CA1251859A

CA1251859A - Signal representation generator

Info

Publication number: CA1251859A
Application number: CA000459937A
Authority: CA
Inventors: Pierre L. Goupillaud; Alexandre Grossmann; Jean P. Morlet
Original assignee: ENELF Inc
Current assignee: ENELF Inc
Priority date: 1983-07-29
Filing date: 1984-07-27
Publication date: 1989-03-28
Also published as: EP0132807A3; ATE43443T1; EP0132807B1; EP0132807A2; JPS6070371A; US4599567A; DE3478347D1

Abstract

SIGNAL REPRESENTATION GENERATOR
Abstract of the Disclosure An apparatus and method for generating a representation of an arbitrary signal wherein the signal is represented as a sum of reference signals derived from a standard wavelet defined on a grid in the frequency-time plane. The apparatus comprises a plurality of octave circuits, each octave circuit, in turn, comprising a plurality of band circuits that divide a frequency range covering the spectrum of the signal into equal intervals of logarithm of frequency. Each band circuit comprises a bandpass filter to select a frequency component of the signal and a correlator to correlate the signal component with a reference signal derived from a standard wavelet to produce a series of correlation values representing the signal for recordation in response to clock pulses received by the octave circuits.
The octave circuits are clocked at rates which increase by a factor of two with increasing octave of frequency with which the octave circuits are associated so that correlation time intervals decreasing by a factor of two for successively higher frequency octaves are defined for all frequency bands within each octave and one correlation value is associated with each octave and one correlation value is associated with each frequency band and with each correlation time interval to effect the representation.

Description

SIGNAL REPRESENTATION GENERATOR
Backg~ound of the Invent-LQn 1. Field of Invention The present invention relates generally to signal 5 decomposition methods and apparatus and, more particularly, but not by way of limitation, to systems for providing representations of transient signals.

2. Brief Description of the Prior Act When a person measures or records a signal of some 10 type; for example, the output of a microphone or geophone, he is generally not înterested in the signal per se.
Ratherp the signal carries information and it is the information carried by the si~nal that he is after.
Very often, the recordation or measurement of the 15 signal is only a first step in obtaining the information.
From an information point of view, the signal, which expresses the variation of a physical quantity with time, is merely one of various representations of the information the signal contains. For example, it is well known that 20 information expressed as a varying signal in the time domain can also be expressed as a combination of pure tones having different amplitudes, frequencies and phases in the frequency domain. In some cases, a time domain representation of the information is well suited to the 25 information gatherer's purposes and in other cases, a frequency domain representation will be better suited to these purposes.
However, there exists situations in which neither of these representations of information which is obtainable by 30 recording a signal is particularly well suited to the purpose for which the information has been gathered. Such is particularly true when the phenomenon underlying the production of the signal is transient in nature. In such circumstances, it may well be the case that neither a pure 35 time domain representation nor a pure frequency ~omain :~ZS~859 representation w 11 provide the information ga-therer with the type of dat~ he needs to make use o~ the information Because of the inadequacy of these epresentations of in ormation carried by a signal to express the information in a useable form, a hybrid representation, having both time and frequency aspects, was proposed by Dennis Gabor in 1946.
Such proposal has been summarized in Technical Report No.
238 by D. Gabor~ published by the ~esearch ~aboratory of ~lectronics at the Massachusetts Institute of Technology on April 3, 1952.
In the hybrid representation suggested by Gabor~ a signal would be decomposed into a plurality of wavelets which would each be associated with a cell in the combined time-frequency domain which Gabor referred to as the information plane. The cells would form a rectangular grid so that each cell would be defined by a fixed interval in each of the time and fre~uency domains. The wavelets proposed by Gabor had the general form of a sinusoid contained within a gaussian envelope and~ in order that the represen-tation, once obtained, would contain the maximum inLormation theoretically possible, each cell had unit area.
~n important aspect of the wavelet proposed by Gabor, as well as wavelets with which the present invention is concerned, is that the wavelets have both time and Ere~uency characteristics. That is, the cell with which the wavelet is associated is defined by a particular re~uency interval and a particular time extent. This characteristic ma~es the wavelets particularly well suited for representing transient signals.
While the hybrid representation proposed by Gabor held great promise for extracting information contained in the transient signal, the proposal was subject -to practical difficulties which have resulted in the proposal not having been implemented prior to the present invention. ~n 35 par-ticular, Gabor recognized the non-orthogonality of his s~

re~resenta3Lion, but indeed, the main diEficulty was due to the fact that f3r the wavelet described by Gabor the higher the fr~uencyr the higher -the number of periods in the wavelet. It follows tha-t the Gabor expansion presents -the same shortcomings as tne Fourier expansion for high frequencies; that is, undersampling. As a result, Gaborls ~slgges3-ion lay dormant until about 13~0 when further theoretical developments were made by Jean Morlet, such developmer.ts having been published in a series of articles beginning in 1981.
To overcome the difficulties associated with time-frequency representation proposed by Gabor, Morlet proposed that a non-rectangular grid be used to effect the decomposition. In particular, MorleL suggested that the frequency scale be divided into suboctave bands, rather than constant frequency intervals, and that a different time scale be used ~or each bandO In particular, the -time scale for each band would be chosen to be a fixed portion of a standard wavelet, expresse~ in cycles, deEined by limiting a s nusoid having the frequency associated with the band by an envelope that has appreciable values for only a few periods of the sinusoid. That isr the general form of the wavelet would be the same for all cells so that the wavelet can be generally defined and~ after such definition, used to establish time scales for the various bands. It will be noted that a wavelet defined for one cell would differ from a wavelet defined for another cell only by a trarslation and a dilation or contraction in the time domain. Morlet was then able to show that the cells so defined would have equal areas in the time-Lrequency domain so that such a representation would be capable of at least approaching the theoretical maximum information contained within the signal so representedr and more particularly would give a good representation of the phase information.
~owever, Morletls proposal introduced a new problem ~'~5~ 9 s with respect to a prac~ical implementation of the tire- requency represen'cation of a signalO To obtain the maximum in~ormation available in a signal, tne bands and waveler port ons used to define the grid n the 5 time-frequency plane give rise to complex re1ationships between the time extents of the cells. Thus/ a mismatch e~ists between time intervals defined on the cycle-octave grid and real times defined by clo-ks which would have to be used to specify a practical time-frequency representation of 10 a signal. The present invention is based on a further theoretical development which has solved this grid time-cloc~i time mismatch. Such development will be discussed below.
5um~lar~ f~the Ir~ventio~
The present invention provides an apparatus ancs method for generatiny a time-frequency representation of an arbitrary signal in order to extract the maximum information carried by the signal. In paLticular, the apparatus comprises a plurality of overlapping bandpass filters which 20 each decompose an aroitrary signal into one o~ a selected set of f requencies defined on a loyarithmic scale. Each of these frequency components is then correlated with a relerence signal derived from a standard wavelet to produce a sequence of correlation 5i gnals which are subsequentl~!
25 recorded~ In particular, the apparatus and method of the present invention provides for the generation or` a plurality o~ recorded signals which are particularly suited for displaying time and frequency information contained within transient signals. Moreoverr the standard wavelet used to 30 generate the correlation siynals expressing the inrormation contained in a signal can be selected with respect to a phenomenon of interest to the user of the present invention to provide the information contained in a signal in a manner wnich is realistic with respect to such phenomenonO

3~ ~dditionally, the present invention provides an apparatus ~251.~9 and method in which the representation of in:Eormation in a signal can stress the time aspects of the information or,conversely can stress the frequency aspect of such information.
Accordingly, the present invention provides a method for generating a representation of an arbitrary signal, comprising the steps of: separating the signal into signal components within different overlapping frequency bands; selecting a correlation time interval for each frequency band; repetitively correlating a segment of the signal component in each band with a reference signal derived from a standard wavelet having both time and fre-quency characteristics to obtain a sequence of correlation signals for each band, wherein the segment oE the signal component of each band that is correlated with the reEerence signal has a time dura-tion equal to thc correlation time interval for tha~ band; and recording said correlation signals.
The present invention also provides an apparatus for generating a representation of an arbitrary signal, comprising: a plurality of octave circuits/ each octave circuit associated with a selected octave of a selected frequency range, for receiving the signal and generating a set of correlation values between dif-ferent frequency componen-ts of the signal and a reference signal derived from a standard wavelet having both time and frequency characteristics in response to reception by the octave circuit of a selected number of clock pulses defining a correlation time interval for the octave circuit; clock means for providing the clock pulses to the octave circuits, the clock means providing ~S~5~3 -5a-clock pulses to dif:Eerent octave circuits at rates that increase by a factor of two for octave circuits associated with succes-sively higher octaves of the fre~uency range, whereby said cor-relation time intervals decrease by a factor of two Eor octave circuits associated with successively higher octaves of the fre-quency range; and means for recording the correlation values at times controlled by said clock.

:~2~ 6 Brief Descrip~ion of the DraT~ings Figure 1 is a schema'cic circuit diagra~ of one ?referred embodiment of a signal representation generator constructed in accordance with the present invention.
Figure 2 is a schematic circuit diagram of an octave circult used in the signal representation generator of Figure 1~
Figure 3 is a schematic circuit diagram of a correlator used in a band circuit forming a portion of an octave 10 circuit of the signal representation generator~
Figure ~ is a timing diagram illustrating the manner in which the octave circuits of the signal representation generator a~e cloc~ed.
Figu.e 5 is a diagram of the even part of a 15 representative standard wavelet useful for generating a representation of a signal.
Figure 6 is a diagram of the odd part of 2 representative standard wavelet useful for generating a representation of a signal.
2Q Figure 7 is a diagram illustrating the selection of passband Lrequencies for overlapping bandpass filters of the signal representation generator.
Figure 8 is a diagram of the information plane divided in accordance with the present invention.
Figure 9 is a diagram illustrating the manner in which re~erence signal values are derived from a standard wavelet for correlation with a signal received by the signal representation generator.
Figure 10 is 2 second form of the even part or a 30 standard wavelet use~wl for generating a representation of a signal.
Figure 11 is a second embodiment of a band circuit for the signal representation generator.

