#  Dim Wihl Gat Tun: The Case for Linguistic Expertise in NLP for
Underdocumented Languages

Clarissa Forbes${}^{\textrm{{Z}}}$ Farhan Samir${}^{\textrm{{Q}}}$ Bruce
Harold Oliver${}^{\textrm{{Q}}}$ Changbing Yang${}^{\textrm{{Q}}}$
Edith Coates${}^{\textrm{{Q}}}$ Garrett Nicolai${}^{\textrm{{Q}}}$ Miikka
Silfverberg${}^{\textrm{{Q}}}$
${}^{\textrm{{Z}}}$Independent Researcher ${}^{\textrm{{Q}}}$University of
British Columbia
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Recent progress in NLP is driven by pretrained models leveraging massive
datasets and has predominantly benefited the world’s political and economic
superpowers. Technologically underserved languages are left behind because
they lack such resources. Hundreds of underserved languages, nevertheless,
have available data sources in the form of interlinear glossed text (IGT) from
language documentation efforts. IGT remains underutilized in NLP work, perhaps
because its annotations are only semi-structured and often language-specific.
With this paper, we make the case that IGT data can be leveraged successfully
provided that target language expertise is available. We specifically advocate
for collaboration with documentary linguists. Our paper provides a roadmap for
successful projects utilizing IGT data: (1) It is essential to define which
NLP tasks can be accomplished with the given IGT data and how these will
benefit the speech community. (2) Great care and target language expertise is
required when converting the data into structured formats commonly employed in
NLP. (3) Task-specific and user-specific evaluation can help to ascertain that
the tools which are created benefit the target language speech community. We
illustrate each step through a case study on developing a morphological
reinflection system for the Tsimchianic language Gitksan.

##  1 Introduction

Progress111Dim wihl gat tun \- “This is what the people should do”222First two
authors contributed equally. in NLP research has primarily manifested in tools
for the world's political and economic superpowers (Blasi et al., 2021), and
it is unclear how we can build more inclusive language technologies. Even
multilingual pretraining methods (e.g., Liu et al., 2020; Artetxe et al.,
2018), capable of producing effective models in the absence of large annotated
training datasets require unannotated corpora that are prohibitively large for
90% of the world's languages (Joshi et al., 2020).

Nevertheless, many languages in this 90% have a body of resources. Language
documentation and linguistic fieldwork are an ongoing task worldwide, and many
resources continue to be developed in these traditions (Bird, 2020). We have
access to wordlists, bilingual dictionaries for over 1000 languages (Wu et
al., 2020), aligned speech recordings for over 700 languages (Black, 2019),
multi-parallel texts for 1600+ languages (McCarthy et al., 2020b), and
knowledge of related languages (Haspelmath et al., 2005). Indeed, researchers
have leveraged these resources to build impressive, useful computational
systems for multilingual morphological analyzers (Nicolai and Yarowsky, 2019),
adapting pretrained language models for over 1000 languages (Ebrahimi and
Kann, 2021), and building massively multilingual speech recognition systems
(Adams et al., 2019), among others.

Figure 1: An example of Gitksan interlinear glossed text (IGT). The text
contains four levels of annotation: (1) An orthographic transcription, (2) A
segmentation into normalized component morphemes (CVC refers to the
reduplicated segment al'), (3) an interlinear gloss and (4) an English
translation.

There are additional language documentation resources which have yet to be
fully leveraged in the aim to produce more inclusive language technology.
Interlinear glossed texts (IGTs) depicted in Figure 1 are semi-structured
texts which comprise not only monolingual corpus data (e.g. al'algaltgathl)
but also morpheme-level segmentations (e.g. CVC~algal-t=gat=hl), glosses for
component-morphemes (e.g. PL~watch-3.II=REPORT=CN), word alignment information
(Zhao et al., 2020), and free translations. IGTs remain a major annotated
datatype produced in the course of linguistic fieldwork: examples are
continuously digitized in large databases for hundreds of languages (Lewis and
Xia, 2010), and entire corpora of IGT are periodically published in volume
series such as Texts in Indigenous Languages of the Americas. They have the
potential to serve as training data for a wide variety of computational
systems including bilingual lexicons, morphological analyzers, dependency
parsers, part-of-speech taggers, and word-aligners (Georgi, 2014). Yet while
they are accessible, they remain severely underutilized for these purposes.

Part of the general hesitancy in adoption of IGT as training data may lie in
the fact that the annotation format is only semi-structured and often
language-specific. While the general IGT format is governed by the Leipzig
glossing rules (Comrie et al., 2015), there remains significant flexibility
for the annotator to customize tags and conventions for any given language.
This makes IGT challenging as a format for training supervised NLP models.

With this paper, we make the case that IGT data can be leveraged in NLP
research and language applications for speech communities, provided that
target language expertise is available. Specifically, we argue that it is
essential to collaborate with documentary linguists who are familiar with the
language-specific annotations in the IGT data in order to leverage the data
for NLP tasks. This may furthermore provide a foundation for co-designing
language technologies with a given speech community (Bird, 2020).

Our paper provides a roadmap, portrayed in Fig. 2, for navigating three areas
of significant uncertainty that arise when incorporating IGT data for
inclusive language technology. First, we need to define what NLP tasks can be
accomplished with a given set of IGT data, and whether they are of value to
the speech community. Second, after selecting useful tasks, we will need to
preprocess the data, potentially by converting it to a structured format
commonly employed in NLP tasks. Finally, we need task-specific and user-
specific evaluation procedures in order to be explicit about the failure modes
of the technology, as it is ultimately being developed for end users like
speakers and linguists rather than solely comparison with other researchers.

Figure 2: A roadmap for incorporating Interlinear Glossed Text (IGT) data for
building more inclusive language technology. (1) We first need to define what
NLP tasks can be accomplished with a given set of IGT data and whether they
are valuable to the speech community (see Section 2.3). (2) Next, we need to
gather the relevant IGT data that was created during linguistic fieldwork with
the speech community (see Section 2.3). (3) Next, the IGT data needs to be
converted to a structured format amenable for NLP formats. (4) The model needs
to be evaluated not only in terms of standard NLP model selection metrics but
also for efficacy for end-users such as efficiency in time-savings and
usability (see Section 4). Crucially, all three stakeholders – speech
community members, NLP researchers, and linguists – should be involved
throughout the process.

We focus on the first two of these areas, forwarding our argument through a
case study on developing a morphological reinflection system for the Gitksan
language (Section 2.3) that has applications in language teaching.

##  2 Background

###  2.1 NLP for Underdocumented Languages

Computational work on underdocumented and low-resource languages has
accelerated in recent years due to increasing recognition of both the role of
NLP in language preservation as well as dedicated workshops like ComputEL
Arppe et al. (2021), AmericasNLP Mager et al. (2021) and SIGTYP Vylomova et
al. (2021). Most of this work aims to assist in language documentation and
revitalization, with machine translation being another important research
area. Mager et al. (2018) and Littell et al. (2018) present surveys of
existing NLP tools for the North American Indigenous languages, many of which
are underdocumented, and discuss core challenges: morphological complexity,
limited training data, and dialectal variation.

Several authors have trained NLP models on IGT to accelerate language
documentation, with automatic glossing being a prominent research direction.
The first approaches simply memorized earlier glossing decisions and enabled
the annotator to re-use these later Baines (2009). Later approaches have
relied on structured models like CRFs McMillan-Major (2020), RNN encoder-
decoders Moeller and Hulden (2018) and transformers Zhao et al. (2020) to
generate glosses for unseen tokens. NLP techniques can also be used to
generate inflection tables from IGT (Moeller et al., 2020). These find
applications both in language documentation and language education, often to
facilitate the production of more IGT data. A related approach is to generate
morphological analyzers using IGT as a starting-point Zamaraeva (2016); Wax
(2014).

