Approximation of the Fourier Series on a finite interval. Converts the signal from the time domain to the frequency domain

Input: 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} $

Discrete Fourier Transform (DFT)

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.

Carleton College

A small unit of extraction of features. From the digitized, quantized representation of the waveform, we need to extract spectral features from a small window of speech that characterizes part of a particular phoneme.

Window

The first step in ASR is to transform the input waveform into a sequence of acoustic feature vectors, where each vector represents the information in a small time window of the signal.

Feature Vector

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

Speech and Language Processing (3rd ed. draft) 

Stationary signal refers to signals whose statistical properties are constant within a certain region (in the case of ASR, inside a small window).

Stationary Signal

Signals whose statistical properties are not constant over time. For example, speech, in general, is a non-stationary signal.

Non-stationary Signal

Frame is the speech extracted from each window.

Frame

The windowing is characterized by three parameters: the window size or frame size of the window (its width in milliseconds), the frame stride, (also called shift or offset) between successive windows, and the shape of the window.

Parameters of Windowing

The rectangular window looks just like the original signal, except being extracted. However, abruptly cutting off the signal at its boundaries will create problems when we do Fourier analysis later.

Rectangular Window

A more commonly used windowing method for acoustic feature creation than rectangular windowing. Hamming window shrinks the values of the signal toward zero at the window boundaries, avoiding discontinuities.

Hamming Window

Spectral Information Extraction

Feature vector that contains the most commonly used features when converting a raw wavefile to a sequence of features.

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