# Sound Source Separation Using Latent Variational
Block-Wise Disentanglement

###### Abstract

While neural network approaches have made significant strides in resolving
classical signal processing problems, it is often the case that hybrid
approaches that draw insight from both signal processing and neural networks
produce more complete solutions. In this paper, we present a hybrid classical
digital signal processing/deep neural network (DSP/DNN) approach to source
separation (SS) highlighting the theoretical link between variational
autoencoder and classical approaches to SS. We propose a system that
transforms the single channel under-determined SS task to an equivalent
multichannel over-determined SS problem in a properly designed latent space.
The separation task in the latent space is treated as finding a variational
block-wise disentangled representation of the mixture. We show empirically,
that the design choices and the variational formulation of the task at hand
motivated by the classical signal processing theoretical results lead to
robustness to unseen out-of-distribution data and reduction of the overfitting
risk. To address the resulting permutation issue we explicitly incorporate a
novel differentiable permutation loss function and augment the model with a
memory mechanism to keep track of the statistics of the individual sources.

Index Terms—  Source separation, Out-of-distribution robustness.

##  1 Introduction

Real-life audio communication signals often capture a complex acoustic scene
where multiple sound sources are active simultaneously. The task of
segregating the sound sources in an audio stream has interested the research
community for the last few decades and the recent development of source
separation systems based on deep neural network (DNN) architectures refueled
the hope of the community in developing reliable real-time systems for this
objective. A pioneering work using DNN architectures for this so-called
cocktail party scenario is the Deep Clustering approach presented in [1] where
the presented architecture is trained to produce spectrogram embeddings that
are discriminative for partition labels given only the spectrogram of the
training data. While this approach was offline, more recent approaches to near
real-time source separation operate online [2] or in a block-online (segment-
wise) manner [3]. The general scheme of most of the state-of-the-art models
follows the encoder, masker, decoder structure e.g., [2, 4, 5, 6]. The input
to these models is the raw time-domain signal and the task of the encoder is
to find a representation where the sources are separable via a masking
operation. Finally, the decoder reconstructs the individual signals in
isolation. The classical line of thought in approaches to blind source
separation (BSS), see e.g., [7, 8, 9], sets relatively strict a priori
assumptions about the stochastic model of the separated signals. While
classical approaches are principled and often formulated to work in an online
manner, the assumptions made about the demixing system and the source
statistics are often limiting factors for these approaches to work in adverse
conditions. On the other hand, DNN-based approaches achieve impressive
performance compared to classical approaches on several datasets, however, the
models are often complex and designed as end-to-end systems, such that there
is limited understanding of the latent representation derived by these models
and their ability to generalize.

In the present study, we introduce a DNN model inspired by classical signal
processing approaches to perform online source separation. We establish the
link between classical source separation approaches and variational auto-
encoders, and show how to modify the auto-encoder architecture to
systematically perform the disentanglement of the sources. The concept of
using a variational auto-encoder as a generative spectrogram model for a
single clean speech source for speech enhancement and separation has been
explored in previous research [10, 11, 12, 13, 14] and for multichannel source
separation in [15] where it has been used for inference of independent factors
in an unsupervised manner assuming a Gaussian latent space. In this study, we
are instead interested in formulating the separation task variationally rather
than approximating individual speech spectrogram densities. More specifically,
we are interested in finding a transformation that decomposes the microphone
signal latent representation into mutually independent multivariate components
following a predetermined stochastic mixture model. The motivation of this
formulation is to make the model more robust to scenarios that were unseen
during training and hence to avoid overfitting.The network accounts for the
permutation problem by introducing an explicit differential permutation loss
over sources and frequencies and the incorporation of a memory mechanism.

##  2 Classical Blind Source Separation

The task of source separation in general for any number of channels is to
adaptively learn a function
$\mathbf{y}(n)=\mathcal{W}(\mathbf{x}(n),\boldsymbol{\theta})$ where
$\mathbf{x}(n)$ represents a frame of the observable sensor signals at time
index $n$, $\boldsymbol{\theta}$ are the separation model parameters,
$\mathbf{x}^{\mathrm{T}}(n)=\left[\mathbf{x}_{1}^{\mathrm{T}}(n),\ldots,\mathbf{x}_{Q}^{\mathrm{T}}(n)\right]$,
and
$\mathbf{y}^{\mathrm{T}}(n)=\left[\mathbf{y}_{1}^{\mathrm{T}}(n),\ldots,\mathbf{y}_{P}^{\mathrm{T}}(n)\right]$,
with
$\mathbf{x}_{q}^{\mathrm{T}}(n)=\left[x_{q}(n),\ldots,x_{q}(n-L+1)\right]$ and
$\mathbf{y}_{p}^{\mathrm{T}}(n)=\left[y_{p}(n),\ldots,y_{p}(n-D+1)\right].$
The number of sources and sensors are $P$ and $Q$ respectively, and $L$ input
frame length required to estimate one sample, and $D$ the time lags defining
the dimensionality of the source model. A general optimization cost function
for BSS has been introduced in [7]. Although the cost function was originally
presented for the determined multichannel linear convolutive case, it can be
generalized to cover the under-determined and non-linear case with a few
modifications. The cost function is given by

$\displaystyle\vspace{-.1cm}\mathcal{J}(m,\boldsymbol{\theta})=$
$\displaystyle-\sum_{i=0}^{\infty}\beta(i,m)\frac{1}{N}$
$\displaystyle\cdot\sum_{j=iN_{L}}^{iN_{L}+N-1}$
$\displaystyle\left\\{\log\left({q}_{s,PD}(\mathbf{y}(j))\right)-\log\left({p}_{y,PD}(\mathbf{y}(j))\right)\right\\},$
(1)

estimated over blocks of $N$ output samples shifted by $N_{L}$ relative to the
previous block, $\beta$ is a window function expressed as a function of time
sample ($i$) and block ($m$) indices, allowing for offline, online and block-
online algorithms. The multivariate distribution of the multichannel signal at
the separation system output is represented as ${p}_{y,PD}(\cdot)$, and the
hypothesized source model ${q}_{s,PD}(\cdot)$ for blind source separation is
often given as

${q}_{s,PD}(\mathbf{y}(j))=\prod_{p=1}^{P}{p}_{y_{p},D}\left(\mathbf{y}_{p}(j)\right),$
(2)

with $D$ denoting the dimensionality of each individual multivariate source.
Typically, non-Gaussian prototypical distributions are used for the density
functions such as multichannel Laplacian with parameters estimated from the
data. For the linear (over)-determined case, the output density in Eq. (2) can
be expressed by means of the input sensor’s signal probability density, which
is independent of the parameters to be estimated, and the log determinant of
the demixing system [7, 16].

