—In this work, we present a framework analysis of a millimeter wave (mmWave) vehicular communications systems. Communications between vehicles take place through a cooper- ative relay which acts as an intermediary base station (BS). The relay is equipped with multiple transmit and receive antennas and it employs decode-and-forward (DF) to process the signal. Also, the relay applies maximal ratio combining (MRC), and maximal ratio transmission (MRT), respectively, to receive and forward the signal. As the vehicles’ speeds are relative high, the channel experiences a fast fading and this time variation is modeled following the Jake’s autocorrelation model. We also assume narrowband fading channel. Closed-form expressions of the reliability metrics such as the outage probability and the mean rate are derived. Capitalizing on these performances, we derive the high signal-to-noise-ratio (SNR) asymptotes to get full insights into the system gains such as the diversity and coding gains. Index Terms —Millimeter wave, vehicular communications, decode-and-forward relaying, cooperative diversity, diversity gain. I. I NTRODUCTION The increase demand for bandwidth has been growing over the last few decades, largely due to the increased number of subscribers and number of communication devices. However, to meet these goals and the need for high-speed wireless data rate are challenging due to the limited spectrum. Millimeter wave (mmWave) communications which refer to the wide spectrum between 28 and 300 GHz become a promising solution to provide high Gbps of bandwidth to emerging 5G network [1], [2]. Such interesting technology has recently attracted enormous attentions both in academia and industry [3]. High deﬁnition (HD) video applications leverage from mmWave frequencies to transmit ultra HD video to HDTV wirelessly [4]. In addition, mmWave can also replace ﬁber optic transmission lines connecting the core network to the backhaul when the cable installation is restricted or expensive [5]. Since mmWave spectrum becomes available for use, industries invested and produced mmWave based commercial products such as IEEE 802.11ad wireless gigabit alliance (WiGig) technology. In fact, this product is dedicated to support wireless display interfaces requiring high resolution and gigabit rate [3], [4], [6]. Furthermore, mmWave frequencies support radar applications for motion sensors, automatic doors, and collision avoidance systems [4], [7]. Moreover, mmWave communications have low interference and high security against hacking and intrusion. In fact, Elyes Balti is with the Wireless Networking and Communications Group, Department of Electrical and Computer Engineering, The University of Texas at Austin, Austin, TX 78712 USA e-mail: ebalti@utexas.edu.mmWave link has narrow beam and short range where the area experiences less interference from neighbor radios. Also, mmWave system is immune since the propagation is restricted to a limited area which enhances the security [4]. In the same context, mmWave technology has tremendous applications in vehicular communications area. Speciﬁcally, self-driving vehicles require a great amount of information from nearby vehicles and road side units (RSU) to enhance the vehicle awareness and avoid the potential accidents. Certainly such application requires a high bandwidth and data rate to support the big data exchanged between the vehicles and RSUs [7], [8]. Although mmWave technology has tremendous advantages, it suffers from many drawbacks. Due to the small wavelength, mmWave has a short propagation range around 20 meters for low power applications [9]. Supporting longer ranges requires large antenna gains and transmit power. Also, since mmWave frequencies are sensitive to blockage, it requires line of sight (LOS) communication as regular objects such as trees, human being, and building completely block the signal [2]. Consequently, the range is reduced and the received signal is weak. Furthermore, due to the high frequencies, mmWave is susbtantially degraded by the path loss as it suffers from the atmospheric attenuations such as the moisture, fog, rain attenuations, and the oxygen absorption [10]. To compensate for the path loss, mmWave systems use high directional array