Chapter 5
Wavelets - an introduction
A wavelet based approach will be used in the following chapter to study respiratory sinus arhythmia during sleep. The present chapter presents a short introduction to wavelet theory.
Decomposition of signals into their frequency components has long been a domain of study in physics and mathematics. Different methods exist to represent a time domain signal or function in the frequency domain, with the Fourier transform being the most commonly used and also a good starting point to introduce and understand wavelets. This chapter will thus start by a short introduction to Fourier analysis, to windowed Fourier transform and to the limitations of this approach, followed by an introduction to wavelets. A comparison between windowed Fourier transform and wavelets from a theoretical point of view concludes the chapter. This chapter is loosely based on [1] and [2].
5.1 Fourier analysis
Fourier analysis is based on the fact that the set of complex functions{eiwt}constitutes an orthonormal basis of the space of square-integrable (L2) functions. Any periodic square- integrable functions(t) can be thus represented as a series of harmonically related sinusoids {eiwt}, and the function is fully determined by the eigenvalues !s(w) associated with each basis function.
5.1.1 Continuous and discrete Fourier transform
An infinite, square-integrable continuous function s(t), can be expressed by its Fourier trans- forms(w), defined by:!
!s(w) =" +∞
−∞
s(t)e−iwtdt (5.1)
For eachw, !s(w) is a complex number that represents the amplitude and phase of the corresponding basis functioneiwtin the development ofs(t) as a linear combination of these functions. The fourier transform!s(w) depends on the values taken bys(t) at all times, for it is calculated usings(t) from−∞to +∞. This mixing of events from different times results in a total loss of time resolution, not allowing for the analysis of local properties ofs(t) from
!s(w).
167
The original function s(t) can be reconstructed from its Fourier transform, using the inverse operation, defined as:
s(t) = 1 2π
" +∞
−∞ s(w)e! iwtdw (5.2)
One important property of Fourier transforms is expressed by the Parseval theorem (also known as the energy theorem):
E=" +∞
−∞ |s(t)|2dt= 1 2π
" +∞
−∞ |!s(w)|2dw (5.3)
with E being called the energy of the signal. The theorem states that the energy of the functions(t) can be calculated either using its time or frequency representation.
So far only continuous infinite signals have been discussed. In practice, however, the signals studied are finite in length, are usually sampled at regular time intervals Ts, and often contain discontinuities.
Sampling Extension of Fourier analysis to regularly sampled signals is easily obtained by considering a sampling functionc(t):
c(t) =
+∞#
n=−∞
δ(t−nTs) (5.4)
Where δis the Dirac function, and c(t) is called a Dirac comb. The Fourier transform ofc(t) is given by:
!c(w) =
+∞#
n=−∞
e−inTsw (5.5)
Convolution of c(t) with a continuous signal s(t), yields a discrete signal sd(t), evenly sampled every Ts seconds. In signal processing it is common use to speak of the sampling frequency (fs) which corresponds to T1
s, and is measured in Hz.
sd(t) =s(t)c(t) =s(t)
+∞#
n=−∞
δ(t−nTs) =
+∞#
n=−∞
s(t)δ(t−nTs) (5.6) withnthe integer index representing thenthsample.
The Fourier transform of the discrete signal sd(t), obtained by sampling the continuous signals(t) at evenly sampledTsis then:
! sd(w) =
+∞
#
n=−∞
s(nTs)e−iwTsn= 1 Ts
+∞
#
k=−∞
!s(w−2kπ Ts
) (5.7)
where n is the integer index representing thenth time sample, and kthe integer index of thekthobservable frequency.
Sampling a continuous infinite signal every Ts seconds (tn =nTs), allows to determine its Fourier transform with a 2πT
s periodicity (fk =k2πTs).
5.1. FOURIER ANALYSIS 169 Finite signals In practice, signals are not sampled over an infinite time, only sampled for a certain period of time T =N Ts. In practicesd(nTs) is thus known only for 0 ≤n < N. Fourier analysis can be extended to finite discrete signals, and this is done by introducing circular convolution, that takes into account the border effects at n = 0 and n =N −1.
Circular convolution assumes the signal has periodicityN, and thatsd(0)≈sd(N Ts).
The family of discrete complex exponentials of the type ek = ei2πknN with 0≤ k < N constitutes an orthogonal basis of the N dimensional space of functions with periodicity N.
