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Quadrat,ure Mirror Filter Banks,M-Band -Extensions andPerfect-ReconstructionTechniques P. P. Vaidyanathan

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linear. Accordingly, it cannot be represented by a transferfunction; we should seek other means of describing thedecimator in he transform-domain. Since a 'decimdtorcauses a compression in the time-domain, we expect a'stretching' in the frequency domain. For example, withM = 2, the quantityX(ei"/') is a stretched versionof X(&").However, since X(e'"'') has a period of4rr rather than2rr,it, s not a valid transform of 'a sequence. It can be verifiedthat Y(ej") in act has two terms. The first is X(ej"/2), and hesecond term is X(-e'"/') which i s a shifted version of thefirst erm shiftedby an amount 2 ~ ) . ore formally,the input-output relation for a two-fo ld decimator can bewri tten in the ransform .domainas [31

Y(ejw)= 0.5[X(ejw/2)+ X(-ei"")]. (1Note that Y(ejw)given as above does have a period of 2rr .This is demonstrated in Fig. l( b) , (c) wherex ( n ) s assumedto be a lowpass type of signal. If the transform of x ( n ) is

I M-fold declmatorExample (M.2)

n 0 1 2 3 4 5 6 7x( n) I 4 8 -I 2 6 3 15 :::y(n1 I ' 8 2 3 e . .x'(n) 0 I 4 8 -I 2 6 3 (shlfted input)y'(n) 0 4 -I 6 - (completely new oulput)

I Example (M.2)I x( n) I 4 8 -I 2 6 3 15 :::y(n1 I ' 8 2 3 e . .n 0 1 2 3 4 5 6 7x'(n) 0 I 4 8 -I 2 6 3 (shlfted input)y'(n) 0 4 -I 6 - (completely new oulput)

I t..W-277 ? 277

Figure 1. A decimator is a system which takes in a se-quence x l n ) and produces a time-compressed sequencey[nI = x [Mn l .The compression ratioM is an integer. Thisoperation is denoted by the downgoing arrow (indicative of'down-sampling): The figure demonstrates the operationof a two-fold decimator. In general the output y[nl of anM-folddecimatar is obtained by retaining only the samplesof the input sequence occurring at times that are integer-multiplesof M. Adecimator is a time-varying system[even hough inear). The figure demonstrates this, byshowing that a shifted version of the input does not pro-duce a shifted version of the output. The effec t ofcompression in the time .domain is anexpansion [orstretching) n he requency domain. The quanti tyI;Xte'"/21 in the figure represents this, while$X[-ej"/21represents aliasingcausedbydownsampling,Forarbi-trary M , equation (21 gives the ransform of the deci-mated sequence.

not bandlimited to -7r/2, 5 w 5 rr /2 , there is an overlapof the two terms in (1 ) as shown by the shaded area in'Flg.",I(c): This.overlap is the aliasing effect, caused by un-dersampling. There i s no way we can get back he originalsignal x ( n ) from y(n), once aliasing has takenplace. Inorder to convince ourself hat the scale factor of 0.5 isrequired in the expression (I),et us imagine, as an exam-ple, that x ( n ) is the unit pulse S(n).Then X(z),is unity forall values of he argument z, hence from ( I )we haveY(ej") = 0.5(1 + 1) = 1.0 for all w , which is consistentwith the fact that y(n) = S(n) n this case.For an M-fold decimator we have y(n) = x ( M n) ,and thetransform-domain elation i s precisely an extensionof ( I ) . Instead of two terms, we now have M.terms; thefirst term is merely a stretched versionof X(ej") (by a factorof M ) ,namelyX(e'"/''"). The remainingtermsare uniformlyshifted versions of the first term (the amount ofhift beinginteger multiples of 2rr) . Thus, it can be shown that

(2)where W = e-'wj'M and z = e'". The scale factor,of 1 / M n(2) can be understood n a manner analogous to the actor0.5 in ( I ) .What happens if we decimate a highpass signal, say by

Figure 2. What. happenswhen a highpasssignal getsdecimated?Fia. 2[al reoresents he ransform X[e'"l[assumed to bgreal for simplicity) of a highpass signal. Ifthis signal is decimated two-fold, the resulting transformhas tw.0 ,components,proportional t o XtJe/"/21 andX[-e'W/21,asshown nFig. 2[bl. Accordingly, the deci-mated signal looks like a regular lowpass signal. In fact ifa lowpass signal with transformBs in Fig. 2[cl were deci-mated by a 'factor of two, the result would again have.appearance as in Fig. 2[bl. Thismeans that given thetransform in Fig. 2[bl, it is impossible t o tell whether iscame fro,m Fig. 2[al or 2[cl.

