2.4 Signal Processing Transformations

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time t/s

time t/s

Fig. 2.23: Morlet wavelet (left) according to Equation 2.80 and Mexican Hat wavelet (right) according

to Equation 2.81.

the complex exponential function. A possible realisation of a Morlet wavelet is e.g. the

function

ψ(t) = e^−t2

2 cos(5t) .

(2.80)

Figure 2.23 (left) shows the composition of the Morlet wavelet according to Equa-

tion 2.80 from harmonic oscillation multiplied by a Gaussian envelope function. The

Mexican Hat wavelet (cf. Figure 2.23, right) has the mathematical form

ψ(t) = e^−t2

2σ (1 −t²) .

(2.81)

Unlike the Morlet wavelet, the Mexican Hat wavelet does not contain a harmonic func-

tion, which leads to differences in the interpretation of the transformation, as will be

discussed later.

The graphical representation of S(a, τ) is either in a three-dimensional figure with

a and τ as x- and y-axis and S as the z-axis, or a two-dimensional figure in which

a is plotted downward in ascending order over τ. The transformation result is then

rendered as a color or brightness in the two-dimensional (a, τ) plane.

The benefit of the wavelet transform for signal analysis lies in the variable wave-

let width. If, for example, a high time resolution is required because very short signal

events occur in the signal that are to be analysed spectrally, the width of the wave-

let can be reduced by means of the scaling value a so that the required time resolu-

tion is achieved. This narrow wavelet then passes through the entire signal by means

of the displacement parameter τ, yielding high S values whenever the wavelet en-

counters the short signal events. However, if the same signal also contains periodic

events with long period durations, such as may be caused by respiration in biosig-

nals, these are captured in the same signal analysis for large a values. This makes

the wavelet transform particularly well suited for the analysis of signals that are com-