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Appendix B

Module by: C. Sidney Burrus. E-mail the author

In this appendix we develop most of the results on scaling functions, wavelets and scaling and wavelet coefficients presented in (Reference) and elsewhere. For convenience, we repeat (Reference), (Reference), (Reference), and (Reference) here

φ ( t ) = n h ( n ) 2 φ ( 2 t - n ) φ ( t ) = n h ( n ) 2 φ ( 2 t - n )
(1)
n h ( n ) = 2 n h ( n ) = 2
(2)

φ ( t ) φ ( t - k ) d t = E δ ( k ) = E i f k = 0 0 o t h e r w i s e φ ( t ) φ ( t - k ) d t = E δ ( k ) = E i f k = 0 0 o t h e r w i s e
(3)

If normalized

φ ( t ) φ ( t - k ) d t = δ ( k ) = 1 i f k = 0 0 o t h e r w i s e φ ( t ) φ ( t - k ) d t = δ ( k ) = 1 i f k = 0 0 o t h e r w i s e
(4)

The results in this appendix refer to equations in the text written in bold face fonts.

Equation (Reference) is the normalization of (Reference) and part of the orthonormal conditions required by Equation 3 for k=0k=0 and E=1E=1.

Equation (Reference) If the φ(x-k)φ(x-k) are orthogonal, Equation 3 states

φ ( x + m ) φ ( x ) d x = E δ ( m ) φ ( x + m ) φ ( x ) d x = E δ ( m )
(5)

Summing both sides over mm gives

m φ ( x + m ) φ ( x ) d x = E m φ ( x + m ) φ ( x ) d x = E
(6)

which after reordering is

φ ( x ) m φ ( x + m ) d x = E . φ ( x ) m φ ( x + m ) d x = E .
(7)

Using (Reference), Equation 21, and Equation 24 gives

φ ( x ) d x A 0 = E φ ( x ) d x A 0 = E
(8)

but φ(x)dx=A0φ(x)dx=A0 from Equation 19, therefore

A 0 2 = E A 0 2 = E
(9)

If the scaling function is not normalized to unity, one can show the more general result of (Reference). This is done by noting that a more general form of (Reference) is

m φ ( x + m ) = φ ( x ) d x m φ ( x + m ) = φ ( x ) d x
(10)

if one does not normalize A0=1A0=1 in Equation 20 through Equation 24.

Equation (Reference) follows from summing Equation 3 over mm as

m φ ( x + m ) φ ( x ) d x = φ ( x ) 2 d x m φ ( x + m ) φ ( x ) d x = φ ( x ) 2 d x
(11)

which after reordering gives

φ ( x ) m φ ( x + m ) d x = φ ( x ) 2 d x φ ( x ) m φ ( x + m ) d x = φ ( x ) 2 d x
(12)

and using Equation 10 gives (Reference).

Equation (Reference) is derived by applying the basic recursion equation to its own right hand side to give

φ ( t ) = n h ( n ) 2 k h ( k ) 2 φ ( 2 ( 2 t - n ) - k ) φ ( t ) = n h ( n ) 2 k h ( k ) 2 φ ( 2 ( 2 t - n ) - k )
(13)

which, with a change of variables of =2n+k=2n+k and reordering of operation, becomes

φ ( t ) = n h ( n ) h ( - 2 n ) 2 φ ( 4 t - ) . φ ( t ) = n h ( n ) h ( - 2 n ) 2 φ ( 4 t - ) .
(14)

Applying this jj times gives the result in (Reference). A similar result can be derived for the wavelet.

Equation (Reference) is derived by defining the sum

A J = φ ( 2 J ) A J = φ ( 2 J )
(15)

and using the basic recursive equation Equation 1 to give

A J = n h ( n ) 2 φ ( 2 2 J - n ) . A J = n h ( n ) 2 φ ( 2 2 J - n ) .
(16)

Interchanging the order of summation gives

A J = 2 n h ( n ) φ ( 2 J - 1 - n ) A J = 2 n h ( n ) φ ( 2 J - 1 - n )
(17)

but the summation over is independent of an integer shift so that using Equation 2 and Equation 15 gives

A J = 2 2 n h ( n ) φ ( 2 J - 1 ) = 2 A J - 1 . A J = 2 2 n h ( n ) φ ( 2 J - 1 ) = 2 A J - 1 .
(18)

This is the linear difference equation

A J - 2 A J - 1 = 0 A J - 2 A J - 1 = 0
(19)

which has as a solution the geometric sequence

A J = A 0 2 J . A J = A 0 2 J .
(20)

If the limit exists, equation Equation 15 divided by 2J2J is the Riemann sum whose limit is the definition of the Riemann integral of φ(x)φ(x)

lim J A J 1 2 J = φ ( x ) d x = A 0 . lim J A J 1 2 J = φ ( x ) d x = A 0 .
(21)

It is stated in (Reference) and shown in Equation 6 that if φ(x)φ(x) is normalized, then A0=1A0=1 and Equation 20 becomes

A J = 2 J . A J = 2 J .
(22)

which gives (Reference).

Equation Equation 21 shows another remarkable property of φ(x)φ(x) in that the bracketed term is exactly equal to the integral, independent of JJ. No limit need be taken!

Equation (Reference) is the “partitioning of unity" by φ(x)φ(x). It follows from (Reference) by setting J=0J=0.

