# An Exploratory Analysis of Multilingual Word-Level Quality Estimation with
Cross-Lingual Transformers

Tharindu Ranasinghe♢, Constantin Orăsan♡ and Ruslan Mitkov♢
♢Research Group in Computational Linguistics, University of Wolverhampton, UK
♡Centre for Translation Studies, University of Surrey, UK
{t.d.ranasinghehettiarachchige<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract

Most studies on word-level Quality Estimation (QE) of machine translation
focus on language-specific models. The obvious disadvantages of these
approaches are the need for labelled data for each language pair and the high
cost required to maintain several language-specific models. To overcome these
problems, we explore different approaches to multilingual, word-level QE. We
show that these QE models perform on par with the current language-specific
models. In the cases of zero-shot and few-shot QE, we demonstrate that it is
possible to accurately predict word-level quality for any given new language
pair from models trained on other language pairs. Our findings suggest that
the word-level QE models based on powerful pre-trained transformers that we
propose in this paper generalise well across languages, making them more
useful in real-world scenarios.

##  1 Introduction

Quality Estimation (QE) is the task of assessing the quality of a translation
without having access to a reference translation Specia et al. (2009).
Translation quality can be estimated at different levels of granularity: word,
sentence and document level Ive et al. (2018). So far the most popular task
has been sentence-level QE Specia et al. (2020), in which QE models provide a
score for each pair of source and target sentences. A more challenging task,
which is currently receiving a lot of attention from the research community,
is word-level quality estimation. This task provides more fine-grained
information about the quality of a translation, indicating which words from
the source have been incorrectly translated in the target, and whether the
words inserted between these words are correct (good vs bad gaps). This
information can be useful for post-editors by indicating the parts of a
sentence on which they have to focus more.

Word-level QE is generally framed as a supervised ML problem Kepler et al.
(2019); Lee (2020) trained on data in which the correctness of translation is
labelled at word level (i.e. good, bad, gap). The training data publicly
available to build word-level QE models is limited to very few language pairs,
which makes it difficult to build QE models for many languages. From an
application perspective, even for the languages with resources, it is
difficult to maintain separate QE models for each language since the state-of-
the-art neural QE models are large in size Ranasinghe et al. (2020b).

Figure 1: Model Architecture

In our paper, we address this problem by developing multilingual word-level QE
models which perform competitively in different domains, MT types and language
pairs. In addition, for the first time, we propose word-level QE as a zero-
shot cross-lingual transfer task, enabling new avenues of research in which
multilingual models can be trained once and then serve a multitude of
languages and domains. The main contributions of this paper are the following:

  1. i

We introduce a simple architecture to perform word-level quality estimation
that predicts the quality of the words in the source sentence, target sentence
and the gaps in the target sentence.

  2. ii

We explore multilingual, word-level quality estimation with the proposed
architecture. We show that multilingual models are competitive with bilingual
models.

  3. iii

We inspect few-shot and zero-shot word-level quality estimation with the
bilingual and multilingual models. We report how the source-target direction,
domain and MT type affect the predictions for a new language pair.

  4. iv

We release the code and the pre-trained models as part of an open-source
framework111Documentation is available on http://tharindu.co.uk/TransQuest/.

##  2 Related Work

#### QE

Early approaches in word-level QE were based on features fed into a
traditional machine learning algorithm. Systems like QuEst++ Specia et al.
(2015) and MARMOT Logacheva et al. (2016) were based on features used with
Conditional Random Fields to perform word-level QE. With deep learning models
becoming popular, the next generation of word-level QE algorithms were based
on bilingual word embeddings fed into deep neural networks. Such approaches
can be found in OpenKiwi Kepler et al. (2019). However, the current state of
the art in word-level QE is based on transformers like BERT Devlin et al.
(2019) and XLM-R Conneau et al. (2020) where a simple linear layer is added on
top of the transformer model to obtain the predictions Lee (2020). All of
these approaches consider quality estimation as a language-specific task and
build a different model for each language pair. This approach has many
drawbacks in real-world applications, some of which are discussed in Section
1.

#### Multilinguality

Multilinguality allows training a single model to perform a task from and/or
to multiple languages. Even though this has been applied to many tasks
Ranasinghe and Zampieri (2020); Ranasinghe and Zampieri (2021) including NMT
Nguyen and Chiang (2017); Aharoni et al. (2019), multilingual approaches have
been rarely used in QE Sun et al. (2020). Shah and Specia (2016) explore QE
models for more than one language where they use multitask learning with
annotators or languages as multiple tasks. They show that multilingual models
led to marginal improvements over bilingual ones with a traditional black-box,
feature-based approach. In a recent study, Ranasinghe et al. (2020b) show that
multilingual QE models based on transformers trained on high-resource
languages can be used for zero-shot, sentence-level QE in low-resource
languages. In a similar architecture, but with multi-task learning, Sun et al.
(2020) report that multilingual QE models outperform bilingual models,
particularly in less balanced quality label distributions and low-resource
settings. However, these two papers are focused on sentence-level QE and to
the best of our knowledge, no prior work has been done on multilingual, word-
level QE models.

##  3 Architecture

Our architecture relies on the XLM-R transformer model Conneau et al. (2020)
to derive the representations of the input sentences. XLM-R has been trained
on a large-scale multilingual dataset in 104 languages, totalling 2.5TB,
extracted from the CommonCrawl datasets. It is trained using only RoBERTa’s
Liu et al. (2019) masked language modelling (MLM) objective. XML-R was used by
the winning systems in the recent WMT 2020 shared task on sentence-level QE
Ranasinghe et al. (2020a); Lee (2020); Fomicheva et al. (2020). This motivated
us to use a similar approach for word-level QE.

Our architecture adds a new token to the XLM-R tokeniser called $<$GAP$>$
which is inserted between the words in the target. We then concatenate the
source and the target with a [SEP] token and we feed them into XLM-R. A simple
linear layer is added on top of word and $<$GAP$>$ embeddings to predict
whether it is ”Good” or ”Bad” as shown in Figure 1. The training
configurations and the system specifications are presented in the
supplementary material.

Language Pair | Source | MT System | Competition | Train Size
---|---|---|---|---
De-En | Pharmaceutical | Phrase-based SMT | WMT 2018 | 25,963
En-Cs | IT | Phrase-based SMT | WMT 2018 | 40,254
En-De | Wiki | fairseq-based NMT | WMT 2020 | 7,000
En-De | IT | fairseq-based NMT | WMT 2019 | 13,442
En-De | IT | Phrase-based SMT | WMT 2018 | 26,273
En-Ru | IT | Online NMT | WMT 2019 | 15,089
En-Lv | Pharmaceutical | Attention-based NMT | WMT 2018 | 12,936
En-Lv | Pharmaceutical | Phrase-based SMT | WMT 2018 | 11,251
En-Zh | Wiki | fairseq-based NMT | WMT 2020 | 7,000

Table 1: Information about the language pairs used to predict word-level
quality. The Language Pair column lists the language pairs we used in ISO
639-1 codes. Source stands for the domain of the sentence and MT System is the
Machine Translation system used to translate the sentences. Competition refers
to the quality estimation competition in which the data was released and the
last column indicates the number of instances the train dataset has for each
language pair respectively.

##  4 Experimental Setup

###  4.1 QE Dataset

We used several language pairs for which word-level QE annotations were
available: English-Chinese (En-Zh), English-Czech (En-Cs), English-German (En-
De), English-Russian (En-Ru), English-Latvian (En-Lv) and German-English (De-
En). The texts are from a variety of domains and the translations were
produced using both neural and statistical machine translation systems. More
details about these datasets can be found in Table 1 and in (Specia et al.,
2018; Fonseca et al., 2019; Specia et al., 2020).

|  | IT | Pharmaceutical | Wiki
---|---|---|---|---
|  Train Language(s) |  En-Cs SMT |  En-De NMT |  En-De SMT |  En-Ru NMT |  De-En SMT |  En-LV NMT |  En-Lv SMT |  En-De NMT |  En-Zh NMT
I | En-Cs SMT | 0.6081 | (-0.09) | (-0.07) | (-0.09) | (-0.15) | (-0.02) | (-0.01) | (-0.10) | (-0.11)
En-De NMT | (-0.17) | 0.4421 | (-0.06) | (-0.02) | (-0.18) | (-0.01) | (-0.02) | (-0.01) | (-0.08)
En-De SMT | (-0.01) | (-0.05) | 0.6348 | (-0.67) | (-0.14) | (-0.06) | (-0.04) | (-0.06) | (-0.09)
En-Ru NMT | (-0.14) | (-0.08) | (-0.16) | 0.5592 | (-0.12) | (-0.01) | (-0.03) | (-0.09) | (-0.08)
De-En SMT | (-0.43) | (-0.23) | (-0.33) | (-0.31) | 0.6485 | (-0.29) | (-0.32) | (-0.25) | (-0.28)
En-LV NMT | (-0.12) | (-0.09) | (-0.14) | (-0.03) | (-0.12) | 0.5868 | (-0.01) | (0.09) | (-0.08)
En-Lv SMT | (-0.04) | (-0.16) | (-0.10) | (-0.09) | (-0.16) | (-0.01) | 0.5939 | (-0.15) | (-0.14)
En-De NMT | (-0.11) | (-0.01) | (-0.08) | (-0.02) | (-0.14) | (-0.02) | (-0.04) | 0.5013 | (-0.06)
En-Zh NMT | (-0.19) | (-0.08) | (-0.17) | (-0.03) | (-0.16) | (-0.03) | (-0.06) | (-0.07) | 0.5402
II | All | 0.6112 | 0.4523 | 0.6583 | 0.5558 | 0.6221 | 0.5991 | 0.5980 | 0.5101 | 0.5229
All-1 | (-0.01) | (-0.01) | (-0.05) | (-0.02) | (-0.12) | (-0.01) | (-0.01) | (-0.01) | (-0.05)
III | Domain | 0.6095 | 0.4467 | 0.6421 | 0.5560 | 0.6331 | 0.5892 | 0.5951 | 0.5021 | 0.5210
IV | SMT/NMT | 0.6092 | 0.4461 | 0.6410 | 0.5421 | 0.6320 | 0.5885 | 0.5934 | 0.5010 | 0.5205
V | Baseline-Marmot | 0.4449 | 0.1812 | 0.3630 | NR | 0.4373 | 0.4208 | 0.3445 | NR | NR
Baseline-OpenKiwi | NR | NR | NR | 0.2412 | NR | NR | NR | 0.4111 | 0.5583
Best system | 0.4449 | 0.4361 | 0.6246 | 0.4780 | 0.6012 | 0.4293 | 0.3618 | 0.6186 | 0.6415

Table 2: Target F1-Multi between the algorithm predictions and human
annotations. Best results for each language by any method are marked in bold.
Sections I, II and III indicate the different evaluation settings. Section IV
shows the results of the state-of-the-art methods and the best system
submitted for the language pair in that competition. NR implies that a
particular result was not reported by the organisers. Zero-shot results are
coloured in grey and the value shows the difference between the best result in
that section for that language pair and itself.

###  4.2 Evaluation Criteria

For evaluation, we used the approach proposed in the WMT shared tasks in which
the classification performance is calculated using the multiplication of
F1-scores for the ‘OK’ and ‘BAD’ classes against the true labels
independently: words in the target (‘OK’ for correct words, ‘BAD’ for
incorrect words), gaps in the target (‘OK’ for genuine gaps, ‘BAD’ for gaps
indicating missing words) and source words (‘BAD’ for words that lead to
errors in the target, ‘OK’ for other words) Specia et al. (2018). In recent
WMT shared tasks, the most popular category was predicting quality for words
in the target. Therefore, in Section 5 we only report the F1-score for words
in the target. Other results are presented in the supplementary material.
Prior to WMT 2019, organisers provided separate scores for gaps and words in
the target, while after WMT 2019 they produce a single result for target gaps
and words. We follow this latter approach.

##  5 Results

The values displayed diagonally across section I of Table 2 show the results
for supervised, bilingual, word-level QE models where the model was trained on
the training set of a particular language pair and tested on the test set of
the same language pair. As can be seen in section V, the architecture
outperforms the baselines in all the language pairs and also outperforms the
majority of the best systems from previous competitions. In addition to the
target word F1-score, our architecture outperforms the baselines and best
systems in target gaps F1-score and source words F1-score too as shown in
Tables 5 and 6. In the following sections we explore its behaviour in
different multilingual settings.

