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Transactions on Networks and Communications - Vol. 9, No. 3
Publication Date: June, 25, 2021
DOI: 10.14738/tnc.93.10214.
Neelakanta, P., & Groff, D. D. (2021). Conceiving Inferential Prototypes of MIMO Channel Models via Buckingham’s Similitude
Principle for 30+ GHz through THz Spectrum. Transactions on Networks and Communicaitons, 9(3). 1-35.
Services for Science and Education – United Kingdom
Conceiving Inferential Prototypes of MIMO Channel Models via
Buckingham’s Similitude Principle for 30+ GHz through THz
Spectrum
Perambur Neelakanta
Department of Computer and Electrical Engineering and Computer Science
Florida Atlantic University, Boca Raton, FL 33431, United States
Dolores De Groff
Department of Computer and Electrical Engineering and Computer Science
Florida Atlantic University, Boca Raton, FL 33431, United States
ABSTRACT
Facilitating newer bands of ‘unused’ segments (windows) of RF spectrum falling in
the mm-wave range (above 30+ GHz) and seeking usable stretches across
unallocated THz spectrum, could viably be considered for Multiple Input Multiple
Output (MIMO) communications. This could accommodate the growing needs of
multigigabit 3G/4G applications in outdoor-based backhauls in picocellular
networks and in indoor-specific multimedia networking. However, in contrast with
cellular and Wi-Fi, wireless systems supporting sub-mm wavelength transreceive
communications in the outdoor electromagnetic (EM) ambient could face
“drastically different propagation geometry”; also, in indoor contexts, envisaging
pertinent spatial-multiplexing with directional, MIMO links could pose grossly
diverse propagation geometry across a number of multipaths; as such, channel- models based on stochastic features of diverse MIMO-specific links in the desired
test spectrum of mm-wave/THz band are sparsely known and almost non-existent.
To alleviate this niche, a method is proposed here to infer sub-mm band MIMO
channel-models (termed as “prototypes”) by judiciously sharing “similarity” of
details available already pertinent to traditional “models” of lower-side EM
spectrum, (namely, VLF through micro-/mm-wave). Relevant method proposed
here relies on the “principle of similitude” due to Edgar Buckingham. Exemplar set
of “model-to-(inferential)-prototype” transformations are derived and prescribed
for an exhaustive set of fading channel models as well as, towards estimating path- loss of various channel statistics in the high-end test spectrum.
Keywords: Channel/propagation-models, mm-wave/THz spectrum, Similitude principle,
Wireless communication
INTRODUCTION
With alarming increase in radio-frequency (RF) spectrum utilization and almost fully occupied
status of available electromagnetic (EM) spectral windows, facilitating newer ranges of RF- spectrum for wireless communication is an explicit necessity, so as to accommodate the
growing needs of broad-bandwidth wireless applications. For example, enabling newer ranges
of ‘unused’ segments (windows) of RF spectrum in the mm-wave band (above 30+ GHz) and
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seeking usable stretches across the unallocated THz spectral range could possibly considered
for Multiple Input Multiple Output (MIMO) communications in accommodating the growing
needs of multigigabit 3G/4G applications enclaving outdoor-based backhauls in picocellular
networks and indoor-specific multimedia networking. However, in contrast with those of
cellular and Wi-Fi systems, usage of smaller, carrier wavelengths (of mm-wave/THz bands) in
outdoor EM ambient could pose “drastically different propagation geometry”; and, in indoor
contexts, pertinent directional mm/THz wave propagations across spatially-multiplexed, line- of-sight (LoS) MIMO links may also face grossly diverse propagation geometry with multiple
number of paths plus nodes of relatively small form-factors. Further, moderately separated
antennas facilitating such directional links, could lead to sparse EM scattering; but, increased
propagation loss in the transreceive paths is inevitable with smaller wavelengths [1] [2].
MIMO techniques, in general, are adopted to combat fading; and, suppose carrier frequencies
falling in the mm/THz wave bands are prescribed for such MIMO systems in order to achieve
multigigabit 3G/4G applications in outdoor-based backhauls (of picocellular networks) and in
indoor-specific multimedia networking. The associated EM propagation artifacts and
interference characteristics of such MIMO-links would pose unique stochastic features and
distinct propagation characteristics thanks to (small) carrier wavelength versus spacing
between the antenna elements and limited EM-scattering involved. Such details could render
the elements of the MIMO-channel matrix not always being independent.
