am-fm energy detection and separation in noise using multiband energy operators

8/7/2019 AM-FM energy detection and separation in noise using multiband energy operators

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IEEE TRANSACTIONS ON SIGNAL PROCESSING. VOL. 41. NO . 12 , DECEMBER 1993 3245

AM-FM Energy Detection and Separation in NoiseUsing Multiband Energy OperatorsAlan C. Bovik, Senior Mem ber, IEEE, Pe t ros Maragos , Senior Member, IEEE,a n d T h o m a s F. Qua t ie r i , Senior Member, IEEE

Abstract-This pape r develops a mul tiband or wavelet ap-proach for capturing the AM-FM components of modulatedsignals immersed in noise. The technique utilizes the recently-popularized nonlinear energy operator Y (s) = (S) - ss o iso-late the AM-FM energy, and an energy separation algorithm(ESA) to extract the instantaneous amplitudes and frequencies.It is demonstrated that the perform ance of the energy operator/ESA approach is vastly improved if the signal is first filteredthrough a bank of bandpass filters, and at each instant ana-lyzed (via Y and the ESA) using the dominant local channelresponse. Moreover, it is found that uniform (worst-case) per-formance across the frequency spectrum is attained by using aconstant-Q, or multiscale wavelet-like filter bank .The elementary stochastic properties of Y and of the ES A ar edeveloped first. The performance of Y and the ESA when ap-plied to bandpass filtered versions of an AM-FM signal-plus-noise combination is then analyzed. The predicted performanceis greatly improved by filtering, if the local signal frequenciesoccur in-band. These observations motivate the multiband en-ergy operator and ES A approach, ensuring the in-band anal-ysis of local AM-FM energy. In particular, the multi-bandsmust have th e constant-Q or wavelet scaling property to ensureuniform performance across bands. The theoretical predictionsand the simulation results indicate that improved practicalstrategies are feasible for tracking and identifying AM-FMcomponents in signals possessing pattern coherencies mani-fested as local concentrations of frequencies.I . INTRODUCTION

ETH OD S for the accurate and efficient extraction ofM mplitude modulation (AM) and frequency modu-lation (FM) information in signals of the forms ( t ) = a ( ? ) co s [4 (03 (1)Manuscript received September 1 , 1992; revised June 10, 1993. TheGuest Editor coordinating the review of this paper and approving it forpublication was Dr. Ahmed Tew fik. This work w as supported in part by aUniversity of Texas Faculty Research Assignment, in part by Texas In-struments under a grant, in part by the National Science Foundation un derGrant MIP-91 -20624, in part by the National Science Foundation Presi-dential Young Investigator Award under Grant MIP-86 -58150 with match-ing funds from Xerox, and in part by the Naval Subm arine Medical Re-search Laboratory.A. C. Bovik is with the Department of Electr ical and Computer Engi-neering, University of Texas at Austin, Austin, TX 78712-1084. Part ofthe research was conducted while he was on sabbatical at the Division ofApplied Sciences, Harvard University, Cambridge, MA 02139.P. M aragos is with the School of Electr ical Eng ineering, Georgia Insti-tu te of Technology, At lanta , GA 30332.T. F . Quatieri is with the Lincoln Laboratory , Massachus etts Instituteof Technology, Lexington, MA 02 173.IEEE Log Number 9212179.

are a topic of increased recent attention, owing to height-ened interest in modulation models for e. g. , speech signalproduction [11-[3] and certain struct ures in optica l images[4] . In ( l ) , s ( t ) has both time-varying amplitude a ( t ) an dtime-varying instantaneous frequencyw l ( t ) = i ( t ) (2)

where 4 d 4 / d t . Generally, the m odel (1) is most use-ful if a ( t ) an d U,(?) o not vary too rapidly, e.g. , in thebandlimited sense [8].The sim ple and elegant nonlinear signal operator\k(s) = (q2- ss (3 )developed by Teager [l ], [2] and systematically intro-duced by Kaiser [5], [ 6 ] , has been shown to be highlyeffective for detecting AM and FM modulation informa-tion in arbitrary AM-FM signals [8], in speech signals[7]-[9] and in its two-dim ensional form, in im age signals[ lo] . Indeed, for AM-FM s ignals of the form (l ) ,\k (s) = a 2 ( t ) w f ( t )\k (s) = a2( t ) w p( t >

with negligible approximation error under general realis-tic cond itions [7]-[9]. This motivatedration algorithm (ESA) :ci2(t)= \k2(s)/\k((s)& ( t ) = \k (s ) /\k (s)

th e energy sepa-

(4 )( 5 )

as estimates of the squared amplitude envelope a 2 ( t )an dsquared instantaneou s frequency w f ( t ) , espectively. Mar-agos, Kaiser, and Quatieri [7]-[9] have analyzed the ef-ficacy of (4) and (5) in detail and have developed bo undson the absolute errors I ci - a1 an d [GI wi , which undergeneral conditions are quite small [8], [9]. Note that inthe case of a mon ochromatic signal (a = constant, w , =constant), (4) and (5) are exact.In the current paper, the deterministic approximationerrors in (4) and (5) are assumed small. Instead, the ef-fects of noise and m ultiscale filtering on the beh avior ofthe operator \k and on the effectiveness of the ESA areconsidered. The effects of significant noise are very con-siderable-rendering \k unpredictable and the ESA highlyunreliable. However, the performance of the energy op-erator/ESA approach is vastly improved if the signal is1053-587X/93$03.000 99 3 IEEE

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3246 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41 , NO . 12, DECEMBER 1993

first filtered through a bank of band pass filters, an d at eachinstant analyzed (via 9 nd the ESA) using the dominantlocal channel response. Optimal performance is obtainedwhen the filters are sufficiently narrowband (thus increas-ing the signal-to-noise ratio), the signal spectrum is sam-pled densely by the filter set (ensuring a high signal re-sponse in the analyzing channe l), and importantly, by us -ing a multiscale wavelet-like filter having the constant-Qproperty. Satisfying all of the prescriptions produces amultiband ESA having a great detal of noise resistance.The remainder of the paper is organized a s follows: InSection 11, the basic statistical properties of the energyoperator (3 ) are developed under the assumption that it isapplied to a signal that is a zero-mean, wide-sense sta-tionary (WSS) Gaussian random process. Th ese results arethen used in Section 111, where approximate expressionsare developed for the statistics of the output of the Teager-Kaiser energy operator 9 when applied to an AM-FM sig-nal of the form (1) immersed in noise. Section IV devel-ops the statistical analysis of the ESA (4) nd ( 5 ) usingthe signal-plus-noise approximation of Section 111. A keyelement of these approximations is that there be an avail-able narrowband channel filter that can effectively capturethe local frequency structure of the noisy signal. In par-ticular, as the analysis window is shifted over the signal,the local frequencies may sw eep across the spectrum. Thisimplies the necessity of a multiband filter implementa-tion, where multiple bandpass filters densely sample thesignal frequencies. It is also shown that consistent per-formance is achieved across low, high, and intermediateinstantaneous frequencies, if the filter bank has the con-stant-Q property. Th ese themes are developed in SectionV, where design criteria for the individual bandpass fil-ters, and also for sampling the signal spectrum w ith mul-tiple bandpass filters, are explored. Section VI developssome important examples, including analysis of the op-eration of the multiband ESA applied to the chirp signalin white noise. Ex tensive simulation results are also givenin Section VI, which demonstrate the dramatic perfor-mance gains obtained using the multiband approach. Thepaper concludes in Section VII.

11. STATISTICSFIn this section the basic statistical (low-order mom ent)properties of the Teager-Kaiser energy operator (3 ) ap -plied to a random signal n ( t ) are developed. These prop-erties prove to be fundam ental in the analysis of systemsthat employ the Teager-Kaiser operator in the presence ofnoise.Assume that n ( t ) s a zero-mean, w ide-sense stationary

(WSS) Gaussian random process, with autocorrelationfunction R (7) nd power spectral density+.(U)= R ( 7 ) e - j w 7 7 .S,

The assumption of (at least) wide-sense stationarity is

necessary to develop nearly all of the properties in a u se-ful form; without the assumption of Gaussianity, the anal-ysis likewise becomes rapidly intractable. The assump-tion of a zero-mean is not critical, s ince the analysis isonly slightly more complicated. In any case, nonzero-mean (and nonstationary) signals expressed as the sum o fa deterministic signal and a zero-mean WSS Gaussianprocess are considered later.Since n( t ) is WSS Gaussian, the processes h(t) an dLi(t) are also WSS Gauss ian. Moreover, h ( t ) is statisticallyindependent of both n ( t ) an d A ( t ) [113. Therefore, the en-ergy operator output

9 ( n ) = (h)2 - nii (6)is the sum of two independent processes. Nevertheless,determining the probability density function of the pro-cess ( 6 ) s difficult, as it is the convolution of two rathercomplicated functions. L etting

Var [n] = R(O)= 7;Var [ 3 = -R '~ ' (o ) = y:Var [ i i ] = ~ ' ~ ' ( 0 )yiE [nn]= R'2'(0)= -7:

where E [e] is the statistical expectation, Var [ a ] is thevariance, andd kd7'k'(7) =7 ( 7 )

the probability density function of[I11

Wll(b) = i o 7 elsewhile the probability density function of the product nLi is

is then given byb > 02 ~ b y : ) - ' / ~ xp [ - b /( 2 y: )] , (7 )

1121

(8 )for every b, where CO2= -&y - y':an d KO ( ) is the mod-ified Bessel function of the second kind and of o rder zero.The density function of 9 ( n ) s then

(9 )where '* ' denotes linear convolution. In general, (9) can-not be expressed in a closed fo rm, although the num ericalevaluation of probabilities involving 9 n) using (7)-(9)is straightforward.

