Skip to content

Advertisement

  • Research
  • Open Access

Performance analysis of generalized QAM modulation under η−μ and κ−μ fading

  • 1, 3,
  • 2Email author,
  • 1, 3 and
  • 1, 3
EURASIP Journal on Advances in Signal Processing20132013:104

https://doi.org/10.1186/1687-6180-2013-104

  • Received: 12 September 2012
  • Accepted: 25 April 2013
  • Published:

Abstract

This paper presents new closed-form expressions for the symbol error probability (SEP) of θ-QAM modulation with maximum ratio combining (MRC) receiver under ημ and κμ fading. The SEP formulae, obtained from the definite integrals of the moment generating function (MGF) of the signal-to-noise ratio (SNR) at the input of the MRC receiver, are written in terms of Lauricella functions. The numerical evaluation of the expressions is carried out for the ημ distribution, which includes important distributions as special cases, such as Hoyt, Nakagami-m, Rayleigh, and one-sided Gaussian, as well as for the κμ distribution, which includes Rice, Nakagami-m, Rayleigh, and one-sided Gaussian as special cases.

Keywords

  • Diversity Order
  • Moment Generate Function
  • Maximum Ratio Combine
  • Quadrature Amplitude Modulation
  • Symbol Error Probability

1 Introduction

Two-dimensional signal constellations with suppressed carrier, such as quadrature amplitude modulation (QAM) schemes, are widely used in communication systems in a variety of applications, such as modems with asynchronous transmission, digital television, and cooperative systems [1, 2].

In [3], the rectangular QAM modulation schemes are evaluated with maximum ratio combining (MRC) receivers and ημ correlated fading, while in [4], these schemes are employed to evaluate the performance of the MRC receiver considering imprecise knowledge of the state of the Rician fading channel. In applications involving asymmetric digital subscriber lines and digital video broadcasting-cable, cross QAM schemes have been adopted. The error probability of cross QAM with MRC reception over generalized ημ fading channel was presented in [5].

Before 2002, the evaluation of the fading effects in the reception of the M-QAM schemes was performed by means of approximate mathematical expressions. In 2002, Cho and Yoon [6] found exact expressions for the bit error probability (BEP) of QAM schemes under additive white noise. An extension of that work for Nakagami fading was presented in [7]. Then, the evaluation of structures such as the MRC receiver with M-QAM was performed by means of exact expressions. The MRC receiver has been proposed in coherent demodulation schemes for environments with Rayleigh and Nakagami fading [810].

Since 1962, when the square QAM (SQAM) modulation was introduced [11], different geometries for the QAM constellation have been proposed. Motivated by aspects such as minimization of the symbol error probability (SEP) and implementation complexity, a new constellation geometry was proposed in [12], the so-called triangular quadrature amplitude modulation (TQAM), for which the symbols associated with the transmitted signals are located in the vertices of contiguous triangles [13].

As an attempt to improve the power gain, Park and Byeon presented in [14] an alternative configuration for the triangular constellation and proposed constellations with irregularly distributed symbols while preserving the triangular structure. By means of theoretical analysis and simulations, they showed how to obtain a 0.62-dB power gain with respect to the SQAM constellation and a 0.20-dB power gain with respect to the TQAM regular constellation for a 10−6 symbol error rate with 64-symbol constellations. The mathematical analysis of the effects of changing the QAM constellation angle θ gave rise to a modulation scheme referred to as parametric θ-QAM, which includes the SQAM and TQAM schemes as special cases [15, 16].

The calculation of exact expressions for the SEP or BEP of a modulation scheme under fading means, in many cases,that the expressions for additive white Gaussian noise are weighted by the probability density function of the signal-to-noise ratio for a given fading. One of the most used structures to improve the BEP and SEP of QAM schemes under different types of fading is the MRC receiver [17]. In [18], the effect of the diversity provided by that structure was evaluated, through the BEP parameter, for square M-QAM and rectangular R-QAM and different types of fading, modeled by distinct distributions.

The analysis of BEP and bit error (BER) in optical wireless links, also known as free-space optics (FSO), has received much attention in recent years. In [19], the authors show that the turbulence-induced fading caused by atmospheric conditions can be modeled as multiplicative random process which follows the K distribution. The authors also present approximated closed-form expressions for the average BER of single-input multiple-output (SIMO) FSO systems. Alternatively, for FSO channels, the fading can be modeled by log-normal or gamma-gamma distributions [20, 21].

