Harmonic Wavelet Analysis David E. Newland Proceedings ...read.pudn.com/downloads81/ebook/314729/harmonic... · forms an orthogonal set. Wavelets of different level (different j)

Harmonic Wavelet Analysis

David E. Newland

Proceedings: Mathematical and Physical Sciences, Vol. 443, No. 1917. (Oct. 8, 1993), pp.203-225.

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Harmonic wavelet analysis

BY DAVIDE. NEWLAND

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 IPZ, U.K.

A new harmonic wavelet is suggested. Unlike wavelets generated by discrete dilation equations, whose shape cannot be expressed in functional form, harmonic wavelets have the simple structure

w(x) = {exp (i4nx) -exp (i2xx)}/i2xx.

This function w(x) is concentrated locally around x = 0, and is orthogonal to its own unit translations and octave dilations. Its frequency spectrum is confined exactly to an octave band so that it is compact in the frequency domain (rather than in the x domain).

An efficient implementation of a discrete transform using this wavelet is based on the fast Fourier transform (FFT). Fourier coefficients are processed in octave bands to generate wavelet coefficients by an orthogonal transformation which is implemented by the FFT. The same process works backwards for the inverse transform.

1. Introduction

I have been interested for some time in the application of multiresolution analysis to engineering problems, particularly in the field of vibration analysis (Newland 1993a,b, 1994). This has related particularly to the study of wavelets derived from two-scale dilation equations (Daubechies 1988, 1989, 1990 ; Strang 1989, 1993, and others). As the number of coefficients increases in a wavelet's dilation equation, two things happen. The wavelet gets progressively longer (a wavelet derived from a dilation equation withNcoefficients extends over N- 1 unit intervals) and the wavelet's Fourier transform becomes more tightly confined to an octave band of frequencies. Although leakage remains, the spectrum becomes more box-like as N increases.

For analysing some engineering problems, it would be convenient to have a wavelet whose spectrum is confined exactly to an octave band. Then the 'level' of a signal's multiresolution would be interchangeable with its frequency band and the interpretation of frequency content, which is inimitable to engineers, would be easier. Also it would not matter much if the length of a wavelet became infinite in the x-domain provided that it were still concentrated locally close to x = 0.

These were the thoughts that led me to look for a wavelet whose Fourier transform was compact and which could, if possible, be constructed from simple functions. This paper is about such a band-limited wavelet and its properties and about an algorithm for its discrete computation.

Proc. R. Soc. Lond. A (1993) 443, 203-225 O 1993 The Royal Society

Printed in Great Britain 203

b i / o 1 2 4 w o i 1 1 -4n -2n 4n w

-i/4n

Figure 1. Fourier transforms of (a)the even, (b) the odd and (c) the complex harmonic wavelet.

2. Harmonic wavelets

Consider a real function we(x) whose Fourier transform is defined by

1/4n for -4n<w<-27 t

and 27t < 0 < 47t,

0 elsewhere,

as shown in figure 1 a. By using the inverse Fourier transform,

m

we(x)= /. K(w)exp (iox) dw = (sin 4nx -sin 27tx)/2nx, (2)

where the subscript e denotes tha t we(x) is an even function of x. Also if (figure 1b)

i/4n for - 4 x < o < - 2 n ,

-i/47t for 27t < w < 47t,

0 elsewhere,

the inverse transform gives

Wo(o) exp (iwx) dw = - (cos 4nx- cos 27tx)/2nx, (4)

where the subscript o denotes an odd function of x. Combining we(x) and wo(x) into a single complex function so that

we define the harmonic wavelet used in this paper by the function

w(x) = {exp (i4nx) -exp (i2nx))/i27tx. (6) The real and imaginary parts of w(x) are shown in figure 2.

Since w(x) is defined by (5), it follows that its Fourier transform W(w) is given by

W(o)= We(@)+i Wo(o)

so that , from (1) and (3) , we have

1/27t for 27t < 0 < 47t,W(w)=

0 elsewhere,

as shown in figure 1 c. By changing the argument in (6)from x to ( 2 ' ~ - k ) where j and k are integers, the

Proc. R. Soc. Lond. A (1993)

205 Harmonic wavelet analysis

-8 -4 0 4 8 X

Figure 2. Real part (a)and imaginary part ( b ) of the harmonic wavelet w(x) = {exp (i4nx) -exp (i2nx)}/i2nx.

shape of the wavelet is not changed but its horizontal scale is compressed by the factor 2j, and its position is translated by k units at the new scale (which is k /2hni ts a t the original scale). The value of j determines the 'level' of a wavelet. At level j = 0, the wavelet's Fourier transform occupies bandwidth 27t to 47t. A t level j , it occupies bandwidth 2x2' to 4x21 which is j octaves higher up the frequency scale.