1 ~ S~ ~ 5e~ 7 Description of the Preferre~ Embo~i~e~
Referring no~ to the drawings~ illustrated ther2in is the circuit diagram for one preferred embodiment of the signal representation generator of t'ne present invention and, further, criteria utilized in the selection of operating characteristics of components or the signal representation generator. In particular, circuit diagrams for the signal representation generator, deslgnated by the general reference numeral 2C in Figure 1, have been 10 illustrated in Figures 1-3 and 11 and Figures 4-10 have been inclu~ed to illustrate criteria by means of which signals produced and recorded by the signal representation generator are related to a representation of an arbitrary signal in terms of a standard form wavelet associated with cells of 15 one prererred grid defined on the information plane.
Referring first to Figure 1, the signal represen-tation generator 20 comprises a plurality of similar octave circuits 20-30 which differ primarily in the selection o fre~,uencies to which the octave circuits respond~ In 20 general, the irst octave circuit, designated 22 in Figure 1~ responds to the highest portion of a frequency range selected for decomposition of an arbitrary siynal; the second octave circuitl designated 24 in Figure 1, responds to a next lower portion of the selectea frequency range; and 25 such decrease in the portions of tne frequency range to which an octave circuit responds is carried out to the fifth octave circuit 30 which responds to the lowest fre~uency portion of the range. As will become clear below, the number of octave circuits included in the signal 30 representation yenerator can be selected to cover substantially any fre~uency range and~ ~urther, the portions of the selected frequency range to which each octave circuit responds can be conveniently identified by an index n that runs serially ~rom zero for the first octave circuit; that ~'~S1~59 isr tne octave ci~cuit responding to the highest fre~uencies in the ranye~ to a number one less -than the selected number o~ octave circuits for the oc-cave circuit responding to -the lowest frequencies in the range. The index n has been 5 indicated f or the octave circuits 22-30 in Figu-e l. Each of the oct~ve circuits 22-30is connected, via signal patns 32-~0, f or the octave circuics 22-30 respectively, and a signal path ~2 to a generator input terminal 4~ which receives a signal to be decomposed.
The signal representation generator 20 further comprises a cloc~ ~6 which periodically provides clock pulses to each of the octave circuits 22-30 in accordance with a timing diagram shown in Figure ~, the clock pluses to octave circuits 22-30 being transmitted on cloc'~ paths 4~-56 15 respectively. In additionr and also in accordance with the timing diagram of Figure 4, the clock 46 supplies clock pulses to the clock input of a multiplexer 58 via clock path 60 and to a counter 62 via the clock path 60 and a clock path 64. The output terminals of the counter 62 are 20 connected to the data selection terminals o the multiplexer 58 to cause the multiplexer 58 to transmit one of a plurality of correlation signals supplied to the multipleYer from the octave circuits 22-~0, on buses 66-74 respectively, to a recorder 76 via a signal transmission channel 78. (As 25 will become clear below, the correlation signals are prererably provided in digitized form on a plurality of parallel signal paths. For clarityr each such plurality of signal paths will be reEerred to herein as a channel and a collection of channels will be referred to nerein as a bus.
30 It will also be clear to those skilled in the art that the multiplexer 58 will include a plurality of sections in order to transmit a digitized siynal e~pressed as a plurality of bits transmitted over a plurality of signal paths comprising a channel.) The recorder includes a conventional coder in 35 order that the signal present at the output terminals of tne ~2~ 19 multipie~er can be codecl and recorded in a suitable recording medium such asl for exampleJ a magnetizable tape or disk. Such recorders are conventional so that the recorder 76 need not be further discussed for purposes of che present disclosure.
~ ererring now to Figure 49 it will be useful to briefly discuss the timlng diagram for the clock 46 be~ore continuing with the description of the construction of the signal representation generator 20 and, for sucn purpose, the timing diagram has been drawn to include a plurality of axes corresponding to the clock paths 48-56 and the clock path 60. For clarity, these axes have been numbered using the same numerical designations as the clock paths to which the a~es correspond and clock pulses on each s;gnal path have been indicated from left to right in the order in which a series of pulses are impressed on the clock pa~hs during the operation of the signal representation generator 20. In particularr the clock 46, and the counter 62r are selected to be resettable and Figuxe 4 illustrates a sequence of pulses, beginning Lrom the left in Figure ~, that would appear on each of the signal paths 6G and 48-56 immediately following the resetting o the clock ~6 and the counter 62.
The salient features of the timing of the clock pulses received by the octave generators 22-30, the multiplexer 58 ancl the counter 62 can best be seen by comparing a series of clock pulses appearing on the clock path 6C with series of clock pulses appearing on the clock paths ~-56. Initially, it will be seen that the multiple~er 58 and counter 62 are clocked at a hlgher rate than any of the octave circuits and that, for every clock pulse appearing on the signal path 60, a clock pulse appears on one~ and only one~ or the clock paths 48-56. Such clocking permits the counter 62 to select a correlation signal from one of the octave circuits~ such octave circuit being the octave circuit clocked coincidently with the multiplexer and counterl for recorcling each time a ~'25~1~S9 clock pulse is delivered to the multiplexer. (In order to phase the generation of correlation signals and the ~ransmission of slch si~nals to the recorder, a multiple pulse clock cn be utilized so that each of the clock pulses 5 lndicated -Eor the ax s 60, a~d the axes 48-56 r would comprise a group of clock pulses as is known in the art. In order to bring out the timing of the signal representation generator, each such group has been represented as a single pulse.) Thus, for example~ a correlation signal is 10 genera~ed by the first octave circuit 22 and recorded in response to the first pulse generated ~y the clock ~6 and impressed on clock pa-th 60 after reset of the clock 46 and counter 62 as indicated by the dashed line ~0 in Figure 4; a correlation signal is generated by the second octave circuit 15 24 and recorded for the second pulse appearin~ on the clock path ~0 to the multiplexer 58 as indicated by the dashed line 82 in Fig~lre ~; and other pulses appearing on the signal path 60 and will be accompanied by the generation of a correlation signal by one of 'che octave circuits 22-30 and 20 the recordation of such signal.
It will also be noticed in Figure 4 that the octave circuits 22-30 are clocked at successively lower rates for increasing values of the index n identifying the octave circuits 22-30. Thus~ the second octave circuit 24 is 25 clocked at one half the rate at which the first octave circuit 22 is clocked; the third octave circuit 26 is clocked at one half the rate of the second octave circuit 24; and so on to the fifth octave circuit 30 which is clocked at one half the rate of the fourth octave circuit 28 30 and at one sixteenth the rate of the first octave circuit 22. In addition, following resettin~ of the clock 46, different periods of delay occur prior to the recep-tion of an initial clock pulse for different octave circuits. Thus, no delay occurs in the transmission of a clock pulse -to the 35 first octave circuit 22 relative to the clocking of the ~518S9 l multiple~er 58; a delay of one mul-tiplexer clock pulse occurs before -the initial clocking oE the second octave circuit, a delay of tnree multi~le~er cloclc pu]ses occurs bet~een the init~al clocking of the third octave circuit;
5 and larger delays occur before initial clocking of the remaining octave circuits. In general, the di~feren~ rates of clocking of the octave circuits and the delays prior to initial clocking o~ each of the octave circuits can be summarized by numbering tne pulses transmitted to the 10 multiplexer 58 following system reset and identifying which of such pulses would also be transmitted to eac'n of the octave circuits. 5uch identification is expressed, using the index n that identiies the octave circuits by the equation:
j = 2n + m 2n~1, m = 07 1~ 2 .... O (1) where j is the number identifying a pulse appearing on the clock path 60. Such selection of pulses appearing on the clock path 60 to also appear on a clock path to an octave circuit can be easily effected by including a plurality of 20 resettable ring counters in the ciock 46 -to be clocked by the pulses appearing on the clock path 60. Such ring counters would each be associated with one of the clock raths ~8-56 and the ring counters would have different numbers of output terminals to effect the halving o~ the 25 ratcs at which the octave circuits are clocked. That is, the ring coun~er associated with the clock path 50 would have twice the number of output terminals as the ring counter associated with the clock path 48. Similarly, the delays in the production of clock pulses on the signal paths 80 48-56 following system reset would be obtained oy the choice of output terminals of the ring counter to which ~he clock path a8-56 associated therewith would be connected. As will be clear from the above description of the timing diagram for the signal representation generator, a selec-ed number 35 of clock pulses appearing on any one of the clock paths 3L'~S~

A8-~, in conjunction wi~h the rate at which the multiplexer 58 is clocked, can be utilized to deine a time interval for each of the octave circuits 22-30 and such time intervals will vary in duration from octave circuit to octave circui-t oy a power of two. Such selection defines an information plane grid in a manner to be discussed below.
Returning now to the description of the construction of the signal representation generator 20 r and with p~rticular reference to Figure 2 wherein is shown a schematic circuit o diagram for the first octave circuit 22, each of the octave circuits is comprised of a plurality of band circuits as indicated by the dashed line boxes in Fiyure 2. In particular, in the one preferred embodiment of the invention presently under discussion, the first octave circuit 22 15 comprises four band circuits which have been indicated by the numerals 84-90 in Figure 4, the other octave circuits similarly have four band circuits in such embodiment.
However, it is contemplated that the octave circuits can have a yreater or lesser number of band circuits; in 20 particular, the number of band circuits to be included in an octave circuit is selected by the user of the present invention to stress the time or frequency aspects o the information contained within a siynal in accordance with a criterion to be discussed below~
In the same manner that the octave circuits respond to different portions of the overall frequency range used to generate a representation of an arbitrary signal, each of the band circuits responds to a different frequency band within the range of frequencies to which the octave circuit 30 in which the band circuit is included responds~ In particular, the band clrcuit 8~ responds to a frequency band near the high frequency limit of the first octave circuit 22; the band circuits 86 and 88 respond to successively lower, intermediate frequency bands~ and the band circui-t 9~
35 responds to a frequency band near the low frequency limit of .. .~.

.~Z~ 35~3 i3 the octave circuit 22. In a manller similar to the identification of portions of the frequenc~ range to ~lhich each octave circuit responæs by the index n, an index k can conveniently be used to identify a frequency band to which 5 each OL the band circuits 84-90 responds. Consistently with the selection oF the index n to increase with response to decreasing frequencies, the value of the index ~ associated with the highest frequency band into which an octave circuit frequency range is divided is selected to be zero and 10 successive bands are consecutively numbered so that the band circuits 84-90 in Figure 2 are associated with band index numbers k = 0 to k = 3 respectivelyv The band circuits in the remaining octave circuits are identically associated with the same values of the band inde~ number 1~. ~ach OL
15 t`ne band circuits 84~90 receive the signal to be decomposed, such signal being present on the signal path 32 that is connected to the generator input terminal ~4, and each of the band circuits 84-90 receives the clock pulses impressed by the cloc~ 45 on the clock path 48, the paths 32 and ~8 20 having been included in Figure 2 to illustrate the manner in which elements of the first octave circuit at 22, as well as element~ of the remaining octave circuits 2~-30, are connected to portions of the circuit of the signal representacion generator 20 illustrated in Figure 1. In 25 particularr each of the band circuits 84-90 includes a substantiall~ distortion free bandpass filter and the signal path 32 provides the signal at the generator input terminal 44 to each of the filters of the band circuits 84-90. Thus, the band circuit 84 includes a bandpass filter 92 which 30 receives the signal to be represented on a signal pat'n 94;
the band circuit 86 includes a bandpass filter 96 which receives the signal to be represented on a signal path 98;
the band circuit 88 includes a bandpass filter 100 which receives the signal on a signal path 102; and the band 35 circuit 90 includes a bandpass filter 104 which receives the s~
1'1 signal on a signal path 106. Similarly~ each o. the filters inciudes an A/D converter which receives the cloc~ pulses impressed on -the cloc~ path 4&o In particuiar~ thQ band circuit ~4 includes an A/D converter 108 whlch recei~-es clock pulses on clock path 110; the ban~ circuit ~6 includes an A/D converter 112 which receives clock pulses on cloc.i path 114; the band circuit 33 includes an A/D converter 116 which receives cloc~ pulses on clock path 118 and the band circuit 90 includes an A/D converter that receives clock pulses on clock path 122. Each of the A/D conver-ters in a band circuit is connected ~o .he output of the filter in the band circuit as indicated by the signal path 124, 126, 128 and 130 for the band circuit ~ 90 respectively and each of the ~/D converters is of the conventional type including a sample and hold c.ircuit and a plurality of comparators so that each of the ~/D converters will generate a dicJital representation of the amplitude of a frequency component passed by the filter to which the ~/~ converter is connected each time a pulse appears on the clock path ~8 These digital representations produced by the A/D converters are transmitted to correlation circuits, for a purpose to be discussed bel.ow, via channels 125r ].277 129 and 131 connected to the multiple output term;nals o. the A/~
converters 108l 112, 116 and 120 respecti~ely~ Prior to discussing the correlation circuits, it will be useful to discuss the signal components provided by t~e filters S2 t 96, 100 and 104 and Figure 7 has been provided for this purpose.
Figure 7 illustrates a graphical method for selecting the passbands of the filters 92, 96, 100 and 10~ of the octave circuit 22 as well as the passbands of filters in -the remaining octave circuits 24-30. To that end, the user of the signal representation generator 20 will select a frequency range for the representation which is consistent 35 with the phenomenon from which the signal being represented ~5~5~

scale, to the base two, as indicatecl in Figure 7 by the mark 132 on the logarithmic freauency axis 134. Such frequency also identiies the logarith~ic center :~requency of the filter in the highest frequency band circuit of the octave 5 circuit associated with the highest frequency portion of the selected frequency range. F or the embodime~t of the invention illus~ratedO the frequency indicaLed by the mark 132 would be the center frequency~ on a logarithmic scale, of the passband of the 'ilter 92 in the k - 0 band circuit 10 8~ of the flrst oc~.ave circuit 22; that is, the octave circuit for which the index n is zero. The logarithmic center frequencies associated with the Ic = 0 band circuits of the remaining octave circuits are then ound by successive halving of the frequency indicated by tne mark 15 132. On a logarithmic scale~ as shown in Figure 7, such frequencies form a succession o~ equally spaced marks as indicated by the marks 136-142 in Figure 7 and the halving is continued until a frequency below the select~d range, indicated by ~he mark 1~4 in Figure 7, is obtained. The 20 numDer of octave circuits to be included in the signal representation generator is then the number of logarithmic center frequencies, within the selected frequency range, found in this manner. Thus, for the case shown in Figure 7, the signal represerltation generator would include five 25 octave circuits wherein the uppermost passband associated with each octave circuit would be logarithmically centered on the frequencies indicated by the marks numbered 132 and 136-1~2. For the special case in which the highest frequency appropriate to the phenomenon that produces the 30 signal being represented is 250 hertz, the frequencies that -"ould be indicated by the marks 136 142 would be 125 hertz, 62-1/2 hertz, 31-1/4 hertz, and 15-5/8 hertz respectively.
The logarithmic scale is then used to select the logarithmic centers of the passbands for all filters in all 35 octave circuits by dividing all of the intervals between the ~z'~s~

