Open access peer-reviewed chapter

Wavelet Filter Banks Using Allpass Filters

Written By

Xi Zhang

Submitted: June 4th, 2020 Reviewed: October 16th, 2020 Published: November 30th, 2020

DOI: 10.5772/intechopen.94519

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Abstract

Allpass filter is a computationally efficient versatile signal processing building block. The interconnection of allpass filters has found numerous applications in digital filtering and wavelets. In this chapter, we discuss several classes of wavelet filter banks by using allpass filters. Firstly, we describe two classes of orthogonal wavelet filter banks composed of two real allpass filters or a complex allpass filter, and then consider design of orthogonal filter banks without or with symmetry, respectively. Next, we present two classes of filter banks by using allpass filters in lifting scheme. One class is causal stable biorthogonal wavelet filter bank and another class is orthogonal wavelet filter bank, all with approximately linear phase response. We also give several design examples to demonstrate the effectiveness of the proposed method.

Keywords

  • wavelet
  • filter bank
  • allpass filter
  • perfect reconstruction
  • symmetry
  • orthogonality

1. Introduction

The discrete wavelet transform (DWT), which is implemented by a two band perfect reconstruction (PR) filter bank, has been applied extensively to digital signal processing, image processing, medical and health care, economy and so on [1, 2, 3, 4]. In many applications such as image processing, wavelets are required to be real since the signal is real-valued in general. We restrict ourselves to real-valued wavelet filter banks in this chapter.

In addition to orthogonality, one desirable property for wavelets is symmetry, which requires all filters in the filter bank to possess exactly linear phase, because the symmetric extension method is generally used to treat the boundaries of images [5, 6]. It is known in [1, 2, 3, 4] that finite impulse response (FIR) filters (corresponding to the compactly supported wavelets) can easily realize exactly linear phase. However, it is widely appreciated that the only FIR solution that produces a real orthogonal symmetric wavelet basis is the Haar wavelet, which is not continuous and the corresponding filter is of order 1 only that is not enough for many practical applications. To obtain wavelet filter banks with higher degrees of freedom, infinite impulse response (IIR) filters have been used to construct wavelet filter banks with some of the desired properties [7, 8, 9, 10, 11, 12]. Among the existing IIR wavelet filter banks, wavelet filter banks composed of allpass filters are attractive [7, 9, 10, 12], which can realize both of orthogonality and symmetry.

Allpass filter is a computationally efficient versatile signal processing building block and quite useful in many applications [13]. Allpass filter possesses unit magnitude at all frequencies (see Appendix) and is a basic scalar lossless building block. The interconnection of allpass filters has found numerous applications in practical filtering problems, such as low sensitivity filter structures, multirate filtering, filter banks and so on [7, 10, 12, 13]. The phase approximation of allpass filters has been also discussed in [13, 14, 15].

The lifting scheme proposed by W. Sweldens in [16, 17] is an efficient tool for constructing second generation wavelets, and has advantages such as faster implementation, fully in-place calculation, reversible integer-to-integer transforms, and so on. It has been proved in [18, 19] that every FIR wavelet filter bank can be decomposed into a finite number of lifting steps, thus this allows the construction of an integer version of the wavelet transform. Such integer wavelet transforms are invertible, and then are attractive in lossless coding applications. Due to these properties, the lifting implemantation has been adopted in the international standard JPEG2000 [5]. Conventionally, the lifting scheme is often used to construct a class of biorthogonal wavelet filter banks. It has been shown in [18] that orthogonal wavelet filter banks can also be realized by the lifting scheme. However, it is not always possible for IIR wavelet filter banks to be decomposed into a finite number of lifting steps.

In this chapter, we discuss several classes of wavelet filter banks by using allpass filters. Firstly, we describe two classes of orthogonal wavelet filter banks composed of two real allpass filters or a single complex allpass filter. We consider design of the proposed orthogonal wavelet filter banks without or with symmetry, respectively, and give the maximally flat solutions, where the orthogonal symmetric wavelet filter banks using real or complex allpass filter are corresponding to half sample symmetric (HSS) and whole sample symmetric (WSS) wavelets, respectively. Next, we present two classes of wavelet filter banks based on the lifting scheme with two lifting steps only. By using real allpass filters in the lifting steps, we can obtain one class of causal stable biorthogonal wavelet filter bank and another class of orthogonal wavelet filter bank, all with approximately linear phase response. In addition, we show some design examples to demonstrate the effectiveness of the proposed method.

