Exact Performance of Wireless Multihop Transmission for M-ary ...

Exact Performance of Wireless Multihop 

Transmission for M-ary Coherent Modulations 

over Generalized Gamma Fading Channels 

Ferkan YILMAZ 1-2 and Oğuz KUCUR 1 

Electronics Engineering Dept., Gebze Institute of Technology, P.K. 141, 41400, Gebze-Kocaeli, Turkey 1 

Source 

Terminal 

Vodafone IT, ITU Teknokent, ARI 2 Binasi, B Blok/1, 34467, Maslak-Istanbul, Turkey 2 

Abstract—Closed-form moment generation function (MGF) 

expressions for the harmonic mean of N independent and not 

identically distributed generalized gamma random variables 

(RVs) are presented and some well known special cases are 

accentuated. The statistical results are also applied in order to 

obtain exact symbol error rate (SER) performance of wireless 

multihop transmission with non-regenerative relays and M-ary 

coherent modulations over generalized gamma flat fading 

channels. 

Index Terms—Symbol error rate (SER), harmonic mean, 

moment generating function (MGF), multihop transmission, 

cooperative/collaborative diversity, generalized gamma fading. 

M 

I. INTRODUCTION 

ULTIHOP relaying technology in which two 

terminals are communicating with each other via 

relaying terminals [1], [2] is an inspiring and expected 

solution for such quality requirements as throughput, high 

data-rate and coverage in bent pipe satellites and microwave 

links, as well as ad-hoc wireless, future cellular (4G and 

5G), WLAN, hybrid wireless communication systems [3] 

without the need to employ large power at the transmitter, 

and to reduce deeply fading or shadowing effects through 

spatial repetition [4]-[6]. The concept of wireless multihop 

transmission is also encountered in cooperative and 

collaborative wireless communications [7]. In multihop 

transmission, when the direct link between source and 

destination terminals is deeply faded, the signal from source 

terminal to the destination terminal propagates through the 

terminals between them, each of which relays a signal 

received from the previous terminal to the next terminal in 

series as shown in Figure 1. Generally, there exist two types 

of such relays. If each relaying terminal in multihop 

transmission just amplifies and re-transmits the information 

signal recovered from previous relaying terminal, it is called 

non-regenerative, while relaying terminals decodes the 

E-mail: ferkan.yilmaz@{vodafone, gmail}.com, okucur@gyte.edu.tr 

Relaying 

Terminal 

Relaying 

Terminal 

signal and then transmits the detected version to the next 

node, it is called regenerative. 

There exist a few published papers concerning wireless 

multihop transmission. An analysis for the performance of 

wireless dual-hop transmission with non-regenerative relays, 

operating over independent but not necessarily identically 

distributed (i.d.) Nakagami-m fading channels, is presented 

in [8] by Zogas et.al. and Hasna and Alouini put forward the 

outage and the error performance of wireless dual-hop 

transmission with regenerative and non-regenerative relays 

over Rayleigh [9] and Nakagami-m [10] fading channels 

[11]. In addition, Karagiannidis et. al. employed the 

geometric mean approximation of harmonic mean in order 

to produce upper performance limit of wireless N-hop 

transmission with fixed gain relays 2 over 

Nakagami-m and Ricean fading channels [8]. To the best of 

our knowledge, the exact performance of wireless multihop 

transmission has not been investigated not only for the case 

of more than one relaying terminals but also for generalized 

gamma fading channels. 

In this paper, we focus on N-hops 2 transmission 

with non-regenerative relays and obtain end-to-end 

performance for coherent M-ary modulation schemes over 

independent, not necessarily identical, generalized gamma 

fading channels. The main contribution of this paper is to 

derive closed form expressions for moment generating 

function (MGF) of normalized harmonic mean of N 

generalized gamma (GG3) random variables (RVs) since 

end-to-end performance in multihop transmission is 

connected with the normalized harmonic mean of signal-tonoise 

ratios (SNR) of relaying / hop links (as it is explained 

in Section II). Besides, other such statistical measures as 

probability density function (PDF), cumulative distribution 

function (CDF) can be obtained easily through the medium 

of employing the linear transformations, derivations and 

linear algebraic manipulations of MGF. Eventually, the endto-end 

exact performance of wireless multihop transmission 

Relaying 

Terminal 

Figure 1. Wireless multihop communication where S communicates with the destination terminal D 

through 1 relaying terminals, , ,…, . 

978-1-4244-2644-7/08/$25.00 © 2008 IEEE 

Destination 

Terminal

over generalized gamma fading channels is obtained. 

