When we plot the magnitude of each frequency from the DFT, we can visualize the spectrum of the given signal

Plotting a spectrum from DFT

https://mathworld.wolfram.com/FastFourierTransform.html

Fast Fourier Transform (Wolfram MathWorld)

An efficient algorithm for computing the DFT of a sequence. Runs in O(n log(n)) time

Fast Fourier Transform (FFT)

Spectral information is extracted on a mel scale, rather than equally across all frequency bands. We do this because humans hearing is more sensitive at lower frequencies than higher ones, so modeling speech recognition the same way helps performance.

Mel Filter Bank and Log

The Discrete Fourier Transform (DFT) is an approximation of the Fourier series on a finite interval. It converts a signal from the time domain to the frequency domain.

Input: A windowed signal $$x[n]... x[m]$$.
Output: For each frequency band, a complex number $$X[k]$$ representing the magnitude and phase of that frequency component in the original signal.

We compute $$X[k]$$ with:
$$X[k] = \sum_{n=0}^{N-1} x[n]e^{-j\frac{2\pi}{N}kn}$$

Carleton College

Google

In feature extraction for ASR, we must extract spectral information from our windowed signal and find out how much energy the signal contains at different frequency bands. This is done with DFT.

Spectral Information Extraction

An on-going but a helpful book resource about NLP
https://web.stanford.edu/~jurafsky/slp3/

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