The discrete Fourier transform (DFT) is then defined as
! sd(k) =
N#−1
n=0
sd(n)e−i2πknN (5.8)
with the inverse DFT defined as
sd(n) = 1 N
N#−1
k=0
!
sd(k)ei2πknN (5.9)
It must be stated thats!d(n) constitutes an approximation to the true Fourier transform of the underlying continuous functions(t). One important condition for this approximation to be valid is that the maximal frequency (wM) present ins(t) must be lower than 2T2πs. This condition is equivalent to imposing that s(t) has no brutal variation between consecutive samples, and can thus be recovered from smooth interpolation. When this condition is not met, the high frequency componentswh, wh > Tπs not satisfying this condition will appear in the calculated spectrum at frequencies $2π
Ts −wh%, distorting the real spectrum in the observable frequency interval.
Time and frequency resolution Applying the DFT to a finite signal decomposes the signal into its frequency components. The time resolution of this analysis equals the time duration of the signal ∆t=N Ts, for the entire signal is used to calculate frequency compo- nents.
The DFT decomposes the signal into N discrete frequencies from [−Tπs,+Tπ
s]. Therefore, the frequency step depends not only on the sampling intervalTs, but also on N, the number of samples taken, for ∆f = N T1
s. The longer the sampling time (higher number of samples), the smaller the increment in frequency. A better frequency resolution is thus attained by looking at the signal for a longer time; of course this is obtained at the expenses of a lower time resolution. The maximal observable frequency (also called the Nyquist frequency fn =2T1s), depends only on the sampling intervalTs. The shorter the sampling interval, the higher this cut-off frequency.
One can thus state that the DFT maps an evenly sampled signal, sampled N times at intervals of Ts seconds, into N discrete frequency components (N/2 are the complex conjugates of the other half), with a frequency step of ∆f = N T1s, limited by the Nyquist frequencyfn.
The fact thatek,0 ≤k < N constitutes an orthogonal basis of the space, imply that the Parseval theorem (equation 5.3) remains valid and can be expressed for discrete, finite signals as:
||s||2=
N#−1
n=0
|sd(n)|2= 1 N
N#−1
k=0
|!sd(k)|2 (5.10)
Equation 5.8 constitutes the practical basis of Fourier signal processing. Faster algo- rithms, such as the Fast Fourier Transform (FFT), have been developed, allowing for a decrease in the computational complexity, with the corresponding decrease in calculating time. The most often used FFT algorithm is the Cooley-Tukey algorithm, which minimizes the calculation complexity by taking as inputs signals whose N is a power of 2.
5.1.2 Windowed Fourier analysis
The DFT applies only to signals which are stationary: signals whose statistical properties remain constant in time. Most phenomena, however, are not stationary, and quite often there is a strong interest in studying changes in the statistical properties of a time series, also in the frequency domain. Signals that are only locally stationary, can be studied by separating the different regimes, and studying each regime independently.
Moreover, as mentioned in 5.1.1, changing from the time to the frequency representation provokes the loss of local time properties of the original signal. All signal is used to compute the frequency content, and the information on when a particular frequency occurred is lost in the process.
It is possible to include time localization and study locally stationary phenomena, by restricting Fourier analysis to local sections of the signal. This is the principle of windowed Fourier analysis. The framework of windowed Fourier was introduced by Gabor’s seminal work on communication theory in 1946 ([3]).
In order to increase the localization in time, the simplest method consists in “cutting”
s(t) into portions, and perform Fourier analysis in these chopped sections of the signal.
Analyzing signals using localized portions allows the extension of Fourier transform to finite signals and also the decomposition of long, non-stationary signals into smaller portions that are locally stationary. This results in increasing the time-localization, at the expenses of reducing the frequency resolution.
Generically, a window function g(t) can be any function that vanishes outside a given interval. The simplest window is a constant functiong= 1 in the interval [−N, N] and zero everywhere else. The product of such a window with the infinite signal reduces the signal to zero everywhere except in the [−N, N] interval, where it remains identical. Such windows is called a rectangular window and corresponds to cutting the signal and looking only at local portions of it. However, discontinuous windows such as the rectangular one are of limited utility, and continuous symmetric real functions, such as those depicted in figure 5.1 are most often used.
