1 Introduction

In recent years, the rapid increase in data-traffic consumption by cellular users will require to implement new connectivity schemes that guarantee acceptable levels of Quality of Service (QoS) and reliability. Multi-hop relaying in next generation cellular networking, with and without the direct intervention of the infrastructure, is foreseen to revolutionize the way connections will be established, combining the traditional centralized schemes of connectivity (where users connect exclusively through their nearest Base Station, or at most from a fixed relay station), with the new paradigm of Device-to-Device (D2D) networking (in which users are allowed to communicate peer-to-peer through direct links that do not necessarily involve the intervention of a Base Station). According to several research studies, the benefits of this new approach are numerous: higher spectrum efficiency, better fairness in resource allocation, energy efficient usage, among others. However, this novel approach for wireless cellular networking faces the challenge of distributed management of communications, subject to mobility, in which end-user terminals will play a more active role in the self-configuration of the network [1–3].

In multi-hop cellular networks (MCN), communication is not established directly between the user equipment (UE) and the base station (BS) [4–7]. Instead, intermediate devices act as repeaters between the BS and a UE. In this way, it is expected to decrease the total required transmission power and possibly mitigate interference and coverage problems and also it is expected to provide service in ‘dead spots’ in a cell, which are not reachable by the BS in direct communication case.

In this paper, by using different path loss models between the mobile (UE or repeater) and base station terminals, the question of how path loss changes by splitting the transmission into a series of sub-paths is investigated. To procure this, first, these path loss models are described. After describing the path loss models, multi-hop and direct communication cases are compared according to these path loss models and finally, some conclusions are drawn.

2 Path Loss Models

In this section, several path loss models are described. These models are (1) the simplest path loss model, (2) COST231-Hata model, (3) Walfisch–Bertoni path loss model, (4) COST231-Walfisch–Ikegami Line-of-Sight model, and (5) Non-Line-of-Sight model. Before describing these models, network cell types are described. The most appropriate propagation prediction technique for a given system is highly dependent on the chosen cell type. Each of the four major cell types is described in this section [7, 8]. Each path loss model is valid for different conditions and cell types. When the path loss models are described, the restrictions of the each model are also expressed.

2.1 Cell types

Different cell types and their restrictions are investigated in this section.

2.1.1 Macro-Cell

In the macro-cell, typical cell radius is about 1–30 km. The BS antenna is mounted above the roof-top level and the heights of the all surrounding buildings are below the base station antenna height [8, 9].

2.1.2 Small Macro-Cell

In the small macro-cell, typical cell radius is about 0.5–3 km. The BS antenna is mounted above the medium roof-top level and the heights of the some surrounding buildings are above the BS antenna height [8–10].

2.1.3 Micro-Cell

In the micro-cell, typical cell radius is up to 1 km. The BS antenna is mounted below the medium of the top level [8–10].

2.1.4 Pico-Cell

In the pico-cell, typical cell radius is up to 500 m. The BS antenna is mounted below the roof-top-level [8–10].

Radio propagation loss basically consists of three components. These components are the distance dependent path loss component, shadowing component, and the fast fading component. Shadowing and the fast fading loss components are generally assumed to be independent of the distance, but de-pendent on the local environment such as the arrangement of the streets and buildings [2]. The propagation loss formula can be expressed by Eq. 1 [2]:

$$L = PL + L_{S} + L_{f} \,{\text{in dB}}$$
(1)

where “PL” is the distance dependent path loss component, “L S ” is the shadowing component, and the “L f ” is the fast fading component. This paper, only focus on the models available for the distance dependent path loss. Fast fading and shadowing effects have not been taken into account. Some path loss models are described as follows.

2.2 Path Loss Models

Different path loss models and their restrictions are investigated in this section.