' ,

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it i s well known that sharp cutoff filters require very'highorder, are highly sensitive o quantization, andoften causeinstability,problems (i f IIR).The philosophy adopted [I], 21, [51 in the QMF bank inorder toovercome thisproblem is to permitaliasingat theoutput of the decimator, y designing theanalysis filters asin Fig. 6(b), and then choosing the synthesis filters Fo(z)and Fl(z) such that he maging produced by he inter-polators cancels the aliasing. In fact exact cancelation ispossible. This observation relieves the designer of a verystringent analysis-filter design problem.Based on the elations (2) and (3 ) for a decimator y d ninterpolator respectively, it is possible to express X(z) inFig. 5 as

k z ) = - [Ho(z)Fo(z) + H1(z)F1(z)]X(z)12+ y [ H o ( - ~ ) F o ( z )+ H I ( - Z ) F I ( Z ) I X ( - ~ )1

(4)Because of he second erm n (4) involving X ( - z ) , wecannot write downan expression for X(z ) /X(z )hat is inde-pendent of X(z) tself. This is not surprising, since the QMFbank is not ime-invariant (as the decimators and intes-polators are time-variant). The second term in (4) repre-sents the effects of aliasing and imaging.This term can bemade to disappear simply by choosing theynthesis filtersto be

F ~ ( z )= H~( -z ) , F ~ ( z )= - H ~ ( - z ) (5)Once the aliasings so canceled, the QMF bank becomesa (linear and) time-invariant system with transfer function

i ( z ) 1T ( z ) =- - [Ho(z )H I ( -z ) - Hl (z )Ho( -z ) ] (6)X(z) 2Ideally, we would like T(z ) o be a delay, i.e., T(z )= z-"O,so that he reconstructed signal is a delayed version ofx(n). Since T(z ) s in general not a delay, it represents adistortion and is called the distortion unction or he over-a l l transfer function. The quantity )T(e'"')) s the ampli tudedistortion and arg[T(e'")l i s the phase distortion. If T(z ) san allpass function, i.e., if IT(e'")l = constant or all w ,then here is no amplitude distortion. If T(z) s a linear-phase FI R function, then a rg[T(ej")] = K w , and there s nophase distor tion. Barnwell [29] has shown that there areseveral interesting ways in which the filters H o( z ) , H l ( z ) ,Fo(z) and Fl(z)can be related,m a s o obtainan appropriatefunctional form for T(z).It is typical to choose H l ( z ) .= Ho( -z ) , so that we have alowpass/highpass pair. Then

1 12 2(z ) = - [ H i ( z ) - H ? ( z ) ]= - - [HZ(z) - HS(-Z)] ( 7 )

which represents the distortion function. Suppose Ho(z )and H l ( z )are linear phase FIR filters, then T(z)give,n by (7 )clearly has linear phase, and phase distortion is easily

eliminated. In summary, the choice of transfer functionsaccording toH, (z ) = Ho(-z), Fo(2) = Ho(z), FI(Z) = -H1(z)

(8)leads tocompleteeliminationof aliasing; if H o ( z )haslinear phase, then phase distortion i s also eliminated.Assuming Ho(z) o be a linear phase owpass FIR filter oforder N - 1, we can write H d e j " ) = e-''(N-l)'ZHO , a (e'")where Ho,.(d") s the real-valued)amplitude response[ 3 2 ] .With this, T(ej") takes on a nice form:

e-jo(N-l)T(ei")'=-IHo(ejW)12 ( -1)N-11Hl(e'") /2] (9)If N - 1 is even, then. referring to Fig. 6(b), at the re-quency w = ~ / 2 , (e'") given by (9) is zero! This impliessevere amplitude distortion. Accordingly, with the choiceof filters as in (8),we must always pick the.orderN - 1 ofthe inear phase FIR filter Ho(z ) o be odd*. Equation (9)then yields 2IT(ej")l ,= Y[/Ho(ej")l2+ IHl(ei")12] (10)

which represents the residual amplitude distortion. SinceT(z )has linear phase,, phase-distortion is absent.Now comes the bad news: if two l inear phase transferfunctions Ho(z)and H 7 ( z )aresuch hat IHo(eiW)/2(Hl(e'")I2 is constant for all w , then H o( z ) nd H I ( z )must betrivial transfer functions [24] with frequency responses ofthe form IHo(ej")J2= cos2(Kw)and IHl(e'")12= sin2(Ko). n

1

'If an even orde r i s req uire d for some other reasons, there is a trickwhich can be employed to avoid the above distortion; see [51.

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other words, for the choice of filters as in (8), there doesnot exist a non-trivia l inear-phase transfer unction Ho(z)such that phase distortion and amplitude distort ion aresimultaneously eliminated!To complement he above discussion, et us now seehow ampli tude d istort ion can be completely eliminated,when one agrees to. tolerate phase distortion. There aremany ways of doing this [ I l l , [27], [29], [431; let us look atone. In the world of II R digital filters, there exists a b6au-tifu l subset of transfer unctions which can be mple-mented as a sum of two allpass functions [331, 341,[391,[40], [43]. If such transfer functions.are used in thenalysis.bank, this automatically forcesT(z ) o be allpass. Withoutgetting into the theoretical etails, let us state the essencecompactly here.Let Ho(z )be a owpass I IR function with numerator poly-nomial.P(z) and denominator polynomial D(z ) of ordersN, i.e.,

N NHo(z)= P( z ) / D ( z )= x , z - " / C .d n z - " . (11)n=O n=O

A typicalmagnitude response of H o ( z ) s indicated nFig. 8(a). Let us define th e highpass function H l ( z ) o besuch that lHo(ejw)12 lHl(e'")12 = 1, for a / / w . Such a pai.r[Ho(z ) , 1(t)] s called a power-complementary pair. GivenHo(z ) uch that JHo(ej")l5 1,we can always find such H l ( z )by defining-it o be H1(z )= Q ( z ) / D ( z )where Q ( z ) is aspectral factor of

IQ(ej")12= lD(ej")l' - P(ejw)12. (12)Usually, the zeros of Ho(z ) re on the unitircle, henceP(z)is a symmetric polynomial, .e., p n = p N T n .Moreover, it is

L

Non-overlappinganalysis filters

( b ) IHo(eiW)[ lHl(elW)I AOverlappinganalysis filters,W

0 lr/2 lr

Ideal "brickwall"characteristics oranalysis filters

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6 _ITypical amplitude-distortion function +

~ Typicalhase-distortion function

distortion unction T(z) = Ao(z)Al(z)/2 which isallpass

is the order of Ho(z)! n other words, he structure ofFig. 8(d) is dramatically efficient.The.,,Perfe,ct-Reconstructionwo Channel QMF Bank

So we see that either phase distortion or amplitude is-tortion can be completely eliminated, according to choThe remaining distortion can be either minimized usingcomputer-aided echniques, or equalized bycascadingwith a filter.Forexample, once phase distortion 'has been elimi-nated, ampli tude,distortion can be minimized by use ofnonlinear optimization oftware [23]. Usually, an objectivefu'nction s.'formulated which i s ' asum of t t i e stop-band error of Ho(z) and the amplitude distortion errorJ[IT(ej")12 - 1 j 2 d o ; the coefficients of Ho(z) are foundsuch that this objective unction is minimized. The re-maining filters Hl(z), Fo(z)nd Fl(z) are found from (8).An

