4. Properties of filter banks for wavelet design

In wavelet designs, filter banks are required to possess three important properties which are fundamental to the realization of a wavelet function. These properties include: perfect reconstruction, orthogonality, and paraunitary condition.

### 4.1 Perfect reconstruction

This property guarantees that the signal at the output of a given filter bank is a delayed version of the signal at the input of the filter bank. Perfect reconstruction is an important property of a filter bank because it cancels the effect of aliasing of the input signal at the output, caused by the downsamplers and upsamplers. To understand this point, consider a two-channel finite impulse response FIR filter bank shown in Figure 5.

The output ^y nð Þ is derived using Eqs. (6) and (10) as follows in terms of the signal component and aliasing component as:

$$\hat{Y}(\mathbf{z}) = \text{signal\\_component} + \text{aliasing\\_component} \tag{11}$$

where the signal\_component and aliasing\_component are defined as:

$$\begin{aligned} \text{signal\\_component} &= \frac{1}{2} [F\_0(\mathbf{z})H\_0(\mathbf{z}) + F\_1(\mathbf{z})H\_1(\mathbf{z})] \mathbf{X}(\mathbf{z})\\ \text{alising\\_component} &= \frac{1}{2} [F\_0(\mathbf{z})H\_0(-\mathbf{z}) + F\_1(\mathbf{z})H\_1(-\mathbf{z})] \mathbf{X}(-\mathbf{z}) \end{aligned} \tag{12}$$

Figure 5. Two-channel FIR filter bank.

To achieve perfect reconstruction, the following condition must be satisfied [1]:

$$\begin{aligned} F\_0(z)H\_0(z) + F\_1(z)H\_1(z) &= 2z^{-1} \\ F\_0(z)H\_0(-z) + F\_1(z)H\_1(-z) &= 0 \end{aligned} \tag{13}$$

The relationships in Eqs. (11) and (13) are possible when the filter bank is constructed as a QMF (quadrature mirror filter) filter bank or CQF (conjugate quadrature filter) filter bank. Both QMF and CQF banks provide a mechanism by which complete cancellation of the aliasing component in Eq. (11) can be accomplished. Using QMF, aliasing cancellation can be achieved by constructing the filters in Figure 5 based on the following relationships [4, 5]:

$$\begin{cases} F\_0(z) = H\_0(z) \\ H\_1(z) = H\_0(-z) \\ F\_1(z) = -H\_1(z) \end{cases} \tag{14}$$

^y n½ �¼ y n½ � � 3 (17)

(18)

The relationship in Eq. (17) states that the output signal ^y n½ � is delayed version of the input signal y n½ � by three samples. We leave the verification of Eq. (16) as an

Having looked at perfect reconstruction as a necessary property for a filter bank in wavelet design, we now look at orthogonality as also an essential property for a

Orthogonality in a filter bank is a situation in which the synthesis filter bank is a transpose of the analysis filter bank. This is a useful property in the sense that it allows for the energy preservation of the signal being processed. This important property is achieved through the imposition of the orthogonality condition on both the analysis and filter bank sections while at the same time preserving the perfect reconstruction condition of the filter bank. The imposition of the orthogonality condition in a filter bank (see Figure 5) occurs when the following relationships are

<sup>~</sup><sup>f</sup> <sup>0</sup>ð Þ <sup>n</sup> � <sup>2</sup><sup>k</sup> ; <sup>h</sup>1ð Þi ¼ <sup>n</sup> � <sup>2</sup> <sup>l</sup> <sup>0</sup> <sup>~</sup><sup>f</sup> <sup>1</sup>ð Þ <sup>n</sup> � <sup>2</sup><sup>k</sup> ; <sup>h</sup>0ð Þi ¼ <sup>n</sup> � <sup>2</sup><sup>l</sup> <sup>0</sup>

ð Þ¼ n gi

ð Þ �n

~gi

exercise for the reader.

CQF two-channel FIR filter bank.

Figure 6.

Figure 7.