Several papers discuss challenges related to IGT as a data type. One of the
principal concerns is the noisiness of the annotations Moeller et al. (2020).
This problem is compounded by the fact that annotation schemas employed by
linguists preparing IGT tend to be idiosyncratic333These systems are well
motivated but unlikely to be easily comparable with other annotation schemas.
and often lack internal consistency Baldridge and Palmer (2009); Palmer et al.
(2009). The design of annotation standards is important: Zhao et al. (2020)
note that this can have an impact on the performance of glossing systems.
McMillan-Major (2020) notes a further challenge: IGT often includes not only
morphological information, but also syntactic, semantic, and pragmatic
annotations, which can be much harder to learn in low-resource settings.

In addition to challenges in the IGT data type itself, there are other
challenges in NLP applications for underdocumented languages. Ward and
Genabith (2003) discuss many problems related to development of computer-
assisted language learning for endangered languages: lack of orthographic
standards, limited resources, and limited documentation of the language. van
Esch et al. (2019) also discuss NLP tools that can be helpful for
documentation of low-resource languages, but they note that restrictive
licenses can often be problematic for engineering.

###  2.2 The Gitksan Language

The Gitxsan are one of the Indigenous peoples of the northern interior region
of British Columbia, Canada. Their traditional territories consist of upwards
of 50,000 square kilometers of land in the upriver Skeena River watershed
area. Their traditional language, called Gitksan in the linguistic literature,
is the easternmost member of the Tsimshianic family, which spans the entirety
of the Skeena and Nass River watersheds to the Pacific Coast.

Today, Gitksan is the most vital Tsimshianic language, but is still critically
endangered with an estimated 300-850 speakers (Dunlop et al., 2018). Community
revitalization efforts are underway but are primarily undertaken by
individuals on an ad-hoc basis. Initiatives include regular in-school language
programming, a few adult language courses, a successful language immersion
camp, and several Master-Apprentice pairs.

Linguistic documentation on Gitksan and the Tsimshianic languages has been
going on intermittently since the 1970s, including the drafting of a never-
published grammar (Rigsby, 1986) and waves of formal phonological, syntactic,
and semantic work over the past thirty years. There are several community-
developed wordlists and workbooks, but no comprehensive dictionary, grammar,
or pedagogical curriculum. There is an accepted orthography (Hindle and
Rigsby, 1973), and a talking dictionary mobile app in active use by the
community (Mother Tongues Dictionaries, formerly Waldayu; Littell et al.
(2017)).

Other computational studies interact with the active documentation efforts
surrounding Gitksan to produce new frameworks and resources. Dunham et al.
(2014) present a database structure for hosting audio and transcribed data in
language documentation contexts, adopted for Gitksan and eight other
underdocumented languages. Littell et al. (2017) present a dictionary
interface which is capable of fuzzy search. They mention this specifically as
a way to increase accessibility in a setting where orthographies have not been
standardized or where many users are language learners. Forbes et al. (2021)
present a finite-state morphological analyzer for Gitksan; they test coverage
across different dialects of Gitksan and use handcrafted rules to increase
coverage for spelling variants.

###  2.3 Constructing a Gitksan Pedagogical Application from IGT Data

Our project generates language learning exercises for Gitksan grammar. The
need for these exercises was identified in discussions with documentary
linguists working on Gitksan (the task definition step in Figure 2).
Specifically, our goal is to automatically generate exercises for noun and
verb inflection. As source material, we use Gitksan IGT data collected by
linguists at the University of British Columbia for language documentation
purposes (the data step in Figure 2). Examples of this data are shown in
Figure 1 and Appendix A.

Due to extensive morphological annotation, IGT provides a valuable starting
point for our work. However, the annotations are far too detailed for our
purposes—many derivational affixes are annotated in the data (further
discussed in Section 3.1). These are irrelevant and can be downright harmful
for grammar exercises. To remedy this misalignment between the raw IGT data
and our NLP task, we collaborate with Gitksan documentary linguists to
identify a set of inflected forms with clearly defined grammatical function,
while discarding derivational morphology. We then convert the IGT data into a
set of inflectional paradigms (the data conversion step in Figure 2). We
further discuss this conversion process in Sections 3.2 and 3.3. Since the
inflectional paradigms sourced from corpora are sparse,444Due to the Zipfian
distribution of language Blevins et al. (2017). we train models to fill in
missing forms (Section 4). This is more widely know as the Paradigm Cell-
Filling Problem (PCFP) (e.g., Silfverberg and Hulden, 2018). We then evaluate
the system on it’s capacity to automatically generate inflections, and discuss
limitations of our current evaluation procedure (the evaluation step in Figure
2).

##  3 Challenges in Incorporating IGT into NLP Research

Because tokens in IGT are already segmented and annotated, it forms an
ostensibly convenient starting-point for further processing and token-based
grouping. In many ways, IGT is, however, a challenging data type for use in
pedagogical and NLP applications. This section presents three specific
challenges posed by IGT data when NLP techniques are applied. First, while IGT
will contain a wealth of useful information for NLP models, it might also
contain information which is far too fine-grained for automatic learning
purposes, at least given the quantity of data which are available. Second, IGT
often contain idiosyncratic or language-specific conventions which may not be
easily converted to or represented in standardized frameworks. Third, because
IGT is used as a device for language documentation, it will often contain
dialectal variation, an important meta-characteristic which in aggregate
cannot be easily distinguished from other types of variation or spelling
errors. We argue that handling these issues for successful data preprocessing
requires consultation with linguistic experts, and exemplify with instances
from the Gitksan IGT and our use-case.

###  3.1 Annotation Granularity

Documentary linguists' goals when annotating IGT is to present an accurate
representation of the surface phonology and morphology of a given utterance,
as well as the syntactic and semantic information contributed by its component
morphemes, with fine attention to detail given the rarity and value of the
data. This goal of providing fine-grained annotations and transcriptions,
however, can be in conflict with the NLP research aim of building models that
can generalize in the real world (i.e., future elicited linguistic data). The
fine-grained details are often extraneous for the purposes of building NLP
models, and can counterproductively act as noise that makes learning
systematic patterns more difficult.

As an example of this mismatch in disciplinary goals, consider the sample IGT
token in .

.maaxwsxwa
maaxws-xw-a
fallen.snow-val-attr
`white'

In this token, the productive stem is deconstructed into a historical root
(maaxws) and a derivational suffix (-xw)—along with an inflectional affix
(-a). It is unclear from the input that the most readily recognizable lexical
stem in this form is the larger unit maaxwsxw `white, snow-colored', and that
the internal boundaries within that stem reference etymological and
derivational information not relevant to the typical NLP task. The
derivational and inflectional affixes are not differentiated in IGT.555For an
English analogue, consider splitting the lexicalized verb enforce into a
prefix en- and root force. The en- prefix is recognizable, but not productive
or relevant to inflection tasks.

At first glance, it might seem reasonable to train an NLP model to
automatically generate such a gloss for Gitksan input words in an effort to
accelerate language documentation. While this remains one of the most common
NLP tasks associated with IGT, it may be difficult for models to deliver high
performance if the IGT input, like Gitksan's, contains a substantial
proportion of derivational and etymological information, since this
information is lexical and unpredictable.

