ch056 Wavelet

Thakor, N.V., Gramatikov, B., Sherman, D. “Wavelet (Time-Scale) Analysis in Biomedical Signal Processing.”The Biomedical Engineering Handbook: Second Edition.Ed. Joseph D. BronzinoBoca Raton: CRC Press LLC, 2000

56Wavelet (Time-Scale)

Analysis in BiomedicalSignal Processing

56.1 Introduction56.2 The Wavelet Transform: Variable Time and

Frequency ResolutionThe Continuous Wavelet Transform • The Discrete Wavelet Transform

56.3 A Multiresolution Theory: Decomposition of Signals Using Orthogonal WaveletsImplementation of the Multiresolution Wavelet Transorm: Analysis and Synthesis of Algorithms

56.4 Further Developments of the Wavelet Transform56.5 Applications

Cardiac Signal Processing • Neurological Signal Processing • Other Applications

Signals recorded from the human body provide information pertaining to its organs. Their characteristicshape, or temporal and spectral properties, can be correlated with normal or pathological functions. Inresponse to dynamical changes in the function of these organs, the signals may exhibit time-varying aswell as nonstationary responses. Time-frequency and time-scale analysis techniques are well suited forsuch biological signals. Joint time-frequency signal analysis techniques include short-term Fourier trans-form and Wigner-Ville distribution, and related reduced interference distribution. Joint time-scaleanalysis includes continuous and discrete, orthonormal and non-orthonormal wavelets. These techniquesfind applications in the analysis of transient and time-varying events in biological signals. Examples ofapplications include cardiac signals (for detection of ischemia and reperfusion-related changes in QRScomplex, and late potentials in ECG) and neurological signals (evoked potentials and seizure spikes).

56.1 Introduction

Digital signal processing uses sophisticated mathematical analysis and algorithms to extract informationhidden in signals derived from sensors. In biomedical applications these sensors such as electrodes,accelerometers, optical imagers, etc., record signals from biological tissue with the goal of revealing theirhealth and well-being in clinical and research settings. Refining those signal processing algorithms forbiological applications requires building suitable signal models to capture signal features and componentsthat are of diagnostic importance. As most signals of a biological origin are time-varying, there is a specialneed for capturing transient phenomena in both healthy and chronically ill states.

Nitish V. ThakorJohns Hopkins School of Medicine

Boris GramatikovJohns Hopkins School of Medicine

David ShermanJohns Hopkins School of Medicine

© 2000 by CRC Press LLC

A critical feature of many biological signals is frequency domain parameters. Time localization of thesechanges is an issue for biomedical researchers who need to understand subtle frequency content changesover time. Certainly, signals marking the transition from severe normative to diseased states of anorganism sometimes undergo severe changes which can easily be detected using methods such as theShort Time Fourier Transform (STFT) for deterministic signals and its companion, the spectrogram, forpower signals. The basis function for the STFT is a complex sinusoid, e j2πft, which is suitable for stationaryanalyses of narrowband signals. For signals of a biological origin, the sinusoid may not be a suitableanalysis signal. Biological signals are often spread out over wide areas of the frequency spectrum. Alsoas Rioul and Vetterli [1] point out, when the frequency content of a signal changes in a rapid fashion,the frequency content becomes smeared over the entire frequency spectrum as it does in the case of theonset of seizure spikes in epilepsy or a fibrillating heartbeat as revealed on an ECG. The use of anarrowband basis function does not accurately represent wideband signals. It is preferred that the basisfunctions be similar to the signal under study. In fact, for a compact representation using as few basisfunctions as possible, it is desirable to use basis functions that have a wider frequency spread as mostbiological signals do. Wavelet theory, which provides for wideband representation of signals [2-4], istherefore a natural choice for biomedical engineers involved in signal processing and currently underintense study [5-9].

56.2 The Wavelet Transform: Variable Time and Frequency Resolution. Continuous Wavelet Transform (CWT)

A decomposition of a signal, based on a wider frequency mapping and consequently better time resolutionis possible with the wavelet transform. The Continuous Wavelet Transform (CWT) [3] is defined thuslyfor a continuous signal, x(t),

(56.1a)

or with change of variable as

(56.1b)

where g(t) is the mother or basic wavelet, * denotes a complex conjugate, a is the scale factor, and τ—atime shift. Typically, g(t) is a bandpass function centered around some center frequency, fo. Scale a allowsthe compression or expansion of g(t) [1, 3, 10]. A larger scale factor generates the same functioncompressed in time whereas a smaller scale factor generates the opposite. When the analyzing signal iscontracted in time, similar signal features or changes that occur over a smaller time window can bestudied. For the wavelet transform, the same basic wavelet is employed with only alterations in this signalarising from scale changes. Likewise, a smaller scale function enables larger time translations or delaysin the basic signal.

The notion of scale is a critical feature of the wavelet transform because of time and frequency domainreciprocity. When the scale factor, a, is enlarged, the effect on frequency is compression as the analysiswindow in the frequency domain is contracted by the amount 1/a [10]. This equal and opposite frequencydomain scaling effect can be put to advantageous use for frequency localization. Since we are usingbandpass filter functions, a center frequency change at a given scale yields wider or narrower frequencyresponse changes depending on the size of the center frequency. This is the same in the analog or digitalfiltering theories as “constant-Q or quality factor” analysis [1, 10, 11]. At a given Q or scale factor,

CWT aa

x at gt

adtx τ τ

, *( ) = ( ) −

∫1

CWT a a x at g ta

dtx τ τ, *( ) = ( ) −

∫


frequency translates are accompanied by proportional bandwidth or resolution changes. In this regard,wavelet transforms are often written with the scale factor rendered as

(56.2)

or

(56.3)

This is the equivalent to logarithmic scaling of the filter bandwidth or octave scaling of the filterbandwidth for power-of-two growth in center frequencies. Larger center frequency entails a larger band-width and vice versa.

The analyzing wavelet, g(t), should satisfy the following conditions:

(1) belong to L2 (R), i.e., be square integrable (be of finite energy) [2];(2) be analytic [G(ω) = 0 for ω < 0] and thus be complex-valued. In fact many wavelets are real-

valued; however, analytic wavelets often provide valuable phase information [3], indicative ofchanges of state, particularly in acoustics, speech, and biomedical signal processing [8]; and

(3) be admissible. This condition was shown to enable invertibility of the transform [2, 6, 12]:

where cg is a constant that depends only on g(t) and a is positive. For an analytic wavelet theconstant should be positive and convergent:

which in turn imposes an admissibility condition on g(t). For a real-valued wavelet, the integralsfrom both –∞ to 0 and 0 to +∞ should exist and be greater than zero.

