Combining Time and Frequency Domain Specifications for Periodic

SignalsAleksandar Chakarov and Sriram Sankaranarayanan

University of Colorado BoulderGeorgios Fainekos

Arizona State University Tempe

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?

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:

Frequency Domain Specifications• Periodic Signals:

– Fourier Series Representation

• General signals:– Fourier Transform Representation

Future Work

Current Work

a1

b1

a2

b2

Fourier Series

• Let be a continuous, periodic signal.– With “finite power”.

• can be written as a Fourier series:

• Amplitude at frequency is given by

General Testing Framework

Model-Based Testing Runtime Verification

Input Specification

Design Output Specification

Main Problems

• Signal Generation Problem

• Signal Recognition Problem

TIME DOMAIN SPECIFICATIONSSignal Generation and Recognition

Time Domain Specifications

Hybrid Automaton H

Continuous State of H

Output Function O

Output Signal O(t)

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

Time Domain Signal Generation

• Use SMT encoding to perform signal generation.

Time Domain Encoder

SMT Solver

Monte Carlo

Hybrid Automaton

Formula

Model

Model

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

FREQUENCY DOMAIN SPECIFICATIONS

Power spectra, signal generation and recognition.

Frequency Domain Specifications

Power Spectral Envelope Function

G(f)

Frequency

Ampl

itude

Signal

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

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

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.

MIXED DOMAIN SPECIFICATIONSCombining time + frequency domain specifications

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

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)

Thank you!