2.2.1 The Simplest Empirical Path Loss Model

The simplest useful form for an empirical path loss model can be expressed by Eqs. 2 and 3 [8, 11].

$$PL_{dB} = A + 10 \cdot n \cdot { \log }\left( {\frac{d}{{d_{o} }}} \right)\quad {\text{for d }} \ge {\text{ d}}_{\text{o}}$$
(2)
$$A_{dB} = 20 \cdot log\left( {\frac{{4\pi d_{o} }}{\lambda }} \right)$$
(3)

where “n” is the path loss exponent, “d” is the distance between the transmitter (Tx) and the receiver (Rx) antennas, in meters, “do” is the reference distance and “λ” is the wavelength of the center frequency of the waveform. In this paper, the reference distance (do) is taken as 1 m. do is often determined empirically. do should be selected such that it is in the far-field of the Tx antenna, but still small relative to any practical distance used in the mobile communication system [12]. For n = 2, this formula gives us the special case of free-space path loss. Path loss exponents are listed in Table 1 for different environments.

Table 1 Path loss exponents, n [10]
Full size table

2.2.2 COST231-Hata Model

COST231-Hata model is the extended version of the Okumura–Hata model to cover the band 1500 MHz ≤ fc ≤ 2000 MHz. COST231-Hata model is given by Eqs. 4, 5 and 6 [9].

$$PL_{dB} = 46.3 + 33.9 \cdot \log \left( f \right) - 13.82 \cdot \log \left( {h_{b} } \right) - a\left( {h_{m} } \right) + \left( {44.9 - 6.55 \cdot \log \left( {h_{b} } \right)} \right) \cdot \log \left( {\frac{d}{1000}} \right) + C_{m}$$
(4)
$$a\left( {h_{m} } \right) = \left( {1.1.\log \left( f \right) - 0.7} \right).h_{m} - 1.56.\log \left( f \right) + 0.8$$
(5)
$$C_{m} = \left\{ {\begin{array}{*{20}l} {0\, dB, \quad for \,medium\, sized \,city \,and \,suburban \,centres \,with \,\,medium\, tree \,density} \hfill \\ {3 \,dB, \quad for\, metropolitan\,\, centres} \hfill \\ \end{array} } \right.$$
(6)

The COST-Hata model is restricted to the following range of parameters [9]: Operating frequency, f (1500–2000 MHz); base station antenna height, hb (30–200 m); mobile terminal antenna height, hm (1–10 m) and the distance, d (1000–2000 m). The application of the COST-Hata model is restricted to large and small macro-cells, i.e. base station antenna heights above the roof-top levels adjacent to the base station. Hata formula and its modification must not be used for micro-cells since the validity of the model is restricted for distance over 1 km [13].

2.2.3 Walfisch–Bertoni Model

The Walfisch–Bertoni path loss model is given by Eqs. 7, 9 and 10 [14].

$$PL_{o} = 32.4 + 20 \cdot \log \left( f \right) + 20 \cdot { \log }\left( {d/1000} \right)$$
(7)

where PLo is the free space path loss formula, d is the distance in meters, and f is the frequency in MHz.

$$PL_{ex} = 57.1 + A + \log \left( f \right) + 18 \cdot \log \left( {\frac{d}{1000}} \right) - 18 \cdot \log \left( {h_{b} - h_{r} } \right) - 18 \cdot log\left[ {1 - \frac{{\left( {d/1000} \right)^{2} }}{{17\left( {h_{b} - h_{r} } \right)}}} \right]$$
(8)

where PLex is the excess path loss. The influence of the building geometry is contained in the term A,

$$A = 5 \cdot log\left[ {\left( {\frac{b}{2}} \right)^{2} + \left( {h_{r} - h_{m} } \right)^{2} } \right] - 9 \cdot \log \left( b \right) + 20 \cdot log\left[ {tan^{ - 1} \cdot \left( {2 \cdot \frac{{\left( {h_{r} - h_{m} } \right)}}{b}} \right)} \right]$$
(9)

hr is the height of a building in meters, hm is the mobile terminal antenna height in meters, hb is the base station antenna height and b is the building separation in meters.