##  3 Relation of Disentangled Variational Autoencoder to Blind Source
Separation

In this section, we will explore the relationship of Variational Auto-Encoders
(VAEs) to the traditional BSS problem. VAEs are prominent deep neural network
models that achieve both nonlinear dimensionality reduction and generative
modeling. A VAE indirectly minimizes the Kullback-Leibler Divergence between a
surrogate density for the latent space and the posterior actual latent space
density by minimizing the following cost function, which can be interpreted as
the negative of the evidence lower bound (ELBO) [17]:

$\displaystyle\mathcal{L}_{r}:=\mathcal{E}_{\mathbf{s}}\Big{[}\mathcal{E}_{\mathbf{z}\sim
q_{\phi}(\mathbf{z}|\mathbf{s})}\left[-\log
p_{\eta}(\mathbf{s}|\mathbf{z})\right]+$
$\displaystyle\operatorname{KL}\left(q_{\phi}(\mathbf{z}|\mathbf{s})\|p(\mathbf{z})\right)\Big{]}$
$\displaystyle+$
$\displaystyle\mathcal{D}\left(q_{\phi}(\mathbf{z})\|p(\mathbf{z})\right),$
(3)

with $\eta$ and $\phi$ denoting the parameters of the decoder and encoder
respectively, and where $\mathbf{z}$ is the latent variable, $\mathbf{s}$ data
sample. $\mathcal{D}$ is a regularization term that is often chosen to
disentangle the latent space as proposed in the so-called DIP-VAE-II by
promoting covariance matrices that are equal to the identity matrix [18].

The variational autoencoder loss function and the classical blind source
separation problem share many similarities. Recent research has focused on the
possibility of finding disentangled data representations in an unsupervised
manner, which has been shown to be theoretically not possible in the general
case without inductive biases [19]. More recently, it has been shown that
unsupervised disentangled representation is only possible in cases where
certain conditions about the encoder and the latent representation are
fulfilled, these conditions being non-Gaussianity and local isometry of the
encoding scheme [20]. For the task at hand of sound source separation, we are
not interested in the classical goal of disentangled VAEs, which is a
disentangled representation of a single sound source belonging to a certain
manifold in order to enable generation of sound sources on this manifold by
few explanatory factors. Rather, we are interested in finding a representation
that disentangles sound sources in a mixture where the individual sound
sources are described by means of multivariate probability densities. To
further illustrate this motivation, consider the second order statistics based
source separation approaches, e.g., [7] where the goal is to block-diagonalize
the autocorrelation matrix of the output signals over multiple time lags
rather than completely diagonalize it. Hence, we allow the latent
representation of each source to have statistical dependencies but at the same
time, we require them to be mutually independent. The inductive biases for our
disentanglement task are defined by the ability to reconstruct the actual
sound sources in the training dataset from the disentangled representation.
Further, instead of requiring the latent space to be Gaussian as is common in
VAEs, we explicitly enforce non-Gaussianity as this is a necessary condition
for decomposing a mixture into independent components [16].

Eq. (2) can be seen as a sample version of the Kullback-Leibler Divergence
between the multivariate prior source distribution and the posterior. However,
a maximum likelihood estimator of the blind source separation task would aim
at maximizing the log likelihood of the sensor signals (mixture) given the
latent variables. The chain rule allows the expression of the sensor signal
log likelihood as

$\displaystyle\log(p(\mathbf{x};\boldsymbol{\theta}))=\int q(\mathbf{z})$
$\displaystyle\left[\log\left(\frac{p(\mathbf{x},\mathbf{z};\boldsymbol{\theta})}{q(\mathbf{z})}\right)\right.$
(4)
$\displaystyle\left.-\log\left(\frac{p(\mathbf{z}\mid\mathbf{x};\boldsymbol{\theta})}{q(\mathbf{z})}\right)\right]\mathrm{d}\mathbf{z},$

with $\boldsymbol{\theta}$ being only the separation parameters, different
from the encoder and decoder parameters ($\eta$, $\phi$). The first term in
the integral is the ELBO as seen before in the VAE context. This can be
written as

$\displaystyle\mathcal{L}_{r}=\mathcal{E}_{\mathbf{x}}\Big{[}\mathcal{E}_{\mathbf{z}\sim
q_{\phi}}\left[-\log
p_{\eta}(\mathbf{x}|\mathbf{z};\boldsymbol{\theta})\right]+\operatorname{KL}\left(q_{\phi}(\mathbf{z}|\mathbf{x};\boldsymbol{\theta})\|p(\mathbf{z};\boldsymbol{\theta})\right)\Big{]},$
(5)

where all of the terms can be computed for a suitable surrogate density
$q_{\phi}$. This cost function leads to a VAE w.r.t. the mixture signal
($\mathbf{x}$). To enforce the disentanglement w.r.t. to the source, we use
the following strategy: First, instead of using the regularization term to
promote disentanglement as in Eq. (3), we use as a surrogate, a distribution
that promotes separation as is common in the classical BSS literature [16,
21], e.g., a non-Gaussian multivariate spherically invariant random process
distribution (SIRP). Second, the reconstruction error as appearing in the ELBO
is replaced by the term

$\displaystyle\mathcal{D}_{\textrm{ind.
bias}}=\sum_{p}\mathcal{E}_{\mathbf{z}\sim q_{\phi}}\left[-\log
p_{\eta}(\mathbf{s}_{p}|\mathbf{A}_{p}\mathbf{z};\boldsymbol{\theta})\right],$
(6)

where $\mathbf{A}_{p}$ is a windowing matrix that selects the $p$-th block of
the latent multivariate vector considered as a structured vector with $P$
blocks each of length $M$, and $\mathbf{s}_{p}$ represeting the ground truth
separated source.An interesting aspect of our formulation is that it uses the
same decoder for the mixture as for the individual isolated sources, making it
inherently resource efficient. For the purposes of this paper and for reasons
described in the next section, we will be replacing the KLD minimization with
the Fisher Divergence as a local approximation.

