BOUNDEDNESS AND APERIODICITY OF COMMERCIAL SIGMA DELTA MODULATORS

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1 BOUNDEDNESS AND APERIODICITY OF COMMERCIAL SIGMA DELTA MODULATORS Heni Huijbets, Alexey Pavlov, Josh Reiss 3 Depatent of Engineeing, 3 Depatent of Electonic Engineeing Queen May, Univesity of London Mile End Road, London, E4NS U.K. H.J.C.Huijbets@qul.ac.u, 3 josh.eiss@elec.qul.ac.u Depatent of Engineeing Cybenetics Nowegian Univesity of Science and Technology Tondhei, NO-749Noway Alexey.Pavlov@it.ntnu.no Abstact: Siga delta odulation is a popula fo of A/D and D/A convesion. This nonlinea device exhibits a high degee of coplex nonlinea behaviou, including chaotic dynaics. One of the ain unsolved pobles in the theoy of siga delta odulation concens the ability to analytically deive conditions fo the boundedness of solutions of a high ode siga delta odulato (SDM). In this wo, we descibe how a siga delta odulato ay be ephased within the context of systes theoy. We pesent seveal theoetical esults concening bounded solutions of geneal high ode SDMs, including necessay and sufficient conditions fo the lac of a finite escape tie, necessay conditions fo bounded solutions based on the natue of the output sequences, and topological popeties of the solutions, which ae a pecuso to the study of chaotic solutions of SDMs. Copyight 5 IFAC Keywods: Analog/Digital Convetes, Modulatos, Nonlinea Systes, Stability, Contol Theoy... Bacgound. INTRODUCTION Siga delta odulation is a popula fo of A/D and D/A convesion. The technique has povided poweful eans fo conveting analog to digital signals and vice vesa with low cicuit coplexity and lage obustness against cicuit ipefections. As a esult of this, -bit siga delta based analogto-digital (A/D) and digital-to-analog (D/A) convetes ae widely used in audio applications, such as cellula phone technology and high-end steeo systes. Siga delta odulation, oiginally conceived by De Jage (95), is a well-established technique. Howeve, theoetical undestanding of the concept is vey liited (Noswothy, et al., 997). Ipotant pogess in the undestanding of the dynaical systes popeties of SDMs has lead to a desciption of thei chaotic behavious (Feely, 997; Dunn and Sandle, 996; Reiss and Sandle, ), a useful lineaization technique (Adalan and Paulos, 987) and a faewo fo descibing thei peiodic behaviou (Reefan, et al., 5). Yet in all these developents, thee is no unified desciption of SDMs. Instead, seveal odels ae povided, each of which descibes soe aspects of an SDM to a cetain accuacy. In this wo, we fae the dynaic behaviou of siga delta odulatos within the context of systes theoy. The focus is on the chaacteization of the boundedness and apeiodicity of solutions. The analysis is intended to descibe a lage nube of feedfowad o intepolative SDM topologies (Noswothy, et al., 997), as used in the design of coecial SDMs... Motivation Figue depicts a 5 th ode, feedfowad SDM. It ay be ipleented in eithe digital o analog cicuity. An input signal u is fed into the syste, and passed though 5 discete-tie integatos. The output of each integato is ultipled by a coefficient c i, the esults ae sued, and then quantised depending on the sign of the su. This quantised output is fed bac to the input. The coefficient vecto C=(c,,c 5 ) seves to shape the noise of quantisation away fo the fequency band of inteest. Thus the SDM acts as a filte, but with