###  5.1 Multilingual QE

We combined instances from all the language pairs and built a single word-
level QE model. Our results, displayed in section II (“All”) of Table 2, show
that multilingual models perform on par with bilingual models or even better
for some language pairs. We also investigate whether combining language pairs
that share either the same domain or MT type can be more beneficial, since it
is possible that the learning process is better when language pairs share
certain characteristics. However as shown in sections III and IV of Table 2,
for the majority of the language pairs, specialised multilingual models built
on certain domains or MT types do not perform better than multilingual models
which contain all the data.

###  5.2 Zero-shot QE

To test whether a QE model trained on a particular language pair can be
generalised to other language pairs, different domains and MT types, we
performed zero-shot quality estimation. We used the QE model trained on a
particular language pair and evaluated it on the test sets of the other
language pairs. Non-diagonal values of section I in Table 2 show how each QE
model performed on other language pairs. For better visualisation, the non-
diagonal values of section I of Table 2 show by how much the score changes
when the zero-shot QE model is used instead of the bilingual QE model. As can
be seen, the scores decrease, but this decrease is negligible and is to be
expected. For most pairs, the QE model that did not see any training instances
of that particular language pair outperforms the baselines that were trained
extensively on that particular language pair. Further analysing the results,
we can see that zero-shot QE performs better when the language pair shares
some properties such as domain, MT type or language direction. For example,
En-De SMT $\Rightarrow$ En-Cs SMT is better than En-De NMT $\Rightarrow$ En-Cs
SMT and En-De SMT $\Rightarrow$ En-De NMT is better than En-Cs SMT
$\Rightarrow$ En-De NMT.

(a) En-Cs SMT IT

(b) En-De NMT IT

(c) En-De SMT IT

(d) En-Ru NMT IT

(e) De-En SMT Pharmaceutical

(f) En-Lv NMT Pharmaceutical

(g) En-Lv SMT Pharmaceutical

(h) En-De NMT Wiki

(i) En-Zh NMT Wiki

Figure 2: Target F1-Multi scores with Few-shot learning

We also experimented with zero-shot QE with multilingual QE models. We trained
the QE model in all the pairs except one and performed prediction on the test
set of the language pair left out. In section II (“All-1”), we show its
difference to the multilingual QE model. This also provides competitive
results for the majority of the languages, proving it is possible to train a
single multilingual QE model and extend it to a multitude of languages and
domains. This approach provides better results than performing transfer
learning from a bilingual model.

One limitation of the zero-shot QE is its inability to perform when the
language direction changes. In the scenario where we performed zero-shot
learning from De-En to other language pairs, results degraded considerably
from the bilingual result. Similarly, the performance is rather poor when we
test on De-En for the multilingual zero-shot experiment as the direction of
all the other pairs used for training is different. This is in line with
results reported by Ranasinghe et al. (2020b) for sentence level.

###  5.3 Few-shot QE

We also evaluated how the QE models behave with a limited number of training
instances. For each language pair, we initiated the weights of the bilingual
model with those of the relevant All-1 QE and trained it on 100, 200, 300 and
up to 1000 training instances. We compared the results with those obtained
having trained the QE model from scratch for that language pair. The results
in Figure 2 show that All-1 or the multilingual model performs well above the
QE model trained from scratch (Bilingual) when there is a limited number of
training instances available. Even for the De-En language pair, for which we
had comparatively poor zero-shot results, the multilingual model provided
better results with a few training instances. It seems that having the model
weights already fine-tuned in the multilingual model provides an additional
boost to the training process which is advantageous in a few-shot scenario.

##  6 Conclusions

In this paper, we explored multilingual, word-level QE with transformers. We
introduced a new architecture based on transformers to perform word-level QE.
The implementation of the architecture, which is based on Hugging Face Wolf et
al. (2020), has been integrated into the TransQuest framework Ranasinghe et
al. (2020b) which won the WMT 2020 QE task Specia et al. (2020) on sentence-
level direct assessment Ranasinghe et al. (2020a)222TransQuest is available on
GitHub https://github.com/tharindudr/TransQuest. In our experiments, we
observed that multilingual QE models deliver excellent results on the language
pairs they were trained on. In addition, the multilingual QE models perform
well in the majority of the zero-shot scenarios where the multilingual QE
model is tested on an unseen language pair. Furthermore, multilingual models
perform very well with few-shot learning on an unseen language pair when
compared to training from scratch for that language pair, proving that
multilingual QE models are effective even with a limited number of training
instances. While we centered our analysis around the F1-score of the target
words, these findings are consistent with the F1-score of the target gaps and
the F1 score of the source words too. This suggests that we can train a single
multilingual QE model on as many languages as possible and apply it on other
language pairs as well. These findings can be beneficial to perform QE in low-
resource languages for which the training data is scarce and when maintaining
several QE models for different language pairs is arduous.

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## Supplementary Material

  1. i

Training Configurations We used an Nvidia Tesla K80 GPU to train the models.
We divided the dataset into a training set and a validation set using 0.8:0.2
split. We evaluated the model while training and performed early stopping if
the validation loss did not improve over ten evaluation steps. We used the
parameter values mentioned in Table 3 and did not change it across the
language pairs. For all the experiments we used the XLM-R large model.

Parameter | Value
---|---
learning rate | 2e-5
maximum sequence length | 128
number of epochs | 3
adam epsilon | 1e-8
warmup ratio | 0.1
warmup steps | 0
max grad norm | 1.0
max seq. length | 140
gradient accumulation steps | 1

Table 3: Parameter Specifications.

  2. ii

Hardware Specifications

In Table 4 we mention the specifications of the GPU we used for the
experiments of the paper.

Parameter | Value
---|---
GPU | Nvidia K80
GPU Memory | 12GB
GPU Memory Clock | 0.82GHz
Performance | 4.1 TFLOPS
No. CPU Cores | 2
RAM | 12GB

Table 4: GPU Specifications.

  3. iii

Other results

In Table 5 and in Table 6 we show the F1 scores for gaps in target and for
words in source. They follow the same format as Table 2. The Marmot baseline
used in WMT 2018 does not support quality prediction for gaps in the target
and words in the source. In addition, after WMT 2019, organisers did not
release scores for gaps in target. For this reason, we do not report them in
Table 5.

|  | IT | Pharmaceutical
---|---|---|---
|  Train Language(s) |  En-Cs SMT |  En-De NMT |  En-De SMT |  De-En SMT |  En-LV NMT |  En-Lv SMT
I | En-Cs SMT | 0.2018 | (-0.10) | (-0.08) | (-0.15) | (-0.02) | (-0.01)
En-De NMT | (-0.17) | 0.1672 | (-0.07) | (-0.18) | (-0.01) | (-0.02)
En-De SMT | (-0.08) | (-0.05) | 0.4927 | (-0.14) | (-0.06) | (-0.04)
En-Ru NMT | (-0.14) | (-0.00) | (-0.15) | (-0.12) | (-0.01) | (-0.03)
De-En SMT | (-0.18) | (-0.14) | (-0.33) | 0.4203 | (-0.29) | (-0.32)
En-LV NMT | (-0.16) | (-0.09) | (-0.15) | (-0.12) | 0.1664 | (-0.01)
En-Lv SMT | (-0.11) | (-0.12) | (-0.11) | (-0.16) | (-0.01) | 0.2356
En-De NMT | (-0.17) | (-0.01) | (-0.09) | (-0.14) | (-0.02) | (-0.04)
En-Zh NMT | (-0.15) | (-0.08) | (-0.16) | (-0.16) | (-0.03) | (-0.06)
II | All | 0.2118 | 0.1773 | 0.5028 | 0.4189 | 0.1772 | 0.2388
All-1 | (-0.03) | (-0.04) | (-0.08) | (-0.14) | (-0.01) | (-0.01)
III | Domain | 0.2112 | 0.1695 | 0.4951 | 0.4132 | 0.1685 | 0.2370
IV | SMT/NMT | 0.2110 | 0.1886 | 0.4921 | 0.4026 | 0.1671 | 0.2289
V | Marmot | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000
Best system | 0.1671 | 0.1343 | 0.3161 | 0.3176 | 0.1598 | 0.1386

Table 5: GAP F1-Multi between the algorithm predictions and human annotations.
Best results for each language by any method are marked in bold. Sections I,
II and III indicate the different evaluation settings. Section IV shows the
results of the state-of-the-art methods and the best system submitted for the
language pair in that competition. NR implies that a particular result was not
reported by the organisers. Zero-shot results are coloured in grey and the
value shows the difference between the best result in that column for that
language and itself.

|  | IT | Pharmaceutical | Wiki
---|---|---|---|---
|  Train Language(s) |  En-Cs SMT |  En-De NMT |  En-De SMT |  En-Ru NMT |  De-En SMT |  En-LV NMT |  En-Lv SMT |  En-De NMT |  En-Zh NMT
I | En-Cs SMT | 0.5327 | (-0.08) | (-0.07) | (-0.09) | (-0.17) | (-0.02) | (-0.01) | (-0.12) | (-0.13)
En-De NMT | (-0.17) | 0.2957 | (-0.07) | (-0.02) | (-0.19) | (-0.01) | (-0.02) | (-0.02) | (-0.08)
En-De SMT | (-0.01) | (-0.05) | 0.5269 | (-0.67) | (-0.14) | (-0.06) | (-0.05) | (-0.08) | (-0.09)
En-Ru NMT | (-0.14) | (-0.08) | (-0.18) | 0.5543 | (-0.14) | (-0.01) | (-0.03) | (-0.09) | (-0.08)
De-En SMT | (-0.42) | (-0.21) | (-0.33) | (-0.31) | 0.4824 | (-0.29) | (-0.32) | (-0.23) | (-0.28)
En-LV NMT | (-0.12) | (-0.09) | (-0.14) | (-0.03) | (-0.12) | 0.4880 | (-0.01) | (0.09) | (-0.08)
En-Lv SMT | (-0.04) | (-0.16) | (-0.11) | (-0.09) | (-0.17) | (-0.02) | 0.4945 | (-0.15) | (-0.14)
En-De NMT | (-0.11) | (-0.01) | (-0.08) | (-0.02) | (-0.15) | (-0.03) | (-0.04) | 0.4456 | (-0.06)
En-Zh NMT | (-0.19) | (-0.08) | (-0.17) | (-0.03) | (-0.18) | (-0.05) | (-0.06) | (-0.07) | 0.4040
II | All | 0.5442 | 0.3021 | 0.5445 | 0.5535 | 0.4791 | 0.4983 | 0.5005 | 0.4483 | 0.4053
All-1 | (-0.02) | (-0.02) | (-0.06) | (-0.03) | (-0.16) | (-0.01) | (-0.01) | (-0.01) | (-0.04)
III | Domain | 0.5421 | 0.2925 | 0.5421 | 0.5259 | 0.4672 | 0.4907 | 0.4991 | 0.4364 | 0.4021
IV | SMT/NMT | 0.5412 | 0.2901 | 0.5412 | 0.5230 | 0.4670 | 0.4889 | 0.4932 | 0.4302 | 0.4012
V | Marmot | 0.0000 | 0.0000 | 0.0000 | NR | 0.0000 | 0.0000 | 0.0000 | NR | NR
OpenKiwi | NR | NR | NR | 0.2647 | NR | NR | NR | 0.3717 | 0.3729
Best system | 0.3937 | 0.2642 | 0.3368 | 0.4541 | 0.3200 | 0.3614 | 0.4945 | 0.5672 | 0.4462

Table 6: SOURCE F1-Multi between the algorithm predictions and human
annotations. Best results for each language by any method are marked in bold.
Rows I, II and III indicate the different evaluation settings. Row IV shows
the results of the state-of-the-art methods and the best system submitted for
the language pair in that competition. NR implies that a particular result was
not reported by the organisers. Zero-shot results are coloured in grey and the
value shows the difference between the best result in that column for that
language and itself.


# Excess Power, Energy and Intensity of Stochastic Fields in Quasi-Static and
Dynamic Environments

Luk R. Arnaut

###### Abstract

The excess power, energy and intensity of a random electromagnetic field above
a high threshold level are characterized based on a Slepian–Kac model for
upcrossings. For quasi-static fields, the probability distribution of the
excess intensity in its regression approximation evolves from $\chi^{2}_{3}$
to $\chi^{2}_{2}$ when the threshold level increases. The excursion area
associated with excess energy exhibits a chi-cubed ($\chi^{3}_{2}$)
distribution above asymptotically high thresholds, where excursions are
parabolic. For dynamic fields, the dependence of the electrical and
environmental modulations of the excess power on the hybrid modulation index
and threshold level are established. The normalized effective power relative
to the quasi-static power increases non-monotonically when this index
increases. The mean and standard deviation of the dynamic excess power are
obtained in closed form and validated by Monte Carlo simulation.

