Simulation Results and Comments

12 12

−

−

+

⎟⎠

⎜ ⎞

⎝⎛

= u

k u

k T

u u k

k

P trace

trace S

μ S

(3.3.3-17)

We can expect that the same results can be obtained from the two-step approach where we use the equivalent MIMO channel

H

^u_k

W

_k^u instead of

H .

^u_k

3.4 Simulation Results and Comments

We will show the performance of above joint Tx/Rx beamforming design for multi-user MIMO-OFDM SDMA downlink system by computer simulations. In this set-up, we first assume that the channel estimations are perfect known at both transmit and receive terminals; the elements of the MIMO channel are independent-identically-distribution (i.i.d) complex Gaussian distribution with zero mean and variance 1 and the channel length is

L .

_C

At BS which is equipped with M transmit antennas, each user’s data are QPSK modulated and de-multiplexed into B parallel paths and each user is equipped with

N

receive antennas. Each path is processed by OFDM modulation. Before passing to OFDM, we have to perform the transmit beamforming for each subcarrier of each user and then process the null-space matrix for each subcarrier of all user. We can see procedures in Figure 3-7. In OFDM, we assume the length of FFT is L and the length of CP is

L which is larger or equal than the channel length

_CP

L in order to

_C

keep the orthogonality between each subcarrier. And then launch the OFDM packets via M transmit antennas. At each receive terminal, thanks to the null-space matrix, each user only receives it own data. After performing OFDM demodulation, the

output signal is processing to the receive beamforming for each subcarrier and then passes to an appropriate interleaver to obtain correct data streams.

Figure 3-9 shows the bit error rate (BER) curves of typical multi-user MIMO-OFDM SDMA systems where the BS equipped with 6 to 8 transmit antennas communicates simultaneously with 3 users. Each user is equipped with 2 receive antennas. The FFT length L is 64. Each OFDM packet contains 640 data symbols and 100 MIMO channel realizations described above are simulated and generated independently for each packet. The total transmit power per symbol period across all antennas is normalized to 1. The SNR is defined as the total transmitted power normalized with the noise variance at each subcarrier.

0 5 10 15 20 25 30

10⁻⁶ 10⁻⁵ 10⁻⁴ 10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM System Under Perfect Channel Estimation

Multi−user case;M=6;N=2;U=3 Single user case;M=6;N=6;U=1 Multi−user case;M=7;N=2;U=3 Single user case;M=7;N=6;U=1 Multi−user case;M=8;N=2;U=3 Single user case;M=8;N=6;U=1

Figure 3-9 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM system under perfect CSI

We call the MIMO system “fully loaded” when the number of parallel streams is equal to the number of BS antennas. When the number of BS antennas is greater than the number of parallel streams, the system is called ”under-loaded”. In this case, the diversity gain can be expected. However, it is impossible to simultaneously transmit more parallel streams (over-loaded case) without inducing irreducible MUI.

In Figure3-9, we also show the single user case which has 6 receive antennas.

From the simulation results, we can summarize the following observations.

z Since the single user case with the same number of receive antennas has more degrees of freedom for spatial processing at the receiver, it has a better performance than the multi-user case.

z When the number of BS antennas increases, we can have of course a better performance because of the diversity gain obtained from the transmit beamforming.

z In multi-user case, adding one antenna at the BS provides a diversity gain of 1 to all users.

z The performance difference between the single user and the multi-user case becomes negligible when the number of BS antennas increases.

3.5 Conclusions

In this chapter, we consider the joint design Tx/Rx beamforming for multi-user MIMO-OFDM downlink communications using SDMA under perfect channel estimation. In multi-user system, the MUI is the major problem which needs to be

handled. We use a null-space technique to decouple the multi-user MIMO SDMA joint design problem into several single user problems, In other words, the product of the MIMO channel and the null-space matrix results a block-diagonal matrix which means the MUI between each user is completely removed. Thus, each user terminal only has to handle its own inter-stream interference. Furthermore, we extend this flat-fading channel case to frequency selective channel environment by using the MIMO-OFDM based system. Using the null-space constraint, the several decoupled single user designs also have the properties that the system structure can be scalable with respect to the number of antennas, size of the coding block, and transmit power.

--- Chapter 4

Robust Design of Joint Tx/Rx MMSE Beamforming with Excellent Channel Estimation Error immunity

for Multi-user MIMO-OFDM SDMA Downlink System

---

We have discussed the joint Tx/Rx MMSE beamforming design for the multi-user MIMO-OFDM SDMA system in chapter 3. The channel information is assumed to be perfectly known at both transmit and receive terminals. However, the channel estimation always contains errors in real communication systems and only imperfect CSI can be obtained. In a practical wireless environment, channel information has to be estimated periodically because of the time-varying characteristic of channel, especially in the mobile environment. The imperfect CSI has a significant impact on the performance. In this chapter, we will focus the robust beamforming design to enhance the system performance.

4.1 Introduction

If the channel estimation is perfect, we can design the optimal solutions of

transmit and receive beamforming matrices in single user MIMO-OFDM case.

Furthermore, for a multi-user MIMO-OFDM SDMA system, a perfect null-space matrix that removes the MUI between each user also can be obtained. However, if the channel estimation contains errors, the null-space and Tx/Rx beamforming matrices will be designed imperfectly in multi-user joint design system (see Figure 4-1).

H

… BS … ^C

¹

C

^U

… …

1

M 2

s

¹

(k)

s

^U

(k) t(k)

…

MS-1

MS-U … … ¹

N

¹

1 N

^U

r

¹

(k)

r

^U

(k)

… …

C

¹

C

^U

s

¹

(k)

s

^U

(k) ( )

( ) ^H

F H W

ˆ ˆ

( ) ^ˆ

1

H G

( ) ^H

^ˆ

G

^U

H ˆ Hˆ

Downlink

The imperfect null-space design will induce MUI between each user and the imperfect transmit and receive beamforming designs will cause the inter-stream interference at each user terminal. That is why the imperfect CSI will cause significant performance degradation. This is an unavoidable problem in realistic wireless communication systems that the channel estimation always contains errors and only the imperfect CSI can be obtained.

The CSI at receiver can be obtained via the training sequence or pilot symbols that allows to estimate the channel. The CSI at transmitter can be obtained by using feedback channels from the receiver to the transmitter. In many cases, a sufficiently

Figure 4-1 Joint Tx/Rx beamforming design for multi-user MIMO downlink system under imperfect CSI

accurate channel information at receiver can be assumed, however, the CSI at transmitter is always far from sufficient accuracy. Hence one can assume that the receiver has perfect CSI to design receive beamforming and the transmitter has imperfect CSI to design transmit beamforming and null-space matrices. But in this chapter, we will consider more general case that both CSI at transmit and receive sides are imperfect to design null-space and Tx/Rx beamforming matrices in multi-user joint design MIMO-OFDM SDMA system.

We are going to apply two robust approaches to our multi-user joint design problem to against the performance degradation caused by MUI and inter-stream interference [6] [18]. Both of these robust methods have the similar performance over fast time-variant and slow time-variant environments. Therefore, we apply the moving average approach that has better performance over slow time-variant environment.

Note that all the robust methods need the statistic properties of the estimation errors.

In the end, we will show the simulation results and give some comments.

4.2 Robust Design of Joint Tx/Rx MMSE Beamforming 4.2.1 Problem Description

We now consider interferences induced by the imperfect null-space matrix and Tx/Rx beamforming due to the channel information error. Recall that for each subcarrier of OFDM system the product of MIMO channel and null-space matrix results a block-diagonalized matrix which means MUI is perfect removed. That is

u

u

i

k i

k

W

=

0

; if ≠

H

(4.2.1-1)

where the subscript notation denotes subcarrier index and the superscript notation represents the user terminal index. And each user’s Tx/Rx beamforming is also designed due to the exact channel information

H . Figure 4-2 illustrates again the

ⁱ_k major block-diagonalized operation at each subcarrier.

1

Moreover, we explain the operation by another point of view in order to derive the robust null-space matrix against MUI. Assume that t is the total transmitted _k signal at subcarrier

k

and it can be formulated as beamforming design MIMO-OFDM SDMA system becomes

u

By the equation (4.2.1-1), the above equation can be rewritten as

Figure 4-2 Joint Tx/Rx beamforming for multi-user MIMO-OFDM downlink system with null-space matrix

u

In (4.2.1-4), it is easy to see that the mutual interferences between co-channel users are removed and leaves only each user’s joint design problem to cope with its own inter-stream interference.

