Summary

A comparison of all transmission protocols mentioned in this chapter is shown in Fig. 2-6 to make the difference of spectral efficiency more clear. The spectral efficiency loss in conventional 2-user cooperation protocol is clear shown in the figure. It uses twice more time slots compared to other protocols. Coded cooperation uses just half of the time but still achieves same diversity as conventional cooperation.

For the proposed modification of coded cooperation protocols, we aim at the relay phase to achieve higher transmission reliability.

In this chapter we review the system model of conventional cooperation and coded cooperation under multi-user schemes. It is shown that by using coded cooperation with appropriate choice of codes, we can avoid the loss of spectral efficiency while still achieving full diversity. Then we point out the potential of extracting more benefits from coded cooperation by using more reliable transmission in relay phase. Two modified protocols which can acquire higher diversity order (>2) without losing spectral efficiency are introduced.

Fig. 2-6. Comparison of channel uses for cooperation protocols (TDMA) time

user1 user2 user3 user4

base codeword broadcast sub-codeword relay sub-codeword

direct transmission 2-user cooperation 2-user coded cooperation Proposed modification of

2-user coded cooperation

? ?

(Protocols)

Chapter 3

Spectrally Efficient Multi-user Coded Cooperation Using

Space-Time Code

In this Chapter we introduce the first modification of coded cooperation:

space-time (ST) coded cooperation. A detailed description of how this protocol works will be given. Performance bounds will be analyzed and showed along with simulation results. Cases when a user does not successfully decode the data from other users are also considered. We focus on 2-user coded cooperation with Alamouti space-time code for simplicity, but extension to more user cases with other types of space-time codes is straightforward.

3.1 Protocols of ST-Coded Cooperation

In coded cooperation, since all users know each other’s data after the broadcast phase, we can treat the users as “virtual antennas”. Since that, applying space-time code in relay phase to achieve transmit diversity is possible. In the following Sections we’ll give an example that shows how 2-user ST-coded cooperation works.

3.1.1 Code Structure and System Model

Since there are two users in the cooperation scheme, we can apply Alamouti code [2] in the transmission of relay phase, the protocol is shown in Fig. 3-1.

Fig. 3-1. Channel use of ST-coded cooperation with two users (TDMA) Assume the same quasi-static fading channel as in Chapter 2, that is, the fading coefficient remains constant during the transmission of the space-time code (one phase). A diversity gain of 2 is extracted by the use of Alamouti code in the relay phase, thus we can expect a total diversity order of 3 for each user data since the broadcast sub-codeword sees independent channel with that seen by relay sub-codeword.

In the broadcast phase, there’s no change to the transmission model comparing with (2.5), so it can be written as

In the relay phase, Alamouti space-time code is applied, the code matrix of the ith element of the sub-codeword is

user1

( )

( )

( )

Thus the received signal during relay phase is

( ) ( ) ( ) ( ) ( ) ( )

Define channel matrix of the relay phase as

( ) ( ) symbols of relay sub-codewords for both users.

Assuming ML detection, from (3.1) and (3.6) we can write the detected signals

for user U (_u ^u∈

{ }

^{1, 2} ) in broadcast phase as space-time block code (OSTBC) can be used in 4-user coded cooperation, the protocol is shown below:

Fig. 3-2. Channel use of ST-coded cooperation with four users (TDMA) The major advantage of ST-coded cooperation is that space-time code with higher diversity order can be applied when more users are involved. There is no loss of

relay phase

spectral efficiency as long as the space-time code used is of rate equal or higher than 1.

3.1.2 Case of Data Exchange Failure

For a wireless cooperative communication system, it is always possible that data exchanged between two users are corrupted due to deep fading and cannot be decoded successfully. We call this event data exchange failure.

The effect of data exchange failure to the coded cooperation can not be ignored.

When it happens, cooperation can not be done since users don’t know each other’s information. Both users have to transmit their own relay sub-codewords by themselves (no cooperation mode). More specifically, if user U₂ can’t decode the broadcast sub-codeword from U₁ correctly, it will notify U₁ by one bit of information to let U₁ transmit its relay sub-codeword itself. Meanwhile, U₂ will also transmit its relay sub-codeword itself. Cyclic redundancy check (CRC) can be used for error detection.

