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# An Alternative Statistical Characterization of TWDP Fading Model

Almir Maric, Pamela Njemcevic, and Enio Kaljic, A. Maric, P. Njemcevic, and E.
Kaljic are with the Department of Telecommunications, Faculty of Electrical
Engineering, University of Sarajevo, Sarajevo, Bosnia and Herzegovina.e-mail:
<EMAIL_ADDRESS>
###### Abstract

Two-wave with diffuse power (TWDP) is one of the most promising models for
description of small-scale fading effects in emerging wireless networks.
However, its current statistical characterization has several fundamental
issues. Primarily, conventional TWDP parameterization is not in accordance
with the model’s underlying physical mechanisms. In addition, available TWDP
expressions for PDF, CDF, and MGF are given either in integral or approximate
forms, or as mathematically untractable closed-form expressions. Consequently,
the existing TWDP statistical characterization does not allow accurate
evaluation of system performance (such as error and outage probability) in all
fading conditions for most modulation and diversity techniques. In this paper,
the existing statistical characterization of the TWDP fading model is improved
by overcoming some of the noticed issues. In this regard, physically justified
TWDP parameterization is proposed and used for further calculations.
Additionally, exact infinite-series PDF and CDF are introduced. Based on these
expressions, the exact MGF of the SNR is derived in form suitable for
mathematical manipulations. The applicability of the proposed MGF for
derivation of the exact average symbol error probability (ASEP) is
demonstrated with the example of M-ary PSK modulation. Therefore, in this
paper, M-ary PSK ASEP is derived as an explicit expression for the first time
in the literature. The derived expression is further simplified for large SNR
values in order to obtain a closed-form asymptotic ASEP, which is shown to be
applicable for SNR > 20 dB. All proposed expressions are verified by Monte
Carlo simulation in a variety of TWDP fading conditions.

###### Index Terms:

TWDP fading channel, MGF, M-ary PSK, ASEP.

##  I Introduction

Due to the tremendous growth of Internet data traffic, bandwidth requirements
have become especially pronounced. To cope with these requirements, the fifth
generation (5G) mobile network is emerging as the latest wireless
communication standard. At the heart of this technology lies the use of
millimeter wave (mmWave) frequency band. However, a signal propagating in
mmWave band exhibits unique propagation properties, making traditional small-
scale fading models inadequate and thus demanding more generalized models. To
address this issue, Durgin et al. [1] proposed the two-wave with diffuse power
(TWDP) model, which assumes that the complex envelope consists of two strong
specular components and many weak diffuse components. As such, it encompasses
Rayleigh, Rician, and two-ray fading models as its special cases [1],
simultaneously enabling modeling of both worse-than-Rayleigh and Rician-like
fading conditions.

In the last twenty years the TWDP model has been extensively studied
theoretically (there are more than 5000 results on Google). Additionally, its
existence is supported by practical evidences both in mmWave communication
systems equipped with directional antennas or arrays [2] and in wireless
sensor networks deployed in cavity environments [3]. However, to the best of
the authors’ knowledge, there are at least two factors that motivate further
studies of TWDP fading and its performance:

  1. 1.

existing TWDP parameterization is not in accordance with the model’s
underlying physical mechanisms,

  2. 2.

analytical forms of the existing expressions for PDF and MGF disallow accurate
evaluation of the effects of TWDP fading on system performance.

To describe TWDP fading, Durgin et al. [1] proposed two parameters, $K\geq 0$
and $0\leq\Delta\leq 1$, which reflect the relationship between specular and
diffuse components and between the specular components themselves. However, it
is striking that for a significant range of values ($0\leq\Delta\leq 0.5$) the
impact of $\Delta$ on the system performance metrics (e.g. ASEP and outage
probability) is negligible (in fact, in some cases the corresponding curves
almost overlap). This is obviously counterintuitive considering the physical
meaning attributed to the parameter $\Delta$. It is thus essential to examine
this problem in depth.

Regarding TWDP PDF expressions, they exist in integral [1, eq. (29)], [1, eq.
(32)], [4, eq. (16)], and approximate [1, eq. (17)] forms. Therefore, the
exact evaluation of system performance metrics based on the existing integral
expressions is not mathematically tractable, disabling direct observation of
TWDP fading effects on system performance. Accordingly, the closed-form
results of performance evaluation (e.g. error and outage probability, ect.)
obtained from PDF are available only in approximate forms [5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16]. However, it has been shown that analysis based on an
approximate PDF expression is accurate only for a narrow range of $K$ and
$\Delta$ values [17, 1], which can only be used for description of limited
fading conditions.

To overcome these limitations, Rao et al. [18] proposed an alternative
approach to statistical characterization of TWDP fading based on the
observation that the TWDP fading model can be expressed in terms of a
conditional underlying Rician distribution. Thus, by invoking the observed
similarities and the existing expressions of Rician fading, Rao et al. derived
a novel form of TWDP MGF expression [18, eq. (25)]. Thereby, in contrast to
previously derived approximate MGF expressions [13, eq. (8)][7, eq. (12)], the
one proposed in [18] is given as a simple closed-form solution. However, this
form is also not suitable for mathematical manipulations, and consequently,
for calculation of the exact ASEP expressions for most modulation and
diversity schemes. The exception is ABEP expressions for DBPSK modulations as
derived in [4]. Accordingly, in order to accurately evaluate the effects of
TWDP fading on ASEP, outage probability, etc., it is of tremendous importance
to provide mathematically tractable PDF and MGF expressions.

Considering the above, our contributions are as follows:

  1. 1.

We proposed alternative TWDP parameterization which is in accordance with
model’s underlying physical mechanisms.

  2. 2.

We introduced the exact convergent infinite-series TWDP envelope PDF and CDF
expressions (previously derived in [19, 20]).

  3. 3.

