INTERNATIONAL JOURNAL OF ENHANCED RESEARCH PUBLICATIONS
Vol.2 Issue X, Month, 2013 ISSN No: XXXX-XXXX
Signal-dependent Orthogonal Transform for ECG Signal Analysis
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH PUBLICATIONS
Vol.2 Issue X, Month, 2013 ISSN No: XXXX-XXXX
H. Baali, R. Akmeliawati and M.J.E. Salami
Department of Mechatronics Engineering, International Islamic University Malaysia (IIUM)
Jalan Gombak, 53100 Kuala Lumpur, MALAYSIA .
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INTERNATIONAL JOURNAL OF ENHANCED RESEARCH PUBLICATIONS
Vol.2 Issue X, Month, 2013 ISSN No: XXXX-XXXX
www.erpublications.com
INTERNATIONAL JOURNAL OF ENHANCED RESEARCH PUBLICATIONS
Vol.2 Issue X, Month, 2013 ISSN No: XXXX-XXXX
www.erpublications.com
INTERNATIONAL JOURNAL OF ENHANCED RESEARCH PUBLICATIONS
Vol.2 Issue X, Month, 2013 ISSN No: XXXX-XXXX
Abstract: Linear prediction is formulated to develop a real valued, signal dependent orthogonal transform with good energy compaction property. In order to appreciate the effectiveness of the technique, applications to Electrocardiographic (ECG) signals compression and interpolation are discussed and benchmarked against the state-of-the-art DCT transform. Results show that the technique achieves competitive results. An insight is given to use the technique for model order selection.
Keywords: ECG, Period normalization, Linear prediction, Signal-dependent transform, SVD.
Introduction
Linear prediction (LP) has been extensively used for signal and spectral analysis [1]. The idea of LP is to consider the observed signal as the output of an all pole filter of order excited by some unknown input. The best prediction coefficients are found by the minimization of the sum-of-squared error (SSE) between the original and the predicted signal with respect to the LPC coefficients.
The original signal can be recovered by using:, where and are respectively the columns vectors of the observed signal samples and the residual error, while is the impulse response matrix of the synthesis filter (also called LPC filter) whose entries are completely determined by the linear prediction coefficients. is a lower triangular and Toeplitz matrix of the form:
(1)
Applying the singular values decomposition (SVD) to gives:
, (2)
where and are orthogonal matrices, and is a real valued diagonal matrix with the singular values of on its diagonal,denotes the transpose. The aforementioned decomposition has been adopted for accurate representation of the excitation signal applied to speech signal in [2] and for ECG period normalization in [3]. In this work, the decomposition is used to build an orthogonal transform that can be applied in a wide range of areas.
We define the forward and inverse transforms respectively as follows: and .
From (2), each component of the residual signal () is mapped onto the space spanned by and then weighted by the corresponding singular value. Knowing that the singular values are always decaying, one can expect that the transform coefficients vector would be sparse.
ECG signal compression
Once the transform coefficients are calculated and sorted in descending order, a compression ratio(CR) is achieved using the proposed transform by keeping the largest coefficients while the remaining are discarded ( LP coefficients are saved as side information to form the matrix ) . The same CR is achieved using the DCT by maintaining the largest transform coefficients [4].
The time domain signal retrieval is accomplished by inserting zeros in the transform domain to replace the discarded coefficients followed by the application of the inverse transforms.
When both DCT and the proposed transform are applied to an ECG signal as shown in Fig. 1, most of the signal energy is packed in a few coefficients where the shape is damped sinusoidal as shown in Fig. 1(b).
Fig. 1 Transform based ECG compression
(a) Original and reconstructed ECGs,
(b) Transform domain vector and DCT coefficients vector
The performance of the two techniques are evaluated on signals with large waveform characteristics variations namely, normal sinus rhythm (N) and premature ventricular contraction (PVC). The data is taken from the MIT-BIH Arrhythmia database. The widely used percent root mean square difference () is employed to measure the distortion between the original and reconstructed signal ( ), that is,
(3)
Different compression ratios are tested using different model orders and the corresponding PRDs obtained after signal reconstruction are summarized in Table 1. Results when the LP coefficients are not taken into account when computing the CR are also reported in TABLE I.
In the case where the LP coefficients are not taken into account, the proposed technique outperforms the DCT for high CR. However, for low CR, DCT achieves better PRD for normal rhythm. When the LP coefficients are taken into account DCT achieves slightly better results with exception to CR 3:1 for PVC rhythm. For both transforms, compression performances are better for PVC rhythm when compared to the normal sinus rhythm.
