Lecture 4 Measurement

Accuracy and Statistical Variation

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

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

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

Quantization Error

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

12lsb

RMS

xx 0avgx

+lsb/2

-lsb/2

x

Test Correlation Tester to Bench Tester to Tester DIB to DIB Day to Day Goal is reproducible

measurements within expected error magnitude

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:

OGvv realmeasured

OGvvm 22

OGvvm 11

12

12

vv

vvG mm

12

1221

vv

vvvvO mm

G

Ovvv measuredcalreal

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.

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:

0

)( dffSvRMS

2)()()( fGfSfS io

0

2)()( dffGfSv io

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:

ifRCV

VfG

i

o

21

1)(

)( fSi

RCV RMS

2

1)(0

)/(4 2

2

HzVV

b

o

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

N

k

knxN

ny1

)1(1

)(

2/)1(2

)2/2sin(

)2/2sin(

Nfie

fN

fNfG

Ndfe

fN

fNV NfiRMS

2/1

2/1

2

2/)1(2

)2/2sin(

)2/2sin(

‘Normal’ Statistics Mean Standard Deviation

Note that this is not an estimate for a total sample set (issue if N<<100), use 1/(N-1)

For large set of data with independent noise sources the distribution is:

Probability

1

0

)(1 N

n

nxN

1

0

2))((1 N

n

nxN

2

2

2

)(

2

1)(

x

exd

b

a

x

ebXaP2

2

2

)(

2

1)(

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