The 12-lead electrocardiogram (ECG) plays an essential role in the diagnosis of cardiac diseases. In wireless body sensor networks, which are state-of-the-art in patient monitoring, the multi-lead ECG signals need to be transmitted for an automated assessment of the patient’s health status. Due to bandwidth limitations and high energy demands of wireless communication, it is inevitable to reduce the amount of ECG data prior to transmission.
In recent years, compressed sensing (CS)  has materialized as a powerful method in the field of signal processing. It is a novel sensing paradigm that allows for sub-Nyquist sampling of signals that are sparsely representable in some orthogonal basis. As such, CS is a methodology that has lately been proposed to solve the problem of energy-efficient, real-time ECG compression in wireless sensor networks.
Related research in this field has proven the applicability of CS methods to multi-lead ECG compression. In , the authors showed that the common support between neighboring ECG leads can be exploited for improved reconstruction. However, their approach incorporates computationally expensive beat detection and period normalization steps, which renders it inapplicable on energetically constraint sensor nodes. Singh, et al.  propose the compressive sensing in Eigenspace for sub-Nyquist sampling of multi-channel electrocardiogram signals. They have shown that by exploiting the correlation structure within ECG signal ensembles, a higher compression ratio can be achieved. Yet, their approach requires the knowledge of the covariance matrix for the Eigenspace transform of the ECG signals. This matrix is generally unknown during reconstruction.
Within the last years, different hardware architectures have been proposed which perform compressed sensing prior to analog-to-digital conversion. In this work, we have conducted a simulative study aimed at exploring the potential of a novel CS architecture for energy-efficient sampling of multi-lead ECG signals - the Filtered Modulated Multiplexer (FM-Mux) as proposed by Ahmed, et al. . For signal reconstruction, we have adopted the greedy sparse approxmitation algorithm Simultaneous Orthogonal Matching Pursuit (SOMP, ) to the FM-Mux compression while targeting the exploitation of the joint sparsity structure of multi-lead ECG signals for improved performance.
The paper is organised as follows. In Section 2. 1, a short introduction into the general CS methodology is given. The compressive multiplexing of multi-channel ECG signals by means of the FM-Mux architecture is outlined in 2.2. Section 2.3 presents the joint sparse reconstruction by of multi-lead ECG signals by an adopted SOMP approach. The results of the simulative study are shown in Section 3 and, finally, a conclusion is given in Section 4.
Notation: Please note that boldfaced typesetting denotes either vectors or matrices.
2.1 Compressed sensing background
Assume that x ∈ ℝN is a signal which is K-sparse in some orthogonal basis Ψ ∈ ℝN×N, i.e. it can be expressed as x =, Ψα where α ∈ ℝN has only K non-zero entries. Let Φ ∈ ℝM×N, where M < N, be the sensing matrix that describes the compressive sampling of x in form of y = Φx = ΦΨα. The task of Compressed Sensing  is the recovery of the original signal x from undersampled linear measurements y.
2.2 Compressive multiplexing of multi-lead ECG signals
The FM-Mux, as it is depicted in Figure 1, was first presented by Ahmed, et al. . It is an extension to the compressive multiplexer as it was proposed by Slavinsky, et al.  to achieve efficient compressed sensing of correlated signal ensembles. In this work, we adopted the proposed FM-Mux architecture to achieve sub-Nyquist sampling of multi-lead ECG signals. Therefore, we will give a short outline of the FM-Mux concept in principle. For a thorough mathematical analysis, the reader is referred to the original publication.
Compressed sensing of multi-lead ECG signals can be achieved by utilizing the FM-Mux as follows. First, each ECG signal xm(t) is modulated with a binary waveform dm(t) at rate Ω > W, where W is the bandwidth of the ECG signals. Second, the signals are convolved with linear time invariant (LTI) filters with impulse responses hm(t) of bandwidth Ω. Finally, the signals are added together onto a single channel and uniformly sampled by an ADC at rate Ω.
The modulation waveforms dm(t) randomly switch their value to ±1 at a rate of Ω. This embeds the signals xm(t) into different subspaces of ℝΩ which facilitates their separation during reconstruction. Since the sampling process can be swapped with the signal addition, the modulation can equivalently be expressed as a multiplication of the signal samples at rate Ω with the modulation sequence. Thus, on the time interval t ∈ [0, 1) the modulation can be written as multiplication of signal xm(t = n/Ω), where 0 n ≤ n ≤ Ω − 1, with the corresponding diagonal modulation matrix Dm, whose non-zero entries are [dm(0), dm(1),..., dm(Ω − 1)]. The binary sequences dm(t) can efficiently be generated, e.g., by linear feedback shift registers.
The LTI filters hm(t) aim at dispersing the signal energy across time, making the FM-Mux effective for correlated signals that otherwise share a similar initial energy distribution. One can write the action of the LTI filter in the mth channel as an Ω × Ω circular matrix Hm, where(1)
with F and F* as the discrete Fourier transform and its inverse respectively, and Ĥm as a diagonal matrix that holds the Fourier series coefficients of the LTI filter hm(t) scaled by a factor of .
