Study of QFactor and Loop Delay Effects of aContinuous-Time ~ L AD Modulator

Haijun Lin, Atushi Motozawa, Pascal Lo Re *, Kunihiko Iizuka*, Haruo Kobayashi, Hao San

A. Transfer function ofdiscrete-time modulator

Fig.l(a) shows the block diagram of a discrete-time band­pass ~ L AD modulator, and its transfer function is given asfollows:

Abstract - This paper describes the design and analysis ofa continuous-time bandpass J .£ AD modulator for RFsampling. We determined the SNDR degradation due to finiteQ value of the loop resonator, and due to excess loop delay(ELD). SNDR was improved by 20dB by adding a digital filterwhich compensates for the effect offinite Q value, and SNDRwas further improved by 20dB by feedforward withparameters optimized to compensate for ELD. We haveconfirmed the effectiveness of the digital filter andfee dforward proposals using Matlab and SPICE simulations.

H{z)y

Index Terms - RF sampling, Continuous-time t::. ~ ADModulator, Q value, ELD, Sub-sampling

I. INTRODUCTION

Weare investigating continuous-time band-pass ~ L ADmodulators in mobile receivers, using sub-sampling to convertRF narrow-band analog signals to digital signals directly withhigh resolution and using low powerj l-B]. To minimizemodulator size, the bandpass filter was implemented as a Gm­C resonator inside the modulator chip. We found that fmitebandpass filter resonator Q value and ELD (Excess LoopDelay) resulted in modulator SNDR degradation, and weproposed ways to compensate for this.

In this paper, (1) we investigate the SNDR degradation dueto finite Qvalue of the Gm-C resonator and due to ELD of themodulator which uses a sub-sampling RF DAC to reduceclock jitter effects. (2) We add a digital filter which introducesnew zeros in the NTF (Noise Transfer Function) of themodulator and compensates for SNDR degradation due to theeffect of fmite resonator Q value. (3) We optimize modulatorparameters and use feedforward to compensate for SNDRdegradation due to ELD.

II. TRANSFER FUNCTION OF CONTINUOUS-TIME a LADMODULATOR

In this section we derive the explicit transfer function of thecontinuous-time ~ L AD modulator which uses an RF DACinside the modulator.

H.J Lin is with the Gunma University ofJapan,(e-mail: lin@el.gunma-u.ac.jp)A.Motozawa is with the Gunma University ofJapan*Pascal Lo Re is with the Sharp Corporation ofJapan* K. Iizuka is with the Sharp Corporation ofJapanH.Kobayashi is the professor of Gunma University of Japan,(e-mail: k_haruo@el.gunma-u.ac.jp)H.San is the vice professor of Tokyo City University ofJapan

978-1-4244-3870-9/09/$25.00©2009 IEEE

230

y

(b)Fig.l (a) Discrete-time bandpass 11 ~AD modulator

(b) Continuous-time bandpass 11 ~AD modulator

Y(z) = H(z) X(z)+ 1 E(z) (1)l+H(z) l+H(z)

Here X(z) is the input signal, Y(z) is the output signal, E(z)is the quantization noise of the loop ADC. The signal transferfunction (STF) and noise transfer function (NTF) can be

written as STF = H(z) NTF = 1l+H(z) l+H(z)

For this 1st order discrete-time modulator,

H(z)=-z-2/1+z-2. Thus STF and NTF can be written

as: STF = _Z-2 ; NTF = 1+z-2 (2)

B. Transfer function ofcontinuous-time modulator with RFDAC.

Fig. 1(b) shows the block diagram of a continuous-timebandpass ~ L AD modulator. Since the NTF determines noiseshaping characteristics of the modulator, we design acontinuous-time ~ L AD modulator whose NTF is equivalentto the discrete-time modulator. We mapped the NTF of thediscrete-time modulator to the equivalent continuous-time oneusing z-transform and modified z-transform techniques[8],[9].The transfer function of the Gm-C resonator can be written as:

H (s) = asOJo +bOJ;f S2 +SOJo/Q+ 0J5

OJo is the center angular frequency of input signal band.

