8/13/2019 The Adaptive Spectrogram - Boashash-Jones, 1992 IEEE DSP-Workshop

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THE ADAPTIVE SPECTROGRAMGraetne J one s and Boualeni Boashash

Centre for Signal Processing Research, Queensland University o TechnologyG P O B o x 2434 Brisbane Q 4001 AUSTRALIATel - G 1 7 864 2484 Fax - +61 7 864 1748

1. INTRODUCTIONTh e spect rogram is used in many non-sta t ionary appl ica-tions of digital signal processing, including speech analy-sis , underwater acoust ics and t ime-varying spect rum est i -mat ion . T he m ajor l imi ta t ion of the spect rog ram, however ,is tha t i t is non-unique . T h e spect ro gram is expressible inthe form

1)

where h t ) is the desired analysis window. Since a spec-trogram may take any valid one-dimensional function as awindow, there are effect ive ly an inf inite num ber th a t m ay begenerated. T h e choice of a certain wiiidow over another maydramatically affect the final form of the spec trog ram . If i ti s desired to genera te the best possible spe ct rogram (unde rsom e given criterio n), window selection mus t necessarily beadaptive. More significantly, due to the non-stationary be-haviour of t ime-varying signals in th e t ime-frequency plane,i t i s also requi red tha t the window be dependent on theanalysis point considered in the plane. Th is pape r wi ll sum-marise a new ada ptiv e spe ctro gram , which uti l ises i irstan-taneous parameters to match appropria te windows in thetime-frequency plane.

I s t , f ) l 2=11Z ( T ) / t ( T ) e - - j 2 r f r d 7 1 2

2. T H E FORM OF THE ADAPTIVESPECTROGRAMThis implementa tion of the ad apt ive spect ro gram employsGaussian windows, of the generic form

2)t ) e- -xa t e - -3xb i2 ,Th e spect rogram may be a l te rna te ly expressed as

m 0 3IS bf)lz= rn1, ( t , f 9 i , f i ) d t i d f i 3)

whereQ f , f , t l . f l ) = w = ( t l , f l ) w / , ( t l k f - f l 4 )

and W , t , f ) s the Wigner-Ville Distribution (W VD ) ofthe signal and W h t , f ) he W VD of the window. Forthe adapt ive spect rogram, the window wi l l be dependenton the t ime-frequency loca tion under considera tion. Th usthe window's cont rol l ing param eters , and b , will now bedependent on t ime and frequency, i .e . u t ,f and b t , f .the parameters) for the t ime frequency points , under somecriterion of optimality. T h e criterion used will be tha t ofma tch ing t h e i ns t an t aneous second o rde r cumulan t s (var i-ances) of the spec t rogram to those of the window (whicha re cons t an t s for a Gaussian) . Before this nie thod is de-tailed, the necessary concept of instantaneous parametersis introduced.

Th e t a sk now i s t o de t e rmine appropr i a t e win ows ( t ha t is ,

3. INSTANTANEOUS PARAMETERS INTIME-FREQUENCYThe concept of an instantaneous parameter i s fami l ia rthrough the widely employed instantaneous frequency mea-sure . I t may be regarded as the instantaneous f i rs t f re -quency moment - instantaneous wi th respect to t ime. Asimi lar quant i ty , also common in geophysics and spec-t roscopy applica t ions, is the time delay i t i s an instanta-neous measure of th e f irs t t ime mom ent , an d dependent onfrequency. It is possible t o generalise such results, an d de-f ine higher order instantaneou s mom ents or cumulan t s ( t heins t an t aneous bandwid th [l] or example) . Th e problemwith such measures, and the i r a t tempts to instantaneouslycharacterise a signal , i s tha t they are only ever instanta-neous wi th respect to t ime or f requency, never both. Whenthere are multiple regions of signal information present inthe t ime-frequency plane, these type of ineasures cannotprovide truly local descriptions of the si nals.to two dimensions by the authors [2]. They have been de-rived in an identical fashion as for the one-dimensional sig-nals, through Parseval 's general Fourier relations 131. Fortwo dimensions, however, the instantaneous power, I z t ) l z ,and the energy densi ty spect rum, l Z f ) I 2 , are replaced byspect rograms. In o the r words , t he i ns t an t aneous pa ram-eters (of the spect rogram) are window dependent in thet ime-frequency plane. When using the WV D, the se t ofinstantaneous parameters may be expressed as

Instantan eous para met ers have recent been general ised

From the equat ion above , i t may a lso be seen th a t these pa-rameters may be te rmed equivalent ly the localm o n i e n t s ofthe WVD defined by the local region concentrated aroundthe WVD of the window. The se param eters a re exten-sions of the one-dimensional measures - for example, ifL = 0 , m = 1 and h t ) 1 in the equation above, thenthe instantaneous parameter der ived from 5)becomes theusual instantan eous f requency. The se imtan taneo us mo-ments, and the i r associa ted cumulants , provide a local de-scr iption of the signal (depe ndent on the window) in the

t ime-frequency plane . In the sam e way tha t the global mo-ments characterise the whole signal, these measures providean instantan eous or local charac ter isa t ion. The y shal l nowbe utilised in a window m atching scheme, to genera te anadapt ive spect rogram.4. GENERATION OF TH E ADAPTIVESPECTROGRAM