Finally, after analog preprocessing, the ECG signals are added together and sampled uniformly at rate Ω which generates the samples of the compressed signal y ∈ ℝΩ.
The compressive sampling of an ECG signal ensemble by the FM-Mux can coherently be expressed by the set of equations:(2)
where ΦFM−Mux is the Ω × MΩ sensing matrix and X is the matrix composed of the discrete representation of M ECG leads sampled at rate, i.e.
The operator vec(B) returns a vector by stacking the columns of a matrix B.
2.3 Joint sparse reconstruction of compressively multiplexed multi-lead ECG signals
The major challenge in the proposed methodology for the compression of multi-lead ECG signals is the exact recovery of the original signals X from the few measurement samples in y. Assuming the ECG signals are sparse in, at least, some domain, this problem can be stated for each signal separately as a minimization problem with relaxed constraints:(4)
where ε bounds the amount of noise in the data and Am = ΦmΨ is the Ω × Ω dictionary matrix that spans the mth signal space. Am is composed of column vectors am,k which are denoted atoms.
The individual signals in a multi-lead ECG recording all share the electrical stimulation of the heart as a common source. Therefore, it is intuitive to impose a strong inter-signal correlation structure within the ECG signal ensemble that can be exploited during signal recovery. In this work, we focus on the joint sparsity of the ECG signals by utilizing the JSM-2 model , which assumes a common support between the ECG leads.
For signal recovery, we adopted the greedy pursuit algorithm to achieve reconstruction of signal ensembles that were compressed with the FM-Mux. The modified SOMP algorithm finds the common support of the ECG signal ensemble by iteratively choosing atoms that best describe the measurement y. Precisely, at step j, the atom set [a1,k, a2,k,...,aM, k] is chosen, that solves the easy optimization problem(5)
where rj−1 is the residual after step j − 1. The residual after the jth step rj is calculated following the equation(6)
as difference between the measurement vector y and the linear projection of the signal estimate in the jth step onto the measurement domain. The iterative selection of atom sets is stopped when the expected sparsity of the signal ensemble is met or when the l2-norm of the residual stops decreasing substantially.
In this section we present the results of the performance evaluation of the proposed method. We carried out simulations on the PTB Diagnostic ECG Database  from the PhysioBank archive which contains 549 15-lead ECG recordings from 290 subjects aged 17 to 87 years. The database incorporates records of healthy subjects as well as those with adverse conditions like for example myocar-dial infarction, cardiomyopathy or dysrythmia. Each signal is recorded with a sample rate of 1 kHz and an ADC resolution of 16 bit.
We evaluated the proposed method on the standard 12-lead ECG recording (which includes the three Einthoven leads I, II and III, the Goldberger leads aVR, aVL and aVF, and the precordial leads V1, V2, V3, V4, V5 and V6) of 100 randomly selected records from the database. Each signal is bandlimited to W = 125 Hz prior to compression. The LTI filters are designed as random phase filters with real impulse responses following . During reconstruction, the maximal estimated sparsity per signal is set to be 200.
The performance is measured as the percentage root-mean-square difference PRD between the original signal x and the reconstructed signal. The compression ratio CR in this paper is defined as CR = (M · 2W) /Ω = M /η, where M is the number of recorded leads (here fixed to 12), Ω is the switching rate of the modulators as well as the ADC sample rate and η = Ω/2W is the oversampling factor.
The results of our conducted experiment are given in Table 1. As it can be seen, the overall reconstruction quality rises with increasing oversampling factor. A mean reconstruction quality that falls below a PRD of 9, which corresponds to the good reconstruction category , is achieved at an oversampling factor of 5. An example for the reconstruction of compressively multiplexed 12-lead ECG signals is presented in Figure 2.
In this paper, we have examined the applicability of the FM-Mux CS architecture for compressive sampling of multi-lead ECG signals. The FM-Mux allows for efficient data compression in ambulatory patient monitoring applications that require wireless transmission of ECG signals, e.g. in body sensor networks. The greedy pursuit algorithm SOMP was adopted to the joint-sparse recovery of multi-lead ECG signals recorded with the FM-Mux. Evaluation of the proposed approach was conducted on the PTB Diagnostic ECG Database which encompasses recordings with a variety of pathologies like myocardial infarction, cardiomyopathy and dysrythmias. The results show a general applicability of the proposed concept. However, the compression ratios achieved are not yet satisfying. It is believed that lower oversampling rates and thus higher compression ratios become feasible when more elaborate reconstruction algorithms that capitalize on the joint signal structure even further are utilized. In future works, we plan to incorporate the low-rank structure of the ECG signal ensemble into the reconstruction for improved performance.
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About the article
Published Online: 2015-09-12
Published in Print: 2015-09-01
Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent has been obtained from all individuals included in this study. Ethical approval: The research related to human use has been complied with all the relevant national regulations, institutional policies and in accordance the tenets of the Helsinki Declaration, and has been approved by the authors’ institutional review board or equivalent committee.