Conventional space time trellis code

x

Although Space time block codes can achieve full diversity on flat fading channels with a simple decoding algorithm, it does not offer any coding gain. In 1998, Space-Time trellis code was first introduced by Tarokh, Seshadri and Calderbank in [5].The system block diagram is shown in Fig2.4. The STTC can simultaneously provide substantial coding gain

and full spatial diversity on flat fading channels. The encoder structure is shown in Fig2.5. transmitted through the n-th transmit antenna at time t is denoted byx_tⁿ.

⊗

The m generator coefficient sets is given by

The total number of states for the trellis encoder is 2^v.The m multiplication coefficient set sequences are called the generator sequences, since they can fully describe the encoder structure.

At the receiver for STTC, the decoder employs the Viterbi algorithm to perform maximum likelihood decoding. Assuming that perfect channel state information is known at the receiver, for a branch labeled by ( ,x x¹_{t t}² x_t^nT), the branch metric is computed as the antenna. The examples such as Tarokh/Seshadri/Calderbank (TSC) codes and Baro/Baush/Hansmann (BBH) codes and performance analysis are discussed in [5].

2.4.2 Space time trellis codes defined on complex field

complex field. As a result, their trellises cannot be combined through simple arithmetic. To overcome this difficulty, a STTC defined on the complex field is developed. The overall scheme is shown in Fig2.6. As seen in Fig2.6, the input sequence c is mapped into the complex field first, we denote c'; afterwards the STTC encoder generates signal mapping for transmitter. The detail of the encoder structure with mapping is shown in Fig2.7.Take two transmit antennas for example, the generator coefficient set is given by

1 1, 1, 1, 1, 1, 1, 1, 1, signal of i-th antenna at time t is given by

⊗

Fig2.7 Encoder of the proposed STTC

, , channel is obtained. The operation of the encoder is now linear. Assume that there are two transmitted antenna and one received antenna, the received signal can be written as

where⊗denotes the convolution operation. To make the combination valid, appropriate complex generating coefficients must be found .In the following, we will present a simple way to design STTC for M-QAM with M=2^m. . Assume that all constellation points are on integer grid point, the generator coefficient are chosen from 2⁰, 2¹,…,2^(m/2-1) and 2⁰j, 2¹j,…,2^(m/2-1)j ;with these coefficients, all grid points can be generated through linear combinations. The operation of the encoder is best illustrated with the following examples.

Example 1:

Code 1.1, code 1.2 and code 1.3 are all complex STTC for 4-QAM (QPSK) system with two input sequence. Memory order of the system is two, which means there are one register in each encoding path.

Code 1.1: ^{g11 =}

(

^{g g101, 111}

)

⁼

^{( )}

^{1,0 , g12 =}

(

^{g g102, 121}

)

⁼

^{( )}

^1,0

^{g12 =}

(

^{g g012, 112}

)

⁼

^{( )}

^{j,0 , g22 =}

(

^{g g022 , 122}

)

⁼

^{( )}

^j,0

Code 1.2: g1¹ =

( )

1,0 , g¹2 =

( )

0,1 , g1² =

( )

j,0 , g2² =

( )

j,0 Code 1.3: g1¹ =

( )

0,1 , g¹2 =

( )

1,0 , g1² =

( )

j,0 , g2² =

( )

0, j

Example 2:

Code 2.1, code 2.2 are complex STTC for 4-QAM system with two input sequence, and memory order of the system is three.

Code 2.1: g1¹ =

(

1,0,0

)

, g¹2 =

(

0,1,0

)

, g1² =

(

0, ,0j

)

, g2² =

(

j,0,0

)

It is found through simulations that coding gains of these codes are similar, but different diversity orders are achieved. The code design criterion for achieving the highest diversity order will be discussed in the following.

The diversity order which a system can achieve is decided by the length of the combined channel in (2.28). Assume 4-QAM for example, the combined channel can be written as

1^'

( )

¹ 1, ¹,1 ¹ 2, ¹,2

in which the total length of the combined channel is L+v-1,where L is length of original channel. The diversity level of the system is L+v-1 theoretically (When L+v-1 is larger than Tx×L, the diversity level be capped by Tx×L). To ensure that the combined channel has potentially the longest length, certain first and last coefficients of the generating polynomials cannot be zero simultaneously. For the 4-QAM example, the following should be observed:

In addition, the power distribution of the combined channel should be as even as possible to achieve a higher diversity order; therefore, the magnitude of the coefficients should be symmetric, i.e,

,1 1 ,2 , 1, 2 , 1, 2,..., 1

c c

i v i

g = g _{− −} c= i= v− . (2.31)

Equations (2.30) and (2.31) are the design criteria for complex-valued STTC to achieve the highest possible diversity order.