3. Orthogonality

Because of the simplicity of the harmonic wavelet's Fourier transform, orthogonality can be demonstrated conveniently in the frequency domain. If W(w) is the Fourier transform of w(x) and V(w) that of v(x), then it is easy to see that the Fourier transform of v(x) = w(2'x- k) is (Appendix A)

V(w) = 1/2j exp ( -iok/2j) W(w/2'). (9)

It follows that the Fourier transforms of successive levels of harmonic wavelets decrease in proportion to their increasing bandwidth, as illustrated in figure 3. For w < 0, they are always zero. This property is why

Proc. R. Soc. L o n d . A (1993)

D. E.Newland

level 0

level 1

level 2 118~1 level 3i i

Figure 3. Fourier transforms of complex harmonic wavelets of different level.

for all j and k. The proof derives from the result (Appendix A) that if w(x) and v(x) are two complex functions of x (which is real) and their Fourier transforms are W(o) and V(o) , then

m

W(X) V(X) dx = 271. W(o)V(- W ) do. (11)

When w(x) and v(x) are two harmonic wavelets, they have the one-sided Fourier transforms in figure 3, so that the product W(o) V(-o) must always be zero. Therefore the right-hand side of (11) is always zero and hence (10) must be true always.

In addition, we need to know the conditions for which

where ~ ( x ) means the complex conjugate of w(x). In this case, since (Appendix A)

CC m1 w(x) u(x) dx = 2n ~ ( o ) (13)~ ( o ) do,

it is immediately clear that wavelets of different level are always orthogonal to each other. This is because their Fourier transforms occupy different frequency bands so that their product W(o) V(w) is again always zero for j # 0.

Wavelet translation within a level is the result of progressive rotation of the wavelet's Fourier transform in the frequency domain. This may be seen from (9).On putting V(w) = exp ( - i d ) W(w) and substituting in (13) for W(w) from (8), we see that the right-hand side of (13) is zero if

1;exp (iok) d o = 0.

The integral will be zero provided that

exp (i471.k) = exp (i271.k), k # 0, (15)

which gives exp (i2nk) = 1, k # 0,

and is satisfied by any integer value of k other than zero. Therefore all wavelets translated by any number of unit intervals are orthogonal to each other.

This result was found for wavelet level zero defined by the Fourier transform in (8).



For other levels, the same result applies except that the unit interval is now that for the wavelet level concerned. For example, for level j the unit interval is 1/2%nd translation must be through any multiple of distance 1/2j.

The conclusion is that the family of wavelets defined by

w(21x-k) = {exp {i4n(2jx -k)}-exp {i2n(2jx -k)})/i2n(2jx-k) (17) forms an orthogonal set. Wavelets of different level (different j) are always orthogonal ;wavelets a t the same level are orthogonal if one is translated with respect to the other by a unit interval (different k).

4. Zero-frequency band

Figure 3 shows the Fourier transforms of successive levels of harmonic wavelets for j k 0. For the octave bands defined by j < 0, the same sequence as in figure 3 may be maintained, in which case the resolution of f(x) embraces wavelets of all orders including vanishingly small octave bands as j-t- co.It is shown in the theory of multiresolution analysis (and demonstrated for harmonic wavelets in Appendix B) that all the negative orders can be rolled together into a single order (which, following standard wavelet terminology, is referred to below as order - 1) that covers the whole of the residual frequency band 0 to 2n in figure 3. This is shown in figure 4.

Following wavelet terminology, the functions generated by calculating the inverses of the Fourier transforms in figure 4 are called scaling functions.

The even Fourier transform

1/4n for - 2 n < w < 2 n , Qe(@) = {0 elsewhere,

gives an even scaling function

$,(x) = sin 2nx/2xx (19)

and the odd Fourier transform

i/4n for - 2 n < w < 0 ,

-i/4n for 0 < w < 2n,

0 elsewhere,

gives an odd scaling function

$,(x) = - (cos 2nx- 1)/2nx

so that , defining the complex scaling function $(x) by

$(x) = $e(x)+i$o(x), we find that $(x) = {exp (i2nx) - l)/i2nx.

I ts Fourier transform is @(w) = Qe(w)+iQo(w)

so that, from (18) and (20):

1/2n for 0 < w < 2n,Q(w)=

0 elsewhere,

as shown in figure 4c.


I-l-il4n

Figure 4. Fourier transforms of (a)the even, (b) the odd and (c) the complex scaling function.

Figure

Graphs of the real and imaginary parts of $(xi are shown in figure 5. It is easy to see that $(x) is orthogonal to its own unit translates (Appendix A)

I t is orthogonal to w(x) because their Fourier transforms are confined to separatc frequency bands.



5. Normalization

From the definitions of w(x) by (6) and $(xi by (23), i t follows (see Appendix A) that

and l$(~-k)1~dx= I .

Also, from ( lo) ,

J:m w'(2jx -k) dx = 0

and, by similar reasoning, mS-,$"x-k)dx = 0. (29)

It is important to have these relations when establishing the expansion of an arbitrary function f(x) (real or complex) in terms of complex harmonic wavelets.

6. Multiresolution off (x)

The expansion formulas of wavelet theory are usually written as

and

I n the first, wavelets of all levels are included; in the second negative levels are replaced by the scaling function (and its translates). These equations assume that f(x) is real, that the wavelets are derived from the solution of two-scale dilation equations with real coefficients and that there is only one wavelet for each j, k pair. For harmonic wavelets there are, as we have seen: two wavelets for each j, k pair, namely an even wavelet w,(2jx -k) and an odd wavelet w,(2jx -k). Since these are combined together by (5)into the single complex wavelet w(2jx- k), the wavelet coefficients aj,, will now also be complex.

