pulse width modu tion power

A.J. Magrath M.B,Sandler

Abstract: A new digital signal procescing technique 15 presented for power digitai-to- dnalogue converters which offers high lineal ity and a cubstantial reduction i i i clock fiequencj compared to conventional pulse widlh moduldtion converter5 The basis of tlie technique is to group together pulses from the output of a siiiglc bit sigma-deltn modulatoi A model i\

derived which shows that the system output is essentially a pulse width modulated sequence Noire and di5tortion are introduced by the pulse grouping but are concidcrably reduced using noise shaped leedback dround the pulce groupei Siinuldtion rewlts are presented M hich vnlidate the model and indicate the performance of thc techiiiquc with ideal dnd nonideal output stages

1 Introduction

In pouci digitdl-to-analogue eoinwcion a pulse code modiilatioii (K”) data stream is convei-ted to a single bit (two-level) signal which dpproximntes to the oiigi- ndl signal in the lorn frequency portion of the zpet;ti um The signal controls d high power switch and the dutput wnveforin is converted to analogue by passive 10% pass filtering (Fig 1 ) The technique i s erpccinlly applicable to dudlo applications, where the analogue arnplifiei can be completely removed from the dudio r e p oductioii chan , dthough applications alco arise where precise inotor control is icquircd High efficiencies are p o w - blc. due to the \witcliing ndture of the power stage, potentially ledding to small power supplies and hent- cinks

Due to the finite switching times of practical switches, energy losses occur on every pulse transition and therefore, foi the power witch to exhibit high effi- cicncq. the average pulse repetition frequency (PRF) of the digital \Ignal must be as I O U as possible The ater-

~ -~ ~ _ - - _ _ _ _ _ _ __ Q IEF, I Y Y G

l’dper f i ic t ieceivrd 27th I d ) 1995 and 111 remcd toir*l 22nd I m u m 10%

The nuthors nre *ith the Ilcpnrtmcnt of Flectronx m d Fle Fngmccring, King’\ College I ondon Strard I mdnn h C 2 R 2LS

age PKI i s dcfined IS t h t ieciprocal of 1he dierage time be?ween coniecuti\r 115ing edges of the pul\e stream Finite switching times a150 infiuence thc time domain propelties of tlie pulse stream and niaj intlo- duce nonlineni dlstoi tloii [ I ]

PresIous recearch hac concenti-ated on the ujc of noise chaped unifoi-nil) caiiiplcd pulse width modul- ation (PWMI [ l 71 4 I?’ bit PC M signal is initiallj oversnmpled 1- Times and noise shnped to reduce the svoi-d length to h bit5 The i inirc chaping redistrihutcs the quantisation noise to higher frequencies $0 f hat d lower wordlength 15 possible. a t rhe eupenhc of a higher sampling rate The noise shaped ddta controls a PR‘M stage. uhere the &bit \a ords dt. iampling fi-equenc.~ L F, are converted to a d a t ~ s t i = m ~ cornpiicing ringle bit ilrordc uitb a PRI’ of I_ F,, in ~ s h i c h each of the possible 2“ amplitude le.iels cori z\p(\nd\ tc, a different pulce uidth Due to the dicrete nature of the piilcc wdtlis i t 1 5 possible tc’ represent them in th r t f ~ g m l domain by a sequence o f w g l e hjt uorci$ at bit rate of 2”1, f’, U m g this repiesentcition it heconics ‘lpp- arent that a sainple ]-ate increnie c ~ f 2’ occuis in the PWM \ t a g On il pracl ic~l !clcl. thic iilciease C ~ U S C S

implernentation difticiiitirs dire to the high frequeiicrei required to clock the output bits l h e noibe iliaping considerably r e d u e s the hit 1 ate (irid Ihe corresponding clock ratc by lowering the wordlength dt thc input of the PWM dage without sncrrficing the basehni-id perforniance of the sqcteni, hawerei t)pical sjiteni parameters of L - Y and 17 - 8 atill yielll cb high bit i-dle of 90 3h.IM7 To achime thi\ hit :ate cuirent implc- mentalions use diccrete logic countet to impiement the PWM stage [2] A lower bit late \ c ~ ~ u i d be desirable t o d l l m a more compact and coit efrrctne ioiiition uwig. for example, ASIC technology