QMF two-channel FIR filter bank.

Analysis of Wavelet Transform Design via Filter Bank Technique

DOI: http://dx.doi.org/10.5772/intechopen.85051

4.2 Orthogonality

satisfied [11]:

where

and

25

filter bank in the design of wavelets.

D D

In Eq. (14), the synthesis filter F0ð Þz has the same coefficients as the analysis filter H0ð Þz ; the analysis filter H1ð Þz has the same coefficients as the analysis filter H0ð Þz , but every other value is negated; the synthesis filter F1ð Þz is a negative copy of the analysis filter H1ð Þz . For example, if the analysis filter H0ð Þz has coefficientsp, q, r, s, then the filter bank in Figure 5 will assume the structure shown in Figure 6.

For the CQF bank, the coefficients of the analysis filter H1ð Þz are a reversed version of the analysis filter H0ð Þz with every other value negated. The synthesis filters F0ð Þz and F1ð Þz are a reversed versions of the analysis filters H0ð Þz and H1ð Þz , respectively. These relationships can be stated mathematically as follows [10]:

$$\begin{aligned} H\_1(z) &= z^{-1} H\_0(-z^{-1}) \\ F\_0(z) &= H\_1(-z) \\ F\_1(z) &= -H\_0(-z) \end{aligned} \tag{15}$$

Based on the relationship in Eq. (15), the filter bank shown in Figure 6 for CQF will assume the structure shown in Figure 7.

Based on the structure of Figures 6 or 7, the output signal ^y n½ � is related to the input signal y n½ � by the expression:

$$
\hat{\jmath}[n] = (pp + qq + rr + ss)\jmath[n-\mathfrak{J}]\tag{16}
$$

If we impose the condition that pp þ qq þ rr þ ss ¼ 1, then Eq. (16) becomes:

Analysis of Wavelet Transform Design via Filter Bank Technique DOI: http://dx.doi.org/10.5772/intechopen.85051

### Figure 6.

To achieve perfect reconstruction, the following condition must be satisfied [1]:

�

(13)

(14)

(15)

<sup>F</sup>0ð Þ<sup>z</sup> <sup>H</sup>0ð Þþ <sup>z</sup> <sup>F</sup>1ð Þ<sup>z</sup> <sup>H</sup>1ð Þ¼ <sup>z</sup> <sup>2</sup> <sup>z</sup>�<sup>1</sup> F0ð Þz H0ð Þþ �z F1ð Þz H1ð Þ¼ �z 0

The relationships in Eqs. (11) and (13) are possible when the filter bank is constructed as a QMF (quadrature mirror filter) filter bank or CQF (conjugate quadrature filter) filter bank. Both QMF and CQF banks provide a mechanism by which complete cancellation of the aliasing component in Eq. (11) can be accomplished. Using QMF, aliasing cancellation can be achieved by constructing the filters

> F0ð Þ¼ z H0ð Þz H1ð Þ¼ z H0ð Þ �z F1ð Þ¼� z H1ð Þz

In Eq. (14), the synthesis filter F0ð Þz has the same coefficients as the analysis filter H0ð Þz ; the analysis filter H1ð Þz has the same coefficients as the analysis filter H0ð Þz , but every other value is negated; the synthesis filter F1ð Þz is a negative copy

coefficientsp, q, r, s, then the filter bank in Figure 5 will assume the structure shown

For the CQF bank, the coefficients of the analysis filter H1ð Þz are a reversed version of the analysis filter H0ð Þz with every other value negated. The synthesis filters F0ð Þz and F1ð Þz are a reversed versions of the analysis filters H0ð Þz and H1ð Þz , respectively. These relationships can be stated mathematically as follows [10]:

> <sup>H</sup>1ð Þ¼ <sup>z</sup> <sup>z</sup>�<sup>1</sup>H<sup>0</sup> �z�<sup>1</sup> ð Þ F0ð Þ¼ z H1ð Þ �z F1ð Þ¼� z H0ð Þ �z

