# Mobile-to-Mobile Uncorrelated Scatter Channels

Michael Walter, Martin Schmidhammer, Miguel A. Bellido-Manganell, Thomas
Wiedemann, and Dmitriy Shutin M. Walter, M. Schmidhammer, M. A. Bellido-
Manganell, T. Wiedemann, and D. Shutin are with the German Aerospace Center
(DLR), Institute of Communications and Navigation, 82234 Wessling, Germany
(e-mail: {m.walter, martin.schmidhammer, miguel.bellidomanganell,
thomas.wiedemann, dmitriy.shutin}@dlr.de).

###### Abstract

In this paper, we present a complete analytic probability based description of
mobile-to-mobile uncorrelated scatter channels. The correlation based
description introduced by Bello and Matz is thus complemented by the presented
probabilistic description leading to a common theoretical description of
uncorrelated scatter channels. Furthermore, we introduce novel two-dimensional
hybrid characteristic probability density functions, which remain a
probability density in one of the variables and a characteristic function in
the other variable. Such a probability based description allows us to derive a
mathematical model, in which the attenuation of the scattering components is
inherently included in these two-dimensional functions. Therefore, there is no
need to determine the path loss exponent. Additionally, the Doppler
probability density function with the inclusion of the path loss leads to a
concave function of the Doppler spectrum, which is quite different from the
Jakes and Doppler spectra and can be directly parameterized by the velocity
vectors and geometry of the scattering plane. Thus, knowing those parameters
permits the theoretical computation of the Doppler spectra and temporal
characteristic functions. Finally, we present a comparison between the
computed probability based theoretical results and measurement data for a
generic mobile-to-mobile channel. The agreement between the two shows the
usefulness of the probability based description and confirms new shapes of the
Doppler power spectra.

###### Index Terms:

Doppler frequency, mobile-to-mobile communication, geometry-based stochastic
channel model, characteristic function, prolate spheroidal coordinate system,
non-stationary.

††publicationid: pubid: This work is planned to be submitted to the IEEE for
possible publication. Copyright may be transferred without notice, after which
this version may no longer be accessible.

##  I Introduction

The significance of mobile-to-mobile (M2M) communication is progressively
growing. Especially vehicle-to-vehicle (V2V) communications – just one area of
modern M2M communications – is becoming increasingly integrated into newly
manufactured cars to mitigate possible accidents via situational awareness.
Moreover, it will become an integral part of future autonomous driving vehicle
solutions [1]. Similar direct communication frameworks are envisioned for
various modes of transportation such as trains, ships, aircraft, and drones.
Historically, the channel models used for the design and testing of
communication systems in the past were wide-sense stationary uncorrelated
scattering (WSSUS). They were largely valid as they were mainly for fixed-to-
mobile channels. However, they are inadequate in an M2M context due to the
inherent non-stationarity and uncorrelated scattering characteristics of the
channel.

Scattering is ultimately a stochastic process. It has been shown in [2] and
[3] that it can be well explained by means of postulating a power spectral
density in the frequency domain, or equivalently with an autocorrelation
function in the temporal domain. Stochastic time variations can also be
described in a similar statistical fashion. Clarke derived in [4] a model that
describes the Doppler spectrum of the propagation channel in the case of a
mobile receiver, e.g., a moving car communicating with a fixed base station.
This type of Doppler spectrum is widely known as Jakes spectrum [5]. In [6]
Bello extended the mentioned models to more general channels, describing
statistical characterizations for both WSSUS and non-WSSUS channels. Later,
Matz in [7] focuses on non-WSSUS channels, which he studies from two
complementary perspectives. First, the time-frequency channel transfer
function can be treated as a non-stationary process in both time and frequency
domains. Second, by assuming that scatterers with distinct delays and Doppler
frequencies are uncorrelated, the resulting channel impulse response can be
studied as non-stationary in time and uncorrelated along the delay. This
allows to introduce the local scattering function (LSF) and the channel
correlation function (CCF), which both describe small-scale channel
statistics. The two functions are naturally related to correlation functions
introduced by Bello in [6] via Fourier transforms.

Bello states in Section IV of his seminal work [6] that it’s difficult to find
an exact statistical description of a time-variant channel in terms of
multidimensional probability functions and that correlation functions are a
more practical approach. In essence, this statement implies that without
further physical assumptions on the propagation environment, an accurate
statistical characterization of non-WSSUS channels can be quite elusive. This
motivates the application of geometric-stochastic modeling approaches, where a
specific propagation geometry is assumed, or at least some additional
assumptions on the structure of the channel are put into place.

One such approach, as exemplified well in [8], reveals that M2M channels
typically violate the wide-sense stationary (WSS) assumption to a greater
extent than the uncorrelated scattering (US) assumption. This, on the one
hand, constrains general non-WSSUS type of channels to more restricted cases,
while on the other hand, building a highly relevant application scenario for
the design of practical M2M communication systems. Our objective in this paper
is therefore to explore such restricted cases, especially non-WSS channels, in
more detail, providing stochastic characterizations describing the non-
stationary behavior of such M2M channels. Thus, we strive to extend our
theoretical non-WSS model to encompass other correlation domains in an attempt
to create a complete stochastic description of a non-stationary M2M channel
characterization. Therefore, we use a prolate spheroidal coordinate (PSC)
system introduced in [9] to derive the time-dependent probability density
functions of delay and Doppler variables. Additionally, we introduce hybrid
characteristic and probability density functions to better capture the complex
dynamics of M2M communication channels. These are in essence (inverse) Fourier
transforms along one of the variables of the corresponding time-varying pdf.
The newly introduced hybrid functions are proportional to the correlation
functions introduced by Bello. We thus extend the proof of the proportionality
between the joint delay Doppler pdf and scattering function for the WSSUS case
[10] to non-WSS channels. In particular, the proportionality between the joint
time frequency characteristic function and the time frequency correlation
function is shown. The time-variant time frequency correlation function shown
in [11] by invoking the US assumption is thus proportional to the joint
characteristic function presented in this paper. We further explain new
insights into the channel structure obtained from newly introduced hybrid
channel functions. By being able to theoretically describe high mobility M2M
channels, new modulation schemes like orthogonal time frequency space (OTFS)
[12], [13] can be employed for integrated sensing and communications (ISAC)
[14].

The rest of the paper is structured as follows. In Section II, we provide a
complete stochastic description of the uncorrelated M2M channel and relate
those functions to the correlation functions, LSF, and CCF. In Section III, we
show, how the hybrid probability characteristic function is calculated in
closed form. The theoretical stochastic description is then verified by
measurement data from an air-to-air measurement campaign in Section IV. The
paper is concluded with Section V.

##  II Characterization of US Channels

In order to obtain analytical solutions for the probability based description,
it is essential to establish a unified mathematical framework. Our goal is to
establish connections between the models developed by Bello in [6] and Matz in
[7] with our probabilistic approach.

Consider a classical communication channel between a transmitter and a
receiver, both possibly mobile. The relationship between the transmitted
signal $s(t)$ and the received signal $r(t)$ can be represented as [7]

$\displaystyle r(t)$ $\displaystyle=$ $\displaystyle\int
h(t,\tau)s(t-\tau)\,\mathrm{d}\tau\,,$ (1)

where the function $h(t,\tau)$ – the time-varying channel impulse response –
fully characterizes the propagation environment between the transmitter and
the receiver. Essentially (1) states that the received signal is a
superposition of differentially delayed copies of the transmitted signal. The
channel can be defined in terms of the spreading function or also known as
Doppler-variant impulse response $S({f}_{\mathrm{d}},\tau)=\int
h(t,\tau)\mathrm{e}^{-\mathrm{j}2\pi{f}_{\mathrm{d}}t}\,\mathrm{d}t$, the
time-varying transfer function $L(t,f)=\int
h(t,\tau)\mathrm{e}^{-\mathrm{j}2\pi f\tau}\,\mathrm{d}\tau$, via a Fourier
transform over the delay variable $\tau$ or the Doppler-variant transfer
function $T({f}_{\mathrm{d}},f)=\int\int
h(t,\tau)\mathrm{e}^{-\mathrm{j}2\pi{f}_{\mathrm{d}}t}\mathrm{e}^{-\mathrm{j}2\pi
f\tau}\,\mathrm{d}t\mathrm{d}\tau$, which is obtained by a double Fourier
transform of $h(t,\tau)$ over the $t$ and $\tau$ variables. Knowledge of
$h(t,\tau)$, or any of the other three functions is thus instrumental for the
design, simulation, or testing of practical communication systems.

###  II-A Correlation Based Description

In practice, an exact form of $h(t,\tau)$ depends on the particular
propagation environment. The environment, however, is rarely known accurately
or in advance at the stage of communication system design. Therefore, as has
been mentioned earlier, statistical properties of $h(t,\tau)$ are of interest.
Similarly, the statistical properties of the other three system functions
$S({f}_{\mathrm{d}},\tau)$, $L(t,f)$, and $T({f}_{\mathrm{d}},f)$ can be
determined. These statistics can be captured by the corresponding
autocorrelation functions of the four system functions according to [6, 7, 15]
as

$\displaystyle R_{h}(t,\tau;\Delta t,\Delta\tau)$
$\displaystyle=\mathbb{E}\\{h(t,\tau+\Delta\tau)h^{*}(t-\Delta t,\tau)\\}\,,$
(2) $\displaystyle R_{L}(t,f;\Delta t,\Delta f)$
$\displaystyle=\mathbb{E}\\{L(t,f+\Delta f)L^{*}(t-\Delta t,f)\\}\,,$
$\displaystyle R_{S}(\tau,{f}_{\mathrm{d}};\Delta\tau,\Delta{f}_{\mathrm{d}})$
$\displaystyle=\mathbb{E}\\{S(\tau,{f}_{\mathrm{d}}+\Delta{f}_{\mathrm{d}})S^{*}(\tau-\Delta\tau,{f}_{\mathrm{d}})\\}\,,$
$\displaystyle R_{T}({f}_{\mathrm{d}},f;\Delta{f}_{\mathrm{d}},\Delta f)$
$\displaystyle=\mathbb{E}\\{T({f}_{\mathrm{d}},f+\Delta
f)T^{*}({f}_{\mathrm{d}}-\Delta{f}_{\mathrm{d}},f)\\}\,.$

Here the operator $\mathbb{E}\\{\cdot\\}$ denotes the expectation
operation.111 Due to equivalence $\mathbb{E}\\{f(x,y+\Delta y)f^{*}(x-\Delta
x,y)\\}=\mathbb{E}\\{f(x,y)f^{*}(x-\Delta x,y-\Delta
y)\\}=\mathbb{E}\\{f(x+\Delta x,y+\Delta y)f^{*}(x,y)\\}$ expressions in (2)
can be modified correspondingly. By taking double Fourier transforms of the
correlation functions, a set of four equivalent, yet different descriptions of
the channel can be obtained.