The overall path loss PL is found by adding PLex to the free space path loss PLo as Eq. 10.

$${\text{PL}}_{\text{dB}} = {\text{ PL}}_{\text{o}} + {\text{ PL}}_{\text{ex}}$$
(10)

The Walfisch–Bertoni model is especially suitable for predicting average path loss in urban environments in the UHF band (300 MHz–3 GHz) [14].

2.2.4 COST231-Walfisch–Ikegami LOS Model

The COST231 Walfisch Ikegami (WI) model allows for improved path loss estimation by consideration of more data to describe the character of the urban environment [9]. The model distinguishes between the line-of-sight (LOS) and the non-line-of-sight (NLOS) situations. The COST231 WI LOS model is given by Eq. 11 [9]:

$${\text{PL}}_{\text{dB}} = \, 42.6 \, + 26 \cdot { \log }\left( {{\text{d}}/1000} \right) + \, 20 \cdot { \log }\left( {\text{f}} \right)\quad {\text{for d }} \ge \, 20\;{\text{m}}.$$
(11)

where PL is in dB, d is distance in meters and f is frequency in MHz.

2.2.5 COST231-Walfisch–Ikegami NLOS Model

In the NLOS case the basic transmission loss is composed of the terms free space loss Lo, multiple screen diffraction loss Lmsd, and roof-top-to-street diffraction and scatter loss Lrts [9]. The path loss formula for this model is given by Eq. 12 [9].

$$PL_{dB} = \left\{ {\begin{array}{*{20}l} {L_{o} + L_{rts} + L_{msd} ,\quad for \quad L_{rts} + L_{msd} > 0} \hfill \\ {L_{o} ,\quad for \,L_{rts} + L_{msd} \le 0} \hfill \\ \end{array} } \right.$$
(12)

The free space path loss is given by Eq. 13.

$$L_{o} = 38.4 + 20 \cdot { \log }\left( d \right)$$
(13)

Lrts is the roof-top-to-street diffraction and scatter loss in dB, and it is given by Eq. 14.

$$L_{rts} = - 16.9 - 10 \cdot \log \left( w \right) + 10 \cdot \log \left( f \right) + 20 \cdot \log \left( {\delta h_{m} } \right) + L_{ori}$$
(14)

Lori is given by Eq. 15,

$$L_{ori} = \left\{ {\begin{array}{*{20}l} { - 10 + 0.354 \cdot \emptyset \quad for \quad 0 \le \emptyset < 35} \hfill \\ {2.5 + 0.075 \cdot \left( {\emptyset - 35} \right) \quad for \quad 35 \le \emptyset < 55} \hfill \\ {4 - 0.114 \cdot \left( {\emptyset - 55} \right) \quad for \quad 55 \le \emptyset < 90} \hfill \\ \end{array} } \right.$$
(15)

where ϕ is the road orientation with respect to the direct radio path in degrees. The Lmsd is the multiple screen diffraction loss in dB, and it is given by Eq. 16.

$$L_{msd} = L_{bsh} + k_{a} + k_{d} \cdot log\left( {\frac{d}{1000}} \right) + k_{f} \cdot \log \left( f \right) - 9 \cdot { \log }\left( b \right)$$
(16)

Lbsh is expressed as in Eq. 17, ka, kd and kf are expressed as in Eqs. 18, 19 and 20 respectively.

