The method used in this study consists of applying a low level rectangular current and high frequency (1 mA, 30 kHz), through a pair of electrodes placed respectively in the front and above the leading edge of the heart [12] -[15] . Another pair of electrodes, placed on the chest of the patient at the level of aorta 2 or 3 cm apart, permit perception of bioimpedance signal [BIS] representing impedance variation ∆Z of the explored thoracic region. Figure 1, shows the electrode configuration for the measurement of the bioimpedance signal.

The aim of this bioimpedance signal analysis is the diagnosis of cardiac diseases by means of cepstral processing of this signal using Fisher theory [16] -[18] .

Cepstral method consists on considering that bioimpedance signal y(t) is the response of left ventricle aorta system to a cardiac excitation signal x(t) and the aorta pulsatile response h(t) (Figure 2):

Then:

(Temporal convolution product)

Cepstral analysis consists on the determination of excitation signal x(t) and pulsatile response h(t), in order to describe, separately, anomalies, respectively, in heart and aorta. Computation is carried out at the minimum phase (Φ = 0).

Let:

where:

Let:

y₁(t) is the Cepstre C1

where:

Let:

Let:

is the Cepstre C2

is the Cepstre C3.

and are considered as the original signal provided, respectively, by heart and aorta.

Figure 3, shows the different steps of the cepstral algorithm.

Early, a statistical study, using the discriminant method analysis, has been performed [17] [18] . This study consists to use 15 parameters: five temporal variables from bioimpedance signal and its derivative (A, C, O, X, S), 3 spectral parameters (r₁, r₂, r₃) and seven cepstrals variables (U, M, N, F, I, G, LF) (Figure 4 and Figure 5) and (Table 1).

Our idea in this study is to use the seven cepstral parameters for the automatic diagnosis of the heart disease

using Fisher’s test. Cesptres C2 and C3 permit to provide these seven relevant parameters: U, M, N, F, I, G, LF (Table 2).

The principle of discriminant analysis is based on FISCHER theory and the criteria of “Step by Step”. The relevant plethysmographic parameters represent the set of parameters which allows having the maximum of matrix product T⁻¹ E. Where T is whole covariance matrix, E is the interclass covariance matrix. The classification of anonymous individuals is based on the use of the FISHER formula [16] :

is the Fisher distance between an anonymous individual and class Y_k, a is the anonymous individual defined by cepstral parameters, y_k is the average of Y_k classes, T_cov is whole covariance matrix.

Computed algorithms are expressed by a MAHAL 3 program [5] . The determination of the best discriminant parameters is carried out at each step from a basic sample (normal and cardiovascular diseases) with a dimension N calculated as follows with an error risk of 5% [8] [17] :

P is the total of average cepstral parameters corresponding to 25 classes (Table 3)

Bioimpedance parameters, proposed for the discrimination between the classes, are in this study 7 cepstral parameters.

After testing the seven parameters during the first step, the program indicates the parameter number 7 which represents the normalized width LF of the aortic cepstral. Therefore, the parameter number 7 is the best discriminant plethismographic parameter. The best classified percentage of individual is then 64.29% (Table 3).

At steps number 2, 3 and 4, the program choose, respectively, parameters number 7, 5, 6 and 4 corresponding respectively to the parameters: LF, I, G and F. The classified percentage is then 86.01%. At step 5, the percentage of classification reaches 93.66% the parameters are 7, 5, 6, 4, and 3 corresponding to the parameters: LF, I, G, F and N. Finally at step number 6 and 7 the program choose parameters 2 and 1 corresponding to M and U respectively with the percentage 94.1% and 95.4%.

The total 7 independent parameters (Table 4), gives 99.4% degree of best classification. Therefore Bioimpedance cepstral parameters with best discrimination are: 7(Lf), 5(I), 6(G), 4(F), 3(N), 6(M), and 7(U).

Automatic diagnosis method consists on preparing, first, a data base with a set of the seven cepstral parameters

(AO.I: aortic insufficiency; AO.S: aortic stenosis; AO.D: aortic diseases; M.I: Mitral Insufficiency; M.S: Mitral stenosis; M.D: Mitral diseases; PS: pulmonary stenosis; IVC: Inter-ventricle communication; IAC: inter-atrium communication; CMP: Cardio-myopathie).

of different bioimpedance signal according to different cardiac diseases and the formula (18). This data base is composed from n classes Yk corresponding to 25 cases (normal and cardiac disease).

The classification of anonymous individuals is based on the use of FISHER formula (8). Minimum dm distance, between a and the Yk, classes provides the kind of cardiac disease. Investigation has concerned a data base of 25 kinds of signal: one normal and 24 pathological cases (Table 3). The number of cross indicates the severity of the disease.

Three cases of anonymous signals are used (a1: AO.S+), (a2: M.S++) and (a3: M.S+++). The diagnosis of these three anonymous cases is confirmed by Echo-Doppler method. Table 5 shows affectation of these cases.

From Table 4, it can be noted that the parameter 7 (LF) has the best discriminating power with a percentage of well class 64.29%, which confirms the results found by Ben Salah et al. [8] [17] .

At step 7 the percentage of well class reaches 95.40%. This result is slightly better than the one we found in previous work using 15 bioimpedance parameters: 94.64% of percentage of correctly classified.

The results found in this work indicate that the seven cepstral parameters defined above are sufficient to perform the automatic diagnosis of the cardiovascular system abnormalities.

The effectiveness of the cepstral parameters classification is confirmed by the exact allocation of 3 anonymous individuals. Indeed our results demonstrate that patients a1, a2, a3 have been allocated respectively to the previous classes: AO.S+ (d = 0.1), M.S.++ (d = 0.1), and D.M.+++ (d = 0.02).