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Notebook[{
Cell[" EKG QRS Detection Algorithm", "Title"],

Cell["About This Notebook", "Subtitle"],

Cell[TextData[{
  "This notebook uses ",
  StyleBox["Mathematica",
    FontSlant->"Italic"],
  " to implement the QRS detection algorithm as described in Tompkins (1993). \
The ability to detect the QRS complex is highly important for a variety of \
clinical applications, in particular, the arrythmia detectors and  the Holter \
tape analysis.  In this notebook, a short introduction to electrocardiography \
is given in the following steps.  First, the recorded EKG is imported and \
displayed. Second, the functions for low- and high-pass digital filters are \
implemented and applied to the signal.  Third, the differentiating and \
integrating functions are implemented and applied. Finally, the QRS complex \
is detected and displayed."
}], "Text"],

Cell["Introduction to Electrocardiography", "Subtitle"],

Cell["\<\
An electrocardiogram is a recording of the electrical activity of the \
heart.The heart can be viewed as a three-dimensional vector. Therefore, its \
electrical activity can, in theory, be recorded by three orthogonal leads. In \
practice, however, a standard clinical EKG is recorded with 12 leads: 6 limb \
leads and 6 precordical leads.\
\>", "Text"],

Cell["\<\
A normal EKG reflects the electrical activity of each of the four heart \
chambers: left and right ventricles and left and right atria (Fig. 1).  \
\>", "Text"],

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Cell[TextData[StyleBox[
"Fig. 1: An Example of a Normal EKG.", "Commentary"]], "Commentary"],

Cell["\<\
The P wave marks the beginning of atrial depolarization. The onset of the Q \
wave is produced by the stimulus from the Purkinje system. The depolarization \
of the ventricle produces the R wave. The S wave is initiated when the \
excitation reaches the base of the ventricle. The T wave is produced by \
ventricular repolarization (Tompkins 1993; Goldman 1992).\
\>", "Text"],

Cell[TextData[
"The frequency range for the signal is 0.05 - 100Hz,  and the signal range is \
10\[Mu]V  (fetal) and 5 mV (adult)."], "Text"],

Cell["Introduction to the Algorithm", "Subtitle"],

Cell[TextData[StyleBox[
"The power spectrum of a recorded EKG (Fig. 2) contains a variety of \
components. There are the complexes of the cardiosignal:  QRS,  P wave, and T \
wave. Also,  there are a number of noise artifacts, such as muscle artifacts, \
the 60Hz powerline interference,  and the noise from motions and other \
electronic instrumentation (Goldman 1992)."]], "Text"],

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Cell["Fig. 2 The Power Spectrum of EKG.", "Commentary"],

Cell[TextData[StyleBox[
"The real-time QRS detection algorithm described in Goldman contains four \
main steps:"]], "Text"],

Cell[TextData[StyleBox[
"Step 1: The bandpass integer filter isolates the QRS energy in 5-15Hz and \
attenuates the artifacts and P and T waves.  It is implemented as a cascade \
of a low-pass filter (the cutoff frequency ~11Hz) and a high-pass filter (the \
low cutoff frequency ~5Hz).\nStep 2: The derivative provides information \
about the slope of the QRS complex.\nStep 3: The squaring function makes all \
data positive and emphasizes the higher frequencies.\nStep 4: The moving \
window integral extracts additional features of the QRS complex. \n\t"]], 
  "Text"],

Cell["Implementation of the Algorithm", "Subtitle"],

Cell[CellGroupData[{

Cell["Importing the EKG Signal and the `QRSdetection Package", "Section",
  FontWeight->"Plain"],

Cell["Set working directory.", "Commentary"],

Cell[BoxData[
    \(SetDirectory["\</WINDOWS/DESKTOP/Biomed/Signals/EKGsamples/\>"]\)], 
  "InputOnly"],

Cell[BoxData[
    \("C:\\WINDOWS\\DESKTOP\\Biomed\\Signals\\EKGsamples"\)], "Output"],

Cell["Import the previously recorded data from the file.", "Commentary"],

Cell[BoxData[
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Cell[TextData[StyleBox[
"The signal (Tompkins 1993) is sampled at 200 samples per second (amplitude \
100 = 0.5 mV; the sample #150 = 0.75 second [the distance between samples is \
5msec])."]], "Text"],

Cell["Display the data.", "Commentary"],

Cell[BoxData[
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Cell["Show the number of samples in the record.", "Commentary"],

Cell[BoxData[
    \(Length[ekglist]\)], "InputOnly"],

Cell[BoxData[
    \(159\)], "Output"],

Cell["Read in the Electrocardiography `QRSdetection package", "Commentary"],

Cell["<<Electrocardiography`QRSdetection`", "InputOnly"],

Cell["Show the names of the functions defined in the package.", "Commentary"],

Cell["Names[\"QRSdetection`*\"]", "InputOnly"],

Cell[BoxData[
    \({"derivative", "highpassfilter", "lowpassfilter", "QRSdetect", 
      "QRSdetection", "realQ", "squarelist", "windowintegral"}\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[TextData[StyleBox["Bandpass Integer Filter",
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Cell[TextData[StyleBox["Low-pass Filter",
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  FontSlant->"Italic"]], "Subsection"],

Cell["\<\
Here is the transfer function of the 2nd order low-pass filter.\
\>", "Text"],

