A Non-Linear Neural Classifier and its Applications in Testing Analog/RF Circuits*
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Transcript of A Non-Linear Neural Classifier and its Applications in Testing Analog/RF Circuits*
A Non-Linear Neural Classifier and its Applications in Testing
Analog/RF Circuits*
Haralampos Stratigopoulos & Yiorgos Makris
Departments of Electrical Engineering & Computer Science
YALE UNIVERSITY
* Partially supported by NSF, SRC, IBM & Intel
Test and Reliability @ YALE
http://eng.yale.edu/trela
Current Research Areas
Machine learning-based approach for test cost reduction Design of on-chip checkers and on-line test methods Neuromorphic built-in self-test (BIST)
Analog/RF circuits
Concurrent error detection / correction methods (FSMs) RTL Concurrent error detection in microprocessor controllers Design transformations for improved soft-error immunity (logic)
Reliability
Fault simulation, ATPG, test compaction for SI circuits Test methods for high-speed pipelines (e.g. Mousetrap) Concurrent error detection and soft-error mitigation
Asynchronous circuits
Outline • Testing analog/RF circuits
• Current practice
• Limitations and challenges
• Summary
• Testing via a non-linear neural classifier• Construction and training
• Selection of measurements
• Test quality vs. test time trade-off
• Machine learning-based testing• General idea
• Regression vs. classification
• Experimental results
• Analog/RF specification test compaction
• Neuromorphic BIST
Analog/RF IC Test – Current Industry Practice
Interface BoardWafer
• Post-silicon production flow
Automatic Test
Equipment (ATE)
Chip
designspecifications
performanceparameters
compare
pass/fail
• Current practice is specification testing
chip
test configurations
ATE
Limitations
• Expensive ATE (multi-million dollar equipment)
• Specialized circuitry for stimuli generation and response measurement
Test Cost:
• Multiple measurements and test configurations
• Switching and settling times
Test Time:
• Fault-model based test – Never really caught on
• Machine learning-based (a.k.a. “alternate”) testing– Regression (Variyam et al., TCAD’02)
– Classification (Pan et al., TCAS-II’99, Lindermeir et al., TCAD’99)
Alternatives?
Machine Learning-Based Testing
• Determine if a chip meets its specifications without explicitly computing the performance parameters and without assuming a prescribed fault model
General idea:
• Infer whether the specifications are violated through a few simpler/cheaper measurements and information that is “learned” from a set of fully tested chips
How does it work?
• Since chips are produced by the same manufacturing process, the relation between measurements and performance parameters can be statistically learned
Underlying assumption:
Regression vs. ClassificationProblem Definition:
simple functions
performanceparameters
specification tests(T1, … , Tk)
designspecifications
compare
pass/fail
unknown,complex,non-linear functions
(no closed-form)
alternate tests(x1, … , xn)
Use machine-learning to approximate these functions
• Explicitly learn these functions (i.e. approximate f:x → )
Regression:• Implicitly learn thes
e functions(i.e. approximate f:x→Y,Y={pass/fail})
Classification:
Overview of Classification Approach
specification tests
pass/fail labels
training set of chips
projection on measurement space -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0.015
0.02
x1
x 2
Nominal patterns
Faulty patterns
simpler measurements
acquisition of measurement patterns
trained classifier
new (untested) chip measurement pattern
pass/fail label
TESTING
learn boundaryLEARNING
Order of Separation Boundary