Based on the relationship in Eq. (15), the filter bank shown in Figure 6 for CQF

Based on the structure of Figures 6 or 7, the output signal ^y n½ � is related to the

If we impose the condition that pp þ qq þ rr þ ss ¼ 1, then Eq. (16) becomes:

of the analysis filter H1ð Þz . For example, if the analysis filter H0ð Þz has

9 >=

>;

9 >=

>;

^y n½ �¼ ð Þ pp þ qq þ rr þ ss y n½ � � 3 (16)

in Figure 5 based on the following relationships [4, 5]:

will assume the structure shown in Figure 7.

input signal y n½ � by the expression:

in Figure 6.

24

Figure 5.

Two-channel FIR filter bank.

Wavelet Transform and Complexity

QMF two-channel FIR filter bank.

Figure 7. CQF two-channel FIR filter bank.

$$
\hat{\mathcal{Y}}[n] = \mathcal{Y}[n-\mathfrak{Z}] \tag{17}
$$

The relationship in Eq. (17) states that the output signal ^y n½ � is delayed version of the input signal y n½ � by three samples. We leave the verification of Eq. (16) as an exercise for the reader.

Having looked at perfect reconstruction as a necessary property for a filter bank in wavelet design, we now look at orthogonality as also an essential property for a filter bank in the design of wavelets.

### 4.2 Orthogonality

Orthogonality in a filter bank is a situation in which the synthesis filter bank is a transpose of the analysis filter bank. This is a useful property in the sense that it allows for the energy preservation of the signal being processed. This important property is achieved through the imposition of the orthogonality condition on both the analysis and filter bank sections while at the same time preserving the perfect reconstruction condition of the filter bank. The imposition of the orthogonality condition in a filter bank (see Figure 5) occurs when the following relationships are satisfied [11]:

$$\left< \tilde{f}\_0(n - 2k), h\_1(n - 2l) \right> = 0 \\ \left< \tilde{f}\_1(n - 2k), h\_0(n - 2l) \right> = 0 \tag{18}$$

where

$$
\bar{\mathbf{g}}\_i(n) = \mathbf{g}\_i(-n),
$$

and

$$
\langle \tilde{\lg}\_0(n - 2k), h\_0(n) \rangle = \delta\_k \left\langle \tilde{\lg}\_1(n - 2k), h\_1(n) \right\rangle = \delta\_k \tag{19}
$$

The conditions in Eqs. (20)–(22) hold true iff Eð Þz and Rð Þz satisfy the following:

where k ¼ L=2, with the first and second condition in Eq. (24) relating to the filter

In the filter bank implementation of a wavelet transform, the paraunitary condition plays the critical role of guaranteeing the generation of orthonormal wavelets, and also perfect recovery of a decomposed signal. The paraunitary condition guarantees that recovered signal will suffer no phase or aliasing effect if a filter bank

Given a polyphase transfer function matrix Eð Þz , the paraunitary condition is

where the H superscript denotes the conjugated transpose, and I denotes the

If the real-coefficient lossless matrix is denoted by Eð Þz ; then the matrix is said to have a special case of lossless degree of one iff it can be characterized by the

where R is an arbitrary M � M unitary matrix and v is an M � 1 column vector

identity matrix. Paraunitary filter banks also have an attractive property of losslessness, which implies that for every frequency, the total signal power is conserved [16]. From this property [17], any M � M real-coefficient lossless matrix with N � 1 degree can be realized using the structure shown in Figure 9 [18].

<sup>E</sup>ð Þ¼ <sup>z</sup> <sup>I</sup> � vv<sup>þ</sup> <sup>þ</sup> <sup>z</sup>�<sup>1</sup>

with unit norm. From Eq. (26), the paraunitary condition for a filter bank is

<sup>k</sup> þ vkv<sup>þ</sup>

I � vkv<sup>þ</sup>

Cascade implementation of Eð Þz as FIR lossless unitary matrices separated by delays.