We shall define the pair of complex wavelet coefficients

and a corresponding pair of complex coefficients for the scaling function terms

When f(x) is real: then aj,, = a',, and is not a new coefficient. But to allow for f(x) being complex, we distinguish between tii,,and aj,,.In terms of these coefficients, the contribution of a single complex wavelet to the function f(x) is

aj,,w(2jx-k)+cij3,m(2jx-k) (34)

Proc. R.Soc. Lond. A (1993)

and the expansion formulas become m m

f ( x ) = C C k ) ){a j , , w(2jx- k ) + ~ ? ~ , ~ a ( 2 j ~ - j=-m

and

m m

+ C C {a j , , w(2jx -k )+a$,,m(2jx-k ) ) . (36) j=O k=-m

Because of the orthogonality of w ( 2 h - k ) and $ ( x - k ) together with their additional properties (26)-(29), i t is straightforward to check that the coefficients in the expansions (35)and (36) can be found by making the convolutions (32)and (33) . When f ( x ) is real, the real part of aj , , is half the amplitude of the even wavelet we(2jx-E) in the expansion; minus the imaginary part of aj , , is half the amplitude of the corresponding odd wavelet wO(2jx- k ) in the expansion. The equivalence of the expansions (35)and (36) is proved in Appendix B.

Using the band-limited structure of the Fourier transform of w(2jx-k) , it can be demonstrated (Appendix C) that

so that the set of functions (w(2 jx -k ) ) is a 'tight frame' in the terminology of wavelet theory. In other words, the band-limited harmonic wavelet defined by (17) provides a complete set of basis functions for expanding an arbitrary function f ( x ) provided that f ( x ) decays to zero as x-t & m so that

7. Implementation

An algorithm to compute $ ( x - k ) and w(2jx-k) from their defining Fourier transforms and then to make the convolutions (32) and (33) can be set up in a straightforward way. but is not efficient numerically and the following alternative approach is better.

We begin by considering a real function f ( x ) represented by the sequence

where N = 2n. By using the discrete Fourier transform, the corresponding Fourier coefficients are

-where FNPm=Fm (41)

and all the Fm are generally complex except for 4 and E",,, which are always real. Now suppose that the corresponding (complex) wavelet coefficients are also known and call these

a,, s = O to N-1.


- - - -

Harmonic wavelet analysis 21 1

As for the Fourier coefficients, they repeat themselves in the second half of the sequence so that

aN-s = as and they are generally complex except for a, and a,/, which are always real. We arrange these in wavelet levels (which are synonymous with octave bands) as follows :

coefficient in wavelet wavelet level transform

Consider level j with 2j coefficients a,r+,, where k = 0 to 2j- 1. Each of these coefficients defines the (complex) amplitude of a wavelet whose Fourier transform is described by Fourier coefficients in the band E12r+k, k = 0 to 21- 1. The first wavelet in the sequence with amplitude a,! has a constant spectral density of relative level 1/21 (see figure 3). Therefore i t contributes a2r/2j to the general coefficient E",, 21 < m < 2jf1. The second wavelet a t level j has amplitude a,l+,. It is translated 1/2* with respect to its neighbour and therefore its Fourier transform is rotated by exp (-iwk/2j) with k = 1 (see Appendix A). Since Fm is the Fourier coefficient for frequency w = 2 ~ m ,the contribution of a,i+? to E", is a,r+, exp ( -i2~m/2~) /2 j . Combining all the contributions from k = 0 to 22- 1, gives

2.I- 1

E", = 2-1 C a2i+, exp ( -i2xmk/2*). k=O

Writing out these relations for the first 9 terms of a 16 term sequence leads to the pattern below :

Fo = 1 level -1 ao Fl 1 level 0 a1 F2 112 1 W2 level I a2 F3 1 w, a3 F4 114 1 W4 W i W J a4 F5 1 W5 W f wz a5 F6 level 2 1 W6 W2 W63 a6 F7 1 W , W$- w; a7

-F8- - - L . - - a 8 - ,

where Wm= exp ( -ixm/2j-l). (47)

This pattern may be continued for the remaining terms F, to F16and a, to a,, when the diagonal blocks continue in reverse order with their elements replaced by

Proc. R. Soc . L o n d . A (1993)

complex conjugates and their rows in reverse order. However, on account of (41) and (43) these blocks do not have to be computed when the input sequence is real and computations need be made only for the AT/2 + 1 terms shown. The first and last of the diagonal blocks are special cases which are discussed shortly.

nTe see that each block of wavelet coefficients is transformed by a unitary matrix (after multiplying by 2jI2 where the order of the matrix is 2j x 2j). In addition each unitary matrix is symmetric. This may be seen for level j = 2 by comparing, for example,

and {exp ( i~6 /2 j - l )}~ exp (in9). W;2 = {exp (in7/2j-1))2 = exp (in7) = =

I ts inverse is therefore just its complex conjugate and so the wavelet coefficients can be calculated easily from the Fourier coefficients.

