Respiratory Signal Extraction from ECG Signal

Abstract

The extraction of respiration from physiological signals such as the electrocardiogram (ECG) and photoplethysmogram (PPG) has been explored for a long time. The proposed methods are mainly based on filters and features. However, the performances among methods are hardly compared and summarized. In this chapter, we focus on the studies of the typical feature-based ECG-derived respirations (EDR). The review of each method is given. The experiment is processed with rest ECG data and reference respiratory data collected synchronously over 60 s. Three parameters, waveform correlation C₁, interval correlation C₂, and respiratory rate RR, are introduced to evaluate each method under conditions of good and poor signal qualities. The results indicate that the optimal method should be determined by applications. For parameter C₁, trough envelope-based method provides the highest similarity with reference waveform (0.8426) when the signal quality is good. However, it is easily affected by the noise, decreasing to −0.3219. For parameter C₂, ECG area mean-based method has the highest similarity with intervals of reference signal (0.8162). Likewise, it performs no better than QRS complex area-based method (0.7013) when the signal quality is poor. In general, signal quality has an effect on the results of these methods.

6. Appendix

1.1 6.1. The Least Square Method

The least square method finds the optimal estimation, which satisfies the minimum sum of squares with original data. It is described as follows.

Assume that a regression equation with two unknown parameters, β₀ and β₁, is defined as follows:

$$ y={\beta}_0+{\beta}_1x $$

(9)

Given a sequence of sample input data {x₁, x₂, …, x_n} and corresponding output data {y₁, y₂, …, y_n}, where n denotes the length of data. The estimated regression equation is as follows:

$$ \hat{y}={b}_0+{b}_1x $$

(10)

where b₀ and b₁ are the estimates of β₀ and β₁, respectively.

The aim is to get the parameter estimates satisfying the condition:

$$ \min \sum {\left({y}_i-{\hat{y}}_i\right)}^2 $$

(11)

where y_i is the corresponding value of original data x, and ${\hat{y}}_i$ is the estimated corresponding value of original data x.

Combining Eq. (10) with Eq. (11), the condition is rewritten as follows:

$$ \min \sum {\left[{y}_i-\left({b}_0+{b}_1{x}_i\right)\right]}^2 $$

(12)

Taking the partial derivative of b₀, we get the expression as follows:

$$ {b}_0=\overline{y}-{b}_1\overline{x} $$

(13)

where $\overline{y}$ and $\overline{x}$ are the average values of y and x, respectively.

Substitute Eq. (13) for b₀ in Eq. (12), and the partial derivative of b₁ is obtained as follows:

$$ {b}_1=\frac{\sum \left({y}_i-\overline{y}\right)\left({x}_i-\overline{x}\right)}{\sum {\left(\overline{x}-{x}_i\right)}^2} $$

(14)