5. Filter bank design of a wavelet transform

The filter bank design of a wavelet transform is usually implemented from the analysis filter bank segment to the synthesis filter bank segment.

### 5.1 Analysis filter bank in wavelet transform design

Given that the expression for a scaling function φ½ � n is the series sum of the shifted versions of φ½ � 2n , then according to [15, 16], φ½ � n can be represented as:

$$\varrho[n] = \sum\_{k} h[k] \sqrt{2} \rho(2n - k), \forall k \in \mathbb{Z} \tag{28}$$

f n½ �¼ <sup>1</sup>

yields [14]:

Figure 10.

29

State chart for wavelet design procedure.

ffiffiffiffiffi M p ∑ ∞ β¼�∞

f n½ �¼ <sup>1</sup>

λαþ1,βφαþ1, <sup>β</sup>½ � n !

Analysis of Wavelet Transform Design via Filter Bank Technique

λα,β2<sup>α</sup>=<sup>2</sup>

λαþ1,<sup>β</sup> ¼ ∑

m

For the next scale, Eq. (34) becomes:

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

ffiffiffiffiffi M p ∑ β

<sup>¼</sup> <sup>1</sup> ffiffiffiffiffi M p ∑ ∞ β¼�∞

<sup>φ</sup> <sup>2</sup><sup>α</sup> ½ �þ <sup>n</sup> � <sup>β</sup> <sup>∑</sup>

β

m

!

Substituting Eqs. (28) and (31) into Eq. (35) and after algebraic manipulations

λα, <sup>β</sup>h½ �þ β � 2m ∑

γα, <sup>β</sup>2<sup>α</sup>=<sup>2</sup>

λαþ1, <sup>β</sup>

ffiffiffiffiffiffiffiffiffi <sup>2</sup><sup>α</sup>þ<sup>1</sup> <sup>p</sup>

<sup>n</sup> � <sup>β</sup> � � !

<sup>ψ</sup> <sup>2</sup><sup>α</sup> ½ � <sup>n</sup> � <sup>β</sup>

γα, <sup>β</sup>g½ � β � 2m (36)

ψ 2<sup>α</sup>þ<sup>1</sup>

(34)

(35)

where h k½ � denotes the scaling coefficients. If n is transformed such that <sup>n</sup> ! <sup>2</sup><sup>α</sup><sup>n</sup> � <sup>β</sup>, then the relationship in Eq. (28) becomes [14]:

$$\log[2^a n - \beta] = \sum\_{k} h[k] \sqrt{2} \rho[2(\mathbb{Z}^a n - \beta) - k] \tag{29}$$

which translates into:

$$\rho[\mathfrak{Z}^a n - \beta] = \sum\_{m=2\beta+k} h[m - 2\beta] \sqrt{2} \rho \left[ \mathfrak{Z}^{a+1} n - m \right] \tag{30}$$

when k ¼ m � 2β.

In a similar consideration to Eq. (28), the wavelet function ψ½ � n can be represented as [19]:

$$\Psi[n] = \sum\_{k} \mathbf{g}[k] \sqrt{2} \rho(2n - k), \forall k \in \mathbb{Z} \tag{31}$$

where g k½ � denotes the wavelet coefficients. Also, if n is transformed such that <sup>n</sup> ! <sup>2</sup><sup>α</sup><sup>n</sup> � <sup>β</sup>, then the relationship in Eq. (31) becomes [14]:

$$\log[\mathfrak{L}^{a}n - \beta] = \sum\_{k} \mathfrak{g}[k] \sqrt{\mathfrak{L}} \rho[\mathfrak{L}(\mathfrak{L}^{a}n - \beta) - k] \tag{32}$$

which translates into:

$$\left[\upmu\left[2^{a}n-\beta\right] = \sum\_{m=2\beta+k} \lg\left[m-2\beta\right] \sqrt{2}\rho\left[2^{a+1}n-m\right] \tag{33}$$

when k ¼ m � 2β.

### 5.2 Synthesis filter bank in wavelet transform design

In the synthesis filter bank, the reconstruction of the original coefficients of a signal can be achieved through the combination of the scaling and wavelet function coefficients at a coarse level of resolution. Given a signal at α þ 1 scaling space f n½ �∈V<sup>α</sup>þ1, then according to [16, 17], the reconstruction is derived as follows:

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

$$f[n] = \frac{1}{\sqrt{M}} \left( \sum\_{\beta = -\infty}^{\infty} \lambda\_{a+1, \beta} \varphi\_{a+1, \beta}[n] \right) = \frac{1}{\sqrt{M}} \left( \sum\_{\beta = -\infty}^{\infty} \lambda\_{a+1, \beta} \sqrt{2^{a+1}} \varphi \left[ 2^{a+1} n - \beta \right] \right) \tag{34}$$