##  4 Estimation by Score Matching

In variational autoencoders, maximizing the ELBO entails minimizing the
Kullback-Leibler Divergence between a prior latent distribution and an
approximate posterior distribution assuming some tractable distribution. In
the context of blind source separation, the prior over the latent variable is
often given in unnormalized manner. For this purpose, a method called Score
Matching (SM) has been developed where instead of matching the distributions,
the derivative of the log densities are matched, which is a variational
calculus approach to circumvent the need to know the normalization terms and
leading to solutions robust to model misspecification [22]. It can be shown
that SM minimizes the Fisher divergence between the data distribution and a
model distribution which is specified by the gradient (with respect to the
multivariate input variable $\boldsymbol{\zeta}$) of its log density function
which is parameterized by $\boldsymbol{\xi}$, [23, 24]. SM introduces the
score function as the derivative of the log density defined as

$\psi(\boldsymbol{\zeta};\boldsymbol{\xi})=\left(\begin{array}[]{c}\frac{\partial\log
q(\boldsymbol{\zeta};\boldsymbol{\xi})}{\partial\zeta_{1}}\\\ \vdots\\\
\frac{\partial\log
q(\boldsymbol{\zeta};\boldsymbol{\xi})}{\partial\zeta_{PM}}\end{array}\right)=\left(\begin{array}[]{c}\psi_{1}(\boldsymbol{\zeta};\boldsymbol{\xi})\\\
\vdots\\\
\psi_{PM}(\boldsymbol{\zeta};\boldsymbol{\xi})\end{array}\right)=\nabla_{\boldsymbol{\zeta}}\log
q(\boldsymbol{\zeta};\boldsymbol{\xi}).$

The cost function of SM reads

$J(\boldsymbol{\xi})=\frac{1}{2}\int_{\boldsymbol{\zeta}\in\mathbb{R}^{PM}}q_{\mathbf{z}}(\mathbf{\zeta})\left\|\boldsymbol{\psi}(\boldsymbol{\zeta};\boldsymbol{\xi})-\boldsymbol{\psi}_{\mathbf{z}}(\boldsymbol{\zeta})\right\|^{2}d\boldsymbol{\zeta},$

which has been shown in [22] to be equal to

$J(\boldsymbol{\xi})=\int_{\boldsymbol{\zeta}\in\mathbb{R}^{PM}}q_{\mathbf{z}}(\boldsymbol{\zeta})\sum_{i=1}^{PM}\left[\partial_{i}\psi_{i}(\boldsymbol{\zeta};\boldsymbol{\xi})+\frac{1}{2}\psi_{i}(\boldsymbol{\zeta};\boldsymbol{\xi})^{2}\right]d\boldsymbol{\zeta}+\text{
const. }$ (7)

where

$\psi_{i}(\boldsymbol{\zeta};\boldsymbol{\xi})=\frac{\partial\log
q(\boldsymbol{\zeta};\boldsymbol{\xi})}{\partial\zeta_{i}}\quad\text{, and}$
$\partial_{i}\psi_{i}(\boldsymbol{\zeta};\boldsymbol{\xi})=\frac{\partial\psi_{i}(\boldsymbol{\zeta};\boldsymbol{\xi})}{\partial\zeta_{i}}=\frac{\partial^{2}\log
q(\boldsymbol{\zeta};\boldsymbol{\xi})}{\partial\zeta_{i}^{2}}.$

This cost function promotes realization of the log density that fits the data
while balancing the curvature and the steepness. When the gradient norm is
zero, the data point is at a local extremum, and the Laplacian determines the
curvature of the log likelihood; if negative, the curvature is convex and the
extremum is a maximum. In this study, we are interested in non-Gaussian
distributions motivated by the condition for the separation into independent
components [16]. Motivated by the choice in [7], we select SIRP as a family of
non-Gaussian distributions. A prominent realization of this family is the
multivariate Laplacian, for which the score function is [21]

$\psi\left(\mathbf{z}_{p};\operatorname{C}_{\mathbf{z}_{p}}\right)=-2\cdot\frac{1}{\sqrt{2u_{p}}}\frac{K_{M/2}\left(\sqrt{2u_{p}}\right)}{K_{M/2-1}\left(\sqrt{2u_{p}}\right)}{\operatorname{C}^{-1}_{\mathbf{z}_{p}}}\mathbf{z}_{p},$
(8)

with $K_{n}$ denoting the n-th order modified Bessel functions of the second
kind, and

$u_{p}=\mathbf{z}_{p}^{\mathrm{T}}{\operatorname{C}^{-1}_{\mathbf{z}_{p}}}\mathbf{z}_{p},$
(9)

with $\operatorname{C}_{\mathbf{z}_{p}}$ representing an estimation of the
covariance matrix of one estimated separated source in the latent space.

##  5 Differentiable Permutation Minimization

In source separation, there are two common permutation problems: source
permutation, in general, if the network has no further conditions on the
statistics of each output, the order of the output streams is random; and
frequency permutation where the order of the separated sources is inconsistent
across different frequency bands. A recurrent architecture can help mitigate
the issue as it helps keeping context of the separated sources. However, for
extended periods of silence, the problem could occur further. Adding dedicated
memory layers on some representation of the embedding or longer sequence
information has been suggested in [6, 3] to circumvent this issue. In the
present work, we introduce a way to incorporate permutation mitigation into
the loss function, further, augment the architecture with a memory function to
keep track of the individual source statistics. The negative of the losses for
the different permutations in a permutation invariant training [25] can be
seen as unnormalized logits of a categorical distribution. The permutation
would be the argmax of the logits which is not differentiable, instead, we use
the softmax across the output channels. An undesirable permutation would occur
if the network exchange the sources across frames or subbands. In order to
minimize such permutations we constrain the derivative of the differentiable
permutation at multiple resolutions. A multiresolutional differential operator
can be obtained as the second to last columns of the Hadamard matrix defined
for $i\in\mathbb{N}$ as

$H_{n=2^{i}}=\left[\begin{array}[]{cc}H_{2^{i-1}}&H_{2^{i-1}}\\\
H_{2^{i-1}}&-H_{2^{i-1}}\end{array}\right].$ (10)

A loss function minimizing the permutation across subbands is established
based on the norm of the softmax output multiplied with the 2nd to last
columns of the Hadamard matrix. The permutation across frames is minimized by
considering the structured norm $\ell_{1,2}$ of the discrete derivative of the
softmax output across frames for each subband, where the $\ell_{1}$ is used
across frames and then an $\ell_{2}$ is taken across frequencies over the
previously calculated $\ell_{1}$ norms.

