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Iterative Bit & Power Allocation for V-BLAST based OFDM MIMO System in Frequency Selective Fading Channel Ka-Wai Ng, Roger S. Cheng, and Ross D. Murch Consumer Media Center Department of Electrical & Electronic Engineering The Hong Kong University of Science & Technology Clear Water Bay, Kowloon, Hong Kong

Abstract  This paper presents an iterative bit and power allocation algorithm for OFDM Multiple-Input Multiple-Output (MIMO) system in frequency selective fading channels and uses V-BLAST as a detection algorithm. Assuming knowledge of the instantaneous channel gains for all spatial channels at all the subcarriers in the receiver, bit and power is assigned to each spatial channel to minimize the total transmission power for a fixed bit error rate (BER). We demonstrate by computer simulation that the proposed iterative bit and power allocation algorithm can significantly improve the system performance compared to the traditional V-BLAST and Singular Value Decomposition (SVD) algorithm for OFDM MIMO system in a frequency selective fading channel model.

I.

and has been implemented and demonstrated with spectral efficiencies of 20 - 40bps/Hz in the real time laboratory. In this paper we investigate the application of the adaptive bit and power allocation scheme, proposed in [7-9], in an OFDM system combined with V-BLAST. We describe a simple iterative algorithm to combine the OFDM system, bit and power allocation and V-BLAST to minimize the total transmission power, increase the transmitted data rate and yet still maintain a reliable communication system. The organization of this paper is as follows. In Section II, we will first introduce the system model of OFDM MIMO system. In Section III, the V-BLAST algorithm is described. Section IV illustrates the proposed iterative bit and power allocation algorithm in detail. Then, we compare the system performance between our proposed algorithm and other traditional algorithms in Section V. Finally, a conclusion is made in Section VI.

INTRODUCTION

Multi-carrier transmission in the form of Orthogonal Frequency Division Multiplexing (OFDM) is regarded as a very promising technique for achieving very high bit rate transmission in many transmission applications including mobile radio channels, high-bit-rate digital subscriber lines (HDSL), asymmetric digital subscriber lines (ADSL), wireless local area network (Wireless LAN), digital audio broadcasting (DAB) and digital video broadcasting (DVB).

II.

An overview of a V-BLAST based OFDM system is shown in Figure 1. At the transmitter, an information bit sequence of length Rb is distributed into sub-channels based on an algorithm described in the next section. In total there are Nc u M sub-channels in which Nc is the number of subcarriers and M is the number of spatial channels in each subcarrier. Therefore,

By dividing the frequency selective fading channel into multiple sub-channels, Inter-Symbol Interference (ISI) can be minimized. Generally in many practical applications, the subcarriers are allocated the same amount of power over the time-varying channel, which may not be the optimal solution. If the receiver has knowledge of the channel transfer function, the water-pouring approach from information theoretic results [1] shows that significantly performance improvement can be obtained by adaptively allocating the information bit and power levels over the sub-channels [2-4].

Nc

Rb

M

¦¦ m

c ,i

(1)

c 1 i 1

where mc,i is the bits allocated in the c-th sub-channel. In general, different sub-channels can contain different numbers of bits and each of them can be independently encoded, interleaved and modulated to form the modulated symbol. For simplicity, we assume that an ordinary M-QAM signal constellation is being used for the modulation.

Theoretical research recently shows that the rich-scattering wireless channel is capable of enormous capacities provided that the multipath effect is properly exploited [5-6]. A simplify wireless communication architecture known as V-BLAST, which utilizes multi-element antenna arrays at both transmitter and receiver

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SYSTEM OVERVIEW

Each modulated symbol of each stream will then translate to different subcarriers which are then passed through an

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at subcarrier c as,

Inverse Fast Fourier Transform (IFFT). A cyclic prefix will be added before transmission to reduce the effect of inter-symbol interference (ISI) and inter-subcarrier interference (ICI).

ac

(a1,c , a 2,c ,..., a M ,c )T

(2)

Then the corresponding N-received vector at the receiver side is, Tx Data Rb

S / P

Nc IFFT 1

Add cyclic prefix and P/S

Tx 1

Nc IFFT M

Add cyclic prefix and P/S

Tx M

Bit and Power Allocation S / P

r1,c

Rx Data Rb

V-BLAST Signal Processing P / S

Figure 1:

Nc FFT 1

V-BLAST diagram.

based

Nc FFT N

OFDM

The detection process is by treating other users as interferers and choosing a linear combination nulling so as to satisfy some performance-related criterion, such as minimum mean-squared error (MMSE) or zero-forcing (ZF).

