j30 dac transisiton distortion, jitter, slew rate

PAPERS

Digital-to-Analog Converterwith Low IntersampleTransition Distortion and Low Sensitivity to Sample Jitter

and Transresistance Amplifier Slew Rate*

MALCOLM OMAR HAWKSFORD, AES Fellow

Department of Electronic Systems Engineering, University of Essex, Colchester C04 3SQ, UK

Multibit digital-to-analog convener technology now claims a sample amplitude accuracyof about 20 bit. However, to achieve commensurate performance in digital audio applications,both sample timing and the complete sample waveform must also have corresponding accura-cies. The errors due to jitter and slew rate are analyzed, as they are treated as a unifiedprocess in the presence of a correlation between the audio signal and sample timing. Theconcept of jitter-equivalent slew-rate-induced distortion is introduced and an enhancedmultibit topology proposed, which offers low sensitivity to both jitter and slew-rate distortionand improves upon waveform reconstruction by exhibiting no waveform discontinuities.

0 INTRODUCTION retiming, this is not always achieved within the con-straints of practical circuitry.

· The fundamentals of sampling theory, uniform ampli- A technique is presented that lowers the sensitivity oftude quantization, and dither are well documented [1]. the DAC to sample timing errors and enables the con-If these processes are implemented correctly, the only verter to operate with band-limited signals, which elimi-errors at the digital-to-analog gateway output are a band nates rapid signal transitions and discontinuities. Jitter

limitation of the input signal and a predictable increase and slew-rate-induced distortion are analyzed and uni-in noise level. However, there is evidence [2] that errors fled, as similar errors result from correlation with the

resulting from imperfect electronics do have a deleteri- intersample values of the audio data. The importance ofous effect on sonic performance, even though the percep- controlling both pulse shape and timing is also empha-tual correlations are not completely understood. For ex- sized in sample reconstruction.ample, although multibit digital-to-analog converters

(DACs) using error-correction techniques can achieve I JITTER IN DIGITAL-TO-ANALOG CONVERSIONprecise level reconstruction, nonlinearity within the

sample transition region resulting from slew-rate and In this paper we define two forms of jitter and use theinduced jitter can produce impairment [3]- [6]. Jitter on terminology random jitter and correlated jitter, the latterthe DAC conversion clock can be nonnoiselike and arises describing a sample time displacement that is correlated

within digital circuits [7] from EMC-related interference with the state of the system. Jitter is strictly a randomand from imperfect phase-locked-loop (PLL) perform- event. However, the foregoing definitions are nowance responding to correlation between the digital audio achieving common usage in this subject area.data and pulse timing in the digital data stream [8]- [10]. In general a reconstructed sample can undergo bothAlthough these problems can be corrected [11 ], [12] by amplitude and time displacement, which together consti-

tute an error vector E, as shown in Fig. 1. However,

* Presented at the 93rd Convention of the Audio Engineering considering only jitter, two classes of sample formatSociety, San Francisco, CA, 1992 October 1-4; revised 1994 are identified:August 20. 1) Samples that are impulsive, of uniform shape, and

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noninteracting, such as switched-capacitor converters timing but also affects the sample weight, as the area2) 100% duration pulses, allowing nonlinear intersam- under a reconstructed pulse changes as it interacts with

pie interaction within the sample sequence the two adjacent samples in the sequence.

Consider a sample of amplitude At, nominal width 1_,1.1 Uniform Sampling with Jitter and and with leading- and trailing-edge jitter ATr and ATr+I,

Noninteracting Pulses respectively. The sample construction is shown in Fig.Consider a uniform and impulsive data sequence of 3. The Fourier transform Pr(f) of the rectangular pulse

sampling ratefs Hz. The jitter model for the rth sample shown in Fig. 3 can be expressed asof weight A r with jitter ATr is shown in Fig. 2, where

the error is the difference between a sample of nominal aj_f{ [ ( 1 )]location and a time-displaced version, and the impulse Pr(f) = exp -j2wf -_ + AT rweight {Ar/fs} of a sample is defined with respect to a

n°minalrectangularpulse°famplitudeArandwidthl/fs" [ ( )]}The Fourier transform Er(f) of the error for the rth - exp -j2xrf 1 + ATr+ 1

sample is 2fs

which, for {2rrfATr} << 1 and {2rtfATr+l} << 1,Ar e-J2_rf ATr)

Er(f) = fss (1 - (la) simplifies to

- arsin(Trf/fs) [cos(Trf_ (ATr AT_+I)which, for 2xrf ATr << 1, approximates Pr(f) fs xrf/fs Ar -L \fs ]

Er(f) = j _ 2_rf ATr . (lb)

Hence for an N-sample cyclic sequence, the errorE_(LfJ

N) is given by However, for AT r = 0 and ATr+ 1 = 0, the target Fouriertransform of a rectangular sample Pr(f) is expressed as

-lr_0EN = j 2"rr _'_ A r ATr e -j 2,,Lr_u (2)

PT(f) -- Ar sin(_f/f_)fs _rf/f_

Eq. (2) shows that the error spectrum is proportionalto the harmonic number L of the sequence repetition

frequency fdN Hz, but that the microstructure of thespectrum depends on intermodulation between the pulseweighting sequence {Ar}/Vand the pulse jitter sequence ',

error ,"{AT,}N. vector _._,. :E Jl .......1.2 Uniform Sampling with Jitter and Samples '.7 6awith Nonlinear Interaction _'-- -.

......Although sample timing errors give rise to the errorspectra described in Section 1.1, it is more common for

a DAC to use 100% duration sample reconstruction. This targetsample relocatedsamplemay arise directly from the DAC output or via a sample- location_ jand-hold circuit used to eliminate glitches during DAC

sampletransitions.Althoughthis strategymaximizes t

signal energy and improves immunity to system noise, Fig. 1. Error vector resulting from simultaneous amplitudethe effect of sample jitter now not only modifies sample and time errors of a sample.

input sequence

Ar 5(0) time displacement _ )-- error sequencef (jitter)

Fig. 2. Elementary model of sampling jitter.

