Consider a system with three cooperating terminals, source, relay, and destination, disposed according to Figure 1. A normalized unity distance has been assumed between source and destination, with the relay at an intermediate distance δ from the source. The relaydestination distance is (1  δ). The channel noise is considered to be AWGN with variance N_{0}/2 per dimension. A longterm quasistatic Rayleigh fading channel [23] is considered, which means that channel coefficients are constant during the retransmissions, changing only for a new block of data. We assume perfect CSI at the receivers.^{a} Due to delay constraints, the system operates within a truncated HARQ assumption, in which only a finite number M of retransmissions is allowed. Moreover, motivated by their practical feasibility, we assume halfduplex nodes operating in an orthogonal fashion.
In the following, we characterize the system in terms of the outage probability and throughput. Sections 2A and 2B describe two known transmission schemes for the above scenario: the direct transmission and the regular cooperative PreHARQ transmission, respectively. These schemes will serve as a benchmark to the proposed scheme, which is presented in Section 2C.
A. Direct transmission
In the direct transmission, the received signal at the destination is:
where P is the transmit power, h_{sd} represents the Rayleigh fading coefficient of the SD link, X is the message to be transmitted representing the information bits u, and w_{sd} is additive noise.
An outage in the direct transmission occurs when I_{sd} < R, where I_{sd} is the mutual information in the SD link and R is the information transmission rate. Supposing real Gaussian inputs and unitary bandwidth, the mutual information can be written as [24]:
where N is the noise power and E_{
b
} is the received energy per information bit, such that P = RE_{
b
} .
The outage probability in the SD link is then:
where is the probability of event ϕ.
We assume that an error free feedback channel between the destination and the other nodes is available. Therefore, after the first source transmission, the destination estimates û from the received code word y_{sd}, detects possible errors, and responds with an ACK/NACK depending on the error verification. If an ACK is received, then the source proceeds with the transmission of the next packet. Instead, in case of a NACK, the retransmission process begins. After each retransmission, the destination applies chase combining, or equivalently maximal ratio combining (MRC), among the received packets. The mutual information in this case, in longterm quasistatic fading and after m retransmissions, is:
where the term (m + 1) is due to the chase combining of each retransmission at the destination, which in fact is seen as an SNR accumulation [21]. The associated outage probability is:
Since we consider the existence of a feedback channel and message retransmission, an important performance metric is the system throughput, which is the effective spectral efficiency seen by the receiver. Based on the outage probability of each transmission, , we define the throughput of the direct communication scheme for at most M retransmissions as:
The first line in Equation 6 represents the case when the transmission from the source was already correctly decoded by the destination, then no retransmissions are necessary. In this case, the throughput will be R, with probability . The following lines represent the contribution of each HARQ round. For instance, considering the first retransmission, the throughput expression must take into account that an error already occurred in the first transmission, which happens with probability . Moreover, since we consider slowfading, the occurrence of a successful retransmission is not independent of an outage in the first transmission, so that a retransmitted packet is correctly decoded with probability: . Finally, for the next retransmissions, the same idea is applied, but the number of terms increases since all the previous erroneous transmissions must be taken into account.
However,
since I_{dir}(m) ≥ I_{dir}(m  1) ≥ ⋯ ≥ I_{
dir
} (0).
Moreover, the conditional probability:
Thus, a closedform expression for the throughput of the direct transmission is given by:
B. Regular cooperative PreHARQ
In the orthogonal cooperative transmission, the communication is carried out in two different time slots. In the first time slot, the transmission from the source is received at both the relay and destination. At the destination, the received packet is described in Equation 1 and the mutual information in Equation 2. At the relay, the received packet is:
where γ_{sr} = δ^{α}, such that α is the path loss exponent, h_{sr} is the channel fading, and w_{sr} is the additive noise vector at the relay.
The mutual information in the SR link is:
while the outage probability is:
Similar to the direct transmission, after the first source transmission, the destination estimates û from the received code word y_{sd}, detects possible errors, and responds with an ACK/NACK using the feedback channel. However, in the case of an NACK, the retransmission process may involve the relay. According to the precooperative ARQ strategy in [18], the retransmissions come from the node with the best nodedestination channel condition, be it either the source or the relay. Since in our topology, the relay is usually with a better channel condition with respect to the destination, due to the shorter distance in the RD link than in the SD link, we assume that all the retransmissions come from the relay, unless when the relay was not able to decode the source message.
In case of a relay retransmission, the signal received at the destination is:
where γ_{rd} = (1  δ)^{α}, h_{rd} is the channel fading coefficient, and w_{rd} is additive noise.
After chase combining of m retransmissions coming from the relay, the mutual information in the regular cooperative scheme equals:
The outage probability is:
The overall outage probability of the PreHARQ scheme can then be written as:
Finally, the throughput of the regular cooperative PreHARQ for a maximum of M retransmissions is:
1) PreHARQ with multiple relays: Now assume that instead of a single relay, a set of K ∈ ℤ^{+} relays is available. Thus, we define the set of relaydestination links as , where is the fading coefficient between the i th relay and the destination. In the case of multiple relays, a controlling protocol must be implemented to coordinate which node will retransmit. As we consider the decodeandforward protocol, this decision of the best node must take into account all the relaydestination links among those relays which were capable of correctly receiving the source message. Therefore, we define the subset , where is the mutual information between the source and the i th relay.
For the sake of simplicity, we assume that all the relays are approximately at the same distance δ from the source, such that . Therefore, we can assume that the outage probability between the source and any of the K relays is given by Equation 11. In the case of a relay retransmission, we assume perfect relay selection. Then, the mutual information in this case is given by:
such that the outage probability is given by .
Then, the throughput of PreHARQ with K relays for a maximum of M retransmissions can be generalized to:
C. Cooperative partial retransmission
The proposed CPR scheme is based on the partial retransmission of the original packet. The packet sent by the source is divided into L ∈ ℤ^{+} equal length subpackets for retransmission. The subpackets can be constructed in practice by puncturing the originally transmitted code word. The method is based on a scheme proposed by the authors for the noncooperative AWGN channel [25].
The proposed scheme can be summarized as:

