1 Introduction

Cardiac medicine has evocative influence on professional, technological, preventive and computational advances. Automatic detection and diagnosis are essential initiatives for identifying different pathologies from an electrocardiogram (ECG), which has been a well-established fundamental diagnostic tool used by cardiologists for the assessment of cardiac arrhythmias in clinical routine [1, 2].

An ECG signal is a combination of three waves viz. P, QRS and T waves [3, 4]. Among these waves, accurate detection of R-peak (QRS complex) is essential for successful automated diagnosis and administration of various cardiovascular diseases. Unfortunately, any acquired ECG signal is generally corrupted by various types of noise/distortion, and therefore, has distinct QRS morphologies. Timely detection of cardiac abnormalities is very important for saving the life of the subjects (patients) [5], which fundamentally depends on correct observations of R-peaks (QRS complexes) [6, 7] in terms of their specific shape and time of occurrence. The heart specialists make use of amplitude, frequency, polarity of P-QRS-T waves for detecting the underlying disease [8]. The normal heart rate lies in the range of 60–100 beats/min. Below 60 beats per minute, it is called bradycardia and above 100 beats per minute, it is known as tachycardia [9, 10].

In view of the above discussion, it can be concluded that due to involvement of various noises/distortions at the time of acquisition of ECG records, effective computational medicine is essential for detecting heart diseases by using a computer aided diagnosis (CAD) system based on an efficient feature extraction algorithm [11, 12].

In the literature, different techniques for R-peaks detection have been reported under such conditions, but it is still a challenge. Several authors have been working on pre-processing of ECG signals, but processing time, proper selection of filter parameters, overlapping of noise and artifacts with QRS complexes, selection of the frequency range of QRS complexes and temperature limits the performance of these techniques. Until now various methods such as self-convolution window (SCW) concept [13], digital differentiator (DD) [13], wavelet transform (WT) [2, 14, 15], empirical mode decomposition (EMD) [16], median filter [17], signal derivatives [18], hilbert transform [19], hidden markov model [20], phase space method [21], first derivative (FD) [22], alexander fractional differential window filter (AFDWF) [23], stockwell transform (s-transform) [24], adaptive signal processing (ASP) [25], support vector machine (SVM) [26], second order difference plot (SODP) [27] and hilbert-huang transform (HHT) [28] etc. have been attempted for pre-processing of ECG signals. In [13], a novel ECG denoising algorithm based on the concept of SCW has been proposed. It relied on Hamming window to have smaller ripples in the stop band. In [22], FD based techniques were solely adopted in a real-time scheme for large databases. These techniques did not demand training of the computational algorithms but needed corrections according to an individual object (patient). Additionally, these offered low computational cost and were faster, but resulted in more detection errors due to presence of the high-frequency noises. These detection errors are not permissible in the present scenario of sophisticated health care assembly. In [16], EMD was considered for analyzing ECG records, but there the selection of a set of intrinsic mode functions (IMFs) was essential. In [27], SODP technique was used to extract HRV features of ECG for diagnosis of different cardiac disorders, but its performance was adversely affected in the presence of noisy ECG records. The methods proposed in [19, 20] are computationally complex. In [29], an optimally designed digital differentiator (DD) has been used for precise detection of the QRS complex. In that paper, Brownian Motion Optimization (BMO) was utilized as optimization algorithm. These techniques (DD and BMO) were utilized in the pre-processing stage of the QRS detector.

Traditional ECG signal analysis techniques relied mainly on the time domain approach, but needed careful attention to analyze all its attributes. However, information about its frequency components is also required for accurate analysis. In [2], an improved QRS complex detection algorithm based on wavelet transform (WT) was presented. But this approach relies on the choice of two important parameters such as wavelet basis function and decomposition level [30,31,32,33]. These two parameters affect the performance and effectiveness of wavelet based approach. Also, WT based methods suffer due to fringing effects and phase shift problems [34].