A generic window function g(t) can be translated byu, to cover a different part of the signal and modulated at a given frequencyξ, and is then expressed as:
gu,ξ(t) =eiξtg(t−u) (5.11)
Real symmetrical window functions, have Fourier transforms that are also real and sym- metrical and are defined by:
!
gu,ξ(w) =e−iu(w−ξ)!g(w−ξ) (5.12) The Fourier transform of a window presents a main lobe centered atw= 0, and decays towards zero, with oscillations. As the window function is limited in time, its Fourier transform has necessarily an infinite support, extending through the entire frequency range from [−∞,+∞]. This is depicted in figure 5.2 for a generic window.
5.1. FOURIER ANALYSIS 171
!!"# ! !"#
!
!"$
!"%
!"&
!"' (
)*++,-.
!!"# ! !"#
!
!"$
!"%
!"&
!"' (
/*011,*-
!!"# ! !"#
!
!"$
!"%
!"&
!"' (
)*--,-.
!!"# ! !"#
!
!"$
!"%
!"&
!"' (
23*45+*-
Figure 5.1: Two types of windowing functionsg(t) used regularly: the Hamming and Gaus- sian windows, presented here with [−12,−12] support (figure adapted from [1]).
Adding a normalization factor so that ||gu,ξ|| = 1 for any (u, ξ) pair, and applying the window to s(t), the windowed signal sw(t) is obtained. Its Fourier transform is expressed by:
!
su,ξ=" +∞
−∞
s(t)g(t−u)e−iξtdt (5.13)
which is also called theshort time Fourier transform (STFT), for the use of a window localizes the Fourier integral aroundt=u.
The energy density function, also calledspectrogram (Ps), is then defined as:
Ps(u, ξ) =|!su,ξ|2=
&
&
&
&
" +∞
−∞
s(t)g(t−u)e−iξtdt
&
&
&
&
2
(5.14) FunctionPs(u, ξ) measures the energy of signals(t) in the time-frequency neighborhood of (u, ξ), neighborhood defined by the window functiongu,ξ.
!"
^
"
"
#"! !
$ $
"
g( )
Figure 5.2: Fourier transform of a generic windowg(t). The energy spread of !gis measured as ∆ω. The amplitudeAof the first side lobes situated at±ω0is a measure of how fast the energy decays towards zero (from [1]).
5.1.3 Limitations of Fourier analysis
From equations 5.3 (infinite signals) or 5.10 (finite, sampled signals), probability density functions for time and frequency can be defined respectively as:
P(t) = 1
||f||2|s(t)|2 and P(w) = 1
2π||f||2|!s(w)|2 (5.15) From the respective probability density functions, the expected mean values of time and frequency, and their corresponding variances σt, σw can be calculated. Using the Cauchy- Schwarz theorem (for a complete demonstration the reader is referred to [1], pages 32-34), one can reach what is usually called the uncertainty principle:
σ2tσw2 ≥ 1
4 (5.16)
Equality in the previous equation is only attained withs(t) a gaussian function function (s(t) =ae−bt2).
This equation establishes an important limitation of Fourier analysis: the precise deter- mination of time and frequency are fundamentally incompatible. In other words, frequency cannot be measured instantaneously, it requires a certain amount of incertitude in time, and vice-versa, time of a specific event cannot be determined without a loss in frequency resolution.
In fact, in order to reduce the frequency variance σw around a determined frequency w, then the signal must be observed for a given time interval ∆t≥w1. Increasing the time interval will increase the precision of the determination ofw, but at the expense of decreasing the time localization. To precisely know a frequency, one can no longer know when the signal presented that specific frequency. The inverse is also true: to improve localization in time, a shorter time window is needed, but a shorter time window implies lower precision in the determination of frequency.
Using windows allows to localize the energy of the Fourier transform in time. However, the uncertainty principle imposes a bound between the time and frequency localization, for more precision in time or frequency corresponds to less variance in the distribution. The product of the square of time and frequency variances can never be smaller than 14. One must thus be aware that increasing the precision in time results in a decrease precision in frequency and vice-versa. The time-frequency localization can thus be seen as a square surface in a two dimensional time-frequency space, called the Heisenberg box (figure 5.3), with a lower limit for its surface established by equation 5.16.