902 J. Audio Eng.Soc., Vol. 42, No. 11, 1994 November

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Thus the error spectrum ERr(f ) = PT(f) - Pr(f), that is,

ERr(f ) =ar[COS(_sf)(ATr-Arr+l) + jsinlXrf_(Arr +[fs,] Arr+l) 1 . (3)

Summing the terms over N samples of a cyclic sequence, the Fourier transform E_(Lf_/N) is

N-IAr (AT r "ff Arr+l)e -j2xtLr/N (4)ERN = cos _- Z Ar(Arr-- Arr+l) e-J2_Lr/N + jsin _- r=_Or=0

Comparing Eqs. (2) and (4) there is now an in-phasecomponent weighted by a cos(wL/N) multiplier that ex- tabulated below and computed over N = 4096 samples,tends the spectrum to dc, where correlation between where the sampling frequencyfs = 44.1 kHz,timing error and signal results in a complicated error

fs H-spectrum that may not be masked by program material, fo = fi = 752f0 HzAlternatively, an impulsive error sequence can be 1o- _/ z

cated at the interface between adjacent samples wherethe impulse weight is proportional to the pulse-area error f2 = 1760fo Hz d = 10 nsresulting from jitter, whereas the impulse timing corres-

ponds to the jitter. This error impulse sequence has si- { (rfxh sin(2wrf2_ }multaneous amplitude and timing modulation, where A r = sin 2w fs] + _, fs /

pulse-area error = {Ar+ l - Ar} {fs ATr+I} ·

ATr=d{sin(2xr_)+ sin (2_r rf2'_ }x fa/ 'An error pulse is assumed rectangular with the leadingedge located at t = (r + 0.5)/f s and the trailing edgeat t = (r + 0.5)/f, + AT_+I. Thus with reference to the Here fo is the sequence repetition frequency, fl and f2

error pulse center the pulse timing error equals AT_+1/2. are the selected signal frequencies, and d is the jitterHence the Fourier transform of the rth error pulse in the noise. In this example fo _ 10.8 Hz, fl _- 8.096 kHz,sequence is

ERr(f) = [ar+l - Ar][ATr+sf_] exp [-j2_ ( r + O'_5+ O'5 ATr+t) ] 'fs

By forming a summation over an N-sample cyclic se-quence, a discrete transform follows as

ERN = r=_0 (Ar+ 1 -- Ar) ATr+ exp -j (2r

To illustrate example error characterizations of jitterwhen mapped onto and correlated with the audio data and f2 _ 18.949 kHz.sequence, sets of error spectra are presented. The first In the following simulated results all spectra are refer-set uses the data and jitter sequences {Ar} N and {ATt}N enced to the input sequence {Ar}N, which is designated 0

erTrea error r_. 1 Ar+l

Ar-1 !_ i _tI I

aT,_ m.41.- £ al;+] .4_2r-0.5 f_ r+0.5

q q

Fig. 3. Construction of 100% rectangular samples with jitter.

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HAWKSFORD PAPERS

dB. The results shown in Fig. 4 correspond to impulsive three-dimensional plot illustrates the effects of varyingsamples as described in Section 1.1, those in Fig. 5 to levels of random jitter noise together with the correlatedthe 100% duration samples described in Section 2.1. displacements of the sample locations for the case ofThe lower frequency components are now predictably 100% samples, where J. is defined asof higher level.

The second set of results uses similar correlated se- Jn = d 100"25(1-x)

quences {At}jr and {ATr}_¢,but now a weighted random for 1 _<x <_ 16 in unit steps of x and d = 10 ns.

noise sequence is added to the jitter function so that Finally, in the third set a modified jitter sequence is

simulated to demonstrate the effect of incorporating a

d_sin(2arrfl_+ sin(2_r rf2_ +J, Rand(r) slowly varying frequency-modulated jitter componentz_Tr=L \ fs/ --J\ fs ] that is superimposed upon the correlated components

already described. The frequency modulation is sinusoi-where Rand(r) is a random function with a triangular dal with a frequency equal to the sequence repetitionprobability distribution function spanning -1 to + 1, frequency f0 Hz, andand/n is the noise weighting factor.

Figs. 6 and 7 show the corresponding results with and ATr=d{sin(2_)+ sin(2_ }without 100% sample reconstruction, where J_ = 10 ns, \ fs ]meaning that the jitter probability distribution function

is triangular and spans - 10 ns to 10 ns. In Fig. 8 a + Jn Rand(r)

-60

-80 ....................,..............................................._............................................................................................._-....................................................................t

-100 ............................................._...........................................................................................................................................................................

- 120 ........................_.........................................................................._...................................................................................................................•..................

- 140 ................................................................................................i.........................................................................................................................................

-160 ........................_..................................................................4....................................................i.............................................

-200 ...............

-220 ............... _........................i ....................

-260 . i i i * i i i i0 200 400 600 800 1000 1200 1400 1600 1800 2000

binnumber

Fig. 4. Output jitter spectrum; no random jitter, impulsive samples.

-60

i

-60 .................._........................._...................................:..'...................................._................_...............] ............_.............

-lOO...............i..................i..................................i....................................i........................i...................]...............i..............-_oI-..............._.................._.........................................i........................................i...................!........I1'................_...............-14o_................i.....................i........................................i.........................................i...............i.......l-li................i.................

..................i.....................i........................................................................................................t Ii........................"................-1.o_.................i......................_,....................................;......................_................!......................i.........,i....................._,................

-240- i i i i i i i i i0 ZOO 400 ;00 800 1000 1200 1400 1600 1800 2000

bin number

Fig. 5. Outputjitter spectrum;no randomjitter, 100%samples.

904 J. Audio Eng. Soc., Vol. 42, No. 11, 1994 November


-60

-80

-100

-120

-140

i-16o , , , i i (

i i i-180 , , i , { , , ,

0 '200 400 600 600 1000 1200 1400 1600 1800 2000

bin number

Fig. 6. Output jitter spectrum; including random jitter, impulsive samples.

