
HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2005 Chapter 16 - Wavelets; Multiscale Activity in Physiological Signals c 2006 Ali Shoeb and Gari Clifford 1 Introduction Physiological signals are dynamic; they exhibit time-varying statistics (such as value of the mean and variance over a given temporal window) in both the time and frequency domain. Physiological signals also exhibit activity that spans a range of time scales. For instance the cycles of an electrocardiogram (ECG) signal contain three components with different time-scales: atrial depolarization represented by the P wave has a duration of 0.1 to 0,15 seconds; ventricular depolarization represented by the QRS complex has a duration around 0.1 seconds; and ventricular repolarization represented by the T-wave has a duration of about 0.2 to 0.4 seconds. However, the inter-beat timing (which when averaged gives the heart rate), changes over much longer time scales; from minutes to hours, to days. Short term variations (activity on the order of seconds and minutes above 0.01 Hz) are due to changes in the sympathetic (fight-and-flight) and parasympathetic (rest-and-digest) activity of the central nervous system acting on the heart. Longer term variations can be partially attributed to changes in activity and intrinsic circadian controls (that lead to sleep for example). The changes in the heart rate over many scales can provide diagnostic information [12]. The spike-and-slow-wave complex observed in the electroencephalogram (EEG) during the evolution of some epileptic seizures is another example of a physiological signal with mul- tiscale activity [8]. In this case the spike component of the waveform represents the short time-scale event, and the slow-wave component of the waveform represents the long time- scale event. Classical signal processing tools such as the Fourier transform are not suited for analyzing dynamic, non-stationary signals because implicit in their formulation is an assumption of signal stationarity. Generalizations of the Fourier transform, such as the Short-Time Fourier transform, can be used to analyze signals with time-varying spectral and temporal character- istics. However, the Short-Time Fourier transform cannot be used to simultaneously resolve activity at different time-scale because implicit in its formulation is a selection of a time- scale. This chapter introduces the wavelet transform, a generalization of the Short-Time Fourier transform that can be used to perform multi-scale signal analysis. 1 ECG Cycle EEG Spike-Wave Figure 1: Physiological Signals with Multiscale Activity: Each cycle of an ECG signal contains three components: The P-wave, QRS complex, and the T-wave. The spike-and- slow wave complex observed in epileptic EEG has two components with different time-scales: A spike, and a slow wave. 2 Short-Time Fourier Transform The Fourier transform is well suited for analyzing stationary signals; these are signals with time-invariant spectral content. The sum of N sinusoids is an example of a stationary signal because at every point it has the same N frequency components. The chirp signal, which is a sinusoid with linearly or quadratically varying frequency, is an example of a non-stationary signal. The Fourier transform cannot capture the spectral evolution of a non-stationary signal for two reasons that are apparent in the Fourier transform analysis/synthesis equation: 1 −j!t 1 j!t F (Ω) = −∞ x(t)e x(t) = −∞ F (Ω)e (1) R R 1. The Fourier Transform synthesizes x(t) from a linear combination of stationary signals; in particular, the sum of time-invariant, ever-lasting sinusoids. A non-stationary signal cannot be accurately represented using a sum of stationary signals. For example, a chirp signal with a frequency that increases linearly from 0-2 Hz over a two second interval cannot be accurately represented using a linear combination of fixed frequency sinusoids. 2. The Fourier Transform of a signal is a mapping from a function of time x(t) to a function of frequency F (Ω). The function F (Ω) tells us the extent to which a signal component with frequency Ω is present in the analyzed signal. It does not indicate how that signal component evolves with time t. For that we need a transform that returns a bivariate function of the form F (t; Ω). Figure 2 shows the time and frequency domain plots of a signal composed from the summing of a 10 Hz, 30 Hz, and 50 Hz frequency sinusoids. The Fourier transform reveals the presence of this stationary signal's three frequency components. 