# A high-performance custom photodetection system to probe the light yield
enhancement in oriented crystals

M Soldani1,2 L Bandiera2 L Bomben3,4 C Brizzolari4,5 R Camattari1,2 D De
Salvador6,7 V Guidi1,2 V Haurylavets8 E Lutsenko3,4 T Maiolino9 V Mascagna3,4
A Mazzolari2 M Prest3,4 M Romagnoni2,10 F Ronchetti3,4 A Selmi3,4 A Sytov2 V
Tikhomirov8 E Vallazza4 1Dipartimento di Fisica e Scienze della Terra,
Università degli Studi di Ferrara, Ferrara, Italy  2Istituto Nazionale di
Fisica Nucleare, Sezione di Ferrara, Ferrara, Italy  3Dipartimento di Scienza
e Alta Tecnologia, Università degli Studi dell’Insubria, Como, Italy
4Istituto Nazionale di Fisica Nucleare, Sezione di Milano Bicocca, Milano,
Italy  5Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano
Bicocca, Milano, Italy  6Dipartimento di Fisica e Astronomia, Università degli
Studi di Padova, Padova, Italy  7Istituto Nazionale di Fisica Nucleare,
Laboratori Nazionali di Legnaro, Legnaro, Italy  8Institute for Nuclear
Problems of Belarusian State University, Minsk, Belarus  9Department of
Physics, Wuhan University, Wuhan, China  10Dipartimento di Fisica, Università
degli Studi di Milano Statale, Milano, Italy<EMAIL_ADDRESS>
###### Abstract

Scintillating homogeneous detectors represent the state of the art in
electromagnetic calorimetry. Moreover, the currently neglected crystalline
nature of the most common inorganic scintillators can be exploited to achieve
an outstanding performance boost in terms of compactness and energy
resolution. In fact, it was recently demonstrated by the AXIAL/ELIOT
experiments that a strong reduction in the radiation length inside PWO, and a
subsequent enhancement in the scintillation light emitted per unit thickness,
are attained when the incident particle trajectory is aligned with a crystal
axis within $\sim 1^{\circ}$. A SiPM-based system has been developed to
directly probe this remarkable effect by measuring the scintillation light
emitted by a PWO sample. The same concept could be applied to full-scale
detectors that would feature a design significantly more compact than
currently achievable and unparalleled resolution in the range of interest for
present and future experiments.

##  1 Axial alignment and coherent effects

The electromagnetic interactions high-energy particles undergo when impinging
on a crystal with small angle with respect to a lattice axis, i.e. a periodic
string of nuclei, heavily differ from the amorphous target case. In
particular, the spectra of bremsstrahlung radiation emission by
electrons/positrons and pair production (PP) by photons in crystals depend on
the angle with respect to the axis, and on the input energy [1, 2].

In case of ultrarelativistic particles (i.e. with an input energy of $\gtrsim
10~{}\mathrm{GeV}$), the axial electromagnetic field in the frame of the
latter is affected by the Lorentz boost: the so-called strong field (SF)
regime is attained for

$\chi=\frac{\gamma E}{E_{0}}>1\mathrm{,}$

where $\gamma=\varepsilon/mc^{2}$ is the Lorentz factor, $\varepsilon$ is the
input energy, $E$ is the axis field in the laboratory frame and $E_{0}\sim
1.32\times 10^{18}~{}\mathrm{V/m}$ is the Schwinger QED critical field, above
which nonlinear field effects occur in vacuum [3]. In this regime, the
interaction between charged particles and the crystalline lattice results in
the emission of quantum synchrotron radiation, which features a strong and
peaked enhancement in the hard part of the photon spectrum with respect to the
Bethe-Heitler, i.e. incoherent, case [4]. Moreover, high-energy photons
experience the SF too: in fact, the PP rate per unit thickness is
macroscopically enhanced with respect to the amorphous case, such an
enhancement being proportional to the input energy [5]. Limited-strength SF-
related effects persist down to $\chi\sim 0.1$; on the other hand, a
saturation is expected to occur at about 100 times the SF threshold energy,
i.e. in the multi-TeV range [6], which is beyond the current experimental
upper limit [7]. The angular range of such effects depends on the continuous
electromagnetic potential associated to the axis $U_{0}$. It has been
evaluated as [1]

$\Theta_{0}=\frac{U_{0}}{mc^{2}}\mathrm{,}$

although a limited enhancement in the bremsstrahlung spectrum by
electrons/positrons and in the number of ${e^{+}e^{-}}$ pairs by photons is
attained also out of the $\Theta_{0}$ threshold and up to approximately
${1^{\circ}}$ [4, 7].