Cell[TextData[{
  Cell[BoxData[
      \(TraditionalForm\`\(H\_lp\)(z)\)]],
  " =",
  Cell[BoxData[
      \(TraditionalForm\`\((1 - z\^\(-6\))\)\^2\/\((1 - z\^\(-1\))\)\^2\)]]
}], "Text",
  TextAlignment->Center],

Cell["Here is the difference equation of the filter. ", "Text",
  TextAlignment->Left],

Cell[TextData[Cell[BoxData[
    \(TraditionalForm
    \`\(y\_lp\)(n\ T)\  = \ 
      2\ \(y(n\ T\  - \ T)\)\  - \ y(\ n\ T\  - \ 2\ T)\  + \ x(n\ T)\  - \ 
        2\ \(x(n\ T\  - \ 6\ T)\)\  + \ x(n\ T\  - \ 12\ T)\)]]], "Text",
  TextAlignment->Center],

Cell["\<\
The filter has a cutoff frequency of approximatelly 11Hz, and the gain is 36.\
\
\>", "Text"],

Cell[TextData[{
  "Apply the ",
  StyleBox["lowpassfilter",
    FontFamily->"Courier"],
  " function to the EKG signal."
}], "Commentary"],

Cell[BoxData[
    \(\(lowpassekg = lowpassfilter[ekglist]; \)\)], "InputOnly"],

Cell[TextData[{
  "Display the result of the application of the ",
  StyleBox["lowpassfilter",
    FontFamily->"Courier"],
  " to the signal together with the original signal."
}], "Commentary"],

Cell[BoxData[
    \(\(Show[
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                FontForm["\<EKG Data Points\>", {"\<Times-Bold\>", 10}], 
                "\<\>"}], \n
          ListPlot[lowpassekg, PlotJoined \[Rule] True, 
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Cell["\<\
Conclusion: The result is attenuation of musle noise and 60Hz powerline \
interference.\
\>", "Text"],

Cell[TextData[StyleBox["High-pass filter",
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Cell["\<\
The high-pass filter is implemented by substracting a first-order low-pass \
filter from an all-pass filter with delay.  Here is its transfer function.\
\>", "Text"],

Cell[TextData[Cell[BoxData[
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Cell["Here is its difference equation. ", "Text"],

Cell[TextData[Cell[BoxData[
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Cell["\<\
The low frequency cutoff of the filter is appproxmately 5Hz and the gain is \
1.\
\>", "Text"],

Cell[TextData[{
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}], "Commentary"],

Cell[BoxData[
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Cell["Display the results.", "Commentary"],

Cell[BoxData[
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                FontForm["\<EKG Data Points\>", {"\<Times-Bold\>", 10}], 
                "\<\>"}], \n
          ListPlot[highpassekg, \ PlotJoined \[Rule] True, 
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            Frame -> True, \ DisplayFunction -> \ Identity, 
            FrameLabel 
              \[Rule] {"\<Sample number\>", "\<Amplitude\>", 
                FontForm["\<Bandpass EKG\>", {"\<Times-Bold\>", 10}], "\<\>"}]
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Differentiation will provide information about the slope of the QRS \
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This derivative is implemented with the following difference equation.\
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Cell["\<\
Conclusion: Squaring makes all samples positive and enhances the higher \
frequencies of the signal.\
\>", "Text"]
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Cell["\<\
In case of some abnormalities with QRS complex, the slope of the R wave might \
not be sufficient criteria to detect the QRS complex. The integration \
provided additional information. It has the following difference equation:\
\>", "Text"],

Cell[TextData[Cell[BoxData[
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    \`\(y\_ntegral\)(n\ T)\  = \ 
      1\/N[\(x(n\ T\  - \ \((N\  - \ 1)\)\ T)\  + \ 
                x(n\ T\  - \ \((N\  - \ 2)\)\ T)\  + \  ... \)\  + x(n\ T)]
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Cell["\<\
where the number N is the number of samples in the moving window. It has to \
be chosen depending on the sampling rate and the maximum possible width of \
the QRS complex. In our case, the rate of 200 samples per second, it is 30.\
\>", "Text"],

Cell[TextData[{
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Cell["Display the results.", "Commentary"],

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Cell["\<\
Conclusion: The width of the QRS complex is equal to the width of the segment \
of the output with the increase of the amplitude (approximately 20 samples in \
this example).  Note that the result of the algorithm shifted right due to \
delays in the inner functions.  This is consistent with the goals of the \
algorithms (which are to detect the QRS complex, show its length and the \
distance between the complexes). If the algorithm is applied to a signal with \
multiple QRS complexes, all detected complexes will have the same shift, \
which will keep the distance between them constant.\
\>", "Text"],

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Cell[TextData[{
  "The QRS detection algorithm was succesfully implemented in ",
  StyleBox["Mathematica.",
    FontSlant->"Italic"],
  " The functions of the low- and high-pass filters and differentiating, \
squaring, and integrating functions were implemented. The example of the \
application was given."
}], "Text"],

Cell["References", "Subtitle"],

Cell[TextData[{
  StyleBox["Tompkins, Willis J., ed. 1993. "],
  StyleBox["Biomedical Digital Signal Processing",
    FontSlant->"Italic"],
  StyleBox[
  ". Englewood Cliffs, N.J.: Prentice Hall. \nGoldman, Mervin J. 1992. "],
  StyleBox["The Principles of Clinical Electrocardiography",
    FontSlant->"Italic"],
  StyleBox[". Los Altos, Calif.: Lange Medical Publications. "]
}], "Text"]
}, Open  ]]
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