# Analogs to Ramanujan’s Master Theorem and Operational Methods

Julius Lehmann111e-mail: julius.lehmann(at)tum.de Physics of Complex
Biosystems, Technical University of Munich, 85748 Garching, Germany

###### Abstract

In this paper, we utilize operational methods to obtain closed-form solutions
for certain classes of integrals in the spirit of Ramanujan’s Master Theorem
and provide several analogs to it. Although the use of operational calculus
makes the proofs formal in nature, they can still yield interesting and
correct results and may stimulate further rigorous investigations in the
future.

##  1 Ramanujan’s Master Theorem

Since the advent of calculus, there has been particular interest in finding
closed-form solutions to general classes of integrals. This problem has been
further stimulated by the fact that derivatives can be computed in an
algorithmic manner, while even simple looking integrals can require a
considerable amount of work and ingenuity to evaluate. This considered, it is
remarkable that it is possible to find a rather simple closed-form solution to
a class of integrals, namely Mellin transforms, using _Ramanujan’s master
theorem_ (RMT), first described by Srinivasa Ramanujan (1887-1920) at the
beginning of the 20${}^{\text{th}}$ century. Before giving the expression of
his master theorem, we note that a similar result in the spirit of the RMT
already appeared in the 19${}^{\text{th}}$ century in a paper by J. W. L.
Glaisher [13, 3]. By integrating term-by-term an identity related to the
Newton interpolation formula of a function, he finds [1]

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sum_{n=0}^{\infty}(-1)^{n}\phi(n)x^{2n}=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}(\phi(0)-\phi(1)x^{2}+\phi(2)x^{4}+\dots)=\frac{\pi}{2}\phi\Big{(}-\frac{1}{2}\Big{)}$
(1)

to which he remarks, “[…] of course, $\phi(n)$ being only defined for $n$ a
positive integer, $\phi\big{(}-\frac{1}{2}\big{)}$ is without meaning. But in
cases where $\phi(n)$ involves factorials, there is a strong presumption […]
that the formula will give correct result if the continuity of the terms is
preserved by the substitution of gamma functions for the factorials. This I
have found to be true in every case to which I have applied (1).” [13] It does
not appear that much further work was done along these lines apart from a
paper by J. O’Kinealy [19] which was well received by J. Glaisher, until
almost 40 years later, Ramanujan gives the following theorem in his First
Quarterly Report [8, 7, 16]:

Ramanujan’s Master Theorem. Suppose that in some neighborhood around $x=0$ we
can expand the function $f(x)$ as a power series of the form

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n},$

with $\phi(0)\neq 0$, then under certain growth conditions on $\phi$ the
following relation holds:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f(x)=\Gamma(s)\phi(-s),\quad
0<s$ (RMT)

where $\phi(-s)$ is interpreted as a “natural” analytic continuation to the
real numbers of the discrete sequence $\\{\phi(n)\\}_{n\in\mathbb{N}_{0}}$.

This theorem was widely used by him as a tool in computing definite integrals
and infinite series. Indeed, as G. H. Hardy puts it in [16], he “was
particularly fond of them, and used them as one of his commonest tools.” Since
many examples of the application of RMT are available in the literature and by
Ramanujan himself, we will refer to Refs. [3, 7, 5] for a collection of nice
examples. It is possible to rigorously prove the RMT using complex analysis
and the inversion formula of the Mellin transform under certain, as G. H.
Hardy describes them, “natural” conditions on $\phi$ [16]. However, similar to
how Ramanujan himself proceeded, it can be proven quite easily using formal
operational methods. This will be shown in the next paragraph.

To formally (albeit non-rigorously) proof the RMT, we employ the formalism of
_operational calculus_ [20], which is a technique for manipulating operators
independently of the function to which they are applied by treating them
algebraically. Although its history goes back to the foundations of calculus
with Gottfried Wilhelm Leibniz, the approach was mainly developed during the
19${}^{\text{th}}$ century by British and Irish mathematicians, including
Sylvester, Boole, Glaisher, Crofton, and Blizard [6, 11], and fully developed
by Oliver Heaviside in the 1890s [18]. The proof is as follows: by using the
forward shift operator $\hat{\lambda}$ with the defining property
$\hat{\lambda}^{n}\phi(\mathbbold{0})=\phi(n)$, we can rewrite the power
series expansion of $f(x)$ as

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}(x\hat{\lambda})^{n}\phi(\mathbbold{0})=\mathrm{e}^{-x\hat{\lambda}}\phi(\mathbbold{0}),$
(2)

where we used $\mathbbold{0}$ instead of just $0$ to further emphasize that
$\phi$ is being acted upon. Inserting this into the integral yields

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\mathrm{e}^{-x\hat{\lambda}}\phi(\mathbbold{0})=\Gamma(s)\hat{\lambda}^{-s}\phi(\mathbbold{0})=\Gamma(s)\phi(-s),$
(3)

where we used the standard representation of the gamma function and, as a key
step, formally treated the operator $\hat{\lambda}$ as a number. It is this
algebraic manipulation, along with the implicit assumption

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\big{(}\mathrm{e}^{-x\hat{\lambda}}\phi(\mathbbold{0})\big{)}=\Bigg{(}\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\mathrm{e}^{-x\hat{\lambda}}\Bigg{)}\phi(\mathbbold{0}),$
(4)

which makes the proof formal: obviously, the coefficients $\phi(n)$ must
satisfy certain conditions for the result to be valid.

By setting $\phi(n)=\psi(n)\Gamma(n+1)$ in RMT we get the slightly different
integral identity

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\sum_{n=0}^{\infty}(-1)^{n}\psi(n)x^{n}=\frac{\pi}{\sin(\pi
s)}\psi(-s),$ (5)

which is given by Ramanujan next to the RMT in his quarterly report [8].
Alternatively, we could have derived it again using the shift operator and
noting that

$\sum_{n=0}^{\infty}(-1)^{n}\psi(n)x^{n}=\sum_{n=0}^{\infty}(-1)^{n}(x\hat{\lambda})^{n}\psi(\mathbbold{0})=\frac{1}{1+x\hat{\lambda}}\psi(\mathbbold{0})$
(6)

and integrate the expression. This also immediately shows that Glaisher’s
relation readily follows from the ansatz

$\sum_{n=0}^{\infty}(-1)^{n}\psi(n)x^{2n}=\frac{1}{1+x^{2}\hat{\lambda}}\psi(\mathbbold{0}).$
(7)

Employing the shift operator $\hat{\lambda}$, we can derive a multitude of
other analogs to Ramanujan’s master theorem (for related works, see [5, 10]).
To begin, consider the function

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{qn+p}=x^{p}\mathrm{e}^{-x^{q}\hat{\lambda}}\phi(\mathbbold{0}),$
(8)

with the last equality following as usual. The resulting Ramanujan-type
relation reads

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f(x)=\frac{1}{q}\Gamma\Big{(}\frac{p+s}{q}\Big{)}\phi\Big{(}-\frac{p+s}{q}\Big{)}.$
(9)

Obviously, setting $\psi(n)=\phi(n)\Gamma(n+1)$ gives the corresponding RMT
for non-exponential power series. A similar result is given in Refs. [5, 14].
A nice application of Eq. (9) is the following integral, where $J_{\alpha}(x)$
is the Bessel function of the first kind of order $\alpha$:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}J_{\alpha}(x^{m}).$ (10)

Recalling the series definition of the Bessel function [2],

$J_{\alpha}(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!\Gamma(n+\alpha+1)}\Big{(}\frac{x}{2}\Big{)}^{2n+\alpha},$
(11)

$J_{\alpha}(x^{m})$ is represented by the choices
$\phi(n)=2^{-2n-\alpha}/\Gamma(n+\alpha+1)$, $q=2m$, and $p=\alpha m$. This
gives us the expression for the integral of

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}J_{\alpha}(x^{m})=2^{\frac{s}{m}-1}\frac{\Gamma(\frac{\alpha}{2}+\frac{s}{2m})}{m\Gamma(1+\frac{\alpha}{2}-\frac{s}{2m})}.$
(12)

This represents a generalization of entry 6.561.14 of [15] to higher powers of
the argument of $J_{\alpha}$. Another interesting analog which was first
derived by Atale et al. [5] involves the Riemann zeta function defined by

$\zeta(s)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}.$ (13)

To derive it, take the RMT and replace $x\to nx$:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}n^{s}f(nx)=\Gamma(s)\phi(-s).$
(14)

Dividing both sides by $n^{s}$ and then summing over all $n$ gives

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}Z(x)=\zeta(s)\Gamma(s)\phi(-s),$
(15)

where we defined

$Z(x)\equiv\sum_{n=1}^{\infty}f(nx).$ (16)

An immediate consequence of this expression is the standard integral
definition of the Riemann zeta function [4]:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{x^{s-1}}{\mathrm{e}^{x}-1}=\zeta(s)\Gamma(s)$
(17)

which follows from $Z(x)=\sum_{n=1}^{\infty}\mathrm{e}^{-nx}$ implying
$\phi(n)=1$.