(3)

(9)

: Re(z),/

"--'

......" ,,

\,: Re(z)

,/"-'

-Ideal

-Sub samp:Q=10 I ~-.- -1

-+-Sub samp:Q=50-Sub samp:Q=10,'.', samp:Q=10

"v" Samp:Q=50"'llI" Samp:Q=100

50

60

,­",,

I,

iii'"tJ 40~CZ 30 ~ , ,~~

en

10 ~ ..., ,

NTFs of the modulators for sampling cr: = 1/4 ·Is 'OJo = 1r/2Ts ) and sub-sampling can be written as :

t Im(z)

20 ~ :.IIiI/f"

0 1 2 3 4 5 6 7 8OSR [2"]

Fig.3 SNDR of the 1st order continuous-time BP a ~AD modulator.

We see from Eq.(7) and Eq.(8) that to obtain equivalentnoise-shaping characteristics when sub-sampling requiresthree times the Q value that sampling would require (in otherwords, for given Q resonator value, using sub-samplingresults in lower SNDR (Fig.(3)).

B. Compensatingfor finite Q value

To reduce or cancel the effects on SNDR of fmite Gm-Cresonator Q value, we propose to add a digital filter betweenthe modulator output and feedback RF DAC to introduce newzeros in NTF. Fig.4 shows the block diagram of the proposedcircuit. The open loop transfer function of the proposed circuitcan be written as:

Heq(z)= Hdf (z)ZlHdac(s)Hf (s)JH df (z), the transfer function of a 2nd order IIR digital

filter, is written as

H (z)=1-fnlZ-l+fn2Z-2df 1 + -1 + -2

- Jdl Z + Jd2 Z

Parameters fnl,fn2,fdl,fd2 of the digital filter are functions

of the Q value. We have derived the NTF of the modulatorplus digital filter as follows.

Fig.2 NTF zeros of continuous-time bandpass a ~AD modulator.

NT'F -1 -2 -tr/2Qsampling - + Z e (7)

NT'F -1 -2 -3tr/2Qsub-sampling - + Z e (8)

SNDR·OSR of 1st order Subsampling and Sampling CT DSM70 r--;======c::=======::I====::::::;---,---------r---,---------,--------,

NTF= 1 11+Z[HlooP(s)] 1+HlooP(z)

Hloop(s), the open-loop transfer function of the

modulator, can be written as: Hloop(s)= H f (s )Hdac (s)HlooP(z) is the open-loop transfer function of the

corresponding discrete-time modulator. To map from the NTFof the continuous-time modulator to the discrete-time

modulator we make Z[Hloop(s)] equal to Hloop(z) .

Z[HlooP(s)] is the z-transform of HlooP(s) . We have

derived the equivalent NTF of a 1st order continuous-timemodulator as follows:

1- 2COS{ROJ T )eaOJo~ Z-1 + e2aOJoT Z-2NTF(z) = pI 0 s (4)

d1 +d2z -1 +d3z -2

a = -lj2Q , fJ = ~1-1/4Q2 dl'd2,d3 are

functions of Gm-C resonator parameters a,b and a, fJ . If we

set the values of a,b so that d1 =I,d2 =0,d3 =0, then

NTF of the 1st order continuous-time ~ L AD modulator canbe written as:

NTF(z) = 1- 2cos(fJOJoTs

)eaOJoTs Z-1 + e2aOJoT Z-2 • (5)

III. cOMPENSATING FoR EFFECT OF FINITE Q VALUE

A. Effect offinite Q value

To reduce the sampling clock frequency in the modulator,

we use sub-sampling .r: =3/4·1s, OJo = 31r/2Ts [1]). If

the Qvalue of the resonator were infmite, then a = 0; fJ = 1,

and Eq.(5) could be written as NTF = 1+ Z-2 ; This is thesame as the NTF of the discrete-time modulator. However, theactual Qvalue is fmite, so Eq.(5) can be expressed as

NTF ~ 1+ Z-2e-OJoTs /Q (6)

Since the Q value is fmite, e-OJoTs/Q <1, which adversely

affects modulator SNDR. The reason is that when e-OJoTs/Q=1,

zeros of NTF are at ± j on the unit circle. But for

e -OJoTs/Q <1, zeros ofNTF are inside the unit circle (Fig.2).

(OJo = 21r ·hn)' Q is the Q factor of the Gm-C resonator.

