1 Introduction
An immense amount of data is created every day from different sensors and peripherals, namely, GPS embedded in vehicles, attached to objects or worn by people, sensors monitoring the environment, real time video streams, radars on roads, social network feeds, etc. Such type of data belongs to real time’s domain, where schedulability is one of the main characteristics of this domain, which means its propensity to respect the expected time constraints. In fact a real time system implies a system ability to ensure that investigated processing produces consistent results, i.e., functionally correct, at the right time. Therefore, to ensure the radio communication for such applications, a low latency as well as extreme reliability are required. In this context, the use of cooperation concept provides spatial and temporal diversity, and constitutes a good alternative to support advanced communications with increased channel capacity ^{1, 2}.
However, in regards to the endtoend latency, this requirement has a significant impact on the system quality and the fluidity of communications, and it is influenced by different features upon the transmission, we mention in particular, the propagation delay as well as the relay delay processing. In fact, depending on the environment and on the application, we can get rid of some supplementary sources of delay, as example, for industrial environments such factories, the distance between two automated robots is not considerable. Hence, the delay propagation can be neglected, and the only generated delay in this case, is that related to the relay processing, which depends mainly on the used relaying technique.
In general, there are various ways of relay processing in cooperative networks, among which we distinct mainly two familiar relaying schemes: amplifyandforward (AF) and decodeandforward (DF) ^{3}. In AF scheme, the relay simply amplifies the received signal and forwards it towards the destination. Thus, in term of the relay processing delay, the AF scheme, does not include a prominent latency ^{4}. However, this relaying scheme suffers from noise amplification. In the DF scheme, the relay first decodes the signal received from the source, reencodes and retransmits it to the destination. This approach suffers from error propagation when the relay transmits an erroneously decoded data block. Selective DF (SDF), where the relay only transmits when it can reliably decode the data packet, has been introduced as an efficient method to reduce error propagation ^{5}.
In the perspective of a low latency, fullduplex (FD) relaying mode allows fast devicetodevice discovery, and hence, contributes on the delay reduction. Furthermore, as the capacity improvement is promoted by the spectral efficiency improvement, the adoption of the FD communication at the relay is more advantageous. Even if fullduplex relaying mode FD generates loop interference from relay input to relay output, it still practical to use on cooperative relaying system due to its spectral efficiency ^{6, 7, 8}. The FD relay requires the duplication of radio frequency circuits to transmits and receives simultaneously in the same time slot and in the same frequency band. It has been shown that the FD mode still feasible even with the presence of a significant loop interference ^{6}, especially with recent advances noted in antenna technology and signal processing techniques. In ^{9}, a novel technique for selfinterference cancellation using antenna cancellation is depicted for FD transmissions. In the same context, through passive suppression and active selfinterference cancellation mechanisms, an experiment study was proposed in ^{10}. Hence, these practical growth incites authors to adopt FD communications in their research, thus, get rid of spectral inefficiency caused by halfduplex relaying mode.

Contributions
Most of previous available works in the literature have investigated the performance analysis of cooperative networks based SDF and AF relaying schemes, with the regard to different purposes ^{11, 12, 13, 14, 15, 16, 17}. In ^{11}, considering the FDAF relaying over Nakagami fading channel, authors cover the performances based on outage probability and ergodic capacity. The authors in ^{12, 13, 14, 15, 16, 17} adopt a FD cooperative scheme with the direct link between the source and the destination nodes is nonnegligible. Still, in ^{12, 13, 14}, to capture the joint benefit of relaying and direct links, at the destination side, authors have assumed a silence period at the source that is equal to the processing delay at the relay. The work in ^{17}, investigates over a Rayleigh fading channel, the optimal mode selection upon a FDAF system and study therefore, the individual impact of the residual selfinterference (RSI) and the direct link on the outage performance. However, none of the cited works have evaluated the relevance in term of latency impact in the context of latency sensitive applications.
In this paper, we address this issue by conducting a refined comparison between AF and SDF relaying schemes. Note that each of them adopts a different block transmission scheme. The pertinence of the direct link effect is also investigated, through the assumption of two different transmission modes, i.e., the non combining mode and the signals combining mode. For that purpose, over the so called Nakagami blockfading channel, we elaborate first the studied transmission schemes communication model, then we derive their outage probability expressions. Theoretical results are represented with Montecarlo simulations and show, on the basis of a low needed latency, the relevance of each relaying technique, according to the operating transmission mode.
The rest of the paper is organized as follows: section 2
presents the studied system model. The outage probabilities are derived
in section 3. In section
4, numerical performance results are
shown and discussed. The paper is concluded in section 5.
Notations

, , and
denote, respectively, a scalar quantity, a column vector, and a matrix.

is the Kronecker symbol, i.e., for and for .

