Elsevier

Signal Processing

Volume 72, Issue 1, 4 January 1999, Pages 15-22
Signal Processing

A QRS estimator using linear prediction approach

https://doi.org/10.1016/S0165-1684(98)00160-1Get rights and content

Abstract

It is shown that the discrete cosine transform (DCT) of a given set of electrocardiographic (ECG) cycles can be approximated to a set of narrow band signals. These signals are estimated by performing singular value decomposition (SVD) on a prediction matrix made up of the DCT coefficients. The QRS pulses are then separated out in the DCT domain from a reduced rank approximation of the prediction matrix. The size of this reduced rank matrix is obtained by means of an optimally established orthogonality relationship between two complementary subspaces derived from the signal space of DCT coefficients.

Zusammenfassung

Es wird gezeigt, daß die diskrete Kosinus-Transformation (DCT) einer gegebenen Menge von Elektrokardiogramm- (ECG) Zyklen durch eine Menge von Schmalbandsignalen approximiert werden kann. Diese Signale werden dadurch geschätzt, daß eine Singulärwertzerlegung (SVD) auf eine Prädiktionsmatrix angewandt wird, deren Einträge die DCT-Koeffizienten sind. QRS-Pulse werden dann in der DCT-Ebene mit Hilfe einer Rang reduzierten Approximation der Prädiktionsmatrix aussortiert. Die Größe der Rang reduzierten Matrix ergibt sich durch einen optimal erstellten orthogonalen Zusammenhang zwischen zwei kompletentären Unterräumen, die vom Signalraum der DCT-Koeffizienten abgeleitet werden.

Résumé

Dans cet article, on montre que la Transformée en Cosinus Discrète (DCT) d'un ensemble donné de cycles d'électrocardiographie (ECG) peut être approximée par un ensemble de signaux à bande étroite. Ces signaux sont estimés par décomposition en valeurs singulières d'une matrice de prédiction formée des coefficients de la DCT. Les impulsions QRS sont ensuite séparées dans le domaine de la DCT grâce à une approximation de la matrice de prédiction. La taille de cette matrice (de rang inférieur) est obtenue grâce à une relation d'orthogonalité établie de manière optimale entre les 2 sous-espaces complémentaires dérivés de l'espace signal des coefficients de la DCT.

Introduction

The electrocardiographic (ECG) signals recorded from the body surface are often used to diagnose abnormalities in the heart's conduction system, which either manifests in the form of rhythm changes or gradual variations in the morphological features. The ECG signals are constituted by a sequence of component-waves separated by regions of zero electrical activity, called iso-electric regions. Under normal conditions, the component-waves repeat themselves in a rhythmic manner with a periodicity determined by the frequency of impulse generation at the sino-atrial (SA) node. The effect of a single impulse excitation as reflected on the body surface is depicted in the ECG cycle shown in Fig. 1. The time evolution of the surface ECG reflects the electro-physiological effects brought about by the propagation of a current-wave resulting from the periodic impulse excitation at the SA node along well defined conduction pathways laid out in the cardiac musculature. The most important of these effects is the concomitant contraction and relaxation of the atrial and ventricular muscles. As shown in Fig. 1, a single ECG beat is made up of three distinct component-waves designated as P-, QRS- and T-waves, respectively. The P-wave characterises the depolarisation of the atrial musculature leading to atrial contraction and occurs at the final third of ventricular filling [5]. The QRS complex corresponds to the ventricular depolarisation followed by the occurrence of the T-wave during ventricular repolarisation, resulting in the consequent relaxation of the ventricular muscles. The interval between the offset point of the QRS complex (called the J-point) and the onset of the following T-wave (Ton) is called the ST-interval. A precise estimation of the ST-segment waveform is often required for the differential diagnosis of Coronary artery disease (CAD) [9]. The extraction of the ST-segment features necessitates a procedure for blanking out the P- and T-waves with an effort to delineate the QRS pulses from the surface ECG.