As a consequence of the aforementioned enhancement in the bremsstrahlung and
PP cross sections, when on axis, a strong acceleration of the electromagnetic
shower started by an input electron, positron or photon in the medium is
attained; the latter corresponds to an overall reduction of the crystal
effective radiation length $X_{0}$ [4, 7]. Moreover, the dependence of the
enhancement factor on the primary particle energy counterbalances the increase
in the shower depth: as a result, the dependence of the shower energy deposit
peak longitudinal position on the input energy is cancelled out almost
completely [8, 9]. In case of scintillating crystals such as lead tungstate
(PbWO4; abb. PWO), the higher number of shower particles per unit thickness
corresponds to an enhancement in the overall scintillation light yield.

##  2 Setup to test the light yield enhancement in an oriented PWO crystal

Recently, a test was performed at the CERN extracted beamlines with the aim of
directly measuring the radiation enhancement in crystalline PWO, demonstrating
a five-fold reduction of $X_{0}$ for $120~{}\mathrm{GeV}$ electrons in an
oriented $[001]$ PWO sample. Here we describe the optimized experimental
setup, similar to that discussed e.g. in [4] and [10], and the development of
a SiPM-based readout system to measure the enhancement in the scintillation
light output.

PWO has a tetragonal, scheelite-like crystalline structure, with lattice
constants $a=b=5.456$ Å and $c=12.020$ Å. The SF is attained at a few tens of
$\mathrm{GeV}$ and $\Theta_{0}\sim 1~{}\mathrm{mrad}$ [9]. The sample under
study in this work is a $4~{}\mathrm{mm}$ (i.e. $0.45X_{0}$) thick strip, with
a transverse surface of $55\times 2~{}\mathrm{mm}^{2}$.

Figure 1: Experimental setup. The trajectory of the input particles is
reconstructed via a silicon microstrip tracking system. Another pair of
microstrip layers are placed downstream with respect to the crystal and
exploited as a multiplicity counter of the output charged particles; the
latter are then separated from the output photons by a dipole. Finally, a lead
glass-based electromagnetic calorimeter measures the total radiated energy.
Adapted from [7].

Figure 1 shows a scheme of the optimized experimental setup for the SPS
extracted lines, where multi-GeV pure electron beams are available. Firstly,
the input particles impinge on a telescope which consists of two tracking
modules. Each module features a double-side, $\sim 2\times
2~{}\mathrm{cm^{2}}$ silicon microstrip detector, $300~{}\mu\mathrm{m}$ thick,
with a readout pitch of $50~{}\mu\mathrm{m}$; an analog readout chain makes it
possible to attain a single-hit resolution of $\sim 10~{}\mu\mathrm{m}$ [11]
and, given the lever arm of $\sim 12~{}\mathrm{m}$, a resolution on the
incident angle of $\sim 8~{}\mu\mathrm{rad}$. Another microstrip module, which
consists of two single-side, $\sim 10\times 10~{}\mathrm{cm^{2}}$,
$410~{}\mu\mathrm{m}$ thick sensors, is placed downstream with respect to the
target, in order to measure the multiplicity of charged particles in the
output state by counting the number of hits. Finally, after the charged
particles are deflected by a dipole magnet, a lead-glass calorimeter measures
the energy of the photons, i.e. the energy loss in the crystalline sample.

A high-precision goniometer allows for the remote control of the crystal
position and orientation in both the horizontal and vertical directions with a
resolution of $\lesssim 5~{}\mu\mathrm{m}$ and $\lesssim 5~{}\mu\mathrm{rad}$
respectively [4, 11, 12]. As shown in Figure 2 Left, the PWO strip is
installed on the latter with the long side parallel to the vertical direction
and one of the $2\times 4~{}\mathrm{mm}^{2}$ faces, which has been polished,
oriented downwards, to face the optical light readout system.

The core of the photodetection system is a Silicon PhotoMultiplier (SiPM). The
ASD-NUV4S-P model by AdvanSiD [13] has been chosen: its $4\times
4~{}\mathrm{mm}^{2}$ square-shaped surface well matches the PWO strip polished
face, and its photodetection efficiency, which has a FWHM of $\sim
190~{}\mathrm{nm}$ that ranges in $380$–$570~{}\mathrm{nm}$ and the peak
sensitivity wavelength at $\sim 420~{}\mathrm{nm}$ (NUV stands for Near
UltraViolet), matches the PWO scintillation spectrum perfectly — the latter
being peaked at $420~{}\mathrm{nm}$ [14]. Moreover, the good timing
performance allows for a detailed study of the signal development.