Now we assume that the MIMO channel information contains errors and can be represented as

k k

k

H H

H ˆ = + Δ

(4.2.1-5) where Δ

H

_k is channel information error and its elements are independent random variables with zero mean and variance

σ

_Δ²_Hk . Since the BS has the imperfect CSI

Hˆ

k, not the exact CSI

H , the null-space matrix is obtained under the condition

_k

u

And we also obtain the imperfect transmit beamforming

Fˆ

_k^u that is designed from the

equivalent channel

H W ^ˆ

ⁱ_k

^ˆ

_k^u. Thus, the total transmitted signal at each subcarrier becomes

And the received signal of user

u

becomes

u be affected by MUI. Furthermore, at user

u

’s terminal, since the first term of

(4.2.1-8), both of transmit and receive beamforming will be designed imperfectly which leads to the inter-stream interference. In the following section, we will first consider the MUI caused by imperfect null-space matrix (the second term effect), and then consider the inter-stream interference caused by imperfectly designed beamformings (the first term effect).

4.2.2 Robust Designs

4.2.2.1 Robust Design Against MUI

In this section, we derive the robust null-space matrix which minimizes the expected power of MUI by using the statistics of the channel estimation error. Before deriving it, we first consider the statistical dependence among the exact channel

H ,

_k the estimated channel

Hˆ

_k and the estimated channel error Δ

H

_k. In general, Δ

H

_k is independent of

H but dependent on

_k

Hˆ

_k. However, if

Δ H

_k is much smaller

than

H

_k , it can be assumed that Δ

H

_k is approximately independent of

Hˆ

_k. Thus, in order to derive the robust null-space matrix, we will assume that

k k

k

k

H H H

H ˆ ⊥ ; ˆ ⊥ Δ

(4.2.2.1-1)

We now investigate the effect that when user

u

’s signal is transmitted, how it will induce interference to affect other users. The received signal at subcarrier

k

and all user terminals when user

u

’s signal is transmitted is

u k k u k u k u k k all

k

H W F s H t

y = ˆ ˆ =

(4.2.2.1-2) Obviously, if the null-space is perfect designed, that is, if we know the exact CSI,

there is no MUI occurred induced by user

u

’s signal and all elements of

y will be

^all_k

zeros except the elements at user

u

’s terminal. We now assume that the transmitted signal of user

u

at subcarrier

k

which minimizes the expected power of MUI is And then (4.2.2.1-2) becomes

( ) ( )( )

received signal, and other elements are all zero values, therefore it does not induce any interference to other users. But the other terms in (4.2.2.1-4) can be consider as the MUI induced by user

u

’s signal. In order to minimize the MUI, we would like to find

a

^u_k to minimize the expected power of these terms

Under the assumptions described in (4.2.2.1-1) and the statistic properties of Δ

H

_k,

that is, its elements are independent random variables with zero mean and variance

2 Hk

σ

Δ . Thus the expectation of equation (4.2.2.1-6) can be rewritten as

{ } {

^uk

}

From above equation, we get the solution as

u

And the transmitted signal of user

u

in (4.2.2.1-3) becomes

u

Thus, we can obtain the robust null-space matrix

W ~

_k^u

which minimizes the expected power of MUI between all users from the equation (4.2.2.1-10)

u

4.2.2.2 Robust Design Against Imperfect Beamforming

After considering the MUI caused by imperfect null-space matrix, we now consider the inter-stream interference induced from imperfect Tx/Rx beamforming at

each user terminal. Recall that for user

u

at subcarrier

k

the transmit and receive beamforming are designed from minimizing the instantaneous MSE matrix

( ) ( )( )

By the robust null-space matrix, we decouple the multi-user joint design problem with worse MUI into a set of parallel single user joint design with minimized MUI. The per-user joint beamforming design with robust null-space matrix can be represented as

( ) ( )( )

In order to enhance the performance, we now consider the minimization of the averaged MSE matrix.