The transfer function of RF DAC [2],[3],[10] can be writtenas :

H () = 1. (1 _ -in.a, \2 4w;dac S 2 e J S(S2 +4w;)

T, is the sampling period, and OJs is the sampling angular

frequency ( OJs = 21r ·Is). The equivalent NTF of the

continuous time modulator can be written as

231

1NTF(z) = ( )

I+Heq z

1+ q1Z-l + q2Z-2 + Q3Z-3 +Q4Z-4

Po + P1Z-1 + P2 Z-

2 + P3Z-3 + P4Z-

4

IV. EFFECT OF ELD AND ITS COUNTERMEASURE

ELD is defmed as the signal delay from the input of

Parameters ql ...P 4 of NTF are determined by parameters

Modulator digital output Dout[n] 1bitADC ......--....------....- ..

y

Hf(s)

SNDR Comparison

Fig.6 First-order continuous-time 1:!1 ~AD modulator with feedforward.

internal ADC to the feedback DAC output in a continuous-time a ~ AD modulator. ELD degrades the SNDR of themodulator. [10], [11]. We propose two methods tocompensate for SNDR degradation due to ELD: (1): adjust(optimize) parameters, depending on ELD values. (2):Feedforward compensation.

A. Solution for Effect ofELD.

We have calculated NTF of the modulator with ELD usingmodified-z transform[8]. The equivalent NTF of 1st-ordercontinuous-time a ~ AD modulator can be written as

~ + z-2e2aOJoTs ~ _ Z-l)NTF(z)~ - - - (10)

1+ glZ-l + g2 Z-2 + g3 Z-3 + g4Z-4

Parameters g 1•••g 4 are determined by parameters of the

resonator and the ELD value. The Z-4 term in Eq.(10) isdue to ELD. The resonator parameter values can be adjusted

depending on ELD value: g 1 = -1, g 2 = 0,

g 3 = 0 ,g 4 = O. Then NTF can be expressed as

NTF(z) ~ 1+ Z-2 e2aOJoTs ,thus we have compensated for

ELD. The proposed feedforward compensation formodulator SDNR degradation is shown in Fig.6.

To compensate for ELD effects, a feedforward path is addedfrom the modulator input to the internal ADC input.

T bl 1 0 . . d fl d·ffi ELD I

Table1 shows optimized values of resonator parameters for

different ELD values. aff ' bff are parameters of the resonator

with feedforward. a,b are parameters in the resonator withoutfeedforward.

For TSMC O.18Jl111 CMOS process, IT ~ 45GHz . If

input signal frequency is 2.4GHz and sub-sampling clockfrequency is 3.2GHz, then ELD~ 50%Ts (Ts is the sampling

a e: 'ptImlze parameter va ues or I erent va uesELD=10% ELD=20% ELD=50%

arr 0.075 0.145 0.356

brr -0.614 -0.542 -0.543

a 0.151 0.171 0.474b -0.454 -0.470 -0.201

ELD=60% ELD=80% ELD=90%

arr 0.502 0.670 0.543

bff -0.971 -0.109 -0.175

a 0.508 0.268 0.138b -0.264 -0.434 -0.441

60

lbit

sel1

o

MUXDFout[n]

Digital Filter

.'.' 1/4Fs Sampling NRZ- .. '3/4Fs Sub RFDAC-3/4Fs Sub RFDAC+DF

40 Q 5030

....................................................... ~ ............

15 .. .., ..

m40

"~35~3en

25

Digital Filter Calculation

for Dout[n]=0

Fig.5 SNDR comparison with different Q values.

Fig.4 Digital filter with low latency.

55,....--------.-------.-------.-------.-------,

501-······················· ~---.

45

a,b of resonator and fnl,fn2,fdl,fd2 of digital filter. The

resultant additional zeros in NTF, outside the unit circle,compensate for SNDR degradation due to fmite Q. Fig.4shows a block diagram of the digital filter.

Since the digital filter is a 2nd-order IIR filter, the digitalcalculation itself might cause delay. However, since theoutput of the modulator (input of digital filter) is lbit, thecalculation result can be used in the digital filter like in acarry-select adder, i.e, when Dout[n-l] is obtained at time n­1, Dout[n] can be calculated for both the cases Dout[n]=Oand 1. Then the value of Dout[n] can be selected bymultiplexer at time n. The output of the multiplexer is theoutput of the digital filter, and MSB of the digital filteroutput is the input of the l-bit RFDAC. In thisimplementation, the latency of the digital filter is just thelatency of the 2-input multiplexer.

Fig.5 shows Matlab simulation results for SNDR (1) forsampling, (2) for sub-sampling, (3) for sub-sampling plusdigital filter. Fig.5 shows that for sub-sampling, SNDR isimproved by 20dB by using the digital filter.

