The reassignment of the spectrogram The original idea of reassignment was introduced in an attempt to improve the spectrogram. Indeed, as any other bilinear energy distribution, the spectrogram is faced with an unavoidable trade-off between the reduction of misleading interference terms and a sharp localization of the signal components. Let us recall the expression of the spectrogram as a 2D-convolution of the Wigner-Ville distribution of the signal by the WVD of the analysis window : $$S_x(t,\nu;h)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} W_x(s,\xi)\ W_h(t-s,\nu-\xi)\ ds\ d\xi.$$ (4.21) Therefore, this distribution reduces the interference terms of the signal's WVD, but at the expense of opposed time and frequency resolutions, and of biased marginals and first order moments. However, a closer look at expression (4.21) shows that $W_h(t-s,\nu-\xi)$ delimits a time-frequency domain at the vicinity of the $(t,\nu)$ point, inside which a weighted average of the signal's WVD values is performed. The key point of the reassignment principle is that these values have no reason to be symmetrically distributed around $(t,\nu)$, which is the geometrical center of this domain. Therefore, their average should not be assigned at this point, but rather at the center of gravity of this domain, which is much more representative of the local energetic distribution of the signal. Reasoning with a mechanical analogy, the local energy distribution $W_h(t-s,\nu-\xi) W_x(s,\xi)$ (as a function of $s$ and $\xi$) can be considered as a mass distribution, and it is much more accurate to assign the total mass (i.e. the spectrogram value) to the center of gravity of the domain rather than to its geometrical center. This is exactly how the reassignment method proceeds : it moves each value of the spectrogram computed at any point $(t,\nu)$ to another point $(\hat{t},\hat{\nu})$ which is the center of gravity of the signal energy distribution around $(t,\nu)$ : $$\hat{t}(x;t,\nu)={ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\inft... ...fty}^{+\infty}\int_{-\infty}^{+\infty} W_h(t-s,\nu-\xi)\ W_x(s,\xi)\ ds\ d\xi}$$ (4.22) $$\hat{\nu}(x;t,\nu)=\frac{\int_{-\infty}^{+\infty}\int_{-\infty}^{... ...fty}^{+\infty}\int_{-\infty}^{+\infty} W_h(t-s,\nu-\xi)\ W_x(s,\xi)\ ds\ d\xi}$$ (4.23) and thus leads to a reassigned spectrogram, whose value at any point $(t',\nu')$ is the sum of all the spectrogram values reassigned to this point : $$S_x^{(r)}(t',\nu';h)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\in... ...)\ \delta(t'-\hat{t}(x;t,\nu))\ \delta(\nu'-\hat{\nu}(x;t,\nu))\ dt\ d\nu \ \$$ (4.24) One of the mostly interesting properties of this new distribution is that it also uses the phase information of the short-time Fourier transform, and not only its squared modulus as in the spectrogram. This can be seen from the following expressions of the reassignment operators : $$\begin{eqnarray*} \hat{t}(x;t,\nu) = -\frac{d\Phi_x(t,\nu;h)}{d\nu}\\ \hat{\nu}(x;t,\nu) = \nu+\frac{d\Phi_x(t,\nu;h)}{dt} \end{eqnarray*}$$ where $\Phi_x(t,\nu;h)$ is the phase of the STFT of $x$ : $\Phi_x(t,\nu;h)=\arg(F_x(t,\nu;h))$. However, these expressions do not lead to an efficient implementation, and have to be replaced by the following ones : $$\begin{eqnarray*} \hat{t}(x;t,\nu)=t-\Re\left\{\frac{F_x(t,\nu;T_h)\ F_x^*(t,\n... ...t,\nu;D_h)\ F_x^*(t,\nu;h)}{\vert F_x(t,\nu;h)\vert^2}\right\} \end{eqnarray*}$$ where $T_h(t)=t\times h(t)$ and $D_h(t)=\frac{dh}{dt}(t)$. Reassigned spectrograms are therefore very easy to implement, and do not require a drastic increase in computational complexity. Finally, it should also be underlined that the reassigned spectrogram, though no longer bilinear, satisfies the time and frequency shifts covariance, the energy conservation (provided that $h(t)$ is of unit energy), and the non-negativity property. It cans also be shown that, since the WVD is perfectly localized on linear chirp signals and impulses, any reassigned spectrogram also satisfies this property : $$\begin{eqnarray*} x(t)=A\ \exp\left\{j\{\nu_0 t+\alpha t^2/2\}\right\}&\Rightarr... ...+\alpha\hat{t}\\ x(t)=A\ \delta(t-t_0)&\Rightarrow&\hat{t}=t_0. \end{eqnarray*}$$ Before presenting the generalization of this method to the Cohen's class and to the affine class, let us have a look at the readability improvement obtained by the reassigned spectrogram on an example of multi-component signal. The reassigned spectrogram is available thanks to the M-file tfrrsp.m. The result is compared to the spectrogram and to the "ideal" representation (tfrideal.m) based on the knowledge of the instantaneous frequency law of each component : >> N=128; [sig1 ifl1]=fmsin(N,0.15,0.45,100,1,0.4,-1); >> [sig2 ifl2]=fmhyp(N,[1 .5],[32 0.05]); >> sig=sig1+sig2; >> tfrideal([ifl1 ifl2]); >> figure; tfrrsp(sig); Figure 4.34: Reassignment of the spectrogram on a synthetic signal composed of a sinusoidal frequency modulation simultaneously with a hyperbolic frequency modulation : comparison with the ``ideal'' time-frequency representation and with the spectrogram The file tfrrsp.m allows you to display the spectrogram itself or its reassigned version. The improvement given by the reassignment method is obvious : the two components are much better localized and almost perfectly concentrated, and there are very few cross-terms.
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