Noise Sources and Switched Cap Filter

8/2/2019 Noise Sources and Switched Cap Filter

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IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. SC-17, NO. 4, AUGUST 198242

[3][4]

[5]

[6]

[7]

[8]

[9]

[10]

F. F. Tsui, JSP-A research signal processor in Josephson tech-nology, IBM J Res. Develop. , vol. 24, pp. 243-2523 Mar. 1980.T. R. Gheewata, Design of 2.5 micrometer Josephson currentinjection logic (CIL), IBM J. Res. Develop. , vol. 24, pp. 130-142, Mar. 1980.P. C. Amett and D. J. Herretl, Regulated AC power for Joseph-son interferometer latching logic circuits, IEEE Trans. Magn.,pp. 554-557, Jan. 1979.M. B. Ketchen, Power supplY regulators for Josephson latchinglogic, in Proc. IEEE Int. Confi Circuits and Comput., ICCC80,Oct. 1980, pp. 874-877.D. J. Herrell and N. Raver, The electrical characteristics of ahigh performance package for Josephson technology, presentedat the High Speed Digital Technol. Conf., San Diego, CA, Jan.1980.H. C. Jones and D. J. Herrelt, The characteristics of chip-to-chipsignat propagation in a package suitable for superconductingcircuits, IBMJ. Res. Develop. , vol. 24, pp. 172-177, Mar. 1980.S. K. Lahiri, P. Geldermans, G. Kolb, J. Sokoloski, and M. J.Patmer, Pluggable contacts for Josephson packaging, J. Eletro-them. Sot. Extend. Abstr., vol. 80, no. 1, p. 216, 1980.J. H. Greiner et al., Fabrication process for Josephson integratedcircuits, IBh!lJ. Res. Develop., vol. 24, pp. 195-205, Mar. 1980.

[11 ] D. E. McCumber, Effect of ac impedance on dc voitagecurrentcharacteristics of superconducting weal-link junctions, J. APP1.Rhys., vol. 39, PP. 3 11 3 - 31 1 8, J u n e 1 9 68 .

Melvin Klein (A42-M53) was born in NewYork, NY, in 1920, He received the M.E. andM.S. degrees from Stevens Institute of Tech-nology, Hoboken, NJ, in 1940 and 1942,respectively, and the M.A. degree from Colum-bia University, New York, NY, in 1948.He worked at IT&T on electronic navigat ionequipment from 1942 to 1946, and at the RadioReceptor Company from 1950 to 1958 on radarequipment and semiconductor devices. In 1958he joined IBM and worked in the Com~onentsDivision on semiconductor devices for logic and memory appli~ations.

He was Manager of Semiconductor Device Development and Packagingfor Magnetic Memories from 1966 to 1969. In 1973 he became a Re-search Staff Member at the IBM Research Center, Yorktown Heights,NY, where he is engaged in the development of Josephson devices andcircuits for high-performance computer applications.

Noise Sources and Calculation Techniques forSwitched Capacitor Filters

JONATHAN H. FISCHER

Absmact-The noise response of switched capacitor networks (SCNS)is reviewed with emphasis on simplifying approximations suitable forSPICE noise simulation. The techniques developed cover all op-arnpnoise sources, as wetl as capacitor switching noise, The close agreementbetween predicted and measured noise responses for several monolithicSCNSbears out the vstklity of these simulation techniques.

I. INTRODUCTION

T HE noise response derivations for switched capacitornetworks (SCNS) developed in this paper differ fromearlier work [1] - [3] by emphasizing approxitrrations thatfacilitate the use of general-purpose programs, such as SPICE,for accurate SCN noise analysis. The derivations will coverideal sampling effects, the development of a suitable SCNintegrator noise model, and computer simulation techniques.Applying the noise model to practical SCNS shows that therelative contributions of op-amp 1/f, foldover flat-band, andcapacitor switching noise are filter topology dependent.

II. MODULATION AND FOLDOVER E FFE CTSTo start the analysis of noise in sampled data systems, we

begin with the effects of the idealized sampling operation ofFig. 1 on a signal band limited to less than half the sampleManuscript received May 6, 1981; revised January 7, 1982.The author is with Belt Laboratories, Hohndel, NJ 07733.

