1 Orthonormal Wavelets with Simple Closed-Form Expressions G. G. Walter and J. Zhang IEEE Trans. Signal Processing, vol. 46, No. 8, August 王隆仁

2 Contents o Introduction o Wavelets and h-Construction A. Orthonormal Wavelets B. Lemarie-Meyer Wavelet and h-Construction o Raised-Cosine Wavelets A. The Raised-Cosine Spectrum B. Raised-Cosine Wavelets o Summary

3 I. Introduction o Recently, there has been much interest in wavelets and their applications in signal processing. o Among the wavelets discover so far, those by Daubechies are popular, especially in image coding, since they are orthonormal, posses finite support (good time-localization), and lead to simple FIR (finite impulse response) filters for the discrete wavelet transform. o They are not sufficiently smooth and decay slowly in the frequency domain, i.e., they do not have good frequency localization. o Lemarie-Meyer’s wavelets are attractive since they are orthonormal and posses good frequency as well as time localization.

4 o These wavelets are bandlimited and can be chosen to have polynomial time decay. o They also have better shift and dilation-invariant properties than Daubechies’ wavelets, which makes them more attractive in some differential and integral equation application. o Lemarie-Meyer’s wavelets generally lead to IIR (infinite impulse response) filters for the discrete wavelet transform. o Almost all known orthonormal wavelets, except for the Harr and the Shannon (the sine function), cannot be expressed in closed form or in terms of simple analytical functions, such as the sine, cosine, exponentials, and polynomials. o Instead, they can only be expressed as the limit of a sequence or the integral of some functions.

5 o We may ask if there are any nontrivial orthonormal wavelets (i.e., not the Harr and Shannon) that have relatively simple analytic forms. o Recently, we have found two classes of such wavelets based on some pulses used in digital communications and signal processing that are characterized by a raised-cosine spectrum. o For this reason, we call them the raised-cosine wavelets. o Interestingly, these pulses themselves are not orthonormal wavelets. o Rather, what we have done is to apply a scheme for constructing Lemarie-Meyer’s wavelets to the square-root of the raised-cosine spectrum. o By imposing an additional constraint on the  parameter of the raised-cosine (  <1/3), we obtain the desired orthonormality.

6 II. Wavelets and h-Construction A. Orthonormal Wavelets o Orthonormal wavelets are functions whose dilations and translations form an orthonormal basis of L 2 (R), which is the space of energy-finite signals. o Specifically, there exists a function  (t) (the “mother wavelet”) such that form an orthonormal basis of L 2 (R). Here, Z is the set of integers.

7 o The wavelet  (t) is often generated from a companion  (t), which is known as the scaling function (or the “father wavelet”), through the following “dilation” equations where {h k } and {g k } are a pair of (quadrature-mirror) lowpass and bandpass filters that are related through

8 o The relationship between the wavelet and scaling function can also be represented in the frequency domain as where we have used to represent the Fourier transform of ; and are the Fourier transforms of the quadrature mirror filters and are both periodic functions with period 2 .

9 o The dilations and translations of induce a multiresolution analysis (MRA) of L 2 (R), i.e., a nested chain of closed subspaces {V m } whose union is dense in L 2 (R). o Here, V m is the subspace spanned by o In particular, is an orthonormal basis of V 0 and satisfies In fact, this is itself the necessary and sufficient condition for  (t) to have orthogonal integer translates.

10 B. Lemarie-Meyer Wavelet and h-Construction o Many L 2 (R) functions may be used as the scaling function to generate the orthonormal wavelet basis of part A. o The ones of particular interest here are the Lemarie-Meyer type, which have compact support in the frequency domain (bandlimited). An example of these is shown in Fig. 1(b). o Let be a non-negative integrable function with support in [-  /3,+  /3] such that o A Lemarie-Meyer scaling function  (t) can then be defined, through its Fourier transform, by the property

11 III. Raised-Cosine Wavelets A. The Raised-Cosine Spectrum o In a digital communication system, the sequence of 0’s and 1’s is mapped into a sequence of “signaling” pulses, which is then transmitted over channel at a rate of R bits (pulses)/s. This process is known as modulation. o To combat channel noise, the received pulse sequence is passed through a correlation filter. The correction filter produces an “output” pulse sequence that is then sampled R times/s to recover the 0’s and 1’s. This process is known as demodulation and is illustrated in Fig. 2(a).

12 o If the channel is bandlimited, as it is in most practical applications, each received pulse will have infinite time support, and this could cause intersymbol interference (ISI). This situation is illustrated in Fig. 2(b). o Since an “output” pulse is the convolution between a “signaling” pulse and the correction filter, the key to eliminating ISI is to select the signaling pulse and the correlation filter in such a way that the output pulse and its shifted versions are orthogonal. That is, at the sampling instant, the magnitudes of neighboring output pulses are zero. This is formalized in a well-known theorem attributed to Nyquist.

13 o Let the output pulse be denoted as x(t), let the sampling period be T, and let T=1/R. The Nyquist theorem states that ISI can be eliminated if and only if where is the Fourier transform of x(t). o A familiar output pulse that eliminates ISI is the Shannon scaling function x(t) = sinc(  t/T) with x(nT) = 0 for n  0. o The Shannon scaling function, however, has slow time decay (O(1/t)) and, since it requires an ideal lowpass filter for its generation, is undesirable.

14 o A better and more practical choice is a pulse with a raised- cosine Fourier spectrum, shown as follows: where  [0,1], and without loss of generality, we have set T = 1.

15 B. Raised-Cosine Wavelets o Comparing the Nyquist’s orthogonality condition with the orthogonality condition for the scaling function, we see a remarkable similarity. o Any Nyquist function can potentially lead to a scaling function through o In order for defined this way to be truly a scaling function, however, it has to satisfy the dilation equations.

16 o If (hence, ) can be expressed by an h-construction. Then the raised cosine for is where is the usual indicator function. o This equation implies that any satisfying is a scaling function; the only question is whether it and its inverse Fourier transform have closed forms.

17 o We give two that satisfy above equation and have closed forms. o The first is the used positive square root of, as below o The second scaling function is complex and given as follows

18 o By direction calculation and taking inverse Fourier transforms, we can find their time-domain representations and o In both cases, the mother wavelet can be found from the standard formula

19 o Again, taking inverse Fourier transfoms, we have and o Plots of,,, and are shown in Figs. 1(d), (e), and 3.

20 o The final part of the results concerns that the h and g filters, which are generally used to perform the discrete wavelet transform. o From a standard result of wavelet theory, we have the discrete Fourier transform relationship o However, notice that o Thus,

21 o From and, we can find the corresponding in closed form and

22 IV. Summary o In this work, we have constructed two classes of Lemarie-Meyer wavelets (and scaling function). o Like most Lemarie-Meyer wavelets, they are orthonormal, bandlimited, and fast-decaying in time. o Unlike most wavelets, these wavelets have relatively simple analytic (closed-form) expressions in terms of sine, cosine, and simple fractions. o The second class of wavelets are particularly interesting in that they are also sampling function.

23 o Both classes of wavelets, called raised-cosine wavelets, are constructed from pulses having the raised-cosine spectrum that are popular in digital communications. o The h construction, which is a scheme for systematically constructing Lemarie-Meyer wavelets, and the properties of the raised-cosine are used to derive the explicit analytic expressions of the wavelets and scaling functions in both the time and frequency domain. o The derivation reveals an interesting relationship between wavelet construction and no-ISI signaling: All Lemarie-Meyer wavelets can be used to construct no-ISI signaling pulses.