decimation matrix D 2 [18], i.e., D = D1D3 = D 2 D 4 , 6
(5)
where D3 , D4 are nonsingular integer matrices. With (5), the set N ( D) can be specified as follows. Lemma 2.1: For matrix D = D1D3 = D2 D4 , the set N ( D) can also be defined as N (D) = {D1p n + u l | u l ∈ N (D1 ), p n ∈ N ( D3 )} ,
(6a)
N (D) = {D2 q s + v m | v m ∈ N (D2 ), q s ∈ N ( D4 )} .
(6b)
The proof is given in Appendix A. In terms of (6a), vectors m k ∈ N (D), k = 0,", D − 1 can be arranged as « k » m k = D1p n + u l , n = « » , l = k − n ⋅ D1 , u l ∈ N ( D1 ), p n ∈ N ( D3 ) . «¬ D1 »¼
(7)
The polyphase decomposition of the analysis and synthesis PFs are given by [1]
H (Ȧ) =
¦
T
e− jȦ mk Ek (DT Ȧ) , G (Ȧ) =
mk ∈N ( D )
¦
T
e jȦ mk Rk (DT Ȧ) ,
(8)
mk ∈N ( D )
where Ek (Ȧ ) and Rk (Ȧ ) , k = 0,", D − 1 are the type-I polyphase components of the analysis PF and the type-III polyphase components of the synthesis PF, respectively. With (1) and (8), the analysis and synthesis filters can be expressed with respect to Ek (Ȧ) and Rk (Ȧ ) as follows
H i (Ȧ) =
¦
e− jȦ
¦
e jȦ
T
mk
T
−1
e j 2π u i D1 mk Ek (DT Ȧ),
m k ∈N ( D )
Gi (Ȧ) =
T
mk
T
(9)
−1
e− j 2π u i D1 mk Rk ( DT Ȧ).
m k ∈N ( D )
As a result, the analysis filter banks can be written as
{
}
[ H 0 (Ȧ), H1 (Ȧ),", H D1 −1 (Ȧ)]T = W1diag E0 (DT Ȧ), E1 (DT Ȧ),", E D −1 (DT Ȧ) e(Ȧ) , where diag denotes diagonal matrix with entries Ek ( DT Ȧ ) and
7
(10)
ª j 2π u T0 D1−1m0 « e « j 2π u T D−1m 0 1 1 W1 = « e « # « T « j 2π u D −1D1−1m0 1 «¬ e
e e
j 2π u T0 D1−1m1 j 2π u 1T D1−1m1
# e
D−1m D1 −1 1 1
j 2π u T
º » j 2π u 1T D1−1m D −1 » " e », » % # » j 2π u T D1−1m D −1 » D1 −1 »¼ " e "
e
j 2π u T0 D1−1m D −1
(11a)
ª e− jȦT m0 º « » « e − jȦT m1 » ». e(Ȧ) = « « » # « » T «e− jȦ m D −1 » ¬ ¼
(11b)
Equation (12.4.37b) from reference [1] presents a property of 2D DFT, i.e.,
¦
m∈N ( MT )
e − j 2π m
T
M −1k
° M , k = 0 . Based on this property and the arrangement in (7), =® °¯0, k ∈ N (M ), k ≠ 0
D1 × D matrix W1 can be factored as
W1 = W∗ ⋅ [I D1 ,", I D1 ] = W∗ ⋅ Mb ,
(12)
where W∗ is the conjugate of the generalized DFT matrix W with WT W ∗ = D1 I D
1
The number of I D
1
[1].
in the D1 × D matrix M b is D3 . From Lemma 2.1 and (7), vector
e(Ȧ) can be expressed as
e(Ȧ) = F(DT2 Ȧ)f (Ȧ) ,
(13)
where T −1 °exp( − jȦ D 2 (m k − v l )),(m k − v l ) mod D 2 = 0,0 ≤ k ≤ D − 1,0 ≤ l ≤ D 2 − 1, (14a) [F (Ȧ )]k ,l = ® °¯0,otherwise.
f (Ȧ) = [exp(− jȦT v0 ),",exp(− jȦT v D2 −1 )]T .
(14b)
In summary, the analysis filter banks can be expressed as
[ H 0 (Ȧ), H1 (Ȧ),", H D1 −1 (Ȧ)]T = W∗E(DT2 Ȧ)f (Ȧ) ,
(15)
where
{
}
E(Ȧ) = I b diag E0 (DT4 Ȧ), E1 (DT4 Ȧ),", E D −1 (DT4 Ȧ) F(Ȧ) . 8
(16)
Similarly, the polyphase decomposition of the synthesis filter banks is
[G0 (Ȧ), G1 (Ȧ),", G D1 −1 (Ȧ)] = f † (Ȧ)R(DT2 Ȧ)WT ,
(17)
where D2 × D1 matrix R (Ȧ) is defined by
{
}
R(Ȧ) = F† (Ȧ)diag R0 (DT4 Ȧ), R1 (DT4 Ȧ),", R D −1 (DT4 Ȧ) MTb .
(18)
In terms of the polyphase decompositions in (15) and (17), the PR condition for two-dimensional DFT modulated filter banks are given in the following theorem. Theorem 2.2 (PR Condition): A two-dimensional DFT modulated filter bank with the modulation matrix D1 and decimation matrix D 2 is PR if and only if R (Ȧ)E(Ȧ) =
1 I . D1 D2
(19)
If synthesis PF satisfies g (−n) = h(n), ∀n ∈ Z 2 , the analysis and synthesis polyphase components are related by Rk (Ȧ) = Ek∗ (Ȧ), k = 0," , D − 1 , which results in
R (Ȧ) = E∗ (Ȧ) .
(20)
In this case, the PR condition is taken as follows. Corollary 2.3: 2D single-prototype DFT modulated filter banks with the modulation matrix D1 and decimation matrix D 2 are PR if and only if the PF satisfy
h(n) = h( −n) E∗ (Ȧ)E(Ȧ) =
1 . I D2 D1
(21)
ª1 −1º
Particularly, for the 2D MDMFBs [12], where D = D1 , D3 = I 2 , D4 = « » . In this case, the ¬1 1 ¼ polyphase matrix E(Ȧ) reduces to ª I D2 E(Ȧ) = diag E0 (DT4 Ȧ), E1 ( DT4 Ȧ)," , E D −1 ( DT4 Ȧ) « − jωx «¬ e I D2
{
}
º ». »¼
(22)
Combining (22) and symmetric PF with h(n) = h(−n) [12], the PR condition of 2D 9
MDMFBs can be stated as follows. Corollary 2.4: 2D modified DFT modulated filter banks with decimation matrix D 2 is PR if and only if the polyphase components of the PF satisfy
h(n) = h( −n) Ek∗ (Ȧ) Ek (Ȧ) + Ek∗+ D2 (Ȧ) Ek + D2 (Ȧ) =
1 . , k = 0,", D2 − 1 D1
(23)
This Corollary is consistent with Theorem 6 in [12]. 2.3 Fast Implementation One of the key features of modulated filter bank is that they can be implemented with very efficient structure, which makes them very popular in practical applications. For instance, 1D 2M-channel DFT modulated filter bank with the decimation factor M allows the fast decomposition and reconstruction of signals via the M-point DFT and inverse DFT [19]. In terms of the polyphase representations in (19), 2D DMFBs also allow fast signal decomposition and reconstruction via the 2D D1 -point DFT and inverse DFT. The flowchart of fast decomposition and reconstruction is shown in Fig.2.
Fig.2. Polyphase structure of two-dimensional DFT modulated filter bank
10
3. DESIGN OF TWO-DIMENSIONAL DFT MODULATED FILTER BANKS As we know, design of 2D filter banks is a highly challenging work, even when it is modulated one. In this section, we present an iterative approach to design 2D DFT modulated filter banks with arbitrary modulation and decimation matrices (under fundamental restriction shown in Section 2.1). The good overall performance of the designed filter bank can be assured by careful formulation of the design problem and efficient algorithm. In view of this point, a spatial domain PR condition is derived for the 2D DMFBs. Based on the condition, an efficient iterative algorithm is proposed to design the filter banks. In this paper, we concentrate on the design of 2D single-prototype DMFBs. 3.1 PR condition in spatial domain With the frequency domain condition (19), PR filter bank can be designed by means of some parametric structures of the polyphase matrices E(ω ) , for instance, the lattice structure [20]. However, due to the highly nonlinear relationship between the performance of the filter bank and the structure parameters, it is difficult to design filter bank with fine performance. Similar to the design of 1D CMFBs [21], the PR condition in (21) is formulized into quadratic equations with respect to the analysis PF, termed as the spatial domain PR condition, shown as follows. The PR condition in (21) contains D2
2
equations with respect to polyphase components.