The third analog is a direct consequence of the RMT together with the relation
$\ln(x)=\frac{\mathrm{d}}{\mathrm{d}s}x^{s}|_{s=0}$. This allows us to
evaluate general logarithmic integrals of the form

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\ln^{m}(x)f(x),$ (18)

where $m$ is a positive integer and the function has a series expansion of
type Eq. (8). By taking successive derivatives with respect to $s$, we obtain
the logarithmic analog of Ramanujan’s master theorem:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\ln^{m}(x)f(x)=\frac{1}{q}\frac{\mathrm{d}^{m}}{\mathrm{d}{s}^{m}}\bigg{[}\Gamma\Big{(}\frac{p+s}{q}\Big{)}\phi\Big{(}-\frac{p+s}{q}\Big{)}\bigg{]},$
(19)

of which we have $m=1$ as a useful special case:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\ln(x)f(x)=\frac{1}{q^{2}}\Gamma\Big{(}\frac{p+s}{q}\Big{)}\bigg{[}\psi_{0}\Big{(}\frac{p+s}{q}\Big{)}\phi\Big{(}-\frac{p+s}{q}\Big{)}-\phi^{\prime}\Big{(}-\frac{p+s}{q}\Big{)}\bigg{]}.$
(20)

$\psi_{0}(x)\equiv\frac{\mathrm{d}}{\mathrm{d}x}\ln(\Gamma(x))$ in the above
equation is the digamma function [2]. A simple application of Eq. (19) is the
well-known expression

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\ln^{2}(x)\mathrm{e}^{-x}=\frac{\mathrm{d}^{2}}{\mathrm{d}{s}^{2}}\Gamma(s)=\Gamma(s)\bigl{(}\psi_{0}^{2}(s)+\psi_{1}(s)\bigr{)},$
(21)

where $\psi_{1}(x)$ is the trigamma function, the derivative of the digamma
function [2]. A more interesting example, however, is the following integral:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{x^{s-1}\ln^{m}(x)}{(1+x^{\alpha})^{\beta}}.$
(22)

Notice that our function can be written as

$\frac{1}{(1+x^{\alpha})^{\beta}}=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\frac{\Gamma(\beta+n)}{\Gamma(\beta)}x^{\alpha
n},$ (23)

we can deduce $\phi(n)=\Gamma(\beta+n)/\Gamma(\beta)$, $q=\alpha$, and $p=0$.
Therefore, using the logarithmic analog to the RMT, we find

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{x^{s-1}\ln^{m}(x)}{(1+x^{\alpha})^{\beta}}=\frac{1}{\alpha\Gamma(\beta)}\frac{\mathrm{d}^{m}}{\mathrm{d}{s}^{m}}\bigg{[}\Gamma\Big{(}\frac{s}{\alpha}\Big{)}\Gamma\Big{(}\beta-\frac{s}{\alpha}\Big{)}\bigg{]},$
(24)

so for example, $m=1$:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{x^{s-1}\ln(x)}{(1+x^{\alpha})^{\beta}}=\frac{\Gamma(\frac{s}{\alpha})\Gamma(\beta-\frac{s}{\alpha})}{\alpha\Gamma(\beta)}\Big{[}\psi_{0}\Big{(}\frac{s}{\alpha}\Big{)}-\psi_{0}\Big{(}\beta-\frac{s}{\alpha}\Big{)}\Big{]},$
(25)

or taking $m=2$, $s=\frac{1}{2}$, and $a=b=1$:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{\ln^{2}(x)}{\sqrt{x}(1+x)}=\pi^{3}.$
(26)

One takeaway from the formal proof of RMT that allows us to generalize the
result fairly easily is that whenever the series expansion of a function is
given by

$f(x)=\sum_{n=0}^{\infty}\phi(n)\gamma(n)x^{n},$ (27)

where $\gamma(n)$ are the expansion coefficients of another function $g(x)$,
so that we can write $f(x)$ in terms of the shift operator as

$f(x)=\sum_{n=0}^{\infty}\psi(n)(x\hat{\lambda})^{n}\phi(\mathbbold{0})=g(x\hat{\lambda})\phi(\mathbbold{0}),$
(28)

the generalized Ramanujan Master Theorem (gRMT) for an integral kernel $K(x)$
is given by

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)f(x)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)g(x\hat{\lambda})\phi(\mathbbold{0}).$
(29)

We can formulate this statement in a different, operational way: by
representing the RHS as an operator, we can write

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)f(x)=\hat{\mathcal{R}}\phi(\mathbbold{0})$
(30)

where now all the relevant information for evaluating the integral is
contained in the operator
$\hat{\mathcal{R}}\equiv\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)g(x\hat{\lambda})$,
which acts on the coefficients of the expansion of $f(x)$. This is in
accordance with similar results given in [10]. The RMT is given as a special
case by the choices $K(x)=x^{s-1}$ and $g(x)=\mathrm{e}^{-x}$ or
$g(x)=\frac{1}{1+x}$, respectively. Now let us consider another choice of
$g(x)$. In particular, take the trigonometric functions
$g_{\mathrm{c}}(x)=\cos(x)$ and $g_{\mathrm{s}}(x)=\sin(x)$. To find the
expansion coefficients $\gamma_{\mathrm{c}}(n)$, consider the power series of
$\cos(x)$ and rewrite it accordingly:

$\cos(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}x^{2n}=\sum_{n=0}^{\infty}\frac{\cos(\frac{\pi}{2}n)}{n!}x^{n},$
(31)

since $\cos(\frac{\pi}{2}n)=(-1)^{n}$ for $n$ even and $0$ for $n$ odd.
Therefore, we find $\gamma_{\mathrm{c}}(n)=\cos(\frac{\pi}{2}n)/n!$, and
similarly, $\gamma_{\mathrm{s}}(n)=\sin(\frac{\pi}{2}n)/n!$. The resulting
trigonometric RMTs that follow from Eq. (28) are therefore given by

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f_{\mathrm{c}}(x)$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n)}{(2n)!}x^{2n}=\Gamma(s)\cos\Big{(}\frac{\pi}{2}s\Big{)}\phi(-s)$
(32)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f_{\mathrm{s}}(x)$
$\displaystyle=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n+1)}{(2n+1)!}x^{2n+1}=\Gamma(s)\sin\Big{(}\frac{\pi}{2}s\Big{)}\phi(-s).$

An interesting observation taken from the gRMT for $\cos$\- or $\sin$-type
series is the following: by using the main assumption of operational calculus,
i. e. treating operators algebraically, we can identify the associated
$\hat{\mathcal{R}}$ as simply being the Fourier cosine/sine transform of
$K(x)$ in $\hat{\lambda}$ [12], namely:

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)\cos(x\hat{\lambda})\phi(\mathbbold{0})$
$\displaystyle=\frac{\pi}{2}\mathcal{F}_{\mathrm{c}}\\{K(x)\\}(\hat{\lambda})\phi(\mathbbold{0})$
(33)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}K(x)\sin(x\hat{\lambda})\phi(\mathbbold{0})$
$\displaystyle=\frac{\pi}{2}\mathcal{F}_{\mathrm{s}}\\{K(x)\\}(\hat{\lambda})\phi(\mathbbold{0}).$

Since there are many transformation pairs available in the literature, this
opens the door to a plethora of Ramanujan-type master theorems involving
$\cos$\- or $\sin$-type functions. Consider the case
$K(x)=\mathrm{e}^{-x^{2}}$, then we have

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-x^{2}}f_{\mathrm{c}}(x)=\frac{\pi}{2}\mathcal{F}_{\mathrm{c}}\\{\mathrm{e}^{-x^{2}}\\}(\hat{\lambda})\phi(\mathbbold{0})=\frac{\sqrt{\pi}}{2}\mathrm{e}^{-\frac{\hat{\lambda}^{2}}{4}}\phi(\mathbbold{0}).$
(34)

By interpreting the RHS as a formal power series in $\hat{\lambda}$, we arrive
at the following power series solution:

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-x^{2}}f_{\mathrm{c}}(x)=\frac{\sqrt{\pi}}{2}\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n)}{4^{n}n!},$
(35)

and analogously for a $\sin$-type series.

In the penultimate part of this section, we apply the general Ramanujan’s
master theorem to integral transformations. The original form of the RMT deals
with Mellin transforms, other very important transforms are the already
mentioned Fourier cosine and sine transforms, the Laplace and Hankel transform
[9, 21]. The former are generated by choosing the kernels $K(x)=\cos(sx)$ and
$K(x)=\sin(sx)$, while the latter have kernels $K(x)=\mathrm{e}^{-sx}$ and
$K(x)=J_{\alpha}(sx)x$. We start with the Fourier-type transforms:

$\mathcal{F}_{\mathrm{c}}\\{f(x)\\}(s)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\cos(sx)f(x),\qquad\mathcal{F}_{\mathrm{s}}\\{f(x)\\}(s)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sin(sx)f(x).$
(36)

By assuming an exponential-type power series expansion for $f(x)$, we find the
solutions in terms of $\hat{\lambda}$ and the expansion coefficients
$\phi(\mathbbold{0})$:

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\cos(sx)f(x)$
$\displaystyle=\frac{\hat{\lambda}}{s^{2}+\hat{\lambda}^{2}}\phi(\mathbbold{0})$
(37) $\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sin(sx)f(x)$
$\displaystyle=\frac{s}{s^{2}+\hat{\lambda}^{2}}\phi(\mathbbold{0}).$

What remains is to define the action of the operators acting on $\phi$. We
again interpret the fractions as formal power series, but we have two ways to
expand them: either as series around the origin or as asymptotic series at
infinity. Both approaches yield series in powers of $s$, one being a small
$s\approx 0$ expansion, the other a large $s\to\infty$ expansion. Taking this
into account, we find

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\cos(sx)f(x)$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}\phi(-2n-1)s^{2n}$ (38)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\cos(sx)f(x)$
$\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(2n+1)}{s^{2n}}.$

and

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sin(sx)f(x)$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}\phi(-2n)s^{2n+1}$ (39)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sin(sx)f(x)$
$\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(2n)}{s^{2n+1}}.$

Without going into too much detail, following the same procedure, we find the
analogous expressions for the Laplace transform

$\mathcal{L}\\{f(x)\\}(s)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-sx}f(x)$
(40)

given by the formulas

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-sx}f(x)$
$\displaystyle=\sum_{n=0}^{\infty}(-1)^{n}\phi(-n-1)s^{n}$ (41)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-sx}f(x)$
$\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(n)}{s^{n+1}}.$

Note that the second expression above is the well-known asymptotic expansion
of the Laplace transform found by term-wise integration of the power series
expansion of $f(x)$. The Hankel transform frequently arises in in the
evaluation of Fourier transforms of radially symmetric functions. It is
defined by

$H_{\alpha}\\{f(x)\\}(s)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}xJ_{\alpha}(sx)f(x)$
(42)

and gives, under the usual assumptions, rise to the following associated
operator in terms of $\hat{\lambda}$:

$H_{\alpha}\\{f(x)\\}(s)=\bigg{(}\frac{s}{\hat{\lambda}+\sqrt{s^{2}+\hat{\lambda}^{2}}}\bigg{)}^{\alpha}\frac{\hat{\lambda}+\alpha\sqrt{s^{2}+\hat{\lambda}^{2}}}{(s^{2}+\hat{\lambda}^{2})^{\frac{3}{2}}}\phi(\mathbbold{0}).$
(43)

Since a general expression for the expansion of the RHS is very cumbersome, we
will focus on the special case $\alpha=0$. There we have

$H_{\alpha}\\{f(x)\\}(s)=\frac{\hat{\lambda}}{(s^{2}+\hat{\lambda}^{2})^{\frac{3}{2}}}\phi(\mathbbold{0}),$
(44)

which leads to the following formulas:

$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}xJ_{0}(sx)f(x)$
$\displaystyle=\sum_{n=0}^{\infty}\binom{-\frac{3}{2}}{n}\phi(-2n)s^{2n}$ (45)
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}xJ_{0}(sx)f(x)$
$\displaystyle\sim\sum_{n=0}^{\infty}\binom{-\frac{3}{2}}{n}\frac{\phi(2n+1)}{s^{2n+3}}.$

The last expression was also found and rigorously derived using the Hardy-
Ramanujan master theorem [16] in Ref. [17].