The admissibility condition along with the issue of reversibility of the transformation is not socritical for applications where the emphasis is on signal analysis and feature extraction. Instead,it is often more important to use a fine sampling of both the translation and scale parameters.This introduces redundancy which is typical for the CWT, unlike for the discrete wavelet transform,which is used in its dyadic, orthogonal, and invertible form.

All admissible wavelets with g ∈ L1(R) have no zero-frequency contribution. That is, they areof zero mean,

or equivalently G(ω) = 0 for ω = 0, meaning that g(t) should not have non-zero DC [6, 12]. Thiscondition is often being applied also to nonadmissible wavelets.

af

f=

0

CWT af

f f fx t g

t

f fdtx

o o o

τ τ, * .=

= ( ) −

∫1

s tc

W aa

gt

a adad

g a

( ) = ( ) −

>

∞

−∞

∞

∫∫1 1 12

0

τ τ τ,

cG

dg =( )

< ∞∞

∫ω

ωω

2

0

g t dt( ) =−∞

+∞

∫ 0


The complex-valued Morlet’s wavelet is often selected as the choice for signal analysis using the CWT.Morlet’s wavelet [3] is defined as

(56.4a)

with its scaled version written as

(56.4b)

Morlet’s wavelet insures that the time-scale representation can be viewed as a time-frequency distributionas in Eq. (56.3). This wavelet has the best representation in both time and frequency because it is basedon the Gaussian window. The Gaussian function guarantees a minimum time-bandwidth product,providing for maximum concentration in both time and frequency domains [1]. This is the best com-promise for a simultaneous localization in both time and frequency as the Gaussian function’s Fouriertransform is simply a scaled version of its time domain function. Also the Morlet wavelet is defined byan explicit function and leads to a quasi continuous discrete version [11]. A modified version of Morlet’swavelet leads to fixed center frequency, fo, with width parameter, σ,

(56.4c)

Once again time-frequency (TF) reciprocity determines the degree of resolution available in time andfrequency domains. Choosing a small window size, σ, in the time domain, yields poor frequency reso-lution while offering excellent time resolution and vice versa [11, 13]. To satisfy the requirement foradmissibility and G(0) = 0, a correction term must be added. For ω > 5, this correction term becomesnegligibly small and can be omitted. The requirements for the wavelet to be analytic and of zero meanis best satisfied for ω0 = 5.3 [3].

Following the definition in Eq. (56.1a,b) the discrete implementation of the CWT in the time-domainis a set of bandpass filters with complex-valued coefficients, derived by dilating the basic wavelet by thescale factor, a, for each analyzing frequency. The discrete form of the filters for each a is the convolution:

with k = τ/Ts, where Ts is the sampling interval. The summation is over a number of terms, n. Becauseof the scaling factor a in the denominator of the argument of the wavelet, the wavelet has to be resampledat a sampling interval Ts/a for each scale a. Should the CWT cover a wide frequency range, a computationalproblem would arise. For example, if we wish to display the CWT over 10 octaves (a change by oneoctave corresponds to changing the frequency by a factor of 2), the computational complexity (size ofthe summation) increases by a factor of 210 = 1024. The algorithm by Holschneider et al. [14] solves thisproblem for certain classes of wavelets by replacing the need to resample the wavelet with a recursiveapplication of an interpolating filter. Since scale is a multiplicative rather than an additive parameter,another way of reducing computational complexity would be by introducing levels between octaves(voices) [15]. Voices are defined to be the scale levels between successive octaves, uniformly distributedin a multiplicative sense [13, 16]. Thus, the ratio between two successive voices is constant. For example,

g t e ej f tot( ) = π −22

2

gt

ae e

jf

ato t

a

=

π −2 2

2 2

g t e ej f to

t

σ σ,( ) = π−

2

2

2 2

S k aa

s i gi k

a as k i g

i

am

i k

k

m

in

n

n

n

,( ) = ( ) ∗ −

= −( ) ∗

= −

+

=−∑ ∑1 1

2

2

2

2


if one wishes to have ten voices per octave, then the ratio between successive voices is 21/10. The distancebetween two levels, ten voices apart is an octave.

The CWT can also be implemented in the frequency domain. Equation (56.1) may be formulated inthe frequency domain as:

(56.5)

where S(ω) and G(ω) denote the Fourier transformed s(t) and g(t), and j = (–1)1/2. The analyzing waveletg(t) generally has the following Fourier transform:

(56.6a)

The Morlet wavelet [Eq. (56.4a,b)] in frequency domain is a Gaussian function:

(56.6b)

From Eq. (56.6a,b) it can be seen that for low frequencies, ω, (larger scales a) the width, ∆ω, of theGaussian is smaller and vice versa. In fact, the ratio ∆ω/ω is constant [1], i.e., Morlet wavelets may beconsidered filter banks of the constant-Q factor.

Based on Eqs. (56.5 and 56.6a,b) the wavelet transform can be implemented in the frequency domain.At each scale, the Fourier image of the signal can be computed as

with S(ω) being the Fourier transform of the signal, Gm(ω, a) being the scaled Fourier image of theMorlet wavelet at scale a, and the operation ● standing for element-by-element multiplication (windowingin frequency domain). The signal at each scale a will finally be obtained by applying the inverse Fouriertransform:

This approach has the advantage of avoiding computationally intensive convolution of time-domainsignals by using multiplication in the frequency domain, as well as the need of resampling the motherwavelet in the time domain [17, 18].