()+- ()s()S(t)= ~8(t-kT)Fig, 1. Ideal sampler.

rate (f~). The baseband (~n) and sideband (~~b) spectraldensities of Fig. 2 are equal as a result of ideal impulse sampl-ing [4], [5]. A convention that will prove useful in the analysisto follow is to number the sidebands so as to associate eachsideband with the sampling frequency harmonic it is centeredabout, as in Fig. 2.For the more general case where the signal bandwidth is

greater than f~/2 (such as the output noise of an op-amp inan SCN), aliasing will occur. As an example, assume thatthe sigmd to be sampled has a bandwidth (Z3Wn) of three timesthe sample rate. The first five sidebands resulting from thesampling operation have been depicted in Fig. 3(a). Referringto the figure, the following are contributors to the frequencyband from dc to f.: the fundamental, *1, +2, and the +3 side-bands. The addition of these sidebands is depicted in thestacked structure of Fig. 3(b). This spectral stacking of anundersampled signal results in an output of larger spectraldensity than the input for the frequency band from dc to f8/2.The noise gain of a track-and-hold circuit is shown in Fig. 4(and is discussed in Section III).

0018 -9200/82/0800 -0742 $00.75 01982 IEEE


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F IS CH ER: NO IS E S OURC ES AND S WITC HE D C AP AC ITO R F ILTE RS 74 3

F(f)I FUND / f

-fin o fin(a)

-~ fs a s

(b)-2 -1 FuND +1 +2 f

-2f, -f* o fs 2f,(c)

Fig. 2. (a) Input spectrum. (b) Sampling function. (c) Modulatedsignal spectrum.

I FU,ND I ~-1 -i -4 -1 -4 -1 -1 -1 [+3 ,+1 +1 +j +4 ,+1 ,+4 I 4-

J-2 -2 -2 -2 -2 -2 -2-21I +2 t2 +2 +2, +2 +2 +2 ~-1-3 -3 -3 -3 -3-3 -3 I1+3, +3, +3, +3. +3, ~.,. ,,

-4-4-4 -4 :4-4 I I +4,+4 ,+4 ,+4 ,+.4 ,+4:-5 -5 -5 -5 -5-51 +5 +,5 +5, +5 +5 ,+5-

-6fs -5fs -4fs -3fs -2fs -fs O f: 2fs 3;s 4;s 5;s 6fs(a)~E13 -8 -2 -2 -2 -2 -2 -2 -2 +4 +4 +4 +4 +4 &

-6fs -5fs -4fs -3fs -2fs -fs O fs 2fs 3fs 4fs 5fs 6fs(b)

Fig. 3. Impulse sampling (a) sidebands and (b) resultant frequencyspectrum.

() IN dBYT/H OUTPUT

INPUT NOISE

B Wn = 4 00 KHzf, = 1 00 HzTT,H= 0.5

Fig. 4. T)H noise response.

For the remainder of this paper, IllVn represents the equivalentnoise bandwidth of a noise source, and will be taken to be thatbandwidth required to contain the same noise power as thesource, but with a uniform spectral density rIn.If the input is a white noise source, the frequency-shifted

sidebands are now uncorrelated with the fundamental or eachother; hence, power rather than voltages are added to computethe total output density (qT). Putting these results intomathematics for .f~< .f< f~,

=on+w?ws (1)

Fig. 5. Integrator topology under study.

-+mvo-b=al-r-L, -~T TTRACK%OLO T H t#T~ - SHSH - ~Fig. 6. Track-and-hold circuit.

OUT

+ TSA~p$EAR CT ON

E-

tTOFF _f_ONSAMPLEOFF tH4 t

CLEAR o: tFig. 7. Track-and-hold equivalent circuit.

where (2BWn/f,) - 1 is th e n um ber of s ideban ds fa llin g in thefrequency range dc < f < f.. To simplify the calculations,BWn/f, will be treated as an integer. If a fraction of a side-band folds into band, BWn/f, is incremented to the nextlarger integer to give an upper bound on the in-band noise.Recalling that ~,1, = qn simplifies (1) to

(2)

III. NOISE MODEL OF A TRACK-AND-H L,DOne of the basic building blocks commonly used in SCNS

is the integrator of Fig. 5. This integrator is a form of atrack-and-hold (T/fI) circuit. While the track operation isnot inherent to all SCN integrator topologies [6] - [8], thisproperty will be included for completeness. The integratoranalysis will begin by describing transmission gate (inputtracking) and S/Ii (hold operation) noise responses using theT/H of Fig. 6. These results will be experimentally verified inSection IV, and a complete integrator model will be developedin Section VI.

