Wavelet Functions

We denote by $W_j$ the difference between the function space $V_{j+1}$ spanned by scaling functions $\varphi_{j+1,k}$ and the function $V_j$ spanned by $\varphi_{j,k}$, i.e.

$$V_{j+1}=V_j \oplus W_j$$

(where $\oplus$ represents the union of the two spaces). $W_j$ is composed of all functions representable in $V_{j+1}$ but not representable in $V_j$ (as the scale of the basis functions of $V_j$ is too coarse for the details of these functions). This can be carried out recursively to get:

$$V_{j+2}=V_{j+1} \oplus W_{j+1}=V_j \oplus W_j \oplus W_{j+1}$$

and finally:

$$L^2(R)=V_{\infty}=V_j \oplus W_j \oplus W_{j+1} \oplus W_{j+2} \oplus \cdots$$

Similar to a function space $V_j$ spanned by the scaling functions $\varphi_{j,k}(t)$, the function space $W_j$ is also spanned by a set of basis function, called the wavelet functions:

$$\psi_{j,k}(t)=2^{j/2}\psi(2^jt-k)$$

We see that the scaling sequence and the wavelet sequence correspond to low-pass filter and band-pass filter, respectively.

Also, as the wavelet functions $\psi_{j,k}(t)$ are members of the space $W_j$ they span as well as all the super-spaces:

$$\psi_{j,k}(t) \in W_j \subset V_{j+1} \subset \cdots$$

they can be expanded in the space $V_{j+1}$ of next higher scale with doubled resolution:

$$\psi_{j,k}(t) =\sum_l h_{\psi}[l] \varphi_{j+1,l}(t) =\sum_l h_{\psi}[l] 2^{(j+1)/2} \varphi(2^{j+1}t-l)$$

where $h_{\psi}$ are the expansion coefficients. Usually we let $j=0$ and drop the subscripts $j$ and $k$ to indicate that any wavelet function $\psi_j(t)$ can be expressed as a linear combination of the basis scaling functions $\varphi_{j+1}(t)$ of the space with the next higher resolution:

$$\psi(t)=\sum_l h_{\psi}[l]\sqrt{2} \varphi(2t-l)$$

This is in the same form for the scaling functions:

$$\varphi(t)=\sum_l h_{\varphi}[l]\sqrt{2} \varphi(2t-l)$$

It can be shown that the coefficients of the two expressions are related

$$h_{\psi}[l]=(-1)^l h_{\varphi}[1-l]$$

Examples: The Haar scaling function $\varphi(t)$ is a square pulse with unit height and width, and the coefficients are $h_{\varphi}[0]=h_{\varphi}[1]=1/\sqrt{2}$. Now the coefficients for the wavelet functions can be obtained as

$$\begin{array}{l} h_{\psi}[0]=(-1)^0 h_{\varphi}[1-l]=1/\sqrt{... ... h_{\psi}[1]=(-1)^1 h_{\varphi}[1-l]=-1/\sqrt{2} \end{array}$$

and the wavelet function is:

$$\psi(t)=\sum_l h_{\psi}[l]\sqrt{2}\varphi[2t-l] =\frac{1}{\s... ...-1 & 0.5 \le t < 1 0 & \mbox{otherwise} \end{array} \right.$$

where the two coefficients $h_{\psi}[0]=-h_{\psi}[0]=1/\sqrt{2}$ are the second row of the Haar matrix ${\bf H}_2$.

The first two panels show the wavelet functions of scale $j=0$: $\psi(t)=\psi_{0,0}(t)$ and $\psi_{0,2}(t)=\psi(t-2)$. Note $\varphi_{1,k}(t)=\sqrt{2}\varphi_{0,k}(2t)$ can be generated by the linear combination of $\varphi_{0,k}(t)$ and $\psi_{0,k}(t)$:

$$\varphi_{1,k}(t)=\frac{\sqrt{2}}{2}[\varphi_{0,k}(t)+\psi_{0,k}(t)]$$

The 3rd panel shows the wavelet functions of scale $j=1$: $\psi_{1,0}(t)=\sqrt{2}\psi(2t)$. The 4th panel shows a function in space $V_0$ spanned by $\varphi_{0,k}(t)$. The 5th panel shows a function in space $W_0$ spanned by $\psi_{0,k}(t)$. The 6th panel shows a function in space $V_1=V_0 \oplus W_0$, a linear combination of $\varphi_{1,k}(t)$, or $\varphi_{0,k}(t)$ and $\psi_{0,k}(t)$.