Use of mm/THz wave spectrum hardly prevails in the state-of-the-art communication areas
(including MIMO systems); as such, relevant neoteric paradigms of spectrum utilization consist
of only a limited/sparse and/or mostly insufficient data detailing the underlying EM wave
propagation characteristics. Thus, a lacuna prevails in the comprehension of channel-models at
this spectral range of interest. As such, this study is motivated to address the sparsity of MIMO
channel modeling vis-à-vis fading and path-loss features; and, the scope of this paper is set to
explore the feasibility of conceiving relevant “inferential prototypes” of channel-models in the
said EM spectral range (for possible adoption in MIMO contexts). The proposed method is based
on judiciously sharing the “similarity” of details between already existing (known) “models” of
traditional, lower-side EM spectrum, (namely, VLF through micro-/mm-wave) availed as ex
post data and the corresponding “prototypes” are conceived as ex ante details inferred for the
test-bands (of mm-wave through THz spectrum). Relevant method relies on the so-called
“principle of similitude” due to Edgar Buckingham [3]. Illustrative “model-to-(inferential)-
prototype” transformations are derived and discussed with reference to an exhaustive set of
channel-models on fading and path-loss characteristics pertinent to diverse channel statistics.
Commensurate with the objectives indicated above, this paper is organized as follows: The next
section (Section 2) reviews the general aspects of wireless channel-models and summarizes the
specific aspects of MIMO-channels. Further, focused considerations on such channel
characterizations in the 30+ GHz through THz spectrum are outlined and associated concerns
are identified. In Section 3, the proposed heuristics of inferring “prototypes” of EM propagation
characteristics required at the high-end spectral range (of mm-wave/THz band) using the prior
known (available) “models” at lower bands (like UHF/microwaves) is described. Relevant
systematic procedure in judiciously projecting the associated “similarity” of details in the
“models” (of low-frequency end) into “inferential prototypes” of high-end RF-channels
(representing the test-bands of interest) is described in Section 4 in terms of the heuristics of
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Neelakanta, P., & Groff, D. D. (2021). Conceiving Inferential Prototypes of MIMO Channel Models via Buckingham’s Similitude Principle for 30+ GHz
through THz Spectrum. Transactions on Networks and Communicaitons, 9(3). 1-35.
URL: http://dx.doi.org/10.14738/tnc.93.10214
the so-called principle of similitude (advocated by Edgar Buckingham [3]). The similitude
considerations involved in the model-to-(inferential) prototype transformations are elaborated
in Section 5 and applied to illustrate fading statistics and path-loss characteristics of channel- models at the test frequency bands of interest, (namely, 30+ GHz through THz EM spectrum).
Section 6 lists the well-known “models” of fading channels with the associated parameters;
hence, the corresponding deduced “inferential prototypes” are presented with descriptive
notes on an exhaustive set of fading statistics. Similarly, “model-to-(inferential)-prototype”
transformations are illustrated to deduce path-loss characteristics the channels in the test
bands of interest. Discussions on the proposed methods are furnished in Section 7 and
pertinent remarks are given in the closure section (Section 8).
WIRELESS- AND MIMO-CHANNEL MODELS: AN OVERVIEW
Modeling wireless communication ambient describing the associated RF propagation and
channel characteristics has been the topic of interest with the advent of mobile communications
and extensive proliferation of cellular networks. Relevant details enclaving transmissions at
traditional HF through UHF/microwave frequencies are addressed exhaustively in the
literature [4-8]. Associated channel-models reflect the propagation characteristics of signals in
pertinent radio environments are essentially needed for evaluating the performances of
wireless communication systems. Typically, commencing from the simple free-space model,
channel characteristics of both outdoor and indoor contexts are indicated for traditional
wireless systems in a gamut of published efforts. To name a few, are the so-called: Lee’s model,
Langley-Rice model, Okumura model, Hata model, Walfish and Bertoni model, Hashemi model,
Rappaport’s SIRCIM model, Saleh and Valenzula model and Seidal and Rappaport model [4-8].
In wireless systems, the EM propagation ambient supporting communication links decides the
integrity of transreceive signals largely specified by the underlying fading and path-loss
characteristics. Fading implies variations (with stochastic attributes) seen in the signal
attenuation versus time, locales of transreceive units and carrier frequency. That is, fading
denotes a random process caused by probabilistic variates on EM propagation due to multipath
and/or shadowing effects from the obstacles. Path-loss is another channel artifact denoting the
EM energy losses encountered in wireless communications that incur on terrestrial paths
through atmospheric environment as well as, in the content-infested indoor ambient.
In all, due to fading and path-loss characteristics, the transmitted wireless signal across its
transceive path would invariably seen morphed at the receiving end. Relevant profile of such
received signals can then be specified in terms of the details on the transmitted signal (known
a priori), if a model for the medium/channel between the transceive entities is also available.
Such characterization of RF signal transit-paths is well-known as a “channel-model”; and, it in
essence, represents the EM propagation profile of the signal and the associated statistics of
signal-strength fluctuations across the channel in question; and, a number of channel-models
have been evolved as listed earlier. The pros and cons of channel considerations in realizing
practical RF communication links are fairly comprehended in the traditional wireless systems
pertinent to classical deployment of VLF through microwave frequencies.