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
-1
-0.5
0
0.5
x1
x 2
Nominal patterns
Faulty patterns
• The boundary that best separates the nominal from the faulty classes in the space of the simple measurements can be highly non-linear
Previous Work: Linear Methods(Lindermeir et al., TCAD’99, Pan et al., TCAS-II’99, Variyam et al., TCAD’00)
i n, allocate a hyper-plane bi that optimally separates
the patterns when labeled with regards to specification i
• Acceptance region is bounded by intersection
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
-1
-0.5
0
0.5
x1
x 2
Nominal patterns
Faulty patternsb2
b3
b1
b4
b5 b6
• Decomposition into n two-class problems solved individually
Our Solution: Non-Linear Neural Classifier
• Allocates a single boundary of arbitrary order
• No prior knowledge of boundary order is required
• The topology is not fixed, but it grows (ontogeny) until it matches the intrinsic complexity of the separation problem
• Constructed using linear perceptrons only
Linear Perceptron
1x dx
iy
1ox
0iw
1iw
diw
perceptron connectivity
synapses
d
jjx
0jiw
iy (x)
0
1
-1
perceptron output
threshold activation function
geometric interpretation
1x
2x
d
jjx
0jiw 0
1
Nominal patterns
Faulty patterns
iy (x) for nominal
-1iy (x)for faulty
adjustsji
wtraining
to minimizeerror
Topology of Neural Classifier
11x dx
1y
ox
0iw1iw
diw
• Every newly added layer assumes the role of the network output
iy
, yi
iy
2i
iy
1
2w
, yiw w
1ox
1y
1y
1
y3iy
2
w, iyi
1i0i
1iw
diw
1diw
1ox
y
d
iid xx
1
21
2
1ox
• Pyramid structure– First perceptron receives
the pattern x Rd
– Successive perceptrons also receive inputs from preceding perceptrons and a parabolic pattern xd+1=xi
2, i=1…d
• Non-linear boundary by training a sequence of linear perceptrons
Training and Outcome• Weights of added layer are adjusted through the
thermal perceptron training algorithm
• Allocated boundary is non-linear in the original space x Rd
• Each perceptron separates its input space linearly
• Weights of preceding layers do not change
Theorem: • The sequence of boundaries allocated by the neurons
converges to a boundary that perfectly separates the two populations in the training set
Boundary Evolution Example (layer 0)
Faulty patterns erroneously classifiedFaulty patterns correctly classified
Nominal patterns erroneously classifiedNominal patterns correctly classified
x1
x 2
-2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Boundary Evolution Example (layer 1)
Faulty patterns erroneously classifiedFaulty patterns correctly classified
Nominal patterns erroneously classifiedNominal patterns correctly classified
-2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x1
x 2
Boundary Evolution Example (layer 2)
Faulty patterns erroneously classifiedFaulty patterns correctly classified
Nominal patterns erroneously classifiedNominal patterns correctly classified
-2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x1
x 2
Boundary Evolution Example (output layer)
Faulty patterns erroneously classifiedFaulty patterns correctly classified
Nominal patterns erroneously classifiedNominal patterns correctly classified
-2 -1.5 -1 -0.5 0 0.5 1 1.5-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
x1
x 2
Matching the Inherent Boundary Order
• Monitor classification on validation set
• Prune down network to layer that achieves the best generalization on validation set
• Evaluate generalization on test set
Finding The Trade-Off Point (Early Stopping)
• No! The goal is to generalize
• Inflexible and over-fitting boundaries
Is Higher Order Always Better?
0 2 4 6 8 10 12 1470
75
80
85
90
95
100
number of layers
rate
(%
)
training setvalidation set
Are All Measurements Useful?
