EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 1EE247Lecture 11• Data converters– Sampling, aliasing, reconstruction– Amplitude quantization– Static converter error sources •Offset• Full-scale error• Differential non-linearity (DNL)• Integral non-linearity (INL)– Measuring DNL & INL• Servo-loop• Code density testing (histogram testing)EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 2Typical Sampling ProcessC.T. ⇒ S.D. ⇒ D.T.ContinuousTimeSampled Data(e.g. T/H signal)ClockDiscrete TimetimePhysicalSignals"MemoryContent"EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 3Discrete Time Signals• A sequence of numbers (or vector) with discrete index time instants• Intermediate signal values not defined(not the same as equal to zero!)• Mathematically convenient, non-physical• We will use the term "sampled data" for related signals that occur in real, physical interface circuitsEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 4Uniform Sampling• Samples spaced T seconds in time• Sampling Period T ⇔ Sampling Frequency fs=1/T• Problem: Multiple continuous time signals can yield exactly the same discrete time signal (aliasing)y(kT)=y(k)t= 1T 2T 3T 4T 5T 6T ...k= 1 2 3 4 5 6 ...EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 5Data Converters• ADC/DACs need to sample/reconstruct to convert from continuous-time to discrete-time signals and back• Purely mathematical discrete-time signals are different from "sampled-data signals" that carry information in actual circuits• Question: How do we ensure that sampling/reconstruction fully preserve information?EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 6Aliasing• The frequencies fxand nfs ±fx, n integer, are indistinguishable in the discrete time domain• Undesired frequency interaction and translation due to sampling is called aliasing• If aliasing occurs, no signal processing operation downstream of the sampling process can recover the original continuous time signal!EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 7Frequency Domain Interpretationfs……..fAmplitudefin2fsAmplitudef/fsSignal scenariobefore samplingSignal scenarioafter sampling Æ DTfs /20.5ContinuousTimeDiscreteTimeÆSignals @ nfS±fmax__signalfold back into band of interest Æ AliasingEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 8Brick Wall Anti-Aliasing FilterSampling at Nyquist rate (fs=2fsignal) Æ required brick-wall anti-aliasing filtersContinuousTimeDiscreteTime0fs2fs... fAmplitude00.5 f/fsFilterEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 9Practical Anti-Aliasing Filter• Practical filter: Nonzero "transition band"• In order to make this work, we need to sample faster than 2x thesignal bandwidth• "Oversampling" ContinuousTimeDiscreteTime0fs...DesiredSignal00.5f/fsfs/2BParasiticToneB/fsAttenuationfs-BfEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 10Data ConverterClassification• fs> 2fmaxNyquist Sampling– "Nyquist Converters"– Actually always slightly oversampled (e.g. CODEC fsigmax =3.4kHz &ADC sampling 8kHz Æfs /fmax=2.35)– Requires anti-aliasing filtering prior to A-to-D conversion• fs>> 2fmaxOversampling– "Oversampled Converters"– Anti-alias filtering is often trivial– Oversampling is also used to reduce quantization noise, see later in the course...• fs< 2fmaxUndersampling (sub-sampling)EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 11Sub-Sampling• Sub-sampling Æ sampling at a rate less than Nyquist rate Æ aliasing• For signals centered @ an intermediate frequency Æ Not destructive!• Sub-sampling can be exploited to mix a narrowband RF or IF signal down to lower frequenciesContinuousTimeDiscreteTime0fs...fAmplitude00.5 f/fsBP FilterEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 12Nyquist Data Converter Topics• Basic operation of data converters – Uniform sampling and reconstruction– Uniform amplitude quantization• Characterization and testing• Common ADC/DAC architectures• Selected topics in converter design– Practical implementations– Compensation & calibration for analog circuit non-idealities• Figures of merit and performance trendsEECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 13Where Are We Now?• We now know how to preserve signal information in CTÆDT transition• How do we go back from DTÆ CT?Analog PostprocessingD/AConversionDSPA/D ConversionAnalog PreprocessingAnalog InputAnalog Output000...001...110Anti-AliasingFilter?Sampling(+Quantization)EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 14Ideal Reconstruction• Unfortunately not all that practical...∑∞−∞=−⋅=kkTtgkxtx )()()(BtBttgππ2)2sin()( =• The DSP books tell us: ⇒x(k) x(t)EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 15Zero-Order Hold Reconstruction• How about just creating a staircase, i.e. hold each discrete time value until new information becomes available? • What does this do to the frequency content of the signal?• Let's analyze this in two steps...0 10 20 30-1-0.6-0.20.21Time [μs]Amplitudesampled dataafter ZOH0.6EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 16DTÆ CT: Infinite Zero PaddingDT sequenceTime Domain Frequency Domain......0.5InfiniteZero paddedInterpolation:CT Signal......f /fsf /fs0.5fs1.5fs2.5fsNext step: pass the samples through a sample & hold stage (ZOH)EECS 247 Lecture 11: Intro. to Data Converters & Performance Metrics © 2009 H. K. Page 17Hold Pulse Tp=TsTransfer FunctionppspfTfTTTfHππ)sin(|)(| =0 0.5 1 1.5 2 2.5 300.20.40.60.81f
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