The T/H can be viewed as two systems in parallel, as in Fig.7, The transmission gate models the feedthrough operationwhen the T/Ii tracks the input. The sample-and-hold (S/n)models the hold operation by sampling at the instant thetransmission gate opens, while resetting to zero during thetrack mode.


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74 4 IE EE J OURNAL OF SOLID-S TATE CIRCUITS , VOL. S C-1 7, NO, 4 , AUGUS T 1 98 2

A. TrackModelIf the input is white noise of bandwidth BWn, the results of

Appendix A can be used to obtain the results below relating IN+th e output to the input power density:

noutm = ~% BWn 10f$ (3b) for Bwn > loffi

where t-~ is the track time/sample period duty cycle. Notethat the output spectral density (in the passband) is nevergreater than that of the input (for a white noise source).B. Sample-and-Hold Noise ModelAn ideal S/H samples instantaneously, and then holds for

the remainder of the sampling period. The hold time is theS/H duty cycle times the sample period (r~~ X T), as in Fig.7. Using the results of Appendix B for a white noise input,

(,)%INdBT /H OUTPUT+INPUT NOISE-

Hz0 200 400 600 800 1000

()SHf BW. = i O KHzTout(f) ~ mkk7 sinc2 BW. < ; f.f.

f, = IOHZTT/H= 0.5

(4a) Fig. 9. SCN noise floor.

ou(f)+fsCase 2) r~~ =l . Ideal S/H results, and the track term drops,

leaving the familiar sine (x) envelope.(4b)

Thus, foldover effects of ideal sampling followed by a hold Tout(f) < zf?n~~fi (~)sin cz[~].opera t ion (th at form s th e ~/ ~ ) can p rodu ce a n in -ba nd respon seth at is s ign ifica ntly la rger th an th e in pu t s pectr al den sit y. ca se 3 ) TSH = 1 / 2 and BW. > 10f~:C Frequency Domain T/H Noise Model ~out(f) =O. [X-%sinc (ik)+lince the transmission gate and the S/H operate during non-overlapping time intervals and the noise is white in nature, Case 4: To investigate the out-of-band response, hold f8their power spectra simply add [3], [4] . The track-and-hold and 7SH constant (let TSH = 0.5 for simplicity), and increase f:operation occupies the entire sampling period:

rT+rsH=l. nout(fl=nn&)sinc2 (*)+~Vn.Recasting the transmission gate response in terms of 7SH and Recalling that \sinc (x)1 < I/lxl,combining with the S/H results, the frequency domain modelof Fig. 8 can be constructed, Writing qout(f) as a function of~n for wide-band white noise, n(%) (%2[2 2[%1+( -4out(f) =n TsH Smc

1 As f in crea s es , th e en velope of t?o~t (~) decrea ses with th eBWn < + f$ (5) square of .f, sofor f >> f$ [see (5) and (6)], the transmissiongate term sets the noise floor in Figs. 4 and 9 at ~n/2.

[ 2 (i+inz[%l-dout(f) ~ V. Z7SH IV. EXPER IMENTAL VERIF ICATIONOF FOLDOVE R THE ORYBWn > lof~. (6) To aid in the isolation of various noise sources, the testcircuit of Fig. 10 was used. To eliminate any effects resulting

To gain physical insight into the above results, let us look at a from finite bandwidth of the op-amps or switches, the noisefew limiting cases. source was selected to produce band-limited white noise withCase 1) rs~ =0: The tracking gate is always closed, hence the cutoff frequency far below fT of th e op-am p. Fu rth er-n o s am plin g, th us th e S/H t erm drops a nd ~Ou t( f ) = Vn. more, the input noise level was set as far above the noise


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F ISCHER : NO ISE SOURCES AND SWITCHED CAPAC ITOR F ILTERS 74s

S*WHITE NOISE + S4GENERATOR

1 1 .CLOCK 1$ IOKGENERATORI 1

w CMOSMG40666)TRANSMISSION GATEFig. 10. Transmission gate and S/H test circuit .

TABLE I(a) TRANSMISSIONATENOISERESPONSEVERSUSDUTYCYCLE (f


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746 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. SC-17 , NO. 4 , AUGUST 1982

N+l--m+izl--EOuTuT(a)

$E1-HEH-EF+zl-Ol(b)

(c)

Fig. 13, Cascaded T/Hs with in-phase (a) and staggered (b) phaseclocking. The resampled wide-band noise spectrum for the staggeredclock case is shown in (c).

