DETECTION OF A LOW LEVEL SIGNAL IN NOISE. Kiefer Aul t. Tobi n BY TEGULOMETRIC METHODS

Transcription

1 DETECTON OF A LOW LEVEL SGNAL N NOSE BY TEGULOMETRC METHODS Kefer Aul t Tob n

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3 NAVAL POSTGRADUATE SCHOOL Monterey, Calforna THESS DETECTON OF A LOW LEVEL SGNAL BY TEGULOMETRC METHODS by Kefer Ault Tobn December, 9 76 N NOSE Thess Advsor Geor ge Marmont Approved for publc release; dstrbuton unlmted. T76085

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5 tmr UNCLASSFED SECURTY CLASSFCATON OF THS PACE (Whmn Dmtm gntmrod) T o»t Nuwaen REPOKT DOCUMENTATON PAGE READ NSTRUCTONS BEFORE COMPLETNG FORM 2. OOVT ACCESSON NO.. RECPENTS CATALOG NUMBER 4. TTLE (and Subttle) Detecton of a Low Level Sgnal n Nose by Tegulometrc Methods 7. AUTHORS». TYRE OF REPORT A PEROO COVERED lasters Thess December, 976 * PERFORMNG ORG. REPORT NUMBER» CONTRACT OR GRANT NUMBERS Kefer Ault Tobn»- PERFORMNG ORGANZATON NAME ANO AOORESS. Naval Postgraduate School Monterey, Ca CONTROLLNG OFFCE NAME ANO AOORESS Naval Postgraduate School Monterey, Ca U. MONTORNG AGENCY NAME AOORESSff/ dlttoront tram Controllng Olllca) Naval Postgraduate School Monterey, Ca PROGRAM ELEMENT. PROJECT, TASK AREA * WORK UNT NUMBERS 2. REPORT OATE Der-Pmhpr, Q76 3. NUMBER OF PAGES 89 B. le. SECURTY CLASS, (ol thla report) UNCLASSFED DECLASSFCATON/ DOWNGRADNG SCHEDULE «. DSTRBUTON STATEMENT (ol thla Rmport) Approved for publc release; Dstrbuton unlmted 7. DSTRBUTON STATEMENT («l tho amattmct ««W n Block 20, // dlllotant trmm Ramott) 0. SUPPLEMENTARY NOTES 9. KEY WOROS (Conthmo on ravorao «<«nacoaaawy ** dentty by bloatt numbar) Tegulometrc deal dgtal flter Sgnal processng Hstogram Sgnal to nose Acoustc data analyss 20. ABSTRACT (Contlmto on ratemaa aldm and dmmtltr by mlmom A novel and senstve method s descrbed for the analyss and detecton of weak sgnals generated by many sources and hghly contamnated by nose. Ths process yelds a spectral dsplay whose elements are related to the probablty of occurrence of a frequency rather than to the power spectral densty of conventonal schemes. Thus, a weak sgnal and stronger sgnals may show equally f they appear over the same percentage of the analyss perod. do, ST* 473 EDTON OP (Page ) S/N NOV» S OBSOLETE UNCLASSFED SECURTY CLASSFCATON OF TMS PAOE (Whan Dmta Bntorowf

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7 UNCLASSFED {lcuwtv classfcaton of ths p»g r>*»>»n r>.e» wwwj. Ths process, called tegulometrc analyss, s tested and mproved n senstvty for detecton of very weak sgnals hdden n gaussan nose. The nvestgaton shows that ths method can be optmzed to produce low level sgnal detectons based upon short observatonal perods. The frequency resoluton capablty of the method s shown to be marked. DD Form 473 Jan 73 UNCLASSFED S/N securty classfcaton of ths PAcer** d«<«en»»r«*)

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9 DETECTON OF A LOW LEVEL SGNAL N NOSE BY TEGULOMETRC METHODS by Kefer Ault Tpbn -. Leutenant Commander B.S., Oregon State Unversty, 960 Submtted n partal fulfllment of the requrements for the degree of MASTER OF SCENCE N ENGNEERNG ACOUSTCS NAVAL from the POSTGRADUATE SCHOOL December 976

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11 ABSTRACT A novel and senstve method s descrbed for the analyss and detecton of weak sgnals generated by many sources and hghly contamnated by nose. Ths process yelds a spectral dsplay whose elements are related to the probablty of occurrence of a frequency rather than to the power spectral densty of conventonal schemes. Thus r a weak sgnal and stronger sgnals may show equally f they appear over the same percentage of the analyss perod. Ths process, called tegulometrc analyss, s tested and mproved n senstvty for detecton of very weak sgnals hdden n gaussan noss. The nvestgaton shows that ths method can be optmzed to produce low level sgnal detectons based upon short observatonal perods. The frequency resoluton capablty of the method s shown to be marked.

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13 TABLE OF CONTENTS LST OF FGURES 7 ACKNOWLEDGEMENT 9. NTRODUCTON 0 A. BACKGROUND 0 B. OBJECTVES 3 C. ADVANTAGES OF regulometrc ANALYSS 3. DESCRPTON OF TEGULOMETRC ANALYSS 5 A. BASS OF THE METHOD 5 B. PROCESSNG SCHEME 8. Defntons 8 2. Memory Map Data nput 2 4. Transform to the Frequency Doman Swept Flter Zero Crossng Algorthm, TEGULO Equalzaton Runnng Average of the Hstogram Contnuty of Data 36. TEST OF THE METHOD 38 A. EQUPMENT 38. Hardware Softwear 39 a. Tme Seres Language (TSL) 39 b. Language, APTEC 39 B. GENERAL APPROACH 40 C. TEST OF KEY ROUTNES 42. Seneral Swept Flter Zero Crossng Routne, TEGULO Equalzaton of the Hstogram 46

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15 5. Contnuty of Fltered Data 47 V. MPROVEMENT OF SENSTVTY 50 A. GENERAL 50 B. ZERO NSERTON REMOVAL 50 C. FREQUENCY COMPONENT LMTNG 50 D. AVERAGNG N THE FREQUENCY DOMAN 54 E. SHAPED FLTER FUNCTON 56 F. FREQUENCY SHFTNG 58 G. RESAMPLNG 58 V. RESULTS 62 V. CONCLUSONS AND RECOMMENDATONS 85 LST OF REFERENCES 88 NTAL DSTRBUTON LST 89