Determining the moments of 9 n ) s much simpler. In-deed, we immediately have

w(b) = Wll(b) * WO2(b)

E [ 9 ( n ) ] = -2R'2'(0) = - u ~ + ( w )dUT RS= 27:.



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BOVIK et al . : AM-FM ENERG Y DETECTION 3241

Thus the expectation of the process 9 ( n ) is twice thevariance of the process h (t)-which is decid edly positiv e.No t so agreeable (although not surprising) is the fact thatthe mean of 9 n) ncreases linearly with the spectral en-ergy variance of n ( t ) . For a signal immersed in a signifi-cant noise element, this may pose severe problems unlesssteps are taken to am eliorate the noise.The autocorrelation func tion R, i n ) (7) f 9 n ) can beeasily found for the case of Gaussian n ( t )by using Isser-lis's formula [131 to reduce all multiple-component mo-ments:R*(,,)(T)= 4[R'2'(0)]2 + 3 [R'2'(~)]2+ 4R"'(7)R'3)(7)

+ R(7)P4)(7) . (1 1)Hence from (10) an d (1 1 )

Var [9n)] = 3 (O)]* + R (O)I?'~' 0)7j, 0 2 + ( w ) dwIj, + ( U ) dw j w4+(w) dw= 37; + &;. (12)Thus, the variance of 9 n ) also increases dramaticallywith the presence of higher frequencies in the process n ( t )Furthermo re, the inequality [ 111

[R(2)(0)]2I (0)R(4)(0)gives interesting bounds on the variance

4[R'2'(0)]2 I ar [ 9 ( n ) ] I R(0)R'4'(0) (13)or equivalently

E2 [9n)] 5 Va r [\k (n)] I Va r [n] Var [ri]. (14)Since the lower bound in (13) is just E2 [9 n) ] , hen,

although the expectation of 9 ( n ) may be positive, it ispossible that 9 ( n ) may take negative values, which ishighly undesirable. Not only does this suggest difficultyin evaluating the ES A of a signal immersed in noise, italso comp licates the interpretation of 9 n) s energy. Forthese reasons, positivity of the output of 9 as been ex-plored in deta il in [8], where sufficient conditions for pos-itivity are given for narrowband AM-FM signals havinglimited amounts of amplitude/frequency modulation, andin [141, where necessary and sufficient conditions aregiven in terms of local geometric (convexity) propertiesof the signal.Note that for an ideal bandpass process ( i . e . , +(a)=0 whenever IwI $ [aI, 2]), the following bounds [from(1 3), (14)] hold

In the extreme case of a monochromatic processR ( 7 ) = A2 cos (w07) , (15)

it easily follows thatE [9n)] = 2A2wi

Va r [9(n)] = E' [9(n)] = 4A4w40.Lastly, in evaluating the ESA (4) and (5) the energiesof signal derivatives will also be required. Th us, the basicstatistics of 9(n) re of use. From the preceding discus-sion, it is easily established that

E [ 9 ( h ) ] = 2R'4'(0) (16)and

( I r ) (7)= 4 [R'~)011 + 3 [P4)7)] + 4R ' ~ ) R ' ~ )4+ R'2' (7) '6'(7)hence,

Var [9h)] = 3 [R'4'(0)]2 + R'2'(0)R(6)(0). (17)The cross-correlation properties of 9 n) nd 9 11) can alsobe developed with a little effort. The cross-correlation is

E [ 9 ( n ) 9 ( h ) ] = -8R'2'(0)R(4)(0),the cross-covariance is

c o v [ 9 ( n ) , ( h ) ] = -8R'2'(0)R'4)(0), (18)and the correlation coefficient between 9 n)and 9 h) is

R(0)R'4'(0) +[Ri2' 0)12p [ * ( n ) , 9 ( h ) ] = 4 * [9 + 3

Thus, for the case of monochromatic process (15) ,an d 9 h ) become linearly relatedP [ * ( n ) , +(h)l = 1.

- 1 1 2

111. ENERG Y F BANDPASS-FILTEREDM-FM SIG NALSI N NOISE

In this section the effect of additive noise on the re-sponse of the Teager-Kaiser energy operator applied toan AM-FM signal of the form ( 1 ) is analyzed. In partic-ular, the degree to which noise effects can be amelioratedby bandpass filtering of the signal-plus-noise process isstudied. Thus, consider the noise-corrupted AM-FM sig-na lf ( 0 = s t ) + n (0 (21)where the deterministic signal s ( t ) s given by (1) an d n ( t )is a zero-mean, W SS Gaussian random process with au-tocorrelation function R (7 ) and power spectral density

+ ( U ) as in the preceding section.Rather than studying the behavior of the energy 9 f)



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3248 IEEE TRANSACTIONS ON SIGNAL PROCESSING, V O L . 4 1 , N O . 12, DECEMBER 1993

of the combined process ( 2 1 ) , instead consider a moregeneral bandpass-filtered version of (21); \k(f) may beanalyzed as a special case using limiting argu ments. Th us,define the linear bandp ass filter with impulse respo nse(22),( O = 2hUW sin (wet),

center freqency U,, and frequency response1JG,(w) = 7 H,(w - U,) - H,(w + w,)l (23)

where1H o ( w ) = - ( w / & (24)&

H ( w ) E L 2 ( R ) s the frequency response of a low-p ass fil-ter with impulse response h : R + R , an d U > 0 is a pa-rameter that scales bandwidth and center frequency. It isalso assumed that h (t) s even-symmetric , so thatG,(O) = 0. (25)

Define the kth-order spread of H ( w ) about the value w =fa o be [ j (U k ) 2 k ( H ( ~ ) ) 2w]l '2* (26)$) (a )= 2" Rwhich is minimized at a = 0

a

= spectral energy variance of H ( w ) .Note also that

VQ ( a ) = a V p ( a / u ) (27)so the bandwidth of H,(w) is U times the bandwidth ofH ( w ) . The filter energy, howe ver, is constant across scalesand is assumed to be unity1 IH,(w)(~w = I H( w) I 2 dw = 1. (28)2" RFor sim plicity, it is assumed that for each combination ofcenter frequency U, and parameter 0, he positive andnegative frequency components of G,(w) do not overlap

I G,WI2 = I H,(w - + IHo(o+ w,)I2. (29)Now denote the filtered signal-plus-noise combination

f,@) = s,( t ) + nu@> (30)where s , ( t ) = s ( t ) * g,(t) is the filtered signal and n , ( t )= n ( t ) * g,(t) the filtered noise process. The remainderof this section is dev oted to analyzing the system depictedin Fig. 1 . In S ection 111-A, approximate expressions arederived for the energy \k (s,) of the filtered sign al; and inSection 111-B, for the mo me nts of the energy \k (nu)of thefiltered noise process. In Section 111-C, the statistics ofthe energy \k(f,) f the filtered combination are ana lyzedutilizing the results of S ection s 111-A and 111-B.

&svPF Tycfo)n

Fig. 1. Diagram of basic single-band energy operator

A. Filtered Signal ApproximationsAn important approximation is made throughout thispaper: if s ( t ) = a ( t ) cos [+ t)] is input to a linear systemwith frequency response G,(w), then the response s , ( t )can be approximated by

s^ 0)= a ( t ) Go [U; (01 co s {+ 0)+ L G, [U; (011 (3 1)In the case of a monochrom atic s ignal, i . e . , a s ingle co-sine, the approximation (31) is exact; indeed, (31) maybe regarded as a quasi-exten sion of the concept of the ei-genfunctions of l inear sys tems. The approximation (3 1)is also exact if g, ( t ) s a unit impu lse function. Otherwise,the error may be bounded according to the following re-sult; the proof is supplied in Appendix A. F irst define,1 12

(32)

Theorem 1 : L et &(t) = I s , ( t ) - s^,(t)I, where S,(t ) isgiven by ( 3 1 ) . T hen ,~ s ( t )I amax ~ 2 ( g o ) ~ ( w i ) 2 A l ( g o ) . 6 ( a ) , (34)

where amax supt I a ( t ) . 0Thus , Theorem 1 bounds the error in terms of the con-centration of g , ( t ) in time (expressed as even momentsof I gu ( t ) 2 ) , and the smoothness of the AM and F M func-tions a ( t ) and wi ( t ) expressed as Sobolev 2-norms [4]. Thebound (34) has another useful interpretation in the specialcase where a ( t ) and w i ( t ) are bandlimited to the frequencyintervals [ - w u , U,] and [-U$, J, respectively [8], [15],[16]. In this case 6(a)I w,I arms nd 6 ( w i )I w + J wJrmS,where arms= ( j a2)lI2 nd similarly for ( w ; ) ~ ~ .Theorem 1 gives additional useful approximations forthe derivatives of the respo nse s , ( t )

+ ( t ) + L G,[w;(t)]+ k - . (35)"1Here, (35) is assumed for k = 1 , 2 , 3. T he error in theseapproximations can be eas i ly bounded by us ing Theorem1; simply stated, the validity of (35) requ ires that the de-rivatives of g,(t) of order up to 3 must be of short dura-tion, in the sense that A,(&) , A, , , (&) , A,( g,) be small ,for m = 1 , 2 given by (32).

