# SIRUP: A DIFFUSION-BASED VIRTUAL UPMIXER OF STEERING VECTORS FOR HIGHLY-DIRECTIVE SPATIALIZATION WITH FIRST-ORDER AMBISONICS

Emilio Picard, Diego Di Carlo, Aditya Arie Nugraha, Mathieu Fontaine, Kazuyoshi Yoshii

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Emilio Picard, Diego Di Carlo, Aditya Arie Nugraha, Mathieu Fontaine, Kazuyoshi Yoshii. SIRUP: A DIFFUSION-BASED VIRTUAL UPMIXER OF STEERING VECTORS FOR HIGHLY-DIRECTIVE SPATIALIZATION WITH FIRST-ORDER AMBISONICS. ICASSP, May 2026, Barcelone, Spain. hal-05516730

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
SIRUP: A DIFFUSION-BASED VIRTUAL UPMIXER OF STEERING VECTORS
FOR HIGHLY-DIRECTIVE SPATIALIZATION WITH FIRST-ORDER AMBISONICS

Emilio Picard¹,² Diego Di Carlo¹ Aditya Arie Nugraha¹ Mathieu Fontaine³,¹ Kazuyoshi Yoshii⁴,¹

¹Center for Advanced Intelligence Project, RIKEN, Japan

²Sorbonne University, France

³LTCI, Télécom Paris, Institut Polytechnique de Paris, France

⁴Graduate School of Engineering, Kyoto University, Japan

ABSTRACT

This paper presents virtual upmixing of steering vectors captured by a fewer-channel spherical microphone array. This challenge has conventionally been addressed by recovering the directions and signals of sound sources from first-order ambisonics (FOA) data, and then rendering the higher-order ambisonics (HOA) data using a physics-based acoustic simulator. This approach, however, struggles to handle the mutual dependency between the spatial directivity of source estimation and the spatial resolution of FOA ambisonics data. Our method, named SIRUP, employs a latent diffusion model architecture. Specifically, a variational autoencoder (VAE) is used to learn a compact encoding of the HOA data in a latent space and a diffusion model is then trained to generate the HOA embeddings, conditioned by the FOA data. Experimental results showed that SIRUP achieved a significant improvement compared to FOA systems for steering vector upmixing, source localization, and speech denoising.

***Index Terms***— Steering vectors, virtual upmixing, latent diffusion model, sound source localization, beamforming.

1. INTRODUCTION

Sound source localization (SSL) and speech enhancement (SE) or denoising remain fundamental tasks in machine listening applications, further used in augmented reality (AR) scenes [1], as well as robotics [2], radar [3], and autonomous driving systems [4]. Beamforming and SSL methods are still widely characterized by acoustic and signal processing [5–7], even though recent work tried to incorporate deep neural networks (DNNs) to address such tasks [8, 9]. These applications call for more accurate spatial information that could be translated as more selective beam patterns, enabling better localization and spatial filtering.

High-order-ambisonic (HOA) microphone arrays [10] are
capable of high spatial resolutions, enabling effective sound

<figure><img src="image_2.png" /><figcaption>Fig. 1: The SIRUP upmixer for downstream tasks.</figcaption></figure>

analysis and synthesis [11, 12]. The spatial resolution of these
systems does relate to the number of ambisonic orders, which
are related to the number of microphones. These limitations
can be alleviated by adopting HOA setup, albeit at the ex-
pense of expensive hardware; consequently, common spatial
recording systems are typically constrained to the first order
(FOA), employing four microphones.

A common remedy is parametric *upmixing*, which esti-
mates scene parameters, e.g., directions of arrival (DOA) via
SSL and source signals via separation or SE, and then render
virtual HOA channels. For example, major methods such as
DirAC [13] and COMPASS [14] encode directional and am-
bient components before ad-hoc HOA spatial rendering [15].
Their performance is tightly coupled to the accuracy of steer-
ing vectors (SVs), the spatial signatures driving classical SSL
(e.g., SRP-PHAT) and SE (e.g., delay-and-sum beamform-
ing) [16]. This cascaded analysis–rendering pipeline is brit-
tle: low-resolution noisy FOA SVs degrade the parameter es-
imates and propagate errors to the HOA rendering.