###  5.1 Memory Block Aided Permutation Solution

The idea of augmenting sequence-to-sequence models with memory is well known,
e.g., [26], for source separation the employment of memory block has been
recently introduced with success in [3]. To robustify our model against
permutation errors in the presence of longer sequences of silence of the
individual sources, we augment the network with a structured memory bank
keeping track of the separated signal latent representations. This information
is fed back to the internal separation as conditioning input. The latent
representation of the output is buffered into a ring buffer with a length $K$
and an adaptive average pooling is performed over the buffered signals, the
pooling output is fed into a gated recurrent layer serving as memory, the
output of the recurrent layer is fed along with the internal separator input
into a fully connected layer to form the new latent separation unit input.
Each separated source latent representation has its own memory. Since the
model includes a feedback loop from the output into the memory and back into
the latent space separator unit, see Fig. 1, during training we forward pass
the signal with frozen parameters to obtain the the memory input and then
train with a copy of the memory output detached from the computational graph
as highlighted in the figure by the dashed arrow. The memory unit is trained
in an alternating manner with rest of the network by incorporating the memory
output within the graph into a contrastive predictive loss function designed
in a manner similar to the contrastive predictive coding concept [27]. To
obtain the targets for this loss, the clean separated ground truth sources are
passed through the network with frozen parameters except those of the memory
unit bypassing the fully connected layer which incorporates the memory input
(see the dashed line skipping the layer before the separation unit). The
output of the latent space separation unit is divided across the time axis
into segments with the same length as the buffer length. The loss is similar
to InfoNCE loss [27] with positive samples being frames from the following
time segment of the same source and negative samples being from the following
segments of other sources in the batch sequences

$\mathcal{L}_{\mathrm{N}}=-\underset{Z}{\mathbb{E}}\left[\log\frac{f\left(\mathbf{z}_{t+k},\mathbf{c}_{t}\right)}{\sum_{\mathbf{z}_{j}\in
Z}f\left(\mathbf{z}_{j},\mathbf{c}_{t}\right)}\right],$ (11)

where $f(\mathbf{e},\mathbf{g})$ =
$\exp(w\cdot\cos({\mathbf{e}},{\mathbf{g}})+b)$ with trainable $w$, and $b$,
and $\mathbf{c}_{t}$ denoting the memory content at a time $t$ and $k$
parameterizes the time horizon of positve samples obtained from the same
source.

##  6 Architecture

The overall architecture of the proposed system is presented in Fig. 1. The
encoder in the model includes recurrent layers (GRUs) to account for long-term
dependencies over the feature map. The encoder transforms the input signal
into a high dimensional representation in the latent space that can be
considered after the concatenation layer in the diagram as analogous to a
virtual microphone array signal. To emulate a non-under-determined source
separation case, the number of virtual microphones in the latent space is
chosen to be larger or equal the number of the desired sources $P$. The latent
space signal branches into a mask calculation unit consisting of a
convolutional layer and a sigmoid activation layer. A skip connection applies
the mask on the “virtual microphone” signals. The output is fed to a fully
connected layer and then a decoder, which decodes one isolated signal at a
time since all output sources share the same decoder. In the proposed network,
the separation in the latent space is performed via a multiplication with an
adaptive matrix ($\mathbf{z}=\boldsymbol{\theta}\mathbf{z^{\prime}}$),
emulating the (over)-determined multichannel separation of $P$-sources
($\mathbf{z}=[\mathbf{z}_{1},\ldots,\mathbf{z}_{P}]^{\mathrm{T}}$) given
$Q$-microphones
($\mathbf{z^{\prime}}=[\mathbf{z^{\prime}}_{1},\ldots,\mathbf{z^{\prime}}_{Q}]^{\mathrm{T}}$)
where $\boldsymbol{\theta}$ has the structure

$\boldsymbol{\theta}=\left[\begin{array}[]{c}\mathbf{F}\cdot\boldsymbol{\theta}_{1}\\\
\ldots\\\ \mathbf{F}\cdot\boldsymbol{\theta}_{Q}\end{array}\right],$ (12)

with $\mathbf{F}$ being a matrix fixed after training with the dimensions
$L_{out}\times L_{in}Q$. Since it is an-element-wise multiplication followed
by a fully connected layer, we are encouraging the network to take the latent
space interdependencies into account, similar to the solutions of classical
DSP approaches to multivariate BSS with the difference that now the separation
itself is done in a latent space.

Fig. 1: Latent space disentangling blind source separation system
architecture.

##  7 Loss function

The loss function in our permutation-invariant training strategy includes
multiple components. We use the disentanglement loss from Eq. (7) and
additionally incorporate a subband-based SI-SDR [28] and $\ell_{2}$
reconstruction loss as a realization of the inductive bias constraint in Eq.
(6), and the mentioned permutation and memory bank losses.

