A New Signal Representation

A New Signal RepresentationAuthor(s): John E. Hershey, Amer A. Hassan and Rao YarlagaddaSource: Proceedings: Mathematical and Physical Sciences, Vol. 449, No. 1936 (May 9, 1995), pp.329-336Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/52700 .

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A new signal representation BY JOHN E. HERSHEY1, AMER A. HASSAN1t AND

RAO YARLAGADDA2

1Electronic Systems Laboratory, General Electric Corp., R&D Center, 1 River Road, Schenectady, New York 12301, U.S.A.

2School of Electrical and Computer Engineering, 202 Engineering South, Oklahoma State University, Stillwater, Oklahoma 74078, U.S.A.

A new method is presented for representing and approximating signals which are made up of odd frequency components. The method uses a 'half-basis' set, all of whose members exhibit identical power spectral densities. In contrast to the con- ventional Fourier approximation, approximations by this new method will spread the approximation error over the odd frequencies in the half basis. Approxima- tions to the full set of signals, i.e. those containing both even and odd frequency components can also be realized by this new method.

1. Introduction

The Fourier sum of sines representation is the most familiar orthogonal basis for representing signals. Many other bases have been offered. In (Sohie & Mara- cas 1988), for example, the authors suggest an orthogonal expansion of appro- priately weighted decaying exponentials which may be a better basis than the Fourier for representing some signals which are of a time decaying nature. Also, a wavelets representation is often considered for representing non-stationary signals and transients (Mallat 1989).

In this paper we discuss a new signal approximation technique which rests on a half-basis whose elements each possess an identical power spectrum. The new half-basis also possesses some interesting attributes per se. It appears to be strongly related to the Weierstrass functions and to the basis used in the classical Walsh transform (Gonzalez & Wintz 1987). In ? 2 we define a basic even frequency/odd frequency partition. In ? 3 we present the underlying new class of functions. In ? 4 we list some of the important properties of these new functions. In ? 5 we develop a continuous signal representation. In ? 6 we give an example. In ? 7 we comment on a related discrete basis. We draw our conclusion in ? 8.

2. A basic even-odd frequency partition

It is customary to speak of a partition of the components of a space or time function as even or odd. It is equally reasonable to posit a partition of the components

t Present address: Ericsson-GE Mobile Communications Inc., Research Triangle Park, North Carolina, U.S.A.

Proc. R. Soc. Lond. A (1995) 449, 329-336 () 1995 The Royal Society Printed in Great Britain 329 TIX Paper



J. E. Hershey, A. A. Hassan and R. Yarlagadda

of a function into parts that comprise only even frequencies and those that comprise only odd frequencies. The characteristic of an odd frequency function, f(x), over the interval [0,1] is f(x +0.5) - -f(x), 0 < x 0.5. Likewise the characteristic of an even frequency function over the same interval is f(x + 0.5) = +f(x), O < x < 0.5.

An example of an even frequency function is f(x) = sin(47rx). An example of an odd frequency function is f(x) = sin(27rx).

3. A new class of functions

The half-basis grows naturally out of a study of a particular set of functions which are defined over an interval [0,1] by the following expression:

n

Wn( ); x) - I sin(2 27rx + ), (3.1) i=O

where n is the 'order' of the function, I = (0o, O1, .. , n) and Xi C {0, 1r}.

By writing

sin(2i - 2-x + i) [ei(2i2rz+) _ e-(2i2rz+)] (3.2) 2j

it is easily seen that the form (3.1) can be expanded into a sum form as

W2 + -1 (-1) 1 (-1)n/2 sin( 2x + A ) n even,

k-13,5 .. 2 (-1)(n-)/2 cos(k. 27x + A ) n odd,

(3.3) where A = (ao, al ,..., a,) is a binary representation of the summation index k with ai E {?l} and k = En= 0a 2'; (0o, X1l, . . . ); A . = Eo i

6(k), k = 0, 1, 2,..., is the Thue-Morse sequence investigated in depth in (Yarlagadda & Hershey 1984) and other papers. The Thue-Morse sequence can be thought of the diagonal elements in the infinite Cartesian product limn,_ Hn, where Hn = Hn_l ( H and

H1= -1)

(The first eight values, 0(0)-e(7), are 1, -1, -1, 1, -1, 1, 1, -1.) For finite order, n, the functions and all of their derivatives are continuous in

the interval (0,1). In figure 1 we plot some representatives of 2(+1)/2Wn('; x), namely 21/2Wo(0; x), 22W3(0; x), and 27/2W6(0; x), where 0 indicates that all of the bis are zero.

4. Properties of the functions

The function set {Wn(; x)} has several properties worth noting. First, by directly integrating, it is evident that the functions of order n form an orthogonal

Proc. R. Soc. Lond. A (1995)

330



A new signal representation 5 (a)

-5 ,. . .

5 "(b)

(c) --5 - - - -.... ; . ? . . . i f, 5-I

u -_ ' ? - -' w' r r-

5 - . . - .,

0 0.2 0.4 0.6 0.8 1.0 x

Figure 1. Graphs of 2(n+ )/2Wn(O; x) for n - 0, 3, 6.

set, i.e.