Before we enter the next section, let us briefly address another relation
given by Ramanujan in his Second Quarterly Report and a double-integral analog
to the RMT. Alongside the other relations, we find in [8] the statement
(adapted for the purpose of the paper)

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}\right)\left(\sum_{n=0}^{\infty}\frac{(-1)^{n}\psi(n)}{n!}x^{n}\right)=\sum_{n=0}^{\infty}(-1)^{n}\phi(n)\psi(-n-1),$
(46)

which can also be given with the roles of $\phi$ and $\psi$ in the RHS
reversed. This relation follows in a straightforward manner from writing the
sums in their respective shift operator form and then expanding the resulting
function of operators. To derive the next relation involving a double
integral, we start with the well-known representation of the 0${}^{\text{th}}$
order modified Bessel function [2]:

$K_{0}(s\xi)=\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{\cos(sx)}{\sqrt{x^{2}+\xi^{2}}}.$
(47)

Given a function $f(x)$ with a $\cos$-type series expansion

$f(x)=\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n)}{(2n)!}x^{2n},$ (48)

we can utilize this integral to arrive at the operational statement

$\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{f(x)}{\sqrt{x^{2}+\xi^{2}}}=K_{0}(\xi\hat{\lambda})\phi(\mathbbold{0}).$
(49)

Multiplying both sides by $\xi^{s-1}$ and then integrating the equation yields
the expression

$\int_{0}^{\infty}\mathrm{d}\xi\\!\mathop{}\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{\xi^{s-1}f(x)}{\sqrt{x^{2}+\xi^{2}}}=\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\frac{\phi(-s)}{2^{2-s}},$

which generalizes Ramanujan’s original master theorem to double integrals.

##  2 Ramanujan’s Master Theorem for Sums

In this brief section we want to present one of Ramanujan’s results on
discrete analogs of his famous master theorem RMT. Ramanujan’s interest not
only covered obtaining formulas for evaluating general classes of integrals,
but also extended to finding similar results for general classes of sums. In
chapter 4 of his first notebook [7], which deals with iterations of the
exponential function and Bell polynomials, he states the following three
relations:

$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}(\phi(2n+1)+\phi(-2n-1))=\frac{\pi}{2}\phi(0)$
(50a) $\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}(\phi(n)+\phi(-n))=\phi(0)$
(50b)
$\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}(\phi(n)-\phi(-n))=\phi^{\prime}(0),$
(50c)

which are valid under conditions elaborated in [7]. We will illustrate
Ramanujan’s derivation using the first equation (50a). As for the master
theorem, the proof is again formal in nature and therefore non-rigorous. We
start by considering the relation $\arctan(x)+\arctan(1/x)=\pi/2$ which holds
for $x>0$. Suppose that in some neighborhood around $x=0$ we can expand the
function $f(x)$ as an exponential power series with expansion coefficient
$\psi(n)$. Replacing $x$ by some power of it in the $\arctan$-relation clearly
does not change its value, therefore Ramanujan writes down the following set
of equations:

$\displaystyle\frac{\psi(0)}{0!}\Big{(}\arctan(x^{0})+\arctan\Big{(}\frac{1}{x^{0}}\Big{)}\Big{)}$
$\displaystyle=\frac{\pi}{2}\frac{\psi(0)}{0!}$ (51)
$\displaystyle\frac{\psi(1)}{1!}\Big{(}\arctan(x^{1})+\arctan\Big{(}\frac{1}{x^{1}}\Big{)}\Big{)}$
$\displaystyle=\frac{\pi}{2}\frac{\psi(1)}{1!}$
$\displaystyle\frac{\psi(2)}{2!}\Big{(}\arctan(x^{2})+\arctan\Big{(}\frac{1}{x^{2}}\Big{)}\Big{)}$
$\displaystyle=\frac{\pi}{2}\frac{\psi(2)}{2!}$
$\displaystyle\mathmakebox[\widthof{{}={}}][c]{\vdots}$
$\displaystyle\frac{\psi(n)}{n!}\Big{(}\arctan(x^{n})+\arctan\Big{(}\frac{1}{x^{n}}\Big{)}\Big{)}$
$\displaystyle=\frac{\pi}{2}\frac{\psi(n)}{n!}.$

In the next step, we take the sum over all $n$ on both sides and conclude:

$\sum_{n=0}^{\infty}\frac{\psi(n)}{n!}\Big{(}\arctan(x^{n})+\arctan\Big{(}\frac{1}{x^{n}}\Big{)}\Big{)}=\frac{\pi}{2}\sum_{n=0}^{\infty}\frac{\psi(n)}{n!},$
(52)

which is still true for all $x>0$. We recognize the right side as the value of
$f$ at $x=1$. Furthermore, by expanding the $\arctan$ as a power series, we
find

$\sum_{n=0}^{\infty}\frac{\psi(n)}{n!}\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}((x^{n})^{2k+1}+(x^{n})^{-2k-1})=\frac{\pi}{2}f(1).$
(53)

The step that follows is what renders this proof formal, since we (and
ultimately Ramanujan) do not check for the conditions that permit it: we
exchange the order of the two summations,

$\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}\sum_{n=0}^{\infty}\frac{\psi(n)}{n!}((x^{2k+1})^{n}+(x^{-2k-1})^{n})=\frac{\pi}{2}f(1),$
(54)

and write the inner sum in terms of $f(x)$:

$\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}(f(x^{2k+1})+f(x^{-2k-1}))=\frac{\pi}{2}f(1).$
(55)

To finish the formal proof, identify $\phi(k)\equiv f(x^{k})$ and note that
$f(1)=f(x^{0})=\phi(0)$:

$\sum_{k=0}^{\infty}\frac{(-1)^{k}}{2k+1}(\phi(2k+1)+\phi(-2k-1))=\frac{\pi}{2}\phi(0).$
(56)

The other equations (50b),(50c) follow in a similar way by using either
$x/(1+x)+(1/x)/(1+1/x)=1$ or $\ln(1+x)-\ln(1+1/x)=\ln(x)$ instead of the
$\arctan$ relation. To arrive at Eq. (50c), one also utilizes the observation
that

$\frac{\mathrm{d}}{\mathrm{d}s}f(x^{s})=x^{s}\ln(x)f^{\prime}(x^{s}).$ (57)

In more general terms, the first two relations rely on suitable functions that
satisfy the functional equation

$h(x)+h\Big{(}\frac{1}{x}\Big{)}=C$ (58)

for some $C$. In this case, assuming that $h(x)$ can be expanded as a power
series around $x=0$ with coefficients $\eta(n)$, we get the general relation

$\sum_{k=0}^{\infty}\eta(k)(\phi(k)+\phi(-k))=C\phi(0),$ (59)

which should hold for some suitable set of conditions on $\phi$. In fact,
there are infinitely many functions that satisfy the functional equation (58).
In particular, we can choose any function $g(x)$ and set $h(x)$ equal to

$h(x)=g(x)-g\Big{(}\frac{1}{x}\Big{)}+\frac{C}{2},$ (60)

therefore $h(1/x)=g(1/x)-g(x)+C/2$ and $h(x)+h(1/x)=C$ [22].

Two nice applications of these theorems follow by considering
$\phi(n)=\mathrm{e}^{\mathrm{i}n\theta}$, from which we have by Eq. (50a)

$\sum_{n=0}^{\infty}\frac{(-1)^{n}\cos((2n+1)\theta)}{2n+1}=\frac{\pi}{4},$
(61)

which is a well-known sum, and further by Eq. (50c)

$\sum_{n=1}^{\infty}\frac{(-1)^{n+1}\sin(n\theta)}{n}=\frac{\theta}{2},$ (62)

which is the familiar Fourier series of the piecewise linear function. A more
complicated sum can be found by substituting $\phi(n)=\sin(n\theta)/n$, where
$-\pi/2<\theta<\pi/2$, in Eq. (50a), from which we obtain another Fourier
series of the piecewise linear function (and, in fact, the Fourier series of
the triangular wave for $\theta\in\mathbb{R}$):

$\sum_{n=0}^{\infty}\frac{(-1)^{n}\sin((2n+1)\theta)}{(2n+1)^{2}}=\frac{\pi}{4}\theta.$
(63)

