Performance evaluation of 4-quadrature amplitude modulation over orthogonal frequency division multiplexing system in different fading channels scenarios

International Journal of Electrical and Computer Engineering (IJECE)
Vol. 12, No. 5, October 2022, pp. 5123~5135
ISSN: 2088-8708, DOI: 10.11591/ijece.v12i5.pp5123-5135  5123
Journal homepage: http://ijece.iaescore.com
Performance evaluation of 4-quadrature amplitude modulation
over orthogonal frequency division multiplexing system in
different fading channels scenarios
Hasan Fadhil Mohammed, Ghanim Abdulkareem Mughir
Electrical Engineering Department, Faculty of Engineering, Mustansiriyah University, Baghdad, Iraq
Article Info ABSTRACT
Article history:
Received Jul 21, 2021
Revised Apr 7, 2022
Accepted May 5, 2022
Orthogonal frequency division multiplexing (OFDM) is a multicarrier
modulation (MCM) technique that divides the wide bandwidth into parallel
narrow bands, each of which is modulated by orthogonal subcarriers.
Currently, OFDM is a high-spectral efficiency modulation technique that is
used in a variety of wired and wireless applications. The transmitted signal
in a wireless communication channel spreads from transmitter to receiver
through multiple reflective paths. This triggers multipath fading, which
causes variations in the received signal's amplitude and phase. Slow/fast and
frequency-selective/frequency-nonselective are the main types of multipath
fading channels. Therefore, in this paper, we proposed new models for
modeling multipath fading channels, such as the exponential fading channel
and the Gamma fading channel. In addition, new bit-error-rate (BER)
derivations have been derived. The performance of the OFDM system over
proposed channel models has been evaluated using Monte-Carlo simulation
and compared to the Rayleigh fading channel model. The obtained results
via simulations show that the exponential fading channel at a rate parameter
(λ=0.5) outperforms the Rayleigh fading channel by 6 dB for all values of
Eb/No, while the Gamma fading channel at (α=2) outperforms the Rayleigh
fading channel by 3 dB for all values of Eb/No.
Keywords:
Bit-error-rate
Exponential fading channel
Gamma fading channel
Orthogonal frequency division
multiplexing
Signal-to-noise ratio
This is an open access article under the CC BY-SA license.
Corresponding Author:
Hasan Fadhil Mohammed
Department of Electrical Engineering, Faculty of Engineering, Mustansiriyah University
Baghdad, Iraq
Email: eema1032@uomustansiriyah.edu.iq
1. INTRODUCTION
The growing interest in wireless broadband communications and interactive media has fueled
intense study on data rate transmission, bandwidth efficiency, and bit-error-rate (BER) performance across
multipath fading channels [1]. Since the information to be sent is modulated into a single carrier in a
conventional communication system, a single fade or interferer may cause the whole connection to fail. To
achieve high bit rates, the symbols must be sent rapidly, using the full bandwidth. When a channel is
frequency selective, its impulse response may span several symbol periods, resulting in what is referred to as
intersymbol interference (ISI). This is a significant issue in wideband transmission across multipath fading
channels, since it makes it more difficult to identify the sent signal, thus degrading the system's performance
significantly [2]. Orthogonal frequency division multiplexing (OFDM) has been explored for a variety of
wireless (applications, standards) and wired because of its popular features of high bandwidth efficiency,
easy channel equalization, and immunity to multipath fading channels [3], [4]. This modulation method may
be utilized effectively and consistently for high-speed digital data transmission across multipath fading

 ISSN: 2088-8708
Int J Elec & Comp Eng, Vol. 12, No. 5, October 2022: 5123-5135
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channels, since it splits the available bandwidth into multiple orthogonal data streams [5]. As a result, fading
will impact a very small proportion of the subcarriers. Each subchannel will occupy a small frequency
interval in the frequency domain, where the channel frequency response will be almost constant and each
symbol will experience a nearly flat fading channel. Therefore, it is seen as one of the most promising
solutions to the ISI problem [2]. Intercarrier interference (ICI) is eliminated when orthogonal subcarriers are
used, and the signals carried on the various subcarriers do not interfere [6]. Hence, a simple frequency
domain equalization method can be used [7]. IEEE 802.11 a, g, and n, IEEE 802.16a, and terrestrial digital
video broadcasting (DVB-T) have all adapted it [8] and ODFM is used in digital subscriber line (DSL) and
asymmetrical digital subscriber line (ADSL) for wired applications [9]. In the last few years, there have been
a lot of studies done on the success of wireless channel models. In [10], the study of the performance
evaluation of the OFDM-based 256 and 1024-QAM under multipath fading propagation circumstances
defines the acceptable signal-to-noise ratio (SNR). In [11], focuses on the performance evaluation and
investigation of OFDM systems operating across multipath fading channels like additive white Gaussian
noise (AWGN), Rician, and Rayleigh. In [12] OFDM is studied and simulated in the presence of Rayleigh
fading channels using channel estimation and equalization. Bit error probability is used to quantify
performance. In [13], it analyzes the bit error rate (BER) performance of discrete cosine transform (DCT)-
based OFDM and compares it to discrete fourier transform (DFT)-based OFDM to determine which
modulation scheme performs better. In [14] The cyclic prefix's effect on the OFDM system's performance
was investigated. The BER for different lengths of the cyclic prefix was compared using an OFDM
transceiver equipped with a 16-QAM modulation method, highlighting the effect of channel noise.