A fiirther di5atlinnlage c x l P R Z 1 i i that i t is fuiada- menrally nonlineal. generatling PK I- Ic;ii ire1 1 hnrtnonrci and sidebands, harmonic diciortiori OT tlie input signal m d intermodula~ion ifoldback I noice [3j Vanoirs Iine-

tron schemec liave been u t i l rmi f 1 2 r~-‘~lucc rh: hai- momc distortion In p$] digit4 h i p a i proce5~ing to emulnte naturallq (analnrue\ sanipled P W M , has no harmonic tlistoiIioi1 I n [SI the nnnbnearrtj i5

inodelled in !!IC frequency domnrn (ind u\ed i o del-iic an adaptive cornpencation filter

IJnf~~rtuiiatclj these scheme\ di i‘ ~ins i ic re r~fu l at coni ple tel y el im mat in g no i ce i n t ei li i L x j :I 1 a t i o 11. in wh 1 ch out-of-band noice coniponents interiiioduia~e with PRF and sideband components to p r d u c e lo noise components ~ h i c h f d l l mto the Bot11 hnrmoiiic diitoirinn and intsin~oi can bt: i educcd uinig reedback [ l hct~*uc\~*r tlie P

,4Y

saniple ralc change prevents it being applied directly and a complex architecture results.

In this paper an alternative method of generating the one bit signal is discussed, using a modified sigma- dclta modulator (SDM) architecture. SDMs achieve high linearity and relatively low bit rates using noise shaped feedback around a one bit quantiser, however, the average PRF of the one bit code is too high for efficient power 13ACs. A method is introduced which rcduces the average PRF of a SDM bitstream. The out- put of the system is modelled as a PWM signal and the PWM distortion is reduced by feedback. The applica- tion ol‘ feedback is now straightforward because there i s no sample rate change introduced in the conversion to PWM.

sise shaping and sigma-delta modulation

The general error feedback (noise shaper) topology of a sigma-delta modulator (SDM) is shown in Fig. 2 [Note 13. The single bit quantiser is a two level nonline- arity, which can be modelled as an additive error sequence e(.). With the exception of first-order SDMs, i t is often assumed that the quantisation error distribu- tion is sufficiently random to be considered a noise sig- nal [6] and this assumption is especia!ly valid when the quantiser is dithered with a random noise source [7].

ldtu rnodLilulol

In the -.-domain, for an oversampled input signal X(z) and feedback filter H(z) the output of the system is given as:

V ( z ) = X ( z ) + E ( Z ) T A V ( Z ) (1)

(2) ~ ; V ( Z ) = I -- w(,j TbJ(z) is the noise transfer function (NTF) of the system and defines the spectral characteristics of the noise in the output signal. The NTF filter is generally designed with a high pass characteristic, so that the quantisation noise is attenuated in the baseband, at the expense of greater noise at higher frequencies.

Note I : The noise shaper topology is used throughout rather than the more cornion signal feedback topology because it is more suitable for the L‘cedback structures described in Section 4.

~.__-_-._______

For implementation, H(z) must include a unit delay, which implies that the first term of the impulse response of T.,.(z) is unity. A general class of FIR noise shaping filters with this property and high pass responses are given by Tewksbury in [ X j , defined by T‘),-(z) =. (1 - z-’F. These have j zeros at z = I and a sin(Oi2Y frequency response, where 8 is angular fre- quency (0 < 8 < 2n). SDMs using these filters are sta- ble for j < 2 and correspond to ‘standard’ first- and second-order modulators [9]. For improved baseband resolution, higher-order NTFs may be used, in which TI.<--) is a high pass filter designed using e.g. a Butter- worth approximation [6, IO]. However, special care must be taken to ensure stability.

