JAES: Synchronized Swept-Sine

# JAES: Synchronized Swept-Sine

A. Novak, P. Lotton & L. Simon (2015), “Synchronized Swept-Sine: Theory, Application, and Implementation”, Journal of the Audio Engineering Society. Vol. 63(10), pp. 786-798.

• ### Motivation

The paper entitled Synchronized SweptSine: Theory, Application and Implementation deals with a design of an
exponential swept-sine signal, that is very often used to analyze nonlinear systems in terms of Higher Order Harmonic Frequency Responses (HHFRF). In the paper, the theory of exponential swept-sine measurements of nonlinear systems is reexamined. First, the synchronization procedure necessary for a proper analysis of higher harmonics is detailed leading to an improvement of the formula for the exponential swept-sine signal generation. Next, an analytical expression of spectra of the swept-sine signal is derived and used in the deconvolution of the impulse response. A Matlab code for generation of the synchronized swept-sine, deconvolution, and separation of the impulse responses is also given with discussion of some application
issues.

### Swept-Sine Design

The exponential swept-sine signal is usually generated using a time-domain formula given by Angelo Farina as

$$x(t) = \sin \left\{ 2 \pi f_1 L \left[ \exp \left( \frac{t}{L} \right) -1 \right] \right\},$$

$L$ being the rate of frequency increase.

The synchronization procedure of the swept-sine, leading to a signal whose higher harmonics are perfectly synchronized with the original signal (see Figure below), is necessary for correct phase estimation of HHFRF.

$$x(t) = \sin \left\{ 2 \pi f_1 L \left[ \exp \left( \frac{t}{L} \right) \right] \right\}$$

Contrary to the original definition the “-1” term is missing and no restriction on parameters $f_1$, $f_2$, or $T$ is necessary.

### Analytical Deconvolution

The exponential swept-sine signal $x(t)$ is then used as the input to the nonlinear system under test and the output signal $y(t)$ is acquired. Then, the output is treated as being passed through a linear system. To obtain the impulse response $h(t)$, we simply deconvolve the output signal $y(t)$ with the original swept-sine signal $x(t)$.

The deconvolution is usually made by inverting the signal $x(t)$ in time and applying an envelope adjustment. In this paper, an analytical formula for spectra of the signal $x(t)$ is derived, and the spectra of the inverse filter $\tilde{X}(f)$ is then expressed as

$$\tilde{X}(f) = 2 \sqrt{\frac{f}{L}} \exp \left\{- j 2 \pi f L \left[1-\ln \left(\frac{f}{f_1} \right) \right] + j\frac{\pi}{4} \right\},$$

that is used to obtain the deconvolved impulse response $h(t)$ as

$$h(t) = \mathcal{F}^{-1} \left[ \mathcal{F}[y(t)] \tilde{X}(f)] \right].$$

• Exponential swept-sine signals are very often used to analyze nonlinear audio systems. A reexamination of this methodology shows that a synchronization procedure is necessary for the proper analysis of higher harmonics. An analytical expression of spectra of the swept-sine signal is derived and used in the deconvolution of the impulse response. Matlab code for generation of the synchronized swept-sine, deconvolution, and separation of the impulse responses is given. This report provides a discussion of some application issues and an illustrative example of harmonic analysis of current distortion of a woofer. An analysis of the higher harmonics of the current distortion of a woofer is compared using both the synchronized and the non-synchronized swept-sine signals.

• @article{novak2015synchronized,
author={Novak, Antonin and Lotton, Pierrick and Simon, Laurent},
title={Synchronized Swept-Sine: Theory, Application, and Implementation},
journal={J. Audio Eng. Soc},
volume={63},
number={10},
pages={786-798},
year={2015},
publisher={Audio Engineering Society}
}