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BOVIK er a l . : AM - F M ENERGY DETECTION 3249

Using reasoning similar to that in the development ofTheorem 1, the following approxim ations for the energy\k of the filtered signal component s , ( t ) and its derivativewill be essential

(36)(37)

% (s,) = a2(t) ; ( t ) - 1 G, [wit ) ] .%(iJ = a2(t)w4(t) - I G,[wi(t)]12.

Note that (36) and (37) are exact for a monochromaticsignal s ( t ) = a. co s (uot), here a,,, uo are constant. Amore general (non monoc hromatic) justification for theseapproxim ations is given next, in Theorem 2 . First define,

Theorem 2: Le t & ( t ) = I*@,) - %(s,,)l, where$(sJ is given by (36). Then,

E* (t) 5 (amaxI2 * [ ~ u 2 ( g u >+ 2 ~ 2 ( g o )+ 2 2 , ~ 2 ( g u > 1 S(u i>+ 2 ama x .+ 2 i u A I W l * S(a>. 0 (39)

~ l ( g u )+ 2uAl(gu)

Theorem 2, while intuitive in view of Theorem 1, re-quires separate proof (given in Appendix B ). However,both in Theorem 2 and the approximations (35)-(37), theerror bounds are reduced by selecting g o ( t ) o be of shortduration and to have derivatives of up to order 2 of shortduration. Slightly looser but intuitive bounds also applywhen a ( t )and/or w i ( t ) are bandlimited (34).

Rb2k(0) nd Rgk(a) re given by (41)-(43). Th en,

(44)where amax supw 9 ( w ) I an d V$ is given by (26). 0Careful examination of (44) reveals that, for arbitrarya , he validity of the appro ximation (41 ) and (42) requirestwo important assumptions. First, the spread

(45)must be small; this occurs when a = U, and when thebandwidth of H ( w ) is small. Thus, the filter G,(u) us tbe narrowband a nd a must fall n ear the center of the pass-band of G u( u) , i .e . , near the filter center frequency u,.Secondly, the quantity

(46)must be small; this requires either that a fall close to thefilter center frequency a,, r at least that a fall within thefilter passband--ifthe in-band amplit ude response 1 G, (U)\is approximately flat. F ig. 2 illustrates these requiremen tson (45) and (46).In the best case cr = U, , the error bound (44) becomes

B . Filtered Noise ApproximationsDenote the autocorrelation of the filtered noise processn , ( t ) in (30) by R , ( 7 ) with associated power spectral den-sity @,,(U)= I G,(w)( *(a). f interest are the values

that the (2k)-th derivatives of R,(T) take at the originR S k ( 0 )= - ( j u ) 2 k ( G , ( ~ ) I ~ dw. (40)

An important, but nontrivial approximation will rou-(41)

which relative to the approximation R b 2 k ( ~ , )iven by (42)can be made constant across frequencies, for each valueof k , by varying the bandwidth parameter (J directly withthe center frequency U,. This observation motivates theconstant-Q, o r wavelet scaling property of the multibandimplementation described in Section V . Of course , smallbandwidths will further reduce the approximation error,particularly when using (42) to approximate high-orderderivatives of R , ( 7 ) at 7 = 0.Specific forms of the approxim ation (41) and (42) willbe of interest. In the sequel, whenever analyzing the fil-tered signal-plus-noise f u ( t ) at time t , we will use (47)

2T Stinely be used

RL2k 0) = R$2k)a),where for a E R we define

= ( - 1 ) k a k I Gu(a)12ru (42)with

1r, = - 1 I c ( w ) ( 2 0 ( w ) dw (43) with a = u j ( t )2a R G(%)the concentration of noise power within the passband of RSk(0 ) = RL2k wi t ) ] .the filter g , ( t ) . The veracity of the approximation (41)requires some assumptions that are made clear in Theo-rem 3 (proved in Appendix C ).Theorem 3: Let E $ j ( a ) = 1 Rb2k(0)- Rb2(a)I, here

This is a novel t ime-varying approximation to the auto-correlation of the filtered noise process. In making suchan approximation, there is a tacit assumption that wher-ever f , ( t ) is being analyzed, it is being done so with a



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3250 IEEE TRANSACTIONS ON SIGNAL PROCESSING, V O L . 4 1 , NO . 12 , DECEMBER 1993

G, 0f;GLr(U)

t b

Fig 2 The validity of the approximation (41) and (42) to the even deriv-atives R,(T) t T = 0 requires both that 101 - w , be small and that 1 C,(or)- G,(w,) l be small These requirements amount to specifying a narrow,flat passband for the filter G,(w)

0 a 0, w

filter that is concentrated in the vicinity of the instanta-neous signal frequency w, t ) .When analyzing filtered noise only, it will be conve-nient to use (47) with CY = U,. This will be used to com-pare the responses of inactive channels (stimulated bynoise only) with active channels (stimulated by signal-plus-noise) in the multiband analysis of Section V.It is desirable that the contribution of the noise elem entto the energy operator output \k (f,) be minimized as muchas possible. By using (4 8), approx imations for the statis-tics of the energy of the filtered noise process can be ob-tained. From ( lo), (16), and (48):E [\k(n,)] = -2Rb2(0) = 2w;(t) I G,[w,(t)]121, (49)E [ * (h, ) ] = 2Rb4(0) = 2w:(t )( G, [~ , ( t )11~I , . (50)

Similarly , from (12 ), (17), and (48)Var [*(nu)] = 3[Ri2(0)I2 + Ru(0)Rb4(O)

= 4 w ~t ) G, (t)l 4r: (51)= 4 4 t ) G, (01 i4r;. (52)

Va r [\k (ti,)] = 3 [Rf (0)l2 + Rf (0)Rb6(0)Clearly, both the means and the variances of * ( n u ) an d* ( n u ) are decreased by making F, small , i . e . , us ing asufficiently narrow filter passband. But regardless of thebandwidth, from (49)-(52)

Var [ * ( nu l l = E2 [*(nu)]Var [*(ti,)] = E2 [ * ( n u ) ]

(53)(54)

which is important within the context of taking ratios ofthese types of quantities. In view of (53) and (54), it isnot unlikely that *(no) o r * ( n u ) will take a value nearzero, or a negative value. In the case of a low signal-to-noise ratio, computing the ESA subsequently becomesunreliable, as found in the next section; fortunately, theuse of an appropriate filtering strategy can greatly im-prove the predicted results.C . Filtered N oisy AM - FM Signal Approx imations

First note that*(f,) = \k(s,) + * ( n u ) + 2S,ti, - Su f i u - f u n ,

the trailing terms of which are zero mean. Using (31),(35), (49), and (5 0), the expected values of the energy \k

of the filtered signal-plus-noise can be approxim ated:E [* f u l l ( t ) Gu[oi t) I 2[a2(t>+ 2ru1 (55)

(56)[* ?,>I = 4 0 G , U ;@>I *[a2@)+ 2ru1.It should be reemphasized that the validity of (55) and(56) utilizes the approximation (48) which requires theassumptions depicted in Fig. 2, i .e. , that the instanta-neous frequency q ( t ) t t ime t falls well within the pass-band of the filter with frequency response G,(w), and thatthe passband is nearly flat. Moreover, within the currentanalysis, without this assumption the signal component ofthe energy signals 9 f,) and * f,)will also become neg-ligible or van ish.Although the ratio of (56) and (55) appears to be anappealing ap proximation of the ex pected value of the ESA(4), such an approximation must be carefully justified,particularly in view of (53) and (54). From (12), (17),

P [ * ( tu) ,(?,)I r , / [ a2 ( t ) + r,i (60)where p denotes correlation coefficient.We shall now reexamine the relative values of the ex-pected values and variances of the energy signals, onlythis time following the filtering operation. Defining theinstantaneous signal-to-noise ratio a s a ratio of in stanta-neous signal pow er to average j i l tered noise power:

SNR,(t) = a 2 ( t ) / r , ( 6 1 )we then find that

an d

In passing, we also note that

Thus for sufficiently large SN R at t ime t , and if all as-sumptions regarding the filter are satisfied (underlying thekey approximations ( 31 ) , (35) and (48)), the energies9 f,) an d \k (fu) are approximately uncorrelated; if theSN R at time t is small, they have nearly a linear relation-ship. In view of (19) and (20), this is most accurate if thebandwidth of the system is sufficiently small.