In this paper we suggested new models to find new simulations and effects as well as to improve the
BER performance of the OFDM system. In addition, two fading channel scenarios: exponential fading channel
and Gamma fading channel have been adopted in this paper. The major contributions to this paper are related
to improving the signal detection and enhancing the BER performance analysis of 4-QAM/OFDM over
Gamma fading channels and exponential fading channels in AWGN based on the following: i) we derive the
real part, the imaginary part of the channel impulse response in the time domain, in addition to the effective
probability distribution and the effective SNR distribution of the Exponential and Gamma fading channels in
the frequency domain at the OFDM demodulator output by using the central limit theorem (CLT), ii) we
derive the variance in each dimension, i.e., the real part and the imaginary part of the exponential and
Gamma fading channels, iii) we derive the BER based on the effective derived SNR distribution in the
Gaussian noise only and in the presence of the exponential and Gamma fading channels in AWGN, and iv)
we detect the received signal by using an optimum detector based on the Euclidean distance depending on the
derived effective probability distribution at the OFDM demodulator output.
The rest of this paper is organized as follows: section 2 system model, while section 3 is the
derivation of the variance in the time domain for exponential distribution and Gamma distribution. Section 4
BER performance analysis of M-ARY digital modulation over AWGN, exponential and Gamma fading
channels, section 5 describes the simulation results and section 6 concludes this paper.
2. RESEARCH METHOD
2.1. System model
In this paper, the binary data bits 𝑑𝑘 are divided into two-bit classes before being converted into the
22
symbols of a 4-QAM constellation to obtain the modulated symbols X=M(c) according to constellation
mapping c shown in Figure 1 [15]. The complex base-band OFDM signal 𝑥𝑛 can then be obtained in the time
domain using an N-points inverse fast fourier transform (IFFT), as seen in [16].
𝑥𝑛 =
1
√𝑁
∑ 𝑋𝑖 𝑒
𝑗2𝜋𝑛𝑖
𝑁 , 0 ≤ 𝑛 ≤ 𝑁 − 1
𝑁−1
𝑖=0 (1)
where 𝑋𝑖 the complex modulated symbol and 𝑁 is is the number of orthogonal subcarriers.
To minimize ISI between consecutive OFDM symbols in multipath fading channels, a time-domain
CP of duration NCP samples is added at the beginning of each OFDM symbol to produce the transmitted
symbol 𝑥
̂ = [𝑥𝑁 − 𝑁𝐶𝑃, 𝑥𝑁 − 𝑁𝐶𝑃 + 1, … , 𝑥𝑁 − 1, 𝑥0, 𝑥1, 𝑥𝑁 − 1], which is programmed to exceed the
cumulative delay spread of the fading channel. Then, 𝑥
̂ is transmitted through the fading channel [17]. This signal
is combined with additive noise in the wireless channel, resulting in the obtained signal as shown in (2) [18]:
𝑦 = ℎ⨂𝑥
̂ + 𝑤 (2)
where ⨂ represents the convolution operation, ℎ represents the complex impulse response coefficients in the
time domain of the multipath fading channel and 𝑤 is the AWGN.