We derived an alternative exact form of SNR MGF based on the adopted CDF
expression and proposed parameterization, which is shown to be suitable for
mathematical manipulations.

  4. 4.

Based on the obtained MGF, we derived M-ary PSK ASEP in exact infinite-series
form, which is, to the best of our knowledge, the first such expression
proposed to date.

  5. 5.

We also derived asymptotic M-ary PSK ASEP as a simple closed-form expression,
which tightly follows the exact one for the practical range of SNR values,
i.e. for SNR > 20 dB.

The rest of the paper is structured as follows. In Section II, the TWDP fading
model is introduced and statistically described using alternative envelope PDF
and CDF expressions, given in terms of newly proposed parameters. The
alternative MGF of the SNR expression is derived in Section III. In Section
IV, the applicability of the proposed MGF for accurate performance analysis is
demonstrated by deriving the exact and asymptotic M-ary PSK ASEP expressions,
which are then verified by Monte Carlo simulation. The main conclusions are
outlined in Section V.

##  II TWDP fading model

In the slow, frequency nonselective fading channel with TWDP statistic, the
complex envelope $r(t)$ is composed of two strong specular components
$v_{1}(t)$ and $v_{2}(t)$ and many low-power diffuse components treated as a
random process $n(t)$:

$\begin{split}r(t)&=v_{1}(t)+v_{2}(t)+n(t)\\\
&=V_{1}\exp{\left(j\Phi_{1}\right)}+V_{2}\exp{\left(j\Phi_{2}\right)}+n(t)\end{split}$
(1)

Specular components are assumed to have constant magnitudes $V_{1}$ and
$V_{2}$ and uniformly distributed phases $\Phi_{1}$ and $\Phi_{2}$ in
$[0,2\pi)$, while diffuse components are treated as a complex zero-mean
Gaussian random process $n(t)$ with average power $2\sigma^{2}$. Consequently,
the average power of a signal $r(t)$ is equal to
$\Omega=V_{1}^{2}+V_{2}^{2}+2\sigma^{2}$.

###  II-A The revision of parameter $\Delta$

Conventional parameterization of TWDP fading, originally proposed in [1],
introduced two parameters:

$K\triangleq\frac{\text{average\leavevmode\nobreak\
specular\leavevmode\nobreak\ power}}{\text{diffuse\leavevmode\nobreak\
power}}=\frac{V_{1}^{2}+V_{2}^{2}}{2\sigma^{2}}$ (2)

and

$\Delta\triangleq\frac{\text{peak\leavevmode\nobreak\
specular\leavevmode\nobreak\ power}}{\text{average\leavevmode\nobreak\
specular\leavevmode\nobreak\
power}}-1=\frac{2V_{1}V_{2}}{V_{1}^{2}+V_{2}^{2}}$ (3)

Parameter $K$, $(0\leq K<\infty)$, like in the Rician fading model,
characterizes TWDP fading severity. Parameter $\Delta$, $(0\leq\Delta\leq 1)$
for $V_{1}\geq 0$, $V_{2}\geq 0$, and $V_{2}\leq V_{1}$, implicitly
characterizes the relationship between the magnitudes of specular components.
However, the physical justification of the relationship between $V_{1}$ and
$V_{2}$, introduced by the definition of parameter $\Delta$ in (3), is
questionable. Namely, according to [21] "for $0<\Delta<1$ there is a nonlinear
relation between the magnitude of the specular components $V_{1}$ and $V_{2}$,
i.e., $V_{2}=V_{1}(1-\sqrt{1-\Delta^{2}})/\Delta$. However, the physical facts
suggest a different conclusion about the relation between $V_{1}$ and $V_{2}$.
In particular, according to the model for TWDP fading, specular components are
constant and they are a consequence of specific propagation conditions. Since
electromagnetic wave is propagating in a linear medium, a natural choice to
appropriately characterize the relation between magnitudes $V_{1}$ and $V_{2}$
is given by $\Gamma\triangleq\ V_{2}/V_{1}$, where $V_{2}\leq V_{1}$.
Seemingly, parameters $\Delta$ and $\Gamma$ are both motivated by physical
arguments. However, they do not have the same level of physical intuition.".

Based on the above citation, it is necessary to further investigate the impact
of nonlinear $\Delta$-based parameterization of TWDP statistics. Accordingly,
parameters $K$ and $\Delta$ are written in terms of $V_{2}/V_{1}$, as:

$\begin{split}K&=\frac{V_{1}^{2}+V_{2}^{2}}{2\sigma^{2}}\\\
&=\frac{V_{1}^{2}}{2\sigma^{2}}\left[1+\left(\frac{V_{2}}{V_{1}}\right)^{2}\right]=K_{Rice}(1+\Gamma^{2})\end{split}$
(4)

and

$\Delta=\frac{2\frac{V_{2}}{V_{1}}}{1+\left(\frac{V_{2}}{V_{1}}\right)^{2}}$
(5)

where $\Gamma=V_{2}/V_{1}$ and $K_{Rice}=V_{1}^{2}/(2\sigma^{2})$ represents
the Rician parameter $K$ of a dominant specular component. Based on the above,
parameter $K$ is also expressed in terms of $\Delta$, as:

$\begin{split}K=&\frac{V_{1}^{2}+V_{2}^{2}}{2\sigma^{2}}\\\
&=\frac{1}{2\sigma^{2}}\frac{2V_{1}V_{2}}{\Delta}\frac{V_{1}}{V_{1}}=\frac{V_{1}^{2}}{2\sigma^{2}}\frac{2}{\Delta}\frac{V_{2}}{V_{1}}\\\
&=K_{Rice}2\frac{1-\sqrt{1-\Delta^{2}}}{\Delta^{2}}\end{split}$ (6)

Fig. 3 illustrates the functional dependence of parameter $\Delta$ versus
$V_{2}/V_{1}$ (5). In the same figure, the linear dependence of $\Gamma$ on
$V_{2}/V_{1}$ is also illustrated as a benchmark. From Fig. 3 it is evident
that for $0<V_{2}/V_{1}<1$, $\Delta$ differs $\Gamma$ not only in value, but
also in terms of the character of their functional dependence on
$V_{2}/V_{1}$. Consequently, when $V_{2}/V_{1}$ changes from $0.6$ to $1$,
$\Delta$ changes only between $0.9$ and $1$. In general, for
$0<V_{2}/V_{1}<1$, $\Delta$ is always greater than $\Gamma$.