, r

Inotherwords,amplitudedistortion i s completelyeliminated (and so i s aliasing, of course). The phase re-sponse of T(z) eads to some phase distortion. A dis-cussion ,on how to design ransfer unctions Ho(z) ndHl(z) satisfying (13) and the condition Hl(z)= Ho(-z), canbe found in [I ll , [431.It is worth pointing out some of the good features ofimplementations based on allpass decompositions, ,suchas (13). The allpass fil ters Adz) and A,(z) can be mple-mented using the Gray-and-Markel lattice structures [381which come in variousconvenient orms.Allof theseforms can be made free from limit cycles [MI (which areparasiticoscillationsunderzero-input,causedbyquantizer-nonlinearities in feedback loops of II R filters).Moreover, nstability problems due to coefficient quan-tization are absent in these latticestructures. It is alsoknown that implementations ased on the decompositionof (13) have low passband sensitivity [341 (even though this

i s not very crucial for QMF applications).The relations (13) imply that Ao(z) = Ho(z)+ Hl(z),andAl(z)*= Ho(z) - Hl(z).Since in addition Hl(z) = Ho(-z),you can verify that Ao(z) inact has the form.ao(z2)nd Al(z)has the formz-1a1(z2). Accordingly, he QMF bankcan beimplemented as in Fig. 8(d); If ao(z) and al(z)'are mple-mented using,a ascade of the one-multipl ier ray-Markellattice structure [381, then the entire QMF bank (i.e., allfour transfer functions Ho(z), H1(z);o(z)and Fl(z)) an beimplemented with a total of about N multipliers, whereN

A( a )

Typical rnagnilude responte.for II R Ho(2I and H,(rI

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alternative to this optimization would be to cas,cade a where d i s ti constant, hencelinear-phase FIR filter to the output 2(n)and equalize heamditude distortion. i -+) [z - l )Ho(z)+ Ho(-z- ')Ho(-z) = d , (15)

cause of symmetry around ?r/2 we have,G + ( z )+ ( - l ) N - l G + ( - z ) = dZ- 'N- l ' (14)

*i.e., funct ion H&) such that G+(z) = Z-'~-~'H~(~-')H~(~).

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*The cutoff frequency of the lowpass filter Ho(z) s not a quarter of2a unless M = 2. Thus, the name quadrature mirror filter is a mis-nomer, when M > 2. However, this has become m ore or less stan-dard, and there is no reason to change the name as long as weremember its origin.

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Fig. I l(b). These identities are exact, and can be proved A General M-Band Alias-Free Systembased on the nput-output relationsofdecimators and For a minute, et us switch our minds to a completelyinterpolators. different picture, viz. Fig. 12. Figure 12(a) shows a set ofNow con'sider Fig.11(c), where S(z) s a transferunction ' transfer functions So(z) , l(z),. .S,&(z) sandwiched be-sandwiched between a decimator andn interpolator.. Let tween M-fold decimatorsand nterpolators. The deci-S(z)= G(z)R(z)be an arbitrary factorization. Then we.can mators are preceded by a chain of delays, whereas themove G(z) and R(z ) around o obtain the equivalent interpolators are followed by a chain ofdelays. Since'deci-diagramshown in he igure. Such manipulations are mation is not preceded by iltering, here is in generalvery useful in obtaining a quick understanding of certain severe aliasing in the structure, for arbitrary x (n ) . How-important issues in the QMF problem. ever, it may be possible to pick the functions Sdz ) suchthat ,the aliasing i s somehow canceled by ?he imaging

effect of the interpolators. Let us probe deeper into thispossibility.