Int J Elec & Comp Eng ISSN: 2088-8708 
Performance evaluation of 4-quadrature amplitude modulation over … (Hasan Fadhil Mohammed)
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Figure 1. The four signal points of 4-QAM modulation in the constellation diagram
The multipath fading channel of wireless communication was modeled using exponential
distribution or Gamma distribution in [19], [20] due to the fact that a large number of distributions used to
model the multipath fading and shadowing fading channels for survival analysis are special instances of the
generalized gamma distribution, such as the exponential and the gamma distributions [19], [21], as explained
in more detail in this section. If the amplitude of the complex random variables h of the channel impulse
response follows the exponential distribution in the time domain, the distribution can be expressed in [20] as
in (3):
𝑓ℎ(ℎ) = 𝜆𝑒−𝜆ℎ
, 0 ≤ ℎ < ∞, 𝜆 > 0 (3)
where λ is the rate parameter of the fading channel.
The main concern with the fading can be stated into real and imaginary parts expressed as ℎ𝑟 and ℎ𝑖,
respectively. Assuming the magnitude of the fading channel ℎ in the time domain, |ℎ| = √ℎ𝑟
2 + ℎ𝑖
2
, exhibits
an exponential distribution where ℎ = ℎ𝑟 + 𝑗ℎ𝑖. Two random parameters are expanded by θ, which is a
randomly distributed random variable between the limits of the bounds π and -π, creating two further random
variables as in (4) and (5).
ℎ𝑟 = |ℎ| cos(𝜃) (4)
ℎ𝑖 = |ℎ| sin(𝜃) (5)
As a result, the PDF of the real part of the multipath fading channel can be expressed as (6) [22].
𝑑ℎ𝑟
𝑑ℎ
= cos(𝜃) (6)
Then, 𝑓(ℎ𝑟)|𝜃 a conditional PDF of ℎ𝑟 subjected to θ, can be written as (7) [22].
𝑓(ℎ𝑟)|𝜃 =
𝑓(ℎ)
𝑑ℎ𝑟 𝑑ℎ
⁄
= 𝜆𝑒−𝜆ℎ 1
𝑑ℎ𝑟/𝑑ℎ
=
𝜆
|cos(𝜃)|
𝑒
−𝜆ℎ
cos(𝜃) (7)
Moreover, 𝑓(ℎ𝑖)|𝜃 a conditional PDF of ℎ𝑖 subjected to θ, can be written as (8):
𝑓(ℎ𝑖)|𝜃 =
𝑓(ℎ)
𝑑ℎ𝑖/𝑑ℎ
= 𝜆𝑒−𝜆ℎ 1
𝑑ℎ𝑖/𝑑ℎ
=
𝜆
|𝑠𝑖𝑛(𝜃)|
𝑒
−𝜆ℎ
sin(𝜃) (8)

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Then the complex received signal, y, processed using the CP removal and fast fourier transform (FFT)
operation [18]. As a result, in the frequency domain, the complex signal can be represented as (9):
𝑌 = 𝐻𝑋 + 𝑊 (9)
𝑌𝑟 + 𝑗𝑌𝑖 = (𝐻𝑟 + 𝑗𝐻𝑖)(𝑋𝑟 + 𝑗𝑋𝑖) + (𝑊
𝑟 + 𝑗𝑊𝑖)
where 𝑌, 𝐻, 𝑋, and 𝑊 represent the complex received signal in the frequency domain, the transfer function
of the complex fading channel, complex modulated signals in the frequency domain, and AWGN,
respectively.
Hence, the distribution of the real part, 𝑓(𝐻𝑟) and the imaginary part, 𝑓(𝐻𝑖) would approach a
Gaussian distribution after the FFT operation with a mean equal to zero, μ=0, and variance 𝜎ℎ𝑟
2
and 𝜎ℎ𝑖
2
computed in the next section. According to the central limit theorem (CLT) [23] as (10).
𝑓(𝐻𝑟) =
1
√2𝜋𝜎ℎ𝑟
2
𝑒
−
𝐻𝑟
2
2𝜎ℎ𝑟
2
(10)
𝑓(𝐻𝑖) =
1
√2𝜋𝜎ℎ𝑖
2
𝑒
−
𝐻𝑖
2
2𝜎ℎ𝑖
2
(11)
As a result, the magnitude of the fading channel in the frequency domain can be computed as |𝐻| =
√𝐻𝑟
2
+ 𝐻𝑖
2
would follow the Rayleigh distribution with variance 𝜎ℎ
2
computed in the next section as (12):
𝑓(𝐻) =
|𝐻|
𝜎ℎ
2 𝑒
−
|𝐻|2
2𝜎ℎ
2
(12)
with uniformly phase distributed in the range [-π, π). Moreover, if the amplitude of the complex random
variables h, |ℎ| = √ℎ𝑟
2 + ℎ𝑖
2
of the channel impulse response follows the Gamma distribution in the time
domain, the PDF can be expressed in [20] as (13):
𝑓ℎ(ℎ) =
1
Γ(α)βα ℎ𝛼−1
𝑒
−
ℎ
𝛽, 0 < h < ∞ , α > 0 , β > 0 (13)
where α is the shape parameter and β is the scale parameter. Following the same mentioned discussions, the
distribution of 𝑓(ℎ𝑟)|𝜃 and 𝑓(ℎ𝑖)|𝜃 can be derived as (14) and (15).