This paper proposes the use of the MRC receiver jointly with θ-QAM modulation and gives new mathematical expressions for the SEP under ημ and κμ fading, modeled by κμ and ημ distributions. The MRC receiver provides an additional degree of freedom for the performance control of the systems, and the probability distributions used to model the fading provide a unification of the mathematical analysis since they characterize different types of fading. One of the contributions of this paper is the derivation of exact expressions for the SEP, written in terms of hypergeometric Lauricella functions.

The paper is organized as follows: In Section 2, the average SEP for an MRC receiver with N branches is presented in terms of the moment generating function (MGF) for ημ and κμ fading. In Section 3, the MGF of the signal-to-noise ratio (SNR) per branch of the MRC receiver is adjusted to the problem under consideration. In Sections 4 and 5, new SEP expressions are provided in terms of Lauricella functions. In Section 6, curves of SEP are presented for different sets of parameters. Section 7 presents the conclusions of the paper.

2 Symbol error probability under generalized fading

This section presents the calculation of the average SEP of a θ-QAM scheme under ημ and κμ fading and MRC diversity. The main objective is to show that this probability can be written in terms of definite integrals of the MGF of SNR per branch of the MRC receiver. In the family of θ-QAM schemes, presented in [15], the authors obtained the expression given in Equation 1 to evaluate the SEP under additive white Gaussian noise:
P s ( γ , θ , M ) = c 1 c 2 Π θ 2 Π + θ 2 exp ( γ δ 2 csc 2 ( ϕ ) ) + c 1 c 3 2 θ Π θ exp ( γ δ 2 csc 2 ( ϕ ) sec 2 θ 2 sin 2 ( θ ) ) + c 1 c 4 Π θ 2 Π exp ( γ δ 2 csc 2 ( ϕ ) ) + c 1 c 5 Π θ 2 Π + 2 θ 2 exp ( γ δ 2 csc 2 ( ϕ ) ) + c 1 c 6 2 θ Π exp ( 4 γ δ 2 csc 2 ( ϕ ) sin 2 ( θ ) ) dϕ.
(1)
In Equation 1, the parameters c 1, c 2, c 3, c 4, c 5, and c 6 are related to the geometry of the constellation θ-QAM and are given by the following [16]:
c 1 = 1 2 ΠM c 2 = 4 ( M 2 ) ( M 1 ) c 3 = 2 ( M 1 ) c 4 = 4 M c 5 = 4 ( M 2 ) c 6 = 2 ( M 2 ) ,
(2)
in which M represents the number of constellation symbols s m,n , whose coordinates (x m ,y n ) are
x m = 2 ( n 1 ) + 1 M d + 2 mod ( m , 2 ) 1 a 2 y n = 2 ( m 1 ) + 1 M b 2 ,
(3)
with m = 1 , , M , n = 1 , , M , a=2d cosθ, and b=2d sinθ, and m o d(x,y) denotes the modulus operation after division of x by y. Half of the Euclidean distance between adjacent symbols of the constellation is given by
d = 6 E av 3 M + ( 4 M ) cos ( 2 θ )
(4)
and is related to the parameter δ and to the average energy per symbol, E av, through the expression
δ = d E av = 6 3 M + ( 4 M ) cos ( 2 θ ) .
(5)

In order to determine the SEP of the θ-QAM modulation scheme under ημ and κμ fading, for an MRC receiver, it is necessary to average the symbol error probability under additive white Gaussian noise conditioned to the instantaneous SNR γ at the MRC input. This conditional probability, denoted by P(E|γ), corresponds to P s (γ,θ,M) for γ = k = 1 N γ k at the MRC output, in which the random variables γ k represent the instantaneous SNR in each of the N branches of the MRC receiver. The MRC structure with N branches makes decisions based on the signals α i e j θ i s(t)+n i (t), for i=1, 2, ,N, in which α i and θ i represent, respectively, the fading attenuations and phase variations in the i th receiver branch, s(t) represents the transmitted signal modulated with θ-QAM, and n i (t) represents the complex AWGN with zero mean and variance N 0/2. The fading attenuation α i can be modeled as ημ distribution as well as κμ distribution. The phase shifts θ i are uniformly distributed in the interval (0,2Π].