With the definitions of matrix F(Ȧ) and block matrix I b , the PR equations can also be expressed as D −1
¦
l0 ,l1 = 0
e
jȦT IJ k ,l ( l0 ,l1 )
η k ,l (l0 , l1 ) El∗0 ( DT4 Ȧ ) El1 ( DT4 Ȧ) = 11
1 δ ( k − l ),0 ≤ k , l ≤ D 2 − 1 , D1
(24)
where IJ k ,l (l0 , l1 ) = D2−1 (m l − m l − v k + v l ) , δ (⋅) is the delta function, and η k ,l (l0 , l1 ) is defined 0
1
as 1,(m l0 − v k ) mod D2 = 0,(m l1 − v l ) mod D2 = 0, ° η k ,l (l0 , l1 ) = ® and (l0 − l1 ) mod D1 = 0, °0,otherwise. ¯
(25)
In what follows, equation (24) is transformed into quadratic equations with the PFs. Let h = {h(n) | n = (nx , ny ), − L ≤ nx , ny ≤ L} be the coefficients matrix of the PF, which is then
stacked into column vector as
h1 = [h(− L, − L),", h(− L, L),", h( L, L)]T .
(26)
For convenience, a set of integer vectors is defined as S = {s = D−1 (n − m k ), n ∈ [− L, L]2 , m k ∈ N (D),(n − m k ) mod D = 0} ,
(27)
where (n − m k ) mod D = 0 means D −1 (n − m k ) = [0, 0]T . For convenience, denote K as the number of vectors in set S . The polyphase component Ek (Ȧ ) can be represented by
Ek (Ȧ) = eTp (Ȧ)U k h1 , k = 0,", D1 − 1 ,
(28)
where T
e p (Ȧ) = [e − jȦ s0 ," , e − jȦ
T
s K −1 T
] ,
°1, t − Dsi = m k , i = 0," , K − 1, j = 0," , 4 L2 + 4 L, [U k ]i , j = ® j °¯0, otherwise. t = {[− L, − L]," ,[− L, L]," ,[ L, L]}.
(29)
(30)
The multiple of vectors e∗p (Ȧ) and eTp (Ȧ) is e∗p (Ȧ )eTp (Ȧ) = ¦ e − jȦ k ī(k ) , T
k∈ψ
where
12
(31)
ψ = {k = s j − si , si , s j ∈ S , i, j = 0," , K − 1} , (32)
1, s − s = k ,0 ≤ k , l ≤ K − 1, [ī(k )]k ,l = ® k l ¯0,otherwise. Substituting (28) and (31) into (24), we obtain D −1
¦
e
jȦT IJ k ,l ( l0 ,l1 )
η k ,l (l0 , l1 ) El∗0 (DT4 Ȧ) El1 (DT4 Ȧ)
l0 ,l1 = 0
=
D −1
¦ ¦e
{
jȦT IJ k ,l ( l0 ,l1 ) − D4k
l0 ,l1 = 0 k∈ψ
}η (l , l )hT UT ī(k )U h . k ,l 0 1 1 l0 l1 1
(33)
Equation (33) can be rewritten as
¦
r∈φk ,l
T
e jȦ r h1T Ĭ k ,l (r )h1 =
1 δ (k − l ) , D1
(34)
where
°r = ηk ,l (l0 , l1 ) IJ k ,l (l0 , l1 ) − D4k , °½ ¾, °¯l0 , l1 = 0,", D − 1, k ∈ψ °¿
φk , l = ® Ĭ k ,l (r ) =
(35)
D −1
¦ ηk ,l (l0 , l1 )UTl ī(D4−1 ( IJ k ,l (l0 , l1 ) − r))Ul . 0
l0 ,l1 = 0
1
From (34), the spatial domain PR condition for the filter bank can be presented as follows. Proposition 3.1: The 2D DFT modulated filter bank with the modulation matrix D1 and decimation matrix D 2 is PR if and only if the coefficients vector h1 satisfies h1T Ĭ k ,l (r )h1 =
1 δ ( k − l )δ ( rx )δ ( ry ), D1
(36)
r = ( rx , ry ) ∈ φk ,l ,0 ≤ k , l ≤ D 2 − 1. T
Up to now, the PR condition in spatial domain has been derived, which is a set of finite quadratic equations with respect to the coefficients vector of the PF. This condition will be used in the design of the filter banks in the next subsection. 3.2 Design Algorithm for 2D DFT modulated filter banks Generally, design of 2D DFT modulated filter banks involves several performance 13
measures, including reconstruction error, stopband attenuation, and passband flatness. Since linear phase is an important property in several applications [22], the PF is assumed to be linear phase. Without loss of generality, the PF is assumed to be symmetric with (0, 0)T , i.e., the PF satisfies h(n) = h(−n), nx ,n y = − L,",L . Its frequency response can be represented using two-variable real-valued cosine polynomial. L
L
H (Ȧ) = h(0,0) + 2 ¦ h(0, n y ) cos([0, n y ]Ȧ ) + 2 ¦ n y =1
L
¦
h(n ) cos(ȦT n)
nx =1 n y =− L
(37)
T
= c (Ȧ , L ) x ,
where
x = [h(0,0),2h(0,1),", 2h(0, L),",2h( L, L)]T , c(Ȧ, L) = [1,cos([0,1]Ȧ)",cos([0, L]Ȧ),",cos([ L, L]Ȧ)]T .
(38a) (38b)
Thus h1 = Bx , where
0.5, ° °0.5, B[m, n] = ® °1, °0, ¯
m + n = 2 L2 + 2 L, m < 2 L2 + 2 L, m − n = 2 L2 + 2 L, m > 2 L2 + 2 L, m = 2 L2 + 2 L, n = 0, otherwise.
(39)
The stopband attenuation of each PF can be controlled by its stopband energy, a convex quadratic function with respect to x
S e ( x) = ³
2
Ȧ∈ȍ S
H (Ȧ) dȦ = xT
{³
Ȧ∈ȍ S
}
c(Ȧ, L)cT (Ȧ, L) dȦx = xT Q s x .
(40)
The passband flatness of the 2D linear phase PF can be controlled by
Pf ( x) = ³
Ȧ∈ȍ p
Qp = ³
Ȧ∈ȍ p
H (Ȧ) −
2
D2 dȦ = xT Q p x − 2bTL x + d ,
c(Ȧ, L)cT (Ȧ, L)dȦ, b L =
D2
³
Ȧ∈ȍ p
c(Ȧ, L)dȦ, d = 4π 2 D2 / D1 .
(41)
On the other hand, the PR condition in (36) can be expressed as a set of quadratic equations with respect to x as 14
xT BT Ĭ k ,l (ri )Bx =
1 δ (k − l )δ (ri0 )δ (ri1 ), J1
(42)
ri = (ri , r ) ∈ φ (k , l ),0 ≤ k , l ≤ J 2 − 1. 0
1 T i
The quadratic equations (42) can be written into compact form A ( x) x = b ,
(43)
where matrix A ( x) and vector b can be easily found from (42) and efficiently calculated by exploiting the high sparsity of matrices U k , ī (k ) , similar to that of 1D case [23]. Note that the elements of A(x) depend linearly on x . Similar to that in [14], good overall performance can be assured by small reconstruction error and high stopband attenuation. Accordingly, the design problem is formulated into the following unconstrained optimization problem
min x
{Φ(x)= A(x)x − b
2 2
}
+ α xT Q s x ,
(44)
where α is a weighted factor to achieve compromise between the two terms in objective function. Before discussing the design algorithm, one property of the objective function is required. Property 3.2: The gradient vector of the objective function in (44) is ∇Φ (x) = 4 AT (x)( A (x)x − b) + 2α Q s x .
(45)
The proof is similar to that in [14] and is neglected. By setting the gradient vector (45) to zero, we have x = ( AT (x) A(x) + 0.5α Q s ) −1 AT (x)b .
(46)
It is difficult to solve x directly from (46) since matrix A ( x) is a also function of x . However, (46) suggests an iterative algorithm similar to that in [14, 21].
15
Algorithm 3.3: Iterative Algorithm for 2D DFT modulated filter banks
Step 1)
Design an initial lowpass filter x 0 by the optimization problem x 0 = arg min{Φ1 (x)}, Φ1 (x) =
{³³
x
2
Ȧ∈ȍ s
cT (Ȧ, L)x dȦ +β ³³
Ȧ∈ȍ p
cT (Ȧ, L)x − 2 D2
2
}
(47)
dȦ .
where β is weighted factor to achieve compromise between the two terms. Step 2)
Given x 0 , solve x with x = ( AT (x0 ) A(x0 ) + 0.5α Q s )−1 ( AT (x0 )b) .