One can see that the correlation functions in (2) are 4D functions. These
functions are difficult to work with, not to mention hard to get insights into
or intuition about their properties. To simplify the analysis, one can often
invoke the WSSUS assumption, see e.g., [6]. Its consequence is that
correlation functions in (2) become dependent only on the corresponding time
and frequency lag variables. They thus collapse to much simpler 2D functions,
see [6] and [7]. Due to this simplification, the WSSUS assumption has
dominated channel modeling over decades, especially for non-mobile
applications or fixed-to-mobile (F2M) channels, i.e., with a fixed base
station. Yet, the mobility of transmitter (TX) and receiver (RX) nowadays
brings in the non-stationarity of the propagation environment, and thus of the
channel. In [7] the LSF was introduced as an alternative second-order channel
statistic to account for non-stationarities in the time and frequency domains.
The LSF generalizes the scattering function for non-WSSUS channels, but, as
mentioned above, it is a 4D function. Many M2M channels, however, in
particular V2V channels, are non-stationary, yet preserve the US property.
This permits describing US channels with only three variables instead of four.

According to [6], US channels were observed for troposcatter communication and
moon reflections. Recently, similar effects were also observed for M2M
channels, where the uncorrelated scattering assumption was validated with
measurements, see e.g., [8]. There the author states that the V2V channel
infringes the WSS assumption much stronger than the US assumption. In the US
case the scatterers can be modeled as a continuum of uncorrelated scatterers
according to [6]. As a result, the general correlation functions in (2) become
independent of $\Delta\tau$, since the correlation between scatterers causing
different delays vanishes. Under the US assumption (2) can be represented, see
also [6, eq. (66)], as222Note that equations in (3) imply that the right-hand
side is independent of $f$ in case of the US assumption.

$\displaystyle R_{h}(t,\tau;\Delta t,\Delta\tau)$
$\displaystyle=P_{h}(t;\tau,\Delta t)\mathrm{\delta}(\Delta\tau)\,,$ (3)
$\displaystyle R_{L}(t,f;\Delta t,\Delta f)$ $\displaystyle=R_{L}(t;\Delta
t,\Delta f)\,,$ $\displaystyle
R_{S}(\tau,{f}_{\mathrm{d}};\Delta\tau,\Delta{f}_{\mathrm{d}})$
$\displaystyle=P_{S}(\tau,{f}_{\mathrm{d}};\Delta{f}_{\mathrm{d}})\delta(\Delta\tau)\,,$
$\displaystyle R_{T}({f}_{\mathrm{d}},f;\Delta{f}_{\mathrm{d}},\Delta f)$
$\displaystyle=R_{T}({f}_{\mathrm{d}};\Delta{f}_{\mathrm{d}},\Delta f)\,,$

where $\delta(\cdot)$ is the Dirac delta distribution and $P_{h}(t;\tau,\Delta
t)$ and $P_{S}(\tau,{f}_{\mathrm{d}};\Delta{f}_{\mathrm{d}})$ are cross-power
spectral densities. Using the former, the LSF is defined [16] as

$\tilde{C}_{\mathbf{H}}(t;\tau,{f}_{\mathrm{d}})=\int P_{h}(t,\tau;\Delta
t)\mathrm{e}^{-\mathrm{j}2\pi\Delta t{f}_{\mathrm{d}}}\,\mathrm{d}\Delta t\,.$
(4)

Furthermore, in [7] Matz defines the corresponding channel correlation
function, which we simplify here to

$\tilde{\mathcal{A}}_{\mathbf{H}}(\Delta t,\Delta
f;\Delta{f}_{\mathrm{d}})=\int R_{L}(t;\Delta t,\Delta
f)\mathrm{e}^{-\mathrm{j}2\pi\Delta{f}_{\mathrm{d}}t}\,\mathrm{d}t\,,$ (5)

which brings the total available functions to describe the US channel
statistically to six, which are given in (3), as well as (4) and (5). The
reduction of dimensionality of the six correlation based functions permits
simpler visualization and interpretation, which in turn provide valuable
insights into the correlation properties of the channel.

Again, the resulting correlation functions can be studied both in time, time
lag, or frequency and frequency lag domains. As a consequence, for the 3D
functions in (3) a total of eight equivalent representations can
mathematically be established. However, in [6] Bello discusses only the four
proper correlation functions, where the time and frequency variables, as well
as the associated lag variables are in the same domain, e.g. either both in
the temporal or both in the frequency domain. Later in [7] Matz proposed the
time-variant LSF and the channel correlation function, which are obtained by
Fourier transforms of the original four correlation functions as shown in (4)
and (5). As we will see, it is also useful to consider the correlation
variables in a mixed temporal frequency domain, thus extending the available
six functions from Bello and Matz to a total of eight. All these functions are
summarized conceptually in Fig. 1 for the probability based description.

$p(t;\xi,{f}_{\mathrm{d}})$ $r(t;\Delta\tilde{f},\Delta t)$ $\rho(t;\xi,\Delta
t)$ $\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$
$\varrho(\Delta{f}_{\mathrm{d}};\xi,\Delta t)$
$\rho(\Delta{f}_{\mathrm{d}};\xi,{f}_{\mathrm{d}})$
$\mathcal{R}(\Delta{f}_{\mathrm{d}};\Delta\tilde{f},\Delta t)$
$r(\Delta{f}_{\mathrm{d}};\Delta\tilde{f},{f}_{\mathrm{d}})$ $\xi$$\Delta
t$${f}_{\mathrm{d}}$$\Delta\tilde{f}$$\Delta{f}_{\mathrm{d}}$$t$$\xi$$\Delta
t$$\Delta{f}_{\mathrm{d}}$$t$$\Delta\tilde{f}$${f}_{\mathrm{d}}$$\Delta
t$$\xi$$\Delta
t$${f}_{\mathrm{d}}$$\xi$$\Delta\tilde{f}$$\Delta\tilde{f}$${f}_{\mathrm{d}}$

Figure 1: Time-variant and Doppler correlated relationships between
characteristic functions, probability density functions, and hybrid
characteristic probability density functions for the US channel. Time-variant
functions, which are referred to in Fig. 2, are framed by a dashed rectangle.

In order to set all correlation based functions in a relationship with the
probability based functions discussed later, we need to make a new definition.
Thus, we define

$P_{\varrho}(t;\Delta
f,{f}_{\mathrm{d}})\triangleq\int\tilde{C}_{\mathbf{H}}(t;\tau,{f}_{\mathrm{d}})\mathrm{e}^{-\mathrm{j}2\pi\Delta
f\tau}\,\mathrm{d}\tau\,,$ (6)

as the Fourier transform of the time-variant LSF with respect to the delay
variable $\tau$. Similarly to [17] and to ease further direct comparison with
probability based descriptions defined in the next section, we define
normalized versions of the correlation based functions as

$\displaystyle\tilde{C}_{\mathbf{H}}(t;{f}_{\mathrm{d}}|\tau)$
$\displaystyle\triangleq\frac{\tilde{C}_{\mathbf{H}}(t;\tau,{f}_{\mathrm{d}})}{P_{h}(t;\tau,\Delta
t=0)}\,,$ (7) $\displaystyle P_{h}(t;\Delta t|\tau)$
$\displaystyle\triangleq\frac{P_{h}(t;\tau,\Delta t)}{P_{h}(t;\tau,\Delta
t=0)}\,,$ $\displaystyle P_{\varrho}(t;\Delta f|{f}_{\mathrm{d}})$
$\displaystyle\triangleq\frac{P_{\varrho}(t;\Delta
f,{f}_{\mathrm{d}})}{P_{\varrho}(t;\Delta f=0,{f}_{\mathrm{d}})}\,,$
$\displaystyle\tilde{R}_{L}(t;\Delta t,\Delta f)$
$\displaystyle\triangleq\frac{R_{L}(t;\Delta t,\Delta f)}{R_{L}(t;\Delta
t=0,\Delta f=0)}\,,$ $\displaystyle\tilde{P}_{h}(t;\tau)$
$\displaystyle\triangleq\frac{P_{h}(t;\tau,\Delta t=0)}{\int
P_{h}(t;\tau,\Delta t=0)\,\mathrm{d}\tau}\,,$
$\displaystyle\tilde{P}_{\varrho}(t;{f}_{\mathrm{d}})$
$\displaystyle\triangleq\frac{P_{\varrho}(t;\Delta f=0,{f}_{\mathrm{d}})}{\int
P_{\varrho}(t;\Delta f=0,{f}_{\mathrm{d}})\,\mathrm{d}{f}_{\mathrm{d}}}\,.$

###  II-B Probability Based Description

In this subsection, we discuss the probabilistic representation of the US
channel more formally. We begin by defining
$\xi\triangleq\tau/\tau_{\mathrm{los}}$ as the normalized delay, where
$\tau_{\mathrm{los}}$ is the line-of-sight (LOS) delay between the transmitter
and receiver. Similarly, we define the normalized frequency lag
$\Delta\tilde{f}\triangleq\Delta f\tau_{\mathrm{los}}$. In the following, we
will largely utilize the notation of our previous works, e.g., [18], where the
focus was on deriving the joint delay Doppler pdf $p(t;\xi,{f}_{\mathrm{d}})$
for the M2M channel.