Therefore, the SEP, represented by P as, can be calculated averaging P s (γ,θ,M) by the joint probability density function p γ 1 , γ 2 , , γ N ( γ 1 , γ 2 , , γ N ) of the variables γ 1, γ 2, , γ N , using the multiple integral
P as = 0 0 0 P s ( γ , θ , M ) · p γ 1 , γ 2 , , γ N × ( γ 1 , γ 2 , , γ N ) d γ 1 d γ 2 d γ N .
(6)
The probability P s (γ,θ,M) can be rewritten as follows:
P s ( γ , θ , M ) = c 1 c 2 Π θ 2 Π + θ 2 k = 1 N exp δ 2 csc 2 ( ϕ ) γ k + c 1 c 3 2 θ Π θ k = 1 N exp δ 2 sec 2 θ 2 sin 2 ( θ ) csc 2 ( ϕ ) γ k + c 1 c 4 Π θ 2 Π k = 1 N exp δ 2 csc 2 ( ϕ ) γ k + c 1 c 5 Π θ 2 Π + 2 θ 2 k = 1 N exp δ 2 csc 2 ( ϕ ) γ k + c 1 c 6 2 θ Π k = 1 N exp 4 δ 2 sin θ csc 2 ( ϕ ) γ k dϕ.
(7)
Substituting Equation 7 into Equation 6, the SEP P as can be written as follows:
P as = c 1 c 2 Π θ 2 Π + θ 2 k = 1 N M γ k δ 2 csc 2 ( ϕ ) + c 1 c 3 2 θ Π θ k = 1 N M γ k δ 2 sec 2 θ 2 sin 2 ( θ ) csc 2 ( ϕ ) + c 1 c 4 Π θ 2 Π k = 1 N M γ k δ 2 csc 2 ( ϕ ) + c 1 c 5 Π θ 2 Π + 2 θ 2 k = 1 N M γ k δ 2 csc 2 ( ϕ ) + c 1 c 6 2 θ Π M γ k 4 δ 2 sin 2 ( θ ) csc 2 ( ϕ ) ,
(8)

in which M γ k ( s ) E e s γ k denotes the MGF of the k th random variable γ k .

From Equation 8, one can notice that if the variables γ k are independent and identically distributed with corresponding MGF M γ (s), then the SEP P as can be written as follows:
P as = c 1 c 2 Π θ 2 Π + θ 2 M γ δ 2 csc 2 ( ϕ ) N + c 1 c 3 2 θ Π θ × M γ δ 2 sec 2 θ 2 sin 2 ( θ ) csc 2 ( ϕ ) N + c 1 c 4 Π θ 2 Π M γ δ 2 csc 2 ( ϕ ) N + c 1 c 5 Π θ 2 Π + 2 θ 2 × M γ δ 2 csc 2 ( ϕ ) N + c 1 c 6 2 θ Π M γ 4 δ 2 sin 2 ( θ ) csc 2 ( ϕ ) N dϕ.
(9)
Under ημ fading, an exact expression for P as can be obtained by adjusting the integration intervals to the interval [0,1] in Equation 9. This is equivalent to modify the integrands of the integral representation of the Lauricella hypergeometric function as follows:
F D ( n ) ( a , b 1 , , b n , c ; x 1 , , x n ) = 1 B ( c a , a ) 0 1 t a 1 ( 1 t ) c a 1 ( 1 x 1 t ) b 1 ( 1 x n t ) b n dt ,
(10)

in which {c}>{a} and {x} denote the real part of x.

On the other hand, for κμ fading, the calculation of P as can be obtained using a series representation of the paremeter s. This representation is taken into account in M γ κ , μ ( s ) , as shown in the next section.

In [15], the BEP was computed using the product of constants and integral expressions. The integrals are related to the SEP of each decision region of the θ-QAM constellation, and the constants are concerned with the number of different bits between decision regions. It was assumed that the decision errors occur very close to the borders of these regions. In [15], these constants were presented as follows:
c 1 = 1 2 ΠM log 2 M c 2 = 4 ( M 2 ) ( M 1 ) c 3 = 2 ( M 1 ) c 4 = 5 ( M 2 ) + 6 c 5 = 3 ( M 2 ) + 2 c 6 = 2 M .
(11)
The BEP approximation obtained so far is highly accurate for medium and high values of SNR. In [16], corrected constants for the BEP were presented
c 4 = 4 M , c 5 = 4 ( M 2 ) , c 6 = 4 ( M 2 ) .
(12)

Using these new constants in Equation 9, one can obtain the BEP with MRC diversity effect.

3 MGF for η−μ and κ−μ fading

The expression presented in Equation 9 is evaluated in this article for the fading models characterized by ημ and κμ distributions. Both these fading models are extensively treated in [22].

The MGFs of these distributions were presented in [23] in a more compact form than their expressions presented in [24],
M γ η μ ( s ) = 4 μ 2 h ( 2 ( h H ) μ + s γ ¯ ) ( 2 ( h + H ) μ + s γ ¯ ) μ ,
(13)
M γ κ μ ( s ) = μ ( 1 + κ ) μ ( 1 + κ ) + s γ ¯ μ exp μ 2 κ ( 1 + κ ) μ ( 1 + κ ) + s γ ¯ μκ .
(14)

To evaluate the SEP in Equation 9, one needs to calculate M γ (s) for s=δ 2csc2(ϕ), 4δ 2sin2(θ)csc2(ϕ) and δ 2 sec 2 θ 2 sin 2 ( θ ) csc 2 ( ϕ ) and perform the integrations in Equation 9.