9 >=

>;

<sup>E</sup><sup>H</sup> <sup>z</sup>�<sup>1</sup> � �Eð Þ¼ <sup>z</sup> <sup>I</sup> (25)

vv<sup>þ</sup> � �R (26)

<sup>k</sup> <sup>z</sup> � �Ekð Þ¼ <sup>z</sup> <sup>E</sup><sup>k</sup>�<sup>1</sup>ð Þ<sup>z</sup> (27)

(24)

<sup>E</sup>^ð Þ<sup>z</sup> <sup>E</sup>ð Þ¼ <sup>z</sup> <sup>I</sup> <sup>R</sup>ð Þ¼ <sup>z</sup> <sup>z</sup>�ð Þ <sup>k</sup>�<sup>1</sup> <sup>E</sup>ð Þ<sup>z</sup> <sup>E</sup>ð Þ¼ <sup>z</sup> <sup>z</sup>�ð Þ <sup>k</sup>�<sup>1</sup> diag 1ð Þ , �<sup>1</sup> <sup>E</sup> <sup>z</sup>�<sup>1</sup> ð Þ<sup>J</sup>

bank orthogonality condition, and the last represents the filter bank symmetry. We now look at the paraunitary condition of a filter bank, which is also a

necessary property in filter bank implementation of wavelets.

Analysis of Wavelet Transform Design via Filter Bank Technique

DOI: http://dx.doi.org/10.5772/intechopen.85051

4.3 Paraunitary condition

satisfies the paraunitary condition [14].

established by the matrix iff [15]:

relationship [18]:

Figure 9.

27

obtained as follows [18]:

In Eq. (18), the inner product of the coefficients of the synthesis filter F0ð Þz and the analysis filter H1ð Þz must be zero and the inner product of the coefficients of the synthesis filter F1ð Þz and the analysis filter H0ð Þz must also be zero for the orthogonality condition to hold.

Also, the low-pass analysis filter H0ð Þz is related to the other three filters through the following expressions [12]:

$$\begin{aligned} H\_1(\mathbf{z}) &= c \mathbf{z}^{-(L-1)} \tilde{H}\_0(-\mathbf{z}) \\ F\_0(\mathbf{z}) &= \mathbf{z}^{-(L-1)} \tilde{H}\_0(\mathbf{z}) \\ F\_1(\mathbf{z}) &= \mathbf{z}^{-(L-1)} \tilde{H}\_1(\mathbf{z}) \end{aligned} \tag{20}$$

where L denotes the length of the filter which must be even, and c is a constant with j j<sup>c</sup> <sup>¼</sup> 1; <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ �<sup>z</sup> is the flipped and conjugated version of <sup>H</sup>0ð Þ<sup>z</sup> , <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ<sup>z</sup> is the conjugated version of <sup>H</sup>0ð Þ<sup>z</sup> , and <sup>H</sup><sup>~</sup> <sup>1</sup>ð Þ<sup>z</sup> is the conjugated version of <sup>H</sup>1ð Þ<sup>z</sup> .

The condition in Eq. (20) also describe the necessary requirement for a filter bank to be paraunitary (which we shall examine in the next section), i.e., the lowpass filter H0ð Þz satisfy the following power symmetry of halfband condition [8, 9]:

$$P(x) + P(-x) = 2 \tag{21}$$

where P zð Þ¼ <sup>H</sup>0ð Þ<sup>z</sup> <sup>H</sup>^ <sup>0</sup>ð Þ<sup>z</sup> . If the low-pass filter <sup>H</sup>0ð Þ<sup>z</sup> satisfies the required symmetry condition:

$$H\_0(\mathbf{z}) = \mathbf{z}^{-(L-1)} H\_0(\mathbf{z}^{-1}) \tag{22}$$

then P zð Þ is said to be a real filter. The implication of the constraint in Eq. (21) is that H1ð Þz and F1ð Þz be antisymmetric filters, and F0ð Þz is a symmetric filter. The relationships in Eqs. (20)–(22) give the necessary and sufficient condition for the characterization of a filter bank with orthogonality and symmetry.