'~ = 0 marks on the logarithmic scale by the selected number Or band circuits to be included in each octave circuit. (As will be ~iscussed below, one advantage of the signal representation generator is that either frequency or time aspects of a signal can be stressed in the representation formed by the signal representation generaLor 20. Tne stress is accomplished by the selection o the n-Lmber of band circuits to be included in an octave circuit.) Thus, the filters in the band circuits 36-90 in the octave circuit 22 would be logarithmically cer.tered at frequencies indicated by the marks 146-150 on the logarithmic scale 134 of Figure 7. The logarithmic centers associated with the filLers in the remaining octave circuits would then be found by similarly dividing the intervals between the marks 13~-144 in Figure 7 as indicated by the unnumbered marks therein.
(The mark 144 establishes an interval that is used to determine the logarithmic centers oE the filters in the octave circuit, such as the octave circuit 30 in the lllustrated embodiment, that is associated with the lowest frequency portion of the selected range and has been included in Figure 7 for this reason.) The filters in each of the band circuits of the octave circuits are then selected to eac'n pass a band of frequencies as illustrated by -tne response curve 141 of the circuit corresponding to the logarithmic center frequency 140. ~ preferred choice for the maximum slope of the response curve is twelve decibels per octave resulting in the responses of the filters overlapping over several bands. ~lternatively to the graphical selection of frequencies to which the filters in the band circuits are to respond, such frequencies can be selected analytically by means of the equation fnrk = fmax 2-n 2 k/K, (2), where fn,k is the logarithmic center f requency selected for the filter in the kth band circuit OlC the nth octave circuit, fmax is the logarithmic center frequency selected 12S~35~ 17 fo- the uppermost band of the uppermost octave of the frequency range Cf interest, n and k are the indices defined above ~or the octave and band circuits respectively and K is the number of band circuits in each octave circuitO (The 5 logarithmic center rrequencies can also be determined from a ~irimum selected ~requency by reversiny the numbering Gf octave and band circuits such that the indices n, k increase Yith increasing frequency and by eliminating the negative signs in the exponents in equation 2.) Referring once again ~o Figure 2, each of the band circuits 8~-90 of the first octave circuit 22, as well as the band circuits and the remaining octave circuits, includes a pair o~ correlators for correlating the signal components passed by the filters in the band circuits with a 15 reference signal tha-t is derived form a standard wavelet into which an arbitrary signal impressed at the generator input 22 is to be expressed. For reasons which ~ill become clear below, one correlator in each band circuit is appropriately termed an "even" correlator and the other 20 correlator in the band circuit is appropriately termed an "odd" correlator. Thusr in Figure 2, the band circuit 84 comprises an even correlator 152 and an odd correlator 154;
the band circuit 86 comprises an even correla-tor 156 and an odd correlator 158; the band circuit ~8 comprises an even 25 correlator 160 and an odd correlator 162; and the band circuit 90 comprises an even correlator 16fi and ~n odd correlator 166. Within an octave circuit, the correlators differ one from another in respects that will be discussed snortly with respect to E`i~ures 3-~ and Figure 9 but 30 correlators are structurally -the same from octave circuit to octave circuit~ Thus, the band circu~ts corresponding to the index k - 0 in the octave circuit 24-30 include even and odd correlators which are similar to the even and odd correlators 152 and 154, the band circuits corresponding to 35 the index k = 1 in the octave circuits 2~-30 contain even 5~3 18 and odd corxelators similar to the even and odd correlators 1~6 and 158; an~ so on for all band circuits corresponding to different values of the inc~e~ k for all octave sircultsO
The general form of each o the correlators, even and 5 odd, and for all band circuits in the signal representation generator 20, is illustrated in Figure 3 wherein is shown the even correlation 152 of the band circuit 84. Such correlator comprises a signal resister 168 whicn receives signal component amplitudes from the A/D converter 108 on a 10 channel 170 (see also Figure 2) and stores a se~uence of values o~ such amplitudes~ corresponding to differen-c -times a~ which a signal is received by the siynal representation generator 20, in a plurality of storage registers of which the signal resister 168 is comprised. (Three such storage 15 registers7 indicted by the numerals 172-176 have been indicted by dashed lines in Figure 3~ However, the signal register 168 will generally contain a larger number of storage registers and, moreover, the number of storage registers will be selected for each band circuit in an 20 octave circuit to relate the result of a correlation by each correlator to an information plane representation o~ a cignal in accordance W7 th a criterion to be discussed below.) The storage registers o-f the signal register 168 are serially interconnected so that, in response to a clock 25 pulse received by the signal register 168, a series of signal amplitudes stored in the storage registers are serially shifted down the sequence of storage registers from storage register 172 toward the storage register 176 with a new signal amplitude being entered in storage register 172 30 and one previously received signal amplitude being lost from storage register 176 with each clock pulse that is received by the signal register 16ao The clock pulses are provided on a clock path 178 which, as shown in Figure 2, connects to the clock path 48 so that each clock pulse supplied to the 35 octave circuit 22 on clock path 48 causes the ~/D converter ..

85'3 19 108 to provide a digital representation of the signal componen`, currently bein~ passed by filter 92 and further causes such representation to be entered into the rirst of the scorage registers of the signal register 168 while shifting previously stored signal components amplitudes along storage registers of the signal register 168.
In addition to the signal register 168, the even correlator 152 comprises a reference register 180 which, like the signal register 168, is comprised of a plurality or storage registers as indicated by dashed lines in Figure 3.
The number of storage registers in the reference register 180 is equal to the number o. storage registers in the signal register 168. Stored within the storage registers of the reference register 180 is a sequence of amplitudes o~ a portion of a reference signal and the storage registers of the reference and signal registers are placed in one-to-one correspondence as indicated by the alignment in Figure 3 of a storage register lS2 in referen~e register 180 with storage register 172 in signal register 168, by the alignment of a storage register 18~ in reference register 180 with a storage register 17~ in signal register 168, and by the alignment of a storage register 186 in reference register 180 with storage register 176 of signal register 168. In addition to the signal register 168 and reference register 180, the even correlator comprises a term~by-term multlplier 188 having a plurality of sections, such as the sections 190-19~ indicated by dashed lines in Figure 3, and an adder 196. Each section of the multiplier 188 is connected to the output terminals of a storage register in 30 the signal register 168 and to the output terminals of the corresponding storage register oE the reference regis~er 180 and each section of the multiplier is a conventional multiplication circuit that multiplies the signal amplitudes in the storage register to which the section is cor.nected in 35 response to a clock pulse delivered to the multiplier 188 on ~L~Sl~3r3~

a clock path 17~ that receives the pulses supplied to the octave circuit 2?. on ciock path 4~ The adder receives -the amplitude products on a bus 200~ adds the products, and outputs the sum on a channel 202 in response to the clock 5 pulse received by the octave circuit 22 on clock path ~8 and transmit--ed to the adder via a clock path 20~. Such sum provides the even part of a correlation signal between the reEerence signa' and an aroitrary signal being received by the signal representation generator 20 to be associated with lO a cell in a time-frequency inormation plane representation of an arbltrary signal in a manner to be discussed below.
The odd correlator 154 differs from the even correlator 152 only wit'n respect to the contents of the storage register of the reference register (not shown) that is 15 included in the odd correlator 154 in the same manner that the reference register 180 is included in the even correlator 152. Where the reference register of t~e even correlator contains a sequence of amplitudes oE an even part of a selected reference signal, the reference register of 20 the odd correlator contains an equal number of amplitudes of an odd part of the reference signal as will now be discussed with reference to Figures 5 and 6 wherein is shown a representative wavelet from which reference signals are derived. As will be discussed below, an arbitrary signal 25 impressed at the generator input terminal ~ is decomposed into a sum of such reEerence signals by the signal representation generator 20.
The wavelet shown in Figures 5 and 6 has been taken from Morlet's publications and is a wavelet particularly 30 suited for representing the information carried by a signal produced by a geophone. Such signal will usually contain short oscillatory pulses, comprised of only a few periods at various frequencies, and such pulses are indicative of reflections of an elastic wave from subterranean strata in 35the region in which the geophone signal is taken. Such form ~L~5~8~ 21 of tne slgnal is carried into the wavelet so that the representation reproduced by the signal representation generator 20 re-~ects the phenomenon to ~hich the geophone is responding.
Tn its general form, the wavelet has two parts7 each of which is a sinusoid limi,ed by an envelope which has an appreciable amplitude for only a few periods of the sinusoid so that 2 wavelet associated with one cell of the information plane on which an arbitrary signal is decomposed lO will contribute only in a minor way to the superposition of wavelets that represent an arbitrary signal in temporally adjacent cells. In Figures 5 and 6, such characteristic has been achieved by selecting -the envelope, designated by the numeral 206 in Figures 5 and 6, to be a gauss an probabilitiy curve having a decay constant selected to effect a large reduction in the amplitude of the envelope over a displacement of two and one half cycles of the sinusoid from the center of the envelope. The two parts of the wavelet are indicated by the curve 20~ in ~igure 5 and the curve 210 in Figure 6. As can be seen in the5e Figuresr the curve 20~ is the result of limiting a cosine curve with the gaussian probablity envelope, hence the curve 208 ls an "even" part o~ the wavelet7 and the curve 210 is the result of limiting a sine curve with the gaussian envelope, hence 25 the curve 210 is an "odd" parc of the ~avelet.
The reference signals associated with the band circuits differ from the wavelet in that the wavelet is defined without regard to any fre~uency while the reference signal associated with each band is defined such that the sinusoidal part of the reference signal has the frequency associated with such band. Thus, each reference signal has the same form as the wavelet but has a time extent that di$fers form band-to-band while the wavelet has a length~
indicted at 209 in Figures 5 and 6, that is defined by the number of oscillations of the sinusoidal part of the wavelet ~L~S1~35~ 2 within the envelope 20~. The use of a wavele-t having even and odd parts defined in this manner, and the derivation of the reference signals by adjustment of the wavelet fo.- band frequencies, preserves phase relationships found in the sigral to be represen,ed as a superposition of reference signals derived from the wavelet.
In the same manner that the frequencies to which the band circuits can be expressed analytically, the reference signals associated with each of tne band circuits can also be expressed analytically for the gaussian limited wavelet shown in Figures 5 and 6. Such analytical expression, for the even and odd parts respectively of the reference signals derived from the gaussian limited cosine and sine curves shown ln Figures 5 and 6, are given by the expression:
ge(t) = exp~-(fn,k t)2 lr.2] cos (2~ fn~k t) (3) and 9o(t) = exp[-(fn,k t)2 ln2] sin (2~ fn k t) (4)-Once a reference signal has been de~ined for each ofthe band circuits, such signal can be used to determine reference signal amplitudes for entry into the reference registers of the correlators for correlation with a signal introduced inko the generator input 44~ For this purpose, the wavelet length 209 is divided into a selected number of intervals and the amplitude of the wavelet at the end points of each Or these intervals is determined. Such amplitudes are selected as the amplitudes of the reference signals to be introduced into the storage registers of the reference registers. (Such method of determining reference signal amplitudes will result in a scaling of the reference signal from band-to band and such scaling must be removed when a signal that has been decomposed is reconstructed. The reconstruction is carried out by multiplying each of the correlation signals produced by the signal representation generator 20 by a constant determined for each frequency by requiring that -the energies of reference signals so defined, ~ 8 59 2 and including -the Erequency of the reference signalJ be the same for all frequency bands. Such energy is defined as the integral over the reference signal; that is, over the wavelet length 209, of the squares o~ the expressions given in equations (3) and (~l) above.) Differing number~ of intervals ror determining reference signal ar.~plitudes are used for band circuits having different values of the index k as will be discussed below.
It will be noted that wavelets and reference signals, defined in this way will have both frequency and time characteristics. The decay of the envelope to a near zero value for a selected number of c~cles provi.des the time characteristic of the wavelet and the sinusoidal variation of the wavelet provides the frequency characteris-tic. The combinati.on of these characteristics in the wavelet makes the reference signals derived therefrom well suited for forming a representation of a signal associated with transient phenomena.
It should be noted that the wavelet shown in Figures 5 and 6 is only an exemplary form of wavelet that might be used in conjunction with the present invention. An important advantage of tne present invention is that the wavelet can be selected to bring out phenomenological characteristics of a signal which is represented by the wavelets. For example, should the signal be that produced by a microphone placed to pick up sound waves produced in a piano recital, a more realistic envelope for the wavelet would have a maximum skewed toward one end of the wavelet to reflect the large initial amplitude, and subsequent slow decay, of a sound wave produced when a piano note is struckO
An example of the even part of such a wavelet has been illustrated in Fiyure 10 and designated by the numeral 207 therein. The skewing of the wavelet envelope 205 in Figure 10 can be achieved by replacing fnr~ t by ln fn,k t in equations (3) and (4) aboveO Similarly, the number of 8~i~ 2'l oscillations contained within the envelope can be selected to meet the characteristics of a phenomenon under considera'cion. Thus, the important characteris'cics of ~he wavelet are not that it has one particular form; rather, the imporLant factor is that it can be selected to realistically reflect the time and frequency characteristics of a signal associated with a transient phenomenon. (Anot,-ler important characteristic of the wavelet is that it must not contain any energy at zero frequency (DC) and in the neighborhood of zero ~requency.) Referring once again to Figures 1-3, the general scheme of tne construction and operation o' the signal representation generator 20 can now be seen. Each octave circuit is associated with a particular octave of a frequency range that includes the frequency components o~ a signal to be represented by reference signals derived from wavelets, such range depending upon the phenomenon that produces the signal~ The range for each octave circuit is further divided into bands, and a band circuit is included in the octave circuit for each of these bands. Each band circuit contains an even correlator and an odd correlator and each correlator includes a reference register that stores a sequence of amplitudes derived from an even or odd part of a standard wavelet; that is, a wavelet that decays to near zero values at end points of the wavelet defined by a fixed number of cycles o sinusoids that enter into the definition of the standard wavelet for all frequencies that might be associated with the waveletO Such frequencies are the frequencies to which the band circuits respond.
Eacn band circuit also contains a bandpass filter to select a particular frequency component of a signal to be represented and an A/D converter to provide a digitized representation of such component each time the octave circuit receives a cloc~ pulseO The digitized representations of the signal components are serially ~ S~5~ 2J

