[Paper Review] Simultaneous measurement of time-invariant linear and nonlinear, and random and extra responses using frequency domain variant of velvet noise

Hideki Kawahara, Ken‐Ichi Sakakibara+3 more|arXiv (Cornell University)|Aug 5, 2020

Advanced Electrical Measurement Techniques26 references4 citations

TL;DR

This paper proposes a novel frequency-domain method using orthogonal sequences of Frequency-domain Variant of Velvet Noise (FVN) to simultaneously measure linear time-invariant, nonlinear time-invariant, and random/time-varying acoustic responses without additional hardware. By leveraging FVN's high design freedom, the method enables real-time, interactive analysis through a custom open-source tool, validated via simulations and real-world loudspeaker measurements.

ABSTRACT

We introduce a new acoustic measurement method that can measure the linear time-invariant response, the nonlinear time-invariant response, and random and time-varying responses simultaneously. The method uses a set of orthogonal sequences made from a set of unit FVNs (Frequency domain variant of Velvet Noise), a new member of the TSP (Time Stretched Pulse). FVN has a unique feature that other TSP members do not. It is a high degree of design freedom that makes the proposed method possible without introducing extra equipment. We introduce two useful cases using two and four orthogonal sequences and illustrates their use using simulations and acoustic measurement examples. We developed an interactive and realtime acoustic analysis tool based on the proposed method. We made it available in an open-source repository. The proposed response analysis method is general and applies to other fields, such as auditory-feedback research and assessment of sound recording and coding.

Motivation & Objective

• To develop a method that simultaneously measures linear time-invariant, nonlinear time-invariant, and random/time-varying responses in acoustic systems.
• To overcome limitations of traditional TSP-based methods that assume ideal linearity and time-invariance in real-world systems with noise and nonlinearities.
• To enable simultaneous decomposition of system responses without specialized equipment or complex post-processing.
• To create an open-source, real-time interactive tool for acoustic analysis based on the proposed FVN method.
• To extend applicability to fields such as auditory feedback research and audio recording quality assessment.

Proposed method

• Uses unit FVNs—derived from two uniformly distributed random sequences—to generate phase-modulated signals with high design flexibility.
• Constructs orthogonal FVN sequences using unit FVNs and binary sequences, enabling separation of multiple system responses.
• Employs two- and four-sequence configurations to isolate linear time-invariant, nonlinear time-invariant, and random/time-varying components.
• Applies a six-term cosine series approximation to design smooth, energy-concentrated phase functions for FVN generation.
• Uses cross-correlation and inverse filtering techniques to extract impulse responses and decompose system behavior in the time and frequency domains.
• Integrates the method into a real-time, interactive analysis tool with visualization of impulse responses, power envelopes, and extra-response spectra.

Experimental results

Research questions

• RQ1 Can FVN-based orthogonal sequences enable simultaneous measurement of linear, nonlinear, and time-varying acoustic responses without additional hardware?
• RQ2 How does the FVN method compare to conventional TSP techniques in resolving nonlinear and time-variant components in real-world acoustic systems?
• RQ3 To what extent can the proposed method decompose background noise, interference, and nonlinear distortions in practical loudspeaker measurements?
• RQ4 Can the method be generalized to other domains such as auditory feedback or audio coding quality assessment?
• RQ5 What is the impact of FVN design parameters on measurement accuracy and robustness in noisy environments?

Key findings

• The FVN-based method successfully isolates linear time-invariant, nonlinear time-invariant, and random/time-varying components in acoustic systems using only orthogonal FVN sequences.
• Simulations and real measurements confirm that the method accurately captures the initial 30 ms of impulse responses and power spectra with minimal distortion.
• The use of four orthogonal FVN sequences enables simultaneous decomposition of all three response types, with the extra component reflecting noise, nonlinearities, and movement effects.
• The developed open-source tool supports real-time measurement and visualization on both macOS and Windows 10, with automatic calibration and file-saving of results.
• The method demonstrates robustness in everyday acoustic environments, including background noise and dynamic system changes.
• The approach is generalizable to other fields, such as auditory feedback research and assessment of lossy audio compression effects on speech signals.