In Chapter 3, the BER performance and diversity order of a cooperative BICM networks are investigated for the S-DF/RT and S-DF/Idle relaying over fast-fading Nakagami-m channels.
Unlike most of the existing works, this chapter considers a packet-by-packet forwarding strategy.
For the BER analysis, the BER based on a given set of active relays is approximated based on extending the expurgating bound proposed in [47]. But, instead of using a Chernoff bound (as in [47]) to evaluate it, a close-form evaluation is proposed and verified to be very accurate in nu-merical results. Based on this approximation and together with a BER-to-PER approximation, approximate the BER at the destination can be obtained. The diversity of the cooperative BICM network is also provided for both S-DF/RT and S-DF/Idle in fast Nakagami-m fading channels through deriving the asymptotic upper and lower bounds of the BER at destination. Numerical results show that, though the BER-to-PER approximation may not be accurate, it does not seri-ously degrade our approximations, especially in some typical case where the S-R links are better than the other links.
Chapter 4
Performance and Diversity Anal-ysis in Block-fading Channels
This chapter proposes the BER analyses and the diversity derivations for BICM-coded co-operative networks with the S-DF/RT and S-DF/Idle relaying over block-fading Nakagami-m channels. Different from the previous chapter, in block-fading environments, the channel gains are constant during the transmission of a packet. Thus, the time index k will be dropped for simplicity. In what follows, the BER analyses are presented first, followed by the diversity anal-yses.
4.1 BER Analysis
In this section, the BER at the destination for BICM-coded cooperative network with the S-DF/RT and S-DF/Idle relaying over block-fading Nakagami-m channels is discussed. Different from the results in fast-fading channels, an effective close-form approximation in block-fading channels is difficult to obtain. As a result, a semi-analytic method is provided.
4.1.1 S-DF/RT
At the destination, the average BER is
realiza-tions, followed by averaging the result w.r.t. the realizarealiza-tions, e.g.,
1
With given the channel realization hj R, 1, hj R, 1
x z, low-to-moderate SNRs. As a result, the expurgation proposed in [47] is adopted to provide a more accurate approximation, i.e.,
0
where the values M , C and i D are listed in Table I for QPSK, 16-QAM and 64-QAM with i Gray mappings.
In [47], (4.9) is evaluated numerically for AWGN channels. However, the approximation in AWGN channels is still loose at low SNRs. In fact, it is exponentially increasing as SNR de-creases, although the actual BER is upper bounded by 0.5. Therefore, when averaging the BER over channel realizations (as in (4.2)), such a loss could leads to a non-trivial gap. To mitigate this gap, some modification is required. The modification adopted is to limit the BER at AWGN such a modification prohibits an analytic expression. As a result, channel averaging is carried out by the Monte Carlo method which evaluates
1
interpolation. Such a table can be pre-built to provide the mapping from the SNR to the resulting BER at AWGN channels, e.g. from E Nb 0 0 to 30 1 with grid spacing 0.01. Then, given theTo obtain an approximation for the PER at relay, i.e., pˆf j, , a commonly-used approxima-tion is adopted to obtain the PER with given the channel state h0, j, i.e.,
1 The maximum Eb N030 is enough for 16-QAM to provide BER lower than 1027 in AWGN channels.
ob-tained according to the results for AWGN channels in [47], or by through a look-up table as de-scribed early.However, a close-form expression of (4.14) is prohibited, and Monte Carlo method is used again to numerically evaluate this expectation. Bringing (4.11) and (4.14) back to (4.1), the average PER at destination for S-DF/RT is obtained. Numerical results will be provided in Section 4.3.
4.1.2 S-DF/Idle
same as those in S-DF/RT so that they can be directly obtained through (4.12)-(4.14). The only difference is pb RIdle, 1
, in which the orthogonal-channels of inactive relays are not used atphase-II.
where the second line is obtained similarly to that for (4.9). Note that the product in (4.17) con-tains only the MGFs corresponding to active relays (and source). Through either a direct numeric integration or a table looking up, pˆb RIdle, 1
can be obtained. Bringing (4.14) and (4.15) back to (4.15), the average PER at destination for S-DF/Idle is obtained. Numerical results will be pro-vided in Section 4.3.4.2 Diversity Analysis
This subsection provides the proof of the diversity orders of the considered BICM systems that characterizes how the average BER behaves with large SNRs. Specifically, upper and lower bounds of BER are derived first, followed by showing that both bounds achieve the same diver-sity order. The diverdiver-sity orders for the S-DF/RT and S-DF/Idle relaying schemes will be denoted by DivRT and DivIdle, respectively. For simplicity, an equal gain power allocation among source and all relays is assumed, i.e., P0 P01 PRP.