-60 [

-70 ............................ r f

_ ,_ _:.............j-100 '

-120

-130

--140 {

-150 - i i i i [ i i i i0 200 400 800 800 1000 1200 1400 1600 1800 2000

bin number

Fig. 7. Output jitter spectrum; including random jitter, 100% samples,

spectrum

an' )litude x_k

\

' 00OFig. 8. Jitter spectral family with varying levels of random jitter noise.


HAWKSFORD PAPERS

where where the integer parameter h is scanned over a range

r\ of 1 _<h _< 16, which in turn addresses the sine-squaredetfm= 1 + Jfm sin 2rr_) amplitude weighting.

Jr,, being the modulation depth, 0 _<Jfm _ 1.2 SLEW-RATE-INDUCED DISTORTIONThis additional jitter component is introduced as a

frequency modulation of the periodic jitter sequence, The performance requirements of multibit DAC elec-

where the modulation depth is Jfm. The results for Jf,_ tronics for digital audio systems are stringent. First the

= 1.8 × 10 -4 and Jr,, = 1 for J, = 0 ns are shown in reconstruction levels must be accurately specified. EdgeFig. 9, whereas the three-dimensional plots in Figs. 10 jitter must be minimized. While it is a primary function

and 11 illustrate the modification in spectral form for a of clock performance, it can result from internal circuitryrange of modulation depths over 1.8 × 10-4 _ Jfm _ exhibiting variability on propagation delay and response1 for Jn = 0 ns and J, = 1 ns, respectively. Finally time as well as electromagnetic interaction between sys-Fig. 12 shows a family of spectra to demonstrate that tern modules. Finally the trajectory of the signal betweenthe total level of noise and distortion varies as a function

adjacent samples should be determined by a linear net-of signal amplitude. Here Jn = 0.1 ns, Jfm = 0, d = work, forming, for example, an exponential curve.10 ns, and the modified signal function is However, because of the rapid response times encoun-

sin2(h_r_sin(2.trrfl_ + sin(2_rrf2_ _ tered, even when sample-and-hold circuitry is used asaAr\ 17 / I. \ f_ ] \ fs / J deglitcher, momentary nonlinearity can result in a small

-60

-80 ..........i ....... :.............f...........i.............. i......... i ...............................i..........

T r,rr'I'r-,20.........................i ............................................

!?:.. .....i .

...........................................-2200 200 400 600 800 1000 1200 1400 11500 1800 2000

bin number

Fig. 9. Jitter spectrum with FM jitter component, used to calibrate Fig. 10.

Jn=0nssp_\l \\\_ N_,_\\\\\_,\\\/\

2000

Fig. 10. Family of jitter spectra for varying FM depths; no random jitter.

906 J. AudioEng. Soc,, Vol. 42, No. 11, 1994 November


change in the pulse area, which is related to intersample wherebyvalues. Because of the small time duration of each non-

linear event, ideal rectangular transitions are assumed %with the error modeled using an equivalent sample jitter ATnr= + 2' (6)component ATnr.

Fig. 13 shows a nonlinear transition between twoideal rectangular pulse

100% rectangular pulses of weight A r and Ar+l, con- Iossofpulsearea lacedAT.r

ConsiderStrainedby a constant slew rate S V/s, transitionWhichresults in Ar+l __r'w.

a loss of pulse area/_kAr, where I_

l_kAr = (Ar+1 -- Ar) Tr ' Ar ii

a rectangularpulsewherethe is dis- iI Im t

placed from its nominal location by ATnr such that {pulse _ aTnr-----_ iarea of Ar+l} -- {pulse area of Ar} = -_,A r that is, I_- t _,

i

r

fs

gr+l ( l_s- Arn r) (j_ q_ Arnr) = _AAr_Ar I Fig. 13. Two adjacent samples linked by dominant slew-ratedistortion.

Jn = 1 ns

spectrumamplitude

Jfm

2000

'0

Fig. l 1. Family of jitter spectra for varying FM depths; with random jitter.

spectrum

mplitude

· [2O00

Fig. 12. Family of jitter spectra with varying levels of input signal.

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But for a constant slew rate, ized by observing distortion products when processingband-limited audio signals. Hence this amplifier stage

S- Ar+_- Ar requires band limitation and slew-rate prevention to% yieldlow correlatedequivalentjitter, where it is sug-

gested that wide-bandwidth, open-loop transresistance

and thus convertersarethepreferredchoicewith possibleprefil-tering to reduce the bandwidth of the DAC output

AT,_ - A_+I - Ar (7) current.2S Also, since most DAC transresistance stages operate

with 100% pulses, the effect of jitter (random or slew-Although this example is idealized, it demonstrates rate equivalent) is increased. To demonstrate this conjec-

how an equivalent correlated jitter component ATnr can ture, the equations presented in Section 1.2 are usedbe assigned to the rth sample, and thus the results in together with Eq. (7) as well as the following data toSection 1.2 can be used. The analysis ignores spectral generate the error spectrum shown in Fig. 14:changes relating to pulse shape, as these events are of Number of samples N = 4096short duration compared with the Nyquist sampling pe- Sampling rate (2 times oversampling) fs = 88.2riod. The use of an equivalent jitter time defined in kHzassociation with slew-rate distortion and other related Transresistance amplifier slew rate S = 50 V/l_Snonlinearities in the current-to-voltage (transresistance) Sample sequence generatorstage of a DAC enables a unification of this class of A r = {sin(2_rrfl/fs) + sin(27rrfJf_)}problem, where Eq. (7) together with Section 1.2 permit Random jitter J, = 1 nsspecifyingthe performance. Correlatedjitter d = 1 × 10-18s

Equivalent jitter resulting from slew-rate distortion is where f0 = fJN Hz, fl = 992f0 Hz, and f2 = 512fo Hz,potentially more serious than random jitter because of that is,fl = 21.36 kHz and f2 = 11.025 kHz. No otherthe natural correlation between slew limiting and the correlated jitter source is included.

data samples. Although with appropriate design tighter Finally using similar data, a three-dimensional plot islimits can be achieved, pulse jitter greater than = 10 ns shown in Fig. 15, where f_ is scanned linearly in 32has been reported to be of audible significance, and steps from 689 Hz to 22.05 kHz,f2 = 11.025 kHz, andthere is anecdotal evidence to support a much tighter d = J, = 1 × 10-18 s. The surface shows tracks ofspecification. However, using the 10-ns criteria and as- intermodulation distortion products, where the calibra-suming by way of example {A_+l -- A_ --- 500 mV}, a tion of the vertical scale can be estimated from Fig. 14transresistance amplifier should exhibit a slew rate that correspond to trace 31 where f2 = 21.359 kHz.greater than 25 V/ixs.

