The Time-Frequency Interference Terms of the Green’s Function for the Harmonic Oscillator

Abstract

The harmonic oscillator is a fundamental prototype for all types of resonances, and hence plays a key role in the study of physical systems governed by differential equations. The time-frequency representation of its Green’s function, obtained through the Wigner distribution, reveals the time-varying frequency structure of resonances. Unfortunately, the Wigner distribution of the Green’s function is affected by strong interference terms with a highly oscillatory structure. We characterize these interference terms by evaluating the ambiguity function of the Green’s function. The obtained result shows that, in the ambiguity domain, the interference terms are localized and separate from the resonance component, and hence they can be reduced by a proper filtering.

Appendix

By using the property (10), the ambiguity function of the Green’s function for the Langevin equation can be obtained from

$$\displaystyle \begin{aligned} A_{h_{L}}(\theta ,\tau )=\int\limits_{-\infty }^{+\infty }\int\limits_{-\infty }^{+\infty }W_{h_{L}}(t,\omega )e^{i\theta t+i\tau \omega }dtd\omega . \end{aligned} $$

(58)

Substituting $W_{h_{L}}(t,\omega )$ from (Fig. 1) gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} A_{h_{L}}(\theta ,\tau ) &\displaystyle =&\displaystyle \int\limits_{-\infty }^{+\infty }\int\limits_{-\infty }^{+\infty }u(t)e^{-2\mu t}\frac{\sin 2\omega t}{\pi \omega }e^{i\theta t+i\tau \omega }dtd\omega , \end{array} \end{aligned} $$

(59)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle \int_{-\infty }^{+\infty }\frac{1}{\pi \omega }\left[ \frac{1}{2i} \int_{0}^{+\infty }e^{\left( -2\mu +i(\theta +2\omega )\right) t}dt\right. \notag \\ &\displaystyle &\displaystyle \left. \hspace{0.55in}-\frac{1}{2i}\int_{0}^{+\infty }e^{\left( -2\mu +i(\theta -2\omega )\right) t}dt\right] e^{i\tau \omega }d\omega , \end{array} \end{aligned} $$

(60)

(61-62)

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle =&\displaystyle \frac{e^{\left( -\mu +i\theta /2\right) \left\vert \tau \right\vert }}{ 2\mu -i\theta }. \end{array} \end{aligned} $$

(63)

which is (42).