Wn( I1; x)Wn(P2; x)dx= {2(n+l) 2- (4.1)

Second, from (3.3) we see that the power spectrum of Wn(P; x) is bi-valued. It has a value of 2-(2n+ ) at the odd frequencies, 1 through 2n+1 - 1, and is zero at the even frequencies.

Third, the autocorrelation, Rwn,wn (T), of a member of {W, (o; x)} will exhibit a very sharp peak about r = 0. This can be seen by noting that the autocorrelation at Tr 1/2n+2 is approximately zero for moderate n. This is clear from the definition of the autocorrelation,

Jo

where ij = (qo, 1, ..., n) and Wn(Pi; x') , specifies the evaluation of Wn(i1; x') at x' = v. When =- 1/2"+2 we can express the term Wn(i; x') l+T as

Wn (P1; x')l x+/2n+2 sin[2 . 2rx + (n + 7r)] E cWn-1_i(); X')l, (4.3)

where the {cai} are coefficients of functions of order n - 1. Clearly,

Wn (1; x') lx+l/2-+2

is orthogonal to Wn(sij; x') x over the interval [0,1] and thus approximately orthogonal over the diminished interval [0,1 - 1/2n+2] for moderate n.

The normalized autocorrelations of the three waveforms presented in figure 1 are plotted in figure 2.

The fourth property is hierarchical. It is possible to find a set {c4} of order n coefficients, such that

cW(Wn( ;x) = sin(k . 27x) or cos(k. 27x), k= 1,3,...,2n+ - 1. (4.4) $i


331



332 J. E. Hershey, A. A. Hassan and R. Yarlagadda 1- (a)

0_

1-

~~lI ' ' ' '

'Q'e~ I - (c)

-1-

-1 0 1

Figure 2. Normalized autocorrelations of 2(n+1)/2Wn (; X) for n 0, 3, 6.

This property may be concluded by expanding

sin(k 2rx + A l) = sin(k 27x) cos(A ? P) + cos(k ? 21x) sin(A ) (4.5)

and noting that the half set of {Wn(P; x)} corresponding to the summation in- dices that have an even number of ones in their binary representations yields a matrix of {cos(A ? i)} that has full rank. Thus any odd frequency cosine term may be isolated. A similar argument obtains for any odd frequency sine term.

This property allows a convenient multiresolution approximation to obtain wherein any approximation of 3 ccWn((';x) is 'upwardly subsumed' by, or embedded in, the set of possible approximations of order n + 1. This property essentially ensures that signal reconstruction based on this method of signal representation will be numerically well behaved.

The fifth property concerns the relationship between the set of functions

{Wn(P; x)}

and a wavelet eigensequence. As Strang (1989) discusses, wavelets are developed from first choosing a basic function, ?(x), and solving for the set of coefficients {ck} which operate in the two-scale dilation equation q(x) =- Eckc(2x - k). If we choose ?(x) to be the 'box' function, which is unity on the interval [0,1] and zero elsewhere, we find that co = 1 and c1 = 1. The wavelet that results from this selection is the Haar wavelet, which is visible on forming q(2x)- ?(2x - 1). The discrete Haar wavelet transform can be easily performed on a set of 2n points by hierarchically performing quadrature mirror filtering as instructed in (Press 1992, ?13.10).

If H is a discrete wavelet transform and s is a data sequence, then an eigentype relation obtains with a condition such as Hs = As. In general, there is only a trivial solution to this equation in the wavelet domain. If, however, we allow the definition to be modified to a recursive format such as Hs2n = (02n- : 2n-1) in which sn is a 2n long sequence and 02n-1 is a 2 -1 long sequence of zeros and




A new signal representation 0.5

-0.5

1.5

- 1.5

1.5

-1.5 t u 0 1

Figure 3. (a) Graph of one period of W6(0; x). (b) Hard quantization of W6(0; x) sampled at the semi-modal points. (c) 1x Haar wavelet transform of the above samples.

the colon denotes concatenation, then we can find some interesting solutions. The solutions will, of course, be intimately related to the particular wavelet. Choosing the Haar wavelet, we can show that the following constitutes such a solution with A = -. We let s2n be the first 2" terms of the Thue-Morse sequence.

If we properly sample Wn(0; x) and hard quantize the samples to {l}, we will obtain the first 2n+1 terms of the Thue-Morse sequence. The proper sampling of the function is at the ever increasing values of x over the interval [0,1] at what we term the semi-modal points {xi}, where

xi = i/2 + i 1 3 5, 2+2 1. (4.6)

The traces shown in figure 3 summarize the results so far using W6(0; x) as an example.

The sixth property is a behavioural similarity to Weierstrass functions. Re- cently, Resnikoff (1990) published an intriguing connection between Weierstrass functions and compactly supported wavelets. Weierstrass functions are of the form

00

E ai cos birx. i=O

This class of functions is interesting in its own right as the functions are ev- erywhere continuous and nowhere differentiable for some values of the defining parameters a and b.

