Approximating the Time-Frequency Representation of Biosignals with Chirplets
© Omid Talakoub et al. 2010
Received: 14 January 2010
Accepted: 29 April 2010
Published: 31 May 2010
A new member of the Cohen's class time-frequency distribution is proposed. The kernel function is determined adaptively based on the signal of interest. The kernel preserves the chirp-like components while removing interference terms generated due to the quadratic characteristic of Wigner-Ville distribution. This approach is based on the chirplet as an underlying model of biomedical signals. We illustrate the method using a number of common biological signals including echo-location and evoked potential signals. Finally, the results are compared with other techniques including chirplet decomposition via matching pursuit and the Choi-Williams distribution function.
Many signals of biological origin are nonstationary in nature. Examples include speech signals, bat calls as well as neuroelectric signals like electroencephalography (EEG) [1, 2], heart rate variability , or event-related potentials (ERPs) . Time-frequency or time-scale representations, in recent years, have found significant application in nonstationary analysis of a wide-range of signals including biomedical signals [5–13]. Constructing a time-frequency representation involves mapping a one-dimensional time-domain signal into a two-dimensional function of time and frequency or time and scale . Time-frequency representations are some of the main tools for nonparametric instantaneous frequency estimation . The position of peaks in the time-frequency representation reveals the main components or structures of the signal.
Among the most commonly used time-frequency distributions are the so-called quadratic distributions. The spectrogram [15, 16] is one of the earliest proposed distributions yet is still commonly used to this day. Nevertheless, the spectrogram has severe drawbacks, both theoretically since it provides biased estimators of the signal instantaneous frequency and group delay , and practically since the Gabor-Heisenberg inequality  makes tradeoffs between temporal and spectral resolution unavoidable. To overcome these shortcomings, other nonstationary representations have been proposed. Among these include the Cohen's class  of bilinear time-frequency energy distributions. The Wigner-Ville distribution , the Margenau-Hill distribution , their smoothed versions [21–23], and others with reduced cross-terms [24–27] are all members of this class. Although Cohen's class distributions tend to reduce the interference between the various signal subcomponents, this reduction can affect the precision by which the instantaneous frequency is estimated. This is mainly due to the predefined smoothing kernel functions which do not distinguish between the signal components and the interference terms. Hence, in the process of reducing or removing cross-terms, the kernel also removes signal components. On the contrary, signal-dependent kernels can provide improved time-frequency representation and have been proposed for various applications [28–31]. An extensive review of the methods proposed for improving time-frequency resolution can be found in .
The nonparametric methods of time-frequency analysis described above can be contrasted with parametric approaches which attempt to model the underlying signal [32, 33]. There has been much debate as to the ideal choice of basis functions to use. Generally speaking, the more similar the basis function is to the signal, the more compact is the decomposition. Many biological signals can be thought of as a sum of more elementary components each of which are relatively narrowband in nature. Common examples include speech which consist of a number of formant frequencies illustrating the resonance of the vocal tract. In such a case, chirplets (or chirp signals of limited time extent) can be thought of as a good model of the underlying signal—any narrowband changes in instantaneous frequency can be described mathematically to first order by linear changes in the time-frequency plane [34–36]. We have been working on ways to decompose biological signals into a sum of chirplets . A time-frequency representation can be obtained from the decomposition by summing up the individual contributions from each chirplet. This provides a clear time-frequency picture of the signal without the cross-term interference. While we have found that this method yields excellent visualization of biomedical signals, there are some significant challenges to overcome because chirplets do not form an orthogonal basis set. In some earlier work, we used matching pursuit to carry out the decomposition process which we found to be prohibitive in terms of computational cost. There is a need to find improved ways to carry out this analysis.
This paper proposes a new class of time-frequency distributions for which the kernel function is determined adaptively based on the signal of interest. This approach can be best characterized as a hybrid approach combining both nonparametric and parametric methods using the chirplet as an underlying model of the biomedical signal. The kernel function preserves the chirping components in the signal while eliminating the interference terms generated by the quadratic characteristic of the time-frequency representation. The proposed method filters out the oscillatory cross-terms and instead preserves the "true'' signal components which are of low spatial frequency.