Step 3)
If x − x0 2 < η ( η is a predefined tolerance, in numerical examples η = 10 −3 ) or the number of iterations exceeds the bound C (a mass set of examples show that C=20 is sufficient), terminate the iteration and output x as the final solutions. Otherwise, set x0 = 0.5(x0 + x) and go to step 2.
In this algorithm, Q s
and b are only required to be computed once. Also,
AT (x0 ) A(x0 ) + 0.5α Q s is a symmetric and positive-definite matrix, x can be obtained by a
computationally efficient method, for instance the Cholesky decomposition. Similar to that in [14], it can be easily shown that the proposed algorithm is one type of modified Newton’s method and the gradient vector satisfies
lim ∇Φ(xk ) = 0 .
k →∞
(48)
Formula (48) implies that the solution sequence generated by Algorithm 3.3 is convergent to a stationary point of the objective function [24].
4 NUMERICAL EXAMPLES In this section, Algorithm 3.3 is utilized to design the 2D DFT modulated filter banks. Generally, the performance of the designed filter banks can be measured by the following 16
items [11]. (1) The transfer function distortion ε t = (2) The aliasing distortion ε a =
max
max
Ȧ∈SPD (π D1− T )
Ȧ∈SPD (π D1− T ), k =1,", D2 −1
{ T (Ȧ) − 1} . 0
{ T (Ȧ) } . k
(3) The reconstruction error ε r , which indicates the mean square error between the input and output signals, in which the input signal is a Gaussian white noise with unit-variance. (4) The perfect-reconstruction distortion (PRD), defined by A(x)x − b 2 . (5) The stopband attenuation (SA) of the PFs, which refers to the maximal amplitude of the normalized frequency response of the PF in the stopband region. Example 1: In this example, a 2D DFT modulated filter bank was designed satisfying the
same specifications as Example 1 in [11]. The specifications and parameters are described as follows: D1 = 4I 2 , D1 = 2I 2 , L = 8, α = 5 × 10 −6 , β = 100 .
It took 8 iterations for the proposed algorithm to converge. The impulse and magnitude responses of the resultant PF are shown in Fig.3 (a) and (b), respectively. Table 1 lists the design results of the bi-iterative second-order cone program (BI-SOCP) method in [11] and proposed algorithm for comparison, where there are two values in SA column for the BI-SOCP method due to the double-prototype. From Table 1, it is observed that the filter bank designed by the BI-SOCP method achieves better overall performance than that by the proposed one owing to the fact that the double-prototype filter bank in [11] possesses double degrees of design freedom as that in proposed method. In optimization problem (43), α serves to achieve trade-off between the reconstruction error and stopband attenuation. In order to verify this viewpoint, the filter bank is redesigned 17
by the proposed algorithm with α = 0.1 . The performance measures of the generated filter bank are illustrated in Table I. The impulse and magnitude responses of the obtained PF are plotted in Fig.3 (c) and (d), respectively. As can be seen from Table 1, the reconstruction error gets worse with increased α , while stopband attenuation is improved. Generally, parameter α can be adjusted according to desired trade-off between the reconstruction property and
stopband characteristic. Besides, parameter β serves to seek compromise between the stopband attenuation and passband flatness. Although there is no rigorous rule to determine the parameter, quite a number of examples suggest that β ∈ [10,1000] is appropriate range. Table 1 Performance comparison in Example 1 Methods BI-SOCP Method [11] Proposed method ( α = 5 × 10 ) −6
Proposed method ( α = 0.1 )
ε t (dB)
ε a (dB)
ε r (dB)
PRD
SA (dB)
-94.67
-94.89
-93.80
2.53 × 10−6
-38.83,-35.19
-68.65
-69.35
-71.31
3.44 × 10−5
-36.94
-56.34
−4
-58.98
-48.73
-60.82
1.91 × 10
Fig.3. Example 1: (a) Impulse response and (b) Magnitude response of the prototype filter designed by the proposed algorithm with α = 5 × 10−6 , (c) Impulse response and (d) Magnitude response of the prototype filter designed with α = 0.1 . 18
Example 2: In this example, a 2D DFT modulated filter bank was designed satisfying the
same specifications as Example 2 in [13]. The specifications and parameters are as follows: ª 4 −4 º ª 2 −2 º −5 D1 = « » , D2 = « 2 2 » , L = 15 , α = 1 × 10 , β = 100 . 4 4 ¬ ¼ ¬ ¼
It took 11 iterations for the proposed algorithm to converge. The magnitude responses of the PF and analysis filters are illustrated in Fig.4. The design results of the modified Newton’s method [13] and proposed algorithm are contrasted in Table 2. It is observed that compared with the method [13], the proposed algorithm can lead to filter bank with much smaller reconstruction error at a slightly degradation of stopband attenuation. The reason is straightforward, reconstruction error is directly minimized in the proposed algorithm while that in [13] is restricted by the attainable stopband attenuation. Table 2 Performance comparison in Example 2 Methods
ε t (dB)
Newton’s method in [13]
-87.27
Proposed method
-94.96
εa
(dB)
-82.30 -88.89
ε r (dB)
PRD
-84.22
5.37 ×10−6
-68.02
-96.46
−6
-67.27
1.34 ×10
SA (dB)
Fig.4. Example 2: (a) Magnitude response of the prototype filter designed by the proposed algorithm. (b) Magnitude responses of the analysis filters.
19
Note that in above two examples, the modulation matrix D1 is integral multiple of the decimation matrix D2 . Accordingly, matrix D3 is an identity matrix and D4 is multiple of the identity matrix, which indicates that the PR constraints are relatively small. If D1 is not multiple of D2 , the PR condition becomes complicate. In this case, the proposed algorithm can also work well, as shown in the following examples. Table 3 Performance comparison in Example 3 Methods
ε t (dB)
BI-SOCP method [11]
-55.20
Newton’s Method [13]
-44.76
Proposed method
-56.60
εa
ε r (dB)
PRD
SA (dB)
-55.27
-53.98
4.43 × 10 −4
-20.93,-26.96
-38.19
-39.04
9.83 × 10−2
-25.79
-59.29
−4
-24.52
(dB)
-60.55
2.44 × 10
Fig.5. Example 3: (a) Impulse response and (b) Magnitude response of the prototype filter designed by the proposed algorithm.
Example 3: This example considers a 2D DFT modulated filter bank satisfying the
specifications as the same as Example 3 in [13]. The specifications and parameters are described as follows D1 = 3I 2 , D2 = 2I 2 , L = 7 , α = 1× 10−5 , β = 100 . 20
The filter bank is with low redundancy ratio 9 / 4 . It took 20 iterations for the algorithm to terminate. The impulse and magnitude responses of the generated PFs are depicted in Fig.5. Table 3 presents the design results of the BI-SOCP method in [11], the modified Newton’s method in [13] and proposed algorithm for contrast. From Table 3, it is observed that the filter bank generated by the proposed algorithm possesses remarkable smaller reconstruction error than the Newton’s method. The reason is similar to that of Example 2. Therefore, when requiring 2D single-prototype DMFBs with low redundancy ratio, the proposed algorithm is a better choice compared with the modified Newton’s method. Besides, the proposed algorithm also achieves smaller reconstruction error than the BI-SOCP method while maintains comparable stopband attenuation.
Fig.6. Example 4: (a) Impulse response and (b) Magnitude response of the prototype filter designed by
the proposed algorithm.
Example 4: In this example, we consider a 2D DFT modulated filter bank whose
specifications and parameters are presented as follows ª 2 −2 º −5 D1 = 5I 2 , D2 = « » , L = 12 , α = 1×10 , β = 100 . 2 2 ¬ ¼
It took 12 iterations for the algorithm to converge. The impulse and magnitude responses of 21
the generated PFs are depicted in Fig. 6. The filter bank achieves performance: transfer function distortion -55.88dB, aliasing distortion -61.97dB, reconstruction error -63.03dB, perfect-reconstruction distortion 8.01 × 10−5 , and stopband attenuation -24.42dB. It is observed that the filter bank is with satisfactory overall performance.