The perspectives of channel-models become even more unique and visibly complex in the
contexts of multiple-input multiple-output (MIMO) technique ushered into wireless
communications in order to achieve a significant increase in data throughput as well as, link
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reliability (mostly without extra bandwidth or boosting transmission power) [9-11]. MIMO
together with orthogonal frequency division multiplexing (OFDM) has proliferated as a key
technique in the long-term evolution (LTE) of 3G/4G cellular networks. In its conception to
achieve higher throughputs within a given bandwidth via space diversity schemes, the MIMO- links were typically modeled in the early theoretical studies (due to Telatar [12] and Foschini
[13]) as narrowband random MIMO channels. Such efforts refer to evolving a simple channel- model represented by a matrix [H] with M-transmit and N-receive antennas.
In the expanded applications, MIMO channel models have been subsequently framed for
different EM environments; and, comparison between channel-models is done on the basis of
the associated theoretical considerations versus measured data. Relevantly, two kinds of
channel models, namely, correlation-based stochastic models (CBSMs) and geometry-based
stochastic models (GBSMs), are prescribed (commensurate with IEEE 802.11n standards) in
order to evaluate the performances of the wireless communication systems; and, the former
being less complex is typically used in theoretical evaluations of MIMO system performance.
However, the accuracy of CBSM is limited for wireless channels exhibiting non-stationary
phenomenon and spherical wave effects. In contrast, GBSM is considered as being accurate in
portraying the realistic channel properties of massive MIMO channel applications (in spite of
the associated higher computation complexity).
MIMO techniques in general are indicated for combating fading. The underlying channel
characterizations, however are unique and mostly differ from those of non-MIMO strategies,
even at lower carrier frequencies. As such, exclusive MIMO-channel models have been
developed. Typically, the MIMO channel characterization is exemplified via a lamp-post based
outdoor deployment model (for example, as in a mesh backhaul), which duly accounts for
fading due to ground and wall reflections; and, in indoor contexts, the MIMO infrastructure with
space diversity and spatial multiplexing (for example, as done in “streaming high-definition
television from a set-top box to a television set”), relevant (MIMO)-channel model becomes
even more unique with a number of eigen modes framing the form-factor of the channel
involved [1].
The MIMO-specific channel-models have been comprehensively studied at traditional lower
end of EM spectrum [9-11]. Though restrictedly addressed (on ad hoc basis), pertinent channel
considerations and channel-modeling in the millimeter wave regime of RF systems [14]
including the MIMO-links [1] also exist. Yet, such a comprehension is rather sparse and even
absent for the EM spectrum ranging 30+ GHz through THz. However, such details are highly
desirable and imminent in any aggressive suite of implementing the current needs of robust
wireless communication links including MIMO systems. Specifically, MIMO channel-modeling
in the high-end spectral range of 30+ GHz through THz is needed towards profitable use of the
large extents of bandwidth available in the said test bands of interest. Pertinent uses as stated
before refer to realizing multi-Gigabit wireless networks with applications ranging from indoor
multimedia networking to accommodating outdoor backhauls for picocellular networks.
However, using the spectral range of 30+ GHz through THz in such applications, the carrier
wavelengths involved are obviously an order (or orders) of magnitude smaller than those in
the existing cellular and Wi-Fi systems. As such, significantly different propagation geometry
can be foreseen in the underlying EM ambient. For example, omnidirectional transmission is
almost ruled out because of the increased propagation losses encountered at smaller
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Neelakanta, P., & Groff, D. D. (2021). Conceiving Inferential Prototypes of MIMO Channel Models via Buckingham’s Similitude Principle for 30+ GHz
through THz Spectrum. Transactions on Networks and Communicaitons, 9(3). 1-35.
URL: http://dx.doi.org/10.14738/tnc.93.10214
wavelengths. So, highly directive transreceive MIMO-links can be envisaged with electronically
steerable beams formed by compact antenna arrays. As a result, only a small number of paths
would dominantly prevail (in such directional small-wavelength links), posing a sparse
scattering ambient unlike, the rich scattering environment at lower carrier frequency
operations. Usage of small wavelengths also implies achieving spatial-multiplexing gains even
in LoS transreceive links or more generally, in the ambient of sparse scattering consistent with
the dispersed of antennas having moderate separations.
As stated earlier, knowledge on EM propagation characteristics (or channel models) in the said
mm/THz bands (henceforth addressed as the “test-bands”) is still primitive and mostly
unknown for applications in MIMO-systems or otherwise. Filling this niche will conform to
transformational improvements in using the radio spectrum of test-bands containing ‘unused’
segments (windows) in mm-band (above 30+ GHz) and usable stretches across the unallocated
THz frequencies. Hence, it will add a dimension of realism in conceiving the exploding
broadband communication needs (including novel MIMO applications), so as to support the
associated traffic intensity of triple-play wireless communication services in the near-future
and beyond.
Hence proposed in this paper is a feasibility consideration on a very practical problem: How
can channel/propagation models be inferentially deduced for the spectral range of test-bands
from the existing (available) knowledge on the lower strata of EM spectrum? And the solution
sought thereof, is to find one-to-one correspondence between the channel characteristics
(dubbed as “models”) known ex post at low-frequencies and corresponding ‘scaled-up’ channel
features (called “prototypes”) attributed ex ante, to higher (test) frequencies of interest.