Figure 2: Left: The PWO strip (orange) coupled to the optical readout system;
the detail of the SiPM (magenta) is shown in the top left corner. Right: PWO
sample detection efficiency map as a function of the electron impact position.

The SiPM is coupled to the evaluation board via a dedicated socket [15]. The
AdvanSiD ASD-EP-EB-N Evaluation Board is used in order to provide easy
interface to the acquisition chain, current to voltage conversion and signal
amplification [16]. The output waveform is then acquired by a fast digitizer,
the CAEN V1730C, which features a 14-bit resolution, a $500~{}\mathrm{MS/s}$
sampling rate and an input dynamic range that can be set either to $0.5$ or to
$2~{}\mathrm{V}_{\mathrm{pp}}$ [17, 18].

This experimental setup allows to directly measure the scintillation light
signal and to study its dependence on the features of the incident particle
trajectory on an event-by-event basis. It was first exploited in a beamtest in
2018 at the SPS H2 beamline. For example, Figure 2 Right shows the PWO strip
detection efficiency, defined as the ratio between the number of events in
which the scintillation light was detected and the total number of crossing
events, as a function of the hit position on the crystal surface, obtained by
projecting the input electron trajectory reconstructed by the upstream
tracking modules. Indeed, it is evident from Figure 2 Right that an efficiency
of $\gtrsim 95\%$ is attained uniformly in the whole sample under test.

##  3 Conclusions and outlook

The prototype of a custom, SiPM-based light readout system to directly measure
the light emitted by oriented scintillating crystalline samples, such as PWO,
and the setup adapted for the CERN SPS hundred-GeV electron beamline have been
described. The outcome of this work might pave the way to the development of
full-scale, ultra-compact and high-resolution homogeneous calorimeters, thanks
to the strong reduction of the effective radiation length due to coherent
effects in oriented crystalline scintillators. This design would prove
particularly beneficial in fixed-target experiments and in space-born $\gamma$
observatories.

## Acknowledgements

We acknowledge the support of the PS/SPS physics coordinator and of the CERN
SPS and EN-EA group technical staff. This work was partially supported by INFN
through the PHOTAG, ELIOT and STORM experiments.

## References

## References

  * [1] Baier V N, Katkov V M and Strakhovenko V M 1986 Zh. Eksp. Teor. Fiz. 90 801
  * [2] Baryshevskiĭ V G and Tikhomirov V 1989 Usp. Fiz. Nauk 159 529 [1989 Sov. Phys. Usp. 32 1013]
  * [3] Buchanam M 2006 Nature Phys. 2 721
  * [4] Bandiera L et al. 2018 Phys. Rev. Lett. 121 021603
  * [5] Moore R et al. 1996 Nucl. Instr. Meth. B 119 149
  * [6] Uggerhoj U I 2005 Rev. Mod. Phys. 77 1131
  * [7] Soldani M et al. 2021 PoS ICHEP2020 872
  * [8] Bandiera L, Haurylavets V and Tikhomirov V 2019 Nucl. Instr. Meth. A 936 124
  * [9] Baryshevsky V G et al. 2017 Nucl. Instr. Meth. B 402 35
  * [10] Bandiera L et al. 2021 Eur. Phys. J. C 81 238
  * [11] Lietti D, Berra A, Prest M and Vallazza E 2013 Nucl. Instr. Meth. A 729 527
  * [12] Bandiera L et al. 2013 Nucl. Instr. Meth. B 309 135
  * [13] www.advansid.com/products/product-detail/asd-rgb-nuv-4s-p
  * [14] Annenkov A A, Korzhik M V and Lecoq P 2002 Nucl. Instr. Meth. A 490 (1) 30
  * [15] www.advansid.com/products/product-detail/sockets-for-csp-sipms
  * [16] www.advansid.com/products/product-detail/asd-ep-eb-n
  * [17] www.caen.it/products/v1730
  * [18] Soldani M et al. 2019 Nucl. Instr. Meth. A 936 327


# Generative Diffusion Models for Radio Wireless Channel Modelling and
Sampling ††thanks: Identify applicable funding agency here. If none, delete
this.