5 CONCLUSION The main contribution of this paper is deriving the necessary and sufficient PR condition for 2D DFT modulated filter banks with arbitrary modulation and decimation matrices. Meanwhile, as another important result, we have derived the spatial domain condition of PR, which facilitates us to propose an efficient iterative algorithm. Several numerical examples have verified the validity of the PR condition and also demonstrated that the proposed algorithm can yield filter banks with good overall performance. Appendix A: Proof of Lemma 2.1 With D = D1D3 , we have
D−1 (D1p n + ul ) = D3−1 (p n + D1−1ul ) = D3−1 (p n + xl ) .
(A1)
Since xl ∈ [0,1) 2 and p n ∈ N (D3 ) , we obtain that p n + xl locates in FPD(D3 ) [1], which implies that D3−1 (p n + xl ) ∈ [0,1) 2 . Therefore, integer vector D1p n + ul belongs to vector set N (D) for n = 0," , D3 − 1, l = 0," , D2 − 1 . Furthermore, in terms of the Division theorem for
integer vectors, vectors D1p n + ul , n = 0,", D3 − 1, l = 0," , D2 − 1 diverse from each other if D1p n ∈ LAT (D1 )
and
.
ul ∈ N (D1 )
In
summary,
vectors
D1p n + ul , n = 0,", D3 − 1, l = 0," , D2 − 1 constitute the set N (D) . Up to now, the proof of (6a)
is completed. The proof of (6b) can be handled in the same way. 22
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24
Short Authors’ Biographies and Key Words (used for publication) Dear Editor, First of all, thank you very much for processing our accepted manuscript entitled by “Theory and Design of Two-dimensional DFT Modulated Filter Bank with Arbitrary Modulation and Decimation Matrices” with ID YDSPR 1781. According to the request, we provide the short biographies of four authors and add key words.
1. Short Authors’ Biographies Jun-Zheng Jiang was born in Zhejiang, China, in 1983. He received the B.S. degree in mathematics from Guilin University of Electronic and Technology, Guilin, China, in 2005 and Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 2011. He is now an associate professor of Guilin University of Electronic and Technology. His research interests include filter bank design and graph signal processing.
Fang Zhou received the B.S. degree and the M.S. degree from Xidian University, Xi’an, China, in 2006 and 2019, respectively, both in electrical engineering. She is currently working toward the Ph.D. degree in signal and information processing in the National Laboratory of Radar Signal Processing of Xidian Univeristy, Xi’an, China. Her research interests include two-dimensional filter bank and nonuniform filter bank design.
Peng-Lang Shui (M’02) received the M.S. degree in mathematics from Nanjing University, China, and the Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 1992 and 1999, respectively. He is now a Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include wavelets and filter bank design, image processing and radar target detection.
Shan Ouyang received the B.S. degree from Guilin University of Electronic Technology (GUET), Guilin, China, in 1986 and the M.S. and Ph.D. degrees from Xidian University, Xi’an, China, in 1992 and 2000, respectively, all in electronic engineering. He is currently a Professor with the School of Information and Communication, GUET. His research interests are mainly in the areas of signal processing for communications and radar, adaptive filtering, and neural network learning theory and applications.
2. Key Words (new) Fully oversampled, Nonlinear optimization, Two-dimensional DFT modulated filter bank.
Perfect-reconstruction
(PR),
PII: DOI: Reference:
S1051-2004(15)00184-0 http://dx.doi.org/10.1016/j.dsp.2015.05.012 YDSPR 1781
To appear in:
Digital Signal Processing
Please cite this article in press as: J.-Z. Jiang et al., Theory and design of two-dimensional DFT modulated filter bank with arbitrary modulation and decimation matrices, Digital Signal Process. (2015), http://dx.doi.org/10.1016/j.dsp.2015.05.012
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Highlights
• The PR condition is derived for the 2D general DFT modulated filter banks. • Spatial domain condition of PR is derived to facility filter bank design. • 2D NPR DFT modulated filter banks with good overall performance are designed.
Theory and Design of Two-dimensional DFT Modulated Filter Bank with Arbitrary Modulation and Decimation Matrices Jun-Zheng Jiang, Fang Zhou∗, Peng-Lang Shui, and Shan Ouyang Abstract: It is well-known that two-dimensional (2D) filter bank is far removed from a straightforward extension of one-dimensional (1D) filter bank. There are many challenging problems on the theory and design methods for the 2D filter bank. Among these problems, the perfect-reconstruction (PR) theory of the 2D DFT modulated filter bank with arbitrary modulation and decimation matrices remains an unsolved difficulty, which is the focus of this paper. The necessary and sufficient condition for perfect reconstruction (PR) is derived by using the polyphase decomposition of the analysis and synthesis filters, as well as the fast implementation structure of the filter bank. Then, the PR condition in frequency domain is transformed into a set of quadratic equations with respect to the prototype filter (PF), which is utilized to formulate the design problem into an unconstrained optimization problem. An efficient iterative algorithm is proposed to solve the problem. Numerical examples are included to verify the validity of the PR condition and the effectiveness of the design method. Index Terms: Nonlinear optimization, Perfect-reconstruction (PR), Two-dimensional DFT modulated filter bank.
Manuscript received September 3, 2014. This work was supported by the National Natural Science Foundation of China under grant 61261032 and Guangxi Natural Science Foundation under grants 2013GXNSFBA019264, 2013GXNSFFA019004. Jun-Zheng Jiang and Shan Ouyang are with the School of Information and Communication, Guilin University of Electronic Technology, Guilin, China.(e-mail: [email protected], [email protected], Telphone: 86-773-2290203, Fax: 86-773-2290203). Fang Zhou (Corresponding author) and Peng-Lang Shui are with the National Lab. of Radar Signal Processing, Xidian University, Xi’an, China (email: [email protected], [email protected], Telphone: 86-29-88201022-8611, Fax: 86-29-88236159). 1
1. INTRODUCTION Among general two-dimensional (2D) filter banks, 2D modulated filter banks are of particular interest due to their ease of design and implementation. (see [1] for fundamentals on 2D filter bank). Furthermore, 2D modulated filter banks can provide 2D fine frequency tiling and good directional selectivity. Generally, 2D modulated filter banks can be categorized into two classes: 2D cosine modulated filter banks (CMFBs) [2-9] and 2D DFT modulated filter banks (DMFBs) [10-15]. During the past two decades, the attention has to a large extent been paid to the theory and design of 2D CMFBs. The support permissibility and perfect reconstruction (PR) condition of the 2D two-parallelogram CMFBs were investigated in [2]. Nevertheless, the 2D two-parallelogram paraunitary CMFBs cannot provide high stopband attenuation due to lack of vertex permissibility. Numerically efficient algorithm was proposed for the design of the two-parallelogram CMFBs [8], which improves the stopband attenuation by relaxing the PR property. In [3], theory and design of 2D four-parallelogram CMFBs were exploited. In [6], 2D PR modulated filter banks with triangular supports were designed from a 2D complex prototype filter (PF) whose passband support is a triangle that is just half of a parallelepiped-shaped passband support defined by the sampling matrix. 2D oversampled PR CMFBs were considered in [7]. On the other hand, there has been increasing interest in the 2D DMFBs. 2D critically sampled DMFBs were preliminarily constructed from one-dimensional (1D) filters [1]. However, 2D critically sampled DMFBs with PR cannot behave high stopband attenuation due to the non-permissible configuration [2]. In [10], 2D oversampled separable DMFBs were
2
constructed by tensors of 1D DMFBs and provided good directional selectivity. However, for 2D separable filter banks, the degrees of design freedom are heavily restricted and the frequency tiling is limited to be rectangular. Compared with separable counterparts, 2D non-separable filter banks offer more flexibility and usually provide better overall performance [5]. Therefore, subsequent research mainly concentrates on 2D non-separable DMFBs. In [11], the permissibility of the 2D DMFBs was analyzed from a pictorial viewpoint. It was shown that the 2D DMFBs can have good PR property and high stopband attenuation only if the filter banks are fully oversampled, a necessary condition for the filter bank to possess good overall performance. Also, the bi-iterative second-order cone program (BI-SOCP) was utilized to design the filter banks with nearly PR (NPR) therein. In [13], the modified Newton’s method was presented to design the 2D DMFBs with single-prototype. The necessary and sufficient condition was derived for the 2D 2× oversampled DMFBs and 2D critically sampled modified DFTDFBs (MDMFBs) to be PR [12]. The equivalence between these two types of filter banks was established in the single-prototype case. A lifting-based method was proposed for designing the 2D 2× oversampled DMFBs with structurally PR and a numerical algorithm was presented to design the 2D critically sampled MDFTDFBs with NPR. In [14], the 2D critically-sampled MDFTDFBs were redesigned by an iterative algorithm, which suppressed the numerical algorithm both in performance and computational efficiency. However, the necessary and sufficient PR condition remains an unsolved problem for the 2D DMFBs with arbitrary modulation and decimation matrices, which heavily restricts the development of the filter bank. In this paper, we investigate the theory and design of the 2D DMFBs with arbitrary 3
modulation and decimation matrices. A necessary and sufficient condition for the PR of the filter banks is derived from the polyphase decomposition of the analysis and synthesis filters. Also, the fast implementation structure of the filter banks is constructed. To the best of our knowledge, the PR condition and fast structure have not been derived for the 2D DMFBs with arbitrary modulation and decimation matrices before. Then, the PR condition in frequency domain is converted into a set of quadratic equations with the PF (termed as spatial domain condition), which leads us to formulate the design of the filter banks into an unconstrained optimization problem whose objective function consists of the PR condition and frequency selectivity of the PF. By setting the gradient vector of the objective function to be zero, the PF is iteratively optimized with closed-form formula. The validity of the PR condition and the effectiveness of the design algorithm are verified by the numerical examples. This paper is organized as follows. In Section II, the general structure of 2D DMFBs is reviewed, the PR condition is derived and fast implementation structure is constructed. In Section ċ, the PR condition is transformed into quadratic equations with the PFs and the design algorithm is presented. In Section IV, several numerical examples are given. Finally, conclusion is presented in Section V. Notations: Most of the notations in this paper are the same as that in [1]. Some frequently used ones are recalled below: a) Boldfaced letters denote matrices or vectors. Superscript T , ∗ and † denote transpose, conjugate, and transpose-conjugate, respectively. Symbol «¬ x »¼ denotes the integer part of x . Matrix I K denotes K × K identity matrix.