INFERRING “PROTOTYPES” FROM “MODELS”: PROPOSED METHOD AND APPROACH
As indicated above, the present study objectively proposes an approach to develop
“prototypes” of EM propagation and channel characterizations of RF signal transmissions in the
test frequency bands covering the new frontiers of (almost) unused 30+ GHz spectral windows
and mostly unexplored terahertz span (above 275 GHz sans spectrum allocations). As briefed
above, relevant concept is illustrated in Figure 1.
The proposed approach towards developing “prototypes” of EM propagation and related
channel characterizations of RF signal transmissions in the spectral bands of (almost) unused
30+ GHz frequencies indicated above (and illustrated in Figure 1) is summarized as follows:
▪ A set of channel details are presumably known a priori at lower frequency bands
(like UHF/microwaves). The set of such ex post features are designated as the
“model”
▪ The said “model” details are listed and judiciously projected to frame a “inferential
prototype” at the desired high-end frequency (test-band); and, the ex ante set of
features so conceived denote an inferred “prototype” that shares the “similarity” of
all the pertinent details in the “model”
▪ The projection of shared similarity indicated above is done via a systematic
procedure based on the so-called principle of similitude advocated by Edgar
Buckingham [3] in terms of dimensional analyses perspectives detailed in the next
section.
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Figure 1. Specifying a “model” function denoting a set {a, b, ...} of known data on channel
attributes at lower-end frequency spectrum (fm) towards formulating a corresponding -
function (for use to infer the “prototype” function as indicated in Step 2). The set {ao, bo...,}
denotes reference entities that render the variable set {a, b, ...,} dimensionless; and, {(m1, m2,
...)} are pertinent exponents prescribed to frame the -relations consistent with Buckingham’s
similitude principle.
The steps in conceiving “inferential prototypes” of RF channel characteristics in a desired test- band using the “model” data available at lower frequencies are as follows:
o Step 1: Construction of the “model”: This is illustrated in Figure 1. The channel
parameters {a, b, ...} known a priori at lower frequencies, fm (say, HF through
microwaves) are first identified and specified as the “model” features denoted
by a functional relation: Func(fm: a, b,...). This model function is then modified
and rewritten, following the so-called as Buckingham’s similitude principle
described in the next section. Relevantly, a Buckingham π-function of the
model is specified as: g[(fm/fo): (a/ao)m1, (b/bo)m2...], where fo is a reference
frequency prescribed to normalize, fm; and, {ao, bo, ...} etc. are normalizing
parameters indicated respectively, for the elements of the set {a, b,...}; and,
the set {m1, m2, ...} depicts appropriate exponents indicated for the aforesaid
-relation. This “model” prescription as above depicts ex post details of the
channel in question.
103
106
109 Hz
“MODEL
”
Model function: Func (f
m
: a, b, ...)
Buckingham π-function of the
model:
g{(f
m
/f
o
): (a/a
o
)
m1
, (b/b
o
)
m2...}
Ex post data of the
“model”
Similitude principle
INPUT
(Known ex post regime)
Available channel-characteristics
HF..., VHF ..., UHF ..., Microwave ... ... mm-Wave ...
. THz
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(p1, p2, ...)} are appropriate exponents imposed on -relations consistent with Buckingham’s
similitude principle (as will be explained in the pursuant section).
BUCKINGHAM’S SIMILITUDE PRINCIPLE
The similitude principle has been deployed extensively in various disciplines, to name a few:
Model-based testing and evaluations of structures in civil, aeronautical/aerospace, naval
architecture [15], reliability predictions [16], radar cross-section (RCS) studies [17] etc. The
underlying considerations of similitude concept can be outlined as follows:
▪ Evolving a “prototype” refers to elucidating a set of new descriptive details formulated
a posteriori and prescribed on an entity (for example, a structure or a phenomenon).
Such inferred (ex ante) details correspond to an existing and available set of ex post data
(referred to as the “model”). Such data is presumably known a priori on the entity in
question. The underlying one-to-one transformation is inferred and interpreted via
similitude considerations based on the theory of modeling consistent with the norms of
dimensional analysis
▪ The underlying theoretics and mathematical description of theory of dimensions
(evolved in terms of dimensions and units) is aptly applied to portray the similitude
aspects of the prior knowledge on the said “model” (depicting the physical entity or
phenomenology) versus the “prototype” posterior information being sought, subject to
the following norms:
o The mathematical description (of transforming a “model” into a “prototype”)
should comply with the laws of nature and specified in a dimensionally
homogeneous form. That is, regardless of the choice of dimensional units (in which
the associated physical variables are measured), the governing equation must be
valid
o All relevant governing equations must be dimensionally homogeneous. That is , an
arbitrary equation of the form, [func(x1, x2, ..., xn) = 0] can be alternatively
expressed as, [g(1, 2,... m) = 0], where the -terms denote dimensionless
products of n physical variables of the set {x1, x2, ..., xn}; and, m = (n – r) with r
depicting the number of fundamental dimensions that are involved in the physical
variables.