## List of Ramanujan’s Master Theorem Analogs

Integral | Function | Closed-Form Solution
---|---|---
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | $\displaystyle\Gamma(s)\phi(-s)$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f(x)$ | $\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\phi(n)x^{n}$ | $\displaystyle\frac{\pi}{\sin(\pi s)}\phi(-s)$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{qn+p}$ | $\displaystyle\frac{1}{q}\Gamma\Big{(}\frac{p+s}{q}\Big{)}\phi\Big{(}-\frac{p+s}{q}\Big{)}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}Z(x)$ | $\displaystyle\sum_{n=1}^{\infty}f(nx)$ | $\displaystyle\zeta(s)\Gamma(s)\phi(-s)$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}\ln^{m}(x)f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | $\displaystyle\frac{1}{q}\frac{\mathrm{d}^{m}}{\mathrm{d}{s}^{m}}\bigg{[}\Gamma\Big{(}\frac{p+s}{q}\Big{)}\phi\Big{(}-\frac{p+s}{q}\Big{)}\bigg{]}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f_{\mathrm{c}}(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n)}{(2n)!}x^{2n}$ | $\displaystyle\Gamma(s)\cos\Big{(}\frac{\pi}{2}s\Big{)}\phi(-s)$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}x^{s-1}f_{\mathrm{s}}(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n+1)}{(2n+1)!}x^{2n+1}$ | $\displaystyle\Gamma(s)\sin\Big{(}\frac{\pi}{2}s\Big{)}\phi(-s)$
Integral | Function | Closed-Form Solution
---|---|---
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\cos(sx)f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | ${\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\phi(-2n-1)s^{2n}}\atop{\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(2n+1)}{s^{2n}}}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\sin(sx)f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | ${\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\phi(-2n)s^{2n+1}}\atop{\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(2n)}{s^{2n+1}}}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\mathrm{e}^{-sx}f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | ${\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\phi(-n-1)s^{n}}\atop{\displaystyle\sim\sum_{n=0}^{\infty}(-1)^{n}\frac{\phi(n)}{s^{n+1}}}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}xJ_{0}(sx)f(x)$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(n)}{n!}x^{n}$ | ${\displaystyle\sum_{n=0}^{\infty}\binom{-\frac{3}{2}}{n}\phi(-2n)s^{2n}}\atop{\displaystyle\sim\sum_{n=0}^{\infty}\binom{-\frac{3}{2}}{n}\frac{\phi(2n+1)}{s^{2n+3}}}$
$\displaystyle\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}f(x)g(x)$ | see (46) | $\displaystyle\sum_{n=0}^{\infty}(-1)^{n}\phi(n)\psi(-n-1)$
$\displaystyle\int_{0}^{\infty}\mathrm{d}\xi\\!\mathop{}\int_{0}^{\infty}\mathrm{d}x\\!\mathop{}\frac{\xi^{s-1}f(x)}{\sqrt{x^{2}+\xi^{2}}}$ | $\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}\phi(2n)}{(2n)!}x^{2n}$ | $\displaystyle\Gamma\Big{(}\frac{s}{2}\Big{)}^{2}\frac{\phi(-s)}{2^{2-s}}$
Sum |  Closed-Form Solution
---|---
$\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2n+1}(\phi(2n+1)+\phi(-2n-1))$ |  $\displaystyle\frac{\pi}{2}\phi(0)$
$\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}(\phi(n)+\phi(-n))$ |  $\displaystyle\phi(0)$
$\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}(\phi(n)-\phi(-n))$ |  $\displaystyle\phi^{\prime}(0)$

## References

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# Outage Rates and Outage Durations of
Opportunistic Relaying Systems

Zoran Hadzi-Velkov, and Nikola Zlatanov Accepted for IEEE CommunLettersZ.
Hadzi-Velkov and N. Zlatanov are with the Faculty of Electrical Engineering
and Information Technologies, Ss. Cyril and Methodius University, Karpos 2 bb,
1000 Skopje, R. Macedonia, Email<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Opportunistic relaying is a simple yet efficient cooperation scheme that
achieves full diversity and preserves the spectral efficiency among the
spatially distributed stations. However, the stations’ mobility causes
temporal correlation of the system’s capacity outage events, which gives rise
to its important second-order outage statistical parameters, such as the
average outage rate (AOR) and the average outage duration (AOD). This letter
presents exact analytical expressions for the AOR and the AOD of an
opportunistic relaying system, which employs a mobile source and a mobile
destination (without a direct path), and an arbitrary number of (fixed-gain
amplify-and-forward or decode-and-forward) mobile relays in Rayleigh fading
environment.

###### Index Terms:

Average outage rate, Average outage duration, Opportunistic relaying, Doppler
effect, Rayleigh fading

##  I Introduction

Cooperative diversity is a highly promising technique for coverage extension
and reliability improvement of wireless networks. It exploits the additional
degrees of freedom of the fading environment, which is introduced by the
spatially distributed multiple relays utilizing either amplify-and-forward
(AF) or decode-and-forward (DF) relaying. Proposed recently in [1], the
opportunistic relaying is a simple yet efficient cooperative diversity
protocol, whose diversity-multiplexing tradeoff is identical to that of the
more complex distributed space-time coding cooperative schemes [2]. By
selecting a single “best” relay among the all available relays, the
opportunistic relaying achieves full spatial diversity while maintaining the
spectral efficiency of a two-hop communication link.

The outage and error probabilities of the opportunistic relaying systems have
been studied in [3]-[5], which clearly demonstrate its excellent performances.
However, there are some design issues for which the outage and error
probabilities’ criteria are not sufficient, such as, packet or slot lengths
and latencies, switching rates [6], power and bandwidth allocation or,
decision criterion for changing adaptive modulation levels.

These issues can be addressed by investigating the system’s second-order
outage statistics. To the best of authors’ knowledge, such statistics that
describe the outage events of cooperative systems, such as, the average outage
rate (AOR) and average outage duration (AOD), have not been studied
previously. We propose the AOR and the AOD be defined with respect to the
capacity outage events derived from information-theoretic capacity of the
opportunistic system. Similar definition of the outage statistics has been
applied over MIMO systems in [7]. In this letter, we derive exact expressions
for the AOR and the AOD of opportunistic systems, employing either AF or DF
relaying in Rayleigh fading environment.

##  II Opportunistic relaying with mobile stations

###  II-A System model

Similarly to [1], we consider a typical half-duplex dual-hop communication
scenario, where the communication between the source $S$ and the destination
$D$ is possible only via $M$ relays (denoted by $R_{k}$, $1\leq k\leq M$), as
the direct path is assumed blocked by an intermediate wall. In the beginning
of each slot (divided into two equal sub-slots), a single “best” opportunistic
relay is selected out of the $M$ possible dual-hop paths for relaying the
communication between $S$ and $D$. During the first sub-slot, $S$ transmits
its signal over the first hop, while the selected relay forwards that signal
toward $D$ over the second hop during the second sub-slot.

The channel is exposed to Rayleigh fading and is assumed to remain constant
during the entire slot duration. Without loss in generality, we assume that
$S$ and the selected “best” relay transmit with equal powers $P_{T}$,
rendering the total available transmission power to $2P_{T}$. Denoting the
Rayleigh-faded channel gains of hops $S\rightarrow R_{k}$ and
$R_{k}\rightarrow D$ during a given slot $t$ by $\alpha_{Sk}(t)$ and
$\alpha_{kD}(t)$, the received signal-to-noise ratios (SNRs) at the relay
$R_{k}$ and at the destination $D$ are expressed as
$\gamma_{Sk}=P_{T}\alpha_{Sk}^{2}(t)/N_{0}$ and
$\gamma_{kD}=P_{T}\alpha_{kD}^{2}(t)/N_{0}$, with $N_{0}$ as the noise power.
Specifying the average squared channel gains as
$E[\alpha_{Sk}^{2}]=\Omega_{Sk}$ and $E[\alpha_{kD}^{2}]=\Omega_{kD}$, the
average received SNRs at $R_{k}$ and $D$ are respectively given by
$\bar{\gamma}_{Sk}=P_{T}\Omega_{Sk}/N_{0}$ and
$\bar{\gamma}_{kD}=P_{T}\Omega_{kD}/N_{0}$.

The relay selection is based on the estimation of the end-to-end performance
over the dual-hop path $k$ using the selection variable $W_{k}(t)$, which is
estimated separately by each relay $R_{k}$ in the beginning of each slot from
its channel state information (CSI) [1]. To facilitate channel state
estimation by the relays, $S$ and $D$ previously exchange short control
packets. In the beginning of slot $t$, the “best” relay $b$ is selected in a
distributed manner by using the selection policy:

$b=\mathop{\arg\max}\limits_{{}^{1\leq k\leq M}}\\{W_{k}(t)\\}.$ (1)

####  II-A1 Decode-and-forward relaying

In the beginning of slot $t$, each DF relay estimates the selection variable

$W_{k}^{DF}(t)=\min\\{\alpha_{Sk}(t),\alpha_{kD}(t)\\}\,,$ (2)

which actually evaluates the minimal instantaneous received SNR among the two
hops, $\min\\{\gamma_{Sk}(t),\gamma_{kD}(t)\\}$ .

####  II-A2 Amplify-and-forward relaying

We assume fixed gain AF relays that amplify the received signal from the first
hop by $\sqrt{P_{T}/(P_{T}\Omega_{Sk}+N_{0})}$ and forward it to $D$ over the
second hop [11]. In the beginning of slot $t$, each AF relay estimates the
selection variable

$W_{k}^{AF}(t)=\frac{\alpha_{Sk}(t)\alpha_{kD}(t)}{\sqrt{C_{k}+\alpha_{kD}^{2}(t)}}\,,$
(3)

where $C_{k}=\Omega_{Sk}+N_{0}/P_{T}$. Actually, each AF relay evaluates the
dual-hop SNR that is relayed over $R_{k}$ and received at $D$. Note that
fixed-gain (e.g., semi-blind) AF relays have considerably simpler but yet
comparably close performance to that of the variable gain AF relays [11].