This also implies that no global optimal window exist. There are many ways of windowing a signal, and each windowing procedure yields a different perspective to the analysis. The windowing must be tailored for the specific signal and the goal pursued.
Choice of window Different choices of windows will result in different compromises on the σ2tσw2 product. Only gaussian windows attain the equality on this relation, all other window functions will represent, from this point of view, a poorer compromise.
For any given window, theσ2tσ2wproduct will be constant, and independent of uand ξ.
In a two dimensional representation of time and frequency, this corresponds, to the surface of a rectangle, of sidesσtandσw. The surface of the rectangle is thus fixed, but a compromise between both variances can still be fine tuned.
5.1. FOURIER ANALYSIS 173
u,!!!
|g (t) |
!"
|g (t) |v
|g |
"
!
0 t
#
"##
"##
u,!!
v
$
$
$
$
t
t
#
#
u v
!"
^
^
|g |
Figure 5.3: Time-frequency representation of the effect of using a window g(t). σtand σw
associated with g(t) determine what is called an Heisenberg box. The surface of these boxes presents a lower limit established by the uncertainty principle, but different compromises can be attained between the time and frequency variance. Once this compromise is set, it is the same for all times (u and v in the figure) and for all frequencies (ξandγ) (from [1]).
However the time-frequency limitation is not the only consequence of applying a window to a signal. If it was the case, gaussian windows would be the optimal choice.
The Fourier transform of a windowed signal corresponds to the convolution of the Fourier transform of the signal with the Fourier transform of the window function.
!
su,ξ=!s(w)∗!gu,ξ(w) (5.17)
Therefore, applying a window distorts the spectral components obtained. The Fourier transform of a window function presents a central peak, whose narrowness is directly re- lated to the frequency resolution, but !g inevitably spreads through the entire frequency range, though tending towards zero. A compromise has to be attained between the desired narrowness of the peak around w= ξ and the rate of decline towards zero of the spectra forw'=ξ. For example, a gaussian window has a larger central peak than the rectangular window (smaller frequency resolution), but decays much faster towards zero (less distortion).
Many different windows can be applied to a signal, and the choice of window has to be performed in function of the signal being analyzed and the goal pursued. Different windows will present narrower or wider central peaks, and different rates of decay towards zero.
The most common used window in analyzing heart-rate variability data is the Hamming window, depicted in the left panel of figure 5.1, and defined as
h(n) =
' w(n) = 0.53836−0.46164 cos(
2πn N−1
) , if 0≤n≤N−1
0 , otherwise
The Fourier transform of the discrete Hamming windowh(n) presents a main lobe with a half-height width ∆ωof 8πN, and secondary lobes with a−43dBattenuation in relation to the main lobe (figure 5.2).
Window scale As mentioned above, the time and frequency resolution of the windowed Fourier transform depend on the spread of the window in time and frequency, which deter- mines the uncertainty productσtσw(the area of the Heisenberg box, figure 5.3).
It is possible to spread a window in time, and thus increase its frequency localization and vice-versa. Window g(t), presenting time and frequency resolution σt and σw can be scaled using a dilation constantd, by lettinggd(t) =√1
dg(dt). Dilation changes the time and frequency resolutions, which become respectivelydσtand σdw
It is important to repeat that for any given window, the area of the Heisenberg box is not altered, but the rectangle is dilated in time by a factord, and compressed in frequency by the same factor. The setting of a parameterddepends on the desired trade-off between time and frequency resolutions. For Fourier transform, the trade off established is everywhere the same, for all times and frequencies. Once the parameterdis set, it applies to the entire range of times and frequencies, as is illustrated in the example of figure 5.3. This constitutes the major practical difference between windowed Fourier analysis and wavelet transforms, and is discussed in detail in the next section (subsection 5.2.3).
5.2 Continuous wavelet transform
Wavelets were first introduced by Grossman and Morlet in 1984 ([4]) to analyze signal structures of very different scales, in the framework of seismic signals. It is a multi-resolution analysis, the time-frequency resolutions of wavelets are different at different positions in the time-frequency surface, unlike what happens with the STFT. The Heisenberg boxes corresponding to wavelets are rectangles, but the time resolution is low for low frequencies, which are determined with a better frequency resolution; on the other hand, for higher frequencies, the time resolution is improved at the cost of lowering the frequency resolution.
This can be seen in figure 5.5, and is fundamentally different from the windowed Fourier approach, depicted in figure 5.3.