cloc~ed into the signal regis~ers of the correla-tors, one new a~.plitude of a signal component being entered into a signal register with each clock pulse delivered to an octave circuit and the oldest amplitude stored in a signal register being clocked out of the register so that~ at any time~ the reference and signal resisters of each correlator will each contain a sequence of amplitudes tnat can be correlated by the multiplier and adder circuits of the correiatorO Such correlations occur with each clock pulse delivered, according to the scheme shown in Figure ~ to each of the octave circuits. The resulting signals, obtained by the correlation, are outputted on a channel to the multiplexer 58 and such channels have been indicated in Figure 2 by the aforementionecl channel 202 from the adder 196 of the e~en correlator 152 and by channels 212-226 for the remaining correlators of the octave circuit 220 These channels form the bus 66 in Figure 1 and the buses 68-74 from -the remaining octave circuits similarly comprise a plurality of channels upon which even and odd parts of correlation signals are impressed with each clocl~ pulse provided to each octave circuit. The counter 62 also receives cloc~ pulses and selects, for each clock pulse delivered to the counter on cloc~ paths ~0 and 6~, a particular pair of even and odd correlation signalsr such signals being provided by one band circuit, to be passed by the multiplexer ~or recordation.
In order to use the signal representation generator 20 as an analytical tool~ the correlation signals generated and recorded by the signal representation generator 20 are referred to a grid on the information~ or time-frequency plane. The grid corresponding to the described embodiment of the signal representation generator has been illustrated in Figure g. An important aspect of the present invention is the structure of this grid, such structure being a major factor which has enabled the concepts of Gabor and Morlet to be embodied in circuitry to make the present invention ~5iL~9 25 possible. Before discussing this grid, it ~ill be useful to consic~er the problem wnich, until tne present invention, prevented the practical application of the theory that .~as cieveloped by ~lorlet upon the suggestion initially made by Gabor. Such problem has been solved by divicling 'he information plane into a grid such as that shown in Figure 8 and by constructing the signal and reference registers of tne correlators in a manner '-hat is consistent with the grid shown in Figure 8.
The information or energy content of a signal is continuously distributed in the time frequency ~information) plane. When a discrete representation of this distribution is desired, proper sampling of this content over the plane is necessary to preserve the information distribution. This sampling is defined over a grid of points which are centers o small domains of the plane, called cellsO In order to represent the information of the signal most compactly and accurately, the uncertainty principle requires that -the sampling of the time-frequency representation of the signal be done at a rate of one sample per cell in a grid made of cells of equal area~ this area being chosen to avoid both over and under sampling.
Snoulcl the information plane be decomposed on frequencies defined by numbers of octaves and times defined 25 by numbers of cycles in a periodic wave, a representation of an arbitrary signal in terms of reference signals associated ith each cycle-octave cell would generally contain much less information than is available in the sic;nal itself because of undersampling~ ~ccordingly, to obtain the 30 maximum information present in the signal, the cells are further subdivided in one or both of the time and frequency domains. In order to minimize oversampling and to obtain a practical grid, it has been found that the maximum information available in the signal can be obtained when the 35 product of the numer of divisions o~ t'ne octaves in the ~ S ~ 27 frequency domain and the numer o- ~ivisions OL tne cycles in the time domain is equal to eight. I;`or example, for the referonce signals s~lown in Fi gures 5 and 6, a mâxir~u~
information representation can be achieved by dividing each octave in the frequency domain into four f.equency bands, as has been done in the described emoodiment of the invention, and by dividing the wavelet length 209 in Figures 5 and 6 into ten units each so that one half cycle of a sinuso:d in a wavelet would be the time unit upon which the representation of the signal is constructed. Thus, there would be two cells per cycle of the wavelet and four bands per oetave of frequency resulting in the desired product 8.
The difficulty with defining an inormation plane grid with cel7s of equal area and equal octave width as shown in Figure 7, indeed, the difficulty that prevented practical implementation of the theory prior to the present invention, is that time scales definecl to be a selected number of periods of reference signals corresponding to different frequency bands deSined on an oc-tave scale cannot be rationalized with clocks necessary to implemen~ a practical device. For example, the ratio between successive logarit'nmic centers of frequency bands in an octave, where four bands per octave are used to establish the information plane ~rid, is proportional to the fourth root of two, an irrational number. This irrationality would be carried in-to the time scales for the different bands by making the time scale for each band extend in time by an amount that is a selected fraction of a reference signal having a sinusoidal portion defined by the freciuency of the band. Thus, should a ti~e scale be developed in real ti~ne to earry out the decomposition of an arbitrary signal in terms of reference signals using the original cycle-octave grid structure developed by Morlet, in which the time extent of each cell in the inforMation plane grid is inversely proportional to the frequency of the band in which the cell is disposed, the ~ ,, ~

5~3 28 esulting irrational numbers would lead to increased computlng time and unacceptable costs which would make this technique impractical. The Eurther ~heoretical development referred to above that enables the theory to be practically implemented eliminates the c~-,ificulties a~ising from the spacing of cells in different frequency bands in irrational ratios.
In the present invention, the modified information plane grid shown in Figure 8 is utilized to effect a 10 decomposition of an arbitrary signal as a sum of reference signals and such grid is made up of cells having time e~,ents which dirfer by factors of two from octave to octave and are equal for all bands in an octave. Such structure of the grid enables a representation o.~ an arbitrary signal in 15 reference signals.to be practically implemented ly the clocking of octave circuits at. different rates, related by powers of two, as has been discussed above with respect to Figure 4~ ~n order to achieve this practical implementation of the theoryl the correlatlon of signal components for 20 different bands of an octave with the reference signal is carried out slightly di-ferently for each of the bands of an octave in a manner that has been indicated in Fiyure ~. The net effect o-f 'che choice of the information plane grid of Figure 8 and the differences in the manner in which a signal 25 component is correlated with a reference signal result in slightly h.igher sampling rate when computing the correlation values in the lower bands of each octave. Nevertheless, these correlation values, in their totality, and with reference to the information plane grid, e~press a 30 representation o. an arbitrary siynal while removing the difficulties in obtaining such a representation. The corresponding discrete representation is valid provided '~hat appropriate compensation for oversampling be applied in the reconstruction of the signal as will be discussed below.
Referring first to Figure 8, shown therein is an 12~ 5~3 information plane grid suitable for use with a signal representation generator which includes five octave circuits, each of which includes four band c~rcuits. The information plane ln such case is defined 'Dy a time axis 211 and a frequency axis 21~. The times plotted on the time axis 211 to define a grid of cells in the information plane are clock times, such as can be determined by the cloc}c 46, and the quan'cities plotted on the frequency axis 21~ are logarit.~ms of the frequencies passed by the collection of filters of the signal representation generator 20 where such logarithms are taken to the base of -cwo. The frequency axis 214 is divided into a pleurality of octavesr corresponding to the octave circuits 22-30~ covering the frequency range of interest and such octaves are numbered, beginning with 0, 1~ from the octave describiny the highest frequency portion of the frequency range of interest consistently with the above-described numbering of the octave circuits using the index n. Each octave is further divided into bandsr corresponding to the band circuits of each octave, which are similarly numbered beginning with 0 at the top of the octave, eonsistently with the above-described numberiny of the band circuits using the inde~ k, and such numbering has been shown in Figure 8 for the case in which a frequency range is divided into five octaves o-f four bands each. The bands and octaves thus divide the information plane into a series of strips parallel to the time axis wich each strip being logarithmically centered on one of the frequencies passed by one of the filters of the signal representation generator 20. Thus, for exampler using the indlces n and Ic to identify a particular band circuit of a particular octave circuit as described above, the loyarithmic center of the frequency band passed by the filter in the third band circuit of the second octave circuit would be the frequency ~2,3 indicated by the dashed line 216 in Figure 8.
Within each octave, the information plane grid is l~S~59 3 o divided into ec~ual time intervals which, unli~e the time intervals descri`oed ~y ~lorle-t, are the same for all bands in the octave. In particular, such time interval is selected to be the time interval defined by Morlet for the llighest frequency band in tne octave. That is, for each band defined by tne indices nl k~ described above, the tl~e in-terval is selected to be given by the equation:
n,k =
C fn,0 (41' where C is the n~ber of cells to be selected for each cycle of a reference signal. For e~ampler if two cells per cycle of the reference signal are selected, as would be the case when each octave is bro~en into four bands as discussed above, the number C would be two. Such time interval differs from the time interval defined for the information plane grid discussed by Morlet in that Morlet selected the time interval for each cell of a band defined by the indices n and k to be proportional to ~he inverse Of fn,k rather than fn,0 Since the frequencies associated with the highest bands in each of the octaves form a sec~uence in which the frequency is halved for a unit increase in the index n ldentifying the octave in which a band is disposed, the time intervals used to define tne information plane grid in Figure 8 double from one octave to another as shown by the tlme intervals indicated by the numerals 218-226 in Figure 8. That is, the time intervals lndicated by these numerals~
for octaves of successively decreasing frequency, are in the ratio of 1.2:4:8:16. Such a scheme would be continued should the signal representation generator 20 contain additional octave circuits so that the information plane would be divided, with respect to frequency, into additional octaves. Thus, the temporal extents or the cells in the information plane, for all bands of the plane, are in ratios 3~ of whole numbers so that times at which correlation signals 125~5~ 31 a.-e generated by the correlators of the octave circuits can be related to particular pulses produced by the clock. In particular, these ratios of whole numbers are incorporated into the clock ~6 by che clocking of di~ferent octave circ~its at different rates, differing by factors of two, as has been shown in Figure 4. Accordingly, the adoption of the information plane grid shown in Figure 8 permits a practical measure of time which is necessary -for the decomposition of a signal into a sum of re~erence signals derived from a standard wavelet.
E~owever, the ef~ect of modifying the information plane grid from the grid proposed by Morlet (that iS9 a grid in which time intervals are defined for each band and difEer for each band in accordance with the frequency associated wi~h the band) to the grid shown in Figure 3 (that is a grid in which time intervals are defined for each octave but are the same for all bands in an octave) is that the correlation values de~ined by Morlet no longer apply. Figure 9 illustrates the manner in which dif~erent sampled values of the reference wavelets corresponding to small changes in sampling rate can be obtained to derive a correct correlation and thereby obtain a valid representation of the signal to be achieved while associating such values wi'h the practical information plane grid shown in Figure S.
In Figure 9, the even part 208 of the standard wavelet shown in Figure 5 has been reproduced along ~.~ith wavelet lengths, similar -to the wavelet length 209 shown in Figure 5, that are associated with individual band circuits in an octave circuit. That is~ the wavelet length 228 shown in Figure 9 is a wavelet length for a band naving an index k =
0; the wavelet length 230 is a wavelet length for a band having an index k - l; the wavelet length 232 is a wavelet length for a band having an index k = 2; and the wavelet length 234 is a wavelet length for a band having an index k 35 = 3. It should be noted that these lengths, when related to l~S~5~ 32 a particular band circuit also derlne a time duration for the reference signal stored in reference registers of the band clrcuit because of the inclusion of the frequencies associated with tne band in the reference signals. These time durations generally differ from time durations o-f cells in the information planeO The time durations of the cells are related to a number of periods of fractions of a period of the reference signal having the frequency associated with the band in which the cell is disposed. As noted above, the correlation signals used to represent an arbitrary signal impressed at the generator input ~ are found by correlating the amplitude of a slgnal component passing through a filter in a band circuit with the amplitude of a reference signal derived from the standard wavelet length and stored in the reerence registers of the correlators of the band circuit.
Correlation signals good enough to provide an accurate representation o a signal as a sum of reference signals can be obtained~ using the practical information plane grid shown in Figure 8, by dividing the wavelet lengths for different bands into different numbers of reference signal intervals, hence the illustration of a plurality of wavelet lengths corresponding to different bands in Figure 9. In particular, the shortening of the cells corresponding to higher band indices that occurs when the practical sampling grid of Figure 8 is adopted in place of the sampling grid proposed by Morlet can be offset by using differin~ numbers of reference signal intervals in correlating signal components in different bands with reference signals derived from the standard wavelet. Figure 9 illustrates the manner in which the use of different numbers of reference signal intervals, for different bands in each octave~ is carried out.
Initially, for a band having an index k = 0, a selected number of reference signal intervals is chosen so that a selected number of wavelet amplitudes can be determined to ~5~3S~3 33 obtain a sequence of reEerence signal amplitudes to be entered into the reEerence registers of the correlators for correlation with different ~requency components of a sigral to be represented. In Fiaure 9, for simpliication, it has been assumed that ten such intervals are selected for each band circuit identiEied with the index k = 0 and sucn intervals have been marked on the k = 0 band wavelet length 228. (This small number of intervals would not be satisfactory for an accurate evaluation of the correlation signal because of undersampling. In a practical embodiment of the signal representation generatorF a larger number of intervals would be defined on the wavelet length 22~ in a manner and for a purpose to be dlscussed below.~ For each end point of each of these intervals, the even part of the reference signal will have a well deEined ampli-tude as indicated, for e~ample, by the point 236 in Figure 9 corresponding to a shift of one interval fxom the center of the reference signalO The amplitudes found by the intervals defined for the wavelet length 228 corresponding to the band index number k = 0, and for the even part oE the reference signal, would be entered into -the storage registers of the even correlators of band circuits assigned the index k = 0O
Similarly~ reEerence signal amplitudes for the odd correlator reference registers in band circuits identified by the index k = 0 would be found by using the breakdown of the wavelet length 228 in Figure 9 with the odd par-t of the standard waveletO
In order to obtain an accurate representation of an arbitrary signal with the practical sampling grid shown in Figure 8, the wavelet lengths associated with remaining fre~uency bands of each octave are divided into a different number o~ reEerence signal intervals as has been shown in Figure 9. Thus, for the case in which the signal representation generator contains four band circuits per octave circuit and ten re-Eerence signal intervals are used ~Z5~ 3 3~