Fig. 3 and Fig. 3 illustrate the dependencies of the normalized parameter $K$
($K/K_{Rice}$) on $\Delta$ (4) and $\Gamma$ (6), which are clearly very
different. For $\Delta\leq 0.8$, $K$ vs. $\Delta$ has a relatively small
slope, while for $\Delta>0.8$, the slope is very sharp. In contrast, for
$0\leq\Gamma\leq 1$, the change in parameter $K$ is relatively uniform. In
other words, parameter $\Gamma$ does not change the character of the
definition expression of parameter $K$ (see (4)), while parameter $\Delta$
completely changes its character (see (6)).

As a consequence, parameterization based on a nonlinear relationship between
$V_{1}$ and $V_{2}$ causes anomalies in graphical representations of PDF and
ASEP expressions. Namely, corresponding ASEP curves are indistinguishably
dense spaced for $\Delta<0.5$, which can be clearly observed from [17, Fig. 3]
and [4, Fig. 7]. In addition, the shapes of the corresponding PDF curves for
$\Delta<0.5$ are almost the same as the shape of a Rician PDF curve obtained
for $\Delta=0$ and the same value of $K$, which is evident from [1, Fig. 7]
and [4, Fig. 3]. Therefore, in using $\Delta$-based parameterization, it is
not possible to clearly observe the effect of the increment of $\Delta$ on PDF
shape and ASEP values. Consequently, in most TWDP literature, PDF and ASEP
curves are plotted only for specific values of $\Delta$, i.e. $\Delta=0.5$ and
$\Delta=1$, for which the mentioned differences can be easily distinguished,
thus avoiding graphical presentation and explanation of the results for
$0\leq\Delta\leq 0.5$.

Accordingly, considering conducted elaboration, TWDP fading in this paper will
be characterized by parameters $K$ and $\Gamma$.

###  II-B Envelope PDF and CDF expressions

To provide a mathematically convenient tool for TWDP performance evaluation,
alternative exact envelope PDF and CDF expressions are proposed. Namely, it is
noticed that assumptions about statistical characteristics of a complex
envelope in a TWDP fading channel given in (1) are the same as those from [19,
20] where the sum of signal, cochannel interference, and AWGN is modeled.
However, unlike the existing approximate TWDP PDF and CDF expressions, PDF and
CDF in [19, 20] are given in the exact form. Accordingly, using [19, eq. (6)]
and [20, eq. (12)] and considering adopted parameterization, we propose the
following TWDP envelope PDF and CDF expressions:

$\begin{split}f_{R}(r)&=\frac{r}{\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}-K\right)}\sum_{m=0}^{\infty}\varepsilon_{m}(-1)^{m}\\\
&\times
I_{m}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{1}{1+\Gamma^{2}}}\right)\\\
&\times
I_{m}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{\Gamma^{2}}{1+\Gamma^{2}}}\right)I_{m}\left(2K\frac{\Gamma}{1+\Gamma^{2}}\right)\end{split}$
(7)

Figure 1: Dependence of $\Delta$ and $\Gamma$ on $V_{2}/V_{1}$

Figure 2: Dependence of $K/K_{Rice}$ on $\Delta$

Figure 3: Dependence of $K/K_{Rice}$ on $\Gamma$

and

$\begin{split}F_{R}(r)&=\frac{r^{2}}{2\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}\right)}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\left(\frac{K}{1+\Gamma^{2}}\right)^{m}\\\
&\times{}_{1}F_{1}\left(1-m;2;\frac{r^{2}}{2\sigma^{2}}\right){}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\end{split}$
(8)

where $0\leq V_{2}\leq V_{1}$, $\varepsilon_{0}=1$, $\varepsilon_{m}=2$ for
$m\geq 1$, $I_{\nu}(\cdot)$ is a modified $\nu$-th order Bessel function of
the first kind, while ${}_{1}F_{1}(\cdot;\cdot;\cdot)$ and
${}_{2}F_{1}(\cdot,\cdot;\cdot;\cdot)$ are confluent and Gaussian
hypergeometric functions, respectively.

####  II-B1 Special cases of a TWDP model

It is easy to show that (7) and (8) can be reduced to Rayleigh and Rician PDF
and CDF expressions.

The Rayleigh model assumes the absence of specular and the presence of only
diffuse multipath components. It can be obtained from TWDP fading for
$V_{1}=V_{2}=0$, i.e. $K=0$. So, by applying $K=0$ into (7) and (8), with
$I_{\nu}(0)=0$ for $\nu\neq 0$ and $I_{0}(0)=1$, (7) and (8) can be reduced to
Rayleigh PDF and CDF expressions:

$\begin{split}f_{R}(r)\bigg{|}_{K=0}&=\frac{r}{\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}\right)}\end{split}$
(9)
$\begin{split}F_{R}(r)\bigg{|}_{K=0}&=\frac{r^{2}}{2\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}\right)}\end{split}$
(10)

respectively.