Consider a 4-user scheme using a 4 4× OSTBC. Note that every user has to know the data of all other users, that is, all users have to successfully decode the data from other users in order to perform cooperation. That gives rather high requirements for inter-user channels. The effect of data exchange failure to the overall performance will be analyzed in next Section.

3.2 Performance Bounds of ST-Coded Cooperation

We present an analytical methodology in this Section for evaluating performance

of the proposed Alamouti ST-coded cooperation protocol. Only the performance of user U₁ data is considered since the error rates of the two user data are statistically equal. The pairwise error probability is calculated using the technique from Simon and Alouini [22], then we determine the union bounds for the overall bit error rate (BER) and frame error rate (FER) using weight enumerating function and the tools given by E. Malkamäki and H. Leib [25]. These bounds will be shown and compared with the simulation result.

3.2.1 Pairwise Error Probability

Since we focus on the performance of single user throughout this Section, some notations in the equations can be simplified. Thus we can rewrite the detected signal for U₁ in broadcast phase as (from (3.7))

where

i = " 1, , N

_r. Sub-codewords are rearranged according to the puncturing pattern and are combined at the destination to rebuild the base codeword. The base codeword can be represented as

( )

respectively. It is straightforward to find that

Since ML detection is used at the receiver, the corresponding symbol error probability is given by [23] separation of the underlying scalar constellation. Since BPSK modulation is assumed, we have ⁱNe =1 and d_min² = 4

Define the transmitted base codeword as x=

[

x x1, 2,",x_N

]

, the probability that

z

i is decided as an erroneous symbol l

i i

x ≠ x conditioned on known channel

( ) ( )

Base on above equation, we have the conditioned pairwise error probability:

(



)

^{1( ) 2 ( ) 2}

that l

i i

x ≠ x . Thus the size of

η

_b are equal to the Hamming distance between the broadcast sub-codeword in

x

and x. In the same way, the size of

η

_r are equal to

the Hamming distance between the relay sub-codeword in

x

^and x. Eq. (3.15) can be further simplified to

Since Q function can be replaced by exponential form:

( )

0^{2 2}2

Hence, the pairwise error probability is

(



)

0^{2 1( )2 2 2 ( )2 2}

The overbar denotes statistical averaging over the random variable. Since channels between the destination and the users are assumed independent, the averaging can be performed separately, thus

(



)

^{( )}

Define a random variable α that is statistically equal to the channel coefficient

( )

In Rayleigh fading channel, the PDF of instantaneous SNR per bit is

( )

^1exp , 0

Substituting (3.25) into (3.23), we have

2 1

Substituting (3.26), (3.27) into (3.20) and note that the average SNR in different channels may not be the same, thus we separate it by γ₁^{( )b} ,γ^{( )}₁^r ,γ^{( )}₂^r , which represent the average SNR between user U and destination in broadcast phase, and the ₁ average SNR between U U and destination in relay phase, respectively. Thus we ₁, ₂ have is, the Hamming distances between broadcast, relay sub-codewords and the transmitted sub-codewords are not zero. The diversity is gained by the use of Alamouti space-time code in the relay sub-codeword, and by the independent channel seen in broadcast phase.

3.2.2 Bit and Block Error Rate

A union bound for the BER and FER can be calculated using weight enumerating functions. In traditional approach [24], the first step is finding the first-error-event probability. Assume all-zero path is the correct path, we want to find the probability that a path through the trellis with Hamming distance d from the all-zero path is the survivor. The second step is taking the summation of the first-error-event probabilities

over all possible Hamming distances. Note the Hamming distance between a path and all-zero path is also the weight of that path. Recall that (3.28) is a function of Hamming distances d d and average SNRs _b, _r γ₁^{( )b} ,γ₁^{( )r} ,γ^{( )}₂^r . Now we assume the SNRs are given, (3.28) can be rewritten as a pairwise error probability function of only the Hamming distances

( ) (

^

)

2 _b, _r ,

P d d =P x x x γ≠ (3.29)

where γ is a SNR vector with elements γ₁^{( )b} ,γ^{( )}₁^r and γ^{( )}₂^r . Denoting the number of paths that the broadcast and relay sub-codewords have weights d d by _b, _r a d d

(

b^, r

)

, we can bound the first-error-event probability by

( ) ( )

where on the right-hand side, we have included all paths through the trellis that merge with the all-zero path.

