# Approximating the Time-Frequency Representation of Biosignals with Chirplets

- Omid Talakoub
^{1}, - Jie Cui
^{1}and - Willy Wong
^{1}Email author

**2010**:857685

https://doi.org/10.1155/2010/857685

© Omid Talakoub et al. 2010

**Received: **14 January 2010

**Accepted: **29 April 2010

**Published: **31 May 2010

## Abstract

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.

## Keywords

## 1. Introduction

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 [3], or event-related potentials (ERPs) [4]. 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 [14]. Time-frequency representations are some of the main tools for nonparametric instantaneous frequency estimation [14]. 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 [17], and practically since the Gabor-Heisenberg inequality [15] 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 [18] of bilinear time-frequency energy distributions. The Wigner-Ville distribution [19], the Margenau-Hill distribution [20], 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 [14].

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 [37]. 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 [40]. 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.

which shows a precise localization of instantaneous frequency and energy. Note however that this is not the case if the changes in instantaneous frequency are not linear [14, 45].

### 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 [37] 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.

## Authors’ Affiliations

## References

- Wang T, Deng J, He B: Classifying EEG-based motor imagery tasks by means of time-frequency synthesized spatial patterns.
*Clinical Neurophysiology*2004, 115(12):2744-2753. 10.1016/j.clinph.2004.06.022View ArticleGoogle Scholar - Miwakeichi F, Martínez-Montes E, Valdés-Sosa PA, Nishiyama N, Mizuhara H, Yamaguchi Y: Decomposing EEG data into space-time-frequency components using Parallel Factor Analysis.
*NeuroImage*2004, 22(3):1035-1045. 10.1016/j.neuroimage.2004.03.039View ArticleGoogle Scholar - Aysin B, Chaparro LF, Gravé I, Shusterman V: Orthonormal-basis partitioning and time-frequency representation of cardiac rhythm dynamics.
*IEEE Transactions on Biomedical Engineering*2005, 52(5):878-889. 10.1109/TBME.2005.845228View ArticleGoogle Scholar - Quiroga RQ, Garcia H: Single-trial event-related potentials with wavelet denoising.
*Clinical Neurophysiology*2003, 114(2):376-390. 10.1016/S1388-2457(02)00365-6View ArticleGoogle Scholar - Van Zaen J, Uldry L, Duchêne C, Prudat Y, Meuli RA, Murray MM, Vesin J-M: Adaptive tracking of EEG oscillations.
*Journal of Neuroscience Methods*2010, 186(1):97-106. 10.1016/j.jneumeth.2009.10.018View ArticleGoogle Scholar - Liu H-L, Li M-L, Shih T-C, Huang S-M, Lu I-Y, Lin D-Y, Lin S-M, Ju K-C: Instantaneous frequency-based ultrasonic temperature estimation during focused ultrasound thermal therapy.
*Ultrasound in Medicine and Biology*2009, 35(10):1647-1661. 10.1016/j.ultrasmedbio.2009.05.004View ArticleGoogle Scholar - Merlo E, Pozzo M, Antonutto G, Di Prampero PE, Merletti R, Farina D: Time-frequency analysis and estimation of muscle fiber conduction velocity from surface EMG signals during explosive dynamic contractions.
*Journal of Neuroscience Methods*2005, 142(2):267-274. 10.1016/j.jneumeth.2004.09.002View ArticleGoogle Scholar - Devuyst G, Vesin J-M, Despland P-A, Bogousslavsky J: The matching pursuit: a new method of characterizing microembolic signals?
*Ultrasound in Medicine and Biology*2000, 26(6):1051-1056. 10.1016/S0301-5629(00)00244-1View ArticleGoogle Scholar - Rankine L, Stevenson N, Mesbah M, Boashash B: A nonstationary model of newborn EEG.
*IEEE Transactions on Biomedical Engineering*2007, 54(1):19-28.View ArticleGoogle Scholar - Celka P, Boashash B, Colditz P: Preprocessing and time-frequency analysis of newborn EEG seizures.
