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Cooperative Strategies for High SNR (Cooperative Diversity)

We consider two classes of strategies by distinguishing whether the sources transmit at the same time and in the same frequency band or not.1 Strategies for which the sources do not interfere are called orthogonal strategies. Such schemes are meant to achieve high diversity gains rather than high rates [112], although orthogonal strategies are good enough for low SNR (see Section 5.5). Non-orthogonal strategies use bandwidth more efficiently and thus generally exhibit better diversity-multiplexing tradeoffs [14, 139, 193]. For example, a cooperative scheme known as dynamic DF achieves the diversity-multiplexing function for low values of multiplexing gains [14]. For larger values, multipath DF (see Section 4.2.7) outperforms dynamic DF [11]. Interestingly, CF achieves the diversity multiplexing function for all multiplexing gains and for any number of antennas at the nodes [196], but recall that it needs extra CSI at the relay and destination.

1 The following material is based mostly on [112].

Fig. 5.3 Channel allocation for: (a) orthogonal direct transmission, (b) orthogonal relaying.

Direct Transmission

Consider the network in Figure 5.2 and the TDM strategy in Figure 5.3(a). This strategy is cooperative in the sense that the users do not interfere with each other, and the destination’s signals are

The outage probability for nodes u = 1,2 is (see (5.7))

where we remark that both nodes can send with twice their average power in their time slots. Observe that, up to a constant, the outage probabilities decrease as

Orthogonal Relaying Strategies

Consider next the cooperative strategy in Figure 5.3(b). During times i = 1,2,…,n/4, node 1 transmits and nodes 2 and 3 receive.

Consider next the cooperative strategy in Figure 5.3(b). During times i = 1,2,…,n/4, node 1 transmits and nodes 2 and 3 receive

During timesnode 2 transmits X2,j as a function ofNode 2 thus acts as a relay and the destination receives

During time i = n/2 + 1,n/2 + 2,… ,n, the roles of nodes 1 and 2 are reversed and the same procedure is repeated. We next consider several ways in which the relays process their received symbols.

Amplify-and-Forward

Suppose node 2 uses AF and forwards

while choosing a to satisfy its power constraint, i.e., we set (see (4.10))

where we recall thatand the relay sends at twice its average power in its time slot. To decode, the destination combines the received signals (5.31b) and (5.31c) using maximum-ratio combining which requires knowledge ofWith Gaussian codebooks, one achieves where we recall thatand the relay sends at twice its average power in its time slot. To decode, the destination combines the received signals (5.31b) and (5.31c) using maximum-ratio combining which requires knowledge ofWith Gaussian codebooks, one achieves

where the factor 1/4 is because nodes 1 and 2 transmit only 1/4 of the time for each message. The outage event I(7) < R is the same as

For high SNR, one can show that the outage probability is given by (see [112])

The diversity gain is thus d = 2, and this is the best possible. To see this, note that the outage probability must be larger than when the relay knows the source message ahead of time, effectively creating a MISO channel with n1 = 2 transmit antennas. We thus have d < 2 by applying (5.15).

Compress-and-Forward

Suppose the relay uses CF with the relay mode M2 acting as a timesharing random variable (see Sections 4.2.4 and 4.3). The rates are

whereWe choosewhen as in Section 4.2.4. The rate (5.37a) is a quarter of (4.28a) and

is the same as (4.28b) withWe

also have

We thus choose

to satisfy (5.37b) with equality. The resulting outage event is

which clearly implies the AF outage event (5.35). The diversity gain is thus again d = 2. Note, however, that the relay and destination need full CSIT and CSIR, respectively.

Decode-and-Forward

We first consider a suboptimal DF strategy that uses a repetition code at the relay, i.e., the relay decodes the source message, re-encodes using the same codebook, and transmits. For Gaussian codebooks, we compute (see (4.32))

The outage event < R is thus the same as

and we have

Observe that d = 1 and this DF strategy offers no diversity gain over non-cooperative communication. In fact, we obtain the same type of result if we use the usual DF strategy, where the only change is that the second logarithm in (5.41) increases to

and a similar change is made in (5.42). The deficiency in both DF strategies is that we force the relay to decode even if |H12|2 is small.