The remainder of this paper is organized as follows: 

Section II introduces wireless multihop transmission and the 

fading channel models. Section III presents MGF of 

normalized harmonic mean of N independent GG3 RVs and 

the obtained results are accentuated for such well known 

special cases as single-hop and dual-hop transmissions in 

literature. Upon applying the obtained results, the end-toend 

performance of wireless multihop transmission with 

nonregenerative relays is evaluated in Section IV. Finally, 

Section V summarizes the main results of the paper. 

II. MULTIHOP TRANSMISSION 

Consider a multihop (N-hop) wireless transmission as 

shown in Figure 1, which operates over independent and not 

identically distributed GG3 fading channels, the source 

terminal S and destination terminal D are too far apart for a 

single link within the sight of power constraints and channel 

fading effects. However, they can be connected using 

multiple wireless links arranged in an end-to-end 

configuration such that the source terminal S communicates 

with the destination terminal D through 1 relaying 

terminals ,,…,. Clearly, these terminals relay the 

information signal only from one hop to the next, acting as 

nonregenerative relays. All relaying terminals concurrently 

receive and transmit in the same frequency band, and no 

latency is incurred in the whole chain of relaying 

transmissions and there is no wireless multihop diversity. 

Assume that the source terminal S transmits a signal , 

which has an average power normalized to one, to the 

destination terminal D through relaying terminals. The 

received signal at the first terminal can be written as 

, (1) 

where is the fading amplitude of the channel between 

source terminal S and the first relaying terminal and 

, is an additive white Gaussian noise (AWGN) signal 

with one sided power spectral density, ,. After the 

received signal at the first terminal is multiplied with which is the gain, it is retransmitted to the next relaying 

terminal (the second relaying terminal) . The received 

signal at the second relaying terminal can be written as 

, , (2) 

where is the fading amplitude of the channel between the 

first relaying terminal and the second relaying terminal 

and , is an AWGN signal with one sided power 

spectral density, ,. In the same manner, the received 

signal at the second terminal is multiplied with and 

retransmitted to the next terminal (the third relaying 

terminal) . The received signal at the third relaying 

terminal can be written as 

 

(3) 

, , , where is the fading amplitude of the channel between the 

second relaying terminal and the third relaying terminal 

and , is an AWGN signal with one sided power 

spectral density, ,. Finally, at the end of whole chain of 

relaying transmissions, the received signal at the destination 

terminal D can be given as [3] 

 

, 

 

 

 

 

with 1. Then, it is straightforward that the end-to-end 

signal-to-noise ratio (SNR), i.e., the SNR at destination 

terminal D, can be written as [4] 

 

(4) 

 

 

, 

(5) 

 

 

 

 

 

It is clear from (5) that the choice of the relay gain 

determines the equivalent SNR between source and 

destination terminals. One choice for the gain is [4] 

1⁄ 

(6) 

where the relay just amplifies the incoming signal with the 

inverse of the channel fading amplitude of the previous hop, 

regardless of the noise of that hop. As mentioned in [4], 

such a relaying technique serves as a benchmark for all 

practical multihop systems employing non-regenerative 

relays. By applying (6) into (5), the end-to-end 

instantaneous SNR (the instantaneous SNR at destination 

terminal D) becomes 

1 

 

1 

 

1 

 

 

1 

(7) 

where the instantaneous SNRs , 1,2,…, are given as 

, 

(8) 

Here, the amplitude fading , 1,2,…, of relays in 

wireless multihop transmission are modeled as GG3 fading 

[12]. Consequently, the PDF of instantaneous SNRs of hops, 

, can be expressed as follows 

Γ 

 

Ω exp 

 

Ω (9) 

over 0 ∞, where Γ 1⁄ ⁄ Γ and 

1⁄ 2, 

0 are such two parameters as the fading 

parameter, the shape parameter, respectively, related to 

fading severity. Ω is the power-scaling parameter of the 

random variable, defined as Ω , with . 

denoting expectation operator. Additionally, GG3 fading 

model covers a wide range of fading conditions for different 

shape parameter and fading figure values. Such commonly 

employed fading models in literature as the Rayleigh 

ξ1, Nakagami-m ξ 1, Weibull 1, lognormal 

∞,ξ0, and AWGN ∞,ξ1 are special 

or limiting cases of the generalized gamma distribution. 

Ricean fading can also be closely approximated by 

generalized gamma fading if the shape parameter ξ1 and 

the relationship 21⁄ 2 1 between the

Ricean factor κ and the fading figure, m exists. As a step 

forward, using definition of MGF / (one-sided) Laplace 

transform given as 

|,,Ω 

| , ,Ω 

 

(10) 

where · | is the Laplace transformation operator 

with respect to x and exploiting Mellin-Barnes integral, we 

can easily derive Moment Generating Function (MGF) as 

follows 

, Ω H, 

1 , 1 

 

,0 

(11) 

0,1 

, 

where H, ·, ,,, is Fox’s H function defined 

in [13], [14]. 