In view of the above ECG signal analysis problems, fractional Fourier transform (FrFT) is proposed in this paper, which can provide better time–frequency localization (i.e. mixed time and frequency components) in fractional domain on the basis of correctly chosen input parameters. The novelty of this paper is to detect the R-peaks directly in raw ECG signal without using any pre-processing/filtering technique. FrFT preserves all its clinical attributes to a significant extent. Next, IPCA is used for detecting R-peak by estimating eigenvalues and eigenvectors belonging to a new orthogonal basis [35]. In IPCA, largest of the estimated variance values helped in the detection of R-peaks resulting in identification of cardiac arrhythmias. Thus a new approach by combining FrFT with IPCA is explored for detecting R-peaks by calculating sensitivity (SE), positive predictive value (PPV), accuracy (ACC), and detection error rate (DER). These parameters are then analyzed to correctly predict the health condition of the heart. The final outcomes establish the application of FrFT combined with IPCA for R-peak detection for low signal-to-noise ratio (SNR), high baseline wander (BLW) and abnormal morphologies.

The whole paper has been organized as; Sect. 2 outlines details of the considered database and methodologies under materials and methods section. Section 3 showcases and discusses some important outcomes. Finally, Sect. 4 concludes the paper.

2 Materials and Methods

Details of the considered ECG database, used feature extraction & detection techniques and performance evaluating parameters are presented in this section.

Figure 1 shows the methodology proposed in this paper.

Fig. 1
figure 1

Proposed methodology

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2.1 ECG Database

Most of the researchers [2, 34, 36, 37] have used MIT-BIH Arrhythmia (MIT-BIH Arr) database for presenting their research outcomes. Therefore in this paper also, MIT-BIH Arr database is used to facilitate appropriate comparison with other existing works.

2.2 Feature Extraction Using Fractional Fourier Transform (FrFT)

Fractional Fourier transform (FrFT) has demonstrated its ability to be a splendid tool for handling the issues arising in the processing of a non-stationary signal among existing signal processing tools [38,39,40]. Mathematically, it is considered as a counter-clockwise rotation of u-axis with respect to the time axis by an angle α [40]. The value of α lies between \( 0 < \alpha < \frac{\pi }{2} \) and it depicts rotated time–frequency description of the signal as shown in Fig. 2. FrFT offers other advantages such as faster computations and better flexibility that are much needed in the domain of biomedical signal processing (BSP) [41].

Fig. 2
figure 2

The angle of rotation of α

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Here, eigenvalues and eigenfunctions are calculated using the basic mathematical expressions of well-known Fourier transform as

$$ {{\rm Y}}\left( {{\rm f}} \right) = \mathop \int \limits_{ - \infty }^{\infty } {{\rm y}}\left( {{\rm t}} \right){{\rm e}}^{{ - {{\rm j}}2\uppi{{\rm ft}}}} {{\rm dt}} $$
(1)
$$ {{\rm y}}\left( {{\rm t}} \right) = \mathop \int \limits_{ - \infty }^{\infty } {{\rm Y}}\left( {{\rm f}} \right){{\rm e}}^{{{{\rm j}}2\uppi{{\rm ft}}}} {{\rm df}} $$
(2)

where y(t) is the time domain ECG signal and Y(f) is its Fourier transform.

Hermite-Gauss functions (HGFs) are used for mathematical expression of eigenfunctions in FrFT [41], expressed as an orthonormal set by

$$ \emptyset_{{\rm n}} \left( {{\rm t}} \right) = \frac{{2^{1/4} }}{{\sqrt {2^{{\rm n}} {{\rm n}}!} }}{{\rm H}}_{{\rm n}} \left( {\sqrt {2\uppi{{\rm t}}} } \right){{\rm e}}^{{ - \uppi{{\rm t}}^{2} }} ;{{\rm Where}}\;n = 0,1,2,3,4,5, \ldots $$
(3)

where \( \emptyset_{{\rm n}} \left( {{\rm t}} \right) \), \( {{\rm H}}_{{\rm n}} \left( . \right) \) denotes orthonormal set and Hermite-Gauss function (HGF) for generalized sequence n, respectively.

It satisfies the standard eigenvalue equation as

$$ {{\rm F }}\left[ {\emptyset_{{\rm n}} \left( {{\rm t}} \right)} \right] =\uplambda_{{\rm n}} \emptyset_{{\rm n}} \left( {{\rm t}} \right) $$
(4)

where F denotes conventional Fourier transform and \( {\uplambda }_{{\rm n}} = {{\rm e}}^{{ - {{\rm jn}}\uppi/2}} \) denotes its eigenvalues, which are further modified for FrFT where all eigenvalues are expressed as \( {{\rm p}}{{\rm th}} \) power of the eigenvalues obtained from the conventional Fourier transform as

$$ {{\rm y}}\left( {{\rm t}} \right) = \mathop \sum \limits_{{{{\rm n}} = 0}}^{\infty } {{\rm X}}_{{\rm n}} \emptyset_{{\rm n}} \left( {{\rm t}} \right) $$
(5)

where eigenfunction \( {{\rm X}}_{{\rm n}} \) is related to time domain ECG signal, expressed as

$$ {{\rm X}}_{{\rm n}} = \mathop \int \limits_{ - \infty }^{\infty } \emptyset_{{\rm n}} \left( {{\rm t}} \right) {{\rm y}}\left( {{\rm t}} \right){{\rm dt}} $$
(6)
$$ {{\rm FrFT}}\left\{ { {{\rm y}}\left( {{\rm t}} \right)} \right\} = \mathop \sum \limits_{{{{\rm n}} = 0}}^{\infty } {{\rm X}}_{{\rm n}} {{\rm e}}^{{ - {{\rm jn}}\uppi/2}} \emptyset_{{\rm n}} \left( {{\rm t}} \right) $$
(7)

Therefore, Eq. (4) is modified for FrFT as

$$ {{\rm F}}^{{\rm p}} \left[ {\emptyset_{{\rm n}} \left( {{\rm t}} \right)} \right] = \uplambda _{{\rm n}}^{{\rm p}} \emptyset_{{\rm n}} \left( {{\rm t}} \right) $$
(8)

The pth (p = 2α/π) order FrFT of an ECG signal y (t)∈L2 (R) is defined as [42]

$$ {{\rm Y}}_{{\rm p}} \left( {{\rm u}} \right) = {{\rm F}}^{{\rm p}} \left\{ {{{\rm y}}\left( {{\rm t}} \right)} \right\}\left( {{\rm u}} \right) = \mathop \int \limits_{ - \infty }^{\infty } {{\rm y}}\left( {{\rm t}} \right){{\rm K}}_{{\rm p}} \left( {{{\rm t}},{{\rm u}}} \right){{\rm dt}} $$
(9)

where

$$ {{\rm K}}_{{\rm p}} \left( {{{\rm t}},{{\rm u}}} \right) = \sqrt {\frac{{1 - {{\rm jcot}\upalpha}}}{{2{{\uppi }}}}} \exp \left( {{{\rm j}}\frac{{{{\rm t}}^{2} + {{\rm u}}^{2} }}{2}{{\rm cot}\upalpha} - {{\rm jut}}\csc {{\upalpha }}} \right),\quad {{\rm when}}\;\upalpha \ne {k\pi } $$
(10)
$$ {{\rm K}}_{{\rm p}} \left( {{{\rm t}},{{\rm u}}} \right) = {{\updelta }}\left( {{{\rm u}} - {{\rm t}}} \right),\quad {{\rm when}}\;{{\upalpha }} = 2{{\rm k}}\uppi $$
(11)
$$ {{\rm K}}_{{\rm p}} \left( {{{\rm t}},{{\rm u}}} \right) = {{\updelta }}\left( {{{\rm u}} + {{\rm t}}} \right),\quad {{\rm When}}\;{{\upalpha }} = \left( {2{{\rm k}} + 1} \right){{\uppi }} $$
(12)

where \( {{\rm K}}_{{\rm p}} \left( {{{\rm t}},{{\rm u}}} \right) \) denotes kernel function associated with FrFT of an ECG signal and \( {{\updelta }} \) denotes delta (impulse) function. The obtained eigenvalues from FrFT are further used in IPCA as discussed in the next subsections. IPCA visualizes the data effectively using less number of components as compared to those considered in ICA and PCA [43]. The crucial step of R-peak detection is completed in four steps as discussed in next subsection.

