Question

Higher-order differentiation filter coefficients in Pan Tompkins algorithm is

• A. \( \text{[-1 -2 0 2 1]} \)
• B. \( \text{[1 2 0 -2 -1]} \)
• C. \( \text{[-2 -1 0 1 2]} \)
• D. \( \text{[2 1 0 -1 -2]} \)

Hint:

The differentiation filter in the Pan-Tompkins algorithm is a critical step for highlighting the rapid changes in the ECG signal characteristic of the QRS complex. The filter is a 5-point moving average derivative approximation. Remember the specific coefficients: $[1, 2, 0, -2, -1]$. This filter is often normalized by a factor like $1/8$. Its antisymmetric nature (e.g., $h[n] = -h[-n]$ for odd functions) ensures it approximates a derivative and has a zero at DC, which is important for suppressing baseline wander.

Answer 1

The Correct Option is B

The differentiation step in the Pan-Tompkins algorithm is crucial for enhancing the QRS complex's steep slopes and reducing the slower components of the P and T waves. While a simple two-point differentiator like $[1, -1]$ could be used, the Pan-Tompkins algorithm employs a specific five-point differentiator to achieve better noise robustness and frequency response. The ideal differentiator has a frequency response that is proportional to $j\omega$. In the time domain, this corresponds to the derivative. For discrete signals, derivatives are approximated using finite difference approximations. The specific differentiation filter coefficients used in the original Pan-Tompkins algorithm are designed to approximate the derivative using a 5-point moving window. The coefficients are: $$y[n] = \frac{1}{8} [2x[n] + x[n-1] - x[n-3] - 2x[n-4]]$$ More commonly, this is seen as coefficients that approximate the derivative over a certain span, often centering around zero. The coefficients are symmetric/antisymmetric. Let's consider the commonly cited coefficients for the differentiator in the Pan-Tompkins algorithm. It's often given as a 5-point filter: $$\frac{1}{T_s} \frac{1}{4} [-1, -2, 0, 2, 1]$$ where $T_s$ is the sampling period. The scaling factor $1/T_s$ and $1/4$ (or $1/8$) are for normalization, but the core shape of the filter is determined by the coefficients. If we look at the options provided, they represent the filter coefficients. Let's analyze the properties of a differentiator filter: 1. It should have a zero at DC (i.e., sum of coefficients should be zero) because the derivative of a constant is zero. 2. It should typically be an odd-length filter, or an even-length filter depending on the specific design. 3. The coefficients should be antisymmetric for a pure derivative. Let's check the sum of coefficients for each option: (A) $[-1 -2 0 2 1]$: Sum = $-1 - 2 + 0 + 2 + 1 = 0$. (Passes DC zero check) (B) $[1 2 0 -2 -1]$: Sum = $1 + 2 + 0 - 2 - 1 = 0$. (Passes DC zero check) (C) $[-2 -1 0 1 2]$: Sum = $-2 - 1 + 0 + 1 + 2 = 0$. (Passes DC zero check) (D) $[2 1 0 -1 -2]$: Sum = $2 + 1 + 0 - 1 - 2 = 0$. (Passes DC zero check) All options pass the DC zero check. The standard Pan-Tompkins differentiator uses a 5-point moving window filter with coefficients: $$\frac{1}{8} [-1, -2, 0, 2, 1]$$ However, the order of coefficients in the options can represent different things (e.g., $h[0], h[1], h[2], h[3], h[4]$ or $h[-2], h[-1], h[0], h[1], h[2]$). Let's consider the most commonly cited version for the differentiation filter in Pan-Tompkins: $y[n] = (1/8) * (2x[n] + x[n-1] - x[n-3] - 2x[n-4])$ This corresponds to a filter with coefficients: $h[0] = 2/8 = 1/4$ $h[1] = 1/8$ $h[2] = 0$ $h[3] = -1/8$ $h[4] = -2/8 = -1/4$ So, the coefficients are $[1/4, 1/8, 0, -1/8, -1/4]$. If we multiply by 8, we get $[2, 1, 0, -1, -2]$. This is option (D). However, the original Pan-Tompkins paper (Pan and Tompkins, 1985) indeed describes a 5-point differentiator. The coefficients they used were: $$y(n) = \frac{1}{8} [ x(n) + 2x(n-1) - 2x(n-3) - x(n-4) ]$$ This set of coefficients gives the filter impulse response: $h[0] = 1/8$ $h[1] = 2/8$ $h[2] = 0$ $h[3] = -2/8$ $h[4] = -1/8$ Multiplying by 8 to get integer coefficients, we get $[1, 2, 0, -2, -1]$. This exactly matches option (B). This specific set of coefficients is widely recognized as the differentiator filter in the Pan-Tompkins algorithm because it balances good noise rejection with accurate QRS slope enhancement. Therefore, the coefficients for the higher-order differentiation filter in the Pan-Tompkins algorithm are $[1, 2, 0, -2, -1]$.