Form II Group-Blind Hybrid Detector

Form II Group-Blind Hybrid Detector

The following result gives the asymptotic distribution of the estimated weight vector of the form II group-blind hybrid detector. The proof is given in the Appendix (Section 3.6.1).

Theorem 3.1: Let the sample autocorrelation of the received signals and its eigen-decomposition be

Equation 3.58

Equation 3.59

Let be the estimated weight vector of the form II group-blind linear hybrid detector, given by

Equation 3.60

Then

with

Equation 3.61

where

Equation 3.62

Equation 3.63

Equation 3.64

Equation 3.65

Define the partition of the following matrix:

Equation 3.66

where the dimension of Y₁₁ is . Note that the left-hand side of (3.66) is equal to [cf. the Appendix (Section 3.6.1)], and therefore it is indeed symmetric. Define further

Equation 3.67

Equation 3.68

The next result gives an expression for the average output SINR of the form II group-blind hybrid detector. The proof is given in the Appendix (Section 3.6.1).

Corollary 3.1: The average output SINR of the estimated form II group-blind linear hybrid detector is given by

Equation 3.69

where

Equation 3.70

Equation 3.71

Equation 3.72

As in Section 2.5, in order to gain insights from the result (3.69), we next compute the average output SINR of the form II group-blind hybrid detector for two special cases: orthogonal signals and equicorrelated signals.

Example 1: Orthogonal Signals In this case w₁ = s₁ and R = I_K. After some manipulations, the average output SINR in this case is

Equation 3.73

where is the SNR of the desired user. On comparing (3.73) with (2.142), we obtain the following necessary and sufficient condition for the group-blind hybrid detector to outperform the subspace blind detector:

Equation 3.74

Since 1, the condition above is always satisfied. Hence we conclude that in this case the group-blind hybrid detector always outperforms the subspace blind detector. On the other hand, based on (3.73) and (2.142), we can also obtain the following necessary and sufficient condition under which the group-blind hybrid detector outperforms the DMI blind detector:

Equation 3.75

It is seen from (3.75) that at very low SNR (e.g., f₁ « 1), the DMI detector will outperform the group-blind hybrid detector. Moreover, a sufficient condition for the group-blind hybrid detector to outperform the DMI detector is f₁ 1 (= 0 dB).

Example 2: Equicorrelated Signals with Perfect Power Control Recall that in this case, it is assumed that for k l; and A₁ = · · · = A_k = A. Denote

Equation 3.76

Equation 3.77

Equation 3.78

Equation 3.79

Equation 3.80

Equation 3.81

It is shown in the Appendix (Section 3.6.1) that the average output SINR of the form II group-blind hybrid detector in this case is given by

Equation 3.82

where

Equation 3.83

Equation 3.84

Equation 3.85

Equation 3.86

Equation 3.87

The average output SINR as a function of SNR and r for the form II group-blind hybrid detector and the subspace blind detector is shown in Fig. 3.1. It is seen that the group-blind hybrid detector outperforms the subspace blind detector. The performance of this group-blind detector in the high cross-correlation and low SNR region is more clearly seen in Figs. 3.2 and 3.3, where its performance under different numbers of known users, as well as the performance of the two blind detectors, is compared as a function of r and SNR, respectively. Interestingly, it is seen from Fig. 3.2 that like the DMI blind detector, the group-blind detector is insensitive to the signal cross-correlation. Moreover, for the SNR values considered here, the group-blind detector outperforms both blind detectors for all ranges of r, even for the case that the numbers of known users = 1. Notethat when = 1, the form II group-blind hybrid detector (3.60) becomes

Equation 3.88

Figure 3.1. Average output SINR versus SNR and r for a subspace blind detector and form II group-blind hybrid detector. N = 32, K = 16, = 8, M = 200. The upper curve represents the performance of the form II group-blind detector.

Figure 3.2. Average output SINR versus r for a form II group-blind hybrid detector and two blind detectors. N = 32, K = 16, M = 200, SNR = 15 dB. (In the figure .)

Figure 3.3. Average output SINR versus SNR for a form II group-blind hybrid detector and two blind detectors. N = 32, K = 16, M = 200, r = 0.4. (In the figure .)

This is essentially the constrained subspace blind detector, with the constraint being . It is seen that by imposing such a constraint on the subspace blind detector, the detector becomes more resistant to high signal cross-correlation. However, from Fig. 3.3, in the low-SNR region, the group-blind detector behaves similarly to the subspace blind detector (e.g., the performance of both detectors deteriorates quickly as SNR drops below 0 dB), whereas the performance degradation of the DMI blind detector in this region is more graceful.

Next, the performance of the group-blind and blind detectors as a function of the number of signal samples, M, is plotted in Fig. 3.4, where it is seen that as the number of known users increases, both the asymptotic SINR (as M ) of the group-blind hybrid detector and its convergence rate increase. Finally, the performance of blind and group-blind detectors as a function of the number of users K is plotted in Fig. 3.5, where it is seen that for the values of SNR and r considered here, when the number of known users > 1 the group-blind hybrid detector outperforms both blind detectors, even in a fully loaded system (i.e., K = N). In summary, we have seen that except for the very low SNR region (e.g., below 0 dB), where the DMI blind detector performs the best (however, such a region is not of practical interest), in general, by incorporating the knowledge of the spreading sequences of other users, the group-blind detector offers performance improvement over both DMI and subspace blind detectors.

Figure 3.4. Average output SINR versus the number of signal samples M for a form II group-blind hybrid detector and two blind detectors. N= 32, K= 16, r= 0.4, SNR = 15 dB. (In the figure .)

Figure 3.5. Average output SINR versus the number of users K for a form II group-blind hybrid detector and two blind detectors. N= 32, M = 200, r= 0.4, SNR = 15 dB. (In the figure .)