Collaboration with documentary linguists, in addition to being important when
a project aims to improve the documentary linguistic workflow, can be useful
for identifying these aspects of the data which may be less valuable to learn.
This information can be applied in data preprocessing to improve model
performance given data scarcity. For the token in , an alternative
segmentation maaxwsxw-a into a word stem and a productive inflectional affix
white-ATTR is more amenable to both automated labeling and inflection tasks,
particularly in low-resource conditions. Furthermore, reference to
derivational information is unnecessary in our use case of performing
automated inflection for use in a pedagogical application. We collaborated
with documentary linguists familiar with Gitksan to manually filter morphology
into derivational versus inflectional, to determine whether an affix should be
classed as part of a lexical stem or should signal a paradigm cell in the
inflectional template. This allowed derivational morphology to be effectively
excluded before we moved to the paradigm cell-filling task. This filtering
process was non-trivial, requiring solid understanding of the target language,
its description, and its vocabulary.

###  3.2 Using Existing Annotation Standards

The annotation schemas employed in IGT are often idiosyncratic Palmer et al.
(2009); Comrie et al. (2015), which typically makes them better suited for
language documentation than NLP tasks. When aiming to leverage IGT data for
use in NLP tasks, we must then consider on a case-by-case basis whether it is
more beneficial to convert the IGT data to an NLP-standard format, or work
with the IGT annotations largely as-is, adapting them to our specific needs.
Relevant to this decision are factors such as how labor-intensive the
conversion will be, how well the standard format accommodates linguistic
information that has been detailed in the IGT, and whether conversion of the
dataset to the standard format aligns with specific project goals and speech
community interests.

The possible format that we consider for annotating inflection tables is the
Unimorph standard McCarthy et al. (2020a); Sylak-Glassman (2016), a popular
schema for annotation of inflectional morphology that can facilitate cross-
lingual transfer by enabling language-independent annotations. Ultimately, we
opted to adapt the Gitksan IGT to our specific needs after determining that
conversion would be extremely labor-intensive, and that several types of
information in the Gitksan IGT could not be represented in the UniMorph
standard. We present three of the most significant issues:

#### 1\. Part-of-Speech

The Unimorph standard relies on part-of-speech (POS) tags as a major component
of word form annotation. However, POS information is frequently not annotated
in IGT Moeller et al. (2020), and no POS information was included in our
Gitksan IGT.

For some underdocumented languages, POS information requires substantial
experience and manual attention to annotate. For example, our target language
Gitksan displays considerable category flexibility, meaning that syntactic and
morphological behavior can cross word class boundaries. In Gitksan, the
inflectional paradigms of nouns and verbs overlap substantially. As an
example, agreement markers can affix to both nouns and verbs, conveying a
number of functions. Some are exemplified in . As a consequence, in Gitksan it
is difficult to use morphological inflection to deduce a lexeme's POS.

. Forms with -'y (1sg series II) . hlguuhlxwi'y \- my child (possr) .̱ yee'y
\- I walked (abs) .̱ t'agi'y \- x forgot me (abs, dependent) .̱ t'agi'y \- I
forgot x (erg)

In addition, Gitksan nouns and verbs are syntactically flexible, meaning that
Gitksan nouns can function as verbs in text, and vice versa. For example, a
noun ganaa'w `frog' can be used predicatively without a copula in main verb
position in the sentence Hlaa ap ganaa'wi'y `I'm a frog now'. It takes
absolutive inflection when it does so. Due to this morphological and syntactic
flexibility, a 1SG-inflected noun like ganaa'wi'y could be annotated two ways
in UniMorph depending on the context (frog;PSS1S versus frog;1SG;ABS666Other
clause type features would be required here but it remains unclear how best to
represent Gitksan’s clause-typing system with UniMorph labels.)—yet in the
IGT, they are uniformly annotated as frog-1SG.II. Reviewing the contextual
function of every noun and verb in the IGT dataset to apply the appropriate
UniMorph tags would require an infeasible amount of expert reannotation.

#### 2\. Inflection vs. derivation

Unimorph postulates a strict division into inflectional and derivational
morphology (and only annotates inflectional morphology). The IGT format has no
such division, because it can be used to represent morphology at any level of
granularity the annotator wishes.

We have mentioned in Section 3.1 that determining the difference between
inflectional and derivational morphology from IGT input is non-trivial. For
example, the Gitksan morpheme -xw has a variety of uses which might be
considered more derivation-like (D) or more inflection-like (I).

  * •

Creating intransitive predicates from nouns: osxw `have a dog' from os `dog'
(D)

  * •

Marking inchoatives: mitxw `be full' vs. causative midin `fill' (D)

  * •

Marking passives: japxw `be made' from transitive jap `do, make' (D?)

  * •

Marking verbs with certain preverbs: sik'ihl huutxw `try to run away' vs. huut
`run away' (I?)

  * •

Optional in some possessives: laxyipxwsi'm `your.pl land' vs. laxyipsi'm
`your.pl land' (?)

This morpheme's uses and degree of productivity are still little-understood,
so its status as inflectional or derivational remains unclear.777Elsewhere,
some linguistic descriptions present cases of morphology which do not fit into
conventional delineations of the inflectional/derivational divide, such as
plural/pluractional markers in Halkomelem Salish (Wiltschko, 2008). For now,
we provisionally exclude this morpheme from our inflection tables as
`derivational'. In a UniMorph system, this morpheme's exclusion or inclusion
in the annotation would constitute a prematurely strong claim about whether it
was inflectional, and the tagset used to annotate it likewise a prematurely
strong claim about its function.

#### 3\. Clitics

Gitksan is rich in clitics, annotated with the equals sign in IGT `='. Their
attachment is determined by prosodic and linear factors. Pre-nominal clitics
are illustrated in example . .Giigwis Maryhl gayt.
giikw-i[-t]=s Mary=hl gayt
buy-tr-3.ii=pn Mary=cn hat
`Mary bought a hat.'

In the example above, the proper noun clitic =s attaches to the verb but is
syntactically associated with Mary. The common noun clitic =hl attaches to
Mary but is associated with gayt `hat'. Since UniMorph does not annotate such
cross-token dependencies (or other clitics), this central feature of Gitksan
cannot be represented.

#### Recommendations

Current computational morphology research relies heavily on standardized
tagsets like UniMorph, in particular for crosslingual transfer (Anastasopoulos
and Neubig, 2019). However, these formats can be either labor-intensive or
impossible to apply to underdocumented language datasets, depending on the
idiosyncratic conventions of a given IGT and language-specific factors. Our
understanding of the language may not be sufficiently mature to implement some
of UniMorph's strict requirements, or important phenomena may fall outside of
the defined scope of UniMorph. We recommend that NLP projects on
underdocumented languages collaborate with language experts to determine where
language-agnostic data formats can be applied, and to design project-specific
data formats as needed.

###  3.3 Dialectal variation

Dialectal variation is a pervasive feature of languages worldwide, from
English (consider African-American English and Standard American English;
Blodgett et al., 2016) to Arabic (consider Modern Standard Arabic and the Doha
dialect; Kumar et al., 2021). Many Indigenous languages of North America also
exhibit vast dialectal variety, with significant variance in the level of
mutual intelligibility between languages and dialects (Mithun, 2001, Ch.6).

Although Gitksan has an estimated fewer than 1K speakers, each village has a
different way of speaking, and the speech community recognizes two salient
dialects (Eastern/Upriver and Western/Downriver). Gitksan dialectal variation
is typically reflected in written materials due to the lack of a widely-
adopted orthographic standard which would `flatten' it.888Linguistic
description frequently aims to record dialectal and even speaker-level
variation. Our datasets are based on IGT data which explicitly annotates such
variation in the orthographic representation. For many underdocumented
languages, written orthographies have been in use for a relatively short
period of time, and communities place different levels of emphasis on literacy
and standardization versus conversational fluency. As a consequence,
orthographic conventions can vary widely across dialects and writers in low-
resource and underdocumented language contexts.