𝑓(ℎ𝑟)|𝜃 =
𝑓(ℎ)
𝑑ℎ𝑟/𝑑ℎ
=
1
𝑒
−
ℎ
𝛽
1
𝑑ℎ𝑟/𝑑ℎ
=
1
Γ(α)βα|cos(𝜃)|
(
ℎ
cos(𝜃)
)
𝛼−1
𝑒
−
ℎ
𝛽cos(𝜃) (14)
and
𝑓(ℎ𝑖)|𝜃 =
𝑓(ℎ)
𝑑ℎ𝑖/𝑑ℎ
=
1
𝑒
−
ℎ
𝛽
1
𝑑ℎ𝑖/𝑑ℎ
=
1
Γ(α)βα|sin(𝜃)|
(
ℎ
sin(𝜃)
)
𝛼−1
𝑒
−
ℎ
𝛽 sin(𝜃) (15)
Thanks to the central limit theorem after FFT operation, which makes the distribution of the real and
imaginary components of the Gamma fading channel approaches Gaussian distribution as expressed in (10)
and (11) in the next section with μ=0 and variance 𝜎ℎ𝑟
2
and 𝜎ℎ𝑖
2
computed in the next section. Therefore, the
magnitude of the fading channel with Gamma distribution will follow Rayleigh distribution as given in (12)
and the phase will follow uniform distribution throughout the frequency domain from the range [-π, π).
Finally, the received signal in (9) can be detected using maximum likelihood (ML) detection. The ML can be
implemented by finding the minimum Euclidean distance between all the possible transmitted symbols over
the channel H and the received signal 𝑌 as [24]:

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𝑑𝑘
̂ = arg 𝑚𝑖𝑛𝑋∈𝐶 ‖𝑌 − 𝐻. 𝑋‖2
(16)
where C=C1, C2, C3, C4 represents a collection of all possible symbols X in a 4-QAM constellation as
presented in Figure 1.
The bit error rate (BER) is determined by comparing the binary data bits generated by the source at
the transmitter, dk, with the eventually obtained bits at the receiver, 𝑑𝑘
̂, and then dividing the result by the
number length of dk [25]. Figure 2 illustrates the configuration of an OFDM transceiver system (transmitter
and receiver) [26], [27].
Figure 2. Baseband OFDM system model over exponential/Gamma fading channel
2.2. Derivation of the variance in the time domain for exponential distribution and gamma distribution
2.2.1. Variance derivation of exponential distribution
Suppose the magnitude of the complex random variable ℎ in the time domain follows the
exponential distribution, that is, it has pdf given by (3). Hence, the variance can be computed as in (17) [20]:
𝜎ℎ
2
= 𝐸(|ℎ|2) − |𝐸(ℎ)|2
(17)
where 𝐸(|ℎ|2) is the second moment and |𝐸(ℎ)|2
is the square of the first moment (expected value).
The variance of an exponential distribution is computed in [20] as 𝜎ℎ
2
=
1
𝜆2. However, the mean and
variance of a complex random variable, ℎ = ℎ𝑟 + 𝑗ℎ𝑖, can be expressed as given in [24] as 𝐸(ℎ) = 𝐸(ℎ𝑟) +
𝑗𝐸(ℎ𝑖) and 𝜎ℎ
2
= 𝜎ℎ𝑟
2
+ 𝜎ℎ𝑖
2
. So, the variance of the real component 𝜎ℎ𝑟
2
, is equal to the variance of the
imaginary component 𝜎ℎ𝑖
2
and the variance in each dimension is:
𝜎ℎ𝑟
2
= 𝜎ℎ𝑖
2
=
𝜎ℎ
2
2
=
1
2𝜆2 (18)
2.2.2. Variance derivation of gamma distribution
If the magnitude of the complex random variable h follows the Gamma distribution as given in (13),
the variance can be expressed as given in [28] as 𝜎ℎ
2
= 𝛼𝛽2
, hence, the variance in each dimension can be
expressed as mentioned above as (19).