Note that by taking Fourier transforms along some of the variables of these
functions, we obtain equivalent representations, yet in the corresponding
frequency domains. This will allow for other, often quite revealing,
interpretations of the channel properties, as will be shown later. These
different transforms are shown for the US case in Fig. 1. Of particular
interest is the time-varying joint delay Doppler pdf
$p(t;\xi,{f}_{\mathrm{d}})$ that we introduced before in [18]. The time-
varying joint delay Doppler pdf $p(t;\xi,{f}_{\mathrm{d}})$ can be computed
analytically, e.g., for M2M US channels. Furthermore, it has a finite support,
since the velocity and sensitivity limit the possible delays and Doppler
frequencies. Since the trajectories of TX and RX are time-variant, it makes
sense to use a time-variant probability density. In our case the time $t$ is a
deterministic variable, whereas the delay $\xi$ and Doppler frequency
${f}_{\mathrm{d}}$ are treated as stochastic variables. Their distributions
are obtained from the assumption that scatterers are uniformly distributed on
the ground. This is followed by a variable transform from the spatial to the
Doppler domain, see [18]. To illustrate the connection between the correlation
functions from [6] and their equivalent forms in [7], we summarize them in the
Table I, using the original notation.333Note that the variable $\xi$ is
defined in [6] as time difference, whereas in this paper it refers to a
normalized delay.

Our objective is to relate the joint delay Doppler pdf
$p(t;\xi,{f}_{\mathrm{d}})$ to the LSF [7] and, what we call, the hybrid
characteristic pdf $\rho(t;\xi,\Delta t)$ to the original correlation function
of Bello [6]. Recall that for the joint pdfs this is equivalent to computing
the characteristic function for one of the two variables. Thus,
$\rho(t;\xi,\Delta t)$ is a characteristic function along $\Delta t$ direction
and a pdf function along the delay direction. Accordingly, this function
presents a time-variant spectral density along the $\xi$ variable and the
temporal correlation along the $\Delta t$ variable.

$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$ $p(t;\xi,{f}_{\mathrm{d}})$
$r(t;\Delta\tilde{f},\Delta t)$ $\rho(t;\xi,\Delta t)$ $p(t;\xi)$
$p(t;{f}_{\mathrm{d}})$ $r(t;\Delta t)$ $r(t;\Delta\tilde{f})$
$\Delta\tilde{f}$$\xi$${f}_{\mathrm{d}}$$\Delta t$${f}_{\mathrm{d}}$$\Delta
t$$\Delta\tilde{f}$$\xi$$\Delta\tilde{f}$$\xi$${f}_{\mathrm{d}}$$\Delta
t$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}\xi\,$$\Delta\tilde{f}=0$$\Delta\tilde{f}=0$$\Delta t=0$$\Delta
t=0$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}{f}_{\mathrm{d}}\,$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}{f}_{\mathrm{d}}\,$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}\xi\,$

Figure 2: Time-variant relationships between characteristic functions,
probability density functions, and hybrid probability density characteristic
function for the US channel.

Since M2M channels are non-stationary, this function is particularly useful,
since it allows observing a time-varying, delay-dependent temporal correlation
of the channel. Using Fourier transforms to convert the time or frequency
variables will lead to other hybrid representations of the channel, where the
function represents a characteristic function along one of the variables and a
probability density along the other. Thus, we consider the Fourier and inverse
Fourier transform as basically the same operation, but with different signs.
In Fig. 1 this is represented with functions, e.g., $\rho(t;\xi,\Delta t)$ and
$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$, which are neither pure
characteristic nor probability density function representations.

For an easier comparison to WSSUS channels, the eight functions in Fig. 1 can
be partitioned into (i) a time-variant (dashed box) and (ii) a Doppler
correlated description. We consider time-variant descriptions as more natural
and easier to interpret, although all descriptions enjoy an equivalence under
the appropriate Fourier transform. The reasoning behind this lies in the fact
that in M2M scenarios both transmitter and receiver move, resulting in time-
dependent velocity vectors. Thus, a time-variant channel description would be
more natural. Yet, via appropriate Fourier transform we can equivalently
obtain Doppler correlated descriptions – the lower part of Fig. 1. These are,
however, less intuitive to interpret.

####  II-B1 Time-variant Functions

We begin with the known time-varying joint delay Doppler pdf
$p(t;\xi,{f}_{\mathrm{d}})$ from [18] as a starting point for further
analysis. An inverse Fourier transform along the variables of delay leads to a
hybrid characteristic pdf representation, as we mentioned above. Let us
consider the joint delay Doppler pdf $p(t;\xi,{f}_{\mathrm{d}})$ and the
corresponding hybrid representation

$\rho(t;\xi,\Delta t)\triangleq\int
p(t;\xi,{f}_{\mathrm{d}})\mathrm{e}^{\mathrm{j}2\pi{f}_{\mathrm{d}}\Delta
t}\,\mathrm{d}{f}_{\mathrm{d}}\,.$ (8)

It is important to note that $\rho(t;\xi,\Delta t=0)=p(t;\xi)$, since (8)
becomes a marginalization integral then. Indeed, by setting the characteristic
variable $\Delta t$ to zero, the exponential function in the integral
vanishes, and the marginal, time-varying pdf $p(t;\xi)$ can be obtained. This
time-variant delay pdf is proportional to $P_{h}(t;\xi,\Delta t=0)$, which is
the power delay profile of the channel, see Conjecture 1. Thus, we have a non-
parametric, geometry-based, time-variant path loss model, which we will
analytically derive for the general M2M channel. With the time-variant delay
pdf $p(t;\xi)$ we can obtain the factorization of the time-varying, delay-
dependent Doppler probability density as
$p(t;\xi,{f}_{\mathrm{d}})=p(t;\xi)p(t;{f}_{\mathrm{d}}|\xi)$, which reveals
the conditional density $p(t;{f}_{\mathrm{d}}|\xi)$, i.e., the Doppler pdf
conditioned on a particular delay $\xi$.

With the time-varying pdf $p(t;\xi)$ the hybrid characteristic pdf can also be
factorized to a delay-dependent characteristic function as

$\rho(t;\Delta t|\xi)=\frac{\rho(t;\xi,\Delta t)}{\rho(t;\xi,\Delta
t=0)}=\frac{\rho(t;\xi,\Delta t)}{p(t;\xi)}\,.$ (9)

The other hybrid characteristic pdf contains a pdf in the Doppler domain and a
characteristic function in the frequency domain, but here the time and Doppler
frequency are in different domains. The function is given by

$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})\triangleq\int
p(t;\xi,{f}_{\mathrm{d}})\mathrm{e}^{-\mathrm{j}2\pi\Delta\tilde{f}\xi}\,\mathrm{d}\xi\,.$
(10)

Similarly, by setting $\Delta\tilde{f}=0$ in (10), we obtain
$\varrho(t;\Delta\tilde{f}=0,{f}_{\mathrm{d}})=p(t;{f}_{\mathrm{d}})$, i.e.,
the time-varying Doppler probability density. The conditional characteristic
function is obtained by

$\varrho(t;\Delta\tilde{f}|{f}_{\mathrm{d}})=\frac{\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})}{\varrho(t;\Delta\tilde{f}=0,{f}_{\mathrm{d}})}=\frac{\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})}{p(t;{f}_{\mathrm{d}})}\,.$
(11)

The time-variant joint characteristic function $r(t;\Delta\tilde{f},\Delta t)$
can be directly obtained by a double Fourier transform as

$r(t;\Delta\tilde{f},\Delta t)=\iint
p(t;\xi,{f}_{\mathrm{d}})\mathrm{e}^{-\mathrm{j}2\pi\left(\Delta\tilde{f}\xi-{f}_{\mathrm{d}}\Delta
t\right)}\,\mathrm{d}\xi\mathrm{d}{f}_{\mathrm{d}}\,,$ (12)

with the property $r(t,\Delta\tilde{f}=0,\Delta t=0)=1$. Note that in (12) we
use one normal and one inverse Fourier transform to compute the joint
characteristic function to be consistent with the channel modeling literature.
This is similar to the description used in[19].

TABLE I: Comparison of correlation and probability based functions for US
channels.

lightgrayFunction Name | Bello [6] | Matz [7],[16] | Walter et.al. | Function Name
---|---|---|---|---
time-varying scattering function | $-$ | $\tilde{\mathcal{C}}_{\mathbf{H}}(t;\tau,\nu)$ | $p(t;\xi,{f}_{\mathrm{d}})$ | TV joint delay Doppler pdf
time correlated delay cross-power spectral density | $P_{{g}}(t,s;\xi)$ | $P_{h}(t,\tau;\Delta t)$ | $\rho(t;\xi,\Delta t)$ | TV hybrid time delay char. pdf
$-$ | $-$ | $-$ | $\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$ | TV hybrid freq. Doppler char. pdf
autocorrelation of time-variant transfer function | $R_{{T}}(\Omega;t,s)$ | $R_{L}(t;\Delta t,\Delta f)$ | $r(t;\Delta\tilde{f},\Delta t)$ | TV joint time frequency char. fun.
DC delay cross-power spectral density | $P_{{U}}(\xi;\nu,\mu)$ | $P_{S}(\tau,\nu;\Delta\nu)$ | $\rho(\Delta{f}_{\mathrm{d}};\xi,{f}_{\mathrm{d}})$ | DC hybrid Doppler delay char. pdf
autocorrelation of output Doppler spread function | $R_{{G}}(\Omega,\nu,\mu)$ | $-$ | $r(\Delta{f}_{\mathrm{d}};\Delta\tilde{f},{f}_{\mathrm{d}})$ | DC frequency Doppler char. fun.
$-$ | $-$ | $-$ | $\varrho(\Delta{f}_{\mathrm{d}};\xi,\Delta t)$ | DC hybrid time delay char. pdf
uncorrelated scatter channel correlation function | $-$ | $\tilde{\mathcal{A}}_{\mathbf{H}}(\Delta t;\Delta f,\Delta\nu)$ | $\mathcal{R}(\Delta{f}_{\mathrm{d}};\Delta\tilde{f},\Delta t)$ | DC time frequency char. fun.

TV: time-variant, DC: Doppler correlated

####  II-B2 Time-variant First and Second Order Moments

The time-variant mean delay and delay spread can be easily calculated by using
the hybrid characteristic pdf and setting $\Delta t=0$. This results in the
first two delay moments as

$\displaystyle\mu_{\xi}(t)=\int\limits_{\xi_{\mathrm{min}}}^{\xi_{\mathrm{max}}}\xi\rho(t;\xi,\Delta
t=0)\,\mathrm{d}\xi\,,$ (13)
$\displaystyle\sigma_{\xi}(t)=\sqrt{\int\limits_{\xi_{\mathrm{min}}}^{\xi_{\mathrm{max}}}\left(\xi-\mu_{\xi}(t)\right)^{2}\rho(t;\xi,\Delta
t=0)\,\mathrm{d}\xi}\,$ (14)

with $\xi_{\mathrm{max}}>\xi_{\mathrm{min}}>\xi_{\mathrm{sr}}$ and

$\xi_{\mathrm{sr}}=\max\left(\sqrt{\frac{A^{2}+B^{2}+D^{2}}{A^{2}+B^{2}+C^{2}}},1\right)\,,$
(15)

being the delay of the specular reflection relative to the line-of-sight
delay. Parameters $A$, $B$, $C$ and $D$ are the orientation coefficients of
the scattering plane. Their geometric interpretation and computation will be
discussed Section III.

