Neural Architectures Learning Fourier Transforms, Signal Processing and Much More....

This report will explore and answer fundamental questions about taking Fourier Transforms and tying it with recent advances in AI and neural architecture. One interpretation of the Fourier Transform is decomposing a signal into its constituent components by projecting them onto complex exponentials. Variants exist, such as discrete cosine transform that does not operate on the complex domain and projects an input signal to only cosine functions oscillating at different frequencies. However, this is a fundamental limitation, and it needs to be more suboptimal. The first one is that all kernels are sinusoidal: What if we could have some kernels adapted or learned according to the problem? What if we can use neural architectures for this? We show how one can learn these kernels from scratch for audio signal processing applications. We find that the neural architecture not only learns sinusoidal kernel shapes but discovers all kinds of incredible signal-processing properties. E.g., windowing functions, onset detectors, high pass filters, low pass filters, modulations, etc. Further, upon analysis of the filters, we find that the neural architecture has a comb filter-like structure on top of the learned kernels. Comb filters that allow harmonic frequencies to pass through are one of the core building blocks/types of filters similar to high-pass, low-pass, and band-pass filters of various traditional signal processing algorithms. Further, we can also use the convolution operation with a signal to be learned from scratch, and we will explore papers in the literature that uses this with that robust Transformer architectures. Further, we would also explore making the learned kernel's content adaptive, i.e., learning different kernels for different inputs.