Note that the CWT is, in the general case, a complex-valued transformation. In addition to itsmagnitude, its phase often contains valuable information pertinent to the signal being analyzed, partic-ularly in instants of transients [3]. Sometimes the TF distribution of the nonstationary signal is muchmore important. This may be obtained by means of real-valued wavelets. Alternatives to the complex-valued Morlet wavelet are simpler, real-valued wavelets that may be utilized for the purpose of the CWT.For example, the early Morlet wavelet, as used for seismic signal analysis [19], had the following real-valued form:

It had a few cycles of a sine wave tapered by a Gaussian envelope. Though computationally attractive,this idea contradicts the requirement for an analytic wavelet, i.e., its Fourier transform G(ω) = 0 for

CWT a a S G a e djτ ω ω ωτω,( ) = ( ) ∗ ( )∫

G a G a eaj

τωτω ω, ( ) = ( )

G em ωω

ω ω( ) = − −( )1

2

0

22

Y a S G amω ω ω, ,( ) = ( )• ( )

CWT a FFT Y aτ ω, ,( ) = ( ) ( )−1

g t t e t( ) = ( ) −cos 52 2


ω < 0. An analytic function is generally complex-valued in the time domain and has its real and imaginaryparts as Hilbert transforms of each other [2, 20]. This guarantees only positive-frequency componentsof the analyzing signal.

A variety of analyzing wavelets have been proposed in recent years for time-scale analysis of the ECG.For example, Senhadji et al. [21] applied a pseudo-Morlet’s wavelet to bandpass filtering to find outwhether some abnormal ECG events like extrasystoles and ischemia are mapped on specific decompo-sition levels:

with the product kf0 defining the number of oscillations of the complex part, and C representing anormalizing constant such that ||g|| = 1. The above function is a modulated complex sine wave that wouldyield complex-valued CWT including phase information. However its envelope is a cosine, rather thana Gaussian, as in the case of the complex Morlet wavelet. It is well known that strictly the Gaussianfunction (both in time and frequency domain) guarantees the smallest possible time-bandwidth productwhich means maximum concentration in the time and frequency domains [22].

The STFT has the same time-frequency resolution regardless of frequency translations. The STFT canbe written as

(56.7)

where g(t) is the time window that selects the time interval for analysis or otherwise known as thespectrum localized in time. Figure 56.1 shows comparative frequency resolution of both the STFT as wellas the wavelet transform. The STFT is often thought to be analogous to a bank of bandpass filters, eachshifted by a certain modulation frequency, fo. In fact, the Fourier transform of a signal can be interpretedas passing the signal through multiple bandpass filters with impulse response, g(t)ej2πft, and then usingcomplex demodulation to downshift the filter output. Ultimately, the STFT as a bandpass filter renditionsimply translates the same low pass filter function through the operation of modulation. The character-istics of the filter stay the same though the frequency is shifted.

Unlike the STFT, the wavelet transform implementation is not frequency independent so that higherfrequencies are studied with analysis filters with wider bandwidth. Scale changes are not equivalent to

FIGURE 56.1 Comparative frequency resolution for Short Time Fourier Transform (STFT) and Wavelet Transform(WT). Note that frequency resolution of STFT is constant across frequency spectrum. The WT has a frequencyresolution that is proportional to the center frequency of the bandpass filter.

g t C f t e t f and ki kf t( ) = + π( )( ) ≤ ∉ −{ }− π1 2 1 2 1 0 102

00cos , , ,integer

STFT f x t g t e dtjftτ τ,( ) = ( ) ∗ −( ) − π

−∞

∞

∫ 2


varying modulation frequencies that the STFT uses. The dilations and contractions of the basis functionallow for variation of time and frequency resolution instead of uniform resolution of the Fourier trans-form [15].

Both the wavelet and Fourier transform are linear Time-Frequency Representations (TFRs) for whichthe rules of superposition or linearity apply [10]. This is advantageous in cases of two or more separatesignal constituents. Linearity means that cross-terms are not generated in applying either the linear TFor time-scale operations. Aside from linear TFRs, there are quadratic TF representations which are quiteuseful in displaying energy and correlation domain information. These techniques, also described else-where in this volume include the Wigner-Ville distribution (WVD), smoothed WVD, the reduced infer-ence distribution (RID), etc. One example of the smoothed Wigner-Ville distribution is

(56.8)

where h(t), is a smoothing function. In this case the smoothing kernel for the generalized or Cohen’sclass of TFRs is

.

These methods display joint TF information in such a fashion as to display rapid changes of energy overthe entire frequency spectrum. They are not subject to variations due to window selection as in the caseof the STFT. A problematic area for these cases is the elimination of those cross-terms that are the resultof the embedded correlation.

It is to be noted that the scalogram or scaled energy representation for wavelets can be represented asa Wigner-Ville distribution as [1]

(56.9)

where

The Discrete Wavelet Transform

In the discrete TFRs both time and scale changes are discrete. Scaling for the discrete wavelet transforminvolves sampling rate changes. A larger scale corresponds to subsampling the signal. For a given numberof samples a larger time swath is covered for a larger scale. This is the basis of signal compression schemesas well [23]. Typically, a dyadic or binary scaling system is employed so that given a discrete waveletfunction, ψ(x), is scaled by values that are binary. Thus

(56.10)

where j is the scaling index and j = 0, –1, –2, …. In a dyadic scheme, subsampling is always decimation-in-time by a power of 2. Translations in time will be proportionally larger as well as for a more sizable scale.

W t f s t e s t h dj f, * *( ) = −

+

− π∫ 1

2

1

2 22τ τ τ ττ

φ τ τ δt h t,( ) =

( )

2

CWT a W u n Wu t

aan dudnx x gτ, , ,( ) = ( ) ∗ −

∫∫2

W t f x t e x t dxj f, *( ) = ∗ −

+

− π∫ 1

2

1

22τ τ ττ

ψ ψ2

2 2j t tj j( ) = ( )


It is for discrete time signals that scale and resolution are related. When the scale is increased, resolutionis lowered. Resolution is strongly related to frequency. Subsampling means lowered frequency content. Riouland Vetterli [1] use the microscope analogy to point out that smaller scale (higher resolution) helps toexplore fine details of a signal. This higher resolution is apparent with samples taken at smaller time intervals.