Explicitly, the second consideration above namely, [func(x1, x2, ..., xn) = 0] [g(1,
2,... m) = 0] implies the following: (i) The -form of physical occurrence can be
deduced via proper use of dimensions of the n physical quantities, {xi}i = 1, 2, ..., n by
resorting to dimensional analysis. (ii) In physical systems that differ only in
magnitudes of the units (used to measure the n quantities, namely {xi}i = 1, 2, ..., n ),
the “model” entity and its reduced or enhanced scale-version (“prototype”) will
have identical functional, g(.); and, (iii) similitude requirements for modeling
result from forcing the -terms, {i}i = 1, 2, ..., m to be equal in the “model” as well as
in the “prototype”. (This is a necessary condition for the full functional
relationships to be equal).
In short, a “model” denotes a representation of the physical system with details (known ex post)
that can be adopted to infer the behavior of a corresponding “prototype” in some respect. That
is, the principle of similitude (due to Buckingham [3]) enables predicting the performance of a
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Neelakanta, P., & Groff, D. D. (2021). Conceiving Inferential Prototypes of MIMO Channel Models via Buckingham’s Similitude Principle for 30+ GHz
through THz Spectrum. Transactions on Networks and Communicaitons, 9(3). 1-35.
URL: http://dx.doi.org/10.14738/tnc.93.10214
new design (called “a prototype”) based on mimicking the data from an existing, similar design
(designated as “the model”).
In such representations, the first consideration is to show that the two systems can be described
in terms of dimensionless numbers as per Buckingham’s heuristics. This can be outlined as
follows: Given any element (m) of the -set of the model and the corresponding element (p)
of the prototype, ideally the similarity principle demands, p/m = 1. But, in practice, at least
some first or second-order deviations may be encountered rendering exact model-to-prototype
extrapolation, somewhat “distorted” [18]. Conceiving an approximated version via piecewise
linearization in similarity relations is, however, feasible as will be indicated later in this paper.
In general, the similitude problems based on Buckingham’s heuristics can be classified as
follows: (i) Geometric similitude, which denotes the “model” in question representing an “exact”
scale-copy of the “prototype” consistent with physical geometry involved; (ii) kinematic
similitude concerned with flow considerations, where the “model” flow is set proportional to
the corresponding flow in the prototype, preserving the associated vector nature of the flowing
entity; and, (iii) dynamic similitude, which refers to forces associated in the “model” being
proportional to the forces seen in the “prototype”. (This is a necessary condition, but not
sufficient for dynamic similitude) [11] [19-23].
Relevant to the present study of applying Buckingham’s similitude principle in the contexts of
channel modeling, first the underlying similitude considerations pertinent to EM theory should
be identified. That is, the associated set of postulations and constitutive relations consistent
with the linearity aspects of Maxwell’s equations [24] should be specified in dimensionless
forms by invoking Buckingham’s -theorem. Then, the EM propagation/channel-modeling is
rendered viable for the similitude approach. Existing studies on applying such similitude
principle in electrodynamics and EM contexts are as follows: (i) Geometrical inferences
(relative to wavelength) such as in antenna designs [25], radar cross-section (RCS) evaluations
[17]; (ii) channel characterizations [26] (studied in a limited extent) and (iii) frequency-scaling
of rain attenuation (such as in ITU-R Rec.P.618-7, 2001 and similar models [27] [28]).
Basically, the kinematic/dynamic similitude is applied in EM contexts considering (i) the flow
of electric current signified by the time-varying states or dynamics of electric charges, (ii) the
rate of change of electric/magnetic flux and (iii) the Poynting vector flow. EM similitude
technique has also been advocated in studying laser scattering [23]. Typically, electrodynamic
similitude is applied to model the nondispersive targets as indicated in [22].
Applying -theorem in EM contexts
In all, the heuristics of adopting Buckingham’s -theorem in the contexts of electromagnetism
(EM) and hence, applying the principle to elucidate the “prototypes” of channel-models vis-à- vis EM propagation characteristics in wireless communications involve the following basics of
EM theory outlined to deduce the associated -relations.
As well known, the EM waves possess a characteristic wavelength length ( in m), which is
related to their frequency (f in Hz) by the relation, (m) = [v in m/s)/f in Hz] where, v denotes
the velocity of propagation of EM wave in the medium of interest and f is the frequency of the
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EM wave. For free-space propagation, v ≡ c (= 3 108 m/s) as decided by the associated
constants of the free-space, namely, the absolute permittivity o = (1/36) 10−9 farad/meter
and absolute permeability, o = (4) 10−7 henry/meter; and, c =1/(oo)
1/2 m/s, (= 3 108 m/s
depicting the speed of the light). And, v = c/(rr)
1/2 m/s [24] expressed in terms of
(dimensionless) relative permittivity and permeability, (r, r) of the medium; hence, for free- space, (r =1; r = 1).