5.2.1 Mother wavelet
The wavelet transform decomposes a signals(t) by projecting it into a family of functions ψd,u(t), all derived from the same original function, called the mother wavelet. There are many different types of mother wavelets (Mexican hat, Morlet, Meyer, Haar, Daubechies, etc) presenting different forms and specificities. Wavelets are oscillations localized in time, which means their energy is also localized in time.
The ensemble of basis functions necessary to represent the signal s(t) in the frequency domain is constructed by translating the mother wavelet in time (factoru) and using a scale factor d that dilates or contracts the mother wavelet. In order to cover the whole time axis, the wavelet is displaced by u, which allows the determination of the local energy of signal. The scale factor, by dilating or compressing the wavelet will result in a change in the underlying frequency, and thus analyze details present at different scales.
5.2. CONTINUOUS WAVELET TRANSFORM 175
!! " !
!"#!
"
"#!
$
!! " !
"
"#!
$
Figure 5.4: Mexican hat mother wavelet (figure represents$#! −Ψ(t)) (from [1]).
A generic waveletψ(t) must satisfy the following criteria:
1. ψ(t) must belong to the space of square integrable functionsL2: E=" +∞
−∞ |ψ(t)|2dt <∞ (5.18) it is usually normalized so that its energyE= 1.
2. ψ(t) must satisfy the admissibility criteria
Cg=" +∞ 0
|ψ(t)|2
w dw <∞ (5.19)
This implies thatψ(t) has no DC component, which is the same as writing thatψ(t) has zero average:
" +∞
−∞
ψ(t)dt= 0 (5.20)
Real and analytical wavelets exist. Analytical wavelets are required to follow the time evolution of a signal. Analytical wavelets allow for the separate determination of amplitude and phase information contained in a signal, and are therefore the type of wavelets used in the next chapter. A function is said to be analytical if its Fourier transform is zero for negative frequencies. An analytical function is necessarily complex, and thus presents a real and imaginary part. In the following ψ∗(t) denotes the complex conjugate of the function ψ(t). The remaining of this chapter will focus on analytical wavelets.
5.2.2 Scale factors
A family of time-frequency wavelets is obtained by scalingψ(t) by a factord, and translating it in time by factoru, withd, u∈ )andd >0:
ψu,d(t) = 1
√dψ(t−u d
) (5.21)
If the mother wavelet is normalized (||ψ(t)|| = 1) and centered in the neighborhood of t= 0, the functionsψu,d(t) remain normalized,||ψu,d||= 1, and centered aroundu.
Translation of the wavelet by u allows it to analyze s(t) locally, in the neighborhood of timeu. The scale factor d, dilates or contracts the original wave. For d <1, the wave is dilated: it then covers a larger interval in time (decrease in time-resolution), the center frequency of the resulting wavelet is decreased, and the frequency resolution increased.
On the other hand, when d > 1, the wavelet covers a smaller time interval (better time resolution), at the expense of a worse frequency resolution, concomitant with an increase in the underlying frequency of the wavelet. This is illustrated in figure 5.5.
The continuous wavelet transform (CWT) of signal s(t) at time u and scale d can be calculated using:
CW T(u, d) =" +∞
−∞
s(t) 1
√dψ∗(t−u d
)dt (5.22)
5.2.3 Time and frequency resolution
If the mother wavelet is centered aroundt= 0, the displacement byuwill result in a wavelet centered aroundu. For a wavelet dilated by factord, its time variance isσ2tu,d =d2σ2t, with σt the time variance of the mother wavelet.
Denoting the center frequency of ψu,d(t) as ξ, and the center frequency of the mother wavelet byξ0the relationship between the two and the dilation factordis expressed byξ=
ξ0
d. In the two dimensional time-frequency representation, ψu,d(t) is thus centered around (u,ξd0), and presents a variance around this point of σtu,d = dσt (time), and σwu,d = σdw (frequency).
The main difference between STFT and CWT results from the dilation of a mother wavelet. Wavelets are adaptive, they correspond to the scale of what is being analyzed.