Even if slew-rate limiting does not occur, an opera- 3 DAC TOPOLOGY WITH LOW JITTER ANDtional amplifier may be close to its open-loop limits SLEW-RATE SENSITIVITYduring periods of rapid signal transition, and this maycontribute momentary "packets" of distortion. There is 3.1 System Topology and Functionlittle doubt that transresistance stages used in DAC sys- To reduce DAC sensitivity to slew rate and jitter, theterns can contribute distortion that is not fully character- rapid signal transition at each sample boundary must be

o

iiii .... •il iiii fillii i-801 .............................................................._......................................................................................................................

-i00 ............................................................. _.................................................................................._..........................................

.................i...............................

-tso ; ] { i , , , i0 ZOO ,too 8oo aoo lOOO 12OO 1400 1600 1800 ZOO0

bin number

Fig. ]4. Distortion spectrum resulting from slew rate used to calibrate Fig. 15.

908 J. AudioEng. Sot., Vol. 42, No. 11, 1994 November


minimized and a degree of intersample isolation intro- value, the reference voltage applied to the other MDACduced to control the distortion described in Section 1.2. is now at zero, at which instant its data are updated inEffectively the operating bandwidth of the DAC should a similar manner. The process then proceeds at a uniformbe lowered so that a more "analoglike" conversion is rate, with data being updated on the corresponding zeroachieved at the gateway of data conversion. The wide of each raised cosine reference signal.bandwidth generally encountered is an artifact of topol- The net result of this process can be summarized asogy that is typified by the current-switching DAC in follows.association with a wide-band transresistance converter. 1) Data conversion only occurs when the reference

The proposed topology consists of two time-inter- to an MDAC is zero. Thus the contribution of jitter isleaved DACs operating with mild oversampling, but minimized. Effectively, the jitter dependence is trans-where the DACs are configured as multiplying converters lated from the digital data to the two raised cosine refer-(MDACs). The reference inputs of the two DACs are ence signals.raised cosine waveforms with low inherent jitter. The 2) Because the current output of each DAC tracks abasic system is shown in Fig. 16 and uses a 4 × oversam- raised cosine, the rate of change is reduced comparedpiing filter to enable initial interpolation of input data. with rectangular switching. Thus slew-rate-induced dis-Fig. 17 shows a series of illustrative waveforms to dem- tortion and other minor nonlinearities within the trans-onstrate operation, resistance converterare virtually eliminated.

The output of the 4 x oversampling filter is multi- 3) Reduction of high-frequency spuriae and the useplexed alternately between two MDACs (MDAC_ and of 4 x oversampling relax the design of the analog re-MDAC2) using sampled latches, where conversion oc- construction filter, and the output signal from the con-curs on the alternating data sequences D_ and D 2. The verter is more "analoglike."DACs therefore run at 2fn_Hz, and output pulses overlap 4) Because a raised cosine consists only of a dc termby 1/f,s, f,_ being the Nyquist sampling frequency, and a single spectral line, noise filtering to reduce jitter

Although the data applied to each DAC are held con- is simplified.stant for two consecutive samples, examination of the 5) Any imbalance in gain between MDAC_ andrespective raised cosine reference waveforms R_ and R2 MDAC 2 is of little consequence and only causes a mildshows each reference voltage to be zero on a conversion increase in the spectral replication at 2fns HZ, whichedge. Consequently assuming that there is no pulse feed- because of 4 x oversampling, is located well above thethrough in the MDAC, any jitter on the data edge is audio band.attenuated. In a practical system, circuitry would ar- 6) Reduced bandwidths of signals within the converterrange for data to be transformed only when the reference mean that circuit layout and parasitic and mutual cou-is zero. However, because of the near zero slope of the pling of circuit elements are less problematic.reference voltage waveform in close proximity to its Cautionary Note. To achieve the performance speci-zero value, the timing of data transition is noncritical, fled in the preceding, the cosine weights for all samplesOnce the data are latched into an MDAC, the reference should be identical. Consequently the frequency re-voltage (controlling the gain of the MDAC) rises from sponse of the MDAC from reference input to currentzero, thus causing the output current I_ or 12 to change output must not be code dependent. This is not a funda-in direct proportion to, but weighted by, the present mental problem, but it does require appropriate attentiondata value. When the cosine waveform reaches its peak in the design of the MDAC.

amplitude

0';___

f2

trace number(scanning fl)

20

Fig. 15. Scanned distortion spectrum resulting from slew rate.

J.AudioEng.Soc.,Vol.42,No.11,1994November 909

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is given by3.2 Estimate of Jitter Suppression

Fig. 18(a) shows sample reconstruction for a single AA r = _ TO+ATrDAC, where the data conversion timing is optimum. J/=0 (Ar+2 - At) [1 - cos(4,rrfnst)] dt.However, if a timing offset TOand a superimposed jitter

component ATr are introduced at the gateway of data After integration,conversion, then the waveform shown in Fig. 18(b) re-sult. In this example only even samples in the over-

sampled data sequence are shown. Z_Ar----(Ar+2_ At)r _T0 + ATr_ sin[4,rrfns(To+ ATr)]_The result of this timing error is a loss in pulse area, 1, 4_rfn_ J

whichcanbe estimatedasfollows.Thepulse-areaerror (8a)

D1 11

NyquistPCM 2 fas(even) I _

dalAf q filter _ reference +_/ O/P'"7 ?I

4 f LATCH _ MDAC2_ 7'

Fig. 16. Basic two-interleaved MDAC topology.