$$L_{{bsh}} = \left\{ {\begin{array}{*{20}l} { - 18.\log \left( {1 + \Delta h_{b} } \right)} \hfill & {for\quad \,~h_{b} > h_{r} } \hfill \\ 0 \hfill & {for~\quad \,h_{b} \le h_{r} } \hfill \\ \end{array} } \right.$$
(17)
$$k_{a} = \left\{ {\begin{array}{*{20}l} {54} \hfill & {for{\mkern 1mu} \quad h_{b} > h_{r} } \hfill \\ {54 - 0.8 \cdot \Delta h_{b} } \hfill & {for{\mkern 1mu} \,\quad d \ge 500{\mkern 1mu} \,{\text{m}}{\mkern 1mu} \,\,and\,\,{\mkern 1mu} h_{b} \le h_{r} } \hfill \\ {54 - 0.8 \cdot \Delta h_{b} \cdot \frac{d}{{500}}} \hfill & {for{\mkern 1mu} \,\quad d < 500{\mkern 1mu} \,{\text{m}}\,\,and\,h_{b} \le h_{r} } \hfill \\ \end{array} } \right.$$
(18)
$$k_{d} = \left\{ {\begin{array}{*{20}l} {18} \hfill & {for\quad ~h_{b} > h_{r} } \hfill \\ {18 - 15.\frac{{\Delta h_{b} }}{{h_{r} }}} \hfill & {for~\quad h_{b} \le h_{r} } \hfill \\ \end{array} } \right.$$
(19)
$$k_{f} = - 4 + \left\{ {\begin{array}{*{20}l} {0.7.\left( {\frac{f}{925} - 1} \right)\quad for\, medium\, sized\, city \,and\, suburban \,centres\, with \,medium\, tree\, density} \hfill \\ {1.5.\left( {\frac{f}{925} - 1} \right) \quad for \,metropolitan\, centres} \hfill \\ \end{array} } \right.$$
(20)

∆hm and ∆hb are expressed as in Eqs. 21 and 22 respectively.

$$\Delta {\text{h}}_{\text{m}} = {\text{ h}}_{\text{r}} {-}{\text{ h}}_{\text{m}}$$
(21)
$$\Delta {\text{h}}_{\text{b}} = {\text{ h}}_{\text{b}} {-}{\text{ h}}_{\text{r}}$$
(22)

hb is the base station antenna height in meters, hr is the height of a building in meters, hm is the mobile station antenna height in meters, d is the distance in meters, f is the center frequency of the carrier in MHz, w is the widths of the roads in meters and b is the building separation in meters. The COST-WI model is restricted to [9]; f (800–2000 MHz), hb (4–50 m), hm (1–3 m) and d (20–5000 m).

3 Simulations

In this section, direct communication and multi-hop communication performances are compared. In the multi-hop communication case, one hop (one repeater) is used. It is assumed that the repeater is in the mid point of the BS and the UE. Multi-hop and direct communication cases are depicted in Fig. 1. In Figs. 2 and 3, equivalent block diagrams are depicted for direct communication and multi-hop communication cases respectively.

Fig. 1
figure 1

Direct communication and multi-hop communication with one hop

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Fig. 2
figure 2

Direct communication equivalent block diagram

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Fig. 3
figure 3

Multi-hop communication equivalent block diagram for one hop

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In Fig. 2, Pr is the BS received power, Pt is the UE transmit power, Gr is the receive antenna gain, Gt is the transmit antenna gain and L is the propagation loss. In Fig. 3, Pr2 is the BS received power, Pt1 is the repeated UE transmit power, Gt1 is the repeated UE transmit antenna gain, Gr1 is the repeater receive antenna gain, G is the repeater gain, Gt2 is the repeater transmit antenna gain, Pr2 is the BS receive antenna gain, L1 and L2 are propagation losses.

The link equation for Fig. 2 is expressed as in Eq. 23.

$$10 \cdot log\left( {P_{r} } \right) = 10 \cdot log\left( {P_{t} } \right) + 10 \cdot log\left( {G_{t} } \right) + 10 \cdot log\left( {G_{r} } \right) - 10 \cdot { \log }\left( L \right)$$
(23)

The link equation for Fig. 3 is expressed as in Eq. 24.

$$10 \cdot \log \left( {P_{r2} } \right) = 10 \cdot \log \left( {P_{t1} } \right) + 10 \cdot \log \left( {G_{t1} } \right) + 10 \cdot \log \left( {G_{r1} } \right) + 10 \cdot \log \left( G \right) + 10 \cdot log\left( {G_{t2} } \right) + 10 \cdot log\left( {G_{r2} } \right) - 10 \cdot log\left( {L_{1} } \right) - 10 \cdot log\left( {L_{2} } \right)$$
(24)

The simulations are based on the comparison between the 10·log(L) (direct communication case) and 10·log(L1) + 10·log(L2) (multi-hop communication case). For this comparison, three different scenarios are drawn. In each scenario, different path loss models are applied between the BS and the UE, BS and the Repeater, and the Repeater and the UE. These path loss models are compared according to different distances. The parameters which are used in simulation are given in Table 2.