antennas. Moreover, due to the large bandwidth, the received mmWave signal experiences high noise degradations at the receiver which substantially limits the received signal-to-noise ratio (SNR). [11]–[27]. To enhance the quality of the received signal, relay based cooperative communication technique has been proposed not only to improve the capacity but also to improve the coverage and assist the received signal power [28]. Related works have proposed various relaying techniques to process the signal such as amplify-and-forward (AF) [29], [30] and decode-and- forward (DF) [30]. The AF consists of amplifying the received signal with a proper gain factor and then forwards it, while DF decodes and denoises the signal and then forwards it. Since mmWaves are susceptible to the high noise power, the AF is not relevant because in this case the relay also ampliﬁes the noise which is incorporated in the received signal. [31]–[38]. In this context, we propose a dual-hop vehicular system wherein the relay employs DF relaying mode. The source and the destination are both a transmit (TX) and receive (RX) vehicles while the relay behaves as an intermediary base station (BS). The rest of this paper is organized as follows: the system model is presented in Section II. The reliability metrics 2 analysis are detailed in Section III, while the discussion of the numerical results is reported in Section IV . Concluding remarks are presented in Section V . II. S YSTEM MODEL Fig. 1 provides an illustration of the relay network. Transmit vehicleReceive vehicleRelay φ1MRC φ2MRT Fig. 1: Dual-hop millimeter wave vehicular relay channel. The proposed system consists of two vehicles communicat- ing through an intermediary relaying base station (BS). Each vehicle is equipped with a single antenna while the relay is equipped with multiple transmit (TX) and receive (RX) antennas, and employs DF protocol to process the received signal. We assume that the RX antennas elements are close enough and the distance between the relay and TX vehicle is long so that the angle of arrival and the time delay are the same for all the signals. The same assumption holds for the second hop, where the transmitted signals at the relay have the same angle of departure 2and time delay to reach the receive vehicle. To estimate the channel state information (CSI), the relay sends periodic feedback to the vehicles. Since the feedback is slowly propagating and the channels are time- varying, the transmitted CSI will be the outdated version. The time correlation between the outdated and updated CSIs can be modeled using the Jake’s autocorrelation model as follows [39, Eq. (8)] i= [J0(2fd;i d;i)]2; (1) whereJ0()is the ﬁrst kind of Bessel function with order zero, the index i= 1;2stands for the ﬁrst and second hop. fd;i=fcvicos( i) cis the doppler frequency, is the angle between vehicle displacement and the relay, fcis the carrier frequency,viis the speed of the vehicle, cis the speed of light, and d;iis the propagation delay. Fig. 2 illustrates the time correlation with respect to the frequency and the vehicle speed. The maximum coherence timeTccorresponding to 28 GHz and a speed of 10 m/s, is 757.61s, while the minimum Tccorresponding to 73 GHz and a speed of 30 m/s is 193.73 s. We observe that the correlation peaks decrease with the speed and the carrier frequency. As the vehicle speed increases mainly in highwaysand rural areas, the correlation decreases and inversely when the vehicle speed is lower, speciﬁcally in urban and sub-urban areas, the correlation improves. In fact, for higher speeds, the channel is rapidly varying and the estimation becomes worse. However, for lower speed, the TX and RX can relatively track the channel variations, and hence the estimation gets much better. 