Algorithm 1 Estimation of Instantaneous RR from PPG signal.

Input PPG

$p\left(t\right)$ ← red or infrared PPG

Remove Base-line wander using SSA

$\mathit{X}$ ← trajectory-matrix [$p\left(t\right)$, M], $M,$ embedding dimension

$[\mathbf{U},\mathbf{\Lambda },\mathbf{V}]$ ← Apply SVD onto $\mathbf{X}$

$\widehat{\mathbf{X}}={\sum }_{j=2,3,...,L}\left[\sqrt{{\lambda }_j}{\mathbf{u}}_j{\mathbf{v}}_j^T\right]$, selection of eigenvectors $L\le M$

$\widehat{p}\left(t\right)$ ← diagonal-averaging$\left(\widehat{\mathbf{X}}\right)$, signal reconstruction

Detect pulse peaks of the $\widehat{p}\left(t\right)$

Estimate Respiratory induced modulation

amplitude modulation (AM) or frequency modulation (FM)

$r\left(t\right)\leftarrow$ Re-sample the modulation into a 4 Hz signal

Estimate average Respiratory Rate (RR)

$r_a\leftarrow$ average RR using Hilbert spectrum or AR-spectral method

Apply SSWT to the 4 Hz modulation

$[\mathbf{f},{\mathbf{W}}_{ssw}]=SSWT\left(r\left(t\right)\right)$, replace $s\left(t\right)$ by $r\left(t\right)$ in Equation (1), implement Equations (3) and (4)

returns $\mathbf{f}$, instantaneous frequencies $\left({\omega }_l\right)$

and ${\mathbf{W}}_{ssw}$, signal energy in time and frequency plan, ($i.e.,T_r({\omega }_l,b$))

Mask SSWT spectrum using averaged RR

$({\mathbf{t}}_i,\mathbf{i})=max\left[{\mathbf{W}}_{ssw}(t:t+\delta t-1,r_f-\omega :r_f+\omega )\right]$

$r_f$ is associated respiratory frequency from average RR ($r_a$)

$\delta t$, time-window size e.g., 32 s, $\omega$, frequency-window size

${\mathbf{t}}_i$, time indices, $\mathbf{i}$, maximum-energy frequency indices

Find instantaneous frequencies ${\mathbf{f}}_{ins}$

$f_{ins}\left({\mathbf{t}}_i\right)=f\left(\mathbf{i}\right)$

Estimate Instantaneous RR in terms of breaths per minute (${\mathbf{f}}_{ins}\times 60$)