Remove cyclic prefix and S/P

Rx 1

Remove cyclic prefix and S/P

Rx N

MIMO

For simplicity, zero-forcing nulling is performed by choosing weighting vectors wi,c, i =1, 2, …, M and c = 1, 2, …, Nc such that,

wiT,c ( H c ) j

G ij

(4)

where (Hc)j is the j-th column of H and G is the Kronecker delta. Thus the decision statistic for the i-th spatial channel at each subcarrier c is,

system

y i ,c

wiT,c r1,c

(5)

The post-detection SNR for the k-th detected component at the c-th subcarrier of the transmitted symbol ac can be easily shown as,

At the receiver side, receivers 1-N will receive the radiated signal from the transmitters 1-M and we assume M d N. The received signal from all the receivers will then pass through a Fast Fourier Transform (FFT) with the removal of the cyclic prefix and we assume that the delay between each path for all the spatial channels are the same. Further, perfect symbol timing synchronization is assumed.

U k ,c

 a k ,c

2

V 2 wk ,c

! 2

(6)

where the expectation in the numerator is taken over the constellation set.

After the process of Fast Fourier Transform, the received signal will pass through the V-BLAST signal processing to retrieve the original transmitted signal.

It should be noticed that V-BLAST detection ordering is trying to detect the strongest signal at each stage and this detection process leads to globally optimum ordering [10]. However, this assumption is valid only when the constellations that are being used are the same. Now, the question is how to assign different number of bit and power to each spatial channel on each subcarrier base on the V-BLAST post detection SNR (6) which is the focus of the next section.

For a more realistic viewpoint, we assume our channel model is a 3-path slow fading channel with an exponential delay profile. Let HcNuM be the channel's transfer function at the c-th subcarrier, where c = 1, 2, …, Nc. And (hc)ij is the complex transfer function from transmitter j to receiver i at the subcarrier c. III. V-BLAST ALGORITHM The original V-BLAST detection algorithm [10] is described as follows. First, we denote the transmitted vector

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(3)

where K is a complex Gaussian noise with zero mean and variance V2.

Channel State Information

P / S

H c ac  K

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IV.

PROPOSED ITERATIVE BIT AND POWER ALLOCATION ALGORITHM

mc , i D

c ,i

c ,i

(7)

(8)

arg min 'Pc ,i

(12) (13)

m c*,i*  'B

(14)

Step 4: Repeat the procedures of Step 1 to Step 3 until there is no other combination such that the total transmission power can be reduced.

and, (9)

The set of {mc,i} is the iterative bit and power allocation that cannot be further reduced the transmission power on the i-th spatial channel of c-th subcarrier and the corresponding transmission power is,

The proposed iterative bit and power allocation algorithm are as follows: Step 1: For each subcarrier c, we first assume the number of bit on each spatial channel is the same such that we can determine the detection ordering based on the V-BLAST detection algorithm described in [10].

Pc ,i

P(mc ,i ) u wi ,c

2

(15)

Then, the post-detection SNR (6) will become,

Step 2: Based on V-BLAST detection ordering for each subcarrier c, a set of nulling vectors wc,i can be obtained. Then the optimal bit and power allocation algorithm [7] for the spatial channel is performed, by assigning zero bits to the spatial channels and then allocating 'B bit at each time, to the spatial channel with the least amount of transmission power. In general, 'B can be any integer value and is fixed for the whole allocation process. The process continues until the bit rate constraint (7) is satisfied. The algorithm can be described by the follow set of equations:

U k ,c '

 a k ,c

V V.

2

!

2

(16)

PERFORMANCE ANALYSIS

The algorithm of iterative bit and power allocation introduced in this paper is investigated for an OFDM MIMO system under a frequency selective fading channel. Perfect channel estimation has been assumed in the computer simulation. The configurations we consider here are for the OFDM system with bandwidth 30MHz and 64 subcarriers with spectral efficiency 12 bits/s/Hz. The set of QAM-constellation used in the simulation is assumed to be {0, 4, 16, 64}, i.e. 'B = 2. And the channel of each link contains three path components with exponential delay

Initialization: (for all c and i)

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2

The essential of doing a full search is to ensure the optimization problem can be converged within a few iterations, although the complexity of doing a full search increases nonlinear with the increases of the number of transmitter.

c 1 i 1

BER d Pe

[ P (m c,i  'B )  P(m c,i )] u wi ,c

Step 3: For the subcarrier c, we do the full search to find the optimal bit and power allocation on each subcarrier c. The process of full search is similar to the one described in Step 2, except we are now looking at a particular c at each time and find the optimal bit and power allocation on each subcarrier for all the possible detection orderings.

M

¦¦ m

(11)

m c*,i*

c 1 i 1

Nc

0

c ,i

subject to, Rb

m c ,i

(c*, i*)

M

¦¦ P

(10)

'Pc,i

Further, we assume the number of bits that can be assigned to each spatial channel is any integer value within [0, B], where B is the maximum number of bits assigned to each spatial channel, so the optimization problem can be written as, Nc

0

Resursion :

In this section, we will present the proposed Iterative Bit and Power Allocation (IB&PA) algorithm for the bit and power optimization. We denote the power required for transmitting mc,i bits as P(mc,i) for a specified BER. While we denote Pc,i as the power that is actually allocated at the i-th spatial subchannel of the c-th subcarrier.