20 22 24 26 28 3001234·10−2 Vehicle Speed v (m/s)Correlation ρfc= 28 GHz fc= 38 GHz fc= 60 GHz fc= 73 GHz Fig. 2: Correlation dependence on the vehicle speed for different values of the carrier frequency. The symbol period Ts= 1 ms, d= 2Ts, and = 4. The relation between the outdated ^ iand updated iin- stantaneous SNRs of the hop iis expressed as follows [39, Eq. (7)] ^ i=pi i+p 1i!i (2) where!iis complex Gaussian distributed with zero-mean and having the same variance as i. A. Millimiter wave channel model of the hop i Since the channels of the ﬁrst and second hops are sym- metric, and the relay transmit and receive strategies are also equivalent (MRC and MRT), we will focus the analysis on a single hop. yi=r AiPi NiNiX n=1hn;is+Mr;iX k=1fk;idk;i+ni; (3) whereAiis the average power gain, Piis the total trans- mitted power equally splitted among the Nichannels,sis the modulated symbol, hn;iis then-th channel, dk;iis the modulated symbol of the k-th interferer with an average power E[jdk;ij2] =Pk;i,fk;iis the fading coefﬁcient of the k-th interferer,Mr;iis the number of the interferers, and the noise niis a zero-mean additive white Gaussian noise (AWGN). The noise variance is given by 2 i[dBm] =B+N0+Nf; (4) The path loss of the link iis given by [10] PLi[dB] =20 log104fc c + 10log10(Li)Gt;iGr;i; (5) whereis the path loss exponent. 3 Table (I) summarizes the values of the system parameters used in the simulation. TABLE I: System Parameters Parameter Symbol Value Transmit antenna gain Gt;i 4 dB Receive antenna gain Gr;i 4 dB Noise spectral density N0 -142 dBm/Hz Noise ﬁgure Nf 0 dB Bandwidth B 700 MHz Speed of light c 3108m/s Link distance Li 50 m B. Statistical analysis of the hop i Given that the single input single output (SISO) channel fading is Nakagami-m distributed, the relative SNR follows the Gamma distribution. Hence the overall SNR, which is the sum ofNiGamma identically distributed SNR per branch with scalemi, is also a Gamma random variable with scale Nimi. The joint probability density function (PDF) of the outdated and updated SNRs is given by [39, Eq. (14)] f^ i; i(x;y) =mi imi+1 xy imi1 2 (1i)(mi)e x+y 1imi i Imi12mipixy (1i) i ;(6) whereI()is the-th order modiﬁed Bessel function of the ﬁrst kind and iis the average SNR. After some mathematical manipulations, the PDF of the outdated CSI is obtained by f^ i(x) =Nimi iNimixNimi1 (Nimi)eNimix (1i) i; (7) Integrating (7) and using [40, Eq. (3.351.1)], the cumulative distribution function (CDF) is given by F^ i(x) =(1i)Nimi (Nimi)  Nimi;Nimix (1i) i ;(8) where (;)is the incomplete lower gamma function. Since Nimiis an integer, the CDF (8) can be reformulated as follows F^ i(x) = 1eNimix (1i) iNimi1X n=0xn n!Nimi (1i) in! (1i)Nimi;(9) We assume that each interferer fading follows Nakagami-m with parameter mr;iand we neglect the impact of the time correlation on each channel. Repeating the same derivation steps, the PDF of the aggregate Mriinterferers’ SNR r;iis given by f r;i(x) =Mr;imr;i r;iMr;imr;ixMr;imr;i1 (Mr;imr;i)eMr;imr;ix r;i; (10) where r;iis the average SNR of the interferers. The effective signal-to-interference-plus-noise-ratio (SINR) is expressed by eff;i=^ i r;i+ 1; (11)Using [40, Eq. (3.351.3)], and after some mathematical manipulations, the CDF of the effective SINR is given by F eff;i(x) = (1i)Nimi 1Mr;imr;i r;imr;i1 (Mr;imr;i) Nimi1X n=0nX p=0n p n!Nimi (1i) in (Mr;imr;i+p) xnMr;imr;i r;i+Nimix (1i) i(Mr;imr;i+p) eNimix (1i) i# ; (12) Differentiating (12) and after some mathematical manipula- tions, the PDF of the effective SINR is given by f eff;i(x) =Nimi iNimiMr;imr;i r;iMr;imr;ixNimi1 (Nimi) eNimix (1i) r;i (Mr;imr;i)mX p=0m p (Mr;imr;i+p) Mr;imr;i r;i+Nimix (1i) i(Mr;imr;i+p) ; (13) Using the identity (s;x) xsv1 sasx!0, at high SNR, (8) can be expressed as F^ i(x)v1 (Nimi)!Nimix iNimi ; (14) After considering (11) and repeating the same derivation steps, the high SNR asymptote of the CDF of the effective SINR is given by F eff;i(x)v1 (Mr;imr;i)(Nimi)!Nimix iNimi NimiX p=0Nimi p (Mr;imr;i+p) r;i Mr;imr;ip ; (15) From (15), the diversity gain per hop Gd;iis Gd;i=Nimi; (16) C. Symbol error probability of hop i For M-QAM modulations, the average symbol error is expressed as Pe;i= E eff;ih Q(p )i ; (17) whereQ()is the Gaussian Q-function, and ; stand for the modulation schemes. Using the integration by part, the symbol error can be reformulated as follows Pe;i= p 2p 21Z 0e 2 p F eff;i( )d ; (18) 4 Inserting (12) in (18) and after using [40, Eq. (7.813.1)], and [41, Eq. (07.43.03.0271.01)], the symbol error is given by Pe;i= p (1)Nimi 2p 2 (0:5)r2 Nimi1X n=0nX p=0n p n!  r;i mr;ip (Mr;imr;i)Nimi (1i) in 2+Nimi (1i i)(n+1 2) G1;2 2;1 Nimi r;i (1i)Mr;imr;i i 2+Nimi (1i) i 1 2!! ; (19) whereGm;n