Lecture 9 Sensors AD sampling noise and jitter
- Slides: 63
Lecture 9 Sensors, A/D, sampling noise and jitter Forrest Brewer
Light Sensors - Photoresistor voltage divider Vsignal = (+5 V) RR/(R + RR) – Choose R=RR at median of intended measured range – Cadmium Sulfide (Cd. S) – Cheap, relatively slow (low current) • t. RC = Cl*(R+RR) Typically R~50 -200 k. W C~20 p. F so t. RC~20 -80 u. S => 10 -50 k. Hz
Light Sensors - Phototransistor Much higher sensitivity - Relatively slow response (~1 -5 u. S due to collector capacitance) -
Light Sensors - Pyroelectric Sensors lithium tantalate crystal is heated by thermal radiation l tuned to 8 -10 m radiation – maximize response to human IR signature l motion detecting burglar alarm l E. g. Eltec 442 -3 sensor - two elements, Fresnel optics, output proportional to the difference between the charge on the left crystal and the charge on the right crystal. l
Other Common Sensors l Force – strain gauges - foil, conductive ink – conductive rubber – rheostatic fluids l – MEMS – Pendulum l – • Charge source – l Field – Antenna – Magnetic • Hall effect • Flux Gate • Usually Piezoelectric – microswitches – shaft encoders – gyros Temperature • Voltage/Current Source – Sonar Position Motor current • Stall/velocity Sound • Both current and charge versions Battery-level • voltage – piezoelectric films – capacitive force – Microphones l Monitoring – • Piezorestive (needs bridge) l Acceleration l Location – Permittivity – Dielectric
Incidence -- photoreflectors
Rotational Position Sensors l Optical Encoders – Relative position – Absolute position l Other Sensors – Resolver – Potentiometer Jizhong Xiao
Optical Encoders mask/diffuser • Relative position light sensor light emitter decode circuitry grating Jizhong Xiao
Optical Encoders • Relative position light sensor - direction - resolution decode circuitry light emitter Phase lag between A and B is 90 degrees (Quadrature Encoder) Ronchi grating A B A leads B Jizhong Xiao
Optical Encoders • Detecting absolute position • Typically 4 k-8 k/2 p • Higher Resolution Available – Laser/Hologram (0. 1 -0. 3” resolution) Jizhong Xiao
Gray Code 0000 Almost universally used encoding l One transition per adjacent number l 0001 0011 – Eliminates alignment issue of multiple 0010 bits 0110 – Simplified Logic 0111 – Eliminates position jitter issues l Recursive Generalization of 2 -bit quadrature code 0101 0100 – Each segment in reverse order as 1100 next bit is added – Preserves unambiguous absolute. . position and direction 2 n-1 1001 Jizhong Xiao
Other Motor Sensors • Resolver • Selsyn pairs (1930 -1960) • High speed • Potentiometer • High resolution • Monotone but poor linearity • Noise! • Deadzone! Jizhong Xiao
Draper Tuning Fork Gyro The rotation of tines causes the Coriolis Force l Forces detected through either electrostatic, electromagnetic or piezoelectric. l Displacements are measured in the Comb drive l
Improvement in MEMS Gyros l Improvement of drift – Little drift improvement in last decade – Controls/Fabrication issue l Improvement of resolution
Piezoelectric Gyroscopes l Basic Principles – Piezoelectric plate with vibrating thickness – Coriolis effect causes a voltage form the material – Very simple design and geometry
Piezoelectric Gyroscope l Advantages – Lower input voltage than vibrating mass – Measures rotation in two directions with a single device – More Robust l Disadvantages – (much) Less sensitive – Output is large when Ω = 0 • Drift compensation
Absolute Angle Measurement Bias errors cause a drift while integrating l Angle is measured with respect to the casing l – The mass is rotated with an initial θ – When the gyroscopes rotates the mass continues to rotate in the same direction l Angular rate is measured by adding a driving frequency ωd