##  8 Experimental results

To prove our concept we train a model based on the proposed architecture with
an encoder consisting of 3 convolutional layers with kernels of 1, 3, and 5, 3
recurrent sections each consisting of 3 GRU layers with skip between the
sections connections and hidden size of 200, a fully connected layer and a
decoder with 3 transposed convolutional layers. The FFT size is 1024, the
window size is 512, and the hop size is 256 from which we calculate 100 Bark
band features which constitutes the network processing domain meaning that the
network output is in the Bark domain but for calculating the losses, the
signal is transformed first back into the full subband resolution. For the
memory, we use an 2 LSTM layers for each source to track the adaptive pooling
output operating on a buffer storing the last 50 frames of the separated
latent representation. We train the network on an in-house acquired dataset
with explicit recording consent. For training we use mixtures of two clean
fully overlapped speech signals without applying any further augmentation and
train to output the separated mixture components. For testing, we use held out
mixtures of unseen speakers with and without background noise. The clean and
noisy conditions have normally distributed SNRs with means of 0dB, -2dB and
standard deviations of 3dB and 3.8dB respectively. For a comparison, we train
SkiM [3] model consisting of 4 SkiM blocks. In each SkiM block,the LSTMs’s
hidden dimension is 256. The segment LSTMs are chosen to be bi-directional and
the segment sequence length is set to 150, the MemLSTM are uni-directional,
and the feature normalization is performed on the feature dimension. Although
the literature reports higher state-of-the-art numbers on synthetic datasets,
in our experiment the test dataset is chosen to cover real-world scenarios
with different conditions, different speakers and speaker styles than the
training set. The results shown in Table 1 confirm that the network
generalizes for the noisy case while performing consistently on the clean
data.

|  SI-SDR
---|---
| Proposed | SkiM
Noisy | 5.3 | 2.8
Clean | 7.2 | 7.1
Table 1: SI-SDR achieved under noisy and clean condition for the online system
trained on only clean mixtures.

##  9 Conclusions

In this paper, we presented an online approach to source separation motivated
from first principles highlighting the links to classical DSP solutions to
source separation. The method aims at a block-wise disentanglement in a latent
space. We account for the permutation problem and long-term dependencies by
using per-source memory augmentation and a novel differentiable permutation
loss.

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# Characterization and Quantum Efficiency Determination of Monocrystalline
Silicon Solar Cells as Sensors for Precise Flux Calibration

Sasha Brownsberger1,2 Lige Zhang2 David Andrade3 and Christopher Stubbs2,4
2Department of Physics, Cambridge, MA 02138, USA,
<EMAIL_ADDRESS>
3School of Engineering and Applied Sciences Cambridge, MA 02138, USA
4Department of Astronomy, Cambridge, MA 02138, USA,
<EMAIL_ADDRESS>
((to be inserted by publisher); (to be inserted by publisher); (to be inserted
by publisher))

###### Abstract

As the precision frontier of astrophysics advances towards the one
millimagnitude level, flux calibration of photometric instrumentation remains
an ongoing challenge. We present the results of a lab-bench assessment of the
viability of monocrystalline silicon solar cells to serve as large-aperture
(up to 125mm diameter), high-precision photodetectors. We measure the
electrical properties, spatial response uniformity, quantum efficiency (QE),
and frequency response of 3rd generation C60 solar cells, manufactured by
Sunpower. Our new results, combined with our previous study of these cells’
linearity, dark current, and noise characteristics, suggest that these devices
hold considerable promise, with QE and linearity that rival those of
traditional, small-aperture photodiodes. We argue that any photocalibration
project that relies on precise knowledge of the intensity of a large-diameter
optical beam should consider using solar cells as calibrating photodetectors.

###### keywords:

instrumentation: detectors – techniques: photometric – methods: laboratory:
solid state

1Corresponding Offer.

; ; ;

##  1 Introduction

Projects at the frontier of precision astrophysics and cosmology, such as the
Large Synoptic Survey of Space and Time (LSST) Ivezić et al. (2019), require
relative photometric calibration of 1% to achieve their science goals. When
combined with precise distances to Galactic stars from the GAIA project Brown
et al. (2018), it is also meaningful to strive for absolute calibration of
flux at a comparable level. Reaching such ambitious calibration targets will
be possible only if the imaging system’s throughput, $T$, is known to better
than 1%. The LSST project’s plan to measure $T$ is based on the methodology
first laid out in Stubbs & Tonry (2006). A monochromatic known dose of photons
is injected into the telescope aperture and measured with both the telescope
imaging system and with a precisely calibrated, reference photodetector.

At present, the most promising arrangement to realize this _in situ_
calibration method uses a collimated beam projector (CBP), an instrument that
emits a collimated monochromatic beam into the aperture of the telescope under
study. Successful CBP-style calibrations have already been performed by
Coughlin et al. (2016, 2018), and the conceptually similar StarDICE
calibration project remains in active development Regnault et al. (2015).

The benefits of CBP-based calibration include:

  * •

direct control of spectral properties of the calibrating light,

  * •

collimated CBP emission mimics the wavefront of collimated stellar emission,

  * •

no need for atmospheric corrections,

  * •

reduced susceptibility to systematic errors from stray light in the optical
system.

However, the accuracy of CBP-based measurements of $T$ are no better than the
accuracy to which the collimated beam is itself calibrated. Many CBP designs
produce collimated beams with large cross sections (5cm in diameter or more),
and calibrated such a large beam with traditional, small aperture photodiodes
is difficult. Optics that focus the collimated beam onto a photodiode
themselves require calibration, and measuring only a section of the beam with
a small photodiode would measure the true intensity of the full beam only if
the beam were perfectly uniform.

Motivated by this and other photo-calibration problems, we are engaged in an
ongoing effort to test whether monocrystalline, back-contact silicon solar
cells (SCs) could serve as large-aperture, high precision photodetectors. The
attractive attributes of these devices include:

  1. 1.

large collecting area, spanning 125 $\times$ 125 mm,

  2. 2.

large thickness, 165 $\mu$m, yielding good NIR quantum efficiency,

  3. 3.

100% fill factor, with no electrodes on the light-facing surface,

  4. 4.

high quantum (QE) efficiency, $>90\%$, across the spectral region of interest
for instruments that use Si sensors,

  5. 5.

affordability,

  6. 6.

portability, allowing for relatively easy deployment in a variety of
observatory settings.

We detailed our initial assessment of these devices, as well as the operating
principle for back-contact monocrystalline SCs, in Brownsberger et al. (2019).
This paper describes progress in our ongoing effort to fully assess these
devices’ advantageous properties for precise flux characterization. Our work
builds on previous high-fidelity characterizations of other SCs, tied to
national-lab calibrated photodiodes Larason & Houston (2008); Xu & Huang
(2012); Roller & Hamadani (2019) . What sets our work (both Brownsberger et
al. (2019) and this paper) apart is our specific and targeted focus on using
characterized SCs to calibrate astronomical imaging systems. We are
principally focused on determining the SC properties that are necessary for
sub-1% determination of telescope throughput and designing SC containment
systems that will allow the calibrated devices to be easily deployed on site.