2 the quantise intoducing a high degee of nonlineaity. u s () s () s () s 3 () s 4 () s 5 () - z - zt - z - z - zt - c c c 3 c 4 Figue. Ipleentation of a coecial fifth ode SDM. It is well-established that a fist ode SDM will poduce bounded behaviou fo input agnitudes less than, and siila esults can be shown fo soe nd ode SDM designs (Faell and Feely, 998). Howeve, highe ode SDMs ay poduce divegent behaviou, such that the agnitude of the input to the quantise becoes exceedingly lage and the output no longe tacs the input. To illustate this, Figue depicts the agnitude of the quantise input fo the ipleentation of a 5 th ode feedfowad SDM (Reefan, et al., 5), intended to be used fo analog to digital convesion in audio applications. The SDM is lowpass, and has a cone fequency of 8Hz, fo a saple ate of 64x44. Hz. We have assued constant input u with all initial conditions set to zeo. It can easily be veified that the dynaics ae bounded fo an input of.7, unbounded fo input.9, and unbounded with finite escape tie (in a sense defined late in the pape) fo an input of y u=.7 u=.9 u= Iteate Figue. The agnitude of the quantise input fo diffeent values of input signal u. Besides boundedness of solutions, the avoidance of low-peiod solutions is also an ipotant design consideation fo coecial SDMs, piaily because they epesent high fequency tones in the output bitstea that wee not pesent in the input signal. It is theefoe ipotant to be able to identify initial conditions that will lead to apeiodic, pefeably chaotic, behaviou. Ou goal in this pape is to pesent a atheatical faewo, based on systes theoy, to descibe the boundedness and apeiodicity of solutions of SDMs. We will focus on DC inputs, since this epesents the ost elevant pactical situation. The oganisation of the est of this pape is as follows. Section intoduces concepts fo contol c 5 theoy, as applied to the geneal class of aps to which ost SDMs belong. Such an intoductoy section is necessay in ode to bidge the gap in teinology and undestanding between the SDM designes and the systes theoists. It also intoduces the concepts of finite escape tie and well-posedness fo SDMs and gives a canonical desciption of the type of SDMs that ae studied in this pape. Section 3 povides seveal inteesting esults concening the boundedness popeties of solutions of abitay SDMs. Notably, we deive a poof that the dynaics of an SDM ae bounded only if the output bitstea is also bounded. This allows the boundedness popeties to be ephased in tes of the constaints on the dynaics iposed by output sequences. Section 4 povides futhe esults which lead to a topological undestanding of the natue of bounded solutions and an indication of the existence of apeiodic and chaotic solutions. Finally, Section 5 povides conclusions and discussion of diections fo futue eseach.. SYSTEM THEORETIC PROPERTIES AND NORMAL FORM FOR SDMS In this section we descibe soe syste theoetic popeties of SDMs which will be of ipotance when studying the boundedness of solutions of SDMs. Fo futhe syste theoetic bacgound we efe the eade to (Benstein, 5; Chen, 998). Conside an n-diensional SDM of the fo σ s = As B( u sgn( y)) () y = Cs whee u denotes the input which is assued to be constant with u <, y denotes the quantise input, and σs denotes the fowad shift of s, i.e., σ s ( ) = s ( ) fo all. We assue that A is a lowe tiangula atix with diagonal eleents equal to and B,C ae atices of appopiate diensions. Recall that the atix pai (A,B) is called contollable if it satisfies one of the following equivalent conditions: an( B AB A B) = n () o an ( λi A B) = n fo all eigenvalues λ of A (3) Futhe, ecall that the atix pai (C,A) is called obsevable if the atix pai (A T,C T ) is contollable. With a slight abuse of teinology, we will call the SDM () contollable if (A,B) is contollable while we will call it obsevable if (C,A) is obsevable. Wite B= col(b,, b n ), C= (c,,c n ), and let a i,j (i,j=,,n) denote the enties of A. By eploying the condition (3) and using the popeties of A, the following esult on contollability and obsevability is alost iediate. Poposition.. The SDM () is contollable if and only if b a i, i, and it is obsevable if and only if cn a i, i. In the est of the pape we will assue that the SDM () is obsevable. This assuption can be ade without loss of geneality, because non-