Table 2 Propagation parameters for simulation [9, 15]
Full size table

Scenarios are drawn according to three different cases;

  1. 1.

    Between a BS and a UE.

  2. 2.

    Between a BS and a repeater.

  3. 3.

    Between a repeater and a UE.

For each case different path loss models and environment conditions are applied. Multi-hop and direct communication system performances are compared based on the following scenarios.

3.1 Scenario I

In scenario I, the simplest path loss model is applied to all cases. Different path loss exponents are used for cases (1), (2) and (3). Simulation results are depicted in Figs. 4 and 5.

Fig. 4
figure 4

Direct communication and multi-hop communication performance comparison for scenario I

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Fig. 5
figure 5

Direct communication and multi-hop communication performance comparison for scenario I

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In all figures, path gain term is used instead of the path loss term (“Path Gain = −Path Loss”, both in dB). In figures, ‘n1’ indicates the path loss exponent for direct communication case, ‘n2’ and ‘n3’ indicate the path loss exponents for multi-hop communication case between the BS-Repeater and the Repeater-UE respectively.

In Fig. 4, square dotted line, star dotted line and circle dotted line show the direct communication case with path loss exponents 3, 5 and 6, respectively. Solid line shows the multi-hop communication case. In the multi-hop communication case, the path loss exponent is used as 3 for both BS-Repeater and Repeater-UE. As can be seen in Fig. 4, the path gain in multi-hop communication case is lower than the direct communication case. If the path loss exponent is 7 or more than this, then the path gain for the multi-hop communication case is lower than the direct communication case. But this is also a non realistic situation.

In Fig. 5, star dotted line, circle dotted line, and square dotted line show the direct communication case with path loss exponents 4, 5 and 6, respectively. Solid line shows the multi-hop communication case with the path loss exponent 2 for both BS-Repeater and Repeater-UE. As seen in Fig. 5, if the path loss exponent is larger than the about five for the direct communication case, the multi-hop com-munication performance is better than the direct communication case. According to simulation results, when the path loss exponent in the multi-hop communication case is much smaller than in the direct communication case, then we can expect better performance for the multi-hop communication system.

For the direct communication case, the path loss is formulated as in Eq. 25.

$$PL1 = 20 \cdot log\left( {\frac{{4\pi d_{o} }}{\lambda }} \right) + 10 \cdot n_{1} \cdot log\left( {\frac{d}{{d_{o} }}} \right)$$
(25)

For the multi-hop communication case with one hop, the path loss is formulated as in Eq. 26.

$$PL2 = 40 \cdot log\left( {\frac{{4\pi d_{o} }}{\lambda }} \right) + 10 \cdot n_{2} \cdot log\left( {\frac{{d_{1} }}{{d_{o} }}} \right) + 10 \cdot n_{3} \cdot log\left( {\frac{{d_{2} }}{{d_{o} }}} \right)$$
(26)

To get lower propagation loss in the multi-hop communication case than in the direct communication case, the Eq. 27 should hold.

$$PL2 < PL1$$
(27)

After some mathematical operations, the Eqs. 28 or 29 can be found.

$$\log \left[ {\left( {\frac{{4\pi d_{o} }}{\lambda }} \right)^{2} \cdot \left( {\frac{{d_{1} }}{{d_{o} }}} \right)^{{n_{2} }} \cdot \left( {\frac{{d_{2} }}{{d_{o} }}} \right)^{{n_{3} }} \cdot \left( {\frac{{d_{o} }}{d}} \right)^{{n_{1} }} } \right] < 0$$
(28)

or

$$\left( {\frac{{d_{1} }}{{d_{o} }}} \right)^{{n_{2} }} \cdot \left( {\frac{{d_{2} }}{{d_{o} }}} \right)^{{n_{3} }} \cdot \left( {\frac{{d_{o} }}{d}} \right)^{{n_{1} }} < \left( {\frac{\lambda }{{4\pi d_{o} }}} \right)^{2}$$
(29)

