By Jaideva C. Goswami, Andrew K. Chan
Such a lot latest books on wavelets are both too mathematical or they specialize in too slender a uniqueness. This ebook presents an intensive therapy of the topic from an engineering viewpoint. it's a one-stop resource of idea, algorithms, purposes, and laptop codes with regards to wavelets. This moment version has been up-to-date via the addition of:
- a part on "Other Wavelets" that describes curvelets, ridgelets, lifting wavelets, and so forth
- a part on lifting algorithms
- Sections on facet Detection and Geophysical functions
- Section on Multiresolution Time area procedure (MRTD) and on Inverse difficulties
Read or Download Fundamentals of Wavelets Theory, Algorithms, and Applications PDF
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Extra info for Fundamentals of Wavelets Theory, Algorithms, and Applications
Sin θ When θ = 0, since the right side is in the indeterminate form 0/0, we apply L H ospital s rule to determine the limit: sin2 nθ 2n sin nθ cos nθ = lim = 2n sin 0 = 0. θ→0 sin θ θ→0 cos θ lim Hence the two sides are equal at θ = 0 by the limit convention. 24) determine the closed-form sums of the following cosine and sine series: n cos θ = 1 + cos θ + · · · + cos nθ =? 25a) =0 n sin θ = sin θ + sin 2θ + · · · + sin nθ =? 25b) as the real and imaginary parts: n n =0 n ej z = =0 θ = n cos θ + j sin θ = =0 n cos θ + j =0 sin θ .
6 9 = 3 3 . It can be easily veri ed that y(t + To ) = y(t). , y(t + T ) = y(t). Since we have uniform spacing f = fk+1 − fk = 1/T , we may still plot Ak and Bk versus k with the understanding that k is the index of equispaced fk ; of course, one may plot Ak and Bk versus the values of fk if that is desired. 7. REVIEW OF RESULTS AND TECHNIQUES 13 3. A non-commensurate y(t) is not periodic, although all its components are periodic. For example, the function √ y(t) = sin(2πt) + 5 sin(2 3πt) √ is not periodic because f1 = 1 and f2 = 3 are not commensurate.
Note that f1 is the GCD (greatest common divisor) of the individual frequencies. In general, fk = k, and we need to distinguish periodic y(t) from non-periodic y(t) by examining its frequency contents. The conditions and results are given below. 1. The function y(t) is said to be a commensurate sum if the ratio of any two individual periods (or frequencies) is a rational fraction ratio of integers with common factors canceled out. 9 Hz, and the ratio fα /fβ = 2/3 is a rational fraction. 2. A commensurate y(t) is periodic with its fundamental frequency being the GCD of the individual frequencies and its common period being the LCM of the individual periods.