V. EXTENS ION TO CASCADED S TAGE SThere are two extreme cases when cascading T/H stages.

One is wh en a ll th e T/Hs have the sam e tim in g (in -ph aseclocks ) a s in Fig. 13(a ). Th e oth er extrem e is wh en on e T/Htracks, and the T/H before an d a fter it a re in th e h old s ta te(s taggered clocks ) a s in Fig. 13 (b). Wh ile pa rts of an SCNm igh t con ta in both clock in g sch emes , SCN rea liza t ion con -s idera t ion s make th e s ta ggered clock ca se th e mos t common[6 ], [7], [11 ].In on e extrem e, a ll th e T/H stages are driven by the same

clock phases. Because the significant foldover effects arepresent only in the hold phase (assuming a memoryless trackphase), the noise a t th e T/H in pu t is s imply th e su m of th eu n sampled n ois e feedin g th rou gh to th e s tage of in teres t .When th e T/H changes to the hold phase, the input noise issampled and folded down to baseband. The resulting outputspectra are described by (5) and (6).For the other extreme (staggered clocks), each successiveT/H samp les th e h eld ou tpu t of th e previou s s tage. Th is

samplin g of a h eld ou tpu t res u lts in TSH = 1 and th e ou tpu tspectru m of Fig. 13 (c) wh ere pb is th e pas sban d edge. It isea s ily s h own th a t th e s in x/ x en velope p rovides effect ivea nt i-a lia s filt er in g of th e fold ed n oise (in crea sin g with la rgerf$/Pb ra tios ) so th at resa mplin g th e h eld T/H ou t p u t m in im a llyin crea s es th e in -ba nd n ois e. As s ta ggered T/Hs a r e c as ca d ed ,th e ou tpu t n oise of a given s ta ge (in th e h old ph as e) is clos elyapproxima ted by th e res amp led n oise of th e p reviou s s tagesadded to th e s am pled wide-ban d n oise of th e s ta ge th e p resen tT/H is sampling.In summary, since most SCNS use staggered clocks, TSH will

be taken to be unity for the rest of the paper, and the effectsof resampling of noise will be ignored.VI. SCN INTEGRATORS AND CHARGE ACCUMULATORSIn this section, a model of an SCN integrator is developed.

The T/H results will then be combined with these results inSection VII where a simple SCN is analyzed.

+(a)

%

Fig. 14. (a) Charge accumulator and (b) simplified accumulator withV~ngrounded.

The topology of the charge accumulator (with the l/~ andflatband op-amp noise modeled by Fn) of Fig. 14 is represen-tative of a basic structure used in SCN filters [6] - [8], so acareful analysis is in order.A. ChargeAccumulator ModelTo aid in visualization, the accumulator is redrawn in Fig.

14(b) with Vk grounded. Assume as initial conditions thatqcr = qc~ = O and that the op-amp voltage gain is infinite.After switch SI has closed, the feedback action of CI willforce the output to()o= 1+% Vn(wh ile d r ivin g th e in vert in g in pu t to equ al Vn ) a n d to t ra ck Vnwith ga in th erea fter . Th e tran sien t res ult in g from C7R ch argin gto V. does n ot trap ch arge on CI becau se on ly th e tra ppedch arge from previou s sample periods will rema in if th e ampli-t ude of Vn retu rn s to zero a t th e in s tan t 111 open s . Th ea dd it ion a l ch a rge tra pped on CI a t th e in s tan t ~1 open s (a tt im e t l) is

AqcI(tl) = qCR = CRUn(tl)with

Avcl(tl) = ~ Vn(tl).With S2 closed and SI open, qc~ is bled off to ground, and

P. tracks Vm with unity gain and a dc offset equal to th evoltage stored on CI. When Sz opens and S1 closes, the in-verting terminal is initially pulled to ground, and the op-ampthen recharges CR to Pn, repea t in g th e above procedu re. Noteth a t th e on ly t ime ch arge is trapped on CI is wh en S1 open s ,r esu lt in g in an accu mu la ted ou tpu t offset ( Vnet) of

Vn et . y ~ V.(iT)i=1 (7)


