Co-designed Turbo TCM Schemes over AWGN and Rayleigh Fading Channels Zhiquan Bai and Dongfeng Yuan Broadband Wireless Mobile Commutation and Transmission Lab., School of Information Science and Engineering. Shandong University, Jinan, China
[email protected] Abstract—In this paper, system performance of Turbo Trellis Coded Modulation (TTCM) is presented and analyzed through computer simulations over two typical channels, AWGN and Rayleigh fading channels. The co-designed TTCM method is introduced and the performance of different mapping strategies based on UP (Ungerboeck Partitioning), MP (Mixed Partitioning) and GP (Gray Partitioning), of the signal constellation of the codesigned TTCM system is given. Our simulation results show that the co-designed TTCM with UP can obtain better performance than TTCM with MP and GP in both AWGN and Rayleigh fading channels.
I.
optimal mapping strategies of the co-designed TTCM system. The organization of this paper is as follows. In Section II, the TTCM system and the different mapping strategies are provided in detail. Section III presents the two typical channel models. Simulation results and performance analysis of TTCM with different mapping strategies are presented in Section IV. Finally, conclusions are presented in Section V. II.
TTCM SYSTEM
This section describes the TTCM system model and the decoding algorithm of TTCM. TTCM systems with 8PSK modulation utilizing five mapping strategies, UP, MP and GP, are presented. The design principle of TTCM is also discussed.
NTRODUCTION
It is well known that Turbo Codes, first proposed by C.Berrou in 1993 [1], [2], can achieve good error-correcting performance. This code has I many advantages: flexible rate design, different concatenation of Recursive Systematic Convolutional (RSC) codes, etc. However, BPSK modulation of the original Turbo Codes gives relatively low bandwidth efficiency.
A. TTCM Structure The system diagram is shown in Fig.1. At each time, information two-tuple dk (here, we assume an RSC encoder with rate 2/3) and its counterpart after interleaving are input to encoder1 and encoder2, respectively. The output of encoder2 is deinterleaved so that the two parity bits correspond to the same information bits. The puncturer here is used to improve the code rate of the system. The information bits and the check bit are mapped to a signal point in the modulation. In TTCM, we use the odd-even random interleaver first described in [4]. This maps even positions to even positions and odd positions to odd positions. d
k
RSC encoder1
We will describe the structure, the basic idea and decoding algorithm of Turbo TCM. TTCM schemes with 8PSK modulation using four mapping strategies, Ungerboeck Partitioning (UP), Mixed partitioning (MP) and Gray partitioning (GP) are presented. The performance over two typical channels, AWGN and Rayleigh fading channels, is evaluated through simulations, and the design principle of TTCM is discussed. It is noted that the mapping strategy is an inherent characteristic of Turbo TCM, but there exist few work to discuss the metrics of selecting the mapping strategies of Turbo TCM. In this paper, we study this metric and the
Puncturer Interleaver RSC encoder2
Deinterleaver
Fig.1 TurboTCM System
B. Mapping Strategies Almost all coded modulation techniques [5] are based on symbol mapping through signal set partitioning introduced by Ungerboeck [3][5].Here we use a set partitioning method proposed by Ungerboeck shown in Fig. 2. In this figure, Ai (i=0, 1, 00,…,11) denotes the signal subset with different set partitioning level and yi (i=0, 1, 2) represents the code bit which is a component of a signal point. We can easily see that
This research was supported by the 2008 Outstanding Youth Scientist Awards 2008 in Shandong Province, China and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.
1-4244-2424-5/08/$20.00 ©2008 IEEE
Signal mapping
In 1982, G. Ungerboeck proposed Trellis Coded Modulation (TCM) in [3]. TCM lets us obtain higher bandwidth efficiencies. TCM codes by themselves combine modulation and coding together by optimizing the Euclidean distance between code words. They can be decoded by a symbol-by-symbol MAP algorithm. Thus, combining Turbo Codes with TCM into Turbo Trellis Coded Modulation (TTCM), we can obtain good performance in both power and bandwidth efficiency. In this way, we can get better BER with respect to traditional TCM and utilize the bandwidth more efficiently with respect to typical QPSK modulation of Turbo Codes.
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the minimum distances between the two signals in different subsets are r1 = 0.586 , r2 = 2 and r3 = 2 . Distance parameters of subsets are greater than before partitioning. This set partitioning is called Ungerboeck partitioning.
For Gray mapping we have the further expression like: y 0 = z 0 + z1 + z 2 , y1 = z1 + z 2 and y 2 = z 2 , (2) 2 where (y , y1, y0) is natural mapping and (z2, z1, z0) is Gray mapping.
Shown in Fig. 3 is another mapping strategy called block partitioning (BP) [5]. Contrary to that in UP, its distance parameter in each level is unchanged, with r1 = r2 = r3 = 0.586 .
A : 8PSK
r1 = 0.586 y =0 0
The third set partitioning method that is often used is Mixed Partitioning (MP) [6]. This is shown in Fig. 4. We can change the turn of BP and UP to make different MP strategies when we use MP. The MP in Fig. 4 consists of UP-BP-UP, the first and the third partitioning are according to the UP rule, and the second partitioning is according to the BP rule.
y1 = 0
0
1
0
2
1
0
2
2
1
1
(100)
( 000 )
A 11 A 01 0
1
0
(010 )
(110)
1
(001)
y =0
1
y1 = 0
1
1
(000 )
(100)
A 11
A 01 1
0
(000)
0
1
(010)
( 010 )
(110)
( 001 )
y0 = 0
r2 = 2 1
0
A 01
(110)
1
0
1
(101 )
(011 )
(111) ( y 2 y1 y 0 )
r1 = 0.586
A1
(100)
2 r3 = 2
Fig.4 Mixed partitioning of 8PSK
1
1
1
0
A 10
A 00 y2 = 0
r2 =
A1
A0
A : 8PSK
A0
0
( y y y0 )
0
y0 = 0
1
1
r1 = 0.586
r1 = 0.586
y2 = 0
(111)
(011)
(101)
0
A : 8PSK
A 10
1
A : 8PSK
where h0 ( D ) = 1 + D 3 , h1 ( D ) = D and h2 ( D ) = D 2 .
A 00
0
Fig.3 Block partitioning of 8PSK
The parity check equation using polynomial notation [3] is h2 ( D ) y 2 ( D ) + h1 ( D ) y1 ( D ) + h 0 ( D ) y 0 ( D ) = 0 , (1)
y1 = 0
r3 = 0.586
2
2
1
1
0
A 10
y2 = 0
( z , z , z ) = ( y , y , y ) for mixed partitioning, ( z , z , z ) = ( y , y + y , y + y ) for Gray partitioning. 1
1
A 00
The mapping strategy is an inherent characteristic of Turbo TCM and co-designing the TTCM structure is the way to get the optimal performance. For the 8PSK UP, we have the TTCM encoder polynomials as [h2, h1, h0] = [0100, 0010, 1001] in binary notation [8]. In the case of GP, we give the generation of encoder polynomials in detail. We assume that (y2, y1, y0) represents the binary representation of an Ungerboeck partitioned 8PSK signal set. Let (z2, z1, z0) be the binary representation of an Ungerboeck partitioned 8PSK signal set. Let (z2, z1, z0) be the binary representation of the other three signal sets. We can easily show that ( z 2 , z1 , z0 ) = ( y 0 , y1 , y 2 ) for block partitioning,
r2 = 0.586
A1
A0
Another set partitioning method used to minimize the BER is Gray Partitioning (GP) [7]. We can see this method in Fig.5.GP maximizes the number of adjacent pairs of signals that differ in only one bit.
2
1
0
(001)
1
(101)
A0
A11
A 00
0
y =0 2
1
(111)
(000 )
1
A 10 1
(100)
1
0
r3 = 0.586
0
1
(010)
(110)
0
(001)
1
(101)
0
(011)
Fig.5 Gray partitioning of 8PSK
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r3 = 2 + 2
A 11
A 01
2
Fig.2 Ungerboeck partitioning of 8PSK
( y y1 y 0 )
r2 = 0.586
A1
y1 = 0
r3 = 2
(011)
1
1
(111) ( y 2 y1 y 0 )
Thus the parity check equation becomes h2 z 2 + h1 ( z1 + z 2 ) + h0 ( z 0 + z1 + z 2 ) = 0 ,
(h
0
Received symbols
‘0’
+ h1 + h 2 ) z 2 + ( h0 + h1 ) z1 + h0 z 0 = 0 ,
(4)җ
3
2
3
1
3
Metrics
Interleaver
(1 + D + D + D ) z + (1 + D + D ) z + (1 + D ) z = 0 . (5)җ 2
Interleaver
decision is expressed as d k = arg max {L ( d k ) = i} , where M, M’
In the decoding process of TTCM, we should pay attention to the received symbols and the a priori information in the first decoding process, since their initial value is different to latter iterations. In TTCM each decoder alternately receives its corresponding encoder’s noisy output symbols, and then the other encoder’s noisy output symbols. Care is taken not to use the systematic information more than once in each decoder. We know that we must have the a priori information before the first decoding process. As seen in Fig.6, the Metrics box calculates the a priori information Liapriori . The Metric1 and
i
are the states of the trellis structure.
γ i ,α k , β k are defined as follows:
p ( sk = M d k = i , sk −1 = M ' ) p ( d k = i sk −1 = M ' )
(7)
2m −1
¦ ¦ γ ( y , M ', M ) ⋅ α ( M ' ) k −1
(8)
¦ γ i ( yk , M ', M )α k −1 ( M ' )
Metric2 boxes calculate Lip& s , the parity and systematic loglikelihood. This term is one of the components of the transition probability γ i ( yk , M ', M )
i =0
2 m −1
βk ( M ) =
¦ ¦γ ( y
k +1
¦γ (y
k +1
i
M ' i =0 2m −1
i
i =0
, M , M ' ) ⋅ β k +1 ( M ' )
.
(9)
III.
, M , M ') α k ( M )
We set the initial conditions below: 1 if s = 0 1 and β N ( s ) = ® α0 ( s ) = ® ¯0 otherwise ¯0
TWO TYPICAL CHANNEL MODELS
A. AWGN Channel Model The AWGN channel is the most common and simplest channel. The parameters of this channel obey the Gaussian distribution. Here, n(t) is a sample function of the Gaussian process with double-sided power spectral density N0/2 . We assume that it is independent of the signal. The values of the mean and variance are zero and N0/2, respectively.
if s = 0 otherwise
In the decoding of binary Turbo codes, we write the loglikelihood L ( d k ) of information bits in this form: L ( d k = i ) = Liapriori + Lichannel + Liextrinsic
(11)җ
We can calculate the extrinsic and systematic information, Lie&s = L ( d k = i ) − Liapriori , this term is transmitted to the other decoder as it is a priori information.
(6)җ
for i = 0,1,...,2 − 1 and a normalizing constant K. The final
,
Decoder2
that affects the parity component also affects the systematic one. We write L ( d k = i ) as:
M'
k
Lp & s
L ( d k = i ) = Liapriori + Lie&s
= log[ K × ¦¦ γ i ( yk , M ', M ) ⋅α k −1 ( M ' ) β k ( M )] ,
i
Lapriori
− + Le & s
L ( d k = i ) = log ( p {d k = i | Y })
M ' i =0 2m −1
Le & s
Metric2 ‘0’
Deinterleaver
+
Fig.6 The Turbo decoder structure
below, where Y = ( y1 , y2 ,..., yN ) is the received symbol.
γ i ( yk , M ', M ) = p ( yk d k = i , sk = M , sk −1 = M ' ) ⋅
Decoder1
−
C. The Algorithm of the Decoder We now introduce the decoding algorithm of TTCM based on the Log-MAP algorithm [9], [11]. In TTCM, the loglikelihood L ( d k ) of the information symbols is calculated as
αk ( M ) =
Lapriori
0
That is, the code polynomials should be [1111, 1011, 1001]. Following the above deduction, we can get the code polynomials as: [0100, 0010, 1001] for UP, [1001, 0010, 0100] for BP (does not exist in systematic form), [0010, 0100, 1001] for MP and [1111, 1011, 1001] for GP.
M m
Metric1
(3)җ
(10)җ
B. Mobile Fading Channel Model
The three parts are the a priori component (the information given by the other decoder for the bit), the systematic component (corresponding to the received systematic value for the bit), and the extrinsic component (the part that depends on all other inputs). In the decoding of TTCM we can write the output into two different components: i) a priori and ii) e&s (extrinsic and systematic), because the information bit and the check bit are mapped to a signal point together and the noise
y (t )
x(t )
a (t )e jθ ( t )
n( t )
Fig.7 Fading channel model
Fig.7 shows the model for the fading channel with additive white Gaussian noise. For slow fading processes, the channel
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gain can be considered constant over the symbol duration Ts. Throughout this work we consider a fully interleaved Rayleigh slow-fading channel. 8PSK signaling with coherent detection is assumed. The discrete representation of the model is: yk = a xk + nk , (12)җ where xk is an 8PSK complex symbol and k n is a complex additive white Gaussian noise component with zero mean and variance N 0 2 . The fading coefficient a is a random variable with probability density function:
We conclude that using TTCM with UP we can have about 0.08dB and 0.09dB gain compared with MP and GP respectively with eight iterations in an AWGN channel.
f ( a ) = 2ae− a , ( a > 0 ) 2
and E ª¬ a 2 º¼ = 1. Perfect phase knowledge is assumed. Knowledge of the channel gain at the receiver will be referred to as channel side information (CSI). CSI may be gained through the use of an auxiliary channel or a direct examination of the SNR. At the receiver, in order to simplify the complexity, we use the average value of CSI with E [ a ] = 0.8862 .
SNR (dB)
Fig.9 TTCM 8PSK with different mapping strategies in AWGN channel
IV.
SIMULATION RESULTS When the number of iterations is eight, we also show the BER curves of data transmission in the flat Rayleigh fading channel in Fig.10. When the BER is around 10-4, the Eb/N0 we need is 9.60dB, 9.90dB and 10.00dB under UP, MP and GP, respectively. We conclude that using TTCM with UP, we have 0.30dB and 0.40dB gain compared with MP and GP, respectively.
In our computer simulation, we performed a code search for an 8-state systematic convolutional encoder. We found that the performance of the encoder structure to be the same as the one that Robertson and Woerz presented in [11]. This obtained the best performance for UP compared with other Turbo TCM encoder structures. We will study the same 8-state Turbo TCM encoder structure used in [11]. Info bit pair Modulation Mapping
T
T
Puncturer
T
Interleaver
Deinterleaver Modulation Mapping
T
T
Fig. 8 Encoder Structure of Turbo TCM (MP)
The TTCM with the optimal RSC encoder is [0100ˈ0010, 1001] for UP [11]. The first two columns are forward polynomials and the last column is the feedback polynomial. For BP, the RSC encoder polynomial is [1001, 0010, 0100]. But since the feedback polynomial is not delay-free, the encoder is not realizable and can not be implemented. The bandwidth efficiency of the system is 2 bit/s/Hz, which is higher than traditional rate 1/2 binary Turbo Codes whose bandwidth efficiency is 1 bit/s/Hz with QPSK. This improves the bandwidth efficiency by two times. The interleaver used is an odd-even random interleaver with the interleaver size being 1024.
SNR (dB)
Fig. 10 TTCM 8PSK with different mapping strategies in Rayleigh channel
V.
CONCLUSIONS
From the simulation results, we conclude that TTCM has good error correcting performance in the regions of low signal to noise ratio. We can get higher bandwidth efficiency by using TTCM, compared with binary Turbo Codes. When using TTCM with the co-designed scheme, we can get better BER performance compared with our precious work [8]. When we co-design the code and the constellation mapping, we can no longer use BP (the encoder is not realizable) and the performances of the different constellation mappings are almost the same since the co-designed schemes make the properties of the TTCM system identical. And under the
When the number of iterations is eight, the simulation results for an AWGN channel are shown in Fig. 9. When the BER is 10-4, the Eb/N0 we need is 3.83dB, 3.84dB under MP and GP respectively, but under UP the Eb/N0 is just 3.75dB.
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[6]
D. F. Yuan, Z. G. Cao, D. W. Schill and J. B. Huber, “Robust Signal Constellation Design for AWGN and Raleigh Fading Channels for Softly Degrading Communication Scheme Using Multilevel Codes,” Chinese Journal of Electronics, vol. 2, no. 2, pp. 115~121, Apr. 2000. [7] G. J. Pottie; D. P. Taylor, “Multilevel codes based on partitioning,” IEEE Trans. on Inform. Theory, vol. 35, issue 1, pp. 87–98, Jan. 1989. [8] Zhiquan Bai and Dongfeng Yuan “Performance analysis of turbo trellis coded modulation in still image transmission over two typical channels”, IEEE VTC 2003-Fall, vol. 4, pp. 2521-2525, Oct. 2003. [9] S. Le Goff, A. Glavieux, and C. Berrou, “Turbo-codes and high spectral efficiency modulation,” Proceedings IEEE ICC’94, vol. 2, pp. 645-649, 1994. [10] Patrick Robertson and Thomas W¨orz, “A Novel Bandwidth Efficient Coding Scheme Employing Turbo Codes,” Proceedings IEEE ICC’96, vol. 2, pp. 962-967, June 1996. [11] Patrick Robertson and Thomas W¨orz, “Bandwidth-Efficient Turbo Trellis-Coded Modulation Using Punctured Component Codes,” IEEE JSAC, vol. 16, issue 2, pp. 206-218, 1998. [12] Jung, Peter; Nasshan and M. Markus, “Comprehensive Comparison of Turbo-Code Decoders,” Proceedings IEEE VTC’1995, vol. 2, pp. 624628, July 1995.
practically simulated conditions, UP can still has the best performance. The Euclidean distance can be one of the metrics to evaluate the error-correcting performance over fading channels. REFERENCES [1]
[2]
[3] [4]
[5]
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