JIMMA UNIVERSITTY

JIMMA INSTITUTE OF TECHNOLOGY

FACULTY OF ELECTRICAL AND COMPUTER ENGINEERING

STREAM OF COMMUNICATION ENGINEERING

WIRELESS AND MOBILE COMMUNICATION ASSIGNMENT

TITLE: NAKAGAMI FADING

GROUP MEMBERS

GROUP NAME ID NUMBER

1. CHALA MEKONNEN RU 4584/09

2. MILIYON WORKU RU 1490/09

3. MERAWI WUBISET RU 1493/09

4. ISRAEL BEKELE RU 2655/06

5. ELIAS KIFLE RT0006/10

Submitted to: MR. Getachew A.

Submission date: Jan 6, 2021
 ABSTRACT
Data communications in wireless networks generally take place over fading channels with time
varying characteristics. Fading is a word used to express rapid fluctuation of the amplitude of a
radio signal over a short period of time or travel distance. There are two major types of fading
that slow fading and fast fading. Also, fast fading has its own types, from them an even wider
class of fading describer is nakagami fading. The study of this topic is very important for
communication that Nakagami fading distribution is suitable for describing mobile radio
transmission in complex medium. In practice it has proved very useful because of easy
manipulation and wide range of applicability of various approximations. Therefore, the main
objective of studying this topic is to understand the topic and use it where its applicable.

Table of Contents
Tables of Figures.........................................................................................................................................iii
1.1 INTRODUCTION.....................................................................................................................................1
1.2. FADING.................................................................................................................................................3
1.2.1 TYPES OF FADING......................................................................................................................3
1.3 Nakagami Fading...................................................................................................................................4
1.4 THE NAKAGAMI DISTRIBUTION.............................................................................................................6
1.5 INTEGRAL CHARACTERISTICS OF NAKAGAMI FADING...........................................................................7
1.6. Nakagami Fading Channel (NFC)...........................................................................................................8
REFERENCE..................................................................................................................................................9
Tables of Figures
Figure 1: PDF of Nakagami.........................................................................................................................4
Figure 2: BPSK Performance of Nakagami.................................................................................................5
 Nakagami Fading 2021

1.1 INTRODUCTION
Data communications in wireless networks generally take place over fading channels with time-
varying characteristics. The extent to which the dynamic nature of the wireless
medium impacts the Quality of Service (QoS) of transmitted data depends on factors such as the
severity of the fading channel and the resource allocation policy being employed to adapt to the
time-varying channel. This is in contrast to wired point-to-point links where QoS is exclusively a
function of the data traffic arrival statistics and the fixed capacity of the transmitter. In the wired
case, QoS attributes, such as delay performance, can usually be studied by using appropriate
queuing models and analyses. The time-varying wireless channel, on the other hand, poses a
challenge in terms of queuing analysis and performance evaluation. The measurement of
achievable performance of wireless communications over fading channels has historically been
relegated to the realm of information theory, where channel capacity is the figure of merit. The
delay component that accounts for the time that data spends in a transmit buffer, as well as other
measures of QoS, are typically decoupled from the information theoretic problem, and often
times simply ignored. This separation is reasonable for wired links where a constant transmit
data rate can be assumed, but results in an inability to capture the important relationship between
physical layer behavior and higher layer performance in a wireless network. Channel state
information (CSI) is essential for receivers performing coherent detection in fading
environments. However, practical channel estimation techniques in wireless communications,
such as pilot symbol assisted modulation (PSAM) or phase lock loop (PLL), only provide a noisy
estimate of the complex fading gain and cause amplitude and phase error. The phase error is
modeled based on the particular phase estimation technique. For instance, the phase error caused
by additive white Gaussian noise (AWGN) in phase-locked loops (PLL) can be modeled by the
Tikhonov distribution. The Tikhonov distribution is characterized by a single parameter, the loop
signal-to-noise ratio (SNR). Its impact on coherent detection over AWGN and fading channels
has been extensively investigated.

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The empirical Nakagami fading distribution was first proposed in the 1940’s by Minoru
Nakagami [2] to model small-scale (rapid) fading in long-distance wireless channels.
Small-scale fading is the short-term variation in the received signal amplitude due to the
constructive and destructive interference of the multiple signal paths between the
transmitter and receiver [1].