Based on this observation, how do we construct someuseful QMF banks?As a possible example, let us imaginettiat T is related to he DFT matrix, i.e.,, T = Wmn/Mwhere W = e-'"j", In particular, this means, , .&(z) = [Go(zM)+ z - ' G I ( z ~ )

+ z-2G2(zM)+ . + Z-(M- l )GM-l (ZM)I (22)and &(z) = Ho(zWk) . n. other words, wehave the fol-lowing situation: suppose somebody gives us a lowpass

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transfer funct ion overcomeshis problem and inddition leads toerfectm reconstruction since S(z) = 1 here; but t does not giveHo(z) = 2 h(n)z-" ( 2 3 ) rise to stable synthesis filters, unless the numerators of the

n=O polyphaseomponents G k ( z )have minimum phase.We can write Ho(z ) n the form (22) simply by defining*

G,(Z) = h ( / ) + h ( / + M ) z - l + h ( / + 2M ) Z + + . . PERFECT RECONSTRUCTION M-CHANNEL QMF BANKS

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on the unit circle of the z-plane, i.e.,ET(z- )E(z)= cl, fo r z = eiw, (27)

where c is a scalar constant. (Note that, assuming E(z) sreal for real z , ET(z-)s precisely the transpose-conjugateof E(z)on the unit circle). f E(r)has complex coefficients,then ET(z-)should be replaced with ES(z-) where sub-script * means coefficient conjugation. As a result, theoutput i ( n ) n Fig. 14(d) continues to be same as that inFi.g. 14(a), except for a scaling constant. We can now n-voke the identities inFig. 11, and rearrange Fig. 14(d) as inFig. 14(e),, which is therefore a perfect reconstructionsystem! As such, unl.ess E(z) s FIR, ET(z-) s unstable, sowe assume E(z) s FIR. To avoid non causal operations, inpractice, we insert a delay in ront of ET(z- ) so thattherearenopositivepowersof z anywhere.FromFig. 14(e) you can deduce that the analysis and synthesisfilters are effectively

andFk ( Z ) = z-PHk(z-) (28b)

where p i s a large enough positive integer to ensure thatthere are no positive powers of z in F k ( z ) .Now, if (27)holds everywhereon the unit circle, then itmust be true for all z , by analytic continuation.Such ma-trices E(z)are said to be paraunikry*. For our discussion,paraunitary will herefore be used as a synonym tounitary on the uni t circle. We can thus state the followingresult: le t Hk(z)be FIR analysis filters w ith polyphase com-ponents k / ( z ) uch that the matrix E(z) = [k/(z) I s para-* Th econcept of parauni tar iness i s we l l - kno wn , , i n c lassica l ,continu ous-tim e netw ork theory; scattering matrices that describelossless multiports satisfy this prope rty [191,[201.

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This uniquenatureof he attice encourages us todesign he wo-channelperfect econstruction analysisfilter Ho(z) n a different way rather hanbyspectral-factorizing a half-band ilter (see Fig. 18). We simplyformulate an objective functionp = iyI H o ( e j m / z d m 8 (29)

* This exa mple was generated b y Truong Q. guyen, at Caltech.

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Speech, Springer Verlag, New York, 1976.P. P. Vaidyanathan, Theory and design of M-channelmaximally-decimated quadrature mirror filters witharbitrary M, having perfect-reconstruction prop-erty, Technical Report, California institute of Tech-nology, April 1986. (Also see IEEE Transactions onASSP, vol. 35, pp. 476-492, April, 1987.)A. H. Gray, Passive cascaded lattice digitalf ilters,/ E Trans. on Circuits andSystems, vol. 27,337-344,May 1980.T. Saramaki, On thedesign of digital filters s a sumof wo allpass filters, / E Trans. on Circuits andSystems, vol. 32, pp..l191-1193, November 1985.R. Ansari and B. Liu, A class of low-noisecompu-tationally efficient recursive digitabfilters with app1.i-cations to sampling rate alterations, / E Trans. onAcoustics, Speech, and Signal Processing, vol. 33,pp. 90-97, February 1985.P. P.Vaidyanathan, Zinnur Doganata, and Truong Q.Nguyen, More esultson heperfect econ- Istruction problem in M-band parallel QMF banks,IEEE international Symp. on Circuits and Systems,Philadelphia, May 1987 (to be published).P. P. Vaidyanathan and Phuong-Quan Hoang, Theperfect-reconstructionQMFbank:newarchitec-tures, solutions, and optimization strategies, IEEEinternational Conference on Acoustics, Speech andSignal Processing, Dallas, pp. 2169-2172, April ,1987(to be published).P. P. Vaidyanathan, P. Regalia and S. K.Mitra, Designof doubly complementary IIR digital filters using asingle complex allpass filter, with multirate applica-tions, IEEE Trans. on Circuits and Systems, vol. 34,