to obtain reference signal amplitudes for the reference registers of the correlators o, the k = 0 band circuits, t~"elve reference signal intervals would be used to obtain wavelet amplitudes, and hence re~erence signal amplitudesr for re erence registers in k = 1 band circuits; fourteen interva's would be used to obtain amplitudes for the reference registers in k = 2 band circuits and sixteen reference signal intervals would be used to obtain amplitudes for the reference regis-ters in k = 3 band circuits. Sucn increase in the number of reference signal intervals associated with different band numbers has been shown in Figure 9 by the division of the wavelet lengths 23~, 232 and 23~, respectively, and the points 23~ 24~ and 2~2 indicate amplitudes that would be selected from the even part of the reference signal shown in Figure 5 Eor a shift of one reference signal interval from Ihe center of the wavelet The sets of amplitudes of the even part of the wavelet 203 corresponding to the divisions of the wavelet lengths 230 7 232 and 234 would be entered into the reference registers of the even correlators of the band circuits identified by -the numbers 1, 2 and 3, respectively, for t'~
band index ko A similar se-t of refererlce signal amplitudes would be determined frorn wavelet amplitudes similarly picked rrom the odd part of the standard wavelet and entered into the reference registers of the odd correlators.
While the above description applies to the special case in which each octave circuit of the signal representation generator includes four bands and ten referense signal intervals are selected to determine, for the k = 0 bands, the wavelet amplitudes to be entered into the reference registers of the correlators as reference signal amplitudes, the scheme can be generalized to provide for the determination of the reference signal amplitudes ko be entered into the reference registers of the correlators of a signal representation generator having any number of band 5~

circuits included in each octave circuit and for any number of reference signal intervals chosen for -the k = 0 band. In general, the number of reference signal intervals used to pic~ amplitudes of the standard wavelet for determina-tion of reference signal amplitudes to be entered into the reference registers is the nearest whole number less than the ~uantity N~ given by the expression Nk = No2k/K ~5) where No is the number of reference signal intervals used to find reference signal amplitudes for the k = 0 band circuits, k is the band circuit inde~ number ~ associated with a band circuit as discussed above7 and K is tne number of band circuits in each of the octave circuits of the signal representation generator.
Figure 9 was drawn for the special case in which it was assumed that the k = O band would be divided into ten reference signal intervals in order to find amplitudes o the standard wavelet for deriving reference signal amplitudes to be entered into the reference registers comprising portions of the band circuits. ~uch number of reference signal intervals was chosen only for clarity of illustration of the manner in which tne reference signal amplitudes are found for each of the band circuits in each of the octave circuits oE the signal representation generator. ~owever~ this small number of intervals is not satisfactory for correlation accuracy and it is contemplated that many more reference signal intervals w~11 be used in order to maximize the accuracy with which the correlations carried out by t'ne correlators is erfected. A particularly useful number of reference signal intervals is related to the number of band circuits to be included in each octave circuit as will now be described.
As noted above, the time extent of each cell of the 35 information plane will correspond to a particular number of . .--~LZS IL~59 cycles or fractions of cycles of a rererence si~nal at the frequency of the band in which the cell is disposed. In the e~ample given above, each octave circuit comprises Eour band circuits so that, as discussed above, each cell of the in~ormation plane grid shown in Figure 8 would have a time duration of one half the period of a periodic wave having the frequency of the band in which the cell is disposed.
For example, if the uppermost frequency band shown in Fisure 8 is centered on a frequency o. ~0 hertzt so that the period associated with such uppermost band is four milliseconds, the time duration of the cells in the uppermost band would be two milliseconds. Thusr by correlating the highest requency component of the signal being represented with the reference signal amplitudes stored in the reference registers of the highest frequency band circuit of the signal representation generator at two millisecond intervals, one could obtain a correlation signal to be associated with each cell o the uppermost row of cells in Figure 8. However, it is also possible to carry out the correlations at a higher rate and select only one of a series of correlation signals thus obtained to be associated with each cell of the uppermost row of cells in the information plane shown in ~igure 8. ~owever, the additional correlation signals are not needed to obtain an adequate sampling o~ tne representationO They would not add to the information content of the representation and therefore it is not essential that they be obtained. This point can be used to facilitate the multiplexing of the correlation signals as will now be explained AS Will be clear from the above description of the construction of the signal representation generator 20, the clock pulses received by the octave circuits define sampling times, separated by selected sampling time intervals, for signal components passed by the filters in tne band 35 circuits. Each time an octave circuit receives a clock 31J~51~ 3 pulser the A/D converters of each of the band circuits in the octave circuit will sample the signal component to which the band circuit responds and for selected pulses of the cloc;; the correlators of the bancl circuit wiil yenerate a correlatlon signal~ ~ormlng a part of the representation and having even and odd components, that is related to a segment of the sic~nal component to whicn the band circuit responds~
This segment is equal in time extent to the product of the sampling time interval for the band circuit and the number 1.0 o~ storage regis-ters in the signal registers of each of the correlators of the band circuit. Thus7 Lhe rate at which correlation sic~nals are produced by che correlators is determined by the races at which the octave circuits are clocked and the appropriate selection of the pulses of the clock as described below.
In order to relate these correlation signals to the grid of the information plane, it is necessary to select one or more of the samplinc~ time intervals to correspond to the time extent of a cell, at the fre~uency to which each particular band circuit responds, in the information plane~
That is? it is necessary to deine a correlation time interval, comprised of one or more sampling time intervals from the rates at ~hich each of the band circui-ts are clocked and then make such correlation time interval the same as tl~e time extent of a cell in the information plane.
T~hile such correlation time interval must include at least one sampling time interval to carry out the representation of an arbitrary signal over the information plane, it can also be any selected number of sampling time intervals.
30 particularly suitable number of sampling time intervals to make up a eorrelation time interval is the number of band circuits in an octave circuit. That is~ the sampling time interval for eaeh oetave circuit is made to be a fraction of the time extents of the cells corresponding to that octave circuit such that K correlation signals are produced by each ~5~ ~3S9 3~

of the band circuits of an octave circuit for each cell of the octave of the information plane to which the octave circuit corresponds, ~here R is the number of band circuits of an octave circuit. The channels from the eorrelators of the band circuits included in the octave circuit are then connected to different input terminals o~ the multiplexer so t'nat, as the multiplexer receives signals rom the counter to select a correlation signal from a partisular oetave eireuit to be passed by the multiplexer~ the even and odd correlation signals from different band circuits of the octave circuit are selected in turn to be transmitted by the multiplexer. ~y this means, one correlation signal from each band circuit is passed by the multiplexer and recordea for each correlation time interval defined by a series of pulses received by an octave circuit and seleeted to be equal to the time extent of a cell in the information plane.
Thus, each time a cloek pulse is delivered to an oetave cireuit, all bancl eircuits in the oe-tave eircuit generate eorrelation signals but only one eorrelation signal is selected for recording so that a series oE elock pulses will resul~ in a correlation signal being generated and recorded for each cell of the information plane by selecting the eorrelation signal to be recorded from each octave to be one of the correlation signals generated by one of the band circuits of that octave. This scheme has been illustrated in Fi~ure 4 for the series of pulses transmitted to ~he first oetave eircuit of the signal representation generator illustrated in Figure 1.
In Figure ~, the correlation time interval for the first octave circuit has been inclicated at 244 and, as shown in the Figure~ four pulses are delivered to the first octave circuit 22 during the interval 244. Sueh number of pulses corresponds to the four band circuits that are included in eaeh oc the octave eircuits of the embodiment of the invention shown and is selected by seleetion of the basic :~51~S~ 39 cloc~ rate indicated on axis 60 of Figu~e 4. That is, once the length OI the correlation time interval has been selected, by selecting the information plane grid Lor representation of a signal~ -the sampling time interval for the first octave circuit is selected to have a period equal to the time extent of the cell in the first octave of the information plane divided by the number of band circuits included in each octave circuit. In the example abovel wherein each cell in the uppermost octave is to have a time extent of two milliseconds, the clock rate for the first octave circuit would thus be 2,000 hertz; i.e., the period of the clock pulses delivered to the first octave circuit would be two milliseconds divlded by four or 0O5 milliseconds. The clock rate for the multiplexer and counter, indicated on axis 60 of Figure ~1, would therefore be ~,000 hertz.
In order to multiplex the correlation signals to related one signal to one cell of the infoxmation plane, the channels 202 and 212 in Figure 2 would be connected to one set of input -terminals of the multiplexer 58 corresponding to one number at the data selection terminal of the multiplexer 58; the channels 214 and 216 would be connected to another set o-F input terminals o~ the multiplexer 5S
corresponding to another number of the data seleckion 25 terminals of the multiplexer 58; the channels 218 and 220 would be connected to a third set of input terminals of the multiplexer 58 corresponding to a third number at the data selection terminals of the multiplexer 58; and the channels 22~ and 226 would be connected to a fourth set of input 30 terminals of ~he multiplexer corresponding to a fourth number at the data selection terminal of the multiplexer.
Moreover, the sets of input terminals to which these channels would be connected would be input terminals that correspond to numbers at the data selection terminals of the 35 multiplexer given in accordance with e~uation (1) above.

