Aleksandar Chakarov and Sriram Sankaranarayanan University of Colorado Boulder Georgios Fainekos Arizona State University Tempe Overview Goal Provide specification formalisms for ID: 408334
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Slide1
Combining Time and Frequency Domain Specifications for Periodic Signals
Aleksandar Chakarov and
Sriram
Sankaranarayanan
University of Colorado Boulder
Georgios
Fainekos
Arizona State University TempeSlide2
Overview
Goal:
Provide specification formalisms for
- Analog Circuits
- Digital Circuits
- Mixed Signal Circuits
- Control Systems
Challenge:
How do we combine
t
ime and frequency
d
omain specifications?Slide3
Time Domain Specifications
( Example Figure1 )
( Example Figure2 )
Two-phase signal:
high (5 ± 0.5V) and low (-5 ± 0.5V
)
Rate of change is in A minimum of 0.5 sec in each phaseTransitions:Initial value of v must be in [-4.6V, 4.6V]Low to High: High to Low: Slide4
Frequency Domain Specifications
Periodic Signals:
Fourier Series Representation
General signals:
Fourier Transform Representation
Future Work
Current Work
a
1
b
1
a
2
b
2Slide5
Fourier Series
Let be a continuous, periodic signal.
With “finite power”.
can be written as a Fourier series:
Amplitude at frequency is given by Slide6
General Testing Framework
Model-Based Testing
Runtime Verification
Input Specification
Design
Output SpecificationSlide7
Main Problems
Signal Generation Problem
Signal Recognition Problem
Slide8
Time Domain Specifications
Signal Generation and RecognitionSlide9
Time Domain Specifications
Hybrid Automaton
H
Continuous State of
H
Output Function
O
Output Signal
O(t)Slide10
Time Domain Encoding
Important primitive for signal generation/ recognition for time domain specifications.
Explore paths in the automaton (bounded depth search)
For each path, perform linear arithmetic encoding.
Time Domain Encoder
Hybrid
Automaton
Linear
Arithmetic
FormulaSlide11
Time Domain Signal Generation
Use SMT encoding to perform signal generation.
Time Domain
Encoder
SMT
Solver
Monte Carlo
Hybrid
Automaton
Formula
Model
ModelSlide12
Time Domain Signal Recognition
Use time domain encoding with run-signal matching.
Matches up generated signal with automaton run.
Time Domain
Encoder
SMT
Solver
Run/Signal Matching
Hybrid
Automaton
Input Signal
LA
Formula
LA
Formula
Accept
RejectSlide13
Frequency Domain Specifications
Power spectra, signal generation and recognition.Slide14
Frequency Domain Specifications
Power Spectral Envelope Function
G(f)
Frequency
Amplitude
SignalSlide15
Frequency Domain Encoding
Input
Signal with
period T
Linear
Program
Power Spectral Envelope
Sample input signal with fixed time period
δ
.
Generate a linear inequality constraint over the coefficients of Fourier series terms with tolerance ε. (linearize)
Frequency Domain EncoderSlide16
Freq. Domain Signal Generation
Use SMT encoding to perform signal generation
Frequency Domain Encoder
Power Spectral Envelope
SMT Solver
Monte Carlo
Linear Program
Model
ModelSlide17
Freq. Domain Signal Recognition
Input
Signal with
period T
Linear
Program
Power Spectral Envelope
SMT
Solver
Accept
Reject
Frequency Domain Encoder
Use SMT encoding to perform signal recognition
Use fixed time period sampling.Slide18
Mixed Domain Specifications
Combining time + frequency domain specificationsSlide19
Mixed Domain Specification
Time Domain
Encoder
SMT
Solver
Monte Carlo
Formula
Model
Time
Domain
Models
Power Spectral Envelope
SMT
Solver
Frequency Domain Encoder
Formula
Monte Carlo
Model
Mixed
Domain
Models
Hybrid AutomatonSlide20
Implementation & Results
We have an implementation that uses
Yices
/Z3 SMT solvers.
Generates a single unified encoding.
Performs well on a set of benchmarks.
More details in paper (available upon request)Slide21
Thank you!