56.3 A Multiresolution Theory: Decomposition of Signals Using Orthogonal Wavelets

One key result of the wavelet theory is that signals can be decomposed in a series of orthogonal wavelets.This is similar to the notion of decomposing a signal in terms of discrete Fourier transform componentsor Walsh or Haar functions. Orthogonality insures a unique and complete representation of the signal.Likewise the orthogonal complement provides some measure of the error in the representation. Thedifference in terms of wavelets is that each of the orthogonal vector spaces offers component signals withvarying levels of resolution and scale. This is why Mallat [24] named his algorithm the multiresolutionsignal decomposition. Each stage of the algorithm generates wavelets with sequentially finer representationsof signal content. To achieve an orthogonal wavelet representation, a given wavelet function, φ(t), at ascaling index level equal to zero, is first dilated by the scale coefficient 2j, then translating it by 2–jn andnormalizing by gives:

The algorithm begins with an operator A2j for discrete signals that takes the projections of a signal,f(t) onto the orthonormal basis, V2j:

(56.11)

where 2j defines the level of resolution. A2j is defined as the multi-resolution operator that approximates asignal at a resolution 2j. Signals at successively lower resolutions can be obtained by repeated applicationof the operator A2j (–J ≤ j ≤ –1), where J specifies the maximum resolution, such that A2j f(x) is the closestapproximation of function f(x) at resolution 2j. Here we note that < > is simply a convolution defined thusly,

(56.12)

Here φ(x) is the impulse response of the scaling function. The Fourier transforms of these functionsare lowpass filter functions with successively smaller halfband lowpass filters. This convolution synthesizesthe coarse signal at a resolution/scaling level j :

(56.13)

Each level j generates new basis functions of the particular orthonormal basis with a given discreteapproximation. In this case, larger j provides for decreasing resolution and increasing the scale inproportional fashion for each level of the orthonormal basis. Likewise, each sequentially larger j providesfor time shift in accordance with scale changes, as mentioned above, and the convolution or inner productoperation generates the set of coefficients for the particular basis function. A set of scaling functions atdecreasing levels of resolution, j = 0, –1, –2, …, –6 is given in [25].

2 j–

2 22

− −−( )j jj t nφ

A f t f u u n t njj

jj

j

n

j2 2 22 2 2( ) = ( ) −( ) −( )− −

=−∞

∞−∑ ,φ φ

f u u n f u u n dujj j( ) −( ) = ( ) −( )− −

−∞

∞

∫,φ φ2 2 2

C f f t t nj jj

2 22= ( ) −( )−, φ


The next step in the algorithm is the expression of basis function of one level of resolution, φ2j by ata higher resolution, φ2 j+1. In the same fashion as above, an orthogonal representation of the basis V2j interms of V2 j+1 is possible, or

(56.14)

Here the coefficients are once again the inner products between the two basis functions. A means oftranslation is possible for converting the coefficients of one basis function to the coefficients of the basisfunction at a higher resolution:

(56.15)

Mallat [24] also conceives of the filter function, h(n), whose impulse response provides this conversion,namely,

(56.16)

where h(n) = 2–j–1 ⟨φ2j (u – 2–j n), φ2 j+1 (u – 2–j–1 k)⟩ and h̃(n) = h(–n) is the impulse response of theappropriate mirror filter.

Using the tools already described, Mallat [24] then proceeds to define the orthogonal complement,O2j to the vector space V2 j at resolution level j. This orthogonal complement to V2 j is the error in theapproximation of the signal in V2 j+1 by use of basis function belonging to the orthogonal complement.The basis functions of the orthogonal complement are called orthogonal wavelets, ψ(x), or simply waveletfunctions. To analyze finer details of the signal, a wavelet function derived from the scaling function isselected. The Fourier transform of this wavelet function has the shape of a bandpass filter in frequencydomain. A basic property of the function ψ is that it can be scaled according to

An orthonormal basis set of wavelet functions is formed by dilating the function ψ(x) with a coefficient2j and then translating it by 2–jn, and normalizing by . They are formed by the operation of convolvingthe scale function with the quadrature mirror filter

where G(ω) = e–jω—H (ω + π) is the quadrature mirror filter transfer response and g(n) = (–1)1–n h(1 – n)

is the corresponding impulse response function.The set of scaling and wavelet functions presented here form a duality, together resolving the temporal

signal into coarse and fine details, respectively. For a given level j then, this detail signal can once againbe represented as a set of inner products:

φ φ φ φ2

1

2 2

1

2

12 2 2 2 21 1j j j jt n u n u k t kj j j j j

k

−( ) = −( ) −( ) −( )− − − − − − − −

=−∞

∞

+ +∑ ,

C f f u t n

u n u k f u t k

j j

j j j

j

j j j

k

j

2 2

1

2 2

1

2

1

2

2 2 2 21 1

= ( ) −( ) =

−( ) −( ) ( ) −( )

−

− − − − −

=−∞

∞− −

+ +∑

,

, ,

φ

φ φ φ

C f f u t n h n k f u t kj j jj j j

k2 2

1

2

12 2 2 21= ( ) −( ) = −( ) ( ) −( )− − − − −

=−∞

∞

+∑, ˜ ,φ φ

ψ ψ2

2 2j t tj j( ) = ( )

2 j–

ψ ω ω φ ω( ) =

G

2 2


(56.17)

For a specific signal, f(x), we can employ the projection operator as before to generate the approxi-mation to this signal on the orthogonal complement. As before, the detail signal can be decomposedusing the higher resolution basis function:

(56.18)

or in terms of the synthesis filter response for the orthogonal wavelet

(56.19)

At this point, the necessary tools are here for a decomposition of a signal in terms of wavelet compo-nents, coarse and detail signals. Multiresolution wavelet description provides for the analysis of a signalinto lowpass components at each level of resolution called coarse signals through the C operators. At thesame time the detail components through the D operator provide information regarding bandpasscomponents. With each decreasing resolution level, different signal approximations are made to captureunique signal features. Procedural details for realizing this algorithm follow.

Implementation of the Multiresolution Wavelet Transform: Analysis and Synthesis of Algorithms

A diagram of the algorithm for the multiresolution wavelet decomposition algorithm is shown in Fig. 56.2.A step-by-step rendition of the analysis is as follows:

1. Start with N samples of the original signal, x(t), at resolution level j = 0.2. Convolve signal with original scaling function, φ(t), to find C1 f as in Eq. (56.13) with j = 0.3. Find coarse signal at successive resolution levels, j = –1, –2, …, –J through Eq. (56.16); Keep every

other sample of the output.

FIGURE 56.2 Flow chart of multiresolution algorithm showing how successive coarse and detail components ofresolution level, j, are generated from higher resolution level, j + 1.