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F ISCHER: NO ISE SOURCES AND SWITCHED CAPAC ITOR F ILTERS 74 7

+0Fig. 15. SCN inte~ator noise model for CR/CI


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748 IE EE JOURNAL OF SOLID -S TATE CIRCUITS , VOL. SC-17, NO. 4, AUGUST 1982

IN

Fco+

soN, F1l kVI?v 1/f

v,/f (f) = ~ v,,, = J-=

=IN co

so

IvN,,/rms Ivfijm ,oLDOv,R F/as,UT UNCORRELATED, USED TO MODEL

co OUTPUT TO NON-SAMPLED PATHSSo OUTPUT TOSAMPLEO PATHS

Fig. 18. Frequency domain SCN op-amp noise simulation model.

approaches unity and 7SH = 1. Then separate the result intoterms related to the CO and SO outputs, respectively:

[12BWnVout(f) ~ % +v. f. 1

with the bracketed term representing the foldover effectsrepresented by lcVnll . Similarly accounting for the l/~ con-tribution (Appendix C),

n l/ f ou t (f) Q $ + (d).Because programs such as SPICE work with noise voltagesrather than powers, the noise voltage gain through samplingis obtained by taking the square root of the bracketed termsto yield

k.i

2BWn-1f,

and the 1/f term [email protected] alternate method of calculating k is as follows. First,

calculate the total output noise power (PT) of the amplifierusing SPICE or other means. The result is doubled to accountfor positive and negative sidebands contributing power, andis divided by the sample rate to yield an average power density.Putting this procedure into mathematics,

where T/FBis the presampled op-amp flat-band power density,The next two subsections will outline in detail how to use

SPICE to model the op-amp noise.C. Flicker Noise ModelBecause SPICE lacks the option of user-defined functional

expressions, a device model must be used to simulate the un-sampled 1/f noise component. To conveniently generate theflicker noise, the boxed circuit of Fig. 19 is used. The deviceis diode-connected for ease of biasing to a predetermined draincurrent by ~&. To assure that the device is in the saturationregion, choose Ml as an enhancement-type device. The noise

v~,,? I

R ~f,.#

RF44B

Fig. 19. Complete SPICE op-amp noise model. The boxed circuit isthe I/f source, md the resistors are uncorrelated white noise sources.

voltage as seen at the drain is

n=is=%To decouple the noise sources so that 1/f and wide-bandnoise effects can be studied separately, select the bias currentand device W/L ra t io to a s su re th a t th e gm term con tr ibu tesmu ch les s n ois e th an th e in pu t-referred op-amp fla t-ban d nois e.With th e device pa rameters s elected , a dju st KF to match th em ea su red op -a mp in pu t-referr ed 1 / f n ois e.

Becau se th e flick er n oise is to be in jected in to th e n on in vert -in g term in a l of a h igh ga in op-amp, th e dc bia s volta ge a t th ed ra in mu s t be b locked. To avoid loa d in g th e d ra in with th e dcb lock in g cir cu it (l?llc, CBLK), a voltage-controlled voltagesource is used as an ideal unity gain buffer, as in Fig. 19.

The next subsection will show how to model the sampled I/fnoise.D. Flat-Band and Foldover Model

The flat-band, folded flat-band, and folded 1/f (Appendix C)can be modeled as white noise sources, which are easily simu-lated by resistors in SPICE.

To model the flat-band noise, the resistor value is selectedto match the measured input referred flat-band noise. Tosatisfy the nodal requirements of SPICE, two resistors (Rnl~and I?nlB of Fig. 19), eac~ twice the value of the input-referred noise resistance, are paralleled so that all nodes haveat least two components connected and a dc path to ground.The source Vnl in jects th e u n sampled fla t-ban d nois e in to th eop-amp, with Vn ll modeling the wide-band foldover effectsand Vjll the 1/f folded noise. Separate resistor sets are usedto drive Vnl, Vnll, and Vll so th at th e s ou rces will b e m u tu a llyuncorre la ted.

Th e ap prop ria te fold ed I/ f res is tor va lu es a re

E. Model LimitationsThis model is accurate for f


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F ISCHER: NO ISE SOURCES AND SWITCHED CAPAC ITOR F ILTERS

TABLE IIIMONOLITHIC FILTER RESULTS. NOISEVOLTAGESARE rms VALUESNTEGRATEDOVERTHEFREQUENCYANGEFROM1 Hz TO10 kHz.

