Associated Discussions

As observation, we can rewrite the optimal U in (4.20) as the following form:

( ) ( )

We can see (4.23) is in a form which is similar to a commonly used robust approach:

the diagonal loading (DL) approach [7][27]. The key idea of DL technique is to regularize the original solution for the weight vector by adding a quadratic term to the objective function. As a result, in the situation of the proposed receiver case, we obtain the DL adaptive matrix is defined as

( )

(

)

DL = γ MT K₋ + I ⁻ I

U I B R B B R D , (4.24)

where γ ≥ is the loading vector. This operation corresponds to injecting an 0 artificial amount of white noise into the main diagonal of B R B^H I . It has been shown that the diagonal loading will guarantee the abilities of inversion of the matrix

( )

H I

γIMT K₋ +B R B , even in the case B R B^H I is a singular matrix. Another meaningful interpretation of the DL is that it is the appropriate tool for combating the unexpected interference like channel mismatch, finite sample effect…etc [7].

Unfortunately, it is not a clear from in reference for what the proper choice of γ is and how it depends on the norm of the channel estimation errors. Until now, the optimal choice of the DL factor is dependent case by case which leads its limited robustness [7], [27], [28]. However, by comparing (4.23) and (4.24), we discover that the optimal γ can be set up as

2 ( )

: _D 1 trace γ =σ ⎛⎜⎜⎜⎝ + K ⎞⎟⎟⎟⎠

J . (4.25)

Our simulation results will confirm this scenario.

4.3 Computer Simulations

In this section, we present the computer simulations for the space-time block coded multi-user MIMO system with the imperfect channel estimation. In all of them, we assume there are Q = transmitters. Here the channel model is assumed as 4 independent Rayleigh fading channel so each element of the channel matrices are independently drawn from a zero-mean complex Guassian distribution. The full-rate Alamouti’s OSTBC (T =2, K = ) is used, in other words, the transmitter is with 2

2

N = transmit antennas. A single receiver of M = receive antennas is assumed. 4 Since the imperfect channel estimation case is assumed, the receivers in this system use the presumed (erroneous) channel matrix H=H+ ΔH rather than the true channel matrix H . In each simulation run, each element of the channel estimation error matrix ΔH is generated by independently drawn from a zero-mean complex Guassian variable with the variance σ due to LS channel estimation assumption. _D² Then the presumed channel matrix H is coming from the addition of every element in ΔH to the corresponding element in H .

At first, Figure 4.2 shows the simulated bit-error-rate (BER) of the GSC-based receiver (3.18) and the robust GSC-based receiver (4.22) versus SNR with imperfect channel estimation. Here the channel estimation error variance σ =_D² 0.01 is simulated. As expected, the robust solution substantially outperforms the non-robust one especially in high SNR region since the optimal robust weighting matrix is designed to handle the channel mismatch. Because of the fixed power of the channel estimation error, the performance is dominated by the channel estimation error rather than channel noise and multi-user interference at high SNR. As a result, the

improvement of the robust receiver becomes more obvious with SNR increasing.

0 5 10 15 20 25 30

10^-3 10^-2 10^-1 10⁰

GSC RGSC

Figure 4.2. BER performance of the GSC-based receiver and the robust GSC-based receiver with imperfect channel estimation

Figure 4.3 illustrates the BER decreasing trend of the GSC-based receiver and the proposed (robust GSC/SIC-based) receiver under the different power (in decreasing form) of the channel estimation error at SNR =35 dB. It can be observed that the proposed receiver is able to combat the different level of channel mismatch from

2 0.1 ~ 0

σ =D . And when the channel information is perfectly known (σ = ), the _D² 0 proposed robust solution reduces to act as the non-robust one. The result can also be confirmed by the derivation of (4.8).

0

Figure 4.3. BER performances of the GSC-based receiver and the proposed receiver under different channel estimation error variance.

Throughout the second part of simulations, the following techniques are examined:

z The proposed (robust GSC/SIC-based ) receiver

z The Stamoulis’s method [5]

z The Naguib’s approaches [6]

z The minimum variance (MV) receiver [7]

z The DL-based MV receiver [7], where γ =5σ_D² is chosen. Note that this is a popular ad hoc choice of γ [7]-[9].

