- 1 -
Space Time Coding in OFDM based on TCM
Chen Hongwei，Ruan Qiaoling，Li Ping’an
School of Information Engineering，Wuhan University of Science and Technology，Wuhan
(430070)
Abstract
Based on the principle of Trellis Code Modulation（TCM）, we suppose a new scheme and Space Time
Coding technology in OFDM system. Space time block coding (STBC) is applied as and inner coding
to ensure the spatial diversity and TCM as the out coding to obtain coding gain. Simulation is carried
on SUI-3 channel model, the method of channel estimation is also improved as well. The results show
that the bit error rate (BER) can be great reduce by this improved hybrid scheme, and greater coding
gain can be obtained without increase the bandwidth.
Keywords：TCM，OFDM，STBC
1. Introduction
Trellis code modulation (TCM), effectively combine the modulation with coding technology, could
develop the performance of the broadcasting system without sacrificing data rate or requiring more
bandwidth. Orthogonal frequency division multiplexing (OFDM) is a multi-carrier modulated
technique, which has been widely regarded as an effective technique for increased spectral efficiency
and combated the effects of frequency selection fading. Spatial diversity is known to be robust against
multi-path fading effectively. The space-time block coding (STBC) scheme proposed by Alamouti
could be simplified realization, which could obtain all transmit diversity gain with high efficiency
decoding algorithm. Combine the two coding schemes together with OFDM could great reduce the bit
error rate (BER) in fading channel.
SUI (Stanford University Interim) channel model, adopted by IEEE [1] to test fixed
wideband wireless access system, is typical slow frequency selective fading channel. This paper select
SUI-3 no light of sight channel to simulate TCM-OFDM-STBC system and analysis its performance
latter.
2. TCM technique
Ungerboeck proposed a TCM scheme in his original paper in and conclude his suggestion
could obtain 3~6dB coding gain without requiring more bandwidth. Instead of separate coding and
modulation apart, TCM break through the conventional model to combine them together. The TCM
encoder is essentially symbol coding, the trellis branches are labeled with redundant nonbinary
modulation signals rather than with binary code symbols. The aim of coding is maximize the free
distance, not the Hamming distance.
This article adopt TCM-8PSK schemes, the 8PSK constellation mapping need to be set partitioning
firstly. Based on Ungerboeck’s paper[2], the 8PSK constellation has been set partitioning as follows:
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Fig 1 TCM-8PSK signal set partitioning scheme
For un-coded 8PSK set, the signals distribute in a circle whose radius is ε , the least Euclidean
distance is
0
12 sin (2 2)
8
d ε π ε ε= = − =
.Eight signals divide into tow
sets( 0 1,B B ) when fist set partitioning, the least distance become 1 2d ε= .Second the
subset( 0 1,B B ) divide into another two subsets( 0 2,C C ),( 1 3,C C ),which enlarge the least distance to
2 2d ε= . The least distance in trellis is called free Euclidean distance, express by fedD .
4 states TCM-8PSK encoder can be described as follows: 2 bits digital data enter the encoder each time
slot, 1 bit coded by a 1/2 rate convolve encoder to 2 bits to entered in the 8PSK modulator with 1
un-coded bit to export one 8PSK symbol. The structure of encoder and state transition diagram are
depicted in figure 2, where the two coded bits 1 2,c c are used to choose one subsets in
( 0 2,C C , 1 3,C C ), the un-coded bit used to choose one of the signals in each subset. As for 8 states
TCM-8PSK scheme, both the input bits are coded by a 2/3 convolve encoder before 8PSK modulation.
These scheme enlarge the
2fedD ε= 。8 states TCM-8PSK encoder and state transition diagram are
depicted in figure 3. In high SNR, the coding gain can be written as::
2
2
( / )
| 10 log
( / )
fed s
SNR
fed s
D E
g g
D E∞ →∞
= = coding
un-code （1）
In the 4 state TCM as figure 2,
2fedD ε= , compared with un-coded 4PSK modulation
0 2d ε= ,4 states trellis diagram obtain 3g dB∞ = coding gain.
- 3 -
Fig 2. 4 states TCM-8PSK encoder and trellis diagram
Fig 3. 8 states TCM-8PSK encoder and trellis diagram
Decoding TCM signals generally need tow steps to apply viterbi soft-decision algorithm[3]. First step
is to confirm the optimal signal in each subset, namely the nearest constellation point to the receive
signal in each subset, this step is called subset decode. The second step is to mapping the signal
selected in each subset and the corresponding Euclidean distance
2 2
{ }
ˆ | min | |
n
n n n na C
r a r a∈− = −∑ to
the branch of viterbi algorithm. Where { }nr is the sampling sequence of received signals, { }na is
transmit sequence, C is the code space ,constituted by all coding signals sequences. In high SNR
with additive noise, the first error-event probability of trellis coding can be approximately written as :
2
02
fed
e fed
D
P N Q
N
⎡ ⎤⎢ ⎥≈ ⎢ ⎥⎣ ⎦ （2）
3. TCM－OFDM－STBC system model
The transmitter and receiver of TCM hybrid STBC-OFDM system model are depicted as figure 4,
figure 5. We adopt 2 transmit antennas and 2 receive antennas’ scheme based on Alamouti’s space-
time coding theory to simulate in SUI-3 channel according to protocol.
- 4 -
Fig4. Transmitter
Fig5. Receiver
The binary bit streams ( )s k have been modulated to 8PSK symbol ( )X k after TCM encoder, we
adopt STBC technique to transform the 8PSK symbol as follows:
*
1
*
2
( ) ( ) ( 1)
( ) ( 1) ( )
S k X k X k
S k X k X k
⎡ ⎤− +⎡ ⎤ = ⎢ ⎥⎢ ⎥ +⎣ ⎦ ⎣ ⎦ （3）
Where the symbols transform to two branches as
*
1( ) [ ( ) ( 1)]S k X k X k= − + ,
*
2 ( ) [ ( 1) ( )]S k X k X k= + .After serial-to-parallel (S/P) conversion and inserting the pilots, IFFT
operation are applied. As a result, the modulated signal in frequency-domain can be expressed as:
1
2 /
0
( ) { ( )} ( )
N
j kn N
i i i
k
x n IFFT S k S k e π
−
=
= =∑
, 1, 2; 0,1, , 1i n N= = −L ,（4）
where i is the number of antenna, N is the total number of sub-carriers. After insert gN as the
guide intervals, the signals can be expressed as:
( ) , 1, , 1
( ) 1, 2
( ) 0,1,2 , 1
i g g
ig
i
x N n n N N
x n i
x n n N
+ = − − + −⎧⎪= =⎨ = −⎪⎩
L
L
（5）
At the receiver, the signals in antennas can be express as:
( ) ( ) ( ) ( )gy n x n h n nη= ⊗ + （6）
After serial-to-parallel (S/P) conversion and the guard interval removal, FFT operation are
applied, the signals transformed to frequent domain can be expressed as:
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[ ] [ ] 11 121 2 1 2 1 2
21 22
( ) ( )
( ) ( ) ( ) ( ) [ ( ) ( )]
( ) ( )
H k H k
Y k Y k X k X k W k W k
H k H k
⎡ ⎤= +⎢ ⎥⎣ ⎦ （7）
Where 1 2[ ( ) ( )]W k W k is additive whiten Gaussian noise（AWGN）.
11 12
21 22
( ) ( )
( ) ( )
H k H k
H k H k
⎡ ⎤⎢ ⎥⎣ ⎦ is
channel state matrix, we apply improved orthogonal code based method to estimate the channel. As 2
×2 antennas scheme, the pilot in fist transmit antenna can be designed as [ ]Xp Xp− ,the pilot in
the second is [ ]Xp Xp , the receive of the pilot can be express as :
1 11 21 11
1 11 21 12
( ) ( ) ( ) ( ) ( )
8
( 1) ( ) ( ) ( ) ( )
p p
p p
Y k Xp k H k Xp k H k n
Y k Xp k H k Xp k H k n
= + +⎧⎪⎨ + = − + +⎪⎩
（）
where ( , 1, 2)ijn i j = are noise, we can get the estimation of channel as follows:
11 1 1 21 1 1
ˆ ˆ( ) ( ( ) ( 1)) / 2 , ( ) ( ( ) ( 1)) / 2p p p pH k Y k Y k Xp H k Y k Y k Xp= − + = + + （10）
12 2 2 22 2 2
ˆ ˆ( ) (( ( ) ( 1)) / 2 ( ) ( ( ) ( 1)) / 2p p p pH k Y k Y k Xp H k Y k Y k Xp= − + = + + （11）
We can decode the space-time code using the channel information above:
* * *
11 1 12 1 21 2 22 2
ˆ ˆ ˆ ˆ ˆ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( 1)X k H Y k H k Y k H k Y k H k Y k ∗= + + + + + （12）
* * * * *
11 1 12 1 21 2 22 2
ˆ ˆ ˆ ˆ ˆ( 1) ( ) ( ) ( ) ( 1) ( ) ( ) ( ) ( 1)X k H k Y k H k Y k H k Y k H k Y k+ = − + + − + （13）
After TCM decode, the decoded space-time signal
ˆ ( )X k ,
ˆ ( 1)X k + can be outputted as ˆ( )s k .
4. Simulation result
This article simulate the proposed TCM-OFDM-STBC scheme in SUI-3 channel model, the channel
power distribution is:
2 2P m σ= +
(14)
where 1
Km P
K
= + ,
2 1
1
P
K
σ = + ,m is the complex constant，, and the 2σ is variance of the
complex Gaussian set. The Doppler Spectrum function can be approximate by:
2 4
0 0 0
0
1 1
( )
0 1
f f f
S f
f
⎧ − + ≤⎪= ⎨ >⎪⎩ where 0 m
ff
f
=
(15)
The total sub-carriers of the OFDM system in simulation is 256, 200 of them used to transmit dates, the
left 56 are set to be zero (1 transmit direct currency). The max delay spread is 1μs, and 3 path in total,
each time delay is 0µs、0. 5µs and1µs, average power of each path is 0 dB、- 5 dB and -10 dB, the max
Doppler frequency shift is 0. 4 Hz. The bandwidth is 20MHz, the length of the cycle preamble is
total transmit power of 2 antennas is equal to 1 transmit antenna. The parameters of the
simulation are set as follows:
2 12 22 21
2 12 22 22
( ) ( ) ( ) ( ) ( )
9
( 1) ( ) ( ) ( ) ( )
p p
p p
Y k Xp k H k Xp k H k n
Y k Xp k H k Xp k H k n
= + +⎧⎪⎨ + = − + +⎪⎩
（ ）
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Tab 1. Parameters in simulation
carriers 256
Guard interval 56
Forward guard interval l28
Back guard interval 27
Antenna relative coefficient
Data sub-carriers 200
Average power each tap [0 -5 -10]
Rice coefficient [1 0 0]
Max Dop shift(Hz) [ ]
CP length 56
Bandwidth 20E+6
3path delay [0 ]
The BER performances of the TCM hybrid STBC-OFDM system in SUI-3 in different encode states
are depict as figure 6, figure with un-coded STBC-OFDM and single TCM-OFDM
systems [5,6], we can see the proposal scheme’s BER have been greatly deduced than others.
Fig 6 4states TCM-OFDM-STBC system performance
Fig 7. 8states TCM-OFDM-STBC system performance
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5. Conclusions
This paper has researched the combination scheme of TCM, OFDM and multi-antenna technique,
verified its validity in based channel. Simulation results show its good ability to robust
against frequency selective fading and time spread. The performance of proposal scheme improve a lot
compared with un-code STBC-OFDM and single TCM-OFDM systems, which can obtain great
coding gain. It is definitely an effective solution in complex mobile environment, but the realization of
this scheme must be a little complex than the other two.
References
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[2]:Trellis-code modulation with redundant signal sets-Part I:Introduction,” IEEE Communication
Magazine,,,
[3] Dariush Divsalar, :Marvink Design of Trellis Coded MPSK for Fading Channels: Performance
Transaction on communications” ,1988
[4],:Channel coding with multilevel/Phase signals, IEEE
Theory”,-28,-67,.
[5] Helard, Floch :Trellis code orthogonal frequency division multiplex for digital video
transmission ,IEEE,1991
[6] Helard, Floch:Trellis Coded orthogoanal frequnecy division multiplexinf for digtal video
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