Time–Frequency Analysis of the Impulse Response

The impulse response is a key function for the analysis and design of systems, therefore understanding its properties is a fundamental problem. To investigate the nonstationary structure of the impulse response, we derive its time-frequency representation both in the Wigner and smoothed Wigner distribution domains. First, we consider the class of single-input single-output (SISO) systems, which models a wide variety of physical phenomena. The obtained Wigner distribution for the class of SISO systems is made by frequency-translated versions of the Wigner distribution of the impulse response for a first-order system, a T-shaped function of time and frequency, plus the interference terms, which are effectively filtered out in the smoothed Wigner domain. Then, we consider the case of multiple-input multiple-output (MIMO) systems, whose impulse response is the collection of the individual responses obtained by applying an impulse to every input separately. Also in this case, the Wigner distribution of the impulse response depends on the T function. We apply our results to a two-dimensional MIMO harmonic oscillator, and to a SISO system with three resonances. Finally, we propose the key idea of a system identification method that operates directly in the time-frequency domain by fitting the smoothed T function. We show that the method works for a nonlinear oscillator with a large cubic term, as well as for a time-varying harmonic oscillator, and that it can tolerate the presence of both stationary and nonstationary noise.