It is desirable in building inclusive language technology to accommodate and
reflect variation, rather than aim to model a homogenous standard form of the
language. In building pedagogical resources for language revitalization, we
furthermore need to mindfully consider potential data biases as well as what
kinds of variation are presented to the user, to avoid implictly suggesting
that certain dialects favored for preservation and teaching, which risks
reinforcing or creating negative social hierarchies (Demszky et al., 2021).

The first step to ensuring dialectal fairness and appropriate handling of
variation in NLP applications is to understand what types of variation are at
play, and in particular what dialect a given token belongs to. This allows us
to proactively control what data is presented to a user and, for example,
ensure that data from different dialects is not mixed together
inappropriately. This task is non-trivial: expertise in the language is
crucial in order to determine what types of variation are dialectal, and which
are idiosyncratic or purely orthographic, including typos and spelling errors.
As an example from Gitksan, gat and get are highly salient East/West dialect
variants, while hun and hon are less-salient variants within the Eastern
dialect; amxsiwaa and amxsiiwaa are two non-dialectal variants of the same
word (spelling error/variant), while sipxw and siipxw are different
lexemes.999In IGT the gloss cannot always be used to differentiate lexemes.
Depending on the convention, the same lexeme may appear with different glosses
in different contexts (e.g. ’wa: ‘find’ or ‘reach’), and different lexemes may
have the same gloss (e.g. yook and gup: ‘eat’, which differ on other grounds –
transitivity). The latter forms which share a gloss must also be
differentiated as lexical variants, not dialectal variants. Presently, we
include all lexeme variants as separate entries in our inflection tables,
enabling us to represent all dialects during training.

#### Recommendations

Distinguishing between different types of variation in the source material is
a challenging task but also a crucial one. Expertise in the target language
and dialects is required for classifying types of variation, and so language
experts are a vital asset for this process. Documentary linguists or community
members may have direct information about the dialectal background of speakers
that are represented in the data, which is useful for modeling, and will
likely have information about how dialectal variation is viewed in the speech
community (e.g. it may be highly politicized), which is important for
application design.

Variation is not only an important issue when constructing datasets. It is
also essential to evaluate the final model's performance according to the
principle of dialectal fairness (Choudhury and Deshpande, 2021) Recently,
measures for dialect fairness have emerged in the NLP community: Faisal et al.
(2021) and Kumar et al. (2021) advocate for computing performance separately
for each dialect rather than computing a single macro average performance
figure over distinct dialects. They also propose to use standard deviation
between system performance on different dialects and the generalized entropy
index Speicher et al. (2018) as measures for dialectal unfairness which we
naturally want to minimize.

##  4 Steps toward Building a Language Learning Application

The inflectional paradigms collected from the adapted IGT corpus are overly
sparse for automatically generating pedagogical exercises. To automatically
fill in these paradigms, an example of which is shown in Appendix B, we train
and evaluate a morphological reinflection system.101010Code and data for this
experiment is available at https://github.com/smfsamir/gitksan-data.

#### Data

We train and test reinflection models on the Gitksan morphological paradigms
described in Section 3. We generate three splits of the data from our complete
set of paradigms: train ($N=858$ word forms), validation ($N=302$ word forms),
and test ($N=124$ word forms) data splits.

#### Training

We form training pairs by using the given forms in each table and learn to
reinflect each given form in a table to another given form in the same table,
following Silfverberg and Hulden (2018). Model parameters are shown in
Appendix C.

#### Evaluation

During test time, we predict forms for missing slots based on each of the
given forms in the table and take a majority vote of the predictions. We
evaluate accuracy on the test set by counting the number of the $124$ forms
that were correctly predicted. We find that the Transformer model generates
87.09% of the test forms correctly.

#### Analysis.

Our model provides strong performance when measured by the standard metric of
accuracy, in particular considering that it is trained on only $858$ examples.
Accuracy, however, only provides one perspective on the efficacy of the model
(Ethayarajh and Jurafsky, 2020). The appropriate evaluation of the system is
highly context dependent: For our goal of generating language learning
exercises, we want to evaluate whether our system and automatically generated
grammar exercises allow for more effective language learning; raw accuracy
gleans little insight to the effectiveness of the system for this goal. If in
contrast our goal was to facilitate language documentation, we would want to
evaluate whether the model gives an overall significant reduction in
documentation effort—this largely depends on whether the automatic annotations
are of sufficient quality that correcting remaining errors takes less time
than annotating all the data from scratch. Further research, in collaboration
with documentary linguists and the speech community, is required to determine
whether our system can achieve the desired goals of building more practical,
inclusive language technology.

##  5 Discussion

#### Incorporating IGT data for NLP

Language documentation provides a valuable data source for many so called
``left-behind'' languages (Joshi et al., 2020), which lack traditional
annotated and unannotated NLP datasets. For example, IGT data can be used to
train systems for morphological inflection, segmentation and automatic
glossing, among other applications. Nevertheless, the annotations in IGT are
rarely ideally suited for typical NLP tasks, and may need to be significantly
adapted. This will typically be hard without extensive knowledge of the target
language and annotation conventions which were employed when the IGT data were
generated. Linguists and community language experts are well-positioned to
address questions related to IGT usability, the structure of the target
language, variation in the data, and other annotations in the source data.
Collaboration with language experts is not only vital for successful data
preprocessing and conversion to the formats required for the typical NLP task,
but can also naturally help define research goals and drive the project toward
them.

Inclusive Research Goals NLP technologies for underdocumented languages have
the capacity to speed up language documentation (e.g., Anastasopoulos, 2019);
assisting language revitalization (e.g., Rijhwani et al., 2020; Lane and Bird,
2020); and creating digital infrastructure (e.g., Anastasopoulos and Neubig,
2019). These high-level goals are only a part of what it may mean to create
inclusive language technology. Equally valuable as a research goal may be
inclusion: for speech communities to be acknowledged and engaged in the course
of the the research project.111111Underdocumented languages are often the
cultural heritage of typically marginalized peoples, sometimes with a history
of their data being exploited for political or commercial purposes. NLP
research without community involvement may feel like a continuation of this
pattern. We encourage NLP projects on low-resource, minoritized, and/or
endangered languages to begin by understanding the speech community context,
proceed with community collaboration or endorsement, and ultimately produce
concrete benefits that speech communities recognize. This might include
outcomes for language teaching and pedagogy, or training opportunities in
technology or research.

Evaluation methods can be compiled which address NLP researchers, linguists,
and communities' overlapping and divergent goals. For example, pedagogical
tools can be directly evaluated for dialect fairness and user/learner
improvement.

Practical Collaboration We suggest seeking out opportunities to collaborate
directly with community members, in order to solicit their specific expertise
when setting the research agenda (i.e. task definition) and conducting
evaluation (Czaykowska-Higgins, 2009; Bird, 2020). When the NLP researcher has
no existing contact or history with the speech community, this can be pursued
via collaboration with a documentary linguist with established community
relationships and a similar desire to engage in this research model. Recognize
that in any collaboration, different individuals contribute different skills
and experience (e.g. pedagogy, annotation, knowledge of community attitudes)
and may have different goals and preferred ways of participating, which should
simply be discussed within the partnership to ensure things run smoothly.

Research accessibility In discussing inclusive language technologies, we also
consider the accessibility of NLP workshops to speech communities, in
particular where venues have a dedicated focus on low-resource languages. We
note that such venues are often inaccessible to communities due to factors
such as the cost of registration. Similarly-oriented workshops in linguistics
(e.g. SAIL, WSCLA, family-specific conferences) typically have a tiered
registration structure enabling community members to attend for free or
minimal cost (e.g. $25). It is worth recognizing that community members are
research stakeholders, and ensuring that venues are open to their
participation.