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17 LST OF FGURES. Typcal Tegules 2 2. Wde and Narrow Band Nose 7 3. Flow Chart Of HSCAN Flter Constructon Zero Crossng Determnaton Sgnal n Whte Gaussan Nose Contnuty of Data Swept Flter Constructon Contnuty of Data test Frequency Component Lmtng Schemes 53. Modfcaton to HSCAN Flowchart for RUSCAN Flter Shapes RESAMPLE Logc 6 4. Sgnal Plus Nose, No Average, 44 Frames Nose, No Average, 44 Frames Sgnal Plus Nose, 6 Ave., 44 Frames Nose, 6 Ave., 44 Frames Resampled Sgnal Plus Nose, No Ave., Frames 7 9. Resampled Nose, No Ave., Frames Resampled Sgnal Plus Nose, 6 Ave., Frames 73

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19 2. Resampled Nose, 6 kve., Frames Resampled Sgnal Plus Nose, 0 Ave., Frames Resampled Nose, 0 Ave., Frames Resampled Sgnal Plus Nose, No Ave., 2 Frames Resampled Nose, No Average, 2 Frames Resampled Sgnal Plus Nose, 6 Ave., 2 Frames Resampled Nose, 6 &ve., 2 Frames Resampled Sgnal Plus Nose, 0 Ave., 2 Frames Resampled Nose, 0 Ave., 2 Frames Averaged DFT of Sgnal Plus Nose, 44 Frames Averaged DFT of Nose, 44 Frames 84

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21 ACKNOWLEDGEMENT Acknowledgement s gven to Professor George Marmont of the 0. S. Postgraduate School who ntated tha method of tegulometrc analyss and contnues to provde the nsght and mpetus to ts contnung development. Hs great patence, nsght and expertse n many areas provded the foundatons for ths work. The Naval Electroncs Systems Command (NAVELET 304) provded support durng the earler phases of the nvestgaton. The unfalng moral support and understandng of my wfe and famly contrbuted mmeasurably to the completon of ths work.

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23 . NTRODUCTON A. BACKGROUND Tegulometrc sgnal processng technques orgnated n the electroencephalograph^ (EEG) work by Dr. George Marmont n 973 at the 0. S. Naval Postgraduate School. n analyzng the narrow band fltered EEG data, he observed that the waveforms n each band appeared as a seres of snusods wthn an envelope lastng about 0. second. These spndle-lke structures were the result of the algebrac summaton of electrcal changes takng place wthn many actvty centers n the bran. Ths waxng and wanng structure was defned as a tegule whch s shown on Fg. The envelope of a typcal tran of tegulss exhbts perods of hgh ampltudes followed by sgnfcant perods of low ampltude. Further analyss revealed that the tegules n one band were seldom concdent wth tegules n other bands, showng that results were not due to the mpulse response of the flter. ntal nvestgaton nto tegulometrc analyss was accomplshed by S. E. Dollar [ Ref 3] and W. E. Stockslager [Ref. 6] wth Dr. Marmont. Separate observatons of narrow band ocean acoustc data n phonograms also revealed a smlar tegular nature. The perods n whch the envelope was low were not as long as n the EEG data but much longer than would be observed from a beatng phenomenon. n the case of acoustc data, however, the tegules resulted from the complex superposton of sgnals from many sources wthn the flter band over many 0

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25 paths and possessng random ampltude and phase. Because of ths, the method of tegulometrc analyss appeared to have potental applcaton n analyzng acoustc data. Such an nvestgaton was conducted by T. J. Colyer [Ref. 2] who demonstrated that weak acoustc sgnals n random nose could be detected and dentfed by ths method. Snce electromagnetc sgnals suffer many of the same nterference problems to whch acoustc sgnals are subected, t s expected that tegulometrc analyss should have equal value to radar, communcatons, ESM, and other low sgnal-to-nose applcatons.

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27 ; l, ; Ml \ \ < ; ;, l ; ;.-l.. lll - " ltmh A 4 r Jlr lr A A tt n t*l J A A A A ^9~V Snnr f- * J f V w t T Ml Fgure - TYPCAL TEGOLES 2

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29 OBJECTVES Prevous research had tested many of the aspects of tegulometrc analyss and appled ths scheme to specfc problems. The results of these nvestgatons demonstrated that ths technque was not only very useful, but also could be a novel method for the detecton and dentfcaton of very weak sgnals bured n non- statonary nose. The purpose of ths nvestgaton was to more thoroughly test each processng step to assure that the results of ths process were accurate and vald. A second obectve, after successful completon of these tests, was establshed to mprove the senstvty of the analyss so that even weaker sgnals could be detected when hdden by many other sources of nose. Because a number of the concepts are novel, ths thess frst provdes a descrpton of both the bass and the processng scheme of tegulometrc analyss. Next, tests of the key routnes are dscussed ncludng equpment used and general test phlosophy. The subsequent secton deals wth the methods used to prove analyss senstvty and ncludes technques whch vared n success from margnal to excellent. Fnally, the most promsng senstvty mprovement methods are combned n a seres of tests and the results are presented. C. ADVANTAGES OF TEGULOMETRC ANALYSS Prevous work ponted out a number of advantages n ths 3

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31 method. The frst s that tegulometrc analyss of acoustcsgnals s more senstve and possess a hgher degree of frequency resoluton than the dscrete fourer transform (DFT). Part of ths effect s beleved due to not fully explotng phase nformaton by the DFT n that only the magntudes of the resultant complex frequency elements are dsplayed. Alternately, tegulometrc analyss depends heavly on the magntude and phase of the sgnal and nose. Because ths technque analyzes relatvely short tme sample wndows, the results are not as affected by non-statonary data resultng from multpath transmsson or fluctuatons of the transmsson medum. Ths modulaton by the non- lnear transmsson medum wll tend to spread the sgnal n frequency and phase, causng conventonal fne-graned frequency technques to be degraded. Further research n ths area remans to be done. 4