Index Terms – excursion area, extreme electromagnetics, immunity testing,
peaks over threshold, regression, reverberation chambers, Slepian model,
threshold level crossings.

##  I Introduction

Random fluctuations of electromagnetic (EM) fields and currents have been
studied for over a century, typically as noise phenomena in a static or quasi-
static EM environment (EME). Dynamic EMEs introduce additional field
fluctuations superimposed onto heterodyne signal modulation of a purely
electrical origin. In this respect, mode-tuned and mode-stirred reverberation
chambers (MT/MSRCs) serve as generators of quasi-static and dynamic spatial
fields in the multipath propagation of time-harmonic or modulated signal
fields.

In modern wireless communications and mobile computing, the signalling systems
and devices typically involve an exceedingly large number of independent
states (degrees of freedom $N$), e.g., in wideband multi-user communications
for multimedia and high-mobility or intelligent transport applications [1].
This results in complex systems that must be verified and operated in real
time under rapidly varying conditions and continuous state transitions, with
minimum latency and uncertainty. In this context, the generation [2] and time-
domain measurement [3]–[6] of ultra-short pulses and peaks are active areas of
research. One important parameter is the peak to average power ratio (PAPR),
which is highly sensitive to $N$.

Specific to testing of complex systems in MT/MSRCs, mechanical stirring may no
longer be feasible to provide sufficiently short cycle times per test state.
Rapid electronic multistirring offers much shorter test cycles, e.g.,
electronic source switching or scanning. In conjunction with increasing rates
and shortened events in dynamic EMEs, quasi-static concepts such as effective
power may need to be reconsidered at microwave and millimeter-wave
frequencies.

To this end, a statistical characterization of excursions for energy and power
above a constant threshold level was made in [7] for mono- and multi-stirred
or -scanned MSRCs. There, the focus was on quasi-static operation – i.e., slow
variations of the field envelope relative to the rate of oscillation of the
excitation field at its carrier frequency – and on temporal characteristics,
i.e., location, duration and rate of threshold crossings and excursions. Such
an analysis complements traditional EM characterization based on field
amplitudes. The rationale is that EM stress and failure of equipment under
test may not only be governed by the instantaneous field amplitude or
intensity, i.e., the “vertical” level, but also by their temporal, i.e.,
“horizontal” sustainment, interruptions, durations and rates for high levels
above a critical threshold.

In this paper, horizontal and vertical metrics are combined into a
comprehensive time-domain characterization. The focus is on the random height
and area of positive excursions above a relatively high threshold (Fig. 1),
from which probability density functions (PDFs) of excess energy and power are
deduced. To this end, the statistics of the energy and its derivative [8] are
fundamental. Although only a single mechanism of variation for the EME will be
considered, e.g., one-dimensional (1-D) monostirring in a MSRC, an extension
to $d$ dimensions of randomization where $d$ stir actions are represented by
multiple time, space and/or state dimensions [7] is possible. The emphasis is
on scalar electric power defined by the electric energy per unit of time, as
opposed to vector EM power density defined by the Poynting vector. Random
variables are denoted by uppercase letters, while their values and constants
are written in corresponding lowercase. An $\exp({\rm j}\omega t)$ time
dependence is assumed.

---
Figure 1:  Two realizations of positive excursions for $U(t)$ (in arbitrary
units), with threshold up- and downcrossing times $t^{+}_{i}$ and $t^{-}_{i}$,
excursion length $\ell^{+}_{i}$, height $h^{+}_{i}$, area $a^{+}_{i}$,
threshold bounded area $a_{{\rm th},i}$ and total area
$a_{i}=a^{+}_{i}+a_{{\rm th},i}$ within time interval $[t^{+}_{i},t^{-}_{i}]$
($i=0,1$).

##  II Slepian–Kac Model for Upcrossings

A Slepian (or Slepian–Kac) model (SKM) [9], [10] represents a decomposition of
a noise process, conditioned on crossing a predetermined threshold level, into
a short-term quasi-deterministic component and a longer-term nonstationary
random component. The former can be conceived as a regression approximation
[11] in the vicinity of an upcrossing, downcrossing, local extremum or other
mark for each positive excursion (exceedance) by an EM quantity or its
temporal or spatial derivative. The SKM is implicit in the modelling of the
excursions’ duration or spatial extent (length or area) [7], as the footprint
of a $(d+1)$-D excursion surface or volume projected onto a $d$-D threshold
contour or surface, respectively. In the present work, a SKM will be developed
for the 1-D excess height and 2-D excess area, with the associated
instantaneous and integrated energy (or power), respectively. To establish
notions and notations, the basic SKM conditioned on level upcrossings is
briefly summarized first.

###  II-A Gauss Normal Field or Current Process

Let $X(t)$ be a sufficiently smooth zero-mean real-valued Gaussian random EM
field or current, assumed to be stationary and ergodic to enable a
probabilistic characterization of $X(t)$ from a single long-run realization
(sample path, sweep) as a time series. Assume that pairs of up- and downward
crossings of a chosen arbitrary threshold level $x_{{\rm th}}$ by $X(t)$ occur
at respective times $t^{+}_{i}$ and $t^{-}_{i}$ ($i=0,1,2,\ldots$) bounding
the $i$th positive excursion $X^{+}(t)$. First, consider a single excursion
starting at $t^{+}=0$, i.e., $X(0)=x_{{\rm th}}$. The second-order SKM for
$X^{+}(t)$ applicable to this $X(t\geq 0)$ is [10]

$\displaystyle X^{+}({t|{x_{{\rm
th}}}})=\left[\frac{r_{X}(t)}{\lambda_{0}}X(0)-\frac{\dot{r}_{X}(t)}{{\lambda_{2}}}\dot{X}(0)\right]+K(t).$
(1)

Here, $\dot{r}_{X}(t)=-\langle{\dot{X}(t^{+})X(t^{+}+t)}\rangle$ is the time
derivative of the ensemble (and, by ergodicity, temporal) autocovariance
function111Autocovariance functions containing an expansion term in $|t|^{3}$
give rise to differences in $f_{L^{+}}(\ell^{+}\rightarrow 0)$, among other
effects [10, sec. 3]. $r_{X}(t)$ of $X(t)$, expandable for $t\ll t^{-}-t^{+}$
as

$\displaystyle r_{X}(t)$
$\displaystyle=\langle{X(0)X(t)}\rangle=\lambda_{0}-\frac{\lambda_{2}}{2!}t^{2}+\frac{\lambda_{4}}{4!}t^{4}+{\cal
O}(t^{6})$ (2)

where $\lambda_{0}=\sigma^{2}_{X}$, $\lambda_{2}=\sigma^{2}_{\dot{X}}$ and
$\lambda_{4}=\sigma^{2}_{\ddot{X}}$ are the finite zeroth, second and fourth
spectral moments of $X(t)$, respectively;
$\dot{X}(0)\stackrel{{\scriptstyle\Delta}}{{=}}Z/\sqrt{\lambda_{2}}>0$ is the
random slope of $X(t)$ at $t^{+}=0$, which in the case of so-called horizontal
windowing [9, eq. (2.1)] exhibits a scaled Rayleigh ($\chi_{2}$) PDF where

$\displaystyle f_{Z}(z)=z\exp\left(-z^{2}/2\right),\,\,\,\,\,\,z>0$ (3)

with sample value $z_{i}$ at $t^{+}_{i}$ for the $i$th excursion; and $K(t)$
is a nonstationary residual Gaussian stochastic field [10] with time average
$\overline{K(t)}=0$ and inhomogeneous autocovariance function

$\displaystyle r_{K}(t_{1},t_{2})$
$\displaystyle\equiv\overline{K(t_{1})K(t_{2})}=\overline{X(t_{1})X(t_{2})\,|\,\left(X(t^{+}),\dot{X}(t^{+})\right)}$
$\displaystyle=r_{X}(t_{1}-t_{2})-r_{X}(t_{1})r_{X}(t_{2})/\lambda_{0}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
-\dot{r}_{X}(t_{1})\dot{r}_{X}(t_{2})/\lambda_{2},\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ t_{1},t_{2}\in[t^{+},t^{-}].$ (4)

The $\dot{X}(0)$ and $K(t)$ in (1) are mutually independent. The bracketed sum
in (1) represents a regression approximation $\tilde{X}^{+}(t|x_{{\rm
th}})\stackrel{{\scriptstyle\Delta}}{{=}}\alpha_{X}(t)+\beta_{X}(t)Z$ for
$X^{+}(t|x_{{\rm th}})$ as a linear combination of two orthogonal basis
functions $X(0)$ and $\dot{X}(0)$ [12], plus a residual fluctuation $K(t)$ for
the actual excursion. For a narrowband $X(t)$, where
$|r_{K}(t,t)|\ll|r_{X}(t)|$ when $t\ll t^{-}-t^{+}$ [10, sec. 1], the
regression $\tilde{X}^{+}(t|x_{{\rm th}})$ dominates $K(t)$. For larger $t$,
the magnitude of $K(t)$ may become comparable to, or exceed
$\tilde{X}^{+}(t|x_{{\rm th}})$. Higher-order extensions of the SKM (1),
conditioned on additional knowledge of higher-order derivatives at $t^{+}$ or
multiple first-order derivatives at $t\not=t^{+}$, are also possible [10, sec.
2], [13, sec. VI].

For economy of notation, we further denote the (dimensionless) normalized
quantities
$\lambda^{\prime}_{j}\stackrel{{\scriptstyle\Delta}}{{=}}{\lambda_{j}}/{\lambda_{0}}$
and define

$\displaystyle
x^{\prime}(t)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{x(t)}{\sigma_{X}},\,u^{\prime}(t)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{u(t)}{\sigma^{2}_{X}},\,r^{\prime}_{X}(t)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{r_{X}(t)}{\sigma^{2}_{X}},\,r^{\prime}_{U}(t)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{r_{U}(t)}{\sigma^{4}_{X}}.$
(5)

With these normalizations, $r^{\prime}_{X}(t)$ is the autocorrelation function
(ACF) of $X(t)$. In particular, from (4), the variance function of $K(t)$
becomes $r^{\prime}_{K}(t,t)=\sigma^{2}_{K}(t)/\sigma^{2}_{X}$, i.e.,

$\displaystyle
r^{\prime}_{K}(t,t)=1-[r^{\prime}_{X}(t)]^{2}-\left[{\dot{r}^{\prime}_{X}(t)}/{\sqrt{\lambda^{\prime}_{2}}}\right]^{2}.$
(6)

Fig. 2 shows $r^{\prime}_{X}(t_{k})$, $-\dot{r}^{\prime}_{X}(t_{k})$ and
$r^{\prime}_{K}(t_{k},t_{k})$ for MSRC data of measured electric field
components ${{\rm Re}}[X(t_{k})]$ and ${{\rm Im}}[X(t_{k})]$, based on
experimental data detailed in Sec. IV and
$t^{+}_{k}\stackrel{{\scriptstyle\Delta}}{{=}}0$. It is seen that, immediately
after upcrossings ($t_{k}\ll 100\,\Delta t_{k}$), $r^{\prime}_{X}(t_{k})$
outweighs $r^{\prime}_{K}(t_{k})$ while for $t_{k}\gg 300\,\Delta t_{k}$, the
opposite holds. Since $r^{\prime}_{X}(t_{k})$ and $r^{\prime}_{K}(t_{k})$
cross at a relatively high level, both contributions are significant at
intermediate times. From Fig. 2(b), contributions by
$-\dot{r}^{\prime}_{X}(t_{k})$ are relatively small compared to
$r^{\prime}_{K}(t_{k})$. Thus, the regression $\tilde{X}(t|x_{{\rm th}})$
dominates the short-term field, i.e., it governs the functional and
statistical dependence within the correlation length $t_{c}$, a fortiori
within the excursion length
$\ell^{+}\stackrel{{\scriptstyle\Delta}}{{=}}t^{+}-t^{-}<t_{c}$. On the other
hand, sufficiently long after an upcrossing at $t^{+}$, the latent field
$K(t)$ tends to become uncorrelated in the long term, well known from coarsely
sampled or unconditioned processes. The conditioned field thus becomes
increasingly less constrained by the earlier field and its derivative at
$t^{+}$ and becomes asymptotically stationary.

Fig. 2(a) also shows instantaneous cross-correlation functions between ${{\rm
Re}}[X(t_{k})]$ and ${{\rm Im}}[X(t_{k})]$. Their low magnitude smaller than
0.1 across the entire time span justifies adopting a SKM for independent
processes [14] in Sec. II-B.

---
(a)
(b)
Figure 2:  (a)-(b) Autocorrelation functions: (a) $r^{\prime}_{X}(t_{k})$,
$r^{\prime}_{U}(t_{k})/4$ (solid) and (b) $-\dot{r}^{\prime}_{X}(t_{k})$,
$-\dot{r}^{\prime}_{U}(t_{k})/4$ (solid) of measured ${{\rm Re}}[X(t)]$
(blue), ${{\rm Im}}[X(t)]$ (red) and $U(t)$ (black) for $n=2$, with comparison
to $r^{\prime}_{K}(t_{k},t_{k})$ for respective $K(t)$ (dashed) and to
theoretical ACF models (dash dotted) $\rho_{{{\rm
Re}}(X)}(t_{k})=1/[1+(t_{k}/105\Delta t_{k})^{2}]$ (cyan), $\rho_{{{\rm
Im}}(X)}(t_{k})=1/[1+(t_{k}/113\Delta t_{k})^{2}]$ (magenta), and
$\rho_{U}(t_{k})=1/[1+(t_{k}/103.5\Delta t_{k})^{2}]^{2}$ (grey). (a) I/Q
cross-correlation functions (dotted): $r^{\prime}_{{{\rm Re}}(X),{{\rm
Im}}(X)}(t_{k})$ (blue) and $r^{\prime}_{{{\rm Im}}(X),{{\rm Re}}(X)}(t_{k})$
(red).

###  II-B $\chi^{2}_{n}$ Field Intensity or Current Intensity Process

Consider the $\chi^{2}_{n}$ distributed $U(t)=\sum^{n}_{j=1}X^{2}_{j}(t)$ for
$n$ independent and identically distributed (i.i.d.) real $X_{j}(t)$ as
defined in Sec. II-A, e.g., a set of $n$ I/Q Cartesian components of the EM
field. It follows that $u_{({\rm th})}/\sigma^{2}_{X_{j}}=n\,u_{({\rm
th})}/\sigma_{U}$ and $r_{U}(t)=2n\,r^{2}_{X_{j}}(t)$. The SKM for
$U^{+}(t\geq 0)$ conditioned on an upcrossing of $u_{{\rm th}}=U(0)$ at
$t^{+}=0$ is [14, Thm. 2]

$\displaystyle{U^{+}}(t|u_{{\rm
th}})=\lambda_{0}\sum^{n}_{j=1}\left[\epsilon_{j}\alpha(t)+\beta(t)W_{j}+K_{j}(t)\right]^{2}$
(7)

with $\epsilon_{1}=1$, $\epsilon_{j\not=1}=0$, and regression coefficient
functions

$\displaystyle\alpha(t)\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{\frac{u_{{\rm
th}}r_{U}(t)}{2n\lambda^{3}_{0}}},\,\,\,\,\,\,\beta(t)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{-\dot{r}_{U}(t)}{\sqrt{8n\lambda_{0}\lambda_{2}r_{U}(t)}}$
(8)

where $W_{1}\equiv Z$ has a standard Rayleigh $\chi_{2}$ PDF;
$W_{2},\ldots,W_{n}$ are mutually independent standard Gauss normal and also
independent of $W_{1}$; all $K_{j}(t)$ are independent nonstationary standard
Gauss normal processes with zero mean ($\overline{K_{j}(t)}=0$) and common
autocovariance function (4). For $t-t^{+}\ll t_{c}$, the autocovariance of
$U(t)$ can be expanded as

$\displaystyle r_{U}(t)$
$\displaystyle=2n\left[\lambda^{2}_{0}-\lambda_{0}\lambda_{2}t^{2}+\frac{3\lambda^{2}_{2}+\lambda_{0}\lambda_{4}}{12}t^{4}+{\cal
O}(t^{6})\right].$ (9)

Examples of non-Gaussian $r_{U}(t)$ were given in [8].

In the remainder, the focus is on a single Cartesian circular complex field
$X(t)={{\rm Re}}[X(t)]+{\rm j}{{\rm Im}}[X(t)]$ associated with $U(t)$, i.e.,
$n=2$. From (7), the normalized mean and variance of $U^{+}$ then follow after
calculation as

$\displaystyle\langle{U^{+}}^{\prime}(t|u^{\prime}_{{\rm th}})\rangle$
$\displaystyle=\alpha^{2}+\sqrt{2\pi}\,\alpha\beta+3\beta^{2}+2\langle
K^{2}\rangle$ (10)
$\displaystyle{\sigma^{\prime}}^{2}_{{U^{+}}}(t|u^{\prime}_{{\rm th}})$
$\displaystyle=(8-2\pi)\alpha^{2}\beta^{2}+2\sqrt{2\pi}\,\alpha\beta^{3}+6\beta^{4}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\left(4\alpha^{2}+4\sqrt{2\pi}\,\alpha\beta+12\beta^{2}+4\langle
K^{2}\rangle\right)\langle K^{2}\rangle$ (11)

where $\langle K^{2}(t)\rangle$ is the normalized variance function (6) for
any $K_{j}(t)$ in (7), which can be expressed equivalently as

$\displaystyle\langle K^{2}(t)\rangle=1-[\alpha^{2}(t)/u^{\prime}_{{\rm
th}}]-\beta^{2}(t).$ (12)

Fig. 3 shows the measured normalized mean and standard deviation of $U^{+}$ as
a function of latency $t_{k}-t^{+}_{k}$ for selected values of
$u^{\prime}_{{\rm th}}$, based on the empirical ACF
$r_{U}(t_{k}-t^{+}_{k})/\sigma^{2}_{U}$ that closely matches222For the present
data set, (13) was found to outperform both Gaussian and ${\rm sinc}^{2}$-type
models of $r_{U}(t_{k}-t^{+}_{k})/\sigma^{2}_{U}$ for $0\leq
t_{k}-t^{+}_{k}<2t_{c}$. the model (cf. Fig. 2 and Sec. IV)

$\displaystyle\rho_{U}(t)=\\{1+[t/(2t_{c})]^{2}\\}^{-2},\,\,\,\,\,2t_{c}\simeq
103.5\,\Delta t.$ (13)

The results show that for $t_{k}-t^{+}_{k}\ll 2t_{c}$ or $\gg 2t_{c}$, the
first or last term inside the brackets in (7) dominates, respectively. For the
former, the regression approximation inside excursions is justified by its
close correspondence. For $t_{k}-t^{+}_{k}\rightarrow+\infty$, both
$\langle{U^{+}}\rangle$ and $\sigma_{U^{+}}$ converge to $\langle
U\rangle=\sigma_{U}=2\sigma^{2}_{X}$ for $\chi^{2}_{2}$, while their
regression approximations approach zero, on average.

Thus, a threshold based characterization has merit in the sense that the
uncertainty of $U(t)$ during an excursion is significantly lower than in
conventional, i.e., unconditioned characterization of $U(t)$ without
thresholding.

---
(a)
(b)
Figure 3:  (a) Normalized mean $\langle{U^{+}}^{\prime}\rangle$ and (b)
normalized standard deviation $\sigma_{{U^{+}}^{\prime}}$ (solid) of intensity
$U^{+}$ with regression approximations $\langle\tilde{U}^{+^{\prime}}\rangle$
and $\sigma_{\tilde{U}^{+^{\prime}}}$ (dashed) for $n=2$ as a function of
$t_{k}-t^{+}_{k}$ (in units $\Delta t_{k}$), after upcrossing of level
$u^{\prime}_{{\rm th}}$ at $t^{+}_{k}$, based on experimental
$r^{\prime}_{U}(t_{k})$. Cross symbols represent values at
$t^{*}_{k}-t^{+}_{k}=\langle L^{+}(u^{\prime}_{{\rm th}})\rangle/2$ situated
near the expected local maxima of $U^{+}(t_{k}-t^{+}_{k})$.

##  III Excess Duration, Intensity and Energy

###  III-A Excursion Length (Excess Duration)

A PDF of the excursion size (hyperlength) $L^{+}_{d}$ along $d$ randomization
dimensions was obtained in [7]. For $d=1$ (cf. Fig. 1), the excursion length
$L^{+}(u_{{\rm th}})\stackrel{{\scriptstyle{\cal
L}}}{{\rightarrow}}2{W_{1}}/\sqrt{{\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}}}$ exhibits asymptotically a Rayleigh PDF

$\displaystyle
f_{L^{+}}(\ell^{+})\rightarrow\frac{\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}}{4}\ell^{+}\exp\left[-\frac{\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}}{8}(\ell^{+})^{2}\right]$ (14)

with mean and standard deviation given by

$\displaystyle\langle
L^{+}\rangle=\sqrt{\frac{2\pi}{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}},\leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{L^{+}}=\sqrt{\frac{8-2\pi}{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}}.$ (15)

###  III-B Excursion Height (Excess Intensity)

####  III-B1 Independently Sampled Field

It is well known [15] that, for i.i.d. samples $X_{i}$ from a general $X(t)$
with PDF $f_{X}(x)$ and $x^{\prime}_{{\rm th}}\rightarrow+\infty$, the
exceedances $H_{X,i}\stackrel{{\scriptstyle\Delta}}{{=}}X_{i}-x_{\rm th}\geq
0$ above $x_{{\rm th}}$ (also known as peaks over threshold (POT)) exhibit a
distribution that belongs to one of three subclasses constituting the
generalized Pareto distribution (GPD) given by

$\displaystyle
G_{X}(x)\stackrel{{\scriptstyle\Delta}}{{=}}F^{*}_{H_{X}}(h_{X})=\lim_{\theta\rightarrow\gamma}\left[1-\left(1+\theta\frac{h_{X}-\mu_{H_{X}}}{\sigma_{H_{X}}}\right)^{-\frac{1}{\theta}}\right].$
(16)

The GPD is a limit distribution, in the sense that it applies to $X_{i}$
approaching the right end point $x_{o}$ of the support of $f_{X}(x)$ as a
limit; here, $x_{o}\rightarrow+\infty$ for the present case of ideal unbounded
fields and energy. In (16), $\sigma_{H_{X}}>0$, $\mu_{H_{X}}$, and $\gamma$
are scale, location, and shape parameters, with $\gamma<0$, $=0$, and $>0$
yielding beta, Gumbel, and simple Pareto distributions for $H_{X}$,
respectively. These three GPD subclasses have a one-to-one correspondence with
the respective Weibull, Gumbel (exponential) and Fréchet distributions in
generalized extreme value (GEV) theory for the sample maximum $M_{X}={{\rm
max}}(X_{1},\ldots,X_{n})$ [16], [17], as defined by

$\displaystyle
F^{*}_{M_{X}}(m_{X})=\lim_{\theta\rightarrow\gamma}\exp\left[-\left(1+\theta\frac{m_{X}-\mu_{M_{X}}}{\sigma_{M_{X}}}\right)^{-\frac{1}{\theta}}\right]$
(17)

i.e., with the same $\gamma$ as in (16). The GEV framework is typically used
in block based characterization of the maximum-value distribution, while GPDs
for POT make more economic use of all available data by retaining also quasi-
extreme values.

For $\chi^{2}_{2}$ i.i.d. $U_{i}$, the GPD of
$H^{+^{\prime}}\stackrel{{\scriptstyle\Delta}}{{=}}{U^{+}}^{\prime}-u^{\prime}_{{\rm
th}}\geq 0$ is exponential. This follows from
$U^{\prime}\sim\exp[-(u^{\prime}_{{\rm th}}+{h}^{+^{\prime}})/2]/2$, yielding
the cumulative distribution function (CDF) of $H^{+^{\prime}}$ as

$\displaystyle F_{H^{+^{\prime}}}(h^{+^{\prime}}|u^{\prime}_{{\rm th}})$
$\displaystyle=\frac{F_{U^{\prime}}(u^{\prime}_{{\rm
th}}+h^{+^{\prime}})-F_{U^{\prime}}(u^{\prime}_{{\rm
th}})}{1-F_{U^{\prime}}(u^{\prime}_{{\rm th}})}$
$\displaystyle=1-\exp(-{{h}^{+^{\prime}}}/{2})$ (18)

i.e., (16) with $\gamma=0$ for $u^{\prime}_{{\rm th}}\rightarrow+\infty$.
Hence

$\displaystyle f_{{H}^{+^{\prime}}}({h}^{+^{\prime}}|u^{\prime}_{{\rm
th}}\rightarrow+\infty)\rightarrow\exp(-{h}^{+^{\prime}}/2)/2.$ (19)

The result (19) can alternatively be derived based on a limit processes and
the asymptotic probability law $H^{+^{\prime}}\stackrel{{\scriptstyle{\cal
L}}}{{\longrightarrow}}W^{2}_{1}=\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}(L^{+})^{2}/4$ for $u^{\prime}_{{\rm th}}\rightarrow+\infty$ [18, Thm.
3.2].