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2. Two band wavelet filter bank

It is well-known [1, 2, 3, 4] that wavelet basis can be generated by two band filter bank shown in Figure 1. In Figure 1, H0z and H1z are analysis filters, and G0z and G1z are synthesis filters. The relationship of input Xz and output Yz of the filter bank is given by

Figure 1.

Two band wavelet filter bank. Xz is input and Yz is output. H0z,H1z are analysis filters and G0z,G1z are synthesis filters.

Yz=12H0zG0z+H1zG1zXz+12H0zG0z+H1zG1zXz.E1

Therefore the PR condition is

H0zG0z+H1zG1z=czIH0zG0z+H1zG1z=0,E2

where c is constant and I is integer.

One desirable property for wavelets is orthogonality, which requires the filter bank is orthogonal, i.e., H0e=G0e=H1ejπω=G1ejπω. Another desirable property is symmetry, i.e., the wavelet basis is symmetric or antisymmetric. It requires all filters in the filter bank to possess exactly linear phase, whose impulse responses are symmetric or antisymmetric.

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3. The proposed orthogonal wavelet filter banks using allpass filters

In this section, we describe several classes of orthogonal wavelet filter banks without or with symmetry. The proposed classes of orthogonal wavelet filter banks are composed of two real allpass filters or a complex allpass filter.

3.1 Orthogonal wavelet filter banks without symmetry

In some applications of signal processing, for example, speech and acoustic signal processing, wavelet filters are required to have minimal phase response rather than exactly linear phase. Therefore, wavelet basis is not necessarily symmetric or antisymmetric. In the following, we discuss two classes of orthogonal wavelet filter banks without symmetry [20].

3.1.1 Filter bank using real allpass filters

We firstly consider a pair of IIR filters H0z and H1z that are based on a parallel connection of two real allpass filters as shown in Figure 2, i.e.,

Figure 2.

Filter bank using real allpass filters. AN1z,AN2z are real allpass filters of order N1 and N2. H0z,H1z are lowpass and highpass filters.

H0z=12AN1z2+z2K1AN2z2H1z=12AN1z2z2K1AN2z2,E3

where K is integer, AN1z and AN2z are real allpass filters of order N1 and N2 respectively. Let the synthesis filters G0z=H0z1 and G1z=H1z1, then the PR condition in Eq.(2) is satisfied.

From Eq.(3), we have

H0z=12AN2z2ANz2+z2K1,E4

where ANz is a real allpass filter of order N=N1+N2, and defined as

ANz=AN1zAN2z=zNn=0Nanznn=0Nanzn,E5

where an is real coefficient, and a0=1.

Let θω be the phase response of ANz, the magnitude responses of H0z and H1z are given by

H0e=cosθ2ω2+K+12ωH1e=sinθ2ω2+K+12ω.E6

It is clear that the magnitude responses satisfy H0e=H1ejπω and the following power-complementary relation;

H0e2+H1e2=1,E7

which means that the filter bank is orthogonal.

For H0z and H1z to be a pair of lowpass and highpass filters, the desired phase response of ANz is given by

θdω=K+12ω=τω,E8

where τ=K+12. From the regularity of wavelets, it is known that an additional flatness condition is required to impose on H0z, i.e.,

kH0eωkω=π=0k=01L1,E9

where L is integer. Hence, the resulting wavelet function will have L consecutive vanishing moments. This flatness condition can be obtained if H0z contains L zeros located at z=1.

For the maximally flat filters, the closed-form formula is given by

an=Nni=1nNτi+1τ+i.E10

Once a set of filter coefficients an are obtained, we compute poles of ANz and then assign the poles inside the unit circle to AN1z as its poles and the poles outside the unit circle to AN2z as its zeros. Therefore, we can obtain causal stable analysis filters H0z and H1z, then the synthesis filters G0z and G1z are anti-causal stable.

In many applications of signal processing, frequency selectivity is also thought of as a useful property from the viewpoint of signal band-splitting. However, regularity and frequency selectivity somewhat contradict each other. For this reason, design of H0z that has the best possible frequency selectivity for the given flatness condition has been also discussed in [20].