We instead propose to directly upmix the *ambisonic* *SVs*.
*SVs* compactly encode the spatial characteristics of sources,
encompassing both the direct paths and early reflections. By
super-resolving FOA *SVs*, one can perform SSL in a highly-
directive HOA space, allowing the DOA estimates to produce
sharper spatial filters. The upmixed *SVs* can also be utilized
to render the source images at a higher spatial resolution.

This work was supported by JST FOREST no. JPMJFR2270, JSPS
KAKENHI nos. JP23K16912, JP23K16913, 24H00742, 24H00748, and
25H01142, and ANR Project SAROUMANE (ANR-22-CE23-0011). All
the code used to produce the results of this paper is available at https:
//github.com/emilio-pcrd/sirup upon acceptance.
Specifically, a SteerIng vectoR UPmixer (SIRUP) learns a latent space of HOA SVs with a VAE and uses a conditional diffusion model to generate HOA embeddings from FOA inputs estimated under an isotropic ambient prior. Once DOAs are inferred from the reconstructed HOA SVs, source signals are recovered via beamforming and subsequently rendered using FOA-consistent or either algebraic free-field SVs, as depicted in Fig. 1.

## 2. BACKGROUND

This section introduces SVs and their applications. We work in the short-time Fourier transform (STFT) domain whose frequency bins and time frames are indexed with $f \in \{1, \dots, F\}$ and $t \in \{1, \dots, T\}$, where $F$ and $T$ are the numbers of frequency bins and frames, respectively.

### 2.1. Steering vectors

Suppose we have an $M$-channel STFT observation $\mathbf{x}_{ft} \in \mathbb{C}^M$ of a static, far-field source [6], which is given by

$$ \mathbf{x}_{ft} = \mathbf{a}_f s_{ft} + \mathbf{n}_{ft}, \quad (1) $$

where $\mathbf{a}_f \in \mathbb{C}^M$ is the acoustic transfer function, $s_{ft} \in \mathbb{C}$ the source STFT, and $\mathbf{n}_{ft} \in \mathbb{C}^M$ diffuse isotropic noise.

For a given array geometry and look direction $\theta$, the algebraic SVs that models the direct path of $\mathbf{a}_f$ is [6]

$$ \tilde{\mathbf{a}}_f(\theta) = [e^{-j2\pi f \tau_1(\theta)}, \dots, e^{-j2\pi f \tau_M(\theta)}]^\top, \quad (2) $$

with relative delays $\tau_m(\theta) = \mathbf{u}_\theta^\top(\mathbf{r}_m - \tilde{\mathbf{r}})/c$, where $\mathbf{r}_m$ is the $m$-th microphone position, $\tilde{\mathbf{r}}$ the array center, $\mathbf{u}_\theta$ the unit vector pointing to $\theta$, and $c$ the speed of sound.

The dominant spatial mode of the spatial covariance matrix (SCM) provides a good estimate of the SVs [17]:

$$ \hat{\mathbf{\Sigma}}_f = \frac{1}{T} \sum_{t=1}^{T} \mathbf{x}_{ft} \mathbf{x}_{ft}^{\mathrm{H}}, \quad \hat{\mathbf{a}}_f = \mathrm{eig}_1(\hat{\mathbf{\Sigma}}_f), \quad (3) $$

where $\mathrm{eig}_1(\cdot)$ denotes the principal eigenvector. These measured SVs capture the direct path plus early reflections, and thus are richer than the algebraic model in Eq. (2).

### 2.2. Applications

For SSL, we evaluate the steered-response-power (SRP) map on a discrete grid of DOAs and pick one $\hat{\theta}$ [18] as follows:

$$ S(\theta) = \sum_f \left| \frac{\tilde{\mathbf{a}}_f(\theta)^\mathrm{H}}{\|\tilde{\mathbf{a}}_f(\theta)\|_2} \frac{\hat{\mathbf{a}}_f}{\|\hat{\mathbf{a}}_f\|_2} \right|^2, \quad \hat{\theta} = \arg \max_{\theta} S(\theta). \quad (4) $$

This corresponds to cross-correlation between the estimated SV and algebraic SVs computed in the frequency domain.