The parameters h and H of Equation 13 can be written in terms of the parameter η, respectively, as h=(2+η −1+η)/4 and H=(η −1η)/4 for format 1 and h=1/(1−η 2) and H=η/(1−η 2) for format 2. In format 1, the parameter η, 0<η<, represents the scattered wave power ratio between in-phase and quadrature components of clusters of multipath. In format 2, the parameter η, −1<η<1, represents the correlation coefficient between scattered wave in-phase and quadrature components of clusters of multipath. The parameter μ can be written as μ = 1 2 V ( α 2 ) 1 + H h 2 for ημ distribution and μ = 1 V ( α 2 ) 1 + 2 κ ( 1 + κ ) 2 for κμ distribution. The parameter κ represents the ratio between the total power of the dominant components and the total power of the scattered waves.

Since the Nakagami-m distribution can be obtained from the distribution κμ for κ→0 and μ=m, it is observed that the MGF M κμ (s) coincides exactly with the MGF expression for the Nakagami-m fading implicit in [15]. For this case, in which κ→0 and μ=m, the expression of the SEP coincides exactly with expression 13 of [15]. The κμ distribution also includes the Rice distribution for μ=1 and κ=k. Hence, the Rayleigh distribution can be obtained from the κμ distribution for κ=0 and μ=1.

The Nakagami-m distribution can also be obtained from ημ distribution for μ=m and η→0 in format 1 or η→±1 in format 2 [22]. It can also be obtained for μ=m/2 and η→1 in format 1 or η→0 in format 2. Given that the Nakagami-m is obtained, the Rayleigh distribution can be obtained for m=1.

Using a similar procedure, the Hoyt distribution can be obtained from the ημ distribution for μ=0.5. In this case, the parameter q of Hoyt is related to η by q 2=η in format 1 or by q 2=(1−η)/(1+η) in format 2. The Rayleigh distribution can be obtained from that result for μ=0.5 and η=1 in format 1 or η=0 in format 2.

4 Symbol error probability under η−μ fading

Applying the expression of the MGF of Equation 13 in Equation 9 and adjusting the ranges of the integrals to Lauricella’s function integrals, it is possible to write the SEP expression as follows:
P as = A 1 F D ( 2 ) β , , , β + 1 2 ; γ 11 , γ 12 + A 2 F D ( 3 ) β , , , 1 2 , β + 1 ; γ 21 , γ 22 , cos 2 θ 2 + A 3 F D ( 2 ) β , , , β + 1 2 ; γ 31 , γ 32 + A 4 F D ( 3 ) β , , , 1 2 , β + 1 ; γ 41 , γ 42 , sin 2 ( θ ) + A 5 F D ( 3 ) β , , , 1 2 , β + 1 ; γ 51 , γ 52 , cos 2 ( θ ) + A 6 F D ( 2 ) β , , , β + 1 2 ; γ 61 , γ 62 + A 7 F D ( 3 ) β , , , 1 2 , β + 1 ; γ 71 , γ 72 , sin 2 ( 2 θ ) ,
(15)
where γ 1 = 2 ( h H ) μ γ ¯ δ 2 , γ 2 = 2 ( h + H ) μ γ ¯ δ 2 , β = 2 μN + 1 2 , ξ = ( 4 μ 2 h ) μN ( δ 2 γ ¯ ) 2 μN ,
A 1 = ξB β , 1 2 ( c 2 + c 4 + c 5 ) c 1 , A 2 = ξB ( β , 1 ) c 2 + c 4 2 + c 5 2 c 1 cos θ 2 4 μN + 1 , A 3 = ξB β , 1 2 c 1 c 3 2 csc θ 2 4 μN , A 4 = ξB ( β , 1 ) c 1 c 3 2 sin ( θ ) cos θ 2 4 μN , A 5 = ξB ( β , 1 ) c 1 c 5 2 cos ( θ ) 4 μN + 1 , A 6 = ξB β , 1 2 c 1 c 6 csc ( θ ) 2 4 μN , A 7 = ξB ( β , 1 ) c 1 c 6 2 cos ( θ ) 2 4 μN sin ( 2 θ ) ,
(16)
and
γ 11 = γ 1 , γ 12 = γ 2 , γ 21 = γ 1 cos θ 2 2 , γ 22 = γ 2 cos θ 2 2 , γ 31 = γ 1 4 csc 2 θ 2 , γ 32 = γ 2 4 csc 2 θ 2 , γ 41 = γ 1 cos 2 θ 2 , γ 42 = γ 2 cos 2 θ 2 , γ 51 = γ 1 cos 2 ( θ ) , γ 52 = γ 2 cos 2 ( θ ) , γ 61 = γ 1 4 csc 2 ( θ ) , γ 62 = γ 4 csc 2 ( θ ) , γ 71 = γ 1 cos 2 ( θ ) , γ 72 = γ 2 cos 2 ( θ ) .
(17)