2. Proposed Method
2.1. Wigner-Ville Distribution and Multicomponent Signals
where is WVD of the th monocomponent autoterm. Cross-terms may lead to an erroneous visual interpretation of the time-frequency representation and are also a hindrance to pattern detection, since the interference can overlap with the signal. Due to the marginal properties of the WVD, i.e., and , the interference terms are oscillatory and zero-mean if the individual components do not overlap at any point in time and frequency . The spatial frequency of the oscillations depends on the distance between the monocomponents in time-frequency plane; that is, the farther apart the components, the higher the oscillation frequency. Although these interferences can be attenuated by time-frequency averaging, this will result in the loss of energy localization.
where is the ambiguity function of th monocomponent and the ambiguity function of the interference terms. While it is not always possible to express in closed form, one can always work with the expression numerically. The transform of (3) gives the multiplication of the signal's ambiguity function with the transform of the kernel. That is, . An ideal kernel should preserve each individual component and its localization in time-frequency domain while removing the cross-terms, that is, .
2.2. The Wigner-Ville and Ambiguity Representation with Gaussian Chirplets
Next we consider the specific case where the monocomponents of a multicomponent function are approximated by Gaussian chirplets.
2.3. Optimal Kernel Determination
where . Based on the superposition property, the Radon transform will show peaks at values of corresponding to the axes orientations of each of the ellipsoids. In order to exclude the effect of the interference terms in the calculation, the Radon transform is carried out only in neighbourhood of the origin. This neighbourhood is defined as the circular region around the origin which includes 50% of the signal energy.
To eliminate artifacts due to sharp cutoffs from kernel filtering (e.g., ringing), the edges of the kernel were smoothed. The smoothing process can be carried out by employing a tapering function like a Hanning or Gaussian function. In Figure 1(d) we show the example of the use of a two-dimensional Gaussian function. A "cleaned'' ambiguity representation is then obtained by multiplying the original ambiguity function with the corresponding mask. Finally, the time-frequency representation of the signal is generated by calculating the inverse Fourier transform of the ambiguity function.
The Expectation-Maximization (EM) algorithm was used for finding maximum likelihood estimation of chirplet parameters in the time-frequency plane. All optimization algorithms suffer from difficulties in parameter initialization and in the selection of the number of parameters or components. However, in this case we make use of the parameters estimated from the Radon transformation in ambiguity space. This significiantly reduces the time required for optimization as well as improves the robustness of the estimation. The number of components can also be set equal to the number of peaks found in the Radon transformation, and then adjusting the number of components from there to minimize the total error.
3. Results and Discussion
We also provide one example where the time-frequency representation is compared with that which was obtained from chirplet decomposition with matching pursuit. Certain dynamic brain mechanisms can be investigated through neuroelectrical brain responses called event-related potentials (EPRs). The visual evoked potential (VEP) is an evoked brain response generated in the visual cortex in response to the presentation of a visual signal. Such signals are noisy and are often averaged before processing. The VEP signal we have analyzed here is equal to an average of 50 trails from a single subject. Three chirplets are estimated for comparison with the results calculated by Cui and Wong  using the matching pursuit algorithm. As can be seen through comparison of both figures, the results are quite similar. The time-frequency representation was also verified through a spectrogram.
Earlier it was shown that the cross-term interference arising from a pair of monocomponents is located between main components. Moreover, the interferences are oscillatory in nature, and the spatial frequency of these oscillations is a function of the distance between the components in time and frequency. That is, the closer the two components, the lower the oscillation frequency. In ambiguity space, this would mean that the interference lies closer to the origin. The low frequency interference also appears as a result of the signal's instantaneous frequency changing nonlinearly with time. Due to the low frequency nature of these oscillations, the cross-terms may not be completely removed by the kernel due to overlap with signal components in the ambiguity space. Although this interference can be removed at the post-processing stage by (say) least-squares fitting to a Gaussian density, it is important to remember that this interference contributes to the signal's overall energy distribution.
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