Adaptive Channel Estimation in Wireless Communications - Adaptive Signal Processing in Wireless Communications - page 52

Choose d mq 's to minimize J . We must have

∂ J

=, 0 (2.53)

∗

∂ d

mq

ˆ

d = mq

d

mq

which leads to

 

Q P − 1 T − 1 T − 1

∑∑ ∑ ()() ′
q


 
 


 
  =

∑

d ( ) = ()()

j
unune

α− α n

j
nu ne

− α n

m m

y .

m

(2.54)

mq ′′ q ′ q q

′= ′= 1 m 0 n = 0



n 0

 

 

= : g

mq

Define

 

1 1 



1  

 
 
  

 
 
 
  

1 e

− j α

e

− j α L 1 1

V := , (2.55)

  



− j α − j α L

1 e e

P − 1 P − 1

 

T H


 


 


 




T T H H

Dd d := ,, , :=, ,  D D  D

 ,

(2.56)

m m 1 mQ 0 P − 1

T H

:= () ,, ()

    , :=, ,


 




T T H H

Hh l  h l H H  H

 ,

(2.57)

l 1 Q 0 L

( ) ⊗.

     

C:= diag c ,,

 VI

c P (2.58)

0 − NQ

By the definition of d mq in (2.56), it then follows that

CH =. (2.59)

It is shown in [57] that if P ≥ L + 1, rank ( C ) = NQ ( L + 1). Hence, we can determine the
h q ( l )'s uniquely by using the estimates of d mq 's.
Define D ˆ as in (2.56) with d mq replaced with ˆ mq , and similarly define G as in (2.56)
with d mq replaced with g mq . Then (2.54) leads to

Ψ ( ) =

ˆ DG,

(2.60)

I N

where the entries of the PQ × PQ matrix Ψ are ( m , m ′ = 0, 1; … , P - 1, q , q ′ = 1, 2, … , Q )