If the influence of the delay is removed, we obtain the temporal correlation
of the channel per delay. Instead of calculating the total mean Doppler and
Doppler spread, we calculate delay-dependent mean Doppler and Doppler spread.
These can again be obtained from hybrid characteristic pdf as

$\displaystyle\mu_{{f}_{\mathrm{d}}|\xi}(t$
$\displaystyle)=\left.\frac{1}{\mathrm{j}2\pi}\frac{\partial}{\partial\Delta
t}{\rho}(t;\Delta t|\xi)\right|_{\Delta t=0}\,,$ (16)
$\displaystyle\sigma_{{f}_{\mathrm{d}}|\xi}(t$ $\displaystyle)=$ (17)
$\displaystyle\frac{1}{2\pi}$
$\displaystyle\left.\sqrt{\left(\frac{\partial}{\partial\Delta
t}{\rho}(t;\Delta t|\xi)\right)^{2}-\frac{\partial^{2}}{\partial\Delta
t^{2}}{\rho}(t;\Delta t|\xi)}\right|_{\Delta t=0}\,,$

where $\rho(t;\Delta t|\xi)$ is the inverse Fourier transform of
$p(t;{f}_{\mathrm{d}}|\xi)$ and is therefore a characteristic function in the
$\Delta t$ variable.

####  II-B3 Proportionality between the Correlation and Probability Based
Functions

The following proposition states that stochastic channel descriptions computed
based on the joint delay Doppler pdf, as shown in Fig. 1, are proportional to
the corresponding correlation based functions derived by Bello and Matz in
their works. Note that the same variables are used.

######  Conjecture 1.

The autocorrelation of the time-variant transfer function $R_{L}(t;\Delta
t,\Delta\tilde{f})$ is proportional to the time-variant joint time frequency
characteristic function $r(t;\Delta\tilde{f},\Delta t)$

$R_{L}(t;\Delta t,\Delta\tilde{f})\propto r(t;\Delta\tilde{f},\Delta t)\,.$

Furthermore, due to the linearity of Fourier transforms, the above stated
proportionality applies for each pair of correlation and probability based
functions as outlined in Table I.

###### Proof.

We follow here the steps similar to those in [10]. The starting point is the
assumption that for the time $t$ the channel can be represented as a linear
combination of $K(t)$ propagation paths

$h(t,\tau)=\sum_{k=0}^{K(t)-1}\alpha_{k}(t)\mathrm{e}^{-\mathrm{j}2\pi
f_{c}\tau_{k}(t)}\delta(t-\tau_{k}(t))\,.$ (18)

Here $\alpha_{k}(t)$ is the complex path weight, $\tau_{k}(t)$ is the time-
varying propagation delay, and $f_{c}$ is the carrier frequency. By taking the
Fourier transform over the delay, a time-variant transfer function can be
constructed as

$L(t,f)=\sum_{k=0}^{K(t)-1}\alpha_{k}(t)\mathrm{e}^{-\mathrm{j}2\pi(f+f_{c})\tau_{k}(t)}\,.$
(19)

We approximate the time-varying channel with a piece-wise linear
approximation. This is done by assuming that for a moment of time
$t=t^{\prime}$ the time-dependent path propagation delay can be locally, over
the interval $\Delta t$, approximated with a MacLauren series. Thus, we can
represent a time-varying delay as

$\tau_{k}(t)=\sum_{n=0}^{\infty}\left.\frac{1}{n!}\frac{\mathrm{d}\tau_{k}(t)}{\mathrm{d}t}\right|_{t=t^{\prime}}t^{n}\approx\,\tilde{\tau}_{k}(t^{\prime})+t\left.\frac{\mathrm{d}\tau_{k}(t)}{\mathrm{d}t}\right|_{t=t^{\prime}}\,.$
(20)

Note that in general, $\tilde{\tau}_{k}(t^{\prime})$ is constant over the
assumed interval $\Delta t$. The derivative
$\left.\frac{\mathrm{d}\tau_{k}(t)}{\mathrm{d}t}\right|_{t=t^{\prime}}$ is
also constant over this interval. These parameters characterize the intercept
and local linear trend of the time-varying delay $\tau_{k}(t)$ at a point
$t^{\prime}$. They do, however, change with time $t$, yet at a lower rate. In
other words, they are piece-wise constant functions of $t$. In the following,
we will make the dependency of these variables on $t$ explicit, keeping the
piece-wise constant nature of these variables in mind.

We note that under the narrow-band assumption, the Doppler frequency
${f}_{\mathrm{d}}(t)$ can be defined as ${f}_{\mathrm{d}}(t)\triangleq-
f_{\mathrm{c}}\,\frac{\mathrm{d}\tau(t)}{\mathrm{d}t}$, i.e., when all
transmitted frequencies experience the same Doppler shift [11]. Using (20) the
time-varying transfer function $L(t,f)$ can be approximated as

$\displaystyle
L(t,f)\approx\sum_{k=0}^{K(t)-1}\tilde{\alpha}_{l}(t)\mathrm{e}^{\mathrm{j}2\pi{f}_{\mathrm{d},k}(t)t}\mathrm{e}^{-\mathrm{j}2\pi
f\tilde{\tau}_{k}(t)}\,,$ (21)

where $\tilde{\alpha}_{k}(t)=\alpha_{k}(t)\mathrm{e}^{-\mathrm{j}2\pi
f_{c}\tilde{\tau}_{k}(t)}$. For the correlation function, we get with
$\xi=\tau/\tau_{\mathrm{los}}$ as shown in [10] and [11] the following

$\displaystyle R_{L}(t,\Delta t,\Delta f\tau_{\mathrm{los}}$ $\displaystyle)=$
$\displaystyle|\tilde{\alpha}(t)|^{2}\mathbb{E}\left\\{\mathrm{e}^{\mathrm{j}2\pi{f}_{\mathrm{d},k}(t)\Delta
t}\mathrm{e}^{-\mathrm{j}2\pi\Delta
f\tau_{\mathrm{los}}(t)\xi_{k}(t)}\right\\}\,,$

where $\tau_{\mathrm{los}}(t)$ is the piece-wise constant LOS delay and the
expectation is taken with respect to the joint distribution
$p(t;\xi,{f}_{\mathrm{d}})$. The result of the latter is the joint
characteristic function $r(t;\Delta\tilde{f},\Delta t)$ as given in (12).
Thus, the correlation function $R_{L}$ is proportional to the joint
characteristic function $r(t;\Delta\tilde{f},\Delta t)$. This relationship
between correlation function and characteristic functions is similarly shown
in [20] if complex exponentials are used for the channel representation. Let
us stress again that the proportionality is valid for an arbitrary, but fixed
$t=t^{\prime}$ and as $\Delta t\to 0$. ∎

####  II-B4 Doppler Correlated Functions

The Doppler correlated functions in Fig. 2 constitute for
$\Delta{f}_{\mathrm{d}}=0$ a temporal average of the functions in the upper
half. The hybrid Doppler delay characteristic pdf
$\rho(\Delta{f}_{\mathrm{d}};\xi,{f}_{\mathrm{d}})$ for example is calculated
as

$\rho(\Delta{f}_{\mathrm{d}};\xi,{f}_{\mathrm{d}})\triangleq\int
p(t;\xi,{f}_{\mathrm{d}})\mathrm{e}^{-\mathrm{j}2\pi\Delta{f}_{\mathrm{d}}t}\,\mathrm{d}t\,,$
(23)

with $\rho(\Delta{f}_{\mathrm{d}}=0;\xi,{f}_{\mathrm{d}})$ being the temporal
mean of the joint delay Doppler probability density function due to the
Fourier properties. This was already implicitly used in describing V2V
scenarios such as two cars driving in opposite directions in [21]. We
normalize the functions in such a way, that the joint pdf
$p(t;\xi,{f}_{\mathrm{d}})$ is a time-variant probability density in the
variables $\xi$ and ${f}_{\mathrm{d}}$. Thus, the time-variant joint
characteristic function $r(t;\Delta f=0,\Delta t=0)=1$. The lower half
functions with $\Delta{f}_{\mathrm{d}}$ deviate from pdfs or characteristic
functions by a factor of $T=1/\Delta{f}_{\mathrm{d}}$. The channel correlation
function $\tilde{\mathcal{A}}_{\mathbf{H}}(\Delta t;\Delta f,\Delta\nu)$ by
Matz thus corresponds to
$\mathcal{R}(\Delta{f}_{\mathrm{d}};\Delta\tilde{f},\Delta t)$. We will focus
our attention in the remaining paper on the time-variant functions since a
time-variant joint pdf as a basis of the description seems more natural with
time-variant trajectories of TX and RX as input to our model.

Finally, as illustrated in Table I, we note that the probability based
functions $r$ and $\rho$ correspond to the correlation functions $R$ and $P$
of Bello, respectively. Further, the joint pdf $p$ corresponds to the time-
variant LSF $\tilde{\mathcal{C}}_{\mathbf{H}}$ and Doppler correlated time
frequency characteristic function $\mathcal{R}$ to the time-variant channel
correlation function $\tilde{\mathcal{A}}_{\mathbf{H}}$. For completeness, we
further presented two new hybrid functions $\varrho$ with mixed variables in
both time-variant and Doppler correlated domains.

##  III Derivation of Hybrid Characteristic PDF

In order to obtain an analytical closed-form solution for the hybrid
characteristic pdf, we have to transform the spatial coordinates into an
adequate coordinate system. We have shown in [9], [22], [18] that a prolate
spheroidal coordinate system is suitable for this purpose. The prolate
spheroidal coordinate system (PSCS) allows for a delay-dependent description
of the M2M channel by exploiting the symmetry of the channel by an ellipsoid-
based delay description. We shortly summarize the corresponding formal steps
from [18] and introduce the coordinate system.