Design consideration Damping needs to be compensated l Irregularities in manufacturing l Angular rate measurement l For angular rate measurement Compensation force
Measurement Accuracy vs. Precision l Expectation of deviation of a given measurement from a known standard – Often written as a percentage of the possible values for an instrument l Precision is the expectation of deviation of a set of measurements – “standard deviation” in the case of normally distributed measurements – Few instruments have normally distributed errors
Deviations l Systematic errors – Portion of errors that is constant over data gathering experiment – Beware timescales and conditions of experiment– if one can identify a measurable input parameter which correlates to an error – the error is systematic – Calibration is the process of reducing systematic errors – Both means and medians provide estimates of the systematic portion of a set of measurements
Random Errors l The portion of deviations of a set of measurements which cannot be reduced by knowledge of measurement parameters – E. g. the temperature of an experiment might correlate to the variance, but the measurement deviations cannot be reduced unless it is known that temperature noise was the sole source of error – Error analysis is based on estimating the magnitude of all noise sources in a system on a given measurement – Stability is the relative freedom from errors that can be reduced by calibration– not freedom from random errors
Model based Calibration Given a set of accurate references and a model of the measurement error process l Estimate a correction to the measurement which minimizes the modeled systematic error l E. g. given two references and measurements, the linear model: l
Noise Reduction: Filtering l Noise is specified as a spectral density (V/Hz 1/2) or W/Hz RMS noise is proportional to the bandwidth of the signal: l Noise density is the square of the transfer function l Net (RMS) noise after filtering is: l
Filter Noise Example l RC filtering of a noisy signal Assume uniform input noise, 1 st order filter l The resulting output noise density is: l We can invert this relation to get the equivalent input noise: l
Averaging (filter analysis) l Simple processing to reduce noise – running average of data samples l The frequency transfer function for an N-pt average is: l To find the RMS voltage noise, use the previous technique: l So input noise is reduced by 1/N 1/2
‘Normal’ Gaussian Statistics Mean l Standard Deviation l – Note that this is not an estimate for a total sample set (issue if N<<100), use 1/(N-1) l For large set of data with independent noise sources the distribution is: l Probability
Issues with Normal statistics l Assumptions: – Noise sources are all uncorrelated – All Noise sources are accounted for – Enough time has elapsed to cover events l In many practical cases, data has ‘outliers’ where non-normal assumptions prevail – Cannot Claim small probability of error unless sample set contains all possible failure modes – Mean may be poor estimator given sporadic noise l Median (middle value in sorted order of data samples) often is better behaved – Not used often since analysis of expectations are difficult
Characteristic of ADC and DAC l DAC – – l Monotonic and nonmonotonic Offset , gain error , DNL and INL Glitch Sampling-time uncertainty ADC – – missing code Offset , gain error , DNL and INL Quantization Noise Sampling-time uncertainty
Monotonic and missing code If DNL < - 1 LSB => missing code. (A/D)
Offset and Gain Error D/A A/D
D/A nonlinearity (D/A) Differential nonlinearity (DNL): Maximum deviation of the analog output step from the ideal value of 1 LSB. Integral nonlinearity (INL): Maximum deviation of the analog output from the ideal value.