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33 . DESCRPTON OF TEGULOMETRC ANALYSS A. BASS OF THE METHOD Observaton of the tme doman representaton of wde band nose reveals ts complex nature. n the narrow band, ths nose exbts the waxng and wanng behavor noted n the ntroducton and defned as tegules. Trace A of Fg 2 s the tme representaton of wde band nose. A narrow band fltered representaton of the same nose over the same tme perod s shown on trace C whch clearly demonstrates the tegular nature of fltered nose. The random frequency elements wthn the flter add to produce perods where the nose not only can be a mnmum, but also can result n phase reversal. Durng these ntervals of low ampltude, a weak sgnal can capture the nose to appear wthn these mnma. Trace B s derved from the data of trace C except that a weak sgnal has been nserted wthn the flter. Note that the sgnal, although very weak n magntude, momentarly captures the surroundng nose and appears. f the presence of ths sgnal durng the nose cancellaton perods could be observed and counted, a potentally very senstve detecton method would result. t s precsely ths phenomenon that tegulometrc analyss explots. As a measure of the probablty of occurrence of the weak sgnal, t s possble to measure the tme between negatve to postve zero crossngs of the fltered data and 5

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35 to convert ths tme to an equvalent frequency. A spectral block s created n the memory n whch each successve word corresponds to an equvalent frequency number. For each neqatve to postve zero crossng, the equvalent frequency s computed and ts respectve memory locaton s ncremented by one count. Ths spectral block s therefore a hstogram whose ampltude, frequency dependent, s a measure of the probablty densty functon (pdf) of the frequences of the nose plus any sgnals. Snce even a weak sgnal wll be present whle the nose vares accordng to ts pdf, t s expected that ths more persstent sgnal wll appear as an ncrease n the hstogram. 6

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39 B. PROCESSNG SCHEME Defnt ons All work for ths thess was performed on a 6 bt dgtal computer for whch each data storage locaton was defned as a "word" contanng an nteger number. Because the computer nstallaton had the capablty of addressng by 8 bt bytes, addresses for words are even numbered (ncrement by 2). A complex number s represented by two successve words, the frst beng the real, nteger part and the second the magnary, nteger part. The complex par s referred to as a complex element or element. The data s handled n much of the programmng as blocks of nformaton. The advantage of ths method s to allow the block of data to be treated as a sngle entty n performng operatons on data. Ths allows data manpulaton by a sngle nstructon wthout resortng to loops. An example mght be a sngle call to ADDBLOCK B0,B whch adds block, BO, to block, B. n order to perform ths, each block has a table (Block dentfcaton Block) whch contans the number of block words, the type of block (real or complex), the absolute address of the frst word of the block, and the block exponent. The block exponent s the bnary exponent whch apples to all words n that block. Ths technque allows processng n nteger arthmetc to reduce core storage requrements whle allowng a greater dynamc range of the data. Ths scheme s requred by the hardware fast fourer transform (FFT) perpheral used n ths nvestgaton. Block length s descrbed n the number of "K" words, where K = 024 = 2 0 8

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41 words accordng to common usage. The rato of sgnal power to nose power requres defnton snce ts magntude depends upon the bandwdth of the nose. For ths report, bandwdth wll be defned to be Hz so that for aose of bandwdth W b, and root mean square (rms) voltage of ff, and sgnal of ampltude A f the sgnal to nose rato wll be S/N " 7Vw n dscussng tha programs and results to follow, t s mportant to defne frequency. f each sample block (tme doman) conssted of N samples whch were sampled at a rate of R, then the tme, T represented by ths block of data was T = N/H Further, f the sampled waveform was lmted to W Hz, the dscrete transform wll have N/2 complex frequency elements whose frequency separaton was af, then Af = /T Snce all data n ths nvestgaton was handled n 4K blocks (4096 samples), N was constant at 4096 words. Further, to prevent alasng n the drect Fourer transform (DFT), the Nyqust samplng rate was always adhered to so that W = H/2 Therefore, once a sample rate was selected, all other parameters were unquely defned. These relatons were adhered to n the recordng of all analog sgnals wth the result that actual tme ceased to be meanngful n dgtal form. The relaton between "real tme" and "dgtal tme". 9

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43 then, was smply the sample rate. Because of ths, t became convenent to defne total sample tme, T, as second so that the frequency separaton n the DFT was Hz. The bandwdth was 2048 Hz and the "sample rate" was 4096 samples per second. To relate to the "real world", all that was needed was the applcaton of the rato of R/4096 to the varous parameters. The conventon descrbed above s adhered to n ths thess for ease of dscusson. Where tme n seconds or frequency n Hertz s noted, the quanttes wll refer to "dqtal" tme. 2. Memory Map. The data memory map for HSCAN, the man program whch performs the tegulometrc analyss, s shown n Fg 3. Block desgnatons are shown as BO, B, etc. The numbers n parentheses ndcate the length of the blocks n "K" words. A bref descrpton of the varous blocks s as follows a. Block BO - used for nput and temporary storage of sampled data. b. Block B8 - ntally contans sampled data, loaded from BO. t subsequently contans the spectral block assocated wth sampled data. c. Blocks B4 and B9 - used n the transfer of sampled data from BO to B8. B9 s also used to flter (zero) the upper half of the spectral block. d. Blocks B0, B2, B5, B - deal flter blocks. B0 contans a copy of a complex^complex form of the spectral block n B8. Blocks B2, B5, and B are used to zero the approprate frequency elements of the flter. 20

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45 e. Block B5 - Overlays B0 so that after the nverse Fourer Transform of B0, block B5 contans the center 2K of the fltered sgnal and nose. The zero crossng algorthm (routne TEGULO) s appled to ths block. f. Block B7 - Contans the hstogram, a functon of frequency, generated by TEGULO. g. Blocks BO, B f B2 - Used for fnal equalzaton and averagng of the hstogram block. The prmes sgnfy that these blocks are not defned and used untl after all data has been analyzed n TEGULO. Block BO* s then the output block. 3 Data nput The followng sectons descrbe the detals of HSCAN. A representaton of the maor data flow and processng steps s contaned n Fg 3. The data nput porton occupes the upper blocks and s descrbed as follows. Sampled data s laaded nto the 2048 word temporary storage block, BO, from ether an analog to dgtal (A/D) converter or a data storage devce such as a dsk. Ths data s subsequently moved from block BO to block B4 by an nstructon M07E B0,B4. the next nput of data s read nto BO and moved to B9. At ths pont, UK samples (words) of a tme seres have been placed nto B8 for further processng startng wth the drect f ourer transform (DFT B8) shown n Fg 3. After the processng loops through TEGULO have been completed, the program returns to enter the next 2K samples. Snce BO stll contans the last 2K samples assocated wth the loop ust completed, ths data s moved to BU. New samples are then read nto BO and moved to B9 for the next processng loop. As wll be descrbed later, ths rather complex method of data nput s necessary 2

