Coherence Analysis

 , (16)

where A and B is the maximum and minimum peak amplitude of QRS complex wave of modulated ECG [23]. The ECG of this subject is demonstrated in Fig. 12(b) and the dash line denotes the amplitude variation of the peaks of QRS complex wave.

Using Eq. (16), the evaluated AM modulation depth is 3.27% (A=1.025 mV, B=0.96 mV).

The FM effect allows the frequency band of carrier signal to broaden due to the frequency swing. Clinically, the most frequently used method for the measurement of frequency swing of ECG is HRV (heart rate variability), and this approach has acquired some achievements [24-26]. We use another parameter defined as

BW

Q f⁰ , (17)

to characterize the frequency swing effect of each spectral harmonic, where f is the₀ central frequency of each harmonic and BW is the 3 dB bandwidth [11,23]. Larger (smaller) Q value indicates a smaller (larger) frequency swing. The calculated Q values for the first five harmonics are 5.5, 11.1, 16.5, 22.3, and 28.1 in sequence.

4.2 Coherence Analysis

For the cardiovascular system, blocked blood vessels, stiff arteries, high peripheral resistance, extruded blood vessel due to damage, blood turbulence, viscous damping, propagation disturbance of arterial pressure pulse, etc., are treated as interferences.

They would affect both the blood flow toward its destination and the pressure pulse contour; therefore, they induce various cardiovascular-related diseases. These perturbation signals also interfere with the resonance and coherent oscillation between

the input and the output of the system. In fact, they are equivalent to the noises generated in transmission channel for electronic communication system.

In our research, the cardiovascular system is regarded as an electrically-driven mechanical-pumping system [11]. Thus, the modulated ECG and blood pressure wave are considered as the input and the output signals of the cardiovascular system, respectively. In other words, in such a system the electrical energy is transformed into mechanical power. Now, a cardiovascular system model described in Fig. 13(a) is established based on the concept of modulated ECG, power coherence and perturbation. Moreover, the spectra of modulated ECG including the inter-modulation effect are demonstrated in Fig. 13(b).

Instead of developing complex mathematical equations to characterize the system, we use the coherence function C_XY( f), which represents the magnitude squared coherence between input signal X and output signal Y, to quantify the macroscopic effect of power coherence for the system. The function is defined as

power respectively, and P_XY( f) is the cross power spectral density between input and output [13]. The coherence function always takes on a value in the interval

1 ) (

0C_XY f , and with a value close to one, if the perturbation (noise) level is low.

When the coherence function is far less than one, it indicates the presence of one or more of the following: (1) a disturbance affecting the output, (2) another input source, or (3) nonlinear relationship between input and output.

Therefore, the function can be used to evaluate the input-output relation of

coherence and extract perturbations between ECG and BPW. Furthermore, the average S of coherence values at the first five harmonic peaks, which constitute most of the energy of the system, is defined as an index for measuring the degree of coherence of the system.

4.3 Experiment and Results

We evaluated the coherence function using the Hamming window with a 600-sample window and an overlap of 500 samples. The estimated coherence functions of a healthy subject and a cardiovascular patient are illustrated in Fig. 14.

The fundamental heart frequencies f are 1.1 Hz and 1.35 Hz for the healthy subject₀ and cardiovascular patient, respectively.

Some remarkable differences for the two subjects are summarized as follows:

(1) The average S of coherence values for the first five harmonic frequencies is close to one for the healthy subject, yet approximately 0.85 (<1) for the patient. This means that, for the healthy subject, the input power is coherently coupled to the output and the output is completely dominated by the input, particularly in the first five resonant frequencies (i.e. harmonic frequencies). However, for the patient, perturbations are generated and the output is not entirely determined by the input.

In other words, the perturbation signals effectively affect the output.

(2) The bandwidth of coherence function for the first five harmonic peaks is more larger and regular for the healthy subject than that for the patient. This implies that, for the healthy subject, more frequency constituents of the system induce a high coherence value.

(3) The coherence function within the frequency range of slow-varying modulation signals (0.4Hz), in synchrony with respiratory rate, also exhibits a peak for both

the healthy subject and patient. This reveals that the power of such modulation signals is also coherently coupled to the output for both the healthy subject and the patient.

Moreover, the above results are observed not only in the two subjects but also in the experiment in which forty-four subjects participated.

The study subjects were composed of two groups. The first group consisted of 22 healthy subjects (12 males and 10 females, at the ages of 18 to 40) whose health checks are normal and without any reported cardiovascular disease. The second group was composed of 22 patients with cardiovascular-related diseases (13 males and 9 females, at the ages of 36 to 55). The disease patterns were not specified for the study of general effect of coherence function on cardiovascular system. Before the measurement, all the subjects were rested for about twenty minutes to get a steady pulse waveform. The statistical results are demonstrated in box plot in Fig. 15. For healthy subjects, the mean value of S (=0.97) is larger than that (=0.92) for cardiovascular patients. In addition, the small p-value (=7.9210^5 0.01) indicates that the coherence values S of healthy subjects and cardiovascular patients are significantly different. Consequently, this index is appropriate for quantifying the coherence characteristic.

Fig. 12. (a) The spectra of ECG and blood pressure wave are shown, where the peak in the range of 0 ~ 0.4 Hz is due to the modulation signals and the side peaks around harmonics (sub-harmonics) are attributed to the inter-modulation effect. (b) The peak amplitude variation of the QRS complex wave of modulated ECG, which results from the AM modulation effect, is plotted with a dashed line.

Powerspectraldensity(dB/Hz)

Frequency (Hz) (a)

AM effect

(Inter-modulation side peaks) Modulation

signal

FM effect

Time (s) (b)

ECG(mV)

f1

f0 f₁ 2 f₁3 f₁ f₀2f₁f0f1f0 f₀f₁f₀2 f₁

Fig. 13. (a) The presented cardiovascular system model is plotted. (b) The schematic diagram of the spectra of original ECG, modulation signal and modulated ECG is illustrated.

Original ECG (Carrier)

Modulated ECG AM and FM Modulation signals

Cardiovascular system

Blood Pressure Wave Perturbation signals

(Noise)

Modulator

Original ECG

Modulation signal

Modulated ECG

‧‧

Modulator

(a)

(b)

Fig. 14. The coherence functions of a healthy subject and a cardiovascular patient are demonstrated.

Frequency (Hz)

Coherencefunction

Fig. 15. The statistical results of power coherence index S of twenty-two healthy subjects and twenty-two cardiovascular patients are demonstrated with box plot.

Healthy subjects Cardiovascular patients

Coherencevalue

CHAPTER 5. BLOOD PRESSURE WAVE UNDER