Rician fading assumes the presence of one specular component and many diffuse
components. It can be obtained from TWDP fading for $V_{2}=0$, i.e.
$\Gamma=0$. In this case, (7) can be reduced to a well-known Rician PDF
expression:

$f_{R}(r)\bigg{|}_{\Gamma=0}=\frac{r}{\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}-K\right)}I_{0}\left(r\sqrt{2K\sigma^{2}}\right)$
(11)

Additionally, by inserting $\Gamma=0$ into (8) and considering that
${}_{2}F_{1}(\cdot,\cdot;\cdot;0)=1$ and ${}_{1}F_{1}(1;2;x)=(e^{x}-1)/x$,
TWDP CDF reduces to:

$\begin{split}F_{R}(r)\bigg{|}_{\Gamma=0}&=1-\exp{\left(-\frac{r^{2}}{2\sigma^{2}}\right)}+\frac{r^{2}}{2\sigma^{2}}\exp{\left(-\frac{r^{2}}{2\sigma^{2}}\right)}\\\
&\times\sum_{m=1}^{\infty}\frac{(-1)^{m}}{m!}K^{m}{}_{1}F_{1}\left(1-m;2;\frac{r^{2}}{2\sigma^{2}}\right)\end{split}$
(12)

which, according to [22, eq. (8.352.1)], [22, eq. (8.972.1)] and [23, eq.
(12)], takes the well-known form of a Rician CDF, expressed in terms of the
first-order Marcum Q-function $Q_{1}(\cdot)$ [4]:

$F_{R}(r)\bigg{|}_{\Gamma=0}=1-Q_{1}\left(\sqrt{2K},\frac{r}{\sigma}\right),$
(13)

####  II-B2 Convergence analysis

It is also easy to show that (7) and (8), as infinite-series expressions, are
convergent.

To prove convergence of (7), d’Alembert’s ratio test is used. According to the
test, the infinite-series $\sum_{k}c_{k}$ is convergent if the limiting
expression $\lim_{k\to\infty}|c_{k+1}/{c_{k}}|$ is smaller than $1$. Thus, the
ratio test applied to (7) yields the following expression:

$\begin{split}&\lim_{k\to\infty}\bigg{|}\frac{c_{k+1}}{c_{k}}\bigg{|}=\lim_{k\to\infty}\Bigg{[}\frac{I_{k+1}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{1}{1+\Gamma^{2}}}\right)}{I_{k}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{1}{1+\Gamma^{2}}}\right)}\\\
&\times\frac{I_{k+1}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{\Gamma^{2}}{1+\Gamma^{2}}}\right)I_{k+1}\left(2K\frac{\Gamma}{1+\Gamma^{2}}\right)}{I_{k}\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{\Gamma^{2}}{1+\Gamma^{2}}}\right)I_{k}\left(2K\frac{\Gamma}{1+\Gamma^{2}}\right)}\Bigg{]}\end{split}$
(14)

which can be calculated using [24, eq. (3.12)] as:

$\begin{split}&\lim_{k\to\infty}\bigg{|}\frac{c_{k+1}}{c_{k}}\bigg{|}=\lim_{k\to\infty}\Bigg{[}\frac{\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{1}{1+\Gamma^{2}}}\right)}{\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{1}{1+\Gamma^{2}}}+k\right)}\\\
&\times\frac{\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{\Gamma^{2}}{1+\Gamma^{2}}}\right)}{\left(2r\sqrt{\frac{K}{2\sigma^{2}}\frac{\Gamma^{2}}{1+\Gamma^{2}}}+k\right)}\frac{\left(2K\frac{\Gamma}{1+\Gamma^{2}}\right)}{\left(2K\frac{\Gamma}{1+\Gamma^{2}}+k\right)}\Bigg{]}=0<1\end{split}$
(15)

The above expression shows that the series in (7) is convergent.

Similarly, the convergence of CDF (8) is also proven using d’Alambert’s ratio
test, with its $k^{th}$ term denoted by $c_{k}$. Since
${}_{2}F_{1}\left(-k,-k;1;\Gamma^{2}\right)$ is $k^{th}$ order polynomial, due
to [22, eq. (8.822-4), (8.911-1), (8.917.1)], it can be written as
$((2k)!(1+\Gamma^{2})^{k})/(2^{k}(k!)^{2})+O(x^{k-1})$. Furthermore, following
[22, eq. (8.970-1)(8.972-1)], it is evident that
${}_{1}F_{1}\left(1-k;2;\frac{r^{2}}{2\sigma^{2}}\right)$ is also a $k^{th}$
order polynomial dominated by $1/k$ when $r^{2}\leq 2\sigma^{2}$ and by
$[((-r^{2})/(2\sigma^{2}))^{(k-1)}]/k!$ when $r^{2}>2\sigma^{2}$. Considering
the above, d’Alambert’s ratio test yields:

$\lim_{k\to\infty}\bigg{|}\frac{c_{k+1}}{c_{k}}\bigg{|}=\begin{cases}\lim_{k\to\infty}\left(K\frac{r^{2}}{2\sigma^{2}}\frac{(2k+1)}{(k+1)^{3}}\right),&r^{2}>2\sigma^{2}\\\
\lim_{k\to\infty}\left(K\frac{(2k+1)k}{(k+1)^{3}}\right),&r^{2}\leq
2\sigma^{2}\end{cases}$

which is always equal to zero and thus smaller than one. Therefore, the series
in (8) is also convergent.

####  II-B3 Graphical results

In order to investigate the accuracy of (7) and (8) and their applicability
for modeling various fading conditions, equations (7) and (8) are plotted for
different sets of TWDP parameters.