D f f t x nj jj

2 22= ( ) −( )−,ψ

D f f u t n

t n u k f u t k Y

j j

j j j

j

j j j

k

j

2 2

1

2 2

1

2

1

2

2 2 2 21 1

= ( ) −( ) =

−( ) −( ) ( ) −( ) ′

−

− − − − −

=−∞

∞− −

+ +∑

,

, ,

ψ

ψ φ φ

D f f u t n g n k f u t kj j j

j j

k

j

2 2

1

2

12 2 2 21

= ( ) −( ) = −( ) ( ) −( )− − −

=−∞

∞− −∑ +, ˜ ,ψ φ


4. Find detail signals at successive resolution levels, j = –1, –2, …, –J through Eq. (56.19); Keep everyother sample of the output.

5. Decrease j and repeat steps 3 through 5 until j = –J

Signal reconstruction details are presented in [24, 26].

56.4 Further Developments of the Wavelet Transform

Of particular interest are Daubechies’ orthonormal wavelets of compact support with the maximumnumber of vanishing moments [27, 28]. In these works, wavelets of very short length are obtained(number of coefficients 4, 6, 8, etc.) thus allowing for very efficient data compression.

Another important development of the multiresolution decomposition is the wavelet packet approach[29]. Instead of the simple multiresolution three-stage filter bank (low-pass/highpass/downsampling)described previously in this chapter, wavelet packets analyse all subsystems of coefficients at all scalelevels, thus yielding the full binary tree for the orthogonal wavelet transform, resulting in a completelyevenly spaced frequency resolution. The wavelet packet system gives a rich orthogonal structure thatallows adaptation to particular signals or signal classes. An optimal basis may be found, minimizing somefunction on the signal sequences, on the manifold of orthonormal bases. Wavelet packets allow for amuch more efficient adaptive data compression [30]. Karrakchou and Kunt report an interesting methodof interference cancelling using a wavelet packet approach [8]. They apply ‘local’ subband adaptivefiltering by using a non-uniform filter bank, having a different filter for each subband (scale). Further-more, the orthonormal basis has “optimally” been adapted to the spectral contents of the signal. Thisyields very efficient noise suppression without significantly changing signal morphology.

56.5 Applications

Cardiac Signal Processing

Signals from the heart, especially the ECG, are well suited for analysis by joint time-frequency and time-scale distributions. That is because the ECG signal has a very characteristic time-varying morphology,identified as the P-QRS-T complex. Throughout this complex, the signal frequencies are distributed:(1) low frequency—P and T waves, (2) mid to high frequency—QRS complex [31, 32]. Of particulardiagnostic importance are changes in the depolarization (activation phase) of the myocardium, repre-sented by the QRS complex. These changes cause alterations in the propagation of the depolarizationwave detectable by time-frequency analysis of one-dimensional electrocardiographic signals recordedfrom the body surface. Two examples are presented below.

Analysis of the QRS Complex Under Myocardial Ischemia

Ischemia-related intra-QRS changes. When the heart muscle becomes ischemic or infarcted, characteristicchanges are seen in the form of elevation or depression of the ST-segment. Detection of these changesrequires an extension of the signal bandwidth to frequencies down to 0.05 Hz and less, making themeasurements susceptible to motion artifact errors. Ischemia also causes changes in conduction velocityand action potential duration, which results in fragmentation in the depolarization front (Fig. 56.3) andappearance of low-amplitude notches and slurs in the body surface ECG signals. These signal changesare detectable with various signal processing methods [33, 34]. Depolarization abnormalities due toischemia may also cause arrhythmogenic reentry [35], which is one more reason to detect intra-QRSchanges precisely.

Identification of ischemia-related changes in the QRS complex is not as well known, and interpretationof the QRS complex would be less susceptible to artifactual errors, as compared to the ST analysis. Thus,time-frequency or time-scale analysis would serve a useful function in localizing the ischemia-relatedchanges within the QRS complex, but would be somewhat independent of the artifactual errors.


Experimental findings. In experimental animals, the response of the heart to coronary artery occlusionand then reperfusion was studied. The Left Anterior Descending (LAD) branch of the coronary arterywas temporarily occluded for 20 min. Subsequent to that, the occlusion was removed, and resultingreperfusion more or less restored the ECG signal after 20 min. The coronary artery was occluded a secondtime for 60 min, and once again occlusion was removed and blood flow was restored. Single ECG cycleswere analyzed using the continuous wavelet transform [33]. Figure 56.4 shows the time-scale plots forthe ECG cycles for each of the five stages of this experiment. The three-dimensional plots give time inthe P-QRS-T complex on one axis, the scale (or equivalent frequency) on another axis, and the normalizedmagnitude on the third axis. First occlusion results in a localized alteration around 100 ms and themidscale, which shows up as a bump in the three-dimensional plot or a broadening in the contour plot.Upon reperfusion, the time-scale plot returns to the pre-occlusion state. The second occlusion bringsabout a far more significant change in the time-scale plot, with increased response in the 0–200 ms andmid-scale ranges. This change is reversible. We were thus able to show, using time-scale technique,ischemia related changes in the QRS complex, and the effects of occlusion as well as reperfusion.

Potential role of ischemia-related intra-QRS changes in coronary angioplasty. The above results are alsoapplicable to human ECGs and clinical cardiology. For example, a fairly common disorder is the occlusionof coronary vessels, causing cardiac ischemia and eventually infarction. An effective approach to thetreatment of the occlusion injury is to open the coronary blood vessels using a procedure called coronaryangioplasty (also known as percutaneous transluminal coronary angioplasty or PTCA). Vessels may beopened using a balloon-type or a laser-based catheter. When reperfusion occurs following the restorationof the blood flow, initially a reperfusion injury is known to occur (which sometimes leads to arrhythmias)[35]. The ST level changes as well, but its detection is not easy due to artifacts, common in a PTCAsetting. In a clinical study, we analyzed ischemia and reperfusion changes before and after the PTCAprocedure. Short-term occlusion and ischemia followed by reperfusion were carried out in a cardiaccatheterization laboratory at the Johns Hopkins Hospital in connection with PTCA) [36]. Figure 56.5shows time-scale plots of a patient derived from continuous wavelet transform. Characteristic midscalehump in the early stages of the QRS cycle is seen in the three-dimensional time-scale plot. Then, 60 minafter angioplasty, the normal looking time-scale plot of the QRS complex is restored in this patient. Thisstudy suggests that time-scale analysis and resulting three-dimensional or contour plots may be usablein monitoring the effects of ischemia and reperfusion in experimental or clinical studies. In another study(4 patients, LAD) we monitored for intra-QRS changes during PTCA. Despite signal noise and availability

FIGURE 56.3 Idealized example of (a) normal propagation and (b) wavefront fragmentation due to an ischemiczone of slow conduction. The superimposed lines are isochrones, connecting points at which the depolarizationarrives at the same time.