In practice the computation (46) can be done directly with the fast Fourier transform (FFT). The expression (45) can be rewritten simply into the standard form of the discrete Fourier transform by substituting n = 21 and s = m-n to get

n-1

F2i+, = ( l l n ) Z a2j+, exp ( - i2xskln) S=O to n-1 k=O

since exp (-i2xk) = 1 for all k. We are left with the remarkable conclusion that the wavelet coefficients can be

obtained by computing the inverse discrete Fourier transform of successive blocks of the Fourier coefficients of the signal f. Obviously this has great computational advantages. A precise comparison with Jlallat's pyramid algorithm (Nallat 1989) for wavelets derived from dilation equations depends on how many coefficients, S,the dilation equations include and on how the calculations are organized. But the algorithm described above and Mallat's algorithm are certainly comparable when N is large. An example is given later.

The block scheme illustrated in (46) includes three blocks with only one element each. The first of these relates to the band of frequencies 0 < w < 2n: (see figure 4), which is the contribution from the scaling function. Because of wrapping round the unit interval, the scaling function is replaced by unity in the discrete model (see Appendix D) and therefore a, =Fo,the mean value off,. The second unit block derives from frequencies in level 0 (see figure 3) and is the FFT of a single term sequence, which is equal to itself. The third block with only one element is FN12= aYl2 which has to be included to pick up the Nyquist frequency components. The wavelet corresponding to ax12 is a saw-tooth sequence

which is a function that is orthogonal to the discrete sequences that describe the harmonic wavelets and exists to pick-up any residual energy a t the Nyquist frequency and ensure that Parseval's theorem applies so that (Appendix D)

where N = P. The algorithm is shown in figure 6.



/ / IFFT IFFT 1

complex conjugates entered

Figure 6. FFT algorithm to compute the harmonic wavelet transform, shown for a sequence of 16 real elements; the algorithm works in reverse for the inverse transform. There is a real input sequence f, ( r = 0 to 15) and an output sequence a,(s = 0 to 15) with a,,_,= 6,(s = 1 to 7).

FFT

Figure 7. FFT algorithm to compute the harmonic wavelet transform of a complex 16-term sequence; the algorithm works in reverse for the inverse transform. There is a complex input sequence f , ( r = 0 to 15) and an output sequence a,(s = 0 to 15).


When the input sequence f, has complex elements, the wavelet sequence a, no longer satisfies a,-, = a,.Instead a,-, = 6,(by analogy with the coefficients in (32)), and the second half of the output sequence can no longer be written across as shown in figure 6. In this case the block pattern of equation (46) has to be continued to fill the complete N x N matrix. Each block of order 21 x 21 in the lower half of the matrix is the complex conjugate of the block of the same order in the upper half of the matrix, with the rows in reverse order. The corresponding version of figure 6 is shown in figure 7.

8. Examples

Figure 8 shows graphs of a level 5 harmonic wavelet alongside a level 5 D20 wavelet with the absolute values of their Fourier transforms plotted below. The latter was derived from a two-scale dilation equation with 20 real coefficients and computed using Nallat's pyramid algorithm. For the harmonic wavelet, its Fourier transform at level 5 is confined to integers Z 5 + 1 = 33 to 26 = 64 (the computer algorithm puts a, = a(1), etc.). I n the case of the D20 wavelet there is a wider spread and, in addition, its Fourier transform covers a generally lower frequency band than the harmonic wavelet covers.

For the harmonic wavelet transform, all frequency components whose Fourier coefficientsFSlie in the band s = 32 to 63 are swept into level 5 of the transform. This may be illustrated by drawing a mean-square wavelet map of a sample function f,of band-limited noise with frequencies in the band s = 32 to 63 only.

The map is constructed as follows (Kewland 1993a; 1994). On a grid base of x (horizontally) and wavelet level (vertically) a surface is drawn whose height above datum is equal to la2j+k12+ la,-23-k12. The construction is devised to ensure that the volume under this surface is equal to the mean-square off, according to Parseval's theorem (50). Along level 0 the surface is flat; along level 1 there is one step (two levels);along level 2 there are three steps (four levels), and so on. A mean-square map is a diagram of this surface showing contours of equal height. For convenience in plotting its discrete representation, the vertical edges of steps are replaced by sloping surfaces, and in that sense the resulting diagram is then approximate.

Returning to the band-limited function f, described above, its harmonic wavelet mean-square map is illustrated in figure 9 a with a three-dimensional mesh diagram of the mean-square surface shown below. The mean-square is confined to level 5 of the transform so the mean-square surface has the shape of a roof of variable height with its ridge running along level 5 and its edges running long levels 4 and 6. For the D20 wavelet, the corresponding diagrams are shown in figure 9b. Kow the mean- square is no longer confined to level 5 and spills over into other levels of the transform. I n this example f,has been generated artificially and is a circular function so that there is no discontinuity between the two ends of the record.