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IV. COMPUTINGHE ES A IN THE PRESENCEF NOISEIn this section, we justify the use of the ES A (4) and(5 ) in the presence of noise, where filtering is applied toreduce the noise contribution. F ig . 3 depicts the ESA sys-tem to be analyzed; compare with Fig. 1 . Extensive useis made of the approximations for the output m oments ofthe energy operator * developed in the preceding section.

In addition, statistical justification of the combined filter/ESA system is developed using approximations for theoutput moments of the ESA. As will be seen , it is difficultto develop statistical ESA approximations unless it is as-

f, -

Fig . 3. Diagram of basic ESA with f iltered sign al-plus-noise input

Howev er, note that_ _sumed that the signal-to-noise ratio is large.A. ESA Approximations for Large SNR

Suppose that SNR, (t) >> 1 is sufficiently large that theratios (62) and (63) are small; for exam ple, using the value0. 1 to signify "small" requires [from (62)] that SNR,(t)following second-order approximations to the expecta-tions of the ESA (4) and (5) are useful [11, p. 2121:

and(69)V ar [d2(t)]E2 [d2( t ) l

4 [5 SNR,(t) + 13[SNR,(t) + 212[SNRi(t) + 12 SNR,(t) + 412> 36.97. U nder this assumption and using (57)-(61), the 5:

an d

Also for the second-order approximation for the variances of the ESA:

4[SNR,(t) - 115: [SNR,(t) + 212

At first glance it may appear that the variances (66) and(67) of the ESA (4) and (5 ) increase dramatically with thefourth powers of the AM and FM funct ions w i t) an d a (t).both of which become negligible at reasonably high val-ues of SNR,(t) . Fig. 4 plots (68) and (69) versus SNR,(t)> 10. As is apparent from the plots, both ratios fall off



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0.

0.010 20 30 40 SO

S N R u ( tFig. 4. Plots of the ratios (68) an d (69). For small values of the plottedfunctions, the approximations (70) an d ( 7 1 ) ma y be considered valid.

fairly rapidly; (68 ) falls to 0.1 at SNR,(t) = 26 (28 db) ,whereas (69) falls to 0.1 at SNR,(t) = 34 (31 db).It subsequently fo llows from (64)-(69) that for SNR,(t)sufficiently large we may confidently assume that&?(t)= w?(t) (70)Li2(l) = a2(t) G, [ U ,(t)] . (71)

If (70) is valid, then in (71), w,(t)may be estimated firstusing ( 5 ) and used to compute G, [a,t ) ] .Another s impleapproach is to use filters with approximately flat in-bandresponses, so that

I GU[W,(t)ll = I G u ( U 0 l .Of course this is just another statement of the constraint(46), which is important in making the key app roxim ation(48) accurate, and is used in the simulation s given in Sec-tion VI.

V . M U L T I B A N DIL T E RINGND ESAEquations (31) and (35) imply that unless the instanta-neous frequency wi ( t ) alls within the passband of the fil-' t e r G,(w), the energy signals \k (s,) an d \k (S,) will becomenegligible or vanish. Th erefore, by the selection of an ap-propriate filter (passband) at time t , it becomes possible

to isolate an AM-FM signal component and to compute itsenergy 9. oreover, it becomes possible to reject noisecomponents not falling within the vicinity of the desiredlocal AM - FM component.These very useful observations about locally coherentsignals (signals with a strong local modulation structure)can be regarded as extend ing the well-known global prop-erties of filtered sinusoidal signals immersed in noise.With it comes a caveat: in ord er to isolate the local mod-ulation energy of an AM-FM signal component of theform (1 ) in the presence of noise, it becom es necessary toestablish a filter passband in advan ce, eit her by estimatingthe local center frequency by a tracking procedure , or byutilizing a bank of bandpass filters that sample the fre-quency domain sufficiently densely. This is particularlyimportant since the instantaneo us signal frequen cies maysweep the signal spectrum, implying that multiple filterswill be involved, at different times, in capturing the mod-ulation signal s ( t ) . Here, the second approach is ex-plored. In particular, criteria for channel filter selectionand for the design of a multiband sampling of the fre-

quency dom ain are discussed in depth. Significantly, it isfound that uniform (worst-case) performance across fre-quency bands is naturally ensured through the use of aconstant-Q or wavelet-like scaling of the channel filters.Fig. 5 depicts a block diagram of the m ultiband energyoperator system that is proposed and analyzed in de tail inthis section. The signal f ( t ) s divided into multiple pass-bands using M filters having impulse responses g, ( t ) , re -quency responses G, ( U ) , with associated cen ter frequen-cies U, and bandwidth parameters U,, producing outputsfm( t ) ;m = 1 , * * , M . Th e filters G,(w) are all assum edto have the form (22)-(29). Section V-A discusses the de-sign of the individual channel filters G,(w), based on theobservations made in the preceding sections, while Sec-tion V-B discusses the multiband implementation. Fol-lowing filtering, the process of energy demodulation us-ing the operator \E is applied to each outpu t. Although notmade explicit in Fig . 5 , at each instant t the response hav-ing the maximum normalized energy

is used as the analysis signal, i . e. , i t is input to the ESA .In this w ay, a filter is made av ailable with large responseto the signal component s ( t ) ,by ensuring that the instan-taneous frequency wi(t) alls within one of the filter sub-bands motivating the multiband implem entations de-scribed in Section V-B. The efficacy of the operation (72)is examined in Section V-C. Thus, a filter g , containingthe instantaneous frequency within its passband will likelybe used in computing the ESA, yielding a stable, noise-resistant result. In this case, depicted by the do tted linesin Fig. 5 , the behavior of the ESA is well described bythe results of the previous sections. Section V-C studiesthe strategy of utilizing the chan nel with maxim um energyoutput \k*(t)as the analysis channel. A similar strategywas employed using Gabor wavelets in a digital imageanalysis application in [4], although the actual filter re-sponses, rather than computed energies, were used to de-termine the analyzing channels.The strategy depicted in F i g . 5 presents a novel andinteresting approach to isolating an AM-FM signal thatmay be modified. T he approach used here yields the chan-nel having the maximum product of channel response andTeager energy. This means that the magnitude of the in-stantaneous frequency 1 wi ( t ) is deem ed equally imp ortantas the amplitude function (a(t)l n detecting the signal,which, while not usual, does have application. However,changing (72) to agree with more con ventional criteria isstraightforward. For example, the ESA may be computedfor each channel, a nd the analysis channel selected o n thebasis of the maximum estimated Iw,(t)( if it is desired tofind high-frequency AM-FM functions), or on the maxi-mum estimated (a(t)l if it is desired to find high-ampli-tude AM -FM fu nctions, which is the most likely goal). Asimple approach which well ap proxim ates the latter strat-egy is to modify the criteria (72) by normalizing also by



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Fig. 5 . Block diagram depicting multiband filtering and energy separationof a noisy AM-FM signal. The dotted lines indicate a single subsystem asanalyzed in the preceding sections, which may be regarded as having thelargest energy reponse 9* t time f.

the squared chan nel center frequency:

eling of the statistical problem be accurate, the spectralenergy duration V#)(O) iven by (27) should also be takensmall. Ignoring momentarily constraints on the deriva-tives of h, , simultaneously optimizing these criteria, byfor example, minimizing the product(73)(h )v#' (0 )

inevitably leads to the minimum uncertainty filters of theform [4]

h,(t) = (2n)-i/4J;; exp {-(nt) '/4 )

which are unit-energy Gaussian functions. Th e functions(22), which are frequency-shifted Gaussians, are then thereal-valued Gabor functions. A couple of points need tobe considered before taking these as one possible basisfor a multi-band implem entation. First, in (29) it was as-H,(u f U,) not have any spectral overlap; clearly, thecriteria. However, by taking the filter bandwidths suffi-

q fm (01q * ( r ) = maxI c m c M I w m G, '

Whichever method is used, the strategy sumed (for simplicity of exposition) that the shifted filtersaffords superior predicted and dem onstrated performan ce.and also novelty, of exposition.In this paper, (72) continue to be used fo r simplicity, functions (74 ), which h ave infinite support, violate thisA . Filter Design

Before exploring the design of the multiband imple-mentation, criteria for selecting the individual filters com-prising the signal decomposition are explored (with themultiband implementation in mind ). There are a varietyof criteria, som e of them conflicting, that affect the designof the individual filters, o r more precisely, th e design ofthe low-pass equivalent filter H ( w ) in (22)-(24). The ar-chitecture of the multi-band implementation will be dic-tated by a set of frequency translated and dilated versionsof the bandpass filter G,(u) defined by (22) and (23) .As will be seen, both low-uncertainty filters [4] nd fil-ters with a flat in-band responses have certain advan tages.Indeed, Theo rems 1 and 2 and (35) suggest that, in orderthat the modeling of the deterministic problem be accu-rate, the channel filters be selected such that the temporalenergy durationsAp p . k ( g o ) Ap z g u , k = 0, 1, 2, 3I d k 1

be made as small as possible, for p = 1, 2 , where A p( )is given by (32). It is enough to consider p = 1 . It is notdifficult to state this in a simple way as constraints on thelow-pass equivalent filters h, . With g, an d h , related by(22), it is a simple matter to show that A , ( g , ) I , ( h , ) ,and that further, for each k = 0, 1, 2, 3, there exists con-stants bk,r = 0 , * * , k , such thatx