4.2.1 S-DF/RT
According to Lemma-1, the expectation in (4.20) is evaluated as
, 1
It is important to note that, in block fading channels, the SNR is multiplied by free distance d , f while in fast-fading case, d appears in the exponent (see Section 3.2). Now, (4.20) becomes f
Furthermore, using Lemma-2, (4.22) is approximated at high SNRs by
squared Euclidean distance D, and the other rows are obtained similar to (4.20).Since the remaining integration in (4.31) is exactly the same with that in (4.20), we have
Denote the diversity order of pb RRT, 1 by DivRT. Similarly, at extremely high SNRs, the
summa-tion is dominated by the terms with the smallest exponent. It turns out that DivRT=DivRT. Since
RT RT RT
, 1 , 1 , 1
b R b R b R
p p p at all ranges of SNR, this implies DivRT DivRT DivRT. Therefore, it
can be concluded that DivRT=DivRT=DivRT, as provided in (4.29). Compared with (3.39), the diversity in fast-fading channel is just d times of that in the block-fading channels. The reason f is rather straightforward. Since all coded bits in a packet suffer the same channel realization, the diversity does not increase with the free distance.
4.2.2 S-DF/Idle
For S-DF/Idle relaying scheme, the BER at destination is first approximated by
Since the upper and lower bounds of pf j, have been provided in the previous subsection. This subsection focuses on pb RIdle, 1
.The derivation of the upper bound of pb RIdle, 1
is very similar to that of pb RRT, 1
. The only difference comes from the fact that the orthogonal-channels of inactive relays are not used for S-DF/Idle relaying scheme. Following the same steps in (4.19)-(4.24), pb RIdle, 1
is upperwhere , which contains all terms that are independent to P N , is slightly different from that 0
in Section 3.2. Now pb RIdle, 1 is upper bounded by
4.3 Numerical Results
This section verifies the numerical results for the BER approximations in Section 4.1 and diversity order proposed in Section 4.2. In the following simulations, a half-rate convolutional code CC(171,133) is considered, the interleaver is S-random with length 1024 and depth 20, and the modulation is Gray-mapped 16-QAM. For simplicity, let P0 P1 PR P (orthogonal channels in the time domain) with PE Rb Cl where E is the bit energy, and b RC 0.5 is the channel code rate.
We first verify the proposed approximations in (4.11), (4.14) and
For simplicity, consider a 3-node network with the setups shown as Network-1 in Table II. In this case, the BER at the destination is (from (4.1))
The simulation results and approximations are plotted in Fig. 4.1 for pbRT,2
1 ,
RT ,2
pb ,pbIdle,2
and in Fig. 4.2 for p and b,1 pf,1. As can be seen in Fig. 4.1, all approxima-tions provide very good predicapproxima-tions of the real BERs. Unfortunately, in Fig. 4.2, the PER ap-proximation pˆf,1, according (4.12), over-estimates pf,1, but p obtained by (4.40) is rather ˆb,1accurate in predicting p . Thus, (4.12) is the main reason that leads to the gap on predicting b,1 PER.
The over-estimation on PER could become negligible when the S-R link is better than the other links. This is the typical case of relay network, e.g., when LOS is possible for S-R links.
Consider Network-2 with setups shown in Table II with n1, 2 and 4 to investigate the ef-fect of different S-R link qualities to our approximations and plotted the results in Fig. 4.3 and Fig. 4.4 for S-DF/Idle and S-DF/RT relaying MODES, respectively. In Fig. 4.3, non-trivial gaps (about 1 dB) are observed between the approximations (App.) and the simulation results (Sim.) for both n1 and 2. For n4, the approximation matches the simulation results with a smaller gap. This is due to the fact that, when the S-R link is better than other links, pf,1 is low enough such that pbIdle,2
pf,1 is relatively smaller than the pbIdle,2
1 and that (4.41) is dominate by only pbIdle,2
1 , which can be well approximated. For S-DF/RT in Fig. 4.4, a 0.5 dB gap is observed only for n1, while approximations for n2 and 4 are very close to the sim-ulations. The approximations for S-DF/RT become more accurate, because pbRT,2
is smaller than pbIdle,2
such that pbRT,2
1 is more likely to dominate the error performance at destina-tion.-5 0 5 10 15 20 25 30 35
Fig. 4.1. Destination BER simulation results and approximations for Network-1
-5 0 5 10 15
Fig. 4.2. Relay BER/PER simulation results and approximations for Network-1
-5 0 5 10 15 20 25 30
Fig. 4.3. BER simulations and approximations of Network-2 with the values of n1, 2 and 4 for S-DF/Idle
Fig. 4.4. BER simulations and approximations of Network-2 with the values of n1, 2 and 4 for S-DF/RT
The results for different relay number R1, 2, 3 and 4 is provided for S-DF/RT and S-DF/Idle in Fig. 4.5 and Fig. 4.6, respectively. Fig. 4.5 considers Network-4 in Table II, and Fig.
4.6 provides another results for Rayleigh fading channels, cf. Network-5 in Table II. As is seen from both figures, the proposed approximations are very close to the simulation results for all R at BER of 105.