𝜎ℎ𝑟
2
= 𝜎ℎ𝑖
2
=
𝜎ℎ
2
2
=
𝛼𝛽2
2
(19)
2.3. BER performance analysis of QAM over AWGN, exponential and gamma fading channels
The influence of fading is measured using 4-QAM modulation. The bit error rate (BER), is a more
accurate performance parameter for measuring modulation schemes. In a slow flat fading path, the following
integral may be used to measure the BER performance of any digital modulation scheme [29].
𝑃𝑏= ∫ 𝑃𝑏,𝐴𝑊𝐺𝑁(𝛾)𝑃𝑑𝑓(𝛾) 𝑑𝛾
∞
0
(20)
The error probability of a specific modulation scheme in an AWGN channel at a specified signal-to-
noise ratio is:

 ISSN: 2088-8708
5128
𝛾 = ℎ2 𝐸𝑏
𝑁𝑜
(21)
where h is the random variable of channel gain, in a non-fading AWGN channel,
𝐸𝑏
𝑁𝑜
is the ratio of the bit
energy to the noise power density. The random variable h2
denotes the fading channel's instantaneous power
and Pdf (γ) is the probability density function of γ due to the fading channel [29].
2.3.1. Derivation of BER of 4-QAM digital modulation in AWGN channel
The cumulative distribution function (CDF) of a continuous random variable X can be expressed as
the integral of its probability density function 𝑝𝑋 follows [20]:
𝐹𝑋(𝑥) = ∫ 𝑝𝑋(𝑡)𝑑𝑡
𝑥
−∞
(22)
The probability of error receiving the constellation symbol C2=1+1j over AWGN channels as shown
in Figure 1 can be computed as (23) [30].
𝑃
𝑒(𝐶2) = 𝐹𝑋
𝐶2(0) + 𝐹𝑌
𝐶2(0) − 𝐹𝑋
𝐶2(0)𝐹𝑌
𝐶2(0) (23)
Because of the symmetry, the marginal probabilities for the real and imaginary components can be calculated
using the Gaussian PDF as (24).
𝐹𝑋
𝐶2(0) = 𝐹𝑌
𝐶2(0) (24)
The equation of Gaussian PDF is:
𝑝(𝑥) =
1
√2𝜋𝜎2
𝑒
−(𝑥−𝜇)2
2𝜎2
, 𝑤𝑖𝑡ℎ 𝜇 = 0 𝑎𝑛𝑑 𝜎2
=
𝑁𝑜
2
(25)
So
𝐹𝑋
𝐶2(0) = ∫
1
√2𝜋𝜎𝑤
2
𝑒
−(𝑥−√𝐸𝑠
2
)2
2𝜎𝑤
2
𝑑𝑥 =
1
2
erf (−√
𝐸𝑠
4𝜎𝑤
2 ) −
1
2
erf(−∞)
0
−∞
𝐹𝑋
𝐶2(0) =
1
2
−
1
2
erf (√
𝛾
2
) , 𝑤ℎ𝑒𝑟𝑒 𝛾 =
𝐸𝑠
2𝜎𝑤
2 (26)
As shown in (26) can be rewritten as (27):
𝐹𝑋
𝐶2(0) =
1
2
𝑒𝑟𝑓𝑐 (√
𝛾
2
) (27)
Now we substitute (27) into (23) we get:
𝑃𝑒(𝐶2) =
1
2
𝛾
2
) +
1
2
𝛾
2
) −
1
2
𝛾
2
)
1
2
𝛾
2
)
= 𝑒𝑟𝑓𝑐 (√
𝛾
2
) −
1
4
𝑒𝑟𝑓𝑐2
(√
𝛾
2
)
≈ 𝑒𝑟𝑓𝑐 (√
𝛾
2
) 𝑓𝑜𝑟 𝛾 ≫ 0 (28)
Due to the constellation symmetry in Figure 1, the error probability of C2 is equal to the error probability of
C0, C1 and C3. The total symbol error probability can be expressed as (29).
𝑃𝑠(𝐶) = 𝑃 (𝐶0) 𝑃𝑒 (𝐶0) + 𝑃 (𝐶1) 𝑃𝑒 (𝐶1) + 𝑃 (𝐶2) 𝑃𝑒 (𝐶2) + 𝑃 (𝐶3) 𝑃𝑒 (𝐶3) (29)
We assume that four symbols are transmitted with equal probability, so P(C0) = P(C1) = P(C2) = P(C3) =
1
4
, hence, the net symbol error probability can be expressed as (30).