In Figure 4.4, the BERs of all the examined receivers are displayed versus SNR. Here

D2

σ is also assumed as 0.01 to test the robustness against imperfect channel state information of the proposed receiver and other techniques. And we assume all

transmitters are equal power. As expected, all tested receivers expect for the proposed one are sensitive to the channel estimation error since they are designed under the perfect channel assumption. Compared to the simulation result in Chapter 3, the stamoulis, the naguib, and the MV approaches all fail to resist the channel mismatch especially in high SNR region. Note that the DL-based MV receiver is a robust version of MV receiver and performs better than non-DL counterparts. However it suffers the finite sample effect of MV receiver and has limited performance. With Figure 4.2, it can be seen the improvement between robust GSC-based receiver and non-robust GSC-based receiver is more manifest than the one between DL-based MV receiver and MV receiver toward same channel estimation error variance. The result also illustrates the derivation of the optimal choice γ in (4.25) do perform better than the popular choice of γ =5σ_D² . Figure 4.5 shows the un-equal power case to confirm the advantage of the SIC mechanism toward the near-far effect.

0 5 10 15 20 25 30

Figure 4.4.BER performances of the proposed receiver and other existing methods (equal-power case)

Figure 4.5. BER performances of the proposed receiver and other existing methods (unequal-power case)

4.4 Summary

In this chapter, we solve the receiver performance degradation problem under imperfect channel estimation in the space-time coded multi-user MIMO system. We analyze the effect and the potential problems due to channel mismatch in the GSC-based receiver (3.18) in Section 4.1. And Section 4.2 is shown that by the perturbation techniques we can estimate the error amount of the estimated blocking matrix. Combining the perturbation analysis result and the distinctive structure of STBC, we can derive the robust solution in a simple closed-form. We also give some associated discussions about the proposed robust solution comparing with other existing robust solutions. Finally the computer simulation results are shown to confirm the robustness of the proposed method in Section 4.3.

Chapter 5

Conclusions and Future Works

In this thesis, receivers for the space-time block coded multi-user MIMO system are proposed. The receiver design process can be divided into two parts. The first part emphasizes on the assumption that channel estimation is perfect. Since the channel state information is not often exactly known at the receiver end due to some limitation of training symbols or channel conditions, the second part relaxes the original assumption and considers the imperfect channel estimation case in order to obtain robust solutions for the system. In Chapter 2, we first give a brief review of STBCs.

Due to the special structure and simple decoding scheme advantage, we focus on the OSTBCs and Alamouti code. Then we incorporate them into MIMO systems to provide a point-to-point and a multi-user space-time block coded MIMO signal model in the space-time signature form. We use this signal model to design the receivers with joint decoding and interference rejection in the following chapters.

In Chapter 3, the interference terms of the multi-user system model is defined first.

Under the goals of combating multi-user interference and noise at the same time, we derive the optimal constrained equalizer to recover the signals-of-interest in the maximum ratio combining sense. In order to avoid the computational efforts in solving the optimization problem, GSC is used to transform the constrained problem into an

unconstrained one. For further performance enhancement, we apply the SIC mechanism to implement the multi-stage detection and interference cancellation by using the multi-group nature of the system. Since Alamouti code is commonly used in OSTBC, we also provide a low computational complexity scheme of the proposed GSC/SIC-based receiver with Alamouti coded employed. Finally, the simulation results shows that the SIC mechanism do improve the performance of the GSC-based receiver and the proposed (GSC/SIC-based) receiver do possess comparable performance with other existing methods for the multi-user system. Note that all the works of this chapter are under the assumption that channel estimation is perfect at the receiver end.

In Chapter 4, we try to seek a solution to ease the receiver performance degradation problem under imperfect channel estimation assumption. Firstly, we analyze the channel mismatch effect and potential problems in the GSC/SIC-based receiver proposed in the previous chapter. By exploiting the perturbation techniques, we can analyze the error effect of the estimated blocking matrix as long as the statistical characteristics of estimation error are known. As a result, a simple closed-form robust solution is obtained thanks to the perturbation analysis and distinctive structure of STBC. We compared it with other existing robust solutions and confirm its robustness by the BER performance with channel estimation error at the end of the chapter.

The study presented in the thesis has addressed the robust GSC/SIC-based equalizer for STBC multi-user MIMO systems. Since the OSIC approach has been a well-recognized solution for STBC systems, we may apply the ordered SIC (OSIC) mechanism instead of the non-ordered one to yield a better performance. By deriving an approximated expression for SNR, the detection order could be accordingly determined.

Appendix A

By the characteristics of the estimation error ΔD is defined in (4.16), the last three terms is equal to zero. The equation (A.1) thus reduces to

{

} ^{

^}

Since the property B D^H =0 and E Δ{ B}=0, the equation (A.3) can be

Proof of (3): The same as previous proof with the linear relationship of B and ΔD defined in (4.13), we have

where the matrix J defined as

1 H 1

D D− D I D D− D

= Σ Σ

J V U R U V . (A.6)

Then the equation (A.5) is reduced as

{ }

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