Blind Channel Estimation
It is seen from the discussion above that unlike the
synchronous case, where the linear detectors can be written in closed form once
the signal subspace components are identified, in multipath channels the
composite channel response vector of the desired user, 
is needed
to form the blind detector. This vector can be viewed as the channel-distorted
original spreading waveform s1. The multipath channel can be estimated by
transmitting a training sequence [32, 64, 112, 442, 581, 612]. Alternatively, the channel can
be estimated blindly by exploiting the orthogonality between the signal and
noise subspaces [31,
272, 484, 548, 551]. We next address the problem of
blind channel estimation. From (2.167),
Equation 2.189
with
Equation 2.190
where pµ is the length of the
channel response {fk[m]},which
satisfies
Equation 2.191
Decimate hk[n] into p subsequences as
Equation 2.192
Note that the sequences fk,q[m] are obtained by down-sampling the sequence {fk[m]} by a factor of p:
Equation 2.193
From (2.192), we
have
Equation 2.194
Denote
Then (2.194) can be
written in matrix form as
Equation 2.195
Finally, denote
Then we have
Equation 2.196
where 
matrix formed from the signature
waveform of the kth user. For instance, when the
oversampling factor p = 2, we have
For other values of p, the
matrix 
is constructed similarly.
Recall that when the ambient channel noise is white, through an
eigendecomposition on the autocorrelation matrix of the received signal [cf. (2.180)], the signal subspace and the noise
subspace can be identified. The channel response fk can
then be estimated by exploiting the orthogonality between the signal subspace
and the noise subspace [31, 272, 484, 548]. Specifically, since Un is
orthogonal to the column space of H and
is in the column space of H [cf.
(2.179)], we have
Equation 2.197
where
Equation 2.198
Based on the relationship above, we can obtain an estimate of
the channel response fk by computing the minimum eigenvector of the
matrix 
. The condition for the channel estimate obtained in such a
way to be unique is that the matrix 
has rank pµ - 1, which necessitates that this matrix be tall
(i.e., [Pm - K(m + I)] 
pµk). Since
µ 
IN [cf. (2.191)], we therefore choose the smoothing factor m to satisfy
Equation 2.199
That is, 
. On the other hand, the condition
(2.199) implies that for fixed m, the total number of users that can be accommodated
in the system is 
Finally, we summarize the batch algorithm for blind linear
multiuser detection in multipath CDMA channels as follows.
Algorithm 2.7: [Subspace blind
linear multiuser detector—multipath CDMA]
Equation 2.200
Equation 2.201
Equation 2.202
Equation 2.203
Equation 2.204
Equation 2.205
Equation 2.206
Equation 2.207
Alternatively, if the linear receiver has the
decimation-combining structure, the noise subspace Un,q is
computed for each q = 0, ..., p - 1. The corresponding channel response fk,q can then be estimated from the
orthogonality relationship
Equation 2.208
Simulation Examples
The simulated system is an asynchronous CDMA system with
processing gain N = 15. The m-sequences of length 15 and their shifted versions are
employed as the user spreading sequences. The chip pulse is a raised cosine
pulse with roll-off factor 0.5. Each user's channel has L = 3 paths. The delay of each path tl,k is uniformly
distributed on [0, 10Tc]. Hence the maximum delay spread is one symbol
interval (i.e., I = 1).
The fading gain of each path in each user's channel is generated from a complex
Gaussian distribution and is fixed over the duration of one signal frame. The
path gains in each user's channel are normalized so that all users' signals
arrive at the receiver with the same power. The oversampling factor is p = 2. The smoothing factor is m = 2. Hence this system can accommodate up to 
users. If the decimation-combining receiver structure is employed, the maximum
number of users is 
. The length of each user's signal
frame is M = 200.
We first consider a five-user system. For the Pm-dimensional implementations, the bit error rates of
a particular user incurred by the exact linear MMSE detector, the exact linear
zero-forcing detector and the estimated linear MMSE detector are plotted in Fig. 2.13. The bit-error rates of the same
user incurred by the three detectors using the decimation-combining structure
are plotted in Fig. 2.14. It is seen that
for the exact linear zero-forcing and linear MMSE detectors, the performance
under the two structures is identical. For the blind linear MMSE receiver, the
Pm-dimensional detector achieves an approximately
1-dB performance gain over the decimation-combining detector for SNR in the
range 4 to 12 dB. Another observation is that the blind linear MMSE detector
tends to exhibit an error floor at high SNR. This is due to the finite length of
the signal frame, from which the detector is estimated. Next, a 10-user system
is simulated using the Pm-dimensional detectors.
The performance of the same user by the three detectors is plotted in Fig. 2.15.
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