Example 1: We consider design of filter banks using two real allpass filters with N1=N2=2 and K=0. By setting L=9,5,1, we have designed H0z by using the design method proposed in [20]. The magnitude responses are shown in Figure 3, and the scaling and wavelet functions are shown in Figure 4, respectively. When L=9, it is seen that H0z is the maximally flat filter, and it is the elliptic filter if L=1. It is clear in Figure 3 that the magnitude error increases with an increasing L, and in Figure 4 that the scaling and wavelet functions decline more rapidly.

Figure 3.

Magnitude responses of H0z in example 1.

Figure 4.

Scaling and wavelet functions in example 1.

3.1.2 Filter bank using complex allpass filter

We consider a pair of H0z and H1z using a single complex allpass filter as shown in Figure 5, i.e.,

Figure 5.

Filter bank using complex allpass filter. ANz,ÂNz are complex allpass filters of order N. H0z,H1z are lowpass and highpass filters.

H0z=12ANz+ÂNzH1z=z12jANzÂNz,E11

where ANz and ÂNz are complex allpass filters of order N, and their coefficients are mutually complex conjugate. Let G0z=H0z1 and G1z=H1z1 similarly. From the orthogonality, ANz and ÂNz must satisfy [7]

ANz=±jÂNz,E12

which means that if α is a pole of ANz, then α is a pole of ANz also. Consequently, ANz has a pair of poles αα or a single pole , where β is real, α is complex and α denotes the complex conjugate of α.

From Eq.(11),

H0z=12ÂNzA2Nz+1,E13

where A2Nz is a complex allpass filter of order 2N, and defined by

A2Nz=ANzÂNz=een=0N1a2nz2n+z2n+jn=0N2a2n+1z2n+1+z2n1n=0N1a2nz2n+z2njn=0N2a2n+1z2n+1+z2n1,E14

where η=±π/4 or ±3π/4, an is real and a0=1/2, N1=N/2 and N2=N/21 if N is even, and N1=N2=N1/2 if N is odd.

Therefore, the phase response θω of A2Nz is given by

θω=2η+2tan1n=0N2a2n+1cos2n+1ωn=0N1a2ncos2,E15

and the magnitude responses of H0z and H1z are

H0e=cosθω2H1e=sinθω2,E16

which satisfies the power-complementary relation in Eq.(7).

The closed-form formula of the maximally flat filters can be given by

an=Cni=1niN1i+N,E17

where C2n=1 and C2n+1=tanη. Therefore, we compute poles of A2Nz and assign the poles inside the unit circle to ANz as its poles to obtain causal stable analysis filters H0z and H1z. Thus, the synthesis filters G0z and G1z are anti-causal stable. Design of H0z having the best possible frequency selectivity for the given degrees of flatness has been also discussed in [20].

Example 2: We consider design of filter banks using a complex allpass filter with N=4 and L=8,4,0. We have designed H0z by using the design method proposed in [20]. The magnitude responses are shown in Figure 6, and the scaling and wavelet functions are shown in Figure 7, respectively. It is seen in Figure 6 that H0z is the maximally flat filter if L=8, and the elliptic filter if L=0 that does not have any zero located at z=1 and is different from that in Example 1.

Figure 6.

Magnitude responses of H0z in example 2.

Figure 7.

Scaling and wavelet functions in example 2.

3.2 Orthogonal symmetric wavelet filter banks

In many applications of image processing, digital filters are required to have exactly linear phase. Therefore, the impulse responses of wavelet filters need to be symmetric or antisymmetric, and the generated wavelet bases are symmetric or antisymmetric also. In the following, we discuss two classes of orthogonal symmetric wavelet filter banks composed of allpass filters: HSS [21] and WSS [22] wavelet filter banks.

3.2.1 Filter bank using real allpass filters

To obtain exactly linear phase, we constitute a pair of H0z and H1z in Figure 2 by using an allpass filter ANz as

H0z=12ANz2+z2K1ANz2H1z=12ANz2z2K1ANz2.E18

Let θω be the phase response of ANz, then the frequency responses of H0z and H1z are given by

H0e=ejK+12ωcosθ2ω+K+12ωH1e=jejK+12ωsinθ2ω+K+12ω.E19

It is clear in Eq.(19) that H0z and H1z have exact linear phase response and satisfy the power-complementary relation in Eq.(7). The filter has a group delay of K+12, and its impulse response is HSS. Therefore, the design problem of the wavelet filter banks becomes the phase approximation of allpass filter ANz. For H0z and H1z to be a pair of lowpass and highpass filters, the desired phase response of ANz is

θdω=K2+14ω=τω,E20

where τ=K2+14. The filter coefficients an of the maximally flat filters can be computed by Eq.(10). Design of wavelet filters having the best possible frequency selectivity for the given degrees of flatness has been also discussed in [21]. It has been pointed out in [21] that we must choose K=,7,6,3,2,1,2,5,6, if N is odd and K=,5,4,1,0,3,4,7, if N is even, in order to obtain a pair of reasonable lowpass and highpass filters to avoid the undesired zero and bump.