2.3 Independent Principal Component Analysis (IPCA)

Independent principal component analysis (IPCA) is used for R-peak detection since it possesses good characteristics of both principal component analysis (PCA) and independent component analysis (ICA) [44]. IPCA technique constitutes by applying firstly PCA using singular value decomposition (SVD) followed by ICA using joint approximate diagonalization of eigenmatrices (JADE) on the features extracted using FrFT in Sect. 2.2. IPCA is applied in the steps shown in Fig. 3 [45] and described as;

Fig. 3
figure 3

Flowchart of IPCA steps

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Step 1: PCA using Singular value Decomposition (SVD)

PCA is applied on extracted features obtained from FrFT i.e. \( {{\rm Y}}\left[ {{\rm i x j}} \right] \) to extract the loading vectors [46, 47]. Mathematically, it can be represented as

$$ {{\rm Y}}\left( {{\rm l}} \right) = {{\rm L}}\left( {{\rm l}} \right) {{\rm D O}}^{{\rm T}} $$
(13)

where \( {{\rm L}}\left( {{\rm l}} \right) \), D, \( {{\rm O}} \) denotes centered data matrix, diagonal matrix & orthogonal matrix, respectively and l denotes

R-peak locations.

Step 2: ICA using JADE algorithm

This algorithm is used on the basis of fourth order statistics to automatically filter out the background Gaussian noises. An original source signal is estimated using JADE algorithm. Additionally, it is more efficient as compared to Fast ICA [48]. The JADE algorithm is applied in the following steps;

  1. (i)

    Pre-whitening process

In this process source signal is assumed to have zero mean and unit variance. Mathematically, it can be expressed as

$$ {{\rm y}}\left( {{\rm t}} \right) = {{\rm Q x}}\left( {{\rm t}} \right) $$
(14)

where Q denotes mixing matrix and x(t) denotes independent components.

  1. (ii)

    Fourth order cumulant process

The fourth order cumulant is expressed as

$$ {{\rm Cumulant}}\;\left( {{{\rm y}}_{{{\rm i}}} ,{{\rm y}}_{{{\rm j}}} ,{{\rm y}}_{{{\rm k}}} ,{{\rm y}}_{{{\rm n}}} } \right) = {{\rm E}}\left[ {{{\rm y}}_{{{\rm i}}} {{\rm y}}_{{{\rm j}}} {{\rm y}}_{{{\rm k}}} {{\rm y}}_{{{\rm n}}} } \right] - {{\rm E}}\left[ {{{\rm y}}_{{{\rm i}}} {{\rm y}}_{{{\rm j}}} } \right]{{\rm E}}\left[ {{{\rm y}}_{{{\rm k}}} {{\rm y}}_{{{\rm n}}} } \right] - {{\rm E}}\left[ {{{\rm y}}_{{{\rm i}}} {{\rm y}}_{{{\rm k}}} } \right]{{\rm E}}\left[ {{{\rm y}}_{{{\rm j}}} {{\rm y}}_{{{\rm n}}} } \right] - {{\rm E}}\left[ {{{\rm y}}_{{{\rm i}}} {{\rm y}}_{{{\rm n}}} } \right]{{\rm E}}\left[ {{{\rm y}}_{{{\rm j}}} {{\rm y}}_{{{\rm k}}} } \right] $$
(15)

In (15), considered four signals are \( {{\rm y}}_{{\rm i}} ,{{\rm y}}_{{\rm j}} ,{{\rm y}}_{{\rm k}} ,{{\rm y}}_{{\rm n}} \) where \( 1 \le {{\rm i}},{{\rm j}},{{\rm k}},{{\rm n}} \le {{\rm N}} \)

$$ [{{\rm C}}_{{\rm y}} \left( {{\rm M}} \right)]_{{\rm ij}} = \mathop \sum \limits_{{{{\rm k}},{{\rm n}} - 1}}^{{\rm N}} {{\rm Cumulant }}\left( {{{\rm y}}_{{\rm i}} ,{{\rm y}}_{{\rm j}} ,{{\rm y}}_{{\rm k}} ,{{\rm y}}_{{\rm n}} } \right){{\rm m}}_{{\rm nk}} $$
(16)

where M and \( {{\rm m}}_{{\rm nk}} \) denotes arbitrary matrix and (n,k)th element, respectively, in fourth order cumulant process. The cumulant is expressed in terms of eigenvalues (\( {{\uplambda }}_{{\rm i}} ) \) as

$$ {{\rm C}}_{{\rm y}} \left( {{{\rm M}}_{{\rm i}} } \right) = {{\uplambda }}_{{\rm i}} {{\rm M}}_{{\rm i}} $$
(17)
  1. (iii)

