An Analytical Multimodulus Algorithm for Blind Demodulation in a Time-Varying MIMO Channel Context

Algorithm 2

Adaptive Analytical-MMA.

Initialize  𝐅,𝐓,𝐩=0,𝐩=0,𝛼=0

for  𝑘=1,2,…

  Update  𝐅, the prewhitened filter,

   by using 𝑦(𝑘) and [12] we obtain 𝐲(𝑘)=𝐅^𝐻𝐲(𝑘)

   Update real vectors

  𝐲(𝑘)=(ℜ(𝐲(𝑘))ℑ(𝐲(𝑘))),𝐲(𝑘)=(𝐰−ℑ(𝐲(𝑘))ℜ(𝐲(𝑘))),_𝑙=()ℜ(𝐰)ℑ(𝐰)

  Update vector  𝐜̃ and 𝐜

  𝐜̃=𝐲(𝑘)⊗𝐲(𝑘)−𝐩/𝛼,𝐜=𝐲(𝑘)⊗𝐲(𝑘)−𝐩/𝛼

  𝐩=𝜆𝐩+(1−𝜆)(𝐲(𝑘)⊗𝐲(𝑘)),𝐩=𝜆𝐩+(1−𝜆)(𝐲(𝑘)⊗𝐲(𝑘))

  𝛼=𝜆𝛼+1−𝜆

  𝐃=𝐓∘𝐓

   Regard 𝐃 as a basis of the nullspace of 𝐂 and 𝐂,

   and update it using 𝐜̃ and 𝐜 (Algorithm 1)

  for  𝑙=1,…,𝑁_𝑡

    𝐃_𝑙=vec⁻1(𝐝_𝑙),𝐭_𝑙̃=𝐃_𝑙𝐭_𝑙̃,𝐭_𝑙̃=𝐭_𝑙̃/‖𝐭_𝑙̃‖

    Obtain complex  𝑁_𝑡×𝑁_𝑡  matrix  𝐓(𝑘)

    for  𝑚=1,…,2𝑁_𝑡

      𝐭_𝑙𝑚=𝐭̃_𝑙𝑚+𝑗𝐭̃_{𝑙(𝑚+𝑁𝑡)}

    end for

end for

  𝐓=recond(𝐓),𝐓=(ℜ(𝐓)ℑ(𝐓))^𝑇

  Estimate the sources

  𝐬(̂𝑘)=𝐓^𝐻(𝑘)𝐲(𝑘)

end for