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The discrete Fourier transform (DFT) is a widely used tool across science and engineering. Nevertheless, the DFT assumes that the frequency characteristics of a signal remain constant over time, and is unable to detect local changes. Researchers beginning with Gabor (1946) addressed this shortcoming by inventing methods to obtain time-frequency representations, and this thesis focuses on one such method: the Sliding Window Discrete Fourier Transform (SWDFT). Whereas the DFT operates on an entire signal, the SWDFT takes an ordered sequence of smaller DFTs on contiguous subsets of a signal. The SWDFT is a fundamental tool in time-frequency analysis, and is used in a variety of applications, such as spectrogram estimation, image enhancement, neural networks, and more. This thesis studies the SWDFT from three perspectives: algorithmic, statistical, and applied. Algorithmically, we introduce the Tree SWDFT algorithm, and extend it to arbitrary dimensions. Statistically, wederive the marginal distribution and covariance structure of SWDFT coefficients for white noise signals, which allows us to characterize the SWDFT coefficients as a Gaussian process with a known covariance. We also propose a localized version of cosine regression, and show that the approximate maximum likelihood estimate of the frequency parameter in this model is the maximum SWDFT coefficient over all possible window sizes. From an applied perspective, we introduce a new algorithm to decompose signals with multiple non-stationary periodic components, called matching demodulation. We demonstrate the utility of matching demodulation in an analysis of local field potential recordings from a neuroscience experiment.

The Sliding Window Discrete Fourier Transform