  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.