The generator matrix set of proposed STTC is a subset of that of the complex-field STTC [6] for QAM modulation. Fig2.8 shows the block diagram of the STTC and the structure of the encoder is the same with that of the proposed STTC shown in Fig2.7. The only difference between two encoders is that the encoder of the STTC has modulo operation and the encoder of proposed STTC does not have. In order to distinguish the two complex-field STTCs, we called the proposed STTC the linear complex field STTC (LCF-STTC) because of the linear operation of the encoder. Take 16-QAM for example, signals generated by the

STTC are

The advantage of the LCF-STTC is that it can combine channels and a joint processing at the receiver can be realized. The performance of the LCF-STTC is better than conventional STTC because the LCF-STTC has the joint processing gain at the receiver. However, the number of generator matrix is constrained by the linear operation of the encoder. It is a trade-off between the joint process and the number of the generator matrix

Fig.2.9 Linear translation mapping

The difference between conventional STTC encoder and LCF-STTC encoder is the generator coefficients. For the conventional STTC, the generator coefficients are defined on finite ring whereas the generator coefficients of LCF-STTC are defined on complex field. However, we can not directly convert the generator coefficients defined on finite ring, into complex field because modulo operation applied in the conventional STTC encoder is a linear operation on finite ring but it is not linearly on complex field.

The STTC can be viewed as delay diversity scheme in some case. For example, let us assume that the generator sequence of a four-state space time trellis coded QPSK scheme with two transmit antennas are

1

note it is actually a delay diversity scheme since the signal sequence transmitted from the first antenna is a delayed version of the signal sequence from the second antenna. The

proposed codes can also be viewed as delay codes. We take codes 1.1-1.3 for example and explain why code 1.3 has better performance than code 1.1 and 1.2 from the point view of delay diversity scheme. Assume that the memory order of the encoder is two. That is each branch of the encoder has one register. The generator matrix for code 1.3 is

0 1 0 and time t+1 slot although they are placed in different symbols. For code 1.2, the generator matrix is

only input information sequence C¹ has delay diversity and it’s performance is worse than code1.3. And for code 1.1, the generator matrix is

1 0 0

1 0 0

G j

= j (2.40) the transmitted antenna are the following

we see that neither input information sequence c'¹ nor c'² extract delay diversity thus the performance of the code is the worst of three with the same input information sequence.

Note that the codes satisfied both design criteria are a form of delay diversity scheme and indeed have good performance.

Another error control code, modulated code, is also defined on complex field [11] -[13].It is a pre-coding technique and the channel condition is known at transmitter. The main advantage of modulated codes is that their encoding arithmetic operation and the multipath channel arithmetic operations are all defined on the complex field and therefore can be algebraically combined together. The joint maximum likelihood decoding is used at receiver. The basic ideal is similar to our proposed codes but their goals are different. The LCF-STTC is designed for exploiting multipath diversity gain whereas the modulated code is designed for exploiting the optimal coding gain form the ISI channel. Both of the two schemes exploit multipath to improve the performance of systems.

Chapter 3 Performance analysis and space time turbo equalization

3.1 Performance analysis

In this section, we find a method to derive an upper bound for the proposed STTC to demonstrate the novel scheme indeed can achieve diversity as we expected. For trellis codes, it is extremely complex to calculate the bit error rate (BER) and no efficient methods exist. Therefore we look for more accessible ways of obtaining a measure of performance.

We consider the probability that a codeword error occurs. Such an error happens when the decoder follows a path in the trellis which diverges from the correct path somewhere in the trellis. The decoder will make an error if the path that follows through its trellis does not coincide with the path taken by the encoder. An error that follows a path in the trellis which diverges and emerges only once from the correct path in the trellis is called the first error

Fig 3.1 An error event

event shown in Fig3.1. In general, the error performance analysis of trellis codes is almost based on a code’s distance spectrum used in union bounding techniques. The distance spectrum for trellis codes is usually found by computer search. In [5], the authors derived design principles for space time channel codes over quasi-static fading channels. The design guidelines were based on maximizing the rank of the codeword difference matrix, in order to achieve the highest possible diversity over the quasi-static fading channel. In the following, we derive the performance bounds for proposed codes with some assumptions.

Because the proposed STTC encoder is defined on complex field, it can be combined with multipath channel directly. Assuming that channel states information is known at receiver.

We obtain the trellis diagram of the combined channels and from it we can get the minimum distance of the first error event assuming that all-zero path is transmitted.

