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Time Frequency Analysis and Wavelet Transform Tutorial TimeFrequency Analysis for Biomedical Engineering ChiaJung Chang (張嘉容) National Taiwan University ABSTRACT Biomedical related research requires lots of mathematical and engineering techniques to analyze data. Among the subfields, electrophysiological research plays the core role. In this tutorial, several tools are examined, including electrocardiogram, and electroencephalogram. These are the most common tools used to diagnose our physiological activities since neural responses carry information. Because biomedical signals are usually non stationary, Fourier transform is not suitable to apply here. Besides, traditional signals are analyzed in frequency domain, separately from time domain, such that extraordinary conditions are hard to be observed. To solve such problem, timefrequency analysis and wavelet transform provide both time and frequency information simultaneously. In the following tutorial, I would like to talk about the theoretical background of both timefrequency analysis and wavelet transform methods, including what properties they have, their common types, and how to operate them. Secondly, I would briefly introduce three common physiological tools such as ECG, and EEG. Where they can be applied, what they are targeting, and what analysis methods can be used, and how they perform will be described. CONTENTS Abstract
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Great progress has been made in applying linear timeinvariant techniques in signal processing. In such cases the deterministic part of the signal is assumed to be composed of complex exponentials, the solutions to linear time invariant differential equations. However, many biomedical signals do not meet these assumptions. Thus, the emerging techniques of timefrequency analysis can provide new insights into the nature of biological signals. There are several different time frequency analysis methods such as short time Fourier transform, S transform, Wigner transform, and Cohen’s distribution. I would focus on Cohen’s class distribution since it is used in biomedical signals analysis more commonly than S transform. As for short time Fourier transform, it is also a popular analysis tool, which will be described in details in the latter chapters.
Cohen’s class distribution is defined as follows where A_{x} is called ambiguity function of x(t) Cohen’s class distribution can be regarded as a series of methods to eliminate crossterm produced by Wigner distribution, and to keep clarity. However, the tradeoff is that it takes large computation, and thus losses some properties. As there are different ways to filter out crossterm, there are several forms.
There are four different distributions being introduced in this section. ChoiWilliamns distribution (CWD) is defined with BornJordan distribution (BJ) is defined with ZhaoAtlasMarks distribution (ZAM) is defined with While Cohen’s class distribution can filter crossterm effectively, resolution on the timefrequency domain is lost at the same time. Another distribution has been brought out, which is called Reduced Interference distribution (RID). It can be regarded as smoothed Wigner distribution. After we use a lowpass filter, the cross term can be easily eliminated, and the resolution is enhanced as well. Reduced Interference distribution (RID) has been defined as where h(t) is a time smooth window, g(v) is a frequency smooth window.
A specific subset of timefrequency distribution is Cohen's class distribution. For these distributions, a time shift in the signal is reflected as an equivalent time shift in the timefrequency distribution, and a shift in the frequency of the signal is reflected as an equivalent frequency shift in the timefrequency distribution. The Wigner distribution, the RID, and ZAM all have this property. Tracing back to history, Kawabata employed the instantaneous power spectrum, a measure of the rate of change of the energy spectrum, to study dynamic variations in the EEG during photic alpha blocking and performance of mental tasks. DeWeerd compared timefrequency analysis of event related potentials obtained by a non uniform filter and the complex energy distribution function. The applicability of the Wigner distribution for the representation of event related potentials was examined by Morgan and Gevins. The RID with a product kernel exhibits the interesting and valuable property of scale invariance. In addition, the RID has information invariance. Cohen's class distribution does not vary with time or frequency automatically. On the other hand, the continuous wavelet transform exhibits scale invariance and timeshift invariance. In Figure 1a, a spectrogram with a 256point window is used. In Figure 1b, a spectrogram with a 32point window is used. Finally, in Figure 1c, an RID is used. Spectrogram and RID are compared, and the first is the original EEG segment. The second is the original EEG segment timescaled to preserve energy. The third segment is a frequencyshifted version of the original EEG segment. Figure 1 The RID provides a highresolution representation of both time and frequency. Furthermore, the RID preserves this structure well even when the original signal is scaled or shifted in frequency. It is clear that these techniques can be applied in neurophysiological research such as EEG analysis.
