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

## 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.

## 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,

Therefore the PR condition is

where

One desirable property for wavelets is orthogonality, which requires the filter bank is orthogonal, i.e.,

## 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

where

From Eq.(3), we have

where

where

Let

It is clear that the magnitude responses satisfy

which means that the filter bank is orthogonal.

For

where

where

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

Once a set of filter coefficients

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

*Example 1*: We consider design of filter banks using two real allpass filters with

#### 3.1.2 Filter bank using complex allpass filter

We consider a pair of

where

which means that if

From Eq.(11),

where

where

Therefore, the phase response

and the magnitude responses of

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

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

where

*Example 2*: We consider design of filter banks using a complex allpass filter with *Example 1*.

### 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

Let

It is clear in Eq.(19) that

where

*Example 3*: We consider design of the maximally flat wavelet filter banks with

#### 3.2.2 Filter bank using complex allpass filter

We consider again

To obtain exactly linear phase,

which means that if

where

where if

and if

Thus, we have

It is clear that

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

where

*Example 4*: We consider design of the filter banks with

## 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

then

### 4.1 Causal stable wavelet filter banks

We use two real allpass filters in lifting steps, i.e.,

Let

For

where

Ideally,

where

where

*Example 5*: We consider design of the maximally flat wavelet filter banks with

### 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

Let

where

whose frequency response is

It is clear that the magnitude responses satisfy

*Example 6*: We consider design of the maximally flat orthogonal wavelet filter banks with

## 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 Class | Sec.3.1 | Sec.3.2 | Sec.4.1 | Sec.4.2 | D-8/8 | D-9/7 |
---|---|---|---|---|---|---|

Filter Type | IIR | IIR | IIR | IIR | FIR | FIR |

Orthogonality | ||||||

Symmetry | ||||||

Causal stablity |

## Acknowledgments

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

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

where

If all poles locate inside the unit circle, then

The phase response

## Abbreviations

DWT | discrete wavelet transform |

PR | perfect reconstruction |

FIR | finite impulse response |

IIR | infinite impulse response |

HSS | half sample symmetric |

WSS | whole sample symmetric |