Adaptive Equalization for Wireless Channels - Adaptive Signal Processing in Wireless Communications - page 255

Another property that can be exploited in multicarrier (but not SCCP) systems is
the guard band in the frequency domain, which was mentioned in section 9.3.1. Often,
the subcarriers on the band edges are left as zeros at the transmitter in order to limit
adjacent channel interference. These are often referred to as null tones. If the channel is
benign, then at the receiver, the null tones should also be zero. Thus, we can form a cost
function as


 


 

∑ β  y

2

   

J = E b , (9.25)

j

j ∈ null tones

where the weights β j would typically be all ones. This is very similar to a decision-directed
algorithm, with only one valid constellation point (i.e., zero). The difference is that (9.25)
is invariant to magnitude and phase distortion, since 0 · Ae j θ = 0. Thus, the effects of the
FEQ can be ignored, which would not be the case for a DD-LMS algorithm operating on
the non-null tones. A gradient descent of (9.25) was proposed in [55] and further analyzed
in [56], and it is sometimes referred to as the carrier nulling algorithm (CNA; not to be
confused with the constant norm algorithm of the same acronym, discussed in the previ-
ous section). It also has an LMS-like structure, although with part of an FFT involved to
relate the values on the null tones to the CSE output:

   

YFrf = b ,

null null

ˆ

    =     −

H H

f b + 1 f b µ rF Y (9.26)

nul l null

ˆ ˆ

    =+        

f b + 1 f b 1 f b + 1 ,

2

where r is the N × L Toeplitz matrix:



( ) +++ … ( ) +++−



       

rb N 1 ν 1 ∆ , , r bN 1 ν 1 ∆ L











:

:

r = , (9.27)

       

rr N + ∆ , … , r bN + − ∆ L

 

and where F null is obtained by taking the rows of the FFT matrix indexed by the null tones,
and Y null is similarly obtained from the FFT output vector.
Now we can reconsider the case in which training is intermittently provided on some
subcarriers. Let S null , S pilot [ b ], and S data [ b ] be the sets of subcarriers containing null tones,
pilot tones, or unknown data (respectively) during block b , with S null ∩ S pilot [ b ] ∩ S data [ b ] =
{0, …, N - 1}. Then we can form the semiblind cost function