A New Algorithm for Estimating a Noiseless, Evenly Sampled, Heart Rate Modulating Signal

Figures

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  Figure 1. An abstract and almost black box model of the heart rate variability signal as a heart rate frequency modulator. (A) A black box model of any physiologic factor (R pulse rate modulator shown upper left) is blindly considered to impose a certain discharge frequency on the cells in the sinus node, which provide an electrical sinus node signal ESN(t) that propagates in the heart, and eventually will be responsible for producing an electrical signal ECG(t) recorded on the skin of the body. An external R-wave detector produces a signal R(t) that is zero at any instant of time, but when an R-wave is detected so its value is set to one (the detected R pulse). (B): The R pulse rate modulator is now black box modelled as an R pulse rate variability signal generator that produces a signal that is equal to the variation of the heart rate (over the average) at each instant of time HRV(t). This signal is then summed to a constant value HR representing the average heart rate in the considered epoch to produce a signal that is equal to the heart rate at each instant of time HR(t). All of the time-domain signals indicated in the figure are assumed to be continuous in time.

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  Figure 2. Upper tracing: the simulation of an artificial instantaneous “heart rate signal”, or HR(t), composed of a constant signal with an amplitude of 70.2 bpm summed to a square-wave signal having a peak-to-peak amplitude of 14.4 bpm and a frequency of 0.02 Hz; the resulting signal is a square-wave signal having an average value of 70.2 bpm alternating between 77.4 bpm and 63 bpm every 25 s. Lower trace: the HR(t) signal imposes the frequency of a sinusoidal signal, creating a “sinusoidal ECG” signal whose frequency is, at each instant of time, precisely that shown in the upper trace. Note that the amplitude of the HR(t) signal is purposely given in beats per minute (bpm), while the units of the simulated “sinusoidal ECG” signal are arbitrary. The frequency of the sinusoidal signal has been modulated by the square signal.

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  Figure 3. The upper trace shows a frequency-modulated signal with the square-wave modulating signal superimposed in red. Zero-crossing detection points (red) on the signal ECG(t) are shown in the middle trace. Bottom tracing shows the created SR(t) signal with a slope depending on the local instantaneous frequency of the simulated ECG(t) signal.

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  Figure 4. Magnitude Bode plot of the moving average filter applied to the cumulative sum signal.

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  Figure 5. Magnitude Bode plot of Holoborodko’s differentiating filter applied to the cumulative sum signal. Please note the differentiating behavior at low frequencies thanks to the almost straight line of the graph (the output is linearly proportional to the frequency of the input signal).

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  Figure 6. Magnitude Bode plot of the moving average filter applied to the recovered first derivative signal (the final recovered heart rate signal).

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  Figure 7. Bottom trace: the recovered heart rate signal. The 2.5-s delay of the recovered signal corresponds to the delays expected from the sequence of FIR filters.

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  Figure 8. Simulation of test signal ECG(t): a sinusoidal signal (bottom trace) that is frequency modulated with the sum of two sinusoidal waves (shown separately in different colors on the upper trace), producing a final modulating signal around the average frequency of 70.2 bpm (middle trace).

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  Figure 9. Upper trace: frequency-modulated signal and modulating signal. Bottom trace: the recovered heart rate signal. The middle trace represents the modulating waveform (cf. Figure 8).

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  Figure 10. A comparison of modulating and recovered simulation signals in the time and frequency domains. The HRV frequencies are perfectly recovered. Different amplitudes of signals in the time domain result from changing the scale from beats per second to beats per minute. Different amplitudes in the frequency domain are also due to the windowing effect (a Blackman-Harris window was used before the FFT calculation).

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  Figure 11. First test of the algorithm on a real ECG trace (top trace). In the second tracing from the top, ECG R-waves are indicated as being detected. In the third tracing, the cumulative summing signal is shown, and in the bottom trace, the filtered first derivative of the latter is plotted showing fast variations of the heart rate during the subject’s activity.

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  Figure 12. Test of the algorithm on a real ECG trace (top trace) during the Valsalva manoeuvre.