1st Ushnish Sengupta  MediaTek research
<EMAIL_ADDRESS>2nd Chinkuo Jao  MediaTek
<EMAIL_ADDRESS>3rd Alberto Bernacchia  MediaTek Research
<EMAIL_ADDRESS>4th Sattar Vakili  MediaTek Research
<EMAIL_ADDRESS>5th Da-shan Shiu  MediaTek Research
<EMAIL_ADDRESS>
###### Abstract

Channel modelling is essential to designing modern wireless communication
systems. The increasing complexity of channel modelling and the cost of
collecting high-quality wireless channel data have become major challenges. In
this paper, we propose a diffusion model based channel sampling approach for
rapidly synthesizing channel realizations from limited data. We use a
diffusion model with a U Net based architecture operating in the frequency
space domain. To evaluate how well the proposed model reproduces the true
distribution of channels in the training dataset, two evaluation metrics are
used: _i)_ the approximate $2$-Wasserstein distance between real and generated
distributions of the normalized power spectrum in the antenna and frequency
domains and _ii)_ precision and recall metric for distributions. We show that,
compared to existing GAN based approaches which suffer from mode collapse and
unstable training, our diffusion based approach trains stably and generates
diverse and high-fidelity samples from the true channel distribution. We also
show that we can pretrain the model on a simulated urban macro-cellular
channel dataset and fine-tune it on a smaller, out-of-distribution urban
micro-cellular dataset, therefore showing that it is feasible to model real
world channels using limited data with this approach.

###### Index Terms:

diffusion models, machine learning, wireless channel sampling

##  I Introduction

Modeling the wireless channel is an essential step in designing and evaluating
the performance of communication systems. In the context of 6G, the wireless
environment has become increasingly complex, and the extension of frequency
bands and the introduction of large-scale multiple-input multiple-output
(MIMO) systems have made it challenging for current channel modeling schemes
to accurately reproduce the characteristics of real-world radio channels.
Therefore, there is a need for more realistic channel modeling approaches. In
addition to accurate channel modeling, the development of deep learning-based
wireless communication systems requires large amounts of high-quality wireless
channel data to support the training of neural networks (NN). However, the
collection of wireless channel data is quite costly and time-consuming, which
motivates the development of novel channel data generating solutions to
support neural network training.

Previous papers have proposed using machine learning techniques as an
alternative to traditional deterministic or stochastic channel models. Markov
models have long been used to model the time-varying behaviour of channels
[1]. Zander [2] showed that a variational autoencoder can be used to generate
coefficients of the channel matrix. In Smith _et al._ [3], a generative
adversarial network (GAN) based channel modeling strategy was introduced and
it was shown that the GAN can reproduce a simple additive white Gaussian noise
channel. Xiao _et al._ [4] proposed ChannelGAN, a Wasserstein GAN (WGAN) that
can generate the multiple-input multiple-output (MIMO) channel matrix. The
MIMO-GAN approach [5] also uses a Wasserstein GAN to model the distribution of
channels, however unlike the ChannelGAN paper, they use pairs of input-output
signals as their data and the channel impulse response is modeled implicitly
by the neural network.

Despite multiple studies that have explored GANs as a way of modeling the
channel from data, they have several significant drawbacks that have been
highlighted in recent research. One of the most significant challenges with
GANs, identified first in the original GAN paper itself [6], is mode collapse,
where the generator learns to generate only a limited set of samples, ignoring
the rest of the data distribution. This results in a lack of diversity in the
generated samples, which can be a major problem for a channel simulator. GANs
are also notoriously difficult to train, with many models failing to converge
or producing low-quality samples. This instability is due to the adversarial
nature of the training process, where the generator and discriminator are
constantly trying to outwit each other. The Wasserstein GAN used in the
ChannelGAN paper [4] employs a different loss function with a gradient penalty
to address this issue and has been shown to be more stable in practice, but
does not resolve it entirely.

Denoising diffusion probabilistic models are a class of generative models that
have recently gained popularity due to their ability to generate high-quality
images, video and audio. Diffusion models are based on the principle of
diffusing noise through a sequence of invertible transformations, where the
noise is initialized as a standard Gaussian distribution and is gradually
transformed into the target data distribution. The process can be reversed to
generate new samples from the target distribution. One advantage of diffusion
models over GANs is that they do not require a discriminator network to guide
the generator network during training. Instead, diffusion models directly
optimize the likelihood of the target data distribution. This is why their
training is also much more stable. Ho _et al._ [7] compared diffusion models
and GANs on several image generation tasks and found that diffusion models
outperformed GANs in terms of image quality and diversity, especially for
high-resolution images. The authors attributed the superior performance of
diffusion models to their ability to capture the full distribution of the
data, while GANs only learn a single mode or a subset of the modes of the
distribution. In this paper, we propose using diffusion models to learn
samples from the distribution of the channel impulse response matrices from
data.