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ve stages of a controlled animal experiment.) of the complex wavelet-transformed signal.

FIGURE 56.4 Time-frequency distributions of the vector magnitude of two ECG leads during fiThe frequency scale is logarithmic, 16 to 200 Hz. The z-axis represents the modulus (normalized

of recordings only from limb leads, superimposed mid-frequency components during ischemic states ofthe heart were observed, which disappeared when perfusion was restored. There was at least one leadthat responded to changes in coronary perfusion. Figure 56.6 shows five different stages of a PTCAprocedure as time plots (lead I) and CWT TFDs (topo-plots). Despite the presence of noise, the WT wasable to unveil elevation of intra-QRS time-frequency components around 20 Hz during balloon inflation(ischemia), and a drop in the same components with reperfusion after balloon deflation. Frequencycomponents 20 to 25 Hz during inflation. No substantial ST changes can be observed in the time-domainplot. The arrows show the zones of change in TFDs with ischemia and reperfusion. Note the representationof power line interference (50 Hz) as “clouds” in (b) and (d) topo plots—far from the region of interest.

Another study analyzed ECG waveforms from patients undergoing the PTCA procedure by the mul-tiresolution wavelet method, decomposing the whole P-QRS-T intervals into coarse and detail compo-nents [26, 37], as can be seen from the analysis of one pre-angioplasty ECG cycle in Fig. 56.7 [26]. ThePTCA procedure results in significant morphological and spectral changes within the QRS complex. Itwas found that certain detail components are more sensitive than others: in this study, the detail com-ponents d6 and d5 corresponding to frequency band of 2.2 to 8.3 Hz are most sensitive to ECG changesfollowing a successful PTCA procedure. From this study it was concluded that monitoring the energy ofECG signals at different detail levels may be useful in assessing the efficacy of angioplasty procedures[37]. A benefit of this approach is that a real-time monitoring instrument for the cardiac catheterizationlaboratory can be envisioned (whereas currently X-ray fluroscopy is needed).

Detection of reperfusion during thrombolytic therapy. Detecting reperfusion-related intra-QRS changes,along with ST changes, in the time-frequency domain would possibly find application in thrombolysismonitoring after myocardial infarction. At present, the ST elevation and its recovery are the mainelectrocardiographic indicators of acute coronary ischemia and reperfusion. Reports using continuousST-segment monitoring have indicated that 25 to 50% of patients treated with intravenous thrombolytictherapy display unstable ST recovery [38], and additional reperfusion indicators are necessary. Signalaveraged ECG and highest frequency ECG components (150 to 250 Hz) have been utilized as a reperfusionmarker during thrombolysis and after angioplasty, but their utility is uncertain, since the degree of changeof the energy values chosen does not appear to be satisfactory [39, 40]. We have analyzed the QRS of thevector magnitude V = of body surface orthogonal ECG leads X, Y and Z during thrombolytic

FIGURE 56.5 Time-frequency distributions of human ECG study using WT. Pre-angioplasty plot (a) shows acharacteristic hump at about 35 Hz, which disappears as indicated by the second plot (b) taken one hour afterangioplasty treatment.

X2 Y 2 Z2+ +


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FIGUR and time-frequency distributions TFD (lower plots) during(a) base ducing stenosis from 95 to 60%, (d) at the end of a 20 mininflatio s show the zones of change in TFDs with ischemia/reperfusion.

E 56.6 Time domain signals (lead I) during PTCA on the LAD (upper plots)line, (b) at the end of a 3 min inflation (7 at), (c) 10 min. after first inflation, re

n (6 at), (e) 10 min after first inflation, reducing stenosis from 95 to 60%. The arrow

therapy of two patients with myocardial infarction. Figure 56.8 shows how TFDs may be affected byreperfusion during thrombolysis. Two interesting trends may be observed on this figure: (1) a mid-frequency peak present during initial ischemia (a) disappears two hours after start of thrombolytic therapy(b) due to smoother depolarization front, and (2) high-frequency components appear with reestablishedperfusion, possibly due to faster propagation velocity of the depolarization front.

Analysis of Late Potentials in the ECG

“Late potentials” are caused by fractionation of the depolarization front after myocardial infarction[35, 41]. They have been shown to be predictive of life threatening reentrant arrhythmias. Late potentialsoccur in the terminal portion of the QRS complex and are characterized by small amplitude and higherfrequencies than in the normal QRS complex. The presence of late potentials may indicate underlyingdispersion of electrical activity of the cells in the heart, and therefore may provide a substrate forproduction of arrhythmias. The conventional Fourier transform does not readily localize these featuresin time and frequency [15, 42]. STFT is more useful because the concentration of signal energy at varioustimes in the cardiac cycle is more readily identified. The STFT techniques suffer from the problem ofselecting a proper window function; for example, window width can affect whether high temporal orhigh spectral resolution is achieved [15]. Another approach sometimes considered is the Wigner-Villedistribution, which also produces a composite time-frequency distribution. However, the Wigner-Villedistribution suffers from the problem of interference from cross-terms. Comparative representations bysmoothed Wigner-Ville, wavelet transform (scalogram), and traditional spectrogram are illustrated inFig. 56.9. This problem causes high levels of signal power to be seen at frequencies not representing theoriginal signal; for example, signal energy contributed at certain frequencies by the QRS complex maymask the signal energy contributed by the late potentials. In this regard, wavelet analysis methods providea more accurate picture of the localized time-scale features indicative of the late potentials [11, 15, 43–45].Figure 56.10 shows that the signal energies at 60 Hz and beyond are localized in the late stage of the QRS

FIGURE 56.7 Detail and coarse components from one ECG cycle. The coarse components represent the lowpassfiltered versions of the signal at successive scales. Detail components, d1 and d2, consist mainly of electrical interference.


and into the ST-segment. For comparison, the scalogram from a healthy person is illustrated in Fig. 56.11.This spreading of the high frequencies into the late cycle stages of the QRS complex is a hallmark of thelate potentials. Time-scale analysis of late potentials may therefore serve as a noninvasive diagnostic toolfor predicting the likelihood of life threatening arrhythmias in the heart.

