Mains notch for an ECG front end with drifting interference
An ECG front end samples at 500 Hz with 16-bit resolution. The QRS amplitude is about 1.2 mV and the ST segment is 0.05 mV; 50 Hz mains pickup on the electrodes is 0.4 mV, and its frequency wanders between 49.5 and 50.5 Hz. The current design uses a second-order IIR notch with Q = 30 centred at 50 Hz. Clinicians report that the ST segment is distorted for about 200 ms after each notch update and that rejection falls off when the mains drifts. A 200-tap linear-phase FIR notch is proposed instead, and an adaptive LMS notch is also on the table. The processor has enough MIPS for either, but coefficient memory is limited to 1 kB.
1. Compute the notch bandwidth and ringing time for Q = 30 and relate them to the reported 200 ms distortion and the mains drift.
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Bandwidth BW = f0/Q = 50/30 = 1.67 Hz, so the −3 dB points are approximately 49.17 Hz and 50.83 Hz.
Ringing time A useful estimate for the settling of a high-Q notch is t ≈ Q/(π f0) = 30/(π × 50) ≈ 0.19 s, which matches the reported 200 ms distortion.
Drift The mains wanders ±0.5 Hz, so the 49.5 Hz line sits close to the notch edge. The deep null is narrow, and a small drift substantially reduces attenuation, allowing residual 50 Hz to appear as baseline movement. The Q = 30 setting buys selectivity at the cost of long ringing and poor tolerance to drift.
Conclusion The two symptoms have the same cause: too high a Q. Lowering Q widens the notch and shortens ringing but starts to remove legitimate ECG content near 50 Hz.
2. Estimate the FIR length needed for a narrow notch at this sampling rate, check the 1 kB memory limit, and decide whether the 200-tap proposal is adequate.
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Transition-width estimate For a Hamming-windowed FIR, the transition width is roughly Δf ≈ 3.3/(N·T), where T = 1/500 Hz = 2 ms.
To keep the notch narrow, say Δf ≈ 2 Hz: N ≈ 3.3/(2 × 0.002) = 825 taps.
Memory
- 200 taps × 2 bytes = 400 bytes, which fits in 1 kB.
- 825 taps × 2 bytes = 1.65 kB, which does not fit.
Adequacy of 200 taps With N = 200, Δf ≈ 3.3/(200 × 0.002) = 8.25 Hz. The notch is therefore about 8 Hz wide, covering roughly 46–54 Hz. That is broad enough to track the ±0.5 Hz drift, but it removes more ECG content than a narrow notch and may attenuate EMG and high-frequency QRS detail. The linear-phase property is attractive, and the delay is 100 samples = 0.2 s, but the selectivity is limited by the memory constraint.
Conclusion A 200-tap FIR is feasible in memory but is not a narrow notch; a proper 2 Hz notch needs roughly 800–1000 taps and exceeds the 1 kB budget.
3. Recommend a filter design and explain the trade-off between ST-segment fidelity, drift tracking and memory.
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Recommendation Use an adaptive second-order IIR notch rather than a fixed high-Q biquad or a long FIR. An LMS notch needs only a couple of adaptive coefficients, so it fits easily in 1 kB, and it tracks the 49.5–50.5 Hz drift without widening the fixed notch.
Why not keep Q = 30 The 0.19 s ringing distorts the ST segment after every disturbance, and the narrow null does not follow drift. A fixed biquad with Q ≈ 8–10 gives a 5–6 Hz bandwidth and about 50–60 ms ringing, which is better for ST fidelity but still not drift-adaptive.
Why not the 200-tap FIR It is linear phase and has no IIR coefficient sensitivity, but an 8 Hz-wide notch is not selective and the 0.2 s delay may be unacceptable in a real-time monitor. Extending it to 800+ taps exceeds memory.
Implementation cautions
- Implement the IIR as a second-order section with at least 16-bit coefficients; a pole radius near 0.99 is sensitive to coefficient quantization.
- Limit the adaptive step size so the notch does not chase QRS energy, and freeze adaptation during large transients if necessary.
- Verify that the adaptive notch does not create a slow baseline oscillation, since that would be more harmful to ST interpretation than the original 50 Hz pickup.