A discussion on the expression proposed in [1]–[3]
for deconvolving the wideband density function is presented. We
prove here that such an expression reduces to be proportional
to the wideband correlation receiver output, or continuous wavelet
transform of the received signal with respect to the transmitted
one. Moreover, we show that the same result has been implicitly
assumed in [1], when the deconvolution equation is derived. We
stress the fact that the analyzed approach is just the orthogonal
projection of the density function onto the image of the wavelet
transform with respect to the transmitted signal. Consequently,
the approach can be considered a good representation of the
density function only under the prior knowledge that the density
function belongs to such a subspace. The choice of the transmitted
signal is thus crucial to this approach.Peer Reviewe

Fernández Rubio, Juan Antonio

Rebollo Neira, Laura

IEEE Signal Processing Letters

English

A discussion on the expression proposed in [1]–[3]

for deconvolving the wideband density function is presented. We

prove here that such an expression reduces to be proportional

to the wideband correlation receiver output, or continuous wavelet

transform of the received signal with respect to the transmitted

one. Moreover, we show that the same result has been implicitly

assumed in [1], when the deconvolution equation is derived. We

stress the fact that the analyzed approach is just the orthogonal

projection of the density function onto the image of the wavelet

transform with respect to the transmitted signal. Consequently,

the approach can be considered a good representation of the

density function only under the prior knowledge that the density

function belongs to such a subspace. The choice of the transmitted

signal is thus crucial to this approach.Peer Reviewe

UPCommons

On wideband deconvolution using wavelet transform

UPCommons. Portal del coneixement obert de la UPC

IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 7, JULY 1997 207On Wideband DeconvolutionUsing Wavelet TransformsL. Rebollo-Neira and J. Fernandez-Rubio, Member, IEEEAbstract—A discussion on the expression proposed in [1]–[3]for deconvolving the wideband density function is presented. Weprove here that such an expression reduces to be proportionalto the wideband correlation receiver output, or continuous wavelettransform of the received signal with respect to the transmittedone. Moreover, we show that the same result has been implicitlyassumed in [1], when the deconvolution equation is derived. Westress the fact that the analyzed approach is just the orthogonalprojection of the density function onto the image of the wavelettransform with respect to the transmitted signal. Consequently,the approach can be considered a good representation of thedensity function only under the prior knowledge that the densityfunction belongs to such a subspace. The choice of the transmittedsignal is thus crucial to this approach.I. INTRODUCTIONIN [1]–[3], wideband processing of acoustic signals isapplied to the problem of source localization, when thereare several scatterers reflecting a transmitted signal .The wideband model for the received acoustic signal, ,reflected by a single scatterer, is assumed to be, where is the time delay, and the scaleis related with the speed of the scatterer. In environmentswith multiple scatterers, the delay and scale are assumedto be distributed according to the density function ,so that the received signal, , is written as(1)By correlating this signal with hypothesized replicas of thetransmitted signal, , the wideband correlation receiveroutput, WC , is formed asWC(2)and is identical to the wavelet transform,, of the signal with respect to the signal ,which is assumed to be admissible as a mother wavelet. Tosimplify notation we denote .Manuscript received December 17, 1996. This work was supported byMinisterio de Educacio´n y Ciencia, Spain, Program Estancias Temporalesde Cientificos y Tecno´logos Extranjeros en Espa~na, and Project TIC 96-0500-C10-01. The associate editor coordinating the review of this manuscript andapproving it for publication was Dr. M. Unser.The authors are with the Departament de Teoria del Senyal i Comunica-cions, Escola Tecnica Superior, d’Enginyers de Telecomunicacio´, Barcelona,Spain (e-mail: laura@gps.tsc.upc.es).Publisher Item Identifier S 1070-9908(97)05233-4.After some manipulation, the density function, , isderived in [1] and presented in [1]–[3] as(3)where is the admissibility constant,, and the Fourier transformof .We shall show the following.1) The constant in (3) must be replaced by .2) Equation (3) (after changing to ) is equivalent to(4)3) The equivalence arises as a consequence of havingimplicitly assumed (4) in [1], when (3) is derived.4) The approach being analyzed is just the orthogonalprojection of the density function onto the image of thewavelet transform, with respect to the transmitted signal.5) If the wavelet transform, is approximatedby a function, say , then, provided that there isno function such that can bewritten as (3) renders a betterapproximation to than itself.II. DISCUSSION ON THE DECONVOLUTION EXPRESSION (3)Let us write the resolution of the identity property of thecontinuous wavelet transform, from which all the propertiesto be used here can be obtained. This property establishes thatfor(5)where is any admissible mother wavelet, and.First, we show that(6)Since(7)1070–9908/97$10.00  1997 IEEE208 IEEE SIGNAL PROCESSING LETTERS, VOL. 4, NO. 7, JULY 1997by performing the change of variables in theintegral, we obtain(8)Thus, the left hand side in (6) can be recast as(9)and replacing by and by in (5), it is provedthat(10)so, the integral in (3) is reduced to .Now we discuss the derivation