-10 -5 0 5 10 15
10-6 10-5 10-4 10-3 10-2 10-1 100
Eb/N0
BER
Sim., R=1 App., R=1 Sim., R=2 App., R=2 Sim., R=3 App., R=3 Sim., R=4 App., R=4
Fig. 4.5. BER simulations and approximations of Network-4 with S-DF/RT for R1, 2, 3 and 4 and i.i.d channel conditions (m2 and 2for all links)
Table III. Network setups
Networks S-D link S-R link R-D link DivRT DivIdle Network-6 m0,2 0,2 1 m0,1 0,1 3 m1,2 1,2 1 2 2 Network-7 m0,2 0,2 1 m0,1 0,1 1 m1,2 1,2 3 3 2 Network-8 m0,2 0,2 1 m0,1 0,1 2 m1,2 1,2 2 3 3
-5 0 5 10 15 20 25 30
Fig. 4.6. BER simulations and approximations of Network-5 with S-DF/Idle for R1, 2, 3 and 4 over Rayleigh fading channels
To verify the diversity results in Section 4.2, we now compare the PER performance of 3 different networks (Network-6, 7 and 8 in Table III) with the channel statistics. All networks have 1 relay and are with m0,2 0,2 1 on the source-to-destination link. Network-6 has a better source-to-relay link (m0,1 0,1 3) , Network-7 has a better relay-to-destination link
1,2 1,2
(m 3), and Network-8 has (m0,1 0,1 m1,2 1,2 2). The PER performances for S-DF/RT are plotted in Fig. 4.7, wherein the PER curves of Network-7 and 8 decrease with the same slope which is steeper than that of Network-6 at high SNRs. This coincides the diversity orders shown in Table III which are calculated according to (4.29). The diversity orders for S-DF/Idle are also provided in Table III according to (4.39) and the verification through simula-tions in Fig. 4.8. It can be clearly seen in Fig. 4.8, Network-8 with diversity order 3 outperforms Network-6 and 7 with diversity order 2 at high SNRs.
-5 0 5 10 15 20 10-6
10-5 10-4 10-3 10-2 10-1 100
Eb/N0
BER
Sim., BER at dest., Network-6 Sim., BER at dest., Network-7 Sim., BER at dest., Network-8
Fig. 4.7. PER performance of Network-6, 7 and 8 in Table III with S-DF/RT
-5 0 5 10 15 20
10-6 10-5 10-4 10-3 10-2 10-1 100
Eb/N0
BER
Sim., BER at dest., Network-6 Sim., BER at dest., Network-7 Sim., BER at dest., Network-8
Fig. 4.8. PER performance of Network-6, 7 and 8 in Table III with S-DF/Idle
4.4 Summary
In Chapter 4, the BER performance and diversity order of BICM-coded cooperative net-works is investigated for the S-DF/RT and S-DF/Idle relaying over block-fading Nakagami-m channels. Unlike most of the existing works, this chapter considers a packet-by-packet forward-ing strategy. For the BER analysis, the BER over block-fadforward-ing channels can be obtained by first obtaining the BER in AWGN channels and then averaging it over channel realizations. Unfortu-nately, a direct integration may introduce non-trivial gap to the exact performance. To overcome this, a modification on the AWGN BER is adopted. But such a modification prohibits a close-form solution. As a result, the Monte Carlo method is used for the channel averaging. The diversity of the cooperative BICM network is also derived for both S-DF/RT and S-DF/Idle in fast Nakagami-m fading channels. The ideas of derivations are the same as those in 3.2, though the details are different. Numerical results show that our approximations are rather accurate for different network setups. An example is also provided to verify our proof of the diversity order.
Chapter 5
Power Allocation
The objective of this Chapter is to determine the transmit power allocation
P P0, , ..., 1 PR
T
P that minimizes BER at the destination under the sum power constraint
0 R
j T
j P P
2. An effective power allocation between transmit nodes could significantly lower the error rate at the destination so as to reduce the probability of re-transmission via (H-)ARQ (which requires additional power/energy and radio resources). Moreover, a good power allocation should also involve relay selection as well, e.g., allocate zero power to the useless relays.This chapter assumes a slowly fading environment so that the channels remain constants for several following frames and assume that full channel state information (CSI) is available. For simplicity, the time index k will be dropped in this chapter. Since every link remains unchanged over the transmission of a packet, it can be treated as an AWGN (additive white Gaussian noise) channel from the power allocation perspective. Note that power allocation can be done either at source or destination depending on the required signaling overhead and where the complexity of
2 The sum power constraint can also be interpreted as a sum energy constraint.
power allocation is to be placed. From the signaling overhead aspect, allocation at destination seems favorable because only the CSIs of the source-to-relay links have to be reported to the des-tination.
In the following sections, power allocation for the AF relaying is firstly discussed, followed by those for the S-DF relaying modes (RT, Idle and AF).