FILTER I/f FLAT CAPACITOR:.:; dihc MEASUREDdBrncHPN 4. V 6.5.. 9.01 12J, = 8kH, -6.5 -6HPN 4.. 3.9 4.1.?/s+4kHz 6.81N -11 -8LPF 12. 25UV

fs-128kHz45.3 531N 6.7 8.5

LPFreduced 12UV 25U 100V* 11o 13 15caps.LPFreduced 2 7UV 25 UVcap s.& 100UV4 11OW 13 16inputdcv.

OdBrn (24.5.v) CM = 0.6PF1 Cw = 0.3rJF1 C.m = L7pF4 C.w = o.3pF

constraint is not severe since most SCNS are designed withthe sample rate much greater than the passband frequency.Further assumptio~s made to simplify the calculations arethat the hold duty cycle of the switched paths (7sH) is unityand that the filter is run at a single sample rate. If stages ofdiffering sample rates are cascaded, the sampled noise spectraof previous stages must be analyzed to account for significantfoldover contributions to the stage under consideration.F. Gdcutation and Measurement ResultsUsing models like the above, SPICE simulations were usedto calculate the output noise for 60 Hz F@N filters sampledat 8 and 64 kHz, and several 128 kHz sample rate fifth-orderelliptic low-pass filters with a cutoff frequency of 3.4 kHz[8]. The calculated C-message weighted (dBrnc) noise resultsare summarized in Table III, and are compared to results ob-tained from monolithic realizations of these circuits. Theop-amp noise was measured to be 50 nV/ @ at 1 kHz, witha flat-band noise component of 35 nV/ @ and a noise band-width of 2 MHz.Fig. 16 shows an actual HPN (sampling at 8 kHz) switched

capacitor filter [8], and its complete noise model is shown inFig, 17, Calculating k for the filter with op-amps having anoise bandwidth (see Appendix C) of 2 MHz,=FT5=-=223Referring to Table V indicates that a = 2.1 for this samplerate.The dominant noise term of the HPN is the capacitor switch-

ing noise. If the falter is followed by an 8 kHz S/H, the wide-band noise of the second amplifier will be boosted by the fold-over factor (22.3) to become the dominant contributor.Similarly, the dominant noise source in the LPFs is the capac-itor switching noise.The measured and predicted noise spectra are compared inFig. 20 for the 1.7 pF minimum capacitor size LPF. If wide-band noise sources are the dominant terms of the LPF, thenoise density should drop by 9 dB for a factor af 8 increase

-410 I 1 1 1 I 1

I , I I 1 I I Jo 2K 4K 6K . 8K 10K

f IN HZFig. 20. Measured versus calculated 1.7 pF unit capacitor LPFresponse.

-80

90

-400

-420

L ! 1 I I 10 2 4 6 8 10I , 1 , 1 i 1 , i Io 46 32 40 64 80

f IN KHz

74 9

noise

Fig. 21. Measured noise response of the 1.7 pF unit capacitor LPF forfs =128MZ (top trace) a nd for f$ =1024 kHz (bottom trace). Allanalyzer settings are the same for both traces except that the lowertrace stop frequency (80 kHz) is eight times that of the top trace(10 kHz).in sample rate. As a test, the noise responses for ~~= 128 and1024 kHz of this LPF are compared in Fig. 21. All the analyzersettings are the same for both traces, with the exception thatthe frequency sweep was changed from 10 ,Wz up to 80 kHz[for ~~= 1024 kHz) so the re~ative densities can be directlycompared. The approximate 10 dB drop in the passband noisedensity clearly indicates that 1/f n oise is definitely not thedominant noise source of these filters.

VIII. CONCLUSIONThe results presented show that the topology of an SCN

determines the significance of the relative contributions ofamplifier flat-band and 1/f, and capacitor switching noise tothe overall filter noise, For the test chip filter designs in ourprocess, the dominant noise source was found to be capacitorswitching noise, followed by op-amp wide-band noise, andlastly, op-amp 1/f noise. A noise model and a simulationtechnique have been develcrped to allow existing simulationprograms (such as SPICE) to be used in accurate SCN noisecalculations.

APPENDIX ATransmissionGate Noise Mod@The transmission gate simply m~dulates the input signal

{x(t)}by a pulse train, as shown in Fig. 22. The Fourier


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750 IE EE J OURNAL OF S OLID-S TATE CIRCUITS, VOL. SC-1 7, NO. 4 , AUGUS T 1 98 2

x(t) --P_y(+) . (+) x Y(t) =x(t) f(t)() rl.--.r

o TT TFig. 22. Transmission gate model.