Al , k ( g u >5 bk.r Al , k ( h u ) .r = OThus, it suffices to take the temporal energy durations ofh , and its low-order derivatives to be small. However, bythe arguments posed by T heorem 3, in order that the mod-

ciently narrow, say one octave, the overlap will be ex-tremely small [4]. In any practical implementation, thefilter bandwidths will be effectively limited to achieve(29). Of course, there exist other low-uncertainty filtersthat have strictly finite support, such as prolate functions[15], which could be used. The difference in perfor-mance, however, would likely be microscopic. Finally,in defining low uncertainty filters that mini mize the prod-uct (73) the durations A l , k ( h o ) f the derivatives have beenignored. Howev er, the low-order derivatives of Gaussiansalso have good time/frequency localization propertiesThere is another important criterion to be considered.Theorem 3 suggests the constraint (46 ), which implies thatwhen a = w,, it is desirable that G , ( a ) = G,(u,). Inother words, the passband of H ( w ) should be approxi-mately flat. Of course, this criterion conflicts with thechoice of a Ga uss iad Ga bo r configuration, since theGaussian frequency characteristic rolls off quickly withinthe passband, and slowly at the transition, rather thanhaving an abrupt transition from a flat passband. While itis an interesting problem to consider the design of filtersthat are both low uncertainty and also have relatively flatpassbands, there is no immediately apparent procedure forsuch a design. In any case, the error bounds described inthe preceding represent worst-case performances. T here-fore , simple multiband filter configurations a re considerednext, consisting either of Gabor functions or ideal (flat)

bandpass filters. In either case the use of a multiscale,dyadic wavelet-like filter bank configuration is motivated.

1171.

B. Multiband DesignOnce a basic low-pass filter H ( w ) , or bas ic bandpassfilter G,(u ) has been selected , the next step is the choice



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of a tesselation of the frequency axis by a set of multiplefrequency translations and dilations of G o U ) . Since in thisdevelopment, signals are being analyzed on continuousdomains, the practical assum ption is made that the systemof interest produces signals that fall within a specific (pos-sibly wide) band of frequencies; of course, any real-worldsystem approximates this hypothesis. Thus, assume theoverall bandwidth of the analysis sy stem to be fixed at anupper limit Q,, . e . , the m agnitude I S ( w ) l of the Fourierintegral s ( t ) is guaranteed to be n egligible for all ( U ( >

Many argum ents have been put forth for the use of time/frequency signal analysis windows having a constantwidth (localization) on a logarithmic scale [181-[2 1 1 . O neof the simplest, yet most com pelling observations, appliesto the local analysis of monochromatic functions; lowerfrequencies require larger analysis windows in order tocapture the signals period. Since we are dealing with sig-nals that may be termed locally quasimonochromatic,similar arguments apply. However, within the currentmodeling framework, the bound in Theorem 3 suppliesadditional motivation, as discussed next.The important approx imations developed for the localnoise response in the presence of an AM-FM s ignal , a l lof which are based o n (48) , can be improved by designingthe basic filter H ( u ) according to certain prescriptions, asin Section V-A. How ever, the modeling error bound (47)is also scaled by

Q C .

2k -/fl [v$k-I)(?)Tso it is necessary that the channel bandwidths be small,but more importantly, in order to maintain consistent(worst-case) predicted perforamnce across the j i l terchannels the error bound can be made con stant across thespectrum by taking the b andwidth (d ilation) parameter U,for each channel G , ( u ) to be the inverse (up to a multi-plicative constant) of the center frequency (translation)parameter U , :

Gm_ -cons tan t ;U m

m = 1, . . . M - Iwhich implies a constant logarithmic spacing betweeenthe filters. In other words, constant+ (wavelet scaled)filter banks will provide uniform worst-cast perfor-mance across channels.Certainly, uniform (worst-case) performance acrosschannels is a design criterion that may be modified. If itis desired to achieve improved performance over somesub-band of frequencies, then this can be accomplishedby narrowing the filters bandwidths (using more filters)ove r that frequency range . Nev erthele ss, it is significantthat the constant-Q property falls naturally out of the anal-ysis. Given the specific goal of estimating a (t) and w i ( t ) ,uniform performance for sm all or large values of o,(t)sobviously desirable unless there is a specific reason oth-erwise. In any case, the (worst-case) performance fromeach channel will depend directly on its Q.

Thus, regardless of the exact filter specification, definethe set of M center frequencies for the subband filters (no-tice the indexing from higher frequencies to lower fre-quencies) by a dyadic relation, yielding one-octave sep-aration:U , = 3 - 2 - ( m + 1 ) ~ 2 , ; = I , * ,M - 1. (75)

The baseband filter will be taken to be an unshifted (low-pass) filter:

while the subband channel filters are given by1G , ( u ) = j J Z El(?) - H(*)];

m = 1 ;* - , M - 1. (77)Gabor Wavelets : The real-valued (sine) Gabor waveletsdefined by (22), (28), (74) an d (75) are of the form

r---

The bandpass functions (78) satisfy the usual wavelet ad-missibility conditions [181-[2 13: complete orthonormaldecomposition of any signal f ( t ) E L2(R) can be con-structed using a basis consisting of frequency translatesand dilates of (78) , from which the signal may b e exactlyreconstructed, in principle. However, from a practicalperspective, in the application being considered here,where highly specific signal components (AM-FM sig-nals) are being extracted in the presence of significantnoise, it is most important that the frequency domain beadequately sampled by the filter set, rather than ensuringthe perfect reconstruction property of the filter responses.Nev erthele ss, if the filter tesselation is defined such thatthe filters intersect at half-peak, then both criteria are sat-isfied. One-octave (half-peak bandwidth) Gabor filtersachiev e this prescription exac tly. In this case the filterbandwidth parameters satisfy

(79)We choose uM = 2uM- for the baseband filter, in orderthat the filters G M - ( ~ ) , M( u ) intersect at the half-peakresponses. Fig. 6depicts a decom position of the unit fre-quency interval [0, 11 (i.e., Q, = 1) using five Gaborwavelets defined by (75) , (77)-(79) .Litt lewood-Paley Wavelets: Littlewood-Paley wavelets



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0.00 0.so I 00w-Fig. 6. Multiband Gabor wavelet f ilter decomposition of the unit fre-quency interval [0 , 11 into f ive subbands of constant (unity) octave band-width (except the baseband).

0.0 0. 5 1 oFig. 7. Multiband f ilter decomposition of the unit frequency interval [O ,11 using f ive ideal subbands of constant (unity) octave bandwidth (exceptthe baseband).

w-

121, p. 1051 satisfyin g (28) have the form (1 ) and assum e that the instantaneous frequency wi ( t ) allswithin the passband of the filter G ,, where for simplicitym # M . Then from ( 5 5 ) ,and defining SNR, ( t )using (61)with U = U,, th e normalized expected signal-plus-noiseenergy is approximatelyG ( w ) = J!& Z ~ - I , I ,(7)

where ZLPl, s the indicator function of the interval [- ,11. Thus (80) represents the ideal case of maximally flatfilters also having good spe ctral localization , but ripple inthe time domain. Although not realizable in practice, thebandpass functions (80) also satisfy the usual wavelet ad-missibility conditions [181-[2 11. He re the prescriptions forperfect reconstruction and complete coverage of the sys-tem bandwidth are in agreement. Thus, defining centerfrequencies U, according to ( 7 5 ) , 76) and the bandwidthparameters according to

= 2-m+ I Q ~ ; m = 1, e . . , M - (81)m = -

For every other channel G q , q # m , from (36) and (lo),and using (41 ) with a = w q , the normalized expected sig-nal-plus-noise energy is approximately

with rq iven by (43). Clearly, in the absence of noiseE I* fm>I >> E [* f q l l

if the filters are properly designed. Otherwise the ratio ofexpected energies:J

E [*(fq)l S N R ~ ( ~ ) . 1 Gm ( o m ) 1nd UM = 2uM- I (baseband filter) once again yields a setof unity-octave filters with unity octave spacing. The E [*(fm)l [ S N R m ( t ) + G m [ w i ( t ) l(positive frequency) filter passbands are the intervals[ 2 - " ~ ~ ,- m + 1 ~ C 1 , = I , . , M - 1 . Fig. 7 depictsa decom positio n of the unit frequency interval [0, 11 usin gfive Littlewood-Paley defined by ( 7 5 ) 7 (77)9 The first term inside the brackets is clearly negligible. T heas ideal is an analytic conv enienc e. In practiv e, the filter to-noise ratio S N R , ( ~ )s large. H ~ ~ ~ ~ ,t is importa nt that(801, and (81). Of course, modelling the bandpass filters second term inside the brackets will be sm all if the signal-characteristics may be approximated using, e.g. , quad-rature mirror filters 1221 related su bband tec hni que s 1231 the bandwidth of each channel filter be taken small toachieve large signal-to-noise ratios across all of the chan-or standard bandpass filter design procedures [26].Comments: Of course, the use of one-octave filters withone-octave separations is somewhat arbitrary in the pres-ence of significant noise, where the signal-to-noise ratiomay be low , it may be advisable to use a large number offilters having narrow bandwidths, such as (1 /2)-octavefilters with (1 /2)-octave separations.C. Channel Selection by Maximum Energy Response