Equation (7) is used to plot the normalized envelope PDF,
$f_{R}(r/\sqrt{\Omega})$, for different fading conditions: Rician with $K=8$
and $\Gamma=0$; Rayleigh with $K=0$; and others, with $K=8$ and $\Gamma=0.5$;
and $K=14$ and $\Gamma=1$. Fig. 4(a) depicts these curves together with
corresponding normalized histograms created by Monte Carlo simulation. All
curves are obtained by limiting truncation error below $10^{-6}$, i.e., by
employing up to $35$ summation terms in all tested cases. Each normalized
histogram, composed of $20$ equally spaced bins, is computed independently by
generating $10^{6}$ samples for the considered fading conditions. Fig. 4(a)
shows matching results between the analytical and simulated approaches, thus
validating the proposed PDF expression in diverse fading conditions.

Fig. 4(b) compares normalized envelope CDF curves $F_{R}(r/\sqrt{\Omega})$
obtained from (8) with normalized cumulative histograms. Similarly, Monte
Carlo simulation is used to generate histograms with the same set of
parameters as in the PDF comparison. Analytically obtained curves are
generated by employing up to $118$ summation terms in order to achieve a
truncation error of less than $10^{-26}$. Normalized cumulative histograms are
created from $10^{6}$ samples divided into $20$ bins. The conducted comparison
shows matching results between the analytical and simulated approaches, thus
demonstrating the applicability of (8) for accurate calculation of CDF values
in different fading conditions.

Rayleigh: ($K=0$)Rice:
($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)($K=14,\Gamma=1$) (a)

Rayleigh: ($K=0$)Rice:
($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)($K=14,\Gamma=1$) (b)

Figure 4: TWDP normalized envelope (a) PDF and (b) CDF curves for various
combinations of $K$ and $\Gamma$

##  III Alternative form of TWDP SNR MGF expression

In this section, the alternative form of the MGF of the SNR is derived based
on the proposed CDF expression. Here, the well-known relationship between CDF
and MGF is used [25, eq. (1.2)]:

$\begin{split}\mathcal{M}_{\gamma}(s)&=\int_{0}^{+\infty}f_{\gamma}(\gamma)\exp{\left(s\gamma\right)}\mathop{}\\!\mathrm{d}{\gamma}\\\
&=\mathcal{L}\left\\{f_{\gamma}(\gamma);\gamma,-s\right\\}\\\
&=\mathcal{L}\left\\{\frac{\mathop{}\\!\mathrm{d}{}}{\mathop{}\\!\mathrm{d}{\gamma}}F_{\gamma}(\gamma);\gamma,-s\right\\}\\\
&=-s\mathcal{L}\left\\{F_{\gamma}(\gamma);\gamma,-s\right\\}-F_{\gamma}(\gamma=0)\\\
&=-s\mathcal{L}\left\\{F_{\gamma}(\gamma);\gamma,-s\right\\}\end{split}$ (16)

where
$\mathcal{L}\\{h(t);t,p\\}\triangleq\int_{0}^{\infty}h(t)e^{-pt}\mathop{}\\!\mathrm{d}{t}$
represents Laplace transform of $h(t)$ from $t$-domain into the $p$-domain,
and $F_{\gamma}(\gamma)$ is the CDF of the SNR. $F_{\gamma}(\gamma)$ is
obtained from (8) according to the random variable transformation
$\gamma=r^{2}\frac{E_{s}}{N_{0}}$, as:

$\begin{split}F_{\gamma}(\gamma)&=\frac{\gamma}{\gamma_{0}}\left(1+K\right)\exp\left({-\frac{\gamma}{\gamma_{0}}(1+K)}\right)\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\\\
&\times\left(\frac{K}{1+\Gamma^{2}}\right)^{m}{}_{1}F_{1}\left(1-m;2;\frac{\gamma}{\gamma_{0}}(1+K)\right)\\\
&\times{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\end{split}$ (17)

where $\gamma_{0}=2\sigma^{2}(1+K)\frac{E_{s}}{N_{0}}$ is the average SNR,
$E_{s}$ denotes symbol energy, and $N_{0}/2$ is the power spectral density of
the white Gaussian noise.

For simplicity, (17) is expressed in the following form:

$\begin{split}F_{\gamma}(\gamma)=\sum_{m=0}^{\infty}A\gamma
B_{m}\exp\left({-A\gamma}\right){}_{1}F_{1}\left(1-m;2;A\gamma\right)\end{split}$
(18)

where
$B_{m}=\left(-K/(1+\Gamma^{2})\right)^{m}{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)/m!$
and $A=(1+K)/\gamma_{0}$. Based on [26, eq. (07.20.16.0001.01)], (18) is
further simplified as:

$F_{\gamma}(\gamma)=\sum_{m=0}^{\infty}A\gamma
B_{m}{}_{1}F_{1}\left(1+m;2;-A\gamma\right)$ (19)

Laplace transform of (19) is then obtained using [27, eq. (3.35.1-2)] as:

$\mathcal{L}\left\\{F_{\gamma}(\gamma);\gamma,s\right\\}=\sum_{m=0}^{\infty}\frac{AB_{m}}{s^{2}}{}_{2}F_{1}\left(1+m;2;2,-\frac{A}{s}\right)$
(20)

which, according to [26, eq. (07.23.03.0080.01)], can be expressed in the
following form:

$\begin{split}\mathcal{L}\left\\{F_{\gamma}(\gamma);\gamma,s\right\\}=\sum_{m=0}^{\infty}\frac{AB_{m}}{\left(A+s\right)^{2}}\left(\frac{s}{A+s}\right)^{m-1}\end{split}$
(21)

Finally, by combining (16) and (21), the MGF is derived as:

$\begin{split}\mathcal{M}_{\gamma}(s)&=\frac{1+K}{1+K-s\gamma_{0}}\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{K}{1+\Gamma^{2}}\right)^{m}\\\
&\times\left(\frac{\gamma_{0}s}{1+K-s\gamma_{0}}\right)^{m}{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\end{split}$
(22)

which represents an alternative form of the exact TWDP MGF of the SNR.

