Algorithm 1. CS-OMP algorithm for breathing signal.

Input: perception matrix $A_{CS}=\mathsf{\Phi }\Gamma$, Observation vector y, Signal spectrum peak frequency $f_{max}$, and the noise boundary $\epsilon$.

Output: 1-sparse reconstructed signal for x, $\widehat{x}$.

1:Initialization residual $r_0=y$, $K=1$, index matrix ${\mathsf{\Lambda }}_0={\mathsf{\Phi }}_0=\varnothing$. 2:Find the index ${\lambda }_K$ corresponding to the residual $r_K$, and the maximum value of the column inner product of the measurement matrix such that ${\lambda }_K=\underset{j=1,\cdots ,N}{argmax}\left|〈r_{K-1},{\varphi }_j〉\right|$; 3:Update index matrix ${\mathsf{\Lambda }}_K=[{\mathsf{\Lambda }}_{K-1},{\lambda }_K]$ and ${\mathsf{\Phi }}_K=[{\mathsf{\Phi }}_{K-1},{\varphi }_{{\lambda }_{K-1}}]$; 4:Calculate the least squares solution, ${\widehat{\alpha }}_K=\underset{{\alpha }_K}{argmin}{∥y-A_{C{S}_K}\alpha ∥}_2$; 5:Update $r_K=y-A_{CS}{\widehat{\alpha }}_K$; 6:If ${∥r_K∥}_2>\epsilon$, return to Step 1. Otherwise, reconstruct the signal x by using $\widehat{x}=\Gamma {\widehat{\alpha }}_K$; 7:If the reconstructed signal spectrum is $f=f_{max}$, stop, output $\widehat{x}$. Otherwise, return to Step 2.