,, and are conjugate, the transpose, and the Hermitian transpose, respectively.

is set of complex number.

For ,
denotes the discrete Fourier transform (DFT) of
, i.e. , with is a unitary matrix whose th element is , . 
denotes the absolute value.

is used to denote the statistical expectation.

is the probability of occurrence of the event .
2 System Model
This section presents a signal model for one relay cooperative system, where a FD relay , assists the communication between two end users, representing respectively, the source and the destination . In this paper, we assume the direct link between the source and the destination nodes is nonnegligible. Since relay operates in FD mode, we take into account the RSI generated from relay’s input to relay’s output. We consider both, the nonregenerative and regenerative relaying schemes, namely, amplifyandforward and selective decodeandforward. Hereafter, we introduce first, the adopted channel model for analysis, then, investigate the system model, covering both, the AF and the SDF schemes, over two different transmission modes, i.e., the noncombining mode where the best link, direct or relay link is decoded and the combining mode, where direct and relay links are combined at the receiver side.
2.1 channel model
The sourcedestination , sourcerelay , the relay selfinterference, and relaydestination channels, are represented by with . In this paper, we assume that , , are modeled by independent Nakagami fading with shape parameter and average power . Thus, the squared magnitudes are Gamma distributed with shape parameter and rate parameter , i.e.,
. The probability density function (PDF) and the cumulative density function (CDF) of a Gamma random variable
are, respectively, given by(1) 
and
(2) 
where denotes the Gamma function and denotes the lower incomplete Gamma function.
2.2 Signal model
At channel use i, the source node broadcasts its signal to both the relay and the destination. Accordingly, the received signal at the relay and the destination, during channel use i, can be expressed, respectively, as:
(3) 
(4) 
with , , denotes the transmit power at the source, is the processing delay at the relay, and is the RSI after undergoing any cancellation techniques and practical isolation at the relay ^{7, 18}, and is assumed to be equivalent to a zero mean complex Gaussian random variable . denotes, a zeromean complex additive white Gaussian noise at the relay. Both and depend on the relaying scheme.

Amplifyandforward: With AF scheme, the relay acts as a repeater which simply amplify the received signal and forwards it to the destination. Thereby,
(5) 
where is the amplification constant factor chosen to satisfy the total power constraint at the relay ^{11}, denotes the transmit power at the relay, is the AF relay processing delay, and denotes a zeromean complex additive white Gaussian noise at the destination.

Selective decodeandforward: In SDF scheme, the relay retransmits the received signal only when the link is not in outage. For this scheme, and
(6) 
From equation (4), we see that the destination node receives the source transmitted signal at different time instances due to the processing delay at the relay. In this work, we consider two transmission modes: Non combining (NC) mode where the receiver is synchronized with the strongest link, direct or relay link, and Signals combining (SC) mode where both direct and relay links are combined at the receiver side.
2.2.1 Non Combining (NC) mode
In this mode, the destination will try to decode the strongest link while, the second one will be considered as interference. Therefore, the system capacity of the NC mode for AF and SDF is expressed respectively, as:
(7) 
(8) 
with and , are respectively, the AF and SDF signaltointerference and noise ratio (SINR), where is the best link and is the worst link considered as interference.
2.2.2 Signals Combining (SC) mode
In SC mode both relay and direct signals are combined at the destination side. Therefore, in order to alleviate the intersymbol interference (ISI) caused by the delayed signal, equalization is performed at the destination. For that purpose, we propose a cyclicprefix (CP) transmission at the source side in order to perform frequencydomain equalization (FDE) at the destination node. Depending on the processing protocol at the relay, AF or SDF, the destination performs signalbased FDE or blockbased FDE. In the following, we assume all channel gains remain constant during channel uses^{1}^{1}1 is less or equal to the channel coherence time . For simplicity, we assume that all links have the same , where is the CP length ().
At the destination side, after the CP removal, the received signal, at channel use , can be expressed as,
(9) 
Equation (9) can be written in vector form to jointly take into account the received signal as:
(10) 
where , , , and is a circulant matrix whose first column matrix is . Note that the circulant matrix , can be decomposed as, . With is a diagonal matrix whose th element is Therefore, the signal can be represented in the frequency domain as,
(11) 
At the destination side, the system capacity is given by,
(12) 
where the factor means that the transmission of useful bits occupies channel uses and represents the overall system average mutual information, and is given by, , where , with , , and .
The system mutual information can be manipulated as below,
(13)  
According to the arithmeticgeometric mean inequality,
, we have Thus, by using the first order Taylor expansion, we have Noting that , the mutual information, in (13), can be approximated as,(14) 

