ECG Signal

• Electrocardiography - The process of producing an electrocardiogram (ECG), a recording of the heart’s electrical activity through repeated cardiac cycles
• ECG records impulses to display how fast the heart is beating, rhythm of the heartbeats, and the timing of electrical impulses as they move through different parts of the heart
• Helpful device in medicine used to detect heart-related issues and other health conditions
• Note: Both projects (Processing Bio ECG Signal and Filtering of an ECG Signal Corrupted by Power Supply Hum) uses the same ECG signal generator MATLAB code¹

Processing Bio ECG Signal²

ECG Signals, Functions, and Filtering Task

• ecg - ECG signal generator to simulate ECG waveforms in MATLAB¹
• $x[n]$ - Input noisy ECG signal modeled as $x[n] = s[n] + sigma$, where sigma is $0<sigma<1$ and sampled at F_s = 500 Hz
• $s[n]$ - Clean ECG signal generated from the function ecg
• $y_{filter}[n]$ - Output ECG signal using the MATLAB function filter
• Task: Reduce the noise fluctuations by filtering or processing $x[n]$ using a moving average Finite Impulse Response (FIR) filter of length $M+1$ by using the MATLAB function conv

Moving Average Filter and Convolution

Adapted from M&I³, the moving average filter is expressed as:

$$ y[n] = \frac{1}{M + 1} \sum_{k=0}^{M} x[n-k], $$

where $M$ is the filter order and $k$ is the index value, is used as a mathematical model to reduce the noise fluctuations from the noisy signal $x[n]$.

The moving average filter is defined by its impulse response $h[n]$:

$$ h[n] = \begin{cases} \frac{1}{M+1}, & \text{if } 0 \leq n \leq M \\ 0, & \text{otherwise} \end{cases} $$

Linear time-invariant (LTI) systems can represented by the convolution sum:

$$ y[n] = x[n] * h[n] = \sum_{k=-∞}^{+∞} x[k]h[n-k] $$

• Ex: $h[n] = [\frac{1}{3}, \frac{1}{3}, \frac{1}{3}]$ is linear and time-invariance

  • Linearity: For any scalar $a$, $S\{ax[n]\} = aS\{x[n]\} = ay[n]$.

    The output is $y_1[n] = \frac{1}{3} (x_1[n] +x_2[n-1] + x_3[n-2])$

  • Time-invariance: Consider the filter $y_1[n]$ above. For a delayed input $x_d[n] = x[n-d]$, the output is delayed by $y_d[n] = y[n-d]$.

    The output is $y_d[n] = y[n-d] = \frac{1}{3} (x[n-d] + x[n-d-1] + x[n-d-2]$

Sigma Values and Filter Order $M$ Parameters

• Sigma values: 0.2, 0.5, and 0.7
• Filter order $M$: 10, 50, and 100
• Results were used to determine the ideal sigma values and filter order $M$ of ECG signals for detecting heart-related issues and other health conditions

Results

Noisy and Clean ECG Signals with Sigma = 0.2, 0.5, and 0.7

Noisy and Filtered ECG Signals with Sigma = 0.2, 0.5, and 0.7 with Filter Order M = 10

Noisy and Filtered ECG Signals with Sigma = 0.2 with Filter Order M = 10, 50, and 100

Filtering of an ECG Signal Corrupted by Power Supply Hum⁴

ECG Signals, Functions, and Filtering Task

• $s[n]$ - Clean ECG signal generated from the function ecg
• $ecg_{1}[n]$ - Four ECG pulses of length 500 samples concatenated to create a 2000 samples long signal
• $s_{1/2}[n]$ - 500 and 2000 length 60 Hz cosine signal with 0.2 amplitude and 0 phase sampled at F_s = 500 Hz, same as the signal $ecg_{1}[n]$ respectively
• $x[n]$ - Noisy ECG signal expressed as $x[n] = ecg_{1}[n] + s[n]$
• $y[n]$ - Output ECG signal using the MATLAB functions designfilt⁵ and filtfilt⁶
• Task: Remove the 60 Hz interference from the noisy signal $x[n]$ by using a IIR notch filter

Results

Clean ECG Signals (500 and 2000 Samples)

60 Hz Cosine Signals Sampled at 500 Hz (500 and 2000 Samples)

60 Hz Interference and Filtered ECG Signals

• A bandstop filter of a 2 Hz bandwidth centered on 60 Hz designed using the MATLAB function designfilt to remove 60 Hz interference

% Filter the messy signal x[n] using 2nd order IIR notch filter of bandwidth = 2 Hz w/ 'designfilt'
d = designfilt('bandstopiir', 'FilterOrder', 2, 'HalfPowerFrequency1', 59, ...
    'HalfPowerFrequency2', 61, 'DesignMethod', 'butter', 'SampleRate', Fs);

Clean and Filtered ECG Signals (2000 Samples)

Footnotes

1. ECG signal generator: MATLAB code ecg.m ↩ ↩²

2. Processing Bio ECG Signal: MATLAB code processing_bio_ecg.m ↩

3. D. G. Manolakis and V. K. Ingle, Applied Digital Signal Processing, 1st ed. Cambridge University Press, 2011, pp. 66–67. ↩

4. Filtering of an ECG Signal Corrupted by Power Supply Hum: MATLAB code iir_ecg.m ↩

5. MATLAB function designfilt documentation ↩

6. MATLAB function filtfilt documentation ↩