*IEEE Engineering in Medicine and Biology Magazine*2001, 20(5):30-39. 10.1109/51.956817View ArticleGoogle Scholar - Monti A, Médigue C, Mangin L: Instantaneous parameter estimation in cardiovascular time series by harmonic and time-frequency analysis.
*IEEE Transactions on Biomedical Engineering*2002, 49(12 I):1547-1556.View ArticleGoogle Scholar - De Cheveigné A: YIN, a fundamental frequency estimator for speech and music.
*The Journal of the Acoustical Society of America*2002, 111(4):1917-1930. 10.1121/1.1458024View ArticleGoogle Scholar - Bonato P, Roy SH, Knaflitz M, de Luca CJ: Time frequency parameters of the surface myoelectric signal for assessing muscle fatigue during cyclic dynamic contractions.
*IEEE Transactions on Biomedical Engineering*2001, 48(7):745-753. 10.1109/10.930899View ArticleGoogle Scholar - Shafi I, Ahmad J, Shah SI, Kashif FM: Techniques to obtain good resolution and concentrated time-frequency distributions: a review.
*EURASIP Journal on Advances in Signal Processing*2009, 2009:-43.Google Scholar - Gabor D: Theory of communication.
*Journal of the Institute of Electrical Engineers*1946, 93(26): 429-457.Google Scholar - Allen JB, Rabiner LR: A unified approach to short-time fourier analysis and synthesis.
*Proceedings of the IEEE*1977, 65(11):1558-1564.View ArticleGoogle Scholar - Martin W, Flandrin P: Wigner-ville spectral analysis of nonstationary processes.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1985, 33(6):1461-1470. 10.1109/TASSP.1985.1164760View ArticleGoogle Scholar - Cohen L: Generalized phase-space distribution functions.
*Journal of Mathematical Physics*1966, 7(5):781-786. 10.1063/1.1931206MathSciNetView ArticleGoogle Scholar - Wigner E: On the quantum correction for thermodynamic equilibrium.
*Physical Review*1932, 40(5):749-759. 10.1103/PhysRev.40.749MathSciNetView ArticleMATHGoogle Scholar - Margenau H, Hill RN: Correlation between measurements in quantum theory.
*Progress of Theoretical Physics*1961, 772-738.Google Scholar - Claasen TACM, Mecklenbrauker WFG: Wigner distribution—a tool for time-frequency signal analysis.
*Philips Journal of Research*1980, 35(4-5):276-300.MathSciNetMATHGoogle Scholar - Flandrin P, Martin W: A general class of estimators for the wigner-ville spectrum of nonstationary processes. In
*Systems Analysis and Optimization of Systems, Lecture Notes in Control and Information Sciences*. Springer, Berlin, Germany; 1984:15-23.Google Scholar - Hippenstiel RD, de Oliveira PM: Time-varying spectral estimation using the instantaneous power spectrum (IPS).
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1990, 38(10):1752-1759. 10.1109/29.60106View ArticleGoogle Scholar - Born M, Jordan P: Zur Quantenmechanik.
*Zeitschrift für Physik*1925, 34(1):858-888. 10.1007/BF01328531View ArticleGoogle Scholar - Choi H, Williams WJ: Improved time-frequency representation of multicomponent signals using exponential kernels.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1989, 37(6):862-871. 10.1109/ASSP.1989.28057View ArticleGoogle Scholar - Papandreou A, Boudreaux-Bartels GF: Distributions for time-frequency analysis: a generalization of Choi-Williams and the Butterworth distribution.
*Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '92), 1992, San Francisco, Calif, USA*181-184.Google Scholar - Jeong J, Williams WJ: Kernel design for reduced interference distributions.
*IEEE Transactions on Signal Processing*1992, 40(2):402-412. 10.1109/78.124950View ArticleGoogle Scholar - Cohen L: Distributions concentrated along the instantaneous frequency.
*Advanced Signal-Processing Algorithms, Architectures, and Implementations, July 1990, Proceedings of SPIE*149-157.View ArticleGoogle Scholar - Kadambe S, Boudreaux-Bartels GF, Duvaut P: Window length selection for smoothing the Wigner distribution by applying an adaptive filter technique.
*Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '89), May 1989*2226-2229.Google Scholar - Jones DL, Baraniuk RG: Adaptive optimal-kernel time-frequency representation.
*IEEE Transactions on Signal Processing*1995, 43(10):2361-2371. 10.1109/78.469854View ArticleGoogle Scholar - Andrieux JC, Feix MR, Mourgues G, Bertrand P, Izrar B, Nguyen VT: Optimum smoothing of the wigner–ville distribution.
*IEEE Transactions on Acoustics, Speech, and Signal Processing*1987, 35(6):764-769. 10.1109/TASSP.1987.1165204View ArticleGoogle Scholar - Doo SH, Ra W-S, Yoon TS, Park JB: Fast time-frequency domain reflectometry based on the AR coefficient estimation of a chirp signal.
*American Control Conference (ACC '09), June 2009*3423-3428.Google Scholar - Jachan M, Matz G, Hlawatsch F: Time-frequency ARMA models and parameter estimators for underspread nonstationary random processes.
*IEEE Transactions on Signal Processing*2007, 55(9):4366-4381.MathSciNetView ArticleGoogle Scholar - Ma N, Vray D, Delachartre P, Gimenez G: Time-frequency representation of multicomponent chirp signals.
*Signal Processing*1997, 56(2):149-155. 10.1016/S0165-1684(96)00163-6View ArticleMATHGoogle Scholar - Akan A, Yalcin M, Chaparro L: An iterative method for instantaneous frequency estimation.
*Proceedings of the 8th IEEE International Conference on Electronics, Circuits and Systems (ICECS '01), 2001*3: 1335-1338.Google Scholar - Wang M, Chan AK, Chui CK: Linear frequency-modulated signal detection using radon-ambiguity transform.
*IEEE Transactions on Signal Processing*1998, 46(3):571-586. 10.1109/78.661326View ArticleGoogle Scholar - Cui J, Wong W: The adaptive chirplet transform and visual evoked potentials.
*IEEE Transactions on Biomedical Engineering*2006, 53(7):1378-1384. 10.1109/TBME.2006.873700View ArticleGoogle Scholar - Hess-Nielsen N, Wickerhauser MV: Wavelets and time-frequency analysis.
*Proceedings of the IEEE*1996, 84(4):523-540. 10.1109/5.488698View ArticleGoogle Scholar - Mann S, Haykin S: Chirplet transform: physical considerations.
*IEEE Transactions on Signal Processing*1995, 43(11):2745-2761. 10.1109/78.482123View ArticleGoogle Scholar - Mallat S:
*A Wavelet Tour of Signal Processing*. 3rd edition. The Sparse Way. Academic Press, London, UK; 2008.MATHGoogle Scholar - Lu Y, Demirli R, Cardoso G, Saniie J: A successive parameter estimation algorithm for chirplet signal decomposition.
*IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control*2006, 53(11):2121-2131.View ArticleGoogle Scholar - Lu Y, Demirli R, Cardoso G, Saniie J: Chirplet transform for ultrasonic signal analysis and NDE applications.
*IEEE Ultrasonics Symposium, September 2005*536-539.Google Scholar - Ioana C, Quinquis A: On the use of time-frequency warping operators for analysis of marine-mammal signals.
*Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '04), May 2004*605-608.Google Scholar - Tang Y, Luo X, Yang Z: Ocean clutter suppression using one-class SVM.
*Proceedings of the 14th IEEE Signal Processing Society Workshop on Machine Learning for Signal Processing, October 2004*559-568.Google Scholar - Shafi I, Ahmad J, Shah SI, Kashif FM: Computing de-blurred time frequency distributions using artificial neural networks.
*Circuits, Systems, and Signal Processing*2008, 27(3):277-294. 10.1007/s00034-008-9027-xView ArticleGoogle Scholar - Borgnat P, Flandrin P: Time-frequency localization from sparsity constraints.
*Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '08), April 2008*3785-3788.Google Scholar - Mallat SG, Zhang Z: Matching pursuits with time-frequency dictionaries.
*IEEE Transactions on Signal Processing*1993, 41(12):3397-3415. 10.1109/78.258082View ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.