We know, in fact, wavelet analysis to decompose signal is "rough" and "fine" in two parts. The "coarse" part change slowly acquires "coarse" is understood to low-pass filtering components; corresponding obtain "fine" component is understood to high-pass filtering.
To be able to break down a level to continue this, we need to define a sub-space sequence $ V_j $ satisfy the following conditions:
(Nestedness) $ V_j \ subset V_ {j + 1} $
(Denseness) $ \ overline {\ cup {V_j}} = L ^ 2 (R) $
(Discreteness) $ \ cap {V_j} = {0} $
(Scale of) $ f (x) \ in V_j \ Longleftrightarrow f (2 ^ {- j} x) \ in V_0 $
(Orthonormal basis) present function $ \ phi \ in V_0 $, $ \ {\ phi (xk); k \ in Z \} $ a $ $ orthonormal basis V_0
From a practical point of view, the most useful of a limited class of scaling function is supported, but this is not a theoretical limit.
Spatial sequence satisfying the above condition $ \ {V_j; j \ in Z \} $ $ and the corresponding function \ Phi $ $ scale function is called by the multi-resolution analysis Phi $ \.
Theorem 1. set $ \ {V_j, j \ in Z \} $ by a scaling function $ \ $ Phi multiresolution analysis, then for any $ j \ in Z $, function set
$\{\phi_{jk} (x) = 2^{j/2} \phi(2^j x-k); k \in Z\}$
$ $ V_J is an orthonormal basis.
Proved ideas: to consider using the scale characteristic v_J $ $ claimant function may be written as $ \ {\ phi (2 ^ {- j} x - k); k \ in Z \} $ linear combination. Then directly demonstrated using the definition of an orthonormal $ \ {\ phi_ {jk} ; k \ in Z \} $ is orthonormal.
Theorem 2. (two-scale relation theorem) set $ \ {V_j, j in Z \ \} $ is in accordance with a scale function $ \ phi $ multi-resolution analysis, the following relationship holds scales:
$ \ Phi (x) = \ sum \ limits_ {k \ in Z} p_k \ phi (2x-k) $, $ p_k = 2 \ int _ {- \ infty} ^ {\ infty} \ phi (x) \ overline {\ phi (2x-k)} dx $
Further, there are $ \ phi_ {j-1, l} = 2 ^ {- 1/2} \ sum \ limits_k p_ {k-2l} \ phi_ {jk} $
Note that some materials will P_K $ $ standardization, this time in front of the equation coefficients corresponding change.
Interpretation and the Proof of: taking into account of the aforementioned theorem of nested space, $ \ phi (x) $ always be written $ \ phi (2x) $, and a linear combination of shifts. Each entry is the linear coefficient of $ \ phi (x) $ orthonormal basis of a projection on the space $ \ {V_1 \} $ a. The $ X $ replace $ 2 ^ {jl} xl $ available license further conclusions. Intuitively may further Conclusion: Basis Function and shift functions $ \ phi (2 ^ jx - k) $ remains unchanged, but it will shift each coefficient to $ p_ {k-2l} $ , to accumulate function obtained after a shift $ \ phi_ {j-1, l} $. Because the $ v_J $ $ V_ {j-1} $ is included with the relation is contained, so that both ends of the shift length equals further conclusions were $ L $ and $ 2l $.
Parseval identity
Let V be a complex inner product space, which is an orthonormal basis $ \ {u_k \} $. If $ f \ in V, g \ in V $, $ f $ $ G $ and formula is represented as follows:
$f=\sum\limits_{k=1}^{\infty} a_k u_k$
$g=\sum\limits_{k=1}^{\infty} b_k u_k$
Then
$\langle f,g \rangle = \sum\limits_{k=1}^{\infty} a_k \overline{b_k}$
Theorem 3. disposed $ \ {V_j; j \ in Z \} $ by a scaling function $ \ $ Phi multiresolution analysis, $ p_k $ preceding theorem. Then the following equation holds:
1 $ \ sum \ limits_ {k \ and Z} P_ {k-2l} \ overline {p_k} = 2 \ delta_ {l0} $
2. $\sum\limits_{k\in Z} |p_k|^2=2$
3. $\sum\limits_{k\in Z} p_k = 2$
4. $\sum\limits_{k\in Z} p_{2k}=1$，$\sum\limits_{k\in Z} p_{2k+1} = 1$
Thinking interpretation and proof: Formula 1 demonstrate the use of the Parseval identity and $ \ {\ phi (xk) \} $ can be orthonormal. Then allowed $ l = 0 $ formula 2. Prove the rest of the equation reference textbooks.
Theorem 4. The set $ \ {V_j; j \ in Z \} $ by a scaling function $ \ $ Phi multi-resolution analysis, and $ \ phi = \ sum \ limits_k p_k \ phi (2x-k) $. Order $ \ psi (x) = \ sum \ limits_ {k \ in Z} (-1) ^ k \ overline {p_ {1-k} \ phi (2x-k)} $, then $ W_j \ subset V_ { j + 1} $ a $ V_ {j + 1} $ $ v_J the orthogonal complement $ and $ \ {\ psi_ {jk} (x) = 2 ^ {j / 2} \ psi (2 ^ jx- k), k \ in Z \ } $ $ a $ w_j an orthonormal basis.
Interpretation and the Proof of: if the $ P_K $ regarded as a low-pass filter coefficient, the signal after the decomposition of $ \ phi (x) $ inner support section for a "smoother" than the original signal, then the corresponding high-pass filter coefficients need to be more " severe jitter "is the original filter, and" full complement. " Multiplied by the coefficient $ (- 1) ^ k $ and reverse order to achieve this effect. $ p $ subscript reverse $ 1-k $ is further such that $ \ langle \ phi, \ psi \ rangle $ during operation may be offset $ p_mp_n $ twenty-two ultimately achieve orthogonal.
Theorem 5. wavelet function set $ \ {\ psi_ {jk} \} $ a $ L ^ 2 (R) $ an orthonormal basis.
Decomposition and Reconstruction:
After obtaining the \ relationship with {jk} $ $ \ phi_ {j-1, k} $ and $ \ psi_ {j-1, k} $ a $ phi_, we can further consider the signal decomposition and reconstruction.
If the function is $ f $ $ $ v_J in, we have: $ f = \ sum \ limits_ {k \ in Z} \ langle f, \ phi_ {jk} \ rangle \ phi_ {jk} $
break down:
$f=\sum\limits_{k\in Z} \langle f, \phi_{j-1,k} \rangle \phi_{j-1,k} + \sum\limits_{k\in Z} \langle f, \psi_{j-1,k} \rangle \psi_{j-1,k} $
$\langle f, \phi_{j-1,l} \rangle = 2^{-1/2} \sum\limits_{k \in Z} \overline{p_{k-2l}} \langle f, \phi_{jk} \rangle $
$\langle f, \psi_{j-1,l} \rangle = 2^{-1/2} \sum\limits_{k \in Z} (-1)^k p_{1-k+2l} \langle f, \phi_{jk} \rangle $
Reconstruction:
$\langle f, \phi_{jk} \rangle = 2^{-1/2}\sum\limits_{l \in Z} p_{k-2l} \langle f, \phi_{j-1,l} \rangle + 2^{-1/2}\sum\limits_{l \in Z} (-1)^k \overline{p_{1-k+2l}} \langle f, \psi_{j-1,l} \rangle $
Explain and demonstrate ideas:
Use Parseval identity and scale relationships.