###  III-A Prolate Spheroidal Coordinates

The transformation between the Cartesian coordinate system (CCS) $(x,y,z)$ and
the PSCs $(\xi,\eta,\vartheta)$ is given by the following equations

$\displaystyle x$ $\displaystyle=$ $\displaystyle
l\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}\cos\vartheta\,,$ (24)
$\displaystyle y$ $\displaystyle=$ $\displaystyle
l\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}\sin\vartheta\,,$
$\displaystyle z$ $\displaystyle=$ $\displaystyle l\xi\eta\,,$

where $\xi\in[1,\infty)$, $\eta\in[-1,1]$, $\vartheta\in[0,2\pi)$ are the new
coordinates and $l$ in (24) is the focus distance of both TX and RX to the
origin of the Cartesian and the prolate spheroidal coordinate system. The
coordinate $\xi$ represents the constant distance or delay, respectively,
between TX and RX via a single-bounce reflection. Geometrically, this
relationship is represented by an ellipsoid.

Consider a scattering plane via which a signal propagates to the receiver. An
arbitrarily oriented scattering plane is given in Cartesian coordinates as

$\displaystyle Ax+By+Cz=lD\,,$ (25)

where the four parameters $\\{A,B,C,D\\}\in\mathbb{R}$ determine its
orientation in space. For our purposes we express (25) in PSCs, which results
in

$\displaystyle Al\sqrt{(\xi^{2}-1)(1-\eta^{2})}\cos\vartheta$ $\displaystyle+$
$\displaystyle Bl\sqrt{(\xi^{2}-1)(1-\eta^{2})}\sin\vartheta$ (26)
$\displaystyle+$ $\displaystyle Cl\xi\eta=lD\,.$

The scattering plane, as any 2D plane embedded in 3D space, can be
parameterized by two independent variables in the selected coordinate system.
Our goal is to obtain a parameterization that allows for a closed-form
derivation of the hybrid time delay characteristic pdf. Since we want $\xi$
for a delay-dependent description, we can choose either $\eta$ or $\vartheta$
as the second variable. In fact, we need both the $(\xi,\eta)$ and
$(\xi,\vartheta)$ parameterizations to cover all possible scattering planes in
3D space. Our main parameterization, however, is in $(\xi,\eta)$-coordinates.
The remaining scattering planes, which cannot be parameterized by $(\xi,\eta)$
since they are orthogonal to the $z$-axis in the local CCS, are described by
the $(\xi,\vartheta)$-coordinates. The $(\xi,\vartheta)$ description actually
complements the $(\xi,\eta)$ parameterization. In the following, we refer to
these cases as _general case_ and _complementary case_ , respectively.

###  III-B Spatial Probability Density

In order to obtain the joint delay Doppler pdf we restrict our analysis to
scatterers lying on the scattering plane. We consider scatterers that lie on
the portion of the scattering plane circumscribed by the intersection ellipse.
The resulting scattering ellipse can be generally described by the implicit
expression $q(\xi,\eta,\vartheta)=0$, which simplifies to $q(\xi,\eta)=0$, if
the parameterization is in $(\xi,\eta)$-coordinates or $q(\xi,\vartheta)=0$,
if the parameterization is in $(\xi,\vartheta)$-coordinates. We assume that
the scatterers lying within $q(\xi,\eta,\vartheta)=0$ are identical and
uniformly distributed. Thus, the two-dimensional density $\mathbf{s}$ of the
scatterers within the scattering ellipse is modeled as

$p(t,q(\xi,\eta,\vartheta);\mathbf{s})=\frac{1}{\mathcal{Y}}\,,$ (27)

where $\mathcal{Y}$ is the equivalent area of the ellipse
$q(\xi,\eta,\vartheta)=0$.

The joint delay Doppler pdf is then obtained by transforming the distribution
of scatterers $\mathbf{s}$ into $(\xi,{f}_{\mathrm{d}})$-coordinates using
(III-C) and rules of probability transformation as

$p\left(t,q(\xi,\eta,\vartheta);\xi,{f}_{\mathrm{d}}\right)=p\left(t,q(\xi,\eta,\vartheta);\mathbf{s}\right)\left|\mathbf{J_{s}}^{-1}\right|\,,$
(28)

where $\mathbf{J_{s}}^{-1}$ is the inverse $2\times 2$ Jacobian matrix of the
variable transformation.

For the joint delay Doppler pdf, the transformation from the spatial domain to
the Doppler domain, i.e., $s\mapsto{f}_{\mathrm{d}}$ or
$\mathbf{s}\mapsto(\xi,{f}_{\mathrm{d}})$ introduces ambiguities in the
mapping. These ambiguities, however, can be resolved by applying the algebraic
curve theory to the Doppler frequency description, see also [22]. Furthermore,
the locations of the extrema and thus the limiting frequencies of the pdfs can
be determined.

In order to use a spatial distribution of the scatterers, we need to calculate
the area enclosed by the delay ellipsoid. Additionally, we need a weighting
function $w(\xi,\eta)$ that takes into account the path loss, which follows
from the radar equation [23]

$\displaystyle w(\xi,\eta)=\frac{1}{\left(\xi^{2}-\eta^{2}\right)^{2}}\,.$
(29)

This essentially states that the received power is proportional to the squared
distances from the scatterer to TX and RX as
$P\propto\left(d_{\mathrm{t}}^{2}d_{\mathrm{r}}^{2}\right)^{-1}$.

For the _general case_ , we obtain the following equation for the weighted
elliptic area

$\displaystyle\mathcal{Y}_{1}=$ (30)
$\displaystyle\int\limits_{\xi_{\mathrm{min}}}^{\xi_{\mathrm{max}}}2\int\limits_{\eta_{1}\left(\xi\right)}^{\eta_{2}\left(\xi\right)}\frac{w\left(\xi,\eta\right)l^{2}\sqrt{A^{2}+B^{2}+C^{2}}\left(\xi^{2}-\eta^{2}\right)\,\mathrm{d}\eta\mathrm{d}\xi}{\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)\left(A^{2}+B^{2}\right)-\left(D-C\xi\eta\right)^{2}}}\,,$

where $\xi_{\mathrm{max}}>\xi_{\mathrm{min}}>\xi_{\mathrm{sr}}$ are the
minimum and maximum normalized delay, which can be set by the user. The other
parameters $\eta_{2}(\xi)>\eta_{1}(\xi)$ are given by (45).

For the _complementary case_ we obtain a weighted circular area as a special
case of the elliptic area, since the semi-major axis of the delay ellipsoid is
orthogonal to the scattering plane. It is given by

$\mathcal{Y}_{2}=\iint\limits_{q(\xi,\eta,\vartheta)=0}w\left(\xi,\frac{D}{C\xi}\right)\,\mathrm{d}S_{1}=\int\limits_{\xi_{\mathrm{min}}}^{\xi_{\mathrm{max}}}\int\limits_{0}^{2\pi}\frac{l^{2}\left(\xi-\frac{D^{2}}{C^{2}\xi^{3}}\right)}{\left(\xi^{2}-\left(\frac{D}{C\xi}\right)^{2}\right)^{2}}\,\mathrm{d}\vartheta\mathrm{d}\xi,$
(31)

where $\mathrm{d}S_{1}=\mathrm{d}\vartheta\mathrm{d}\xi$ is the differential
scattering area with $\xi_{\mathrm{max}}>\xi_{\mathrm{min}}>\xi_{\mathrm{sr}}$
defined similarly to the _general case_.

For deriving the hybrid time delay characteristic pdf, we use the spatial
density of the scatterers instead of transforming the Doppler frequency
${f}_{\mathrm{d}}$, as was done in previous works, e.g., [18, 22, 10]. We thus
either transform over the variable $\eta$ for the _general case_ or over the
variable $\vartheta$ for the _complementary case_.

$\eta_{\mathrm{1},\mathrm{2}}(\xi)=\frac{DC\xi\pm\sqrt{D^{2}C^{2}\xi^{2}-(A^{2}\xi^{2}+B^{2}\xi^{2}+C^{2}\xi^{2}-A^{2}-B^{2})(A^{2}+B^{2}+D^{2}-A^{2}\xi^{2}-B^{2}\xi^{2})}}{A^{2}\xi^{2}+B^{2}\xi^{2}+C^{2}\xi^{2}-A^{2}-B^{2}}$
(45) $\displaystyle
f_{\mathrm{d},i}^{\star}(t;\xi,\eta)=\frac{1}{\left(A^{2}+B^{2}\right)\left(\xi^{2}-\eta^{2}\right)}\Bigg{(}\left(D-C\xi\eta\right)\left(A\left({v}_{\mathrm{r}x}\left(\xi+\eta\right)+{v}_{\mathrm{t}x}\left(\xi-\eta\right)\right)+B\left({v}_{\mathrm{r}y}\left(\xi+\eta\right)+{v}_{\mathrm{t}y}\left(\xi-\eta\right)\right)\right)$
$\displaystyle\pm\sqrt{\left(\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)\left(A^{2}+B^{2}\right)-\left(D-C\xi\eta\right)^{2}\right)\left(B\left({v}_{\mathrm{r}x}\left(\xi+\eta\right)+{v}_{\mathrm{t}x}\left(\xi-\eta\right)\right)-A\left({v}_{\mathrm{r}y}\left(\xi+\eta\right)+{v}_{\mathrm{t}y}\left(\xi-\eta\right)\right)\right)^{2}}$
$\displaystyle+\left(A^{2}+B^{2}\right)\left({v}_{\mathrm{r}z}\left(\xi\eta-1\right)\left(\xi+\eta\right)+{v}_{\mathrm{t}z}\left(\xi\eta+1\right)\left(\xi-\eta\right)\right)\Bigg{)}\frac{f_{\mathrm{c}}}{c}$
(46)

Note that we do not provide the derivation with the simpler delay-dependent
description based on the length of the intersection as in previous
publications. Thus, we present the more realistic case of the area of the
intersection ellipse, where the differential scatterers have a two-dimensional
displacement.

###  III-C Hybrid Time Delay Characteristic Probability Density

In this subsection, we derive the hybrid characteristic pdf $\rho(t;\xi,\Delta
t)$ in delay $\xi$ and time lag $\Delta t$ domains for general M2M scattering
channels as discussed in Section II. Since our starting point is the joint
delay Doppler pdf, we obtain the hybrid characteristic probability density
function by an inverse Fourier transform in the Doppler frequency variable. We
show that in the limiting case, those newly derived hybrid characteristic pdfs
converge to known results of correlation functions in the literature.

