Input parameters: observation y, measurement matrix $\Phi$, threshold $\alpha$, initial step size $L\ne 0$, ${\delta }_k$.

Output parameters: sparse signal $\stackrel{\wedge }{\theta }$.

Initialization: signal margin $r=y$, sparsity $k_0=1$, sparse signal $\stackrel{\wedge }{\theta }=0$, index set ${\Lambda }_0=\varnothing$, $F_0=\varnothing$, support set $A_0=\varnothing$, maximum atomic number $S_{max}=M/2$.

(1) Calculate the correlation coefficient u by the formula $u=\left\{u_i|u_i=|\langle r,{\phi }_i\rangle |,i=1,2,\cdots ,N\right\}$, and select an index corresponding to the $k_0$ maximum values from u to be stored in the index set ${\Lambda }_0$.

(2) If ${\left|\right|{\Phi }_{{\Lambda }_0}^{T}Y\left|\right|}_2\le \frac{1-{\delta }_K}{\sqrt{1+{\delta }_K}}{\left|\right|Y\left|\right|}_2$, then $k_0=k_0+L$; turn to Step (1).

(3) Select T in $u=\left\{u_i|u_i=|\langle r,{\phi }_i\rangle |,i=1,2,\cdots ,N\right\}$ that are greater than $Th=\alpha \underset{i=1,2,\cdots ,N}{argmax}|\langle r,{\phi }_i\rangle |$. If $T<k_{0 }$, decrease the threshold value α; turn to Step (3). If $T>S_{max}$, increase the threshold value $\alpha$; turn to Step (3).

(4) Sort the selected atoms in Step (3) in descending order and select the first $k_0$ values. The corresponding serial numbers of the selected values constitute the set $J_0$, so $F_0=F_0{{\displaystyle \cup }}^{\text{}}{J}_0$ and $A_0=A_0{{\displaystyle \cup }}^{\text{}}{\phi }_{J_0}$.

(5) Determine the initial residual $r=y-{\left(A_0^T{A}_0\right)}^{-1}{A}_0^{T}y$ for initialization stage $stage=1$.

(6) Select T values greater than $Th=\alpha \underset{i=1,2,\cdots ,N}{argmax}|\langle r,{\phi }_i\rangle |$ in $u=\left\{u_i|u_i=|\langle r,{\phi }_i\rangle |,i=1,2,\cdots ,N\right\}$, combine the sequence numbers corresponding to the selected values into a set $J_t$, and judge the selected atoms in descending order. If the number in set $J_t$ is less than or equal to the current step length L, turn to Step (7); if the number is greater than L, select the first L values and turn to Step (7).

(7) Update the index set and the support set, let $F_t=F_{t-1}{{\displaystyle \cup }}^{\text{}}{J}_0$, and check the number in ${\mathrm{F}}_t$. If it is greater than $S_{max}$, terminate; otherwise, $A_t=A_{t-1}{{\displaystyle \cup }}^{\text{}}{\phi }_{J_t}$.

(8) Calculate the least-squares solution of $y=A_t\stackrel{\wedge }{\theta },$ $\stackrel{\wedge }{{\theta }_t}={\left(A_t^T{A}_t\right)}^{-1}{A}_t^{T}y ,$ and update the residuals $r_t=y-A_t{\stackrel{\wedge }{\theta }}_t=y-{\left(A_t^T{A}_t\right)}^{-1}{A}_t^{T}y$.

(9) Check the margin $r_t$. If $r_t\le {\epsilon }_1\left({\epsilon }_1\approx {10}^{-3}\right)$, terminate; otherwise, $stage=stage +1$, $L=stage\cdot L$, and turn to Step (6);

(10) Output sparse signal $\stackrel{\wedge }{\theta }$.