Comparison of time-frequency methods

brought a wide variety of signal analysis methods into applications in biomedical research. The most widespread are the spectral methods which make possible the identification of the basic rhythms present in the signal. Conventional methods of the analysis assumed stationarity of the signal, in spite of the fact that interesting processes are often reflected in fast dynamic changes of signal. This implied the application to the analysis of the signals methods operating in time-frequency space. The available time-frequency methods can be roughly divided into two categories:

• Those that give directly continuous estimators of energy density in the time-

frequency space

• Those that decompose the signal into components localized in the time-

frequency space, which can be described by sets of parameters, and at the sec- ond step the components can be used to create the estimators of time-frequency energy density distribution

An example of time-frequency energy distribution obtained by means of different methods is shown inFigure 2.21.

In the first category—the Cohen’s class of time-frequency distributions—one ob- tains directly the time-frequency estimators of energy density without decomposing the signal into some predefined set of simple elements. This allows for maximal flex- ibility in expressing the time frequency content of the signal. However, there are two consequences:

• The first consequence is the lack of parametric description of the signals struc-

tures

• The second consequence is that, no matter how much the signal structures are

separated in time or frequency, they interfere and produce cross-terms. The problem of compromise between time and frequency resolution manifests when one selects the proper filter kernel to suppress the cross-terms.

In the second category the most natural transition from spectral analysis to the analysis in time-frequency space is the use of short time Fourier transform (STFT) and a representation of energy density derived from it—the spectrogram. The pos- itive properties of this approach are the speed of computations and the time and frequency shift invariance, which makes the interpretation of the resulting time- frequency energy density maps easy to interpret. The main drawbacks are: (1) the a priori fixed compromise between time and frequency resolution in the whole time- frequency space, which results in smearing the time-frequency representation, (2) the presence of cross-terms between the neighboring time-frequency structures.

Another common choice in the second category is the CWT. From the practical point of view the main difference from the STFT relies on another compromise be- tween the time and frequency resolution. In case of CWT, one sacrifices the time resolution for the better frequency resolution of low frequency components and vice

FIGURE 2.21: Comparison of energy density in the time-frequency plane ob- tained by different estimators for a signal e): a) spectrogram, b) discrete wavelet transform, c) Choi-Williams transform, d) continuous wavelets, f) matching pursuit. Construction of the simulated signal shown in (e), the signal consisting of: a sinusoid, two Gabor functions with the same frequency but different time positions, a Gabor function with frequency higher than the previous pair, an impulse. From [Blinowska et al., 2004b].

versa for higher frequency components; also the change of the frequency of a struc- ture leads to the change of the frequency resolution.

STFT and CWT can be considered as atomic representations of the signal, and as such give a certain parametric description of the signals. However, the representation in not sparse; in other words there are too many parameters; hence they are not very informative.

The sparse representation of the signal is provided by DWT and MP, which leads to efficient parameterization of the time series. The DWT can decompose the signal into a base of functions, that is a set of waveforms that has no redundancy. There are fast algorithms to compute the DWT. Similar to CWT, the DWT has poor time resolution for low frequencies and poor frequency resolution for high frequencies. The DWT is very useful in signal denoising or signal compression applications. The lack of redundancy has a consequence in the loss of time and frequency shift invariance. DWT may be appropriate for time-locked phenomena, but much less for transients appearing in time at random, since parameters describing a given structure depend

on its location inside the considered window.

The decomposition based on the matching pursuit algorithm offers the step-wise adaptive compromise between the time and frequency resolution. The resulting de- composition is time and frequency invariant. The time-frequency energy density esti- mator derived from the MP decomposition has explicitly no cross-term, which leads to clean and easy-to-interpret time-frequency maps of energy density. The price for the excellent properties of the MP decomposition is the higher computational com- plexity.

The sparsity of the DWT and MP decompositions has a different character which has an effect on their applicability. DWT is especially well suited to describing time locked phenomena since it provides the common bases. MP is especially useful for structures appearing in the time series at random. The sparsity of MP stems from the very redundant set of functions, which allows to represent the signal structures as a limited number of atoms. The MP decomposition gives the parameterization of the signal structures in terms of the amplitude, frequency, time of occurrence, time, and frequency span which are close to the intuition of practitioners.

2.4.2.2.9 Empirical mode decomposition and Hilbert-Huang transform The Hilbert-Huang transform (HHT) was proposed by Huang et al. [Huang et al., 1998]. It consists of two general steps:

• The empirical mode decomposition (EMD) method to decompose a signal into

the so-called intrinsic mode function (IMF)

• The Hilbert spectral analysis (HSA) method to obtain instantaneous frequency

data

The HHT is a non-parametric method and may be applied for analyzing non- stationary and non-linear time series data.

Empirical mode decomposition (EMD) is a procedure for decomposition of a sig- nal into so called intrinsic mode functions (IMF). An IMF is any function with the same number of extrema and zero crossings, with its envelopes being symmetric with respect to zero. The definition of an IMF guarantees a well-behaved Hilbert trans- form of the IMF. The procedure of extracting an IMF is called sifting. The sifting process is as follows:

1. Between each successive pair of zero crossings, identify a local extremum in the signal.

2. Connect all the local maxima by a cubic spline line as the upper envelope

Eu(t).

3. Repeat the procedure for the local minima to produce the lower envelope El(t).

4. Compute the mean of the upper and lower envelope: m11(t) =1₂(Eu(t)+El(t)).

5. A candidate h11for the first IMF component is obtained as the difference be- tween the signal x(t) and m11(t): h11(t) = x(t) − m11(t).

In a general case the first candidate h11doesn’t satisfy the IMF conditions. In such case the sifting is repeated taking h11as the signal. The sifting is repeated iteratively:

h1k(t) = h1(k−1)(t) − m1k(t) (2.120) until the assumed threshold for standard deviation SD computed for the two consec- utive siftings is achieved. The SD is defined as:

SD= T

∑

Authors of the method suggest the SD of 0.2–0.3 [Huang et al., 1998]. At the end of the sifting process after k iterations the first IMF is obtained:

c1= h1k (2.122)

The c1mode should contain the shortest period component of the signal. Subtracting it from the signal gives the first residue:

r1= x(t) − c1 (2.123)

The procedure of finding consecutive IMFs can be iteratively continued until the variance of the residue is below a predefined threshold, or the residue becomes a monotonic function—the trend (the next IMF cannot be obtained). The signal can be expressed as a sum of the n-empirical modes and a residue:

x(t) =

n

∑

i=1

ci− rn (2.124)

Each of the components can be expressed by means of a Hilbert transform as a product of instantaneous amplitude aj(t) and an oscillation with instantaneous fre-

quency ωj(t) (Sect. 2.4.1): cj= aj(t)ei

R

ωj(t)dt_{. Substituting this to (2.124) gives}

representation of the signal in the form:

x(t) = n

∑

Equation (2.125) makes possible construction of time-frequency representation—the so-called Hilbert spectrum. The weight assigned to each time-frequency coordinate is the local amplitude.