III. STATISTICAL BACKGROUND 

In order to study important statistical metrics of the end-toend 

SNR, MGF of the normalized harmonic mean of the 

GG3 RVs given in (7) can be given as follows 

 

 

 

 

1⁄ 1⁄ 1⁄ 

 

|,, Ω … 

(12) 

After algebraic manipulations based Mellin-Barnes integrals 

and Fox’s H functions [14], we have obtained exact MGF of 

the end-to-end SNR for wireless multihop transmission as 

follows 

 

2√H , 

where H is defined as 

 

 

, Δ,∞ H, H, 

Δ1,1, Δ, 

 

 

Ω 

, Δ,∞ H, 

Δ0,1, Δ, 

 

 

Ω 

(13) 

(14) 

where Δ, and are the parameters of Fox’s H 

function defined as 

1 

Δ, , 

 

1 

(15) 

 

 

 

 

Ω 

 

(16) 

Finally, let us consider some special cases in order to check 

analytical simplicity and accuracy: 

A. Special Case I (Single-hop special case): Most of the 

investigated communications systems in literature are based 

on single-hop, which is the special case of wireless multihop 

transmission such that there is no relaying terminal in the 

transmission (total number of hops 1). Besides, most of 

the results for single-hop communications systems are given 

for Nakagami-m fading channels. Then, we can simplify 

MGF given in (13) for single-hop Nakagami-m 

0.5, 1 fading channels as follows 

 

ΩΓ 

, 

2√H, 

Ω 

 

0,1, 1,1 

 

(17) 

 

Using Fox’s H function equality [13, eq. (2.9.19)], 

, 

H, 

4 

 

 

2 

,1 , 

2 

,1 2 

2 

 

, (18) 

where , and is the nth-order modified Bessel 

function of second kind, we simplify the MGF as 

2 

Γ 

Ω 

 

 

 

 

2√ 2 

Ω 

 

(19) 

 

Then, substituting the integral equality [15, eq. (3.16.11.7)] 

into (19) gives MGF for Nakagami-m fading channels as 

follows 

1 Ω 

 

 

(20) 

which is well-known in literature [16, eq. (5.15)], [17]. 

B. Special Case II (Balanced double-hop special case): For 

the case in which there is one relaying terminal in the 

transmission 2, we can simplify MGF given in (13) 

for Nakagami-m 0.5, 1 

fading channels as follows 

2 

ΩΓ 2√ 

 

 

, 

H, 

Ω 

 

1 

2 ,1,1 

2 ,1 

, 

H, 

Ω 

 

1 

2 ,1,1 

2 ,1 

(21) 

in which replacing Fox’s H equality given in (18) gives 

 

 

 

 

8 

Γ 

Ω 

2√ 

 

 

2 

Ω 2 

Ω 

(22) 

Finally, upon employing [15, eq. (8.4.23.29)], MGF for 

Nakagami-m fading channels is obtained as follows 

, 2; 1 Ω 

; 

2 4 

(23) 

which has been already proposed in [10, eq. (5)], where 

·,·;·;· is the Gauss’ hypergeometric function defined in 

[18, eq.(15.1.1)]. If the fading figure 1 (i.e. it is the 

balanced double-hop transmission over Rayleigh fading 

channels), then (23) reduces into [9, eq. (21)].

C. Special Case III (Balanced triple-hop special case): For 

the case in which there exists two relaying terminals in 

transmission (total number of hops is 3), we can easily 

simplify MGF given in (13) for Nakagami-m 

0.5, 1 fading channels as 

follows 

 

 

24 

Γ 

Ω 

 

 

2√ 2 

 

 

Ω 2 

Ω 

(24) 

This integral can be easily taken by means of substituting 

power series of Bessel functions into (24). Then, the MGF, 

having not proposed in literature yet, is obtained as follows 

3√ 

 

2 

Γ 1 

 

 

Ω 

 

 

∞ 

 

! ! 

 

 

, 

G, 4 

23 

, 

3 

133 

3 

, 6 1 

6 

2 3 2 

3 

, 

3 

, 6 2 

(25) 

 

3 

, 

where G, ·, ,,, is the MeijerG function 

defined in [19, eq. (9.301)]. However, this obtained MGF 

has singularity while expressing performance expression for 

M-PSK modulation schemes. In order to avoid this problem, 

we can apply Laguerre Transformation [20] onto MGF. 

Consequently, we obtain the MGF as follows 

3√ 

 

2 

Γ 

 

Ω Ω 

 

 

∞ 

L (26) 

! 

 

where L is defined as 

L 1 

 

, 

G, 4 

! 