, . ;~

~.~s~s~ ~o Thus, with the first pulse delivered to the counter 60 by the clock 45, such pulse indicated at 246 in Figure 4~ the correlation signal produced by the k = 0 band circuit 84 of the n = 0 octave circuit would be transmitted by the multiplexer and recorded. The third, fifth, and seventh pulses, indicated at 248-252, respectlvely, in Figure 4 ;-ould similarly result in the recordation of the correlation signals produceæ by the band circuits 86, 8~ and 90, respectively, in Figure 2. Since each of the clock pulses indicated at 2~6-252 in Fic~ure 4 occurs within one correlation time interval corresponding to the time extent of one cell in the information plane grid, the series of correlation signals so recorded would be correlation signals for the first column of information plane grid cells in the uppermost octa~e.shown in Figure 8. That is, the correlation signal recorded with reception of the pulse 246 by the counter 62 would be an appropriate correlation signal for the cell 254 in Flgure 8; the correlation signal recorded for the pulse 2~8 would be appropriate for the cell 260 in Figure 8; the correlation signal recorded for the pulse 250 would be appropriate for the cell 262; and the correlation signal recorded for the pulse 252 would be appropriate for the cell 264 in Figure 8. ~he channels from the correlators of the hand circuits in the first octave 25 circuit would a].so be connected to input terminals of the multiplexer corresponding to higher odd numbers to be impressed at the data selection terminals of the multiplexer, in accordance with e~uation (1) abover so that a se~uence set of clock pulses delivered to the counter 30 ~lould cause correlation signals corresponding to the next column of information plane cells to be transmitted to the recorder and recorded.
In the same manner that one set of pulses delivered to the first octave circuit over a correlation time interval 35 equal to the time extent of a cell in the first octave of ~S~35"3 ~,1 the information plane causes each of the correlation signals delivered by cach oE the band circuits in the first octave circuit to be recordedr the same number of pulses delivered to tne second octave circuit would resul c in recorda-tion of 5 correlation signals for one column of cells in the second octave; that is, the octave for which .he index n is equal to one corresponding to a similarly indexed octave circuit;
and the band circuits of the second octave circuit would be connected to appropri.a~e sets of input terminals of the 10 multiplexer 58, again in accordance with equation (1) above, so that the second, sixth, tenth and fourteenth pulses delivered to the counter, such pulses indicated at 255-272 in Figure 4 would result in the recordation of correlation signals produced by the band circuits of the second octave circuit and appropriate to the cells 274-280 of the information plane shown in Figure 8. The correlation signals produced by the remainirlg octave cireuits would be similarly recorded.
It will be noted that only one correlation signal 20 produced by the second octave circuit would be recorcied for every two correlation signals from the firs-t octave cireuit that are recorded because of the halvin~ of the rate at which pulses are clelivered to the second octave circuit with respect to the rate at which pulses are deliverecl to the 25 first octave circuit. Such halving of the clock rate to the second octave circuit thus defines a correlation time interval for the second octave circuit which is twice the correlation time interval 244 for the first octave circuit as indicated at 282 in Figure 4. Such doubling of the 30 correlation time interval between octave circuits corresponds to the doubling of the time extents of the cells in different octaves for -the information plane as inclicated by the time extents of the cells 274-2~30 with respect to the time extents of the cells 258-264 in Fic~ure 8. Thusl by 35 utilizing the cloclcing scheme shown in Figure 4 and ~z5~S~3 ~2 e~pressed analytically in ec~uation (1), the signal representation ~enerator 23 will generate and record on correlation signal between one frequency component of an aEbitrary signal and a rererence signal derived from the 5 star.~ard ~avelet for every cell of the information plane.
Dy this ~eans, the siynal representation generator generates and records correlation signals for all cells of a selected information plane grid and it can be shown that such correlation signals arer after scaling as discussed above, appropriate coefficients for summing wavelets, each having frequency and time characteristics associated with a particular cell in the information plane, taken over all cells of the plane.
As has been noted above, the number of reference signal inte~vals into which the wavelet length corresponding to the index k = 0 is divided; that is, the value of No in equation ~5), must be much larger than the ten intervals that were shown in Figure 9, to ensure accuracy of the correlation. A
suitable number for the wavelet shown in Figures 4 and 5 is 20 forty. Such number is chosen on the basis that, generally~
the period of the sinusoidal part of the wavelet has to be divided into eight intervals to obtain accurate correlation signals for the representationO Thus, for the wavelet shown in ~igures 4 and 5 in which the envelope 206 contains five 25 periods, the number of intervals into which the wavelet length has to be divided is forty, leading co forty-one wavelet amplitudes to be entered into the reference registers of the k = 0 band circuits. This division of the length of the wavelet defines the rate at which clock pulses 30 should be sent to the octave circuits to control the rates at which the A/D converters provide amplitudes of signal components to be entered in the signal registers. Such division corresponds to the octave circui-t which contains the registers being clocked at a rate of four clock pulses 35 per cell of the information plane. In such case, since the 8S~
~3 corre'ation time interval extends over four clock pulses, four new amplitudes of a si~nal component would be entered into the signal regiscer for each cell o:E the information plane so thatt if the wavelet is forty cells long, forty values of the amplitude of a signal component would be required to effect a correlation of the signal component witn the entire reference signal. Thus, the signal register in a correlator, for this case, would have to contain at least forty storage registers and at least forty amplitudes of the reference signal would be contained in the corresponding reference register. Such number of amplitudes could be obtained by dividing the wavelet into forty reference signal in-tervals and selecting the amplitude of the wavelet at the center of each interval for determination 1~ of a sequence of reference signal amplitudes. EIoweverr it is preferred to use the end points o the reference signal intervals to determine the reference signal values to be entered into the reference register in order to include a point at the center of the wavelet for correlation. Thus, forty one, rather than forty, storage registers would be included in the reference and signal registers of a k = 0 band circuit. A larger number of intervals, as given by equation (5) above, is then used for each of the other band circuits to efect the approximate correlation of signal components associated with the other bands that has been described above and that is used to permi,- the information plane to be divided into cells of equal extents, in clock time rather than wavelet cycle time, in each octave of the information plane.
QPeration o ch~_~L~n~_ Repr ~en_a ion Gener t2~
An important advantage of the signal represencation generator ~0 is that it can be utilized to provide a signal representation that is consistent with the phenomenon that produces the signal to be represented~ ~n particularr the signal representation generator can include substantially r ~.~S~15't3 ~I~

any number of octave circuits so that a signal having frequeIlcy components in substantially any range can be represented and, the form of the representation can be varied via the selection of the form of the standard wavelet and the number of band circuits included in eacn OL the octave circuits~ In order to provide a complete understanding of the invention, it will be useful to briefly discuss the manner in whicn the number of octave circuits!
the number of band circuits per octave circuit and the form of the wavelet would be selected to carry out the operation of the signal xepresentatior-. generator.
Initially, for any given signal to be represented, the phenomenon that produces the signal woulc'. determine the frequency range of the signal and such requency range would be utilized to select the number of octave clrcuits to be included in the signal representa~ion yenerator 20 as follows. The highest frequency component of t'ne signal would be utilized to determine the logarithmic center frequency of the highest frequency band in the octave circuit associated with tne highest frequency portion of the frequency range oE the signal. T'nis hiyhest frec~uency would be divided successively by the number two until a frequency below the range of interest is obtained and the number of division required to obtain such low Erequency would specify the number of octave circuits to be included in the sigr.al representation generator. Thus, if the frequency range extends from approximately 250 hertz to approYimately 15 hertz, five octave circuits would be utilized as has been illustrated in the clrawinys.
In addition to knowing the frequency range of interest from the phenomenon that produces the signal, the user of the signal representation generator would also know the form of transients that occur in the signal and such form would be utilized to select a standard wavelet in terms of which the signal representation is to be carried out. Such 1;25~135~
~5 wavelet ~ill ha~e both time ancl frequency characteristics ~hich duplicate the time and frequ2ncy characteristics of transients occurring at cli ferent fre~uencies i~ the signal and such characteristics are incorporated into the wavelet by seiecting a wavelet envelope and a number of oscillations of a sinusoid to be included within the envelope. The resulting oscillatory character of the wavelet provides the wavelet with a requency characteristic and the number of oscillations included within the envelope provides the wavelet with a time characteristic that corresponds to the frequency characteristic of the wavelet. Thus, in the situation in which the signal is produced oy a geophone, a wavelet shape having both time and frequency characteristics that would be well suited to representing the geophone signal would be a wavelet in ~hich the ervelope is a gaussian probability curve that has appreciable values over appro~imately five periods of a sinusoid as has been illustrated in Figures 5 and 6. The distribu-tion in time and frequency of the selected standard wavelet is the main determining factor for selection of the grid because it is clear that the shape o each cell should be close to that of this distribution. In other words, the grid used Lor the representation must be fitted to the time frequency distribution of the selected standard wavelet. But, in all cases, the s-tandard wavelet must contain no eneryy in the neighborhood of zero frequency.
Once the number OL octave circuits to be included in a signal representation generator 20 has been determined and the shape of the wavelet has been selected, the user of the signal representation generator will select an information plane grid which will enable the representation to be meaningfully interpreted once the representation has been obtained. In making sucn selectionsr the user can stress either time or frequency characteristics of the signal by ~5 selecting the num~er of band circuits to be included in each 35~3 ~6 of the octave circuits or the signal representation gererator. Thus, if the frequency characteristics are to be stressed, each octave circuit would include a lar~e number of band circuits to divide the information plane into a large nwnber o~ frequency bands which are divided into cells having relatively long extents in time. To stress the time characteristics, fe~er frequency bands per octave would be used and the time extents of the cells would be correspondingly shorter. Such division of the information plane is carried out by requiring that the product of the num~er of bands per octave ancl the number of cells per period of the wavele-t be equal to each for reasons that have been noted above. For example, should tne frequency characteristics be a primary concern and the wavelet contains forty cycles, the time extents of the cell could be made equal to four cycles o, the wavelet so rhat the information plane would contain one ourth cell per cycle.
Thirty-two band circuits would then be used in eacn octave circuit so that the product of thirty-two band circuits and one fourth cell per cycle would equal to the number eight.
Conversely, shouid it be desired to stress the time characteristics of the signal, the user might choose three bands per octave so that the time extents of the cells in the inEormation plane would be equal to two and two thirds of a wavelet cycle to again obtain the product eightO Once the number o band circuits per octave circuit has been chosen to stress one or the other of these characteristics~
the logarithmic center frequencies of all of the bands of the information plane are determined as has been discussed above with reference to Figure 7. In addition, -the time extents of the cells oE the information plane in each octave are determined from the frequency associated with the uppermost band in the information plane and the number of cells per cycle of the standard wavelet. Such determination is carried out by finding the period corresponding to the ~.~S~5~3 ~7 uppermost requenc~ band in the information plane and then dividing such period by the number of cells to be included within each period of the wavelet. For example, if the maximum frequency to be utilized in carrying out the representation is 250 hertz, corresponding to a period of four millisecondsr and two cells are to be established for each oc,ave of the information plane, the cells in the uppermost octave would each have a tlme duration of two milliseconds. Cells in successively lowering .r~quency octaves of the information plane would have time extents round by successively doubling the time extent so found for the uppermost octave of the information plane.
Once the information plane has been established, the rate at which the counter and, accordingly, the rates at 15 which the octave circuits are to be clocked is determined from the time extent of the cells in the uppermost octave of the information plane and the n~ber oE band circuits to be included in each octave circuit. The rate at which the counter is to be clocked is found by multiplying the num~er 20 of band circuits per octave circuit by the inverse of the time extent of the cells in the uppermost octave, to obtain the appropriate clocking rate for the octave circuit associated with the highest portion of the frequency band, and such number is then further multiplied by two to provide 25 the doubling of the rate at whi.ch the counter and multiplexer are clocked with respect to the clocking of such octave circuit that has been illustrated in Figure 4. This defines a master clock rate; that is, the rate at which the multiplexer is to be clocked as indicated on the axis ~0 of 30 Figure 4. Thus, for example, if the siynal representatlon generator includes four band circuits per octave circuit and the time extent of the cells in the uppermos-t octave of the information plane is two milliseconds, the basic clock rate for the signal representation generator 20; that is, the 35 clock rate for the counter 62 and multiplexer 58, would be :~5~5~

