Lecture 9 Sensors, A/D, sampling noise and jitter

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Lecture 9Sensors, A/D, sampling noise and jitter

• Forrest Brewer

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Light Sensors - Photoresistor

• voltage divider Vsignal (5V) RR/(R RR)
• Choose RRR at median of intended measured range
• Cadmium Sulfide (CdS)
• Cheap, relatively slow (low current)
• tRC Cl(RRR)
• Typically R50-200kW C20pF so tRC20-80uS gt
  10-50kHz

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Light Sensors - Phototransistor

• Much higher sensitivity
• Relatively slow response (1-5uS due to collector
  capacitance)

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Light Sensors - Pyroelectric Sensors

• lithium tantalate crystal is heated by thermal
  radiation
• tuned to 8-10 ?m radiation maximize response to
  human IR signature
• motion detecting burglar alarm
• 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.

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Other Common Sensors

• Force
• strain gauges - foil, conductive ink
• conductive rubber
• rheostatic fluids
• Piezorestive (needs bridge)
• piezoelectric films
• capacitive force
• Charge source
• Sound
• Microphones
• Both current and charge versions
• Sonar
• Usually Piezoelectric
• Position
• microswitches
• shaft encoders
• gyros

• Acceleration
• MEMS
• Pendulum
• Monitoring
• Battery-level
• voltage
• Motor current
• Stall/velocity
• Temperature
• Voltage/Current Source
• Field
• Antenna
• Magnetic
• Hall effect
• Flux Gate
• Location
• Permittivity
• Dielectric

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Incidence -- photoreflectors
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Rotational Position Sensors

• Optical Encoders
• Relative position
• Absolute position
• Other Sensors
• Resolver
• Potentiometer

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Optical Encoders
mask/diffuser

• Relative position

light sensor
decode circuitry
light emitter
grating
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Optical Encoders

• Relative position

- direction
light sensor
- resolution
decode circuitry
light emitter
Phase lag between A and B is 90 degrees
(Quadrature Encoder)
Ronchi grating
A
B
A leads B
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Optical Encoders

• Detecting absolute position
• Typically 4k-8k/2p
• Higher Resolution Available Laser/Hologram
  (0.1-0.3 resolution)

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Gray Code
0000 0001 0011 0010 0110 0111 0101 0100 1100 .. 10
01

• Almost universally used encoding
• One transition per adjacent number
• Eliminates alignment issue of multiple bits
• Simplified Logic
• Eliminates position jitter issues
• Recursive Generalization of 2-bit quadrature code
• Each 2n-1 segment in reverse order as next bit is
  added
• Preserves unambiguous absolute position and
  direction

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Other Motor Sensors

• Resolver
• Selsyn pairs (1930-1960)
• High speed

• Potentiometer
• High resolution
• Monotone but poor linearity
• Noise!
• Deadzone!

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Draper Tuning Fork Gyro

• The rotation of tines causes the Coriolis Force
• Forces detected through either electrostatic,
  electromagnetic or piezoelectric.
• Displacements are measured in the Comb drive

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Improvement in MEMS Gyros

• Improvement of drift
• Little drift improvement in last decade
• Controls/Fabrication issue
• Improvement of resolution

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Piezoelectric Gyroscopes

• Basic Principles
• Piezoelectric plate with vibrating thickness
• Coriolis effect causes a voltage form the
  material
• Very simple design and geometry

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Piezoelectric Gyroscope

• Advantages
• Lower input voltage than vibrating mass
• Measures rotation in two directions with a single
  device
• More Robust
• Disadvantages
• (much) Less sensitive
• Output is large when O 0
• Drift compensation

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Absolute Angle Measurement

• Bias errors cause a drift while integrating
• Angle is measured with respect to the casing
• The mass is rotated with an initial ?
• When the gyroscopes rotates the mass continues to
  rotate in the same direction
• Angular rate is measured by adding a driving
  frequency ?d

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Design consideration

• Damping needs to be compensated
• Irregularities in manufacturing
• Angular rate measurement

For angular rate measurement
Compensation force
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Measurement Accuracy vs. Precision

• Expectation of deviation of a given measurement
  from a known standard
• Often written as a percentage of the possible
  values for an instrument
• 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

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Deviations

• 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

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Random Errors

• 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

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Model based Calibration

• Given a set of accurate references and a model of
  the measurement error process
• Estimate a correction to the measurement which
  minimizes the modeled systematic error
• E.g. given two references and measurements, the
  linear model

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Noise Reduction Filtering

• Noise is specified as a spectral density
  (V/Hz1/2) or W/Hz
• RMS noise is proportional to the bandwidth of the
  signal
• Noise density is the square of the transfer
  function
• Net (RMS) noise after filtering is

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Filter Noise Example

• RC filtering of a noisy signal
• Assume uniform input noise, 1st order filter
• The resulting output noise density is
• We can invert this relation to get the equivalent
  input noise

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Averaging (filter analysis)

• Simple processing to reduce noise running
  average of data samples
• The frequency transfer function for an N-pt
  average is
• To find the RMS voltage noise, use the previous
  technique
• So input noise is reduced by 1/N1/2

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Normal Gaussian Statistics

• Mean
• Standard Deviation
• Note that this is not an estimate for a total
  sample set (issue if Nltlt100), use 1/(N-1)
• For large set of data with independent noise
  sources the distribution is
• Probability

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Issues with Normal statistics

• Assumptions
• Noise sources are all uncorrelated
• All Noise sources are accounted for
• Enough time has elapsed to cover events
• 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
• Median (middle value in sorted order of data
  samples) often is better behaved
• Not used often since analysis of expectations are
  difficult

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Characteristic of ADC and DAC

• DAC
• 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

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Monotonic and missing code
If DNL lt - 1 LSB gt missing code. (A/D)
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Offset and Gain Error
D/A
A/D
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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.
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D/A nonlinearity (A/D)

• Differential nonlinearity (DNL) Maximum
  deviation in step width (width between
  transitions) from the ideal value of 1 LSB
• Integral nonlinearity (INL) Maximum deviation of
  the step midpoints from the ideal step midpoints.
  Or the maximum deviation of the transition points
  from ideal.