Predictions of noise performance for a given NTF depend on a reliable estimate of quantisation noise. A general framework for this analysis is quasilinear mod- elling [ I 1, 121, in which the quantiser is modelled as a gain term K followed by a stationary white random process with variance on = var{e(n)} (Fig. 3). In [ I l l a model has been derived based on two gain terms, one for the signal and one for the noise circulating in the loop; however, for noise calculations it is adequate to use only a single AC gain term [12]. For a given steady- state input signal, the value of K is a constant chosen so that the variance on is minimised and the error sig- nal e(.) becomes uncorrelated with the quantiser input. In [ l l j expressions have been derived for K and on based upon the statistical properties of the quantiser input u(n) and output v(n) and these are repeated here:

CJ; = 1 - K2E{a2(n ) } - E{v(n) t2 (4)

U(.) = U(.) - E{u(n ) } (5) where E{ } is the expectation operator. A straightfor- ward way of evaluating K and 0,” is to run a short sim- ulation of the modulator and measure the signal expectations above. An expression for the baseband power of the quantisation noise Pb, is then found by including K into the noise transfer function:

1 - H ( z ) 1 + ( K - ~ ) H ( z ) T.Y(Z, K ) =

2. ’I sig m a-delta modulator In power DACs, a low PRF is essential to ensure low power dissipation in the power switch. The PRF of the output of a SDM depends on the oversampling ratio L and the composition of the limit cycles in the output. For a stable SDM with a DC input d,, the output sig- nal v(n) has the property E{v(n)} = d,. For zero input level E{y(n)} = 0 and the output oscillates with limit cycles such as 1, -1, 1, -1, 1, -1, ... With increased input level the ‘density’ of Is increases and the average PRF falls. The maximum possible PRF of a SDM is LFsl2, which occurs for the repeating limit cycle 1, -1, 1, -1, 1, -1, ... In practice this limit cycle rarely occurs, even for zero input and 1, 1, -1, -1, 1, 1, -1, ... patterns are more common. To achieve 16-bit resolution with a

Pulse repetition frequency of a one-bit

IEE Proc.-Circuits Devices Syst , Voi 143, No. 3, June 1996 1 so

second-order modulator, an oversampling ratio of L = 256 is required [13]. The average PRF of this modula- tor with input amplitude is plotted in Fig. 4 which indi- cates a maximum PRF of 4 MHz for F, = 44.11tHz. Using third- or fourth-order modulators, the ratio L = 64 is required and experiments have shown that typical modulators with this oversampling ratio reach maxi- mum PRFs in the order of 1.2MHz, too high for cffi- cient power switching. Furthermore, as will be seen in Section 7.3, the dcpendence of the PRF on input amplitude can give rise to intermodulation noise when a nonideal power switch is used. It will be seen that a constant PRF is desirable to reduce these effects.

C

c 0 c

g2.5-

2. 2 -

E 5 - 5 0

0 01 0 2 0.3 0.4 0.5 0.6 0.7 0.8 0 9 1 input amplitude

Fig. 4 A wuge puke riperitron fiquiJncF of .seconihrdw, r.=25h, .si~gmu cleltu niod~ilutor

3 Pulse group modulation

A straightforward method of reducing the PRF of the SDM bitstream and forcing it to be constant is to group together samples with the same sign, so that after the sample-and-hold the transitions are reduced. This technique will bc termed pulse group modulation (PGM). The output is divided into frames of length N and the samples in each frame are relocated so that all ihe Is occur in a single group. The group size is given by:

v- I

IL=O

where v(n) are the output samples of the SDM and p is the group number.

1 I PGM output

a singlesided 1 1 0 0 O] 1 1 1 011 1 0 0 1 1 1 1 0 ]

b doublesidedE 1 0 010 1 1 110 1 1 0 1 1 1 1 0 1

c twosample ( 1 o o 010 I 1 1 1 1 1 o o 10 1 1 1 j consecutive

Fig. 5 Pulse grouping sc,hmcs, N = 4

Various grouping schemes are possible, with classifi- cations borrowed from PWM (Fig. 5) . In single-sided trailing edge (SS) PGM, the group begins at the start of the frame. In double-sided alternate-odd (DS) PGM, an attempt is made to locate the group in the centre of the frame. For N even and S odd or vice versa, the group cannot be Kocatcd at the frame centre, and com-

pensation is made by locating such groups alternately to the left then right of centre. In two sample consecu- tive (TSC) PGM alternate groups are positioned at the beginning then end of the frame and this results in a reduction in PRF by a factor of two For SS and DS PGM, the PRF IS L F J N , and for TSC PGM ihe PRF is LF,i%N.

