EE 435. Lecture 28. Data Converters Linearity INL/DNL Spectral Performance
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1 EE 435 Lecture 8 Data Converters Linearity INL/DNL Spectral Performance
2 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
3 Linearity A data converter (ADC or DAC) can be viewed as an amplifier that interfaces between the analog and digital domains Linearity is of considerable concern in amplifiers irrespective of whether the I/O is analog:analog, analog:digital, digital:analog, or digital:digital The seemingly simple concept of linearity is challenging to accurately characterize
4 Performance Characterization of Data Converters Spectral Characterization 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 Linearity Metrics
5 Spectral Characterization
6 INL Often Not a Good Measure of Linearity Four identical INL with dramatically different linearity X OUT X OUT X REF X REF X IN X IN X REF X REF X OUT X OUT X REF X REF X IN X IN X REF X REF
7 Linearity Issues INL is often not adequate for predicting the linearity performance of a data converter Distortion (or lac thereof) is of major concern in many applications Distortion is generally characterized in terms of the harmonics that may appear in a waveform
8 Spectral Analysis If f(t) is periodic f(t) alternately f(t) A A T 0 A sin ω t θ 1 0 a sin t 1 1 ω t b cos ω ω π T A a b Termed the Fourier Series Representation of f(t)
9 Spectral Analysis X IN (t) Nonlinear System (wealy) X OUT (t) Often the system of interest is ideally linear but practically it is wealy nonlinear. Often the input is nearly periodic and often sinusoidal and in latter case desired output is also sinusoidal Wea nonlinearity will cause harmonic distortion (often just termed distortion) of signal as it is propagated through the system Spectral analysis often used to characterize effects of the wea nonlinearity
10 Spectral Analysis X IN (t) Nonlinear System (wealy) X OUT (t) Distortion Types: Frequency Distortion Nonlinear Distortion (alt. harmonic distortion) Frequency Distortion: Amplitude and phase of system is altered but output is linearly related to input Nonlinear Distortion: System is not linear, frequency components usually appear in the output that are not present in the input Spectral Analysis is the characterization of a system with a periodic input with the Fourier series relationships between the input and output waveforms
11 Spectral Analysis X IN (t) Nonlinear System X OUT (t) If X t X sinωt θ IN m All spectral performance metrics depend upon the sequences A 0 1 Spectral performance metrics of interest: Alternately sin cos X t A a t b t OUT SNDR, SDR, THD, SFDR, IMOD A a b 1 b tan a
12 A A Often termed the DFT coefficients (will show later) Spectral lines, not a continuous function A 1 is termed the fundamental A is termed the th harmonic
13 A A Often ideal response will have only fundamental present and all remaining spectral terms will vanish
14 A A For a low distortion signal, the nd and higher harmonics are generally much smaller than the fundamental The magnitude of the harmonics generally decrease rapidly with for low distortion signals
15 A Assume f(t) is periodic with period 1 T f f(t) is band-limited to frequency π f X if A =0 for all > x Where A 0 are the Fourier series coefficients of f(t)
16 Total Harmonic Distortion, THD THD RMS voltage in harmonics RMS voltage of fundamental THD A A3 A 1 A4... THD A A 1
17 Spurious Free Dynamic Range, SFDR The SFDR is the difference between the fundamental and the largest harmonic A SFDR SFDR is usually determined by either the second or third harmonic
18 In a fully differential symmetric circuit, all even harmonics are absent in the differential output! A
19 Theorem: In a fully differential symmetric circuit, all even-order terms are absent in the Taylor s series output for symmetric differential excitations! Proof: V ID + - V O1 V O + V OD Expanding in a Taylor s series around V ID =0, we obtain - V f V ID h 1 VID 0 V f -V h -V ID ID 0 V =V V h V h -V OD 1 ID ID 0 0 V = h V -V OD ID ID 0 1 V = h V V OD ID ID 0 When is even, term in [ ] vanishes