For SE, a beamformer steered to $\hat{\theta}$ is given by

$$ \mathbf{w}_f = \frac{\mathbf{a}_f(\hat{\theta})}{\|\mathbf{a}_f(\hat{\theta})\|_2}, \quad \hat{s}_{ft} = \mathbf{w}_f^{\mathrm{H}} \mathbf{x}_{ft}, \quad (5) $$

where $\mathbf{a}_f$ can be either the algebraic or measured SV pointing at $\hat{\theta}$ direction. Assuming spatially white noise, choosing

$w_f \propto \hat{a}_f$ yields the MaxSINR beamformer [19].

### 2.3. Latent diffusion models for image out-painting

Given a cropped image $\mathbf{x}^L \in \mathbb{R}^{I_L \times J_L}$, out-painting aims at recovering the missing regions of an original image $\mathbf{x}^H \in \mathbb{R}^{I_H \times J_H}$ with $I_H > I_L, J_H > J_L$. This approach naturally extends for audio super-resolution [20]. State-of-the-art performances on these tasks are obtained with latent diffusion models [21–24], where the diffusion is processed in a latent space learned by a variational autoencoder (VAE).

Let $E_\phi$ and $D_\psi$ be the VAE encoder and decoder, i.e. $\mathbf{z}_0 = E_\phi(\mathbf{x}^H), \hat{\mathbf{x}}^H = D_\psi(\mathbf{z}_0)$. In diffusion models, the forward process represents the variational posterior $q(\mathbf{z}_{1:T}|\mathbf{z}_0)$ as a Gaussian Markov chain, where noise is gradually added to the latent vector $\mathbf{z}_0$ according to a fixed schedule. In latent diffusion, the reverse process $p_\theta(\mathbf{z}_{0:T}|\mathbf{c})$ is similarly modeled as a Gaussian Markov chain, conditioned on a context embedding $\mathbf{c} = C(\mathbf{x}^L)$ injected via concatenation or cross-attention into the denoising network $\epsilon_\theta(\mathbf{z}_t, t, \mathbf{c})$. The function $\epsilon_\theta$ is used to predict the mean of each $p_\theta(\mathbf{z}_{t-1}|\mathbf{z}_t, \mathbf{c})$. The training score is $\mathbb{E}[\|\epsilon - \epsilon_\theta(\mathbf{z}_t, t, \mathbf{c})\|^2]$ where $\epsilon \sim \mathcal{N}(0, \mathbf{I})$ and $\mathbf{z}_t$ is a linear combination of $\mathbf{z}_0$ and $\epsilon$.

## 3. PROPOSED METHOD

This section introduces the proposed SV upmixing method that extends a latent diffusion models to SV spatial super-resolution, and discuss its application for audio analysis.

### 3.1. Steering vector upmixing

The proposed SIRUP model is a conditioned latent diffusion model that uses the first $M$ channels to generate the missing $M' - M$ ambisonic channels, yielding the upmixed SVs $\hat{\mathbf{A}}^{\mathrm{up}} \in \mathbb{C}^{F \times M'}$ with $M' > M$. Let $\hat{\mathbf{A}} \in \mathbb{C}^{F \times M}$ be the concatenation of measured FOA SVs across frequency and define the conditioning tensor by zero-padding channels as

$$ \mathbf{c} = [\hat{\mathbf{A}}, \mathbf{0}_{F \times (M'-M)}] \in \mathbb{C}^{F \times M'}. \quad (6) $$

Optionally, the algebraic SVs may fill $\mathbf{c}_{:,M+1:M'}$ if steering directions are known.

