RCPC-Encoded V-BLAST MIMO With MMSE-Based Detection

1Lydia Sari, 2Gunawan Wibisono, and 3Dadang Gunawan Electrical Engineering Department

University of Indonesia Depok, Indonesia

1lydia.sari@ui.ac.id, 2gunawan@eng.ui.ac.id, 3guna@eng.ui.ac.id

Abstract— A Rate-Compatible Punctured Convolutional (RCPC)-encoded Vertical Bell Laboratories Layered Space-Time (V-BLAST) MIMO system to provide a robust wireless communication link is proposed. The V-BLAST detection is based on Minimum Mean-Squared Error (MMSE) criterion, to alleviate the noise enhancement commonly found in the original V-BLAST scheme based on Zero Forcing criterion. Different code rates and transmit powers are allocated to data streams emitted from the multiple transmit antennas, according to the attenuation levels of the subchannels. This scheme is efficient in terms of transmit power and bandwidth usage, as the data stream entering a destructive subchannel is given low transmit power and high code rate. Simulations show that the proposed system can attain good performance at a reasonably low Eb/No.

Keywords – V-BLAST, MIMO, RCPC, MMSE, singular value decomposition

I. INTRODUCTION The multiple-input multiple-output (MIMO) wireless

communication scheme has been a subject of rigorous researches for the past decade. A vast body of theoretical and simulation works has stated its potential of realizing substantial capacity without requiring extra bandwidth [1-3], making this scheme particularly suitable for today’s multimedia communication needs.

MIMO scheme exploits multipath scattering to increase capacity, and therefore suitable for an indoor environment with rich scattering pattern. A processing architecture for a MIMO system known to be capable of realizing the theoretical high capacity nature of MIMO scheme is Vertical Bell laboratories layered space-time (V-BLAST) [2]. V-BLAST MIMO can achieve spectral efficiencies of 20-40 bps/Hz at 24-34 dB average SNR without coding [2]. An appropriate coding scheme is needed to provide the scheme with a high diversity gain.

In this paper Rate-Compatible Punctured Convolutional (RCPC) codes are used to protect the data streams. By using RCPC code, different coding rates can be given to different data streams transmitted by the multiple transmit antennas. The different code rates are adapted to the attenuation levels present in the different subchannels of the system. A data stream entering a subchannel with low level of attenuation will be given a low code rate. On the contrary, a data stream entering a subchannel with high level of attenuation will be given a high code rate. Reference [4] reports that an RCPC-

encoded MIMO system outperforms the MIMO system without RCPC encoding. However the research does not cover a layered architecture to boost the capacity of the MIMO system. Reference [5] reports that RCPC-encoded V-BLAST MIMO system has robust performance. The V-BLAST scheme in [5] is based on Zero Forcing (ZF) criterion, which is prone to noise enhancement in several cases

In this paper an RCPC-encoded V-BLAST MIMO system with Minimum Mean Squared Error (MMSE) criterion is proposed. An MMSE detector chooses the linear filter to minimize the average mean-square value of the output signals with added noise. Other than its low complexity, MMSE-based detection has significant practical advantage because it does not need training sequences [6]. A V-BLAST MIMO scheme based on MMSE criterion has been shown to have better performance compared to one based on ZF criterion [7]. The transmitter block in the proposed system allocates different code rates for the data streams based on the Channel State Information (CSI). The CSI is firstly used by the transmitter to differentiate the transmit powers given to the data streams emitted by the multiple transmit antennas. A data stream entering a subchannel with a low level of attenuation will be given a higher transmit power compared to a data stream entering a subchannel with a high level of attenuation. A low rate code will be given to the data stream with high transmit power. Therefore, the data stream entering a subchannel with low level of attenuation will benefit from both a high transmit power and low rate code. This way, the transmit power and system bandwidth will not be wasted on the data stream which will most likely be deteriorated by a high level of attenuation.

This paper is organized as follows. Section 2 gives the system model and the analytical probability of error for the proposed system. Simulation results are given in Section 3 while Section 4 concludes the paper.

II. SYSTEM MODEL The proposed system model is given in Fig. 1. It is

assumed that the channel undergoes Rayleigh fading and there exists a number of scatterers between the transmit and receive antennas. The number of transmit antennas is equal to the number of receive antennas, and is denoted as m.

2010 Second International Conference on Communication Software and Networks

978-0-7695-3961-4/10 $26.00 © 2010 IEEE

DOI 10.1109/ICCSN.2010.21

73

1~s

2~s 2λ

ms~

1~y

2~y

my~

1s

ms

1y

2y

mymλ

1λ

2s

/1y

/2y

/my

Figure 1. System model

The transmitter block consists of a V-BLAST

demodulator and an RCPC encoder. Codes with rates Rci, i=1,..,m are given to the data streams according to the subchannel attenuation level.