To examine the accuracy of modelling a sharp edge, figure 10 shows the reconstruction of a step function by harmonic and D20 wavelets. After computing both transforms for a 512 term sequence, only the first 64 (complex) terms of the harmonic wavelet transform were retained (together with their complex conjugates) and only the first 128 terms of the D20 wavelet were retained. This means that levels up to level 5 of the harmonic wavelet transform and up to level 6 of the D20 wavelet transform were included. Reconstruction from the harmonic wavelet is shown as the solid curve and from the D20 wavelet as the dashed curve. Evidently there is little between the accuracy of the two reconstructions.

Proc R.Soc. Lond. A (1993)


Figure 8. (a)D20 wavelet and (b) even harmonic wavelet, both a t level 5 and for 2048-term sequences, with (c) the absolute values of their Fourier transforms.

Figure 9. (a) Mean-square map and mesh diagrams computed by the harmonic wavelet transform for noise in the frequency band s = 32 to 63 for a 512 term sequence. (i) Function f,, (ii) wavelet map off, with contours log spaced, (iii) mesh plot of wavelet map to log scale. ( 6 ) The same diagrams as (a) but computed using the D20 wavelet transform.

To compare their numerical efficiency, figure 11 shows the ratio of the number of floating point operations (flops) of the D20 wavelet transform, using my MATLABBT program for this transform based on Mallat's pyramid algorithm (Mallat 1989) and the corresponding number of flops for a version of the harmonic wavelet algorithm in figure 6 using MATLAB's FFT function. The numbers of flops are computed by MATLAB's flops function which counts 2 for complex additions and subtractions and 6 for complex multiplications and divisions, and 1 for real operatians. With

t MATLAB@ is a registered trademark of The Mathworks Inc.


Figure 10. Reconstruction of the corner of a box by truncated harmonic and D20 (dashed) wavelet transforms; for a 512-term sequence, the first 64 (complex) terms of the harmonic wavelet transform and the first 128 (real) terms of the D20 wavelet transform have been retained.

Figure 11. Ratio R of the number of floating point operations, D20 wavelet transform/harmonic wavelet transform, when computing the transforms of a real sequence of length 2".

increasing sequence length (212= 4096) the ratio falls but for my programs the harmonic wavelet transform has a definite computational superiority. The comparison in figure 11 was made for a real input sequence in each case.

Two-dimensional (and higher) harmonic wavelet transforms can be assembled from the one-dimensional transform in the same way that the two-dimensional FFT

is assembled. I t is essential to use a one-dimensional transform based on figure 7 even though the input array may always be real. This is because there are two stages of FFT computation in the two-dimensional transform and the input sequence to the second stage is complex.

9. Conclusions

Mallat's pyramid algorithm for computing the discrete wavelet transform (Mallat 1989) involves a sequential filtering operation. At each level of the transform, a high- pass filter separates fine structure to give the wavelet coefficients a t this level ; a low- pass filter compresses the signal by halving its sequence length, ready for the next level of processing. Dioul & Duhamel (1992) have shown how these filtering operations may be implemented using the FFT. But the computation remains a sequential process and the pyramid structure of the algorithm remains. The algorithm described in this paper and illustrated in figures 6 and 7 is a parallel



algorithm in the sense that all the wavelet coefficients a,, 5 = 0 to N- 1, can be computed simultaneously. Mallat's pyramid is replaced by a flat sandwich with FFT

operations a t only two levels of the sandwich. In essence the computation is extremely simple. After computing a signal's FFT,

octave bands are reconstructed by the IFFT (or its complex conjugate, which is the FFT). Each reconstructed octave band is the wavelet transform a t this level of the resolution.

The only disadvantage appears to be that the rate of decay of harmonic wavelets is relatively low (proportional to x-l) so that localization is not very precise. However, that is the penalty of having a wavelet whose Fourier transform is restricted to a specified frequency band.

Appendix A. Orthogonality and normalization (a) Fourier transform of w(2jx- k)

Let W(w) be the Fourier transform of w(x) and B(w) the Fourier transform of w(2jx-Ic). Then

Putting z = 2jx- k, so that dz = 2jdx and x = ( z + k) 2-j, (A 1) gives

1 " V(w)=2rr[-a:~ ( z )exp {-iw(z+ k) 2')2-1dz

= 2-j exp ( -iwk/2j) W(w/21). (A 2)

Compression of w(x) in the x-domain by factor 2j, spreads out (or dilates) W(o) in the w-domain by replacing W(w) by W(w/2j). Translation of w(x) by k units involves rotation of its Fourier transform in the complex plane of W(w), the amount of rotation depending on the frequency w and the compression factor 2j according to (A 2).

(6) Conditions for orthogonality

If w(x) and v(x) are orthogonal functions in L2(R)with Fourier transforms W(w) and B(w) so that

a:

W(X) = \ exp (iwx) dw, (A 3)~ ( w )

a:

v(x) = r V(W) exp (iwx) dw. (A 4)

then

From the theory of generalized functions, the Fourier transform of the function 6(x- c) is

1S(x- C) exp ( -iwx) dx =-exp ( -iwc) (A 6)27t


218 D. E.Newland

so that , by the inverse transform,

SYa:exp {iw(x -c)) dw = 211S(x-c).