AmplifyandForward
: AF is classified as memoryless scheme in which the relay processes the received signal in a symbolbysymbol manner. Therefore, the processing delay
is in term of channel uses and thus, the equalization, at the destination side, is a signalbased equalization. 
Selective Decode and Forward: Unlike AF, SDF is a memory scheme where the entire received block need to be decoded, before deciding to retransmit or not the reencoded block through link. This results in a blockbased processing delay . Thereby, as depicted in Fig.1, to deal with interblock interferences, the communication takes place assuming one superblock transmission of blocks, each gathering symbols and the SDF CP prefix is constructed using D blocks of , i.e., .
3 Outage Probability
In this section, we present the outage analysis of different schemes investigated in section 2. The system outage occurs when the received SINR at the destination side is below a target SNR threshold, whether
for SC mode, where both relay and direct signals are combined at the
destination side, or NC mode, where the destination will try to decode
the strongest link while, the second one will be considered as interference.
Note that, in this work, the packet retransmission is not considered.
Hereafter, we derive first for each mode, i.e. NC and SC, the overall fullduplex outage probability
of the AF scheme as well as that of the SDF ^{13}. For the purpose
of investigating the analysis, let’s first introduce the instantaneous
SINRs for each link.
The received SINR of the , the and the links are denoted, respectively, as,
(15) 
Note that , and are the result of a Gamma random variable scaled by a constant. Therefore, , and are Gamma distributed with shape parameter and rate parameter , where , and .
Herein, the outage probability is denoted and expressed as:
(16)  
where , with is the bit rate per channel use, and is the CDF of .
3.1 Non combining mode
Herein, for the rest of NC mode analysis, we consider .

Amplifyandforward
For AF relaying scheme, using (15), we extract the corresponding endtoend SINR as,
(17) 
Herein, for the case where the link is stronger than the , after some manipulations, the equation (16), tends to an integral form which doesn’t generate a closed form expression, and can be evaluated numerically using matlab software. Otherwise, using ^{13} Eq 12, the outage probability can be derived as,
(18) 
where denotes, the Whittaker function, , , , , and .

Selective Decode and Forward
The SDF relay system outage probability, is generally, defined as:
(19) 
where and denote respectively, the outage probability of link and link, and can be expressed as in ^{13},
(20)  
(21) 
denotes the outage probability of the best link , i.e., or , when the relay correctly decodes the received signal, and it can be derived as follows,
(22) 
and it is given by the following expression ^{13} Eq. 12:
(23) 
where denotes, the Whittaker function, , , , , and .
3.2 Signals Combining mode

Amplifyandforward
The outage probability of FD AF combining system is derived as follows:
(24)  
where . By substituting (15) into , we get,
(25) 
Thus, the CDF of can be derived as,
(26) 
where presents the lower incomplete Gamma function ^{19}.
Hereafter, to solve this double integral, we need to decompose the integration into two steps. Therefore, in order to proceed, let’s denote . First, while treating as constant, we have to integrate with respect to the limits . Using the serie form, i.e. ^{19} 8.352.6 and the polynomial expansion, i.e. , we get therefore, the first integral resolution, i.e., as represented in (27).
(27) 
Now, the resulting expression, i.e., is integrated accordingly with respect to bounds, as represented in (28).
(28)  
(29)  
The integrals generally, do not generate a closed form expression, thus, it can be evaluated numerically using matlab software.