23 

, 

3 

133 

3 

, 6 1 

6 

2 3 2 

3 

, 

3 

, 6 2 

 

(27) 

 

3 

D. Special Case IV (Balanced N-hop special case): For the 

case that there are 1 relaying terminals in the system, 

(the total number of hops is ), we can simplify MGF for 

Nakagami-m 0.5, 1 fading channels, which has not 

proposed in literature yet, as follows 

2 

Γ 

Ω 

∞ 

 

2√ 

 

2 

Ω 2 

Ω 

(28) 

MGF for different hop counts is depicted in Figure 2 by 

means of using (28). If the total number of relaying 

terminals in transmission goes to infinity ∞, the 

MGF becomes 

lim 

∞ 1, 0 

0, (29) 

which means that the performance of wireless multihop (Nhop) 

transmission deteriorates when increases, i.e., lower 

limit symbol error rate (SER) performance of wireless 

multihop transmission for M-PSK signaling is ⁄ 1 

error/symbol, where is the modulation level, for DPSK / 

BPSK is 0.5 error/symbol. As a result, total number of hops 

is limited with respect to the desired performance and 

coverage. 

Figure 2. The variation of MGF for different hop counts (fading figures 

, 2, shaping parameters , 1 and average powers , Ω 

1 , 1,2, … , ) 

IV. PERFORMANCE EVALUATION 

Having the MGF of the end-to-end instantaneous SNR, and employing the MGF-based approach for the 

performance evaluation of digital modulations over fading 

channels [16], the obtained MGF result in (13) enables us to 

obtain SER performance for various M-ary modulations 

[such as M-ary phase shift keying (MPSK), M-ary 

differential phase shift keying (MDPSK), and M-ary 

quadrature amplitude modulation (MQAM)] as follows 

 

 

 

V 2Λ (30) 

 

where , , , V, Λ and are the performance 

parameters for M-ary modulation which are given in Table 

I. For instance, the performance expression for coherent 

MDPSK modulations is easily reduced to [16] 

 

 

2 

 

2 

 

1 

 

2 

 

(31) 

 

In literature, the most of the performance results 

concerning wireless multihop transmission are given for 

binary DPSK (BDPSK). On account of that, Figure 3 

compares the bit error rate (BER) performances of wireless 

multihop transmission with non-regenerative relays and 

DPSK modulation for different hop counts. Similar to the 

MGF results obtained from Figure 2, it is clear that the 

number of hops increases then the performance deteriorates. 

Therefore, total number of hops in transmission is limited 

with respect to the desired performance and coverage under

TABLE I 

PARAMETERS FOR SPECIFIC MODULATIONS 

M-ary 

Modulation 

Schemes 

V Λ MPSK 1 

MDPSK 1 

MQAM 2 

MPAM 1 

1 

 

2 

 

4√ 4 

√ 

44√ 

√ 

2 2 

 

1 

0 

2 

 

 

1 

 

0 1 

2 

0 1 

2 

0 1 

2 

the assumption that there is no multihop diversity in the 

transmission. Besides, if the fading figure increases (i.e. 

the channel fading conditions improve), the performance of 

wireless multihop transmission with nonregenerative relays 

will converge to the performance of BDPSK over AWGN 

channels. 

Figure 3. BER performance of wireless multihop transmission with nonregenerative 

relays (fading figures , 1,2, shaping parameters 

, 1, 1,2, … , ). 

V. CONCLUSION 

 

 

 

 

 

 

3⁄ 2 

1 

3⁄ 2 

1 

3 

1 

The parameters for such common coherent modulations as M-ary phase shift 

keying (MPSK), M-ary differential phase shift keying (MDPSK), M-ary square 

quadrature amplitude modulation (MQAM) with and even, M-ary pulse amplitude 

modulation (MPAM). 

In this paper, we presented not only the exact and 

generalized performance of wireless multihop (N-hop) 

transmission 2 with nonregenerative relays for M-ary 

modulations over generalized gamma (GG3) fading 

channels but also the MGF of the normalized harmonic 

mean of N GG3 random variables by means of Mellin- 

Barnes integrals and Fox’s H functions. Such special cases 

of multihop transmission as single-hop, balanced doublehop, 

balanced triple-hop and balanced N-hop are 

accentuated for Nakagami-m and Rayleigh fading channels. 

 

2 

 

4 

 

2 

Furthermore, the exact generalized performance expression 

is presented for M-ary modulation schemes and the 

performance for BDPSK modulation is depicted on the 

purpose of comparing the obtained results for wireless 

double-hop transmission with the literature. Besides, it is 

shown that there is a threshold between number of hops and 

the desired performance and coverage when there is no 

multihop diversity in the transmission. 

[1] 

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