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1 Uiversity of Est Lodo Istitutiol Repository: This pper is mde ville olie i ccordce with pulisher policies. Plese scroll dow to view the documet itself. Plese refer to the repository record for this item d our policy iformtio ville from the repository home pge for further iformtio. To see the fil versio of this pper plese visit the pulisher s wesite. Access to the pulished versio my require suscriptio. Author(s): Lot, Jswider. Al-Ji, Mohmmed., le, Izzet Article Title: Stility Alysis of Higher-Order Delt-Sigm Modultors for Dul Siusoidl Iputs Yer of pulictio: 7 Cittio: Lot, J. Al-Ji, M., le, I. (7) Stility lysis of higher-order delt-sigm modultors for siusoidl iputs. I: Proceedigs of the IEEE Istrumettio d Mesuremet Techology Coferece, Wrsw, Pold, My - 3, 7. IEEE, Los Almitos, USA, pp. -5. ISBN Lik to pulished versio: DOI: (ot stted) Pulisher sttemet:

2 Stility Alysis of Higher-Order Delt-Sigm Modultors for Dul Siusoidl Iputs Jswider Lot*, MIEEE, Mohmmed Al-Ji*, MIEEE, Izzet le*, MIEEE *Applied DSP d VLSI Reserch Group, Deprtmet of Electroic Systems, Uiversity of Westmister, Lodo, U Applied DSP d VLSI Reserch Cetre, Ester Mediterre Uiversity, Gzimgus, Mersi, Turkey jsi@ieee.org, M.Al-Ji@wmi.c.uk, klei@wmi.c.uk Astrct- The im of this pper is to determie the stility of higher-order -Σ modultors for siusoidl iputs. The olier gis for the sigle it qutizer for dul siusoidl iput hve ee derived d the mximum stle iput limits for fifth-order Cheyshev Type II sed -Σ modultors re estlished. These results re useful for optimisig the desig of higher-order -Σ modultors. I. INTRODUCTION The stle iput mplitude limits for -Σ modultors is complicted to predict due to the o-lierity itroduced y the qutizer i the feedck loop. Vrious pproches hve ee employed to expli this olier ehviour. Usig qusilier modelig, ew iterprettio of the istility mechism for -Σ modultors sed o the oise mplifictio curve is give i []. This is restricted for DC iputs d uity qutizer gis. The qusilier method c e exteded to more th oe iput with ech iput represeted y seprte equivlet gi. This cocept forms the sis for the Descriig Fuctio (DF) method []. I [3] the stility lysis for higher-order -Σ modultors sed o the oise mplifictio curve ws performed usig the DF method for DC d (sigletoe) siusoidl iputs for o-uity qutizer gi vlues. I this pper the lysis is exteded for multiple (dul) toe siusoidl iputs. II. QUASILINEAR STABILITY ANALYSIS OF -Σ MODULATORS A geeric -Σ modultor hvig its qutizer replced y gi fctor followed y dditive qutiztio oise q(k) [] is show i Figure. x(k) G(z) - e(k) Figure. Qusilier -Σ modultor Qutizer Model. The output of the modultor i the z-domi is give y : Y ( z) = STF ( z) X ( z) NTF ( z) Q( z) () q(k) y(k) where, Y(z), X(z) d Q(z) re the z-trsforms of the output, iput d qutizer oise sigls respectively. Also, STF(z) d NTF(z) re the Sigl d Noise Trsfer fuctios of the -Σ modultor derived from Figure. STF ( z) =. G( z) (). H ( z) NTF ( z) = (3). H ( z) Sice the poles of the deomitor () determie the stility of the modultor, for give, there will e certi itervl [ mi, mx ] for which the modultor is stle [4]. Assumig q(k) to e Gussi white stochstic G(, σ q ) d the trsfer fuctio etwee q(k) d y(k) to e kow, the the output oise vrice is give y jπf Vr{ y( k) } = σ q NTF ( e ) df = σ q A( ) (4) where, σ q is the vrice of q(k) d A() is the totl output oise power mplifictio fctor. Usig Prsevl s reltio, A() c e foud i the time domi s []: A( ) = tf ( k) tf (5) k= where tf(k) is the impulse respose correspodig to NTF(z) d A(k) is the squred two-orm of NTF(z). The A() curves of the loop-filter re crucil for the stility lysis of the -Σ modultors. Typicl curves for Type II Cheyshev 3 rd d 4 th order re show i Figure. A() Figure. A() Curves for Type II Cheyshev NTF. The A mi vlue is the glol miimum of the curve. It hs ee show i [] tht for stle opertio A(k)>A mi.