To analyse the effect of the pulse grouping, consider a sequence v(n) of (1-bit) data from the SDM, assumed for clarity to have the values CO, 1) rather than ( - 1 1 , l}. Suppose at each sample instant, the sum of the present sample and previous N-1 samples is taken:

2 -1

k=O In the z-domain:

(9)

and so the summation is equivalent to a moving aver- age filter (MAF) of length N . Every Nth sample of the summation corresponds in amplitude to the group size of each PGM pulse (Fig. 6). The operation of taking every Ntli sample and discarding the remaining samples is that of decimation, and the conversion to a pulse group is uniformly sampled PWM, which involves a sample rate increase by a factor N . The decimation produces aliassing and the PWM introduces harmonic distortion, carrier and sideband tones and intermodula- tion noise.

MAF

I output

I I

3.7 Levels of aliassing noise The baseband noise power introduced by aliassing in the PGM system, ignoring the noise component due to the initial signal quantisation is given by:

, ~ - ~ ""+E P" = g 4 'Y/ jn~(B)i"lT'~(B,K)J~de (12) ._TAT

n=O Znr, K

X L

The summation includes the frequency bands which alias into the baseband and the 1/N2 term compensates for a gain of N in the MAF. This equation predicts that, as long as the aliassing noise dominates the I'WM intermodulation noise, the deterioration in signal to noise ratio will depend on the group size N, the oversampling ratio L and the SDM NTF but not the grouping type. The results of the numerical solution of this equation are comlpared to experimental results in Section 7, and the;se results confirm the above assertion.

3.2 ~~~ distortion 1 he approxirrintr theoretical tone spe;ti n for uniform PWM hnxe bren deiived in [14] for sinewave Inputs dnd these reiiills aic applied heie tn the equnnlent PGM process The tone spectiuni depeiida on the grouping type I n the followng equations D 15 the

(0, ic quency of the input tone. IJ), = h L F , firquencj of the cniiiei. J , ( ) I$ an 17th ordei Re,sel funrtiori of Ihe f i r i t kind niid a - 01, w,

3.2. I Single side trailing edge PGM:

Il~ndulnlloa depth (0 < D ' 1 I

7 1 1 -: I

3-22? Double sided alternate odd PGM:

The spcctra contain the phase distorted input frequency and harmonics. the carrier frequency- and carrier har- monics with sic!eband terms separated b j the input fre- quency. The harmonics and sidebands decrease monotonically with signal aniplitude and also with increasing carrier (pulse repetition) frequency or reduc- ing input frequency. For the DS and TSC modulation spectra ai l evcn harmonics of the carrier are zero, For TSC niodulatioii. all even harmonics of the input tone are also zero and niultiples of the sidebands are zcro when M Z + I I is even. DS and TSC modulation types offer considerably lower levels o f !iariiionjc distortion t h a n SS nioduiation [ 151, making them bettei- candi- dates for high resolution audio.

152

4

To iiiipro\~e the distortion and noise performance of PGM, error feedback can be applied around the PGM process. taking advantage of the property that the input and output data rates are the same. This tech- nique is equivalent to noise shaping, but here the non- linear error consists of aliassing noise, intermodulation noise and harmonic distortion, rather than quantisa- tion noise and distortion. The PGM nonlinearity, Ivhich introduces the noise and distortion, is modelled in the r-domain as an additive sequence W(z):

Pulse group modulation with feedback

(16) = 2- (N-l)p- z ( 1 + M7(.) The delay OCCUTS because the PGM block can only start outputting data when the Nth sample of the present block has been read in. The modified system structure is shown in Fig. 7 and the model is shown in Fig. 8. E(:) is derived and fed back to the system input 1.ia the loop filter G(z). The riN--') delay is required to compensate for the delay in the PGM block. The noise shaper is required in the loop to requantise the input to the PGM block to one-bit. It now becomes clear the noise shaper has an advantage over the more common signal feedback structure in that the signal transfer function is unity, which simplifies the design of G(z).