####  III-B2 Correlated Continuous Field

For a time-continuous $U(t)$ instead of discrete samples $U_{i}$, discrete
instances of exceedance are replaced by continuous excursions of duration
$\ell^{+}_{i}$. Within each excursion $i$, i.e., for
$t^{+}_{i}<t<t^{+}_{i}+\ell^{+}_{i}$, the excess height $H^{+}_{i}(t)$ is
random but strongly correlated among nearby time points because
$\ell^{+}_{i}\ll t_{c}$, a fortiori when $u^{\prime}_{{\rm th}}\gg 1$. The POT
can then be restricted by retaining only a single representative value
$h^{+}_{i}$ at some $t_{i}$ per excursion,333 Note that (19) applies to
independent samples, viz., for $t_{k}\gg t_{c}$ spread across well separated
excursions where the $K_{j}(t)$ in (7) are dominant, rather than within
excursions. e.g., the local maximum per excursion for the purpose of EMC
immunity testing. For convenience, the midpoint
$t_{i}\stackrel{{\scriptstyle\Delta}}{{=}}t^{+}_{i}+\ell^{+}_{i}/2$ may be
chosen, which is close to the instance of the local maximum of $H^{+}$ in the
SKM when $u^{\prime}_{{\rm th}}\gg 1$ (cf. Sec. III-D). Nevertheless, in
principle any choice of $t_{i}\in\leavevmode\nobreak\
]t^{+}_{i},t^{+}_{i}+\ell^{+}_{i}[$ is permissible, yielding (18).

In order to characterize $H^{+}$ nonasymptotically, the general PDF of
$U^{+}(t|u^{\prime}_{{\rm th}})$ for $t-t^{+}\not\rightarrow+\infty$ can be
sought from (7). In a first approximation, the asymptotic $\chi^{2}_{2}$ PDF
(19) can be maintained but replacing
$\sigma_{H^{+^{\prime}}}=\sigma_{U^{\prime}}=\langle U^{\prime}\rangle$ with
$\langle H^{+^{\prime}}\rangle$ at
$t^{*}\stackrel{{\scriptstyle\Delta}}{{=}}t^{+}+\langle L^{+}(u^{\prime}_{{\rm
th}})\rangle/2\ll t_{c}$. Denoting
${H^{*^{\prime}}}\stackrel{{\scriptstyle\Delta}}{{=}}{H^{+^{\prime}}}(t^{*})$,
then

$\displaystyle\langle{H^{*^{\prime}}}\rangle=\frac{\pi}{2}+\frac{3\pi}{2u^{\prime}_{{\rm
th}}},\,\,\,\,\sigma^{2}_{H^{*^{\prime}}}=({4-\pi}){\pi}+\frac{\pi^{2}}{u^{\prime}_{{\rm
th}}}$ (20)

for $u^{\prime}_{{\rm th}}\gg 1$ to leading order in $1/u^{\prime}_{{\rm
th}}$, as follows from (8)–(10) and (15). Hence $\langle
H^{*^{\prime}}\rangle<\langle
U^{\prime}(t-t^{+}\rightarrow+\infty)\rangle\equiv 2$ for $u^{\prime}_{{\rm
th}}>3\pi/(4-\pi)$. Note that (20) is independent of the bandwidth
$\sqrt{\lambda^{\prime}_{2}}$. As a further refinement, the PDF of the
regression $\tilde{U}^{+}(t|u_{{\rm th}})$ can be used, upon setting all
$K(t)=0$ in (7). For $n=2$, the PDF of the regressed height
$\tilde{H}^{+}=\tilde{U}^{+}-u_{{\rm th}}$ follows from (7) after variate
transformation as

$\displaystyle\leavevmode\nobreak\ f_{\tilde{H}^{+}}(\tilde{h}^{+}|u_{{\rm
th}})=\frac{\exp\left[-(\alpha^{2}+u_{{\rm
th}})/(2\beta^{2})\right]}{\sqrt{8\pi}\,|\beta|^{3}}\exp\left(-\frac{\tilde{h}^{+}}{2\beta^{2}}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\int^{\sqrt{\tilde{h}^{+}+u_{{\rm
th}}}}_{\alpha}\frac{(y-\alpha)\exp\left({\alpha}\,y/{\beta^{2}}\right)}{\sqrt{\tilde{h}^{+}+u_{{\rm
th}}-y^{2}}}{\rm d}y$ (21)

for $0\leq t-t^{+}<t_{c}$, evaluated in $t=t^{*}$. Fig. 4 shows the
numerically computed PDF (21) for selected $u^{\prime}_{{\rm th}}$, indicating
a transition from a $\chi^{2}_{2}$ PDF to a $\chi^{2}_{3}$ PDF when
$u^{\prime}_{{\rm th}}$ decreases from $+\infty$ to $0$. Both these limit PDFs
are easily retrieved analytically from (21). Note that the PDF of
$\tilde{H}^{+}|0+\equiv\tilde{U}^{+}|0+$ differs from the PDF of the
unconditioned $U$.

---
Figure 4:  PDF of standardized ${\tilde{H}^{+}}$ for selected values of
$u^{\prime}_{{\rm th}}$.

The normalized regression mean and variance of $\tilde{H}^{+^{\prime}}$ are

$\displaystyle\langle{\tilde{H}^{+^{\prime}}}\rangle=\langle{\tilde{U}^{+^{\prime}}}\rangle-u^{\prime}_{{\rm
th}},\,\,\,\,\,\,\,\
\sigma^{2}_{{\tilde{H}^{+^{\prime}}}}=\sigma^{2}_{{\tilde{U}^{+^{\prime}}}}$
(22)

where $\langle{\tilde{U}^{+^{\prime}}}\rangle$ and
$\sigma^{2}_{{\tilde{U}^{+^{\prime}}}}$ are obtained by setting $\langle
K^{2}(t)\rangle=0$ in (10) and (11), respectively. Fig. 5 shows
$\langle\tilde{H}^{+^{\prime}}(t-t^{+})\rangle$ for the data with $n=2$. When
$u^{\prime}_{{\rm th}}<2$, the mean increases from zero at $t_{k}=t^{+}_{k}$
to the unconditioned mean $\langle U^{\prime}\rangle=2$ for the steady state
($t_{k}-t^{+}_{k}\rightarrow+\infty$) of the $\chi^{2}_{2}$ PDF, except for an
offset $-u^{\prime}_{{\rm th}}$. The approximate maximum
$\langle{\tilde{H}^{*^{\prime}}}\rangle$ decreases for increasing
$u^{\prime}_{{\rm th}}$ even when $u^{\prime}_{{\rm th}}\not\gg 1$. For
$u^{\prime}_{{\rm th}}>2$, the parabolic shape of
$\langle{H^{+}}^{\prime}(t_{k}-t^{+}_{k})\rangle$ (cf. Sec. ) remains
approximately constant. Thus, for $u_{{\rm th}}>\langle U\rangle\equiv
2\sigma^{2}_{X}$, long-term correlation causes $\langle
H^{+}(t_{k}>t^{*}_{k})\rangle$ to bend back toward zero, whereas for $u_{{\rm
th}}<\langle U\rangle$ the effect of correlation on $\langle H^{+}\rangle$
persists as a long-term offset. Recall that these graphs represent the time
evolution and dependence of the mean
$\langle{H^{+}}^{\prime}(t-t^{+})\rangle$, rather than
${h^{+}_{i}}^{\prime}(t-t^{+}_{i})$ for an individual excursion $i$.

---
Figure 5:  Normalized mean excess intensity $\langle{H}^{+^{\prime}}\rangle$
(solid) and its regression approximation
$\langle\tilde{H}^{+^{\prime}}\rangle$ (dashed) for $n=2$ as a function of
$t_{k}-t^{+}_{k}$ (in units $\Delta t_{k}$), after upcrossing of level
$u^{\prime}_{{\rm th}}$ at $t^{+}_{k}$ and based on experimental
$r^{\prime}_{U}(t_{k})$. Cross symbols represent values at
$t^{*}_{k}-t^{+}_{k}=\langle L^{+}(u^{\prime}_{{\rm th}})\rangle/2$ situated
near the expected local maxima of $H^{+^{\prime}}(t_{k}-t^{+}_{k})$.

###  III-C Excursion Shape (Excess Intensity Profile)

Based on (9), for $u^{\prime}_{{\rm th}}\gg 1$ an excursion ${U^{+}}^{\prime}$
spanning $[t^{+},\,t^{+}+L^{+}]$ asymptotically approximates a parabolic cap
that is deterministic in curvature but random in location and height specified
by its apex [18], i.e., with upcrossing ${U}^{\prime}(t^{+})=u^{\prime}_{{\rm
th}}$ and apex $U^{\prime}_{{\rm max}}(t_{{\rm max}})=u^{\prime}_{{\rm
th}}+Z^{2}$ at $t_{{\rm
max}}=t^{+}+Z/\sqrt{\lambda^{\prime}_{2}u^{\prime}_{{\rm th}}}$. Hence the
asymptotic form of ${U^{+}}^{\prime}(t)$ conditioned on its upcrossing
$u^{\prime}_{{\rm th}}$ at $t^{+}$ is

$\displaystyle{U^{+}}^{\prime}(t|u^{\prime}_{{\rm th}})\rightarrow
u^{\prime}_{{\rm th}}-\left[\sqrt{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}\,(t-t^{+})-Z\right]^{2}+Z^{2}.$ (23)

From the local maximum associated with an upcrossing-apex pair, the sample
value of $Z$ follows by differentiating (23) as

$\displaystyle z_{i}=\sqrt{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}\,(t_{i,{{\rm max}}}-t^{+}_{i})\simeq\sqrt{u^{\prime}_{i,{{\rm
max}}}-u^{\prime}_{{\rm th}}}.$ (24)

The second equality in (24) is an approximation because, in this SKM,
conditioning is based on upcrossings, as opposed to, e.g., the local maximum
during the excursion [11].

In general, for $d$ field randomizations ($d\geq 1$), the parabolic excursion
generalizes to a paraboloid cap with a $(d+1)$-D (hyper)volume of excess above
a $d$-D (hyper)plane threshold.

###  III-D Excursion Area (Excess Energy)

Integration of the instantaneous excursion height across the excursion length
$L^{+}=t^{-}-t^{+}$ yields the random excursion area
$A^{+}=\int^{t^{-}}_{t^{+}}h^{+}(t){\rm d}t$. In dynamic EMEs, the rate of the
induced field fluctuations $\sqrt{\lambda^{\prime}_{2}}$ varies (e.g., stir
speed in MSRCs [7, sec. VI-B2]), thus affecting the profile of $U(t)$.
Therefore, the dependence of $A^{+}$ on $\lambda^{\prime}_{2}$ is of specific
interest.