Equation (Reference) is generalization of (Reference) by noting that the sum in (Reference) is independent of a shift of the form

φ ( 2 J - L 2 M ) = 2 J φ ( 2 J - L 2 M ) = 2 J
(23)

for any integers MJMJ and LL. In the limit as MM, L2ML2M can be made arbitrarily close to any xx, therefore, if φ(x)φ(x) is continuous,

φ ( 2 J - x ) = 2 J . φ ( 2 J - x ) = 2 J .
(24)

This gives (Reference) and becomes (Reference) for J=0J=0. Equation (Reference) is called a “partitioning of unity" for obvious reasons.

The first four relationships for the scaling function hold in a generalized form for the more general defining equation (Reference). Only (Reference) is different. It becomes

k φ ( k M J ) = M J k φ ( k M J ) = M J
(25)

for MM an integer. It may be possible to show that certain rational MM are allowed.

Equations (Reference), (Reference), and (Reference) are the recursive relationship for the Fourier transform of the scaling function and are obtained by simply taking the transform (Reference) of both sides of Equation 1 giving

Φ ( ω ) = n h ( n ) 2 φ ( 2 t - n ) e - j ω t d t Φ ( ω ) = n h ( n ) 2 φ ( 2 t - n ) e - j ω t d t
(26)

which after the change of variables y=2t-ny=2t-n becomes

Φ ( ω ) = 2 2 n h ( n ) φ ( y ) e - j ω ( y + n ) / 2 d y Φ ( ω ) = 2 2 n h ( n ) φ ( y ) e - j ω ( y + n ) / 2 d y
(27)

and using (Reference) gives

Φ ( ω ) = 1 2 n h ( n ) e - j ω n / 2 φ ( y ) e - j ω y / 2 d y = 1 2 H ( ω / 2 ) Φ ( ω / 2 ) Φ ( ω ) = 1 2 n h ( n ) e - j ω n / 2 φ ( y ) e - j ω y / 2 d y = 1 2 H ( ω / 2 ) Φ ( ω / 2 )
(28)

which is (Reference) and (Reference). Applying this recursively gives the infinite product (Reference) which holds for any normalization.

Equation (Reference) states that the sum of the squares of samples of the Fourier transform of the scaling function is one if the samples are uniform every 2π2π. An alternative derivation to that in Appendix A is shown here by taking the definition of the Fourier transform of φ(x)φ(x), sampling it every 2πk2πk points and multiplying it times its complex conjugate.

Φ ( ω + 2 π k ) Φ ( ω + 2 π k ) ¯ = φ ( x ) e - j ( ω + 2 π k ) x d x φ ( y ) e j ( ω + 2 π k ) y d y Φ ( ω + 2 π k ) Φ ( ω + 2 π k ) ¯ = φ ( x ) e - j ( ω + 2 π k ) x d x φ ( y ) e j ( ω + 2 π k ) y d y
(29)

Summing over kk gives

k | Φ ( ω + 2 π k ) | 2 = k φ ( x ) φ ( y ) e - j ω ( x - y ) e - j 2 π k ( x - y ) d x d y k | Φ ( ω + 2 π k ) | 2 = k φ ( x ) φ ( y ) e - j ω ( x - y ) e - j 2 π k ( x - y ) d x d y
(30)
= φ ( x ) φ ( y ) e j ω ( y - x ) k e j 2 π k ( y - x ) d x d y = φ ( x ) φ ( y ) e j ω ( y - x ) k e j 2 π k ( y - x ) d x d y
(31)
= φ ( x ) φ ( x + z ) e j ω z k e j 2 π k z d x d z = φ ( x ) φ ( x + z ) e j ω z k e j 2 π k z d x d z
(32)

but

k e j 2 π k z = δ ( z - ) k e j 2 π k z = δ ( z - )
(33)

therefore

k | Φ ( ω + 2 π k ) | 2 = φ ( x ) φ ( x + ) e - j ω d x k | Φ ( ω + 2 π k ) | 2 = φ ( x ) φ ( x + ) e - j ω d x
(34)

which becomes

φ ( x ) φ ( x + ) d x e j ω φ ( x ) φ ( x + ) d x e j ω
(35)

Because of the orthogonality of integer translates of φ(x)φ(x), this is not a function of ωω but is |φ(x)|2dx|φ(x)|2dx which, if normalized, is unity as stated in (Reference). This is the frequency domain equivalent of (Reference).

Equations (Reference) and (Reference) show how the scaling function determines the equation coefficients. This is derived by multiplying both sides of Equation 1 by φ(2x-m)φ(2x-m) and integrating to give

φ ( x ) φ ( 2 x - m ) d x = n h ( n ) φ ( 2 x - n ) φ ( 2 x - m ) d x φ ( x ) φ ( 2 x - m ) d x = n h ( n ) φ ( 2 x - n ) φ ( 2 x - m ) d x
(36)
= 1 2 n h ( n ) φ ( x - n ) φ ( x - m ) d x . = 1 2 n h ( n ) φ ( x - n ) φ ( x - m ) d x .
(37)

Using the orthogonality condition Equation 3 gives

φ ( x ) φ ( 2 x - m ) d x = h ( m ) 1 2 | φ ( y ) | 2 d y = 1 2 h ( m ) φ ( x ) φ ( 2 x - m ) d x = h ( m ) 1 2 | φ ( y ) | 2 d y = 1 2 h ( m )
(38)

which gives (Reference). A similar argument gives (Reference).

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