noise shaper r - - - - I - - - - - -

noise shaper

The output sequence IS given by I ( z ) = z - i ' - ' ){x(z) + E(z)T1x ( 2 ) + W ( X ) T E ( Z ) } (17)

\+here the quantiser noire shaping tran5fer function T I ( = ) nnd the PGM error transfer function (ETF) T,(z) die gil en by

7\ ( 7 ) = 1 ~ N ( z )

TF ( 2 ) = 1 - z-(;"-1)6 0 z

(1x1

(19) The loop filtei G(z) is designed so that Tc(z) has d high dttenuation over the baseband, and so the error power in the baseband due to the pulse grouping is reduced For implementation G(z) must include a unit delay and thc totnl I ' delay prevents the system responding immediately to error disturbancec To design coefti- cients for G(z), the class of 'Tewksbury' filters can be extended to zero-interleaved filters [ 1.51 by replacing z mith z ' to give T,(z) = ( I - z "')I These filters are

JEb P o( C i i ( u i t i Lk1i(ec . 5 i \ l VI) / 143 bo 3 Junc 1996

essentially comb filters with 7eros at 12,iIV and the effective euppresmn bandwidth reduced bq a Fdctor Y The first order filter 0 z which can be impiemented b j setting G(7) = 7 ' In the frequency domain.

1) ha\ transfer function 1

The delay limits the performance of the ey\tem wlth feedback (Section 7) and consequently it I S decirnhle to use low values of N and obtain high SNRs at low PRFs by using low overcampling I atios dnd higher- order loop filters inctead

5 Extension to higher order

An optimisation algorithin based on Timulated anneal- ing [16] has been used to find suitable higher order IIR transfer functions for H(z) dnd G(z), foi PGM syrtems uscd with feedback The motivation for ucing optimisa- tion is that due to the PGM delay, the first N terms in the impulse response of the ETF die fixed (eqn 19) and so conventional 11R filter decign methods cannot be used The optimisation algorithm searches the multidi- mensional coefficient space of H ( z ) or G(z) to find coefficient sets which minimise a cet of condition\ or cost functions, in an attempt to find the moot vtable modulator which meets a specified baseband iioi\e requirement.

For a general NTF, 7 ( z ) , the algorithm target is tcr minimise the infinity norm

(21) 7r IT(H)lL for - < Q < ~i 1, sublect to

IT(H)lA < IR(B)/L for all 0 5 e 5 E (72 ) L The constraint Imposed by eqn 21 iestricts the maxi- nium level of out-of-band iioise produced by the noise shaping, which IS a requirement of stability 111 higher- order modulatorc [lo] The Function B(0)I2 defines an upper bound on the shaping function in the baseband and to achieve a flat noise floor in the baceband, lB(0)12 has an inverse shape to the baseband error spectrum The functions T(z) and iB(0)12 are defined a\ follow (I) For the SDM NTF, T(z) = TA(z) = 1 ~ H(L) (from eqn 18) The quantiser error cpectrum is accumed to he white. therefore lB(8)I2 is a constant B, (11) For the PGM ETF, T(z) - TF(z) = 1 (from eyn 19). It can be shown that baceband power spectral density of the aliassed PGM error scquence follows a O2 function, and so a suitable defk t ion o f IB(8)l2 after appropriate normalisation is

(23 )

where Re is a constant. Due to interaction between the two nonlinearities in

the loop, it is not possible to independently define tar- get attenuations Bn and B, for a given performance using linear analysis. Nonlinear techniques are beyond the scope of this paper; however, for given system parameters N , L, a set of filters can be optiniised, then simulations performed to evaluate the performance of specific filter pair. An example PGM system designed using this technique is described in Section 7.3.