In Section 2, we discuss the differences between the previous generation and
the new generation of studied SCs. In Section 3, we provide measurements of
the shunt resistances of 176 SCs, describe how we selected which devices
warranted further characterization, and how we prepared selected devices for
test. In Section 4, we measure the small-scale spatial variations of the QE of
the new SCs, noting that the effect is much reduced relative to the same
behavior in the previous cell generation. In Section 5, we show our 1%
measurement of the absolute QE of one mounted SC, and we describe the
measurement setup that we used to arrive at our result. In Section 6, we
determine that the frequency response of an SC is dominated by its intrinsic
electrical characteristics (shunt resistance and shunt capacitance), and
provide an example of measuring the RC time constant for an SC under test. We
conclude in Section 7 and discuss some expected short term applications of
these calibrated SCs.

##  2 C60 Solar Cell Generations

Since we published our initial assessment of 2nd generation C60 SCs (Gen 2) in
Brownsberger et al. (2019), the Sunpower corporation released their improved
3rd generation C60 SCs (Gen 3). The Gen 3 devices differ from the Gen 2
devices in the following ways:

  * •

More finely spaced backside electrode, now only 0.48 mm apart, compared to the
0.59 mm of Gen 2 SCs (see Figure 4).

  * •

Improved AR coating, with less reflected blue light, giving the Gen 3 SCs good
QE even into the ultraviolet (see Figure 6).

  * •

Different connection bond pads, with smaller footprints.

  * •

Systematically larger shunt capacitance and systematically smaller shunt
resistance (see Figure 1), presumably due to tighter spacing of the
alternating positive and negative electrodes.

We concluded that the new Gen 3 SCs are, on balance, better photometric
calibrators than Gen 2 SCs. Where we show results of Gen 2 SCs, we explicitly
say so.

##  3 The Solar Cell Shunt Resistances, Device Selection, and Device
Preparation

###  3.1 The Distribution of Solar Cell Shunt Resistances and Selection
Criterion

As we described in Section 3.2 of Brownsberger et al. (2019), the ratio of an
attached load’s impedance, $R_{L}$, to the SC shunt resistance, $R_{sh}$,
determines what fraction of SC photocurrent is discharged in the cell, going
undetected by the load. Therefore, when assessing an individual SC’s viability
as a photometric calibrator, $R_{sh}$ is a vital figure of merit.

To better understand the variance of this important detector property in mass-
produced batches of C60 SCs, we measured $R_{sh}$ of 176 SCs from both Gen 2
and Gen 3. We performed this measurement using the basic method first
implemented by Chan & Phang (1984), by covering the front and back of the SC
under test with black foil and applying an ohmmeter to its external wire bond
mount points. We show each of the 176 measured $R_{sh}$ values in Figure 1.
Cell IDs were assigned in order in which $R_{sh}$ was measured.

Figure 1: The distribution of shunt resistances, $R_{sh}$, of Gen 2 (batch 1
and 2) and Gen 3 (batch 3) C60 solar cells (SCs), 176 in total. Note that the
third generation SCs (magenta) have systematically smaller shunt resistances.
The horizontal line at $R_{sh}=300\Omega$ defined our cut off for cell
selection. Those SCs marked ‘BAD’ were damaged either before or during
characterization.

We found that the Gen 3 SCs had systematically lower $R_{sh}$ than the Gen 2
SCs, presumably because the electrodes are more tightly packed, facilitating
electron leakage between electrodes of the sort described by Banerjee &
Anderson (1986). In all SC batches that we studied, most devices had
unacceptably small $R_{sh}$ values. Setting our selection criterion at
$R_{sh}>300\Omega$, 30% of Gen 2 cells and 8% of Gen 3 cells were acceptable.
We selected only SCs that met this selection criterion for further study. The
results in this paper are mostly based on measurements of Gen 3 solar cell ED.

###  3.2 Device Preparation

After selecting SCs with acceptably large shunt resistances (see Section 3.1),
we experimented with various soldering methods and electrically conductive
adhesives to connect wire leads to the electrodes of an SC. We settled on a
method of soldering a solid piece of 24-gauge wire to the central solar cell
electrode, and then connecting a pigtail BNC connector to a ground-isolated
bulkhead BNC feed-through connector. This electrical connection is shown in
Figure 2. This method produces more stable electric bonds than adhesive-based
connection methods.

We designed specialized housing to hold a single SC under study.
Characteristics of these housings include:

  * •

a robust, metal casing to protect the delicate SC,

  * •

a removable front pane to make inserting the SC easy,

  * •

a metal snout extending 5 cm in front of the cell to block background light
contamination,

  * •

internal 1/4”-20 threads to mount the SC casing to optical stands,

  * •

a mounted BNC connector so that the SC wire leads can be connected to external
electronics.

To mount an SC in this housing, the device was first glued to a piece of
electrically insulating plexiglass, which was then glued to the interior of
the aluminum SC casing. We show an image of the fully packaged cell in Figure
3.

Figure 2: Electrical connection and packaging of a Gen 3 C60 solar cell. A
piece of solid wire is temporarily held in place by Kapton tape, and is
soldered to a larger gauge BNC pigtail. The solar cell, facing down in this
image, is glued to an insulating plexiglass spacer that is in turn glued to
the aluminum frame shown. This frame is then mounted inside the detector
housing shown in Figure 3. The temporary Kapton tape is removed before
operation. Figure 3: Image of packaged solar cell. An upstream aperture limits
illumination to the central area of the solar cell.

##  4 Spatial Variations in Solar Cell Quantum Efficiency

In Brownsberger et al. (2019), we noticed a 10% variation in the QE of a Gen 2
SC as we moved a small spot of light across the cell’s front face, and we
noted that such a substantial change in SC QE would introduce significant
systematic error to any calibration effort involving small diameter light
beams. These QE variations are due to the charge traps produced by the
corrugated electric fields of the alternating positive and negative SC
electrodes. Because the electrodes are more tightly spaced in Gen 3 SCs, we
repeat the QE response measurements, scanning a 100$\mu$m spot across the
surface of Gen 3 cell ED. We show the results of both the old Gen 2 scan and
this new Gen 3 scan in Figure 4.