We obtain the hybrid characteristic pdfs for the _general case_ , as it was
defined above, by using the spatial variable $\eta$ instead of the Doppler
frequency ${f}_{\mathrm{d}}$. By using relationship (28), we obtain with
(III-B)

$\displaystyle\rho(t;\xi,\Delta t)=\sum_{i=1}^{2}\int
p\left(t,q(\xi,\eta,\vartheta);\xi,{f}_{\mathrm{d}}\right)\mathrm{e}^{\mathrm{j}2\pi\Delta
tf_{\mathrm{d},i}^{\star}}\,\mathrm{d}f_{\mathrm{d},i}^{\star}$
$\displaystyle=\sum_{i=1}^{2}\int
p\left(t,q(\xi,\eta,\vartheta);\xi,{f}_{\mathrm{d}}(\eta)\right)\mathrm{e}^{\mathrm{j}2\pi\Delta
tf_{\mathrm{d},i}^{\star}(\eta)}\left|\mathbf{J_{s}}\right|\,\mathrm{d}\eta\,.$
(32)

Thus, we can directly insert the weighted spatial scatterer density
$p(t;\xi,\eta)$ and perform the inverse Fourier function over $\eta$ as

$\displaystyle\rho(t;\xi,\Delta t)=\frac{1}{\mathcal{Y}_{1}}\sum_{i=1}^{2}$
(33)
$\displaystyle\int\limits_{\eta_{1}\left(\xi\right)}^{\eta_{2}\left(\xi\right)}\frac{w\left(\xi,\eta\right)l^{2}\sqrt{A^{2}+B^{2}+C^{2}}\left(\xi^{2}-\eta^{2}\right)\mathrm{e}^{\mathrm{j}2\pi\Delta
tf_{\mathrm{d},i}^{\star}(t;\xi,\eta)}}{\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)\left(A^{2}+B^{2}\right)-\left(D-C\xi\eta\right)^{2}}}\mathrm{d}\eta\,,$

with the time-variant Doppler frequency $f_{\mathrm{d},i}^{\star}(t;\xi,\eta)$
according to (III-B) and the integral limits $\eta_{1}(\xi)$ and
$\eta_{2}(\xi)$ according to (45) with $\eta_{2}(\xi)>\eta_{1}(\xi)$.

For the _complementary case_ we insert the weighted circular area given in
(31) and can calculate the hybrid characteristic pdf by an inverse Fourier
transform of the Doppler variable ${f}_{\mathrm{d}}$ as

$\displaystyle\rho(t;\xi,\Delta
t)=\frac{1}{\mathcal{Y}_{2}}\int\limits_{0}^{2\pi}\frac{l^{2}\left(\xi-\frac{D^{2}}{C^{2}\xi^{3}}\right)}{\left(\xi^{2}-\left(\frac{D}{C\xi}\right)^{2}\right)^{2}}\mathrm{e}^{\mathrm{j}2\pi{f}_{\mathrm{d}}\left(t;\xi,\vartheta\right)\Delta
t}\,\mathrm{d}\vartheta$
$\displaystyle=\frac{2l^{2}\left(\xi-\frac{D^{2}}{C^{2}\xi^{3}}\right)}{\mathcal{Y}_{2}\left(\xi^{2}-\left(\frac{D}{C\xi}\right)^{2}\right)^{2}}\int\limits_{-f_{\mathrm{l}}}^{f_{\mathrm{l}}}\frac{\mathrm{e}^{\mathrm{j}2\pi{f}_{\mathrm{d}}\Delta
t}}{f_{\mathrm{l}}\sqrt{1-\left(\frac{{f}_{\mathrm{d}}-f_{\mathrm{o}}}{f_{\mathrm{l}}}\right)^{2}}}\,\mathrm{d}{f}_{\mathrm{d}}$
(34) $\displaystyle=\frac{1}{\mathcal{Y}_{2}}\frac{2\pi
l^{2}\left(\xi-\frac{D^{2}}{C^{2}\xi^{3}}\right)}{\left(\xi^{2}-\left(\frac{D}{C\xi}\right)^{2}\right)^{2}}\mathrm{J}_{0}\left(2\pi
f_{\mathrm{l}}(t;\xi)\Delta t\right)\mathrm{e}^{\mathrm{j}2\pi
f_{\mathrm{o}}(t;\xi)\Delta t}\,$

where $\mathrm{J}_{0}$ is a zeroth-order Bessel function of the first kind,

$f_{\mathrm{o}}(t;\xi)=\frac{f_{\mathrm{c}}}{c}\left(\frac{\textstyle{\frac{D}{C}}+1}{\xi+\textstyle\frac{D}{C\xi}}{v}_{\mathrm{t}z}+\frac{\textstyle{\frac{D}{C}}-1}{\xi-\textstyle\frac{D}{C\xi}}{v}_{\mathrm{r}z}\right)\,,$
(35)

is the offset frequency caused by the movement of TX and RX along the
$z$-axis, and

$\displaystyle f_{\mathrm{l}}($ $\displaystyle t$
$\displaystyle;\xi)=\frac{f_{\mathrm{c}}}{c}\sqrt{\left(\xi^{2}-1\right)\left(1-\textstyle\left(\frac{D}{C\xi}\right)^{2}\right)}\times$
$\displaystyle\sqrt{\left(\frac{{v}_{\mathrm{t}x}}{\xi+\textstyle\frac{D}{C\xi}}+\frac{{v}_{\mathrm{r}x}}{\xi-\textstyle\frac{D}{C\xi}}\right)^{2}+\left(\frac{{v}_{\mathrm{t}y}}{\xi+\textstyle\frac{D}{C\xi}}+\frac{{v}_{\mathrm{r}y}}{\xi-\textstyle\frac{D}{C\xi}}\right)^{2}}\,,$

is the limiting frequency. The basis for our calculations above is the Doppler
frequency in [18, (3)] as

$\displaystyle{f}_{\mathrm{d}}$
$\displaystyle(t;\xi,\eta,\vartheta)=\frac{f_{\mathrm{c}}}{c}\Bigg{(}$
$\displaystyle\frac{\xi\eta+1}{\xi+\eta}{v}_{\mathrm{t}z}+\frac{\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}}{\xi+\eta}\left({v}_{\mathrm{t}x}\cos\vartheta+{v}_{\mathrm{t}y}\sin\vartheta\right)$
$\displaystyle+$
$\displaystyle\frac{\xi\eta-1}{\xi-\eta}{v}_{\mathrm{r}z}+\frac{\sqrt{\left(\xi^{2}-1\right)\left(1-\eta^{2}\right)}}{\xi-\eta}\left({v}_{\mathrm{r}x}\cos\vartheta+{v}_{\mathrm{r}y}\sin\vartheta\right)\Bigg{)}$

where
$\mathbf{v}_{\mathrm{t}}=\left[{v}_{\mathrm{t}x},{v}_{\mathrm{t}y},{v}_{\mathrm{t}z}\right]^{\mathrm{T}}$
and
$\mathbf{v}_{\mathrm{r}}=\left[{v}_{\mathrm{r}x},{v}_{\mathrm{r}y},{v}_{\mathrm{r}z}\right]^{\mathrm{T}}$
are the velocity vectors of TX and RX in the local CCS, respectively. Since we
obtain the relationship $\eta=D/(C\xi)$ for the _complementary case_ , the
Doppler frequency reduces to ${f}_{\mathrm{d}}\left(t;\xi,\vartheta\right)$.
Note that in the _complementary case_ the delay and Doppler pdfs factor, and
thus are independent of each other.

###  III-D Limiting Value Consideration

By studying the delay-dependent Doppler pdf and the delay-dependent
characteristic function in the asymptotic regime, as $\xi\to\infty$, we obtain
several expressions that are well-known in the literature. Specifically, we
derive

$\displaystyle\lim_{\xi\to\infty}\rho(t;\Delta
t|\xi)=\mathrm{J_{0}}\left(2\pi\Delta
t\frac{\lVert\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}+\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}\rVert}{c}f_{\mathrm{c}}\right)\,,$
(38) $\displaystyle\lim_{\xi\to\infty}p(t;{f}_{\mathrm{d}}|\xi)=\frac{1}{\pi
f_{\mathrm{l}\infty}(t)\sqrt{1-\left(\frac{{f}_{\mathrm{d}}}{f_{\mathrm{l}\infty}(t)}\right)^{2}}}\,,$
(39)
$\displaystyle\lim_{\xi\to\infty}\mu_{{f}_{\mathrm{d}}|\xi}(t;{f}_{\mathrm{d}})=0\,,$
(40)
$\displaystyle\lim_{\xi\to\infty}\sigma_{{f}_{\mathrm{d}}|\xi}(t;{f}_{\mathrm{d}})=\frac{\lVert\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}+\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}\rVert}{\sqrt{2}c}f_{\mathrm{c}}=\frac{f_{\mathrm{l}\infty}(t)}{\sqrt{2}}\,,$
(41)

with parallel velocity vectors and limiting Doppler frequency given by

$\displaystyle\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}=\frac{\mathbf{n}_{\mathrm{E}}\times\left(\mathbf{v}_{\mathrm{t}}\times\mathbf{n}_{\mathrm{E}}\right)}{\lVert\mathbf{n}_{\mathrm{E}}\rVert^{2}}=\mathbf{v}_{\mathrm{t}}-\frac{\left(\mathbf{v}_{\mathrm{t}}\cdot\mathbf{n}_{\mathrm{E}}\right)\mathbf{n}_{\mathrm{E}}}{\lVert\mathbf{n}_{\mathrm{E}}\rVert^{2}}\,,$
(42)
$\displaystyle\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}=\frac{\mathbf{n}_{\mathrm{E}}\times\left(\mathbf{v}_{\mathrm{r}}\times\mathbf{n}_{\mathrm{E}}\right)}{\lVert\mathbf{n}_{\mathrm{E}}\rVert^{2}}=\mathbf{v}_{\mathrm{r}}-\frac{\left(\mathbf{v}_{\mathrm{r}}\cdot\mathbf{n}_{\mathrm{E}}\right)\mathbf{n}_{\mathrm{E}}}{\lVert\mathbf{n}_{\mathrm{E}}\rVert^{2}}\,,$
(43)
$\displaystyle\lim_{\xi\to\infty}f_{\mathrm{l}\infty}(t)=\frac{\lVert\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}+\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}\rVert}{c}f_{\mathrm{c}}\,.$
(44)

The velocity vectors $\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}$ in (42) and
$\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}$ in (43) of TX and RX are parallel
to the scattering plane. The limiting frequencies $f_{\mathrm{l}\infty}(t)$ in
(44) for $\xi\to\infty$ are given by the solution of the $\eta$ variable
$\pm\left(v_{\mathrm{t}z\parallel\mathrm{E}}+v_{\mathrm{r}z\parallel\mathrm{E}}\right)/\left(\lVert\mathbf{v}_{\mathrm{t}\parallel\mathrm{E}}+\mathbf{v}_{\mathrm{r}\parallel\mathrm{E}}\rVert\right)$
of the polynomial in [18, eq. (31)]. The result in (39) matches the classical
Jakes result. The width of the spectrum, however, is determined by the
velocity vector components of TX and RX, which are parallel to the scattering
plane. The reason for this is that for large $\xi$ the eccentricity of the
ellipsoid reduces toward $0$, thus, approaching a sphere. The intersection
with the scattering plane thus results in a scattering circle on which the
scatterers are uniformly distributed. The corresponding Fourier transform of
the delay-dependent pdf in (39) results in the typical Bessel function in (38)
as the delay-dependent characteristic function $\rho(t;\Delta t|\xi)$.