The orthogonality condition for a filter bank can also be examined from a polyphase perspective. Consider the polyphase representation of the filter bank in Figure 5 as illustrated in Figure 8 [13].

If Eð Þz in Figure 8 is type-I analysis polyphase matrix, and Rð Þz is type-II synthesis polyphase matrix, then [13]:

$$\begin{aligned} \begin{bmatrix} H\_0(\mathbf{z}) \ H\_1(\mathbf{z}) \end{bmatrix}^T &= \mathbf{E}(\mathbf{z}^2) \begin{bmatrix} \mathbf{1} & \mathbf{z}^{-1} \end{bmatrix}^T\\ \begin{bmatrix} F\_0(\mathbf{z}) & F\_1(\mathbf{z}) \end{bmatrix} &= \begin{bmatrix} \mathbf{z}^{-1} & \mathbf{1} \end{bmatrix} \mathbf{R}(\mathbf{z}^2) \end{aligned} \tag{23}$$

Figure 8. Polyphase implementation of filter bank.

The conditions in Eqs. (20)–(22) hold true iff Eð Þz and Rð Þz satisfy the following:

$$\begin{aligned} \hat{\mathbf{E}}(z)\mathbf{E}(z) &= \mathbf{I} \\ \mathbf{R}(z) &= z^{-(k-1)}\mathbf{E}(z) \\ \mathbf{E}(z) &= z^{-(k-1)}\text{diag}(\mathbf{1}, -\mathbf{1})\mathbf{E}(z^{-1})\mathbf{J} \end{aligned} \tag{24}$$

where k ¼ L=2, with the first and second condition in Eq. (24) relating to the filter bank orthogonality condition, and the last represents the filter bank symmetry.

We now look at the paraunitary condition of a filter bank, which is also a necessary property in filter bank implementation of wavelets.

## 4.3 Paraunitary condition

~g0ð Þ n � 2k ; h0ð Þi ¼ n δ<sup>k</sup> ~g1ð Þ n � 2k ; h1ð Þi ¼ n δ<sup>k</sup>

In Eq. (18), the inner product of the coefficients of the synthesis filter F0ð Þz and the analysis filter H1ð Þz must be zero and the inner product of the coefficients of the synthesis filter F1ð Þz and the analysis filter H0ð Þz must also be zero for the orthog-

Also, the low-pass analysis filter H0ð Þz is related to the other three filters

<sup>H</sup>1ð Þ¼ <sup>z</sup> cz�ð Þ <sup>L</sup>�<sup>1</sup> <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ �<sup>z</sup> <sup>F</sup>0ð Þ¼ <sup>z</sup> <sup>z</sup>�ð Þ <sup>L</sup>�<sup>1</sup> <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ<sup>z</sup> <sup>F</sup>1ð Þ¼ <sup>z</sup> <sup>z</sup>�ð Þ <sup>L</sup>�<sup>1</sup> <sup>H</sup><sup>~</sup> <sup>1</sup>ð Þ<sup>z</sup>

where L denotes the length of the filter which must be even, and c is a constant with j j<sup>c</sup> <sup>¼</sup> 1; <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ �<sup>z</sup> is the flipped and conjugated version of <sup>H</sup>0ð Þ<sup>z</sup> , <sup>H</sup><sup>~</sup> <sup>0</sup>ð Þ<sup>z</sup> is the conjugated version of <sup>H</sup>0ð Þ<sup>z</sup> , and <sup>H</sup><sup>~</sup> <sup>1</sup>ð Þ<sup>z</sup> is the conjugated version of <sup>H</sup>1ð Þ<sup>z</sup> .