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47 to prevent the loss of data whle allowng for the vrtual elmnaton of end effects resultng from the dgtal processng. 22

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49 Data BO B8 B4 (Tme) B9 V DFT B8 B8 (Freq.) For each 2K of data. ZERO B9 n> FT B8 B8 B8 (Tme) V (Freq.) DFT B8* MOVE B8,B0 * complex- complex BO (Freq.) B2 B5 3 FT BO* Swept Flter BO (Tme) B5 TEGULO B7 (Hstogram) EQUAL BO,B7 t BO* Bl B2< RUNAVG BO BO PLOT BO* Fgure 3 - FLOW CHART OF H SCAN 23

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51 4. Transform to the F r eguency. Doman Once the tme seres s loaded n to block B8, the correspondng frequency spectrum, consstng of 2048 complex elements, s computed usng a hardware FFT mcroprocessor wth the results stored back n block 38. At ths pont, an deal flter s constructed by smply settng all frequency elements outsde the desred flter bandpass equal to zero. The nverse Fourer Transform (FT) of ths block s the deally fltered tme sgnal. Ths waveform s tegular n nature and can be analyzed by zero crossng technques. Because tegulometrc methods requre consderable numercal calculaton and manpulaton, t was necessary to search for methods to speed up the computatonal process. Snce many of the tme-consumng operatons are related to the DFT-FT routnes, Professor Marmont modfed the drver program of the hardware FFT devce to store all requred sne and cosne values n a table nstead of usng more lengthy nterpolaton. A second sgnfcant ncrease resulted by reducng the number of operatons necessary to compute a sequence of DFT, flter, and FT. The algorthm used n the F4 mcroprocessor makes provson for computng ether the real to complex or the complex to complex transform. The real to complex DFT performs the followng general steps complex fold, unscramble, and unfold. The correspondng FT does a fold, complex fold and unscramble. The fnal step of the DFT s to unfold the frequency components whereas the frst step of the FT s to perform the reverse operaton of foldng before proceedng. Snce the analyss method uses the DFT/FT to produce only a narrow band fltered tme sgnal for zero crossng analyss, the fold-unfold steps 24

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53 represent extra computatonal steps whch Professor Marmont realzed could be elmnated. The complex to complex form of the DFT defnes the tme functon as complex where the even words (the frst sample s defned as the zero sample) and odd sample numbers as magnary. The DFT then forms a complex fold, and unscramble wth no unfoldng. The complex to complex FT then wll accept the unfolded data, perform a complex fold, and unscramble to obtan the tme functon. Ths latter form of the DFT/FT resulted n sgnfcant speed ncreases n that the real to complex takes mllseconds whle the complex to complex transform requres mllseconds. Because the frequency components are not unfolded, however, t does ncrease the complexty of the flter. 5- Swep t Flter Because the bass for ths method les n frequency determnaton by measurement of the zero crossngs of the narrow band fltered nose, ths mples that the flter must be "moved" ncrementally through the frequency block. The flter could be moved n steps such that t does not overlap wth the prevous settng. Although ths results n lower computatonal requrements, t also results n a very poor frequency determnaton because a. The predomnant frequency found by the zero crossng algorthm wll be weghted toward the center frequency of the flter. b. Strong frequences wthn the flter passband wll "pull" other frequences toward the strong element. c. Weak sgnals wll be captured by strong sgnals wthn the flter band. 25

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55 As a result of the above effects, the flter s "swept" through the range of nterest, ncrementng the flter locaton by one frequency element after each zero crossng frequency analyss. Ths tends to average the above effects and produce a lnear frequency scale. Snce the same DFT s used for an entre flter sweep, a copy s mantaned n block B8 and moved to block B0 for successve flterng. Upon completon of the flter sweep across the frequences of nterest, a new set of tme samples s read nto BO and B9 for the next flter and TEGDLO cycle. Snce the DFT used for the flter s the result of a complex to complex cpmputaton, the frequences are not unfolded. As a result, the values for a specfc frequency component, X (n) = Xr (n) + X(n), are located n two addresses whch are symmetrc around the mdpont of the frequency block as shown n Fg 4. The frst block depcts a real to complex DFT n whch two frequences X(n), low n the block, and X(m), hgh n the block, are located. A flter whch detects the low frequency, X (n), would be constructed by settng all other elements outsde the bandwdth to zero. Because of the lack of unfoldng n the complex to complex case, t s seen that locaton s a combnaton of low frequency,x(n), and hgh frequency, X (N-m) ; locaton 2 s a combnaton of X(m) and X(N-n). By necessty, the flter must be structured to allow both the X(n) and X (N-n) components through for the complex to complex FT whch follows. To prevent X(m) and X (N-m) from nterferng n the FT, the upper frequences are set equal to zero by zerong block B9 pror to performng the complex to complex DFT. The resultng complex to complex FT wll then contan only the frequences wthn the desred bandpass. Although the complex to complex DFT flterng 26

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57 requres settng all frequences greater than X (N/2) equal to zero, where X(N) equals the maxmum frequency n the real to complex DFT, t s stll possble to analyze ths hgher block of frequency elements. By shftng the upper half of B8 to the lower half pror to zerong block B9, the hgher frequences may be analyzed wth no other effect on the programmng. 27

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59 Real to Complex DFT Flter ZERO ZERO ; 4 ( L X(0) X(n) X(N/2) X(m) X(N) A Complex to Complex DFT Flter ZERO,» ZERO A ZERO X(n) X(N-m) X(m) X (N-n) Complex to Complex Flter Formaton 4K Tme Samples t REAL TO COMPLEX DFT ZERO T COMPLEX TO REAL FT T 4K Tme Samples (Low Pass Fltered) ZERO XCn) V COMPLEX TO COMPLEX DFT vr ZERO t X(.N-n) ZERO Fgure 4 - FLTER CONSTRUCTON 28