To extract information from signals and reveal the underlying dynamics that corresponds to the signals, proper signal processing technique is needed. Typically, the process of signal processing transforms a time domain signal into another domain since the characteristic information embedded within the time domain is not readily observable in its original form. Mathematically, this can be achieved by representing the time domain signal as a series of coefficients, based on a comparison between the signal x(t) and template functions {_{n}(t)}. The inner product between the two functions x(t) and _{n}(t) is The inner product describes an operation of comparing the similarity between the signal and the template function, i.e. the degree of closeness between the two functions. This is realized by observing the similarities between the wavelet transform and other commonly used techniques, in terms of the choice of the template functions. A non stationary signal is shown in Figure 2 as an example. The signal consists of four groups of impulsive signal trains. In these groups, the signals are composed of two major frequencies, 650 and 1500 Hz. Figure 2
Using the notation of inner product, the Fourier transform of a signal can be expressed as Assuming that the signal has finite energy The Fourier transform is essentially a convolution between the time series x(t) and a series of sine and cosine functions that can be viewed as template functions. The operation measures the similarity between x(t) and the template functions, and expresses the average frequency information during the entire period of the signal analyzed as shown in Figure 3. Figure 3 However, it does not reveal how the signal’s frequency contents vary with time; that is, it does not reveal if two frequency components are present continuously throughout the time of observation or only at certain intervals, as is implicitly shown in the timedomain representation. Because the temporal structure of the signal is not revealed, the merit of the Fourier transform is limited; it is not suited for analyzing non stationary signals.
In Figure 4, the STFT employs a sliding window function g(t). A timelocalized Fourier transform performed on the signal within the window. Subsequently, the window is removed along the time, and another transform is performed. The signal segment within the window function is assumed to be stationary. As a result, the STFT decomposes a time signal into a 2D timefrequency domain, and variations of the frequency within the window function are revealed. Figure 4 STFT can be expressed as According to the uncertainty principle, the time and frequency resolutions of the STFT technique cannot be chosen arbitrarily at the same time. As shown in Figure 5, the products of the time and frequency resolutions of the window function (i.e., the area of f) are the same regardless of the window size ( or 0.5). Figure 5
Wavelet transform is a tool that converts a signal into a different form. This conversion reveals the characteristics hidden in the original signal. The wavelet is a small wave that has an oscillating wavelike characteristic and has its energy concentrated in time. The first reference to the wavelet goes back to the early twentieth century. Harr’s research on orthogonal systems of functions led to the development of a set of rectangular basis functions. The Haar wavelet (Figure 6) was named on the basis of this set of functions, and it is also the simplest wavelet family developed till this date. Figure 6 In contrast to STFT, the wavelet transform enables variable window sizes in analyzing different frequency components within a signal. By comparing the signal with a set of functions obtained from the scaling and shift of a base wavelet, it is realized as shown in Figure 7. Figure 7 Wavelet transform can be expressed as As in Figure 8, variations of the time and frequency resolutions of the Morlet wavelet at two locations on the time–frequency plane, (_{1}, s_{1}) and (_{2}, s_{2}) are illustrated. Figure 8 Through variations of the scale and time shifts of the base wavelet function, the wavelet transform can extract the components within over its entire spectrum, by using small scales for decomposing high frequency parts and large scales for low frequency components analysis.
The definition of continuous wavelet transform where a shifts time, b modulates the width (not frequency), and(t) is mother wavelt. It has superposition property. If the continuous wavelet transform of x(t) is X_{ }(s,) and of y(t) is Y(s,), then the continuous wavelet transform of z(t) = k1x(t) + k2y(t) can be expressed as 