Lecture 4 Measurement

1
Lecture 4 Measurement

• Accuracy and Statistical Variation

2
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

3
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

4
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

5
Quantization Error
lsb/2
x
-lsb/2

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

6
Test Correlation

• Tester to Bench
• Tester to Tester
• DIB to DIB
• Day to Day
• Goal is reproducible measurements within expected
  error magnitude

7
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

8
Multi-tone Calibration

• DSP testing often uses multi-tone signals from
  digital sources
• Analog signal recovery and DIB impedance
  matching distort the signal
• Tester Calibration can restore signal levels
• Signal strength usually measured as RMS value
• Corresponds to square-law calibration fixture
• Modeling proceeds similarly to linear calibration
  as long as the model is unimodal. In principle,
  any such model can be approximated by linear
  segments, and each segment inverted to find the
  calibration adjustment.

9
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

10
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

11
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

12
Normal 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

13
Issues with Normal statistics

• Assumptions
• Noise sources are all uncorrelated
• All Noise sources are accounted for
• 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