To calculate BER, first define b d d

(

b^, r

)

as the total number of bit errors in paths that the broadcast and relay sub-codewords have weights d d , then the union _b, _r bound of BER is

However, it is shown in [25] that the union bound approach was found to provide quite loose bounds. This is because there is no dominant error event in (3.31), even at high SNR region. Therefore, a modification to this method is proposed by [25] to obtain a much tighter bound. It is done by limiting the conditioned union bound on the bit error probability before averaging over the fading matrix. Thus we need the pairwise error probability conditioned on a given channel, which is given in (3.18).

Rewrite (3.18) as

( )

^{2 12 22 32} obtain the bound of bit error probability which is a function of known channel coefficients:

The bit error probability is limited to 1/2 because in practice, the maximum error rate using Viterbi decoder is 1/2. By averaging over the channel coefficients, the new bound can be obtained:

( ) ( ) ( ) ( )

A much tighter bound can be obtained this way, but the order of the integrations and the summations can not be exchanged in (3.34) due to the min operator, thus numerical integral is needed to calculate the results. The bounds will be compared with simulation results in Section 3.3.

Frame error rate can be evaluated in similar way. For a data block of K bits, given the event error probabilityP , the FER can be bounded as [26] _e

( )

1 1 ^K

f e e

P ≤ − −P ≤KP (3.35)

The traditional approach gives loose bound for the FER, thus the limit-before-averaging technique is applied. We first calculate the bound of event error probability conditioned on given channel coefficients:

(

1, 2, 3

)

min 1,

(

,

) (

2 1, 2, 3, ,

)

e b r b r

P α α α ≤ ^⎡⎢

∑ ∑

^{∞ ∞} a d d P α α α d d ^⎤⎥ ^(3.36)

This time the event error rate is limited to 1 because the maximum error rate is 1 in practice. Apply (3.36) to (3.35) and average over the channel coefficients we get

( )

Again, the above equation needs to be carried out numerically in this case due to the min operator.

3.2.3 Impact of Data Exchange Failure

Now consider the impact to the performance due to data exchange failure. We focus on 2-user case first. Since the transmissions from user U to ₁ U and from ₂ U to 2 U are on the same frequency band and the same coherence interval, we ₁ assume they see equal channels due to the reciprocity theorem [27]. Denoting P_{f u,} as the rate of the data exchange failure between U and ₁ U , the probability of ₂ successfully cooperation between two users can be written as

1 ,

C f u

P = −P (3.38)

Denote the FER of 2-user cooperation with no data exchange failure as P_f,2user and the FER which data exchange always fails as P_{f nocoop,} . We have shown how to calculate P_f,2user in previous two Sections. The calculation of P_{f nocoop,} can be done

the frame error rate will be dominated by P_{f nocoop,} and (3.39) can be bounded as

, ,

f f nocoop f u

P >P P (3.40)

It can be seen that the diversity gain decreases at high SNR region.

Now consider a 4-user scheme. Note that all users have to decode data from others correctly to apply 4 4× OSTBC. Assuming the rates of data exchange failure are equal between all users, the probability of successfully cooperation can be written as

( )

⁶

,4 1 ,

C f u

P = −P (3.41)

Thus we have the overall FER in 4-user case as

( )

where P_f,4user is the FER with 4-user coded cooperation and no data exchange failure.

