The diagnostic content of an electrocardiogram lies in a few brief waveforms per heartbeat: the P wave, the QRS complex and the T wave. In a recording from a wearable device these sit among disturbances that overlap them in both time and frequency. Muscle noise is broadband. Electrode motion produces slow excursions and abrupt steps, and respiration and posture move the baseline. Linear filters with fixed passbands remove what lies outside the cardiac band and can do nothing about what lies inside it.

Wavelet shrinkage acts differently. The signal is expanded in a wavelet basis, small coefficients are suppressed and the remainder is transformed back. Because a QRS complex is represented by a few large coefficients while noise spreads thinly over all of them, thresholding can separate the two even where their spectra overlap. Donoho and Johnstone (1994) proved that a single threshold proportional to the noise standard deviation and to the square root of twice the logarithm of the sample count comes within a logarithmic factor of ideal performance, and Donoho (1995) analysed the soft-thresholding form. The result assumes white Gaussian noise of known level. Wearable recordings violate that assumption, and this note describes a rule adapted to them.

Three properties of ambulatory noise defeat the textbook rule. The noise is coloured: motion artefact concentrates at low frequencies and muscle noise at high ones, so its variance differs from one decomposition level to the next and a threshold fixed from the finest level is too small for the coarse levels. It is also non-stationary, rising when the wearer moves and falling at rest. And the universal threshold is conservative by design, which with soft thresholding shrinks every retained coefficient by the full threshold and so lowers the amplitude of the R wave. Since clinical measurements and beat detectors both depend on QRS amplitude and slope, that bias matters more here than in generic denoising.

The statistical literature supplies the remedies. Donoho and Johnstone (1995) showed that thresholds chosen level by level adapt to unknown smoothness, and the practice of estimating the noise scale from the median absolute deviation of the coefficients, divided by 0.6745, makes the estimate insensitive to the few large coefficients that carry signal (Donoho and Johnstone, 1994). Gao (1998) proposed the non-negative garrote as a shrinkage function that is continuous like soft thresholding yet approaches the identity for large coefficients like hard thresholding. The rule below assembles these with one element specific to the electrocardiogram.

The rule has four parts and a closing step for the baseline, summarised in Table 1. Figure 1 shows how the treatment differs across the sub-bands of the decomposition.

Efficiency follows from the structure and can be stated without measurement. The lifting transform and its inverse need a fixed, small number of integer additions and shifts per sample. Thresholding adds one comparison and, for retained coefficients only, one division. The median is the single operation that is not linear in block length when done by sorting, and it is needed once per level per block on arrays that halve in size at each level; a selection algorithm or a running histogram of the integer coefficients brings it to linear time as well. Memory is one block of samples, since lifting works in place. The decimated transform is not translation invariant, and the cycle-spinning remedy of Coifman and Donoho (1995) would multiply the cost by the number of shifts averaged. The rule leaves it out and relies on the garrote function to limit the artefacts that cycle spinning was designed to remove.

No recordings were processed for this note, and the rule should be regarded as untested. A fair assessment is nonetheless easy to specify, because the field has shared materials. The MIT-BIH Arrhythmia Database provides annotated two-channel recordings (Moody and Mark, 2001), and the noise stress test of Moody et al. (1984) supplies separate records of baseline wander, muscle artefact and electrode motion artefact that can be added to clean segments at chosen signal-to-noise ratios. With a known clean reference, the improvement in signal-to-noise ratio can be computed directly.

Waveform fidelity is not the only criterion. What the user of a wearable needs is that beats are still detected and that intervals and amplitudes are still measured correctly. An assessment should therefore also report the sensitivity and positive predictivity of a standard QRS detector before and after denoising, and the shift in wave boundaries found by a delineator such as that of Martínez et al. (2004). The rule should be compared with the universal soft threshold, with a level-dependent threshold lacking the R-peak relaxation, and with a conventional band-pass filter, so that the contribution of each part can be seen. Records added from noise databases remain an approximation to the artefacts of a specific device, and final judgement needs recordings from that device.

A thresholding rule suited to wearable electrocardiography can be put together from components that are individually well understood: integer lifting, level-wise robust noise estimation, a threshold relaxed around the R peak and garrote shrinkage. The combination respects the coloured, changing noise of ambulatory recording, guards the QRS complex against the amplitude loss of the universal rule and stays within the arithmetic budget of a small processor. Its two free parameters and its dependence on a preliminary R-peak detector are the points at which it may fail, and they are the first things an experimental study should examine.

To be declared by the author(s) before publication.

To be declared by the author(s) before publication.