##  6 Conclusion

Although a majority of the world's languages lack the kind of large annotated
and massive unannotated datasets which are used to train modern NLP models for
high-resourced languages like English (Joshi et al., 2020; Blasi et al.,
2021), many languages have other potential data sources such as language
documentation data, which so far have remained under-explored. However, care
must be taken when applying this type of data, which originally is not
intended for NLP use. This is important to ensure that the resulting
technologies actually achieve their intended goals like accelerated language
documentation or genuinely helpful computer-assisted language learning.

Collaboration with linguists can provide the expertise necessary to engage in
modeling with IGT data for underdocumented languages. Linguists can help
define an NLP task with good value propositions, given their familiarity and
connections with the speech community. They can provide guidance on navigating
the IGT format so that we can extract the most useful information for the task
at hand. Finally, they can assist in evaluating whether the model achieves
appropriate performance on the speech community use cases, and provide
feedback on metrics for model success and fairness across dialects. Throughout
the development process, documentary linguists and speech community members
should be consulted. This will further a greater understanding of the source
data and lead to more equitable and effective technologies.

##  7 Acknowledgements

We want to thank Henry Davis, Lisa Matthewson and the Gitksan research lab at
the Department of Linguistics at UBC for generous help with this project and
access to Gitksan IGT data. We also want to thank for anonymous reviewers for
valuable comments. We also want to thank Samantha Quinto for assisting with
visual design. This research was supported by funding from the National
Endowment for the Humanities (Documenting Endangered Languages Fellowship) and
the Social Sciences and Humanities Research Council of Canada (Grant
430-2020-00793). Any views/findings/conclusions expressed in this publication
do not necessarily reflect those of the NEH, NSF or SSHRC.

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##  Appendix A Sample IGT data

The first four lines of a sample text from the Gitksan interlinear glossed
text corpus. This example is revised from initial publication in Forbes et al.
(2017).

.[]Dim mehldi'y wila wilhl win hii hagun bekwhl mismaaxwsxum get go'ohl
ts'ebim Gitwinhlguu'l gik'uuhl.
dim mehl-T-i-'y wila wil=hl win hii hogun bekw=hl CVC~maaxws-xw-m get go'o=hl
ts'ep-m Gitwinhlguu'l gi-k'uuhl
PROSP tell-T-TR-1SG.II MANR be/do=CN COMP initially toward arrive.PL[-3.II]=CN
PL~fallen.snow-VAL-ATTR people LOC[-3.II]=CN community-ATTR Gitwinhlguu'l
prior-year
I will tell about when the white men first came to Kitwancool long ago.

.[]Ha'on dii 'nekw hlidaa bekwhl get dipun, ii sagaytgoodindiithl hli gedihl
Gitwinhlguu'l.
ha'on dii 'nekw hli=da bekw=hl get dip=un ii sagayt-gooda-in-diit=hl hli
get-T=hl Gitwinhlguu'l
not.yet FOC long PART=SPT arrive.PL[-3.II]=CN people ASSOC=DEM.PROX CCNJ
together-all.gone-CAUS2-3PL.II=CN PART people-T[-3.II]=CN Gitwinhlguu'l
Not long after these people arrived, they gathered together the people of
Kitwancool.

.[]Hasakdiit dimt mehldiit win hlaa dim sii ha'niijokt go'ohl win t'aahl
galts'ephl Gitwinhlguu'l.
hasak-diit dim=t mehl-T-diit win hlaa dim sii ha-'nii-jok-t go'o=hl win
t'aa=hl gal-ts'ep=hl Gitwinhlguu'l
desire-3PL.II PROSP=3.I tell-T-3PL.II COMP INCEP PROSP new INS-on-dwell-3.II
LOC[-3.II]=CN COMP sit[-3.II]=CN container-community[-3.II]=CN Gitwinhlguu'l
They wanted to tell about the new place where the village of Kitwancool is to
be.

.[]'Nit sagootxwhl "government" siwatdiit, ii dim 'nii wenhl dim jokhl
aluugiget go'ohl lax "reserve" siwatdiit.
'nit si-goot-xw=hl *government si-wa-T-diit ii dim 'nii wen=hl dim jok=hl
aluu-CV~get go'o=hl lax *reserve si-wa-T-diit
3.III CAUS1-heart-VAL[-3.II]=CN *government CAUS1-name-T[-TR]-3PL.II CCNJ
PROSP on sit.PL[-3.II]=CN PROSP dwell[-3.II]=CN clearly-PL~people
LOC[-3.II]=CN on *reserve CAUS1-name-T[-TR]-3PL.II
The plan of the so-called government was that they will have Indian people
live on a so-called reserve.

##  Appendix B Sample inflection table

A Gitksan inflection table for 'wa (`to find, reach') generated from IGT and
displayed in TSV format. Many cells in the table are empty since they were
unattested in the IGT data.

    ROOTΨfindΨ’waΨ’waΨ’wa
    ROOT-SXΨ_Ψ_Ψ_Ψ_
    ROOT-PLΨ_Ψ_Ψ_Ψ_
    ROOT-3PLΨ_Ψ_Ψ_Ψ_
    ROOT-ATTRΨ_Ψ_Ψ_Ψ_
    ROOT-3.IIΨfind-3.IIΨ’wa-tΨ’watΨ’wa-3.II
    ROOT-PL-SXΨ_Ψ_Ψ_Ψ_
    ROOT-1SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-2SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-2PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-3PL.IIΨfind-3PL.IIΨ’wa-diitΨ’wadiitΨ’wa-3PL.II
    ROOT-1PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-3PLΨ_Ψ_Ψ_Ψ_
    ROOT-TR-3.IIΨfind-TR-3.IIΨ’wa-i-tΨ’wayitΨ’wa-TR-3.II
    ROOT-PL-3.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-ATTRΨ_Ψ_Ψ_Ψ_
    ROOT-PL-2SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-TR-1SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-3PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-1SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-TR-1PL.IIΨfind-TR-1PL.IIΨ’wa-i-’mΨ’wayi’mΨ’wa-TR-1PL.II
    ROOT-PL-1PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-TR-2PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-TR-3PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-TR-2SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-3.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-2SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-3PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-1SG.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-1PL.IIΨ_Ψ_Ψ_Ψ_
    ROOT-PL-TR-2PL.IIΨ_Ψ_Ψ_Ψ_

##  Appendix C Fairseq parameters

#### Model

We use the Fairseq Ott et al. (2019) model implementation of Transformer
Vaswani et al. (2017). Both the encoder and decoder have $4$ layers with 4
attention heads, an embedding size of $256$ and hidden layer size of $512$. We
train with the Adam optimizer starting of the learning rate at $0.001$. We
chose the batch size ($400$) and maximum updates ($20000$) based on the
highest accuracy on the development data.


# Characteristic Modes of Nonreciprocal Structures

Niklas Wingren, , Daniel Sjöberg, ,
Mats Gustafsson, , Johan Lundgren, ,
Miloslav Capek, , Lukas Jelinek, and Kurt Schab Manuscript received ; revised
. This work was supported by the Czech Science Foundation under project No.
21-19025M, and the Swedish Governmental Agency for Innovation Systems (Vinnova
NFFP7-2023-01204).N. Wingren, D. Sjöberg, M. Gustafsson, and J. Lundgren are
with Lund University, Lund, Sweden, (e-mails: {niklas.wingren, daniel.sjoberg,
mats.gustafsson, johan.lundgren}@eit.lth.se).L. Jelinek and M. Capek are with
Czech Technical University in Prague, Czech Republic (e-mails:
{lukas.jelinek,miloslav.capek,}@fel.cvut.cz).K. Schab is with Santa Clara
University, Santa Clara, USA (e-mail: kschab@scu.edu).