The existing literature on this topic also does not focus too much on
measuring how much the generated channels overlap with the distribution of the
original data. To compare the distribution of generated channel matrices with
the real ones, we first compute the approximate Wasserstein distance between
the distributions of real and generated channel power spectra. We also
consider precision and recall as a measure to better separately capture the
fidelity and diversity of the generated images.

Simulations can generate copious amounts of channel data but the datasets
collected from real-world experimental campaigns are limited in size. This is
a problem for data-hungry machine learning approaches to channel modelling.
Our proposed solution involves transfer learning: we show that we can pre-
train the diffusion model on a simulated dataset and fine-tune it on a
smaller, out-of-distribution dataset, therefore showing that it is feasible to
model real world channels using limited data with this approach.

##  II Methods

###  II-A Comparison with Wasserstein GAN

Figure 1: The ChannelGAN architecture [4]

As discussed in the introduction, WGANs have so far shown promising results in
being able to model the distribution characteristics of real channel matrices,
so we compare our diffusion models with a WGAN. We borrow the ChannelGAN
architecture [4] with minor modifications. For the generator $G$, it maps a
noise vector $z$ to a generated channel $H\in\mathbb{R}^{N_{a}\times
N_{f^{\prime}}\times 2}$, where $N_{a}=N_{t}\times N_{r}$ denotes the number
of antenna pairs, $N_{f^{\prime}}$ denotes the number of frequency bins
retained (the high frequency portion of the response is cropped off). Firstly,
in order to convert from input noise vector to the feature maps, the noise
vector $z$ is processed by a dense layer along with a reshape. Next, five up-
sampling blocks are stacked up. Specifically, each up-sampling block is
composed of a $2\times 2$ up-sampling layer (nearest neighbor interpolation),
a $3\times 3$ convolutional layer (Conv) with $M$ filters and a batch
normalization layer. Two optional activation functions are also attached at
the tail of each block with the opting flag $V\in{0,1}$, where $V=0$ and $V=1$
indicate utilization of leaky rectified linear unit (LeakyReLU) and hyperbolic
tangent function (tanh), respectively. Here five up-sampling blocks are
configured with parameters $M=\\{1024,512,256,128,2\\}$ and
$V=\\{0,0,0,0,1\\}$ sequentially. Note that the tanh in the last Up-Sampling
Block limits the amplitude of the elements in fake channel in $[-1,1]$ which
matches the real channel, and the LeakyReLU in remaining blocks brings the
non-linear transformation to the network. Finally, the two-dimensional
cropping layer (Cropping2D) is used to maintain the shape of generated fake
channel $H$ consistent with the real channel $H$.

For the discriminator $D$, the input is firstly padded by a two-dimensional
zero padding layer (ZeroPadding2D). Then, six down-sampling blocks are stacked
up, where the $5\times 5$ Conv with the strides of $2\times 2$ and
$M=\\{32,64,128,256,512,1024\\}$ filters, a LeakyReLU and a dropout layer are
deployed sequentially. In the end, a flatten operation and dense layer with
single output are utilized.

###  II-B Diffusion models

Denoising diffusion probabilistic models (DDPMs) are a new class of generative
models introduced by Sohl-Dickstein and colleagues [8] following a line of
research on Markov-chain based generative models [9, 10]. These methods are
based on iteratively corrupting the data to transform the data distribution
into a known simple distribution, often an isotropic normal distribution, and
learning to reverse the process by estimating the mean for the reverse
iterations using a neural network. References [11, 12] proposed a similar
algorithm that learns the score of the data distribution in the forward
process and uses the Langevin dynamic to reverse the process and sample from
the distribution.

The diffusion model generates a sequence of samples by iteratively adding
noise to the previous samples. At each step, the forward diffusion process
adds Gaussian noise with a variance determined by a noising schedule to the
previous sample. Given a data-point $\textbf{x}_{0}$ sampled from the real
data distribution $q$, one can define a forward diffusion process by adding
noise stepwise. The forward diffusion process is performed for $T$ steps:

$\textbf{x}_{t+1}=\sqrt{1-\beta_{t}}\textbf{x}_{t}+\sqrt{\beta_{t}}\epsilon_{t}$

where $\textbf{x}_{t}$ is the noisy datapoint at step $t$, $\textbf{x}_{t+1}$
is the noisy datapoint at step $t+1$, $\beta_{t}$ is the noise level at step
$t$ which is determined according to a cosine noising schedule [13] and
$\epsilon_{t}$ is a Gaussian noise with zero mean and unit variance.