It can be proven that (22) can be easily transformed into the well-known TWDP
MGF expression form [18, eq. (25)] (originally given in terms of $K$ and
$\Delta$). Namely, by using the identity between the Gaussian hypergeometric
function and the Legendre polynomial given by [28, eq. (15.4.14)], as well as
the identity between the Legendre polynomial and the first-kind zero-order
Bessel function given by [29, eq. (0.6)], and after some simple manipulations,
it can be shown that:

$\sum_{m=0}^{\infty}\frac{1}{m!}a^{m}{}_{2}F_{1}(-m,-m;1;b)=\exp{\left(a+ab\right)}I_{0}\left(2a\sqrt{b}\right)$
(23)

Therefore, by using (23), (22) can be written as:

$\begin{split}\mathcal{M}_{\gamma}(s)&=\frac{1+K}{1+K-\gamma_{0}s}\exp{\left(\frac{\gamma_{0}Ks}{1+K-\gamma_{0}s}\right)}\\\
&\times
I_{0}\left(\frac{2\Gamma\gamma_{0}Ks}{(1+K-\gamma_{0}s)(1+\Gamma^{2})}\right)\end{split}$
(24)

which is the same expression as the verified SNR MGF from [18], only expressed
in terms of $K$ and $\Gamma$.

Although simple, the analytical form of MGF expressed by (24) has not been
often used for error rate performance evaluation in TWDP fading channels. The
main disadvantage with this expression is its unfavorable analytical form for
mathematical manipulations. In contrast, the analytical form of MGF as
expressed by (22) enables derivation of the exact expressions for the
performance evaluation in a variety of TWDP fading conditions.

##  IV Error probability of M-ary PSK receiver in TWDP fading channel

###  IV-A The exact M-ary PSK ASEP expression

This section demonstrates the applicability of the proposed TWDP SNR MGF (22)
for derivation of the exact M-ary PSK ASEP expression, where M represents the
order of PSK modulation.

M-ary PSK ASEP in a TWDP fading channel can be determined from [25, eq.
(5.78)]:

$P_{s}(\gamma_{0})=\frac{1}{\pi}\int_{0}^{\pi-\frac{\pi}{M}}\mathcal{M}_{\gamma}\left(-\frac{\sin^{2}{\frac{\pi}{M}}}{\sin^{2}{\theta}}\right)\mathop{}\\!\mathrm{d}{\theta}$
(25)

where $\mathcal{M}_{\gamma}\left(\cdot\right)$ represents the MGF of the SNR
given in (22). Accordingly, equation (25) can be expressed as
$P_{s}(\gamma_{0})=2\mathcal{I}|_{0}^{\frac{\pi}{2}}-\mathcal{I}|_{0}^{\frac{\pi}{M}}$,
where $\mathcal{I}$ represents the indefinite integral defined as:

$\begin{split}\mathcal{I}&=\frac{1}{\pi}\int\mathcal{M}_{\gamma}\left(-\frac{\sin^{2}{\frac{\pi}{M}}}{\sin^{2}{\theta}}\right)\mathop{}\\!\mathrm{d}{\theta}\\\
&=\frac{1}{\pi}\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{K}{1+\Gamma^{2}}\right)^{m}{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\\\
&\times\int\left(\frac{1+K}{1+K-\gamma_{0}s}\right)\left(\frac{\gamma_{0}s}{1+K-\gamma_{0}s}\right)^{m}\mathop{}\\!\mathrm{d}{\theta}\Bigg{|}_{s=-\frac{\sin^{2}{\frac{\pi}{M}}}{\sin^{2}{\theta}}}\end{split}$
(26)

which can be solved using Wolfram Mathematica as:

$\begin{split}\mathcal{I}&=\frac{1}{3\pi}\frac{1+K}{\gamma_{0}}\sum_{m=0}^{\infty}\frac{(-1)^{m}}{m!}\left(\frac{K}{1+\Gamma^{2}}\right)^{m}\frac{\sin^{3}{\theta}}{\sin^{2}{\pi/M}}\\\
&\times{AF}_{1}\left(\frac{3}{2};\frac{1}{2},1+m;\frac{5}{2};\sin^{2}{\theta},-\frac{1+K}{\gamma_{0}}\frac{\sin^{2}{\theta}}{\sin^{2}{\pi/M}}\right)\\\
&\times{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right),\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{for }0\leq\theta\leq\frac{\pi}{2}\end{split}$ (27)

where $AF_{1}(\cdot;\cdot,\cdot;\cdot;\cdot,\cdot)$ is an Appell
hypergeometric function. Considering the above, the integral in (25) can be
solved as:

$\begin{split}P_{s}&(\gamma_{0})=\frac{\sin{\frac{\pi}{M}}}{3\pi}\frac{1+K}{\gamma_{0}}\sum_{m=0}^{\infty}\frac{1}{m!}{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\\\
&\times{\left(\frac{-K}{1+\Gamma^{2}}\right)^{m}}\left[\frac{3\pi}{2\sin^{3}{\frac{\pi}{M}}}{}_{2}F_{1}\left(\frac{3}{2},1+m;2;-\frac{1+K}{\gamma_{0}\sin{\frac{\pi}{M}}}\right)\right.\\\
&\left.-AF_{1}\left(\frac{3}{2};\frac{1}{2},1+m;\frac{5}{2};\sin^{2}{\frac{\pi}{M}},-\frac{1+K}{\gamma_{0}}\right)\right]\end{split}$
(28)

which represents M-ary PSK ASEP given as the exact analytical expression.

###  IV-B Asymptotic expression of M-ary PSK ASEP

To gain further insight into the TWDP M-ary PSK ASEP behavior, the asymptotic
ASEP for large values of $\gamma_{0}$ is derived. Furthermore, this allow us
to relax the computational complexity which occurs for large values of $K$.

