In awesome processing, the Discrete Fourier Transform (DFT) is nary uncertainty the astir important method. But the mathematics progressive is highly complex, literally, involving a summation complete a analyzable number word e^(-iwt), wherever e is the Euler number, one is the imaginary unit, w is the angular frequency, and t is time.
I developed this workout to show that underneath specified complexity, DFT is conscionable a bid of matrix multiplications you tin cipher by hand. ✍️ Once you spot that, it should not astonishment you that a heavy neural network, which is besides a bid of matrix multiplications, pinch activation functions in-between, tin study to execute DFT to process and analyse signals truthful effectively.
💡 Learned vs. Fixed: U-Net learns its filters from information to process a awesome successful the spatial domain. The DFT is the classical opposite, a fixed transform, designed by manus alternatively than learned, that views the aforesaid awesome successful the wave domain arsenic a operation of cosine waves.
How does DFT work?
Signals A, B, and C successful the 🟧 wave domain:
A = cos(w) + 2cos(2w)
B = cos(w) + cos(3w) + cos(4w)
C = -cos(2w) + cos(3w)
Each awesome is simply a weighed sum of 4 cosine waves astatine frequencies 1w, 2w, 3w, and 4w.
We will use Inverse DFT to person the signals to clip domain representations, and past show DFT tin person backmost to their original wave domain representations.
Signal X successful the 🟩 clip domain. X is sampled astatine 10 clip points 1t, 2t, …, 10t:
X = [-2.5, -1.8, 3, -0.7, -1.0, -0.7, 3, -1.8, -2.5, 5]
Suppose X is besides a weighted sum of the aforesaid 4 cosine waves, but we don’t already cognize their weights. We will use DFT to observe them.
Write the coefficients of A, B, C arsenic a matrix F. Each awesome is simply a row. Each wave is simply a column.
A → [1, 2, 0, 0]
B → [1, 0, 1, 1]
C → [0, -1, 1, 0]
Sample from the continuous cosine waves astatine discrete clip points 1t, 2t, 3t, to 10t.
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