TInput data |

(Nyquist pcm' fns) T / T T

4 fimesinterpolateddata T T T T T T T T I I T T T _ T T T T T

Rlraisedc°sinel_ference j_/'_/_/'_/_/-_/'_j_/'_j_

R2raisedc°sinerefevence /-_j_/-_/_/_/_/_/'_/'_/-_

onM_AtC_u_Snetdc_a¢_lweighled/_/,l__/_j_/_/_ _

MDAC_ laised cosine weighled /_/_°utput_'urrent:dataO2 /_/_ /_/_/_]_/_/

composite signal

cosine interpolation

Fig. 17. Illustrative waveforms showing raised cosine interpolation in time-interleaved DAC topology.

910 d.Audio Eng. Soc., Vol. 42, No. 11, 1994 November

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which, for 4.rrf.,(T0 + ATr) << 1, simplifies to The Fourier transform Fc(f) then follows as

AAr = (Ar+ 2 - A r) (_fns) 2 (TO+ATr) 3 . (8b) Fo(f) = 0.5 [l + cos(4xrfnst) ] e -j2'_ftdt- l/4fos

Defining an equivalent time offset and jitter error re' r forthe rectangular pulse shown in Fig. 18, that is,

(Ar+ 2 -- Ar ) rer _ _u_ r

1

yielding Fc(f) = 8-_ [ _,2f.s \2fJ

8rer = 5 ('Irfns)2 (r° + Arr)3 (9)

+ I_ _,2f.s+that is,

rer 8 where sine(x) = sin(x)/x.TO + ATr 3 [xrf"s (TO + ATr)]2 (10) The time and the corresponding Fourier transform of

the time-limited raised cosine pulse are shown in Fig.

Eq. (9) estimates the equivalent timing error using the 19. The Fourier transform shows that there is a signifi-raised cosine sample format compared with the case us- cant response to 3f.s, but for 3f. s and above there ising rectangular samples, whereas Eq. (10) reconfigures attenuation. However, because a 4 × oversampling filterthis result to show the corresponding reduction in depen- is prescribed, the first spectral replication in the finaldence of the timing errors. For example, let reconstructed signal is centered on 4fns Hz and extends

-fas/2 Hz. Consequently the inherent attenuation offered

(TO + ATr) = 200 ns, fas = 44.1 kHz by the raised cosine waveform at 4f_s (noting that atexactly 4f_s Hz the Fourier transform is zero) signifi-

whereby the effective reduction in timing error _ 2 x cantly suppresses the first spectral replication and thus10 -3. This demonstrates that a DAC topology with a relaxes the design of the analog recovery filter. Indeed,response or settling time of only 200 ns is adequate, the spectrum in Fig. 19 suggests that the analog filter

The analysis demonstrates a remarkable reduction in can be designed to have a band-reject response centeredsensitivity to jitter within the digital data stream. This on 4f_s Hz or, alternatively, a twin resonant circuit withimprovement is partially dependent on low jitter in the rejection bands centered on 3.5fa s and 4.5f.s, respec-raised cosine waveform and a reference voltage that ac- tively, followed by a mild low-pass filter response tocurately attains a zero value at the minima of the raised reduce out-of-band noise and spuriae.cosine function. However, the form of the raised cosine Finally, because of the form of the raised cosine trans-waveform with only two spectral lines, dc and 2fas Hz, form Fc(f) described by Eq. (12) and to enable a flatmeans that band-pass filtering can achieve a low inherent audio passband, mild linear-phase equalization shouldjitter performance that is considerably more effective be included in the oversampling filter to match the in-than smoothing a high-frequency rectangular clock, as verse response over the frequency band of 0-0.5f_s Hz.the equivalent noise bandwidth can be made lower be- It may also be expedient to include a minor correctioncause of the lower number of contributing harmonics, for the analog reconstruction filter transfer function, al-Also the band-limited raised cosine reference waveforms though in practice this will be small.can be applied directly to the MDACs without additionalnoise-inducing counters and logic circuits. These fac- 3.4 MDAC Nonlinearity in Reference Signal Path

tors, together with an almost total independence of digi- Nonlinearity within an MDAC can be modeled bytal data jitter, are the principal attributes of this new assuming a perfect DAC combined with a dynamic non-DAC topology, linear network in the reference input, as shown in Fig.

20. Because the modified MDAC has now been desensi-

3.3 Spectral Response of Raised Cosine tized to distortion components generated at the data tran-Modulated Samples and Requirements on Analog sition, the residual errors are solely dependent on theReconstruction Filters accuracy of pulse amplitude reconstruction and the non-

The requirements for analog signal recovery subse- linearity within the circuitry associated with the refer-quent to digital-to-analog conversion can be determined ence (gain-defining) input. If the MDAC is assumedby analyzing the combination of 4 x oversampling and ideal, then nonideality can be modeled by a nonlinearthe overlapping raised cosine weighting that is associ- network in the reference channel where pulse amplitudeated with each sample. The impulse response he(t) of the errors can be accounted for by allowing the data inputtime-limited raised cosine generator can be expressed as to modulate this network. We thus identify two possibly

interrelated error mechanisms, which can cause distor-

hc(t) = 0.5 {1 + cos(4_rfr,st }rectl/2f_ (t) . (11) tion in the reconstructed output. However, this model


HAWKSFORD PAPERS

is also useful as it allows us to define succinctly the and consider adjacent samples of V0 such that the ampli-conditions that prevent distortion from entering the criti- tude separation of A,+ _ and Ar is maximized when usingcal audio band. If the only nonlinearity is in the reference 4 x oversampling (that is, 4f. s Hz). Thusinput channel, and the data input in no way alters this

nonlinearity, then the result is only an addition of hat- ()rrmonic distortion to the raised cosine waveform. Thus at A, = -A msinthe output of the MDAC we would expect to observe amodest level of spectral replication of the input about

these harmonics, which can readily be removed by the (_r)output analog reconstruction filter and thus is of little Ar+ 1 -- A m sin _- .consequence. It is only when the data sequence modifiesthe nonlinearity that level reconstruction errors occur,

Hence from Eq. (13) the minimum slew rate Smi n of thewhich will result in output distortion. However, this istransresistance converter is

fundamental to all multilevel DACs, and this system isno exception to this class of distortion.