They present a large time base for analyzing the low frequency components (large scale), and present a better time resolution for analyzing phenomena that are more transitory (small scale). This is graphically represented in the comparison between figures 5.3 and 5.5
From the continuous wavelet transform, and taking into account the previous paraghraph the time frequency energy can be calculated for every (u, d) pair:
Pwf(u, ξ) =|CW T(u, d)|2 (5.23) This energy density function is called ascalogram. Pwf(u, ξ) is proportional to the energy ofs(t) aroundt=u, around the frequencyw= wd0, neighborhood defined by the variances σtu,d, andσwu,d.
The scalogram can be integrated alonguandd, yielding the total energy of the signal.
Morlet wavelet In the work presented in the following chapter, the Morlet wavelet was selected. The Morlet wavelet is an analytical wavelet, resulting from the product of a complex exponential and a gaussian envelope. It can be expressed by [5]:
ψ(t) =π−1/4
*
eiw0t−e−(w0)
2 2
+
e−12t2. (5.24)
5.2. CONTINUOUS WAVELET TRANSFORM 177
0!t
s
!"
s
s!t
!"
s0
u ,s0 0
u ,s0 0
#
$
0
"
u u0 t
#u,s u,s
s0 s
!#"""""#"$!
!#"""#"$!^
^
$
Figure 5.5: Time-frequency representation of wavelet analysis. σt andσw associated with the scaling factor ‘s’ determine different Heisenberg boxes at different frequencies. The boxes have the same surface, but different time-frequency compromises are established for different frequencies. Contrary to STFT, CWT produces different time and frequency resolutions at different frequencies: higher frequencies are discriminated with a better time resolution and lower frequency resolution, while lower frequencies are less finely determined in time, but present a better frequency resolution (from [1]).
This mother wavelet is then dilated and translated, as discussed above. The parameter w0 is the central frequency of the mother wavelet. It determines the number of significant oscillations contained inside the gaussian envelopee−12t2. The Morlet wavelet is not, in the strict sense, a localized wavelet for it extends indefinitely. However, the amplitude of the oscillations becomes quickly negligible. As the Morlet wavelet is contained in a Gaussian envelope, it minimizes the uncertainty inequality,σt2σt2= 14.
The choice of the central frequency (w0) of the mother wavelet has a direct impact on the time and frequency resolution. Multiplying w0 by a factork, increases the time resolution at all frequencies by the same factor, and decreases the frequency resolution by 1/k. The choice of the central frequency of the mother wavelet is a compromise between time and frequency resolution desired. When the goal of the algorithm being implemented is clear, a compromise can be established.
5.3 Comparison between wavelet and windowed Fourier analysis
Windowed Fourier analysis (STFT) is used is many fields, from signal analysis to de-noising, from compression algorithms to clinical studies. In some biomedical approaches however, the stationarity condition for the application of this method is not met, sometimes it is even the non-stationarity (transients, changes in operational point) that constitutes the main interest of the analysis itself.
As discussed in sections 5.1.2 and 5.1.3, STFT analysis is implemented by applying consecutively the chosen window to the signal at different timesuand frequenciesξ, with or without overlap. The consecutive application allows to cover the entire 2 dimensional time- frequency surface, thus allow the determination of the energy contained by the signal at all times and frequencies. Each time-frequency determination corresponds to a Heisenberg box, centered at the time and frequency point (u, ξ). For windowed Fourier, the dimensions of the Heisenberg boxes are the same for each point where it is calculated (independent of the values of (u, ξ)). The size of the box depends on the length and time-frequency spread of the windowg. The plan is thus covered by translating one Heisenberg box over a uniform grid, covering the surface with identical rectangles.
This constant time-frequency resolution, identical for all frequencies, imposes a com- promise: a low time resolution is required to analyze low frequencies, while a higher time resolution could be used for the higher frequencies. The compromise is often performed in light of what is being pursued: for example, if the problem requires the determination of high frequencies with a high time resolution, a short window is chosen, resulting in a poor frequency resolution for the low frequencies contained in the signal. If low frequencies are not relevant for the analysis at hand, then an optimal solution has been attained. The com- promise is not trivial and requires a trade-off when the signal contains two or more ranges os frequency of interest.
This is the case with the analysis of the cardio-respiratory interactions. The heart rate time series comprises oscillations in three different frequency bands (chapter 1, section 1.4.3), and in this case the compromise for implementing STFT is not trivial. A good time resolution for the high frequency (HF) band ([0.15; 0.4] Hz) will result in a poor frequency resolution in the low frequency (LF) band ([0.04; 0.15] Hz); on the other hand, if one chooses to implement a high frequency resolution in the LF band, the time resolution in the HF band is necessarily poor.