4
b) The Fourier transform of 2D signal x(n) is denoted by X (Ȧ) with Ȧ = [ωx , ω y ]T . c) For a 2×2 nonsingular integer matrix
M , its coset N (M ) is defined by
N (M ) = {m k = Mx k , x k ∈ [0,1) 2 , k = 0,1," , M − 1} . The number of vectors in set N (M ) is equal to
M , which is the absolute value of the determinant of matrix M . The parallelepiped FPD(M )
is
defined
FPD(M ) = {Mx, x ∈ [0,1) 2 }
by
.
Region
SPD(π D−T ) ≡ {Ȧ ∈ R 2 : −π ≤ ȦT d1 , ȦT d 2 < π } for D = [d1 , d 2 ] .
2 TWO-DIMENSIONAL DFT MODULATED FILTER BANKS 2.1 Preliminary of 2D DFT modulated filter banks
As 1D DFT modulated filter banks [16, 17], 2D DMFBs are generated by one or two 2D lowpass PFs. Let the modulation matrix D1 be a 2 × 2 nonsingular integer matrix. And denote H (Ȧ) and G (Ȧ) as the lowpass analysis and synthesis PFs, respectively. All the analysis and synthesis filters H i (Ȧ ) and Gi (Ȧ) are DFT modulated versions of the PFs as H i (Ȧ ) = H (Ȧ − 2π D1−T u i ), u i ∈ N ( D1T ), Gi (Ȧ ) = G (Ȧ − 2π D1−T u i ), i = 0,1," , D1 − 1.
(1)
Fig.1. Structure of 2D DFT modulated filter bank
Assume D 2 as another 2 × 2 nonsingular integer matrix. Then, a 2D DFT modulated filter bank with the modulation matrix D1 and decimation matrix D2 is shown in Fig.1, where 5
X (Ȧ), Yi (Ȧ) and X (Ȧ) are the 2D Fourier transforms of the input signal, subband signals, and
output signal, respectively. The Fourier transforms of subband signals and the output signal are written as [11] Yi (Ȧ) =
1 D2
D2 −1
¦ H (D i
−T 2
(Ȧ − 2π v k )) X ( D2−T (Ȧ − 2π v k )) ,
(2)
k =0
X (Ȧ) = T0 (Ȧ) X (Ȧ) +
D2 −1
¦ T (Ȧ) X (Ȧ − 2π D k
−T 2
v k ), v k ∈ N ( DT2 ) ,
(3)
k =1
where Tk (Ȧ) =
1 D2
D1 −1
¦ H (Ȧ − 2π D i
−T 2
v k )Gi (Ȧ), k = 0," , D2 − 1 with v 0 = [0, 0]T . In addition, the
i =0
stopband, transition-band, and passband of 2D DMFBs are specified as ȍ s = {Ȧ ∈ [−π , π ) 2 and Ȧ ∉ SPD (π D−2 T )}, ȍt = {Ȧ ∈ SPD(π D−2 T ) and Ȧ ∉ SPD(π D1−T )},
(4)
ȍ p = SPD(π D ). −T 1
When ȍ p ⊂ SPD(π D−2 T ) , the filter bank is oversampled, in this case D1 > D2 . Further, if ȍ p ⊂ SPD(π D−2 T ) and there is no intersection between ȍ p and ȍ s , the filter bank is fully
oversampled. It is shown in [11] that only 2D fully oversampled DMFBs are permissible, that is, the filter banks can simultaneously possess small reconstruction error and high stopband attenuation. Nevertheless, the permissibility is just the necessary condition for the filter banks to possess good overall performance. In what follows, we derive the necessary and sufficient condition for the PR of the filter bank using the polyphase decomposition. 2.2 Perfect Reconstruction Condition
In this subsection, we derive a necessary and sufficient condition for the PR of the 2D DMFBs with arbitrary modulation and decimation matrices. Assume D as the least common right multiple of the modulation matrix D1 and decimation matrix D 2 [18], i.e., D = D1D3 = D 2 D 4 , 6
(5)
where D3 , D4 are nonsingular integer matrices. With (5), the set N ( D) can be specified as follows. Lemma 2.1: For matrix D = D1D3 = D2 D4 , the set N ( D) can also be defined as N (D) = {D1p n + u l | u l ∈ N (D1 ), p n ∈ N ( D3 )} ,
(6a)
N (D) = {D2 q s + v m | v m ∈ N (D2 ), q s ∈ N ( D4 )} .
(6b)
The proof is given in Appendix A. In terms of (6a), vectors m k ∈ N (D), k = 0,", D − 1 can be arranged as « k » m k = D1p n + u l , n = « » , l = k − n ⋅ D1 , u l ∈ N ( D1 ), p n ∈ N ( D3 ) . «¬ D1 »¼
(7)
The polyphase decomposition of the analysis and synthesis PFs are given by [1]
H (Ȧ) =
¦
T
e− jȦ mk Ek (DT Ȧ) , G (Ȧ) =
mk ∈N ( D )
¦
T
e jȦ mk Rk (DT Ȧ) ,
(8)
mk ∈N ( D )
where Ek (Ȧ ) and Rk (Ȧ ) , k = 0,", D − 1 are the type-I polyphase components of the analysis PF and the type-III polyphase components of the synthesis PF, respectively. With (1) and (8), the analysis and synthesis filters can be expressed with respect to Ek (Ȧ) and Rk (Ȧ ) as follows
H i (Ȧ) =
¦
e− jȦ
¦
e jȦ
T
mk
T
−1
e j 2π u i D1 mk Ek (DT Ȧ),
m k ∈N ( D )
Gi (Ȧ) =
T
mk
T
(9)
−1
e− j 2π u i D1 mk Rk ( DT Ȧ).
m k ∈N ( D )
As a result, the analysis filter banks can be written as
{
}
[ H 0 (Ȧ), H1 (Ȧ),", H D1 −1 (Ȧ)]T = W1diag E0 (DT Ȧ), E1 (DT Ȧ),", E D −1 (DT Ȧ) e(Ȧ) , where diag denotes diagonal matrix with entries Ek ( DT Ȧ ) and
7
(10)
ª j 2π u T0 D1−1m0 « e « j 2π u T D−1m 0 1 1 W1 = « e « # « T « j 2π u D −1D1−1m0 1 «¬ e
e e
j 2π u T0 D1−1m1 j 2π u 1T D1−1m1
# e
D−1m D1 −1 1 1
j 2π u T
º » j 2π u 1T D1−1m D −1 » " e », » % # » j 2π u T D1−1m D −1 » D1 −1 »¼ " e "
e
j 2π u T0 D1−1m D −1
(11a)
ª e− jȦT m0 º « » « e − jȦT m1 » ». e(Ȧ) = « « » # « » T «e− jȦ m D −1 » ¬ ¼
(11b)
Equation (12.4.37b) from reference [1] presents a property of 2D DFT, i.e.,
¦
m∈N ( MT )
e − j 2π m
T
M −1k
° M , k = 0 . Based on this property and the arrangement in (7), =® °¯0, k ∈ N (M ), k ≠ 0
D1 × D matrix W1 can be factored as
W1 = W∗ ⋅ [I D1 ,", I D1 ] = W∗ ⋅ Mb ,
(12)
where W∗ is the conjugate of the generalized DFT matrix W with WT W ∗ = D1 I D
1
The number of I D
1
[1].
in the D1 × D matrix M b is D3 . From Lemma 2.1 and (7), vector
e(Ȧ) can be expressed as
e(Ȧ) = F(DT2 Ȧ)f (Ȧ) ,
(13)
where T −1 °exp( − jȦ D 2 (m k − v l )),(m k − v l ) mod D 2 = 0,0 ≤ k ≤ D − 1,0 ≤ l ≤ D 2 − 1, (14a) [F (Ȧ )]k ,l = ® °¯0,otherwise.
f (Ȧ) = [exp(− jȦT v0 ),",exp(− jȦT v D2 −1 )]T .