    Diagonalization process

The diagonalization process is expressed as

$$ {{\rm C}}_{{\rm y}} \left( {{{\rm M}}_{{\rm i}} } \right) = {{\rm U}}^{{\rm T}} {{\rm C}}\left( {{{\rm M}}_{{\rm i}} } \right){{\rm U}} = {{\rm Diag}}\left[ {{{\rm k}}_{4} \left( {{{\rm y}}_{1} } \right), {{\rm k}}_{4} \left( {{{\rm y}}_{2} } \right), \ldots ,{{\rm k}}_{4} \left( {{{\rm y}}_{{\rm N}} } \right)} \right] $$
(18)

where U is unitary matrix and Diag denotes approximate diagonalization of fourth order cumulant process.

  1. (iv)

    Calculation of Weight matrix (W)

The weight matrix is computed using

$$ {{\rm W}} = {{\rm U}}^{{\rm T}} {{\rm Q}} $$
(19)

where U is unitary matrix and Q is mixing matrix.

Step 3: Projection of centered data matrix

In this step, mean of all independent components are calculated and subtracted from the respective components.

Here, four signals viz. \( ({{\rm y}}_{{\rm i}} - \overline{{{{\rm y}}_{{\rm i}} }} ), \) \( ({{\rm y}}_{{\rm j}} - \overline{{{{\rm y}}_{{\rm j}} }} ), \) \( ({{\rm y}}_{{\rm k}} - \overline{{{{\rm y}}_{{\rm k}} }} ), \) and \( ({{\rm y}}_{{\rm n}} - \overline{{{{\rm y}}_{{\rm n}} }} ), \) are projected.

Step 4: Estimation of kurtosis and variance values

In IPCA, kurtosis and variance values are used to choose number of independent components (ICs) and principal components (PCs), respectively [49]. The kurtosis (Ku) and variance (Va) for IPCA are given by-

$$ {{\rm K}}_{{\rm u}} = \frac{{\mathop \sum \nolimits_{{{{\rm i}} = 1}}^{{\rm D}} \left( {{{\rm y}}_{{\rm i}} - {{\rm y}}_{{\rm m}} } \right)^{4} /{{\rm D}}}}{{{{\upsigma }}^{4} }} $$
(20)
$$ {{\rm V}}_{{\rm a}} = \frac{1}{{\rm N}}\mathop \sum \limits_{{{{\rm i}} = 1}}^{{\rm D}} \left( {{{\rm y}}_{{\rm i}} - {{\rm y}}_{{\rm m}} } \right)^{2} $$
(21)

where \( {{\rm y}}_{{\rm m}} \) denotes mean value of data, D denotes data size, and \( \sigma \) denotes standard deviation (calculated at N data points).

The proposed methodology has been evaluated on the basis of various parameters such as sensitivity, positive predictive value, accuracy, and detection error rate. These are defined in next subsection.

2.4 Performance Evaluating Parameters

The performance of R-peak detection in an ECG dataset has been evaluated on the basis of different parameters [26, 36, 50,51,52] defined as

$$ {{\rm Sensitivity}}\left( {{\rm SE}} \right) = \frac{{\rm TP}}{{{{\rm TP}} + {{\rm FN}}}} $$
(22)
$$ {{\rm Positive Predictive Value }}\left( {{\rm PPV}} \right) = \frac{{\rm TP}}{{{{\rm TP}} + {{\rm FP}}}} $$
(23)
$$ {{\rm Accuracy}}\left( {{\rm ACC}} \right) = \frac{{\rm TP}}{{{{\rm TP}} + {{\rm FP}} + {{\rm FN}}}} $$
(24)
$$ {{\rm Detection Error Rate }}\left( {{\rm DER}} \right) \, = \frac{{{{\rm FN}} + {{\rm FP}}}}{{\rm Actual beats}} $$
(25)

where TP (True Positive) denotes correctly detected R-peaks, FN (False Negative) denotes number of R-peaks that were not detected by the proposed method, and FP (False Positive) denotes falsely detected R-peaks.