Furthermore we assume the error probability is dominated by the minimum distance, d_min. Because the distribution of channel random variable h_{j i,} is complex Guassian distributed with zero mean and unit variance, its amplitude is Rayleigh distributed. Then we use the Rayleigh sum distributions and densities [7] to find the probability density function (PDF) of the random variable d_min. The upper bound for the proposed STTC can be calculated from PDF of d_min. Although there is no closed form for the upper bound so far, the upper bound derived by our method is an evidence to explain that the proposed codes can achieve diversity orders as we expected. The method to derive the upper bound can simply be described as following steps:

1. Obtain d_min from the trellis of combined channels.

2. Find pdf of d_minusing Rayleigh sum distribution.

3. Calculate upper bound for the proposed code.

1 2 1 2

In the following, we take code 1.3 for example, two transmitted antenna and one receive antenna are used and channel length is two. From (2.28), the combined channel will be

(3.1) combined channels could be considered as tap delay line and the trellis diagram is shown in Fig3.2. We assume all-zero path is transmitted and define the Euclidean distance between two branches asd_ij = r r_i− _j ² , where r_i and r_j are received signals of i-th and j-th

1' [ ,21 11 22, 12] h = h h +h h

2' [ 11, ( 12 21), 22] h = jh j h +h jh

Fig3.3 Minimum distance in the trellis diagram

branch respectively. From the trellis diagram in Fig3.3, we can obtain thed_min:

(3.2)

There are three terms in (3.2), each term presents an event error. We can easily see that the third term of (3.2) is always larger than the others, thus it does not need to be considered.

(3.2) can be simplified as:

X= h₁₁ + h₁₂ +h₂₁ + h₂₂

Y= h₂₁ + h₁₁+h₂₂ + h₁₂ , (3.4)

Figure 3.4 PDF of random varialve X

X ,Y are Rayleigh sum distributed ,and also they are independent and identical distribution (i.i.d) under this condition. Fig3.4 shows the pdf of the random variable X. For Rayleigh sum distribution, the approximated probability density function and cumulative density function (CDF) of N i.i.d. Rayleigh random variables are as following :

PDF:

CDF:

(3.6) .

Based on probability theorem, the probability density function of d_min can be calculated as the form:

Because the error probability is dominated by random variabled_min, we derive the upper bound conditioned on channel as:

min

where Es is the energy per symbol at each transmit antenna and Q function is the complementary error function defined by

2

in order to get an upper bound on the unconditional error probability, we need to average the channel with respect to the random variabled_min. The error probability can be upper bounded by

min2 bound, some assumptions are needed to simplify calculation. Although the upper bound is not a tied bound, it provides an evidence to demonstrate that the proposed codes indeed can achieve diversity order we expected.

3.2 Space time turbo equalization

We use joint STTC/channel decoder based on MLSE to deal with ISI and exploit the multipath diversity gain. However, the computational complexity of the receiver is determined by the channel length and the number of trellis states. It grows exponentially when the channel length or the number of trellis states increase. Because of the high complexity, optimal receiver might be infeasible in most practical systems. For complexity reasons, the equalizer and decoder of most practical systems are separated. The straightforward way to implement this separate equalization and decoding process is for the equalizer to make hard decisions as to which sequence of channel symbols were transmitted and for these hard decisions to be mapped into their constituent binary code bits.

The process of making hard decisions on the channel symbols actually destroys information pertaining to how likely each of the possible channel symbols might been, however. To mitigate the performance degradation made by hard decision, soft decision is considered. The ‘soft’ information can be converted into probabilities that each of the

received code bits takes on the value of zero or one that is precisely the form of information that can be exploited by a decoding algorithm. Many practical systems use this form of soft-input error control decoding by passing soft information between an equalizer and decoding algorithm. In order to approach the remarkable performance of optimal receivers with lower complexity, turbo equalization [14] is proposed. It is a receiver that the iterative process is between equalization and decoding. The transmitter and receiver structures for two transmit antennas are illustrated in Fig3.4. The BCJR algorithm [15] is used both in the equalizer and the decoder to estimate the soft information. At the center of the turbo equalizer are two BCJR algorithms that can operate on observations and prior information about individual bits or symbols. Only the extrinsic information is fed back in the iterative loop.

