# Download Ordinary Differential Equations (Classics in Applied by George F. Carrier PDF

By George F. Carrier

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Example text

5 Signal compression 21 to produce the next higher resolution averaged signal (5, 5, 11, 11, 7, 7, 5, 5), and the second addition then supplies enough further details to produce the full resolution signal f . In general, if the number N of signal values is divisible k times by 2, then a k-level MRA: f = Ak + Dk + · · · + D2 + D1 can be performed on the signal f . Rather than subjecting the reader to the gory details, we conclude by describing a computer example generated using FAWAV. 1(a). This signal has 210 values so 10 levels of MRA are possible.

3 This © 2008 by Taylor & Francis Group, LLC ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ 2. 6). 3 For each of the signals f given below, compute an approximate signal e f by inverse transforming the compressed Haar transform (a1 | 0, . . , 0) obtained by setting all the ﬂuctuation values equal to zero. In each case, ﬁnd the largest error between each value of f and e f. 3 above. When will there be a diﬀerence between a value of f and a value of the approximate signal e f , and when will the two signals’ values be the same?

1 VN/2 = 1 1 0, 0, . . , 0, √ , √ 2 2 . 11) Using these Haar scaling signals, the values a1 , . . 12) for m = 1, 2, . . , N/2. The Haar scaling signals are quite similar to the Haar wavelets. They all have energy 1 and have a support consisting of just two consecutive timeindices. In fact, they are all translates by an even multiple of time-units of the ﬁrst scaling signal V11 . Unlike the Haar wavelets, however, the average values of the Haar √scaling signals are not zero. In fact, they each have an average value of 1/ 2.