TURBO SPACE TIME CODES FOR TAILBITING

TRANSMISSION

Piotr Remlein and Filip Łęcki

Faculty of Electronics and Telecommunications, Poznan University of Technology, Piotrowo 3a; 60-965 Poznan, Poland

Keywords: Space time encoding, turbo encoding, MIMO channel, codes over ring.

Abstract: This paper studies a novel technique that combines turbo tailbiting codes over the ring with symbol

interleaved space-time modulation. After initial theoretical introduction and analysis, the simulation results

are presented. It is shown that for different antenna array sizes, different packet sizes and two kinds of

channels the Turbo space time tailbiting coding have a clear advantage over the transmission method with

direct truncation.

1 INTRODUCTION

Driven by the increasing need for capacity and

system availability whilst using the same bandwidth,

multiple-input-multiple-output (MIMO) technique

has become an interesting extension to wireless

communication systems (Vucetic, 2003). With the

introduction of multiple antenna arrays came the

need for proper channel coding and transmission

method. Recently, in order to improve the error

performance of wireless systems techniques

combining space-time (ST) and turbo coding have

been proposed. Typically in such systems packet

data transmission is used. However, in order to use

convolutional codes in the packet transmission we

must convert these codes to block codes. There are

some well known methods for this conversion (Ma,

1986). One of them is called Direct Truncation. The

most popular method is called Tailbiting (TB). In

this method we transmit the convolutional coded

data in a block form without known tail. By framing

coded data we are not adding known bits to the end

of the data information stream (Cox, 1994). Simply

the binary encoder starts and finishes the encoding

process in the same state. This state is not known to

the decoder.

In this paper, we propose a novel technique that

combines turbo tailbiting codes over the ring with

symbol interleaved space-time modulation. The

turbo encoder consists of two feedback systematic

convolutional encoders over the ring of integers

modulo-M. This encoder is connected to s-random

symbol interleaver and next to space-time

modulator. We investigate the system which

contains a turbo space time encoder over ring

modulo-4 and QPSK modulator.

The performance of the proposed system for

various packet sizes, different number of transmit

antennas and MIMO quasi-static or fast fading

channel is evaluated by simulation.

This paper is organised as follows. Section 2

describes the TB encoding procedure which uses the

feedback systematic convolutional encoders over

ring Z

M

. In Section 3, we present the structure of

space time turbo encoder over ring Z

M

. In Section 4,

the MIMO channel model is described. Section 5

explains iterative decoding with symbol interleaver.

Section 6 presents the simulation results. Finally, a

conclusion is drawn in Section 7.

2 TAILBITING

CONVOLUTIONAL CODES

OVER RING

In this paper we concentrate on the usage of non-

binary codes (Massey, 1989). There are a few

reasons for using them. They have larger Euclidean

distances than codes over GF(2). Moreover, they

make the usage of mapping and set partitioning not

necessary. In the analyzed method we encode and

decode a block of N (M-ary) symbols without a

know tail, thus keeping the effective rate of

transmission.

140

Remlein P. and ŁÄ

´

Zcki F. (2007).

TURBO SPACE TIME CODES FOR TAILBITING TRANSMISSION.

In Proceedings of the Second International Conference on Wireless Information Networks and Systems, pages 140-145

DOI: 10.5220/0002146101400145

Copyright

c

SciTePress

The encoding procedure to achieve this is not

difficult if the structure of the used encoder is

feedforward. Then the starting state depends only on

the m last information symbols in the transmited

packet (m is the number of memory cells in the

encoder). In case of using a recursive systematic

convolutional encoder (RSC) with feedback (Figure

1) the ending state depends on all of the information

symbols in the packet. Finding initial state, wherein

the encoder should start its work and after

processing N symbols to end encoding in the same

state is complex and not always possible (Weiß,

2001).

Figure 1: RSC encoder over ring ℜ=Z

4

.

In Figure 1, we show an example of the RSC

encoder over the ring ℜ=Z

4

of integers modulo-4

(Remlein, 2003). The code rate is R=1/2 (k=1, n=2)

and the transfer function matrix is:

At instant t an information vector with M-ary

symbols belonging to the ring Z

M

={0, 1, 2, ... , M-

1}, (ℜ=Z

M

; M=4) inputs the encoder. The

convolutional encoder generates encoded vector

which contains sequence of the symbols elements

belonging to the same ring ℜ=Z

4

.

The encoder coefficients in Fig. 1 are from the

set {0,...,M-1}, M=4. The memory cells are capable

of storing the ring elements. Multiplications and

additions are performed in the ring of integers

modulo-M.

3 TURBO SPACE TIME

ENCODER OVER RING

Model of the system analyzed in this article joins

packet transmission tailbiting convolutional

encoding over the ring with space time encoding.

The convolution codes over the ring Z

M

considered

in (Remlein, 2003) were not optimized for the multi

antennas systems. Authors of this article used these

codes in the system shown in Figure 2. This is a

modification of solution considered in (Stefanov,

2001). We propose placing the tailbiting turbo

encoder over the ring in the transmitter. In our

proposal there is no need to consider set-partitioning

and symbol mapping. The usage of M-symbol

alphabet in the encoder in a natural way maps

symbols into M-PSK signal elements. Every

constellation point has its equivalent symbol.

Figure 2: Transmitter diagram.

The transmitter (Fig. 2) is constructed as

follows:

1. Construct encoder for M symbol alphabet.

2. Fed the turbo encoder with uninterleaved

and interleaved data sequence.

3. Turbo encode the uniterleaved and

interleaved input sequence.

4. Interleave the encoder output sequence in

S-random interleaver.

5. Modulate the signals.

6. Input symbol to different antenna in the

array.

In Figure 3 it is shown the turbo encoder

(Berrou, 1993) used in the transmitter.

Figure 3: Structure of the turbo encoder.

The encoder I and encoder II are RSC encoders

shown in Figure 1. The interleaver is an UMTS type

interleaver (3GPP TS 25.212). The interleaver is

responsible for decorrelating inputs to encoder I and

encoder II.

The S-random interleaver (Dolinar, 1995)

reorders the turbo coded sequence to further

decorrelate the symbols before feeding the antenna

inputs.

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

++

++

=

2

2

331

23

1)(

DD

DD

DG

TURBO SPACE TIME CODES FOR TAILBITING TRANSMISSION

141

The S-random interleaver (S=1,2,...) is a

semirandom interleaver. Each randomly selected

integer is compared with S previously selected

numbers. If the difference between them is smaller

then S, the current selection is rejected and a new

integer is generated. The process repeats until N

distinct integers are selected.

4 CHANNEL MODEL

The telecommunication system with n

T

transmit and

n

R

receive antennas we consider can be described by

the following formula:

ν

+

= Hsy

(1

)

where

1×

∈

T

n

Cs defines the transmission vector,

1×

∈

R

n

Cv defines the additive white Gaussian noise

vector,

TR

nn

CH

×

∈ is the channel matrix, that

describes the connections between the transmitter

and receiver and can be expressed as:

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎣

⎡

=

MNMM

N

N

H

ααα

ααα

ααα

L

MOMM

L

L

21

22221

11211

(2

)

Where

mn

α

is the complex transmission

coefficient between element m at the transmitter

(TX) and element n at the receiver (RX). To

generate channel matrix H, we used the narrowband

Kronecker model presented in (Ozcelik, 2003). In

this model it is assumed that the receive correlation

matrix is independent of the corresponding transmit

matrix and vice-versa. The channel model used in

this paper was further simplified according to the

assumptions in (Ozcelik, 2003). To calculate the

correlation coefficients between the antennas in the

transmitting as well as in the receiving array, we

followed the approach in (Loyka, 2002).

Figure 4 depicts the influence of distance

between array elements on capacity. The simulations

were made for the case of 2 element arrays. The

solid line presents the idealized case, when no

correlation can be observed among the antennas at

the mobile station (MS). As we can see, the line

representing the distance of 0.5 wavelength, lies

very close to the solid line. Hence, we can assume

that this distance is sufficient for normal operation.

The distance between elements at the Base Station

(BS) is normally much larger than in the Mobile

Station, and it does not have much influence on the

capacity. Therefore, we used for the simulations the

following parameters: BS azimuth spread = 18

o

, BS

distance=10 λ. MS azimuth spread = 90

o

, MS

distance=0.5 λ.

Figure 4: Capacity vs distance between array elements for

2 antennas. BS azimuth spread = 18

o

. MS azimuth spread

= 90

o

.

5 RECEIVER

The block scheme of the receiver used in the

analyzed system is shown in Figure 5.

Figure 5: Receiver diagram.

The demodulator performs two operations: it

demodulates the signal and it calculates the

aposteriori values of the received signal. The

demodulated symbols are deinterleaved and fed to

the turbo decoder.

The scheme of the turbo decoder is shown in the

Figure 6. The decoder uses a well known from the

literature connection of two SISO (Soft Input Soft

Output) decoders (Hagenauer, 1996) with feedback.

The crucial and most interesting part of the

decoding process is the iterative exchange of

information between decoders. This is also the

reason for calling this method “turbo”. With each

iteration, the Bit/Symbol error rate drops. Now we

can see, why there is so much research on proper

interleavers. Let us consider a situation when the

channel undergoes a deep fade. With great

probability part of the transmitted signal is lost, but

0 5 10 15 20 25 30

0

2

4

6

8

10

12

14

SNR [dB]

Capacity [bit/sec/Hz]

Decorelated

0.5

λ

0.1

λ

WINSYS 2007 - International Conference on Wireless Information Networks and Systems

142

due to interleaving and trellis decoding, it might be

reconstructed.

Figure 6: Turbo decoder. EXT- extrinsic information,

s

k

y

- systematic information,

p

k

y

1

- parity information from

the first encoder,

p

k

y

1

- parity information from the second

decoder.

The technique that allows the turbo codes to

almost achieve the Shannon bound is in fact the

exchange of information between the decoding

blocks. To process the information in the receiver so

called Soft Input Soft Output Decoder (SISO)

(Hagenauer, 1989) was developed. Our SISO

decoder uses the improved Soft Output Viterbi

Algorithm which is able to decode the non-binary

codes (Cong, 1999). The processing steps of SISO

decoder are explained after the notation is described.

u

i

is the i-th symbol from the alphabet {0,1,...,M-

1};

()

k

sΓ is cumulative metric value at state s

k

;

)](

ˆ

[)(

ˆ

, kjk

sLsL

μ

=

is δ x M reliability measure

matrix stored at each state, where j=k − δ + 1, ..., k,

μ = 0, 1, ...,M − 1, and δ is the size of the decoding

window;

Δ is reliability difference between the survivor at

state s

k

and the most likely path terminating in the

state s

k

with u

j

=μ.

At each state s

k

corresponding to the decoding

time k, SOVA calculates and stores the

()

k

sΓ as in

the normal Viterbi Algorithm, additionally it stores

)(

ˆ

k

sL

. At each step (k+1) for every state s

k+1

the

algorithm evaluates cumulative metric candidates

),(),...,,(

1

1

1

0

+

−

+

ΓΓ

k

M

kkk

ssss originating from s

k

and selects the minimum cumulative metric

)(

1+

Γ

k

s . However, it does not neglect the rest, they

are being stored as competitive paths. After finding

)(

1+

Γ

k

s , the reliability differences Δ can be

calculated in the following way:

{}

),(min),(

1

1,...,0

1 +

−∈

+

Γ−Γ=Δ

k

m

k

Mm

k

m

km

ssss

(3

)

where Δ

m

=0 for the surviving path at state s

k+1

.

We first set

μμ

Δ=

++

)(

ˆ

1,1 kk

sL , because Δμ is the

reliability difference between the survivor at s

k+1

and

the most likely path terminating in state s

k+1

with

u

k+1

= μ. The next step is to update the remaining

values in

)(

ˆ

1, +kj

sL

μ

, j=k-δ+1,...,k based on the

following rule:

{}

{

}

m

m

j

Mm

k

m

j

LsL Δ+=

−∈

+

μμ

,

1,...,0

1,

ˆ

min)(

ˆ

(4

)

where )(

ˆˆ

,,

m

kj

m

j

sLL

μμ

≡ . The modified SOVA

algorithm, shows performance equal to the of Max-

Log-MAP algorithm at a lover computational

expense (Berrou, 1993).

6 SIMULATION RESULTS

The computer simulations were performed in

MATLAB environment. The simulations were

conducted for QPSK modulation and RSC encoder

over ring Z

4

with the transfer function matrix:

In this paper, both quasi-static and fast fading

channels were considered. For quasi static fading, it

is assumed that the fading coefficients are constant

during a packet transmission. These coefficients

vary from one packet to another independently. For

fast fading channels the fading coefficients vary

from one symbol to another independently.

Basic performance of the turbo decoder for

symbol tailbiting over ring transmission after 6 inner

iterations with respect to the block length is showed

in Figure 7. We observe that using packet of length

500, for SER=10

-2

, it is possible to achieve a gain of

2dB with respect to the transmission with packet

size equal to 40 symbols.

At first we consider 2 transmit and 2 receive

antenna system with a fast fading channel. The

simulations were conducted for different block

lengths. As expected, the longer the block, the better

the SER.

Next simulation was made for 2 different block

lengths and for 2 different decoding techniques.

⎥

⎥

⎦

⎤

⎢

⎢

⎣

⎡

++

++

=

2

2

331

23

1)(

DD

DD

DG

TURBO SPACE TIME CODES FOR TAILBITING TRANSMISSION

143

1 2 3 4 5 6 7

10

-2

10

-1

10

0

SNR [dB]

SER

40

160

500

1 2 3 4 5 6 7

10

-2

10

-1

10

0

SNR [dB]

SER

40 Tailbiting

40 Direct truncation

160 Tailbiting

160 Direct truncation

Figure 8 shows a comparison between two

decoding methods; tailbiting and direct truncation.

As expected, the later method delivers worse results.

Another important observation is that for higher

SNR, the difference between those two methods is

getting larger in favor of tailbiting. We can observe

that the tailbiting method improves the performance

about 0.3 dB for SER=10

-2

with respect to the direct

truncation method, for transmission with packet size

equal to 500.

Figure 7: Different block sizes for turbo decoder and

tailbiting trellis ending. 2TX, 2RX,

2-symbols/Hz/antennas pair, QPSK, fast fading channel.

Figure 8: Tailbiting decoding vs Direct truncation trellis

ending for 2 block sizes. 2TX, 2RX,

2-symbols/Hz/antennas pair, QPSK, fast fading channel.

For burst channels (with burst length of 130

space-time symbols), (2TX × 2RX) scenario and

different block lengths, the situation differs. As is

shown in Figure 9, the performance of the system

drops significantly.

Figure 9: Quasi-static channel (130 space symbols),

different block sizes for turbo decoder and tailbiting trellis

ending. 2TX, 2RX, 2 symbols/Hz/antennas pair, QPSK.

Figure 10: Quasi-static channel (130 space symbols),

tailbiting decoding vs direct truncation trellis ending for 2

block sizes. 2TX, 2RX, 2 symbols/Hz/antennas pair,

QPSK.

The waterfall region moved from 4 dB in fast

fading channel to 6-7 dB in quasi-static channel. For

burst channels both decoding methods, tailbiting and

direct truncation, yield almost the same performance

(still tailbiting performs better), see Figure 10.

Figure 11, shows the relation between SER and

different array sizes in a quasi-static channel for 500

block size. At 4 dB there is observable a break-

through, thus more element antenna arrays deliver

better performance. We observe that using 4TX/4RX

antennas, for SER=10

-2

, it is possible to achieve a

gain about of 0.5 dB with respect to the transmission

with 3TX/3RX antennas and the gain about of 1 dB

with respect to the system with 2TX/2RX antennas.

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144

1 2 3 4 5 6 7

10

-3

10

-2

10

-1

10

0

SNR [dB]

SER

2TX 2RX

3TX 3RX

4TX 4RX

Break-Through

Figure 11: SER performance vs different array sizes, 500

symbols block, quasistatic channel (130 space symbols),

for turbo decoder and tailbiting trellis ending.

2-symbols/Hz/antennas pair, QPSK.

7 CONCLUSIONS

New structure of Turbo space time encoder over ring

for tailbiting transmission is proposed. This paper

aimed at investigating the performance of space time

symbol interleaved non-binary turbo tailbiting coded

systems. A quasi-static and fast fading channels

were considered. In the receiver, were used iterative

soft input soft output (SOVA) algorithm.

Analysis of the obtained results shows that the

use of tailbiting coding over ring improves the

quality of the transmission in comparison to the

direct truncation method. An improvement is

observed for all packet lengths. We have to mention

that the encoding process in tailbiting method is

more complex than in direct truncation method.

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