of (3) given in [1], andwe show that such a derivation assumes.In that publication, the property thatis a reproducing kernel in a subspace of the Hilbert spaceis applied for deconvolving .That is a reproducing kernel in a sub-space of readily follows because[see (8)] and from (5), with and, we have(11)However, (11) does not hold for all the Hilbert space, but onlyfor the subspace generated by [4]. Let us call sucha subspace , i.e.,. Equation (11) provides the necessary and sufficientcondition for a function to belong to . This meansthat if a function satisfies(12)then there exists a function such that. Nevertheless, not every function sat-isfies (12), i.e., not every can be expressed asfor some . Therefore, to assume, asin [1], that the density function satisfies (12) is equivalentto assuming that . Theconstant arises due to the fact that it is not present inthe reproducing kernel equation (17) in [1]. Thus, in thedeconvolution equation (18) of [1], written here as (3),should be replaced by and, according to (10), the equationreduces to (which is consistentwith the way it has been derived).Certainly, there is no reason, per se, why the densityfunction should satisfy (12). Denoting to be the or-thogonal complement of in , we have that any functionfulfills ,because ,due to the fact that and isorthogonal to every function in . Consequently, the mostgeneral solution of the inverse problem (1) can be expressed as, which implies that thedetermination of the density function has no unique solution,if only one transmitted signal is used. The hypothesis thatsatisfies (12) is tantamount to setting .Otherwise, in general, the right-hand side of (12) gives rise tothe orthogonal projection of onto . It is pertinentto stress then, that can be accepted as a goodrepresentation of the density function only in those exceptionalcases in which one has the prior knowledge that the densityfunction must lie in , or be close to it. As is generatedby the transmitted signal, , it is obvious that the selectionof , far from being arbitrary, is of crucial importance.In order to discuss the effect of noise in the approach beinganalyzed, let us assume that the received signal, ,is affected by additive random noise and one is able toobtain only . If the random noise, then can be evaluated and the deconvolutionequation (3) is equivalent to writing .In practice, however, the exact computation of (orin the noiseless case) may not be possible but ratheran approximation of is available. If is therealizable approximation of there is no longer anyguarantee that , and so, whena better approximation to than itself is, atleast in theory, possible. Indeed, using we can buildan artificial signal as(13)and, taking the inner product of with, we have(14)If , it will not satisfy the reproducingkernel equation (12), and from (14) we have:, with . Therefore, asit follows from the inequality,is a better approximation tothan . The proof is immediate: Let us denote, then. [Theprevious equality holds for being orthogonal to .]By comparing (3) and (14) we can conclude that when thewavelet transform, is approximated by a function, the deconvolution equation (3) gives a betterREBOLLO-NEIRA AND FERNANDEZ-RUBIO: WIDEBAND DECONVOLUTION 209approximation to than itself. Notice that,in practice, the statement may not be true if the computationof (3) introduces large errors.III. CONCLUSIONSWe have discussed the deconvolution equation for thewideband density function given in [1]–[3]. We have provedthat this equation is equivalent to .We have shown that this equality has been used implicitly inthe derivation of the deconvolution equation and that is whythe equation is reduced to it.We have remarked that, in fact, what the analyzed approachprovides is the orthogonal projection of the density functiononto the subspace generated by the image of the wavelettransform, with respect to the transmitted signal . Thus, theapproach makes sense only in those cases for which one hasthe prior knowledge that the density function belongs to sucha subspace. As a consequence, the selection of the transmittedsignal can not be arbitrary.Finally, we have proved that when the wavelet transformis approximated by a function that is notin the image of the wavelet transform, i.e., when no signalsuch that exists, thenwhat the analyzed deconvolution equation yields is a betterapproximation to than itself.ACKNOWLEDGMENTAt the acceptance of this paper, one of the reviewers madenote that some ideas presented here have also been recognizedby other authors in related papers [5] and [6].The authors wish to thank to R. J. Meziat Velez for usefuldiscussions, and A. B. Maclean and Dr. J. Goldberg forcorrections of the manuscript.REFERENCES[1] L. G. Weiss, R. K. Young, and L. H. Sibul, “Wide-band processing ofacoustic-signals using wavelet transforms 1: Theory,” J. Acoust. Soc.Amer., vol. 96, pp. 850–856, 1994.[2] L. G. Weiss, “Wide-band processing of acoustic-signals using wavelettransforms 2: Efficient implementation and examples,” J. Acoust. Soc.Amer., vol. 96, pp. 857–866, 1994.[3] , “Wavelets and wideband correlation processing,” IEEE SignalProcessing Mag., vol. 11, pp. 13–32, 1994.[4] I. Daubechies, “Ten lectures on wavelets,” in CBMS-NSF, SIAM,Philadelphia, 1992.[5] T. L. Dixon and L. H. Sibul, “A parameterized Hough transformapproach for estimating the support of the wideband spreading functionof a distributed object,” Multidimen. Syst. Signal Processing, vol. 7, pp.75–86, 1996.[6] , “Wideband imaging and parameter estimation of distributedobjects using the continuous wavelet transform,” Signal Process., vol.54, pp. 207–223, 1996.

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On wideband deconvolution using wavelet transform

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