###  II-B Stations’ mobility

We consider 2-dimensional isotropic scattering around source $S$, relays
$R_{k}$ and destination $D$, all of which are assumed to be mobile and have no
line-of-sight with other stations. Thus, each $S\rightarrow R_{k}$
($R_{k}\rightarrow D$) hop behaves as a mobile-to-mobile Rayleigh channel. Its
channel gain, $\alpha_{Sk}(t)$ ($\alpha_{kD}(t))$, is a time-correlated
Rayleigh random process with known statistical properties (e.g., the Doppler
spectrum) [9]. If a station at one end of a hop is fixed, the mobile-to-mobile
Rayleigh-fading hop is transformed into the “classic” fixed-to-mobile
Rayleigh-fading hop [8]. The time derivative $\dot{\alpha}_{Sk}$
($\dot{\alpha}_{kD}$) is independent from the channel gain $\alpha_{Sk}$
($\alpha_{kD}$), and follows the Gaussian probability distribution function
(PDF) with zero mean and variance [References, Eq. (A5)] [References, Eq.
(39)]

$\displaystyle\sigma_{\dot{\alpha}_{Sk}}^{2}$ $\displaystyle=$
$\displaystyle\pi^{2}\Omega_{Sk}(f_{mS}^{2}+f_{mk}^{2})\,,$ (4)
$\displaystyle\sigma_{\dot{\alpha}_{kD}}^{2}$ $\displaystyle=$
$\displaystyle\pi^{2}\Omega_{kD}(f_{mk}^{2}+f_{mD}^{2})\,.$ (5)

In (4)-(5), $f_{mS}$, $f_{mD}$ and $f_{mk}$ denote the maximum Doppler rates
of $S$, $D$ and relay $R_{k}$ $(1\leq k\leq M)$, respectively. The Doppler
rate (and consequently the AOR) is expressed in the unit $slot^{-1}$.
Expressing the slot duration in seconds, both the Doppler rate (now becoming
the Doppler frequency) and the AOR are expressed in Hz.

###  II-C Capacity outage events

In a given slot $t$, the opportunistic relaying system experiences a capacity
outage event when the mutual information of the dual-hop path over the “best”
relay $b$ drops below some predefined spectral efficiency $R$ [2],

$I(t)=\frac{1}{2}\log_{2}\left(1+\frac{P_{T}}{N_{0}}(W_{\max}(t))^{2}\right)\leq
R\,,$ (6)

where

$W_{\max}(t)=\max\\{W_{1}(t),W_{2}(t),\dots,W_{M}(t)\\}\,.$ (7)

Thus, the resulting time-varying capacity $I(t)$ suffers from the random
occurrence of capacity outage events, during which the channel is unable to
support the specified $R$. Transforming (6), the capacity outage event at slot
$t$ occurs if

$W_{\max}(t)\leq Z\,,$ (8)

where the outage threshold is $Z=\sqrt{(2^{2R}-1)/(P_{T}/N_{0})}$. Note, $Z$
can be varied by varying $R$ (as in [7]) or $P_{T}/N_{0}$ (as in this work),
but the functional dependencies of the increasing $R$ or the decreasing
$P_{T}/N_{0}$ (in dB) have almost same shapes.

##  III Average Outage Rates and Outage Durations

###  III-A General expression

Using a similar approach to that presented in [10] for deriving level crossing
rates (LCR) of “classic” selection diversity systems, the joint PDF of
$W_{\max}$ and $\dot{W}_{\max}$ is expressed as

$f_{W_{\max}\dot{W}_{\max}}(w,\dot{w})=\sum_{k=1}^{M}f_{W_{k}\dot{W}_{k}}(w,\dot{w})P_{k}(w)$
(9)

where $f_{W_{k}\dot{W}_{k}}(w,\dot{w})$ denotes the joint PDF of selection
variable $W_{k}(t)$ and its time derivative $\dot{W}_{k}(t)$. Assuming $R_{k}$
is selected “best” relay, $P_{k}(w)$ denotes the conditional probability that
the selection variable $W_{k}$ of $R_{k}$ drops below $w$,

$P_{k}(w)={\rm{Pr}}\\{W_{k}\leq w\big{|}W_{k}\textrm{ is
max}\\}=\prod_{i=1,i\neq k}^{M}F_{W_{i}}(w)\,,$ (10)

where $F_{W_{i}}(w)=\textrm{Pr}\\{W_{i}\leq w\\}$ denotes the cumulative
distribution function (CDF) of $W_{i}$. The AOR is evaluated based on the
standard LCR definition [References, Chapter 1], yielding

$\displaystyle N(Z)\stackrel{{\scriptstyle\rm
def}}{{=}}\int_{0}^{\infty}\dot{w}f_{W_{\max}\dot{W}_{\max}}(Z,\dot{w})d\dot{w}\qquad\qquad\qquad\qquad$
$\displaystyle=\sum_{k=1}^{M}P_{k}(Z)\int_{0}^{\infty}\dot{w}f_{W_{k},\dot{W}_{k}}(Z,\dot{w})=\sum_{k=1}^{M}P_{k}(Z)N_{k}(Z),$
(11)

where $N_{k}(Z)$ denotes the AOR of the dual-hop path over the relay $R_{k}$.
The AOD is then given by

$\displaystyle T(Z)\stackrel{{\scriptstyle\rm
def}}{{=}}\frac{\textrm{Pr}\\{W_{\max}\leq
Z\\}}{N(Z)}\qquad\qquad\qquad\qquad\qquad\qquad$
$\displaystyle=N(Z)^{-1}\prod_{k=1}^{M}F_{W_{k}}(Z)=\left(\sum_{k=1}^{M}\frac{N_{k}(Z)}{F_{W_{k}}(Z)}\right)^{-1}.$
(12)

###  III-B Decode-and-forward relaying

We now focus on the random process $W_{k}^{DF}(t)$, defined by (2).
Communication through the relay $R_{k}$ falls in outage if either one of the
two hops fail. Thus, the PDF of $W_{k}^{DF}(t)$ is given by

$f_{W_{k}}(w)=f_{\alpha_{Sk}}(w)\textrm{Pr}\\{\alpha_{kD}>w\\}+f_{\alpha_{kD}}(w)\textrm{Pr}\\{\alpha_{Sk}>w\\}.$
(13)

Since both hops follow the Rayleigh PDF, $W_{k}^{DF}$ is also determined to
follow the Rayleigh PDF,

$f_{W_{k}}(w)=2\Lambda_{k}we^{-\Lambda_{k}w^{2}}\,,$ (14)

with $\Lambda_{k}=\Omega_{Sk}^{-1}+\Omega_{kD}^{-1}$, and the respective CDF
$F_{W_{k}}(w)=1-\exp(-\Lambda_{k}w^{2})$. Using

$\dot{W}_{k}=\left\\{\begin{array}[]{c}\dot{\alpha}_{Sk},\quad\alpha_{Sk}\leq\alpha_{kD}\\\
\dot{\alpha}_{kD},\quad\alpha_{kD}<\alpha_{Sk}\end{array}\right.$ (15)

and the independence of the channel gains and their respective time
derivatives, the required joint PDF is found as

$\displaystyle
f_{W_{k}\dot{W}_{k}}(w,\dot{w})=f_{\dot{\alpha}_{Sk}}(\dot{w})f_{\alpha_{Sk}}(w)\textrm{Pr}\\{\alpha_{kD}>w\\}$
$\displaystyle+f_{\dot{\alpha}_{kD}}(\dot{w})f_{\alpha_{kD}}(w)\textrm{Pr}\\{\alpha_{Sk}>w\\}.$
(16)

In Rayleigh fading, (III-B) specializes to

$f_{W_{k}\dot{W}_{k}}(w,\dot{w})=f_{W_{k}}(w)f_{\dot{W}_{k}}(\dot{w}),$ (17)

thus rendering $W_{k}$ and $\dot{W}_{k}$ as independent RVs, where
$f_{W_{k}}(w)$ is given by (14) and

$f_{\dot{W}_{k}}(\dot{w})=\frac{\Omega_{kD}}{\Omega_{Sk}+\Omega_{kD}}f_{\dot{\alpha}_{Sk}}(\dot{w})+\frac{\Omega_{Sk}}{\Omega_{Sk}+\Omega_{kD}}f_{\dot{\alpha}_{kD}}(\dot{w})$
(18)

with $f_{\dot{\alpha}_{Sk}}(\cdot)$ and $f_{\dot{\alpha}_{kD}}(\cdot)$
denoting zero mean Gaussian PDFs with variances
$\sigma_{\dot{\alpha}_{Sk}}^{2}$ (4) and $\sigma_{\dot{\alpha}_{kD}}^{2}$ (5),
respectively. Thus, the AOR of dual-hop path over $R_{k}$ is obtained as

$N_{k}(Z)=\frac{\Omega_{kD}\sigma_{\dot{\alpha}_{Sk}}+\Omega_{Sk}\sigma_{\dot{\alpha}_{kD}}}{\Omega_{Sk}+\Omega_{kD}}\frac{f_{W_{k}}(Z)}{\sqrt{2\pi}}$
(19)

with $f_{W_{k}}(\cdot)$ given by (14). Inserting (19) into (III-A) and
(III-A), we obtain the AOR and the AOD of DF relaying system.

Figure 1: Normalized Average Outage Rate in DF and AF opportunistic relaying
systems

###  III-C Amplify-and-forward relaying

The exact expressions for the CDF and the AOR of the random process
$W_{k}^{AF}(t)$, defined by (3), are respectively given by [References, Eq.
(9)] and [References, Eq. (19)], as

$\displaystyle
F_{W_{k}}(Z)=1-2Z\sqrt{\frac{C_{k}}{\Omega_{Sk}\Omega_{kD}}}\exp\left(-\frac{Z^{2}}{\Omega_{Sk}}\right)\qquad$
$\displaystyle\times\,K_{1}\left(2Z\sqrt{\frac{C_{k}}{\Omega_{Sk}\Omega_{kD}}}\right)\,,$
(20) $\displaystyle
N_{k}(Z)=\sqrt{\frac{2}{\pi}}\frac{2Z\exp(-Z^{2}/\Omega_{Sk})}{\Omega_{Sk}\Omega_{kD}}\qquad\qquad\qquad\quad$
$\displaystyle\times\int_{0}^{\infty}\sqrt{\sigma_{\dot{\alpha}_{Sk}}^{2}(y^{2}+C_{k})+\frac{1}{y^{4}}\sigma_{\dot{\alpha}_{kD}}^{2}C_{k}^{2}Z^{2}}\quad$
$\displaystyle\times\exp\left(-\frac{\Omega_{Sk}y^{4}+C_{k}\Omega_{kD}Z^{2}}{y^{2}\Omega_{Sk}\Omega_{kD}}\right)dy\,,$
(21)

where $K_{1}(\cdot)$ is the first-order modified Bessel function of the second
kind. Note that (21) can be efficiently and accurately evaluated by applying
the Gauss-Hermite quadrature rule [References, Eq. (25.4.46)]. Combining (20)
and (21) into (III-A) and (III-A), we obtain the AOR and the AOD of AF
relaying system.