##  IV Comparative Analysis of Theory and Measurement

In our previous works [10] and [22], we primarily examined the scattering
contributions in terms of delay and Doppler frequency shift. However, the
decrease in the scattering power with increasing delay, reflecting the
influence of the channel’s power delay profile (PDP), has not yet been
addressed and verified. The typical approach to account for the PDP is to
empirically adjust the scattering power behavior using a specific path loss
exponent, e.g., as in [21]. In contrast to these empirical approaches, the
analytical description of the hybrid characteristic pdf of the channel, as
presented in this work, allows the calculation of the time-variant PDP for any
scenario, taking into account the geometry of the environment and the
velocities of the transceivers.

In the following, we examine the air-to-air (A2A) channel as a representative
example of a US M2M channel, being the most general channel where both TX and
RX are not confined to the scattering plane. We compare data obtained from a
measurement campaign [24] and the data obtained from the numerical evaluation
of the hybrid characteristic pdf from (33) from Section III.

###  IV-A Measurement and Simulation Scenario

$x$$y$$z$$\mathbf{v}_{\mathrm{t}}$$\mathbf{v}_{\mathrm{r}}$$h_{\mathrm{t}}$$h_{\mathrm{r}}$$d_{\mathrm{los}}$$0x+1y+0z=l\cdot 1.8486=$580\text{\,}\mathrm{m}$$$h_{\mathrm{t}}=h_{\mathrm{r}}=$580\text{\,}\mathrm{m}$$$d_{\mathrm{los}}=2l=$627.5\text{\,}\mathrm{m}$$$\mathbf{v}_{\mathrm{t}}={[0,0,247.3]}^{\mathrm{T}}$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$ $\mathbf{v}_{\mathrm{r}}={[0,0,245.4]}^{\mathrm{T}}$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$ Figure 3: Aircraft positions, velocity vectors, distance and placement of the scattering plane in a local coordinate system for simulation. TABLE II: Measurement and evaluation parameters lightgrayParameter Name | Variable | Value
---|---|---
Carrier frequency | $f_{\mathrm{c}}$ | $250\text{\,}\mathrm{MHz}$
Bandwidth | $B$ | $20\text{\,}\mathrm{MHz}$
Frequency resolution | $\Delta f$ | $39.1\text{\,}\mathrm{kHz}$
Normalized maximum frequency | $\tilde{f}_{\mathrm{max}}$ |  $\pm$ 20.92
Normalized frequency resolution | $\Delta\tilde{f}$ | 0.082
Signal period | $\tau_{\mathrm{max}}$ | $25.6\text{\,}\mathrm{\SIUnitSymbolMicro s}$
Delay resolution | $\Delta\tau$ | $50\text{\,}\mathrm{ns}$
Normalized signal period | $\xi_{\mathrm{max}}$ | 12.24
Normalized delay resolution | $\Delta\xi$ | 0.024
Time period | $t_{\mathrm{max}}$ | $2.1\text{\,}\mathrm{s}$
Measurement time grid | $\Delta t$ | $2.048\text{\,}\mathrm{ms}$
Max. Doppler frequency | $f_{\mathrm{d,max}}$ |  $\pm$$244\text{\,}\mathrm{Hz}$
Doppler resolution | $\Delta{f}_{\mathrm{d}}$ | $0.5\text{\,}\mathrm{Hz}$

We consider a scenario in which two aircraft fly at the same altitude above
ground and are positioned $d_{\mathrm{los}}=\tau_{\mathrm{los}}c$ behind each
other. An overview of the scenario including positions of the aircraft,
velocity vectors $\mathbf{v}_{\mathrm{t}}$ and $\mathbf{v}_{\mathrm{r}}$ in a
local CCS, and the parameters of the scattering plane for the simulation is
shown in Fig. 3.

The measurement parameters that are given by the channel sounding equipment
are provided in Tab. II. The normalized parameters $\xi$ and $\tilde{f}$ can
be derived from the physical parameters and the line-of-sight delay
$\tau_{\mathrm{los}}$.

In order to show the advantages of the probability based channel description
functions with respect to real world measurements and the corresponding
correlation functions, we first compare normalized versions of both
probability based and correlation based functions. In a second step, we use
the non-normalized probabilistic 2D functions and marginalize them to obtain
the delay and Doppler spectra and the correlation in the time and frequency
domain. We discuss the 2D functions in the same order as in Section II-B.

###  IV-B Normalized Probability and Correlation Based Functions

For continuity with our previous paper [18], we begin by examining the
factorized pdf $p(t;{f}_{\mathrm{d}}|\xi)=p(t;\xi,{f}_{\mathrm{d}})/p(t;\xi)$
in Fig. 4 and compare it with the real part of the time-variant, delay-
dependent LSF $\tilde{\mathcal{C}}_{\mathbf{H}}(t;{f}_{\mathrm{d}}|\xi)$ in
Fig. 5. Additionally, we illustrate the delay-dependent mean Doppler
$\mu_{{f}_{\mathrm{d}}|\xi}(t)$ from (16) and Doppler spread
$\sigma_{{f}_{\mathrm{d}}|\xi}(t)$ from (17) in Fig. 4. Since the spectra are
symmetric, the mean Doppler stays zero, but the Doppler spread is increasing
with delay $\xi$ and approaches, according to (41), the value
$\sigma_{{f}_{\mathrm{d}}|\xi\to\infty}(t)=$80.65\text{\,}\mathrm{Hz}$$. In
the limiting case, the delay-dependent spectrum conforms to a Jakes spectrum
according to (39), consistent with the literature. The analysis of the
theoretical results in Fig. 4 reveals that the shape and the values of both
the probability based and the correlation based functions are the same. The
scattering power in the measurement data is very weak and close to the noise
threshold. Furthermore, the scattering does not occur uniformly on the ground
as in our assumption. Thus, we can observe gaps in Fig. 5. This will lead to
slight differences in the marginalized one-dimensional functions. The
structure of the measured channel, however, is well captured with the delay-
dependent Doppler pdf $p(t;{f}_{\mathrm{d}}|\xi)$.

Figure 4: Theoretical time-variant, delay-dependent Doppler pdf
$p(t;{f}_{\mathrm{d}}|\xi)$ with mean Doppler $\mu_{{f}_{\mathrm{d}}|\xi}(t)$
and Doppler spread $\sigma_{{f}_{\mathrm{d}}|\xi}(t)$. Figure 5: Measured
time-variant, delay-dependent local scattering function
$\Re\left\\{\tilde{\mathcal{C}}_{\mathbf{H}}(t;{f}_{\mathrm{d}}|\xi)\right\\}$.

We continue our comparison with the real part of the delay-dependent hybrid
time delay characteristic pdf $\rho(t;\Delta t|\xi)$ in Fig. 6. The real part
of the normalized temporally correlated delay-dependent delay cross power
density $P_{h}(t;\Delta t|\xi)$ is given in Fig. 7. Both theoretical results
and measurement data demonstrate a strong agreement in the temporal
correlation of the channel. The correlation decreases noticeably with
increasing delay. For large delays, $\xi\to\infty$, the delay-dependent
characteristic function converges to a Bessel function as described in (38)
aligning well with theoretical expectations. Since both the delay-dependent
Doppler pdf and delay-dependent, time-variant LSF get wider with increasing
delay, the correlation in the $\Delta t$ variable naturally decreases, as can
be seen in Figs. 4 and 5. The influence of the LOS signal and the specular
reflection (SR) reflection, which was eliminated from the measurement data for
comparison reasons, is still slightly observable for delays $\xi$ close to
$\xi_{\mathrm{sr}}=2.1$ according to (15).

Figure 6: Theoretical time-variant, delay-dependent temporal characteristic
function $\Re\left\\{\rho(t;\Delta t|\xi)\right\\}$. Figure 7: Measured time-
variant, delay-dependent temporal correlation function
$\Re\left\\{P_{h}(t;\Delta t|\xi)\right\\}$.

Next, we compare the real parts of the newly introduced time-variant hybrid
Doppler-dependent frequency characteristic pdf
$\varrho(t;\Delta\tilde{f}|{f}_{\mathrm{d}})$ with
$R_{\varrho}(t;\Delta\tilde{f}|{f}_{\mathrm{d}})$. The time-variant hybrid
frequency Doppler characteristic pdf
$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$ is thus divided by its Doppler
spectral density $p(t;{f}_{\mathrm{d}})$. Therefore the correlation in
normalized frequency lag $\Delta\tilde{f}$ becomes visible. The theoretical
results in Fig. 8 show that the correlation is largest for the Doppler
frequency ${f}_{\mathrm{d}}=$0\text{\,}\mathrm{Hz}$$. Both with increasing and
decreasing Doppler frequency, the correlation symmetrically diminishes along
the frequency axis. The correlation of the measurement data in Fig. 9 shows a
similar behavior.