SUMMARY.OFKEY RESULTSThe M-channel Quadrature Mirror Filter (QMF) bank inFig,..IO,,is,Called the maximally decimated, parallel QMF

bank..in order to avoid spectral gaps while splitting thesignal x(n ) in toM bands, the frequency responses of theanalysis filters Hk(z) are permitted tooverlap.Con-sequently, there i s aliasing at the output of he decimators.This aliasing can be canceled by the imaging effects of theinterpolators, if he synthesis fi l ters Fdz) are chosenappropriately. Some schemes [IO], [I ll for perfect cancel-ationof aliasing are shown in Fig. 12 and Fig. 13, andtypically require high orders for F k ( z ) . Approximate can-celation of aliasing can be achieved by use of suitablesynthesis filters of low order [8], [9].

Once aliasing hasAbeen canceled, the econstructedsignal i s givenby X(z) = T(z)X(z) where T(z) is theoverall transfer function or the distortion transfer func-tion. i f If(e)l.is.constant independent of w (i.e., if T(z) san, allpass function) here is no amplitude distortibn; ifarg[T(ej)] = Kw , (i,e,, if T(z) s a linear-phase (FIR) func-tion) then there s no, phase distortion. In fact it has beenpossible in he past to thus eliminate either amplitudedistortion or phase distortion completely [I ], [5], [1.0],

[Ill,[291. Simultaneous elimination of a l l three distortions(i.e., aliasing, amplitude and phase distortions) is difficultbut can be done. Such a QMF structurewil l be a perfect-reconstruction structure and satisfies9(n)= cx(n - no) . f k , n ( ~ ) , 0,I I - 1 represent the M polyphase com-ponents of the analysis filters Hk(z),0 . k I - 1, (seediscussions around equations (23), (24) for meaning ofpolyphase components) and if the matrix function

E(z) = [ k n z)l (A1April 1987 (to be published).[MI P. P. Vaidyanathan and V. Liu, An mproved suf-ficient condition for absence of limi t cycles in digital ing the synthesis filters to befilters, IEEE Trans. on Circuits and Systems, vol.CAS-34, pp. 319-322, March 1987. Fk(Z) = z-PHk(Z-) ( B 1

is unitary on the unit circle of the z-plane Fig. 141, then itis possible to obtain perfect reconstruction simply by tak-

where p is an integer large enough so that there are no

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Y(z) = E(z)X(z), hen the energy in the vector sequencey ( n ) is equal to that in the, vector sequence x ( n ) . In thecontinuous-time world, such paraunitary systems arewell-known; scattering, matrices of lossless multiports areknown to have this property [ 19 ] , 20].

A second way to ook at the perfect-reconstructionscheme is through the Alias Cancelation (AC-) matrix H(z)in Eqn. (18). If this matrix i s unitary, then we can solvefor he synthesis filter vector f(z) simply by akingf(z) = Z-~H'(Z-')Vwhere p is large enough so that thereare no positive powers of z in the expressions for F&)).

It turns out that, wi th this viewpoint, the same solutionsviz. Eqn. (B) results and the (paralunitariness of E(z) isequivalent to (para)unitariness of H(z).

This work was supported in part by the National Science Foundationgrants ECS 84-04245 and DCI 8552579, and in part by Caltech's Pro-grams in Advanced Technology grant, sponsored by Aerojet GeneraGeneral Motors, GTE and TRW.

ACOUSTICS, SPEECH AN D SIGNAL PROCESSING SOCIETYCall for Adcom Nominations

George W. JohnsonIBMFederal Systems Division9500 Godwin DriveManassas, VA 221 10(703) 367-4708

Stephen E. LevinsonRoom 2D-528AT&T Bell Laboratories600 Mounta in AvenueMurray Hill, NJ07974(201) 582-3503James SnyderAT&T Bell Laboratories6 0 0 Mounta in AvenueMurray Hill, NJ07974(201 ) 582-6403

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