List of Figures

1. (a): left--components of the simulated signal: sine A, Dirac's delta B and Gabor functions C, D and E. Right--signals, labelled b, c and d, constructed as sum of structures A-E and white noise, and decomposed in corresponding panels (b), (c) and (d). (b): time-frequency energy distribution (eq. 12) obtained for sum of structures A-E; in 3-D representation on the left energy is proportional to the height, in right panel--to the shades of gray. Panels (c) and (d): decompositions of signals with linear addition of noise, S/N = 1/2 ($-3$ dB) in (c) and $-6$ dB in (d), the same realization of white noise was used in both cases. Exact parameters of presented time-frequency structures are given in Table 1.
2. Histograms of frequencies of sleep spindles detected in one overnight EEG recording. Plots are placed on page according to relative positions of corresponding derivations (Table 2 on page )--front of head towards the top of page. We observe sparse occurences in peripheral (Fp*, O*, T* and F[7-8]) electrodes, therefore in the following multi-derivations plots (figures 3, 8, 9) results only for the central 9 derivations (O*, C* and F[3,4,z]) will be presented.
3. Amplitudes of detected spindles (vertical) plotted versus their frequencies (horizontal) for the nine central derivations from Figure 2.
4. Time-frequency energy distribution (equation 12) of 20 seconds of sleep EEG; structures corresponding to sleep spindles are marked by letters A-F. Structures C and D , as well as E and F, were classified as one spindle, i.e. their centers fell within a time section marked by expert as one spindle's occurrence.
5. Spindles F (upper plot) and E (lower plot) from Figure 4 across channels. In each box: frequency [Hz], amplitude [$\mu$V], relative position in time [s], phase. Shades of gray proportional to the amplitude. Front of head towards top of page.
6. Hypnogram a) and time course--in the same horizontal scale--of: b) frequencies and c) amplitudes of detected spindles, d) spindles density [1/min], e) SWA power, f) and g) frequencies and amplitudes of structures classified as SWA.
7. Statistical properties of MP decomposition of 50 epochs of sleep EEG (a, b) and white noise (c-f) over dyadic (a, c, e) and stochastic (b, d, f) dictionaries. Histograms of frequency centers of atoms fitted by MP decomposition over dyadic dictionary to EEG (a) and noise (c) reveal additional structure, absent in corresponding decompositions performed over stochastic dictionaries b and d, respectively. Maximum in the middle of frequency range in panel d results from convention of assigning half of Nyquist frequency to Diras'c delta. In the top panel centers of atoms fitted to white noise are given in the time-frequency plane for dyadic (left, e) and stochastic (right, f) dictionaries.
8. Histograms of frequencies of sleep spindles detected in the same EEG recording as in Figure 2, decomposed in stochastic dictionaries. Nine central derivations are presented.
9. Amplitudes of sleep spindles (vertical) plotted versus their frequencies (horizontal) detected in the same EEG recording as in Figure 3 (as well as 2 and 8), based upon MP decomposition over stochastic dictionaries.
10. Decomposition of signal composed of two chirps--sines of linearly changing frequency--presented in (a). In 3-dimensional plots on the left side energy is proportional to height, on flat pictures on the right--to the shades of gray. (b) presents results of a single decomposition over dictionary consisting of 500.000 atoms, and (c)--time-frequency representation averaged over 50 realizations of smaller dictionary (15.000 atoms).