mohatta2015.pdf

90 LINEAR EQUALIZATION
where emj(n) can be interpreted as a chip estimate during symbol period m»
for transmitter i () . We can stack these estimates into an N c x 1 vector β^" =
[e ( m,}(0) ... e {^{N c - l)] r , which is obtained by
where
e = W H v, (4.122)
W = C" 1^"' (4.123)
F(¿„) = [f(¿n)( 0 ) ... f( ¿ ")(iV c -l)l . (4.124)
We can interpret the operation in (4.122) as a matrix equalizer (matrix multiply),
in which the nth column of W is a weight vector used to obtain an estimate of the
nth chip transmitted from transmitter ¿ ( > during symbol period mo.
While the elements of F**"' do not depend on fco or mo, the elements of C„ do.
As a result, the entries in W will be code-specific , a function of the spreading codes
used during symbol period mo.
We can trade performance for reduced complexity by approximating C„ with
its average, averaged over the possible spreading codes. Then W would be the
same for each symbol period, requiring fewer weight computations. However, even
with code-averaging, we would still need a separate weight vector for each chip
period. We can reduce complexity further by constraining the equalizer to use a
sliding window of receive samples when computing different chip estimates. Such a
form of equalization is called transversal equalization. Specifically, when forming
em„ (no), we use
v(m 0 7' + n„T c + djT s ), j = 0, ...,J-1, (4.125)
where dj is a relative processing delay, relative to both the symbol period and the
chip period within the symbol. Complexity is reduced because with code averaging,
the weight vector is the same for each chip period no and each symbol period mo.
Before performing code averaging, let's look at the code-specific transversal solution.
To obtain the weight vector for forming eS„(no), we can use the analysis
above, assume
Aj = n t) T c + d/F a , (4.126)
and examine the noth column of W in (4.123) (denoted w). From (4.123),
C v (m 0 )w = ¥*\n 0 T c ). (4.127)
where C„(mo) is a JN r x JN r matrix of the form given in (4.111) and (4.113) and
f (i ">(n7' c ) is defined in (4.120). From (4.120) and (4.126), we see that f' io >(n 0 ï' c )
is really independent of no and can be denoted f**"'.
The elements of f' 1 "' are code-independent, but the elements of C„(mo) are not.
To obtain a code-averaged solution, we average C„(mo) over the spreading codes
and use the result in (4.127) to solve for the weights. Thus,
C„w = f (i,l) , (4.128)