CP-Based SBHT-RLS Algorithms for Tracking Channel Estimates in Multicarrier Modulation Systems

Table 2

CP-based Extended SBHT-RLS algorithm (CPE3).

Input: 𝐲^𝑘(𝑓), 𝐲^(𝑓)𝑘−1,…, 𝐲^(𝑓)𝑘−2(𝑣) and 𝐘𝑘
Known parameters: Γ𝑖 and 𝜎𝑖
Selecting parameters: 𝜆 (with 0<𝜆≤1)
Initialization: 𝑘=0, an initial training process is used to initialize 𝐡̂0, 𝐑0=𝛿𝐈,
𝐑^0−𝑇=𝛿^−1𝐈 (with 0<𝛿≪1 is small positive scalar).
Algorithm: 𝑘=1,2,3,…
(1) ℋ𝑖,𝑘−1=(1/√𝑚)∑^𝑣𝑙=0ℎ𝑙,𝑘−1𝑒^−𝑗(2𝜋𝑖𝑙)/𝑚, 𝑖=0,…,𝑚−1
(2) 𝑤𝑖,𝑘−1=(Γ^𝑖1/2ℋ^∗𝑖,𝑘−1)/(Γ𝑖‖ℋ𝑖,𝑘−1‖^2+𝜎^2𝑖), 𝑖=0,…,𝑚−1
(3) 𝑋𝑖,𝑘=𝑌𝑖,𝑘𝑤𝑖,𝑘−1, 𝑖=0,…,𝑚−1
(4) 𝑥𝑖,𝑘=(1/√𝑚)∑^𝑚−1𝑙=0𝑞(𝑋𝑙,𝑘)𝑒^𝑗(2𝜋𝑖𝑙)/𝑚, 𝑖=𝑚−𝑣,…,𝑚−1
(5) 𝐀̇𝑘=⎡⎢⎢⎣√𝜆𝐑𝑘−1𝐀𝑘⎤⎥⎥⎦,𝐁̇𝑘=⎡⎢⎢⎣𝐑^−𝑇𝑘−1𝟎𝑇/√𝜆⎤⎥⎥⎦,
𝐲̇^𝑘(𝑓)=⎡⎢⎢⎢⎢⎣𝜆𝐲^(𝑓)𝑘−2√(𝑣)𝜆𝐲^(𝑓)𝑘−1𝐲^𝑘(𝑓)⎤⎥⎥⎥⎥⎦=⎡⎢⎢⎣√𝜆𝐲̇^(𝑓)𝑘−1𝐲^𝑘(𝑓)⎤⎥⎥⎦
(6) 𝐇𝑘𝐀̇𝑘𝐁̇𝑘𝐲̇^𝑘(𝑓)=⎡⎢⎢⎣𝐑^𝑘𝐑^𝑘−𝑇𝐮𝑘𝟎𝐖^𝑇𝑘𝐯𝑘⎤⎥⎥⎦,
where 𝐇𝑘=𝐇^𝑘(𝑟)𝐇^𝑘(𝑟−1),…,𝐇^𝑘(1),
with 𝐇^𝑘(𝑖)=𝐈𝑝−𝑖+1−𝛽𝐯𝐯^𝑇, 𝑝=2𝑣+1, 𝛽=2/𝐯^𝑇𝐯,
𝐯=𝐱±‖𝐱‖2𝐞1,
𝐞1=[1,0,…,0]^𝑇, and 𝐱=𝐀̇𝑘(𝑖∶𝑝,𝑖).
(7) 𝐡𝑘=𝐡𝑘−1−𝐖𝑘𝐯𝑘