Adaptive Receiver Structures
We next consider adaptive algorithms for sequentially estimating the blind linear detector. First, we address adaptive implementation of the blind channel estimator discussed above. Suppose that the signal subspace Us is known. Denote by z[i] the projection of the received signal r[i] onto the noise subspace:
Since z[i] lies in the noise subspace, it is orthogonal to any
signal in the signal subspace. In particular, it is orthogonal to 
. Hence
f1 is the solution to the following
constrained optimization problem:
In order to obtain a sequential algorithm to solve the optimization problem above, we write it in the following (trivial) state-space form:
The standard Kalman filter can then be applied to the system
above, as follows (we define 
Note that (2.213) contains a normalization step to satisfy the constraint || f1[i]|| = 1.
Since the subspace blind detector may be written in closed form as a function of the signal subspace components, one may use a suitable subspace tracking algorithm in conjuction with this detector and a channel estimator to form an adaptive detector that is able to track changes in the number of users and their composite signature waveforms. Figure 2.16 contains a block diagram of such a receiver. The received signal r[i] is fed into a subspace tracker that sequentially estimates the signal subspace components (Us,Ls). The signal r[i] is then projected onto the noise subspace to obtain z[i], which is in turn passed through a linear filter that is determined by the signature sequence of the desired user. The output of this filter is fed into a channel tracker that estimates the channel state of the desired user. Finally, the linear MMSE detector is constructed in closed form based on the estimated signal subspace components and the channel state. The adaptive receiver algorithm is summarized as follows. Suppose that at time (i - 1), the estimated signal subspace rank is r[i - 1] and the components are Us[i - 1], Ls[i - 1], and s2[i - 1]. The estimated channel vector is f1[i - 1]. Then at time i, the adaptive detector performs the following steps to update the detector and estimate the data.
Figure 2.16. Blind adaptive multiuser receiver for multipath CDMA systems.
Algorithm 2.8: [Adaptive blind linear multiuser detector based on subspace tracking—multipath CDMA]
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Update the signal subspace: Use a particular signal subspace tracking algorithm to update the signal subspace rank r[i] and the subspace components Us[i] and Ls[i].
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Update the channel: Use (2.212)–(2.214) to update the channel estimate f1[i].
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Form the detector and perform differential detection:
Simulation Example
We next give a simulation example illustrating the performance of the blind adaptive receiver in an asynchronous CDMA system with multipath channels. The processing gain N = 15 and the spreading codes are Gold codes of length 15. Each user's channel has L = 3 paths. The delay of each path tl,k is uniformly distributed on [0, 10Tc]. Hence, as in the preceding example, 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 for all simulations. The path gains in each user's channel are normalized so that all users' signals arrive at the receiver with the same power. The smoothing factor is m = 2. The received signal is sampled at twice the chip rate (p = 2). Hence the total number of users that this system can accommodate is 10. Figure 2.17 shows the performance of subspace blind adaptive receiver using the NAHJ subspace tracking algorithm discussed in Section 2.6.3 in terms of output SINR. During the first 1000 iterations there are eight total users. At iteration 1000, four new users are added to the system. At iteration 2000, one additional known user is added and three existing users vanish. We see that this blind adaptive receiver can closely track the dynamics of the channel.
Figure 2.17. Performance of a subspace blind adaptive multiuser detector in an asynchronous CDMA system with multipath.
We note that there are many other approaches to blind multiuser detection in multipath CDMA channels, such as constrained optimization methods [59, 60, 80, 187, 300, 305, 306, 427, 485, 487, 490, 498, 583, 584, 605], the auxiliary vector method [364],other subspace methods [10, 31, 252, 258, 272, 287, 446, 484, 548, 551, 564], linear prediction methods [69, 117, 207, 606], the multistage Wiener filtering method [156, 186], the constant modulus method [79, 218, 582], the spreading code design method [435], the maximum-likelihood method [56], the parallel factor method [447], the least-squares smoothing method [483, 610], a method based on cyclostationarity [351], and more general methods based on multiple-input/multiple-output (MIMO) blind channel identification [78, 214, 261, 298, 462, 494–497].