An example of elucidating dimensionless parameters, functions, equations etc. in the contexts
of electrodynamics as required in similitude applications is as follows: Considering the well- known Coulomb’s law of force (|F|) between given two charges of q1 and q2 units, separated by
a distance d units in a dielectric medium characterized by a (dimensioned) constant, k, the said
law is specified by the equation, {func(x1, x2, ..., xn) = 0} func[|F|, k, (q1, q2), d] = 0; explicitly,
this Coulomb’s law can be written as: {[|F| − k q1q2/d2]} = 0. In terms of the associated units
term-by-term, Coulomb’s law can be rewritten as: func{[K1 force unit] − [K2 (capacitance
unit/length unit)−1 K3 (charge unit)2]/[K4(length unit)2] = 0}, where, the func{.} is expressed
in terms of corresponding fundamental measures that describe the associated quantities; and
K1, K2 and K3 are relevant proportionality constants.
The above relation holds for any system of units, (CGS, SI or FPS) for force, capacitance, charge
and length; and as such, it is dimensionally homogeneous. Now, in terms of the notions of
Buckingham [3], the implicit functional relation of Coulomb’s law, namely, func[|F|, k, (q1, q2),
d] = 0 can also be represented as a product of powers in a form as follows: |F| = CD × [k
(q1,
q2)
d
] newton, where CD is a dimensionless parameter, (which by itself could be a function of
dimensionless groupings of some pertinent physical entities; or, it can be a simple constant,
otherwise); and, the exponents = + 1, = − 2 and = + 2 are explicitly pertinent to the
Coulomb’s law under discussion.
Next, the functional relation, F = CD[k
(q1, q2)
d
] can be cast in the -form, {g( m) = 0}
using similitude principle. Relevantly, consistent with Buckingham’s -theorem, the
dimensionally homogeneous equation, {func[|F|, k, (q1, q2), d] = 0}n = 4 is reduced to an
equivalent equation involving a set of dimensionless products, namely, {g(1, 2, ..., m) = 0}m =
2 ≡ {g(kq2/Fd2, q2F/kd2) = 0}. In other words, the given equation, {func(x1, x2, ..., xn) = 0}
{func[F, k, (q1, q2), d] = 0}n = 4 implies the same Coulomb’s law specified in a dimensionless
format as: {g( m) = 0} {g(kq2/Fd2, q2F/kd2) = 0}m = 2 .
Likewise, all the postulations and constitutive relations of EM theory based on the linearity
aspects of Maxwell’s equations [24] can be specified in dimensionless forms by invoking
Buckingham’s -theorem via relevant algorithmic suites as indicated above. However,
distortions may not be uncommon in such similitude comparisons of EM contexts due to
dissimilarities that may prevail in the overall geometry, boundary/initial conditions and/or
(nonlinear) material (EM medium) implications. In such cases, still a piecewise linearization
can be considered [18] as will be illustrated later.
Similitude considerations on EM propagation channel models
Following the similitude notions narrated above relevant to the contexts of electromagnetics,
relevant -relational details can be developed towards deducing EM propagation profiles and
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Figure 3. A hypothetical set of -functions, {g[.]} specified across several stretches of frequency
bands, (Band-1, Band-2, ..., Band-j ...); and, each g[.] is approximated by an asymptotic
(piecewise linear) representation. Each frequency band is specified with a value of the
normalized frequency set: {(fm/fo)m = 1, 2, 3,...}.
The variable set {fm; a, b, ...} can further be expressed in a normalized form and rendered
dimensionless with a corresponding set of reference values, {fo; ao, bo, ...}. Each of the resulting
dimensionless variable is also specified with a dimensionless exponent. In all, FUNC(fm; a, b, ...)
g[fm/fo; (a/ao)
, (b/bo)
, ...]. The function, g[.] denotes the Buckingham’s −function
designated with the exponent set { ...}.
Now, suppose a hypothetical set of -functions {g[.]} is presumably known ex post relevant to a
spectral range of frequencies. The function g[.] is then approximated by an asymptotic
(piecewise linear) representation and attributed to represent each segmented stretch of
frequency bands across the EM spectral range as shown in Figure 3. (Note: The bands shown
need not however, be continuously and successively adjacent. There could be a window of
unknown features between any two bands wherein, g[.] is unspecified).
Considering a set of successive frequency bands as illustrated in Figure 2, the EM details
prescribed for each band via the function g[.]. That is, g[.] denotes the “models”, namely, model- 1, model-2, model-3 etc. respectively for the bands, Band-1, Band-2, Band-3 etc. Now, for each
frequency segment of the set {(fm/fo)m = 1, 2, 3,...} depicting {(Band)m = 1, 2, 3,...}, the associated
“model” prescription can be transformed into a corresponding “prototype” of the next,
(successive) higher frequency band. This is done via a successive iteration method using
Buckingham’s similitude principle as described in the pseudocode presented in Table 2.