3.5 Estimate of Maximum Rate of Change of Smi n = 4"rrf._Amsin _- . (14)Signal at Output of Transresistance Converter

Consider two adjacent samples Ar and Ar+ 1 separated By way of an example, let A m = 2N/2 V andf, s = 44.1by the oversampled time interval 1/4f, s. The DAC at- kHz, whereby Stain = 1.12 V/IJ-S.tempts to edit these samples by a half-cosine wave inter-

This basic analysis shows that for a standard 2 V,_spolation, as shown in Fig. 17. The amplitude of this output signal the maximum rate of change of the outputhalf-cosine segment is therefore {0.5(Ar+ | - At) } suchthat over the sample interval 1/4f,_ the reconstructed signal for the transresistance converter is constrained.signal is Consequentlyaperformancecommensuratewithlowin-

band distortion is simpler to achieve and should be com-

V0 = Ar + 0.5(Ar+ 1 - Ar) I1 - cos(4_rfnst)] . pared with the example given in Section 2.

The maximum slope of the output signal is therefore 4 EXPERIMENTAL RAISED COSINE DAC

dV° max= 2_rfns(Ar+l -- Ar) • (13) An experimental raised cosine DAC is shown in Fig.dt 21, which uses commercially available parts. The designemployes a Micro Power Systems MP7616 16-bit CMOS

Assume a maximum amplitude-coded sine wave of fre- four-quadrant multiplying DAC having the basic archi-quencyfJ2 Hz that has the analog form tecture shown in Fig. 22. (Although this is a 16-bit

device, it is only of marginal performance for high-V o -- A m sin(_rf.st) quality applications offering a current settling time of 2

Ar+2 -- 2At+ 2-

At- II 2Ar--_ I /t _, t/

rectangular data reconstruction raised cosine data reconstruction

(a)

2A_+2- -f'_ error areaerror area I

2At_ LXA_\ ] I A__

AT

(b)

Fig. 18. (a) Rectangular and raised cosine sample reconstruction (samples weighted to have same area). (b) Effect of timingoffset and jitter on reconstructed raised cosine samples.

912 J. AudioEng. Soc., Vol. 42, No. 11, 1994 November


!xs to 0.01% of FSR. However it was the only device set to form the two raised cosine waveforms. The PLLavailable at the time of experimentation.)To lower toler- and the bandpass filter are also instrumental in reduc-ance on resistor matching, the MP7616 features 15 equi- ina jitter.valued current sources with inputs decoded from the This system validated the operation of the raised co-four MSBs. The remaining 12 bit are then converted sine DAC, and measurement confirmed cosinusoidal in-using a binary weighted tree. A key feature of this DAC terpolation between adjacent samples, thus relaxing thein the present application is the bipolar reference input, slew-rate requirements for the current-to-voltage con-which is driven by one of the two raised cosine wave- verter. In this sense the DAC can be seen to filter theforms. MDACoutputcurrentwaveform,but withoutcompro-

SPDIF serial digital data are decoded by a Yamaha mising noise performance, which occurs when a filterreceiver (YM3623), and the sampling rate is increased is placed between the DAC and the transresistance con-eight times using a Burr Brown DF1700. The over- verter. Also such filters do not reduce the jitter presentsampled data are next converted to a parallel format and, in the source data. However, in this experimental modelvia two alternately clocked 16-bit latches, input to the the resolution of the DAC and its settling time weretwo 16-bit MDACs. Complementary parallel data allow limiting factors.the use of in-phase raised-cosine waveforms as defined

in Fig. 21, where complementary dc offsets result in a 5 CONCLUSIONzero-mean output current when the two MDAC outputsare summed, thus simplifying the design of the transre- This paper has described a technique of time interleav-distance converter as no dc correction is required, ina two MDAC converters using complementary raised

Raised cosine waveform phase alignment is main- cosine generators applied to the reference input. Thetained by including the cosine waveform generator effect of this process is to replace the normalrectangularwithin a PLL, as shown in Fig. 23. The voltage-con- pulses in a 4 x oversampled converter with raised cosinetrolled oscillator (VCO) of the PLL drives two analog weighted samples that are time limited to span two con-gates to produce a symmetrical square wave, which is secutive samples in the oversampled data stream. Thesubsequently band limited by a second-order bandpass effect of this process is a reduced sensitivity to datafilter.' Two in-phase signals are formed on secondary timing jitter and transition distortion as well as a reduc-windings coupled to the tuned circuit, which are dc off- tion in the slew-rate requirement of the transresistance

amplifier. It was also shown that the requirements of theanalog recovery filter at the DAC are relaxed.