For the next scale, Eq. (34) becomes:

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

The filter bank design of a wavelet transform is usually implemented from the

Given that the expression for a scaling function φ½ � n is the series sum of the shifted versions of φ½ � 2n , then according to [15, 16], φ½ � n can be represented as:

<sup>p</sup> <sup>φ</sup>ð Þ <sup>2</sup><sup>n</sup> � <sup>k</sup> , <sup>∀</sup>k<sup>∈</sup> <sup>ℤ</sup> (28)

<sup>p</sup> <sup>φ</sup> 2 2<sup>α</sup> ½ � ð Þ� <sup>n</sup> � <sup>β</sup> <sup>k</sup> (29)

<sup>p</sup> <sup>φ</sup>ð Þ <sup>2</sup><sup>n</sup> � <sup>k</sup> , <sup>∀</sup>k<sup>∈</sup> <sup>ℤ</sup> (31)

<sup>p</sup> <sup>φ</sup> 2 2<sup>α</sup> ½ � ð Þ� <sup>n</sup> � <sup>β</sup> <sup>k</sup> (32)

<sup>n</sup> � <sup>m</sup> � � (33)

<sup>n</sup> � <sup>m</sup> � � (30)

h k½ � ffiffi 2

where h k½ � denotes the scaling coefficients. If n is transformed such that

h k½ � ffiffi 2

h m½ � � <sup>2</sup><sup>β</sup> ffiffi

2 <sup>p</sup> <sup>φ</sup> <sup>2</sup><sup>α</sup>þ<sup>1</sup>

k

m¼2βþk

In a similar consideration to Eq. (28), the wavelet function ψ½ � n can be

g k½ � ffiffi 2

k

m¼2βþk

where g k½ � denotes the wavelet coefficients. Also, if n is transformed such that

g k½ � ffiffi 2

g m½ � � <sup>2</sup><sup>β</sup> ffiffi

In the synthesis filter bank, the reconstruction of the original coefficients of a signal can be achieved through the combination of the scaling and wavelet function coefficients at a coarse level of resolution. Given a signal at α þ 1 scaling space f n½ �∈V<sup>α</sup>þ1, then according to [16, 17], the reconstruction is derived as follows:

2 <sup>p</sup> <sup>φ</sup> <sup>2</sup><sup>α</sup>þ<sup>1</sup>

in the design of a wavelet.

Wavelet Transform and Complexity

which translates into:

when k ¼ m � 2β.

which translates into:

when k ¼ m � 2β.

28

represented as [19]:

5. Filter bank design of a wavelet transform

5.1 Analysis filter bank in wavelet transform design

φ½ �¼ n ∑ k

<sup>n</sup> ! <sup>2</sup><sup>α</sup><sup>n</sup> � <sup>β</sup>, then the relationship in Eq. (28) becomes [14]:

<sup>φ</sup> <sup>2</sup><sup>α</sup> ½ �¼ <sup>n</sup> � <sup>β</sup> <sup>∑</sup>

<sup>φ</sup> <sup>2</sup><sup>α</sup> ½ �¼ <sup>n</sup> � <sup>β</sup> <sup>∑</sup>

ψ½ �¼ n ∑ k

<sup>n</sup> ! <sup>2</sup><sup>α</sup><sup>n</sup> � <sup>β</sup>, then the relationship in Eq. (31) becomes [14]:

<sup>φ</sup> <sup>2</sup><sup>α</sup> ½ �¼ <sup>n</sup> � <sup>β</sup> <sup>∑</sup>

<sup>ψ</sup> <sup>2</sup><sup>α</sup> ½ �¼ <sup>n</sup> � <sup>β</sup> <sup>∑</sup>

5.2 Synthesis filter bank in wavelet transform design

analysis filter bank segment to the synthesis filter bank segment.

$$f[n] = \frac{1}{\sqrt{M}} \left( \sum\_{\beta} \lambda\_{a,\beta} \mathfrak{L}^{a/2} \varrho[\mathfrak{L}^a n - \beta] + \sum\_{\beta} \chi\_{a,\beta} \mathfrak{L}^{a/2} \mathfrak{w}[\mathfrak{L}^a n - \beta] \right) \tag{35}$$

Substituting Eqs. (28) and (31) into Eq. (35) and after algebraic manipulations yields [14]:

$$
\lambda\_{a+1,\beta} = \sum\_{m} \lambda\_{a,\beta} h[\beta - 2m] + \sum\_{m} \chi\_{a,\beta} \mathbf{g}[\beta - 2m] \tag{36}
$$

Figure 10. State chart for wavelet design procedure.