IEE Prnc.-Circuit.s l1ev1reS L ~ l J . S r . . Vo/ 143, /\o. 3. June 1996

6 detection

Nonideal power switches and pulse callisiian

The analy\is precentcd \o far hac 'tssuincd the output pouer witch 15 ideal (I e the lransitronc occur mctmta- ncoucl>). In this Sectioii we conrider the effects of finite rice and fall tirner, I n terms of liiicar errors, finite ~ I T itching times caure the magnitude re\ponse to he cittcnualed at high fri-quencies [ I ] , however. wc nre mainly concerned here with nonlinear m o r s (I e how the linearity of the conversion to analogue I $ dffccted)

A cirnple model for the nonlinear artefacts o f Finite switching times has been proposed in [F]. in which the change in area under ihe output waveform 15 modelled in the digital domain as erior inipulves occiirring on every transition A simplified approach to anal] sing the effect? or there errni-\ IC I O considei how the DC com- ponent varies with the inpnt cignal level If the PRF paries nonlinearly with the input amplitude, ac is the case with S D M system\ (Seclion 2 11, the DC compo- nent of the enol bill also wry nonlinearly with the inpii t. caiisiiig hnrrnc>nic a n d iiiterniod ulat ion prritluctc to occui

An advantage o f the PGM cycteiii 15 that the PRF is general13 constant gicing immunity to theqe efrectc. however, at Tery high input levelc overloading may occur, causing the puI\c framec to hecome full and adjacent pulces to 'collitfe' When thic ocriirc, thc PRF dccreasec A detailed discussion is not possible due to space limitatiams, howcver, a solution to this problem i s to introduce a n artditional rans sit ion hy a n appropri- ate quaiitiser invercion when 21 collision i \ detected The inve~cion is madc inside the rioise shaping loop so tha t the error 15 compencated foL in sulxeqrient samples F o r further details of relntcd dlgorithmq. rcfcr til [ I X . 191 117 Section 7 3 the tolcrnncc o l thih r7ew scheme lo unequal rise and fall t i m p c ir romparcd to standard PGM and SIIM syctems

7 Experimental and theoretical results

In Sections 7 1 and 7 2. a systen1 with an Ideal pouer \witch is considered For the majority o f the eaailnplev. a \rconcl c r d u rnodu1:itor is uced tn deinonctr nlr- thc technique and zinipirti, d m l ~ z i s I n Section 7 3 high- ordci systems are invectigated. will1 performance tar- getted dt audio quality conversion, and the effeclc of a nonide<zl putput \witch i i r ~ a150 cclnsidered

i I

7.7 Open loop PGM The performance of open-loop PGM system has been evaluated by computer simulation using a second-order SDM dithered with rectangular PDF white noise span- ning the yuantiser step. The dither is used to reduce the level of idle tones present in the one bit output of the SDM 171, which may alias back into the baseband with pulse-gro uping .

In Fig. 9, a -6dB 5kHz sinewave is applied to a SS PGM system with parameters L = 64, N = 8. Second harmonic distortion and increased baseband noise are observable due to PWM and aliassing. The lower plot also shows the direct output of the SDM. Here; third harmonic distortion is present due to overloading in the SDM.

The experimentally measured levels of noise intro- duced for SS PGM for various L and N are plotted in Fig. 10. On the same graph the numerical solution of eqn. 12 is plotted, using values of K and o,, computed in a short simulation of the SDM using eqns. 3 and 4. It can be seen that there is a close correspondence between theoretical and simulated results. In Fig. 11 . the experimental results are repeated for DS and TSC grouping types and almost identical noise figures are produced. These results indicate that the aliassing noise dominates over PWM intermodulation noise, and vali- dates the conjecture that the introduced noise is inde- pendent of the grouping type.