The condition in Eq. (20) also describe the necessary requirement for a filter bank to be paraunitary (which we shall examine in the next section), i.e., the lowpass filter H0ð Þz satisfy the following power symmetry of halfband condition [8, 9]:

where P zð Þ¼ <sup>H</sup>0ð Þ<sup>z</sup> <sup>H</sup>^ <sup>0</sup>ð Þ<sup>z</sup> . If the low-pass filter <sup>H</sup>0ð Þ<sup>z</sup> satisfies the required

then P zð Þ is said to be a real filter. The implication of the constraint in Eq. (21) is that H1ð Þz and F1ð Þz be antisymmetric filters, and F0ð Þz is a symmetric filter. The relationships in Eqs. (20)–(22) give the necessary and sufficient condition for the

The orthogonality condition for a filter bank can also be examined from a polyphase perspective. Consider the polyphase representation of the filter bank in

If Eð Þz in Figure 8 is type-I analysis polyphase matrix, and Rð Þz is type-II

<sup>T</sup> <sup>¼</sup> <sup>E</sup> <sup>z</sup><sup>2</sup> ð Þ <sup>1</sup> <sup>z</sup>�<sup>1</sup> � �<sup>T</sup>

<sup>½</sup> <sup>F</sup>0ð Þ<sup>z</sup> <sup>F</sup>1ð Þ<sup>z</sup> � ¼ <sup>z</sup>�<sup>1</sup> <sup>1</sup> � �<sup>R</sup> <sup>z</sup><sup>2</sup> ð Þ (23)

characterization of a filter bank with orthogonality and symmetry.

½ � H0ð Þz H1ð Þz

onality condition to hold.

Wavelet Transform and Complexity

symmetry condition:

Figure 8.

26

Polyphase implementation of filter bank.

Figure 5 as illustrated in Figure 8 [13].

synthesis polyphase matrix, then [13]:

through the following expressions [12]:

� � (19)

9 >=

>;

P zð Þþ Pð Þ¼ �z 2 (21)

<sup>H</sup>0ð Þ¼ <sup>z</sup> <sup>z</sup>�ð Þ <sup>L</sup>�<sup>1</sup> <sup>H</sup><sup>0</sup> <sup>z</sup>�<sup>1</sup> � � (22)

(20)

In the filter bank implementation of a wavelet transform, the paraunitary condition plays the critical role of guaranteeing the generation of orthonormal wavelets, and also perfect recovery of a decomposed signal. The paraunitary condition guarantees that recovered signal will suffer no phase or aliasing effect if a filter bank satisfies the paraunitary condition [14].

Given a polyphase transfer function matrix Eð Þz , the paraunitary condition is established by the matrix iff [15]:

$$\mathbf{E}^H(z^{-1})\mathbf{E}(z) = \mathbf{I} \tag{25}$$

where the H superscript denotes the conjugated transpose, and I denotes the identity matrix. Paraunitary filter banks also have an attractive property of losslessness, which implies that for every frequency, the total signal power is conserved [16]. From this property [17], any M � M real-coefficient lossless matrix with N � 1 degree can be realized using the structure shown in Figure 9 [18].

If the real-coefficient lossless matrix is denoted by Eð Þz ; then the matrix is said to have a special case of lossless degree of one iff it can be characterized by the relationship [18]:

$$\mathbf{E}(z) = \left[\mathbf{I} - \mathbf{v}\mathbf{v}^+ + z^{-1}\mathbf{v}\mathbf{v}^+\right]\mathbf{R} \tag{26}$$

where R is an arbitrary M � M unitary matrix and v is an M � 1 column vector with unit norm. From Eq. (26), the paraunitary condition for a filter bank is obtained as follows [18]:

$$\left[\mathbf{I} - \mathbf{v}\_k \mathbf{v}\_k^+ + \mathbf{v}\_k \mathbf{v}\_k^+ z\right] \mathbf{E}\_k(z) = \mathbf{E}\_{k-1}(z) \tag{27}$$

Figure 9. Cascade implementation of Eð Þz as FIR lossless unitary matrices separated by delays.

Having looked at the filter bank and its three important properties for the design of a wavelet, we will in the next section examine the application of these properties in the design of a wavelet.
