Download Engineer's Mini-Notebook: Magnet and Magnet Sensor Projects by Forest M. Mims II PDF
By Forest M. Mims II
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Additional resources for Engineer's Mini-Notebook: Magnet and Magnet Sensor Projects
Example text
41). 9. 9 Solution of the dilation equation for N = 1, that is, L = 2, resulting in the Haar wavelet. 53) −∞ For N = 3, we obtain the case presented earlier, the third-order Daubechies filter. 38). 10. 11 the scaling function and the wavelet for N = 1, 3, 5, 7, and 9 are illustrated. We are not going to delve further into the solution of the dilation equation because for the electromagnetics problems in which we are interested namely, solution of large matrix equations the scaling functions and the wavelets are really not necessary because the discrete wavelet representation can be carried out from the knowledge of only h(m)!
However, for the discrete wavelet transform, as we shall see, φ(t) and t) are not at all required in the numerical computation! 10 Scaling function and wavelet and their transforms corresponding to the third-order Daubechies filter. 11 Scaling functions and wavelets for different orders of Daubechies filters. 9 38 Wavelet Applications in Engineering Electromagnetics In summary, the mathematical basis of wavelets has been presented from a filter theory perspective. We have shown how to construct scaling functions φ and Z DYHOHW V VW DUW LQJ IURP ILOW HUV h (m) and utilizing the perfect reconstruction argument presented in subband filtering techniques.
Therefore we approximate the HG DQG VKLIW HG YHUVLRQ RI W KH VDP H IXQFW LRQ t). 64) where δpq represents a Dirac delta function, so that its value is unity when p = q and zero otherwise. ,WLVLQW HUHVW LQJ W R QRW HW KDW n,k(t KDV] HURGF YDOXH^ n,k(ω = 0) = 0}, whereas x(t) may not! 61), we can only talk about mean square convergence because the constant term is missing. 61) is finite, as we will see later. 65) where < •,• > defines the inner product. The terms XDWT(n,k) are the kth discrete wavelet coefficients at scale n of the function x(t) and are symbolically denoted Wavelets from an Electrical Engineering Perspective 39 by dk,n.