The different convolutional code rates are derived from a mother code with rate 1/N. For a puncturing period of L bits, the resulting punctured convolutional code rates are [8]

miLNL

LR

ic ,...,1,)1(,,1, =−=+

= …δδ

(1)

There are no systematic methods available to construct good rate L/L+δ codes [9]. Consequently, the search for good codes in this paper is based on the best-known generator polynomial of rate 1/3. This generator polynomial yields a “mother code”. The bits in the mother code are the punctured to attain codes of rates ½, using different puncturing matrices [10]. For the proposed system model with L = 6, the puncturing matrices are as follows [10]

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

111111000000111111

A ,⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

000000111111111111

B

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

111111000111000111

C ,⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

000111111111000111

D

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

000111000111111111

E , ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

100111100111100111

F

For L = 6, two simulations will be done, one with puncturing matrices A and B and another with puncturing matrices C, D, E, and F. This is to show the performance difference between the use of puncturing matrices where the

bits in one whole row are punctured, as opposed to the use of puncturing matrices where the bits in more than one rows are punctured.

For L = 5 the puncturing matrices are as follows

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

111111111100000

G , ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

111110000011111

H

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡=

000001111111111

I

Using BPSK modulation, the elements of a signal vector

s = [s1, s2,…,sm]T are transmitted simultaneously from first to the m-th transmit antennas, and the signal arriving at the receive antennas y = [y1, y2,…,ym]T can be expressed as

y = Hs + n (2)

where H is the matrix channel of a MIMO system which elements are the channel gains between the transmit and receive antennas, and n is a noise vector with complex Gaussian distribution, zero mean and variance σ2. Using SVD, the matrix channel H is decomposed into [3]

H = U D V* (3)

where U and V are complex unitary matrices which dimensions are m × m, (.)* denotes conjugate transpose and D is diagonal matrix which can be expressed as [3]

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

mλ

λ

λ

…

…

…

00

0

00

00

2

1

D (4)

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where λ1,…,λm are the eigenvalues of HH*. Taking the definition [3]

yUy ~⋅= (5a) sVs ~⋅= (5b) nUn ~⋅= (5c)

and substituting (3) and (5) into (2), the signal arriving at the receiving antennas can be stated as

nsD y ~~~ +⋅= (6)

The V-BLAST demodulator extracts s from y using MMSE criterion. This criterion is based on matched filter (MF) to estimate the signal output yMF, where

y*Hy =MF (7)

The MMSE approach is to find a coefficient r which

minimizes the criterion

( )( ){ }∗s-rys-ryE (8)

Reference [6] solves the coefficient r which gives

( ) ∗⋅+∗=−

HInHHr1

(9)

The term n⋅I is the noise term where I is the identity matrix which size matches H. The noise vector n needs to be multiplied with I to preserve the matrix convention. The extracted symbols in the receiver block can be stated as

MFyry' ⋅= (10)

Subtituting (2), (7) and (9) into (10) will yield

( ) n*HsInH*Hy'1

+⋅+=−

(11)

where the first term on the right hand side of the equation sign can be stated as

( ) ( )( ) sInI

sIn*VDU*VDU

sInH*Hs1

1

1*

ˆ

−

−

−

⋅+=

⋅+⋅⋅⋅⋅=

⋅+

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞⎜

⎝⎛=

(12)

The performance of the system is affected by s-s . A

cancellation process is needed to extract a particular data

stream from the mixed streams arriving at the receiver block. The cancellation process can be stated as

∑−

=

−=1

1

/i

jjMFiy sy (13)

However to preserve optimal ordering of the V-BLAST

system, for m = 2 the receiver should be able to choose whether to cancel out the effect of s1 or s2 first. The decision is made based on the powers of s1 and s2, which are λ1 and λ2, respectively. The received power at the receiving antennas generated by transmission of s1 can be stated as

2

21

2

111 hh +=λ (14)

Whereas the transmission of s2 will generate

2

22

2

122 hh +=λ (15)

at the receiving antennas. If λ1 > λ2 the receiver will remove the effect of s1 first

from the received vector. On the contrary, if λ2 > λ1 then the effect of s2 will be removed first from the received vector.

The system performance is bounded by the probability that the detected symbol is not equal to the transmitted

symbol, P(yi’≠ si). This can be stated as

( ) ( )γγβ i

m

iibP Β∑=

=1 (16)

where γ is the instantaneous signal to noise ratio (SNR),

βi(γ) is the SNR pdf at the i-th detection step, and Bi(γ) is the instantaneous BER which depends on the modulation type and the coding rate used. As in the simulation m = 2, the number of detection steps is 2 and the SNR pdf can be stated as [12]

( ) ⎟⎠⎞

⎜⎝⎛

∂∂

=0

1 2γγ

γγβ (17)

( )2

2

02 ⎟

⎠⎞

⎜⎝⎛

∂∂

=γγ

γγβ (18)

where γ0 is the average pre-processing SNR per transmit

antenna which is assumed to be equal for both transmit antennas.