TABLE IVTRANSMISS1ONATENOISE RESPONSEAS A FUNCTIONOF DUTYCYCLE AND BWJf,

)UTY CYCLE

T

1.00

0950 soO 600,500350200100.050

10 0

1.000.950 soO 600.500.3502001000490

/owv .B Wn

as a function of ~

10

I 0009 40790.590.490340190.0900.0400

6

I .000.930 7s0590.4s0330180 0S60.0290

J .1

I .000910710540.450.2s011002900070

( fdd. v. r eiTccts absent)

s er ies of th e m odu la tin g pu lse is(- ~)if(t)=T+; -g ()

27rtsin (i7rr)cos i=~ i T

1Y2

1,000.9006 403 70.2501 20.040.01000030

(Al)

where ~ is the duty cycle and T is th e sample period. Workingout the details for a white noise input of bandwidth BWn,[21F)H+in(+] A2tlt(.f)=n. T + j ~where h = BWn/j_~ (number of positive sidebands falling inband).Since (A2) is not in a convenient form for h, qout(~) has been

tabulated in Table IV for various BWn / f~ and ~. An importantapproximation can be made for h >10:

BW.%Nlt(f)= w?? >10,or f, (A3)while at the other extreme, h = O(sin terms are not present),

%ut(f) = ~% for 2BWn


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FISCHER: NOISESOURCES AND SWITCHED CAPACITOR FILTERS 75 1

Calculating lX(ei2zfT)12,

For the case at hand, x(f) is white noise; hence, the aliasesare uncorrelated, with the result that power, rather thanvoltage, terms are added. Using (2),()X(e~2fT)12 + r?. ~ for f, G 2BWn (B3a)

1l?n~ for f,> 2BWn, (B3b)

and for the S/H, [01W)12r2T2 sinc2 r ~ .f. (B4)Combining (B3) and (4) yields

Jy(fJ2=2w%2sinc2 [(;)1s 2BWn. (B5b)The usual application of sampled data systems is the processingof information in the passband (f< f~/2); accordingly, the in-band signal-to-noise ratio is of interest. Since signal-to-noiseis given by

s _ Iinput (2nf)12 [G(2nf)]2 Iinput (2mf)12X - ]X(ej2TfT)[2 lG(27rf)12 = ]X(ei2nfT)]2

it is independent of the reconstruction filter in the passbandfor white noise input, but does depend on the undersamplingof noise.

APPENDIX CFoldover Effects and Equivalent Noise BandwidthThis analysis will treat the op-amp 1/f and wide-band noise

separately to clearly show the different effects of undersamplingof these noise sources and to point out useful approximations.Wide-BandNoiseFig. 26 depicts a characteristic voltage-follower output noise

spectrum (of the wide-band component only) that has beendivided into frequency bands of width f,. Settling time con-straints of SCNS usually restrict f~ to be much less than theop-amp unity gain frequency. Additionally, the switch band-widths are usually set much wider than that of the op-amp.With the wide-bandwidth switches, effectively all the outputnoise of the op-amp is folded back in to the baseband from dc

OUTPUTNOISEDENSITY II

Fig, 26. Voltagefoltower output noise spectrum with test configuration.

h

~f- fs -fof fs

Fig. 27. Fundamental and fust two sidebands of a sampled I/f noisesource.

to f.. The tota l op-amp ou tpu t n ois e is ca lcu la ted by addin gup the power in each frequency band of width f~,

The equivalent noise bandwidth (BWn) of an op -a mp willbe taken to be th a t ba n dwid th requ ired to con ta in th e samenoise power a s th e op-am p, bu t with a u n iform spectra ld en sit y qn .