Figure 8: Theoretical time-variant Doppler-dependent frequency characteristic
function $\Re\left\\{\varrho(t;\Delta\tilde{f}|{f}_{\mathrm{d}})\right\\}$.
Figure 9: Measured time-variant Doppler-dependent frequency correlation
function
$\Re\left\\{R_{\varrho}(t;\Delta\tilde{f}|{f}_{\mathrm{d}})\right\\}$.

Finally, Fourier transforms of both hybrid characteristic pdfs lead to the
joint time-variant characteristic function. The real part of the time-variant
joint time frequency characteristic function $r(t;\Delta\tilde{f},\Delta t)$
and the real part of the time frequency correlation function
$\tilde{R}_{L}(t;\Delta\tilde{f},\Delta t)$ are shown in Fig. 10 and Fig. 11.
They both have a peak at zero time and zero frequency shift. Along the time
and frequency axes, both functions further exhibit the typical decreasing
correlation behavior.

Figure 10: Theoretical time-variant, joint time frequency characteristic
function $\Re\left\\{r(t;\Delta\tilde{f},\Delta t)\right\\}$. Figure 11:
Measured time-variant, joint time frequency correlation function
$\Re\left\\{\tilde{R}_{L}(t;\Delta\tilde{f},\Delta t)\right\\}$.

###  IV-C Time-Variant Probability Based Functions

In this subsection, we examine the full time-variant 2D probabilistic
description of the channel. In reference to Fig. 2, the block diagram in Fig.
12 shows the relationship between the different channel descriptions and their
corresponding mutual transformations.

We begin with the joint delay Doppler pdf $p(t;\xi,{f}_{\mathrm{d}})$ in Fig.
12(I). Note the similarity to the conditional pdf in Fig. 4. Yet, the key
difference is the obvious drop of the probability mass, or equivalently signal
power, with increasing delay. This drop of probability with increasing delay,
as captured by the marginal $p(t;\xi)$ in Fig. 12(i), is explicitly
transferred to the hybrid time delay characteristic pdf $\rho(t;\xi,\Delta
t)$, see Fig. 12(II). Notably, the joint pdf descriptions correctly account
for the weighting of the functions in the delay domain. A comparable weighting
occurs in the Doppler domain with the Doppler pdf $p(t;{f}_{\mathrm{d}})$
affecting $\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$ as shown in Fig.
12(III). Naturally, the joint characteristic function
$r(t;\Delta\tilde{f},\Delta t)$ in Fig. 12(IV) accounts for both of these
weightings implicitly through the Fourier transform.

The clear advantage of these joint descriptions is their computability from
environmental models and location information of transceivers, as demonstrated
in [25]. Moreover, they enable the calculation of four time-varying
marginalized descriptions: the delay pdf $p(t;\xi)$, the Doppler pdf
$p(t;{f}_{\mathrm{d}})$, as well as the temporal characteristic function
$r(t;\Delta t)$ and the frequency characteristic function
$r(t;\Delta\tilde{f})$ following the properties of pdfs or corresponding
characteristic functions. In Fig. 12, the respective relationships of these
marginalized descriptions are depicted by arrows.

The time-variant probability densities $p(t;\xi)$ and $p(t;{f}_{\mathrm{d}})$
are computed by integrating the joint delay Doppler pdf
$p(t;\xi,{f}_{\mathrm{d}})$ of Fig. 12(I), where the integration variable is
${f}_{\mathrm{d}}$ for $p(t;\xi)$ and $\xi$ for $p(t;{f}_{\mathrm{d}})$.
Alternatively, the same result can be obtained by setting the Delta variables
$\Delta t=0$ in $\rho(t;\xi,\Delta t)$, Fig. 12(II) and $\Delta\tilde{f}=0$ in
$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$, Fig. 12(III), respectively.
This is a general property of a characteristic function. As illustrated in
Figs. 12(i)-(ii), the analytically computed pdfs $p(t;\xi)$ and
$p(t;{f}_{\mathrm{d}})$ align remarkably well with those derived from
measurement data. Note again, as mentioned above, that in order to reveal the
scattering in the channel, the signal components from LOS and SR are
eliminated. Therefore, the decreasing behavior of the scatter channel caused
by path loss can be clearly observed in Fig. 12(i) and is well reflected by
the delay pdf. Regarding the Doppler pdf shown in Fig. 12(ii), it is evident
that the shape deviates from the traditional Jakes spectrum. While the
theoretical model shows a concave shape, the measurements exhibit a higher
probability at zero Doppler due to the imperfect elimination of LOS and SR,
along with a slight increase at the limiting Doppler frequencies.

Finally, let us analyze the two time-variant characteristic functions
$r(t;\Delta\tilde{f})$ and $r(t;\Delta t)$. The former, shown in Fig. 12(iii),
can be obtained via marginalization of the hybrid frequency Doppler
characteristic pdf $\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$ or by setting
$\Delta t=0$ in $r(t;\Delta\tilde{f},\Delta t)$. The temporal characteristic
function, shown in Fig. 12(iv), $r(t;\Delta t)$ can be similarly computed from
$\rho(t;\xi,\Delta t)$ or $r(t;\Delta\tilde{f},\Delta t)$. We observe that the
zero crossings and sidelobes of the theoretical curves closely match those
computed from measurement data both in Fig. 12(iii) and Fig. 12(iv). Further,
we can determine the coherence bandwidth as a solution to
$\Re\left\\{r(t;\Delta\tilde{f}=0,\Delta t)\right\\}=1/2$. The normalized
coherence bandwidth is about $B_{\mathrm{C}}=0.126$, which corresponds to a
physical bandwidth of $B_{\mathrm{C^{\prime}}}=$60.239\text{\,}\mathrm{kHz}$$.
Note that in Fig. 12(iv) the empirical evaluations show a slight elevation of
the sidelobes. This discrepancy can again be attributed to the imperfect
elimination of the LOS and SR components. Equivalently to the coherence
bandwidth, we derive the channel coherence time as a solution to
$\Re\left\\{r(t;\Delta\tilde{f},\Delta t=0)\right\\}=1/2$, resulting in
$T_{\mathrm{C}}=$6.4\text{\,}\mathrm{ms}$$.

(III)Y1.0e0]pt$\varrho(t;\Delta\tilde{f},{f}_{\mathrm{d}})$(II)Y1.0e0]pt$\rho(t;\xi,\Delta
t)$(I)Y1.0e0]pt$p(t;\xi,{f}_{\mathrm{d}})$(IV)Y1.0e0]pt$r(t;\Delta\tilde{f},\Delta
t)$(i)Y1.0e0]pt$p(t;\xi)$ $\tilde{P}_{h}(t;\xi)$
(ii)Y1.0e0]pt$p(t;{f}_{\mathrm{d}})$
$\tilde{P}_{\varrho}(t;{f}_{\mathrm{d}})$(iv)Y1.0e0]pt$r(t;\Delta t)$
$\tilde{R}_{L}(t;\Delta t)$(iii)Y1.0e0]pt$r(t;\Delta\tilde{f})$
$\tilde{R}_{L}(t;\Delta\tilde{f})$$\Delta\tilde{f}$$\xi$${f}_{\mathrm{d}}$$\Delta
t$${f}_{\mathrm{d}}$$\Delta
t$$\Delta\tilde{f}$$\xi$$\Delta\tilde{f}$$\xi$${f}_{\mathrm{d}}$$\Delta
t$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}\xi\,$$\Delta\tilde{f}=0$$\Delta\tilde{f}=0$$\Delta t=0$$\Delta
t=0$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}{f}_{\mathrm{d}}\,$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}{f}_{\mathrm{d}}\,$$\displaystyle\int\boldsymbol{\cdot}\hskip
1.0pt\mathrm{d}\xi\,$

Figure 12: Time-variant probability based functions from Fig. 2 for an
aircraft-to-aircraft scenario with plane parameters $A=0$, $B=1$, $C=0$,
$D=1.8486$, and $l=$313.75\text{\,}\mathrm{m}$$, with velocity vectors of the
transmitter
$\mathbf{v}_{\mathrm{t}}={[0,0,247.3]}^{\mathrm{T}}$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$
and the receiver
$\mathbf{v}_{\mathrm{r}}={[0,0,245.4]}^{\mathrm{T}}$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h}$$
according to Fig. 3.

##  V Conclusion

We have presented a complete analytic probability based description of the
mobile-to-mobile uncorrelated scatter channel. The probability based
description is proportional to the correlation based description introduced by
Bello for wide-sense stationary, uncorrelated scattering channels and by Matz
for the general case.

A new set of functions, which we term the hybrid characteristic probability
density function is introduced and derived. These functions have hybrid
properties in the sense that in one variable they behave like a probability
density function, while in the other they act as a characteristic function.
The hybrid time delay characteristic probability density functions are shown
to be proportional to the temporally correlated delay cross-power spectral
density as introduced by Bello. The complete two-dimensional description
allows for the scattering-based path loss to be naturally included in the
model.

The verification of the probability based description is done by using the
measurement data from an air-to-air measurement campaign. This scenario is the
most general mobile-to-mobile channel, since the transmitter and receiver are
arbitrarily located in 3D space. Through appropriate normalization of both
probability based and correlation based functions, the two approaches can be
directly compared without determining the proportionality constant. The
comparison shows that the proposed new probabilistic description closely
aligns with the measurements.

## Acknowledgment

Special thanks go to Rohde $\&$ Schwarz GmbH $\&$ Co. KG. for providing the
air-to-air measurement data.

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  *[M2M]: mobile-to-mobile
  *[V2V]: vehicle-to-vehicle
  *[WSSUS]: wide-sense stationary uncorrelated scattering
  *[LSF]: local scattering function
  *[CCF]: channel correlation function
  *[WSS]: wide-sense stationary
  *[US]: uncorrelated scattering
  *[PSC]: prolate spheroidal coordinate
  *[pdf]: probability density function
  *[OTFS]: orthogonal time frequency space
  *[ISAC]: integrated sensing and communications
  *[F2M]: fixed-to-mobile
  *[TX]: transmitter
  *[RX]: receiver
  *[LOS]: line-of-sight
  *[pdfs]: probability density function
  *[PSCS]: prolate spheroidal coordinate system
  *[CCS]: Cartesian coordinate system
  *[PSCs]: prolate spheroidal coordinate
  *[PDP]: power delay profile
  *[A2A]: air-to-air
  *[SR]: specular reflection