Selection Decode-and-Forward

To overcome the drawback of DF, we compare the measured channel gainat the relay with a pre-specified threshold. If is above the threshold, then the source remains silent and the relay forwards the information (either via a repetition code or another code). Otherwise, the source repeats its transmission. Observe that this method requires the relay to send one feedback bit to the source after every other transmission block. We call this a selection DF strategy.

We remark that the source must limit its transmit power to less than 2P in its time slots. However, if Po is small, as we shall assume, then the source rarely needs to transmit twice. Hence, we have the source transmit with power 2P in its first time slot and with a small amount of extra power in the second time slot if need arises (alternatively, the source never transmits in the second time slot).

Suppose the relay uses a repetition code. We compute

Observe that we chose the threshold so that the direct transmission is repeated if there is an outage on the source-relay link. The overall outage event iswhere

and so we have

which is identical to (5.36) except for the factor of 2 in front of This factor arises because A has twice the probability of B for large Other DF strategies achieve similar performance [82, 84, 86, 125, 126, 172, 173]. The results are summarized in Table 5.1 where we set for all

Table 5.1 Diversity gains of orthogonal cooperative strategies for

Cooperative	High SNR	Comment	Extra CSI
strategy	performance
Direct trans.		No diversity
DF		No diversity
Selection DF		Full diversity	1 bit feedback
AF		Full diversity
CF		Full diversity

Incremental Relaying

An inspection of the selection DF strategy reveals that if |H13|2/da3 is above the threshold g (7,2R)/2 then the relay need not decode the message. Instead, the source can improve its rate by sending fresh information in the next time slot (as in hybrid ARQ or incremental redundancy ARQ). The resulting strategy operates at source power P and rate R/2 when the destination reception is successful, and at source power P/2 and rate R/4 otherwise. Let

where = P/P1 and P1 is the power the source can use when transmitting, i.e., P1 satisfies

We wish to compute the expected rate

A complication arises in that several values of R can give the same R; we choose the smallest R that satisfies (5.50). One can show that incremental DF achieves full diversity, and that somewhat better results are possible with incremental AF (see [112, Sec. IV.E]). We remark, however, that this analysis implicitly requires using many fading blocks rather than just one.

Non-orthogonal Cooperative Strategies

The TDM constraint limits the rate more than necessary. We next consider non-orthogonal strategies that give good diversity multiplexing tradeoffs; for AF see also [139, 14, 193], for DF see [14], and for CF see [196].

Amplify-and-Forward

Consider two consecutive symbol transmissions from the source. During the first symbol transmission, the relay listens. In the next symbol period, the source transmits a new symbol and the relay uses AF. Thus, the relay effectively creates an intersymbol-interference channel (see Section 4.2.2). One can show that this AF strategy achieves the diversity-multiplexing pairs (d,r) satisfying (see [14])

whereThis strategy achieves a better diversity-multiplexing tradeoff than orthogonal AF. However, for this strategy is only as good as non-cooperative transmission because of the half-duplex constraint. This drawback is removed with the next strategy.

Dynamic Decode-and-Forward

Suppose the relay listens until it collects sufficient energy to decode the source message. It then re-encodes the message with its own codebook and transmits for the remaining time that the source transmits [93, 134, 136]. One can show that this dynamic DF scheme achieves the diversity-multiplexing pairs (d, r) satisfying

Observe that the diversity gain is the same as for asystem forunfortunately, the relay cannot help enough of the time to achieve the MISO upper bound.

Compress-and-Forward

Suppose the relay listens half the time and uses CF. One can show that CF achieves the MISO upper bound forso we have[196, Sec. VI]. In fact, CF achieves the MISO upper bound for any number of antennas at the source, relay, and destination nodes. Note again, however, that the relay and destination need full CSIT and CSIR, respectively. The results are summarized in Table 5.2 and the diversity-multiplexing pairs are plotted in Figure 5.4.

Table 5.2 Diversity-multiplexing tradeoff of non-orthogonal cooperative strategies.

Cooperative strategy			Extra CSI
No cooperation
Orthogonal AF			\|Hl2 \|²/d^a₂ to dest.
Orthogonal DF
Non-orthogonal AF
Dynamic DF			Relay wait time
CF
MISO bound			Message to relay.