The instantaneous BER of the system Bi(γ) is upperbounded by the Viterbi’s upperbound on error event probability [12]. As an RCPC code is used instead of a convolutional code, Bi(γ) can be stated as

( ) ( ) dfreedd

diii PcssL

∑∞

=−=Β ˆ1γ (19)

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where cd is the the total number of error bits contained in the incorrect paths in the convolutional trellis and Pd is the probability of selecting a wrong path at distance d. For a Rayleigh fading channel with soft decision on y and full CSI, Pd can be upperbounded by [8]

d

RP

c

d ⎟⎟⎠

⎞⎜⎜⎝

⎛+

≤γ1

1

2

1 (20)

III. SIMULATION RESULTS AND DISCUSSION For all simulations the number of Tx antennas is equal to

the number of Rx antennas used, m = 2. The low code rate given to a substream of information entering a non-destructive subchannel is 6/18 or 5/15 for L = 6 or 5, respectively. These low code rates are attained using mother codes with no puncturing. A substream entering a destructive subchannel is given high code rate, which is 6/12 or 5/10 for L = 6 or 5, respectively. The generator polynomial for L = 6 is [75 53 47] while for L = 5, the generator polynomial is [36 25 23].

Fig.2 shows the simulation results for an RCPC-encoded V-BLAST MIMO system using MMSE criterion in a Rayleigh fading channel with period L = 6. The puncturing matrices used for the high code rate are matrices A and B. It is shown that the substream of information with a low code rate Rc1 = 6/18 outperforms the substream with high code rate Rc2 = 6/12 by more than 2 dB for BER 10-10. However the substreams coded with Rc2 are still able to attain a BER level of 10-10 at less than 8 dB Eb/No, showing that the proposed system is capable of providing a robust performance with moderate Eb/No requirement. It is also shown that the the different puncturing patterns used for the substream entering a subchannel with high level of attenuation give similar performances.

1 101 .10 10

1 .10 9

1 .10 8

1 .10 7

1 .10 6

1 .10 5

1 .10 4

1 .10 3

Eb/No (dB)

Pb

Figure 2. Results for RCPC-encoded V-BLAST MIMO system with L =

6, puncturing matrices A and B

Fig. 3 shows the simulation results for the proposed system using puncturing matrices C, D, E and F with period L = 6. Consistent with the simulation results depicted in Fig.

2, Fig. 3 shows that the system can attain a BER level of up to 10-10 with Eb/No level of less than 8 dB. It is also shown that the puncturing matrices C, D, E and F, while providing similar results to matrices A and B in high Eb/No levels, yields better performance in low Eb/No levels. This implies that using the same code rate with different puncturing pattern will affect the system performance at low Eb/No, while maintaining the performance at high Eb/No levels.

Fig. 4 shows the simulation result for the proposed system with puncturing matrices G, H and I. The puncturing period is L = 5. It is shown that Rc2 = 5/10 provides slightly poorer performance compared to Rc2 = 6/12 which confirms that the system performance will degrade when the puncturing periode is decreased. It is also shown that the different puncturing matrices used will give similar performances. This is due to the comparable dfree and cd values of the resulting codes.

All simulation results show that the use of RCPC codes provide a gain variation by up to 4 dB for L = 5.

1 101 .10 10

1 .10 9

1 .10 8

1 .10 7

1 .10 6

1 .10 5

1 .10 4

1 .10 3

0.01

Eb/No (dB)

Pb

Figure 3. Results for RCPC-encoded V-BLAST MIMO system with L =

6, puncturing matrices C, D, E and F

1 101 .10 10

1 .10 9

1 .10 8

1 .10 7

1 .10 6

1 .10 5

1 .10 4

1 .10 3

0.01

Eb/No (dB)

Pb

Figure 4. Results for RCPC-encoded V-BLAST MIMO system with L = 5, puncturing matrices G, H, and I

____ No puncturing - - - - Puncturing matrix A __ __ Puncturing matrix B

____ No puncturing - - - - Puncturing matrix C _ _ _ Puncturing matrix D _ . _ Puncturing matrix E

o Puncturing matrix F

____ No puncturing - - - - Puncturing matrix G _ _ _ Puncturing matrix H _ . _ Puncturing matrix I

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IV. CONCLUSION The performance of an RCPC-encoded V-BLAST

MIMO system with MMSE criterion has been simulated and analyzed. It is shown that the proposed system can provide reliable performance at modest Eb/No requirement. The different patterns used to create different puncturing matrices for the RCPC encoder yield similar performances in high Eb/No values. However it is noted that different puncturing patterns affect the system performance in low Eb/No values. In addition to a robust performance, RCPC codes also provides a significant bandwidth preservation. Overall, the MMSE criterion boosts the performance of the original V-BLAST MIMO system which uses ZF criterion.

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