As $T$ becomes large, the latent $\textbf{x}_{T}$ approaches an isotropic
Gaussian distribution. Therefore, if we manage to learn the reverse
distribution $q(\textbf{x}_{t-1}|\textbf{x}_{t})$ using a neural network, we
can sample $\textbf{x}_{T}$ from the unit normal distribution, run the reverse
process and acquire a sample from $q(\textbf{x}_{0})$, generating novel data
points from the original data distribution.

Figure 2: The U-Net architecture used for denoising in our diffusion models

A neural network with a U-Net architecture (Figure 1) is used to predict the
added noise $\epsilon_{t}$ given a noisy input. The U-Net is trained to
minimize the mean squared error between the predicted noise and the true
noise. The architecture consists of a contracting path and an expanding path.
The contracting path consists of a series of convolutional layers with max-
pooling. The expanding path consists of a series of convolutional layers with
up-sampling. The contracting and expanding paths are connected by skip
connections. The skip connections enable the network to learn both low-level
and high-level features.

###  II-C Fine-tuning the pretrained model

Few-shot generation seeks to generate more data of a given domain, with only
few available training examples. As it is unreasonable to expect to fully
infer the distribution of real channels from just a few observations, we seek
to leverage a large, related source domain (simulated urban macro data) for
pre-training. Thus, we wish to preserve the diversity of the source domain,
while adapting to the appearance of the target. We adapt our pre-trained
model, without introducing any additional parameters, to the fewer examples of
the target domain, by training on the new data using a lower learning rate.
Crucially, we regularize the changes of the weights during this adaptation, in
order to best preserve the information of the source dataset, while fitting
the target. We demonstrate the effectiveness of this approach by generating
high-quality samples from the urban microcellular scenario.

###  II-D Performance metrics

For generative models, it is important to quantify both the quality and the
fidelity of the generated data individually. Let $P_{g}$ denote the generated
channels and $P_{r}$ denote the test dataset. We compute the normalized power
spectra of the channels in the frequency domain and the antenna domain
representations. We fit a multivariate Gaussian to the generated and real
power spectra. Let ($\mu_{g}$, $\Sigma_{g}$) and ($\mu_{r}$, $\Sigma_{r}$)
denote the mean and covariances of the two Gaussians. The metric we will use
to measure how close the real and generated distributions are will be the
Wasserstein-2 distance between these two distributions:

$W_{2}(P_{g},P_{r})=||\mu_{g}-\mu_{r}||^{2}+\text{Tr}(\Sigma_{g}+\Sigma_{r}-2(\Sigma_{r}^{1/2}\Sigma_{g}\Sigma_{r}^{1/2})^{1/2})$
(1)

Figure 3: Diagram illustrating precision and recall for distributions [14] We
denote the distribution of real data with $P_{r}$ (blue) and the distribution
of generated data with $P_{g}$ (red). Precision is the probability that a
random sample from $P_{g}$ falls within the support of $P_{r}$. Recall is the
probability that a random sample from $P_{r}$ falls within the support of
$P_{g}$.

However, this approximate Wasserstein distance isn’t the best measure, because
it uses a Gaussian approximation to a non-Gaussian distribution and it also
ignores the phase of the complex channel response. It also fails to
disentangle the accuracy and variety of generated channels. A metric that
better captures the fidelity and diversity of the generated channels is the
improved precision and recall metric proposed by Kynkäänniemi _et al._ [14].
The key idea is to draw an equal number of samples from real and generated
distributions and embed them into a lower dimensional feature space using a
pre-trained autoencoder network. We then calculate the pairwise Euclidean
distances between all feature vectors in the set and, for each feature vector,
form a hypersphere with radius equal to the distance to its nearest neighbor.
This hypersphere defines a volume in the feature space that serves as an
estimate of the true manifold. To determine precision, we query for each
generated channel whether it is within the estimated manifold of real
channels. For recall, we query for each real channel whether it is within
estimated manifold of generated channels.

##  III Dataset description

The channel dataset is generated from a system-level simulator (SLS) with a 3D
stochastic channel model. The channel generation procedure follows the steps
described in TR 38.901 [15] A total of 120 User Equipments (UEs) are dropped
in a hexagonal network layout and associated with the serving Base Station
(BS) with maximum reference signal received power (RSRP). For each UE-BS link,
the small-scale parameters of the channel angle (AoD, AoA, ZoD, ZoA) and delay
information are created independently. Based on these parameters, channel
impulse responses are generated depending on BS and UE antenna settings. We
also considered the spatial consistency procedure defined in TR 38.901.
Therefore, the channel data of adjacent UEs are correlated. Two scenarios UMa
and UMi channels are used for different training purposes. 32000 samples of
channel impulse response matrices from 96 UEs in each scenario are used as
training data and 8000 channel matrices from the remaining 24 UEs are used for
testing and validation.

