JAES: Synchronized SweptSine

Motivation
The paper entitled Synchronized SweptSine: Theory, Application and Implementation deals with a design of an
exponential sweptsine 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 sweptsine 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 sweptsine signal generation. Next, an analytical expression of spectra of the sweptsine signal is derived and used in the deconvolution of the impulse response. A Matlab code for generation of the synchronized sweptsine, deconvolution, and separation of the impulse responses is also given with discussion of some application
issues.SweptSine Design
The exponential sweptsine signal is usually generated using a timedomain 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 sweptsine, 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 sweptsine 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 sweptsine 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 sweptsine 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 sweptsine signal is derived and used in the deconvolution of the impulse response. Matlab code for generation of the synchronized sweptsine, 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 nonsynchronized sweptsine signals.
@article{novak2015synchronized,
author={Novak, Antonin and Lotton, Pierrick and Simon, Laurent},
title={Synchronized SweptSine: Theory, Application, and Implementation},
journal={J. Audio Eng. Soc},
volume={63},
number={10},
pages={786798},
year={2015},
publisher={Audio Engineering Society}
}