EE 435. Lecture 31. Absolute and Relative Accuracy DAC Design. The String DAC

Transcription

1 EE 435 Lecure 3 Absolue and Relaive Accuracy DAC Design The Sring DAC

2 . Review from las lecure. DFT Simulaion from Malab

3 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

4 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

5 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

6 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

7 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

8 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

9 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

10 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 / d -.5 X LSB XLSB Q T E RMS X LSB 2

11 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 nlog2 =6.02n db Noe: db subscrip ofen negleced when no concerned abou confusion n

12 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 nlog2 =6.02n

13 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

14 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

15 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

16 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 ()

17 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 ]

18 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 V = n d = σ +μ RMS T n kts n kt S

19 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 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

20 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

21 ENOB based upon Quanizaion Noise Solving for n, obain SNR = 6.02 n +.76 ENOB = SNRdB 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

22 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 Almos no difference for n 3 SNR = 6.02 n +.76

23 Quanizaion Noise Effecs of quanizaion noise can be very significan, even a high resoluion, when signals are no of maximum magniude X REF X IN X REF X IN Quanizaion noise remains consan bu signal level is reduced The desire o use a daa converer a a small fracion of full range Is one of he major reasons high resoluion is required

24 Quanizaion Noise Effecs of quanizaion noise can be very significan, even a high resoluion, when signals are no of maximum magniude X REF X IN

25 Quanizaion Noise Example: If a 4-bi audio oupu is derived from a DAC designed for providing an oupu of 00W bu he normal lisening level is a 50mW, wha is he SNR due o quanizaion noise a maximum oupu and a he normal lisening level? Wha is he ENOB of he audio sysem when operaing a 50mW? A 00W oupu, SNR=6.02n+.76 = 90.6dB 2 V =00W R L V R 2 L =50mW V V= log 0 V =20log 0 V-20log =20log 0 V -33dB A 50mW oupu, SNR reduced by 33dB o 57.6dB SNRdB ENOB = = = Noe he dramaic reducion in he effecive resoluion of he DAC when operaed a only a small fracion of full-scale.

26 ENOB Summary Resoluion: INL: ENOB = log N log 2 0 ACT 0 R 2 ENOB = n -log - log N 2 ACT n R specified res, ν INL in LSB DNL: VMAX VMIN ENOB log2 MAX V MAX and V MIN are max and min oupus and Δ MAX is maximum absolue sep (HW problem) Quanizaion noise: ENOB = ENOB = SNR db 6.02 SNRdB rel o riangle/sawooh rel o sinusoid

27 Performance Characerizaion of Daa Converers Saic characerisics Resoluion Leas Significan Bi (LSB) Offse and Gain Errors Absolue Accuracy Relaive Accuracy Inegral Nonlineariy (INL) Differenial Nonlineariy (DNL) Monooniciy (DAC) Missing Codes (ADC) Quanizaion Noise Low-f Spurious Free Dynamic Range (SFDR) Low-f Toal Harmonic Disorion (THD) Effecive Number of Bis (ENOB) Power Dissipaion

28 Absolue Accuracy Absolue Accuracy is he difference beween he acual oupu and he ideal or desired oupu of a daa converer The ideal or desired oupu is in reference o an absolue sandard (ofen mainained by he Naional Bureau of Sandards) and could be vols, amps, ime, weigh, disance, or one of a large number of oher physical quaniies) Absolue accuracy provides no olerance o offse errors, gain errors, nonlineariy errors, quanizaion errors, or noise In many applicaions, absolue accuracy is no of a major concern bu scales, meers, ec. may be more concerned abou Absolue accuracy han any oher parameer

29 Relaive Accuracy In he conex of daa converers, pseudo-saic Relaive Accuracy is he difference beween he acual oupu and an appropriae fi-line o overall oupu of he daa converer INL is ofen used as a measure of he relaive accuracy In many, if no mos, applicaions, relaive accuracy is of much more concern han absolue accuracy Some archiecures wih good relaive accuracy will have very small deviaions in he oupus for closely-spaced inpus whereas ohers may have relaively large deviaions in oupus for closely-spaced inpus DNL provides some measure of how oupus for closely-spaced inpus compare X OUT X OUT X REF X REF X IN X IN X REF X REF

30 End of Lecure 3