**3.6 Optimisation of the parameters of the back-propagation algorithm**

Fig. 3.2 shows the effect of changing the learning rate (), the momentum constant () and the middle layer size on the classification of the maternal QRS-complexes and the fetal heartbeats using the BIC template matching technique. The effect of changing the learning rate on the classification rate is shown in Fig. 3.2 (a). Small values of are not able to track the variations in the bispectral contours. For classification of the bispectral contours, reaches its optimum value at 0.2. For values larger than 0.2, the output values are too large so that the difference with respect to the reference signal (template) will increase. This leads to larger error that will be fed back to the network, which will lead to slower convergence. The network will take long time to converge, or it might not converge at all. The optimum value of the momentum constant is found to be 0.2, as depicted in Fig. 3.2 (b). Smaller values are not enough to push the adaptations to avoid local minima. While larger values tend to affect the routine detrimentally by bypassing the global minimum. The performance deteriorates significantly as the learning rate and the momentum constant diverge from their optimum values. The number and size of the middle layers were investigated by trial and error. There is a trade off between networks that should be small enough to allow faster implementation, and larger networks in size and number of hidden layers which are very slow and can not be implemented on-line using the current technology. Large networks could have complex relationships that represent nonlinearities that might not exist in the real signals at all. The optimum parameters indicated in Fig. 3.2 are calculated without considering the CPU time factor which might render those parameters undesirable for real-time applications. The CPU time for training is in the range of 17 to 60 sec. The average mean-squared error (MSE) is 0.04. The worst error is 0.1, which is the criterion for convergence. The implemented neural network has a single middle layer size of 6 x 6 as shown in Fig. 3.2 (c). The number of passes (epochs) required for training varied from 6 to 14.

Detection and Classification of Adult and

parallelise (Zgallai, 2012).

Fetal ECG Using Recurrent Neural Networks, Embedded Volterra and Higher-Order Statistics 239

The bispectral contour template matching technique works with only 10 templates of maternal transabdominal QRS-complexes and 20 templates of fetal heartbeats with maternal contribution. The classification of the four segments involves a pattern-by-pattern updating rather than batch updating for the weight adjustments. This is more suitable to speed up the performance. Pattern-by-pattern updating tends to be orders of magnitude faster than batch updating. However, it should be noted that pattern-by-pattern updating is harder to

Table 3.1 shows a top classification rate of 100% for maternal QRS-complexes using bispectral contours for signal processing and single-hidden-layer perceptron classification. The 100% maternal QRS-complex classification rate has been achievable with or without linearisation. It makes no difference to the results. A brief description of the optimised parameters required for the linearisation process is given. To calculate the maternal heart rate an auxiliary method to pinpoint the R-wave is needed. For this application an adaptive thresholding method has been employed. Note that this is not very accurate when one deals with deformed QRS-complexes in heart patients. All maternal ECGs, however, exhibit normal-to-the-patient QRS-complexes. The instantaneous maternal heart rate is calculated by dividing 60 by the R-to-R interval (in seconds). The application of this auxiliary routine

*Parameters:* The second-order Volterra parameters are: filter length = 6, step-size parameters = 0.005, and 0.0004 for linear and quadratic parts, respectively, delay = 3. The third-order Volterra parameters are: filter length = 6, step-size parameters = 0.001, 0.0002, and 0.0004 for linear, quadratic and cubic parts, respectively, delay = 4. A dual-band-pass filter is applied to the bispectrum, the first has a band-pass of 10 Hz to 20 Hz and the second has a bandpass of 25 Hz to 40 Hz. Optimised Kaiser windows centred at frequencies of 15 Hz, 16 Hz, 17 Hz, 18 Hz, and 19 Hz for the maternal spectrum, and at frequencies of 28 Hz, 29 Hz, 30 Hz, 31 Hz, 32 Hz, 33 Hz, 34 Hz, 35 Hz, 36 Hz, 37 Hz, and 38 Hz for the fetal spectrum are

**3.8.2 Fetal heartbeat detection quality and classification rate for the bispectral** 

The classification rate of non-coincident maternal and fetal QRS-complexes is 99.21%.

Table 3.2 summarises the results of the fetal heartbeat detection using the power spectrum method (second-order statistics), and the bispectrum contour template matching technique. Optimised adaptive LMF-based second- and third-order Volterra synthesisers are employed. The power spectrum method has a classification rate of 71.47%. Using the second hybrid system, the classification rate increased to 87.72% without linearisation, and to 88.28% and 90.12% using second- and third-order Volterra synthesisers with LMF update, respectively. The second hybrid method has an improvement of 19% and 4% in the classification rate over and above that achieved with the second-order statistics and the TOC template matching technique, respectively. The classification rate of the coincident maternal and fetal QRS-complexes is 0%.

**3.7 Bispectral contour matching of the maternal QRS-complex and fetal heartbeat** 

**3.8 The maternal QRS-complex and the fetal heartbeat classification rates** 

**3.8.1 The maternal QRS-complex classification rate** 

leads to a maternal heart rate with an accuracy of 99.85%.

used in both the power spectrum and the BIC.

**contour template matching technique** 

Fig. 3.2. The effect of changing (a) the learning rate, (b) the momentum constant, and (c) the middle layer size on the classification rate of the maternal QRS-complexes (l.h.s.) and fetal heartbeats (r.h.s.) from transabdominally-measured ECG signals using bispectral contours and their templates to be matched using a single-hidden-layer perceptron backpropagation algorithm with momentum. Data length = 250 msec. The optimised parameters for the BIC classification are: learning rate = 0.2, momentum constant = 0.2, and middle-layer size = 6 x 6. (Code: 5-1-100).
