Download Lecture 3 Mixed Signal Testing

Lecture 4 Measurement
Accuracy and Statistical Variation
Accuracy vs. Precision
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Expectation of deviation of a given
measurement from a known standard
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Often written as a percentage of the possible
values for an instrument
Precision is the expectation of deviation of a
set of measurements
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“standard deviation” in the case of normally
distributed measurements
Few instruments have normally distributed errors
Deviations
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Systematic errors
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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
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The portion of deviations of a set of
measurements which cannot be reduced by
knowledge of measurement parameters
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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
+lsb/2
x
-lsb/2
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Deviations produced by digitization of analog measurements
For random signal with uniform quantization of xlsb:
xRMS 
xlsb
12
xavg  0
Test Correlation
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Tester to Bench
Tester to Tester
DIB to DIB
Day to Day
Goal is reproducible measurements
within expected error magnitude
Model based Calibration
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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:
vmeasured  Gvreal  O
vm 2  Gv2  O
vm1  Gv1  O
vm 2  vm1
G
v2 v1
vm1v2  vm 2 v1
O
v2  v1
vreal  vcal 
vmeasured  O
G
Multi-tone Calibration
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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
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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
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Noise is specified as a spectral density (V/Hz1/2) or W/Hz
RMS noise is proportional to the bandwidth of the signal:

vRMS 
 S ( f )df
0
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Noise density is the square of the transfer function
S o ( f )  Si ( f ) G ( f )
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Net (RMS) noise after filtering is:

vo 
 Si ( f ) G( f ) df
2
0
2
Filter Noise Example
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RC filtering of a noisy signal
Assume uniform input noise, 1st order filter
Si ( f )  

G( f ) 
The resulting output noise density is:
V0( RMS )
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Vo
1

Vi 1  2ifRC
1 

2 RC
We can invert this relation to get the equivalent input noise:

4Vo2
b
(V 2 / Hz )
Averaging (filter analysis)
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Simple processing to reduce noise – running average of data
samples
1 N
y (n)   x(n  k  1)
N k 1
The frequency transfer function for an N-pt average is:
 sin( 2fN / 2)  i 2 f ( N 1) / 2
e
G f   
 N sin( 2f / 2) 
To find the RMS voltage noise, use the previous technique:
VRMS  
1/ 2

1 / 2

2
sin( 2fN / 2) i 2 f ( N 1) / 2
e
df 
N sin( 2f / 2)
So input noise is reduced by 1/N1/2

N
‘Normal’ Statistics
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N 1
 x ( n)
Mean
Standard Deviation  
1

N
n 0
1 N 1
( x ( n)  ) 2

N n 0
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:
( x  )

1
2
2
d ( x) 
b

 2
e
1
e
Probability P(a  X  b)  
a  2

2
( x  ) 2
22
Issues with Normal statistics
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Assumptions:
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In many practical cases, data has ‘outliers’ where
non-normal assumptions prevail
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Noise sources are all uncorrelated
All Noise sources are accounted for
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
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Not used often since analysis of expectations are difficult