POZNAN UNIVERSIT Y OF TECHNOLOGY ACADEMIC JOURNALS
No 54
Electrical Engineering
2007
Filip ŁĘCKI*
Piotr REMLEIN*
TAILBITING TRANSMISSION
WITH TURBO SPACE TIME ENCODING
In the article, the turbo space time encoding method with tailbiting transmission is
proposed. The turbo encoder consists of two feedback systematic convolutional encoders
over ring ZM. Performance study of this encoder and decoder is analyzed. The paper
presents the simulation results of the system with turbo space time encoding over ring ZM
with QPSK modulation and transmission over quasi-static fading channel.
Keywords: space time encoding, turbo encoding, MIMO channel
1. INTRODUCTION
The modern wireless communication systems must be increasingly infallible.
Simultaneously these systems must realize services with the increasing
transmission speed. In recent years, multiple-input-multiple-output (MIMO)
technique has become an interesting extension to the wireless communication
systems [8]. In many wireless systems data are transmitted in packets [8]. When
the packet is created often additional bits (called the tail) are appended to the
information bits. There is thus a loss of rate, but for large block lengths the rate
reduction is negligible. It is also possible to transmit the packet without tail bits. In
this case the effective transmission rate is equal to the code rate. In a so called
tailbiting case [1, 2, 3] no additional bits are appended to the information bits but
the encoder starts and ends the encoding process in the same state, unknown by the
decoder. However, the decoder knows that the starting and ending states are the
same. Fig. 2 illustrates this situation.
In this paper, we propose a novel technique that combines non-binary turbo
tailbiting codes with space-time modulation. The turbo encoder consists of two
feedback systematic convolutional encoders over ring of integers modulo-M (ZM).
The performance of the proposed system with various packet sizes and transmit
antennas in MIMO quasi-static channel is evaluated by simulation.
__________________________________________
* Poznan University of Technology.
Filip Łęcki, Piotr Remlein
142
This paper is organised as follows. Section 2 describes the TB encoding
procedure which uses the feedback systematic convolutional encoders over ring
ZM. In Section 3, we present the structure of space time turbo encoder over ring ZM.
In Section 4, the MIMO channel model is described. Section 5 explains iterative
decoding with symbol interleaver. Section 6 presents the simulation results and
finally, a conclusion is drawn in Section 7.
2. TAILBITING CONVOLUTIONAL CODES OVER RING
The structure of the analysed in this article convolutional encoder over ring of
integers modulo-M is given in Fig. 1. At time t, the information vector with M-ary
symbols belonging to the ring ZM={0, 1, 2, ... , M-1}, (=ZM; M=4) is inputted
into the encoder. The convolutional encoder produces a coded sequence of the
symbols which belong to the same ring =Z4.
The coefficients of the encoder structure (Fig. 1) are taken from the same set
{0,...,M-1}; (M=4). The memory cells are capable to store the ring elements.
Multipliers and adders perform multiplication and addition respectively in the ring
of integers modulo-M [2, 3].
Fig. 1. The structure of the feedback convolutional encoder over ring Z4.
In this paper we encode and decode the blocks of N (M-ary) symbols without a
know tail, thus keeping the effective rate of transmission equal to the code rate.
This is done by letting the encoder start and end in the same state, unknown for the
decoder. The encoding procedure is not difficult if we use the feedforward
encoder. Then the starting state depends only on the m last information symbols in
the transmitted packet (m is the number of memory cells in the encoder).
starting sate, unknown
to the decoder
(information dependent)
ending state
identical to
starting state
N * n coded information symbols
Fig. 2. Data packet created by tailbiting method
Tailbiting transmission with turbo space time encoding
143
In the case of using the recursive systematic convolutional encoder (RSC) with
the feedback (Fig. 1) the starting state depends on all of the information symbols in
the packet. Finding the initial state, wherein the encoder should start the work, is
complex. One of the methods to find this initial state is proposed for binary
encoding in [1] and extended for multilevel codes in [2].
3. TURBO SPACE TIME ENCODER OVER RING ZM
The model of the system analyzed in this article links packet transmission
tailbiting convolutional encoding codes over the ring with space time encoding.
The convolution codes over a ring ZM designated in [3] were not optimized for the
multi antennas systems. From this reason authors of the article used these codes in
the system shown on the Fig. 3. This is the modification of solution [4]. We
propose to place the tailbiting turbo encoder over ring ZM in the transmitter.
Data Source
Modulator
„Tailbiting“
Turbo
encoder
S-random
Interleave
S/R
Modulator
Fig. 3. Transmitter diagram
Our turbo encoder is known form literature and it is parallel concatenation of
simple convolutional encoders through an interleaver [4], shown in Fig. 4.
Fig. 4. Structure of the turbo encoder
The encoder I and encoder II are previously mentioned RSC encoders shown in
Fig. 1. The interleaver is an UMTS type interleaver [5]. This UMTS interleaver is
responsible for decorrelating inputs to encoder I and encoder II.
Filip Łęcki, Piotr Remlein
144
The next block, after the turbo encoder in the transmitter (see Fig. 3) is Srandom interleaver [6]. S-random interleaver reorders the turbo coded sequence to
further decorrelate the symbols before feeding the antenna inputs.
The interleaved encoded symbols input the serial to parallel converter and next are
sent through MIMO fading channel to the receiver.
4. CHANNEL MODEL
Every telecommunication system can be simplified to a relationship between the
transmitted and received symbol for nT transmit and nR receive antennas:
y  Hs  n
(1)
n 1
n 1
where s  C T defines the transmission vector, n  C R defines the additive
n n
white Gaussian noise vector, H  C R T is the channel matrix, that describes the
connections between the transmitter and receiver. Channel matrix can be expressed
as:
 11

H   21
 

 M 1
12
 22

M 2
 1 N 
  2 N 

 

  MN 
(2)
where  mn 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 [7]. The principle of this
model lies in the assumption that the receive correlation matrix is independent of
the corresponding transmit matrix and vice-versa. To calculate the correlation
coefficients between the antennas in the transmitting as well as in the receiving
array, we followed the approach in [7].
5. RECEIVER
The simplified view of the receiver structure is showed in Fig. 5. As we can
see, the demodulator is responsible for calculating the a-posteriori probabilities for
each symbol. The demodulator is followed by the de-interleaver block ,which
forwards the symbols to the turbo decoder.
The turbo-decoder is shown in Fig. 6. This scheme exploits the parallel interconnection of two SISO (Soft Input Soft Output) decoders [9] with the feedback.
The algorithm which is used in the SISO block is a modified version of the
well known SOVA (Soft Output Viterbi Algorithm) proposed by Hagenauer in [9].
It was modified to decode non-binary symbols according to [10,11].
Tailbiting transmission with turbo space time encoding
Demodulator/
a posteriori
calculation
S-random
deinterleaver
145
Turbo
decoder
Fig 5. Receiver structure
At each state of every trellis state SOVA calculates and stores the best path metric.
Additionally it stores the reliability measure value. The reliability is the difference
between the survivor metric and the competitor [10].
EXT 2
UMTS
Interleaver
EXT 1
SISO 1
ys
k
y
p
k1
EXT 2
UMTS
Interleaver
ys
k
UMTS
Deinterleaver
SISO 2
y
p
k2
UMTS
Deinterleaver
demux
Fig 6. Turbo decoder. EXT- extrinsic information,
information from the first encoder,
ykp1
y ks
- systematic information,
ykp1
- parity
- parity information from the second decoder.
The crucial and most interesting part of the decoding process is the iterative
exchange of information between decoders (Fig. 6), this is also the reason for
calling this method “turbo”. With each iteration, the bit/symbol error rate drops. To
calculate the extrinsic values, which will be the a-priori values for the next
decoder, one is only interested in how sure the decoder decisions actually are. This
means that before passing the information to the next block, we need to subtract the
influence of the channel and previous a-priori values.
6. SIMULATION RESULTS
To estimate the transmission quality of the system analyzed in this article,
MATLAB simulations were performed. We transmitted data blocks of different
lengths: 40, 160, 500 symbols. A quasi-static Rayleigh fading channel was used.
The time of a single fade was equal to 130 space-time symbols. It was also
assumed that the receiver has the perfect knowledge of the channel. In Fig. 7 the
relation between the symbol error rate (SER) and the iterations count of the turbodecoder is shown. We transmitted packets that were 500 symbols large without the
Filip Łęcki, Piotr Remlein
146
additional tail bits. The system was constructed from 2 transmit and 2 receive
antennas (2Tx/2Rx). The modulation scheme was QPSK. As expected, the higher
the iterations count, the lower the SER. We can observe that there is a high
difference between the 1-th and 3-rd iterations. Further increase of the iterations
count does not decrease the error rate much. The difference between 3 and 12
iterations improves the performance only about 0.5 dB for SER=10-3.
0
10
-1
SER
10
-2
10
1 Iteracja
2 Iteracje
3 Iteracje
4 Iteracje
6 Iteracji
9 Iteracji
12 Iteracji
-3
10
-4
10
1
2
3
4
5
6
7
8
SNR [dB]
Fig. 7. SER vs. Iteration count for space-time turbo-coding system Z4, for a quasi-static Rayleigh
fading Channel with 2Tx/2Rx.
In Figure 8 symbol error rates for space-time turbo coded systems with
2Tx/2Rx for varying packet sizes are shown. The decoder was set to perform 6
iterations before making the final decision. As is shown in Fig. 8 for SNR higher
than 6 dB the system performed better for longer packets. In the interval between 3
and 5 dB the difference in the decoding quality between the packets is
insignificant.
SER
10
10
10
0
-1
-2
1
40
160
500
2
3
4
SNR [dB]
5
6
7
Fig. 8. SER vs. packet length for space-time turbo-coding system Z4, for a quasi-static Rayleigh
fading Channel with 2Tx/2Rx
Figure 9 shows SER obtained for the same system with packet length of 500
symbols but with varying antenna array size. As expected, the best results were
Tailbiting transmission with turbo space time encoding
147
obtained for 4 transmit and 4 receive antennas (4Tx/4Rx). For higher SNR (5dB
and more), the difference in the transmission quality between 4Tx/4Rx systems and
2Tx/2Rx, 3Tx/3Rx is significant.
Fig. 10 depicts SER relation between different decoding methods. We analyzed the
system for packet lengths equal to 40 and 160 symbols. The turbocoder was using
either tailbiting method or direct truncation method. In the direct truncation
method, the encoder does not need to calculate the beginning state, as it begins
encoding always from the same state. Nevertheless it can finish in any state. The
decoder does not have the information about the ending state. Figure 10 shows that
better results were obtained for the tailbiting case for both packet lengths.
10
-1
SER
10
0
10
10
Break-Through
-2
2TX 2RX
3TX 3RX
4TX 4RX
-3
1
2
3
4
5
6
7
SNR [dB]
Fig. 9. SER vs. array size for space-time turbo-coding system Z4, for a quasi-static Rayleigh fading
Channel with 2Tx/2Rx
SER
10
10
10
0
-1
-2
1
40 Tailbiting
40 Direct truncation
160 Tailbiting
160 Direct truncation
2
3
4
5
6
7
SNR [dB]
Fig. 10. SER vs. different decoding methods for packets length of 40 and 160 symbols, for space-time
turbo-coding system Z4, for a quasi-static Rayleigh fading Channel with 2Tx/2Rx
7. CONCLUSIONS
New structure of turbo space time encoder over ring for tailbiting transmission
are proposed. This paper is aimed at investigating the performance of space time
symbol interleaved non-binary turbo tailbiting coded systems. The computer
148
Filip Łęcki, Piotr Remlein
simulations were performed in MATLAB enviroment. The simulations were
conducted for QPSK modulation and RSC encoder over ring Z4. We considered 2
transmit and 2 receive antenna system with a quasi-static fading channel. It was
shown that for every simulation scenario tailbiting method provided better
performance than direct truncation independently from the packets length . We
have to remember that the tailbiting method is more complexity than in direct
truncation method.
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