As presented in sections 5.2.2 and 5.2.3 wavelets cover the time-frequency plan using different Heisenberg box dimensions at different frequency scales. The uncertainty rectangle is adapted to the local characteristics of the signal: high frequencies are analyzed with a higher time resolution and a lower frequency resolution, while low frequencies are analyzed with a better frequency resolution but a lower time resolution. This is depicted in figure 5.6.
In heart rate time series analysis, the poorer frequency resolution at higher frequencies is tolerable, for usually only one spectral peak is present in this frequency band (usually corresponding to the respiratory frequency, section 1.4.3), while a higher time resolution is certainly very important in order to follow transients and fast changes. On the other hand, the LF band is analyzed with a better frequency resolution, which is important to distinguish LF from very low frequency oscillations, and also to distinguish LF from HF oscillations close to the established 0.15 Hz border between LF and HF ([6]). The precise
5.3. WAVELET COMPARED TO WINDOWED FOURIER ANALYSIS 179
(t)
!
t
t
"j,n "j+1,p(t)
Figure 5.6: Coverage of the time-frequency surface using wavelets. Higher frequencies are analyzed with better time resolution, and lower frequency resolution, while lower frequencies analysis is performed at higher frequency resolution, but lower time resolution (from [1]).
values of time and frequency resolutions at each scale can be set, for the Morlet wavelet, by the parameterω0(equation 5.24).
In conclusion, wavelet based analysis of heart rate time series presents several advantages compared to STFT, the major being that it offers a multi-resolution approach, as discussed in the previous paragraphs, thus allowing a better compromise between time and frequency resolution at different scales. It is also applicable to the analysis of non stationary signals ([7]), allowing the study of transients and sudden changes. Wavelet analysis is not limited to sinusoidal functions, it can use different mother wavelets corresponding more closely to the signal being analyzed.
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Chapter 6
A method for the analysis of respiratory sinus arrhythmia using continuous wavelet
transforms
This chapter is based on the article of the same title, by Laurence Cnockaert, Pierre-Fran¸cois Migeotte, Lise Daubigny, G. Kim Prisk, Francis Grenez and Rui Carlos S´a, published in IEEE Transactions on Biomedical Engineering, Vol. 55, No. 5, May 2008.
6.1 Abstract
A continuous wavelet transform-based method is presented to study the non-stationary strength and phase delay of the respiratory sinus arrhythmia (RSA). RSA is the cyclic vari- ation of instantaneous heart rate at the breathing frequency. In studies of cardio-respiratory interaction during sleep, paced breathing or postural changes, low respiratory frequencies and fast changes can occur. Comparison on synthetic data presented here shows that the proposed method outperforms traditional short-time Fourier-transform analysis in these conditions. On the one hand, wavelet analysis presents a sufficient frequency-resolution to handle low respiratory frequencies, for which time-frames should be long in Fourier-based analysis. On the other hand, it is able to track fast variations of the signals in both amplitude and phase, for which time-frames should be short in Fourier-based analysis. A continuous wavelet transform-based analysis presents thus a better compromise for the study of respi- ratory sinus arrhythmia during sleep.
6.2 Introduction
The heart rate variability (HRV) is traditionally divided in very low frequency (VLF), low frequency (LF), and high frequency (HF) components, the frequency bands of which are respectively [0.003,0.04] Hz, [0.04,0.15] Hz, and [0.15,0.4] Hz ([1]). In the HF band, an os- cillation can usually be observed at the breathing frequency. It is called the respiratory sinus arrhythmia (RSA). The study of RSA has produced extensive literature both concerning its
183
causes ([2], [3]), and the methodology to determine it ([1]). It is generally accepted that RSA amplitude is a non-invasive marker of the activity of the parasympathetic nervous system ([3], [4]) and can therefore be used to infer relative changes in parasympathetic cardiac tone.
RSA phase delay is a measure of the time delay between respiratory cycles and RSA. It is an indirect, non-invasive measure of the integration time of the cardio-respiratory interaction, and can be estimated by frequency ([5], [6]) as well as time domain methods ([7]).