Example 3: We consider design of the maximally flat wavelet filter banks with N=4 and L=9. We have designed ANz with K=0 and K=1. The magnitude responses of H0z are shown in Figure 8. It is seen in Figure 8 that H0z with K=1 has the undesired zero and bump nearby ω=π/2. The generated scaling and wavelet functions are shown in Figure 9 respectively. It is seen in Figure 9 that the scaling functions are symmetric, while the wavelet functions are antisymmetric. Although H0z with K=0 and K=1 have the same degrees of flatness, it is seen that the scaling and wavelet functions of K=1 decline more slowly than that of K=0, because of the undesired zero and bump. Therefore, we should not choose K=1 in this case.

Figure 8.

Magnitude responses of H0z in example 3. H0z has an undesired zero and bump when K=1.

Figure 9.

Scaling and wavelet functions in example 3. The symmetric point is dependent on the group delay of K+12.

3.2.2 Filter bank using complex allpass filter

We consider again H0z and H1z using a complex allpass filter in Figure 5,

H0z=12ANz+ÂNzH1z=z12jANzÂNz.E21

To obtain exactly linear phase, ANz and ÂNz must satisfy

ÂNz=1ANz,E22

which means that if α is a pole of ANz, then 1/α is a pole of ANz too. In addition to orthogonality, ANz has a quadruplet of poles αα1/α1/α or a pair of poles 1/. Therefore, we have

ANz=ezNa0+ja1z+a2z2++a2zN2+ja1zN1+a0zNa0ja1z1+a2z2++a2zN+2ja1zN+1+a0zN,E23

where N is even, an is real and a0=1. The phase response θω of ANz is given by

θω=η+2φω,E24

where if M=N/2 is even,

φω=tan1n=0M/21a2n+1cosM2n1ωaM2+n=0M/21a2ncosM2nω,E25

and if M=N/2 is odd,

φω=tan1aM2+n=0M3/2a2n+1cosM2n1ωn=0M1/2a2ncosM2nω.E26

Thus, we have

H0e=cosθωH1e=esinθω.E27

It is clear that H0z and H1z have exactly linear phase responses and satisfy the power-complementary relation in Eq.(7). Its impulse response is WSS. Therefore, the design problem of wavelet filter banks becomes the phase approximation of ANz in Eq.(23).

For the maximally flat filters, the closed-form formula is given in [22] by

an=CnNn,E28

where C2n=1 and C2n+1=tanη2. Design of wavelet filters having the best possible frequency selectivity for the given degrees of flatness has been also discussed in [22]. It has been pointed out in [22] that we must choose η=±π/4 if M is even and η=±3π/4 if M is odd.

Example 4: We consider design of the filter banks with N=6 and η=3π/4. We have designed ANz with L=0,2,4,6 by using the design method proposed in [22]. The magnitude responses of H0z are shown in Figure 10, and the scaling and wavelet functions are shown in Figure 11, respectively. It is seen in Figure 10 that H0z with L=6 corresponds to the maximally flat filter, and H0z with L=0 is the minimax filter that has no zero located at z=1. In Figure 11, the scaling and wavelet functions are not continuous because the regularity condition is not satisfied when L=0, and become more smooth with an increasing L. Both the scaling and wavelet functions are symmetric.

Figure 10.

Magnitude responses of H0z in example 4.

Figure 11.

Scaling and wavelet functions in example 4.

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4. Lifting-based wavelet filter banks using allpass filters

The lifting scheme proposed in [16] and [17] is an efficient tool for constructing second generation wavelets, and has advantages such as faster implementation, fully in-place calculation, reversible integer-to-integer transforms, and so on. It has been proved in [18] and [19] that every FIR wavelet filter bank can be decomposed into a finite number of lifting steps, thus this allows the construction of an integer version of the wavelet transform. Such integer wavelet transforms are invertible, and then are attractive in lossless coding applications. Conventionally, the lifting scheme is often used to construct a class of biorthogonal wavelet filter banks. It has been shown in [18] that the orthogonal wavelet filter banks can also be realized by the lifting scheme. However, it is not always possible for IIR wavelet filter banks to be decomposed into a finite number of lifting steps. For example, it is difficult to realize the IIR orthogonal wavelet filter banks discussed in Section 3 by using a finite number of lifting steps.

