Despite recent advances in the treatment of life-threatening ventricular arrhythmias, sudden cardiac death (SCD; an unexpected death due to cardiac causes occurring within an hour of symptom onset in a person with known or unknown cardiac disease) remains one of the leading causes of death in developed countries [1] . At the present time, patients are selected for clinical evaluation and treatment of ventricular arrhythmias only after they have experienced and survived a major cardiac event. Thus, from a public health viewpoint, identification and treat- ment of such high-risk subjects before the occurrence of a cardiac event are expected to have a great impact on the problem of SCD [2] .

The simplest way to diagnose most of the cardiac abnormalities that can lead to SCD is to perform an elec- trocardiogram (ECG) test, which is noninvasive, painless and affordable, and provides a recording of the elec- trical activity of the heart by means of surface electrodes disposed on the skin in standardized positions. The ECG signal is, by its nature, an analog pseudo-periodical signal constituted by the repetition of a typical wave- form-complex, representing a cardiac cycle (heartbeat) (Figure 1). At rest cardiac cells are polarized (electrical- ly negative inside) but, if electrically stimulated, can temporally depolarize to generate the myocardium contrac- tion. Under normal conditions, the electrical impulse is spontaneously generated by the sinoatrial node. Such impulse is then propagated throughout the right and left atria via the internodal tracts to stimulate the atrial myocardium contraction, generating the P wave. The internodal tracts end in the atrioventricular node which delays ventricular contraction, as represented by the PR segment. From the atrioventricular node the electrical impulse is propagated to the ventricles through the Bundle of His that splits into the left and right bundle branches, which further taper out into numerous Purkinje fibers to stimulate contraction of individual groups of myocardial cells. The spread of electrical activity through the ventricles is represented by the QRS complex, which also ob- scure most of atrial repolarization. Eventually, the ventricles repolarize, generating the electrocardiographic J wave, ST segment, and T and U waves. Nowadays most ECG recorders acquire the ECG signal in digital format, so that it can be stored in a computer memory and automatically processed for information extraction.

Among all the possible causes of SCD there are the abnormalities in the repolarization phase of the heart, which are known to be associated to susceptibility to malignant ventricular arrhythmias and SCD [3] . The stan- dard indicator of cardiac safety in clinical trials is the QT interval [4] , which is measured as the time distance between the onset of the Q wave and the offset of the T wave, thus representing the total duration of the contrac- tion and subsequent relaxation of the ventricles. The QT-interval measure, however, is strongly method- [5] and lead-dependent [6] , and thus little reliable. The difficulties in accurately measuring the QT interval, together with the clinical observation that not all pronged QT intervals necessarily lead to ventricular arrhythmias [7] , has stimulated the interest in identifying new alternative markers of abnormal repolarization [8] , among which mi- crovolt T-wave alternans (TWA) is the most promising one [2] [9] -[16] .

T-wave alternans (TWA) consists in an alternation of the ECG T-wave amplitude on every-other-beat bases. Macroscopic (i.e. visible) forms of TWA are quite rare and may even show alternating T-wave polarity (Figure 2). Instead, microvolt forms of TWA are more common, but require specifically designed algorithms to be iden- tified [17] -[19] because invisible to the naked eye. Both macroscopic and microvolt TWA have been associated to malignant ventricular arrhythmias and SCD [2] [9] -[16] .

The heart-rate adaptive match filter (AMF) based method is a popular technique for automatic TWA identifi- cation [20] . It is has been used in several comparative [18] [19] [21] and clinical studies [10] [22] -[26] , each of which has identified a specific peculiarity of the technique. Nevertheless, a detailed technical description of the filter, which allows a clear understanding of its methodological principles, has never been reported. Thus, aim

of the present study was to provide such a detailed AMF description. A simulation study was then used to dem- onstrate the goodness of the AMF-based method in identifying TWA in the typical noisy conditions usually af- fecting real ECG tracings.

Ideally, at fixed heart rate (HR), TWA is characterized by a single frequency, by definition equal to half heart rate. However, in real clinical ECG tracings some physiologic HR variability always occurs. Consequently, the AMF-based method supposes TWA to be characterized by a small frequency band centered in half mean heart rate (f_TWA), and conceives the AMF as a HR (and, thus, f_TWA) adaptive narrow-band passing filter (ideally a match filter) with its passing band centered in f_TWA. Its implementation consists of a 6^th order bidirectional Butterworth band-pass filter characterized by a 0.12 Hz wide passing band centered in f_TWA, which can be thought as cascade of a low-pass filter (LPF; cut-off frequency f_LPF = f_TWA + df_TWA, with df_TWA = 0.06 Hz) and a high- pass filter (HPF; cut-off frequency f_HPF = f_TWA − df_TWA) [20] . The squared module of the AMF is expressed by the following equation:

In the practical computerized applications involving digital ECG tracings the AMF has to be designed in the digital domain. To compute the digital AMF filter coefficients b_i and a_i (i = 0, 2… 6), bilinear transformation with frequency prewarping was used to convert the analog filter into a digital filter. Careful frequency adjust- ment guarantees that the analog filter and the digital filter will have the same frequency response magnitude at the cut-off frequencies. The AMF transfer function is reported in Equation (2), where the actual values of b_i and a_i parameters depend on HR and sample frequency (a₁ = a₃ = a₅ = 0):

An example of digital AMF magnitude and phase responses relative to a mean HR of 80 bpm (or 1.33 Hz) and a sampling rate of 200 Hz is displayed in Figure 3. The magnitude response is a narrow passing band around 0.67 Hz, which corresponds to half HR, while the phase response is a non-linearly descending curve.