Then we encode the condition with a VAE encoder $E_\phi$ and run latent diffusion conditioned on this embedding. Starting from pure noise $\mathbf{z}_T \sim \mathcal{N}(0, \mathbf{I})$, the denoiser $\epsilon_\theta(\mathbf{z}_t, t, E_\phi(\mathbf{c}))$ iteratively produces $\mathbf{z}_{t-1}$ until $\mathbf{z}_0$. The decoder $D_\psi$ then produces the super-resolved ambisonic SVs:

$$ \mathbf{z}_{t-1} = \epsilon_{\theta}(\mathbf{z}_t, t; E_{\phi}(\mathbf{c})), \quad \hat{\mathbf{A}}^{\mathrm{up}} = D_{\psi}(\mathbf{z}_0). \quad (7) $$

During inference, the latent diffusion model is used to sample from pure noise and the condition, to generate a suitable latent candidate for the decoder. The VAE thus acts as a training-time regularizer establishing a well-behaved latent manifold, through a small KL-divergence penalty.

To enhance spatial reconstruction, we employ a composite loss function that combines a cosine similarity term with
both feature-matching and mean squared error (MSE) objec-
tives. This significantly improves both learning stability and
overall performance. Furthermore, dilated convolutions are
introduced along the frequency axis of the network architec-
ture. This modification enforces spatial coherence across dif-
ferent frequency bands, while the microphone axis coherence
is already implicitly handled by the cosine similarity term.

**3.2. Downstream tasks with up-mixed steering vectors**

Figure 1 outlines the proposed upmixing pipeline for sound-
scene analysis and synthesis. First, measured SVs $\hat{a}_f \in \mathbb{C}^M$
for the target source are estimated from the multichannel mix-
ture $\mathbf{x}_{ft} \in \mathbb{C}^M$ over a short time window using Eq. (3). Next,
the $M$-channel ambisonic representation of $\hat{a}_f$ is upmixed to
a higher ambisonic order with $M'$ channels via SIRUP (cf.
Section 3.1), and then mapped back to the signal domain as
SVs of a virtual $M'$-channel spherical array.

SSL is subsequently performed in the upmixed domain
using Eq. (4) as a grid search using algebraic SVs projected
onto the upmixed counterparts. For enhancement and ren-
dering, the upmixed SVs can be truncated to the original $M$-
channel subspace to obtain denoised spatial images. Note that
SIRUP can also be trained with a denoising objective to re-
fine the first $M$ ambisonic coefficients, thereby improving the
accuracy of the estimated SVs.

4. EVALUATION

This section presents the performance of the proposed upmix-
ing approach in terms of SSL and beamforming metrics.

**4.1. Experimental settings**

We first explain the experimental data, model configuration,
and evaluation metrics.

4.1.1. *Experimental data*

Room impulse responses (RIRs) were simulated using the
image source model (ISM) in the pyroomacoustics li-
brary [25]. For each investigation, we generated 30 distinct
acoustic scenes with various signal-to-noise ratios (D<sub>SNR</sub>) and
reverberation times (D<sub>RT60</sub>). The microphone array was cen-
trally placed in a 6×4×3 m room, with its horizontal position
randomized by up to 1 m. Source locations were randomly
selected within the azimuth plane, with a fixed elevation of
approximately 28°. This specific elevation was chosen to sim-
ulate a typical desktop condition, where the microphones are
placed on top of a table. All sources were modeled as point
speech sources, randomly selected and cropped to 4 seconds
from the dev-clean folder of the LibriSpeech database [26].

For SSL experiments and testing the upmixing method,
we simulated rooms containing a single speech source with
reverberation and noise. The SVs were directly estimated
from the mixture using Eq. (3). For SE experiments, we simu-
lated mixtures comprising two speech sources across various

acoustic environments. To compute their SVs, we introduced
a 2-second delay between the start times of the two 4-second
sources. This configuration allowed us to use the first 2 sec-
onds and the last 2 seconds of the overall mixture for the SV
estimation of the first and second sources, respectively.

All data were sampled at *f*<sub>s</sub> = 16 kHz. The short-time
Fourier transform (STFT) was computed using a 512-sample
frame size with 50% overlap and a Hamming window. For the
ambisonic representation, third-order data (*M*′ = 16) were
designated as the target HOA data, corresponding to first-
order ambisonics (*M* = 4) as the low-resolution input.