3 Results and discussion

R-peaks are detected using IPCA by estimating the eigenvalues and eigenvectors matrices alongwith variances. The average PC1 eigenvalue variance was 97.33%, which had a kurtosis value lying between 11.256 to 36.298 for the MIT-BIH Arr database. Analysis of an ECG signal heavily relies on variety of noises and artifacts that creeps in during its acquisition. Therefore, an efficient pre-processing stage is required under such circumstances which in-turn increases the computational cost of the automated ECG signal analysis further. In this paper, the proposed methodology skips the pre-processing stage and the whole operation has been performed on features extracted using FrFT from raw ECG signal followed by detection using IPCA. The raw ECG signal is shown in Fig. 4 and real & imaginary parts of FrFT are shown in Fig. 5 for 233 m and 215 m records. These datasets are selected to make suitable comparisons as most of the researchers have reported their results by considering them [3, 4, 26, 29, 46]. The peaks of the real & imaginary parts of FrFT lie at the centre (t = 4000 s.) of the duration of the recorded signal (t = 8000 s). This is similar to the characteristics of a sinc function for a rectangular input [39, 53]. Due to the noise offsets, average peak amplitudes of 87.77 mV and 329.4 mV are observed for real and imaginary FrFT parts, respectively for entire MIT-BIH Arr database. Figure 6 shows inverse FrFT where average peak amplitude of 347.77 mVolt is obtained for entire MIT-BIH Arr database. Here, it is observed that BLW noise/distortion still exists due to which even manual interpretation of R-peaks may result in wrong analysis. Therefore, IPCA is proposed to be used for R-peak detection in this paper. Finally, Fig. 7 shows detected R-peaks using IPCA for MIT-BIH Arr database (233 m and 215 m data sets) represented by light green plus sign. Using the proposed algorithm, TP of 3077 & 3376, FN of 1 &1, and FP of 0 &1 for 233 m and 215 m data sets, respectively are obtained.

Fig. 4
figure 4

Raw ECG signal-a 233 m, b 215 m

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Fig. 5
figure 5

Real and imaginary part of FrFT-a on 233 m, b on 215 m

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Fig. 6
figure 6

Inverse FrFT-a on 233 m, b on 215 m

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Fig. 7
figure 7

R-peak detection –a on 233 m, b on 215 m

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Tables 1 and 2 summarize the findings and comparison with other state-of-the-art methods, respectively. For instance, obtained detection parameters for 105 m are TP = 2587, FN = 1 & FP = 1, and for 201 m are TP = 1971, FN = 0 & FP = 1. In [13, 54, 55], FN/FP values for 105 m data set were 5/4, 12/33, 0/1, respectively, and for 201 m data set, FN/FP values were 12/9, 5/0, 25/2 in that order. Table 2 shows comparison of the evaluation parameters viz. SE, PPV and ACC obtained by the proposed methodology with other existing techniques. It is observed that detection results in terms of these parameters are approximately 100%. This clearly indicates the effectiveness of the proposed methodology.

Table 1 TP, FN, and FP values obtained using FrFT + IPCA
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Table 2 Comparison of the proposed and existing methods
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Table 3 and Table 4 present the comparison between proposed and existing methods on the basis of FN + FP for whole dataset and total beats, TP, (FN + FP), SE, PPV and ACC, respectively.

Table 3 Comparison of the proposed and existing methods on the basis of false detection (FN + FP) in whole data sets
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Table 4 Comparison of the proposed and existing methods on the basis of total beats, TP, (FN + FP), SE, PPV and ACC
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It has been observed that the proposed method performs better than existing methods and yields more accurate and fast results. This makes the proposed method a good candidate to be used as an effective tool for automatic ECG signal interpretation.