Fig3.5 The structure of transmitter and turbo equalizer

The MAP equalizer computers the LLR log-likelihood ratio of each group of information

The decoder operates on a trellis with M_s states. The forward recursive variables can be computed as follows

and the backward recursive variables can be computed as

1

, ,

attenuation between n-th transmit antenna and j-th receive antenna , x_tⁿ is the modulated symbol at time t, transmitted from n-th antenna and associated with the transition

1 '

t t

S₋ = → = , and l S l p i_t( )is the a priori probability of x_t = . i

The iterative process between equalizer and decoder is that the soft information from equalizer is interleaved and taken into account in the decoding process and similarly the soft information from decoder is entered the equalizer, creating a feedback loop between equalizer and decoder. The performance of the BCJR algorithm can be greatly improved if good prior information is available. With iterative process, the performance of turbo equalizer would approach the optimal receiver.

3.3 Simulation results

In our simulation, a system with two transmit antennas and one receive antenna on the Rayleigh fading channel is used and 4-QAM modulation is used. We assume that the channel estimation is perfect.

Fig3.6 shows the comparison of STBC with Alamouti scheme, STTC and space time delay code (STDC) over flat fading channel. The system with two transmitted antennas and one received antenna and 4-QAM modulation is used. We assume that channel estimation is perfect. That is the channel state information is known at the receiver. We use the

conventional STTC with the coefficientsg¹ =[(02),(20)],g² =[(01),(10)] and the STDC with one delay. Note that the STTC has better coding gain than STBC.

Fig3.7 shows the BER of codes 1.1, code 1.2 and code 1.3.The signals are transmitted over frequency selective channel with length of two. Note that code 1.3 has the highest diversity order. Two upper bounds for code1.1 and 1.3 calculated from our method are shown in Fig3.8 and Fig 3.9 respectively. In Fig3.10, simulation results shows that different upper bounds exactly have the diversity order of two, there and four.

Fig3.6 Bit error rate of space time codes on flat fading channel

Fig3.7 Bit error rate of code 1.1, 1.2 and 1.3

Fig3.8 Upper bound for code 1.1

Fig3.9 Upper bound for code 1.3

Fig3.10 Upper bound with different diversity order

Fig3.11 Performance of LCF-STTC and conventional STTC

over flat fading channel

Fig3.12 Performance of LCF-STTC with code 1.1 and conventional STTC over frequency selective channels.

Fig3.13 Performance of LCF-STTC with code 1.3 and conventional STTC over frequency selective channels.

Fig3.14 Bit error rate of the turbo equalizer

Fig3.11 shows the performance comparison between LCF-STTC and conventional STTC over flat fading channel. In Fig3.11-Fig3.13, data1 is the code with coefficients g¹=[(02),(20)] , g² =[(01),(10)] ; data2 is the code with coefficients g¹ =[(02),(10)],g² =[(22),(01)] ,and data3 is the code with coefficients

1 [(22),(10)], 2 [(02),(30)]

g = g = ;data4 and data5 are the same complex code. In Fig3.11 and Fig3.13, code1.3 is used, and code 1.1 is used in Fig3.12. Data5 is encoded with complex coefficients and decoded separately at the receiver whereas data4 is encoded with complex coefficients and jointly decoded at receiver. We see that Both two schemes have the same diversity order and the LCF-STTC has better performance than conventional STTC because the LCF-STTC has joint processing gain at the receiver. The performances of above two schemes over frequency selective channels are shown in Fig 3.12 and Fig3.13.The complex code satisfied designed criterion can combined with channel thus the receiver will extract mutipath diversity gain. With joint processing at receiver, the LCF-STTC has better performance than conventional STTC. Fig3.14 shows that with turbo equalizer, the performance of conventional STTC approaches the LCF-STTC after 0,1,2 and 5 iterations over frequency selective channels.

The complex-value coefficients of encoder can combine multipath channels directly and joint decoding based on MLSE receiver is used to exploit the multipath diversity gain. It shows that the proposed codes obtain the spatial diversity provided by multiple antennas and the temporal diversity provided by multipath. The conventional STTC designed for flat fading scenarios are guaranteed to extract spatial diversity at least if used in a frequency selective environment.

3.4 Summary

Space time trellis coding has been proposed as an effective approach to support high data rate transmission over fading channels. It is shown that the LCF-STTC with joint decoding based on MLSE receiver has better performance than the conventional STTC defined on finite ring over flat fading channel or frequency selective channels. Space time codes defined on complex field can combine multipath channels thus the receiver will extract the path diversity gain. Particular attention has been paid on the analytical performance evaluation of space-time coding. One method is to compute the code distance spectrum and apply the union bound technique to calculate the average pairwise error probability [16]

[17]. A more accurate performance evaluation can be obtained with exact evaluation of

[17]. A more accurate performance evaluation can be obtained with exact evaluation of

在文檔中 在頻率選擇通道下定義在複數域的時空柵欄碼之研究 (頁 19-0)