The overall FER can be bounded as follow in high SNR region

( )

(

⁶

)

, 1 1 ,

f f nocoop f u

P >P − −P (3.43)

Again, we can expect the loss in diversity order at high SNR region, what’s more, note that

(

^{1− −}

(

^{1 Pf u,}

)

⁶

)

is much higher than P_{f u,} , the degradation of performance will be larger compared to 2-user case. It will be shown in Fig. 3-4 in next Section.

To this problem, an adaptive scheme can be used to compensate the huge performance loss in 4-user case. It is unnecessary to make all users back to no cooperation mode if some of them still received others’ data correctly. Instead, other space-time codes can be applied to the users who is capable for cooperation. For example, if there are only two users successfully exchanged information with each other, an Alamouti space-time code is applied instead of 4 4× OSTBC; if there are

Alamouti code and 4 4× OSTBC in the adaptive scheme, the probability of frame error becomes

,4 ,4 ,2 ,2 , ,0

f f user C f user C f nocoop C

P =P P +P P +P P (3.44)

where P is the probability that all four users exchanged information successfully, _C,4 which is equal to (3.41); P is the probability of any user successfully exchanged _C,2 data with U ; ₁ P is the probability of no user successfully exchanged data with _C,0

U . They are 1

respectively. Thus (3.44) can be written as

( )

⁶

( ( )

^{6 3}

)

³

,4 1 , ,2 1 1 , , , ,

f f user f u f user f u f u f nocoop f u

P =P −P +P − −P −P +P P (3.46)

The FER at high SNR region can be bounded as

3

, ,

f f nocoop f u

P ≈P P (3.47)

Note the difference between (3.43) and (3.47). The probability that users go back to no cooperation mode is much lower comparing to the probability in non-adaptive scheme, thus provides a better performance. Simulations of the adaptive scheme will be presented in Fig. 3-5 in next Section.

3.3 Computer Simulations

In this Section we simulate the proposed space-time coded cooperation protocols and compare with other protocols mentioned in Chapter 2. All systems are with equal code rate R and hence equal data rate. We use a rate-1/4 base code with generator [15

17 13 15] used by [18]. For the conventional coded cooperation and the proposed ST-coded cooperation, the puncture patterns are [1 1 0 0] for the broadcast sub-codeword, [0 0 1 1] for the relay sub-codeword. Binary phase-shift keying (BPSK) modulation is used. The frame size is 260 bits. We consider the case that both nodes communicate with the same destination. Each user and the destination are equipped with a single antenna. The channel is slow Rayleigh fading channel with AWGN.

Fig. 3-3 plots the analytical bound and simulation results of FER as a function of the transmit SNR. Similar results could be obtained in terms of BER. Perfect inter-user channel is assumed so there is no data exchange failure. Investigating the FER at high SNR region and comparing with the analytical bounds (dash line with diamonds), it is clear that the proposed ST-coded cooperation (line with diamonds) achieves full diversity as we expected in Section 3.2. Comparing the performance of the proposed protocol with conventional coded cooperation (line with squares) at FER of 10⁻³, the proposed ST-coded cooperation using Alamouti code achieves nearly 4dB gain over the conventional coded cooperation. If more users join the cooperation, higher order space-time code can be used to gain more diversity. For the case of four users (line with down triangles), additional gain of 2.5dB is achieved comparing to 2-user case.

0 2 4 6 8 10 12 14 16 18 20

Conventional coded cooperation (union bound) Conventional coded cooperation

Proposed ST-coded (Alamouti)(union bound) Proposed ST-coded (Alamouti)

Proposed ST-coded (4x4 OSTBC)(union bound) Proposed ST-coded (4x4 OSTBC)

Fig. 3-3. Simulations and bounds of frame error rate (FER) in ST-coded cooperation. Equal uplink SNR, base code [15 17 13 15]

Fig. 3-4 shows the impact of data exchange failure to the overall FER. We assume that the rate of data exchange failure between users is 0.1. Analytical bounds based on Section 3.2.3 are shown in the figure. It can be seen that the simulation result is consistent with the analytical bound.