p;q(j)is the Meijer- Gfunction,1=1 2n;1Mr;imr;ip , and2= 0. At high SNR, the error probability can be expressed as Pe;i=(Gc;i i)Gd;i; (20) whereGc;iis the coding gain. After inserting (15) in (18), the high SNR error is derived as follows Pe;i= (Nimi+1 2) 2p(Nimi)!2Nimi iNimiNimiX p=0Nimi p (Mr;imr;i+p (Mr;imr;i)) r;i mr;ip ;(21) After identifying (21) with (20), the coding gain is given by Gc;i= 2Nimi2(Mr;imr;i)(Nimi)!p (Nimi+1 2)1 m  NimiX p=0Nimi p (Mr;imr;i+p) r;i Mr;imr;ip!1 m ; (22)D. Capacity of hop i The average rate Ci, expressed in (bps), is deﬁned as the maximum error-free data rate transmitted by the system. Assuming Gaussian signaling, the mean rate can be written as follows Ci=BE eff;i[log2(1 + )]; (23) The derivation of the closed-form expression involves three main steps. The ﬁrst one is to insert (13) in (23). Sec- ondly, we convert each function of the integral into Fox- Hfunction form. Finally, after applying the identity [42, Eq. (2.3)], the capacity is given by (24). Note that the function Hm1;n1:m2;n2:m3;n3 p1:q1;p2;q2:p3;q3(jjj;)is called the bivariate Fox- Hfunction. An efﬁcient MATLAB implementation of this function is given by [43, Appendix B]. E. Blockage model MmWave communications are very sensitive to blockage. Consequently, it is important to consider this factor in the analysis to get consistent results. In this work, we adopt the exponential blockage model ( Plos=ed=), wherePlosis the probability of line of sight (LOS). According to 3GPP, = 200 m in suburban area [10]. III. R ELIABILITY METRICS ANALYSIS In this section, we provide the performance analysis of the proposed system, in particular, we consider the end-to- end outage probability, the probability of symbol error, and the ergodic capacity for the overall relay system. Basically, the statistical results derived in the previous section will be considered to get the reliability metrics and to quantify the performance losses and the system gains. A. Outage probability analysis The outage probability is deﬁned as the prob- ability that the overall SINR falls below a given threshold x. This threshold is a standard value designed to target a desirable quality of service. Ci=B(1i)Nimi log(2)(Nimi)(Mr;imr;i)NimiX p=0Nimi p r;i Mr;imr;ip H0;1:1;1:1;2 1;0:1;1:2;2 (1Nimi; 1;1)  (1Mr;imr;ip;1) (0;1) (1;1);(1;1) (1;1);(0;1) r;i Mr;imr;i;(1i) i Nimi! ;(24) Since LOS and NLOS propagations are assumed, the aver- age CDF per hop is given by F eff;i(x) =PlosFlos eff;i(x) + (1Plos)Fnlos eff;i(x); (25) whereFlos eff;i()andFnlos eff;i()are the CDF of hop ievaluated respectively, for LOS and NLOS. The overall SINR efor the relay system employing DF protocol is given by e= min ( eff;1; eff;2); (26)Consequently, the CDF of eis given by F e(x) =F eff;1(x) +F eff;2(x)F eff;1(x)F eff;2(x);(27) Finally, the outage probability is obtained by Pout(x) =Prob[ e<x] =F e(x); (28) B. Symbol error probability Closed-form expression of the symbol error for the overall relay system is too long. Due to the space limitation, we will 5 consider the following CDF approximation F e(x)=F eff,1(x) +F eff,2(x); (29) Then, the overall symbol error can be obtained by Pe=Pe;1+Pe;2; (30) C. Ergodic capacity Since DF is assumed, the overall capacity of the relay channel can be expressed as follows C= min (C1;C2); (31) D. System gains The overall diversity gain Gdis given by Gd= min (Gd;1; Gd;2); (32) while the overall coding gain Gcis obtained by Gc=8 >< >:Gc;1 Gd;1<Gd;2 GGd c;1+GGd c;21 GdGd;1=Gd;2 Gc;2 Gd;1>Gd;2(33) IV. N UMERICAL RESULTS In this section, we present the analytical results of the reliability metrics following their discussion. The Nakagami- m parameters are 4 and 2, respectively, for LOS and NLOS propagations. For the interferers, the Nakagami-m parameter is equal to 1. The path loss exponents are 2 and 4, respectively, for LOS and NLOS. 12108642 0 2 4 6 8100 101 102 103 104 105 106 107 108 109 Average SNR [dB]Outage Probability1= 0.01 1= 0.75 1= 0.85 1= 0.90 High SNR Fig. 3: Outage probability versus the average SNR for various time correlation values. The carrier frequency is 60 GHz and the link range is 50 m. The number of transmit and receive antennas are 5 and the time correlation for the second hop 2 = 0.7. Two interferers are assumed