232

SNDROSR60

50

40iii'"C

£t30CZen

20

10

00 2 3 4 5 6 7 8

OSR[2n]98

00 3 4 5 6 7

10'~·········· .. , .....--

40'~·········· , ..~iii'"C

£t30'~ : : : : .Czen

20'~··············· ~

50'~····· L_---.,.--_....,...--------' I

period). Fig7 shows Matlab simulation results withSNDROSR50% ELD

OSR[2n]

Fig.7 SNDR result for modulator with feedforward.Output Power Spectrum

0

-10

-20

-30

ai' -40~~

CD -50~0a. -60

-70

-80

-90

-100L..--------L...------'--------:.--.1...---------'------'0.5 0.6 0.7 0.8 0.9

Frequency(Fin/Fs)Fig.8 Output power spectrum of a ~AD modulator (SPICE results).

ELD=50%Ts. (l)Without feedforward, where resonatorparameters are not optimized. (2)Without feedforward, butwith optimized resonator parameters. (3)With feedforward,and optimized resonator parameters. This confrrms that usingthe modulator in Fig.7 with feedforward and optimizedresonator parameters improves SNDR by 20dB whenELD=50%Ts.

V. CMOS MODULATOR DESIGN AND ANALYSIS

We have designed a modulator with a low-power Gm-Cresonator using TSMC 0.18um CMOS[ll], internal ADC andRFDAC were modeled using Verilog-A. Fig.8 shows SPICEsimulation results with input signal of 2.4GHz and samplingclock frequency of3.2GHz.

Fig.9 compares SPICE and Matlab simulation results withresonator Q of 40. We see that SPICE and Matlab simulationresults are almost the same. The SPICE and Matlabsimulations thus validate the proposed methods to compensatefor continuous-time Ii L AD modulator SNDR degradationdue to ELD and fmite resonator Q.

Fig 9. SNDR comparison between SPICE and Matlab simulations

VI. CONCLUSIONS

In this paper, we have clarified the SDNR degradationeffect of fmite resonator Q factor in a continuous-time Ii LAD modulator with sub-sampling, and we have proposed adigital filter to compensate for this. We have also proposed amethod to compensate for SNDR degradation caused by ELD.Matlab and SPICE simulations validate the effectiveness ofour proposed methods.

We would like to thank K. Wilkinson for improving theEnglish in this manuscript.

References

[I] M.Uemori,H.Kobayashi,T.Ichikawa,A.Wada,K.Mashiko,T.Tsukada,M.Htta "High-Speed Continuous-Time Sub-sampling Bandpass t!1 ~ ADModulator Architecture." IEICE Trans. Fundamentals ,April 2006

[2] A.Motozawa, K.Shimizu, M. Uemori, Pascal Lo Re, K. Iizuka, H.Kobayashi, H. San " Design of RF Sampling Continuous-TimeBandpass t!1 ~ AD Modulator. The 20nd Workshop on Circuits andSystems in Karuizawa", April 2007.

[3] A. Motozawa, Pascal Lo Re, H.Lin, T. Tanabe, H. Kobayashi, H. San"Study of RF Sampling Continuous-Time bandpass t!1 ~ ADModulator Architecture." The Conference of Electronic Circuit ofIEEJ. March. 2008

[4] J. Cherry, W. Snelgrove. " Continuous Time Delta-Sigma Modulatorsfor High Speed AID Conversion." Kluwer Academic Publishers (2002).

[5] R. Schreier, G.Times, "Understanding Delta-Sigma Data Converters,"Willy-IEEE Press (2004).

[6] Julien Ryckaert,J.Borremans,B.Verbruggen,G.Van der Plas,"A 2.4GHz40mW 40dB SFDR 60MHz Bandwidth Mirrored-Image RF Bandpasst!1 ~ ADC in 90nm CMOS", IEEE Asian Solid-State CircuitsConference pp361-364,Nov.2008.

[7] C. L. Phillips, H. T. Nagle, "Digital Control System". Prentice­Hall,lnc.(1990).

[8] J. Engelen "Stability Analysis and Design of Bandpass Sigma DeltaModulators,"(1999).

[9] J, Engelen, R. Plassche, Bandpass Sigma Delta Modulators, KluwerAcademic Publishers (1999).

[10] H.Lin, T. Tanabe, H. Kobayashi, H.San " Design and Analysis ofInverter Type Gm-C bandpass Filter". The 22nd Workshop on Circuitsand Systems in Karuizawa, April 2009.

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