M'ed.&Biol.Eng.& Comput.,1977,15,423-430

Measurement of heart.rate variability: Part 2-Hardware digital device for the assessment of heart-rate variability A. J. R. M . Coenen

O. Rornpelman

R. I. K i t n e y

Biomedical Electrical Engineering, Delft University of Technology, Delft, The Netherlands

Departments of Biophysics & Physiology, Chelsea College, London University, London, England

A b s t r a c t - - P a r t 1 of the paper was dedicated to some general aspects of h.r.v, measurements. In Part 2, the spectral properties of the integral pulse frequency modulator (i.p.f.m.) are shown. The choice of a hardware device for Iowpass filtering the cardiac event series by means of a stepwise convolution is discussed. A method to overcome the problems of distortion due to spurious spectral components in the filter passband is shown. The hardware realisation of the filter is described and results are given. Keyword--Heart-rate variability

1 Introduction IN PART I of the paper (ROMPELMANet al., 1977) the concept of heart-rate variability (h.r.v.) was discussed with reference to the optimum method of obtaining a variability signal from the e.c.g. The conclusion was that heart-rate variations arise from a process known as integral pulse frequency modulation (i.p.f.m.) (BAYLY, 1968). Demodulation of the signal can be achieved by a digital computer. However, this procedure is inappropriate for a number of reasons: first, the calculations are time consuming; and, secondly, the approach is unsuitable for online applications. These problems can be overcome when

a hardware demodulator is used. The design and building of such a device is the subject of this paper.

1.1 Resum# of theory of integral-pulse-frequency modulation Referring to Fig. 1, which is a diagram of the i.p.f.m, model, BAYLY (1968) derived the spectrum of the generated event series when the input signal re(t) is given by

m(t) = me+m1 cos (21rft t+v)

(1)

The output from the model is a pulse train, which is described by the equation

RESE~ ml

m(t)

~

]

y(t}

oo

p(t) = lfo+l~R-cos(2rtfl t+v)+2lfo E

+co

N

k= l I1=--oo

p(t) R

,

kfo] d, \-R-~z ] re(t) I "--~t

x cos [2zr(kfo + nfl) t + nv + 2kr~fo a

--~t p{t)r

I

I

I

I

I

I I I ~1111

I

I

I

I

I

1 | ----t

Fig. 1 Integral pulse frequency modulator (i.p.fm.). modulation signal applied to the integrator input Received 23rd August 1976

Medical and Biological Engineering & Computing

kml sin (v--2nfl a)] Rf,

. . . .

(2)

J

where I is the impulse content, R being the constantlevel input in Fig. 1, fo = mo/R is the unmodulated pulse-repetition frequency, ~ is a random variable, such that y ( - a ) = O, and J, is a Bessel function of order n.

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We may conclude from eqn. 2 that: (a) The signal contains a d.c. component Ifo. (b) The signal contains a component proportional to the a.c. component of the modulating signal I

mx c o s

(2~rfl t + v).

(c) The signal contains higher harmonics of the unmodulated repetition frequencyfz (note that the side frequencies are spreaded asymmetrically in amplitude around the harmonics of the unmoduluted repetition frequency). Therefore demodulation can be performed by lowpass filtering; however, distortion may occur owing to side frequencies appearing in the passband of the filter. KOENDERINK and VAN DOORN (1973) described another form of distortion which is said to occur if the modulation signal is a linear combination of two sinusoidal signals of different frequencies. This distortion is supposed to consist of a combination of multiples of the input frequencies. The established spectral components are most probably due to mixing products of the unmodulated frequency and a multiple of one of the modulation frequencies, as can be predicted from Bayly's formula.

1.2 Example of the Lp.f.m. model applied to heart-rate variability Assuming that the mean heart rate is 60 beats/min, and the h.r.v, is mainly due to respiration with a periodicity of about 3s (0-3 Hz), it follows that fo = 1 Hz and f l = 0-3 Hz. Assuming further that h.r.v, consists of fluctuations of about 1 0 ~ of the mean heart rate, then ml/mo = 0.1. The spectral components [P(f)[ are then (if I = 1) [P(O)l = lfo = 1 I e ( / 0 [ = fP(0.3)l = Ifo m~ = 0.1 mo

The amplitude of the sideband components follows from k=x,=-oc

kfo]

~ mof,]l

The amplitude spectrum of the pulse series p(t) with this modulation signal is shown in Fig. 2. In this Figure, three possible lowpass-filter characteristics are shown, each of which might be used to demodulate the signal. The choice of the appropriate filter will be discussed later. 2 Design of a filter