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61 6- Zero Cros sng Algorthm Usng TEG ULO The fundamental program for ths analyss s the zero crossng measurement of frequency, computed n a routne called TEGULO. For each step n the flter sweep, TEGULO detects the sample pont separaton between seccessve negatve to postve zero crossngs. Ths separaton s converted to frequency and the approprate locaton of the hstogram block s ncremented by one. The hstogram block s a UK word block whose addresses are proportonal to the frequences determned by TEGULO. As the zero crossngs are- encounterd and the respectve frequences are measured and counted, the hstogram block becomes a measure of the relatve occurrence of the varous frequences. The dscrete fourer transform s a dscrete representaton of a contnuous spectrum. f the Nyqust sample rate s always satsfed, all frequences wthn the passband are represented by these dscrete frequency elements. Snce the narrowband fltered tme sgnal wll contan all frequences wthn the flter, a hgher degree of frequency resoluton s possble. Ths s accomplshed by performng an nterpolaton between the samples as though there are 2 addtonal ponts between successve sample ponts. A typcal sampled waveform s shown n Fg 5. t s seen that the negatve to postve zero crossng par occurrs between samples one and two and samples n and n+. Snce the negatve value w and postve value w,, are 3 n r n+ known, the actual tme of the zero crossng can be determned usng smlar trangle relatons. Ths lnear nterpolaton may be used f the maxmum frequency analyzed s lmted to some fracton of the maxmum avalable n the DFT. n TEGULO, a compromse s made to lmt the analyss 29

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63 frequency such that there are a mnmum of 0 samples per cycle. Ths allows foe reasonable accuracy n the lnear approxmaton. By countng the number of nteger and fractonal samples multpled by the 2048 constructed ponts per sample nterval, the frequency may be determned by the nverse relaton between perod and frequency. Ths value found s converted to a frequency address n the hstogram such that there s an ncrease n frequency determnaton by a factor of 0. Then the hstogram block, consstng of U096 words, wll represent successve frequences. The deal flterng results n dstorton of the fltered tme doman block because of convoluton wth the Sne functon whch s the fourer transform of the flter aperture. Because of the recprocal relaton between frequency and tme, a narrower flter mples greater dstorton n the tme doman. n HSCAN, the optmum flter has been found to be 0 frequency elements n wdth so that the zeroes of the Sne functon occur every samples. Ths effect s prmarly notced at tha ends of the tme block. To mnmze ths end effect, only the center 2K of the fltered tme data s analyzed for zero crossngs. The algorthm of TEGULG detects the frst negatve to postve zero crossng and defnes ths as the ntal zero crossng so that there wll be one less cycle counted n a gven block than the total number of cycles. Reducton of end effects dctates that only half the FT block should be analyzed by TEGULO so that the total number of cycles counted s /2 that of the frequency. As seen n the lower porton of Fg 5, whch represents blocks B0 and TEGULO block B5, only four zero crossngs out of the eght n B0 are avalable for countng. Snce TEGULO marks the ntal negatve to postve zero crossng as the ntal pont, the analyss results n producng a fnal output of 3 counts. 30

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65 Extenson of ths analyss to lower frequences reveals that no counts wll result for frequences less than 4 Hz. Therefore, the hstogram block s establshed to contan frequences from 4.0 to 43.5 Hz. (t should be remembered that those frequency elements are "dgtal" frequency. The actual frequences depend upon sample rate and any shftng of the DFT elements pror to TESULO.) Because phase reversals can occurr n the tegules, zero crossng analyss can produce frequency measurements whch le outsde the flter settng. Snce these frequences are extraneous, TEGULO reects all zero crossngs outsde the current flter settng and ncrements only those hstogram frequency locatons correspondng to the current passband. 3

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67 Typcal Sampled Waveform Lnear nterpolaton n = w = d = sample number value of waveform at pont n zero crossng to next sample pont d = 2048 w, n+ K + w * 2048 ponts» addtonal Zero Crossng Counts Block BO Block B5 2 Fgure 5 - ZERO CROSSNG DETERMNATON 32

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69 7 Equalzaton After all data has been analyzed and the results placed n. the hstogram block, t becomes necessary to equalze the hstogram data. n a gven block of data, the number of zero crossngs, hence the number of counts n the hstogram, wll be greater for a hgh frequency than a low frequency. A program, called EQOAL, multples each frequency dependent word n the hstogram block by a scalng factor whose value decreases wth ncreasng frequency. Ths results n a hstogram whose ampltude s now a measure of the percentage of tme that a frequency was observed durng the analyss perod. f the total analyss perod s short, there may be only a few counts n the unequalzed hstogram. To compensate for ths and to allow for a readable plot, an addtonal arbtrary scalng multpler s appled to all words of the block. Usng a fxed multpler, however, can lead to overscalng due to the potentally large dynamc range of ths method. To preclude ths from occurrng, an overflow detecton scheme was ncorporated durng ths work to dynamcally reduce the scalng multpler when requred. 8. Runnng Averag e of the H sto gra m Pror to fnal dsplay, t becomes necessary to smooth the equalzed but hghly rregular hstogram. Ths rregularty results from two sources. A real sgnal wll typcally vary slghtly n frequency over short tme perods due to modulatng effects of the transmsson medum. Ths results n a spread of counts to adacent frequency locatons n the hstogram and can reduce the det actablty 33

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71 of a sgnal n a fne-graned analyss. Addtonally, at hgh "dgtal" frequences, roundoff n the F4 FFT mcroprocessor and n zero crossng nterpolaton can result n placng frequency counts up to 5 hstogram addresses from the actual frequency. Ths s not a sgnfcant loss n resoluton snce t s restrcted to the upper frequences analyzed by TEGULO and snce the nherent accuracy of TEGULO s a factor of 0 greater than the parent DFT. The equalzed hstogram s therefore smoothed n routne RDNAVG by addng successve and adacent hstogram words to produce a runnng average. The number of frequences ncluded n ths runnng average can be vared wth an external nput parameter to the program. n addton, ths routne scales the block to maxmum numercal sgnfcance wthout producng overscalng (overflow) of the largest value n the hstogram. The fnal results appear as a smoothed hstogram whose ampltudes, frequency dependent, are a measure of the probablty densty or persstence of each frequency. An example of a typcal output s contaned n Fg 6. Examnaton of Fg 6 shows two maor characterstcs of ths type of analyss. The frst s the appearance of capture valleys adacent to the frequency component. The second shows that the sgnal wll buld upon the nose hstogram value to produce a hgher peak at the sgnal locaton. 34

