Time-frequency distribution of signal's energy

Expansion (3) can be used to construct a time-frequency map of signal's energy, which is a priori free of cross terms.

Energy is a quadratic function of the signal. If we have signal composed of structures $a$ and $b$, then its square gives $(a+b)^2=a^2+b^2+2ab$. In most of the time-frequency distributions, cross-terms ($2ab$) may have up to twice the energy of terms directly related to signal's structures, but appear at time and frequency positions where no activity occurs in the signal. Sophisticated mathematics is being applied to reduce the cross-terms, resulting in a multitude of distributions: each of them may produce a better picture for certain signal. Their variety, combined with the complex multi-component character of EEG, requires a careful choice of a proper analysis method and its parameters for each experimental setup.

But once we managed to decompose the signal into linear expansion (3), total elimination of cross-terms is trivial. We simply omit all the mixed elements appearing in the square of the right side, leaving sum of Wigner distributions of $M$ auto-terms $g_{\gamma_n}$ [6]:

$$E f (t, \omega) = \sum_{n=0}^M \vert<R^n f, \;g_{\gamma_n}>\vert^2 \; W g_{\gamma_n} (t, \omega)$$ (5)

This estimate is derived from a general procedure, which does not require a signal-dependent tuning, and conserves high resolution of the underlying parametrization (eq. 3). Figure 1 (f) presents this magnitude calculated for a simple signal simulated as a sum of typical components.

Figure 1: Time-frequency maps of energy density (eq. (5)) of a 500-points simulated signal (e) composed of four sine-modulated Gaussians, i.e. Gabor functions (a-b), sine wave and one-point discontinuity (c) and sine wave with linear frequency modulation--chirp (d). Distribution (f) was obtained from a single decomposition in a large ($2*10^6$ waveforms) dictionary: signal is described by only 30 functions, but changing frequency of the chirp is represented as a series of functions, since in the Gabor dictionary we have only constant frequency modulations. (g) presents average of 100 decompositions of the same signal in different realizations of smaller ($5*10^4$ atoms) stochastic dictionaries. This smoothes the representation of the chirp, but underlying parametrization is no more compact. (h)--the same as (g), presented in 3 dimensions. Square root of energy proportional to the height of the surface or "temperature" on 2-dimensional plots.

Figure 2 presents time-frequency distributions of energy of a simulated signal from Figure 1, obtained by means of several different methods. Spectrograms (short-time Fourier transform) from plots (b) and (c) indicate the tradeoff between time and frequency resolutions, inherent to this distribution. It is regulated by the length of the time window--its average setting gives usually rather low overall resolution, with minimized cross-terms. Plot (d) represents the continuous wavelet transform, indicating its most typical property: time resolution changing proportionally with the frequency, e.g. good time and bad frequency resolutions in the high frequency region. Finally, plot (e) and its 3-dimensional representation in (f) are obtained from a pseudo smoothed Wigner-Ville distribution, with separable parameters governing time and frequency resolutions. These parameters were chosen to optimize the representation of all the structures present in this signal, at a cost of incorporating relatively low-energy interferences (cross-terms).

Figure 2: Time-frequency distributions of energy of the simulated signal (here plotted as a) from Figure 1. (b) and (c) present spectrograms (i.e. windowed or short-time Fourier transforms) calculated, respectively, for long (125 points) and short (21 points) windows. (d) is a scalogram (continuous wavelet transform), (e)--smoothed pseudo Wigner-Ville distribution, (f)--the same as (e) in 3 dimensions. Presentation as in Figure 1.

Neither this figure alone, nor its comparison with Figure 1, is intended as a proof of a total superiority of one method. For each type or even instance of a signal, different distributions may exhibit specific advantages. The conclusion, important in the context of this paper, is that results provided by time-frequency distributions heavily depend on the particular choice of the method and its parameters. On the contrary, MP decomposition offers one of the highest possible resolutions in most of the cases--provided that Gabor functions describe well the analyzed signal, assumption which usually holds for EEG. Performance of the procedure can be significantly distorted only if we apply an unproportionally small dictionary. Although there is still no golden rule for a proper choice of ``large enough'' dictionary's size⁵, practical experience in this respect is rapidly gained and applicable to a wide classes of signals. Another free parameter in the MP decomposition is the number of iterations (i.e. waveforms in expansion (3)), but this can be regulated by a requirement of explaining a given percentage of signal's energy.

Footnotes

... size⁵

Further increasing the size of a dictionary used for MP decomposition does not deterior the quality of decomposition--on the contrary, in theory the resolution grows with its size, but starting from certain values this effect is hard to notice. The problem is that the algorithm requires large computational resources--time and memory--proportional to the size of applied dictionary.