Draw the FWT filter bank required to compute the transform in Problem 7.16. Label all inpu...

Draw the FWT filter bank required to compute the transform in Problem 7.16. Label all inputs and outputs with the appropriate sequences.

7.16 The DWT in Eqs. (7.3-5) and (7.3-6) is a function of starting scale j0.

(7.3-5)

(7.3-6)

(a) Recompute the one-dimensional DWT of function f(n) = {1, 4, −3, 0} for 0 ≤ n ≤ 3 in Example 7.8 with j0 = 1 (rather than 0).

EXAMPLE 7.8: Computing a one-dimensional discrete wavelet transform.

To illustrate the use of Eqs. (7.3-5) through (7.3-7), consider the discrete function of four points: f(0) = 1,f(1) = 4, f(2) = −3, and f(3) = 0. Because M = 4, J = 2 and, with j0 = 0, the summations are performed over x = 0, 1, 2, 3, j = 0, 1, and k = 0 for j = 0 or k = 0, 1 for j = 1. We will use the Haar scaling and wavelet functions and assume that the four samples of f(x) are distributed over the support of the basis functions, which is 1 in width. Substituting the four samples into Eq. (7.3-5), we find that

(7.3-7)

because φ0,0(n) = 1 for n = 0, 1, 2, 3. Note that we have employed uniformly spaced samples of the Haar scaling function for j = 0 and k = 0. The values correspond to the first row of Haar transformation matrix H4 of Section 7.1.3. Continuing with Eq. (7.3-6) and similarly spaced samples of ψj,k (x), which correspond to rows 2,3, and 4 of H4, we get

Thus, the discrete wavelet transform of our simple four-sample function relative to the Haar wavelet and scaling function is , where the transform coefficients have been arranged in the order in which they were computed.

Equation (7.3-7) lets us reconstruct the original function from its transform. Iterating through its summation indices, we get

for n = 0, 1, 2, 3. If n = 0, for instance,

As in the forward case, uniformly spaced samples of the scaling and wavelet functions are used in the computation of the inverse.

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(b) Use the result from (a) to compute f(1) from the transform values.