6.3 Signals of the Cardiovascular System

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follow-up of chronic heart failure. The relationship of this parameter to the phonocar-

diogram becomes clear when the causes of the two heart sounds are recalled. The first

heart sound is produced by the rapid contraction movement of the ventricles, which

is associated with the onset of the expulsion phase. At the end of the expulsion phase,

the aortic and pulmonary valves close, which in turn causes the second heart sound.

Thus, LVET corresponds quite closely to the time between the first and second heart

sounds.

When designing a suitable algorithm for the determination of LVET, it must be

taken into account that phonocardiograms are in practice overlaid by artifact noise

and that the amplitude of heart sounds in cardiac patients is often weak. A simple

algorithm for determining maximum values and their location would quickly lead to

incorrect results. Instead, an algorithm based on the autocorrelation function (AKF)

(cf. subsubsection 5.3.1.2) is presented here. The AKF offers the advantage of highlight-

ing signal components that occur systematically. First, the phonocardiogram is recti-

fied and subjected to low-pass filtering using a moving-average filter. The Figure 6.39

shows in each case the processing result for the sequence from Figure 6.37 (top).

Like the Fourier or Laplace transform, the AKF belongs to the class of integral

transforms. In the continuous-time domain the transformation rule is

ρxx(τ) = ∫x(t)x(t + τ)dt ,

(6.30)

where x(t) is the function to be transformed, here the function from Figure 6.39 (be-

low), and τ is a shift parameter in the function x(t), which becomes the variable of

the AKF. Equation 6.30 can be translated as the function x(t) is shifted against itself

by τ, the shifted function and the original function are multiplied together, and then

the area of the multiplication result is calculated by integration. This is repeated for

all shift values τ, resulting in the AKF ρ(τ). In the discrete-time domain, the integral

becomes a sum with the discrete-time variable m and the displacement value n:

ρxx(n) = ∑(x(m)x(m + n)) .

(6.31)

Applying the AKF to the processed phonocardiogram from Figure 6.39 (below), the

AKF value will be becomes large whenever the function x(t) or x(m) is shifted exactly

one heart cycle. Then the first heart sound falls back to the first and the second heart

sound to the second. Product and integral in Equation 6.30 take large values in that

case. Smaller maxima occur when the first heart sound is switched to the second or the

second is shifted onto the first. The first of these two cases is particularly interesting,

because the value for the shift of the first to the second heart sound just corresponds

to the LVET we are looking for. Figure 6.40 (above) shows the result of AKF formation

from the signal in Figure 6.39 (bottom) and a magnification of the AKF around the

maximum value (bottom). The LVET we are looking for corresponds to the distance

in τ of a main maximum to the next right side maximum. The LVET, i.e. the distance

between the main maximum and the first right minor maximum, can be calculated

using the algorithm given in Listing 6.3.2.