###### Abstract

The scattering formulation of characteristic mode decomposition is utilized to
extend modal analysis to lossless scatterers breaking time-reversal symmetry.
This enables characteristic modes analysis on devices containing gyrotropic or
moving media. The resulting nonreciprocity introduces features not observed in
reciprocal scenarios, such as asymmetric phase progression in characteristic
far fields. These new phenomena are carefully discussed using examples of
varying complexity. Indicators of nonreciprocity based on modal data are also
introduced.

###### Index Terms:

Antenna theory, eigenvalues and eigenfunctions, computational
electromagnetics, characteristic modes, scattering.

##  I Introduction

Characteristic modes are an intensively studied branch of classical
electrodynamics with applications in antenna and scattering theory [1].
Commonly, characteristic mode analysis is associated with the method of
moments and the study of perfectly conducting bodies [2, 3], but its original
scattering formulation [4, 5], recently expanded and formalized in [6, 7, 8],
significantly broadens the application of characteristic modes to scatterers
comprised of arbitrary materials. This immediately raises questions about the
properties of characteristic modes in the presence of exotic materials, such
as those with bianisotropic response and/or those in which nonreciprocity
stems from breaking time-reversal symmetry [9]. These scenarios emerge when
studying antennas on gyrotropic substrates [10, 11, 12] or material bodies in
motion [13], and give characteristic modes additional parameterization in
quantities such as the intensity of a biasing field or velocity. This letter
initiates a discussion on the characteristic modes of lossless nonreciprocal
scatterers by showing their fundamental properties and examining indicators of
modal nonreciprocity.

##  II Scattering-based characteristic modes

This paper uses a formulation of characteristic modes based on an object’s
scattering matrix [4] recently formalized in [6, 7, 8]. In contrast to
impedance formulations [2, 14], the scattering approach avoids problems
reported in [6] in cases where the radiation matrix is not equal to the real
(Hermitian) part of the impedance matrix, see [2, Eqs. (10)–(11)].

Using the scattering matrix $\mathbf{S}$ of a linear material object [4],
characteristic modes are defined by the eigenvalue problem [15, 6]

$\mathbf{S}\mathbf{a}_{n}=s_{n}\mathbf{a}_{n},$ (1)

where the complex eigenvalue $s_{n}=1+2t_{n}$ is commonly parametrized by a
real characteristic number $\lambda_{n}$ via relation
$s_{n}=-(1-\mathrm{j}\lambda_{n})/(1+\mathrm{j}\lambda_{n})$ and where
eigenvectors $\mathbf{a}_{n}$ collect the expansion coefficients of the fields
in the selected basis, such as spherical vector waves commonly used in the
case of finite scatterers in free space [16]. The modal significance of an
eigenvector is denoted as
$\left|t_{n}\right|=\left|1+\mathrm{j}\lambda_{n}\right|^{-1}=\left|s_{n}-1\right|/2$.
Other characteristic quantities, such as the characteristic currents, can be
computed by exciting the system with the vector $\mathbf{a}_{n}$ and
evaluating the desired quantity [6].

For lossless scatterers, the scattering matrix is unitary,
$\mathbf{S}^{\mathrm{H}}\mathbf{S}=\mathbf{1}$, and pairs of characteristic
excitations can be made orthonormal, i.e.,

$\mathbf{a}_{m}^{\mathrm{H}}\mathbf{a}_{n}=\delta_{mn},$ (2)

where $\delta_{mn}$ is the Kronecker delta and H denotes Hermitian transpose.

##  III Characterizing Reciprocity and Nonreciprocity

Characteristic modes of a lossless scatterer satisfy [6, App. B]

$\mathbf{S}^{\mathrm{T}}\mathbf{a}_{n}^{*}=s_{n}\mathbf{a}_{n}^{*}$ (3)

due to the unitarity of $\mathbf{S}$. The scattering matrix is symmetric
$\mathbf{S}^{\mathrm{T}}=\mathbf{S}$ for reciprocal scatterers, in which case
both $\mathbf{a}_{n}$ and $\mathbf{a}_{n}^{*}$ are eigenvectors of
$\mathbf{S}$ with the same eigenvalue. Hence, the eigenvectors can always be
made real-valued111Eigenvectors for non-degenerate cases are generally
equiphase (with orthonormal $\mathbf{a}_{n}$ satisfying
$|\mathbf{a}_{n}^{\mathrm{T}}\mathbf{a}_{n}|=1$). For a degenerate case, the
modes can exhibit phase progression, but linear combinations of vectors
$\mathbf{a}_{n}$ and $\mathbf{a}_{n}^{*}$ can be used to produce real-valued
eigenvectors., akin to the treatment of characteristic currents [2]. Due to
symmetries of vector spherical waves [16, Sec. 7.8], the real-valued
characteristic vectors $\mathbf{a}_{n}$ imply conjugate inversion symmetry
[17, Eqs. (29)–(30)] of the characteristic far fields

$\boldsymbol{F}_{n}(-\hat{\boldsymbol{r}})=-\boldsymbol{F}_{n}^{\ast}(\hat{\boldsymbol{r}}),$
(4)

where $\hat{\boldsymbol{r}}$ is an observation direction
$\hat{\boldsymbol{r}}=\boldsymbol{r}/r$ and the far field $\boldsymbol{F}$ is
defined by the scattered field $\boldsymbol{E}_{\mathrm{s}}$ as
$\boldsymbol{F}(\hat{\boldsymbol{r}})=r\,\mathrm{exp}\left\\{\mathrm{j}kr\right\\}\boldsymbol{E}_{\mathrm{s}}(\boldsymbol{r})$
for $r\to\infty$. This property also follows from selecting real-valued
currents in reciprocal EFIE formulations of characteristic modes222The
integral relationship between a real-valued current distribution and its
radiated far-field [18, §3.5-3.6] illustrates this property..

The above-mentioned properties of reciprocal systems (real-valued
eigenvectors, far-field inversion symmetry) are lost in nonreciprocal systems
where $\mathbf{S}\neq\mathbf{S}^{\mathrm{T}}$, and their absence can be
utilized as distinguishing features to characterize nonreciprocity. To detect
nonreciprocity of characteristic modes, the scattering matrix $\mathbf{S}$ is
decomposed into its symmetric and antisymmetric parts as
$\mathbf{S}=(\mathbf{S}+\mathbf{S}^{\mathrm{T}})/2+(\mathbf{S}-\mathbf{S}^{\mathrm{T}})/2$.
This implies that the cycle-mean outgoing power $P_{\mathrm{o}}$ can be
written as

$P_{\mathrm{o}}=\dfrac{1}{2}|\mathbf{S}\mathbf{a}|^{2}=\dfrac{1}{8}\left|(\mathbf{S}+\mathbf{S}^{\mathrm{T}})\mathbf{a}\right|^{2}\\\
+\dfrac{1}{8}\left|(\mathbf{S}-\mathbf{S}^{\mathrm{T}})\mathbf{a}\right|^{2}+\dfrac{1}{4}\left(\left|\mathbf{S}\mathbf{a}\right|^{2}-\left|\mathbf{S}^{\mathrm{T}}\mathbf{a}\right|^{2}\right).$
(5)

We denote the first term on the right-hand side as the reciprocal outgoing
power $P_{\mathrm{o}}^{\mathrm{r}}$, while the second row corresponds to the
nonreciprocal outgoing power $P_{\mathrm{o}}^{\mathrm{nr}}$. We then propose
the ratio

$\nu=\dfrac{P_{\mathrm{o}}^{\mathrm{nr}}}{P_{\mathrm{o}}}\in\left[0,1\right]$
(6)

to characterize the presence of nonreciprocity for a given excitation at a
particular frequency. Furthermore, modal far fields $\boldsymbol{F}_{n}$
violating (4), such as those with asymmetric phase progression, are an
indicator of nonreciprocal behavior. Together, the study of the quantity $\nu$
and modal phase progression serve to classify individual characteristic modes
as either reciprocal or nonreciprocal. Another classification of
nonreciprocity based on the extinction cross section has recently been
proposed in [19], but is not readily applicable to characteristic modes.