4.1.2. *Model configuration*

The training of SIRUP was conducted in a two-stage pro-
cess. In the first stage, a VAE was trained to reconstruct HOA
SVs conditioned on FOA SVs. The training data consisted
of measured SVs obtained from single-source noisy mixtures
utilizing Eq. (3). The autoencoder (3.1M parameters) was
optimized for 40 epochs using the AdamW optimizer with a
learning rate of 3 × 10<sup>-4</sup>. The objective function (Section 3)
combined an l<sub>2</sub> reconstruction loss, a cosine loss objective, a
perceptual loss, and a small KL term.

The second stage involved freezing the encoder and fine-
tuning only the decoder for an additional 20 epochs. This
fine-tuning utilized a combination of MSE and cosine simi-
larity losses, applied with an exponential learning-rate sched-
ule. We trained an UNet-based latent diffusion model (4.1M
parameters) as in [24]. This model operates within the VAE
latent space to perform the upmixing task, conditioned on the
FOA inputs. Prior to inputting the latent values to the UNet,
the encoder outputs were scaled to the range [-1, 1] [24].

Conditions were injected into the UNet in two ways: (i)
concatenation of the FOA tensor with the noisy latent repre-
sentation at the UNet input layer and (ii) application of cross-
attention within each architectural block. Training was con-
ducted for 100 epochs with AdamW at a learning rate of 3 ×
10⁻⁴. The total number of diffusion steps, *T*, was set to 1, 000
during training and 200 during inference.

For training, 3,000 pairs of measured FOA and HOA SVs
($\hat{\mathbf{A}} \in \mathbb{C}^{F \times M}$, $\hat{\mathbf{a}}^{\text{up}} \in \mathbb{C}^{F \times M'}$) were used. The target HOA
data were generated by convolving the source signals with the
simulated RIRs. The corresponding FOA data were obtained
by retaining only the first $M = 4$ channels of the HOA data.
We represent complex SVs as stacked real/imag parts, hence
the leading dimension “2” in tensors of shape $(2, F, M)$ and
$(2, F, M')$, respectively. The remaining $(M' - M)$ channels
required for the model input were addressed by zero-padding
the FOA conditioning tensor.

4.1.3. *Evaluation metrics*

To assess performance, we utilized two distinct evaluation
sets: D<sub>SNR</sub> and D<sub>RT60</sub>. In the D<sub>SNR</sub> set, the SNR was varied
within the range [5, 20] dB while the reverberation time was
fixed at RT60 = 0.2 s. Conversely, in the D<sub>RT60</sub> set, RT60
was varied within [0.2, 0.7] s with a fixed SNR = 20 dB.
<table><caption>**Table 1:** Performances for different spatial representations, averaged over 30 simulated rooms. Ground-truth HOA values (gray) are provided for reference. Beamwidth is measured at -3dB (lower is better).</caption><thead><tr><th rowspan="2"></th><th colspan="3">DI [dB] ↑</th><th colspan="3">3-dB BW [°] ↓</th><th colspan="3">SL [dB] ↓</th></tr><tr><th>FOA</th><th>Upmixed</th><th>HOA</th><th>FOA</th><th>Upmixed</th><th>HOA</th><th>FOA</th><th>Upmixed</th><th>HOA</th></tr></thead><tbody><tr><th>D<sub>RT60</sub></th><td>10.0 ± 2.6</td><td><b>19.8 ± 2.3</b></td><td>20.0 ± 2.2</td><td>30 ± 6</td><td><b>24.0 ± 3.3</b></td><td>24 ± 2</td><td>-0.9 ± 0.7</td><td><b>-9.5 ± 3.1</b></td><td>-11.2 ± 2.8</td></tr><tr><th>D<sub>SNR</sub></th><td>8.1 ± 2.7</td><td><b>17.1 ± 2.1</b></td><td>17.7 ± 2.0</td><td>48.0 ± 6.7</td><td><b>27.0 ± 3.5</b></td><td>26.0 ± 2.2</td><td>-1.2 ± 0.9</td><td><b>-9.6 ± 3.4</b></td><td>-11.7 ± 2.7</td></tr></tbody></table>