-11or

group size,N Fig.10 ~\ioi.~e/loor oj"sin,q/e sided PGIM Experiiiiental and theoretical theor, L = 64 -0- enp, L = 64 - + - theoi-. L 128 . 0 exp, L = 128 - x theor. L = 256 -A- cxp, L = 256 -3:-

_*-_-----

a s *

-40

* ....... -- -3

. ..... __._ ~ ~ ~ _ ~ _ ----- ~ - . ~ ~ -

m

__.- +-*-----~-- . Y-

-70 -

-8O.l / ~

-100 1,; -1 10

1 2 3 4 5 5 7 0 group size,N

1Voi.w /loor oj'doouble sided und tbljo suniple coiisecurive PGhf Fig. 1 1 Expei-iiiiental theor. L = 64 -0- exp, L = 64 ~ + ~

tilcor, L = 128 r i cxp, L = 12x x theor, L = 256 -A-

I54

cxp, L = 256 -W-

In Fig. 12, the variation of the second harmonic dis- tortion with input level, for a SS PGM system with L = 64, N = 8 is compared to theoretical results. Results are presented for 1 kHz and 5kHz input tones. The validity of the PGM model is again demonstrated by the close correspondence between experimental and theoretical results. The curves diverge where the harmonic tones reach the rising noise floor of the converter. Results are also plotted on the same graph for DS and TSC PGM, but here experimental comparisons are not possible because the tones fall below the noise floor produced by aliassing. In Fig. 13, third harmonic levels are plot- ted for the three modulation types.

input arnplitude,Log scale Fig. 12 Secoiiil harmonic drstorlion j b r SS and DS PGM, Ilcl fz and 5kH; input Theoretical aiid experimental exp 5 k SS -0- theor 5 k SS - - - theor 5 k - exp 1 k SS ~ + - theor 1 k SS theor I k ~ ~~

0.05 0.25 0.45 0 5 5 085 input amplitude,Log scale

Third 17cirnionic ~Iistortioiz for SS aiid D S sided und TSC PGM, Fig. 13 1kHz md 5kHz iiiput Thcoretical 5 k. SS -. 5 k . DS and TSC ~ ~ -: 1 k. SS ~ ~ -; I k, DS and TSC

Considerably lower levels of distortion are produced by DS and TSC modulation types, with TSC modula- tion offering the added advantage of having only odd order harmonics.

7.2 Closed loop PGM The feedback system described in Fig. 4 with Tnr(z) = (1 - and T,-(z) = 1 - z-~' has been simulated. The baseband noise power for different N and L with the three grouping types is shown in Fig. 14, and it can be seen that grouping types offer similar performance. The gain in SNR obtained using feedback compared with open loop for SS PGM is plotted in Fig. 15. As N increases, the SNR advantage decreases. This is due to the delay in the feedback signal which reduces the

IEE Pruc.-Circuit.i Devices Syst., Vu1 143. No 3, Junr 1996

effective suppression due to eqn. 20. Furthermore, the delay forces the input node error (U(z) in Fig. 7) to be larger, causing the quantiser gain K to reduce and the noise power of the SDM to increase. These effects are plotted in Fig. 16. Consequently it is desirable to use low values of N and obtain high SNRs at low PRFs using low oversampling ratios with higher-order loop filters. The restriction on N also suggests that TSC modulation is optimal as it achieves half the PRF of other modulation types with the same delay.

-40 i

- l o o ! i -1lOY '

1 2 3 4 5 6 ' 7 8 group size,N

Bu.reband noise aower o f PGM with feedback Fiq. 14 Expcrimenlal SS L = 286 -0- TSC L = 256 - 0 - S S L z 1 2 X - - + - - D S L = 1 2 X - A - T S C L = 1 2 8 - + - S S L = 6 4 -0- D S L = 6 4 -%- T S C L = 6 4 -U -

DS L = 286 - X -

35 I I

10 '51

I

1 2 3 4 5 6 7 0 group s1ze.N

SNR gain of SS PGM wrthjkdback Fig. 15 Experimentdl L = 256-0-, L = 128- + -, L = 6 4 - 0 -

0.8

0.1 1 2 3 4 5 6 7 0

group size,N Quantik gaiiderror variance 55ariution with group size N Fig. 16

Experimental K -, var - - -

7.3 High order PG M with ideal and nonideal power switches In this Section the performlance of two high-order PGM systems are compared to a high order SDM sys- tem. Both ideal and nonideal power switches are con- sidered. The optimisation technique described in Section 5 has been used in the design of three syslems, detailed in Table 1, with paraimeters F, = 44.1 kHz. L = 128.