Note, if source $S$ and destination $D$ are fixed, the approach presented in
this section can be applied to derive analogous analytic expressions for AORs
and AODs for more general fading channels (such as, Rice and Nakagami-$m$
models), because such fixed-to-mobile hops have known second-order statistical
properties.

Figure 2: Normalized Average Outage Duration in DF and AF opportunistic
relaying systems

##  IV Numerical and simulation examples

In this section, we present illustrative examples for the normalized AOR (Fig.
1) an the normalized AOD (Fig. 2) in function of $P_{T}/N_{0}$ of
opportunistic relaying system employing either DF or AF relays. The source $S$
and the destination $D$ are fixed ($f_{mS}=f_{mD}=0$), whereas all the relays
are mobile and introduce same maximum Doppler rates $f_{mk}=f_{m0}$, $1\leq
k\leq M$. The AOR and the AOD are normalized with respect to the Doppler rate
$f_{m0}$ as $N/f_{m0}$ and $T\cdot f_{m0}$. The average squared channel gains
of all hops are equal to $0.5$ (i.e., $\Omega_{Sk}=\Omega_{kD}=0.5$, $1\leq
k\leq M$), thus rendering total available transmission power equal to $P_{T}$.
The spectral efficiency is set to $R=1$ bps/Hz. The Monte Carlo simulations
clearly validate our derived analytical results.

## References

  * [1] A. Bletsas, A. Khisti, D. P. Reed and A. Lippman, “A simple cooperative diversity method based on network path selection,” _IEEE J. Select. Areas. Commun._ , vol. 24, no. 3, pp. 659-672, Mar. 2006
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  * [6] D. Michalopoulos, A. Lioumpas, G. K. Karagiannidis and R. Schober, “Selective cooperative relaying over time-varying channels”, _submitted to IEEE Trans. Commun._ , http://arxiv.org/abs/0905.0564v1
  * [7] B. O. Hogstad, M. Patzold, N. Youssef, V. Kontorovitch, “Exact closed-form expressions for the distribution, the level-crossing rate, and the average duration of fades on the capacity of OSTBC-MIMO channels”, _IEEE Trans. Veh. Technol._ , vol. 58, no. 2, pp. 1011-1016, Feb. 2009
  * [8] W. C. Jakes, Microwave Mobile Communications, 2nd ed. Piscataway, NJ: _IEEE Press_ , 1994.
  * [9] A. S. Akki and F. Haber, “A statistical properties of mobile-to-mobile land communication channel,” _IEEE Trans. Veh. Technol._ , vol. 43, no. 4, pp. 826-831, Nov. 1994
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# Energy Consumption in Wireless Systems Equipped with RES, UAVs, and IRSs

Adam Samorzewski

###### Abstract

The paper considers the characteristics of the energy budget for mobile base
stations (BSs) in the form of Unmanned Aerial Vehicles (UAVs) equipped with
Radio Frequency (RF) transceivers, Intelligent Reconfigurable Surfaces (IRSs),
and Renewable Energy Sources (RESs). The obtained results highlight the
benefits and challenges related to using the aforementioned mobile base
stations from the energy side. The research cases took into account two types
of UAV devices – multirotor and fixed-wing (airplane-like).777The work has
been realized within research project no. 2021/43/B/ST7/01365 funded by the
National Science Center in Poland.888Copyright © 2023 National Institute of
Telecommunications in Poland. Personal use is permitted. For any other
purposes, permission must be obtained from the National Institute of
Telecommunications in Poland by emailing<EMAIL_ADDRESS>This is the author’s
version of an article that has been published in the Journal of
Telecommunications and Information Technology (JTIT) by the National Institute
of Telecommunications in Poland. Changes were made to this version by the
publisher before publication, the final version of the record is available at:
https://dx.doi.org/10.26636/jtit.2023.170923. To cite the paper use: A.
Samorzewski, ”Energy Consumption in Wireless Systems Equipped with RES, UAVs,
and IRSs,” Journal of Telecommunications and Information Technology (JTIT),
2023, no. 2, pp. 35–40, doi: 10.26636/jtit.2023.170923 or visit
https://jtit.pl/jtit/article/view/1175/1161.

###### keywords:

6G, power consumption, Renewable Energy Sources, Intelligent Reconfigurable
Surfaces, Unmanned Aerial Vehicles, wireless systems

##  1 Introduction

In recent years, obtaining energy from solar radiation has become a very
common solution that finds application not only in industry but also in many
different areas of everyday life. The process of obtaining solar energy is
carried out using generators in the form of so-called photovoltaic panels (PV
– Photovoltaics). The current market attractiveness of photovoltaics results
not only from the ecological approach to obtaining energy resources but also
from the prices of solar modules, which have been on a downward trend for the
last few years [1]. Therefore, it is not surprising that photovoltaic panels
are now also used in telecommunications systems. Particular benefits can be
seen from the perspective of heterogeneous wireless systems such as mobile
networks.

Wind energy has also gained momentum. Generating energy from wind power is a
process that depends mainly on the location, current weather conditions, and
altitude above sea level. A generator that extracts energy from the wind is
called a wind turbine (WT). A wind turbine converts the flowing wind speed
into mechanical energy and then into electrical energy. Wind energy can be a
good alternative to solar energy, and for some locations or seasons, it may
turn out to be an even more reliable solution [2], [3].

Current wireless systems are powered mainly by conventional energy sources,
the utilization of which is still progressing. Due to the increasing
transmission (service) requirements of Information and Communication
Technology (ICT) networks, with each subsequent year, they show an increasing
energy demand compared to previous years. Equipping wireless network access
points with Renewable Energy Sources (RESs) could reduce or completely
eliminate the demand for electricity from conventional sources. This, in turn,
would not only reduce carbon dioxide $\left(\text{CO}_{2}\right)$ emissions to
the atmosphere but would also slow down the consumption of non-renewable
resources. However, despite the benefits of RESs, they also have one basic
disadvantage – the supply of energy resources is not guaranteed in an even and
continuous manner due to changing weather conditions. Therefore, it may be
necessary to engage more than one type of renewable source at the same time or
to develop algorithms to optimize the use of available energy [2]–[4].

In addition, one of the paradigms in present and future telecommunications
systems is the provision of services in hard-to-reach areas. Such areas are
characterized by unfavorable conditions for the development of
telecommunications infrastructure, which result, for example, from the terrain
or the inability to connect to the existing telecommunications and energy
networks. In a situation where the first problem could be solved by using
mobile access nodes (e.g., drones) and involving wireless connectivity for
backhaul communication, the second case is extremely difficult to cope with
[5]. However, considering the off-grid approach in the context of powering
individual system components, RES generators (e.g., photovoltaic panels, wind
turbines) in combination with an autonomous battery system could be able to
meet the energy needs of the Radio Access Network (RAN) [6]. However, this
requires appropriate modeling of the energy cycle of the system, i.e.,
predicting the results of production and the use of energy resources in time
in such a way as to ensure their adequate amount for each type of load on
network links [7].

In the context of increasing the efficiency of wireless systems, the
Intelligent Reflective Surface (IRS) is proving to be a very promising
concept. The IRS is a device in the form of a flat surface containing a large
number of passive reflecting elements. Each of these elements is able to
independently cause a controlled change in the amplitude and/or phase of the
incident radio signal. The dense distribution of IRS surfaces in the wireless
network and intelligent coordination of their reflections can make the
propagation of the radio signal (wireless channels) between transmitters and
receivers flexibly reconfigurable. This, in turn, can solve the problems of
wireless channel dropout and interference. In addition, thanks to the use of
IRS, improvements in the throughput and reliability of wireless communication
can be achieved [8].

This paper proposes models of energy consumption for wireless access networks
whose nodes are Unmanned Aerial Vehicles (UAVs) equipped with Intelligent
Reconfigurable Surfaces (IRSs) and powered by Renewable Energy Sources (RESs).
In the contribution, the characteristics of the energy balance for the
aforementioned system depending on the type of equipment and weather
conditions have been also investigated.

The paper has been organized as follows: first, the mathematical models for
energy consumption and harvesting have been described; next, the simulation
setup configuration has been presented. After the third section, the results
of performed simulation runs have been overviewed. Finally, the last part is
focused on conclusions summarizing the contribution of the paper.

##  2 Energy Models

In order to estimate the energy efficiency of the considered wireless system,
an extremely important step is to precisely configure the study scenario and
to model the energy cycle for the components of the network. The simulation
scenario considered in the paper assumes the use of a UAV device as a base
station (BS), which is equipped with photovoltaic panels and wind turbines for
energy generation. In addition, the mobile access node has one IRS device to
extend the range of the emitted radio signal and improve the performance of
the wireless system. The used UAV base station may be of the multirotor or
fixed-wing type, which will allow comparing the energy balance characteristics
for both kinds. The UAV node is able to maintain a connection with the major
station on the backhaul link implemented in accordance with microwave (MW)
technology and on the access link realized in the Radio Frequency (RF) band.
The UAV base station hovers/flies above the ground in free space in the city
of Poznan. Where a multirotor mobile station can maintain a fixed position in
the sky, a fixed-wing type station is forced to follow some flight trajectory.
The work considers a static case in which drones do not change their position
(multirotor) or trajectory (fixed-wing) – optimization of the location of the
UAV station and its impact on the functioning of the network is the subject of
parallel activities. However, the paper takes into account the impact of
atmospheric factors such as wind or cloud cover on both the consumption of
energy resources and their harvesting from RESs. The power consumption (PC)
characteristics take into account the utilization related to the
hovering/flying of the UAV station and the provision of services by it in the
considered area using the MIMO technique and IRS device. However, in the
context of obtaining energy resources from RESs, the generation processes were
analyzed both for different times of the day and year, as well as for
different wind speeds, solar radiation densities, and types of cloud cover,
i.e., different cloud thicknesses. Telecommunications traffic in the network
is assumed to be fixed.