1)
In the first transmission, the source operates as usual, transmitting the whole packet;

2)
If a retransmission is required, then the relay sends only the first fraction of symbols pertaining to that packet. If the relay was not able to decode the source message, then the source retransmits as usual, the entire packet;

3)
If more retransmissions are required, the relay sends the next fraction. If the L th fraction of symbols was already sent, the method repeats from step 2.
Note that, in the above method, the length of the retransmitted fractions is constant. Since the retransmissions from the relay are always shorter than in the regular cooperative scheme, it is reasonable to expect that this would reflect in an improved throughput. In addition, it is also reasonable to expect a lower energy consumption due to the shorter retransmissions. Moreover, the received fractions are always combined with the previously received symbols, just as in the other methods.
The mutual information after chase combining of m partial retransmissions coming from the relay is:
where the function ⌊ϕ⌋ returns the largest integer not larger than ϕ. The outage probability associated with the above mutual information is:
The solution for the outage probability is particular for any given L and m, as it involves a sum of multiple random variables. As an example, next we particularize the solution for m = L = 4:
which we compare to PreHARQ in Section 2D. More details about can be found in the Appendix.
The overall outage probability of the proposed CPR scheme is:
while the throughput for the maximum of M retransmissions is:
1) CPR with multiple relays: Similar to PreHARQ, the mutual information of the partial cooperation is:
while the outage probability is given by , and the throughput of CPR with K relays for a maximum of M retransmissions is:
D. Performance comparison
Figure 2 compares the outage probabilities of the direct transmission, PreHARQ, and the proposed CPR scheme with L = 4, given by Equations 3, 15, and 22. We assumed unitary transmit power (P = 1 W), the relay is exactly in the midpoint between the source and the destination (δ = 0.5) in the cooperative schemes, the path loss exponent is α = 4 (dense urban area), the information transmission rate is , and the number of retransmissions in this case is m = 4. For all the transmission schemes, we plot the theoretical expressions and the simulation results, obtained via the Monte Carlo method, which show the accuracy of the theoretical analysis. From the figure, we can notice that the outage probability of the proposed CPR scheme is worse than that of PreHARQ for L = 4 and m = 4. However, as the relay retransmits packets which are shorter, a positive impact in throughput is expected.
The throughput performance of the presented schemes is shown in Figure 3 for a maximum of M = 4 retransmissions.^{b} From the figure, we can see that CPR outperforms considerably the regular PreHARQ, achieving a higher throughput in the medium and high SNR regions, which are the regions of interest for communication. Only in the low SNR region, and by a very small margin, the proposed CPR scheme is outperformed by PreHARQ. Moreover, we can also notice that the throughput performance increases with L. When comparing the proposed scheme with PreHARQ, up to 2.1 dB of throughput gain is achieved when L = 2, up to 2.8 dB when L = 3, and up to 3.3 dB when L = 4. However, by increasing L, other issues must be taken into account. By retransmitting a fraction which is very small, the number of required retransmissions could increase and consequently compromise the energy efficiency of the scheme. The issue of the energy consumption for each of the above cooperative schemes will be handled in the next section.
Figure 4 extends these results to the case when multiple relays are available for the cooperative schemes. We compare the performance of the direct transmission, CPR with L = 4 and PreHARQ, both with K = 1, 2, and 4 relays. In the case of multiple relays, all relays were assumed to be at the midpoint between the source and the destination.^{c} From the figure, we can see that the proposed CPR scheme with a single relay already outperforms considerably the regular PreHARQ in the high SNR region, even when K = 4 relays are available in the latter. The introduction of multiple relays in PreHARQ increases the system performance only in the low SNR region, which was already noted in [26]. This is due to the fact that in the low SNR region, the sourcedestination link is very noisy, and more relays provide alternative paths with a higher reliability. On the other hand, while the SNR increases, the number of retransmissions requests is low, then the throughput performance of PreHARQ is not affected by increasing the number of relays. When multiple relays are also available in CPR, we can notice similar throughput increase in the low to medium SNR region, presenting a large advantage when compared to PreHARQ with the same number of relays. The throughput gain when K = 4 relays are available can be up to 6 dB when comparing CPR with PreHARQ. However, due to its simplicity and to its practical feasibility, next we consider only the case of K = 1 relay.