The existing techniques have SE of 99.29%, 99.90%, 99.39%, 99.80%, 99.93%, 99.90%, PPV of 99.89%, 99.10%, 99.49%, 99.90%, 99.95%, 99.88%, & ACC of 99.63%, 98.11%, 98.89%, 99.60%, 99.87%, 99.77% in Kaya and Pehlivan [56], Martis et al. [8], Qin et al. [57], Rai et al. [58], Gupta and Mittal [59] and Thakor and Zhu [51], respectively. The proposed technique has SE of 99.97%, PPV of 99.98%, & ACC of 99.94%. It is clearly observed that the proposed low computational cost CAD system results (due to eliminating the need of pre-processing altogether) are consistently higher as compared to those obtained with the existing methodologies, all of which depend upon pre-processing to produce the reported detection results. The main advantage of the proposed technique is its less detection error (FN + FP) as shown in Fig. 8. In the existing techniques, (FN + FP) of 500, 2100, 1229, 1774, 138, 249 in Kaya and Pehlivan [56], Martis et al. [8], Qin et al. [57], Rai et al. [58], Gupta and Mittal [59] and Thakor and Zhu [51], respecitlvey have been reported. In [8, 56], ACC is given but number of (FN + FP) is not mentioned, which is obtained by assuming TP of 1,10,000 using (24) for performing comparison in this paper. Rai et al. [58] have not mentioned total number of (FN + FP) in the whole database considered in their study, hence, it is scaled here for the purpose of comparison.

Fig. 8
figure 8

Comparison of detection parameters between proposed and previous state-of-the-art techniques

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The average CPU processing time of 1.2 s, 0.872 s, 0.833 s, 13.7 s, 0.737 s, 0.677 s and DER of 0.44%, 1.11%, 0.49%, 0.163%, 0.125%, 0.058% have been reportedin Kaya and Pehlivan [56], Qin et al. [57], Nguyen et al. [60], Park et al. [61], Gupta and Mittal [59], and in the proposed technique, respectively. It can be observed from Fig. 9 that the proposed technique has least detection error rate and computational cost among all the techniques considered for comparison establishing its prowess.

Fig. 9
figure 9

Comparison of DER and CPU time between proposed and previous state-of-the-art techniques

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In this paper, 17 real-time recordings are also examined using the proposed technique for validating this research work. These recordings are obtained at a sampling rate of 360 Hz using BIOPAC@MP36 equipment (@BIOPAC Systems, Inc.) [62] at NIT, Jalandhar, India. Different age-groups subjects volunteered and these records were then analyzed using BIOPAC Acqknowledge 4.0 software (@BIOPAC Systems, Inc.) [62]. Total 18,677 beats were captured in all 17 real time recordings. The results are verified by using the proposed technique. R-peaks are detected in terms of various detection parameters viz. TP of 18,669, FN of 5, FP of 4 yielding SE of 99.97%, PPV of 99.98% and ACC of 99.95%.

4 Conclusion

The proposed methodology has been implemented successfully on mostly used (in the literature) datasets of MIT-BIH Arr database to obtain SE of 99.97%, PPV of 99.98%, and ACC of 99.94%. The average processing time of CPU and detection error rate (DER) are 0.677 s and 0.058%, respectively for the proposed method. These results signify the effectiveness of the proposed method in analyzing the time-varying signals alongwith its low computational cost without compromising its accuracy. Hence, this method can be used for human stress related arrhythmia detection, in practical arrhythmia monitoring systems, in electronic cardiac pacemakers, for treatment of patients in intensive care units (ICU), to distinguish between similar cardiac arrhythmias even when the ECG signal is corrupted with noise and baseline wandering. This algorithm may also be utilized in intelligent remote health caring systems, telemedicine applications, cardiac arrhythmia measurement in varying R–R intervals/QRS complex (wide or narrow, large or small amplitudes, negative R peaks and highly noisy ECG signals) and mass screening for cardiac health. Due to low computational cost, the proposed technique (FrFT + IPCA) is likely to enhance the strength of real-time arrhythmia analysis system. It can speed up the diagnosis process without compromising with the accuracy. Finally, it may be concluded that the use of computerized analysis proposed in this paper can reduce workload of the cardiologists significantly.

As FrFT provides an alternative representation of an ECG signal in time fractional Fourier domain at selected rotation angles, it can find major applications in the domain of biomedical signal and image processing. In future, this work can be extended to use Fractional Wavelet Transform (FrWT) making best use of inherit multiresolution property of the wavelets along with its capability of representing the signal in the fractional domain similar to FrFT.