Marques and Almeida

Marques and Almeida [1] use a slightly more general model in their publication, namely

$$f_M(t) = A e^{j(\phi_0 + \omega_0 t + \beta t^2)}e^{-t^2\frac{1}{2\sigma^2}}$$ (12)

where $\phi$ is the phase offset, $\omega_0$ is the center frequency, $\beta$ is the chirp parameter, and $\sigma^2$ is the variance of the Gaussian time domain window used. The authors give a closed form expression for the Fourier Transform of the signal:

$$F_M(\omega) = A e^{j\phi_0} Z(w)$$ (13)

with

$$Z(\omega) = \alpha(\omega)e^{jB(\omega)}$$ (14)

$$\alpha(\omega) = \sqrt{\frac{1+2j\beta\sigma^2}{1+4\beta^2\sigma^4}} \exp\left(-\frac{\sigma^2}{2}\frac{(\omega-\omega_0)^2}{1+4\beta^2\sigma^4} \right)$$ (15)

$$B(\omega) = \frac{\beta \sigma^4}{1+4\beta^2\sigma^4} (\omega-\omega_0)^2.$$ (16)

To estimate the parameter $\beta$,the authors first fit a parabola to the log magnitude spectrum:

$$a\omega^2 + b\omega + c \approx \ln(\alpha(\omega))$$ (17)

They then estimate the center frequency via

$$\omega_0 = -\frac{b}{a}$$ (18)

and the chirp parameter as

$$\beta = \pm\frac{1}{2\sigma^2}\sqrt{\frac{-\sigma^2}{2a} - 1}.$$ (19)

Making the trivial substitution for $x$ that we did above for the JOS method, we obtain:

$$\beta = \pm\frac{1}{2\sigma^2}\sqrt{\frac{-\sigma^2}{x} - 1}.$$ (20)

which we see is equivalent to the ``abbreviated'' JOS method. We may now observe:

• It may also be shown that the closed form phase expression given above is identical to that derived for the JOS estimator. Nonetheless, Marques and Almeida do not exploit this information in this portion of their estimator.
• Since this model deals with only the magnitude spectrum we will have an interesting problem. When $\beta$ becomes very small, we would require a very large $x$ (or $2a$) value. This cannot be done ad infinitum, because the curvature will be limited as the window width approaches that of the window transform.
• The above constraint may not be a problem, however, as the limit is approached more slowly for the Gaussian window than for the Hann window (see plots).