###  2.1 Energy Consumption

The power consumption for a fixed-wing UAV mobile base station
$\left(P_{\text{UAV,FW}}\right)$ in time step $t$ can be described by the
following formula [9]:

$\displaystyle P_{\text{UAV,FW}}\left(t\right)=$ (1)
$\displaystyle\left|c_{1}v_{\text{UAV}}^{3}\left(t\right)+\frac{c_{2}}{v_{\text{UAV}}\left(t\right)}\left(1+\frac{a_{\perp}^{2}\left(t\right)}{g^{2}}\right)+m_{\text{ALL}}a_{\parallel}\left(t\right)v_{\text{UAV}}\left(t\right)\right|,$

where $v_{\text{UAV}}$ is the forward flight velocity of the UAV, $a_{\perp}$
and $a_{\parallel}$ are the centripetal acceleration when the base station
moves along the circular trajectory, and the forward acceleration,
respectively. The $m_{\text{ALL}}$ is the sum of the masses of all the
components of a particular UAV, i.e., the weights of the UAV itself
$\left(m_{\text{UAV}}\right)$, its battery system
$\left(m_{\text{BATT}}\right)$, RES generators $\left(m_{\text{PV}},\text{
}m_{\text{WT}}\right)$, and additional equipment $\left(m_{\text{RF}},\text{
}m_{\text{IRS}},\text{ }m_{\text{PKG}}\right)$. Next, the $g$ parameter is the
gravitational acceleration, and $c_{1}$ and $c_{2}$ are the parameters
described by the following formulas [9]:

$\displaystyle c_{1}\overset{\Delta}{=}\frac{1}{2}\rho C_{\text{D}_{0}}S,$
$\displaystyle\qquad c_{2}\overset{\Delta}{=}\frac{2W^{2}}{\pi
e_{0}A_{\text{R}}\rho S},$ (2)

where $\rho$ is the air density (calculated according to the formula from
[10]), $C_{\text{D}_{0}}$ is the zero-lift drag coefficient and $S$ is the
reference area. The parameter of $W=m_{\text{ALL}}\cdot g$ is the weight force
of the UAV device. Whereas $e_{0}$ and $A_{\text{R}}$ are wingspan efficiency
(Oswald efficiency) and UAV wing aspect ratio, respectively.

As for the power consumption of multirotor base stations
$\left(P_{\text{UAV,MR}}\right)$ in the current time step $t$, its
mathematical model is as follows [11]:

$\displaystyle P_{\text{UAV,MR}}\left(t\right)=\frac{d_{0}}{2}\rho
nsA_{\text{UAV}}v_{\text{UAV}}^{3}\left(t\right)+P_{0}\left(t\right)\left(1+\frac{3v_{\text{UAV}}^{2}\left(t\right)}{\Omega^{2}\left(t\right)R^{2}}\right)$
$\displaystyle+P_{\text{i}}\tilde{\kappa}\left(t\right)\left(\sqrt{\tilde{\kappa}\left(t\right)+\frac{v_{\text{UAV}}^{4}\left(t\right)}{4v_{0}^{4}}}-\frac{v_{\text{UAV}}^{2}\left(t\right)}{2v_{0}^{2}}\right)^{\frac{1}{2}},$
(3)

where $\Omega$, $R$, $v_{0}$ are the angular velocity, radius and mean induced
velocity in the hover of a single rotor of a UAV, respectively. The latter can
be expressed as $v_{0}=\sqrt{\frac{W}{2\rho A_{\text{UAV}}}}$, where
$A_{\text{UAV}}$ is the area of a rotor of the UAV. The parameters of $n$ and
$s$ are the number of UAV rotors and the solidity of a single one. In turn,
$\tilde{\kappa}$ and $d_{0}$ are the ratio of thrust to weight forces of the
UAV and the fuselage drag ratio. The values of the parameters of $P_{0}$
$\left(\text{in time step }t\right)$ and $P_{\text{i}}$ are described by the
following formulas [11]:

$\displaystyle P_{0}\left(t\right)=\frac{\delta}{8}\rho
nsA_{\text{UAV}}\Omega^{3}\left(t\right)R^{3},\quad
P_{i}=\left(1+k\right)\left(\frac{W^{3}}{2\rho nA_{\text{UAV}}}\right),$ (4)

where $\delta$ is the profile drag coefficient and $k$ is the incremental
correction factor to induced power.

The power utilized by the IRS device connected to the UAV
$\left(P_{\text{IRS}}\right)$ and RF transceiver
$\left(P_{\text{MIMO}}\right)$ has been evaluated in accordance with the
mathematical models contained in [12] and [13], respectively. Those models are
described by the equations (5) and (6) attached below:

$\displaystyle P_{\text{IRS}}\left(t,b\right)=N\left(t\right)\cdot
P_{\text{n}}\left(b\right),$ (5) $\displaystyle
P_{\text{MIMO}}\left(t\right)=P_{\text{FIX}}+P_{\text{TC}}\big{(}M\left(t\right)\big{)}+P_{\text{CE}}\big{(}M\left(t\right),K\left(t\right)\big{)}$
(6)
$\displaystyle+P_{\text{C/D}}\big{(}\text{TR}_{\text{UL}}\left(t\right),\text{TR}_{\text{DL}}\left(t\right)\big{)}+P_{\text{BH}}\big{(}\text{TR}_{\text{UL}}\left(t\right),\text{TR}_{\text{DL}}\left(t\right)\big{)}$
$\displaystyle+P_{\text{SP}}\big{(}M\left(t\right),K\left(t\right)\big{)}+\frac{P_{\text{TX}}\left(t\right)}{\mu_{\text{PA}}},$

where $P_{\text{FIX}}$ is the fixed power demand of a particular mobile base
station, $P_{\text{TC}}$ is the consumption of transceiver chains, and
$P_{\text{CE}}$, $P_{\text{C/D}}$, and $P_{\text{SP}}$ are the power
components utilized for operations performed within a base station such as
channel estimation, data coding/decoding, and signal processing. Next,
$P_{\text{BH}}$ and $P_{\text{TX}}$ are the powers related to mobile data
flow, i.e., load-dependent MW backhaul and RF fronthaul, respectively. The
parameters of $M\left(t\right)$ and $K\left(t\right)$ are the numbers of
active antenna elements and served users in the current time step $t$. Next,
$\mu_{\text{PA}}$ is the efficiency of the power amplifier. Finally, the
$\text{TR}_{\text{UL}}$ and $\text{TR}_{\text{DL}}$ components are uplink (UL)
and downlink (DL) total data throughputs, which can change over time. In
addition, for channel estimation and signal processing the use of the minimum
mean-squared error (MMSE) scheme has been assumed.

Eq. (5) consists of only two parameters – $N$ and $P_{\text{n}}$, which are
the number of identical reflecting elements that effectively perform phase
shifting on the impinging signal, and the power consumption of each phase
shifter. The value of the second one is dependent on the bit resolution $b$ of
the used phase shifter.

###  2.2 Energy Generation

Models of harvesting energy resources by the applied generators of Renewable
Energy Sources are presented below. The power generated by the photovoltaic
panel of the UAV base station $\left(P_{\text{PV}}\right)$ can be described by
the following mathematical formula [14]:

$\displaystyle
P_{\text{PV}}\left(t\right)=P_{\text{r,PV}}f_{\text{PV}}\left(\frac{\overline{G}_{\text{T}}\left(t\right)}{\overline{G}_{\text{T,STC}}}\right)\left[1+\alpha_{\text{P}}\left(T_{\text{c}}\left(t\right)-T_{\text{c,STC}}\right)\right],$
(7)

where $P_{\text{r,PV}}$ is the rated capacity of the PV array (its output
power under Standard Test Conditions – STC), $f_{\text{PV}}$ is the PV
derating factor, and $\alpha_{\text{P}}$ is the temperature coefficient of
power, which indicates how strongly the PV array power output depends on the
cell temperature. In turn, $\overline{G}_{\text{T}}$ and
$\overline{G}_{\text{T, STC}}$ are the solar incident radiation on the PV
array in the current time step $t$ and at STC, respectively. Whereas
$T_{\text{c}}$ and $T_{\text{c,STC}}$ are also the PV cell temperature in the
current time step $t$ and under STC, respectively. The PV cell temperature can
be calculated with the equation contained in [14].

The power generated by the wind turbine $\left(P_{\text{WT}}\right)$ can be
described as follows [15]:

$P_{\text{WT}}\left(t\right)=\begin{cases}0,&\text{if
}v_{\text{w}}\left(t\right)<v_{\text{in}}\\\
F\big{(}v_{\text{w}}\left(t\right)\big{)},&\text{if }v_{\text{in}}\leqslant
v_{\text{w}}\left(t\right)<v_{\text{r}}\\\ P_{\text{r,WT}},&\text{if
}v_{\text{r}}\text{ }\leqslant v_{\text{w}}\left(t\right)\leqslant
v_{\text{out}},\end{cases}$ (8)

where $v_{\text{in}}$, $v_{\text{out}}$, $v_{\text{w}}$ are the cut-in and
cut-out speed of the wind turbine (defining the range of wind speed values for
which the wind turbine is able to generate energy resources), and the
instantaneous wind speed, respectively. Next, $F$ is the power curve function
of the WT, which determines its energy harvesting characteristic. Finally,
$P_{\text{r}}$ and $v_{\text{r}}$ denote the rated output power of the used
wind turbine and minimum wind speed
$\left(v_{\text{in}}<v_{\text{r}}<v_{\text{out}}\right)$, for which this power
can be achieved. The power curve function normalized in the relation to the
maximum value of the WT’s output power has been presented in Fig. 1. The
corrections of wind speed (to get its correct value at the altitude of the
generator) and WT output power (which depends on the current air density) were
realized in accordance with [14].

