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Table 1 Algorithm of the structured lattice reduction

From: Low complexity lattice reduction scheme for STBC two-user uplink MIMO systems

Structured Lattice Reduction (SLR)
Input: H H R = h R , 1 , , h R , 4 , h ̄ R , 1 , , h ̄ R , 4
[1st Stage of the SLR]
   1. Go to OLR-block with H R , α = h R , 1 , h R , 3 , h ̄ R , 1 , h ̄ R , 3
   2. Output: H R = h R , 1 , h R , 2 , h R , 3 , h R , 4 , h ̄ R , 1 , h ̄ R , 2 , h ̄ R , 3 , h ̄ R , 4
[2nd Stage of the SLR]
   3. Go to OLR-block with H R , α = h R , 1 , h R , 4 , h ̄ R , 1 , h ̄ R , 4
   4. Output: H R = h R , 1 , h R , 2 , h R , 3 , h R , 4 , h ̄ R , 1 , h ̄ R , 2 , h ̄ R , 3 , h ̄ R , 4
Function of OLR-block
Input: Hin, α= [hα, 1, hα, 2, hα, 3, hα, 4]
[Initial Sorting with Hin,α]
1. Hin,αS α = [s1, s2, s3, s4]
s.t. s i = hα, θ(i)and |s i | ≤ |si+1|, for 1 ≤ i < 4
[Conventional LLL-LR with S α ]
2. S α = [s1, s2, s3, s4] → G α = [g1, g2, g3, g4]
Order of columns is changed during LLL
s.t. π(i) → i, for 1 ≤ i ≤ 4, (i.e. g i sπ(i))
[Re-ordering]
3. Re-ordering caused by the LLL
s.t. U α = [u1, u2, u3, u4], (uπ(i)= g i )
   4. Re-ordering caused by the initial sorting
   s.t. H in , α = h α , 1 , h α , 2 , h α , 3 , h α , 4 , h α , θ ( i ) = u i
[Remaining Basis Generation]
   5. H in , α H in , β = h β , 1 , h β , 2 , h β , 3 , h β , 4
[Output]
At 1st stage: H out , 1 = h α , 1 , h β , 1 , h a , 2 , h β , 2 , h α , 3 , h β , 3 , h α , 4 , h β , 4
At 2nd stage: H out , 2 = h α , 1 , h β , 1 , h β , 2 , h a , 2 , h α , 3 , h β , 3 , h β , 4 , h α , 4