Selective Decode and Forward
In the following, we briefly introduce the FD SDF relay system outage probability. In SC mode, if the relay correctly decodes the received packet, and decides to retransmit the reencoded block through the link, both relay and direct links are combined at the destination side. This mode’s outage probability is generally, denoted as the same form as (19). However, the threshold will be redefined accordingly as, ^{2}^{2}2The factor means that the transmission of useful bits occupies channel uses. , and in will be denoted, , which represents the outage probability of the combined signal, i.e., direct and relayed signals, at the destination side, and it can be derived as follows,
(30) 
Hence, by referring to ^{12, 13}, the endtoend SINR, can be approximated to . Thus, using (14) and (15), the expression of is approximated and given by,
Therefore, can be derived as:
(31) 
Hereafter, while developing the integral form, we got the expression of , as given in ^{13},
(32) 
where denotes the Kummer’s confluent hypergeometric function.
4 Numerical Results
In this section, the theoretical findings derived in section 3, are numerically verified and confirmed using Montecarlo simulations. The variation of outage probability for various transmission schemes investigated in section 2, is represented. Moreover, for an exhaustive comparison, simulations include the direct transmission mode with no relay cooperation. To assess SCSDF performances in term of super block length, we consider two cases: 1) the case where the superblock length is very large compared with the relay processing delay, , 2) and the case of short superblock where ^{3}^{3}3The superblock length is set while respecting a low latency requirement less than 1 . In this section, the first case is denoted SCSDF while the second case is denoted SCSDF3. For all simulations, we consider a packet length of symbols, and a relay processing delay of symbol for symbolbysymbol transmissions and ^{4}^{4}4According to a 3rd Generation Partnership Project 3GPP study on latency reduction techniques for LTE, the latency induced for encoding and decoding processing is proportional to the block size, and it represents 3 times the block size ^{20}. Therefore, , with represents the duration of one symbol. In this work, we consider which represents a typical symbol duration in Millimeterwave (mmWave) bands with subcarrier spacing of 120 ^{21}. for blockbyblock transmissions. and .
First, we compare the studied relaying schemes performances in term of the spectral efficiency level. For that purpose, in Fig.3 and Fig.3, we plot the outage probability versus the transmission bit rate R. First, we notice that the simulation results confirm the accuracy of the analytical expressions, obtained in section 3. In both figures, we note that, for transmissions that support very long superblock, i.e., , the SCSDF offers the best performances. However, for transmissions with low latency requirements less than ms, the superblock length must be less than . Thus, using SCSDF is not anymore the obvious choice. Thereafter, in term of the low latency purpose, we see that, in Fig. 3, when the direct link gain is very low compared to the first and second hop gains, the low processing delay scheme, SCAF scheme, offers the best performance. However, as the direct link gain increases, we start to notice that SCSDF3 becomes more desirable for low transmission rate, while the SCAF still the best choice for high transmission rate. This is mainly due to the SDF rate penalty of that impacts banefully the spectral efficiency, whenever the super block size decreases.
Hereafter, to point out the impact of the RSI level on performances, Fig. 5 and Fig. 5 illustrate the outage probabilities as function of . In one hand, we see clearly that SCSDF still provides the best performance. However, this scheme can not be practically adopted for low latency transmissions. In the other hand, we notice that there are three transmission schemes that outperform each other depending on the RSI level at the relay and can be practically adopted for low latency transmissions: SCAF, NCSDF, and direct transmission. In fact, SCAF seems to be the most suitable scheme for low latency transmissions with low RSI at the relay, i.e., dB. However, for moderate and high RSI, i.e., , we can either use NCSDF if the direct link is not strong enough (Fig. 5) or just switchoff the relay if the direct link gain is as good as the relay link (Fig. 5). In fact, in Fig. 5, we see that the direct transmission clearly outperforms NCSDF scheme. This is due to the fact that, in NC mode, the destination will try to decode the strongest received signal while the remaining signal will be considered as interference. Accordingly, at low RSI, where the relay can correctly decode and forward the reencoded block, the destination will receive a useful signal as strong as the interfering signal, which dramatically deteriorates the system performances. As the RSI gain increases, i.e., , the relay fails to correctly decode the received packet. Therefore, the only received signal at destination is the direct link signal. That is clearly seen in Fig. 5 where the NCSDF curve improves, as the increases, to be similar to the direct transmission curve.
Now, we consider the scenario where the link is much better than the link, i.e., . Fig.7 and Fig.7 plot the outage probability versus a range of link gains. We see clearly that, for moderate link quality, i.e., , SCAF scheme offers better outage performance than all other studied schemes. On the other hand, as the link variance increases, i.e., , we notice that the performance of NC modes are enhanced for low link gain, i.e.,
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