Role of Wavelets in Arrhythmia Analysis

Detection of altered QRS morphology. Further applications to the generalized field of arrhythmia classi-fication can be envisaged [46]. When arrhythmias, such as premature ventricular contractions (PVCs)and tachycardia do occur, the P-QRS-T complex undergoes a significant morphological change. Abnormalbeats, such as PVCs, have different time-scale signatures than normal beats. Often the QRS complex maywiden and sometimes invert with PVCs. As the QRS complex widens, its power spectrum shows dimin-ished contribution at higher frequencies and these are spread out over a wider body of the signal [47, 48].This empirical description of the time-domain features of the ECG signal lends itself particularly well toanalysis by time-frequency and time-scale techniques. A more challenging problem is to distinguishmultiform PVCs.

Use of the orthogonal wavelet decomposition to separate dynamical activities embedded in a time series.Another interesting application of the discrete wavelet transform to arrhythmia research is the dynamicalanalysis of the ECG before and during ventricular tachycardia and ventricular fibrillation. Earlier workson heart rate variability have shown that heart rhythm becomes rigidly constant prior to the onset oflife threatening arrhythmias, whereas the correlation dimension as a measure of randomness increasesto values above 1 during disorganized rhythms like fast ventricular tachycardia, ventricular flutter, andventricular fibrillation. This fact was used to identify patients at risk, and is promising with regard toprediction of arrhythmic episodes. Encouraging results were recently obtained by combining multires-olution wavelet analysis and dynamical analysis, in an attempt to find a decorrelated scale best projectingchanges in low-dimensional dynamics [49]. The authors used records 115 and 207 from the MIT

FIGURE 56.8 Change of the time-frequency distributions (QRS complex) of the vector magnitude of orthogonal leadsX, Y and Z during thrombolysis: (a) 5 minutes after start, and (b) 2 hr after initiation of therapy. A mid-frequency peakhas disappeared due to smoother depolarization, and high-frequency components appear due to faster propagation.


Arrhythmia Database to study nonlinear dynamics preceding ventricular flutter. Distinct changes in thecorrelation dimension (D2 ≈ 3) in frequency band 45 to 90 Hz were observed before the onset of arrythmia,indicating the presence of underlying low-dimensional activity.

Neurological Signal Processing

Evoked Potentials

Evoked potentials are the signals recorded from the brain in response to external stimulation. Evokedresponses can be elicited by electrical stimulation (somatosensory evoked response), visual stimulation(visual evoked response), or auditory stimulation (brainstem auditory evoked response). Usually thesignals are small, while the background noise, mostly the background EEG activity, is quite large. Thelow signal-to-noise ratio (SNR) necessitates use of ensemble averaging, sometimes signal averaging asmany as a thousand responses [50]. After enhancing the SNR, one obtains a characteristic wave patternthat includes the stimulus artifact and an undulating pattern characterized by one or more peaks atspecific latencies beyond the stimulus. Conventionally, the amplitude and the latency of the signal peaksis used in arriving at a clinical diagnosis. However, when the signals have a complex morphology, simpleamplitude and latency analysis does not adequately describe all the complex changes that may occur asa result of brain injury or disease. Time-frequency and wavelet analysis have been shown to be useful inidentifying the features localized within the waveform that are most indicative of the brain’s response [51].

In one recent experimental study, we evaluated the somatosensory evoked response from experimentalanimals in whom injury was caused by oxygen deprivation. The evoked response signal was decomposed

FIGURE 56.9 Comparison of time-frequency representations of sinusoids with specific on-off times: (a) shows truetime-frequency representation of 40 and 60 Hz sinusoids, (b) shows representation of smoothed Wigner-Ville trans-form, (c) spectrogram representation, (d) shows wavelet transform version of signal.


into its coarse and detail components with the aid of the multiresolution wavelet analysis technique(Fig. 56.12) [25]. The magnitude of the detail components was observed to be sensitive to the cerebralhypoxia during its early stages. Figures 56.13a and 56.13b show a time trend of the magnitude of thedetail components along with the trend of the amplitude and the latency of the primary peak of thesomatosensory evoked response. The experimental animal was initially challenged by nitrous gas mixturewith 100% oxygen (a non-injury causing event), and late by inspired air with 7 to 8% oxygen. As expected,the amplitude trend shows an initial rise because of the 100% oxygen, and later a gradual decline inresponse to hypoxia. The magnitude of the detail component shows a trend more responsive to injury:while there is not a significant change in response to the non-injury causing event, the magnitude of thedetail component d4 drops quite rapidly when the brain becomes hypoxic. These data suggest that detailcomponents of the evoked response may serve as indicators of early stages of brain injury. Evoked responsemonitoring can be useful in patient monitoring during surgery and in neurological critical care [52, 53].Other applications include study of cognitive or event-related potentials in human patients for normalcognitive function evaluation or for assessment of clinical situations, such as a response in Alzheimer’sdisease [54]. Proper characterization of evoked responses from multiple channel recordings facilitateslocalization of the source using the dipole localization theory [55].

EEG and Seizures

Electroencephalographic signals are usually analyzed by spectrum analysis techniques, dividing the EEGsignal into various arbitrary frequency bands (α, β, θ, δ). Conventional spectrum analysis is useful whenthese events are slowly unfolding, as when a person goes to sleep, the power in the EEG shifts from higherto lower frequency bands. However, when transient events such as epileptic seizures occur, there are oftensharp spikes or a bursting series of events in the recorded waveform. This form of the signal, that istemporally well localized and has a spectrum that is distinctive from normal or ongoing events, lends

FIGURE 56.10 Healthy person: (a) first recorded beat, (b) 3-D representation of the modified WT for the first beat,(c) contour plot of the modified WT for the first beat, and (d) contour plot of the modified WT for the second beat.


itself to wavelet analysis. A patient’s EEG recorded over an extended period, preceding and following theseizure, was recorded and analyzed. Figure 56.14 shows a short segment of the EEG signal with a seizureburst. The multiresolution wavelet analysis technique was employed to identify the initiation of the seizureburst. Figure 56.15 shows a sudden burst onset in the magnitude of the detail components when theseizure event starts. The bursting subsides at the end of the seizure, as seen by a significant drop in themagnitude of the detail components. Wavelet analysis, thus, may be employed for the detection of onsetand termination of seizures. Further possibilities exist in the use of this technique for discriminatinginterictal spikes and classifying them [56]. Certain seizures, like the petit mal and the grand mal epilepsyseizures, have very characteristic morphologies (e.g., spike and dome pattern). These waveforms wouldbe expected to lend themselves very well to wavelet analysis.