For $u^{\prime}_{{\rm th}}\rightarrow+\infty$, in the deterministic second-
order, i.e., parabolic limit SKM, the product of the duration (chord length)
and the maximum height multiplied by $2/3$ provides an estimate of $A^{+}$.
The total area $A$ below $U(t)$ spanning $L^{+}$ (cf. Fig. 1) is then the sum
of $A_{{\rm th}}$ for the rectangle of fixed height $u_{{\rm th}}$ and random
width $L^{+}$ plus $A^{+}$ for the parabolic cap of random height $H$ and
width $L^{+}$, i.e.,

$\displaystyle A$ $\displaystyle=A_{{\rm th}}+A^{+}=u_{{\rm
th}}L^{+}+{2H^{+}L^{+}}/{3}.$ (25)

For $H^{+^{\prime}}$, the asymptotic PDF (19) can be used. With
${A^{+}}^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}A^{+}/\sigma^{2}_{X}\stackrel{{\scriptstyle{\cal
L}}}{{\rightarrow}}{4W^{3}_{1}}/(3\sqrt{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}})$ and [19, eq. (3.326.2)], the asymptotic PDF of $A^{+^{\prime}}$
follows by variate transformation as

$\displaystyle\,f_{A^{+^{\prime}}}(a^{+^{\prime}})\rightarrow\left(\frac{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}{48\,a^{+^{\prime}}}\right)^{\frac{1}{3}}\exp\left[-\left(\frac{9\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}{128}(a^{+^{\prime}})^{2}\right)^{\frac{1}{3}}\right]$
$\displaystyle=\left(\frac{\pi}{6\langle
A^{+^{\prime}}\rangle^{2}a^{+^{\prime}}}\right)^{\frac{1}{3}}\exp\left[-\left(\frac{9{\pi}(a^{+^{\prime}})^{2}}{16\langle
A^{+^{\prime}}\rangle^{2}}\right)^{\frac{1}{3}}\right]$ (26)

with the asymptotic mean and standard deviation of $A^{+^{\prime}}$ as

$\displaystyle\langle
A^{+^{\prime}}\rangle=\sqrt{\frac{8\pi}{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}},\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\sigma_{A^{+^{\prime}}}=\sqrt{\left(\frac{256}{3}-8\pi\right)\frac{1}{\lambda^{\prime}_{2}\,u^{\prime}_{{\rm
th}}}}.$ (27)

The coefficient of variation $\nu_{A^{+}}=\sigma_{A^{+}}/\langle
A^{+}\rangle=\sqrt{32/(3\pi)-1}\simeq 1.55>1$ indicates a considerably larger
relative spread of fluctuations of $A^{+}$ than those of $\tilde{H}^{+}$,
viz., $\nu_{\tilde{H}^{+}}=\sqrt{2/q}<1$ for $2<q<3$ depending on
$u^{\prime}_{{\rm th}}$.

Eq. (26) represents a $\chi^{3}_{2}$ (chi-cubed with two degrees of freedom)
asymptotic PDF for the scaled product of a $\chi_{2}$-distributed $L^{+}$ and
a fully dependent $\chi^{2}_{2}$-distributed $H^{+}$. Fig. 6 shows the scaled
(26) for selected $u^{\prime}_{{\rm th}}$. Because of the asymptotic
Gaussianity of $U$ above $u^{\prime}_{{\rm th}}\gg 1$ [8], (26) also holds
(with a scaled mean and variance) for the excursion area of the Gaussian
$X(t)$ itself and for more general $\chi^{2}_{n}$ intensities, including
$\chi^{2}_{1}$ for the intensity of random static fields.

---
Figure 6:  Asymptotic $\chi^{3}_{2}$ PDFs of $A^{+}$ (26) (solid) and
regression PDFs of $\tilde{A}^{+}$ (29) (dashed) for selected values of
$u^{\prime}_{{\rm th}}$.

For high but non-asymptotic $u^{\prime}_{{\rm th}}\gg 1$, excursions remain
sparse and short whence the terms $K_{j}(t)$ dominate the sum in (7). For
$n=2$, the asymptotic $\chi^{2}_{2}$ PDF of $H^{+}$ tends increasingly toward
$\chi^{2}_{3}$ when $u^{\prime}_{{\rm th}}$ is lowered (cf. Sec. III-B2). The
corresponding perturbation and evolution of $A^{+}$ depends on $r_{U}(t)$ –
which affects the correlation between the $H^{+}$ of excursions and between
their $L^{+}$ [20] – and on the decreasing correlation between $H^{+}$ and
$L^{+}$ (reduced curvature). In a first approximation, the parabolic
approximation $A^{+}=2H^{+}L^{+}/3$ can be maintained on account of a similar
shape of $\langle H^{+}\rangle$ for $u^{\prime}_{{\rm th}}>2$ (cf. Fig. 5),
with $\sigma_{U^{+}}\ll\langle U^{+}\rangle$ and $\rho_{U}(t-t^{+}\ll
t_{c})\simeq 1$ when $u^{\prime}_{{\rm th}}\gg 1$ (cf. Fig. 3). Thus,
individual excursions are also approximately parabolic within each excursion
$[t^{+}_{i},\/t^{+}_{i}+\ell^{+}_{i}]$, while $H^{+}$ and $L^{+}$ are still
strongly correlated. Application of (20) then yields

$\displaystyle\langle
A^{*^{\prime}}\rangle\stackrel{{\scriptstyle\Delta}}{{=}}\left\langle\frac{2H^{*^{\prime}}L^{+}}{3}\right\rangle$
$\displaystyle\simeq\sqrt{\frac{\pi^{3}}{2\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}}}$ (28)

as a nonasymptotic estimate444Note that $\langle L^{+}(u^{\prime}_{{\rm
th}}\not\gg 1)\rangle>\langle L^{+}(u^{\prime}_{{\rm
th}}\rightarrow+\infty)\rangle$, which counters the diminishing effect of
$\langle H^{*}\rangle<\langle U\rangle$ on $A^{+}$. replacing $\pi\langle
A^{+^{\prime}}\rangle/4$ in (27). As a further refinement, application of the
variate transformation $A^{+}\stackrel{{\scriptstyle\cal
L}}{{\rightarrow}}2(H^{+})^{3/2}/3$ to (21) results in

$\displaystyle\leavevmode\nobreak\ f_{\tilde{A}^{+}}(\tilde{a}^{+}|u_{{\rm
th}})\simeq\frac{C_{\tilde{A}^{+}}}{(\tilde{a}^{+})^{{1}/{3}}}\exp\left(-\frac{h^{+}(\tilde{a}^{+})}{2\beta^{2}}\right)$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\times\int^{\sqrt{h^{+}(\tilde{a}^{+})+u_{{\rm
th}}}}_{\alpha}\frac{(y-\alpha)\exp\left({\alpha\,y}/{\beta^{2}}\right)}{\sqrt{h^{+}(\tilde{a}^{+})+u_{{\rm
th}}-y^{2}}}{\rm d}y$ (29)

where
$h^{+}(\tilde{a}^{+})=(3\tilde{a}^{+}\sqrt{\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}}}/4)^{2/3}$, with

$\displaystyle C_{\tilde{A}^{+}}=\frac{(\lambda^{\prime}_{2}u^{\prime}_{{\rm
th}})^{1/3}\,\exp[-(\alpha^{2}+u_{{\rm
th}})/(2\beta^{2})]}{{2^{11/6}\,3^{1/3}\,\sqrt{\pi}\,|\beta|^{3}}}.$ (30)

Combining (28) and (29) yields
$f_{\tilde{A}^{*}}(\tilde{a}^{*})\rightarrow(4/\pi)f_{\tilde{A}^{+}}(4\tilde{a}^{*}/\pi)$
when $u^{\prime}_{{\rm th}}\rightarrow+\infty$. Fig. 6 shows close
correspondence between the regression PDF (29) and the asymptotic PDF (26)
when $u^{\prime}_{{\rm th}}\gg 1$, a fortiori for the pertinent range
$a^{+}/\sigma_{A^{+}}\ll 1$.

The approximation in (29) reflects the departure from ideal full statistical
dependence between $L^{+}$ and $H^{+}$ in writing $A^{+}\simeq
2(H^{+})^{3/2}/3$. When $u^{\prime}_{{\rm th}}\not\rightarrow+\infty$, the
actual dependence can be represented by a copula density linking
$f_{H^{+}}(h^{+})$ and $f_{L^{+}}(\ell^{+})$ [21]. The departure of
nonasymptotic $f_{L^{+}}(\ell^{+})$ from a $\chi_{2}$ PDF can be evaluated
numerically using a high-order Rice series for multiple crossings of
$u^{\prime}_{{\rm th}}$ [22, sec. 3.4], [23].

##  IV Experimental Results

The following results are based on $40\,000$ data points measured for uniform
continuous stirring in a MSRC as a dynamic EME [6], [7]. A pulse amplitude
modulated (PAM) source with carrier frequency $f_{c}=2.5$ GHz and 40% duty
cycle provides the excitation. Each $k^{{\rm th}}$ post-processed data point
$X(t_{k})$ is a 998-point sample average over the late steady-state ON part
received from the PAM input. The sample times are
$t^{(j)}_{k}\in[1000\,\delta{t},1997\,\delta{t}]$ within each $k^{{\rm th}}$
pulse period $k\cdot[0,T]=k\cdot[0,5000\,\delta{t}]$, with spacing $\Delta
t_{k}=T=100\,\mu$s. Data were collected across one full revolution at a
constant rotational speed of 0.25 r/s, for which
$\sqrt{\lambda^{\prime}_{2}}=147.9$ rad/s.

Fig. 7 shows $U(t_{k})/\sigma_{U}\equiv U^{\prime}(t_{k})/2$ and its SKM
parabolic approximations above a level $u_{\rm th}/\sigma_{U}\equiv
u^{\prime}_{{\rm th}}/2=4.1$. Fig. 7(b)–(d) indicates that the SKM is accurate
for this threshold provided that the local maxima are sufficiently widely
separated in time. Although the SKM is strictly valid only in excess of
$u_{{\rm th}}$, the model appears to be adequate also well below $u_{{\rm
th}}$. The differences between the data and the SKM estimates, shown in blue
in Fig. 7(d1), increase on average with $t_{k}-t^{+}_{k}$ and demonstrate the
nonstationarity of $K(t_{k}-t^{+}_{k})$. At the downcrossing $t^{-}_{k}$, the
error is $0.034$, i.e., $0.8\%$ of $u_{\rm th}/\sigma_{U}$.

---
(a)
(b)
(c) (d)
Figure 7:  (a) Experimental full stir sweep data $U(t_{k})/\sigma_{U}$.
(b)–(d) Comparison of experimental data $U(t_{k})/\sigma_{U}$ (black dots) vs.
SKM parabolic model (red curves) conditioned on upcrossings $t^{+}_{i}$ of
$u_{{\rm th}}/\sigma_{U}=4.1$: (c) for $15\,150\leq t_{k}\leq 15\,650$, with
$(t^{+}_{3},t_{3,{\rm max}},z_{3})=(15\,284,15\,299,0.449)$ and
$(t^{+}_{4},t_{4,{\rm max}},z_{4})=(15\,531,15\,543.5,0.374)$; (d1) data
(black dots), SKM conditioned on $t^{+}_{4}$ (red solid), and residual
difference raised by $u_{{\rm th}}/\sigma_{U}$ (blue solid); (d2) SKMs
conditional on local maximum for noisy crossings of $u_{{\rm th}}/\sigma_{U}$
for $15\,650\leq t_{k}\leq 17\,690$, at $(t^{+}_{i},t_{i,{\rm
max}},z_{i})=(17\,664,17\,664,0)$, $(17\,666,17\,667.5,0.045)$,
$(17\,671,17\,671,0)$ and $(17\,681,17\,681,0)$ with $i=6,7,8$ and $9$,
respectively. All $t_{k}$ in units $\Delta t_{k}$.

The small bias of the SKM parabola towards lower $t_{k}$ is a result of
conditioning on upcrossings as opposed to, e.g., local maxima [11], [13]. A
slightly improved approximation can be obtained using a more elaborate doubly
conditioned SKM that uses both up- and downcrossings [10, eq. (20)].

Fig. 7(d2) shows that noise in data – even after averaging within each pulse
period – affects the accuracy of the SKM, a fortiori near a local maximum.
Here, the SKM was conditioned on the estimated height and location of such a
maximum. Such conditioning is prone to larger uncertainty and requires precise
knowledge of $\lambda^{\prime}_{4}$ in (9) [7, Figs. 6(c) and 7(c)], compared
with conditioning on upcrossings. Clearly, the overlapping SKM parabolas fail
to provide an accurate approximation. This is seen in Fig. 7(b) and (d2) from
the overlap of adjacent parabolas, each one being characterized by
$z_{i}\simeq 0$ on account of $u^{\prime}_{{{\rm th}},i}\simeq
u^{\prime}_{{{\rm max}},i}$, whence successive crossings now resemble a noise
process themselves.

The approximation of the next local maximum at $t_{{{\rm max}},10}$ [cf. Fig.
7(b)] is also affected, because at $t^{+}_{10}$ the data near
$u^{\prime}_{{\rm th}}$ lack the smoothness of threshold crossing required for
an accurate SKM. Comparison of Fig. 7(d1) with Fig. 2(b) confirms that, for
this excursion at this $u_{{\rm th}}$, the excursion length
$\ell^{+}=25\,\Delta t_{k}\ll(103.5/2)\,\Delta t_{k}=t_{c}$ is sufficiently
short compared to $t_{c}$ in order that the regression term dominates the
$K_{j}(t)$ in (7).

