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This module shows how to compute the scaling function. It also has a section with a proof for an assumption made for the computation.

Given coefficients h n that satisfy the regularity conditions, we can iteratively calculate samples of φ t on a fine grid of points t using the cascade algorithm . Once we have obtained φ t , the wavelet scaling equation can be used to construct ψ t .

In this discussion we assume that H z is causal with impulse response length N . Recall, from our discussion of the regularity conditions , that this implies φ t will have compact support on the interval 0 N 1 . The cascade algorithm is described below.

  1. Consider the scaling function at integer times t m 0 N 1 : φ m 2 n 0 N 1 h n φ 2 m n Knowing that φ t 0 for t 0 N 1 , the previous equation can be written using an N x N matrix. In the case where N 4 , we have
    φ 0 φ 1 φ 2 φ 3 2 h 0 0 0 0 h 2 h 1 h 0 0 0 h 3 h 2 h 1 0 0 0 h 3 φ 0 φ 1 φ 2 φ 3
    where H h 0 0 0 0 h 2 h 1 h 0 0 0 h 3 h 2 h 1 0 0 0 h 3 The matrix H is structured as a row-decimated convolution matrix . From the matrix equation above ( [link] ), we see that φ 0 φ 1 φ 2 φ 3 must be (some scaled version of) the eigenvector of H corresponding to eigenvalue 2 -1 . In general, the nonzero values of φ n n , i.e. , φ 0 φ 1 φ N 1 , can be calculated by appropriately scaling the eigenvector of the N x N row-decimated convolution matrix H corresponding to the eigenvalue 2 -1 . It can be shown that this eigenvector must be scaled so that n 0 N 1 φ n 1 .
  2. Given φ n n , we can use the scaling equation to determine φ n 2 n :
    φ m 2 2 n 0 N 1 h n φ m n
    This produces the 2 N 1 non-zero samples φ 0 φ 1 2 φ 1 φ 3 2 φ N 1 .
  3. Given φ n 2 n , the scaling equation can be used to find φ n 4 n :
    φ m 4 2 n 0 N 1 h n φ m 2 n 2 p p even h p 2 φ m p 2 2 p p h 2 p φ 1 2 m p
    where h 2 p denotes the impulse response of H z 2 , i.e. , a 2-upsampled version of h n , and where φ 1 2 m φ m 2 . Note that φ n 4 n is the result of convolving h 2 n with φ 1 2 n .
  4. Given φ n 4 n , another convolution yields φ n 8 n :
    φ m 8 2 n 0 N 1 h n φ m 4 n 2 p p h 4 p φ 1 4 m p
    where h 4 n is a 4-upsampled version of h n and where φ 1 4 m φ m 4 .
  5. At the th stage, φ n 2 is calculated by convolving the result of the 1 th stage with a 2 1 -upsampled version of h n :
    φ 1 2 m 2 p p h 2 1 p φ 1 2 1 m p
For 10 , this gives a very good approximation of φ t . At this point, you could verify the key properties of φ t , such as orthonormality and the satisfaction of the scaling equation.

In [link] we show steps 1 through 5 of the cascade algorithm, as well as step 10, using Daubechies'db2 coefficients (for which N 4 ).

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Source:  OpenStax, Dspa. OpenStax CNX. May 18, 2010 Download for free at http://cnx.org/content/col10599/1.5
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