TABLE I: Channel statistics and parameters setting Scenario | Urban Macro (UMa) | Urban Micro (UMi)
---|---|---
LOS | NLOS | LOS | NLOS
Number of clusters | $12$ | $20$ | $12$ | $19$
Delay spread (ns) | $103.8$ | $455.6$ | $55.7$ | $113.6$
AoD spread (degree) | $12.6$ | $29.2$ | $15.4$ | $26.3$
AoA spread (degree) | $64.6$ | $99.7$ | $49.2$ | $59.1$
Inter-site distance (ISD) | $500m$ | $200m$
BS antenna height | $25m$ | $10m$
Career frequency |  $2$ _GHz_
UE speed |  $3$ and $30$ _km/hr_
BS antenna setting | $(M,N,P,M_{p},N_{p})=(4,4,2,1,4)$
UE antenna setting | $(M,N,P,M_{p},N_{p})=(1,2,2,1,2)$

To demonstrate that our model can be pre-trained on a corpus of simulated data
and fine-tuned on a limited real-world dataset, we fine tune the diffusion
model trained on urban macro-cell scenarios on a dataset of 20000 (different
proportions of the dataset are used in the fine-tuning experiments) channel
impulse responses from a urban micro-cell simulation. Compared to the
simulated urban macro channels, the urban micro channels have a different
distribution of arrival/ departure angles as well as delay spreads, and acts
as a stand-in for real data.

As shown in Table 1, the inter-site distance (ISD) and BS height of UMa are
greater than those of UMi. With wider coverage, UMa is a richly scattered
environment with larger channel delay and angle spread characteristics. Both
LOS and NLOS conditions are considered in our datasets. At the BS side, we
consider a 2D antenna array, $(M,N,P,M_{p},N_{p})=(4,4,2,1,4)$, where $N$ is
the number of columns, $M$ is the number of antenna elements with the same
polarization in each column. $M_{p}$ and $N_{p}$ is the number of antenna
ports in row and column. $P=2$ means that polarization antennas are used.
Therefore, four vertical antenna elements are virtualized to one antenna port
in each column. A total of eight antenna ports are used in the channel model.

The MIMO channel impulse response in the frequency domain can be denoted as a
complex tensor $\mathbf{H}\in\mathbb{C}^{N_{t}\times N_{r}\times N_{f}}$,
where $N_{t}$ is the number of transmitting antennae, $N_{r}$ is the number of
receiving antennae and $N_{f}$ represents the number of frequency bins. For
training our generative models, we reshape and crop the tensor so that it
becomes a real-valued tensor with two channels $\in\mathbb{R}^{N_{a}\times
N_{f^{\prime}}\times 2}$, where $N_{a}=N_{t}\times N_{r}$ denotes the number
of antenna pairs, $N_{f^{\prime}}$ denotes the number of frequency bins
retained (the high frequency portion of the response is cropped off) and the
last dimension with a value of 2 represents the real and imaginary parts of
original complex channel, respectively. Our objective is to generate samples
of channel matrices $\mathbf{H}\sim q(\mathbf{H})$ from the true distribution
of $\mathbf{H}$, given a limited number of channel samples as our training
data.

##  IV Results

###  IV-A Performance and comparison with WGAN

Figure 4: Normalized power spectra of urban macrocellular channel impulse
response samples generated by the diffusion model (top row) and the GAN
(bottom row). The diffusion model clearly generates more diverse channel
samples and this is confirmed by our metrics.

In this section, we walk through the evaluation of our diffusion model
approach and the Wasserstein GAN. Both models were trained and evaluated using
the same urban macrocell channel dataset described in Section 3. In figure 4,
we compare the training progress of the diffusion model and the Wasserstein
GAN by plotting the 2-Wasserstein distance of the antenna-domain power spectra
between generated channels and real channels from the test dataset against the
number of epochs the model has trained for. As discussed in the methods
section, this metric is an approximate measure of how well the distribution of
generated channels matches the distribution of real channels. We observe that
the diffusion model converges stably to a fairly low Wasserstein distance, but
the GAN experiences episodes of periodic instability throughout the training
process due to the adversarial nature of the training. As a result, even the
GAN with the best set of hyperparameters cannot outperform in terms of
2-Wasserstein distance (refer to Table II).