Fig. 8(a) shows sample values of $Z$ from the SKM for excursions $i$ in Fig.
7(a) at selected threshold levels. When local maxima of $U(t_{k})$ are closely
spaced, the $z_{i}$ become more volatile, leading to peaks of $z_{i}$, e.g.,
for $t_{k}\simeq 17\,000\,\Delta t_{k}$ to $23\,000\,\Delta t_{k}$. Except for
such peaks, most values of $z_{i}$ are notably similar for a chosen $u_{{\rm
th}}/\sigma_{U}$.

For an arbitrary excursion $i$, the value of $z_{i}$ decreases for increasing
$u_{{\rm th}}/\sigma_{U}$. Further analyzing this decrease, Fig. 8(b)
demonstrates that the functional dependence $z_{i}(u_{{\rm th}}/\sigma_{U})$
exhibits the typical square-root roll-off to zero (rounded tip) as the local
maximum is approached sufficiently closely, i.e., for $t_{i,{{\rm
max}}}-t^{+}_{i}\rightarrow 0$. Sharp notches in $z_{i}(u_{{\rm
th}}/\sigma_{U})$ indicate excursions that degenerate to multiple peaks as
$u_{{\rm th}}$ is increased, e.g., near $u_{{\rm th}}/\sigma_{U}=4.1$ for
$t_{7}=17\,670\,\Delta t_{k}$. Such degeneracies typically start at relatively
large $z_{i}$ (long excursion) for low $u_{{\rm th}}/\sigma_{U}$. A large
cluster of local maxima breaks up into smaller excursions when $u_{{\rm th}}$
is increased. Such clustering may have important consequences in an EM test
device with a relatively large time constant or small recovery times. In this
respect, the notches in $z_{i}(u_{{\rm th}}/\sigma_{U})$ are indications of
changes in connectivity (topology) of the excursions.

Fig. 9 shows experimental sample values of $m_{A^{+}}$ and $s_{A^{+}}$ for
$\langle A^{+}\rangle$ and $\sigma_{A^{+}}$ as a function of $u_{{\rm
th}}/\sigma_{U}$. Each value of $a^{+}_{i}$ was estimated from a discrete
staircase approximation of $U_{i}(t)$ by rectangles of height $U^{+}(t_{k})$
and width $\Delta t_{k}$. For $u_{{\rm th}}/\sigma_{U}\gg 1$, the estimated
values of $a^{+}_{i}$ ultimately suffer from inevitable depletion of data and
extreme narrowness of excursions, whereby the accuracy of the staircase
approximation suffers.

---
(a)
(b)
Figure 8:  Sample values $z_{i}$ of $Z$ in the SKM parabolic approximation of
$U(t)/\sigma_{U}$: (a) for $i^{{\rm th}}$ positive excursion at selected
threshold levels $u_{{\rm th}}/\sigma_{U}$ from 3.5 to 5.9; (b) sample paths
$z_{i}(u_{{\rm th}}/\sigma_{U})$ for selected excursions in Fig. 7(a).
---
Figure 9:  Experimental mean $m_{A^{+}}$ and standard deviation $s_{A^{+}}$
(solid red resp. blue) in dimensionless units $1^{2}\cdot\Delta{t_{k}}$ vs.
theoretical asymptotic $\langle{A^{+}}\rangle$ and $\sigma_{A^{+}}$ (dashed
magenta resp. black) as a function of $u_{{\rm th}}/\sigma_{U}$.

##  V Excess Energy and Power in Dynamic EMEs

###  V-A Hybrid Modulation

In this second part of the paper, the interest is shifted to the
characterization of energy and power exceedance in a rapid dynamic, i.e.,
continuously varying EME. Its variation can occur with respect to the
electronic state (e.g., transitions of ambient EM sources or biasing fields)
or the mechanical configuration (e.g., motion of EM cavity boundaries or
scatterers). Configurational changes induce secondary modulation in amplitude
[5], phase or frequency [6] to an unmodulated continuous wave (CW) or to a
primary (i.e., electric heterodyne) modulated source signal field. The
occurrence of both modulation types produces hybrid modulation (HM). The
overall rate of fluctuation of the field in HM is governed by both the source
and the time-varying EME [24], [25].

In focusing on dynamic effects, the source is here assumed to be unmodulated
(CW), including the limiting case of a narrowband modulation, while the EME
provides additional harmonic variation at a constant rate. In a MSRC context,
overlap of cavity modes causes a CW source field to become narrowband
modulated, while mode stirring produces modulation of each modal component.
Thus, the spectra reduce to a single line in each case. The HM can be
characterized by the HM index

$\displaystyle
m\stackrel{{\scriptstyle\Delta}}{{=}}\sqrt{\lambda^{\prime}_{2}}/\omega$ (31)

representing the ratio of the EME’s dynamic (stir) rate of fluctuation
$\sqrt{\lambda^{\prime}_{2}}$ to the rate of the static (unstirred) EM
oscillation $\omega$ of the CW excitation source.

For wideband electric modulation of the source signal, HM applies to each
quasi-static frequency component of the source and produces a widened dynamic
spectrum. HM by the EME results in an increased bandwidth. Similarly, for an
EME changing at a nonuniform rate [6], its effect on the EM field can be
characterized by the configurational Fourier spectrum, in which the $i$th
spectral component of the EME transfer function can be analyzed in conjunction
with the $j$th component of the EM source. If
$\sqrt{\lambda^{\prime}_{2}}\sim\omega$ or if either spectrum is sufficiently
wideband such that $\sqrt{(\lambda^{\prime}_{2})_{i}}\sim\omega_{j}$, then HM
coupling occurs.

###  V-B Effective Power

The previous exposition for quasi-static operation can be extended to dynamic
EMEs. Recall the circuit definitions of deterministic instantaneous power
$p(t)$, energy $u(T)$ and quasi-static effective (time-averaged) power
$p_{{\rm eff}}$, i.e.,

$\displaystyle p(t)=v(t)i(t),\leavevmode\nobreak\ \leavevmode\nobreak\
u(T)=\int^{T}_{0}p(t){\rm d}t,\leavevmode\nobreak\ \leavevmode\nobreak\
p_{{\rm eff}}=\frac{u(T)}{T}$ (32)

for a voltage $v(t)$ and current $i(t)$ with arbitrary time dependence.
Corresponding definitions apply to random $P(t)$, $U(T)$ and $P_{{\rm eff}}$,
which pose no issue provided that the rate of dynamic variation does not
exceed that of the electrical fluctuation for the field
($\sqrt{\lambda^{\prime}_{2}}/\omega<1$). This is the case for classical
mechanical stirring, viz., $\sqrt{\lambda^{\prime}_{2}}/\omega=9.4\times
10^{-9}$ in the present experiment, but may approach $1$ more closely in rapid
electronic stirring. The power associated with excursion intervals is then
obtained by dividing the excess energy by the corresponding excursion length.
For rapid variations ($\sqrt{\lambda^{\prime}_{2}}/\omega\not\ll 1$), the
fluctuation of the field and its time-varying envelope must be taken into
account, which is considered next.

For a time-invariant amplitude (peak voltage) $v$, the electric energy
accumulated across $[0,T]$ is

$\displaystyle u(T)=\int^{T}_{0}[v\cos(\omega t)]^{2}{\rm
d}t=\frac{v^{2}}{2}\left(T+\frac{\sin(2\omega T)}{2\omega}\right).$ (33)

The effective power $p_{{\rm eff}}$ with reference to $[0,T]$ follows as

$\displaystyle p_{{\rm eff}}=\left(\frac{v}{\sqrt{2}}\right)^{2}\left[1+{{\rm
sinc}}(2\omega T)\right]\stackrel{{\scriptstyle\small\omega
T\rightarrow\infty}}{{\longrightarrow}}\left(\frac{v}{\sqrt{2}}\right)^{2}$
(34)

whence $v_{{\rm eff}}=v/\sqrt{2}$ in the quasi-stationary low-frequency (LF)
limit is retrieved. In analogy with (5), the corresponding random quantities
are further normalized as

$\displaystyle{P^{(+)}_{({\rm
eff})}}^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{P^{(+)}_{({\rm
eff})}}{\sigma^{2}_{V}},\leavevmode\nobreak\ \leavevmode\nobreak\
{U^{(+)}_{({\rm
eff})}}^{\prime}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{U^{(+)}_{({\rm
eff})}}{\sigma^{2}_{V}},\leavevmode\nobreak\ \leavevmode\nobreak\
V^{\prime}_{({\rm eff})}\stackrel{{\scriptstyle\Delta}}{{=}}\frac{V_{({\rm
eff})}}{\sigma_{V}}.$ (35)

As is well known (cf., e.g., [26, sec. 6.2.2]), the definition of the real
envelope $V(t)$ – as used in defining the effective voltage and effective
power – holds regardless of the bandwidth of the process. Therefore, the
following exposition is valid regardless of the value of
$\sqrt{(\lambda^{\prime}_{2})_{i}}/\omega_{j}$ for the spectral components of
the motion vs. excitation.

For $u^{\prime}_{{\rm th}}\gg 1$, the SKM (23) holds and the duration is

$\displaystyle L^{+}(u^{\prime}_{{\rm th}})=2(t^{+}_{{\rm
max}}-t^{+})=\frac{2Z}{\omega\kappa^{+}}$ (36)

in which the dimensionless parameter $\kappa^{+}$ is

$\displaystyle\kappa^{+}\stackrel{{\scriptstyle\Delta}}{{=}}m\sqrt{u^{\prime}_{{\rm
th}}}=\sqrt{\lambda^{\prime}_{2}u^{\prime}_{{\rm th}}}/{\omega}$ (37)

and where $m$ is the HM index (31). The accumulated energy follows from (33)
by integrating $P(t)=V^{2}(t)/R$ across $T=L^{+}$, with $V^{2}(t)$ given by
(23) and referencing to a unit resistance $R$, yielding

$\displaystyle U^{\prime}(Z)$ $\displaystyle=\left(\frac{u^{\prime}_{{\rm
th}}}{\omega\kappa^{+}}-\frac{\kappa^{+}}{4\omega}\right)Z+\frac{2\,Z^{3}}{3\omega\kappa^{+}}-\frac{\kappa^{+}Z}{4\omega}\cos\left(\frac{4Z}{\kappa^{+}}\right)$
$\displaystyle\leavevmode\nobreak\ +\left(\frac{u^{\prime}_{{\rm
th}}}{4\omega}+\frac{{\kappa^{+}}^{2}}{8\omega}\right)\sin\left(\frac{4Z}{{\kappa^{+}}}\right).$
(38)

With $Z$ taking a value $z_{i}$, this results in $u^{\prime}(z_{i})$ and
$p^{\prime}_{i}=u^{\prime}(z_{i})/\ell^{+}_{i}$. Finally, the effective power
across $L^{+}$ follows as

$\displaystyle P^{\prime}_{{\rm eff}}(Z)$
$\displaystyle=\frac{U^{\prime}(Z)}{T}=\frac{\omega{\kappa^{+}}}{2Z}U^{\prime}(Z)$
$\displaystyle=\frac{u^{\prime}_{{\rm
th}}}{2}-\frac{{\kappa^{+}}^{2}}{8}+\frac{Z^{2}}{3}-\frac{{\kappa^{+}}^{2}}{8}\cos\left(\frac{4Z}{{\kappa^{+}}}\right)$
$\displaystyle\leavevmode\nobreak\ +\left(\frac{u^{\prime}_{{\rm
th}}}{2}+\frac{{\kappa^{+}}^{2}}{4}\right){\rm
sinc}\left(\frac{4Z}{{\kappa^{+}}}\right).$ (39)

In (38) and (39), the harmonic terms contain amplitudes that depend on
$\kappa^{+}$, while the phase $4Z/\kappa^{+}$ is Rayleigh distributed, i.e.,
nonuniform. The ratio $P^{\prime}_{{\rm eff}}/(u^{\prime}_{{\rm th}}/2)$
measures the relative contribution of the EME dynamics to the power in the
excursion.