<table><caption>**Table 2:** Source synthesis enhancement from mixture across varying for mixtures of two speech sources.</caption><thead><tr><th rowspan="2">Metric</th><th colspan="2">Measured SVs</th><th colspan="2">Algebraic SVs following SSL</th></tr><tr><th>SV-FOA</th><th>SV-SIRUP-M</th><th>SV-alg FOA</th><th>SV-alg SIRUP</th></tr></thead><tbody><tr><th>SDR [dB]</th><td>17.2 ± 3.2</td><td><b>17.4 ± 3.1</b></td><td>12.6 ± 7.4</td><td><b>13.0 ± 7.2</b></td></tr><tr><th>SIR [dB]</th><td>38.8 ± 3.6</td><td>38.8 ± 3.3</td><td>33.5 ± 7.8</td><td><b>34.0 ± 7.5</b></td></tr><tr><th>SAR [dB]</th><td>17.3 ± 3.2</td><td><b>17.4 ± 3.1</b></td><td>12.6 ± 7.3</td><td><b>13.0 ± 7.2</b></td></tr></tbody></table>

SSL performance was primarily evaluated in terms of the angular error on the azimuthal plane, benchmarked against an SRP baseline derived from FOA mixtures. To assess the quality of the upsampled SVs, we considered the directivity index (DI), the 3dB-beamwidth (3dB BW), and sidelobe (SL) metrics. Finally, the output of the beamforming process was evaluated using common source separation metrics, as detailed in Tab. 2 and computed using the bss_eval toolkit [6].

## 4.2. Experimental results

Fig. 2 reports the average DOA estimation error of the sound source by pickpeaking on the SRP angular map (shown in Fig 3, against different noise and reverberation ratios. SIRUP was shown to have outperformed the FOA in noisy conditions, begin closed to the topline using HOA setup. We however reports comparable results increasing the reverberation.

Next, we investigate the spatial quality of the upmixed SVs compared to the FOA baseline in terms of beampattern metrics computed azimuthal plane. As reported in Table 1, the proposed method SIRUP yields an average beamwidth improvement of +10° and sidelobes suppression of -9 dB. Qualitatively, this performances translate in narrower beam shapes comparable to the one of a HOA setup, as shown in 3.

Finally, we studied the beamforming performance for mixtures of 2 sources. The results reported in Table 2 compare the denoised output when using the following SVs model for beamforming as in Eq. (5): the SVs compute for FOA mixture (SV-FOA), the first *M* channels of the SIRUP's output (SV-SIRUP-M), the algebraic SVs from DOA estimation of both the measured FOA SVs (SV-alg FOA) and SIRUP (SV-alg SIRUP), respectively. The performances of SV-SIRUP-M are expected to be comparable the one of SV-FOA denoting effective training of the diffusion model. However, we observed improvements, which can be explained by the capacity of the model to denoise SVs during inference. In addition, as we have better DOA estimation using SIRUP, we also note

<figure><img src="image_3.png" /><figcaption>**Fig. 2:** Average angular errors across conditions using different localization methods and SV models, using SRP (Eq. (5)).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Fig. 3:** 2D Heatmap comparaison of estimated SV for FOA and ground truth HOA setup.</figcaption></figure>

improvements for the SV-alg experiments. In most practical cases, SIRUP is outperforming the FOA conditions with equal information, leading to spatial-resolution improvements in every experiences.

## 5. CONCLUSION

This paper presented a novel diffusion-based approach named SIRUP, designed to significantly enhance the spatial information derived from FOA setups, allowing it to emulate the performance of HOA microphone arrays. SIRUP directly addresses the inherent limitations of FOA systems, such as lower spatial resolution and imprecise SSL, by virtually upmixing SVs using a two-step latent diffusion model. The experimental results demonstrated that SIRUP achieved significant improvements in steering vector upmixing, sound source localization, and speech denoising compared to FOA systems.

Future work will focus on comprehensively utilizing the upmixed SVs for downstream tasks like source separation and rendering. We also plan to make the model capable of learning from noisier SVs. These advancements could significantly enhance machine listening applications requiring high spatial accuracy, such as augmented reality, robotics, radar, and autonomous driving systems.
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