3.3 Performance of Group-Blind Multiuser Detectors


3.3 Performance of Group -Blind Multiuser Detectors

In this section we consider the performance of group-blind linear multiuser detection. Specifically, we focus on the performance of the group-blind linear hybrid detector defined by (3.9). As in Section 2.5, for simplicity, we consider only real-valued signals. The analytical framework presented in this section was developed in [197].

3.3.1 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

graphics/03equ058.gif


Equation 3.59

graphics/03equ059.gif


Let graphics/w1.gif be the estimated weight vector of the form II group-blind linear hybrid detector, given by

Equation 3.60

graphics/03equ060.gif


Then

graphics/125equ01.gif

with

Equation 3.61

graphics/03equ061.gif


where

Equation 3.62

graphics/03equ062.gif


Equation 3.63

graphics/03equ063.gif


Equation 3.64

graphics/03equ064.gif


Equation 3.65

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Define the partition of the following matrix:

Equation 3.66

graphics/03equ066.gif


where the dimension of Y 11 is graphics/120fig01.gif . Note that the left-hand side of (3.66) is equal to graphics/120fig02.gif [cf. the Appendix (Section 3.6.1)], and therefore it is indeed symmetric. Define further

Equation 3.67

graphics/03equ067.gif


Equation 3.68

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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

graphics/03equ069.gif


where

Equation 3.70

graphics/03equ070.gif


Equation 3.71

graphics/03equ071.gif


Equation 3.72

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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 w 1 = s 1 and R = I K . After some manipulations, the average output SINR in this case is

Equation 3.73

graphics/03equ073.gif


where graphics/121fig02.gif 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

graphics/03equ074.gif


Since graphics/ktilde.gif 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

graphics/03equ075.gif


It is seen from (3.75) that at very low SNR (e.g., f 1 « 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 f 1 1 (= 0 dB).

Example 2: Equicorrelated Signals with Perfect Power Control Recall that in this case, it is assumed that graphics/121fig01.gif for k l ; and A 1 = · · · = A k = A . Denote

Equation 3.76

graphics/03equ076.gif


Equation 3.77

graphics/03equ077.gif


Equation 3.78

graphics/03equ078.gif


Equation 3.79

graphics/03equ079.gif


Equation 3.80

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Equation 3.81

graphics/03equ081.gif


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

graphics/03equ082.gif


where

Equation 3.83

graphics/03equ083.gif


Equation 3.84

graphics/03equ084.gif


Equation 3.85

graphics/03equ085.gif


Equation 3.86

graphics/03equ086.gif


Equation 3.87

graphics/03equ087.gif


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 graphics/ktilde.gif = 1. Notethat when graphics/ktilde.gif = 1, the form II group-blind hybrid detector (3.60) becomes

Equation 3.88

graphics/03equ088.gif


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, graphics/ktilde.gif = 8, M = 200. The upper curve represents the performance of the form II group-blind detector.

graphics/03fig01.jpg

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 graphics/124fig01.gif .)

graphics/03fig02.gif

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 graphics/124fig01.gif .)

graphics/03fig03.gif

This is essentially the constrained subspace blind detector , with the constraint being graphics/123fig01.gif . 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 graphics/ktilde.gif 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 graphics/ktilde.gif > 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 graphics/124fig01.gif .)

graphics/03fig04.gif

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 graphics/124fig01.gif .)

graphics/03fig05.gif

3.3.2 Form I Group-Blind Detectors

Define

Equation 3.89

graphics/03equ089.gif


Equation 3.90

graphics/03equ090.gif


and let graphics/125fig04.gif . The following result gives the asymptotic distribution of the estimated weight vector of the form I linear group-blind hybrid detector and that of the form I linear group-blind MMSE detector. The proof is given in the Appendix (Section 3.6.2).

Theorem 3.2: Let

Equation 3.91

graphics/03equ091.gif


be the sample autocorrelation matrix of the received signals based on M samples. Define

Equation 3.92

graphics/03equ092.gif


Let graphics/lambarcs.gif and graphics/ubarcs.gif contain, respectively, the largest graphics/110fig05.gif eigenvalues of graphics/125fig01.gif and the corresponding eigenvectors. Let graphics/w1.gif be the estimated weight vector of the form I group-blind linear detector, given by

Equation 3.93

graphics/03equ093.gif


where graphics/125fig02.gif for the group-blind linear hybrid detector and graphics/125fig03.gif for the group-blind linear MMSE detector. Then

graphics/125equ01.gif

with

Equation 3.94

graphics/03equ094.gif


where

Equation 3.95

graphics/03equ095.gif


Equation 3.96

graphics/03equ096.gif


As before, the SINRs for the form I group-blind detectors can be expressed in terms of R , s 2 , and A . However, the closed-form SINR expressions are too complicated for this case and we therefore do not present them here. Nevertheless, the SINR of the group-blind linear hybrid detector for orthogonal signals can be obtained explicitly, as in the following example.