The propagation of electromagnetic waves between a transmitter and a receiver through a

complex environment is accompanied by typical wave phenomena such as diffraction, scattering,
reflection, and absorption [3]. Therefore, the received signal is composed of various components
with different delays and amplitudes; these commonly include a direct component and scattered
and reflected components. The total received electric field can be interpreted as a vector sum in
the complex plane [4]. Due to the existence of a great variety of fading environments; several
statistical models are used to describe the probability distribution of the received signal
amplitude. Small-scale fading models include Rayleigh [5], Rice [6]. Rayleigh distribution is the
simplest fast-fading model. It is based on the assumption that there are a large number of
statistically independent reflected and scattered components with random amplitudes and delays
(i.e. pure diffuse scattering). The instantaneous amplitude is obtained as the modulus of a
complex Gaussian process; r (t) X (t) jY (t), where X and Y are uncorrelated zero mean
Gaussian processes with equal variances. In general, other fading distributions can be
theoretically derived by assuming that the random vector phases are not uniformly distributed, or
that the vectors are correlated [7]. For example, when there exists a certain communication path
(usually, a line-of-sight) that results in a strong component besides the multiple independent
multipath components, Rician distribution provides a better model [8].
Making no assumptions on the statistics of the amplitudes and phases of the multiple
received versions (i.e., allowing X and Y to have different variances or being correlated
[9]) led to the more general Nakagami-m distribution. Nakagami (and independently,
Beckmann) has derived the m-distribution as an approximate form of the distribution of
the sum of large number of vectors allowing correlated components with different mean
values and variances [10]. The Nakagami-m distribution has gained a lot of attention due
to its ability to model a wide class of fading channel conditions and to fit well the
empirical data. It was originally proposed because it matched empirical data better
than other distributions (i.e. Rayleigh, Rice, or lognormal distributions.

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1.2. FADING
Fading is used to express rapid fluctuation of the amplitude of a radio signal over a short period
of time or travel distance. Fading channel models are often used to model the effects of
electromagnetic transmission of information over the air in cellular networks and broadcast
communication. Fading channel models are also used in underwater acoustic communications to
model the distortion caused by the water.
1.2.1 TYPES OF FADING
There are two types of fading are there. Those are:
1 Slow fading
2. Fast fading
SLOW FADING
Arises when the coherence time of the channel is large relative to the delay requirement of the
application. In this regime, the amplitude and phase change imposed by the channel can be
considered roughly constant over the period of use. Slow fading can be caused by events such
as shadowing, where a large obstruction such as a hill or large building obscures the main signal
path between the transmitter and the receiver. The received power change caused by shadowing
is often modeled using a long normal distribution with a standard deviation according to
the long-distance path loss model.
FAST FADING
Occurs when the coherence time of the channel is small relative to the delay requirement of the
application. In this case, the amplitude and phase change imposed by the channel varies
considerably over the period of use.
The Nakagami fading model is capable of modeling a wide class of fading channel conditions
and it fits well the empirical data.

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1.3 Nakagami Fading

The Nakagami-m distribution is the best-fit for urban radio multipath channels.
If α ≡ Nakagami-m fading,
f(α)=Γ(2m) (mΩ)mα2m-1e-mα2/Ω,whereΩ=E[α2].
m<1:WorsethanRayleighfadinginperformance
m=1:Rayleighfading
m>1:BetterthanRayleighfadinginperformance
Notably, m = Var E2[[αα22]] = E[(αΩ2-2Ω)2] is called the fading figure.
Probability density function of Nakagami-m with Ω=1

Figure 1: PDF of Nakagami

BPSK performance under Nakagami-m fading is

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Figure 2: BPSK Performance of Nakagami

In some channel, the system performance may degrade even worse, such as Rummler’s model.
Where deep fading occurs at some frequency.

FΩ = 1 density function

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1.4 THE NAKAGAMI DISTRIBUTION

The Nakagami distribution describes an even wider class of fading. Let the useful signal be
described with the dependence
st rtcosot t
In order to describe the signal envelope the Nakagami distribution having the density

Where:
= mean signal power defined as
  Er 2

m=depths of fading, is the inverse of the standardized variance of the useful signal
envelope square, i.e. the inverse of the standardized mean signal power.
Parameter m is calculated with the dependence:

The Nakagami distribution is a chi distribution, in which parameter m can also take non integer
values. The Nakagami distribution is often referred to as distribution m. A signal having the
Nakagami envelope distribution has the following moment value of the k-order

 one-sided standard distribution, when m=0.5;

 the Rayleigh distribution, when m = 1;

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1.5 INTEGRAL CHARACTERISTICS OF NAKAGAMI FADING

Nakagami fading distribution is suitable for describing mobile radio transmission in complex
medium. In practice it has proved very useful because of easy manipulation and wide range of
applicability of various approximations. Since the Nakagami-m random process is defined as
envelope of the sum of 2m independent Gauss random processes, the Nakagami fading
distribution is described by pdf:

Where z is the received signal level, Γ() is gamma function, m is the parameter of
fading depth (fading figure), defined as:

While Ω is average signal power:

In the case when the parameter of fading depth is m=1 Nakagami-m distribution reduces to the
familiar Rayleigh distribution, while the case m=0.5 corresponds to the unilateral Gauss
distribution. Case m  describes the channel without fading. With certain restrictions
Nakagami-m distribution can approximate Rice distribution. Nakagami-m channel model in the
analytical sense is simpler than Rice’s, in which appears Bessel function, so that using the above
approximation calculation of statistical characteristics significantly simplifies. We choose to
analyze Nakagami-m channel model for reasons of generality, and because the other models can
be described by the Nakagami-m distribution by appropriate choice of relevant parameters.

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1.6. Nakagami Fading Channel (NFC)

Nakagami fading formulated a parametric gamma function to describe rapid fading in high
frequency long-distance propagation. Although empirical, the formula is rather elegant and has
proven useful. In any atmosphere in which fading is a present, wireless engineer must effort to
predict the effect of such fading on the transmitted signal as it passes through the transmission
medium to the receiver. A variety of statistical models derived from probability theory and actual
field observations help us model channel behavior and in so doing provide us with information
essential to system design. If received signal amplitude levels can be predicted based on these
models, then required transmitter power, system architecture, modulation technique, and a host
of other parameters can be adjusted to compensate for the channel.
The Nakagami-m distribution is one such model, which can be used to emulate fading channel
conditions is called as Nakagami Fading Channel (NFC). There is no direct way to generate an
efficient and high quality Nakagami fading channel with the correct temporal correlation
properties. There are two general approaches to generate the Nakagami fading with a known
time correlation function. The first method maps a Rayleigh fading channel into the required
Nakagami channel using some transfer function. The Rayleigh fading channel could be
generated using one of the well-established methods mentioned below. However, the time
correlation function for the Nakagami process should be also mapped to the one required for the
Rayleigh process or for each of the in-phase and quadrature components of the Rayleigh process.
The second method decomposes the Nakagami process into a sum of Gaussian Processes. The
probability density function of a Nakagami process is the amplitude of the sum of squared
independent Gaussian processes when m is an integer or half integer. In the conducted simulation
scenario, the receiver is constantly moving away from the transmitter. Thus plotting the power
envelope versus distance is equivalent of plotting it versus the time. Each component of the
power envelope is observed separately to confirm that each part of the model is correct. It shows
clearly how the path loss exponent changes at the critical distance at226 m (rural environment).
The power at distance1 m (reference distance) is calculated using the free space path loss model.
From the graph it is shown to be around -53 dB.

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REFERENCE
[1] D. Tse, and P. Viswanath, Fundamentals of wireless communications. Cambridge
University Press, 2005.
[2] M. Nakagami, “The m-distribution – A general formula of intensity distribution of
rapid fading,” in Statistical Methods in Radio Wave Propagation. New York: Pergamon,
1960, pp. 3-36.
[3] W. R. Braun, and U. Dersch, “A physical mobile radio channel model,” IEEE Trans.
Veh.Technol.,vol.40,pp.472–482,May1991.
[4] U. Dersch, and R. J. Ruegg, "Simulations of the time and frequency selective
outdoor mobile radio channel", IEEE Trans. Veh. Technol., vol. 42, no. 3, pp. 338-344,
Aug. 1993.
[5] L. Rayleigh, “On the Resultant of a Large Number of Vibrations of the Same Pitch
and of Arbitrary Phase,” Philosophical Mag., vol. 27, no. 6, pp. 460–69, Jun. 1889.
[6] S. O. Rice, “Statistical properties of a sine wave plus random noise,” Bell Syst. J.,
vol. 27, pp. 109–157, Jan. 1948.
[7] G. E. Pollon, "Statistical parameters for scattering from randomly oriented arrays,
cylinders, and plates," IEEE Trans Trans. Antennas Propagat., vol. 18, no. 1, pp. 68-75,
January 1970.
[8] D. Tse, and P. Viswanath, Fundamentals of wireless communications. Cambridge
University Press, 2005.
[9] J. M. Romero-Jerez, and F. J. Lopez-Martinez, "A new framework for the
performance analysis of wireless communications under Hoyt (nakagami-q)
fading", IEEE Trans. Inf. Theory, vol. 63, no. 3, pp. 1693-1702, 2017.
[10] B. Chytil, “The Distribution of Amplitude Scintillation and the Conversion of
Scintillation Indices,” J. Atmospheric and Terrestrial Physics, vol. 29, no. 9, Sept. 1967,
pp. 1175–77.