To avoid distortions due to the non-linearity of the phase response that causes the group delay not to be con- stant, the AMF is applied in a bidirectional fashion, that consists in a zero-phase digital filtering procedure per- formed by processing the input ECG data in both forward and reverse directions. The overall filter transfer func- tion equals the squared magnitude of the original filter transfer function and doubles its order, so that actual fil- tering order becomes 12.

Each time the AMF is fed with an ECG, it first computes HR and corresponding f_TWA, and then filters out every ECG components but the one relative to TWA. Thus, the output of the AMF is an amplitude-modulated sinusoidal signal, called the TWA signal, having the same length of the input ECG and characterized by a fre- quency which matches f_TWA. If really pertaining to TWA (and not, for example, to QRS oscillations which match f_TWA), the TWA signal maxima and minima have to fall inside in the ECG repolarization segment (ST segment and T-wave) (Figure 4). The local amplitude of the TWA signal in correspondence of the i^th beat provides a quantification of the TWA amplitude (TWAA_i, in µV) characterizing that beat. If the TWA signal has its maxima and minima outside the repolarization segment (as in the presence of QRS alternans, for example), the TWAA_i values are set to zero (Figure 4). Eventually, in case of an ECG tracing affected by no alternans (of any kind), the TWA signal at the output of the AMF reduces to a zero constant signal. Consequently, all the TWAA_i values are equal to zero. A global measure of TWA amplitude (TWAA) relative to an ECG tracing can be ob- tained by averaging all TWAA_i values over the number of beats.

This study describes how adaptive match filtering can be used in a biomedical application finalized to TWA identification from digital ECG signals. TWA is a promising indicator of SCD ideally characterized by a fre- quency that, by definition, matches half HR (i.e. f_TWA). Although TWA is an electrophysiological phenomenon which implies stable sinus rhythm, some small variations of the HR are usually acceptable, so that the pheno- menon is more properly characterized by a very narrow band around f_TWA. Thus, a heart-rate adaptive filter, with a narrow passing band appears as a useful tool to extract the frequency components characterizing TWA from an ECG tracing. While extracting the TWA signal, the AMF simultaneously filters out any other ECG component, including noise and artifacts. Because of this, the AMF-based method for TWA identification is a technique particularly robust to noise and interferences likely affecting real ECG tracings [21] . Still, other kinds of alterna- tions, such as QRSA and some noise components, may fall in the narrow frequency band characterizing TWA, jeopardizing its correct identification. To minimize the probability of such occurrence, the phase of the TWA signal is analyzed before measuring its amplitude to determine TWAA. The maxima and the minima of the TWA signal, which is a pseudo-sinusoid, occur in correspondence of the center of mass of the alternations at f_TWA present in the ECG signal. If such alternations mostly pertain to the T-waves, such maxima and minima must fall in correspondence of such waves. If this is not the case, TWA is not identified.

Usually, the AMF is realized as a 6^th order Butterwort filter, characterized by a magnitude response that is maximally flat in the pass-band and monotonic overall. The Butterworth filter sacrifices roll-off steepness for monotonicity in the pass- and stop-bands, but this is does not represents a limitation for the AMF. Indeed, it is applied bidirectionally, so that the final filtering order is 12, which is quite high for ECG applications (roll-off steepness increases with increasing filter order).

In the simulation study reported in this work, a MATLAB implementation of the AMF-method provided by B.M.E.D. Bio-Medical Engineering Development SRL (http://www.bmed-bioengineering.com) was used for TWA identification from synthetic ECG tracings affected by the most typical and representative noise factors. Results confirm the goodness of this technique in both avoiding false-positive as well as false-negative identifi- cations, even though the presence of noise and interferences, such as HRV, may compromise perfect quantifica- tion of TWAA. The noise level and the baseline amplitude considered here were both 100 µV, i.e. equal to TWAA. Thus, the signal-to-noise ratio was low, around 1. In the worst corrupted conditions, the maximum error reported was 20% which does not represent a significant practical problem from a clinical point of view, since it is usually much more important to know if TWA is present or not and its order of magnitude rather than its exact amplitude. Very small levels of TWA (for example under 10 µV) are usually considered negligible, and could even be eliminated using a threshold criterion.

Heart-rate adaptive match filtering represents a useful technique for a reliable TWA identification from the digi- tal ECG. The filter can efficiently be implemented using a digital 6^th order narrow-passing-band Butterworth fil- ter, which has to be specifically designed for each ECG tracing, since the passing band has to centered around the f_TWA, which is by definition, equal to half heart-rate. Given the non-linearity of the phase response, the filter has to be applied in a bidirectional fashion to avoid signal distortions.