Over the past two decades, much of the effort, both in hardware and software, has understandably been concentrated on the detection of QRS complexes alone 6, 8, 11. However, the problem of accurately delineating the QRS pulse shape remained elusive until Murthy et al. proposed a rational function representation of the ECG signal in the DCT domain and used a pole-zero model to carry out the delineation of component-waves [7]. Murthy's work brings out the inherent difficulties involved in a direct time domain parametric modelling of the ECG. The advantage gained by performing a DCT is that, the higher order energy coefficients are brought closer to the origin and consequently, the time domain signal gets mapped into a decaying set of coefficients. As a result of this transformation, the DCT of an individual component-wave is approximated by a set of damped cosinusoids with closely spaced frequencies. Each damped cosinusoid, referred to as a fractional component is approximated by a biphasic rational function with second order numerator and denominator polynomials. The delineation of component-waves is accomplished by decomposing the rational function into partial fractions with second order numerator and denominator coefficients. The pole angles in each partial fraction expression is shown to have a one to one relationship with the peak location of each one of the component-waves present. The desired set of component-waves are then reconstructed in the DCT domain by grouping together the subset of those partial fraction expressions which contain poles with their corresponding angles related to the peak locations of the component-waves to be separated out. A major drawback of this approach is that the selective separation of the component-waves require a prior knowledge of their respective temporal locations in the ECG record. Moreover, the subset of parameters needed for the reconstruction is chosen from a partial fraction expansion of the rational function model. This involves lengthy computations when a large number of ECG beats are involved. In view of these difficulties, we now propose a QRS delineation method which works independently on each lead and performs the task of separating out the QRS morphology. The advantage of using the present method lies in the fact that it is well suited not only for noisy ECGs but also for those containing ectopic and missed beats. Some of the other important features of the proposed method are that it does not require a prior knowledge of the position of QRS pulses and is capable of providing a more accurate description of the QRS morphology irrespective of its temporal location in the recorded ECG.

Section snippets

Theory

Considering N samples of an ECG record y(n), a sequence ỹ(n) of length 2N is constructed with ỹ(n)=y(n)forn=0,1,…,N−1, and zero elsewhere. A 2N-point discrete Fourier transform (DFT) of the sequence ỹ(n) is defined asYF(k)=n=02N−1ỹ(n)expjnkπN,fork=0,1,…,2N−1.The sequence obtained by even extending y(n) is denoted by ye(n)=2y(0),n=0,y(n),1⩽n⩽N−1,0,n=N,y(2N−n),N+1⩽n⩽2N−1.The discrete cosine transform (DCT) of the sequence y(n) is defined as the first N points of the 2N-point DFT of ye(n),

QRS estimator

We refer to the term `QRS subspace' to denote a set of narrow band signals represented by the largest set of eigenvalues λ1>λ2>⋯>λu, with up and the complementary subspace containing the P and T wave information, as represented by the remaining set of the pu eigen values. We start with a given prediction matrix XRL of size L×Np (L=p+1) which is full rank and spanned by the columns of any unitary matrix in RL×L, the Euclidean space of square and real valued L dimensional matrices. The same

Summary of the procedure

A closed procedure including all steps beginning with the data record (ECG) with N samples and ending with the reconstructed QRS cycles is as follows:

  • 1.

    Find the N-point DCT of the data record in accordance with the definition for DCT as given in Eq. (3).

  • 2.

    Decide the value of p: As discussed earlier, choosing the best value for p is crucial to achieving the best performance of the estimator. If the choice of p is too low (lower than the number of peaks in the Fourier spectrum of the DCT

Conclusions

It is demonstrated that the component-waves present in the time domain ECG record get mapped into a set of narrow band signals in the DCT domain. The narrow band signals are selectively estimated by performing an SVD operation on a prediction matrix of size as determined by the appropriate number of these signals with which the component-waves are approximated in the DCT domain. Since the time domain features of the QRS components are manifested by pulses having very short duration with sharp

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