Other Applications

Wavelet, or time-scale analysis is applicable to problems in which signals have characteristic morphologiesor equivalently differing spectral signature attributed to different parts of the waveform, and the eventsof diagnostic interest are well localized in time and scale. The examples of such situations and applicationsare many.

In cardiac signal processing, there are several potential applications. Well localized features of ECGsignals, such as the P-QRS-T lend themselves well to wavelet analysis [57]. The application of waveletanalysis to ischemic-reperfusion injury changes and the late potentials has been illustrated above. Thisidea has been extended to the study of body surface maps recorded using numerous electrodes placedon the chest. In a preliminary study [58], spatio-temporal maps can been constructed and interpretedusing time-scale analysis techniques.

FIGURE 56.11 Patient with ventricular tachycardia diagnosis: (a) first beat, (b) 3-D representation of the modifiedWT for the first beat, (c) contour plot of the modified WT for the first beat, and (d) contour plot of the modifiedWT for the second beat.


In many situations noise and artifact result in inaccurate detection of the QRS complex. Waveletanalysis may prove to be helpful in removal of electrical interference from ECG [59]. Wavelet techniqueshave successfully been used in removing the noise from functional MRI data [8]. A more challengingapplication would be in distinguishing artifact from signal. Since wavelet analysis naturally decomposesthe signals at different scales at well localized times, the artifactual events can be localized and eliminated.Fast computational methods may prove to be useful in real-time monitoring of ECG signal at a bedsideor in analysis of signals recorded by Holter monitors.

Other cardiovascular signals, such as heart sounds may be analyzed by time-frequency or time-scaleanalysis techniques. Characteristic responses to various normal and abnormal conditions along withsounds that are well localized in time and scale make these signals good candidates for wavelet analysis[60]. Normal patterns may be discriminated from pathological sound patterns, or sounds from variousblood vessels can be identified [61]. Blood pressure waveform similarly has a characteristic patternamenable to time-scale analysis. The dicrotic notch of the pressure waveform results from blood flowthrough the valves whose opening and closing affects the pressure signal pattern. The dicrotic notch canbe detected by wavelet analysis [62]. Sounds from the chest, indicative of respiratory patterns are beinginvestigated using wavelet techniques. Applications include analysis of respiratory patterns of infants [63]and respiration during sleep [64].

Two applications in neurological signal processing, evoked response and seizure detection, aredescribed above. Other potential applications include detection and interpretation of signals from mul-tiple neurons obtained using microelectrodes [65]. Since waveforms (called spikes) from individual

FIGURE 56.12 Coarse (a) and detail (b) components from somatosensory evoked potentials during normal,hypoxic, and reoxygenation phases of experiment.


neurons (called units) have different patterns because of their separation and orientation with respectto the recording microelectrode, multiunit spike analysis becomes a challenging problem. Time-scaleanalysis techniques may be employed to localize and analyze the responses of individual units and fromthat derive the overall activity and interrelation among these units so as to understand the behavior ofneural networks. An analogous problem is that of detecting and discriminating signals from “motorunits,” that is, the muscle cells and fibers [66]. Signals from motor units are obtained by using small,micro or needle electrodes, and characteristic spike trains are obtained, which can be further analyzedby time-scale analysis techniques to discriminate normal and abnormal motor unit activity.

Discussion and Conclusions

Biological signals with their time-varying nature and characteristic morphologies and spectral signaturesare particularly well suited for analysis and interpretation using time-frequency and time-scale analysistechniques. For example, the P-QRS-T complex of the ECG signal shows localized low frequencies in theP- and the ST-segments and high frequencies in the QRS complex. In time-scale frame, the ischemiarelated changes are seen in certain detail components of the QRS complex. The late segment of the QRScycle exhibits the so-called late potentials more easily localized by means of time-scale analysis. Othercardiovascular signals, such as pressure waves, heart sounds, and blood flow are being analyzed by thenewly developed wavelet analysis algorithms. Other examples of time-scale analysis include neurological

FIGURE 56.13 (a) Amplitude and latency of major evoked potential peak during control, hypoxic, and reoxygen-ation portions of experiment; (b) mean amplitude of respective detail components during phases of experiment.


FIGURE 56.14 Example of epilepsy burst.

FIGURE 56.15 Localization of burst example with wavelet detail components.


signals with potential applications in the analysis of single and multiunit recordings from neurons, evokedresponse, EEG, and epileptic spikes and seizures.

The desirable requirements for a successful application of time-scale analysis to biomedical signals isthat events are well localized in time and exhibit morphological and spectral variations within thelocalized events. Objectively viewing the signal at different scales should provide meaningful new infor-mation. For example, are there fine features of signal that are observable only at scales that pick out thedetail components? Are there features of the signal that span a significant portion of the waveform sothat they are best studied at a coarse scale. The signal analysis should be able to optimize the trade-offbetween time and scale, i.e., distinguish short lasting events and long lasting events.

For these reasons, the signals described in this article have been found to be particularly useful modelsfor data analysis. However, one needs to be cautious in using any newly developed tool or technology. Mostimportant questions to be addressed before proceeding with a new application are: Is the signal well suitedto the tool, and in applying the tool, are any errors inadvertently introduced? Does the analysis provide anew and more useful interpretation of the data and assist in the discovery of new diagnostic information?

Wavelet analysis techniques appear to have robust theoretical properties allowing novel interpretationof biomedical data. As new algorithms emerge, they are likely to find application in the analysis of morediverse biomedical signals. Analogously, the problems faced in the biomedical signal acquisition andprocessing world will hopefully stimulate development of new algorithms.

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