Algorithm 1. Summarize: The main procedures of the TC-AOMP algorithm.
0	Parameter specification: wk is the residual; the iteration number k; $\varnothing$ denotes the empty set; ${\Lambda }_k$ is the index set in k iterations; zk is the selected index in k iterations; ${\mathsf{\Omega }}_k$ is the optimal atomic set selected from the sensing matrix Θ; the iteration termination threshold ε; θ is the estimation CIR; <∙,∙> denotes the inner product operator;
1	Initialization: set the residual w0 = y; k = 1; the total iteration number K = Lpre (Lpre is preamble length of PN sequence) ${\mathsf{\Omega }}_0=\varnothing$; ${\Lambda }_0=\varnothing$; ${\mathbf{\theta }}_0=\varnothing$;
2	Optimal sparse CIR estimation: Go through each k in [1 K]with interval 1;
3	fork = 1,⋯,K do Calculate the inner product of the sensing matrix Θ and w0, and then find the index corresponding to the maximum inner product value, which given as $z_k=\arg \underset{l}{\max }\left|<{\mathsf{\Theta }}_l,{\mathit{w}}_{k-1}>\right|, l=1,2,\cdots ,L_{tap}, l\ne z_{k-1}$;
4	Update the index set ${\Lambda }_k={\Lambda }_{k-1}\cup z_k$ and the atomic matrix ${\mathsf{\Omega }}_k={\mathsf{\Omega }}_{k-1}\cup {\mathsf{\Theta }}_{z_k}$;
5	Calculate the CIR via least squares at k iterations, as ${\mathbf{\theta }}_k({\Lambda }_k)={({\mathsf{\Omega }}_k^H{\mathsf{\Omega }}_k)}^{-1}{\mathsf{\Omega }}_k^H\mathit{y}$;
6	Equalize the signal rcs-m via the estimated CIR ${\mathbf{\theta }}_k$, obtain the equalized signal req-m;
7	Perform temporal correlation of the equalized signal req-m, and then calculate its ISLR;
8	If $ISLR\ge \epsilon$ or k > K, jump out of the loop, output the equalized signal req-m; otherwise continue iteration;
9	Update the residual ${\mathit{w}}_k=\mathit{y}-{\mathsf{\Omega }}_k{\theta }_k({\Lambda }_k)$;
10	End