Figure 5: The evolution of the 2-Wasserstein distance in antenna domain
between the real and generated power spectra in the antenna domain for the
diffusion model (top) and the WGAN (bottom) during training. The large spikes
in the training curve of the WGAN even in later epochs indicate that it
suffers from instabilities throughout the training process.

In table II, we also compare the performance of the WGAN with the diffusion
models in terms of precision and recall, to understand how faithfully each
model captures real channels and if the variety of the generated samples match
that of the data. Unsurprisingly, we find that the WGAN is fairly competitive
with the diffusion model when it comes to precision, which measures the
fidelity of generated samples. However, the WGAN is much worse when it comes
to recall, which corresponds to the diversity of the generated data. This
indicates that the Wasserstein GAN has suffered from at least partial mode
collapse. This is also apparent from the channel samples generated by each
model (Figure 4); the diffusion model generates channels that are visibly more
diverse.

TABLE II: Comparison of the Wasserstein distance, precision and recall computed on the validation dataset for the diffusion model and WGAN. |  $W_{2}$ (antenna) |  $W_{2}$ (frequency) | Precision | Recall
---|---|---|---|---
WGAN | 0.0017 | 0.0029 | 0.8350 | 0.5750
DDPM (ours) | 0.0010 | 0.0023 | 0.8210 | 0.8950

###  IV-B Performance of pretrained model after fine-tuning

For the fine-tuning experiments, we use different proportions of urban
microcellular data for fine-tuning ($5\%$, $10\%$, $25\%$ and $100\%$ of the
dataset) and compare its performance with a model trained from scratch with
the same amount of urban microcellular data. We find that the fine-tuned
models achieve considerably better precision and recall, and this effect is
particularly visible when using a small fraction ($5\%-10\%$) of the data.
Even when using all of the data from the out-of-distribution, the fine-tuned
model maintains a small advantage over the trained-from-scratch problem, which
indicates that there is no significant negative transfer for this problem. We
also observe that fine-tuning has a more pronounced effect on recall
(diversity) than precision (fidelity). The plots of precision and recall in
figure 6 summarizes the results of the fine-tuning experiments. These results
are very encouraging and indicates that a model pre-trained on simulated
channel data can learn to generate channels from a related but different
distribution, such as real world channels, using much less data.

Figure 6: Comparison of precision (top) and recall (bottom) metrics between a
diffusion model pre-trained on the urban macro dataset/ fine-tuned on urban
micro dataset and a model trained from scratch on the urban micro dataset. The
fine-tuned model clearly has better precision and recall, especially when a
low proportion of the target dataset (urban microcellular) is used for
training.

##  V Conclusions

In this paper, a diffusion model based wireless channel modeling framework has
been proposed and analyzed. Unlike traditional methods which involve detailed
theoretical analysis and data processing to derive key channel parameters from
real measurement data, our new method does not require any domain-specific
knowledge or technical expertise, and can obtain the target channel model by
directly learning from raw MIMO channel data with a diffusion model, which
iteratively denoises isotropic Gaussian noise to generates samples from the
true data distribution. Using synthetic channel data from a 3D stochastic
model as our proving ground, the distribution of generated channel samples has
been compared with that of the real channel data using approximate Wasserstein
distance between power spectra distributions as well as precision and recall
metrics from the generative machine learning literature. These metrics help us
measure the fidelity and diversity of the generated channels seperately. We
find that the diffusion models not only generate high-fidelity channels, but
also manage to capture the diversity of the data very well, unlike the WGAN
used in previous studies for channel modelling. This is an important property
for a channel emulator to have, since it needs to be able to emulate all
possible channel conditions. We also learned that diffusion models are much
less temperamental to train compared to the GANs, which suffer from frequent
instabilities during training.

Using a simulated out-of-distribution dataset (the Urban Micro scenario) as
our stand-in for real channel data, we demonstrated that we can use fine-
tuning to help a model pre-trained on simulated data generalize to real world
channel data, which have a different distribution. We showed that reaching the
same level of precision and recall is possible with the pretrained model using
2-3 times less data compared to training from scratch. This is promising as it
is much more expensive to collect real channel data, compared to simulations.

Our future work will include more experiments with real MIMO channel datasets
and validation of our synthetic channel model on downstream tasks such as CSI
feedback, position estimation from channel data and channel equalization.
Another interesting challenge is modelling the time-variation of channels.
Advances in video diffusion models [16] might hold the key to modelling time-
variations in statistical channels.

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