Considering that
$AF_{1}\left(a;b_{1},b_{2};c;z_{1},z\right)\sim{}_{2}F_{1}\left(a,b_{1};c;z_{1}\right)$
and ${}_{2}F_{1}\left(a,b;c;z\right)\sim 1$ when $z\xrightarrow{}0$, equation
(28) for large values of $\gamma_{0}$ can be expressed as:

$\begin{split}P_{s}(\gamma_{0})&\approx\frac{\sin{\frac{\pi}{M}}}{3\pi}\frac{1+K}{\gamma_{0}}\sum_{m=0}^{\infty}\frac{\left(\frac{K}{1+\Gamma^{2}}\right)^{m}}{m!}{}_{2}F_{1}\left(-m,-m;1;\Gamma^{2}\right)\\\
&\times\left[\frac{3\pi}{2\sin^{3}{\frac{\pi}{M}}}-{}_{2}F_{1}\left(\frac{3}{2},\frac{1}{2};\frac{5}{2};\sin^{2}{\frac{\pi}{M}}\right)\right]\end{split}$
(29)

Equation (29) can be further simplified using the identity [30, p. 24] and
equation (23), as:

$\begin{split}P_{s}(\gamma_{0})\approx&\frac{1+K}{2\pi\gamma_{0}}\frac{\pi-\frac{\pi}{M}+\frac{1}{2}\sin{\frac{2\pi}{M}}}{\sin^{2}{\frac{\pi}{M}}}e^{-K}I_{0}\left(\frac{2\Gamma
K}{1+\Gamma^{2}}\right)\end{split}$ (30)

which represents a simple, closed-form asymptotic M-ary PSK ASEP expression.

|
---|---
|

Rayleigh: ($K=0$) ..Rice: ($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)
…($K=14,\Gamma=1$).5 … (a)

Rayleigh: ($K=0$) ..Rice: ($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)
…($K=14,\Gamma=1$).5 … (b)

Rayleigh: ($K=0$) ..Rice: ($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)
…($K=14,\Gamma=1$).5 … (c)

Rayleigh: ($K=0$) ..Rice: ($K=8,\Gamma=0$)($K=\phantom{1}8,\Gamma=0.5$)
…($K=14,\Gamma=1$).5 … (d)

Figure 5: Exact (solid line) and asymptotic (dashed line) expressions of TWDP
ASEP for (a) 2-PSK, (b) 4-PSK, (c) 8-PSK and (d) 16-PSK modulations compared
with Monte Carlo simulation results (dots)

###  IV-C Numerical results

In order to validate the conducted error performance analysis and to justify
the proposed parameterization, this section provides graphical interpretation
of analytically derived M-ary PSK ASEP and its comparison to results obtained
by Monte Carlo simulation. Different modulation orders and TWDP parameters are
investigated.

Fig. 5(a) \- 5(d) illustrate the exact (28) and the asymptotic (30) ASEP for
$2$-PSK, $4$-PSK, $8$-PSK, and $16$-PSK modulations for a set of previously
adopted TWDP parameters. ASEP curves, obtained from (28) by limiting
truncation error to $10^{-6}$, i.e., by employing up to 78 summation terms,
are compared with those obtained using Monte Carlo simulations generated with
$10^{6}$ samples. Matching results between the exact and simulated ASEP, as
well as between the exact and high-SNR asymptotic ASEP, can be observed for
the considered modulation orders and the set of TWDP parameters. Accordingly,
derived ASEP expressions can be used to accurately evaluate the error
probability of the M-ary PSK receiver for all fading conditions implied by the
TWDP model.

Based on the above, a comparison of error performance of channels with
different fading severities is also performed following Fig. 5(a) \- 5(d).
Clearly, the signal in the fading condition characterized with $K=14$,
$\Gamma=1$ exhibits worse performance compared to the Rayleigh fading channel
($K=0$), thus representing signal in near hyper-Rayleigh fading conditions. It
also can be observed that ASEP in fading conditions described with the same
value of $K$ increases with increasing $\Gamma$, indicating that signal
performance significantly degrades in channels with $\Gamma=0.5$ with respect
to those in typical Rician channels ($\Gamma=0$).

Fig. 6 illustrates the effect of proposed parameterization on 2-PSK ASEP
curves in TWDP fading channel with $K=6$. Obviously, $\Gamma$-based
parameterization solved the problem of densely-spaced ASEP curves observed for
the entire range of $\Delta$ between $0$ and $0.5$.

(a)

(b)

Figure 6: BPSK ASEP in TWDP channel for $K=6$ and different values of
parameter (a) $\Delta$ (b) $\Gamma$

##  V Conclusion

This paper proposed a novel analytical characterization of TWDP fading
channels achieved by introducing physically justified TWDP parameterization
and exact PDF and CDF expressions, and by deriving the alternative form of the
exact SNR MGF expression. Benefits of the proposed parameterization are
demonstrated on TWDP PDF and ASEP graphical interpretations. A derived MGF is
used for derivation of the exact M-ary PSK ASEP expression, which can be used
to accurately evaluate the error performance of M-ary PSK in various fading
conditions.

## Acknowledgment

The authors would like to thank Prof. Ivo M. Kostić for many valuable
discussions and advice.