Theeliminationof edge-transitiondistortionandthe

i!_i i'ii ZlZll transformation of jitter dependence from the DAC data

i' III'IIIIii i i ii.i. }ii1.11117iii i clocks to the raised cosine generators are seen as pivotal

....i 4 ..i _ i i in the design, as is the reducedbandwidthof the signal

, i.: ...... ; : : presented to the transresistance stage. A signal (the ref-

ii ii iiii ii erencegeneratoroutput)that consistsof onlytwo spec-i tral lines (dc and 2JnsHz) is simpler to filter to suppressnoiseandspuriae,thesourceofjitter, thanin thecaseof a broad-band square wave. Even if a square wave

,_oo timingsignalis combfilteredto containonlythefunda-

_ii_ mentaland its harmonics,there is still a finitenoise

bandwidthassociatedwith each harmonicwhich is

_0 greaterthanthatof theraisedcosine.To support the proposal for a DAC with low jitter

sensitivity,ananalysiswaspresentedthatdescribesthe

0 mechanismbywhichjitter canintroducedistortioninto! - ::i:: ! : :_ :::: : ':: : ::i:: 5 :: the audio band. It was shown that jitter that mapped to

? -2 o 2 pulse-area distortion as well as producing timing errors

Fig. 19. Raised cosine waveform in both time and frequen- resulted in greater low-frequency distortion. For jittercy domains, that onlymistimeda sampleevent, thedistortionspec-

Data input _,l MDAC)

'1 Id esi current output

Raised cosine Nonlinear l

generator _ network "_rererence

input

Fig. 20. Basic model of MDAC nonlinearity.

d.AudioEng.Soc.,Vol.42,No.11,1994November 913

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trum was proportional to the frequency, whereas the form of the distortion remains the same, but in direct

inclusion of area modulation extended this response to proportion to the signals. This is true whether or notdc. Analysis and computer simulation revealed that the there is correlation between signal and jitter. To illus-distortion was more problematic where a correlation ex- trate this feature, Fig. 12 included both random andists between timing error and program material. Indeed, correlated components.in many practical digital systems the use of PLLs with A powerful extension of the jitter analysis was toinadequate timing recovery can yield correlation, even consider edge transition distortion resulting from slew-though this process can be highly nonlinear. For exam- rate limiting in the transresistance converter, or indeedple, where a PLL responds to a change in the data se- inherent within the DAC, and to translate this to an

quence, even though this is in coded form, correlation equivalent edge jitter when using 100% rectangular sam-can exist and a highly complicated jitter spectrum result, ples. The transformation revealed that edge jitter, slewwhich even though of very low level, does not necessar- rate, and related transition distortion fall into a common

ily fall under the auditory mask causing perturbations regime and that a similar analysis procedure can be used.of the detection thresholds within a number of critical However, with DAC transition distortion the correlation

bands [13]. It can be argued that an inherent design with the signal will almost certainly be higher, implyinglimitation of the SPDIF serial code is the nonscrambling a greater subjective significance. Although slew rate isof serial data by coding to break the correlation between a dominant distortion, it should be recalled that the oper-audio data and bit-pattern-induced jitter. If scrambling ational amplifier, at the edge transition, is operating nearwere used, any resulting jitter that is related to the serial open loop, so although the transition may appear wellbit pattern would be decorrelated, and therefore wouldproduce only a noiselike residue of benign character.

This is a conjecture developed in a supporting paper [8]. v c

Example results for both correlated and random jitter --_,'_Iil

were included as well as the effect of adding low-fre-

quency modulation. Fig. 8 demonstrated that the corre- r- -I raised cosineo/p

lated and random jitter components did not intermodu- 01 band-passfilter _ 5(1 + cos(2_f0)late and can be considered essentially additive for a , _o..r-I =352kHzcfanalogconstant-amplitudeinput sequence. However,the inclu- gat_ , sv

o ±sion of low-frequency (sinusoidal) modulation of the ,z ,

jitter sequencemimickinga low-frequencyerror in a ] +PLL, for example, produced significant levels of inter-

modulation. Figs. 10 and 11 demonstrated this interac- ',_.. raised cosineo/ption both with and without a random jitter sequence.However, because all the jitter-induced distortion spec- 'Vc ' $(] ' cosa_ft))

tra are dependent on the amplitude of the signal se- I i.21

quence, the spectral levels should all be read with respect fp Phase-lockloop

Ito the signal level, whereby if the signal is reduced, the ipl[ 8timesNyqist(=352kHz)distortion changes in direct proportion. To demonstrate

this inherent characteristic, Fig. 12 showed the distor- Fig. 23. Raised cosine generation using bandpass filter andtion spectrum as a function of signal level, where the phase-locked loop.

VDD

Ireference Equally weightedcurrentsourcesinput

_._ O l°ut2I I I I I I I I I I I I I I tTo switches '0 aFB

I* To 12-bit DAC

1 2 3 4 5 6 7 8 9 lO 11 12 13 14 15 16 GNDMSB LSB

Fig. 22. MP7616 four-quadrant multiplying DAC.


HAWKSFORD PAPERS

behaved, there may nevertheless be some distortion in-duction. In this paper the results presented are relatively 6 ACKNOWLEDGMENTextreme and do not take account of any softening of the The author would like to thank Andrew McCarthyDAC transitions, either internally or by using a trans- and Phillipe Dolman for their work on constructing aresistance stage with a capacitive feedback element or prototype DAC as part of their B.Eng. program in thecurrent input filter. However, the simulations do indicate Department of Electronic Systems Engineering at thecomplicated distortion spectra that are increased in mag- University of Essex.nitude due to the fundamental correlation between signal

and jitter equivalence, as illustrated in Fig. 15, where 7 REFERENCESit can also be observed that the distortion terms are

not well matched to psychoacoustic masking thresholds, [1] M. O. J. Hawksford, "An Introduction to Digitalthus possibly gaining in subjective significance. Audio," in Proc. lOth Int. AES Conf. (London, 1991

It should be noted that when correlation between sig- Sept.), pp. T3-T42.nal and jitter was considered and the distortion calculated [2] J. R. Stuart, "Estimating the Significance of Er-by Eq. (5), the differential of the signal sequence was rors in Audio Systems," presented at the 91st Conven-multiplied by the jitter AT r, which was made directly tion of the Audio Engineering Society, J. Audio Eng.proportional to the signal. However, for slew-rate-in- Soc. (Abstracts), vol. 39, p. 1011 (1991 Dec.), pre-duced distortion the jitter equivalence described by Eq. print 3208.(7) is proportional also to the differential of the input [3] S. Harris, "The Effects of Sampling Clock Jittersequence. Therefore the resulting distortion is propor- on Nyquist.Sampling Analog-to-Digital Converters, andtional to the square of the differential of the input, a on Oversampling Delta-Sigma ADCs", J. Audio Eng.subtle but possibly significant difference that implies a Soc., vol. 38, pp. 537-542 (1990 July/Aug.).greater intermodulation distortion dependence on high- [4] P. van Willenswaard, "Industry Update," Stereo-frequency signal components, phile, vol. 13, pp. 78-83 (1990 Nov.).

