#### Abstract

In this paper, we optimize the throughput of millimeter wave communications using relay selection techniques. We study opportunistic amplify and forward (OAF), opportunistic decode and forward (ODF), and partial and reactive relay selection (PRS and RRS). Our analysis is valid for interference-limited millimeter wave communications. We suggest a new optimal power allocation (OPA) strategy that offers significant performance enhancement with respect to uniform power allocation (UPA). The proposed OPA offers up to 2 dB gain with respect to UPA. Our analysis is confirmed with extensive simulation results for Nakagami fading channels.

#### 1. Introduction

Millimeter wave (mmWave) communications offer high data rates of several Gb/s [1–3]. Millimeter wave communication operates on an important bandwidth from 3 to 300 GHz [1–6]. Cooperation is mandatory in mmWave communications because the mmWave signals cannot penetrate through walls [1–7]. Dual relaying can be used where the signal goes from a source *S* to relay and then to the destination *D*. Multihop multibranch relaying allows to extend the coverage of mmWave communications [7–12]. Amplify and forward relays use an adaptive or fixed gain. When the gain is adaptive, these are called nonblind relays. Otherwise, relays with fixed gain are less complex and are called blind relays. AF can be implemented with relay selection techniques such as OAF, PRS, and RRS.

In decode and forward (DF) relaying, each relay decodes the signal and is allowed to transmit only if it has correctly decoded the packet [13–15]. In opportunistic DF (ODF), the chosen relay offers the best SINR of the second hop between relays and destination *D* [15–18].

Optimal power allocation has not been yet proposed for mmWave communication with an analysis at the packet level. In previous studies, power allocation has been optimized to minimize the symbol or bit error probabilities [19]. The innovations of the paper are as follows:(i)We derive the throughput and packet error probability (PEP) of millimeter wave communications for OAF, ODF, PRS, and RRS. Previous papers [1–10] studied only the symbol and bit error probabilities. To the best of our knowledge, the PEP has not been yet derived in closed form for mmWave communication with OAF, ODF, PRS, and RRS.(ii)We propose a new optimal power allocation (OPA) strategy that offers up to 2 dB gains with respect to uniform power allocation (UPA). Our optimization is performed at the packet level.(iii)Our analysis is valid for quadrature amplitude modulation (QAM), amplitude-shift keying (ASK), and phase-shift keying (PSK) modulations with any number of relays having arbitrary positions.

The paper contains seven sections. The signal to interference plus noise ratio (SINR) is analyzed in Section 2. Section 3 derives the cumulative distribution function (CDF) of SINR of different relay selection techniques. In Section 4, we derive the throughput of mmWave communications. Section 5 suggests an optimal power allocation strategy, while Section 6 gives some simulation and theoretical results. The last section summarizes the obtained results.

#### 2. SINR Statistics

The system model illustrated in Figure 1 contains a source *S*, *M* relays , and a destination *D*. It is assumed that the received signal relay is affected by interferers as shown in Figure 1. In interference-limited millimeter wave communications, the SINR at *k*-th relay is equal to [9]where is the transmitted energy per symbol of the source *S*, is the channel coefficient between *S* and , is the power spectral density (PSD) of additive white Gaussian noise (AWGN), and is the interference term at written as follows [9]:where is the number of interferers at , is the transmitted energy per symbol of *p*-th interferer, and is the channel coefficient between *p*-th interferer and relay .

We derive the statistics of an upper bound of SINR:

For Nakagami fading channels, follows a gamma distribution defined aswhere is the *m*-fading figure of link ( corresponds to Rayleigh channels),

The interference term is expressed aswhere

It is assumed that is the sum of independent and identically distributed (i.i.d) gamma random variables (r.v) that follows a where and

The sum of i.i.d gamma r.v. is a gamma r.v. . Therefore, in (3) is the quotient of two gamma r.v. that follows a general prime distribution with PDF [20]:where

The CDF of SINR is obtained by a simple primitive of PDF. We use the following result [21]:where , , and is the hypergeometric function.

We use (13) to express the CDF of SINR as

Using (8), the asymptotic PDF is expressed as

By a primitive, we deduce the asymptotic CDF expressed as

These asymptotic expressions will be used to derive the optimal power allocation (OPA) strategy for cooperative mmWave communications.

It is assumed that the interference term at *D* is the sum of i.i.d gamma r.v. with distribution . The CDF of SINR of the second hop between and *D* is written similarly to (8)where is the *m*-fading figure of link,

#### 3. Performance Analysis of Relay Selection Techniques

In this section, we assume that a single relay is selected. There is no interference between the signals transmitted by different relays as a single relay is selected to amplify or decode the source packet. Therefore, our analysis is valid for any number of relays.