( )

can have following equation

{ }

Using properties (4.2.2.2-4) and (4.2.2.2-4), the equation (4.2.2.2-3) becomes

( )

Therefore, using the two-step approach, the optimal receive beamforming that minimizes the averaged MSE matrix can be obtained as the Wiener solution

( )

u k^u

And we can obtain the transmit beamforming which has the similar form

u Combining these two robust methods, we can obtain a better performance than the naive design which designs the null-space matrix and Tx/Rx beamforming by the imperfect channel information

Hˆ

^u_k. By the way, for single user joint design problem

under channel estimation error, the null-space matrix becomes an identity matrix and we just need to apply the robust Tx/Rx beamforming to improve the performance.

4.2.2.3 Robust Design Using Moving Average Method

We have introduced the approach that combines robust null-space matrix which minimizes the MUI between each user and robust Tx/Rx beamforming which copes with the inter-stream interference of each user terminal. However, the combination of

these two robust methods uses the instantaneous estimated channel information

Hˆ

^u_k

to calculus the robust null-space matrix and Tx/Rx beamforming. In other words, once we receive an OFDM packet and estimate the channel by the long preamble of the packet, we use the instantaneous estimated channel information to apply to our combined robust approach to improve the performance. Thus, we can expect that the similar performance can be obtained under fast- and slow-fading channel caused by the Doppler effect of moving user terminals.

However, under the slow time-variant environment we can apply the moving average approach (the same throughput) to improve the channel estimation error instead of using the instantaneous estimated channel information. Using the moving average approach in such multi-user joint design system, we calculus the null-space matrix and Tx/Rx beamforming by the same approach described in chapter 2 and chapter 3. The only difference is that we replace the instantaneous estimated channel information by the moving average of the estimated channel. These robust approaches to improve the system performance are simulated by computer and shown as next section. We also give some comments to these simulation results.

4.3 Simulation Results and Comments

we consider a multi-user MIMO-OFDM SDMA where the BS equipped with 6 to 7 transmit antennas communicates simultaneously with 3 users. Each user is equipped with 2 receive antennas. The other parameters are set up in the same way as that in section 3-4.

0 5 10 15 20 25 30 10⁻³

10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM SDMA System Under CSI Error 0.075

M=6;N=2;U=3; Naive design

M=6;N=2;U=3; Robust null−space matrix only

M=6;N=2;U=3; Robust null−space matrix and Robust beamforming M=7;N=2;U=3; Naive design

M=7;N=2;U=3; Robust null−space matrix only

M=7;N=2;U=3; Robust null−space matrix and Robust beamforming

0 5 10 15 20 25 30

10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM SDMA System Under CSI Error 0.05

M=6;N=2;U=3; Naive design

M=6;N=2;U=3; Robust null−space matrix only

M=6;N=2;U=3; Robust null−space matrix and Robust beamforming M=7;N=2;U=3; Naive design

M=7;N=2;U=3; Robust null−space matrix only

M=7;N=2;U=3; Robust null−space matrix and Robust beamforming

Figure 4-3 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM SDMA system under CSI error 0.075

Figure 4-4 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM SDMA system under CSI error 0.05

0 5 10 15 20 25 30 10⁻³

10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM SDMA System Under CSI Error 0.025

M=6;N=2;U=3; Naive design

M=6;N=2;U=3; Modified null−space matrix only

M=6;N=2;U=3; Robust null−sapce matrix and Robust beamforming M=7;N=2;U=3; Naive design

M=7;N=2;U=3; Modified null−space matrix only

M=7;N=2;U=3; Robust null−sapce matrix and Robust beamforming

Figure 4-3, Figure 4-4 and Figure 4-5 are the simulation results of joint Tx/Rx beamforming design for multi-user MIMO-ODFM SDMA downlink system when the variances of channel estimation error are 0.075, 0.05 and 0.025 respectively.

Comparing these simulation results, we give the following comments:

z The naive design of course has worst performance due to the imperfect design of null-space matrix and beamforming which induces the MUI and inter-stream interference respectively. However, using the robust null-space matrix to resist the MUI, we can obtain the performance improvement.

z Furthermore, combining the robust null-space matrix with the robust Tx/Rx beamforming to simultaneously resist the MUI and inter-stream interference,

Figure 4-5 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM SDMA system under CSI error 0.025

we certainly obtain a better performance than above two designs.

z When the variance of channel estimation error increases, the performance becomes worse. The improvement of the combined robust approach is observable.

z The effect of improvement of the robust null-space matrix is larger than the robust beamforming design since the MUI will cause the more performance loss than the inter-stream interference.

z

For the under-loaded case, we can see the performance improvement caused by the increase of diversity gain.