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Glitch (D/A)

• I1 represents the MSB current
• I2 represents the N-1 LSB current
• ex01111 to 10000

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Sampling Theorem

• Perfect Reconstruction of a continuous-time
  signal with Band limit f requires samples no
  longer than 1/2f
• Band limit is not Bandwidth but limit of
  maximum frequency
• Any signal beyond f aliases the samples

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Aliasing (Sinusoids)
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Aliased Reconstruction

• Reconstruction assumes values on principle branch
  usually lower frequency
• Nyquist Theorem assumes infinite history is
  available
• Aliasing issues are worse for finite length
  samples
• Dont crowd Nyquist limits!

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Alaising

• For Sinusoid signals (natural band limit)
• For Cos(wn), w2pkw0
• Samples for all k are the same!
• Unambiguous if 0ltwltp
• Thus One-half cycle per sample
• So if sampling at T, frequencies of fe1/2T will
  map to frequency e

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Quantization Effects

• Samples are digitized into finite digital
  resolution
• Shows up as uniform random noise
• Zero bias (for ideal A/D)

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Quantization Error
lsb/2
x
-lsb/2

• Deviations produced by digitization of analog
  measurements
• For white, random signal with uniform
  quantization of xlsb

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Quantization Noise (A/D)
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Quantization Noise

• Uniform Random Value
• Bounded range VLSB/2, VLSB/2
• Zero Mean

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Sampling Jitter (Timing Error)

• Practical Sampling is performed at uncertain time
• Sampling interval noise measured as value error
• Sampling timing noise also measured as value
  error

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Sampling-Time Uncertainty

• (Aperture Jitter)
• Assume a full-scale sinusoidal input,
• want
• then

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Jitter Noise Analysis

• Assume that samples are skewed by random amount
  tj
• Expanding v(t) into a Taylor Series
• Assuming tj to be small

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Sampling Jitter Bounds

• Error signal is proportional to the derivative
• Bounding the bandwidth bounds the derivative
• For tRMS, the RMS noise is
• If we limit vRMS to LSB we can bound the jitter
• So for a 1MHz bandwidth, and 12 bit A/D we need
  less than 100pS of RMS jitter

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DAC Timing Jitter

• DAC output is convolution of unit steps
• Jitter RMS error depends on both timing error and
  sample period

Dv
tj
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DAC Timing Jitter

• Error is
• Energy error
• RMS jitter error
• Relating to continuous time

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DAC Jitter Bounds

• We can use the same band limit argument as for
  sampling to find the jitter bound for a D-bit
  DAC
• So a 10MHz, 5-bit DAC can have at most 85pS of
  jitter.

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Decoder-Based D/A converters

• Inherently monotonic.
• DNL depend on local matching of neighboring R's.
• INL depends on global matching of the R-string.

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Decoder-Based D/A converters

• 4-bit folded R-string D/A converter

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Decoder-Based D/A converters

• Multiple R-string 6-bit D/A converter
• interpolating

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Decoder-Based D/A converters

• R-string DACs with binary-tree decoding.
• Speed is limited by the delay through the
  resistor string as well as the delay through the
  switch network.

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Binary-Scaled D/A Converters

• Monotonicity is not guaranteed.
• Potentially large glitches due to timing skews.

Current-mode converter
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Binary-Scaled D/A Converters
Binary-array charge-redistribution D/A converter

• 4 bit R-2R based D/A converter
• No wide-range scaling of resistors.

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Thermometer-Code Converter
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Flash (Parallel) Converters

• High speed. Requires only one comparison cycle
  per conversion.
• Large size and power dissipation for large N.

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Feedback in Sensing/Conversion

• High Resolution and Linearity Converters
• Very expensive to build open-loop (precision
  components)
• Aging, Drift, Temperature Compensation
• 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

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Nyquist-Rate A/D converters
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Integrating converters

• Low conversion rate.

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Successive-Approximation Converters

• Binary search

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Successive-Approximation Converters

• 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.

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Sigma Delta A/D Converter
en
Decimation Filter
fs
fs
2 fo
Sampler
Modulator
x(t)
xn
yn
16 bits
Bandlimited to fo
Digital
Analog
Over Sampling Ratio 2fo is Nyquist
frequency Transfer function for an Lth order
modulator given by
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Modulator Characteristics

• Highpass character for noise transfer function
• In-band noise power is given by

• no falls by 3(2L1) for doubling of Over Sampling
  Ratio
• L0.5 bits of resolution for doubling of Over
  Sampling Ratio
• no essentially is uncorrelated for
• Dithering is used to decorrelate quantization
  noise