The Gen 2 and Gen 3 same-polarity electrodes are respectively spaced $1.18\pm
0.01$ mm and $0.96\pm 0.01$mm apart. Consistent with these spacings, we find
that the largest Fourier peaks of the spatial scans are located at wavenumbers
of $0.83\pm 0.01$mm-1 and $1.03\pm 0.01$mm-1. The size of the Fourier peak in
the Gen 3 cell is only 20% the height of the Fourier peak in the Gen 2. cell,
relative to the continuum. Owing to the tighter spacing of their conducting
electrodes, the QE of the Gen 3 cells varies by only 2% on small spatial
scales. Solar cell flux measurement errors due to spatial QE variations will
therefore be smaller in Gen 3 cells.

Figure 4:  The normalized photocurrent as we move a 0.1mm spot of light across
the front of a solar cell orthogonal to the backside electrodes (left column),
the Fourier amplitude of that normalized photocurrent (middle column), and a
magnified image of the back-side spatial structure with a ruler included for
scale (right column) of a Gen 2 (top row) and Gen 3 (bottom row) SunPower C60
solar cell. The locations of the first Fourier peaks of the Gen 2 and Gen 3
SCs are respectively at wavenumbers of $0.83\pm 0.01$mm-1 and $1.03\pm
0.01$mm-1. These measurements are consistent with the measured spacing of the
backside electrodes: in the Gen 2 and Gen 3 cells, we respectively measure
electrode spacings of $0.58\pm 0.01$mm and $0.48\pm 0.01$ mm, corresponding to
same-voltage electrode spacings of $1.18\pm 0.01$mm and $0.96\pm 0.01$mm. As
we described in Brownsberger et al. (2019), the alternating voltages of the
back contact electrodes produce a corrugated electric field, resulting in
charge traps that reduce the cells’ effective QE. This charge trapping effect
is a factor of five less pronounced in the Gen 3 SCs, likely due to their
tighter electrode spacing and a reduction in the relative size of differently
charged electrodes.

##  5 Absolute Quantum Efficiency Determination

###  5.1 Method and setup

We designed an experimental setup to measure the absolute quantum efficiency
(QE) of a single C60 SC to better than 1% accuracy by tying the measured SC
photocurrent to that of a NIST-calibrated Larason & Houston (2008) photodiode
(PD). Figure 5 shows our experimental setup. The PD is placed at the waist of
a diverging beam on a mechanical flipper, and the SC is placed further back in
the beam path. We position both the PD and SC so that the beam underfills
their photosensitive areas.

To avoid systematic errors associated with the readout signal chain, we wire
both detectors in parallel and feed their aggregate signals to the same series
ammeter and parallel voltmeter. An order-blocking filter eliminates second
order light contamination when operating at wavelengths longer than 650 nm.
The variable ND filter is used to adjust the photocurrent in the calibrated
photodiode to a target of 0.9$\mu$A when possible.

Table 1 lists the details and part numbers of the devices used.

Part | Manufacturer | Product Number
---|---|---
Continuum light source | Thorlabs | SLS302
Monochromator | Newport | Corner Stone 260
Longpass filter | Edmund Optics | 46 063 OG590
Variable ND Filter | Edmund Optics | 41960
Optical Fiber | Thorlabs | M93L02
Reflective Collimator | Thorlabs | RC08SMA-F01
Focusing lenses (2) | Thorlabs | LA1050-A-ML & LA1417-A-ML
Photodiode | Hamamatsu, NIST calibrated | S2281
Solar cell | SunPower | C60
Cell enclosure | Custom | N/A
Ammeter | Keysight | B2987A
Table 1: Table 1 - Components of the setup used to measure the QE of a C60
solar cell using a NIST calibrated Hamamatsu photodiode shown in Figure 5.
Figure 5: The experimental setup used to tie the quantum efficiency (QE) of a
solar cell (SC) under test to that of a NIST-calibrated photodiode (PD). The
monochromator, with an optical bandwidth of 5nm, illuminates a variable ND
filter and then the tip of an optical fiber. The optical fiber feeds a
reflective collimator, the collimated output of which is focused by a pair of
lenses. We place the PD on a mechanical flipper, allowing us to remotely move
it into and out of the waste of the focused beam. The SC rests at the end of
this optical chain, with $\simeq$50% of its photoconducting surface
illuminated by the monochromatic light when the PD is flipped out of the beam
path. Both SC and PD are wired in parallel, their generated photocurrents
aggregated and fed to a single high-fidelity current meter. A Windows laptop
controls the moving components of the measurement setup and stores the
acquired data. We mount the variable ND filter to a digitally controlled
translation stage, allowing us to adjust the photodiode photocurrent to a
target value of 0.9$\mu$A over most of the wavelength range (see the upper
left panel of Figure 6). For wavelengths above 650nm, we insert a 590nm
longpass filter in the beam path to block second order light contamination.
Not shown are baffles and stray light controls that are essential to arrive at
the desired accuracy.

###  5.2 Results

To measure the QE of the solar cell, we repeatedly scan the monochromator from
350-1050nm, in 1nm steps. At each nm, we measure the photocurrent with the
photodiode in the beam path with the monochromator shutter open and closed,
then we flipped the photodiode out of the beam path and measure the
photocurrent in the SC with the shutter open and closed. Each photocurrent
measurement consists of 200 individual current readings, at a sampling
frequency of 60 Hz. To measure the photocurrent in the SC and the PD, we
discard the first 50 components of each sample chain to avoid shutter and
flipper artifacts and then measure the median of the remaining samples. The
median dark current (shutter closed) value is subtracted from the
corresponding shutter open median value. Running this measurement continuously
for approximately three days, we densely and repeatedly scan the SC QE curve.

We show the results of this long scan in Figure 6. The C60 SC that we study
(cell ED) has high QE over the spectral range of primary interest - greater
than $90\%$ between 450nm and 950nm, and above $70\%$ between 400 and 1000nm.
Relative to the NIST-calibrated PD, the SC’s QE is particularly high between
375 and 400nm, where the effect of the improved AR coating is made especially
clear. Our results are consistent between scans, showing no drift. We report
the absolute QE of a C60 solar cell to better than $1\%$.