D/A nonlinearity (A/D) Differential nonlinearity (DNL): Maximum deviation in step width (width between transitions) from the ideal value of 1 LSB l Integral nonlinearity (INL): Maximum deviation of the step midpoints from the ideal step midpoints. Or the maximum deviation of the transition points from ideal. l
Glitch (D/A) I 1 represents the MSB current l I 2 represents the N-1 LSB current l ex: 0111… 1 to 1000… 0 l
Sampling Theorem l Perfect Reconstruction of a continuous-time signal with Band limit f requires samples no longer than 1/2 f – Band limit is not Bandwidth – but limit of maximum frequency – Any signal beyond f aliases the samples
Aliasing (Sinusoids)
Aliased Reconstruction assumes values on principle branch – usually lower frequency l Nyquist Theorem assumes infinite history is available – Aliasing issues are worse for finite length samples – Don’t crowd Nyquist limits! l
Alaising For Sinusoid signals (natural band limit): l For Cos(wn), w=2 pk+w 0 l – Samples for all k are the same! – Unambiguous if 0<w<p – Thus One-half cycle per sample l So if sampling at T, frequencies of f=e+1/2 T will map to frequency e
Quantization Effects l Samples are digitized into finite digital resolution – Shows up as uniform random noise – Zero bias (for ideal A/D)
Quantization Error +lsb/2 x -lsb/2 Deviations produced by digitization of analog measurements l For white, random signal with uniform quantization of xlsb: l
Quantization Noise (A/D)
Quantization Noise Uniform Random Value l Bounded range: –VLSB/2, +VLSB/2 l Zero Mean l
Sampling Jitter (Timing Error) l Practical Sampling is performed at uncertain time – Sampling interval noise – measured as value error – Sampling timing noise – also measured as value error
Sampling-Time Uncertainty (Aperture Jitter) l Assume a full-scale sinusoidal input, l want l then l
Jitter Noise Analysis l Assume that samples are skewed by random amount t j: l Expanding v(t) into a Taylor Series: l Assuming tj to be small:
Sampling Jitter Bounds l Error signal is proportional to the derivative l Bounding the bandwidth bounds the derivative l For t. RMS, the RMS noise is: l If we limit v. RMS to LSB – we can bound the jitter l So for a 1 MHz bandwidth, and 12 bit A/D we need less than 100 p. S of RMS jitter
DAC Timing Jitter l DAC output is convolution of unit steps – Jitter RMS error depends on both timing error and sample period Dv tj
DAC Timing Jitter l Error is: l Energy error: l RMS jitter error: l Relating to continuous time:
DAC Jitter Bounds l We can use the same band limit argument as for sampling to find the jitter bound for a D-bit DAC: l So a 10 MHz, 5 -bit DAC can have at most 85 p. S of jitter.
Decoder-Based D/A converters Inherently monotonic. l DNL depend on local matching of neighboring R's. l INL depends on global matching of the R-string. l
Decoder-Based D/A converters l 4 -bit folded R -string D/A converter
Decoder-Based D/A converters Multiple Rstring 6 -bit D/A converter l interpolating l
Decoder-Based D/A converters R-string DACs with binary-tree decoding. l Speed is limited by the delay through the resistor string as well as the delay through the switch network. l
Binary-Scaled D/A Converters Monotonicity is not guaranteed. l Potentially large glitches due to timing skews. l Current-mode converter
Binary-Scaled D/A Converters Binary-array charge-redistribution D/A converter 4 bit R-2 R based D/A converter l No wide-range scaling of resistors. l
Thermometer-Code Converter
Flash (Parallel) Converters High speed. Requires only one comparison cycle per conversion. l Large size and power dissipation for large N. l
Feedback in Sensing/Conversion l High Resolution and Linearity Converters – Very expensive to build open-loop (precision components) – Aging, Drift, Temperature Compensation l Closed-Loop Converters – Much higher possible resolution – Greatly improved linearity – Can use inexpensive components by substituting amplifier gain for component precision But – Higher Measurement Latency – Decreased Bandwidth – Eg. Successive Approx, Sigma-Delta
Nyquist-Rate A/D converters
Integrating converters l Low conversion rate.
Successive-Approximation Converters l Binary search
Successive-Approximation Converters l DAC-based successive-approximation converter. – Requires a high-speed DAC with precision on the order of the converter itself. – Excellent trade-off between accuracy and speed. Most widely used architecture for monolithic A/D.
Sigma Delta A/D Converter e[n] x(t) Bandlimited to fo Sampler fs x[n] Analog Modulator fs y[n] Decimation Filter 2 fo 16 bits Digital Over Sampling Ratio = 2 fo is Nyquist frequency Transfer function for an Lth order modulator given by
Modulator Characteristics Highpass character for noise transfer function: l In-band noise power is given by l no falls by 3(2 L+1) for doubling of Over Sampling Ratio l L+0. 5 bits of resolution for doubling of Over Sampling Ratio l no essentially is uncorrelated for l Dithering is used to decorrelate quantization noise l
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- Jitter
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- Smokeping jitter
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