In the limit, as n -- oo, the function, 2nWn(0;x), remains continuous over the interval (0,1) but becomes non-differentiable within the interval. To see this, consider one of the identities due to Euler given in Resnikoff (1990), namely

1 = 1 cosv + cos2v + cos3v + cos4v ... (4.7)

for 0 < v < 7. From this we can see that

0 = ? cos v - cos 3v i cos 5v i cos 7v + ... (4.8)

As @( ) takes on only the values {1l}, we conclude that, in the limit, 2nW,(0; x) remains continuous but cannot be differentiated and is thus a Weierstrass-like


333




function. In the limit, the eigensequence becomes an eigenprocess which can be set in the framework of the continuous Haar wavelet transform.

5. Signal representation

The set of functions {Wn ('; x)} can be used to represent by approximation an odd frequency function on the interval [0,1]. To do this we form

z(x)= - ZcWn(^x; x), (5.1)

where

c = 2n+ z(x)Wn({; x)dx. (5.2)

What is especially interesting about this approach to continuous signal representation is that the basis functions, unlike the elements of the usual bases, all have the same power spectral density as is evident from (3.3).

Asymptotically, as n -- 00o, the functions {Wn({; x)} form a basis capable of representing odd frequency functions. The functions are orthogonal and belong to the Hilbert space of finite energy functions with a norm defined by

(Z, Wn(P; x)) = 2n+1 z(x)Wn(4; x) dx. (5.3)

(See, for example, Theorem 5.17.8 (the Fourier series theorem) in Naylor & Sell (1982).)

The signal representation can be made to include even functions also, and thus all functions, by a number of means. One such way is simply to decompose the function to recover the odd frequency components, as per (5.2), and then invert the right half of the function and decompose again for the even frequency components.

6. An example As an example of continuous signal representation, we represent by approxi-

mation, z*(x), the odd frequency function,

(6.1) z(x) -+ sin(47rx), 0 ? x ? 0.5,

-sin(47rx), 0.5 < x < 1.0,

by (5.1) for order six. The results are shown in figure 4.

7. A discrete basis

For a discrete signal representation, we can form a discrete basis for representing a sampled signal in the following manner. We form the basis for order n, Bn(X), consisting of 2" vectors of 2n values each. Each basis vector, Bn,~(X), corresponds to one of the 2" unique values of (P. The vector values are obtained for each vector by sampling Wn-_I(Q; x) at the modal points {Xi}, where

Xi - (2i + 1)/2, 0, 1, 2,.1. (7.1)


334



A new signal representation 335

2 -

0

2 2-

-2

0.010

0 o -......

-0 010 0.2 0.4 0.6 0.8 1.0 x

Figure 4. An odd function, its order six approximation, and the approximation error.

The basis vectors have the convenient property that B 2(xi) = 1 (7.2)

i

and therefore the discrete signal, Z = (zo, z1,. .., z2n-l), can be represented as

Z = ZcBn,(X), (7.3)

where c= -

z(i)Bn,xi). (7.4)

The orthonormal basis, B (X), is related to the Walsh transform which consists solely of plus ones and minus ones in a traditional order. The basis, Bn(X), can be arranged to exhibit the same ordering of signs but the coefficient magnitudes will not be unity. The discretized basis, Bn(X), also differs from the continuous basis in that the discretized basis vectors do not exhibit identical power spectra.

8. Conclusion

We have developed a new representation method for signals that proceeds by first dividing them into an even and odd frequency synthesis and then proceeds to represent both odd and even components with a half-basis all of whose members exhibit identical power spectral densities. This new method of approximation will distribute the approximation errors over a relatively broad spectrum in contrast to the traditional Fourier approach to approximation by spectrum truncation. The authors thank Dr Aiman Abdel-Malek and Dr Max H. Costa for valuable advice and counselling.

References

Gonzalez, R. C. & Wintz, P. 1987 Digital image processing, 2nd edn. Addison-Wesley. Mallat, S. G. 1989 A theory for multiresolution signal decomposition: the wavelet representation.

IEEE Trans. Pattern Analysis Machine Intell. 11, 674-693.


I




Naylor, A. W. & Sell, G. R. 1982 Linear operator theory in engineering and science. Springer. Press, W. H., Teukolsky, S. A., Vetterling, W. T. & Flannery, B. P. 1992 Numerical recipes in

Fortran: the art of scientific computing, 2nd edn. Cambridge University Press.

Resnikoff, H. L. 1990 Weierstrass functions and compactly supported wavelets. Aware Tech. Rep. no. AD900810.

Sohie, G. R. L. & Maracas, G. N. 1988 Orthogonality of exponential transients. Proc. IEEE 76, 1616-1618.

Strang, G. 1989 Wavelets and dilation equations: a brief introduction. SIAM Rev. 31, 614-627.

Yarlagadda, R. & Hershey, J. E. 1984 Spectral properties of the Thue-Morse sequence. IEEE Trans. Commun. COM-32, 974-977.

Received 7 February 1994; accepted 24 May 1994


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A New Signal Representation