TABLE I. COMPRESSION PERFORMANCES FOR DIFFERENT COMPRESSION RATIOS COMPARED TO THE STATE-OF-THE-ART DCT TECHNIQUE
CRPRD / 3:1 / 4.5:1 / 6:1 / 9:1
Normal sinus rhythm
DCT / 0.9609 / 3.0815 / 7.5394 / 22.5177
Proposed ( =1) / 1.2999 / 3.2173 / 8.2514 / 24.1802
P=1, LPC not maintained / 1.2754 / 3.1279 / 7.7170 / 23.0390
Proposed ( =2) / 1.6787 / 3.5586 / 8.3577 / 24.8113
P=2, LPC not maintained / 1.6276 / 3.3526 / 7.5413 / 22.3715
Proposed ( =3) / 1.9316 / 4.0540 / 8.2409 / 23.6260
P=3, LPC not maintained / 1.8421 / 3.6212 / 7.2555 / 19.7117
Proposed ( =4) / 1.7023 / 4.1814 / 9.2708 / 25.3171
P=4, LPC not maintained / 1.5814 / 3.5539 / 7.6041 / 20.5481
PVC
DCT / 0.6142 / 1.0441 / 1.6895 / 4.3642
Proposed ( =1) / 0.6262 / 1.0949 / 1.8725 / 4.4706
=1,LPC not maintained / 0.6213 / 1.0755 / 1.8275 / 4.2998
Proposed ( =2) / 0.6054 / 1.1357 / 1.9606 / 4.7059
=2, LPC not maintained / 0.5969 / 1.0890 / 1.8532 / 4.3381
Proposed ( =3) / 0.6089 / 1.1590 / 2.0187 / 4.9107
=3, LPC not maintained / 0.5962 / 1.0867 / 1.8518 / 4.3391
Proposed ( =4) / 0.6016 / 1.1429 / 2.1006 / 5.1777
=4, LPC not maintained / 0.5835 / 1.0497 / 1.8816 / 4.4091
ECG signal Interpolation
In the special case of first order linear prediction, time compression and stretching can be accomplished using the following steps:
1) Compute the left singular vectors of the impulse response matrix to construct an orthonormal transform (matrix U).
2) Map the ECG period () from the time domain to the SVD domain (the transform domain) by using: .
3) Zero pad (when the signal is to be stretched) or truncate (when the signal is to be shrunk) the transformed ECG vector () to match the desired length). The resulting vector is called.
4) Multiply by the matrix to form a normalized ECG period (. Note that represents the impulse response matrix of the desired size and is given by (4), and is its left singular vectors matrix.
(4)
5) Apply reverse steps to recover the heartbeats with their original lengths. The recovered signal is shown in Fig. 2 along with the residual error (this last step might be needed in some applications but not always).
Similarly to the DCT, the method is reversible when the signal is expended (All the transformations are isometric). On the other hand, the distortion due to time compression (shrinking) of the signal is minimized due the energy packing efficiency of the technique.
The basis functions of the transform matrix can be found analytically as follows:
, for = 1, ..., and = 1, ..., .
The constant is chosen so that the norm of is unity.
, k = 1,..., N, are the solutions of the equation:
(5)
is the LP coefficient
Similarly, the bases functions of are given by:
, for = 1,..., and = 1, ..., .
The constant is chosen so that the norm of is unity.
k = 1,...,N*, are the solutions of the equation:
(6)
Fig. 2 Distortion-free recovering of a normal signal (stretched by 50% of its original length)
Model order selection (MOS)
Accurate MOS has been a challenging problem in many areas of signal processing where several approaches have been reported in the literature [5]. For ECG signals, the reported prediction orders varies between one and four [6][7]. Using the proposed transformation, it is observed that the transform efficiency keeps improving when the model order increases from one to three then decreases for a model order of four as shown in Fig. 3. This observation holds for both normal and PVC rhythms analyzed in this paper. Consequently, the proposed technique provides an alternative solution to MOS which would be further investigated in our future research work. The transform efficiency is defined as the ratio between the energy retained in the first mth transform coefficients and the energy of the entire ECG period (or equivalently all the transform coefficients).
Fig. 3 Transform efficiency for PVC rhythm
Acknowledgments
This work was supported by the ministry of higher education (MOHE) of Malaysia under the fundamental research grant scheme FRGS.
Conclusion
A signal dependent transform that has the ability to adjust to the signal characteristics is introduced and applied to some fundamental signal processing problems. The overhead side information needed to generate the basis functions is reduced when compared to other signal dependent transforms. An analytical model has been developed to generate basis functions in the case of first order linear prediction.
References
[1]. Vaidyanathan, P. P, “The Theory of Linear Prediction” : Morgan & Claypool, London, U.K, 2008.
[2]. Atal,B., “A model of LPC excitation in terms of eigenvectors of the autocorrelation of the impulse response of the LPC filter,” ICASSP 1989,1,45-48.
[3]. Baali,H, Akmeliawati,R, Salami,M.J.E, Aibinu, M., and Gani,A, “ Based Approach for ECG Period Normalization,” Computing in Cardiology, 38,2011, pp. 533-536.
[4]. Ranjeet, K. , Kumar, A., and Pandey, R. K. , “ECG Signal Compression Using Different Techniques,” Communications in Computer and Information Science,” 125, 2011, pp. 231-241.
[5]. Stoica, P., and Selen, Y, “Model-order selection: A review of information criterion rules,” IEEE Signal Processing Magazine, 21(4 ) , 2004,pp. 36-47.
[6]. Jalaleddine, S.M.S, Hutcgens, C.G, Strattan, R.D, and Coberly, W. A, “ECG data compression techniques-A unified approach,” IEEE Transactions on Biomedical Engineering,” 37( 4), 1990, pp. 329-343.
[7]. Arnavut, Z, “ECG Signal Compression Based on Burrows-Wheeler Transformation and Inversion Ranks of Linear Prediction,” IEEE Transactions on Biomedical Engineering, 54(3), 2007,pp. 410-418.
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