EE 435 Lecture 26 Data Converters Differential Nonlinearity Spectral Performance
. Review from last lecture. Integral Nonlinearity (DAC) Nonideal DAC INL often expressed in LSB INL = X k INL= max OUT OF 0kN-1 X k - X LSB INL k k INL C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 INL is often the most important parameter of a DAC INL 0 and INL N-1 are 0 (by definition) There are N-2 elements in the set of INL k that are of concern INL is almost always nominally 0 (i.e. designers try to make it 0) INL is a random variable at the design stage INL k is a random variable for 0<k<N-1 INL k and INL k+j are almost always correlated for all k,j (not incl 0, N-1) Fit Line is a random variable INL is the N-2 order statistic of a set of N-2 correlated random variables
. Review from last lecture. Integral Nonlinearity (ADC) Nonideal ADC Break-point INL definition XTk -X INL FTl k= 1k N-2 X LSB C 7 C 6 C 5 C 4 INL max 2kN-2 INL k C 3 C 2 C 1 C 0 INL 3 X T1 X T2 X T3 X T4XT5 X T6XT7 X FT1 X FT2 X FT3 X FT4 X FT5 X FT6 X FT7 INL can be readily measured in laboratory but often dominates test costs because of number of measurements needed when n is large INL is a random variable and is a major contributor to yield loss in many designs Expected value of INL k at k=(n-1)/2 is largest for many architectures Major effort in ADC design is in obtaining an acceptable yield
. Review from last lecture. INL-based ENOB ENOB = nr-1-log 2 Consider an ADC with specified resolution of n R and INL of ν LSB ½ ENOB n 1 n-1 2 n-2 4 n-3 8 n-4 16 n-5
Performance Characterization of Data Converters Static characteristics Resolution Least Significant Bit (LSB) Offset and Gain Errors Absolute Accuracy Relative Accuracy Integral Nonlinearity (INL) Differential Nonlinearity (DNL) Monotonicity (DAC) Missing Codes (ADC) Low-f Spurious Free Dynamic Range (SFDR) Low-f Total Harmonic Distortion (THD) Effective Number of Bits (ENOB) Power Dissipation
Differential Nonlinearity (DAC) Nonideal DAC Increment at code 4 (k)- (k-1) C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 DNL(k) is the actual increment from code (k-1) to code k minus the ideal increment normalized to X LSB X k -X k-1 -X DNL k = X OUT OUT LSB LSB
Differential Nonlinearity (DAC) Nonideal DAC Increment at code 4 (k)- (k-1) C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 Increment at code k is a signed quantity and will be negative if (k)< (k-1) X k -X k-1 -X DNL k = X DNL= OUT OUT LSB max 1k N-1 LSB DNL k DNL=0 for an ideal DAC
Monotonicity (DAC) Nonideal DAC C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 Monotone DAC Non-monotone DAC Definition: A DAC is monotone if (k) > (k-1) for all k Theorem: A DAC is monotone if DNL(k)> -1 for all k
Differential Nonlinearity (DAC) Nonideal DAC Increment at code 4 (k)-(k-1) C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 Theorem: The INL k of a DAC can be obtained from the DNL by the expression INL = k k i=1 DNL i Caution: Be careful about using this theorem to measure the INL since errors in DNL measurement (or simulation) can accumulate Corollary: DNL(k)=INL k -INL k-1
Differential Nonlinearity (DAC) Nonideal DAC Increment at code 4 (k)-(k-1) C 0 C 1 C 2 C 3 C 4 C 5 C 6 C 7 Theorem: If the INL of a DAC satisfies the relationship then the DAC is monotone 1 INL < X 2 LSB Note: This is a necessary but not sufficient condition for monotonicity
Differential Nonlinearity (ADC) Nonideal ADC C 7 C 6 Code width for code C 3 C 5 C 4 C 3 C 2 C 1 C 0 X T1 X T2 X T3 X T4XT5 X T6XT7 DNL(k) is the code width for code k ideal code width normalized to X LSB T(k+1) - Tk - LSB DNL k = X X X X LSB
Differential Nonlinearity (ADC) Nonideal ADC C 7 C 6 C 5 Code width for code C 3 C 4 C 3 C 2 C 1 C 0 X T1 X T2 X T3 X T4XT5 X T6XT7 T(k+1) - Tk - LSB DNL k = X X X X LSB DNL= max 2kN-1 DNL k DNL=0 for an ideal ADC Note: In some nonideal ADCs, two or more break points could cause transitions to the same code C k making the definition of DNL ambiguous
Monotonicity in an ADC Nonideal ADCs C 7 C 7 C 6 C 6 C 5 C 5 C 4 C 4 C 3 C 3 C 2 C 2 C 1 C 0 C 1 C 0 X T1 X T2 X T3 X T4XT5 X T6XT7 X B1 X B2 X B3 X B4 X B5 X B6XB7 Monotone ADC Nonmonotone ADC Definition: An ADC is monotone if the X ( X ) X ( X ) whenever X X OUT k OUT m k m Note: Have used X Bk instead of X Tk since more than one transition point to a given code Note: Some authors do not define monotonicity in an ADC.