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Ps(C) =
1
4
(Pe(C0) + Pe(C1) + Pe(C2) + Pe(C3)) =
1
4
(4𝑒𝑟𝑓𝑐 (√
𝛾
2
) )
= 𝑒𝑟𝑓𝑐 (√
𝛾
2
) (30)
Therefore, the BER of the 4-QAM constellation can be computed as (31):
Pb(C) =
Ps(C)
log2 𝑀
=
Ps(C)
2
Pb(C) =
1
2
𝐸𝑏
2𝜎𝑤
2 ) (31)
Leads to the
BER4QAM,AWGN =
1
2
𝐸𝑏
𝑁𝑜
) (32)
2.3.2. Derivation of BER of 4-QAM digital modulation in exponential fading channel
For the exponential fading channel as expressed in (3) in the time domain, the H follows the
Rayleigh distribution in the frequency domain after FFT operation based on CLT as computed in (12), and
|H|2
has a chi-square distribution of two degrees of freedom as expressed in (33) [29]:
𝑃𝑑𝑓(𝛾) =
1
𝛾
̅
exp ( −
𝛾
𝛾
̅
) (33)
where 𝛾̅ =
𝐸𝑏
𝑁𝑜
𝐸[|𝐻|2] is the signal-to-noise ratio on average for 𝐸[|𝐻|2] = 1/𝜆2
, substituting in (33), gets
1
(1/𝜆2)𝐸𝑏 𝑁𝑜
⁄
𝑒
−𝛾
(1/𝜆2) 𝐸𝑏 𝑁𝑜
⁄
(34)
Substituting (32) and e (34) in (20) we get:
𝑃𝑏= ∫
1
2
𝑒𝑟𝑓𝑐(√𝛾) ∗
1
⁄
𝑒
−𝛾
⁄
𝑑𝛾
∞
0
=
(
−
(1/𝜆)√𝐸𝑏 𝑁𝑜
⁄ erf(
√𝛾(1/𝜆2)(𝐸𝑏 𝑁𝑜
⁄ )+1)
⁄
)
2√(1/𝜆2)(𝐸𝑏 𝑁𝑜
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(√𝛾) 𝑒
−𝛾
(
1
𝜆2)𝐸𝑏 𝑁𝑜
⁄
)
∞
0
The equation after substituting the integral values becomes:
= [−
(
1
𝜆
)√𝐸𝑏 𝑁𝑜
⁄ erf(∞)
2√(
1
𝜆
2
)(𝐸𝑏 𝑁𝑜
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(∞) 𝑒
−∞
(
1
⁄
] − [−
⁄ 𝑒𝑟𝑓(0)
2√(1/𝜆2)(𝐸𝑏 𝑁𝑜
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(0)𝑒
0
(
1
⁄
]
𝑤ℎ𝑒𝑟𝑒 𝑒𝑟𝑓(∞) = 1, 𝑒𝑟𝑓𝑐(∞) = 0 , 𝑒𝑟𝑓(0) = 0, 𝑒𝑟𝑓𝑐(0) = 1
= −
⁄
2√(1/𝜆2)(𝐸𝑏 𝑁𝑜
⁄ )+1
+
1
2
(35)
As shown in (35) can be rewritten as (36).
𝑃𝑏 =
1
2
(1 −
√1/𝜆2𝐸𝑏 𝑁𝑜
⁄
√1/𝜆2(𝐸𝑏 𝑁𝑜
⁄ )+1
) (36)
2.3.3. Derivation of BER of 4-QAM digital modulation in gamma fading channel
Using the same steps that are used to find the 𝑃𝑑𝑓(𝛾) of the exponential fading channel, the 𝑃𝑑𝑓(𝛾)
for the Gamma fading channel is:

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1
𝛼𝛽2𝐸𝑏 𝑁𝑜
⁄
𝑒
−𝛾
⁄
(37)
Sub (32) and (37) in (20) we get:
𝑃𝑏= ∫
1
2
𝑒𝑟𝑓𝑐(√𝛾) ∗
1
⁄
𝑒
−𝛾
⁄
𝑑𝛾
∞
0
=
(
−
√𝛼𝛽2∗𝐸𝑏 𝑁𝑜
⁄ erf(
√𝛾(𝛼𝛽2(𝐸𝑏 𝑁𝑜
⁄ )+1)
⁄
)
2√𝛼𝛽2(𝐸𝑏 𝑁𝑜
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(√𝛾) 𝑒
−𝛾
𝛼𝛽2∗(𝐸𝑏 𝑁𝑜
⁄ )
)
∞
0
The equation after substituting the integral values becomes:
= [−
⁄ erf(∞)
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(√∞) 𝑒
−∞
⁄ )
] − [−
⁄ erf(0)
⁄ )+1
−
1
2
𝑒𝑟𝑓𝑐(√0)𝑒
0
⁄ )
]
= −
⁄
2√(𝛼𝛽2∗𝐸𝑏 𝑁𝑜
⁄ )+1
+
1
2
(38)
As shown in (38) can be rewritten as (39).