3 III. NOISE AMPLIFICATION CURVES DF METHOD The qusilier qutizer model i Figure c e exteded usig seprte gis x d for the DF model s show i Figures 3 d 4 [5]. x(k) G(z) Figure 3. -Σ modultor Qutizer Sigl-Model Figure 4. -Σ modultor Qutizer Noise-Model Figure 3 descries the model for the iput sigl with lier gi x wheres Figure 4 descries the oise sigl model with lier gi. The comied output sigl is give y: y( k) = yx ( k) y( k) (6) The lierised gis for two siusoidl iput sigls x (t)=cos(ω (t)φ ), x (t)=cos(ω (t)φ ) (where, re costts, ω, ω the siusoidl frequecies, φ d φ rdom phses) d rdom Gussi sigl represetig the feedck compoets hve ee solved for the cse of oe-it qutizer with output ± i Appedix A where the fil expressios re show elow: 5 = 3 F,, ρ ψ (7) π σ ρ 5 = 3 F,, ψ (8) π σ ρ ρ = e e ζ (9) π σ where ψ ρ ρ ρ ρ () = = ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ = ρ ρ q(k) ψ () ζ () d ρ =(/)( /σ ), ρ =(/)( /σ ). F(.) is the cofluet hypergoemetric fuctio [6]. The output oise vrice is give y: Vr - e x(k) y (k) x { y ( k )} σ σ = (3) e q e (k) y x(k) where σ q is the qutiztio oise power for the two ucorrelted siusoidl iputs x (t) d x (t). Therefore from (4), (9) d (3) the oise mplifictio fctor is give y: ρ ρ e e ζ σ π q A ( ) = (4) σ q Sice x (t) d x (t) re ucorrelted, the power of the output sigl is give y: E{ y ( k) } = σ σ σ e q e σ e (5) where σ e d σ e re the powers of the siusoidl iputs t the qutizer iput which re give y: σ = σ d σ = σ (6) e e From (9), (5) d (6) we get: = { e e } ζ σ (7) π q Rerrgig (7), the qutiztio oise is give y: σ = { e e } ζ (8) q π From (8) d (6) we get: 5 ρ 3 F,, [ ] ψ = (9) π Similrly from (7) d (6) for the siusoid x (t) we hve: 5 ρ 3 F,, [ ] ψ = () π The two simulteous equtios (9) d () were solved y deployig the MATLAB symolic toolox i order to get the vlues of ρ d ρ for vrious vlues of d. IV. RESULTS & SIMULATIONS From (9) d (), the vlues of ρ hve ee plotted i Figure 5. It is see tht ρ gets igger s the mplitude icreses. However, the icrese i ρ gets tteuted s the sigl mplitude icreses from. to Amplitude () Figure 5. Vritio of ρ versus for differet mplitudes.

4 Usig (8) the qutiztio oise σ q is plotted i Figure 6. The σ q i the regios <., <.4 d <.6 for the curves I ( =.), II ( =.4) d III ( =.6) respectively icreses mily due to ρ. As ρ ecomes igger whe the mplitude icreses from. to.6, so does σ q. The icrese i σ q i the regios >., >.4 d >.6 for the curves I, II, d III respectively is mily ttriuted to the icrese i ρ. As ρ icreses with reductio i the mplitude from.6 to. i Figure 5, so does σ q. Qutiztio Noise Sigl Amplitude () Figure 6. Vritio of qutiztio oise versus the two sie mplitudes. Figure 7 shows the oise mplifictio curves otied from (4) for =.,.4 d.6. Noise Amplifictio A() Sigl Amplitude () Figure 7. A(k) vritio versus the two sie mplitudes. Usig the vlues otied for A (k), the stle mplitude limits for hve ee plotted for the 5 th - Cheyshev Type II sed NTF for =. i Figure 8. Stle Amplitude Limit () Qutizer Gi() Stop Bd Atteutio (db) Figure 8. Stle limits of mplitude of 5 th -order for =.. Simultios for the 5 th -order Cheyshev Type II sed - Σ modultor show i Figure 9 were performed for 6384 smples where the iput mplitude ws icresed i steps of.. The mximum stle mplitude limits were otied d compred with simultios s show i Figure 9. Results otied i [3] were used for the DC d sigle siusoidl iputs. Stle Amplitude Limit () dcd c P r e d ic te d S im u l t e d si e s i e ( =. ) s i e ( =.4 ) Figure 9. Simultio results for dc, sie & two siusoidl iputs. The reso for vritio c e ttriuted to the fct tht the derivtio of the three gis (i.e. siusoids d oe Gussi) is sed o the modified o-lierity cocept. I order to compute the gi for y of the 3 iputs, it is ssumed tht the o-lier fuctio hs ee modified i tur y ech of the remiig iputs. However, i rel-time this my ot e the cse s ll the 3 iputs coexist simulteously. V. CONCLUSION The stility of higher-order -Σ modultors for dul toe siusoidl iputs usig the Descriig Fuctio Method hs ee predicted. The olier gis for the sigle it qutizer for dul siusoidl iput hve ee derived d the mximum stle iput limits for 5 th -order Cheyshev Type II sed -Σ modultor hve ee estlished. Accurte results for the stle mplitude curves c e otied for rge of vlues of qutizer gi i which the -Σ modultors re likely to operte. APPENDIX A I this Appedix, the derivtio of the gis for two iputs ( dul-toe siusoidl oe Gussi) for sigleit qutizer is mde. If the iputs to the olierity re of differet (Proility Desity Fuctios) PDFs or of differet mgitudes of similr wveforms, the output compoet from oe of these iputs depeds ot oly o the mgitude of this prticulr iput ut lso o the mgitudes of ll the other iputs. The cocept used here is the modified lierity cocept [7], wherey to determie the respose to prticulr iput, the olier chrcteristic is modified i tur y ech of the iput sigls preset to oti modified olierity to which the iput is pplied.

