EE 435 Lecure 35 Absolue and Relaive Accuracy DAC Design The Sring DAC
Makekup Lecures Rm 6 Sweeney 5:00 Rm 06 Coover 6:00 o 8:00
. Review from las lecure. Summary of ime and ampliude quanizaion assessmen Time and ampliude quanizaion do no inroduce harmonic disorion Time and ampliude quanizaion do increase he noise floor
. Review from las lecure. Quanizaion Noise DACs and ADCs generally quanize boh ampliude and ime If convering a coninuous-ime signal (ADC) or generaing a desired coninuousime signal (DAC) hese quanizaions cause a difference in ime and ampliude from he desired signal Firs a few commens abou Noise
. Review from las lecure. Noise We will define Noise o be he difference beween he acual oupu and he desired oupu of a sysem Types of noise: Random noise due o movemen of elecrons in elecronic circuis Inerfering signals generaed by oher sysems Inerfering signals generaed by a circui or sysem iself Error signals associaed wih imperfec signal processing algorihms or circuis
. Review from las lecure. Noise We will define Noise o be he difference beween he acual oupu and he desired oupu of a sysem All of hese ypes of noise are presen in daa converers and are of concern when designing mos daa converers Can no eliminae any of hese noise ypes bu wih careful design can manage heir effecs o cerain levels Noise (in paricular he random noise) is ofen he major facor limiing he ulimae performance poenial of many if no mos daa converers
. Review from las lecure. Noise We will define Noise o be he difference beween he acual oupu and he desired oupu of a sysem Types of noise: Random noise due o movemen of elecrons in elecronic circuis Inerfering signals generaed by oher sysems Inerfering signals generaed by a circui or sysem iself Error signals associaed wih imperfec signal processing algorihms or circuis Quanizaion noise is a significan componen of his noise in ADCs and DACs and is presen even if he ADC or DAC is ideal
Quanizaion Noise in ADC (same conceps apply o DACs) Consider an Ideal ADC wih firs ransiion poin a 0.5X LSB X IN ADC n X OUT X REF If he inpu is a low frequency sawooh waveform of period T ha goes from 0 o X REF, he error signal in he ime domain will be: ε Q.5 X LSB T 2T 3T 4T T -.5 X LSB where T =T/2 n This ime-domain waveform is ermed he Quanizaion Noise for he ADC wih a sawooh (or riangular) inpu
Quanizaion Noise in ADC ε Q.5 X LSB T 2T 3T 4T T -.5 X LSB For large n, his periodic waveform behaves much like a random noise source ha is uncorrelaed wih he inpu and can be characerized by is RMS value which can be obained by inegraing over any inerval of lengh T. For noaional convenience, shif he waveform by T /2 unis E RMS T T /2 T /2 2 Q d
Quanizaion Noise in ADC ε Q ε Q.5 X LSB T 2T 3T 4T T -0.5T.5 X LSB 0.5T -.5 X LSB -.5 X LSB E RMS T T /2 T /2 2 Q d In his inerval, ε Q can be expressed as XLSB Q T
Quanizaion Noise in ADC E RMS T T /2 T /2 2 Q d -0.5T ε Q.5 X LSB 0.5T T /2 X E LSB RMS - T T E X T /2 T /2 RMS LSB 3 T 3 -T /2 3 2 2 d -.5 X LSB XLSB Q T E RMS X LSB 2
Quanizaion Noise in ADC E RMS X LSB 2 The signal o quanizaion noise raio (SNR) can now be deermined. Since he inpu signal is a sawooh waveform of period T and ampliude X REF, i follows by he same analysis ha i has an RMS value of X RMS Thus he SNR is given by or, in db, X REF 2 X SNR = RMS X RMS 2 E X RMS LSB SNR =20 n log2 =6.02n db Noe: db subscrip ofen negleced when no concerned abou confusion n
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? X IN X REF SNR =20 n log2 =6.02n
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? X REF X IN Time and ampliude quanizaion poins
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? X REF X IN X QIN Time and Ampliude Quanized Waveform
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? X REF X IN X QIN ε Q X LSB Error waveform
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? ε Q X LSB Appears o be highly uncorrelaed wih inpu even hough deerminisic Mahemaical expression for ε Q very messy Excursions exceed X LSB (bu will be smaller and bounded by ± X LSB /2 for lower frequency signal/frequency clock raios) For lower frequency inpus and higher resoluion, a any ime, errors are approximaely uniformly disribued beween X LSB /2 and X LSB /2 Analyical form for ε QRMS essenially impossible o obain from ε Q ()
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? 0.5X LSB ε Q -0.5X LSB For low f SIG /f CL raios, bounded by ±XLB and a any poin in ime, behaves almos as if a uniformly disribued random variable ε Q ~ U[-0.5X LSB, 0.5X LSB ]
Recall: Quanizaion Noise in ADC If he random variable f is uniformly disribued in he inerval [A,B] f : U[A,B] hen he mean and sandard deviaion of f are given by A+B B-A μ f = σ= f 2 2 Theorem: If n() is a random process and <n(kt S )> is a sequence of samples of n() hen for large T/T S, +T 2 2 2 V = n d = σ +μ RMS T n kts n kt S
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? 0.5X LSB ε Q -0.5X LSB ε Q ~ U[-0.5X LSB, 0.5X LSB ] A+B μ = Q 2 0 B-A X σ= LSB f 2 2 +T 2 2 2 V = n d = σ +μ RMS n n T V = RMS Q X LSB Noe his is he same RMS noise ha was presen wih a riangular inpu 2
Quanizaion Noise in ADC How does he SNR change if he inpu is a sinusoid ha goes from 0 o X REF cenered a X REF /2? 0.5X LSB ε Q -0.5X LSB X V LSB RMS = 2 Bu XREF V INRMS= 2 2 Thus obain XREF 2 2 n 3 SNR = = 2 XLSB 2 2 Finally, in db, n 3 SNR db = 20log2 =6.02 n +.76 2
ENOB based upon Quanizaion Noise Reference Solving for n, obain SNR = 6.02 n +.76 ENOB = SNRdB-.76 6.02 Noe: could have used he SNR db for a riangle inpu and would have obained he expression SNR db ENOB = 6.02 Bu he earlier expression is more widely used when specifying he ENOB based upon he noise level presen in a daa converer
ENOB based upon Quanizaion Noise For very low resoluion levels, he assumpion ha he quanizaion noise is uncorrelaed wih he signal is no valid and he ENOB expression will cause a modes error n 4 3 corr from van de Plassche (p3) SNR 2-2+ π 2 Table values in db Res (n) SNR corr SNR 3.86 7.78 2 2.06 3.8 3 9.0 9.82 4 25.44 25.84 5 3.66 3.86 6 37.79 37.88 8 49.90 49.92 0 6.95 6.96 Almos no difference for n 3 SNR = 6.02 n +.76
End of Lecure 35