four times five hundred (the inverse of two milliseconds) times two. That is~ the cloc~ rate selected for the counter ~2 and mul~iplexer ~& would be 4,000 hert~
~e~t, an appropriate subset of the pulses transmitced to the multiplexer is selected for cransmission to each of the octave circuits, in accordance with the principles discussed above, to control the sampling rate of the A/D
converters in order to obtain the appropriate num~er of divisions of each period of the reference signal for the k =
0 band circuitO Preferably, such selection will provide eight divisions per period of the reference signal. The number of storage registers in each of tne reference and signal registers of the band circui-ts for the index k = 0 is then the selected number of divisions ~er period of the reference signal multiplied by the number of periods of the sinusoid under the envelope of the standard wavelet plus one. ~hus, for example~ if the information plane is to contain two cells per cycle of a five cycle wavelet, the wavelet would be divided into forty reference signal intervals for all k = 0 bands. Accorclingly~ the number oE
storage registers in the reference and signal registers oE
each of the highest frequency band circuits in the octave circuits would be forty-one. The reference signal amplitude to be stored in such registers would then be determined, as described abover by determining the amplitude of each of the even and odd parts of tne wavele-t at the ends of each of the reference signal intervals cnd entering sucn amplitudes into the storage registers for such highest frequency band.
The number of storage regis-ters in the signal and reference registers of the remaining band circuits, and the reference siynal amplitudes to be stored in the storage registers of the reference registers of such circuits would then be found by choosing a different n~ber of reference signal intervals for each of the other band circuits according to equation (5) above. Once the storage registers 35~3 of the reference registers of all of -the band circuits have been loaded with the reference signal amplitudes so determined, the user o~ the signal representation generato~
is ready to obtain a representation of tne signal that is impressed at the generator input ~a as will now be described.
Once the number of octave circuits to be used in the signal representation generator has been included therein~
along with the selected number of band circuits in each of the octave circuits, and the clock rates and reference signal amplitudes have been chosen and the latter entered into the reference reyisters of the correlators of the band circuits~ the slgnal representation generator is ready for use. Such use is commenced by connec-ting the device tha-t produces the sign~1 to be represented to the generator input 44, starting the recorder, and resetting the clock 46 and counter 62 to synchronize the signal representation generator to the selected information plane grid~
As the signal appears at the generator input 4~, it is transmitted to the filters of all the band circuits so that the signal is separated into a plurality of signal components, each or which lies within a different frequency band to which one of the filters responds. These signal components are transmitted to the A/D converters so that digitized representations can be produced by each of the A/D
converters each time the octave circuit of which the A/D
converter is a part receives a clock pulse from the clock ~6.
As time proceeds, each of the octave circuits will receive a sequence of clock pulses on the clock paths ~-56 so that the A/D converters included in the band circuits of the octave circuits will periodically generate a digital signal expressing the amplitude of 2 particular signal component of the signal to be represented by the signal ~5 representation generator. Each of these digital signals is :lZ5~

transmitted to the even and odd correlators to which the A/D
converters are connected and clocked~ one--by-one, into the signal registers of the correlators. Thus, at some time after the signal representation generator has been start~d, each of the signal registers of each of the even an~ odd correlators of all band circuits will contain a segment of a signal component within a particular frequency band and such segment will be correlated, with reception of a clock pulse by each octave circuit of the signal representation generator. The resul'ing correlation signals are outputted to the multiplexer so that one such correlation signal, corresponding to one particular cell of the information plane identified by the count provided to the select terminals of the multiplexer by the counter, will be transmitted by the multiplexer to the recorder for recording. Since the reEerence and signal registers in a correlator contain the same number of storage registersr the lergths of the siynal segments correlated will have different time durations from band-to-band and such time durations reClect the different numbers of sampling time intervals for the bands of each octave and the different clocking rates for the octave circuitO Such number is made inversely proportional to the logarithmic center frequencies of the bands of each octave by the selection of a number of storage registers in the reference registers in accordance with equation (5) as discussed above and duplicated from octave to octave by the halving of clocking rates of tne octave circuits.
An important aspect of the present invention is that a representation generated by a signal representation generator constructed as described above wi:Ll correspond to a substantially constant information content for each cell of the information plane so that the representation provided by the signal representation generator will present the 35 maximum information that can be obtained from a signal~

~5~35~

This beneIit was the basis for the Gabor and Morlet proposals that a signal be decomposed on an information plane and it was to obtain this benefit that Gabor and Morlet susgested that the information plane grid be divided into cells of equal area. ~lowever, neither the Gabor nor Morlet proposals could be practically implemented with the information plane grids that were proposed. In the grid proposed by Gabor, lower frequency bands would have to be ~-osslv oversampled to provide reasonably p~ecise correlation signals for hign frequency bands, particularly where phase information abou-t a signal is desired and, in the information plane grid proposed by Morlet, the time extents of the cells would be related by irrational numbers so that a series of cells could no~ be corresponded with clock times necessary to carry out an implementation of Morlet's proposal without extreme difficulties in obtaining the representation. The present invention meets the constant information content per cell requirement necessary for achieving a ma~imum information representatlon, while at the same time dividing the information plane into cells having rationally related time ex-tents permitting practical implementation of a signal representation generator, by causing the segments of the signal components correlated with the reference signals to have different time durations selected by clocking all band circuits in each octave circuit at the same rate while dividing the standard wavelet into different numbers of correlation signal intervals for different band circuits and entering reference signal amplitudes derived from the ~avelet amplitude at the end 30 points of the intervals into different n~mbers of storage registers of which the reference reyisters of the band circuits are comprised. That is, for a band circuit that responds to the highest frequency band of an octave of the information plane, the number of storage registers in the 35 reference registers of the band circuit correlators will be 59~ 9 the number No which is selected as has been described above.
Thus the time duration of a segment of a signal component correlated with a refexence signal for a band having a band index number k = 0 will be the product of Mo and the sampling time interval at which the oc-tave circuit whic'n includes such band circuit is clocked. For the other bands in the octave circuit, the number of reference signal intervals used to obtain wavelet amplitudes for deriving reference signal values to be entered into the reference registers of the correlators of the band circuits are increased with increasing band circuit index number so that increasing numbers of storage registers are included in the correlators of the band circuits for increasing values of the band index number l~ so that the duration of a segment of a signal component correlated with the reference signal for increasing value of the index k increases in accor~ance with equation (5) for different band circuits of each octave circuit. That is, since 'che sampling time interval defined by the rate at which an octave circuit is clocked is the same for all band circuits in an octave circuit and the reference and signal registers in the correlators of each band circuit contain the same number of storage registers so that the number of sampling time intervals for each band is the same as the number of reference signal intervals for the band, the duration of the signal component segment correlated with a reference signal increases with increasing band index number in accordance with equation (5). Since the logarithmic center fre~uencies to which the filters in the band circuits respond decrease with increasing band 30 index number in accordance with equation (2), the decrease in logarithmic center of the frequency band to which each band circuit responds is just offset by such increasing duration of the segments of -the signal components correlated with the reference signal with increasing band index number, 35 as can be seen by comparing equations (2) and (5). Thus, ~'~5~359 ~3 all cells for each octave of the information plane associated with the siynal representation generator wiil have constant information content. The constant information content characteristic of the cells is then carried from octave to octave by clocking each octave circuit at a successively lower rate with increasing octave circuit index number n on factors of iwo and repeating ~he pattern of reference signal intervals used to derive the reference signals for the band circuit correlator reference registers from octave circuit to octave circuit. That is, as the frequency associated with two bands having the same band index number k decreases from one octave circuit to a successive octave circuit that responds to lower portions of the frequency range used to carry out the representation, the clocking rate also decreases by a factor of two to double the duration of a signal componen-t segment usecl to carry out the correlation in -the octave circuit corresponding to the lower frequency range~ That is, the doubling of the sampling time interval defined by the clocking rate of an octave circuit while maintaining the number of amplitudes to be entered into a register, one amplitude being entered into a register in an octave circuit per sampling time interval, doubles the duration of a signal component segment correlated with a reference signal as the 25 octave index number increases by one while the band index number is held constant. The doubling of the durat on of a segment of a signal component in the lower frequency octave combined with the halving of the frequencies of the bands from octave to octave with increasing octave circuit index n 30 then causes two information plane cells having the same band index number k to have the same information content without regard to the octave index number of the cell. Thus, since the information content of all cells in each octa~e of the information plane have constant information content and 35 cells in different octaves of the information plane havins i2S~~S~
5a the same band index number k n~ve the same information con~ent~ all cells of the information plane ~`7ill contain the same amount of information about the signal being represented.
In the preferred embodiment shown, the information generator 20 is comprised of reference registers, each of which is comprised of a plurality of storage registers, multiplier circuits, adder circuits and multiplexers which are related to each other only through their inclusion in the signal representation generator 20. ~owever, it is recognized that these registers and other components can be related to each other in an additional way. In particular~
the reference registers can be registers that are found in the memory of a general purpose digital compu~er and, in such case, the multiplier circuits, adder circuits and multiplexers would be comprised of the central processing unit of -the computer and registers in the memory o the computer that would suppl~ appropriate instructions ~o the central processing unit of the computer to carry out the correlations of the signal components of the signal that is represented by the signal representation generatorO In such case, the A/D converters would be connected to appropriate input ports of the computer and the recorder would be connected to an appropriate output port of the computer as 25 is conventional in the computer art.
~-~LiE~io-~-Q~ ~L~
Referring now to Figure 11, shown therein and designated by the general reference numeral 300 is a second form of band circuit particularly suited for use in the 30 signal representation generator 20 when the generator 20 is used to represent signals lying in relatively high fre~uency ranges. Like the band circuits ~4-90~ the band circuit 300 receives a signal to be represented, on a signal path 302, and provides digitized outputs to the multiplexer 53 on 35 channels 304 and 306, the output on channel 30~ providing an :1~51 ~35~