Now, we restrict ourselves to the lifting scheme with two lifting steps [10] as shown in Figure 12. Let H0z and H1z be a pair of lowpass and highpass filters,

Figure 12.

Lifting scheme with two lifting steps [10]. Xz is input and Yz is output. Pz,Qz are filters.

H0z=12z2K11+Pz2H1z=z2K2Qz2H0z,E29

then G0z=H1z and G1z=H0z. It is clear in Figure 12 that the PR condition is structurally satisfied. Therefore, the design of H0z and H1z becomes how to determine Pz and Qz. In the following, we describe two classes of near symmetric wavelet filter banks by using real allpass filters in the lifting scheme: causal stable biorthogonal wavelet filter bank [23] and orthogonal wavelet filter bank [24].

4.1 Causal stable wavelet filter banks

We use two real allpass filters in lifting steps, i.e., Pz=AN1z and Qz=AN2z, and thus,

H0z=12z2K11+AN1z2H1z=z2K2AN2z2H0z.E30

Let θ1ω be the phase response of AN1z, the frequency response of H0z is given by

H0e=ejθ12ω2K1+12ωcosθ12ω2+K1+12ω.E31

For H0z to be lowpass filter, the desired phase response of AN1z is

θ1dω=K1+12ω=τ1ω,E32

where τ1=K1+12. According to Appendix, the order of AN1z is required to be N1=K1 or N1=K1+1 to obtain causal stable allpass filter.

Ideally, H0e=0 in the stopband of H0z, then H1e=ej2K2ω, having linear phase response from Eq.(30). In the passband of H0z, H0e=ej2K1+1ω ideally, thus,

H1e=ej2K2ωejθ22ωej2K1+1ω=2jejθ22ω2K1+K2+12ωsinθ22ω2K1K2+12ω,E33

where θ2ω is the phase response of AN2z. Therefore, in the stopband of H1z, the desired phase response of AN2z is

θ2dω=K1K2+12ω=τ2ω,E34

where τ2=K2K112. Similarly, the order of AN2z is required to be N2=K2K1 or N2=K2K11 to obtain causal stable allpass filter. Therefore, once N1 and N2 are given, we can obtain causal stable wavelet filter banks by appropriately choosing K1 and K2. The maximally flat filters can be designed by using Eq.(10). H0z and H1z have approximately linear phase response.

Example 5: We consider design of the maximally flat wavelet filter banks with N1=N2=6. We have designed AN1z with K1=5, and the magnitude response of H0z is shown in Figure 13. We then designed AN2z with K2=11 and K2=12, and the magnitude responses of H1z are shown also in Figure 13. It is seen in Figure 13 that H1z with K2=11 has a large overshoot nearby ω=π/2. To avoid this overshoot, we should choose K2=N2+K1+1 if K1=N11 and K2=N2+K1 if K1=N1. The scaling and wavelet functions generated by analysis and synthesis filters with K1=5 and K2=12 are shown in Figure 14 respectively.

Figure 13.

Magnitude responses of H0z and H1z in example 5.

Figure 14.

Scaling and wavelet functions generated by analysis and synthesis filters in example 5. It is because the wavelet filter bank is biorthogonal, but not orthogonal.

4.2 Orthogonal wavelet filter banks

The above-mentioned causal stable wavelet filter banks are biorthogonal (not orthogonal). Here we discuss a class of orthogonal wavelet filter banks using the lefting scheme. We use Pz=ANz and Qz=ANz1, then,

H0z=12z2K11+ANz2H1z=z2K2ANz2H0z.E35

Let θω be the phase response of ANz, the frequency response of H0z is the same as in Eq.(31), thus the desired phase response of ANz is

θdω=K1+12ω=τω,E36

where τ=K1+12. To be orthogonal, we set K2=0 and have

H1z=1ANz212z2K11+ANz2=121z2K11ANz2,E37

whose frequency response is

H1e=jejθ2ω2+K1+12ωsinθ2ω2+K1+12ω.E38

It is clear that the magnitude responses satisfy H0e=H1ejπω and the power-complementary relation in Eq.(7). Therefore, this class of wavelet filter banks is orthogonal and both H0z and H1z have approximately linear phase response. Design of this class of orthogonal wavelet filter banks has been discussed and applied to lossy to lossless image coding in [24].