(14b)
In summary, the analysis filter banks can be expressed as
[ H 0 (Ȧ), H1 (Ȧ),", H D1 −1 (Ȧ)]T = W∗E(DT2 Ȧ)f (Ȧ) ,
(15)
where
{
}
E(Ȧ) = I b diag E0 (DT4 Ȧ), E1 (DT4 Ȧ),", E D −1 (DT4 Ȧ) F(Ȧ) . 8
(16)
Similarly, the polyphase decomposition of the synthesis filter banks is
[G0 (Ȧ), G1 (Ȧ),", G D1 −1 (Ȧ)] = f † (Ȧ)R(DT2 Ȧ)WT ,
(17)
where D2 × D1 matrix R (Ȧ) is defined by
{
}
R(Ȧ) = F† (Ȧ)diag R0 (DT4 Ȧ), R1 (DT4 Ȧ),", R D −1 (DT4 Ȧ) MTb .
(18)
In terms of the polyphase decompositions in (15) and (17), the PR condition for two-dimensional DFT modulated filter banks are given in the following theorem. Theorem 2.2 (PR Condition): A two-dimensional DFT modulated filter bank with the modulation matrix D1 and decimation matrix D 2 is PR if and only if R (Ȧ)E(Ȧ) =
1 I . D1 D2
(19)
If synthesis PF satisfies g (−n) = h(n), ∀n ∈ Z 2 , the analysis and synthesis polyphase components are related by Rk (Ȧ) = Ek∗ (Ȧ), k = 0," , D − 1 , which results in
R (Ȧ) = E∗ (Ȧ) .
(20)
In this case, the PR condition is taken as follows. Corollary 2.3: 2D single-prototype DFT modulated filter banks with the modulation matrix D1 and decimation matrix D 2 are PR if and only if the PF satisfy
h(n) = h( −n) E∗ (Ȧ)E(Ȧ) =
1 . I D2 D1
(21)
ª1 −1º
Particularly, for the 2D MDMFBs [12], where D = D1 , D3 = I 2 , D4 = « » . In this case, the ¬1 1 ¼ polyphase matrix E(Ȧ) reduces to ª I D2 E(Ȧ) = diag E0 (DT4 Ȧ), E1 ( DT4 Ȧ)," , E D −1 ( DT4 Ȧ) « − jωx «¬ e I D2
{
}
º ». »¼
(22)
Combining (22) and symmetric PF with h(n) = h(−n) [12], the PR condition of 2D 9
MDMFBs can be stated as follows. Corollary 2.4: 2D modified DFT modulated filter banks with decimation matrix D 2 is PR if and only if the polyphase components of the PF satisfy
h(n) = h( −n) Ek∗ (Ȧ) Ek (Ȧ) + Ek∗+ D2 (Ȧ) Ek + D2 (Ȧ) =
1 . , k = 0,", D2 − 1 D1
(23)
This Corollary is consistent with Theorem 6 in [12]. 2.3 Fast Implementation One of the key features of modulated filter bank is that they can be implemented with very efficient structure, which makes them very popular in practical applications. For instance, 1D 2M-channel DFT modulated filter bank with the decimation factor M allows the fast decomposition and reconstruction of signals via the M-point DFT and inverse DFT [19]. In terms of the polyphase representations in (19), 2D DMFBs also allow fast signal decomposition and reconstruction via the 2D D1 -point DFT and inverse DFT. The flowchart of fast decomposition and reconstruction is shown in Fig.2.
Fig.2. Polyphase structure of two-dimensional DFT modulated filter bank
10
3. DESIGN OF TWO-DIMENSIONAL DFT MODULATED FILTER BANKS As we know, design of 2D filter banks is a highly challenging work, even when it is modulated one. In this section, we present an iterative approach to design 2D DFT modulated filter banks with arbitrary modulation and decimation matrices (under fundamental restriction shown in Section 2.1). The good overall performance of the designed filter bank can be assured by careful formulation of the design problem and efficient algorithm. In view of this point, a spatial domain PR condition is derived for the 2D DMFBs. Based on the condition, an efficient iterative algorithm is proposed to design the filter banks. In this paper, we concentrate on the design of 2D single-prototype DMFBs. 3.1 PR condition in spatial domain With the frequency domain condition (19), PR filter bank can be designed by means of some parametric structures of the polyphase matrices E(ω ) , for instance, the lattice structure [20]. However, due to the highly nonlinear relationship between the performance of the filter bank and the structure parameters, it is difficult to design filter bank with fine performance. Similar to the design of 1D CMFBs [21], the PR condition in (21) is formulized into quadratic equations with respect to the analysis PF, termed as the spatial domain PR condition, shown as follows. The PR condition in (21) contains D2
2
equations with respect to polyphase components.
With the definitions of matrix F(Ȧ) and block matrix I b , the PR equations can also be expressed as D −1
¦
l0 ,l1 = 0
e
jȦT IJ k ,l ( l0 ,l1 )
η k ,l (l0 , l1 ) El∗0 ( DT4 Ȧ ) El1 ( DT4 Ȧ) = 11
1 δ ( k − l ),0 ≤ k , l ≤ D 2 − 1 , D1
(24)
where IJ k ,l (l0 , l1 ) = D2−1 (m l − m l − v k + v l ) , δ (⋅) is the delta function, and η k ,l (l0 , l1 ) is defined 0
1
as 1,(m l0 − v k ) mod D2 = 0,(m l1 − v l ) mod D2 = 0, ° η k ,l (l0 , l1 ) = ® and (l0 − l1 ) mod D1 = 0, °0,otherwise. ¯
(25)
In what follows, equation (24) is transformed into quadratic equations with the PFs. Let h = {h(n) | n = (nx , ny ), − L ≤ nx , ny ≤ L} be the coefficients matrix of the PF, which is then
stacked into column vector as
h1 = [h(− L, − L),", h(− L, L),", h( L, L)]T .
(26)
For convenience, a set of integer vectors is defined as S = {s = D−1 (n − m k ), n ∈ [− L, L]2 , m k ∈ N (D),(n − m k ) mod D = 0} ,
(27)
where (n − m k ) mod D = 0 means D −1 (n − m k ) = [0, 0]T . For convenience, denote K as the number of vectors in set S . The polyphase component Ek (Ȧ ) can be represented by
Ek (Ȧ) = eTp (Ȧ)U k h1 , k = 0,", D1 − 1 ,
(28)
where T
e p (Ȧ) = [e − jȦ s0 ," , e − jȦ
T
s K −1 T
] ,
°1, t − Dsi = m k , i = 0," , K − 1, j = 0," , 4 L2 + 4 L, [U k ]i , j = ® j °¯0, otherwise. t = {[− L, − L]," ,[− L, L]," ,[ L, L]}.