2 Sum of Three Sinusoids (10,30,50 Hz) Spectrum of Sum Of Three Sinusoids (10,30,50 Hz) 140 2 1.5 120 1 100 0.5 80 0 Amplitude −0.5 Magnitude 60 −1 40 −1.5 20 −2 −2.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 10 20 30 40 50 60 70 80 90 100 Time (seconds) Frequency (Hz) Figure 2: Fourier Analysis of Stationary Signals: The Fourier transform is well-suited for the analysis of stationary signals such as the sum of three sinusoids. In this case, the Fourier transform is used to reveal the presence of three frequency components (10 Hz, 30 Hz, 50 Hz) in the signal. Figure 3 shows the time and frequency domain representations of a chirp signal with a frequency that increases linearly from 0 to 250 Hz. The Fourier transform of the chirp signal implies the presence of frequencies between 0-250 Hz for the entirety of the signal. In reality, each of these frequency components is only present for a short duration of the chirp signal. The generalization of the Fourier transform that allows one to study non-stationary signals is the Short-Time Fourier Transform (STFT). The STFT maps x(t) into a bivariate function F (t; Ω). This function can be used to determine the extent to which a signal component with frequency Ω is present at time t = t0. The construction of F (t; Ω) involves: 1. A segment of the signal x(t) that begins at time t = t0 is extracted using a window w(t) with duration L. The segment is given by s(t) = x(t)w(t t ) and has a duration − 0 L. 2. The Fourier Transform of the segment s(t) is computed to give S(Ω) = F (t = t0; Ω). The function F (t = t0; Ω) reveals the spectral content of the signal segment in an interval of time beginning at t = t0 and ending at t = t0 + L. 3. The window w(t) is shifted so that it can be used to extract a new signal segment s(t) = x(t)w(t t0 ∆t). As in the previous step, the Fourier Transform is again used to reveal the sp−ectral− content of the new segment. A plot of the magnitude of the signal segment spectra is known as a Spectrogram. The spectrogram has time on the x-axis; frequency on the y-axis; and magnitude of the spectra on the z-axis. The spectrogram illustrates visually how different frequency components evolve over the duration of a signal. Figure 4 is graphical illustration of the process leading to the construction of a Spectrogram. 3 Chirp Signal with Linear Frequency Increase Spectrum of Chirp Signal 1.4 0.8 1.2 0.6 0.4 1 0.2 0.8 0 0.6 Amplitude −0.2 Magnitude −0.4 0.4 −0.6 0.2 −0.8 −1 0 0.1 0.2 0.3 0.4 0.5 0.6 0 50 100 150 200 250 300 350 400 450 500 Time (sec) Frequency (Hz) Figure 3: Time and Frequency Plots of Chirp Signal: The Fourier Transform is not well- suited for the analysis of non-stationary signals like the chirp signal. The Fourier Transform in the case of this chirp signal demonstrates the presence of its frequency components for all time, when in actuality each component is present for only a brief duration of time. Windows Signal F F F F F F T T T T T T F r e q u e n c y Time Figure 4: Graphical Construction of the Spectrogram: A spectrogram is constructed by aligning the spectra of adjacent, overlapping signal segments in the time-frequency plane. The spectrogram can be used to study the evolution of a signal's frequency components. 4 The mathematical expressions for the continuous and discrete short-time Fourier transforms mirror the graphical construction outlined above in Figure 4. F (t; Ω) = x(τ)w(t τ)e−jΩτ F (n; !) = x(k)w(n k)e−j!k (2) R − P − As an example, consider constructing the spectrograms of two chirp signals. The chirp signal c1 has linearly increasing frequency, while the chirp signal c2 has quadratically increasing frequency. c(t)1 = sin(2π(α + βt)t); (3) 2 c(t)2 = sin(2π(α + βt + γt )t); (4) As expected, the spectrogram of c1 shows that as time progresses the frequency present in signal segments increases linearly. The spectrogram of c2 shows a quadratic increase in frequency with time. Sinusoid with Linearly Varying Frequency Sinusoid With Quadraticlly Varying Frequency 500 500 450 450 400 400 350 350 300 300 250 250 Frequency 200 Frequency 200 150 150 100 100 50 50 0 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Time Time Figure 5: Spectrogram of Chirp Signals With Linearly and Quadratically Increasing Fre- quency The window w(t) determines the spectral and temporal resolution of the Short-Time Fourier Transform.
Details
-
File Typepdf
-
Upload Time-
-
Content LanguagesEnglish
-
Upload UserAnonymous/Not logged-in
-
File Pages29 Page
-
File Size-