....
Frequency bands: (fm
/f
o
)m = 1, 2, 3,...
Band-2 ...
f
2
Band-j
f
j
Band-1
f
1
g[.]
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g[(f1/fo); (a1/ao)
1, (b1/bo)
1;(c1/co)
1, ...]
≡ g[(f2/fo); (a2/ao)
2,(b2/bo)
2;(c2/co)
2, ...]
Next
Deduce
→ [k2×P1(f2/fo)]C
2
This is a scaled version of [k1×G1(f1/fo)]C
1 with (k1/k2) as a
scaling constant and {C1 and C2} being the exponents that decide
a power-law relation
Prototype-2 of Band-2
Next
→ Use Prototype-2 as a pseudo- “model” for prototyping Band-3
→ Consider: Band-2 − −format of the g[.] function
P2: g[(f2/fo); (a2/ao)
2, (b2/bo)
2;(c2/co)
2, ...]
→ Rename P2(.): Prototype - 2 as: Pseudo-model-2
→ Consider: Band-3 - -format of the g[.] function
P3: g[(f3/fo); (a3/ao)
3, (b3/bo)
3;(c3/co)
3, ...]
→ Designate P3(.) as: Prototype-3
Next
Equate
→ Band-2 - -format of the g[.] function: P2(.) and Band-3 - -format of the g[.]
function: P3(.)
→ g[(f2/fo); (a2/ao)
2, (b2/bo)
2;(c2/co)
2, ...]
≡ g[(f3/fo); (a3/ao)
3, (b3/bo)
3;(c3/co)
3, ...]
Deduce
→ [k3×P3(f3/fo)]C
3
→ This is a scaled version of [k2×P2(f2/fo)]C2 with (k2/k3) being a scaling
constant; and, {C2 and C3} are the exponents that decide the associated
power-law relation
Prototype-3 of Band-3
Next
→ Use Prototype-3 as a pseudo-model for prototyping Band-4
(and so on ...)
Continue
→ Iteration until the entire spectrum of interest is exhausted
List
→ “Prototypes” evolved across the entire spectrum of interest
%% Comment: The above procedure allows elucidating the features in a given
band, say, Band- M, by knowing the features in the previous band, Band (M −
1) via scaling and similitude principle.
→ That is, the features of Band (M – 1) depict the known ex post
profile or the “model”; hence, the unknown profile of the Band- M inferred ex ante depicts the desired “prototype”
End
------------------------------------------------------------------------------------------------------------------------
Ascertaining “prototypes” of channel models in the desired spectral range (of mm-wave
through THz band stretching into suboptical regions) following the procedure described above,
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suggested here is to invoke the thematic of Buckingham’s similitude principle to deduce the
“prototypes” of path-loss characteristics at the desired test bands vis-à-vis corresponding
known details of the “models” in the available stretches of lower-end frequency bands.
Hence, commensurate with the objective as above, the model-to-prototype transformation as
needed, a compatible set of path-loss functions (PL1 through PL9) can be formatted in
dimensionless forms as listed in Table A.1 wherein, the pertinent details include both LoS and
non-LoS transmissions across single- and/or multi-path signal transits, each path possibly
posing unique angle-of-arrival (AoA).
Constructing a “prototype” of a path-loss-model
The underlying concept of similitude-based scaling of path-loss up into higher frequency
spectra can be specified via “model”-to-prototype transformation. Using the relevant path-loss
ratio, PLR (= PR/PT) functions indicated in dimensionless forms in Table A.I, the underlying
similitude consideration can be illustrated with the following illustrative example (of Figure 4):
Figure 4 A typical, relative path-loss ratio profile expressed as RF power attenuation (per km)
versus frequency (from 5 GHz through 1000 GHz with fo: Normalization (reference frequency) =
900 MHz [52]. (A: Actual measured data as in [52] and B: Inferred data as per linearized
similitude principle; Band-1 and Band-2 are designated spectra on either side of the corner
frequency, fc/fc = 575 and similitude-based inference is made in Band-2).
Suppose as an example, the data available in [52] on attenuation (per km) versus frequency
range up to 1000 GHz (normalized with a reference frequency, fo = 900 MHz) is plotted as
illustrated in Figure 4 (pursuing band-by-band segmentation strategy indicated in Figure 3). In
Figure 4, the profiles of path-loss versus decades of frequency changes are shown for the bands,
(Band-1 and -2). There are two asymptotes marked (in dotted-lines) with a point of intersection
at a corner frequency, fc/fo = 575. Now for analysis, the left-side of fc/fo taken as Band-1
representing the ex post details of the “model”: FUNC(Band-1) [(Path-loss)Band-1]; and, the
0
0.2
0.6
1
0 200 400 575 800 1000 1200
g1
= m1
(f1
/fo
)
g2
= c2
+ m2
(f2
/fo
)
(f/fo
): Normalized frequency (Dimensionless
)
Path-loss
ratio
Band-1 Band-2
fc
/fo
0.2
600 800 1000 1200
1.0
Path-loss
ratio
Band-2
0.6
A B
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Table 3. Relative path-loss ratio: Measured empirical data [52] versus values inferred via
similitude principle
Path-loss ratio
f2/fo
Actual
data [52]
Inferred
values
600 0.31 0.29
650 0.36 0.29
700 0.42 0.31
750 0.49 0.34
800 0.55 0.37
850 0.63 0.42
900 0.70 0.48
950 0.78 0.60
1000 0.87 0.64
1050 0.96 0.74
1100 1.05 0.90
Exemplars of lower frequency band “models” of path-loss characteristics
Pursuing the procedure indicated with an example above, the art of similitude adopted towards
framing the model-to-prototype transformations of path-loss characteristics, the following
examples of lower frequency band “models” are indicated to obtain their corresponding
prototypes.