In this subsection the technique (72) of selecting ananalysis band for the extraction of signal frequency mod-ulations by finding the channel having the largest energyresponse is studied. Suppose that the signal is given by

nels, in order that the strategy of using the channel withthe maximum normalized response (72) as the analysisband will be most effective. H ~ ~ ~ ~he scaling by thefactor does indicate that the ESA will be sensi..tive wh& it is desired to detect low-frequency AM -FMsignals immersed in high-frequency noise. This is be-cause of the strategy (72) that is used. As discussed below(72), it is possible to ameliorate this effect by computingthe ESA for every channel.The only other question concerns the behavior of theoverall ES A as the instantaneous frequency w i ( t ) sweepsacross channels. Conceptually, there is a discontinuity inthe flow of the algorithm when there is a transition from



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one maximizing channel to another according to (72). Inview of the abov e discussion this occurs when the instan-taneous frequency makes a transition from one passbandto another. How ever, unde r rather general conditions, theESA (4) and (5) gives continuously-varying results .In applying the energy operator f to the signal (1 ) asin ( 3 ) , it must be assumed that the signal s ( t ) is a twice-differentiable function. However, if the signal is filteredfirst, then it is more conven ient to impose additional con-ditions on the filter impulse response such that the ESAwill be regular. Ind eed, it is sufficient to assum e that thelow-pass equivalent function h ( t ) s continuously three-times-differentiable, a con ditio n clearly met by the ex-ample wavelet configurations discussed in Section V-B.Then, the energy responses f [f, t)]; m = 1, , Mwill all be continu ous functions [see (B.6)] , as wil l be theassociated en ergies f jm t ) ]necessary fo r the ESA com -putations. Since the maxima of continuous functions iscontinuous, the m aximum Teager response f * t ) will alsobe continuous. Accordingly, the associated energyf* f(t)] should also be continuous. In order to assurethis, the energy-of-derivatives could also be comp uted forevery channel and the maximum taken, although accord-ing to the approximation (56) this is likely unnecessary inpractice. Finally, since the ESA (4) nd ( 5 ) s defined interms of cont inuous mappings , the ESA responses shownin Fig. 5 will be continuous functions.Comments: The abov e analysis could, in principle, be ex-tended in a natural and interesting way by casting thechannel selection problem, currently implemented via(72), as an M-ary hypothesis testing problem. Such anapproach would lead to a probabilistic description of th eoverall multiband ESA performance as a function of thenoise, the filter bandwidths, and the number of filters.Complicating this approach, however, is the necessity ofexpressing the necessary output conditional distributionsof the Teager-Kaiser operator for a signal-plus-noise in-put. As indicated in Section 11, this problem is compli-cated even for the case of zero signal; for the case of sig-nal-plus-noise, it is much m ore difficult (Section II- C). Itis for this reason that th e stoch astic analysis in this paperhas been restricted to the co mpu tation of useful m omen tsof the energy o perator output.

VI. EXAMPLESND S I M U L A T I O N SIn this section, examples ar e provided of the predictedESA performance for a sinusoidal signal (Section VI-A)and fo r a chirp signal (Section VI-B). Section VI-C pro-vides actual MATLAB simulation results , using an ap-proximate implementation of the Littlewood-Paley filter

bank described in Section V - B .A . Sinusoidal Signal-Monochromatic Noisesinusoidal signalAs an instructive example consider the case of a pure

(82)( t ) = a0 cos (wot )

where the signal frequen cy falls in an assume d frequencyinterval: w o E [0, 11 ( i . e . , Q, = 1 for simplicity). Furtherassume that the signal is embedded in zero-m ean, add itiveGaussian monochromatic noise n ( t )with autocorrelationR (T )= A2 co s ( ~ ~ 7 ) . (83)

We shall now study the efficacy of the overall system d e-picted in Fig. 5 , using the Littlewood-Paley wavelet con-figuration (80). This example (82) and (83) is canonical,since nearly all of the approximations [except (64)-(67)]made prior to this point hold exactly, provided that wo an dol all into different frequency bands, although that casewill be considered as well. Of course, this example rep-resents the simplest possible case, where there is no in-formation in the AM-FM signals. It is an important ap-plication, since it is often of interest to cap ture a dom inantsinusoidal signal in the presence of o ther random periodicsignals(s). The example also provides insights into thegeneral performance of the ESA in the presence of filterednoise; for signals having slo wly-vary ing modulating fu nc-tions, this example is largely applicable. Of course, ifeither or both of the signal am plitude and the signal phasevary with extreme rapidity, then the approxim ations, par-ticularly those developed in Theorems 1 an d 2 , are notguaranteed to be close. One interpretation of this, how-ever, is that it is simply quite difficult to track a signalwith rapidly fluctuating AM or FM information.Assume that the signal frequency w o > 0 falls w ithinthe passband of the filter Gm(w), nd that the noise fre-quency w , > 0 falls out of band, viz. , in the band of thefilter Gq(w),where for simplicity m, # M . The casewhere the signal and noise frequencies fall into the sameband is really the same as the case where no bandpassprefiltering is applied; that is considered next. F rom (36) ,(37), (80), an d (81) the exac t normalized in-band filteredsignal energies are

(84)since the noise falls out-of-ban d.For the out-ofband energies, the filter/energy o peratorsystem responds to the noise only. The exact normalizedenergy moments are found by directly evaluating (40) in(49)-(52), (80), an d (81)

The responses of all other channels will be zero. T he ratioof (84) to (85):

makes it clear that for sufficiently large signal-to-noiseratio a&: > 2A2w:, the correct channel G,,,(w) will pro-



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I I signal frequencies (the in-band channel) will b e selectedas the analysis channel with a high probability. H owever,from (97) and (98) it may again be observed that the en-ergy operator has increased sensitivity w hen the signal tobe detected is lower frequency, and immersed in high-frequency noise [see the discussion below (72)].ES A Computation: If the correct channel is selected,then since the ratio of the square of (89) to (91), and ofthe square of (90) to (92) are both bounded below by

2" snro. .2 ~ 5 .4 2 ~ 2 2 ' 2~

W0then for reasonably high snro the normalized energies sat-isfy

' ( ' m ) = ai(2wot)2

'(m) = ai(2wot)4.

Fig. 8. Division of chirp signal ~ ( f ) iven by (86) into time intervals overwhich different Gabor wavelets from the set of five depicted in Fig. 6 willbe "active." I Gm (wm)Iv a r ( f m ) ~= 27am(2wot)s(u; 27am) I Gm ( w m > II Gm (wm) Hence, when snro is large, the computation of the ESAusing the channel that yields the maximum normalized en-

' exP [ 4 (2wotaim ] (92) ergy response will yield the desired ESA compu tation sOut-of-Band Energy Moments: Suppose q # m* ( t ) t t imet (although it is possible that gq(t) may be selected as theanalyzing wavelet). To determine the normalized out-of-band energy moments (energy m omen ts from band q),usethe approximation (41) with a = w q

and

(93)

(94)

(95)

For narrowband channel filters, these approximationsyield accurate approximations for the mom ents of the re-sponses of any channel to a pure noise stimu lus.Channel Selection: Observing from (88) that 2 w o t >(9 /8 )2 -" fo r m = m*(t), he ratios of (89) to (93) and(90) to (94) are respectively bounded below by

( 2 ) 2 3 4 - 2 m J l n 26 snro (97)

wherea;snro = -.7

F or snro arge, the in-band channel will dominate the other(noise) channels. Thus, the channel containing the local

c$(t) = (2W01)*Ci2(t)= ai

for each t E [ 1 / 2 M ~ 0 ) ,/ (2w0)] .C. Simulation Results

In order to verify the predicted results developed in thepreceding, the multiband ESA was implemented and ap-plied to several interesting signals, using the MATLABSignal Processing Toolkit [25]. In all of the examples thefilters in the Littlewood-Paley filter bank (80) were ap-proximated by equiripple (Chebyshev) FIR bandpass fil-ters numerically designed using the Parks-McC lellan al-gorithm [26]. The filters are all of order 101. As in thepreceding analysis, five channel filters were used (includ-ing baseband). The frequency resp onse magnitudes of thefive filters are shown in Fig. 9; note that unity-magnitudepassbands w ere used to simplify the computation of (72).The ESA results in the simulations were computed usingthe discrete-time Teager-Kaiser energy operato r [9];whilethe theoretical developm ent of the discrete-time operator/ESA is nontrivial (and hence not developed here), imple-mentation of it requires only minor modifications relatingprimarily to sampling approx imations that do not presentan issue in the context of these simulations.In the first simulation example, a digital implementa-tion of the multiband ESA was applied to a signal com-posed of a sum of two sinusoidal functions, on e with m ag-nitude 1 .0 and 1000 Hz frequency, and the other withmagnitude 0.2 and a 2000 Hz frequency. T he signal (andall other simulated signals) were sampled at a rate of10,000 Hz. This example corresponds to the analysis inSection VI-A , where one of the sinusoid com ponents maybe considered to be noise. Fig. 10(a) and (b) show thecomputed ESA results using the multiband ESA . Both the



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0.5I'im20 0 400"0 20 0 400

Frequency (DFT samples)(a)

"0 20 0 400Frequency ( D I T samples)

(C)

1 5

200 4000-F r eq w n c) (DFT amples)

(b)

Frequency ( D I T samples)(d )

1.5 , I

Frequency ( D F T samples)(e )

Frequency (DFT samples)( f )

Fig. 9 . (a)-(e) Magnitude responses of the filters composing the multibandimplementation used in the simulations. The filters were designed usingthe Parks-McClellan FIR equiripple (Ch ebyshev) algorithm. ( f ) Plot of allfive magnitude responses showing the small am ount of band overlap.

