Linear Multiuser Detectors
Suppose that we are interested in demodulating the data of user 1. Then (2.168) can be written as
where 
denotes the kth column of 
. In (2.170), the first term contains the data bit of the
desired user at time i; the second term contains
the previous data bits of the desired user [i.e., we have intersymbol
interference (ISI)]; and the last term contains the signals from other users
[i.e., multiple-access interference (MAI)]. Hence, compared with the synchronous
model considered in previous sections, the multipath channel introduces ISI,
which together with MAI must be contended with at the receiver. Moreover, the
augmented signal model (2.169) is very
similar to the synchronous signal model (2.5). We proceed to develop
linear receivers for this system.
A linear receiver for this purpose can be represented by a
Pmdimensional complex vector 
, which is
correlated with the received signal r[i] in (2.169)
to obtain
The coherent detection rule is then given by
and the differential detection rule is given by
As before, two forms of such linear detectors are the linear decorrelating detector and the linear minimum mean-square-error (MMSE) detector, which are described next.
Linear Decorrelating Detector
The linear decorrelating detector for user 1 has the form of (2.171)–(2.173) with the weight vector w1 = d1, such that both the multiple-access interference (MAI) and the intersymbol interference (ISI) are eliminated completely at the detector output.[3]
[3] In the context of equalization, this detector is known as a zero-forcing equalizer, as noted in Chapter 1.
Denote by el the K(m + I)-vector with all-zero entries except for the lth entry, which is 1. Recall that the smoothing factor
m is chosen such that the matrix H in (2.169) is a tall matrix. Assume that H has full column rank [i.e., 
]. Let
H
be the Moore–Penrose generalized inverse of the matrix H:
The linear decorrelating detector for user 1 is then given by
Using (2.169) and (2.175), we have
It is seen from (2.176) that both the MAI and the ISI are eliminated completely at the output of the linear zero-forcing detector. In the absence of noise (i.e., n[i] = 0), the data bit of the desired user, b1[i], is recovered perfectly.
Linear MMSE Detector
The linear minimum mean-square-error (MMSE) detector for user 1
has the form of (2.171)–(2.173) with the weight vector w1 = m1, where 
is chosen
to minimize the output mean-square error (MSE):
where
Subspace Linear Detectors
Let l1 
l2 ··· lPm be the
eigenvalues of Cr in (2.178). Since the matrix H has full column rank 
the signal
component of the covariance matrix Cr
(i.e., HHH) has rank r. Therefore, we have
By performing an eigendecomposition of the matrix Cr, we obtain
where Ls = diag(l1, ..., lr) contains the r largest eigenvalues of Cr in descending order, Us = [u1 · · · ur] contains the corresponding orthogonal eigenvectors, and Un= [ur+1 ··· uPm] contains the Pm-r orthogonal eigenvectors that correspond to the eigenvalue s2. It is easy to see that range(H) = range(Us). As before, the column space of Us is called the signal subspace and its orthogonal complement, the noise subspace, is spanned by the columns of Un.
Following exactly the same line of development as in the synchronous case, it can be shown that the linear decorrelating detector given by (2.175), and the linear MMSE detector given by (2.177), can be expressed in terms of the signal subspace components above as [548]
Decimation-Combining Linear Detectors
The linear detectors discussed above operate in a Pm-dimensional vector space. As will be seen in the
next section, the major computation in channel estimation involves computing the
singular value decomposition (SVD) of the autocorrelation matrix Cr of
dimensions Pm x Pm, which has computational complexity 
. By
down-sampling the received signal sample vector r[i] by a factor
of p, it is possible to construct the linear
detectors in an Nm-dimensional space and to
reduce the total computational complexity of channel estimation by a factor of
. (Recall that p is the chip
oversampling factor.) This technique is described next.
For q = 0, ..., p - 1, denote
Similar to what we did before, we can write
Assume that Nm K(m+I) (i.e.,
the matrix Hq is a tall matrix), and rank(Hq) =
K(m + I) (i.e., Hq has
full column rank). For each down-sampled received signal rq[i], the
corresponding weight vectors for user 1's linear decorrelating detector and the
linear MMSE detector are given, respectively, by
where 
. By computing the subspace
components of the autocorrelation matrix Cq,
subspace versions of the linear detectors above can be constructed in forms
similar to (2.181) and (2.182).
To detect user 1's data bits, each down-sampled signal vector rq[i] is correlated with the corresponding weight vector to obtain
The data bits are then demodulated according to
In the decimation-combining approach described above, since the
signal vectors have dimension Nm, the complexity
of estimating each decimated channel response hk,q,
q = 0, ..., p–1,
is 
. Hence the total complexity of channel estimation is 
[i.e., a
reduction of 
is achieved compared with the Pm-dimensional detectors]. However, the number of users
that can be supported by this receiver structure is reduced by a factor of p. That is, for a given smoothing factor m, the number of users that can be accommodated by the
Pm-dimensional detector is 
, whereas
by forming p Nm-dimensional detectors and then combining their
outputs, the number of users that can be supported is reduced to 
.