Example 6: We consider design of the maximally flat orthogonal wavelet filter banks with N=2,4,6. We have designed ANz with K1=N1. The magnitude responses of H0z are shown in Figure 15. The generated scaling and wavelet functions are shown in Figure 16, respectively. It is seen in Figure 16 that the wavelet functions are near symmetric.

Figure 15.

Magnitude responses of H0z in example 6.

Figure 16.

Scaling and wavelet functions in example 6.

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5. Conclusions

In this chapter, we have proposed several new classes of wavelet filter banks with some properties of orthogonality, symmetry and causal stablity by using allpass filters, which are potential options for readers to choose wavelet basis in practical applications. As shown in Table 1, first class of wavelet filter banks in Section 3.1 is orthogonal, but asymmetric, its analysis filters is causal stable. Second class of wavelet filter banks in Section 3.2 is orthogonal and symmetric, but not causal. Third and fourth classes of wavelet filter banks are based on the lifting scheme. Third class in Section 4.1 is biorthogonal, causal stable and near symmetric, while fourth class in Section 4.2 is orthogonal and near symmetric, but not causal. There is no solution to all of orthogonality, symmetry and causal stablity. The wavlet filter banks using allpass filters have been extended to Hilbert transform pair of wavelets [25], 2D wavelet filter banks [26], and applied to lossy to lossless image coding [27, 28, 29, 30] and scalable video compression [31]. It is possible also to extend them to higher dimension and irregural signal processing and to apply them to wavelet denoising, image fusion and so on.

Filter Bank ClassSec.3.1Sec.3.2Sec.4.1Sec.4.2D-8/8D-9/7
Filter TypeIIRIIRIIRIIRFIRFIR
Orthogonality××
Symmetry×ΔΔ×
Causal stablityΔ××

Table 1.

Comparison of the proposed classes of wavelet filter banks with the conventional wavelets D-8/8, D-9/7 in [1].

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Acknowledgments

This work was supported by JSPS KAKENHI Grant Number 18 K11260.

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Appendix

Digital allpass filter is a computationally efficient signal processing building block and quite useful in many signal processing applications. One of the most widely used applications is phase or delay equalizer. Allpass filter possesses unit magnitude at all frequencies and is a basic scalar lossless building block. The interconnection of allpass filters has found numerous applications in practical filtering problems, such as low sensitivity filter structures, multirate filtering, filter banks and so on [13].

The transfer function of an Nth-order allpass filter is defined as

ANz=zNn=0Nanznn=0Nanzn,E39

where an=anr+jani is a complex coefficient in general, and an denotes the complex conjugate of an. When ani=0, an is a real coefficient and ANz is a real allpass filter. Thus the real allpass filter is a special case of complex allpass filter. All poles and zeros of ANz occur in mirror-image pairs with respect to the unit circle, and then the frequency response ANe exhibits unit magnitude at all frequencies, i.e., ANe1 for all ω. The phase response of ANz is given by

θω=+2tan1n=0Nanrsin+anicosn=0Nanrcosanisin.E40

If all poles locate inside the unit circle, then ANz is causal stable. The phase response decreases monotonically with an increasing frequency and θπ=θπ2. If ANz is real allpass filter, θ0=0 and θπ=. When one pole locates at the origin, it is seen that ANz=z1AN1z due to aN=0. Then zN is a special case of ANz if all poles locate at the origin. When k poles locate outside the unit circle, we can divide ANz into two causal stable allpass filters ANkz and Akz, i.e.,

ANz=ANkzAkz.E41

The phase response θω of ANz is the phase difference between ANkz and Akz, and θπ=θπ2N2kπ. The design problem of allpass filters to approximate the specified phase response in the Chebyshev sense has been discussed in [14] and [15].

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Abbreviations

DWTdiscrete wavelet transform
PRperfect reconstruction
FIRfinite impulse response
IIRinfinite impulse response
HSShalf sample symmetric
WSSwhole sample symmetric

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Written By

Xi Zhang

Submitted: June 4th, 2020 Reviewed: October 16th, 2020 Published: November 30th, 2020