Time (samples) Time samples)(a) (b)

1.5 I I 3000, 1

. 0 " 1 3 2000d c$ 0.5 [ o0 000 loo 200 300 400 OO 100 200 300 400

Time samples) Time (samples)(C) (d)

Fig. 10. ESA applied to sum of two sinusoidal components with (magni-tude, frequency) = ( 1 . 0 . lo00 Hz) and (0.2 , 2000 Hz). (a), (b) Amplitudeand frequencies computed using the multiband ESA. (c), (d) Amplitudeand frequencies computed using the ES A wirhour filtering.

signal amplitude and the signal frequency of the compo-nent having the largest energy were approximated with ahigh accuracy, although a small ripple in the computed

result may be observed. By compa rison, F ig. 1O(c) and(d) show the computed ESA results without filtering. Inthis case, the ESA result is extremely poor and cannoteven be considered to be useful. Fig. ll(a)-(d) depicts asimilar result, only this time the signal is a unity-magni-tude 1000 Hz sinusoid added to a 1200 Hz sinusoid withmagnitude 0.3. Similar high-quality results were ob-tained, demonstrating the capability of the multiband ESAto provide excellent resolution, provided that the signalcomponents do not fall within the same band.Fig. 12(a) depicts a chirp signal having a 3000Hz/second sweep rate (1000 Hz initial frequency), im-mersed in additive Gaussian white noise. The signal-to-noise ratio (SNR) is 15 dB. F ig. 12(b) and (c) depict theresult of the multiband ESA computation; again, there-sults are highly accurate, although there is noticeable fluc-tuation in the estimate of the amplitude. If plotted, theestimated instantaneous frequencies could not be dis-cerned, since they fall directly on the plot of the computedresult. The root-mean-squared (rms) error relative to theideal was computed for both the AM and FM estimates,and found to be 0.039 (AM rms error) and 11.4 Hz (FMrms error). By comparison, Fig. 12(d) and (e) shows theESA computation obtained without any filtering. Thecomputed rms values were 0.77 (AM rms error) and594.17 Hz (FM rms error). Clearly, the multiband ap-proach affords a vast improv ement. Both results (with andwithout filtering) could be impro ved by som e kind of post-processing, e.g. , low-pass or median post-smoothing, but,in any case far superior results are clearly obtained byusing the multiband approach to isolate the signal infor-mation.Fig. 13(a) depicts another chirp signal, with initial fre-quency 2000 Hz and a 3000 Hz/se c sweep, only this timewith a 20 Hz amplitude modulation. The signal was im-mersed in noise as depicted in Fig . 13(b) (SNR = 15dB).The ESA results with filtering are shown in F ig. 13(c) and(d) (AM rms error = 0.046, FM rms error = 32 . 4 Hz)and without filtering in Fig. 13(e) and (f ) (AM rms error= 0 . 27 , FM rms error = 543.41 Hz). A gain, the mult i-band ESA results present enormous improvem ent over thesimple ESA computation.Finally, Fig. 14(a) depicts an AM (20 Hz) noisy chirpsignal (SNR = 15 dB) with initial frequency 2400 Hz anda 3000 Hz /sec sweep rate. Fig. 14 (b) and (c) depict theresults of the multiband ESA compu tation. The computederrors were found to be 0.044 (AM rms error) and 5 9.8Hz (FM rms error). This last example is presented sincethe instantaneous frequencies of the chirp signal compo-nent sweep well across a transition from one channelpassband to ano ther, demonstrating the sustained, contin-uous performance of the multiband ESA as predicted inSection V-C. Fig. 14(d) depicts the transitions betweenthe two highest frequency channels (labeled as channels2 and 1) that were automatically selected by the m ultibandESA algorithm according to the maximum criterion in(72). Clearly, (72) oscillates quite a bit between channels2 an d 1 near the theoretical cross-over point of 2500 H z.



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3260 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41 . NO. 12, DECEMBER 1993

1.5- 8" 00- 0 2 1B 1 -5 0.5 - p 500 - - 2 p C000.5

0 0. 00 100 200 30 0 400 0 100 200 300 400 OO loo 200 300 400 0 100 200 300 400Time (samples) Time (samples) Time (samples) Time (samples)

(a ) (b ) (C) (d)F ig . 11 . ESA applied to sum of two sinusoidal components with (magnitude, frequency) = ( 1 O, 1000 Hz) and (0.3, 1200 Hz) .(a) , (b) Amplitude and frequen cies compu ted using the multiband ESA. (c ) , (d) Amplitude and frequencies compute d using theES A withour filtering.

-, .. , . I , , . I . . I,., .. I, . , .I1 .-20 200 400 600

Time (samples)(a )

0 ' I0 200 400 600Time (samples)

(b )

01 I0 200 400 600Time (samples)

(c )5000

I

Time (samples)(d )

Time (samples)(e )

Fig. 12. ESA computation applied to the noisy chirp signal depicted in (a) . (b), (c) Amplitude and frequencies computed usingthe multiband ESA. (d ) , (e) Amplitude and frequen cies computed using the ESA without filtering.

-2 I0 200 400 600Time (samples)

(b )

0' I0 200 400 6 00Time (samples)

(C)2.5

22 1. 54

hi 3000Ug 2000 3

;;" lo00 E 1. 0.50 00 200 400 600 0 200 400 600

Time (samples) Time (samples) Time (samples)z4m0ig . 13. (d )SA computation applied to AM chirp signal depictede) in (a) immersed in noise as in (b) . ( c) , (d) Amplitudef ) andfrequencies computed using the multiband ESA . (e) , (f) Amplitude and frequencies computed using the ESA withour filtering.Authorized licensed use limited to: Nanyang Technological University. Downloaded on December 22, 2009 at 21:04 from IEEE Xplore. Restrictions apply.


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by an interchange of integrals and the Cauchy-Schwarzinequality.Furthermore, from (A.2), ( A . 6 ) , an d (2) we haveP

- 44 1 The proof is completed by combining (A.5) , ( A . 6 ) , an d(A. 8) . 0+- a(t)I dr.

j R g,(x)l l a 0 - x) exp [- jQ+(t , x) lA P P E N D I X(A5 ) The proof of Theore m 2 requires the following Lemm a:Lemma 1: Suppose that the AM-FM signal s ( t ) = a( t )cos [+ (t)] is applied to two linear systems gl : R + R an dg2: R -+ R. et

Both terms in (A.5 ) may b e further bound ed identicallyj R g,(x)l l a 0 - exp [*jQ+(t , x) l - a@) ( X

k (6 = [g ,(0 * s @)I 2 W * s (01andj, IS,(x)l lQ( t - Iexp [*jQ+(t, 4 1 - 11

= E , , + ( t ) + & s , a ( t ) . (A.6 )How ever, by again applying Taylors theorem with re-mainder

we obtain



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However, from (A.6) and (A . 8 ) we have that

where

We also have from (A.6) and (A.7) that

so that (B.3) becomesE k , + ( t ; m, n) 5 2amax[El Al(g2) + E 2 A I ( ~ I ) I ( a ) .(B.5)

proof. 0\k(s,) = ( S J 2 - s,$, = (s * g ) 2 - (s * g)(s * g).