If the path loss exponent n1 is large enough in the direct communication case because of the bad environment condition, then if we choose the optimum repeater position such away that the path loss exponents n2 and n3 are very low and d1 and d2 distances are optimum so the above equation holds, then we can expect the some path loss reductions in multi-hop communication case according to the direct communication case. The important issue here is to estimate the path loss exponents between the terminals and to determine the optimum number of repeaters and the distances between them.

3.2 Scenario II

In scenario II, the proposed propagation models for multi-hop communication and the direct communication systems are: For case (1) and case (2), the COST-Hata model is used. For case (3), the simplest path loss model is used for different path loss exponents. In Fig. 6, comparison between the direct communication and the multi-hop communication is shown.

Fig. 6
figure 6

Direct communication and the multi-hop communication performance comparison for scenario II

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3.3 Scenario III

In scenario III, the proposed propagation models for multi-hop communication and the direct communication systems are: For case (1), the COST-WI-NLOS model is used. For case (2), the COST-WI-LOS model is used and for case (3), the simplest path loss model is used for different path loss exponents. In Fig. 7, comparison between the direct communication and the multi-hop communication is shown.

Fig. 7
figure 7

Direct communication and the multi-hop communication performance comparison for scenario III

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3.4 Scenario IV

In scenario IV, the proposed propagation models for multi-hop communication and the direct communication systems are: For case (1) and (2), the Walfisch Bertoni model is used. For case (3), the simplest path loss model is used for different path loss exponents. In Fig. 8, comparison between the direct communication and the multi-hop communication is shown.

Fig. 8
figure 8

Direct communication and the multi-hop communication performance comparison for scenario IV

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Based on Figs. 6, 7 and 8, it might be never possible to reduce the total transmit power in the multi-hop communication system according to scenario II, III and IV. To decrease the total transmit power, by splitting a communication path into two parts, large increases in the path gains are needed, and these might be difficult to come by in these scenarios. Path gain curves are going to be almost flat after the about 1 km distance. If the path loss exponent changes drastically between the BS and the UE because of the different environment conditions, in this case it is possible to expect the big path loss changes between the BS and the UE. In such a case, splitting the path between the BS and the UE into a number of hops allows to reduce the path loss.

Most propagation models assume that the transmit antenna is high and the receive antenna is low. However, the repeater and the UE antennas are low placed antennas. This also shows that it is needed more elaborated path loss models for multi-hop communication systems, especially between the repeater and the UE.

4 Conclusions

In this paper, multi-hop and direct communication cases are compared by using the different scenarios. According to comparisons, in some cases it is possible to reduce the path losses by using the multi-hop communication but generally it is not possible to reduce the path losses. To reduce the path loss between the BS and the UE by using the multi-hop communication system, large changes (reductions) in the path loss between the BS and the UE are needed.

For multi-hop communication, to investigate the more accurate path loss models is one of the important issues. The other important issue in multi-hop communication systems is to define the optimum number of repeaters and the optimum distances. This will also affect the multi-hop system performance.

By using the multi-hop communication, almost each individual mobile terminal’ transmit power reduces even though total transmit power increases except for some cases. Reduction in the transmit power of the each individual mobile terminal gives the opportunity to reduce the interference problems and perhaps also reduce the potential health risk. In the multi-hop communication system, using the some user equipments as repeaters will increase the coverage area of the network and also reduce the dead spots in the cell. These benefits of the multi-hop communication systems are also important even though it is not possible to reduce the path losses except for some cases. Yet another issue is that the repeater gain between Rx and Tx has not been included. This gain helps on the performance of multi-hop networks.

Multi-hop communication systems hold promise for next generation mobile systems. However, multi-hop systems need to overcome some challenging topics. To develop an accurate and more elaborated path loss model and to develop optimum routing algorithms are some of them.