2.1 General principles It has been shown that an h.r.v, signal is band limited to about half the mean heart rate (RoMPELMAN et al., 1977). In the analysis of these signals it is therefore sufficient to choose a relatively low sampling rate of a few hertz, as far as the output signal is concerned. The e.c.g, has a bandwidth of about 250Hz (Cox and OLtVER, 1969), hence it is clear that the h.r.v, signal will give a large data reduction. Since it is very desirable in a practical analysis to obtain a data reduction before computer processing, it was decided to develop a hardware device able to produce a h.r.v, signal according to the lowpassfiltering principle. There are two ways of performing this lowpass filtering, namely by an analogue method or a digital method. These methods will now be discussed with reference to Fig. 3. In the analogue method, the R-wave event pulses are converted to rectangular pulses, their constant width being half the mean R - R interval. Variability in the original event series causes a variation in the duty cycle of the generated rectangular waveform. By means of an analogue lowpass filter, this variableduty cycle is converted to a corresponding variability

2. frequency component at f= [Hzl fl .3 fo -2fT ./' to - fl .7 fo 1 fo § fl 1.3 f~ * 2 fl 1.6 2f o - 2f/ l.& 2fo- fl 1.7 2fo 2 2fo* fl 2.3 Fig. 2Example of a spec2f o * 2f I 2.6 trum o f a modulated pulse series for a practical case : three possible low-passfilter characteristics are shown

ampl.l 1.5-

Ifo

curve:

i

0.5

.3 .4 .5 fl

424

.7

1 fo

1.3 1.-=/.

1.~ 17

2

2.3

2.6

2f o Medical

and Biological

=- f (Hz) Engineering

& Computing

July 1977

signal. It should be noted that the amplitude and phase characteristics of this filter are of importance. In the digital method (discussed with reference to a hardware realisation), filtering is based on a stepwise convolution of the event series with the impulse response of the filter.

pml

1

t

o

2

I__ . . . . T1 . _1_

ANALOGUE WAY

-

I

I

1

T2

of an event series with an arbitrary impulse response is shown. The upper half of the diagram shows the R-wave event series p(t), the broken line is the impulse response of the filter. Note that this curve is symmetrical, causing a linear phase-frequency relation; in other words a fixed time delay.

_1_ r,.

I ULSE

3 T3 =1_

_r-LI

FORMER

q *++>

-I

I

T4

/. _1

==t

LOW PASS FILTER

~

Xcl(t}

_p (t) /

\\\~DG IT IALWAY

_I "~

DIGITAL CONVOLUTION

I= Xc2(t)

Fig. 3 Reconstruction of the modulating signal xc(t) from an event series p(t), according t o t h e i.p.f,m, model by means of either an analogue or a digital method

The convolution formula used is therefore

Xc(7:) = +i. p(t) h(z- t) dt --a

- a is the starting point of signal p(t) (Fig. 4). Note that the formula usually given is d

Xc(z)= I p(t) h(z-t)dt

Referring to Fig. 4, the value at, say, 2 T of the filtered signal is found by taking the values of the impulse-response curve at the event occurrence times and summing them. The next value of the filtered signal is obtained by shifting the impulse-response curve with respect to the event process and performing the same calculations as before. The shifting increment T is chosen according to the cutoff frequency of the filter; when this cutoff frequency is fc, the shifting increment has to be

01

--a

which is only valid for physically realisable filters with a finite impulse response (LEE, 1960). In Fig. 4, the principle of a stepwise convolution

pit) I

=1

I

I

h(t-t) I

I

I

I

..3"

T

T

t

2T

3T

4T

5T

---=.- "~

Fig. 4 Filtering of the event series p(t) by means of stepwise convolution

Medical and Biological Engineering & Computing

1

T = ~ (Nyquist criterion) zJc In a hardware realisation, the cutoff frequency is only dependent upon the clock frequency of the digital filter and can be easily adjusted. Because the digital filter is based on a stepwise convolution: (a) the impulse response can be stored in a memory; and (b), in principle, any impulse response may be used. Analogue filters operating in the frequency range under discussion ( < 1 Hz) suffer from the well known problems of large components, such as drift etc. F o r these reasons, a digital realisation of the lowpass filter was chosen (CoENEN, 1975). The choice of impulse responses is mainly limited by the available storage capacity.

2.2 Choice of the impulse response Theoretically, the best way of filtering is by means of an ideal lowpass filter (i.l.p.f.), see curve 1 in Fig. 2.

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The impulse response hc(t) of this filter is

The spectral shape of this filter is given by Hsi(to) = 1

0 ~< to toc

sin x sinx sinz hc(t)=-+ +-x ~ 2z

.

.

.

.

(4)

where toc is the cutoff frequency yielding an impulse response given by 2toc(t + 89

sin toc t hs~(t)- - -

- o o < t < oo

COc t

2to~(t- 89 --oo

This is a physically nonrealisable filter. In practice, the following impulse response would be appropriate as this impulse response is similar to the ideal case and it is physically realisable:

hst(t)

-

sin ~ ( t - - ~') toAt-r)

hsi(t) =- 0

0