Example 1: Orthogonal Signals We consider the form I linear hybrid detector. In this case w 1 = u = s 1 , graphics/127fig02.gif , and graphics/lambars.gif = diag graphics/127fig01a.gif graphics/127fig01b.gif . Moreover, graphics/127fig03.gif , so that graphics/127fig04.gif , and graphics/dtilde.gif = . Hence graphics/127fig05.gif . Substituting these into (2.119), and after some manipulation, we get

Equation 3.97

graphics/03equ097.gif


Comparing (3.73) and (3.97), we see that for the orthogonal-signal case, the form I group-blind hybrid detector always outperforms the form II group-blind hybrid detector.

In Fig. 3.6 the output SINR of the two blind detectors and that of the two forms of group-blind hybrid detectors [given, respectively, by (2.142), (3.73), and (3.97)] are plotted as functions of the desired user's SNR, f 1 . It is seen that in the high-SNR region, the DMI blind detector has the worst performance among these detectors. In the low-SNR region, however, both the form II group-blind detector and the subspace blind detector perform worse than the DMI blind detector. The form I group-blind detector performs the best in this case.

Figure 3.6. Average output SINR versus f 1 of blind and group-blind detectors for the orthogonal signal case. N = 32, K = 16, M = 200. (In the figure graphics/124fig01.gif .)

graphics/03fig06.gif

Example 2: Equicorrelated Signals with Perfect Power Control Although we do not present a closed-form expression for the output SINR for the form I group-blind detector, we can still evaluate the SINR for this case as follows . As noted above, the SINR is a function of the user spreading sequences S only through the correlation matrix graphics/127fig06.gif . In other words, with the same A and, s 2 , systems employing different set of spreading sequences S and S ' will have the same SINR as long as graphics/127fig07.gif [even if the spreading sequences take real values rather than the form graphics/128fig01.gif , s j,k {+1, “1}]. Hence given R , A , and s 2 , we can, for example, designate S to be of the form

Equation 3.98

graphics/03equ098.gif


(where graphics/rsquare.gif denotes the Cholesky factor of R ) and then use (2.119) and (3.94) to compute the SINR. Note that each column of S in (3.98) has unit norm since the diagonal elements of R are all 1s. Our computation shows that the performance of the form I group-blind hybrid detector is similar to that of the form II group-blind hybrid detector, with the exception that the form I detector behaves similarly to the DMI blind detector in the very low SNR region ”namely, it does not deteriorate as much as do the form I group-blind and subspace blind detectors. This is shown in Fig. 3.7. (The performance of the form I group-blind MMSE detector is indistinguishable from that of the form I group-blind hybrid detector in this case.)

Figure 3.7. Average output SINR versus SNR for form I and form II group-blind detectors and blind detectors. N = 32, K = 16, M = 200, r = 0.4. (In the figure graphics/124fig01.gif .)

graphics/03fig07.gif

In summary, we have seen that the performance of the subspace blind detector deteriorates in the low-SNR and high-cross-correlation region, the form II group-blind detector is resistant to high cross-correlation but not to low SNR, and the form I group-blind detector is resistant to both high cross-correlation and low SNR. Although the DMI blind detector is also insensitive to both high cross-correlation and low SNR, its performance in other regions is inferior to all the subspace-based blind and group-blind detectors. Hence we conclude that the form I group-blind detector achieves the best overall performance among all the detectors considered here.

Finally, we compare the analytical performance expressions given in this section with the simulation results. The simulated system is the same as that in Section 2.5 ( N = 13, K = 11). The analytical and simulated SINR performance of the form I and form II group-blind detectors is shown in Fig. 3.8. For each detector, the SINR is plotted as a function of the number of signal samples ( M ) used for estimating the detector at some fixed SNR. The simulated and analytical BER performance of these estimated detectors is shown in Fig. 3.9. As before, the analytical BER performance is based on a Gaussian approximation of the output of the estimated linear detector. It is seen that as is true for the DMI blind detector and the subspace detector treated in Section 2.5, the analytical performance expressions discussed in this section for group-blind detectors match the simulation results very well. Performance analysis for the group-blind detectors in the more realistic complex-valued asynchronous CDMA with multipath channels and blind channel estimation can be found in [196]. Some upper bounds on the achievable performance of various group blind multiuser detectors are obtained in [192, 195]. Moreover, large-system asymptotic performance of the group-blind multiuser detectors is given in [604].

Figure 3.8. Output average SINR versus the number of signal samples M for form I and form II group-blind hybrid detectors. N =13, K = 11. The solid line is the anyalytical performace, and the dashed line is the simulation performace.

graphics/03fig08.gif

Figure 3.9. BER versus the number of signal samples M for form I and II group-blind hybrid detectors. N = 13, K = 11. The solid line is the analytical performance, and the dashed line is the simulation performance.

graphics/03fig09.gif



Wireless Communication Systems
Wireless Communication Systems: Advanced Techniques for Signal Reception (paperback)
ISBN: 0137020805
EAN: 2147483647
Year: 2003
Pages: 91

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