-3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-3
-2
-1
0
1
2
3
4
Nominal patterns
Faulty patterns
Non-Discriminatory
x1
x 2
-4 -3 -2 -1 0 1 2 3 4-4
-3
-2
-1
0
1
2
3
4
Nominal patterns
Faulty patterns
Linearly Dependent
x1x 2
Curse of Dimensionality
• Several possible boundaries exist – choice is random
• By increasing the dimensionality we may reach a point where the distributions are very sparse
New patterns
• Random label assignment to new patterns
Nominal training patterns
Faulty training patterns
Nominal training patterns
Faulty training patterns
Nominal training patterns
Faulty training patterns
Genetic Measurement Selection• Encode measurements in a bit string, with the k-th bit
denoting the presence (1) or the absence (0) of the k-th measurement
0 1 1 0 1 1 1 0 0 1
0 0 0 0 1 1 1 0 0 1
1 1 1 0 1 1 0 0 0 0
0 1 1 1 1 0 1 0 0 0
1 0 1 0 1 1 1 0 0 0
0 1 0 0 1 0 1 0 1 1
0 0 1 0 1 1 1 0 1 1
0 0 1 0 1 1 1 0 0 1
0 1 1 0 1 1 0 0 0 0
0 1 1 1 1 1 0 0 0 0
1 0 1 0 1 1 1 0 0 0
0 1 0 0 1 0 1 0 0 0
ReproductionCrossover &
Mutation
GENERATION
tGENERATION
t+1
NSGA II: Genetic algorithm with multi-objective fitness function reporting pareto-front for error rate (gr) and number of re-tested circuits (nr)
Fitness function:
Two-Tier Test Strategy
Low-Cost Tester
All ChipsSimple
AlternativeMeasurements
Neural Classifier
Most Chips
Highly AccuratePass/FailDecision
MeasurementPattern in
Guard-band
Few Chips
High-Cost Tester
SpecificationTest
Measurements
Compare
Highly Accurate Pass/Fail Decision
DesignSpecs
Inexpensive Machine Learning Test
Expensive Specification Test
Reasons for Non-Zero Classification Error
2.25 2.3 2.35 2.4 2.45 2.5 2.551.65
1.7
1.75
1.8
1.85
1.9
x1
x 2
Nominal patterns
Faulty patterns
test hypersurface
• Continuity of analog signals and parametric drifts
• Finite training set
Guard-bands
2.25 2.3 2.35 2.4 2.45 2.5 2.551.65
1.7
1.75
1.8
1.85
1.9
nominal guard-band
faulty guard-band
x1
x 2
Nominal patterns
Faulty patterns
test hypersurface
Guard-banded region
• Introduce a trichotomy in the measurement space in order to assess the confidence of the decision
Testing Using Guard-bands
• Examine measurement pattern with respect to the guard-bands
2.25 2.3 2.35 2.4 2.45 2.5 2.551.65
1.7
1.75
1.8
1.85
1.9
nominal guard-band
faulty guard-band
x1
x 2
Nominal patterns
Faulty patterns
test hypersurface
Guard-banded region
Classify chip to dominant class
Re-test viaspecification test
• Identify circuits that need to be re-tested via specification tests
Guard-band Allocation
D
• Identify the overlapping regions
• Guard-bands are allocated separately
• The ontogenic classifier allocates the guard-band
D
• Clear the overlapping regions
D
Nominal patterns
Faulty patterns
Guard-band Allocation
D
D D
D
• The guard-banded region can been varied by controlling D
Nominal patterns
Faulty patterns
Experiment 1: Switch-Capacitor Filter
C1A
C2A
C2B
C4A
C4B
C5A
C1
C2
C3
C4
C5
C1C
C3A
C3B
C5B
C5C
C3D
C1D
C3C
C1BVin
Vout
1
2 2
2
1
1
1
1
2
2
2
2
1
1
1
1
1
1
2
2
2
2
Specifications considered
• Ripples in stop- and pass-bands
• Gain errors• Group delay• Phase response• THD
Experiment Setup
• N=2000 instances• N/2 assigned to the
training set, N/4 to the validation set and N/4 to the test set
White-Noise Test Stimulus
CLK
Din QnQm
LP filter
Type-1 LFSR
DUTsignature
DUT digitizer
pseudo-random bit sequence white noise
Time (ms)0
-2.0
0
2.0
4.0
Vol
tage
(V)
2 4 6 8 10 12 14 16 18
DUTsignature
pseudo-random bit sequence
white noise
t1 t2 t30
tr
ts
Test Time vs. Test Accuracy Trade-Off
(0%,4%)
(8.1%,2%)
(15.6%,0.6%) (20.7%,0%)
0 5 10 15 20 25 30 35 40 450
1
2
3
4
5
6
7
8
retested circuits (%)
test
err
or
(%)
f2
f1
f3
guard-bands test time=15msspecification test time >0.5s
Experiment 2: RFMD2411
Gain 3rd Order Intercept
Noise Figure