(29)
(30)
The multiple of vectors e∗p (Ȧ) and eTp (Ȧ) is e∗p (Ȧ )eTp (Ȧ) = ¦ e − jȦ k ī(k ) , T
k∈ψ
where
12
(31)
ψ = {k = s j − si , si , s j ∈ S , i, j = 0," , K − 1} , (32)
1, s − s = k ,0 ≤ k , l ≤ K − 1, [ī(k )]k ,l = ® k l ¯0,otherwise. Substituting (28) and (31) into (24), we obtain D −1
¦
e
jȦT IJ k ,l ( l0 ,l1 )
η k ,l (l0 , l1 ) El∗0 (DT4 Ȧ) El1 (DT4 Ȧ)
l0 ,l1 = 0
=
D −1
¦ ¦e
{
jȦT IJ k ,l ( l0 ,l1 ) − D4k
l0 ,l1 = 0 k∈ψ
}η (l , l )hT UT ī(k )U h . k ,l 0 1 1 l0 l1 1
(33)
Equation (33) can be rewritten as
¦
r∈φk ,l
T
e jȦ r h1T Ĭ k ,l (r )h1 =
1 δ (k − l ) , D1
(34)
where
°r = ηk ,l (l0 , l1 ) IJ k ,l (l0 , l1 ) − D4k , °½ ¾, °¯l0 , l1 = 0,", D − 1, k ∈ψ °¿
φk , l = ® Ĭ k ,l (r ) =
(35)
D −1
¦ ηk ,l (l0 , l1 )UTl ī(D4−1 ( IJ k ,l (l0 , l1 ) − r))Ul . 0
l0 ,l1 = 0
1
From (34), the spatial domain PR condition for the filter bank can be presented as follows. Proposition 3.1: The 2D DFT modulated filter bank with the modulation matrix D1 and decimation matrix D 2 is PR if and only if the coefficients vector h1 satisfies h1T Ĭ k ,l (r )h1 =
1 δ ( k − l )δ ( rx )δ ( ry ), D1
(36)
r = ( rx , ry ) ∈ φk ,l ,0 ≤ k , l ≤ D 2 − 1. T
Up to now, the PR condition in spatial domain has been derived, which is a set of finite quadratic equations with respect to the coefficients vector of the PF. This condition will be used in the design of the filter banks in the next subsection. 3.2 Design Algorithm for 2D DFT modulated filter banks Generally, design of 2D DFT modulated filter banks involves several performance 13
measures, including reconstruction error, stopband attenuation, and passband flatness. Since linear phase is an important property in several applications [22], the PF is assumed to be linear phase. Without loss of generality, the PF is assumed to be symmetric with (0, 0)T , i.e., the PF satisfies h(n) = h(−n), nx ,n y = − L,",L . Its frequency response can be represented using two-variable real-valued cosine polynomial. L
L
H (Ȧ) = h(0,0) + 2 ¦ h(0, n y ) cos([0, n y ]Ȧ ) + 2 ¦ n y =1
L
¦
h(n ) cos(ȦT n)
nx =1 n y =− L
(37)
T
= c (Ȧ , L ) x ,
where
x = [h(0,0),2h(0,1),", 2h(0, L),",2h( L, L)]T , c(Ȧ, L) = [1,cos([0,1]Ȧ)",cos([0, L]Ȧ),",cos([ L, L]Ȧ)]T .
(38a) (38b)
Thus h1 = Bx , where
0.5, ° °0.5, B[m, n] = ® °1, °0, ¯
m + n = 2 L2 + 2 L, m < 2 L2 + 2 L, m − n = 2 L2 + 2 L, m > 2 L2 + 2 L, m = 2 L2 + 2 L, n = 0, otherwise.
(39)
The stopband attenuation of each PF can be controlled by its stopband energy, a convex quadratic function with respect to x
S e ( x) = ³
2
Ȧ∈ȍ S
H (Ȧ) dȦ = xT
{³
Ȧ∈ȍ S
}
c(Ȧ, L)cT (Ȧ, L) dȦx = xT Q s x .
(40)
The passband flatness of the 2D linear phase PF can be controlled by
Pf ( x) = ³
Ȧ∈ȍ p
Qp = ³
Ȧ∈ȍ p
H (Ȧ) −
2
D2 dȦ = xT Q p x − 2bTL x + d ,
c(Ȧ, L)cT (Ȧ, L)dȦ, b L =
D2
³
Ȧ∈ȍ p
c(Ȧ, L)dȦ, d = 4π 2 D2 / D1 .
(41)
On the other hand, the PR condition in (36) can be expressed as a set of quadratic equations with respect to x as 14
xT BT Ĭ k ,l (ri )Bx =
1 δ (k − l )δ (ri0 )δ (ri1 ), J1
(42)
ri = (ri , r ) ∈ φ (k , l ),0 ≤ k , l ≤ J 2 − 1. 0
1 T i
The quadratic equations (42) can be written into compact form A ( x) x = b ,
(43)
where matrix A ( x) and vector b can be easily found from (42) and efficiently calculated by exploiting the high sparsity of matrices U k , ī (k ) , similar to that of 1D case [23]. Note that the elements of A(x) depend linearly on x . Similar to that in [14], good overall performance can be assured by small reconstruction error and high stopband attenuation. Accordingly, the design problem is formulated into the following unconstrained optimization problem
min x
{Φ(x)= A(x)x − b
2 2
}
+ α xT Q s x ,
(44)
where α is a weighted factor to achieve compromise between the two terms in objective function. Before discussing the design algorithm, one property of the objective function is required. Property 3.2: The gradient vector of the objective function in (44) is ∇Φ (x) = 4 AT (x)( A (x)x − b) + 2α Q s x .
(45)
The proof is similar to that in [14] and is neglected. By setting the gradient vector (45) to zero, we have x = ( AT (x) A(x) + 0.5α Q s ) −1 AT (x)b .
(46)
It is difficult to solve x directly from (46) since matrix A ( x) is a also function of x . However, (46) suggests an iterative algorithm similar to that in [14, 21].
15
Algorithm 3.3: Iterative Algorithm for 2D DFT modulated filter banks
Step 1)
Design an initial lowpass filter x 0 by the optimization problem x 0 = arg min{Φ1 (x)}, Φ1 (x) =
{³³
x
2
Ȧ∈ȍ s
cT (Ȧ, L)x dȦ +β ³³
Ȧ∈ȍ p
cT (Ȧ, L)x − 2 D2
2
}
(47)
dȦ .
where β is weighted factor to achieve compromise between the two terms. Step 2)
Given x 0 , solve x with x = ( AT (x0 ) A(x0 ) + 0.5α Q s )−1 ( AT (x0 )b) .
Step 3)
If x − x0 2 < η ( η is a predefined tolerance, in numerical examples η = 10 −3 ) or the number of iterations exceeds the bound C (a mass set of examples show that C=20 is sufficient), terminate the iteration and output x as the final solutions. Otherwise, set x0 = 0.5(x0 + x) and go to step 2.
In this algorithm, Q s
and b are only required to be computed once. Also,
AT (x0 ) A(x0 ) + 0.5α Q s is a symmetric and positive-definite matrix, x can be obtained by a
computationally efficient method, for instance the Cholesky decomposition. Similar to that in [14], it can be easily shown that the proposed algorithm is one type of modified Newton’s method and the gradient vector satisfies
lim ∇Φ(xk ) = 0 .
k →∞
(48)
Formula (48) implies that the solution sequence generated by Algorithm 3.3 is convergent to a stationary point of the objective function [24].
4 NUMERICAL EXAMPLES In this section, Algorithm 3.3 is utilized to design the 2D DFT modulated filter banks. Generally, the performance of the designed filter banks can be measured by the following 16
items [11]. (1) The transfer function distortion ε t = (2) The aliasing distortion ε a =
max
max
Ȧ∈SPD (π D1− T )
Ȧ∈SPD (π D1− T ), k =1,", D2 −1
{ T (Ȧ) − 1} . 0
{ T (Ȧ) } . k
(3) The reconstruction error ε r , which indicates the mean square error between the input and output signals, in which the input signal is a Gaussian white noise with unit-variance. (4) The perfect-reconstruction distortion (PRD), defined by A(x)x − b 2 . (5) The stopband attenuation (SA) of the PFs, which refers to the maximal amplitude of the normalized frequency response of the PF in the stopband region. Example 1: In this example, a 2D DFT modulated filter bank was designed satisfying the
same specifications as Example 1 in [11]. The specifications and parameters are described as follows: D1 = 4I 2 , D1 = 2I 2 , L = 8, α = 5 × 10 −6 , β = 100 .
It took 8 iterations for the proposed algorithm to converge. The impulse and magnitude responses of the resultant PF are shown in Fig.3 (a) and (b), respectively. Table 1 lists the design results of the bi-iterative second-order cone program (BI-SOCP) method in [11] and proposed algorithm for comparison, where there are two values in SA column for the BI-SOCP method due to the double-prototype. From Table 1, it is observed that the filter bank designed by the BI-SOCP method achieves better overall performance than that by the proposed one owing to the fact that the double-prototype filter bank in [11] possesses double degrees of design freedom as that in proposed method. In optimization problem (43), α serves to achieve trade-off between the reconstruction error and stopband attenuation. In order to verify this viewpoint, the filter bank is redesigned 17
by the proposed algorithm with α = 0.1 . The performance measures of the generated filter bank are illustrated in Table I. The impulse and magnitude responses of the obtained PF are plotted in Fig.3 (c) and (d), respectively. As can be seen from Table 1, the reconstruction error gets worse with increased α , while stopband attenuation is improved. Generally, parameter α can be adjusted according to desired trade-off between the reconstruction property and
stopband characteristic. Besides, parameter β serves to seek compromise between the stopband attenuation and passband flatness. Although there is no rigorous rule to determine the parameter, quite a number of examples suggest that β ∈ [10,1000] is appropriate range. Table 1 Performance comparison in Example 1 Methods BI-SOCP Method [11] Proposed method ( α = 5 × 10 ) −6
Proposed method ( α = 0.1 )
ε t (dB)
ε a (dB)
ε r (dB)
PRD
SA (dB)
-94.67
-94.89
-93.80
2.53 × 10−6
-38.83,-35.19
-68.65
-69.35
-71.31
3.44 × 10−5
-36.94
-56.34
−4
-58.98
-48.73
-60.82
1.91 × 10
Fig.3. Example 1: (a) Impulse response and (b) Magnitude response of the prototype filter designed by the proposed algorithm with α = 5 × 10−6 , (c) Impulse response and (d) Magnitude response of the prototype filter designed with α = 0.1 . 18
Example 2: In this example, a 2D DFT modulated filter bank was designed satisfying the
same specifications as Example 2 in [13]. The specifications and parameters are as follows: ª 4 −4 º ª 2 −2 º −5 D1 = « » , D2 = « 2 2 » , L = 15 , α = 1 × 10 , β = 100 . 4 4 ¬ ¼ ¬ ¼
It took 11 iterations for the proposed algorithm to converge. The magnitude responses of the PF and analysis filters are illustrated in Fig.4. The design results of the modified Newton’s method [13] and proposed algorithm are contrasted in Table 2. It is observed that compared with the method [13], the proposed algorithm can lead to filter bank with much smaller reconstruction error at a slightly degradation of stopband attenuation. The reason is straightforward, reconstruction error is directly minimized in the proposed algorithm while that in [13] is restricted by the attainable stopband attenuation. Table 2 Performance comparison in Example 2 Methods
ε t (dB)
Newton’s method in [13]
-87.27
Proposed method
-94.96
εa
(dB)
-82.30 -88.89
ε r (dB)
PRD
-84.22
5.37 ×10−6
-68.02
-96.46
−6
-67.27
1.34 ×10
SA (dB)
Fig.4. Example 2: (a) Magnitude response of the prototype filter designed by the proposed algorithm. (b) Magnitude responses of the analysis filters.
19
Note that in above two examples, the modulation matrix D1 is integral multiple of the decimation matrix D2 . Accordingly, matrix D3 is an identity matrix and D4 is multiple of the identity matrix, which indicates that the PR constraints are relatively small. If D1 is not multiple of D2 , the PR condition becomes complicate. In this case, the proposed algorithm can also work well, as shown in the following examples. Table 3 Performance comparison in Example 3 Methods
ε t (dB)
BI-SOCP method [11]
-55.20
Newton’s Method [13]
-44.76
Proposed method
-56.60
εa
ε r (dB)
PRD
SA (dB)
-55.27
-53.98
4.43 × 10 −4
-20.93,-26.96
-38.19
-39.04
9.83 × 10−2
-25.79
-59.29
−4
-24.52
(dB)
-60.55
2.44 × 10
Fig.5. Example 3: (a) Impulse response and (b) Magnitude response of the prototype filter designed by the proposed algorithm.
Example 3: This example considers a 2D DFT modulated filter bank satisfying the
specifications as the same as Example 3 in [13]. The specifications and parameters are described as follows D1 = 3I 2 , D2 = 2I 2 , L = 7 , α = 1× 10−5 , β = 100 . 20
The filter bank is with low redundancy ratio 9 / 4 . It took 20 iterations for the algorithm to terminate. The impulse and magnitude responses of the generated PFs are depicted in Fig.5. Table 3 presents the design results of the BI-SOCP method in [11], the modified Newton’s method in [13] and proposed algorithm for contrast. From Table 3, it is observed that the filter bank generated by the proposed algorithm possesses remarkable smaller reconstruction error than the Newton’s method. The reason is similar to that of Example 2. Therefore, when requiring 2D single-prototype DMFBs with low redundancy ratio, the proposed algorithm is a better choice compared with the modified Newton’s method. Besides, the proposed algorithm also achieves smaller reconstruction error than the BI-SOCP method while maintains comparable stopband attenuation.
Fig.6. Example 4: (a) Impulse response and (b) Magnitude response of the prototype filter designed by
the proposed algorithm.
Example 4: In this example, we consider a 2D DFT modulated filter bank whose
specifications and parameters are presented as follows ª 2 −2 º −5 D1 = 5I 2 , D2 = « » , L = 12 , α = 1×10 , β = 100 . 2 2 ¬ ¼
It took 12 iterations for the algorithm to converge. The impulse and magnitude responses of 21
the generated PFs are depicted in Fig. 6. The filter bank achieves performance: transfer function distortion -55.88dB, aliasing distortion -61.97dB, reconstruction error -63.03dB, perfect-reconstruction distortion 8.01 × 10−5 , and stopband attenuation -24.42dB. It is observed that the filter bank is with satisfactory overall performance.
5 CONCLUSION The main contribution of this paper is deriving the necessary and sufficient PR condition for 2D DFT modulated filter banks with arbitrary modulation and decimation matrices. Meanwhile, as another important result, we have derived the spatial domain condition of PR, which facilitates us to propose an efficient iterative algorithm. Several numerical examples have verified the validity of the PR condition and also demonstrated that the proposed algorithm can yield filter banks with good overall performance. Appendix A: Proof of Lemma 2.1 With D = D1D3 , we have
D−1 (D1p n + ul ) = D3−1 (p n + D1−1ul ) = D3−1 (p n + xl ) .
(A1)
Since xl ∈ [0,1) 2 and p n ∈ N (D3 ) , we obtain that p n + xl locates in FPD(D3 ) [1], which implies that D3−1 (p n + xl ) ∈ [0,1) 2 . Therefore, integer vector D1p n + ul belongs to vector set N (D) for n = 0," , D3 − 1, l = 0," , D2 − 1 . Furthermore, in terms of the Division theorem for
integer vectors, vectors D1p n + ul , n = 0,", D3 − 1, l = 0," , D2 − 1 diverse from each other if D1p n ∈ LAT (D1 )
and
.
ul ∈ N (D1 )
In
summary,
vectors
D1p n + ul , n = 0,", D3 − 1, l = 0," , D2 − 1 constitute the set N (D) . Up to now, the proof of (6a)
is completed. The proof of (6b) can be handled in the same way. 22
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Short Authors’ Biographies and Key Words (used for publication) Dear Editor, First of all, thank you very much for processing our accepted manuscript entitled by “Theory and Design of Two-dimensional DFT Modulated Filter Bank with Arbitrary Modulation and Decimation Matrices” with ID YDSPR 1781. According to the request, we provide the short biographies of four authors and add key words.
1. Short Authors’ Biographies Jun-Zheng Jiang was born in Zhejiang, China, in 1983. He received the B.S. degree in mathematics from Guilin University of Electronic and Technology, Guilin, China, in 2005 and Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 2011. He is now an associate professor of Guilin University of Electronic and Technology. His research interests include filter bank design and graph signal processing.
Fang Zhou received the B.S. degree and the M.S. degree from Xidian University, Xi’an, China, in 2006 and 2019, respectively, both in electrical engineering. She is currently working toward the Ph.D. degree in signal and information processing in the National Laboratory of Radar Signal Processing of Xidian Univeristy, Xi’an, China. Her research interests include two-dimensional filter bank and nonuniform filter bank design.
Peng-Lang Shui (M’02) received the M.S. degree in mathematics from Nanjing University, China, and the Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 1992 and 1999, respectively. He is now a Professor with the National Laboratory of Radar Signal Processing, Xidian University. His research interests include wavelets and filter bank design, image processing and radar target detection.
Shan Ouyang received the B.S. degree from Guilin University of Electronic Technology (GUET), Guilin, China, in 1986 and the M.S. and Ph.D. degrees from Xidian University, Xi’an, China, in 1992 and 2000, respectively, all in electronic engineering. He is currently a Professor with the School of Information and Communication, GUET. His research interests are mainly in the areas of signal processing for communications and radar, adaptive filtering, and neural network learning theory and applications.
2. Key Words (new) Fully oversampled, Nonlinear optimization, Two-dimensional DFT modulated filter bank.
Perfect-reconstruction
(PR),