5 Symbol error probability under κ−μ fading

The expression of the SEP for κμ fading can be obtained by writing the exponential function of Equation 14 in terms of its series representation and following the same procedure applied for ημ fading. In this case, after adjusting the interval of integration to [0,1], it is possible to represent each of the integrals of P as by means of sums of Lauricella functions:
P as = n = 0 P as ( n ) ,
(18)
P as ( n ) = A 1 ( n ) F D ( 1 ) ( β + n , μN + n , β + 1 2 + n ; γ 11 ) + A 2 ( n ) F D ( 2 ) ( β + n , μN + n , 1 2 , β + 1 + n ; γ 21 , γ 22 ) + A 3 ( n ) F D ( 1 ) ( β + n , μN + n , β + 1 2 + n ; γ 31 ) + A 4 ( n ) F D ( 2 ) ( β + n , μN + n , 1 2 , β + 1 + n ; γ 41 , γ 42 ) + A 5 ( n ) F D ( 2 ) ( β + n , μN + n , 1 2 , β + 1 + n ; γ 51 , γ 52 ) + A 6 ( n ) F D ( 1 ) ( β + n , μN + n , β + 1 2 + n ; γ 61 ) + A 7 ( n ) F D ( 2 ) ( β + n , μN + n , 1 2 , β + 1 + n ; γ 71 , γ 72 ) ,
(19)
where β = μN + 1 2 , γ 1 = μ ( 1 + κ ) γ ¯ δ 2 , γ 2 = μ ( 1 + κ ) 4 γ ¯ δ 2 , ξ 1 = e κμN γ 1 μN , ξ 2 = e κμN γ 2 μN ,
A 1 ( n ) = ξ 1 ( κμN γ 1 ) n B β + n , 1 2 n ! ( c 2 + c 4 + c 5 ) c 1 , A 2 ( n ) = ξ 1 ( κμN γ 1 ) n B β + n , 1 n ! ( c 2 + c 4 2 + c 5 2 ) c 1 cos θ 2 2 ( β + n ) , A 3 ( n ) = ξ 2 ( κμN γ 2 ) n B β + n , 1 2 n ! c 1 c 3 2 csc θ 2 2 ( κμ + n ) , A 4 ( n ) = ξ 1 ( κμN γ 1 ) n B β + n , 1 n ! c 1 c 3 2 sin ( θ ) cos ( θ ) 2 n , A 5 ( n ) = ξ 1 ( κμN γ 1 ) n B β + n , 1 n ! c 1 c 5 2 cos ( θ ) 2 ( β + 1 2 + n ) , A 6 ( n ) = ξ 2 ( κμN γ 2 ) n B β + n , 1 2 n ! c 1 c 6 csc ( θ ) 2 ( μN + n ) , A 7 ( n ) = ξ 1 ( κμN γ 1 ) n B β + n , 1 n ! c 1 c 6 2 sin ( 2 θ ) cos ( θ ) 4 μN ,
(20)
and
γ 11 = γ 1 , γ 21 = γ 1 cos 2 θ 2 , γ 22 = cos 2 θ 2 , γ 31 = γ 2 csc 2 θ 2 , γ 41 = γ 1 cos 2 ( θ ) , γ 42 = sin 2 ( θ ) , γ 51 = γ 1 cos 2 ( θ ) , γ 52 = cos 2 ( θ ) , γ 61 = γ 2 csc 2 ( θ ) , γ 71 = γ 1 cos 2 ( θ ) , γ 72 = sin 2 ( 2 θ ) .
(21)