Figure 1: Power curve function of wind turbine (based on [16]).

##  3 Simulation Setup

The examined base stations were UAVs of the multirotor and fixed-wing type.
Furthermore, each UAV BS was also equipped with a single RF transceiver and
IRS device. The impact of RES generators enabling on the overall power
consumption of the mobile base station has been also taken into consideration.
Thus, four study cases have been simulated. The first investigates PC in
different seasons of the year, which is caused by the UAV and its additional
equipment (IRS device and RF transceiver) with no RES generators on board. The
second, third, and fourth cases take into account as well the power
consumption resulting from powering the mobile BS with a PV, WT, and both of
them, respectively. The examination of energy generation from RESs has been
considered based on the real products, the specifications of which can be
found in [16] and [17].

Simulations have been prepared using the Python programming language. As the
outcomes, the characteristics for energy consumption of the mobile base
stations as well as for energy harvesting of the RES generators have been
evaluated. The results have been prepared for four different dates starting
various seasons of the last year – vernal equinox $\left(20^{\text{th}}\text{
March }2022\right)$, summer solstice $\left(21^{\text{st}}\text{ June
}2022\right)$, autumn equinox $\left(23^{\text{rd}}\text{ September
}2022\right)$, and winter solstice $\left(21^{\text{st}}\text{ December
}2022\right)$. All the weather data used in the simulations are the real
historical data taken from the following source [18] for the city of Poznan
(Poland). The one exception is the global solar radiation density which was
estimated by using the MAC model [19].

Table 1: VALUES OF THE SIMULATION PARAMETERS [7], [9], [11], [12], [16], [17],
[20]–[23]

Symbol | Quantity | Value | Unit
---|---|---|---
$m_{\text{UAV}}$ | mass of a UAV | $5.0$ | $\left[\text{kg}\right]$
$m_{\text{BATT}}$ | mass of a battery | $0.94$ | $\left[\text{kg}\right]$
$m_{\text{RF}}$ | mass of a RF trans. | $2.0$ | $\left[\text{kg}\right]$
$m_{\text{IRS}}$ | mass of an IRS dev. | $1.0$ | $\left[\text{kg}\right]$
$m_{\text{PV}}$ | mass of a PV panel | $2.78$ | $\left[\text{kg}\right]$
$m_{\text{WT}}$ | mass of a WT | $6.0$ | $\left[\text{kg}\right]$
$m_{\text{PKG}}$ | mass of an addi. package | $0.0$ | $\left[\text{kg}\right]$
$\Omega$ | rotor angular velocity | $300.0^{*}$ | $\left[\text{rad}/\text{s}\right]$
$v_{\text{UAV}}$ | velocity of a UAV | $0.0^{*}$ | $\left[\text{m}/\text{s}\right]$
$10.0^{**}$
$v_{\text{in}}$ | cut-in speed of a WT | $2.0$ | $\left[\text{m}/\text{s}\right]$
$v_{\text{out}}$ | cut-out speed of a WT | $20.0$ | $\left[\text{m}/\text{s}\right]$
$v_{\text{r}}$ | rated speed of a WT | $16.0$ | $\left[\text{m}/\text{s}\right]$
$a_{\parallel}$ | UAV’s forward accel. | $0.0^{**}$ | $\left[\text{m}/\text{s}^{2}\right]$
$a_{\perp}$ | UAV’s centripetal accel. | $0.0^{**}$ | $\left[\text{m}/\text{s}^{2}\right]$
$g$ | gravitational acceler. | $9.81$ | $\left[\text{m}/\text{s}^{2}\right]$
$\delta$ | profile drag coefficient | $0.012^{*}$ |
$k$ | incre. correction factor | $0.1^{*}$ |
$\tilde{\kappa}$ | thrust-to-weight ratio | $1^{*}$ |
$d_{0}$ | fuselage drag ratio | $14.52^{*}$ |
$C_{\text{D}_{0}}$ | zero-lift drag coeffi. | $0.01^{**}$ |
$e_{0}$ | Oswald efficiency | $0.85^{**}$ |
$A_{\text{R}}$ | UAV’s wing aspect ratio | $118.81^{**}$ |
$S$ | UAV’s ref. (wing) area | $1.0^{**}$ | $\left[\text{m}^{2}\right]$
$A_{\text{UAV}}$ | UAV’s rotor area | $0.071^{*}$ | $\left[\text{m}^{2}\right]$
$R$ | UAV’s rotor radius | $0.15^{*}$ | $\left[\text{m}\right]$
$s$ | UAV’s rotor solidity | $0.067^{*}$ |
$n$ | number of rotors | $8^{*}$ |
$M$ | number of antenna ele. | $16$ |
$K$ | number of users | $10$ |
$N$ | number of reflect. ele. | $16$ |
$b$ | phase shifter resolution | $6$ | $\left[\text{bit}\right]$
$\text{TR}_{\text{UL}}$ | UL data throughput | $50.0$ | $\left[\text{Mbps}\right]$
$\text{TR}_{\text{DL}}$ | DL data throughput | $100.0$ | $\left[\text{Mbps}\right]$
$f_{\text{PV}}$ | PV’s derating factor | $0.72$ |
$\alpha_{\text{P}}$ | tem. coeffi. of power | $-0.5$ | $\left[\%/^{\text{o}}\text{C}\right]$
$T_{\text{c,STC}}$ | STC solar cell temper. | $25$ | $\left[{}^{\text{o}}\text{C}\right]$
$G_{\text{T,STC}}$ | STC solar incident radi. | $1000$ | $\left[\text{W}\right]$
$P_{\text{r,PV}}$ | PV panel’s rated power | $20.0$ | $\left[\text{W}\right]$
$P_{\text{r,WT}}$ | rated power of a WT | $30.0$ | $\left[\text{W}\right]$
$P_{\text{n}}$ | power of a phase shifter | $7.8$ | $\left[\text{W}\right]$
$P_{\text{TX}}$ | transmit power of a UAV | $15.0$ | $\left[\text{W}\right]$
$\mu_{\text{PA}}$ | power amplifier efficiency | $0.35$ |

111*

only for the multirotor type 777* only for the fixed-wing type

All the prepared energy characteristics charts have been normalized in
relation to the maximum values obtained for them. Furthermore, in Tab. 1 the
values for the parameters of the mathematical energy models aforementioned in
Section 2 have been highlighted. The values of the parameters related to the
power consumption model for RF transmission $(\text{not contained in
Tab.~{}\ref{tabSimParams}})$ were taken from Tab. 5.3 in [13].

##  4 Results

Below are the charts with the characteristics of energy consumption by the
mobile base stations as well as of energy generation by their RES generators
have been shown. A particular color of the plots and bars is strictly
associated with a specific season of the year: blue – the vernal equinox,
yellow – the summer solstice, green – the autumn equinox, and red – the winter
solstice.

Figure 2: Energy consumption character. for multirotor and fixed-wing UAVs.

In Fig. 2, the normalized energy utilization process for the different UAV
types has been presented. The chart at the top denotes the characteristics of
the multirotor UAV, and the second one placed below describes the consumed
energy by the fixed-wing type. The values presented in the figure were not
normalized in the relation to the same value, but each one was scaled
according to the maximum value obtained for a particular study case. Thus,
lower consumption that can be observed for the multirotor does not mean this
type of UAV uses less energy than the fixed-wing type. The main goal of the
investigation was to highlight the impact of adding extra equipment to the UAV
weight on the energy utilization itself in different seasons of the year. For
a hovering multirotor UAV the amount of its total mass has a crucial meaning
to its energy demand. By adding a PV panel and wind turbine (around $9$ kg) to
the base station, its energy consumption increases almost twice. For different
seasons, some deviations in the use of energy resources by the multirotor UAV
can be observed as well. The reason for that is fluctuating air density value
during the year. For the UAV of the fixed-wing type, aforementioned weight and
weather dependencies can be noticed as well, but their impact on PC is not so
significant as for the multirotor BS. On the other hand, within this paper,
only one situation was considered, in which the fixed-wing UAV is flying with
a constant velocity equal to $10$ m/s. The real impact might be seen by
studying changes in energy utilization of the drone lifting various weight
loads from the ground, or studying changes in its aerodynamic characteristics.
Both cases will be considered in future work.

Figure 3: Energy generation characteristics for PV panel and wind turbine.

Fig. 3 presents the energy harvesting processes of the PV panel and wind
turbine used in the contribution to power the UAV BSs. The characteristics
have been prepared based on the real equipment products, the specifications of
which can be found in [16] and [17]. The charts contained in Fig. 3 highlight
the energy gain that can be achieved for different seasons of the year. Based
on that it is possible to assess when equipping the mobile base station with a
particular RES generator is beneficial. For instance, in the summer season,
the PV panel is the most effective and WT the least. In the wintertime of the
year, the situation is the opposite. Interestingly, the peak of energy
harvesting for both generators can be observed for the vernal equinox although
it is not the most effective season for producing power by PV panels or wind
turbines. However, the reason for that is the limitations of RES generators,
i.e., each type has its temperature or wind speed ranges, in which they can
produce renewable energy resources. In addition, the output power of the PV
panel is greatly influenced by the current temperature of its solar cells
(dependent, e.g., on the current ambient temperature) and the opacity of the
clouds in the sky. Thus, an increase in air temperature and the amount of
clouds can cause a decrease in energy harvesting efficiency.

##  5 Conclusions

In this paper, the characteristics of energy utilization and generation by the
two types of UAV mobile base stations equipped with RF transceivers, IRS
devices, and RES generators (PV panels and wind turbines) have been
investigated. Obtained results highlight the impact of the UAV’s equipment on
the energy balance for different seasons of the year. The paper’s outcomes
seem to be valuable for the process of network planning for future wireless
systems considering the use of UAVs, IRSs, and RESs.

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