EE 435. Lecture 29. Data Converters. Linearity Measures Spectral Performance

Transcription

1 EE 435 Lecture 9 Data Converters Linearity Measures Spectral Performance

2 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)

3 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 3 hits per code

4 Linearity Measurements (testing) Code density testing: X ˆ OUT, C N- i=1 C C = N- i Ci-C DNL i= C 0 i=0,n- i i ˆ INL = C -ic =1 1i N-3 C DNL = max 1 i N- INL = max 1 i N-3 DNLi DNLi C 0 C N-1 This measurement is widely used Does not eep trac of order bins are filled Some weird things can occasionally happen with this approach

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

6 Spectral Characterization

7 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

8 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

9 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)

10 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 distortion of signal as it is propagated through the system Spectral analysis often used to characterize effects of the wea nonlinearity

11 Spectral Analysis If X IN (t) X IN Nonlinear System t X sinωt θ m X OUT (t) All spectral performance metrics depend upon the sequence A 0 Spectral performance metrics of interest: SNDR, SDR, THD, SFDR, IMOD

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 f(t) is band-limited to frequency π f if A =0 for all > x

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 harmonics are absent in the differential output for symmetric differential excitations! Proof: V ID V OD Expanding in a Taylor s series around V ID =0, we obtain V 1 V - V OD V f V ID OD ID 0 Assume V ID =Ksin(ωt) W.L.O.G. assume K=1 V h sin ω t V h -sinω t V O1 O1 V 0 O 0 h h O V 0 sin ωt -sin ωt h sin ωt 1 sin ω t Observe the even-ordered harmonics are absent in this last sum 0

20 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)

21 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

22 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

23 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 MAX 1 0 m h -1 Χ mn N f = P for all not defined above 1 is the DFT of the sequence xt 0 S f N N 1 N, and h = Int -1 P NP N1 0

24 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 f N f MAX = N 1 is the DFT of the sequence xt 0 S P, and h = Int f MAX f N1 0

25 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

26 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

27 End of Lecture 9