Using this result, the integral over x in (A 5) is

exp {i(w, +w,) x) dx = 2nS(wl+w,). (A 8) S:, On substituting (A 8) into (A 5) and integrating over w,, we are left with the result tha t

W(X) v(x) dx = 271 (A 9)

after replacing w, by w. Also, by replacing v(x) in (A 1.5) by ~ ( x ) , the corresponding result is

W(X) ~ ( x ) 211 (A 10) dx =

(c) Orthogonality of $(x) and $(x-k)

If @(w) is the Fourier transform of $(x), then tha t of $(x-lc) is, from (a) above, e-iwk@ ( ) . On substituting into (A 9) and (A lo),the conditions for orthogonality are

$(x)$(x- k) dx = @(w) @( -W ) exp (iwk) dw = 0S:, S-,a:

(A 11)

and $(x)$(x- k) dx = @(w) &(w) exp (iwk) d o = 0. (A 12)

Since @(w) is the one-sided function (25)) i t follows immediately that the right-hand side of (A 11) is always zero so that the first condition is always satisfied whatever the value of k. On substituting for @(w) from (25) into (A 12), the second orthogonality condition is satisfied if

1; (&)'eXp (iwk) dw =

which gives exp (i2nk) = 1, k # 0 (A 14)

and is satisfied by k = & 1, f2, etc.

Therefore $(x) and $(x- k) are orthogonal for all integer values of k except k = 0.

(d) Normalization From (A lo) ,

so tha t , using (A 2),

w(2jx-k) m(2'x -k) dx = 2n2-,j W(w/2j) V(w/29 dw (A 17)



and, from (8), W(w/2j)= 1/2n, 2n2j < w < 4x2j.

Therefore

l ~ ( 2 j x - k ) 1 ~ d x = 2 ~ 2 - ~ j ( 1 / 2 ~ ) ~ d w = 2 - j

Similarly

so that, from (25),

Appendix B. Equivalence of the wavelet expansion formulas (35) and (36)

The formulas are identical provided that

and similarly with a"$,,, 6(x-k), Gj,, and w(2jx-k). The definition of a,,,, from (33) is

and, substituting f(x) ~ ( w )= exp (iwx) dw

m

and $(x-k) = ( C(w) exp ( -iwk) exp (iwx) dw, (B 4)

we have m

a$, = r m dx r m dwl ( do, F ( w l ) @w,) exp (io, k) exp (i(ol -w,) x). (B 5 )

Csing the result that

exp {i(w, -w,) x) dx = 2n6(w1-w,),lrm this gives a,, ,= dwF(w)6 ( w ) exp (iwk). 2n Sra: From (B 4) and (B 7), the left-hand side of (B 1) becomes

= 2 m

2n dw, [a:

dw, ~ ( w , )q w , ) a(,,)exp {i(wl -w,) k) exp (iw, x). (B 8)


The summation over k involves only the term exp {i(w, -w,) k). This can be evaluated by using Poisson's summation formula (see, for example, Lighthill 1962)

m m

1/21 C exp ( -ixkx/l) = C 6(x-2ml). (B 9)k=-m m=-m

Putting x = w, -w,, x/l = 1, we get a: m

C exp {i(wl -w,) k) = 2x C 6(w2-w, -2xm) (B 10) k=-m m=-m

and so (B 8) becomes

a: m

= C Cm dw1 ( dw, F ( w , ) q w , ) @(w,) exp (iw, x) 6(w, -w1 -2nm)( 2 ~ ) ~ (B 11)

I t turns out that only one term in the new summation needs to be included since the product G(w,)@(w,) is otherwise zero. Prom (25), the product 6 ( w 1 ) @(w,) is zero unless w, and w, both lie in the band 0 < w < 2n, which is the case when m = 0 in (B 11). Then w, = w, and G(w,) @(w,) = 1/(27c),. Hence (B 11) gives

m

C a#, $(x- k) = F ( w ) exp (iwx) dw. (B 12) k=-m

The same analysis may be applied to the right-hand side of (B 1). Beginning with, from (33),

aj, = 21 I:rnf@) la(2jx-k) dx (B 13)

and transferring this to the frequency domain by using (B 3) and, from (A 2), m

V(w/2j) exp (iwk/2j) exp ( -iwx), (B 14)

we have

jl m

a;, = dw, dw, F(ol)bV(w2/2j) exp ( i 4 k/2j) exp {i(w, -w,) 2). (B 15)

Again using (B 6), gives

Then, substituting for aj,, from (B 16) and for w(2jx-k) from (B 14) into the right- hand side of (B 1) gives

= C C 2x2-j J-a dw, F ( w l ) W(wl/2j) W(w2/2j) exp (iw, x) j=-= ,=- m !-mdwl x exp {i(w, -w,) k/2j). (B 17)


Harmonic wavelet analysis 22 1

Taking the summation over k first, Poisson's summation (B 9) gives

m m

C exp {i(wl -w,) k/2j} = 2n2j C S(w,-wl -2n2jm). (B 18) k=-m m=-a:

From (8)the product W(w1/2j) W(w2/2j) is zero unless m = 0 in the summation ( B IS), when i t is 1/(2n),. The conclusion is tha t ( B 17) gives

-1 m

C C C F ( w ) exp (iwx) dw, ~ ~ , ~ w ( 2 j x - k ) = F ( w ) exp (iwx) dw = k=-=

(B 19)

which is the same as (B 12). The band-limited character of the Fourier transforms of $(x) and w(x) therefore

provides the basis of a formal proof of the equivalence of the expansions (35) and (36).