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73 Fgure 6 - SGNAL N WHTE GAUSSAN NOSE 35

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75 9 Cont nuty Of Data t was seen n the secton on on TEGULO that only one half of the fltered data n the tme doman was used for the zero crossng analyss n order to reduce the effects of the narrow band deal flter n the tme doman. Unless care was taken n the data nput ths would have resulted n a loss of nformaton. To prevent such a loss, all data nputs were made va block BO whch s a temporary storage block seen on the flow chart of HSCAN. Before brngng n new data, BO, contanng the last 2k samples from the prevous frame, s moved to BU. New data s then loaded nto BO and B9 so that these blocks represent contnuous data. Ths cycle s shown n Fg 7 where the numerals refer to the nput samples and the letters refer to the central 2K fltered data analyzed by TEGULO. The bottom secton of Fg 7 demonstrates that the analyss block, B5, does represent contnuous data. 36

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77 (B5J Data nput Scheme Data n BO B4 B8/B0 B5 B9 Relaton of nput to TEGULO Data for Each Loop (B4) * (B9) 2 (B5) A * Loaded nto B4 from BO. BO contans samples from prevous loop. CB4) 2 (B9) 3 (B5) B (B4) 3 (B9) 4 C Relaton of nput Data to TEGULO Analyzed Data nput Data Frames n A B C D E "-= TEGULO Data Frames Fgure 7 - CONTNUTY OF DATA 37

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79 . TEST OF THE METHOD A. EQUPMENT. Hardwa re All tegulo metrc analyss methods were performed dgtally. Ths resulted from the nature of the analyss method whch requres complex but flexble data handlng equpment. The followng equpment was used n the present nvestgaton. a. Dgtal computer (PDP-/40), a hgh speed 6 bt central processor wth 32K words of core memory. b. FPE4 fast fourer transform (FFT) hardware mcroprocessor manufactured by the Tme/Data Corporaton. Ths processor transforms a 4096 pont tme seres to a spectral seres n mllseconds. The nverse fourer transform (FT) s performed n the same tme. Softwear support was provded by Tme/Data and later entrely replaced by Professor Marmont to ncrease processor speed to mllseconds for 4K of data. t should be noted that the FFT operates usng nteger arthmetc vce floatng pont. t keeps track of and adusts the exponent for the block of data so that maxmum sgnfcance of the nteger (mantssa) s mantaned n the data. c. Dsk Memory (RK-D) manufactured by Dgtal Equpment Corporaton, holdng.2 mllon words per dsk pack, was used for program and data storage. d. Other equpment such as an X-T plotter, a storage 38

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81 osclloscope, and varous sgnal generators were used as requred for data output and nput. d. analog to dgtal converson utlzed a hgh speed, 2 bt A/D converter. Selectable sample rates were hghly accurate snce the sample pulses were generated by a crystal controlled oscllator. 2. Softwear a. Tme Seres Language (TSL) The ntal softwear was wrtten n lme Seres Language (TSL), a form of BASC whch was provded by Tme/Data. TSL was specfcally desgned to manpulate blocks of data, to utlze the nstalled hardwear FFT mcroprocessor, and to dsplay results on ether an X-Y plotter or a memory osclloscope. Although relatvely easy to use, TSL was found to have a number of dsadvantages, among whch were slow speed, a lack of adequate flexblty, and nablty to perforn the specalzed routnes necessary for tegulometrc analyss. As a result, dedcated machne language routnes were lnked to TSL to ncrease speed and program smplcty. b. Language, APTEC As more routnes were wrtted n machne language and as the requrement for absolute control over program and data locaton n memory became pressng, all tegulometrc programs were wrtted n machne language. Ths necesstated the development of nput/output routnes and general data handlng routnes prevously used n TSL. Thus, a new operatng system, called APTEC, was developed. 39

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83 Ths system, developed by Professor Marraont, provded mproved speed n the FFT, absolute control of program and data locaton, and control of the system montor program of the Dgtal Equpment Corporaton. ts greatest assets were great flexblty of programmng and smplcty of use. Ths language s stll under actve expanson. B. GENERAL APPROACH n order to demonstrate the accuracy of the method and to measure mprovements n senstvty, t was obvous that nput data had to be consstent from one computer run to the next. Snce the maor nput was to be approxmately whte gaussan nose, a lengthy recordng of nose was fltered wth a fourth-order Butterworth flter, sampled n dgtal form at the Nygust rate, and placed on a memory dsk. Test sgnals were generated n softwear and also stored on the dsk. Test programs read the nose and sgnal off the dsk, scaled the sgnal to the desred sgnal to nose rato, and added the scaled sgnal to the nose n the data nput block, BO. The desred parameter control and reproducblty was acheved n ths manner. Two sets of nose were recorded wth an A/D full scale settng of *.0 volts. The frst conssted of 60 2K samples at 0.24 volts rms wth a flter settng of 00 Hz and a sample rate of Hz. Ths corresponded to 0 mnutes of actual data. The second sample contaned 20 2K samples recorded at 0.24 volts rms wth a flter settng of 2000 Hz and a sample rate of Hz. Ths represented one mnute of data. The voltage levels were chosen to obtan maxmum numercal sgnfcance of the nose data whle reducng the probablty of numercal overflow to a mnmum. 40

84

85 Snce both the sgnal and nose was n dgtal form, accurate measurement of the mportant nose parameters could be made. The frst test conssted of checkng for any overrangng of the analog to dgtal converson process n the recorded data. Of the 60 frames of 2048 samples per frame n the frst sample, there were only 7 A/D overranges, no two of whch were contguous. The DC value was determned to be -30 (out of a total range of 32767) Ths corresponded to an analog voltage of 0.92 mllvolts. Analyss of the second set of nose yelded smlar results. From the standpont of overflow and low DC, the test nose was udged as satsfactory. The dgtal form of the nose data was also analyzed to determne ts root lean square value. Ths could have bsen computed by the averaged tme nterval of the nose squared over the tme of samplng. m v(t) 2 = ±f [v(t)] 2 dt Snce the recorded nose was bandwdth lmted to W Hz, the average nose power, N, could be computed by usng the relatonshp TW N-» c(n)c*(n) -TW where c (n) and c* (n) are the complex Fourer coeffcents and ther complex conugates. Further, t was assumed that the nose was gaussan wth a zero mean so that the varance of the nose could be descrbed as, 2 2 = vct) 2 - v(t) but v(t) = TW a 2 = v(t) 2 = c(n)c*(n) -TW Snce the computer returns the Fourer seres components a and b, the actual algorthm used to determne n n 3 4

86

87 the value of a 2 was. M-l M- r N-l N- -» a -z L L[? K + ^] where M=total number of sample frames N=total number of frequency elements per frame The square root of ths result s the rms value of the nose. To obtan the nose power per Hz, N N = 2 a /W Ths technque permtted the determnaton of the nose power and rms value for any frequency range for sgnal to nose calculatons. C. TEST OF KEY ROUTNES General Ths secton s devoted to the testng of the maor and more complex routnes used n the man program. Those routnes not descrbed here, but used n the analyss, such as movng data from block to block, performng square roots, defnng data blocks, etc., were tested thoroughly n other work. 2 Swep t Flter The detaled structure of the complex to complex DFT form of the flter s shown n Fg 8. The set of numbers above the block descrbes the locaton of the frequency elements. The numbers below the block represent consecutve 42

88

89 complex element numbers and those wthn the block the consecutve word numbers n memory. t s seen that element number 3 contans the data for frequences 3 and ts symmetrc element locaton hgh n core s number Pror to formng the complex to complex DFT, all frequences above 024 were set equal to zero so that the flter shown n Fg 8 would allow only frequences 3,4, and 5 to appear n the FT. The logc whch determned the locaton and sze of flter blocks B2, B, and B5 was verfed n a test run. Addtonally, a tme doman test sgnal of 256 Hz was nput, the flter set at 256 (no sweep) wth the bandwdth set to one element, and the FT examned. The FT showed a cosne wave of 256 Hz wth no dstorton. 43

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91 00 o CN o CN CO -vt O CN o CN Q «H W CQ N ^r o CN CN < > ^r o CN to *) o CM n n o CN o cn ^r O CN ro o n ^ ^ CN CN O O rh rh 00 D CN CN O O rh rh 4Q CO o CN W +J C a> "^ g CN cd rh O H ro CN O W H U rh 4-> CD > CO > rh D cd C H CD -P 4J d G G tf 0) CD s () 6 3 o J- CD cr H M-l rh CD CD CQ T B D cu X <D o$ C o 4J H H rh f«co A a O s F, r-( p Tl > u O G H «CU CD w 4-> CQ CD E-t CC5 A. ^ S J o U rh H H rc G lh T) CQ H H H C CD 4J Sk.G Eh H T5 g 4-> & 0) G H W CQ tn d H 5 5 S-l u d ^ CO LD CO o CN O CN 0 n cu CD CD 4-) 5- S Xl CO H CD f ^»» S 3 CO p c CO c 4- G CO CQ 3 c S-l 5- g (D m CD 5- (D CD M CD d CD A X T d & CD 5 g g 5-( en &.H o 9 9 H D CD J c a 5 fo CN ro ro CN n o CN o CN en CN CQ CD 4J a r^ o CN CO o CN QOS CN W 03 N 44

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93 3» Zero Crossng Routne^ TEGULO TEGULO was tested for accuracy of zero crossng measurement, for accuracy of converson from zero crossng spacng to frequency, foe adequacy of the logc to exclude extraneous frequences, and for preventon of overflow. The frst test nserted a number of dscrete frequency elements nto the DFT block after settng the block equal to zero. The flter was allowed to sweep from frequency element 4 to 406 wth a bandwdth of one element. The results n the hstogram block showed that a. Frequency elements below 320 Hz were counted accurately and placed n the correct frequency locaton n the hstogram block. b. At the hgher frequences, the correct number of zero crossngs were counted but some of the counts were placed n hstogram locatons adacent to the proper hstogram ste. The worst case condton occurred for 320 Hz for whch 950 counts appeared at locaton 39.9, and 320 counts at both locatons and The sum agreed wth the theoretcal total of 590 counts. The effect s beleved caused by roundoff n the lnear nterpolaton of frequency correspondng to a zero crossng par. Ths was determned to be of neglble mportance snce the problem was restrcted to the hgh hstogram frequences and snce the routne, runnng average, effectvely added the separated counts pror to fnal output. The spurous frequency elmnaton logc was tested by fxng the flter at a sngle center frequency wth varous bandwdths and then analyzng the hstogram data. Ths examnaton revealed that frequency counts appeared only wthn the flter bandwdth *5 hstogram elements (# /2 DFT elements) as requred. 45

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95 n the whte, gaussan nose samples tested, t was seen that the hstogram counts were farly unformly dstrbuted through the block and low n numercal value. The presence of a strong sgnal, however, could result n a large number of counts n the sgnal hstogram locaton. n a lengthy analyss, ths could have resulted n overflow of the hstogram. To preclude overflow, logc was added to prevent any hstogram locaton from exceedng the maxmum value of (77777, octal). Ths was tested by countng a 256 Hz sgnal and observng the buldup n counts n the hstogram. The maxmum value was acheved as expected and no overflow resulted. Because EQUAL and RUNAVG both scaled the hstogram data based on the largest value n the block, a very strong sgnal, counted n double precson, would result n a small scalng factor. Ths n turn would reduce the smaller varatons n the hstogram to the pont where small sgnals could not be detected. Ths scheme therefore had the addtonal advantage of lmtng the effect of strong sgnals when searchng for weak sgnals n nose. Hm Equalza ton of the Hstog ram The test conssted of loadng the hstogram block wth the theoretcal counts correspondng to each frequency. After equalzaton wth EQUAL, the data n all locatons should have been equal f roundoff was neglected. As expected, frequences through 20 of the 406 n the hstogram were low by about 5 percent. Ths was not due to the naccuracy of TEG0LO but a reflecton of the large scalng necessary to ncrease the small number of low frequency zero crossng counts to a large number usng 6 bnary bts. because Ths effect was not consdered a sgnfcant problem 46