Other S Parameters
LNA x x x x x xMixer x x x x - -
LNA + Mixer (Cascade)
x x x - - -
• 541 devices (training set 341, validation set 100, test set 100)• Considered 13 specs @850MHz (7 different test configurations)
Test Configuration
Stimulus
DUT Response
0 1 2 3 4 5 6 7 8 9 10
x 107
0
50
100
150
200
250
300
350
400
450
frequency
FF
T o
f D
UT
re
spon
se
noise floor
28 tones
FFT
DUT
TestSignal
xt(t)
f1 f2
Mixer
DUTsignature
xs(t)LPF
0 5 10 15 20 25 30 35 40 45 500
1
2
3
4
5
6
7
8
retested circuits (%)
test
err
or
(%)
f2 (gr<2)f2 (gr<4)f3
f1
(0%,3.92%)
(11.67%,1.96%)
(27.62%,0%)
Test Time vs. Test Accuracy Trade-Off
guard-bands test time=5msspecification test time ~1s
Analog/RF Specification Test Compaction
• Instead of “other, alternate measurements” use existing inexpensive specification tests to predict pass/fail for the chip and eliminate expensive ones
Non-disruptive application:
• RFCMOS zero-IF down-converter for cell-phones
• Fabricated at IBM, currently in production
• Data from 944 devices (870 pass – 74 fail)
• 136 performances (65 non-RF – 71 RF)
• Non-RF measurements assumed much cheaper than RF, aim at minimizing number of measurements
Case-Study:
Setup and Questions Asked
• How accurately can we predict pass/fail of a device using only a subset of non-RF measurements?
Question #1:
• How does this accuracy improve by selectively adding a few RF measurements to this subset?
Question #2:
• Split 944 devices into training and test set (472 devices each, chosen uniformly at random.
• Repeat 200 times and average to obtain statistically significant results
Setup:
Results
Only non-RF performances:
• 15 measurement suffice to predict correctly over 99% of the devices (in our case 4 out of 472 are on average mispredicted
Adding RF performances:
• By adding to these 15 non-RF performances 12 RF performances, error drops to 0.38% (i.e. 1 out of 472 devices mispredicted)
What’s Next? Neuromorphic BIST
• On-chip generation of simple test stimulus
• On-chip acquisition of simple measurements
• On-chip implementation of trainable neural classifier
A stand-alone BIST method for mixed-signal/RF circuits:
On-ChipStimulus
Generator
AnalogMux
Mixed-Signal
/RF
Circuit
ProgrammableResponseAcquisition
TrainableOn-Chip Neural
Classifier
Input
Output
BIST Input
BIST Output
BIST Control
Summary
• A non-linear neural classifier supporting a novel paradigm for testing analog/RF circuits – Achieves significant reduction in test cost
without sacrificing accuracy of specification test– Enables exploration of trade-off between
test cost and test accuracy via guard-bands
Contribution:
• Non-disruptive: Specification test compaction• Disruptive: Machine learning-based test • Futuristic: Neuromorphic BIST
Applications:
More Information
• E-mail: [email protected], [email protected]
Contact:
• H-G. D. Stratigopoulos, Y. Makris, “Constructive Derivation of Analog Specification Test Criteria,” Proceedings of the IEEE VLSI Test Symposium (VTS), pp. 245-251, 2005
• H-G. D. Stratigopoulos, Y. Makris, “Non-Linear Decision Boundaries for Testing Analog Circuits,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (T.CAD), pp. 1360-1373, Nov. 2005
• H-G. D. Stratigopoulos, Y. Makris, “Bridging the Accuracy of Functional and Machine-Learning-Based Mixed-Signal Testing,” Proceedings of the IEEE VLSI Test Symposium (VTS), pp. 406-411, 2006
• H-G. D. Stratigopoulos, P. Drineas, M. Slamani, Y. Makris, “Non-RF to RF Test Correlation Using Learning Machines: A Case Study,” Proceedings of the IEEE VLSI Test Symposium (VTS), pp. 9-14, 2007
• H-G. D. Stratigopoulos, Y. Makris, “Error Moderation in Low-Cost Machine Learning-Based Analog/RF Testing ,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems (T.CAD), pp. 339-351, Feb. 2008
Publications: