Low Complexity Submatrix Divided MMSE Sparse-SQRD Detection for MIMO-OFDM with ESPAR Antenna Receiver

Algorithm 1

MMSE sparse-SQRD.

1: Input: 𝐇, Hnz

2: 𝑐𝑜𝑙𝑠← # of columns of 𝐇

3: 𝑟𝑜𝑤𝑠← # of rows of 𝐇

4: 𝐑 = 0, 𝐐 = 𝐇, p = (1,…, cols)

5: for  𝑖=1,…,𝑐𝑜𝑙𝑠  do

6: for  𝑗=1,…,5  do

7:  𝐧𝐨𝐫𝐦_𝑖∶=𝐧𝐨𝐫𝐦_𝑖+‖𝑞_{𝐻𝑛𝑧(𝑗,𝑖),𝑖}‖²

8:  end for

9: end for

10: for  𝑖=1,…,𝑐𝑜𝑙𝑠  do

11:  𝑘_𝑖= argmin_{𝑙=𝑖,…,𝑐𝑜𝑙𝑠}𝐧𝐨𝐫𝐦_𝑙

12: exchange columns 𝑖 and 𝑘_𝑖 in 𝐑, 𝐐, norm, p

13: 𝑟_𝑖,𝑖=√𝐧𝐨𝐫𝐦_𝑖

14: nz ← indices of the non-zero elements of 𝐪_𝑖

15: for  𝑗=1,…,𝑙𝑒𝑛𝑔𝑡ℎ(nz) do

16:  𝑞_{𝑛𝑧(𝑗),𝑖}∶=𝑞_{𝑛𝑧(𝑗),𝑖}/𝑟_𝑖,𝑖

17: end for

18: for  𝑙=𝑖+1,…,𝑐𝑜𝑙𝑠  do

19:   for  𝑗=1,…,𝑙𝑒𝑛𝑔𝑡ℎ(nz)  do

20:   𝑟_𝑖,𝑙∶=𝑟_𝑖,𝑙+𝑞^∗_{𝑗𝑛𝑧,𝑖}𝑞_{𝑛𝑧(𝑗),𝑙}

21:  end for

22:   if  𝑟_𝑖,𝑙≠0  then

23:   for  𝑗=1,…,𝑙𝑒𝑛𝑔𝑡ℎ(nz)  do

24:    𝑞_{𝑛𝑧(𝑗),𝑙}∶=𝑞_{𝑛𝑧(𝑗),𝑙}−𝑟_𝑖,𝑙𝑞_{𝑛𝑧(𝑗),𝑖}

25:   end for

26:   𝐧𝐨𝐫𝐦_𝑙∶=𝐧𝐨𝐫𝐦_𝑙−‖𝑟_𝑖,𝑙‖²

27:  end if

28: end for

29: end for

30: 𝐐₁←𝐐(1∶𝑟𝑜𝑤𝑠−𝑐𝑜𝑙𝑠,∶)

31: 𝐲=𝐐^𝐻₁𝐮

32: for  𝑘=𝑐𝑜𝑙𝑠,…,1  do

33: 𝑑=∑^{𝑐𝑜𝑙𝑠}_𝑖=𝑘+1𝑟_𝑘,𝑖𝐱_𝑖

34: 𝐱_𝑘=𝒬(𝐲_𝑘−𝑑)/𝑟_𝑘,𝑘

35: end for

36: Permutate 𝐱 according to p