EE247 Lecture 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 1 Typical Sampling Process C T S D D T time Continuous Time Physical Signals Sampled Data e g T H signal Clock Memory Content Discrete Time EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 2 Discrete 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 circuits EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 3 Uniform Sampling y kT y k t 1T k 1 2T 2 3T 3 4T 4 5T 5 6T 6 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 EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 4 Data 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 5 Aliasing The frequencies fx and 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 6 Amplitude Frequency Domain Interpretation Signal scenario before sampling Continuous Time fin fs 2 2fs f fs Discrete Time Amplitude Signal scenario after sampling DT 0 5 f fs Signals nfS fmax signal fold back into band of interest Aliasing EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 7 Brick Wall Anti Aliasing Filter Amplitude Filter Continuous Time 0 fs 2fs f Discrete Time 0 0 5 f fs Sampling at Nyquist rate fs 2fsignal required brick wall anti aliasing filters EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 8 Practical Anti Aliasing Filter Parasitic Tone Desired Signal Attenuation Continuous Time 0 B fs 2 0 B fs 0 5 fs B fs f Discrete Time f fs Practical filter Nonzero transition band In order to make this work we need to sample faster than 2x the signal bandwidth Oversampling EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 9 Data Converter Classification fs 2fmax Nyquist Sampling Nyquist Converters Actually always slightly oversampled e g CODEC fsigmax 3 4kHz ADC sampling 8kHz fs fmax 2 35 Requires anti aliasing filtering prior to A to D conversion fs 2fmax Oversampling Oversampled Converters Anti alias filtering is often trivial Oversampling is also used to reduce quantization noise see later in the course fs 2fmax Undersampling sub sampling EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 10 Sub Sampling Amplitude BP Filter Continuous Time 0 fs f Discrete Time 0 0 5 f fs Sub sampling sampling at a rate less than Nyquist rate aliasing For signals centered an intermediate frequency Not destructive Sub sampling can be exploited to mix a narrowband RF or IF signal down to lower frequencies EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 11 Nyquist 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 trends EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 12 Where Are We Now Analog Input We now know how to preserve signal information in CT DT transition Analog Preprocessing Anti Aliasing Filter Sampling Quantization A D Conversion 000 001 110 DSP D A Conversion How do we go back from DT CT Analog Postprocessing Analog Output EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 13 Ideal Reconstruction x k x t The DSP books tell us x t x k g t kT k g t sin 2 Bt 2 Bt Unfortunately not all that practical EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 14 Zero Order Hold Reconstruction How about just creating a staircase i e hold each discrete time value until new information becomes available 1 Amplitude 0 6 0 2 0 2 What does this do to the frequency content of the signal 0 6 1 sampled data after ZOH 0 10 20 30 Let s analyze this in two steps Time s EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 15 DT CT Infinite Zero Padding Time Domain DT sequence Frequency Domain f fs 0 5 Infinite Zero padded Interpolation CT Signal f fs 0 5fs 1 5fs 2 5fs Next step pass the samples through a sample hold stage ZOH EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 16 Hold Pulse Tp Ts Transfer Function 1 sin T fsin Ts fT p p H f f Ts Ts fT p H f abs H f 0 8 0 6 0 4 0 2 0 0 0 5 1 1 5 2 2 5 3 f fs EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 17 ZOH Spectral Shaping 1 Continuous Time Pulse Train Spectrum X k 0 5 0 0 0 5 1 1 5 2 2 5 3 1 ZOH Transfer Function Sinc Shaping ZOH 0 5 0 0 0 5 1 0 5 1 1 5 2 2 5 3 1 5 2 2 5 3 1 ZOH output Spectrum of Staircase Approximation 0 5 0 0 f fs EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 18 Smoothing Filter 1 Order of the filter required is a function of oversampling ratio Filter out the high frequency content associated with staircase shape of the signal 0 8 0 6 High oversampling helps reduce filter order requirement 0 4 0 2 0 0 0 5 1 1 5 2 2 5 3 f fs EECS 247 Lecture 11 Intro to Data Converters Performance Metrics 2009 H K Page 19 Summary Sampling theorem fs 2fmax usually dictates anti aliasing
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