6 Results

Firstly, two important aspects of Equations 15 and 19 are highlighted. Despite the expression of SEP in terms of Equations 15 and 19 seems at a first glance more complex than Expression 9, it is not true since the Lauricella functions can be easily calculated by the integral representation in the interval [0,1]. For real numbers, Lauricella functions, F D r ( a , b 1 , , b r , c , x 1 , , x r ) , can be written in terms of a sum of a finite number of terms using the quadrature Gauss-Legendre, for instance. In this case,
F D ( a , b 1 , , b r ; c ; x 1 , , x r ) = Γ ( c ) 2 Γ ( c a ) Γ ( a ) × l = 1 N t w l f ( 0 . 5 ξ l + 0 . 5 ) ,
(22)
in which
w l = 2 ( 1 ξ l 2 ) P N t ( ξ l ) 2 ,
(23)
ξ l , for l=1,2,,N t , are the roots of the Legendre polynomials P N t of order N t , and
f ( x ) = x a 1 ( 1 x ) c a 1 ( 1 x x 1 ) b 1 ( 1 x x 2 ) b 2 ( 1 x x r ) b r .
(24)
The second aspect regards Equation 19. Although it is written in terms of an infinite series, its convergence is fast. In the present work, it was observed that n ranging from 0 to 10 suffices. Indeed, for the first term, one has
A 1 ( n ) 0 1 t β 1 ( 1 t ) 1 2 ( 1 + t γ 11 ) μN t 1 + t γ 11 n dt 0
(25)
since
t 1 + t γ 11 < 1
(26)

and A 1(n) is directly proportional to Γ ( 2 ( β + n ) ) 2 2 n B ( β + n , 1 ) n ! Γ 2 ( β + n ) .

6.1 SEP as a function of θ

Figure 1 presents SEP curves as a function of the angle θ for the ημ model. Due to the symmetry of the geometric regions of the θ-QAM constellation, it suffices to take the angle θ between 45° and 90°. The red curves were obtained considering the diversity order of the MRC receiver equal to 2, while the blue curves were obtained for diversity order equal to 3. In both cases the SNR was maintained equal to 10 dB. The assignment of parameter values η and μ in Figure 1 provided the following types of fading: Nakagami-m for η=0 and μ=1.5, Rayleigh for η=0 and μ=0.5, one-sided Gaussian for η=0 and μ=0.25, and Hoyt for η=0.65 and μ=0.5.
Figure 1
Figure 1

SEP curves for θ -QAM with M=16 under η−μ fading.

It is observed in Figure 1 that the spatial diversity of the MRC has a stronger influence in the SEP when compared to that of the optimum angle θ, for which the curves attain a minimum. This is an important aspect since it is not possible to perform an optimum Gray mapping for every value of θ. All curves attain their minimum value for θ close to 60°, and as one can see, there is a small difference in the SEP values when θ ranges from 45° to 60° compared to the difference obtained increasing the diversity order by one.

In Figure 2, the curves were obtained from the model κμ. Similarly to Figure 1, red curves were plotted for N=2 and blue curves for N=3. The SNR was set at 10 dB, and the choice of the parameters κ and μ provided the following types of fading: Nakagami-m for κ=0 and μ=3.0, Rayleigh for κ=0 and μ=1.0, one-sided Gaussian for κ=0 and μ=0.5, and Rice for κ=0.8 and μ=1.0.
Figure 2
Figure 2

SEP curves for θ -QAM for M =16 under κ−μ fading.

The parameters used in the curves of Figure 2, except for Rice fading, provide the same curves presented in Figure 1. These parameters were properly chosen to show that both the results of Equations 15 and 19 can be used in the calculation of the SEP under the main categories of fading. The difference is that Hoyt (Nakagami-q) fading can only be modeled by the ημ distribution, and Rice fading can be modeled by the κμ distribution.

6.2 SEP as a function of SNR for a fixed θ

The curves in Figure 3 show the behavior of the SEP as a function of SNR, under ημ fading, for the parameter values used in Section 6.1. The curves were obtained for the order of the constellation M=64 and diversity orders N=1 and N=3.
Figure 3
Figure 3

SEP curves for θ -QAM with M =64 under η−μ fading.

As can be seen in Figure 3, regarding Nakagami fading obtained from the ημ model with η=0 and μ=1.5, the increase of the diversity order from N=1 to N=3 provides a gain in SNR of about 7.5 dB for a SEP fixed at 10−2. For all types of fading under consideration (Nakagami for η=0 and μ=1.5, Rayleigh for η=0 and μ=0.5, one-sided Gaussian for η=0 and μ=0.25, and Hoyt for η=0.65 and μ=0.5), one can note that a diversity order N=3 does not assure the SEP lower than 10−3 for SNR <20 dB. The curves in Figure 4 present the behavior of the SEP as a function of SNR under κμ fading for the constellation order M=64 and diversity orders N=1 and N=3.
Figure 4
Figure 4

SEP curves for θ -QAM with M =64 under κ−μ fading and θ=62° .