##  IV Examples

###  IV-A Ideal circulator

As an initial example demonstrating characteristic modes for nonreciprocal
systems, we consider the scattering matrix of an ideal $N$-port circulator
[20, §7.1]

$\mathbf{S}=\begin{bmatrix}0&1&0&\cdots&0\\\ 0&0&1&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&1\\\
1&0&0&\cdots&0\end{bmatrix}.$ (7)

The characteristic modes (in this case, eigenmodes of the matrix $\mathbf{S}$
[6]) associated with this system [21] can be classified as progressive phase
modes

$\displaystyle\mathbf{a}_{\pm p}$
$\displaystyle=N^{-1/2}\left[1,\mathrm{e}^{\pm\mathrm{j}2p\pi/N},\cdots,\mathrm{e}^{\pm\mathrm{j}2p\pi(N-1)/N}\right]^{\mathrm{T}},$
(8) $\displaystyle s_{\pm p}$
$\displaystyle=\mathrm{e}^{\pm\mathrm{j}2p\pi/N},\quad 1\leq p\leq(N-1)/2,$

with $|\mathbf{a}_{\pm p}^{\mathrm{T}}\mathbf{a}_{\pm p}|=0$, an equiphase
even mode

$\mathbf{a}_{\mathrm{e}}=N^{-1/2}\left[1,1,\cdots,1\right]^{\mathrm{T}},\quad
s_{\mathrm{e}}=1,$ (9)

with $|\mathbf{a}_{\mathrm{e}}^{\mathrm{T}}\mathbf{a}_{\mathrm{e}}|=1$, and an
equiphase odd mode (even $N$ only)

$\mathbf{a}_{\mathrm{o}}=N^{-1/2}\left[1,-1,\cdots,1,-1\right]^{\mathrm{T}},\quad
s_{\mathrm{o}}=-1$ (10)

with $|\mathbf{a}_{\mathrm{o}}^{\mathrm{T}}\mathbf{a}_{\mathrm{o}}|=1$.
Application of the definition in (6) to each class of characteristic modes
gives

$\nu_{\pm
p}=\sin^{2}(2p\pi/N),\quad\nu_{\mathrm{e}}=0,\quad\nu_{\mathrm{o}}=0.$ (11)

Similarly, the modal significance of the modes reads

$\left|t_{\pm
p}\right|=\left|\sin(p\pi/N)\right|,\quad\left|t_{\mathrm{e}}\right|=0,\quad\left|t_{\mathrm{o}}\right|=1.$
(12)

These results show that characteristic modes of nonreciprocal devices (such as
this ideal circulator) can exhibit a range of values of the parameter $\nu$,
including particular excitations which, to the outside observer, cause the
system to behave in a completely reciprocal manner with $\nu=0$, cf., the
equiphase even / odd modes in (11). In contrast, the progressive phase modes
$\mathbf{a}_{\pm p}$ cannot be normalized to be real-valued and exhibit non-
zero values $\nu$, both indicators of non-reciprocal behavior.

Scattering or transition matrices analogous to (7) can be used to describe
hypothetical objects exhibiting nonreciprocal coupling between the incident
and scattered spherical waves, and a scattering dyadic can be constructed to
describe non-reciprocal scattering on a direction-by-direction basis [8].

###  IV-B Nonreciprocal subcircuit

The second example deals with a free-space scatterer in which nonreciprocity
is induced by the presence of an ideal lumped circulator. Specifically, the
circulator is connected to delta-gap ports of three thin-strip dipoles of
length $L$ and width $L/50$, see the inset in Fig. 1. The dipoles are made of
a perfect electric conductor, are parallel to the $z$-axis, and are positioned
so that their centers lie on the circle with radius $L/8$. The dipoles are
distributed uniformly along this circle. The circulator is placed at the
origin and is connected with the dipole ports via radial transmission lines of
length $L/8$ and characteristic impedance $Z_{0}=50\,\mathrm{\Omega}$ matching
the system impedance of the ideal circulator. The scattering scenario is
modeled using a combination of the method-of-moments [22] and the scattering
matrix description of a three-port circuit [20].

Figure 1: (Top) Modal significances of three parallel dipoles connected to an
ideal circulator. (Bottom) Nonreciprocity ratio $\nu$ corresponding to modes
depicted in the top panel.

Modal significances of this system are depicted in the top panel of Fig. 1,
while the nonreciprocity ratio $\nu$ is shown in the bottom panel of Fig. 1.
The color coding of both figures is identical. Most of the modal spectrum
shows negligible values of the ratio $\nu$. Exceptions are peaks at
$ka\approx\left\\{1.40,1.54,3.73\right\\}$, see black circle markers at the
top of the plots, also exhibiting high modal significance in the same modes.
The other modes with high modal significance, such as those belonging to the
red dotted line in Fig. 1, show reciprocal behavior with $\nu=0$ and symmetric
equiphase radiation patterns satisfying (4).

The nonreciprocal effects associated with the two peaks at $ka\approx 1.40$
and $ka\approx 1.54$ are best observed in the modal far fields, which are
depicted in Fig. 2 as a directivity pattern and in Fig. 3 as a phase of the
$\theta$ component of the electric far field in the $xy$-plane
($\theta=\pi/2$). As a reference, the right panel in Fig. 2 shows the
radiation pattern of an omnidirectional (inversely symmetric) significant
reciprocal mode at $ka\approx 1.68$. The dashed red trace in Fig. 3 also shows
that this radiation pattern is equiphase. It can be seen that both
nonreciprocal examples share features which contrast with the reciprocal mode.
Notably, their radiation diagrams lack inversion symmetry (4) and exhibit
$120^{\circ}$ rotational $\mathrm{C}_{3z}$ symmetry and phase progression
along angular direction $\varphi$. These modal radiation patterns represent
field rotation around the $z$-axis, a property that is not present around
reciprocal scatterers. Note that because of the $\mathrm{C}_{3z}$ symmetry of
this particular system, its characteristic modes bear a strong resemblance to
the $p=\pm 1$ modes of the ideal circulator discussed in Sec. IV-A, with
$120^{\circ}$ phase offsets between radiated fields at locations separated by
that same angular separation in the $xy$ plane.

Figure 2: Directivity patterns corresponding to solid orange peak at
$ka\approx 1.40$, dashed blue peak at $ka\approx 1.54$, and dotted red peak at
$ka\approx 1.68$ in Fig. 1, see black circle markers at the top of the plot.
Figure 3: Phase of electric far-field corresponding to Fig. 2. Only
$\theta$-component in the $xy$ plane is shown.

###  IV-C General nonreciprocal media

The last example presents characteristic modes of a penetrable body made of a
nonreciprocal medium, showing the generality of the scattering-based
formulation of characteristic modes for systems not readily analyzed using
impedance-based methods. Consider a cylinder of radius $\rho$ and height $h$,
centered at the origin with its axis lying along axis $z$, filled with a
homogeneous, real-valued permittivity
$\varepsilon_{\mathrm{r}}\varepsilon_{0}$ and permeability
$\mu_{\mathrm{r}}\mu_{0}$. Let the cylinder rotate around its axis with
angular frequency $\omega$ such that the velocity of a point of the cylinder
reads
$\boldsymbol{v}=\omega\left(\hat{\boldsymbol{y}}x-\hat{\boldsymbol{x}}y\right)$.
The maximum speed is assumed to be much smaller than the speed of light
$c_{0}=1/\sqrt{\mu_{0}\varepsilon_{0}}$, i.e.
$\beta_{\rho}=\omega\rho/c_{0}\ll 1$.