min

Pc,i

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Figures 2 and 3 demonstrate the effectiveness of our proposed iterative bit and power allocation. Compared to V-BLAST with bit and power allocation algorithm, the proposed iterative scheme has about 1dB at BER = 4u10-4 and 0.5dB gain at BER = 3u10-4 in a (4,4) and (4,6) antenna array system respectively. A close observation is the difference between IB&PA and SVD with bit and power allocation, IB&PA has only 1.8dB at BER = 4u10-4 and 0.8dB at BER = 3u10-4 away from the SVD with bit and power allocation in a (4,4) and (4,6) antenna array system respectively. As the SVD with bit and power allocation is the theoretical bound for the OFDM MIMO system, we show that our proposed iterative algorithm can tend to the theoretical limit.

profile and Wrms = 20ns. For the simulation to be shown below, the horizontal axis is the average received SNR, which is defined as, 1 N

N

¦ SNR

(17)

i

i 1

where SNRi is the ratio of the received signal power to the noise power which is received from all M transmitter at the i-th receiver. As a reference we compare our results to Singular Value Decomposition (SVD) [9]. The main problem of the SVD is the heavy loading of channel feedback for the singular values and the singular vectors. However, it can provide an optimum solution for us to compare for.

Another point to raise is that the performance of one-stage iteration and three-stage iteration of the proposed scheme are almost the same. It means that the speed of convergence for our proposed iterative scheme is very fast.

SVD B&PA VB IB&PA 1 VB IB&PA 3 VB B&PA VB

-1

10

Figures 2 and 3 also show the significant improvement of the system performance for V-BLAST with bit and power allocation by about 17dB gain compared to the traditional V-BLAST algorithm in a (4,4) antenna array system at BER = 4u10-4, and about 4dB gain is achieved in a (4,6) antenna array system at BER = 3u10-4. It should be noticed that increasing the number of receivers can significantly improve the system performance because increasing the spatial dimension can reduce the magnitude of the weighting vectors wi,c and lower the transmission power defined in (15) in result.

Bit Error Rate

-2

10

-3

10

-4

10

5

Figure 2

10

15

20 SNR (dB)

25

30

35

VI. CONCLUSION

Performance of using different allocation schemes for (4,4) antenna array system.

In this paper, we propose an Iterative Bit and Power Allocation algorithm (IB&PA) for assigning the bit and power on the spatial channel of each subcarrier in the OFDM MIMO system based on the V-BLAST detection algorithm to detect the signal under a frequency selective fading channel. The optimization problem is to minimize the total transmission power for a given information rate and fixed BER for an OFDM MIMO system. We show via a computer simulation that our proposed iterative bit and power allocation algorithm can significantly improve the system performance compared to the traditional V-BLAST based OFDM MIMO system.

SVD B&PA VB IB&PA 1 VB IB&PA 3 VB B&PA VB

-1

10

Bit Error Rate

-2

10

-3

10

REFERENCES [1]

-4

10

5

Figure 3

7

9

11 13 SNR (dB)

15

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[2]

Performance of using different allocation schemes for (4,6) antenna array system.

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T. M. Cover, J. A. Thomas, Elements of Information Theory, Wiley, 1991 C. Y. Wong, R. S. Cheng, K. B. Letaief and R.D. Murch, “Multiuser OFDM with Adaptive Subcarrier, Bit, and Power Allocation,” IEEE, pp.1747-1758 Vol.17, Oct, 1999

[3]

Czylwik, “Adaptive OFDM for wideband radio channels,” Proc. IEEE GLOBECOM, pp.713-18, November 1996 [4] H. Rohling and R. Grunheid, “Performance of an OFDM-TDMA Mobile communication system,” Proc. VTC, pp.1589-93, May 1996 [5] G. J. Foschini, “Layered Space-Time Architecture for Wireless Communication in a Fading Environment When Using Multiple Antennas”, Bell Laboratories Technical Journal, Vol. 1, No. 2, Autumn, 1996, pp.41-59 [6] G. G. Raleigh, and J. M. Cioffi, “Spatio-Temporal Coding for Wireless Communications”, Proc. 1996 IEEE Globecom, Nov. 1996, pp.1809-1814 [7] B. S. Krongold, K. Ramchandran and D. L. Jones, “Computationally efficient optimal power allocation algorithm for multi-carrier communication systems,” Proc. ICC, pp.1018-1022, May, 1998 [8] S. K. Lai, R. S. Cheng, K. B. Letaief and R. D. Murch, “Adaptive Tracking of Optimal Bit and Power Allocation of OFDM Systems in Time-Varying Channels,” WCNC IEEE, pp.776-780 Vol.2, 1999 [9] K. K. Wong, S. K. Lai, R. S-K. Cheng, K.B. Letaief, and R.D. Murch, “Adaptive Spatial-Subcarrier Trellis Coded MQAM and Power Optimization for OFDM Transmission,” VTC2000 IEEE, pp.2049 -2053 vol.3 [10] P. W. Wolniansky, G. J. Foschini, G. D. Golden, R. A. Valenzuela, “V-BLAST: An Architecture for Realizing Very High Data Rates Over the Rich-Scattering Wireless Channel,” Signals, Systems, and Electronics, 1998. ISSSE 98

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