0 5 10 15 20 25 30

10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for MIMO−OFDM SDMA System Under CSI Error 0.05

M=6;N=2;U=3; Naive design

M=6;N=2;U=3; Robust null−space matrix only

M=6;N=2;U=3; Robust null−space matrix and Robust beamforming M=7;N=2;U=3; Naive design

M=7;N=2;U=3; Robust null−space matrix only

M=7;N=2;U=3; Robust null−space matrix and Robust beamforming M=6;N=2;U=3; Moving average approach for 10 times

M=7;N=2;U=3; Moving average approach for 10 times

In the sequel, we are going to apply the moving average approach to the

Figure 4-6 Joint Tx/Rx MMSE beamforming design for MIMO-OFDM SDMA system under CSI error 0.05

slow-fading environment. Total 100 MIMO channel realizations are simulated and, we apply the moving average approach 10 times to each channel realization. We can see that the moving average approach is superior to the original robust methods in Figure 4-6. But this approach is only suitable for the slow-fading channel. That means only the slight Doppler Effect caused by slow-moving user terminals exists.

We also consider the single user case (Figure 4.7, 4.8 and 4.9), where the null-space matrix is an identity matrix and only the robust beamforming matrix is used.

0 5 10 15 20 25 30

10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for Single−user MIMO−OFDM System Under CSI Error 0.1

M=6;N=6;U=1; Naive design M=6;N=6;U=1; Bobust beamforming M=7;N=6;U=1; Naive design M=7;N=6;U=1; Bobust beamforming

Figure 4-7 Joint Tx/Rx MMSE beamforming design for single user MIMO-OFDM system under CSI error 0.1

0 5 10 15 20 25 30 10⁻³

10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for Single−user MIMO−OFDM System Under CSI Error 0.075

M=6;N=6;U=1; Naive design M=6;N=6;U=1; Bobust beamforming M=7;N=6;U=1; Naive design M=7;N=6;U=1; Bobust beamforming

0 5 10 15 20 25 30

10⁻³ 10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for Single−user MIMO−OFDM System Under CSI Error 0.05

M=6;N=6;U=1; Naive design M=6;N=6;U=1; Bobust beamforming M=7;N=6;U=1; Naive design M=7;N=6;U=1; Bobust beamforming

Figure 4-8 Joint Tx/Rx MMSE beamforming design for single user MIMO-OFDM system under CSI error 0.075

Figure 4-9 Joint Tx/Rx MMSE beamforming design for single user MIMO-OFDM system under CSI error 0.05

0 5 10 15 20 25 30 10⁻³

10⁻² 10⁻¹ 10⁰

Total Tx Power / Received Noise Power

BER

Joint Tx−Rx MMSE Beamforming Design for Single−user MIMO−OFDM System Under CSI Error 0.05

M=6;N=6;U=1; Naive design M=6;N=6;U=1; Bobust beamforming M=7;N=6;U=1; Naive design M=7;N=6;U=1; Bobust beamforming

M=6;N=6;U=1; Moving average approach for 10 times M=7;N=6;U=1; Moving average approach for 10 times

Figure 4-10 shows that the moving average approach enhances the performance when the channel error variance is 0.05. We can see that the moving average approach is also superior to the original robust methods. However this approach is only suitable for the slow-fading channel.

4.4 Conclusions

In this chapter, we address the problem of joint Tx/Rx beamforming design for multi-user MIMO-OFDM SDMA downlink system with channel estimation errors.

Imperfect CSI will cause significant performance degradation. The performance degradation is caused by the erroneous null-space matrix and Tx/Rx beamforming

Figure 4-10 Joint Tx/Rx MMSE beamforming design for single user MIMO-OFDM system under CSI error 0.05

which will induce the MUI among users and the inter-stream interference of each user respectively. To handle these two kinds of interference, we combine two robust approaches to improve the performance. The combination of these two robust

which will induce the MUI among users and the inter-stream interference of each user respectively. To handle these two kinds of interference, we combine two robust approaches to improve the performance. The combination of these two robust

在文檔中 多重輸入多重輸出正交分頻多工空間分割存取下鏈系統在通道估計錯誤下之共同波束設計 (頁 53-0)