For arbitrary $u^{\prime}_{{\rm th}}$, two limiting cases can be identified:

  * •

quasi-static LF excitation or exceedingly “high” EME velocity: this implies
$1/{\kappa^{+}}\rightarrow 0$, i.e.,
$\sqrt{\lambda^{\prime}_{2}}/\omega\rightarrow+\infty$ for finite
$u^{\prime}_{{\rm th}}$ and $Z$. In this case, a series expansion of (39)
leads to

$\displaystyle P^{\prime}_{{\rm eff}}(Z)\simeq u^{\prime}_{{\rm
th}}+\frac{2\,Z^{2}}{3}-\frac{4\,Z^{4}}{5\,(\kappa^{+})^{2}}+{\cal
O}\left[\frac{Z^{6}}{(\kappa^{+})^{4}}\right];$ (40)

  * •

quasi-optical HF excitation or quasi-statically “low” EME velocity: this
corresponds to $\kappa^{+}\rightarrow 0$. In this classic regime, it follows
that

$\displaystyle P^{\prime}_{{\rm
eff}}(Z)\rightarrow\frac{1}{2}\left(u^{\prime}_{{\rm
th}}+\frac{2\,Z^{2}}{3}\right).$ (41)

In this case, $P^{+}_{{\rm eff}}$ maps to half of the area of the parabolic
cap $A^{+}/2$, for which the PDF (26) applies. Together with the area $A_{{\rm
th}}$ and the root mean square factor $1/2$ for effective scaling, this yields
$P^{\prime}_{{\rm eff}}$ as expected.

The PDF $f_{P^{\prime}_{{\rm eff}}}(p^{\prime}_{{\rm eff}})$ can be obtained
from the inverse variate transformation $Z\rightarrow P^{\prime}_{{\rm eff}}$
in (39). Alternatively, and more conveniently, this PDF is also obtained by
Monte Carlo (MC) simulation through generating $N$ Rayleigh distributed values
of $Z$. Fig. 10 shows the thus obtained complementary CDF (CCDF)
$1-F_{P^{\prime}_{{\rm eff}}}(p^{\prime}_{{\rm eff}})$ of Cartesian power
($n=2$) for $u^{\prime}_{{\rm th}}=10$ at selected values of $m$, based on
$N=10^{5}$ independent replicas. The tail dependence for $m\ll 1$ indicates
that quasi-static $P^{\prime}_{{\rm eff}}$ as an area-to-length ratio also
follows a GPD distribution with $\gamma=0$.

---
Figure 10:  CCDF of Cartesian $P^{\prime}_{{\rm eff}}$ for $u^{\prime}_{{\rm
th}}=10$ at selected values of HM index $m$, based on $N=10^{5}$ MC replicas.

The mean of $P^{\prime}_{{\rm eff}}$ is obtained after calculation as

$\displaystyle\langle P^{\prime}_{{\rm eff}}\rangle$
$\displaystyle=\int^{+\infty}_{0}P^{\prime}_{{\rm eff}}(z)f_{Z}(z){\rm
d}z=\frac{2}{3}+\frac{u^{\prime}_{{\rm th}}}{2}-\frac{{\kappa^{+}}^{2}}{4}$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
+\left[\frac{\sqrt{2}}{2}\left(1+\frac{u^{\prime}_{{\rm
th}}}{4}\right){\kappa^{+}}+\frac{{\kappa^{+}}^{3}}{8\sqrt{2}}\right]D\left(\frac{2\sqrt{2}}{{\kappa^{+}}}\right)$
(42)

where $D(\cdot)$ is Dawson’s integral [27], defined by

$\displaystyle
D(x)\stackrel{{\scriptstyle\Delta}}{{=}}\int^{x}_{0}\exp(t^{2}-x^{2}){\rm
d}t=\frac{\sqrt{\pi}}{2}\exp(-x^{2}){{\rm erfi}}(x)$ (43)

in which ${{\rm erfi}}(x)\stackrel{{\scriptstyle\Delta}}{{=}}(-{\rm j}){{\rm
erf}}({\rm j}x)$, with ${{\rm erf}}(\cdot)$ denoting the (real) error
function. Asymptotic expansions up to second order in ${\kappa^{+}}$ follow as

$\displaystyle\langle P^{\prime}_{{\rm
eff}}\rangle=\left\\{\begin{array}[]{ll}\frac{2}{3}+\frac{u^{\prime}_{{\rm
th}}}{2}+\frac{{\kappa^{+}}^{2}}{4}(3+u^{\prime}_{{\rm
th}})+\ldots,&\kappa^{+}\ll 1\\\ \\\ \frac{4}{3}+u^{\prime}_{{\rm
th}}-\frac{8}{3{\kappa^{+}}^{2}}\left(\frac{2}{5}+u^{\prime}_{{\rm
th}}\right)+\ldots,&{\kappa^{+}}\gg 1\end{array}\right.$ (47)

confirming that $\langle P^{\prime}_{{\rm
eff}}({\kappa^{+}}\rightarrow+\infty)\rangle=2\langle P^{\prime}_{{\rm
eff}}({\kappa^{+}}\rightarrow 0)\rangle$. It follows that for $u^{\prime}_{\rm
th}\rightarrow+\infty$, the ratio of the average above-threshold excess power
$\langle P^{+}_{{\rm eff}}\rangle$ to the average below-threshold power
$\langle P_{{{\rm eff}},{\rm th}}\rangle$ for ${\kappa^{+}}\ll 1$ equals this
ratio for ${\kappa^{+}}\gg 1$, to second order in $\kappa^{+}$, viz.,
$(2/3)/(u^{\prime}_{{\rm th}}/2)=4/(3u^{\prime}_{{\rm th}})$. Specifically,

$\displaystyle\frac{\langle P^{+}_{{\rm eff}}\rangle}{\langle P_{{{\rm
eff}},{\rm th}}\rangle}=\left\\{\begin{array}[]{ll}\frac{4}{3u^{\prime}_{{\rm
th}}}\left(1+\frac{5{\kappa^{+}}^{2}}{8}+\ldots\right),&\kappa^{+}\ll 1\\\ \\\
\frac{4}{3u^{\prime}_{{\rm
th}}}\left(1+\frac{28}{15{\kappa^{+}}^{2}}+\ldots\right),&\kappa^{+}\gg
1.\end{array}\right.$ (51)

For arbitrary ${\kappa^{+}}$, this ratio is frequency dependent, as is also
apparent from (42). Since $P^{+}_{{\rm eff}}$ arises as the difference of
effective powers $P_{{\rm eff}}$ and $P_{{\rm eff},{\rm th}}$ from ground
level up, $P^{+}_{{\rm eff}}$ represents an effective power quantity as well.

The second moment $\langle{P^{\prime}_{{\rm
eff}}}^{2}\rangle\stackrel{{\scriptstyle\Delta}}{{=}}\int^{\infty}_{0}[P^{\prime}_{{\rm
eff}}(z)]^{2}f_{Z}(z){\rm d}z$ follows by a similar integration. Combining
with (42) yields the variance of $P^{\prime}_{{\rm eff}}$ as

$\displaystyle\,\sigma^{2}_{P^{\prime}_{{\rm
eff}}}=\frac{16}{9}+\frac{u^{\prime}_{{\rm
th}}}{3}+\frac{{\kappa^{+}}^{2}}{6}-\sqrt{2}\left[\frac{16}{3{\kappa^{+}}}+\frac{{\kappa^{+}}}{3}-\frac{{\kappa^{+}}^{3}}{12}-\frac{{\kappa^{+}}^{5}}{64}\right.$
$\displaystyle\leavevmode\nobreak\
\left.+\left(\frac{4}{3{\kappa^{+}}}+\frac{{\kappa^{+}}}{12}-\frac{{\kappa^{+}}^{3}}{32}\right)u^{\prime}_{{\rm
th}}\right]D\left(\frac{2\sqrt{2}}{{\kappa^{+}}}\right)$
$\displaystyle\leavevmode\nobreak\
-\sqrt{2}\left(\frac{{\kappa^{+}}^{3}}{16}+\frac{{\kappa^{+}}^{5}}{128}+\frac{{\kappa^{+}}^{3}u^{\prime}_{{\rm
th}}}{64}\right)D\left(\frac{4\sqrt{2}}{{\kappa^{+}}}\right)$
$\displaystyle\leavevmode\nobreak\
-\frac{1}{2}\left[\left(1+\frac{u^{\prime}_{{\rm
th}}}{4}\right){\kappa^{+}}+\frac{{\kappa^{+}}^{3}}{8}\right]^{2}D^{2}\left(\frac{2\sqrt{2}}{{\kappa^{+}}}\right)$
$\displaystyle\leavevmode\nobreak\
+\left(\frac{{\kappa^{+}}^{4}}{16}+\frac{{u^{\prime}_{{\rm
th}}}^{2}}{4}+\frac{u^{\prime}_{{\rm
th}}{\kappa^{+}}^{2}}{4}\right)\/_{2}F_{2}\left(1,1;\frac{3}{2},2;-\frac{32}{{\kappa^{+}}^{2}}\right)$
(52)

where $\/{}_{2}F_{2}\left(\cdot,\cdot;\cdot,\cdot;y\right)$ is a generalized
hypergeometric series.

Fig. 11 shows $\langle P^{\prime}_{{\rm eff}}(m)\rangle/u^{\prime}_{{\rm th}}$
and $\sigma_{P^{\prime}_{{\rm eff}}}(m)/u^{\prime}_{{\rm th}}$ together with
$\nu_{P^{\prime}_{{\rm eff}}}(m)=\sigma_{P^{\prime}_{{\rm
eff}}}(m)/\langle{P^{\prime}_{{\rm eff}}}(m)\rangle$, for selected values of
$u^{\prime}_{{\rm th}}$. These characteristics were calculated from (42) and
(52), and subsequently validated by MC simulation of (39), thus also
validating the CCDFs in Fig. 10 in the process. The dependencies for $m>1$ are
hypothetical, shown as dashed curves. The increase of the power is more
pronounced for increased $u^{\prime}_{{\rm th}}$. At the same time, the
interval with resonance-like variation shifts towards lower $m$, with this
transition region showing first relatively large positive, then negative
variation compared to the quasi-static values for $m\rightarrow 0$. For slow
dynamics and LF excitation, typical values of the maximum-to-mean value of
$P^{\prime}_{{\rm eff}}$ are of the order of $u^{\prime}_{{{\rm th}},{{\rm
max}}}\ll 10$ and indicate that the quasi-static approximation for the mean
and standard deviation of the power is excellent, i.e., valid for non-extreme
speeds.

(
---
(a)
(b)
(c)
Figure 11:  (a) Normalized mean, (b) normalized standard deviation, and (c)
coefficient of variation of dynamic power $P^{\prime}_{eff}$ as a function of
modulation index $m$ for selected values of $u^{\prime}_{{\rm th}}$.

##  VI Conclusion

EM fields can become modulated at the source, using traditional electrical
modulation techniques, or through propagation effects in a dynamic EME. In
this work, the relative contribution of both mechanisms in the power of the
received field above high threshold levels was investigated, based on a
second-order parabolic SKM conditioned on upcrossings for the excursions of
intensity. Asymptotic and regressed PDFs of the excursion height and area were
obtained as (19), (21), (26) and (29), representative of quasi-static field
intensity and energy. For dynamic EMEs, the mean (42) and variance (52) of the
effective excess power (39) were obtained, whose CCDF was calculated
numerically (Fig. 10).

For sufficiently high threshold crossing levels, the decomposition provided by
a SKM permits the extraction of the quasi-deterministic short-term
contribution to the field, which dominates during level excursions. This
greatly simplifies its representation, in particular for the PDF of the
regressed early energy. For the field intensity, an SKM yields a more accurate
estimation of local and global maximum values with a reduced uncertainty
(Figs. 3 and 7), compared with a traditional statistical characterization
based on unconditioned sampled fields (i.e., without thresholding). The latter
is retrieved as the long-term asymptotic in the SKM. For relatively low
thresholds, the interdependence between lengths of consecutive positive and
negative excursions [20] and the partial correlation between excursion height
and length make accurate modelling for such thresholds more complicated.

Compared to other approaches to conditional statistical characterization,
e.g., based on prior PDFs in Bayesian inference [28], the SKM only uses
timings of upcrossings (or another choice of marks) together with the Rayleigh
PDF (3) for the slope, thus simplifying the modelling. Using a prior PDF for
the bandwidth–level product $\sqrt{\lambda^{\prime}_{2}u^{\prime}_{{\rm th}}}$
as parameter, the likelihood obtained in the SKM can be used to yield
posterior PDFs, in particular for sampling distributions for short intervals.

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