Dynamic analysis of HRV is traditionally performed by means of short-time Fourier transform (STFT), following established guidelines ([1]). However, STFT analysis is limited by its time-frequency resolution trade-off: a long window gives a better frequency-resolution, and a short window a better time-resolution. The window-length optimization is difficult for short or non-stationary time series. In these cases, outputs are averaged over different states or conditions, and transients blurred. Therefore, non-stationary spectral methods have been implemented, based on time-variant autoregressive models ([8]), Wigner-Ville Distribution ([8], [9]), selective discrete Fourier transform ([10]), or discrete or continuous wavelet transform, using various mother wavelets (e.g., Daubechies 4 [11], harmonic wavelet [12], Gaussian wavelet [13], windowed sine and cosine [14]).
The aim of this chapter is to present an analysis method of the cardio-respiratory inter- action based on continuous wavelet transforms (CWT). It is compared to STFT analysis on synthetic data, with a view to handle low breathing frequencies and fast variations. This is of particular relevance in the study of RSA dynamics during sleep, where respiratory frequen- cies below 0.2 Hz (e.g [15]) occur simultaneously to fast variations in RSA, due to changes in sleep stage and corresponding sympatho-vagal balance ([16]). Sudden RSA changes, ac- companied by low breathing frequencies, are also present in paced breathing protocols ([7]), and postural changes (tilt) experiments ([8], [10]).
6.3 Methods
6.3.1 Algorithm
The algorithm calculates the gain and phase delay of the RSA, based on CWTs. The analysis is limited to time-intervals where the estimated respiratory frequency and the estimated frequency of the main peak in the HF band of HRV are equal within±0.02 Hz.
The overall algorithm is schematically described in figure 6.1. It consists of two blocks, the first performing the necessary data preprocessing of the respiratory and heart-rate time series and the second using wavelets to decompose the preprocessed signals into their fre- quency components, calculating the gain and phase delay between the two signals. Not shown in the schema are a set of exclusion rules, that eliminate unwanted portions of the signal where noise and other artefacts are present. The algorithm is described in detail below.
Preprocessing
TheRR-interval time series (the instantaneous heart period) is calculated by detecting the occurrence of the R-peak in the electrocardiogram. Premature ventricular contractions, ectopic beats, as well as occasional detection errors are removed from the RRtime series, which is then interpolated using cubic splines and resampled at 8 Hz, in order to obtain a uniformly sampledRR-interval time series (U RRI).
The respiratory volume signal, recorded at 32 Hz, was downsampled to 8 Hz, after low- pass filtering by a fifth order Butterworth filter, with a 4 Hz cut-off frequency. Forward and
6.3. METHODS 185
Figure 6.1: Schematic view of the algorithm. The algorithm consists of a preprocessing block and the main block. Preprocessing consist in low-pass filtering and down-sampling (to 8 Hz) the respiratory signal, detecting R-peaks in theECGsignal, and building a uniformly sampled (8 Hz) instantaneous heart beat interval time series (U RRI). CW T is applied to these signals. The respiratory frequency and the frequency component of heart rate variability are determined, and gain and phase delay computed.
backward filtering was performed to obtain a zero phase lag.
Continuous wavelet transform analysis The time-dependent power spectra of the respiratory and HRV signal are calculated by means of CWTs.
The continuous wavelet transform of a signalx(t) is defined as CW T(t, λ) =" +∞
−∞
x(u) 1
√λψ∗
,u−t
λ -
du, (6.1)
whereψ(t) is the mother wavelet, andCW T(λ, t) the wavelet transform coefficient for scaleλat timet. The time- and frequency-resolution of the waveletψλ(t) are defined as 4σt
and 4σf, respectively. σtandσf are the standard deviation of|ψλ(t)|2and&&&ψˆλ(f)&&&
2, where ψλ(t) =√1
λψ$u−t
λ
%andˆsymbolizes the Fourier transform.
The amplitude and phase of the complex CWT coefficients obtained using an analytical mother wavelet are estimates of the envelope and instantaneous phase of the spectral com- ponents of the signal in the frequency-band centered on the central frequency of the wavelet ([17]). Here, the complex Morlet wavelet was selected because it is a Gaussian-shaped ana- lytical wavelet. This shape optimizes the product of the time- and frequency-resolutions of the wavelet. Moreover, the analytical property of the wavelet allows the determination of phase information.