Appendix C. Proof of Parseval's theorem

The following proves equation (37) without making any assumption about the wavelet expansion (35). Beginning with the definitions (32)

and using the result, from (B 14),

w(2jx-k) = 2" [m

exp ( -iwk/2') W(w/2') exp (iwx) dw,

where W(w) is the Fourier transform of w(x), we find that , as in Appendix B ,

where (c4)

Hence

x 1:

x 1:

dw, 2nF(w2) W(w2/29 exp {i(wl -w,) k/2j},

dw, 2xP( -w,) W(w2/2j) exp { -i(w, -w,) k/2j}.


222 D. E.Newland

We now have to sum these expressions over all j and k. The key summation is that over k because this involves only the term

and its complex conjugate. According to Poisson's summation (Lighthill 1962) a: m

1/21 C exp ( -ixkxll) = C 6(x- 2ml) (C 7) Ic=-m m=-m

so tha t , substituting x = w,- w,, 1= 2jx, we get m m

C exp {i(w, -w,) k/2j} = 2x2j C 6(w,-w, -m2n:2j). (C 8) Ic=-m m=-a:

I n each of (C 5 ) this summation is multiplied by the product W(w1/2j) W(w2/2j) or its complex conjugate which, because of the band-limited spectrum of harmonic wavelets, is zero unless w, and w, both fall within an interval 2n2j < w < 4x2j. I t follows tha t W(w1/2j) W(w2/2j) will be zero for all values of m in the summation in (C 8) except for m = 0. Because of this result,

and (c9)

Xow from (8) the Fourier transform of the complex harmonic wavelet (6) is zero except for W(w) = 1/2x for 2x < w < 4x, so tha t W(w/2j) W(wl2.I) = 1 / ( 2 7 ~ ) ~for 2x2j < w < 4n2' and its summation for all j is just a constant 1 / ( 2 5 ~ ) ~for 0 < w < a.Adding the two expressions (C 9) then gives

and we find, finally, tha t

from Appendix A, thereby proving (37).

Appendix D. Continuous and discrete comparison (a) Circular scaling and wavelet functions

Comparison between the continuous and discrete wavelet transforms depends on the assumptions that (i) the discrete transform covers a unit interval of x, and (ii) tha t the function being analysed is periodic in x with period 1. Where continuous functions overlap the unit interval, they are considered to have been wrapped round to reappear a t the opposite side of the interval. Each continuous function which overlaps the sides of the unit interval has to be replaced by a corresponding circular continuous function.

Proc. R. Soc . L o n d . A (1993)


I n the case of the scaling function,$(x) this becomes the circular scaling function $,(XI, where

m

$c(x) = C $(x-k). (D 1) k=-a:

By substituting $(x -k) with its Fourier transform we have

m m

$,(x) = C @(w)exp ( -iwk) exp (iwx) dw (D 2) k=-m

and, from (B lo) , m m

C exp ( -iwk) = 2x C S(w-2xm). k=-m m=-m

Since, from (25), @(w) is zero except in the band 0 6 w < 2x, only the term with m = 0 needs to be retained in (D 3), so that

$,(x) = @(w)exp (iwx) 2n S(w) do,

which with (25) gives $,(XI = 1.

The same argument applies when $(x-k) in (D 1) is replaced by its complex conjugate.

Expansion of a circular function f,(x) by the discrete transform has only one term derived from the contribution of the scaling function. Prom (36) this may be seen to be a# $,(x) +d$$,(x) which, from (D 5), is just the single constant a, = a$+d#.

For the general wavelet function w(2jx- k), its circular equivalent is

a:

= Z 2-' I:mexp ( -iwk/2j)exp ( -iwm) W(wl2') exp (iwx) dw (k = 0 to 21- 1) m=-m

(D 6) from (A 2). The summation over m is given by (D 3). Since W(w/2j) = 1/2x for 2n2j < w < 2n2j+l and is zero elsewhere, the only values of m that have to be considered in (D 3) are m = 21 to 2.1"- 1. Substituting for (D 3) in (D 6) and completing the integration gives

zl+l-l

w,,j(21x-k) = 2-j C exp {i2xm(x -k/2j)) (k = 0 to 2j -1). (D 7) m=z3

For the zero-level circular wavelet (one wavelet per unit interval), j = 0 and then

w,,,(x) = exp {i2xx). (D 8) The two circular wavelets a t level 1 are given by

w,, ,(2x- k) = 1/2[exp {i2x(2x -k))+exp {i2n(3x- 3k/2))] (k = 0 , l ) (D 9) and similarly for higher levels.