Missing Codes (ADC) C 7 C 6 Nonideal ADCs C 7 C 6 C 5 C 5 C 4 C 4 C 3 C 3 C 2 C 2 C 1 C 0 C 1 C 0 X T1 X T2 X T3 X T4XT5 X T6XT7 X T1 X T2 X T3 X T6XT7 X T4 No missing codes One missing code Definition: An ADC has no missing codes if there are N-1 transition points and a single LSB code increment occurs at each transition point. If these criteria are not satisfied, we say the ADC has missing code(s). Note: With this definition, all codes can be present but we still say it has missing codes Note: Some authors claim that missing codes in an ADC are the counterpart to nonmonotonicity in a DAC. This association is questionable.
C 7 C 6 Missing Codes (ADC) Nonideal ADCs C 7 C 6 C 5 C 5 C 4 C 4 C 3 C 3 C 2 C 2 C 1 C 0 X T1 X T2 X T3 X T4 X T5 C 1 C 0 X T1 X T2 X T3 X T5 X T4 X T6XT7 Missing codes Missing code with all codes present
Weird Things Can Happen Nonideal ADCs C 7 C 7 C 6 C 5 C 4 C 3 C 2 C 1 C 0 C 6 C 5 C 4 C 3 C 2 C 1 X B1 X B2 X B3 X B4 X B5 X B6XB7 C 0 X T1 X T2 X T3 X T5 X T4 X T6XT7 Multiple outputs for given inputs All codes present but missing codes Be careful on definition and measurement of linearity parameters to avoid having weird behavior convolute analysis, simulation or measurements Most authors (including manufacturers) are sloppy with their definitions of data converter performance parameters and are not robust to some weird operation
Linearity Measurements (testing) Consider ADC V IN (t) DUT X IOUT V REF Linearity testing often based upon code density testing Code density testing: V IN (t) V IN (t) V REF V REF t t Ramp or multiple ramps often used for excitation Linearity of test signal is critical (typically 3 or 4 bits more linear than DUT)
Linearity Measurements (testing) Code density testing: VIN(t) VREF t V IN (t) DUT X IOUT X ˆ OUT, C V REF C 0 C N-1 First and last bins generally have many extra counts (and thus no useful information) Typically average 16 or 32 hits per code
Linearity Measurements (testing) Code density testing: X ˆ OUT, C N-2 i=1 C C = N-2 i Ci-C DNL i= C 0 i=0,n-2 i i ˆ INL = Ck -ic k=1 1i N-3 C DNL = max 1 i N-2 INL = max 1 i N-3 DNLi INLi C 0 C N-1 This measurement is widely used Does not keep track of order bins are filled Some weird things can occasionally happen with this approach
Performance Characterization of Data Converters Static characteristics Resolution Least Significant Bit (LSB) Offset and Gain Errors Absolute Accuracy Relative Accuracy Integral Nonlinearity (INL) Differential Nonlinearity (DNL) Monotonicity (DAC) Missing Codes (ADC) Low-f Spurious Free Dynamic Range (SFDR) Low-f Total Harmonic Distortion (THD) Effective Number of Bits (ENOB) Power Dissipation
Spectral Characterization
INL Often Not a Good Measure of Linearity Four identical INL with dramatically different linearity
Linearity Issues INL is often not adequate for predicting the linearity performance of a data converter Distortion (or lack thereof) is of major concern in many applications Distortion is generally characterized in terms of the harmonics that may appear in a waveform
Spectral Analysis If f(t) is periodic f(t) alternately f(t) A A T 0 A ksin kω t θ k k 1 0 a ksin t k 1 k 1 kω t b kcos kω ω 2π T A k a 2 k b 2 k Termed the Fourier Series Representation of f(t)
Spectral Analysis (t) Nonlinear System (weakly) (t) Often the system of interest is ideally linear but practically it is weakly nonlinear. Often the input is nearly periodic and often sinusoidal and in latter case desired output is also sinusoidal Weak nonlinearity will cause distortion of signal as it is propagated through the system Spectral analysis often used to characterize effects of the weak nonlinearity
Spectral Analysis If (t) X IN Nonlinear System t X sinωt θ m (t) All spectral performance metrics depend upon the sequence A k k0 Spectral performance metrics of interest: SNDR, SDR, THD, SFDR, IMOD
Distortion Analysis A k A k k 0 1 2 3 4 5 6 k Often termed the DFT coefficients (will show later) Spectral lines, not a continuous function A 1 is termed the fundamental A k is termed the kth harmonic
Distortion Analysis A k A k k 0 1 2 3 4 5 6 k Often ideal response will have only fundamental present and all remaining spectral terms will vanish
Distortion Analysis A k A k k 0 1 2 3 4 5 6 k For a low distortion signal, the 2 nd and higher harmonics are generally much smaller than the fundamental The magnitude of the harmonics generally decrease rapidly with k for low distortion signals
Distortion Analysis A k 1 2 3 4 5 6 k f(t) is band-limited to frequency 2π f k if A k =0 for all k>k x
Distortion Analysis Total Harmonic Distortion, THD THD RMS voltage in harmonics RMS voltage of fundamental THD 2 2 2 2 A 2 A3 2 A 1 2 A4 2... THD k2 A A 1 2 k
Distortion Analysis Spurious Free Dynamic Range, SFDR The SFDR is the difference between the fundamental and the largest harmonic A k SFDR 1 2 3 4 5 6 k SFDR is usually determined by either the second or third harmonic
Distortion Analysis In a fully differential symmetric circuit, all even harmonics are absent in the differential output! A k 1 2 3 4 5 6 k
End of Lecture 26