The unidirectional movement of the cylinder breaks time-reversal symmetry and
introduces nonreciprocity analogous to the motion of fluid studied in [9, 23].
To study the electromagnetic scattering from such an object and to evaluate
its characteristic modes, the rotating cylinder can be replaced by a
stationary cylinder with constitutive relations [13, 24]

$\displaystyle\boldsymbol{D}$
$\displaystyle=\boldsymbol{\varepsilon}\cdot\boldsymbol{E}+c_{0}^{-1}\boldsymbol{\xi}\cdot\boldsymbol{H}$
(13) $\displaystyle\boldsymbol{B}$
$\displaystyle=c_{0}^{-1}\boldsymbol{\varsigma}\cdot\boldsymbol{E}+\boldsymbol{\mu}\cdot\boldsymbol{H}$

where
$\boldsymbol{\varepsilon}=\varepsilon_{\mathrm{r}}\varepsilon_{0}\mathbf{1}$
and $\boldsymbol{\mu}=\mu_{\mathrm{r}}\mu_{0}\mathbf{1}$ with $\mathbf{1}$
denoting the identity tensor and (using the Cartesian coordinate system)

$\boldsymbol{\xi}=\omega\dfrac{\varepsilon_{\mathrm{r}}\mu_{\mathrm{r}}-1}{c_{0}}\matrixquantity(0&0&x\\\
0&0&y\\\ -x&-y&0),\quad\boldsymbol{\varsigma}=-\boldsymbol{\xi}.$ (14)

The constitutive relations above describe a lossless
($\boldsymbol{\varepsilon}=\boldsymbol{\varepsilon}^{\mathrm{H}}$,
$\boldsymbol{\mu}=\boldsymbol{\mu}^{\mathrm{H}}$,
$\boldsymbol{\varsigma}=\boldsymbol{\xi}^{\mathrm{H}}$), but inhomogeneous and
nonreciprocal ($\boldsymbol{\varsigma}\neq-\boldsymbol{\xi}^{\mathrm{T}}$)
medium [16].

A common benchmarking example for characteristic modes is a dielectric
cylinder of radius $\rho=5.25\,\mathrm{mm}$ and height $h=4.6\,\mathrm{mm}$
with $\varepsilon_{\mathrm{r}}=38$, and $\mu_{\mathrm{r}}=1$ [7]. Applying the
transformation in (13)–(14), the characteristic modes of this system are
computed for various relative speeds $\beta_{\rho}\leq 0.01$ using the
iterative method [25], which employs a finite element-boundary integral hybrid
solver capable of computations involving general bianisotropic media [26].
Speeds of this order have been shown to significantly impact the scattering
characteristics compared to the stationary case [24].

Modal significances of the cylinder at three different $\beta_{\rho}$ are
shown in Fig. 4. The resonant frequencies of the modes at different
$\beta_{\rho}$ are shown in Table I for clarity. For the stationary case
($\beta_{\rho}=0$), the results match those in [7]. There are six modes
displayed, two of which are rotationally symmetric (TE01 and TM01) and four of
which are degenerate pairs (HEM11 and HEM12). All modes have $\nu=0$ and
exhibit symmetric far fields satisfying (4). As rotation increases
($\beta_{\rho}=0.005$ and $\beta_{\rho}=0.01$), the symmetric TE01 and TM01
modes are relatively unaffected and preserve their symmetry properties, as
their symmetry axis matches the axis of rotation. In contrast, the mode pairs
which are degenerate at $\beta_{\rho}=0$ show level splitting with their
frequency separation increasing for higher $\beta_{\rho}$, similarly to the
case studied in [9].

For $\beta_{\rho}=0.01$, modal significances and $\nu$ are shown in Fig. 5 for
the 10 first modes. It is seen that non-zero $\nu$ is present in all modes
which exhibit level splitting. It is, however, also seen that the mode drawn
as a blue line has significant $\nu$ at $f>7.62\,\mathrm{GHz}$. The modal
significance of this mode matches the TM01 at $f<7.62\,\mathrm{GHz}$, but is
swapped with the solid purple line at $f>7.62\,\mathrm{GHz}$ due to crossing
avoidance, which occurs in nonsymmetric meshes [27]. This forces the far field
of the blue line to abruptly change from rotationally symmetric to non-
symmetric, which is accompanied by a similarly abrupt change in $\nu$. The
spikes in $\nu$ for the solid green and red lines at $f=6.67\,\mathrm{GHz}$
are also related to the tracking as they correspond to a point where crossing
avoidance does not happen due to coarse frequency resolution. Together, these
observations show how the characterization of nonreciprocity in characteristic
modes is sensitive to tracking.

Figure 4: Modal significances of a rotating cylinder for different
$\beta_{\rho}$. The first 6 modes are displayed, corresponding to TE01
(black), HEM11 (red), HEM12 (green) and TM01 (blue). Both HEM modes are
degenerated at $\beta_{\rho}=0$.

Figure 5: (Top) Modal significance of a rotating cylinder with $\beta_{\rho}=0.01$. (Bottom) Nonreciprocity ratio $\nu$ corresponding to modes depicted in the top panel. Solid/dashed lines of the same color are degenerate modes at $\beta_{\rho}=0$. TABLE I: Resonant frequencies in GHz of modes in Fig. 4 at different $\beta_{\rho}$. $\beta_{\rho}$ | TE01 | HEM11 | HEM12 | TM10
---|---|---|---|---
0 | 4.86 | 6.35 | 6.35 | 6.63 | 6.63 | 7.53
0.005 | 4.86 | 6.30 | 6.39 | 6.59 | 6.68 | 7.53
0.01 | 4.86 | 6.27 | 6.44 | 6.55 | 6.73 | 7.54

The far field corresponding to one of the nonreciprocal modes is illustrated
in Fig. 6 for the cylinder stationary and rotating with $\beta_{\rho}=0.01$.
It is seen that the far field can be selected to be purely real with inversion
symmetry (4) in the stationary case, but not in the rotating case. In the
stationary case, the HEM mode is degenerated with the far field of the other
degenerate mode being shifted $90^{\circ}$ in the $\varphi$ coordinate. While
the two far fields in Fig. 6 look very different, it is noted that a far field
very similar to that at $\beta_{\rho}=0.01$ can be obtained by linearly
combining the two degenerate modes at $\beta_{\rho}=0$ with one mode phase
shifted $\pi/2$.

Figure 6: Far field for the HEM11 mode corresponding to the solid red line in
Fig. 4 at two different $\beta_{\rho}$. Color images show the amplitude of the
far field normalized to its peak value, while arrows indicate the directions
of real (black) and imaginary (blue) parts.

##  V Discussion and Conclusions

This letter describes how characteristic mode analysis can be applied to
nonreciprocal devices, thereby extending the established theoretical and
computational state-of-the-art. This advance enables the analysis of problems
involving gyrotropic materials, moving media, and lumped non-reciprocal
devices. The decomposition employs scattering matrices, which can be computed
numerically in arbitrary full-wave solvers. Certain cases, such as a spherical
shell filled with bianisotropic material, even retain the potential for
analytical solutions [16].

Examples discussed throughout the letter illustrate how specific modal
properties are lost when a system exhibits nonreciprocal behavior. Modal
nonreciprocity varies with frequency, hindering straightforward single-
frequency classification of a mode as being (non)reciprocal. One remaining
challenge lies in defining a measure that effectively quantifies modal
nonreciprocity.

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