𝑃𝑏 =
1
2
(1 −
⁄
√(𝛼𝛽2∗𝐸𝑏 𝑁𝑜
⁄ )+1
) (39)
3. SIMULATION RESULTS AND DISCUSSION
The BER performances are investigated using the MATLAB simulation program for
4-QAM/OFDM over two different channel distributions such as the multipath exponential fading channel and
the multipath Gamma fading channel in AWGN, and the results are compared with the 4-QAM/OFDM over
the Rayleigh fading channel. The following simulation parameters were set, the number of sub-carriers was
set as N=64 and CP=16, the constellation size was set as 4-QAM and the system bandwidth was set at
20 MHz for the ITU pedestrian standard channel for multipath exponential fading channel and multipath
Gamma fading channel are used, as shown in Table 1 [31].
Table 1. ITU channel profile
ITU Pedestrian Ch.103
Path n Power (dB) Delay (µs)
1 0 0
2 -0.9 0.2
3 -4.9 0.8
4 -8 1.2
5 -7.8 2.3
6 -23.9 3.7
3.1. Performance of 4-QAM/OFDM system over exponential fading channel
We are interested in this section to show the matching of the derived distributions with simulated
results in the time domain based on (3) and the frequency domain based on e (12), for 𝜎ℎ
2
computed in (18).
In Figure 3, we see the exponential fading histograms, where the parameters are set to λ=0.5, λ=1 and λ=1.5,
respectively. It should be noted that (3) and the derived closed-form PDF in (12) have similar analytical
distributions to those of the simulation results in the probability domain.
The impact of changing λ parameter on the BER performance of the 4-QAM/OFDM system over
exponential fading channels and AWGN compared to the Rayleigh fading channel is shown in Figure 4.
Simulations were run over different scenarios of the channel effect by changing the scale parameter λ to 0.5,
1, and 1.5 for severe impact, normal impact, and low impact, respectively. For instance, in the case of the

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normal impact, the exponential fading channel has the same effect as the Rayleigh fading channel, and in this
case, the parameter of λ is chosen as follows:
The PDF of the signal to noise ratio distribution at the OFDM demodulator output can be derived
based on CLT as given in (34), while in the Rayleigh fading channel it can be expressed as in [32].
1
(2𝜎2)𝐸𝑏 𝑁𝑜
⁄
𝑒
−𝛾
(2𝜎2) 𝐸𝑏 𝑁𝑜
⁄
(40)
Hence, both distributions have the same effect when
1
𝜆2 = 2𝜎2
By take the square root for both sides:
1
𝜆
= √2𝜎
The standard deviation 𝜎 =
1
√2
in both directions of the Rayleigh distribution. Hence,
1
𝜆
= √2
1
√2
𝑠𝑜 𝜆 = 1
Based on the first value (λ =0.5), we determined that the BER performance of the 4-QAM/OFDM
method over an exponential fading channel is superior to that of the system over a Rayleigh fading channel.
At the second value (λ =1), the BER performance of the exponential fading channel is close to or similar to
that of the Rayleigh fading channel. But on the third value (λ =1.5) the performance will be worse. Table 2
shows the BER performance of the proposed system compared to the conventional system at Eb/No=20 dB.
The Table 2 shows that the BER improved by decreasing the rate parameter λ, and the performance is
matched to the conventional system over Rayleigh fading channel when λ=1.