## References

  * [1] G. D. Durgin, T. S. Rappaport, and D. A. de Wolf, “New analytical models and probability density functions for fading in wireless communications,” _IEEE Transactions on Communications_ , vol. 50, no. 6, pp. 1005–1015, 2002.
  * [2] T. Rappaport, R. Heath, R. Daniels, and J. Murdock, _Millimeter wave wireless communications_. Upper Saddle River, NJ, USA: Prentice Hall, 2015.
  * [3] J. Frolik, “A case for considering hyper-Rayleigh fading channels,” _IEEE Transactions on Wireless Communications_ , vol. 6, no. 4, pp. 1235–1239, 2007.
  * [4] M. Rao, F. J. Lopez-Martinez, and A. Goldsmith, “Statistics and system performance metrics for the two wave with diffuse power fading model,” in _2014 48th Annual Conference on Information Sciences and Systems (CISS)_ , 2014.
  * [5] S. H. Oh and K. H. Li, “BER performance of BPSK receivers over two-wave with diffuse power fading channels,” _IEEE Transactions on Wireless Communications_ , vol. 4, no. 4, pp. 1448–1454, 2005.
  * [6] H. A. Suraweera, W. S. Lee, and S. H. Oh, “Performance analysis of QAM in a two-wave with diffuse power fading environment,” _IEEE Communications Letters_ , vol. 12, no. 2, pp. 109–111, 2008.
  * [7] D. Dixit and P. R. Sahu, “Performance of QAM signaling over TWDP fading channels,” _IEEE Transactions on Wireless Communications_ , vol. 12, no. 4, pp. 1794–1799, 2013.
  * [8] S. Singh and V. Kansal, “Performance of M-ary PSK over TWDP fading channels,” _International Journal of Electronics Letters_ , vol. 4, no. 4, pp. 433–437, 2016.
  * [9] B. S. Tan, K. H. Li, and K. C. Teh, “Symbol-error rate of selection combining over two-wave with diffuse power fading,” in _2011 5th International Conference on Signal Processing and Communication Systems (ICSPCS)_ , 2011.
  * [10] S. Haghani, “Average BER of BFSK with postdetection switch-and-stay combining in TWDP fading,” in _2011 IEEE Vehicular Technology Conference (VTC Fall)_ , 2011.
  * [11] W. S. Lee and S. H. Oh, “Performance of dual switch-and-stay diversity NCFSK systems over two-Wave with diffuse power fading channels,” in _2007 6th International Conference on Information, Communications & Signal Processing_, 2007.
  * [12] S. Haghani and H. Dashtestani, “BER of noncoherent MFSK with postdetection switch-and-stay combining in TWDP fading,” in _2012 IEEE Vehicular Technology Conference (VTC Fall)_ , 2012.
  * [13] S. H. Oh, K. H. Li, and W. S. Lee, “Performance of BPSK pre-detection MRC systems over two-wave with diffuse power fading channels,” _IEEE Transactions on Wireless Communications_ , vol. 6, no. 8, pp. 2772–2775, 2007\.
  * [14] R. Subadara and A. D. Singh, “Performance of M-MRC receivers over TWDP fading channels,” _International Journal of Electronics and Communications_ , vol. 68, no. 6, pp. 569–572, 2014.
  * [15] Y. Lu and N. Yang, “Symbol error probability of QAM with MRC diversity in two-wave with diffuse power fading channels,” _IEEE Communications Letters_ , vol. 15, no. 1, pp. 10–12, 2011.
  * [16] W. S. Lee, “Performance of postdetection EGC NCFSK and DPSK systems over two-wave with diffuse power fading channels,” in _2007 International Symposium on Communications and Information Technologies_ , 2007.
  * [17] D. Kim, H. Lee, and J. Kang, “Comprehensive analysis of the impact of TWDP fading on the achievable error rate performance of BPSK signaling,” _IEICE Transactions on Communications_ , vol. E101.B, no. 2, pp. 500–507, 2018.
  * [18] M. Rao, F. J. Lopez-Martinez, M. Alouini, and A. Goldsmith, “MGF approach to the analysis of generalized two-ray fading models,” _IEEE Transactions on Wireless Communications_ , vol. 14, no. 5, pp. 2548–2561, 2015\.
  * [19] I. M. Kostic, “Envelope probability density function of the sum of signal, noise and interference,” _Electronics Letters_ , vol. 14, no. 15, pp. 490–491, 1978.
  * [20] ——, “Envelope probability distribution of the sum of signal, noise and interference,” in _TELFOR 96 Conference_ , 1996, pp. 301–303.
  * [21] ——, private communication, Aug. 2020.
  * [22] I. S. Gradshteyn and I. M. Ryzhik, _Table of Integrals, Series, and Products_ , 7th ed. Academic Press, 2007\.
  * [23] S. András, A. Baricz, and Y. Sun, “The generalized Marcum Q-function: An orthogonal polynomial approach,” _Computing Research Repository - CORR. 3._ , 2010.
  * [24] C. M. Joshi and S. K. Bissu, “Some inequalities of Bessel and modified Bessel functions,” _Journal of the Australian Mathematical Society_ , vol. 50, no. 02, pp. 490–491, 1991.
  * [25] M. K. Simon and M.-S. Alouini, _Digital communication over fading channels_ , 2nd ed. Wiley-IEEE Press, 2005\.
  * [26] _Wolfram Mathematica_ , accessed October, 2020. [Online]. Available: wolfram.com
  * [27] A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev, _Integrals and series: Direct Laplace transforms_. Gordon and Breach Science Publishers, 1992, vol. 4.
  * [28] M. Abramowitz and I. A. Stegun, _Handbook of mathematical functions: with formulas, graphs, and mathematical tables_ , 10th ed. US Government Printing Office, Washington, 1972.
  * [29] W. Koepf, _Hypergeometric summation: An algorithmic approach to hypergeometric summation and special function identities_ , ser. Advanced Lectures in Mathematics. Springer Fachmedien Wiesbaden, 1998.
  * [30] K. D. Steidley, _Table of ${}_{2}F_{1}(a,b;c;z)$ for $a=0(1/2)7$, $b=0(1/2)7$ and $c=0(1/2)5/2$ with comments on closed forms of ${}_{2}F_{1}(a,b;c;z)$_. National Aeronautics and Space Administration, 1963.