A. Lee’s model [5]
This path-loss model suggests an asymptotic variation in path-loss across decades of frequency
changes as illustrated in Figure 5. For example, suppose, the attenuation versus frequency is
specified and known in the range up to 1000 MHz (denoted as Band-1).
Figure 5. Lee’s model [5]: An example depicting the change in signal attenuation (expressed in
terms of the slope dB/decade) over a range of: 40 MHz through 4000 MHz observed in the path- loss profile. The data is specified within the radius of coverage based on radio horizon distance
and base-station antenna height under non-obstructive point-to-point RF transits.
The signal attenuation (in dB/decade) shown represents a “model”: FUNC(f1-band; 1)
[(PR/PT)f1-band]
1. Relevantly, the EM power-loss is regarded mainly due to free-space, far-field
EM wave proliferation; as such, it follows the (1/R)-law (as limited by antenna-beaming
considerations). (However, in the event of EM absorption versus frequency due to path- 0
40
80
120
10 1000 100000
Frequency in Hz
Signal
attenuation
in dB/decade
Band-1
Band-2
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m
m
o m o o
1m "Model" at f 2 1/2 1/2
o m o o
2m "Model" at f T m o o
(2 / ) [0.5 ( / ) ( / ) {1
[a ]
[( / ) / (2 f / f ) ( / )] } 1]
and
[a ] [(d / d ) / (d / )]
+
= −
=
(13)
where do and dT denote the Rayleigh-criterion specified on terrain irregularity (shape) across
Ro and RT respectively; and, depicts the normalized conductivity responsible for the EM
absorption in the medium.
Now, the scaling of equation (13) can be done via similitude principle to get the “prototype” at
a desired elevated frequency fp >> fm as follows: For the “prototype” being elucidated at fp:
1 2 3
4 A 4 T
R p o o p p o p T p R p o p
( ) ( )
1p p o p 2p p o p
[P (R / ) / P (R / )] [(R / ) / (R / )] [(h / ) / (h / )] [( / )]
[{exp( a (R / ) / (R / )} {exp( a (R / ) / (R / )} ]
− − −
=
− −
(14)
where the subscripted index ‘p’, as stated before denotes the “prototype” considerations
applicable at fp with the prototype exponent set being: The normalized set of
parametric data in equations (13) and (14) are:
{PR(R/λm or p), Po(Ro/λm or p), (hT/λm or p), (hR/λm or p), (λo/λm or p), a1 m or p, a2 m or p}
(15)
These entities and the exponents indicated enable formulating the required similitude of
“model” -to- “prototype” function transformation concerning the path-attenuation under
consideration.
With reference to a typical Okumura path-loss model presented in Figure 6, there are three
observable aspects on path-loss characteristics. Relevant path-loss slope varies over the
decades of frequency, the path distance (R) and EM propagation across more than one type of
environment. In essence, the variation of path-loss versus decades of frequency is indicated by
varying path-loss slope for a given antenna height ratio and transceive distance. Thus, the
results in Figure 6 depict segmented asymptotic variations across some (Band-1)-to-(Band-2).
Hence, corresponding pair of, “model”: FUNC(f1-band; 1) [(PR/PT)f1-band]
1 and a “prototype”:
FUNC(f2-band; 2) [(PR/PT)f2-band]
2 can be inferred via similitude considerations.
C. Medium-specific absorption losses – Crane model/JPL model on rain-attenuation [53-55]
Precipitation (rain, snow and fog specific) losses are highly dependent on seasonal variations,
rain structure and rain-rate statistics [43], and conform to outdoor EM absorption resulting
from dielectric losses. The extent of such losses per unit distance will depend on the density
and dispersive pervasion of absorbing (lossy) contents being present; and, such lossy dielectric- based EM absorption is also approximately proportional to the square of the frequency of the
propagating wave. Further, the vertical structure as well as, the rain cell-size would play
significant role in making of a relevant global attenuation model. Typically, the so-called Crane
attenuation model is an exemplary depiction, which duly accounts for the parameters on the
path-length (R), rain-rate, rain-height, rain- volume etc. over typical sets of terrains.