Time-Frequency Domain

STFT¶

Signals can be represented in the so called time-frequency-domain. This can be useful for calculations and signal manipulations, as well as for visualization, since this shows the evolution of a signal's spectrum over time. This visual representation is referred to as spectrogram. The short-time Fourier transform (STFT) devides a signal into overlapping segments - or frames - of equal length and calculates a Fourier transform for each frame.

Typical framesizes $L_f$ range from $10$ to $40\ \mathrm{ms}$. Overlaps vary from $0$ percent to $90$ percent.

The larger the frames, the better the frequency resolution. The larger the overlap, the better the time resolution.

Violin¶

For musical sounds with a pitch, the spectrogram shows the individual partials as prominent horizontal lines. The following spectrogram of a violin sound, recorded with a sampling rate of $f_s = 96\ \mathrm{kHz}$ is calculated with a framesize of $4096$ samples and an overlap of $2048$ samples:

Exercise: Calculate the framesize and overlap in seconds. What is the maximum represented frequency?

Kick Drum¶

The essential part of a typical electronic kick is a single sine wave, dropping in frequency from about $200\ \mathrm{Hz}$ to about $60\ \mathrm{Hz}$:

Drums¶

Other percussive elements contain so called transients - short regions of noise or rapid fluctuations. Snare drums thus appear in a spectrogram as vertical lines, as this short drum solo shows: