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   -> Volume 5, Issue 3


Answer: Mistake in table in Chui's book? (WD 5.2 #24)
 
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goswami@decwa.ece.uiuc.edu (Jaideva C. Goswami)
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PostPosted: Wed Dec 04, 2002 9:51 am    
Subject: Answer: Mistake in table in Chui's book? (WD 5.2 #24)
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Answer: Mistake in table in Chui's book? (WD 5.2 #24)

This is in response to a question regarding accurarcy of values of
the decomposition sequence for linear spline case in Chui's book
(WD 5.2 #24). There is no mistake in the table. Here is a simple way
to verify the accuracy (equation numbers mentioned below are from
Chui's book).

(1) Coefficients {a_k} and {g_k} are related (see, for example,
eqn. 6.5.2).

(2) Consider 6.2.1 (this equation is equivalent to 6.5.1).

(3) Expression for E_phi and P(z) are known, e.g., for m = 2 (linear
spline)

6z E_phi(z) = 1 + 4z + z^2 ; 4P(z) = 1 + 2z + z^2.

(4) Write 1 / E_phi(z) = sum_k r_k z^k. Expressions for r_k for a spline
of order m can be seen in [1,2]. In particluar, for m = 2

r_k = r_{-k} = (-1)^k sqrt(3) (2 - sqrt(3))^k ; k = 0, 1, ....

(5) The coefficients {g_k} and hence {a_k} can be found by comparing
the powers of z on both sides of 6.2.1.

(6) Finally, observe that sum_k a_k = 1 (can be verified using 6.2.1 and the
partition of unity property of spline). If we simply add all the
coefficients in the table given in Chui's book, we obtain 1. While
adding we must take care of the negative indices. More precisely,
for the present case

sum_k a_k = a_1 + 2 sum_{k=2}^{k=21} a_k = 1.0

From the above also it is clear that a_1 could not be 0.4330127 as
mentioned in the question if the rest of the coefficients are correct.

[1] Jaideva C. Goswami, Applications of Semi-Orthogonal Spline Wavelets in
Electromagnetics and Microwave Problems, Ph.D. Dissertation, Texas A&M
University, College Station, August 1995.

[2] J. C. Goswami, A. K. Chan, and C. K. Chui, ``On a spline-based fast
integral wavelet transform algorithm," in Ultra-wideband short pulse
electromagnetics 2, L. Carin and L. B. Felsen, Eds., New York: Plenum
Press, pp. 455-463, 1995.

Jaideva C. Goswami

Jaideva C. Goswami, Ph.D.
Electromagnetic Communication Laboratory
Department of Electrical and Computer Engineering
University of Illinois
1406 W. Green Street, Urbana, IL 61801-2991

Phone (217)-244-0178 (Work) (217)-344-8480 (Home)
Fax (217)-333-8986 e-mail: goswami@decwa.ece.uiuc.edu
All times are GMT + 1 Hour
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