combining time and frequency domain specifications for periodic signals aleksandar chakarov and...
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Combining Time and Frequency Domain Specifications for
Periodic SignalsAleksandar Chakarov and Sriram Sankaranarayanan
University of Colorado BoulderGeorgios Fainekos
Arizona State University Tempe
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Overview
• Goal: Provide specification formalisms for- Analog Circuits- Digital Circuits- Mixed Signal Circuits- Control Systems
• Challenge: How do we combine time and frequency
domain specifications?
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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 phase
• Transitions:– Initial value of v must be in [-4.6V, 4.6V]– Low to High:
– High to Low:
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Frequency Domain Specifications• Periodic Signals:
– Fourier Series Representation
• General signals:– Fourier Transform Representation
Future Work
Current Work
a1
b1
a2
b2
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Fourier Series
• Let be a continuous, periodic signal.– With “finite power”.
• can be written as a Fourier series:
• Amplitude at frequency is given by
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General Testing Framework
Model-Based Testing Runtime Verification
Input Specification
Design Output Specification
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Main Problems
• Signal Generation Problem
• Signal Recognition Problem
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TIME DOMAIN SPECIFICATIONSSignal Generation and Recognition
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Time Domain Specifications
Hybrid Automaton H
Continuous State of H
Output Function O
Output Signal O(t)
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Time Domain Encoding
Important primitive for signal generation/ recognition for time domain specifications.1. Explore paths in the automaton (bounded depth search)2. For each path, perform linear arithmetic encoding.
Time Domain EncoderHybrid Automaton
Linear Arithmetic Formula
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Time Domain Signal Generation
• Use SMT encoding to perform signal generation.
Time Domain Encoder
SMT Solver
Monte Carlo
Hybrid Automaton
Formula
Model
Model
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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
HybridAutomaton
Input Signal
LAFormula
LA
Formula
Accept
Reject
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FREQUENCY DOMAIN SPECIFICATIONS
Power spectra, signal generation and recognition.
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Frequency Domain Specifications
Power Spectral Envelope Function
G(f)
Frequency
Ampl
itude
Signal
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Frequency Domain Encoding
Input Signal with
period T Linear ProgramPower
Spectral Envelope
1. Sample input signal with fixed time period δ.
2. Generate a linear inequality constraint over the coefficients of Fourier series terms with tolerance ε. (linearize)
Frequency Domain Encoder
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Freq. Domain Signal Generation
• Use SMT encoding to perform signal generation
Frequency Domain Encoder
Power Spectral Envelope
SMT Solver
Monte Carlo
Linear
Program
Model
Model
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Freq. Domain Signal Recognition
Input Signal with
period TLinear
Program
Power Spectral Envelope
SMT Solver
Accept
RejectFrequency Domain
Encoder
• Use SMT encoding to perform signal recognition– Use fixed time period sampling.
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MIXED DOMAIN SPECIFICATIONSCombining time + frequency domain specifications
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Mixed Domain Specification
Time Domain Encoder
SMT Solver
Monte Carlo
Formula
ModelTime
DomainModels
Power Spectral Envelope
SMT SolverFrequency
Domain Encoder
Formula
Monte Carlo
Model
Mixed DomainModels
Hybrid Automaton
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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)
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Thank you!