5 Siusoidl Gis The two siusoidl iputs cosidered here re x (t)=cos(ω (t)φ ) d x (t)=cos(ω (t)φ ) where, re costts, ω, ω the siusoidl frequecies, ssumed to e icommesurte, φ d φ re RVs ech hvig uiform PDF i the itervl [, π]. The secod iput is the qutiztio oise ssumed to e Gussi G(, σ) i.e. with zero me d vrice σ.the modified olierity of sigle-it qutizer with rdom iput is give y [8]: ( ) = q ( y dy (A) ) where ± is the output of the qutizer d q(y) is the PDF of the rdom iput. Therefore for Gussi iput: y = σ ( ) e dy σ π (A) O itegrtio (A) simplifies to: ( ) = erf( ) (A3) σ The o-lierity () further modified to () y oe of the siusoidl sigls sy x (t) which is give y [7]: ( ) = p( x) ( x ) dx (A4) where p(x) is the PDF of x (t). Therefore: x ( ) = erf dx (A5) π x σ () is ow the olierity of the qutizer which hs ee modified y the siusoidl iput x (t) d the qutiztio oise G(,σ). The ext step is to evlute the gi for x (t) to this modified olierity. The gi of the siusoidl iput x (t) to this o-lierity () is give y [8]: = x x r x dx ( ) ( ) σ (A9) where σ = /, is the vrice d r(x) the PDF of (t). O itegrtig (A9) we get the gi for s: x 5 = 3 F,, ψ (A7) π σ where, ψ = ρ ρ ρ ρ.. (A8) I order to oti the gi for x (t), we proceed s i ove to get: 5 = 3 F,, ψ (A9) π σ Noise Gi The modified olierity of order for Gussi iput to sigle it qutizer is give y [8]: y (, ) = ( y ) H q( y) dy σ σ (A) where H is the Hermite Polyomil of the first order. Sustitutig for q(y) d (y) i (A): y σ σ ( σ, ) = σ π ye dy = π e (A) The oise gi i the presece of other rdom iput with PDF p(r) is give y [8] = ( σ, r ) p( r ) dr σ (A) Here we cosider the dditiol rdom iput s comitio of two ucorrelted siusoidl iputs. The joit PDF p(r) of the two siusoidl sigls hvig mplitudes d, with icommesurte frequecies is: p(r)=(r/π)(/siθ), where θ=cos - {[ -r ]/}. Solvig the itegrl ove we get the oise gi s: where, = e e ζ (A7) π σ ρ ρ ρ ρ ρ ρ = ρ ρ ζ (A8) REFERENCES [] Riso, L., Stility Predictios for Higher-Order Sigm-Delt Modultors Bsed o Qusilier Modelig, IEEE Itertiol Symposium o Circuits & Systems, Volume 5, pge , 994. [] Gel, A., Vder Velde, W., E., Multiple-Iput Descriig Fuctios d Nolier System Desig, New York McGrw-Hill, 968. [3] Lot, Jswider., Al Ji, Mohmmed., le, Izzet., Stility Alysis of Higher-Order Sigm-Delt Modultors usig the Descriig Fuctio Method, IEEE Itertiol Symposium o Circuits & Systems, Volume 5, pge , 6. [4] Stikvoort, E., F., Some Remrks o the Stility d Performce of the Noise Shper or Sigm-Delt Modultor IEEE Trs. O Commuictios, Volume 36, o., pge 57-6, Oct 988. [5] Ardl, S., H., Pulos, J., A Alysis of Nolier Behvior i Delt-Sigm Modultors, IEEE Trsctios o Circuits & Systems, Volume CAS-34, No.6, Jue 987. [6] Hddd, A., H., Nolier Systems, Bechmrk Ppers i Electricl Egieerig d Computer Sciece, vol., Dowde, Hutchiso & Ross, Ic d Hlsted Press, pge 97, 975. [7] D.P. Atherto, G.F. Turull, Respose of Nolier Chrcteristics to Severl Iputs d the use of the Modified Lierity Cocept i Cotrol Systems, Proc IEE, vol., No., pges 57-64, Jury 964. [8] Atherto, D., P., Nolier Cotrol Egieerig-Descriig Fuctio Alysis & Desig, V Notsrd Reihold Lodo, pge , 98. [9] A.. Mhlis, A.. Nth, O Dul-Iput Descriig Fuctios of Nolier Elemet, IEEE Trs. Automtic Cotrol, vol., issue, pges 3-4, 965.