Figure 6: The data sets from which we measure the quantum efficiency (QE) of
one Gen 3 C60 solar cell (SC) ED. Starting with the photocurrents in the NIST-
calibrated photodiode (PD) and SC ED (upper left), we take the ratio of the
photocurrents (upper right) and divide that ratio by the reported QE of the PD
(lower left plot, blue curve) to acquire our measured QE of cell ED (lower
left plot, red curve). We show the high-QE section of the cell ED’s measured
QE curve in the lower right panel. The SC has excellent QE, peaking at 96% and
staying above 90% between about 400nm and 900nm. The scatter in the measured
points was due principally to instability in the SC dark current and small
drifts in the continuum source intensity. Even with these systematic
uncertainties, our measured SC QE is stable to better than 1%.

##  6 Bandwidth and Frequency Response

To understand the effect that a possibly slow photoelectric response and a
relatively large internal shunt capacitance might have on our ability to use
chopped illumination with SC-based photocalibration, we re-purpose the QE
setup described in Section 5.1 to measure the time-response of solar cell ED.
We change the setup in three ways:

  * •

switch from the monochromator to a brighter, broad-band light source,

  * •

place an optical chopper in the path of the converging beam, before the
calibrated photodiode,

  * •

exchange the ammeter with a voltmeter, inferring photocurrent response by
measuring the voltage as the photocurrent was discharged over the SC internal
shunt resistance, $R_{sh}$.

Running the chopper wheel at various frequencies, $f$, we measure the time-
varying voltage induced across cell ED, in photovoltaic mode. Under the
assumption that the RC shunt circuit of the C60 solar cell dominates the SC
time response, we fit the measured output voltage with the standard RC
‘sharkfin’ waveform. The results are shown in Figure 7.

We find that, for chopper frequencies above 200Hz, an RC waveform with a time
constant of 580$\mu$s effectively characterizes the photocurrent-induced
voltage. This suggests that the time response of a C60 solar cell is dominated
by the internal electronic properties of the cell itself. For photocalibration
purposes, the strobe period of a calibration light source should be at least
five times longer than the RC time constant. For cell ED with an RC value of
580$\mu$s, this implies a strobe frequency of $<1/(5\times
580\mu\textrm{s})\simeq 340$Hz. This is comfortably above both power line
harmonics and 1/f noise corners for well-chosen electronics.

Figure 7: The time response of the C60 solar cell ED to chopped illumination
(blue, dashed), the best-fit RC circuit response (red, dotted), and the
difference between the data and model (solid, purple) at increasing
frequencies. An RC model with a time constant of 550-600$\mu$s characterized
the SC time response curve well, though not perfectly. To first order, the
time response of a C60 solar cell is dominated by the internal shunt
capacitance and resistance of the cell.

##  7 Conclusion and a Possible Photocalibration Application

We have continued our assessment of monocrystalline back-contact silicon solar
cells (SCs) as possible high-precision photometric calibrators, begun in
Brownsberger et al. (2019).

In this new work, we discussed the differences between 2nd generation (Gen 2)
and 3rd generation (Gen 3) SCs (Section 2), and argued that Gen 3 cells should
be used in calibration efforts. Following our conclusion in Brownsberger et
al. (2019) that the internal SC shunt resistance, $R_{sh}$, is a vital figure
of merit when assessing these devices’ viability as photocalibrators, we
measured $R_{sh}$ for 176 SCs of both generations. Setting our minimum
acceptable $R_{sh}$ vale at 300$\Omega$, we found that about 30% of Gen 2
cells and 8% of Gen 3. cells could be used in further calibration efforts. We
detailed our improved method for mounting these devices in Section 3.2.

Because Gen 3 cells have more tightly spaced back-contact electrodes than Gen
2 cells, we repeated in Section 4 the small-scale QE variation measurement
that we developed in Brownsberger et al. (2019). Consistent with our
expectations, we found that the Gen 3 cells had more rapidly varying and
smaller amplitude QE variations than the Gen 2 cells. For photo-calibration
purposes, this smaller spatial variation will introduce less systematic error
into flux measurements.

For the first time in our analysis of these SCs, we devised and implemented a
measurement to tie the QE of a SC under study to the QE of a NIST-calibrated
photodiode (Section 5.1). Using this setup for several uninterrupted days, we
determined the absolute QE of one SC to better than 1% precision. We found
that these SCs exhibit excellent QE, peaking around 96% at 600nm and remaining
above 90% between 400 and 900nm. We also noted that the Gen 3 cells have
particularly good QE in the blue, relative to the reference PD, presumably
owing to their improved AR coating.

Considering our intention to use SCs as photometric calibrators with chopped
light sources, we measured in Section 6 the frequency-response of a Gen 3 SC.
We determined that a SC’s time response is dominated by its internal
electronic properties (shunt resistance and shunt capacitance). In particular,
we found that the time-response of cell ED to a chopped beam was well
described by a standard RC waveform, with a time constant of 580$\mu$s. As
with the shunt resistances of the individual SCs (Figure 1), we fully expect
that the distribution of RC time constants will vary across cells. Each cell
that is slated for use in photocalibration should similarly have its RC time
constant characterized.

Considering the combined results of our initial analysis in Brownsberger et
al. (2019) and those we present here, we are confident that solar cells are
one of the best, and perhaps the best, tools available for achieving 1%
precision measurements of large-diameter optical beams. We are actively using
and intend to use the SCs that we have characterized in various high-precision
calibration projects, including the full characterization of the StarDICE
determination of telescope throughput Regnault et al. (2015) already underway.

The next generation of advanced telescopes demands exquisite calibration to
realize their ambitious science goals. The use of terrestrial, large-diameter
light sources, built to mimic collimated starlight, will play a crucial in
these calibration efforts. We expect that large-area, high-fidelity
photodetectors, such as the C60 SCs that we have studied, will play an
important role in accurately characterizing such calibration beams.

## Acknowledgments

We are grateful to the US Department of Energy for their support of our
optimization and precision calibration efforts for the Rubin Observatory,
under DOE grant DE-SC0007881, and to Harvard University for its support of our
program.

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