In a number of data-driven applications such as detection of arrhythmia, interferometry or audio compression, observations are acquired indistinctly in the time or frequency domains: temporal observations allow us to study the spectral content of signals (e.g., audio), while frequency-domain observations are used to reconstruct temporal/spatial data (e.g., MRI). Classical approaches for spectral analysis rely either on i) a discretisation of the time and frequency domains, where the fast Fourier transform stands out as the \textit{de facto} off-the-shelf resource, or ii) stringent parametric models with closed-form spectra. However, the general literature fails to cater for missing observations and noise-corrupted data. Our aim is to address the lack of a principled treatment of data acquired indistinctly in the temporal and frequency domains in a way that is robust to missing or noisy observations, and that at the same time models uncertainty effectively. To achieve this aim, we first define a joint probabilistic model for the temporal and spectral representations of signals, to then perform a Bayesian model update in the light of observations, thus jointly reconstructing the complete (latent) time and frequency representations. The proposed model is analysed from a classical spectral analysis perspective, and its implementation is illustrated through intuitive examples. Lastly, we show that the proposed model is able to perform joint time and frequency reconstruction of real-world audio, healthcare and astronomy signals, while successfully dealing with missing data and handling uncertainty (noise) naturally against both classical and modern approaches for spectral estimation.

中文翻译：

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傅立叶对的贝叶斯重建

在许多数据驱动的应用中，例如心律失常检测、干涉测量或音频压缩，在时域或频域中的观察是模糊的：时间观察使我们能够研究信号（例如音频）的频谱内容，而频率-域观察用于重建时间/空间数据（例如，MRI）。经典的频谱分析方法依赖于 i) 时域和频域的离散化，其中快速傅立叶变换作为\textit{de facto} 现货资源脱颖而出，或者 ii) 具有封闭的严格参数模型-形成光谱。然而，一般文献无法满足缺失的观察和噪声破坏的数据。我们的目标是以一种对缺失或嘈杂的观察结果鲁棒的方式解决在时域和频域中模糊获取的数据缺乏原则性处理的问题，同时有效地对不确定性进行建模。为了实现这一目标，我们首先为信号的时间和频谱表示定义一个联合概率模型，然后根据观察执行贝叶斯模型更新，从而联合重建完整的（潜在）时间和频率表示。从经典光谱分析的角度分析了所提出的模型，并通过直观的例子说明了其实现。最后，我们表明所提出的模型能够对现实世界的音频、医疗保健和天文信号进行联合时间和频率重建，