Local window matching in the t ime-frequency plane is em-ployed here. It may b e perceived as a complicated variationon the m atched f i l te r . In th e case of a matched f i l te r the

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bes t ou t put i s ach ieved w hen th e s tep r e spo n e of the f il -t e r is the t ime-reversed incoming s i nal I t is l o t feasible,however , to s imply extend the matc%edf i l ter doncept , anduse the s ignal i tself as the spec t r ogr am w indo+. This w i llgive the result

which is only a squared and scaled WV D . D ue to th e non-s ta t ionar i ty in the p lane , the match ing i s done a t t h e l o c alor ins tan taneou s leve l. Th e ins tan taneou s pa amete r s a r ef or mat ion is then used to match the appr opr ip te w indow .S ince the match ing w indow i s a Gaussian, which is com-used to ob ta in a par t ia l s ignal chara cter isa t iop this in-

show n to conver ge thr ough the allowing lo ca l h c e r t a i n t y relat ionship [2, 41I

1T2FO t T I P t , ) 2 +TOP , )- TOP , 2 = 7 )47rwhere the symbols in the above equation refed t o those ofequation 5 ) . E q u a t i o n 7 ) is satisfied for ahy ar b i t r a r ysignal under analysis with a window of t h e f o r k

S imula t ion s have show n tha t good convergdnce may b eachieved in as l i t t l e as f our i t e r a t ions . T he co lnputa t iona lload of the adap t ive spec t r ogr am is approximately propor-t ional to 7 5 1 N 3 [4] I is th e num be r of iterations1 used), coni-p a r e d t o N 2 1 0 g 2 N f or the convent iona l spec t r bgr am ( overT h e m e t h o d m a y b e m a d e m o r e e ff ic ie nt , a t t h e e x p e n s ea n N x N point t ime-f requency ar ray) .

of creat ing a s u b o p t i m a l s p e c t ro g r am , b u t o e tha t s t i l lpossesses excellent resolution. An ins tanta neo ds character-is t ic func tion m a y b e c r e a t e d [2], th r ough w hic li the ins tan-t a n e o u s m o m e n t s a n d c u m u l a n t s of t h e s p e c t r o r a m c a n b eexpressed (w ith so me ef for t) in t e r m s of t h e g l o f a l m o m e n t sor cum ula nts of the s ignal and window themsel$es . Explici texpr ess ions f or the ins tan tan eous cu mulants mdy b e de r ivedif , for example, only the moments to the foukth order ofboth s igna l and w indow a r e cons ide red . By eqba t ing theseappr oximate expr es s ions f or the ins tan taneouf second or -de r cumulants to the cumulants of the G auss ian w indow ,appr o pr ia te w indow par amete r s m ay be de te r r d ined f or anygiven t ime-f requency point .5. FIGURES

I

Tw o f igures are included which give som e ideaformance of the adaptive spectrogram. Data of humpbackwhale song is analy sed, , compr is ing 64 d at a saxnples with anequivalent sampling rate of 5000 Hz. Figure 1 Shows a nor-m a l s p e c t o g r a m ( w i t h a G auss ian w indow w hbse WV D iscircular in the t ime-f requency region consideied) . F igure2 show s the adapt iv e spec t r ogr am, taken a f te r ~4 t e r a t ions .I t is a much c lea r e r and be t te r r e so lved spec t r ogr am thanthe f ir s t , and r equi red no as sumpt ion s on the f o lm th e da ta .

REFERENCES[l] L. Cohen and C. L e e , I n st a n t a n e o u s B a n d w i d t h ,C h a p t e r 4, i n Methods and Applications of Tinie-Frequency Signal Analysis, B. Boa5hash Eb., HalsteadPress (Wiley) , 1992. I

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E:.-e

ooo - e o . Q I. 1-92 a . af t equency ut x 1 3 3 )Figure 1: N or mal Spec t r ogr am

,000 . eo .960 I. - 1.92 2 no

f r equeney Hz X IO 3 )F i g u r e 2: A d a p t i v e S p e c t r o g r a m

[2] G . Jones and B. Boashash , T he Theor y of I ns tan ta -neous Window Matching for Time-Frequency Dis tr ibu-t ions, Proc. Int. Synip. on Signal Processing and itsApplications, G old C oas t , A us t r al ia , A u gus t , 1992.[3] A . W. Rihaczek , Principles of High Resolution Radar,McGraw-Hill, New York, 1969.[4] G. Jone s, 1nstantai:eous Frequency , Time -Frequ ency

na ls , Ph.D hesis, Queensland Univers i ty of Technol-ogy, 1992.Distributions and the Analysis of Multicomponent Sig-

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