Combining (B. l ) , (B .4 ) , and (B.5) completes theProof of 7heorem 2: From (3) and (A. 1 ) we have

(B .6 )The proof follows by applying Lemm a 1 to the first termof (B.6) with g, = g2 = g,, by also applying Lemma 1 tothe second term of (B .6 ) with g, = g,, g, = g, and bytaking the differences of the respective approximations toyield (3 5). The bound (39) follows easily by applying thetriangle inequality to the sum of the errors of these

Note: Equation (B .6 ) has also been used in [27] as analternative way of applying the energy operator to sam-pled, bandpass-filtered speech signals, as a me ans for re-ducing the discretization effects introduced by using dis-crete derivative operators to approxim ate 9.

approximations. 0



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3264 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 41 , N O . 12 , DECEMBER 1993

APPENDIXProof of Theorem 3: Clearly

where the equality in (C .3) is obtained from ( 2 6 ) .Finally,the simple inequality

. dw I + B ,where

Now A = 0 when k = 0, hence assume k L 1. Using the 2 k - Iidentities, I q , , a x v p ( T ) [2 k - I

( C . 4 )where the simple inequality JbT+II 61 + 1 is usedto obtain (C.4). The kth-order spread V$ in (C.4) is de-fined in ( 2 6 ) .

w * k - c y 2 k = (a - a ) c J k - I - r Q ! r = + CY)r = O2k - 1. C w 2 k - 1 - r ( - 4 r = O

and from (29) it follows that In addition,

2 k - 1

where the equality follows from the symmetry of H ( w ) .Simp le algebraic inequalities yield2 k - 1

CY I I k ( ( w ( 2 k - 1 ( C Y ( 2 k - 1 ) . (C.2)i = OSubstituting (C .2) into (C . 1) and applying the C auchy-Schwarz inequality yields

ll?

from (28) and (29). Combining (C.4) and ( C.5) yie lds thebound (44). 0ACKNOWLEDGMENTS

The authors thank the reviewers for their careful read-ing of the paper, an d for providing several excellent com-ments that improved the quality of the expo sition.REFERENCES

[ I ] H . M. Teager, Some observat ions on oral air flow during phona-t ion, IEEE Trans. Acous t. , Speech, Signal Proces sing, vol . ASSP-28, pp. 599-601, Oc t . 1980.[2 ] H . M. Teager and S . M. Te ager, Eviden ce for nonlinear speechproduction mechanisms in the vocal tract, NATO Advanced StudyInstitute on Speech Production and Speech Mo deling, Bonas , France ,July 24, 1989.[3 ] P . Maragos , T . F . Quat i er i, and J . F. Kaiser, Speech nonlineari t ies ,modulat ions, and energy operators , in Proc. IEEE ICASSP 91, To -ronto, Ontario, Canada, May 1991.[4] A. C. Bovik, N. Gopal , T. Emmoth, and A. Restrepo, Local izedmeasurement of emergent image frequencies of Gabor wavelets ,IEEE Trans. Informat. Theory, vol . 38, pp. 691-712, Mar. 1992.[ 5 ] J . F. Kaiser, O n a s imple algori thm to calculate the energy of asignal , in Proc. IEEE ICASSP 90,Albuquerque , NM, Apr . 1990 .[6] -, On Teagers energy algori thm and i ts generalizat ion to contin-uous s ignals , in Proc. IEEE Digital Signal Processing Workshop,New Paltz, NY, Sept . 1990.[7] P. Maragos , T . F. Quatieri , and J . F. Kaiser, On separat ing ampli-tude from frequency modulations using energy operators, in Proc .IEEE ICASSP 92 , San Francisco , CA, Mar . 1992 .

Boston, MA: Kluw er, 1990, pp. 241-261.



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[8 ] -, On amplitude and freuqency demodulation using energy op-erators, IEEE Trans. Signal Processing, vol. 41, pp. 1532-1550,Apr. 1993.[9] -, Energy separation in signal modulations with application tospeech analysis, Harvard Robotics Laboratory, Harvard University,Cambr idge , MA, Tech. Rep. 91-17, Nov. 1991.[ IO ] P. Maragos , A. C . Bovik , and T. F . Qua t ie r i , A mul t id imensionalenergy operator for image processing , in Proc. SPIE Symp. VisualCommun. Image Processing, Boston, MA, Nov. 1992.[ I 11 A. Papoul is , Probabili ty, Random Variabl es, and Stochastic Pro-cesses , First edition. New York: McG raw-Hill, 1965.[ 121 K. S . Mille r , Mulridimensional Gaussian Distribu tions. New York,Wiley , 1964.(131 L. Isserlis, On a formula for the product-moment coefficient of anyorder of a normal frequency distr ibution in any num ber of variables,Eiometrika, vol . 12 , pp . 134-139, 1918.[I41 A . C. Bovik and P. Marago s, Conditio ns for positivity of an energyoperator , IEEE Trans. Signal Processin g, to appear Feb. 1994.(151 A. Papoulis, The Fourier Integral and ifs Applications. New York:McGraw-Hil l , 1962.[I61 -, Limits on bandlimited signals, Proc. IEEE, vol . 55 , pp .1677-1686, Oct. 1967.[ I71 J . -B . M ar tens , The Hermite t r ansform-Theory , IEEE Trans.Acoust. , Speech, Signal Processing, vol. 38, pp. 1 595-1606, Sept. ,1990.[le] S . G. Mallat, Multifrequency channel decompositions of images andwavelet models, IEEE Trans. Aco ust. , Speech, Signal Processing ,vol . 37 , pp . 2091-21 10, Dec . 1989.[191 I . Daubechies, The wavelet transform, time frequency localizationand signal analysis, IEEE Trans. Informat. Theory , vol. 36, pp.

961-1005, Sept . 1990.[20] -, Orthonormal bases of compactly supported wavelets, C o m -mun. Pure Appl. Math. , vol . 41 , pp . 909-996, 19 88.(211 M. B . Ruska i , G. Beylkin , R . R . Coifman, I . Daubechies , S . G. Mal-lat, et a l . , Wavelets and Their Applications. Boston: Jones and Bar-tlett, 1992.1221 A. Crosier , D. Esteban, and C. Galand, Perfect channel splittingby use of interpolation, decimation, and tree decomposition tech-niques, in Proc. ICISS, Pa tros , Greece , Aug. 1976.1231 E. P . Simoncelli and E. H . Adelson, Su bband transforms, inJ . W. Woods, Ed. , Subbandlmage Cod ing . Boston: Kluwer, 1991,

Alan Conrad Bovik (SEO-M84-SM90) wasborn in Kirkwood, MO on June 25 , 1958. He re-cieved the B.S. degree in Computer Engineeringin 1980 , and the M.S . and Ph.D. degrees in Elec -tr ical and Computer Engineering in 1982 and1984, respective ly, all from the University of 11-l inois , Urbana-Champa ign.He is currently the Hartwig Endowed Fellowand Professo r in the Department of Electr ical andCompute r Enginee r ing , the Depar tment of Com-puter Sciences, and the Biomedical EngineeringProgram at the University of Tex as as Austin, w here he is the Director ofthe Laboratory for Vision Systems. During the Spring of 1992, he held avisiting position in the Division of Applied Scienc es, Harvard U niversity,Cambridge, Massachusetts. His current research interests include imageprocessing, computer vision, three-dimensional microsco py, and compu-tational aspects of biological visual perception. He has published over 170technical articles in these areas, and holds two U.S. patents.Dr. Bovik is a winner of the University of Texas Engin eering FoundationHalliburton Faculty Excellence Award, an Honorable M ention winner ofthe international Pattern Recognition Society Award for Outstanding Con-tr ibution, and was a National Finalist for the 19 90 Eta Kappa Nu Outstand-ing Young Electr ical Engineer Award. He has been involved in numerousprofessional society activities, including: Associate Editor , IEEE Trans-ac t ions on S IGNALROCESSING,989-presen t; Associate Editor of the in-ternational journal Pattern Recognition, 1988-prese nt; Editorial Boardmember for the Journal of Visual Communication and Image Representa-t ion, 1993-present ; S teer ing Com mit tee , IEEE TRANS ACTION SN I MA G EPROCESSING,992-present ; Genera l Cha irman, First IEEE InternationalConference on Image Processing, to be held in Austin, Texas, in Novem-ber, 199 4; Program Chairman, SPIE/SPSE Symposium on Electronic Im-ag ing , February 1990; and Conference Chairman, SPIE Conference onBiomedical Image Processing, 1990 and 199 1. He is a registered Profes-sional Engineer in the State of Texas and is a frequent consultant to indus-try and academic institutions.

pp. 143-192.1241 I . S . Gradshteyn and I. M. Ryzhik , Tables of Integrals, Series, andProducts. Orlando, Florida: Academic, 1980.[25] J. N. Little and L. Schure, Signal Processing Toolbox User s Guide ,The Mathworks , Inc . , 1992.[26] L . R . Rabine r , J . H. M cCle l lan , and T. W . Parks, FIR digital f ilterdesign techniques using weighted Chebyshev approximation, Proc.[27] H. M. Hanson, P . M aragos , and A. Potamianos , A sys tem for find-ing speech formants and modulations in energy separation, HarvardRobotics Laboratory, Harvard University, Cambridge, MA, Tech.Rep., 1992.

Petros Maragos (S81-M85-SM91), for a photograph and biography ,see page 1550 of the Apr i l 1993 issue of th is TRANSA CTIONS.

IEEE, vol. 63, pp. 595-610, 1975.

Thomas F. Quatieri (S 73-M79-SM87) . for a photograph and biog-raphy. see page 1550 of the April 1993 issue of this TRAN SAC TION S.

am-fm energy detection and separation in noise using multiband energy operators

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