The time-interleaved dual DAC topology is potentially [5] J. A. Atkinson, "Jitter, Bits and Sound Quality,"less sensitive to many of these problems. Provided the Stereophile, vol. 13, pp. 179-181 (1990 Dec.).MDACs can achieve accurate level reconstruction and [6] R. Harley, "Industry Update," Stereophile, vol.

their output/reference input frequency response is not 14, pp. 38-45 (1991 Sept.); vol. 16, p. 65 (1993 Feb.);code dependent, then even if jitter exists and data transi- vol 16, pp. 47-91 (1993 Sept.).tion distortion would normally occur with rectangular [7] E. Meitner and R. Gendron, "Time Distortionssamples, these errors are of little consequence, as was within Digital Audio Equipment Due to Integrated Cir-demonstrated by the equivalent timing displacement es- cuit Logic Induced Modulation Products," presented attimated in Section 3.2. Thus the relatively low band- the 91st Convention of the Audio Engineering Society,width excitation of the transresistance converter together J. Audio Eng. Soc. (Abstracts), vol. 39, p. 992.with the relaxation of the analog filter topology in associ- [8] C. Dunn and M. O. J. Hawksford, "Is the AES/ation with a standard 4 × or 8 × oversampling filter in EBU/SPDIF Digital Audio Interface Flawed?," pre-the digital domain should result in near theoretic per- sented at the 93rd Convention of the Audio Engineeringformance. Society, J. Audio Eng. Soc. (Abstracts), vol. 40, p.

Section 4 presented an experimental system to confirm 1040 (1992 Dec.)., preprint 3360.operation, although performance restrictions of the [9] J. Dunn, "Jitter: Specification and Assessment inavailable four-quadrant MDAC should be noted. It is Digital Audio Equipment," presented at the 93rd Conven-possible that an MDAC may exhibit code-dependent dis- tion of the Audio Engineering Society, J. Audio Eng. Soc.tortion, although the relaxation of slew-rate-dependent (Abstracts), vol. 40, p. 1040 (1992 Dec.), preprint 3361.distortion is potentially of greater benefit. Also, al- [10] J. Dunn, "Considerations for Interfacing Digitalthough analog filters can be used to band-limit the output Audio Equipment to the Standard AES-3, AES-5, AES-waveform of a DAC prior to the transresistance stage, 11 ," in Proc. lOth Int. AES Conf. (London, 1991 Sept.),hence reduce slew-rate-induced distortion, this does not pp. 115-126.address jitter or distortion present within the DAC set- [11] R. D. Fourre, "Jitter, Jitter, Jitter .... "Appli-tling period after conversion. Such circuits usually imply cation Note AP-03, Ultra Analog Inc., Fremont, CAa high-frequency noise penalty due to a shunt impedance (1992 Sept.).at the transresistance input. [12] M. O. J. Hawksford, Letter in response to R.

There are DACs having 20-bit amplitude resolution Adams, "Comments on 'Chaos, Oversampling, andthat were designed for high-quality digital audio and Noise-Shaping in Digital-to-AnalogConversion,'" J.military systems. If these can be modified to include Audio Eng. Soc. (Letters to the Editor), vol. 38, pp.access to the reference input and thus allow operation as 767-768 (1990 Oct.).a precision MDAC, there now exists a means to virtually [13] J.R. Stuart, "Predicting the Audibility, Delecta-eliminate the vestiges of a number of inherent imperfec- bility, and Loudness of Errors in Audio Systems," pre-tions, which, although of low level, can still pervade sented at the 91st Convention of the Audio EngineeringDAC systems and represent an ultimate performance Society, J. Audio Eng. Soc. (Abstracts), vol. 39, pp.bound. 1010-1011(1991Dec.),preprint3209.

916 J. Audio Eng.Soc., Vol. 42, No. 11, 1994 November

PAPERS DIGITAL-TO-ANALOGCONVERTER

THE AUTHOR

Malcolm Omar Hawksford is director of the Centre which has led to an advanced digital and active loud-for Audio Research and Engineering and a professor in speaker system being produced by the University Corn-the Department of Electronic Systems Engineering at pany, Wivenhoe Enterprises, under the name Essexthe University of Essex, where his interests encompass Audio. Research has also encompassed oversamplingaudio engineering, electronic system design, and signal and noise shaping as a means of analog-to-digital andprocessing. Professor Hawksford studied electrical en- digital-to-analog conversion that includes digital linear-gineering at the University of Aston in Birmingham ization of PWM encoders.where he gained a First Class Honours B.Sc. and Ph.D. Professor Hawksford has published in the Journal ofThe Ph.D. program was supported by a BBC Research the Audio Engineering Society on topics that include errorScholarship and investigated the application of delta correction in amplifiers, oversampling techniques, andmodulation for color television and the development of a MLS techniques. His supplementary activities includetime-compression/time-multiplex system for combining writing for Hi-Fi News and Record Review and designingluminance and chrominance signals. During his time at high-end analog and digital audio equipment. He is a char-Essex University, he has undertaken research on ampli- tered engineer and is a Fellow of the AES, the Institutiontier studies, digital signal processing, and loudspeaker of Electrical Engineers, and the Institute of Acoustics. Hesystems. Since 1982 research into digital crossover sys- is also technical adviser to HFN and Record Review andtems and loudspeaker equalization has been pursued a technical consultant to LFD audio, UK.

J. AudioEng.Soc.,Vol. 42,No. 11,1994November 917

j30 dac transisiton distortion, jitter, slew rate

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