3 obsevability would iply the pesence of intenal dynaics that do not contibute to the behavio of the quantise input y. Define the tansfe function of the SDM () by Gz ( ) = CzI ( A) B and ecall that when () is obsevable we have that Gz ( ) = qz ( )/ pz ( ) whee p(z)=det(zi-a)= (z) n and deg(p)> deg(q). If we then wite qz ( ) = qz, we have that the quantise input y satisfies the following diffeence equation: n n n ( ) y ( ) = q ( u sgn( y ( )) We will say that a solution of () has finite escape tie if thee exists a such that sgn(y())=sgn(y( )) fo all. Futhe, we will call the SDM () well-posed if it has no solutions with finite escape tie. We then have the following esult on well-posedness. (4) Poposition.. The SDM () is well-posed if and only if β : = q >. Poof. (Setch) Assue that β. Conside a solution of (4) with y()> fo =,...,N, N>n. Define y :=y() fo =,...,n-. Define ε : = β ( u ). It ay then be shown that y( ) = P( ) y Q( ) ε ( = n,..., N) (5) whee n n ( ) P ( ) = ( ),!( n)! = n (6) Q ( ) = ( ) n! = Thus we see that the solution is polynoial in with coefficients of,, n- depending linealy on y,,y n- and coefficient of n equal to ε. This eans that we can choose y,...,y n- in such a way that all coefficients of y() ae positive. This gives that y()> fo all, and hence y() has finite escape tie. This establishes ou clai. To chec well-posedness, the following esult can be used. Lea.3. If () is obsevable, we have that n ai, i β = bc. Poof. Recall that if () is obsevable, it follows fo Cae s Rule that qz ( ) = Cadj( zi AB ) whee adj(m) denotes the adjoint (see, e.g. (Benstein, 5)) of the squae atix M. This then gives that β = q() = Cadj( I A) B. Using the popeties of A, it is staightfowadly checed that adj( I - A) n = a i, i, while all othe enties of adj( I - A ) ae zeo. This iediately establishes ou clai. As a consequence of Poposition. and Lea.3 we have that the SDM () is well-posed only if it is contollable. In studying geneal popeties of systes, it is often useful to conside canonical fos that ae equivalent to a whole class of systes up to a coodinate tansfoation. In linea systes theoy two of the ost well-nown canonical fos ae the so-called contolle canonical fo and the obseve canonical fo. Howeve, fo SDMs these canonical fos ae pehaps not the ost insightful canonical fos in tes of the physical intepetation of the SDM. We theefoe define a contolle canonical SDM fo as σ sc = As c c Bc( usgn( yc)) (7) yc = Ccsc whee A c is a atix with diagonal eleents equal to, (A c ) i,i = (,, n-) and all othe enties zeo, B c = col(,,,), and C c is abitay. Poposition.4. Evey contollable SDM () adits a contolle canonical SDM fo in the sense that thee exists an invetible atix V such that s c :=V - s satisfies (7). Poof. Assue that (A,B) is contollable and define the atix V=(v,,v n ) whee v i =(A-I) i- B(,,n). By pefoing eleentay colun opeations, it is staightfowadly shown that an(v) = an(b A n- B), which iplies that V is invetible. Note that v =B, which iplies that V B = Bc. It is futhe staightfowadly shown that Av i = v i v i (i =,, n-). Also, it follows fo the Cayley- Hailton Theoe (see, e.g. (Benstein, 5)) that (A-I) n- =, which allows one to show that Av n =v n. All identities obtained above then staightfowadly iply that V - AV = A c and hence that s c := V - s satisfies (7). 3. RESULTS ON BOUNDEDNESS OF SOLUTIONS OF SDMS Assuing y()>, fo a well-posed SDM it follows fo Section that thee ae infinite sequences N, N such that sgn(y())= fo { P,..., P N } and sgn(y())=- fo { P N,..., P }, whee and = P : ( N N ). Siila sequences can be defined fo y()<. In this section we will give necessay conditions fo boundedness of solutions in tes of these sequences. Theoe 3.. Assue that the SDM () is wellposed. Then a solution of () is bounded only if the lengths of bit steas associated with the solution ae bounded. Poof. Conside a solution of (4) with y()> fo =,...,N, N>n. Define y :=y() fo =,...,n-. Conside the expession fo the solution given in (5) and (6), and define the polynoial P ( ) = P ( ). Note that fo all we

4 have that P ( ) > P ( ). Then the fact that y(n)> iplies by (5) that QN ( ) ε < P( N) y (8) P ( N) y < P( N) y which gives that QN ( ) y > PN ( ) ε (9) = Since deg(q)=n, deg(p)=n-, this gives that y as N. In a siila way it ay be shown that solutions becoe unbounded if the lengths of negative bit steas becoe unbounded. Thus ou clai is established. The following esult and its elated coollaies povide oe stingent necessay conditions fo bounded solutions. Theoe 3. Conside a solution of a well-posed SDM with u <. Define M : = ( u ) N ( u ) N () Then the solution is bounded only if the sequence M ( ) is bounded. Moeove, fo a onediensional SDM, this condition is also a sufficient condition fo boundedness. Poof. Conside a bounded solution s() of the SDM. Then obviously we have that s () is bounded. It is staightfowadly checed that s ( N N) () = s () ( u ) N ( u ) N which establishes ou clai. Since fo a onediensional SDM s () is the only state space vaiable, sufficiency fo one-diensional SDMs is iediate. Coollay 3.. Conside a solution of a well-posed SDM with u <. If the solution is bounded and u, we have that ( )(, : < < ) () ( u ) N ( u ) N = Poof. Fo the Bolzano-Weiestass Theoe it follows that the bounded sequence M has a convegent subsequence, i.e., thee exists a sequence ( ρ l ) l with ρl as l such that lil M ρl exists. As a consequence, we have that ( ε > )( l )( στ, > l )( Mρ M ) σ ρ < ε (3) τ Assue that u, and wite u = a/b, whee a, b and a < b. Choose ε< /b and let be given. Then accoding to (3) thee exist < :=ρ σ < :=ρ τ such that M M < ε, which gives that bm bm < bε <. Noting that bm, bm, this iplies that in fact bm bm =, which establishes (). Coollay 3.3. Conside a bounded solution of a well-posed SDM with u < and u. Define σ : =, and define ρ, σ ecusively in the following way: ρ : = in > σ ( : < < ) σ { ( ( u ) N ( u ) N = )} { : = in > ρ u N u N = ρ ( ) ( ) = } (4) Note that fo Coollay 4. the sequences ρ, σ ae well-defined and infinite. Then thee exists a such that fo all we have that ρ =σ. Poof. Define the index sets I : = { ρ,..., σ }, J := sequence ( ) I. Due to the boundedness of the M, the sequence ( M ) J is also bounded. Assue that ou clai does not hold. This iplies that the index set J is infinite. Using a siila aguent as in the poof of Coollay 3., this iplies that thee exist j,j with j < j <, j such that ( u ) N ( u ) N =. Since j the sequences ρ, σ ae infinite, we have that thee exists a such that σ <j < ρ. Howeve, this contadicts the definition of ρ. This establishes ou esult. 4. TOPOLOGICAL RESULTS CONCERNING THE BOUNDEDNESS OF SDMS Thoughout this section, we conside an SDM in the contolle canonical fo (7). Fo bevity s sae, we will oit the c subscipts in the desciption of the siga delta odulato. We denote the vecto col(,,...,) by e n. Theoe 4. Let s ( ), be a bounded solution of the SDM and b ( ) : = sgn( Cs ( )), be the coesponding bit sequence. Then fo any othe bounded solution s ( ) with the sae bit sequence, thee exists α such that s ( ) s( ) = αen fo all. Poof: By the conditions of the theoe, s( ) and s ( ) satisfy σ s = As B( u b) (5) σ s = As B( u b) Theefoe, the diffeence ξ = s s satisfies the diffeence equationσξ = Aξ, which gives that ξ( ) = A ξ(). It can be veified (Reefan, et al., 5) that fo n, A = I β ( ) T whee = j j

5 ! β ( ) = and the atices T!( )! satisfy (T ) ij =δ i,j (i,j =, n) with δ ij denoting the Konece delta. Thus the second coponent of the vecto ξ () is given by, ξ( ) = β( ) ξ() ξ(). Notice that βi ( ) as, fo,,n-. Theefoe, if ξ (), then ξ ( ) is unbounded. At the sae tie, ξ () is bounded because by the conditions of the theoe both s( ) and s ( ) ae bounded. Hence, we conclude that ξ () =. Repeating this easoning fo the eaining coponents, we conclude that ξ i () = fo,,n-, while the last coponent ξ n () can be abitay. Hence, ξ () = αen fo soe α. Since A en = en, we obtain ξ ( ) = αen fo all. This copletes the poof. Theoe 4. Conside an SDM in the contolle canonical fo. Let s(), be a bounded solution of the SDM and b() := sgn(cs()), be the coesponding bit sequence. Suppose thee is δ> such that Cs() >δ fo all. Then thee exists ε> such that fo all α <ε the sequence s ( ): = s( ) αen is a solution of the SDM and the coesponding bit sequence equals b(). Poof: We will show that s ( ) satisfies (7). Fist note that due to the fact that Ae n =e n we have that A(() s αen) = As () αen (6) Next, assue that y() = Cs()>. It then follows fo the condition of the theoe that y()>δ. Theefoe, it follows fo the choice of α and ε that C(() s αen) > δ α Cen > δ ε Cen >. Thus, sgn( Cs ( ( ) αen )) = sgn( y ( )) (7) Cobining (6) and (7) we then obtain As ( ) B( u sgn( Cs ( ))) = As( ) αen Bu ( sgn( y ( ))) (8) = s ( ) αen = s ( ) fo all. Hence, s ( ) is a solution of (7) and it has the bit sequence b(). The conditions of Theoe ae satisfied, fo exaple, fo peiodic solutions s that satisfy Cs(),. Thus, we can foulate the following coollay. Coollay 4. Let s() be a peiodic solution of an SDM such that Cs(),. Then fo all sufficiently sall α, s() αe n is a peiodic solution of the SDM with the sae bit sequence. We note that a diffeent poof of this coollay was povided in (Reefan, et al., 5), but that wo only consideed peiodic solutions. Next we pesent esults on geneic boundedness popeties of solutions of SDMs. Theoe 4.3 Conside an SDM with n. The set of all bounded solutions is isoophic to a subset of. Poof: We fist intoduce soe notations. By bold font lettes we will denote sequences. Fo exaple, a solution of the SDM s(), =,,, is denoted by s. The bit sequence b(); =,,, is denoted by b. Λ denotes the set of all bounded solutions of (7), and B Λ denotes the set of all bit sequences b coesponding to the solutions s Λ. By the constuction of the set B Λ, fo any b B Λ thee is a solution s b Λ which has the bit sequence b. By Theoe 4., any othe bounded solution of (7) s b with the sae bit sequence b can be epesented as s b = s αe n fo soe α. (Hee, by adding the b vecto αe n to the sequence s b we ean that this vecto is added to evey eleent of the sequence.) Denote A b to be the set of all α such that s= s b αe n is a bounded solution of (). Constuct the set M : = { sb αen, α A b, b BΛ}. By the constuction, M=Λ. Denote P : = {( α, b): α A b, b BΛ}. Thus Λ P. Fo a bit sequence b, define the function (b):= b ( ). In othe wods, the bit = sequence b is a binay epesentation of the decial nube d=(b). By D Λ we denote the set of all nubes d = (b) such that b B Λ, i.e., DΛ = { d : d = ( b), b BΛ}. Notice that thee is a one-to-one coespondence between a nube and its binay epesentation. Theefoe, BΛ DΛ. Hence, Λ P P : = {( α, d): α A, d D } ( d) Λ whee - (d) is the binay epesentation of d. Notice that P {( α, d) : α, d } =. This poves the clai of the theoe. Roughly speaing, Theoe 4.3 states that the set of all initial conditions coesponding to bounded solutions is not thice than. This allows us to foulate the following coollay. Coollay 4. Conside an SDM with n 3. Then fo alost all initial conditions the coesponding solutions of SDM ae unbounded. The esult of Coollay 4. explains why fo twodiensional SDMs the bounded solutions ae elatively abundant, while fo highe-diensional SDMs it is uch oe difficult to find bounded solutions. It also has the iplication that fo higheode SDMs sall petubations ay lead to an othewise bounded solution becoing unbounded. The next esult concens bounded solutions of SDMs with peiodic o asyptotically peiodic bit sequences. It is said that a bit sequence b is peiodic if thee exists T > such that b() =b(t) fo all We say two sets ae isoophic (denoted by ), when thee is a one-to-one coespondence between the sets.

6 . A bit sequence b is called asyptotically peiodic if it is peiodic afte soe tie instant N >, i.e., if b() = b( T) fo all N. Theoe 4.4 Conside an SDM, with n. The set of all bounded solutions with bit sequences that ae eithe peiodic o asyptotically peiodic is isoophic to a subset of. Poof: Siila to the poof of Theoe 4.3, we intoduce the following notations. Let denote the set of all bounded solutions of (7) with bit sequences that ae eithe peiodic o asyptotically peiodic. By B denote the set of all bit sequences b coesponding to the solutions s, and define D = { d : d = ( b), b B}. As in the poof of Theoe 4.3, it can be shown that P : {(, ):, } = α d α A d D ( d). By the constuction of the set D, any nube d D has a peiodic (afte soe digit ode nube) binay epesentation. This can happen if and only if d is a ational nube. Theefoe, D. Thus P : {(, d ):, d D = α α A } ( d) (9) {( α, d) : α, d } This copletes the poof. Coollay 4.3 Conside an SDM with n =. The set of solutions of the SDM stating in any open set of initial conditions contains eithe an unbounded solution o a bounded solution with a bit sequence that is neithe peiodic no asyptotically peiodic. Poof: Conside soe open set of initial conditions Θ. Suppose that the set of all solutions of an SDM stating in Θ does not contain unbounded solutions (if it does, then the clai is poved). Since Θ is open, it contains an open subset Ω that is isoophic to. Theefoe, the set of solutions stating in Ω is isoophic to. Assue that all solutions stating in Ω have bit sequences that ae asyptotically peiodic. By Theoe 4.4, we would then have that this set of solutions would be isoophic to a subset of Howeve, this would iply that Ω is also isoophic to a subset of, which gives a contadiction. It follows fo Coollay 4.3 that if fo a twodiensional SDM one can identify an open set of initial conditions that lead to bounded solutions, thee exist apeiodic solutions aongst these solutions. It is well-nown (Faell and Feely, 998; Scheie, et al., 997) that indeed one can identify these sets of initial conditions. Thus, this indicates that thee ay be chaotic solutions fo twodiensional SDMs. 5. CONCLUSION The wo descibed heein is concened with the boundedness of solutions of SDMs. Ou appoach has been to ephase the siga delta odulato as a contollable and obsevable discete tie syste. We have shown that typical SDM designs fall into such a categoy. We futhe define an SDM as wellposed if it has no solutions which divege to infinity in finite escape tie. Fo an SDM design point of view, the input signal u is always confined to a agnitude less than, and the coefficient vecto C is stictly positive fo a lowpass SDM. Thus typical SDM designs ae iplicitly well-posed. This allows us to show that an SDM yields bounded behaviou only if the length of the associated output bit steas ae also bounded. To the best of the authos nowledge, this esult has neve befoe been deived. Its stength lies in that it allows the boundedness popeties to be ephased in tes of the constaints iposed by the output bits. Futue eseach along this diection is concened with poving the convese, and with identifying bounded solutions fo coecial SDM designs. ACKNOWLEDGMENTS Reseach conducted by the second autho was patly pefoed while visiting Queen May, Univesity of London and suppoted by the EPSRC NETCA Netwo. REFERENCES Adalan S.H. and J. J. Paulos (987). An Analysis of Nonlinea Behavio in Delta-Siga Modulatos, IEEE Tans. Cic. Syst. I, 34, pp Benstein, D.S. (5). Matix Matheatics. Pinceton Univesity Pess, Pinceton, NJ. Chen C.-T. (998). Linea Syste Theoy and Design. Oxfod Univesity Pess, New Yo. De Jage, F. (95). Delta odulation - a ethod of {PCM} tansission using the one unit code, Philips Res. Rep., 7, pp Dunn, C. and M. Sandle (996). A copaison of Ditheed and Chaotic Siga-Delta Modulatos, J. Audio Eng. Soc., 44, pp Faell, R. and O. Feely (998). Bounding the integato outputs of second ode siga-delta odulatos, IEEE Tans. Cic. Syst. II, 45, pp Feely O. (997). A tutoial intoduction to nonlinea dynaics and chaos and thei application to siga-delta odulatos, Int. J. Cic. Theoy Appl., 5, pp Noswothy S., R. Scheie and G. Tees (997). Delta-Siga Data Convetes, IEEE Pess. Reefan D., J. Reiss, E. Janssen and M. Sandle (5). Desciption of liit cycles in sigadelta odulatos, IEEE Tans. Cic. Syst. I, 5, pp. -3. Reiss J. and M.B. Sandle (). The Benefits of Multibit Chaotic Siga Delta Modulation, CHAOS,, pp Scheie R., M. Goodson and B. Zhang (997). An algoith fo coputing convex positively invaiant sets fo delta-siga odulatos, IEEE Tans. Cic. Syst. I, 44, pp