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97 a. Analyss of actual data wll normally occur over many frames of nput data. Ths allows the counts, low n the hstogram, to buld up to a value where the nteger arthmatc of the computer has only a mnor nflusnce. b. The effect s lmted to only a small part of the fnal hstogram and appears as a mnor reducton of the low frequency ampltudes. Snce detecton depends upon the relatve heghts of the hstogram, a sgnal wll stll show up as more persstent than the neghborng frequences. The errors dscussed n these last two sectons were found to have no effect on the bulk of the fnal hstogram outputs. The mdrange from about 20 to about 300 Hz was accurate n frequency and ampltude to wthn one percent. From 300 to 406, the hghest element, the ampltudes were wthn 0.2 percent. 5. Cont nuty of Fltered Data t was asserted n the ntal descrptons that end effects n the tme doman, due to the deal flterng, could be reduced n the legulo block, B5, by dscardng the frst and last 024 samples n the analyss block by TEGOLO. By careful nput of data t was also possble to process wthout loss of data. n order to evaluate ths, HSCAN was modfed to plot the fltered tme doman sgnal and nose nstead of computng the hstogram. Ths program, called TEGPLT, was run over successve frames of nput data at a constant flter settng. t was found that the fltered data was ndeed contnuous from one frame to the next wth no dscontnutes n the data. Ths test was performed over a wde range of data and flter settngs. t was noted that small dstortons dd appear f the bandwdth of the flter was reduced much below 6 elements. Ths effect was expected snce narrowng the flter expands the effect n 47

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99 the tme doman. From ths t was concluded that the data used by TEGULO was essentally free from effects normally assocated wth samplng and deal flterng usng the dscrate Fourer transform. An example of ths s seen n Fg 9. The breaks n the graph between data frames are an artfact of the plottng routne and not an ndcaton of a problem n the data. 48

100

101 Fgure 9 - CONTNUTY OF DATA TEST 49

102

103 V. MPROVEMENT OF SENSTVTY A. GENERAL Because the method of zero crossng analyss s hghly non-lnear, the fundamental approach was expermental rather than theoretcal. Usng current theory as a gude, varous schemes were selected as havng the potantal for senstvty mprovements. Snce the test nose and sgnal were recorded on the dsk, the nput data was constant for each computer run. Any mprovements n detectblty could be related to the method and parameters used. B. ZERO NSERTON REMOVAL ntal versons of HSCAN used tme doman zero nserton. Zero nserton, however, was found to produce artfact frequences n the DFT. Snce ths method depended strongly upon narrow-band flterng and zero crossng analyss, ths resulted n potental dstorton of the fnal output. After removal of the zero nserton, tests wth snusodal sgnals showed sgnfcant mprovement n the hstogram data. C. FREQUENCY COMPONENT LMTNG 50

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105 t was found that a typcal DFT of nose contaned a number of strong frequency elements. f any strong frequency was wthn the flter bandwdth of a weak sgnal, the sgnal was "captured" and possbly would not appear n the fltered tme representaton. To allevate ths stuaton, the effect of Lmtng the magntude of both very hgh and very low frequency elements n the ntal DFT was nvestgated A program, called SLCE, computed the magntude of each DFT element and modfed those frequences whose magntudes exceeded ether a preset laxmum or mnmum lmt. Ths s shown n Fg 0 whch depcts a DFT block and the two SLCE. levels. The upper or LMT level represents the cutoff of those frequences whose magntudes are very large. Values exceedng the LMT level were ether set equal to zero or ther complex components were scaled so that ther magntude equalled the LMT level wthout affectng the phase. The lower or CLP level represents those frequency elements whch are due entrely to nose. The complex frequency elements below the CLP level were set equal to zero. Ths method suffered from a maor dffculty n that the actual magntude of the sgnal element plus nose had to be known n order to set the levels nvolved. Snce only the fact that a sgnal was present wthn a band of frequences was assumed, no apror knowledge as to ts level could be used. f these levels were ncorrectly set, a strong sgnal could be ether removed or ts effect reduced. A very weaksgnal, whch adds coherently to the nose, could combne wth the nose such that the fnal magntude was below the CLP level and thus be lost. Despte these obvous problems, testng of the method was stll fearlessly accomplshed. The SLCE levels were 5

106

107 set such that the sgnal element magntude was always wthn the SLCE band and therefore not modfed. Wth a sgnal to nose rato of -8.0 db, based on a nose bandwdth of one hertz, lttle or no mprovement n the detectablty was seen. Because of the dffcultes descrbed above n settng the SLCE levels, a thrd method, called WHTEN, was developed. As seen n the thrd graph of Fg 0, all frequency elements were set to a sngle magntude level. All scalng was accomplshed so that the phase relatons were unaffected by the magntude change. Tests of ths method also ndcated a lack of sgnfcant mprovement. t was concluded that ths method had lttle value at ths pont n the development of Tegulometrc Analyss. t may be possble, however, that future tests wth real data contanng known nterference sources may yet show that smlar technques are useful. 52

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109 General Scheme Lmt Level Clp Level SLCE Frequency LL CL WHTEN Frequency Whten Level Frequency Fgure 0 - FREQUENCY COMPONENT LMTNG SCHEMES 53

110

111 D. AVERAGNG N THE FREQUENCY DOMAN A proven method n reducng the effects of random nose s to average successve drect Fourer transforms. Wth gaussan nose, each frequency component wll have an rms value, a, and a mean of zero. f there are M averages, then the resultng rms value "cf wll be a = ct/vm The rms of the sgnal wll not be changed by the average so that potental sgnal to nose would be mproved. A modfed form of HSCAN, called RDSCAN, (runnng DFT average wth HSCAN) was developed to take runnng averages of the DFT and analyze the resultng frequency block. Ths modfcaton s shown n Fg. nput data s read nto block BO and moved to B8 n the same manner as HSCAN. After DFT B8 and ZERO B9 f the lower half of B8 s added to B6, whch contans the current runnng sum of the drect fourer transforms. After ntal memory load, the oldest DFT s subtracted from B6 and the newest DFT replaces the old data n the DFT memory blocks. The current averaged DFT s moved nto B8 and processng contnues as n HSCAN. Results of the averagng technque showed a sgnfcant mprovement n sgnal detecton. Sgnals that were prevously dstngushed wth dffculty at -8 db could be easly seen on the hstogram. 54