##### 3.1. Opportunistic AF Relaying

For AF relaying, the SINR between the source *S*, relay , and destination *D* is expressed as [22]

The SINR can be tightly upper bounded by [22]

The CDF of SINR is expressed aswhere

Assuming that and are independent, we have

In OAF, the chosen relay offers the best end-to-end SINR

Using (27) and assuming that the SINRs for different relays are independent, the CDF of SINR is expressed as

##### 3.2. Partial Relay Selection

In PRS, the selected relay offers the best SINR of the first hop. Let be the probability that relay is selected and the corresponding SINR is expressed aswhere is the SINR at relay which is the highest SINR between S and relays,

The SINR can be tightly upper bounded by [22]

If and are independent, we have

We have

The CDF of SINR is expressed as

The probability to select relay is expressed as follows:

Let we can writewhere is the PDF of

Assuming that the SINRs of the first hop are independent, the CDF of *X* is written as

We have

For PRS, the CDF of SINR at *D* is written as (35) with given in (37).

##### 3.3. Reactive Relay Selection

In RRS, the chosen relay offers the best SINR between relays and destination. Let be the probability that relay is selected and the corresponding SINR is expressed aswhere is the highest SINR of the second hop

The SINR can be tightly upper bounded by [22]

If and are independent, we have

We have

The CDF of SINR is expressed as

The probability to select relay is expressed as follows:

Let , we can writewhere is the PDF of

Assuming that the SINRs of the second hop are independent, the CDF of *Y* is written as

We have

For RRS, the CDF of SINR at *D* is written as (46) with given in (48).

##### 3.4. Performance Analysis of Opportunistic DF

In ODF, we activate the relay with largest SNR of the second hop. This relay should correctly decode *S* packet. The PEP is written aswherewhere is the PEP at relay .

is the PEP when is the set of relays having correctly decoded expressed as [23]where is a waterfall thresholdwhere is the PEP for SINR *x*. It is detailed in the next section for different modulations.

#### 4. Throughput of Cooperative mmWave Communications

The PEP is expressed aswhere is the PDF of SINR and is the PEP for SINR *x.*

For K-QAM, we havewhere *N* is packet length in symbols and *K* is the constellation size.

For K-ASK, we have

For K-PSK modulation, we have

The PEP can be upper bounded by [23]where is the CDF of SINR and is a waterfall threshold [23]:

In order to compute the PEP, we have to only derive the CDF of SINR and use (51). The throughput is written aswhere coefficient 0.5 is due to the fact that half the frame duration is used for transmission by the source and the other half by the selected relay.

The throughput of OAF, PRS, and RRS are given in (61) where the PEP is deduced from CDF of SINR as (59). The CDF of SINR of OAF, PRS, and RRS are respectively given in (28), (35), and (46). The throughput of ODF is given in (61) with PEP expressed as (51).

#### 5. Optimal Power Allocation

For each candidate relay, we suggest an optimal power allocation that minimizes the asymptotic packet error probability expressed as

Using the definitions of (20) and (12), we have

Let be the fraction of power allocated to the source. is the transmitted energy per symbol of *S* and is the transmitted energy per symbol. is the transmitted energy per symbol of . Similarly, is the fraction of power allocated to . We deduce

Therefore, the outage probability (63) to be minimized is expressed asunder the constraint that and where

The Lagrangian of the problem is expressed as

The derivative of *J* is expressed as

When the fading figure of link is the same as that of , i.e., , we obtain the OPA strategy

If , we deduce from (68) and (69) that we have to solve with the Newton algorithm the following equation:

We deduce the fraction of power allocated to relay .

#### 6. Theoretical and Simulation Results

We plot theoretical curves with MATLAB and made some simulations for 64-QAM modulation. We have varied the number of relays . The path loss exponent is equal to 3. The results are valid for Nakagami channels with and . The fading figure of interference terms is .

Figures 2 and 3 show the PEP and throughput of OAF for different number of relays *M*. There are two interferers. The distance between *S* and is . The distance between and *D* is . We notice that the throughput improves as *M* increases due to cooperative diversity. A good accordance between theoretical and simulation results is observed.

Figure 4 shows the throughput of ODF in the same context as Figures 2 and 3. We notice a good accordance between theoretical and simulation results. Also, the throughput improves as the number of relays is increased.

Figure 5 compares the throughput of OAF, PRS, and RRS for . There are three relays and two interferers. We observe that OAF offers the highest throughput. RRS offers better performance than PRS because the relays are close to S. Figure 6 shows that PRS offers a higher throughput than RRS when relays are close to *D*, . OAF offers the highest throughput since it selects the relay with the largest end-end SINR.

Figures 7–10 show the throughput of OAF, PRS, RRS, and ODF for optimal or uniform power allocation (OPA or UPA) with two interferers and three relays. Figures 7–10 show that the proposed OPA allows up to 2 dB gain with respect to UPA.

#### 7. Conclusion

In this paper, we have optimized the throughput of millimeter wave communications using OAF, ODF, PRS, and RRS. Our analysis is valid for interference-limited millimeter wave communications with any number of relays with arbitrary positions. The proposed optimal power allocation (OPA) strategy offers up to 2 dB gain with respect to uniform power allocation (UPA). Our analysis was confirmed with extensive simulation results in the presence of Rayleigh fading channels. The main contribution of our paper is to suggest an optimal power allocation to enhance the throughput at the packet level, while previous studies perform the optimization to minimize bit or symbol error probabilities [19]. Besides, our results showed that OAF offers higher throughput than PRS and RRS as it selects the relay with highest SINR at destination, while PRS and RRS use only the SINR of the first and second hops.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.