Discrete Fourier Transform by Hand

In signal processing, the Discrete Fourier Transform (DFT) is no doubt the most important method. But the math involved is extremely complex, literally, involving a summation over a complex number term e^(-iwt), where e is the Euler number, i is the imaginary unit, w is the angular frequency, and t is time.

I developed this exercise to demonstrate that underneath such complexity, DFT is just a series of matrix multiplications you can calculate by hand. ✍️ Once you see that, it should not surprise you that a deep neural network, which is also a series of matrix multiplications, with activation functions in-between, can learn to perform DFT to process and analyze signals so effectively.

💡 Learned vs. Fixed: U-Net learns its filters from data to process a signal in the spatial domain. The DFT is the classical opposite, a fixed transform, designed by hand rather than learned, that views the same signal in the frequency domain as a combination of cosine waves.

How does DFT work?

Setup

Step 1 of 12: Given

Signals A, B, and C in the 🟧 frequency domain:

A = cos(w) + 2cos(2w)

B = cos(w) + cos(3w) + cos(4w)

C = -cos(2w) + cos(3w)

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