Combining Time and Frequency Domain Specifications for Periodic Signals Aleksandar Chakarov and Sriram Sankaranarayanan University of Colorado Boulder Georgios 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 a1a1 b1b1 a2a2 b2b2

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 TestingRuntime Verification Input Specification Design Output Specification

Main Problems Signal Generation Problem Signal Recognition Problem

TIME DOMAIN SPECIFICATIONS Signal 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 Encoder Hybrid 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

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 Reject

FREQUENCY DOMAIN SPECIFICATIONS Power spectra, signal generation and recognition.

Frequency Domain Specifications Power Spectral Envelope Function G(f) Power Spectral Envelope Function G(f) Frequency Amplitude Signal

Frequency Domain Encoding Input Signal with period T Linear Program Power Spectral Envelope 1.Sample input signal with fixed time period δ. 1.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

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.

MIXED DOMAIN SPECIFICATIONS Combining time + frequency domain specifications

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 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)

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