Lines with diamonds and down triangles are the same as the simulation results in Fig. 3-3, that is, ST-coded cooperation with no data exchange failure. Diamonds and down triangles with no lines are the simulation result when data exchange failure is considered. From the figure we can see that the proposed ST-coded cooperation protocols lose their diversity at high SNR region. At that region, they seem to have diversity of order 2 because the two sub-codewords still experience independent

channels in broadcast and relay phase.

Despite the loss in diversity, the proposed 2-user ST-coded cooperation still has advantages over the conventional one. But the performance of 4-user scheme is even worse than 2-user case. As mentioned in Section 3.2.3, cooperation among four users with imperfect inter-user channel experiences severe performance degradation since the data exchange between 4 users is hardly all successful. To this problem, we’ll show in Fig. 3-5 the performance improvements of 4-user cooperation with adaptive protocol.

Conventional coded cooperation (failure rate=0.1) ST-coded (4x4 OSTBC) (failure rate=0.1) (union bound)

ST-coded (4x4 OSTBC) (failure rate=0.1) ST-coded (Alamouti) (failure rate=0.1) (union bound)

ST-coded (Alamouti) (failure rate=0.1) ST-coded (Alamouti) (perfect cooperation) ST-coded (4x4 OSTBC) (perfect cooperation)

Fig. 3-4. Frame error rate (FER) with imperfect inter-user channels. Equal uplink SNR, generator [15 17 13 15], inter-user FER=0.1

Comparing the performance of 4-user case (down triangles) in Fig. 3-4 and Fig.

3-5, we can see clearly the improvement by applying adaptive protocol mentioned in

Section 3.2.3. The algorithm assures the performance under 4-user scheme not worse than the case when only 2-user Alamouti code is applied. Meanwhile, it still benefits from the use of 4 4× OSTBC when the data is exchanged successfully. For the case of data exchange failure rate=0.1, the average probability of 4-user cooperation is about 0.531, 2-user cooperation is about 0.468 and the probability of no cooperation is only 0.001.

Conventional coded cooperation (inter-user FER=0.1) ST-coded (Alamouti) (failure rate=0.1)

ST-coded (Alamouti) (perfect cooperation) ST-coded (4x4 OSTBC) (failure rate=0.1) (union bound)

ST-coded (4x4 OSTBC) (failure rate=0.1) ST-coded (4x4 OSTBC) (perfect cooperation)

Fig. 3-5. Frame error rate (FER) with imperfect inter-user channels. Equal uplink SNR, generator [15 17 13 15], inter-user FER=0.1, adaptive algorithm

3.4 Summary

In this Chapter we give a detailed description of the proposed space-time coded

cooperation protocol. We show the diversity gain in the case of 2-user coded cooperation with Alamouti space-time code by evaluating the pairwise error probability. Extension to other space-time code is straightforward. The proposed protocol can utilizes full diversity gain from the used space-time code. Tight union bounds for the BER as well as the FER are given by using the weight enumerating function and the limit-before-averaging technique. Both analytical and simulation results have been shown to prove the performance gain. We also consider the impact of imperfect inter-user channel to the proposed protocol and give an adaptive way to reduce the performance loss.

Chapter 4

Spectrally Efficient Multi-user Coded Cooperation using Code Partitioning

In this Chapter we introduce the second modification of coded cooperation:

code-partition (CP) coded cooperation. The proposed protocols still achieve great system reliability while maintaining equal spectral efficiency as non-cooperation protocols. The CP-coded cooperation has similar performance to the protocols in Chapter 3. Besides, it has the advantages of less complexity and lower requirements for inter-user channel.

4.1 Protocols of CP-Coded Cooperation

Look at the performance bound in eq. (3.28), we can see that the diversity gain comes from the independent channels of user U in the broadcast phase and relay ₁ phase. Further more, additional diversity is gained from U channel by using ₂ Alamouti space-time code in relay phase. Thus the total diversity gain compared to direct transmission protocol is 2 1 3+ = . There is another factor in (3.28) that contributes to the overall performance: the Hamming weight of the sub-codewords

在文檔中 應用於多用戶編碼協力式通訊之高頻譜效率空時協定 (頁 25-0)