Time-frequency distributions averaged over dictionaries

In the preceding section we have demonstrated how to avoid the bias of dictionary structure for averages estimated from multiple realisations. We can use averaging technique in case of single data epoch with the aim of improvement of the time-frequency characteristics. A signal, which is particularly difficult to represent by means of MP, is a chirp (a sine with linearly changing frequency), since in the dictionary there are no waveforms of changing frequency. Such a signal has to be approximated by means of several waveforms. We can overcome this difficulty by decomposing the signal several times, each time over a dictionary with a different, randomly chosen grid. This procedure is presented on a signal composed from two chirps.

Figure 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).

Figure 10b illustrates a decomposition of the signal (shown in 10a) over a dictionary containing 500,000 waveforms. We notice that structures of changing frequency are represented by a series of Gabor functions, since in the dictionary there are only structures of constant frequency. Figure 10c shows an average of 50 time-frequency maps, constructed from decompositions of the same signal over different realizations of smaller stochastic dictionaries, each of them containing 15,000 atoms. This result is closer to the expected representation than the single decomposition given in 10b. The computational cost need not to be higher than for single decomposition, since in case of repetition of the procedure, dictionaries containing smaller number of atoms can be used. The problem of the density of the dictionary in the context of a quality of the decomposition is dicussed in [Durka et al., 2001]. The repetition of the decomposition over several stochastic dictionaries is recommended when a very accurate estimation of the frequency changes is of interest. In this particular case of chirp, better representation could have been obtained by a method aimed at chirps detection (e.g. [Qian et al., 1998]) or some of the time-frequency distributions discussed in the Introduction. Nevertheless, proposed approach is a general one--it is by no means limited to a particular kind of structures and gives unbiased and free of cross terms estimates.