It is clear from (D 7) that a circular wavelet of level j has 2j discrete harmonics whose frequencies are 2n2j, 2x(2j+ I ) , 2x(23+ 2), . .. . . . ,2x(2j+'-1).

Proc. R. Soc . L o n d . A (1993)

(b) Speciul case of the Nyquist frequency

For a discrete transform of sequence length N = 2n , only wavelet levels up to n -2 can be included. Because the transform has to compute wavelet coefficients cij,, as well as aj:k (see (35), (36)), there is no space for level n - 1. Only one harmonic may be fitted in with the Nyquist frequency Nx. This defines a wavelet

N-1

f r = FN12 r = o to S- 1,c Frexp (i2nkr/A7) = exp (ixr), k=O (D 10)

since all the other Fkare zero, and gives

when FN12= 1. This saw-tooth function is orthogonal to all the discrete sequences tha t represent

wavelets. The proof is as follows. Each discrete wavelet is a summation of harmonics. Consider only one harmonic defined by

This will be orthogonal to the saw-tooth sequence (D 11) if

On substituting for f r from (D 12), the orthogonality condition is

By taking an example and drawing the different directions of each of the exponential terms, it can be seen tha t these form a symmetrical star so that their summation vanishes unless k =S / 2 when

Hence (49) is orthogonal to all the discrete sequences tha t describe harmonic wavelets.

The contribution of the Nyquist term uN12 to the mean-square ensures that Parseval's theorem in the discrete form (50) applies. This may be seen as follows. With reference to figure 7, all the elements of the discrete harmonic wavelet transform are obtained by operation of the IFFT (or its complex conjugate) on octave groups of the frequency coefficients F,, s = 1 to A72-1 and s =A7/2+ 1 to S- 1. Because Parseval's theorem applies individually to each of the octave groups, the summation of terms

21-1

2-' c la2f+kI2 k=O

in each group is equal to the corresponding summation



By including the first element a, = F, and a,,, = FNi2in the summation, the right- hand side of (50) is equal to CE: IFJ2which by Parseval's theorem for the complete sequence is equal to l/NCF:O' ( f T ( 2 .Hence inclusion of the Syquist element a,/, is necessary to ensure that the discrete harmonic wavelet transform constitutes the discrete equivalent of a tight frame. Then equation (50) describes the distribution of the mean-square of the input sequence f .

References Daubechies, I . 1988 Orthonormal bases of compactly supported wavelets. Comm. pure appl. Math.

XLI, 909-996. Daubechies, I . 1989 Orthonormal bases of wavelets with finite support - connection with discrete

filters. In Wavelets, time-frequency methods and phase space (ed. J . M. Combes, A. Grossmann & Ph. Tchamitchian). Berlin : Springer-Verlag.

Daubechies, I . 1990 The wavelet transform, time-frequency localization and signal analysis. I.E.E.E. Trans. Info. Theory 36, 961-1005.

Lighthill, 11. J. 1962 A n introduction to Fourier analysis and generalised functions, pp. 67, 68. Cambridge University Press.

Mallat, 8. 1989 A theory for multiresolution signal decomposition: the wavelet representation. I.E.E.E. Trans. Pattern Analysis ,%fachine Intell. 11, 674-693.

Newland, D. E. 1993a Random vibrations, spectral and wavelet analysis, 3rd edn. Harlow: Longman, and New York: John Wiley.

Newland, D. E . 19933 Wavelet analysis of vibration. I n Proc. Structural Dynamics and Vibration Symp., ASME Energy-Sources Technology Conference, Houston, U.S.A., vol. 52, pp. 1-12. American Society of Mechanical Engineers.

Newland, D. E . 1994 Some properties of discrete wavelet maps. Prob. Engng ,%fech. 9. (Issue to mark the 65th birthday of Professor T . K. Caughey.) (In the press.)

Rioul, 0 . & Duhamel, P . 1992 Fast algorithms for discrete and continuous wavelet transforms. I.E.E.E. Trans. Info. Theory 38, 569-586.

Strang, G. 1989 Wavelets and dilation equations: a brief introduction. S I M I Rev. 31, 614-627. Strang, G. 1993 Wavelet transforms versus Fourier transforms. Bull. A m . ~ V a t h . Soc. 28, 288-305.

Receiced 28 January 1993; accepted 16 April 1993

Note added in proof (27 July 1993). In my sequel paper (Harmonic and musical wavelets, Proc. R. Soc. Lond. A (In the press.)), the concept of a harmonic wavelet is extended to describe a family of mixed wavelets. Each wavelet is band-limited in the frequency domain but its spectrum is not confined necessarily to an octave. This has the advantage that localization in frequency or in the physical domain can be adjusted within one family by choosing the bandwidths of individual wavelets appropriately. For example, wavelets whose frequency content ascends according to the musical scale can be generated which provide greater frequency discrimination than wavelets whose frequency interval is an octave; correspondingly, their localization in the physical domain is less precise than for wavelets whose bandwith is wider.

Proc. R . Soc. Lond. A (1993)

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