Loop-Reduction LLL Algorithm and Architecture for Lattice-Reduction-Aided MIMO Detection

Algorithm 2

The proposed loop reduction LLL algorithm using look-ahead check technique.

INPUT: 𝐐,𝐑,suchthat𝐇=𝐐𝐑

OUTPUT: 𝐐,𝐑,𝐓suchthat𝐇𝐓=𝐐𝐑

(1) Initialize  𝐐=𝐐,𝐑=𝐑,𝐓=𝐈_𝑁

(2)  𝑘=2

(3)  while  𝑘≤𝑁

(4) for 𝑝=𝑘−1,…,1

(5)   𝜇=⌈𝐑(𝑝,𝑘)/𝐑(𝑝,𝑝)⌋

(6)   if  𝜇≠0

(7)   𝐑(1∶𝑝,𝑘)←𝐑(1∶𝑝,𝑘)−𝜇𝐑(1∶𝑝,𝑝)

(8)   𝐓(∶,𝑘)←𝐓(∶,𝑘)−𝜇𝐓(∶,𝑝)

(9)   end

(10) end

(11) if 𝛿|𝐑(𝑘−1,𝑘−1)|²>|𝐑(𝑘,𝑘)|²+|𝐑(𝑘−1,𝑘)|²

(12)   𝐝𝐨{

(13)   swap columns𝑘−1and𝑘in𝐑and𝐓

(14)   calculate Givens rotation matrix Θ=⎛⎜⎜⎝𝑎−𝑏𝑏𝑎⎞⎟⎟⎠

𝑎=𝐑(𝑘−1,𝑘−1)‖𝐑(𝑘−1∶𝑘,𝑘−1)‖and𝑏=𝐑(𝑘,𝑘−1)‖𝐑(𝑘−1∶𝑘,𝑘−1)‖

(15)   𝐑(𝑘−1∶𝑘,𝑘−1∶𝑁)←Θ𝐑(𝑘−1∶𝑘,𝑘−1∶𝑁)

(16)   𝐐(∶,𝑘−1∶𝑘)←𝐐(∶,𝑘−1∶𝑘)Θ^𝑇

(17)   𝑘←𝑘−1

(18)  }while (𝛿|𝐑(𝑘−1,𝑘−1)|²>|𝐑(𝑘,𝑘)|²+|𝐑(𝑘−1,𝑘)|²&&𝑘≠1)

(19)   if  ⌈𝐑(𝑘,𝑘+1)/𝐑(𝑘,𝑘)⌋≠0

(20)𝑘←𝑘+1

(21)   else

(22)𝑘←k+2

(23)  end

(24)  else

(25)𝑘←𝑘+1

(26) end

(27) end