Figure 3. Simulated Vs histogram plots in the time domain and the frequency domain for
exponential fading channel

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Figure 4. Performance of 4-QAM/OFDM system over exponential fading channel
Table 2. BER values with different scenarios of λ parameter
λ parameter BER (Exponential) BER (Rayleigh)
0.5 0.6238 ×10-3
0.2481×10-2
1 0.2481×10-2
0.2481×10-2
1.5 0.5532×10-2
0.2481×10-2
3.2. Performance of 4-QAM/OFDM system over gamma fading channel
A comparison of the histogram plots of the derived distributions with simulated results for Gamma
distribution as presented in the time domain in (13) and in the frequency domain will follow the Rayleigh
distribution as given in (12), for 𝜎ℎ
2
computed in (19). In Figure 5, we see the Gamma fading histograms,
where the parameters are set to α=0.5, α=1 and α=2 for β=1. It should be noted that (13) and the derived
closed-form PDF in (12) have similar analytical distributions to those of the simulation results in the
probability domain.
Figure 5. Simulated Vs histogram plots in the time domain and the frequency domain for
Gamma fading channel

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The impact of changing α and β parameters on the BER performance of 4-QAM/OFDM systems
over Gamma fading channel and AWGN compared to Rayleigh fading channel is shown in Figure 6.
Simulations were run for various values of the channel effect by changing the α parameter (0.5, 1, 2) for
severe impact, normal impact and low impact, respectively, and for the β parameter equal to 1, by using the
same method to find the normal effect in the exponential fading channel, we found the normal effect in the
Gamma fading channel at α=1.
Figure 6. Performance of 4-QAM/OFDM system over gamma fading channel
In this figure, the BER performance of the 4-QAM/OFDM system over a Gamma fading channel is
lower than that of the system over a Rayleigh fading channel based on the first value (α=0.5). Moreover, the
BER performance of the Gamma fading channel is equal to or equivalent to that of the Rayleigh fading
channel at the second value (α=1). Furthermore, the result will be better on the third value (α=2). Table 3
shows the BER performance of the proposed system compared to the conventional system at Eb/No=20 dB.
The Table 3 shows that the BER improved by increasing the shape parameter α, and the performance is
matched to the conventional system over Rayleigh fading channel when α=1.
Table 3. BER values with different scenarios of Shape parameter (α) at β=1
α parameter BER (Gamma) BER (Rayleigh)
0.5 0.4899 ×10-2
0.2481×10-2
1 0.2481×10-2
0.2481×10-2
2 0.1245×10-2
0.2481×10-2
4. CONCLUSION
The BER of the 4-QAM/OFDM system in AWGN over different channel scenarios, such as
exponential and Gamma fading channels, in addition to the BER performance, has been investigated for both
channels in this paper. The simulation results obtained via the MATLAB program have confirmed the
feasibility of the exact derived PDFs at the OFDM demodulator output in the frequency domain based on
CLT, leading to derived exact new theoretical BER equations matching to the Monte-Carlo simulation for
both proposed channel distributions. The 4-QAM OFDM system in AWGN over exponential fading channel
with different λ parameters (0.5, 1, and 1.5) has been simulated. It has been shown from the results that the
exponential fading channel outperforms the Rayleigh fading channel when (λ<1) for all values of Eb/No.
Additionally, the 4-QAM OFDM system in AWGN over Gamma fading channel with different values for α
(0.5, 1, and 2) and for β=1 has been simulated. It has been shown from the results that the Gamma fading
channel outperforms the Rayleigh fading channel when (α >1) for all values of Eb/No.

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BIOGRAPHIES OF AUTHORS
Hasan Fadhil Mohammed was born in Diyala, Iraq, in 1985. He received
the B.Sc. In Electronic Engineering from Faculty of Engineering of Diyala University,
Iraq in 2008. Currently, he is a master student in Electronics and Communications
Engineering Department, Mustansiriyah University, Baghdad, Iraq. His recent research is
mitigation of performance degradation by different fading channels for convolutional
coded QAM/OFDM system over gaussian noise. He can be contacted at email:
eema1032@uomustansiriyah.edu.iq.
Ghanim Abdulkareem Mughir is a Lecturer in the Electrical Engineering
Department, Faculty of Engineering, Mustansiriyah University, Baghdad, Iraq. He
received his M.Sc. in Electronics and Communications from Al-Mustansiriya University,
Baghdad, Iraq, in 1999. He received his PhD from the School of Electrical and Electronics
Engineering, Newcastle University, Newcastle Upon Tyne, U.K, in 2017. His research
focuses on wired/wireless communications on coded systems, OFDM systems, fading and
power line communication channels, channel modelling and receiver design. He can be
contacted at email: g.a.m.al-rubaye@uomustansiriyah.edu.iq.

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