Algorithm 1 The detection and timing estimation algorithm for random access sequence $X$

Input: received signal $y$, random access sequence $\mathit{X}$, threshold $T$, maximum integer CFO $\zeta$

Output: detection indicator $d$, estimated timing $\widehat{\tau }$

1: Calculate $q_1$ and $q_2$ using (25), $v_1$ and $v_2$ using (27), and $\lambda \left(\epsilon \right)$ using (26).

2: if $\lambda \left(\epsilon \right)<T$ then $d=0$, $\widehat{\tau }=\varnothing$

3: else

4:       calculate $Q{V}_{r_1,r_2}$ using (37), $Z_{r_1,r_2}$ using (41)

5:         if $Z_{r_1,r_2}\cap Q{V}_{r_1,r_2}=\varnothing$ then $d=0$, $\widehat{\tau }=\varnothing$

6:         else

7:                d = 1

8:                for i = 0 to 3 do

9:                      if $Z_{r_1,r_2}\left(i\right)\in Q{V}_{r_1,r_2}$ then

10:                          identify index $n$ ($0\le n\le L+1$) where $Q{V}_{r_1,r_2}\left(n\right)=Z_{r_1,r_2}\left(i\right)$,

11:                          and calculate $\widehat{\tau }$ using (42)

12:                          break

13:                    end if

14:              end for

15:       end if

16: end if