44

|

2 Fundamentals of Information, Signal and System Theory

posed of sequences with different bandwidths and are sometimes of short duration.

Thus, the wavelet transform finds wide application in the analysis of electroenceph-

alograms, for example, to detect short epileptic episodes in the low-frequency funda-

mental waves of electrical neuron activity.

Finally, the filter property of wavelets should be pointed out. The Equation 2.77

ψ(t) can also be understood as a bandpass filter. The bandwidth of the bandpass is

directly related to the scaling value a. The smaller a is, the larger is the bandwidth

of the bandpass filter. This suggests building a filterbank from a single wavelet where

only the scaling value a is changed. The Morlet wavelet is particularly suitable as a

bandpass filter because it is based on a harmonic function (cf. Equation 2.80). Varying

the scaling value a, the frequency of the harmonic function of the Morlet wavelet is

modulated. The signal s(t) to be filtered is correlated with the harmonic function of

the Morlet wavelet during the transformation, which means a spectral decomposition

of s(t) according to the frequencies determined by the scaling value a.

2.4.4 Continuous Linear Convolution

In the section on the Fourier transform, it was already mentioned that for a linear

time-invariant system, in the time domain the output signal y(t) is calculated from

the mathematical convolution of the input signal s(t) with the impulse response of

the system h(t) (cf. Figure 2.21). Equation 2.82 gives the mathematical operation for

this:

y(t) = ∫s(τ)h(t −τ)dτ .

(2.82)

Here, the variable t is renamed to τ: s(t) →s(τ), h(t) →h(τ). The second function

(here h) is mirrored at the ordinate by the negative sign of the variable τ. The value t in

the argument of the second function is on the one hand the displacement parameter of

the function h and at the same time the variable of the output function y(t). Thus, the

function h is mirrored, shifted by t, then multiplied by s and the result is integrated

over the time interval τ. This process is repeated for all displacement values t, from

which the new function y(t) is formed.

The convolution integral, given by Equation 2.82, is identical to the correlation

function except for the negative sign in the argument of the second function, which

thus also belongs to the class of integral transformations. The correlation function

provides a measure for the similarity of two signals or functions. In this sense, in the

Fourier transform the signal s(t) was correlated with the complex exponential func-

tion, and the convolution integral provides a measure for the similarity of s(τ) and

h(−τ) at the respective shift values t. The symbol for the notation of the convolution

is a star between the functions to be convolved:

y(t) = s(t) ∗h(t) .

(2.83)