I/f Foldover EffectsIdealized sampling of a 1/f noise source produces the spectrum

of Fig. 27 where only the first sidebands have been shown forsimplicity, In the analysis that follows, it is assumed that thenoise spectral density follows an ampl~ude envelope of theform A/f, and that the sidebands are mutually uncorrelatedso that power, rather that voltage, is summed.Evaluating the foldover effects in the baseband from dc to fs,

(cl)where N is the number of sidebands folding into the baseband.To understand the foldover effects described in (Cl), let us

assume that the voltage-follower op-amp attenuates the l/fnoise for frequencies above 20 MHz, A = 1000 for computationease, and that the frequency range of interest is the familiartelephone voiceband from 300 to 4 kHz. Fig. 28 comparesthe presampled and postsampled spectra for several commonclock rates. An important aspect of Fig. 28 is that all theprocess dependence is lumped into the constant A, so thesecurves are simply multiplied by the same scale factor to fitany process. Evaluating just the foldover terms of (Cl) (andcalling that sum a) indicates that the 1/f foldover effects canbe closely approximated by adding a constant to the pre-sampled spectrum. Also note that for typical sample rates of64 kHz or higher, the foldover contribution is less than 20percent of the 1/f baseband density at 1 kHz. A simplifyingapproximation is to neglect the 1/f foldover term entirely fora sample rate of 100 kHz or higher. For lower sample rates,


11/11

752 IE EE JOURNAL OF SOJ . ID- STATE CIRCUITS , VOL. SC- 17 , NO. 4 , AUGUST 1 98 2

7,,,

0 400 iK 4OKf IN Hz

Fig. 28. Pre- and postsampled l/~ spectra for several sample rates. FO I t h es e c u rve s, V~/f =A/f with A = 1000.

TABLE V [5]1/y FOLDOVER FACTORIN UNITS OF HZ-lrfil. L-,(Jz), = 8kHz 16kHz 32kHz 64Hz 12SkHz 256kHz

300 2.1 0.96 0.44 0 .20 0.09 0.041000 2. 1 0.96 0.44 0.20 0.09 0.042000 2.1 0.97 0.44 0 .20 0.09 0,04 [7]3000 2.2 0.97 0.44 0.20 0.09 0.043500 2.2 0.97 0.44 0.20 0.09 0.044000 2.2 0.97 0.44 0.20 0.09 0.04 [8]

place a white noise source in series with the sampled op-amp [9]output (Fig. 16) of magnitude cd as summarized in Table V. [10]ACKNOWLEDGMENTThe author is grateful to P. E. Fleischer, A. Ganesan, D. G. [11]

Marsh, and K. R. Laker for their helpful discussions and ideas.Special thanks go to A. A. Schwarz for the excellent layoutand laboratory work performed.

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REFERENCESC.-A. Gobet and A. Knob, Noise analysis of switched capacitornetworks, in ISCAS Proc., Apr. 1981, pp. 856-859.B. Furrer and W. Guggenbuhl, Noise analysis of sampled-datacircuits, in ISCASF%oc., Apr. 1981, pp. 860-863.M. L. Llou and Y-L. Kuo, Exact analysis of switched capaci torcircuits with arbitrary inputs, IEEE Trans. Circuits Syst., vol.CAS-26, pp. 213-223, API. 1979.A. B. Carlson, Communication Systems. New York: McGraw-Hill, 1968.

R.A. Gable and R. A. Roberts, Signals and Linear Systems.New York: Wiley, 1973.G. M, Jacobs, Practical design considerations for MOS switchedcapacitor ladder ffiters, Electron. Res. Lab. , Univ. California,Berkeley, Memo. UCB/ERL M77/69, Nov. 1977.D. J. Allstot, MOS switched capacitor ladder filters, Ph.D.dissertation, Electron. Res. Lab., Univ. California, Berkeley,Memo. UCB/ERL M79/30, May 1979.P. E. Fleischer and K. R. Laker, A family of active switchedcapacitor biquad building blocks, Bell Syst. Tech. J., vol. 58,pp. 2235-2269, Dec. 1979.C.-A. Gobet and A. Knob, Noise generated in switched capacitornetworks, IEE Electron. Lett., vol. 16, pp. 734-735, Sept.1980.A. Peled and B. Liu, Digital Signal Processing. New York: Wiley,1976.P. E. Fleischer, A. Ganesan, and K. R. Laker, Effects of finiteop amp gain and band-width on switched capacitor falters,presented at the IEEE Int. Symp. Circuits and Syst., Apr. 1981.

Jonathan H. Fischer received the B.S. and M.S.degrees in electrical engineering from the Univer-sity of Colorado, Boulder, and the Universityof California, Berkeley, in 1978 and 1979,respectively.In 1978 he joined Bell Laboratories, Holmdel,NJ, as a member of the Technical Staff. Sincethen he has been involved in various aspects ofcharge-redistribution codec and filter work.

Noise Sources and Switched Cap Filter

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