20 Theorem: In a fully differential symmetric circuit, all even harmonics are absent in the differential output for symmetric differential excitations! V ID + V O1 + V OD Proof: - V O - Recall: sin n x n 1 0 n 0 h sin n x for n odd g sin n x for n even where h, g, and θ are constants That is, odd powers of sin n (x) have only odd harmonics present and even powers have only even harmonics present
21 Theorem: In a fully differential symmetric circuit, all even harmonics are absent in the differential output for symmetric differential sinusoidal excitations! V OD Proof: V ID + V f V h V - V O1 V O + V OD Expanding in a Taylor s series around V ID =0, we obtain Assume V ID =Ksin(ωt) W.L.O.G. assume K=1 V h sin ω t V h -sinω t V O1 O1 and O1 ID ID 0 V 0 O 0 h O - O ID ID 0 ) V f -V h (-V 0 sin ωt -sin ωt h sin ωt 1 sin ω t Observe the even-ordered powers and hence even harmonics are absent in this last sum 0
22 How are spectral components determined? By integral a or ωt t 1 T A 1 ωt t 1 T t 1 f t e jωt dt t 1 T t 1 f t e jωt dt f tsin tωdt b f tcos tωdt t 1 ωt Integral is very time consuming, particularly if large number of components are required t 1 T t 1 By DFT (with some restrictions that will be discussed) By FFT (special computational method for obtaining DFT)
23 How are spectral components determined? T T S Consider sampling f(t) at uniformly spaced points in time T S seconds apart This gives a sequence of samples N f T s =1
24 Consider a function f(t) that is periodic with period T f(t) A T 0 A sin ω t θ 1 =f = T Band-limited Periodic Functions Definition: A periodic function of frequency f is band limited to a frequency f max if A =0 for all f max f
25 T NOTATION: T S T: Period of Excitation T S : Sampling Period N P : Number of periods over which samples are taen N: Total number of samples N P NT T S N 1 h = Int -1 N P Note: N P is not an integer unless a specific relationship exists between N, T S and T Note: The function Int(x) is the integer part of x
26 T f MAX, then Am N and Χ 0 T S THEOREM: If N P is an integer and x(t) is band limited to where f = 1/T, Χ 1 0 m h -1 Χ mn N P for all not defined above 1 is the DFT of the sequence xt S 0 f N f MAX = N, and N 1 h = Int -1 N P P Key Theorem central to Spectral Analysis that is widely used!!! and often abused N1 0
27 T 0 T S THEOREM?: If N P is an integer and x(t) is band limited to f MAX, then Am ΧmNP 1 0 m h N and Χ for all not defined above where f = 1/T, Χ N 1 is the DFT of the sequence xt S 0 f N f MAX = N P, and h = Int f MAX f N1 0
28 Why is this a Key Theorem? T f MAX, then Am N and Χ 0 T S THEOREM: If N P is an integer and x(t) is band limited to where f = 1/T, Χ 1 0 m h -1 Χ mn N P for all not defined above 1 is the DFT of the sequence xt S 0 f N f MAX = N, and N 1 h = Int -1 N P P DFT requires dramatically less computation time than the integrals for obtaining Fourier Series coefficients Can easily determine the sampling rate (often termed the Nyquist rate) to satisfy the band limited part of the theorem N1 0
29 How is this theorem abused? T f MAX, then Am N and Χ 0 T S THEOREM: If N P is an integer and x(t) is band limited to where f = 1/T, Χ 1 0 m h -1 Χ mn N P for all not defined above 1 is the DFT of the sequence xt S 0 f f = N, and N 1 h = Int -1 MAX NP NP Much evidence of engineers attempting to use the theorem when N P is not an integer Challenging to have N P an integer in practical applications Dramatic errors can result if there are not exactly an integer number of periods in the sampling window N1 0
30 T T S If the hypothesis of the theorem are satisfied, we thus have A 1 A 0 A A 3 A4 N P +1 N P +1 3N P +1 4N P +1
31 If the hypothesis of the theorem are satisfied, we thus have A 1 A A 3 A4 A 0 N P +1 N P +1 3N P +1 4N P +1 FFT is a computationally efficient way of calculating the DFT, particularly when N is a power of
32 End of Lecture 8
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