with an average receive SNR of 0 dB and the outage threshold is - 50 dB. Fig. 3 shows the dependence of the outage probability with respect to the time correlation of the ﬁrst hop. The results show that the outage performance gets much worse when the channel estimation is bad, while the performance improves as the system becomes more or less able to track the channel variations and gets a better version of the estimated channel.15 10 5 0 5 10 15 20100 101 102 103 104 105 106 107 108 109 Average SNR [dB]Outage Probability r= 0 dB r= 5 dB r= 10 dB r= 20 dB High SNR Fig. 4: Outage probability versus the average SNR for different receive SNR values of the interferers. The carrier frequency is 60 GHz and the link range is 50 m. The number of transmit and receive antennas are 5 and the time correlation is 1=2 = 0.1. The outage threshold is - 50 dB. Fig. 4 illustrates the impacts of the power interferers on the outage performance. We observe that when the system is saturated by a large received power of the interferers, the outage deteriorates. Inversely, when the receiver becomes less susceptible to interferers (less power), the system performance gets much better. 10 5 0 5 10 15 20 25105104103102101100 Average SNR [dB]Symbol Error ProbabilityBPSK QPSK 16-QAM High SNR Fig. 5: Probability of symbol error versus the average SNR for different modulation schemes. The carrier frequency is 38 GHz and the link range is 50 m. The time correlation ais 1=2 = 0.5. We assume only LOS propagation in this scenario. Fig. 5 shows the variations of the symbol error for various modulation schemes. This result is expected as BPSK modulation outperforms QPSK and 16-QAM. Also, this result is considered as a benchmarking tool to verify the accuracy of the symbol error derivation. Fig. 6 shows the outage performance for various number of transmit and receive antennas. These results show how much diversity gain achieved by the system scaled by the number of antennas. As the number of antennas increase, the system becomes robust enough to combat the fading and improve the error performance. To achieve a given outage of 103, the system setting for 10 antennas ahieves an array gain around 15 dB compared to the single antenna conﬁguration. 6 15 10 5 0 5 10 15 20100 101 102 103 104 105 106 107 108 109 Average SNR [dB]Outage ProbabilityN1=N2= 1 N1=N2= 3 N1=N2= 5 N1=N2= 10 High SNR Fig. 6: Outage probability versus the average SNR for different number of transmit and receive antennas. The carrier frequency is 73 GHz and the link range is 50 m. The time correlation is 1=2= 0.1, and the outage threshold is -50 dB. 0 2:5 5 7:5 10 12:515 17:520 22:525 27:53000:511:522:533:544:5 Average Transmit Power [dB]Capacity [Gbps]Blockage Density ( = 50 m) Blockage Density ( = 100 m) Blockage Density ( = 150 m) Blockage Density ( = 200 m) Fig. 7: Capacity versus the average SNR for different blockage densities. The carrier frequency is 28 GHz, the link range is 50 m, and the time correlation is 1=2=103. Fig. 7 illustrates the effects of the blockage density on the capacity. As the environment becomes more dense (lower ) with physical objects, the capacity substantially decreases. This is explained by the fact that mmWaves are very sensitive to blockage. As the blockage becomes severe, the signal is completely attenuated. In addition, the system achieves high data rate of Gbps even though for severe blockage constraints. V. C ONCLUSION In this work, we proposed a global framework analysis of a mmWave vehicular relay channel. We investigated the dependence of the reliability metrics on different parameters such as the number of antennas, the time correlation, the modulation, the power interferers, and the blockage density. We showed that the system becomes more resilient to fading by increasing the number of antennas and having a better channel estimation. Also, the performance improves when the receiver becomes less vulnerable to the interferers powers. Moreover, the capacity depends to a large extent on the blockage density, but, it is still in the order of Gbps even for severe blockage. As future directions, we intend to extend this work by considering wideband fading with transmit and receive beamforming designs to improve the channel capacity.