The equivalence of ημ and κμ models, for appropriated choices of the corresponding parameters, can be observed in Figures 3 and 4, for which the SEP curves for Nakagami, Rayleigh, and one-sided Gaussian fading are exactly the same.

In Figure 4, the results for Rice fading are obtained using κ=0.8 and μ=1.0. For all curves, a diversity order N up to 3 does not guarantee a SEP lower than 10−3 for SNR <20 dB.

The curves in Figure 5 are obtained for ημ fading, constellation order M=64, and diversity orders N=1 and N=3. The angle θ was considered 62° near the angle 60° of the triangular QAM constellation. Considering only the dashed curves for η=0.1, one can observe that the SEP increases when μ decreases from 2.5 to 1.5. This is expected because the corresponding fading intensity also increases. A similar behavior occurs for η=0.8. For the parameters ranging from (η=0.1, μ=2.5) to (η=0.80, μ=1.5) and N=3, about 3 dB of SNR is necessary to maintain the SEP at 10−4.
Figure 5
Figure 5

SEP curves for θ -QAM with M=64 under η−μ fading and θ=62° .

The curves in Figure 6 were obtained for κμ fading, modulation order M=256 and diversity order N=3 and N=4. The angle θ was maintained at 62°. As one can note, even for diversity order of N=4, the curves of SEP remain above 10−2 for values of SNR lower than 20 dB. This occurs because the higher-order θ-QAM constellations are more susceptible to fading effects. For the parameters ranging from (κ=2.50,μ=2.50) to (κ=0.60,μ=1.50) and N=4, about 1.25 dB must be invested in terms of the SNR to keep the SEP at 10−2.
Figure 6
Figure 6

SEP curves for θ -QAM with M =256 under κ−μ fading.

7 Conclusions

This paper presents new and exact expressions for the SEP of θ-QAM modulation considering the MRC receiver under ημ and κμ fading, modeled by ημ and κμ distributions. An important aspect of those distributions is the fact that they can provide a unified analysis of the influence of different types of fading in the performance of the communications system, such as Hoyt, Rice, Nakagami-m, Rayleigh, and one-sided Gaussian. The SEP expressions, obtained from the definite integrals of the MGF function of the SNR at the input of the MRC receiver, are written in terms of Lauricella functions. Expressions obtained in the paper show that the spatial diversity introduced by the MRC receiver has a strong influence for reducing the SEP, even for high-order constellations. As an example, considering 256-QAM constellation and a SEP of 10−2, savings of about 1.7 dB can be obtained when the diversity order is inscreased from N=3 to N=4. The SEP expressions can be useful to design and evaluate the performance of wireless communication systems, since they alleviate the need for Monte Carlo simulations. Future works include the effects of fading correlation and the impact of channel estimation errors on the performance of wireless systems.

Declarations

Acknowledgements

The authors would like to express their thanks to CNPq for the financial support on this research.

Authors’ Affiliations

(1)
Department of Electrical Engineering, Federal University of Campina Grande, Campina Grande, Paraíba, Brazil
(2)
Polytechnic School of Pernambuco, University of Pernambuco, Recife, Pernambuco, Brazil
(3)
Institute for Advanced Studies in Telecommunications, Campina Grande, Paraíba, Brazil