Table 1: System parameters

NTF

Grouping type Order Bn

System Modulator

(dB) _ _ _ ~

A SDM - 4 104

B PGM TSC 6 100 standard N = 4

collision detect N = 4

C PGM TSC 6 84

ETF

Order Be

- - uniform

Dither

(dB)

PDF

5 70 none 0 . 4 P P

5 62.5 none

The filters have been designed to achieve stable oper- ation with a 0.2 p ~ p sinewave input. The differing NTF and ETF attenuations reflect the comparative sta- bility of the three systems ( A > B > C). System C is more difficult to stabilise because the additional yuan- tiser inversions increase the instantaneous quantiser error, causing the quaritiser gain to reduce.

Dither is used in the SDM system ( A ) to attenuate idle tones; however, it has been found that the two PGM systems require ino dithering, because the combi- nation of the use of high-order filters and the addi- tional feedback loop tends to increase the complexity of the quantisation error.

The systems have been simulated with a 1 kHz sine- wave input of amplitude 0 . 2 ~ ~ p using an ideal power switch and a switch with simulated rise and fall time mismatch of 1nS. The baseband responses of the three systems are plotted in Figs. 1'7 and 18 for the ideal and nonideal cases, respectively. The SNRs are given in Table 2. With ideal switches, the SNR of the systems reflect the differing filter attenuations, with the SDM system achieving the best performance. With nonideal switches the trend is reversed, and the PGM system with collision detection (C) offers the best SNR.

0 7 - '1

-140

-" -160

-1 80

0,

~

0 5 1 0 15 20 frequency, kHz

Fig. 17 with ideal power JM itch

Baxband FTT of qatimited high ordu SDM and PGM iystem;\

C- B p p p A - - -

155 IEE Proc.-Circuits Devices Sys . , Vol. 143, No 3, .Jmr 1996

Table 2: System performance

SNR (dR) SNR (dBi PRF ikWz) ideal 11 on i d ea i System

~. __ . . ~~ . _____-~ A 122.6 62,6 7697.4 - 2043.7 R 108.0 78.7 687.3 - 753 6

C 98.4 95.6 755.6 l_l.-. I.-..-- -ll__lll_r_ll--..~__l_Il--oc-sl---~.-

The range of PRFs exhibited bj, the s!;stems ha le also been determined for a I k H r sincnra:-e input \ar>.- ing in a.mplitude hetween zero and the niodulator o\;er- load point. 'l'hese results are given in Table 2. Thc PRF

'tern is greater than the PGM systenis inptit amplitude. causing noise inter-

modulation when nonideal power switches are used (refer to Secl ion 6). The PRF of ihe standard PGYl

wer than the theoretical 7O5.6kHr aiid also t drrc to pulzc group o\-e:-loadiiq. The PG34

systeni with collision detection acliiei.es a conrtant PRF, explaining its high tolerance to inisniarched rise and fall times. In conclusion. the combination o f a ion- and constant PRF makes the PGM system n.itli crlli- sioii dejection [he hest candidate for powei- DhCs.

8 ~~~~~~~~~~~

A digital signal proecssing technique has been described for power digital-to-analoue conrwters which offers lower bit clock rates anti low distortion wheil compared to conventional pulse n r i c t t l i 171odula- lion converte!~. ' f i e system i, based upmi a niodificd SIIM, whei-e low PRFs are obtained bq grvuyiiir togetlux output pulses. The grouping hac beer mod- elled a s a linear filtering, decimation anti pulse IT idth modulation process. The noise and distortinn iritro- duced by this process are reduced b>- feedback. The performancc potential has hccn insrc:ixd usihg opti-

m i d hi@ oi dei filtei s The considerably reduced bit clock rates ledd tn thc possibility of a full DSP colution \\itliou~ the need for dixrete logic counter?, iiial<ing the

tern dn rtttidctixe alternative to conventional PWM based con\citerc Soine ot the effects of nonideal povei snit i l i babe bcen reported and n modification to the PGt l \!,tern has been bi iefl) described, nhich con- bidel abl) ieduces the senc i f i~ / i t y to mismatched rise nnd fnli i imej

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