even correlation value for a particular frequeney band of the information plane and the output on ehannel 306 p~-ovicing an ~e~ cor~elatio~ value for such Erequ~nel band.
As the band circuits 84-90, such frequency band is seleeted 5 by a bandpass -Filter whieh has been indieated at 304 in ~igure 11.
However, in the band cireuit 307 correlation is carried out prior to digitization as will now be discussecd. ~s shown in Figure 11, the output of ~ilter 30-1 ls transmitted via signal paths 306-310 to even and odd correlators 312 and 31~ which are analog eircuits designed to provide continuous, as opposed to discrete, eorrela-tion signals, on sisnal paths 316 and 31&~ respeetively, to ~/D eonverters 320 and 322~ Sueh eorrelators 312 and 314 can be any of the known analog correlators which have been cleveloped, for example, for use in seismie prospeeting. In order to use sueh eorrelators in the signal representation generator 20 the transfer Eunction of the correlators would be seleeted in a eonventional manner to cause the signal component provided by the filter 304 to be continuously correlated with referenee signals having the form expressecl in equations (3) and (4). Thus, it will be noted, segments of the signal eomponents having different time durations~
eorresponding to the time durations of signals digitally expressed in the signal registers o~ the band eircuits 8~90 ~ will be continuously eorrelated with reference signals derived from the standard wavelet so that the correlators 312 ~ 314 will provide eontinuous eorrelation signals analogous to the correlation signals produeed by the band circuits 8~-9Oo These eontinuous eorrelation sigrals are then digitized by the ~/D eonverters 3201 322 in response to periodic eloek pulses received on the clock paths 324-328 sO that the ~/D eonverters provide eorrelation signals that are the same as the correlation signals provided by the eve~ and odd correlators of the band ~25i~59 circuits 84-9Q for recording. The operation of a sic,nal representation generator using band circuits constructed in accordance with the band circuit 300 shown in Figure 11 ill, accordingly, be identical to the operation of the signal representation cenerator 20 that has been discussecl above.
An important quality cri~erion of the representation of a signal is the fidelity with whicn it is possible to recover the initial signal from its representation.
This new representation makes it simple and precise by allowing an accurate reconstruction of the signal derived from:
1) the wavelets related to every cell of the grid of Figure 7;
2) the sampled values of the representation, related to the same cells of the grid of Figure 7, obtained by erosscorrelation as described above;
and 3) a scaling factor which compensates for the gain variationsl the wavelet normali~ation and the oversampling used during the diecomposition. Such sealing factor is determined by requiring a constant energy per reference signal and for constant cell area.
More particularly, the signal can be reconstructed by interpolating in time the representations in each frequency band and summing over the whole frequency range the narrow band signals thus obtained.
It is clear that the present invention is well adapted 30 to carry out the objects and attain the encls and advantages mentioned as ~ell as those inherent therein. While a presently preferred embodiment of the invention has been described for purposes of this diselosure, numerous changes may be made which will readily suggest themselves to those 35 skilled in the art and which are encompassecl within the ~S~ 15~

spirit of the invention disclosed and as defined in the appended claims=

Claims

THE EMBODIMENTS OF THE INVENTION IN WHICH AN EXCLUSIVE
PROPERTY OR PRIVILEGE IS CLAIMED ARE DEFINED AS FOLLOWS:

1. A method for generating a representation of an arbi-trary signal, comprising the steps of:
separating the signal into signal components within different overlapping frequency bands;
selecting a correlation time interval for each frequency band;
repetitively correlating a segment of the signal com-ponent in each band with a reference signal derived from a stan-dard wavelet having both time and frequency characteristics to obtain a sequence of correlation signals for each band, wherein the segment of the signal component of each band that is correla-ted with the reference signal has a time duration equal to the correlation time interval for that band; and recording said cor-relation signals.

2. The method of claim 1 wherein said bands are selected to divide a selected frequency range into equal intervals of logarithm of frequency; and wherein the durations of the segments of the signal components correlated with the reference signal are in inverse proportion to the logarithmic center frequencies of the bands.

3. The method of claim 1 wherein the standard wavelet is characterized as having an even part formed by limiting a cosine wave with a preselected envelope and an odd part formed by limiting a sine wave with said preselected envelope; and wherein the step -58a-of correlating a segment of the signal component in each band with the reference signal is further characterized as correlating the segment of the signal component in each band with each of an even part of the reference signal derived from the even part of the standard wavelet and an odd part of the reference signal derived from an odd part of the standard wavelet.

4. The method of claim 2 wherein the standard wavelet is characterized as having an even part formed by limiting a cosine wave with a preselected envelope and an odd part formed by limiting a sine wave with said preselected envelope; and wherein the step of correlating a segment of the signal component in each band with the reference signal is further characterized as correlating the segment of the signal component in each band with each of an even part a reference signal derived from the even part of the standard wavelet and an odd part of the reference signal derived from an odd part of the standard wavelet.

5. The method of claim 2 wherein the logarithms of frequency selected to divide said frequency range into bands are to the base two, whereby the frequency bands are grouped into a sequence of octaves each associated with a portion of said frequency range; wherein the selected correlation time intervals are the same for all bands in each octave; and wherein the selected correlation time intervals for octaves associated with successively higher frequency portions of said frequency range decrease by a factor of two from octave to octave.

6. The method of claim 5 wherein the step of correlating a segment of the signal component within a frequency band with the reference signal comprises the steps of:
storing a succession of amplitudes of the reference signal corresponding to a sequence of sampling points dividing the standard wavelet into a selected number of equal reference signal intervals;
storing a succession of amplitudes of the signal component corresponding to a sequence of sampling times dividing said segment of the signal component into a selected number of equal sampling time intervals, wherein the number of sampling time intervals into which said segment of the signal is divided is equal to the number of reference signal intervals into which the reference signal is divided; and correlating the stored amplitudes of the signal component with the stored amplitudes of the reference signal to obtain a correlation signal.

7. The method of claim 6 wherein the sampling time intervals for frequency bands in each octave are the same for all bands in the octave; wherein the sampling time intervals for octaves associated with successively higher frequency portions of said frequency range decrease by a factor of two from octave to octave; wherein the number of sampling time intervals selected for the bands in each octave varies from band to band in substantially inverse proportion to the logarithmic center frequencies of the bands, and wherein the set of numbers of sampling time intervals selected for the bands of one octave are selected for the bands of all octaves.

8. The method of claim 4 wherein the logarithms of frequency selected to divide said frequency range into bands are to the base two, whereby the frequency bands are grouped into a sequence of octaves each associated with a portion of said frequency range; wherein the selected correlation time intervals are the same for all bands in each octave; and wherein the selected correlation time intervals for octaves associated with successively higher frequency portions of said frequency range decrease by a factor of two from octave to octave.

9. The method of claim 8 wherein the steps of correlating a segment of the signal component within a frequency band with the even and odd parts of the reference signal comprises the steps of:
storing a succession of amplitudes of each of the even and odd parts of the reference signal corresponding to a sequence of sampling points dividing each of the even and odd parts of the standard wavelet into a selected number of equal reference signal intervals;
storing a succession of amplitudes of the signal component corresponding to a sequence of sampling times dividing said segment of the signal component into a selected number of equal sampling time intervals, wherein the number of sampling time intervals into which said segment of the signal is divided is equal to the number of reference signal intervals into which of the even and odd parts of the reference signal is divided, and correlating the stored amplitudes of the signal component with the stored amplitudes of each of the even and odd parts of the reference signal to obtain a correlation signal.

10. The method of claim 9 wherein the sampling time intervals for frequency bands in each octave are the same for all bands in the octave; wherein the sampling time intervals for octaves associated with successively higher frequency portions of said frequency range decrease by a factor of two from octave to octave; wherein the number of sampling time intervals selected for the bands in each octave varies from band to band in substantially inverse proportion to the logarithmic center frequencies of the bands; and wherein the set of numbers of sampling time intervals selected for the bands of one octave are selected for the bands of all octaves.

11. An apparatus for generating a representation of an arbitrary signal, comprising:
a plurality of octave circuits, each octave circuit associated with a selected octave of a selected frequency range, for receiving the signal and generating a set of correlation values between different frequency components of the signal and a reference signal derived from a standard wavelet having both time and frequency characteristics in response to reception by the octave circuit of a selected number of clock pulses defining a correlation time interval for the octave circuit;
clock means for providing the clock pulses to the octave circuits, the clock means providing clock pulses to different octave circuits at rates that increase by a factor of two for octave circuits associated with successively higher octaves of the frequency range, whereby said correlation time intervals decrease by a factor of two for octave circuits associated with successively higher octaves of the frequency range; and means for recording the correlation values at times controlled by said clock.

12. The apparatus of claim 11 wherein each octave circuit comprises a plurality of band circuits, each band circuit, associated with a selected frequency band of the octave of the frequency range with which the octave circuit is associated, generating one member of the set of correlation values for each correlation time interval of the octave circuit.

13. The apparatus of claim 12 wherein the frequency bands associated with the band circuits of each octave circuit divide the octave of frequency associated with the octave circuit into equal intervals of logarithm, to the base two, of frequency.

14. The apparatus of claim 13 wherein each band circuit comprises:
a bandpass filter receiving said signal and passing a selected frequency component of the signal within the frequency band with which the band circuit is associated;
an A/D converter connected to the bandpass filter to receive the signal component passed thereby, the A/D converter receiving the clock pulses received by the octave circuit of which the A/D converter is a part to provide a digital representation of the amplitude of the signal component received by the A/D converter in response to each of said clock pulses; and signal correlation means for correlating a succession of amplitudes provided by the A/D
converter with a succession of amplitudes of the reference signal derived from the standard wavelet.

15. The apparatus of claim 14 wherein the signal correlation means comprises:
a reference register comprising a plurality of storage registers, each storage register containing a digital representation of one amplitude of the reference signal;
a signal register, connected to the A/D converter and the clock means to receive the digital representations of the signal amplitudes provided by the A/D converter and the clock pulses received by the octave circuit of which the signal register is a part, the signal register comprising a plurality of serially connected storage registers for storing a succession of signal component amplitudes and shifting the stored signal component amplitudes serially through the storage registers in response to a clock signal received by the signal register, the signal register having a number of storage registers equal to the number of storage registers of said reference register and each storage register of the signal register corresponding to a selected storage register of the reference register;
a term-by-term multiplier circuit connected to the output terminals of the storage register of said reference and signal registers, the multiplier circuit connected to the clock means to receive the clock pulses received by the octave circuit of which the multiplier circuit is a part for multiplying the contents of each storage register of said reference register by the contents of the corresponding storage register of said signal register in response to a clock pulse received by the multiplier circuit; and an adder circuit, connected to the multiplier circuit and to the clock means so as to receive the products formed by the multiplier circuit and clock pulses received by the octave circuit of which the adder is a part, for adding said products in response to a clock pulse received by the adder circuit.

16. The apparatus of claim 15 wherein the number of storage registers in each of the reference and signal registers varies from band circuit to band circuit within each octave circuit substantially in inverse proportion to the logarithmic centers or the frequency bands with which the band circuits are associated.

17. The apparatus of claim 14 wherein the reference signal is characterized as having an even part and an odd part; and wherein the signal correlation means comprises:
even signal correlation means for correlating a succession of amplitudes provided by the A/D
converter with a succession of amplitudes of the even part of the reference signal; and odd signal correlation means for correlating a succession of amplitudes provided by the A/D
converter with a succession of amplitudes of the odd part of the reference signal; and wherein each of the even signal correlation means the odd signal correlation means comprises:
a reference register comprising a plurality of storage registers, each storage register containing a digital representation of one amplitude of the reference signal;
a signal register, connected to the A/D converter and the clock means to receive the digital representation of the signal amplitudes provided by the A/D converter and the clock pulses received by the octave circuit of which the signal register is a part, the signal register comprising a plurality of serially connected storage registers for storing a succession of signal component amplitudes and shifting the stored signal component amplitudes serially through the storage registers in response to a clock signal received by the signal register, the signal register having a number of storage registers equal to the number of storage registers of said reference register and each storage register of the signal register corresponding to a selected storage register of the reference register;
a term-by-term multiplier circuit connected to the output terminals of the storage register of said reference and signal registers, the multiplier circuit connected to the clock means to receive the clock pulses received by the octave circuit of which the multiplier circuit is a part for multiplying the contents of each storage register of said reference register by the contents of the corresponding storage register of said signal register in response to a clock pulse received by the multiplier circuit; and an adder circuit, connected to the multiplier circuit and to the clock means so as to receive the products formed by the multiplier circuit and the clock pulses received by the octave circuit of which the adder is a part, for adding said products in response to a clock pulse received by the adder circuit.

18. The apparatus of claim 17 wherein the number of storage registers in each of the reference and signal registers of the even correlation means of a band circuit is equal to the number of storage registers in each of the reference and signal registers of the odd correlation means of the band circuit and wherein the number of storage registers in each of the reference and signal registers of the band circuits varies from band circuit to band circuit within each octave circuit substantially in inverse proportion to the logarithmic centers of the frequency bands with which the band circuits are associated.