References

  1. Lee Y, Tsai MH: Performance of decode-and-forward cooperative communications over Nakagami-m fading channels. IEEE Trans. Vehicular Technol 2009, 58(3):1218-1228.View ArticleGoogle Scholar
  2. Mahinthan V, Mark JW, Shen X: Performance analysis and power allocation for M-QAM cooperative diversity systems. IEEE Trans Wireless Commun 2010, 9(3):1237-1247.View ArticleGoogle Scholar
  3. Asghari V, da Costa DB, Aissa S: Symbol error probability of rectangular QAM in MRC systems with correlated η−μ fading channels. IEEE Trans. Vehicular Technol 2010, 59(3):1497-1503.View ArticleGoogle Scholar
  4. López-Martínez FJ, Martos-Naya E, Paris JF, Fernández-Plazaola U: Generalized BER analysis of QAM and its application to MRC under imperfect CSI and interference in Ricean fading channels. IEEE Trans. Vehicular Technol 2010, 59(5):2598-2604.View ArticleGoogle Scholar
  5. Yu H, Wei G, Ji F, Zhang X: On the error probability of cross-QAM with MRC reception over generalized η−μ fading channels. IEEE Trans. Vehicular Technol 2011, 60(6):2631-2643.View ArticleGoogle Scholar
  6. Cho K, Yoon D: On the general BER expression of one and two dimensional amplitude modulations. IEEE Trans. Commun 2002, 50(7):1074-1080. 10.1109/TCOMM.2002.800818View ArticleGoogle Scholar
  7. Queiroz WJL, Lopes WTA, Madeiro F, Alencar MS: An alternative method to compute the bit error probability of modulation schemes subject to Nakagami- m fading. EURASIP J. Adv. Signal Process 2010, 2010: 1-12.View ArticleGoogle Scholar
  8. Kim C, Kim v Jeong G, Lee H: BER analysis of QAM with MRC space diversity in Rayleigh fading channels. In Proceedings of the IEEE International Symposium on Personal, Indoor and Mobile Communications (PIMRC’95). Toronto, ON, Canada; 1995:482-485.Google Scholar
  9. Lu J, Tjhung TT, Chai CC: Error probability performance of L- branch diversity reception of MQAM in Rayleigh fading. IEEE Trans. on Commun 1998, 46: 179-181. 10.1109/26.659476View ArticleGoogle Scholar
  10. Femenias G, Furió I: Dual MRC diversity reception of TCM-MPSK signals over Nakagami fading channels. Electron Lett 1996, 32: 1752-1754. 10.1049/el:19961222View ArticleGoogle Scholar
  11. Campopiano C, Glazer B: A coherent digital amplitude and phase modulation scheme. IRE Trans. Commun 1962, 10: 90-95.View ArticleGoogle Scholar
  12. Park SJ: Triangular quadrature amplitude modulation. IEEE Commun. Lett 2007, 11(4):292-294.View ArticleGoogle Scholar
  13. Foschini G, Gitlin R, Weinstein S: Optimization of two-dimensional signal constellations in the presence of Gaussian noise. IEEE Trans. Commun 1974, 22: 28-38. 10.1109/TCOM.1974.1092061View ArticleGoogle Scholar
  14. Park SJ, Byeon MK: Irregularly distributed triangular quadrature amplitude modulation. In Proceedings of the 19th Annual International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC’08). Cannes; September 2008:1-5.Google Scholar
  15. Pappi KN, Lioumpas AS, Karagiannidis GK: θ-QAM: A parametric quadrature amplitude modulation family and its performance in AWGN and fading channels. IEEE Trans. Commun 2010, 58(4):1014-1019.View ArticleGoogle Scholar
  16. Lee J, Yoon D, Cho K: Error performance analysis of M -ary θ-QAM. IEEE Trans. Vehicular Technol 2012, 61(3):1423-1427.View ArticleGoogle Scholar
  17. Ropokis GA, Rontogianis AA, Mathiopoulos PT, Berberidis K: An exact performance analysis of MRC/OSTBC over generalized fading channels. IEEE Trans. Commun 2010, 58(9):2486-2492.View ArticleGoogle Scholar
  18. de Queiroz WJL, Alencar MS, Lopes WTA, Madeiro F: Error probability in multichannel reception with M-QAM, M-PAM and R-QAM schemes under generalized fading. IEICE Trans. Commun 2010, E93-B(10):2677-2687. 10.1587/transcom.E93.B.2677View ArticleGoogle Scholar
  19. Tsiftsis TA, Sandalidis HG, Karagiannidis GK, Uysal M: Optical wireless link with spatial diversity over strong atmospheric turbulence channels. IEEE Trans. Wireless Commun 2009, 8(2):951-957.View ArticleGoogle Scholar
  20. Nistazakis HE, Tombras GS: On the use of wavelength and time diversity in optical wireless communication systems over gamma-gamma turbulence channels. Optics Laser, Technol 2012, 44: 2088-2094. 10.1016/j.optlastec.2012.03.021View ArticleGoogle Scholar
  21. Navidpour SM, Uysal M, Kavehrad M: BER performance of free-space optical transmission with spatial diversity. IEEE Trans. Wireless Commun 2007, 6(8):2813-2819.View ArticleGoogle Scholar
  22. Yacoub MD: The κ−μ distribution and the η−μ distribution. IEEE Antennas and Propagation Mag 2007, 49: 68-81.View ArticleGoogle Scholar
  23. Ermolova NY: Moment generating functions of the generalized η−μ and κ−μ distributions and their applications to performance evaluations of communication systems. IEEE Commun. Lett 2008, 12(7):502-504.MathSciNetView ArticleGoogle Scholar
  24. da Costa DB, Yacoub MD: Moment generating functions of generalized fading distributions and applications. IEEE Commun. Lett 2008, 12(2):112-114.View ArticleGoogle Scholar

Copyright

© Queiroz et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advertisement