When a signal passes through the Fast Fourier Transform, we obtain its frequency spectrum. This is an array of numbers that show the amplitude of each frequency present in the signal. For an audio recording a few seconds long, with a sampling rate of 44.1 kHz and a window size of 2048 points, the spectrum contains about 1024 values per frame. If frames shift every 10 milliseconds, we get roughly 100 spectra per second, or over 100,000 numbers in total. For an entire song, this amounts to millions of values.
Working with such a large amount of data is impractical. When we want to compare two signals, track changes over time, or detect anomalies, we need a more compact representation. Instead of working with entire spectra, we can extract a few numbers from them that capture the essential properties of the signal. These numbers are called features.
But features are not just a means of compression. Their primary purpose is to turn the signal into something that can be compared, recognized, and classified. When a speech recognition system needs to determine which word was spoken, it does not compare the recording directly with thousands of other recordings – that would be too slow and noisy. Instead, it extracts features from the signal (most commonly MFCCs) and compares this compact set of numbers with precomputed features of different words. Similarly, when we want to determine whether a machine is operating normally, we can monitor its vibration features – shifts in the spectral centroid, changes in the crest factor, or the appearance of new peaks. These numbers become a "fingerprint" of the system's state.
Some features are easy to interpret. The spectral centroid, for example, shows where the mid-frequency balance of the signal lies – whether the sound is darker or brighter. The spectral roll-off gives an idea of where the significant energy ends. Other features require more explanation. MFCCs, for instance, pass the spectrum through filters that mimic the human ear and return coefficients that have proven exceptionally useful for speech recognition. Chroma features group energy according to musical semitones and are used for harmonic analysis.
In this article, we will examine the main types of features – spectral, cepstral, chroma, modulation, and some time-domain ones. We will focus on definitions, computational steps, and the meaning of the resulting values. Machine learning remains aside – this is about pure DSP mathematics and how a signal can be described by numbers that carry meaning.
From Spectrum to Features – Graphic Example
The plot below shows a single spectral peak centred at 500 Hz. From this spectrum, we extract three compact features:
- Spectral centroid (orange line) – the centre of mass of the spectrum. With a single, symmetric peak, it aligns exactly with the peak frequency.
- Spectral spread (shaded region) – the width of the spectrum around the centroid.
- Spectral roll-off (red line) – the frequency below which 85% of the energy is contained.
These three numbers describe the entire spectrum in a compact, meaningful way.
Observation: The centroid is exactly at 500 Hz. The spread captures the width of the peak. The roll-off sits where the tail becomes insignificant. This is the simplest and clearest demonstration of feature extraction.
Read the full article: Signal Features in the Frequency Domain →